Optical tomography and light irradiation to reduce bacterial loads in

oral health application

by

Feixiao Long

A Thesis Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

DOCTOR OF PHILOSOPHY

Major Subject: Biomedical Engineering

Approved by the

Examining Committee:

_________________________________________

Xavier Intes, Thesis Adviser

_________________________________________

Shiva P. Kotha, Member

_________________________________________

Ge Wang, Member

_________________________________________

Fengyan Li, Member

_________________________________________

Juergen Hahn, Member

Rensselaer Polytechnic Institute

Troy, New York

April 2016

(For Graduation May 2016)

ii

© Copyright 2016

by

Feixiao Long

All Rights Reserved

iii

CONTENTS

LIST OF TABLES ........................................................................................................... vii

LIST OF FIGURES .......................................................................................................... ix

ACKNOWLEDGMENT ................................................................................................. xv

ABSTRACT .................................................................................................................... xv

1. Introduction .................................................................................................................. 1

1.1 Dental caries and current clinical practice ......................................................... 1

1.2 Structure of teeth ................................................................................................ 3

1.2.1 Enamel.................................................................................................... 3

1.2.2 Dentin ..................................................................................................... 3

1.2.3 Pulp ........................................................................................................ 4

1.2.4 Cementum .............................................................................................. 5

1.3 The necessity of detecting pulp functions .......................................................... 5

1.4 The importance of detecting the situation of dental fillings and reducing the

population of bacteria ......................................................................................... 6

1.5 The challenge of detecting pulp activity and status of fillings........................... 8

1.5.1 Lack of other suitable imaging modalities to assay pulp function (X-ray)

and status of fillings ............................................................................... 8

1.5.2 Optical imaging of soft tissue or fillings covered by hard tissues ......... 9

1.5.3 Limited volume of pulp and root canal ................................................ 10

1.5.4 The curvature and size of teeth ............................................................ 10

1.5.5 Blood flow of pulp and sampling rate .................................................. 10

1.6 Optical based imaging modalities .................................................................... 10

1.6.1 Forward problem .................................................................................. 12

1.6.2 Inverse problem .................................................................................... 14

1.6.3 Hardware settings of optical imaging system ...................................... 15

1.7 Reducing bacteria after surgery and monitoring the situation of fillings ........ 17

1.8 Structure of thesis ............................................................................................. 18

iv

2. Algorithms to solve the radiative transfer equation as a forward problem ................ 19

2.1 Algorithms ....................................................................................................... 22

2.1.1 Radiative transfer equation .................................................................. 22

2.1.2 Discrete ordinate method (DOM) ........................................................ 23

2.1.3 Phase function normalization technique .............................................. 27

2.1.4 DOM with continuous Galerkin finite element method ....................... 28

2.1.5 Diffusion approximation (DA) ............................................................. 30

2.1.6 Monte Carlo simulation (MC) .............................................................. 31

2.2 Settings for numerical simulations ................................................................... 31

2.2.1 Ideal pencil beam simulation ............................................................... 31

2.2.2 Gaussian shape beam simulation ......................................................... 33

2.2.3 Teeth model .......................................................................................... 35

2.2.4 Delta-Eddington phase function simulation ......................................... 37

2.3 Results .............................................................................................................. 38

2.3.1 3-D rectangle simulations with ideal pencil beam ............................... 38

2.3.2 3-D rectangle simulations with Gaussian modeled intensity beam ..... 43

2.3.3 Teeth model simulations with Gaussian beam irradiation ................... 45

2.3.4 Delta-Eddington phase function simulation ......................................... 46

2.3.5 Importance of phase function normalization technique ....................... 51

2.4 Discussion and future work .............................................................................. 52

2.5 Conclusion ....................................................................................................... 54

3. Mesoscopic fluorescence molecular tomography applied in dental imaging ............ 55

3.1 Materials and methods ..................................................................................... 55

3.1.1 Samples preparation and experiments procedure ................................. 55

3.1.2 Optical settings of MFMT .................................................................... 58

3.1.3 Optical reconstruction .......................................................................... 59

3.1.4 Image registration of multimodal data sets .......................................... 61

v

3.2 Results .............................................................................................................. 62

3.3 Discussions ....................................................................................................... 65

4. Upconverting nanoparticles and their application in dental imaging ........................ 68

4.1 Materials and methods ..................................................................................... 69

4.1.1 Upconverting nanoparticles (UCNPs).................................................. 69

Teeth phantom preparation .................................................................. 73

4.1.3 Optical settings ..................................................................................... 74

4.1.4 Optical reconstruction .......................................................................... 75

4.1.5 In silico experiment design .................................................................. 77

4.1.6 Procedure of experiments ..................................................................... 80

4.2 Results .............................................................................................................. 80

4.2.1 In silico simulations ............................................................................. 80

4.2.2 Determination of power index of UCNPs with emission of blue light 87

4.2.3 Ex vivo experiments ............................................................................. 88

4.3 Discussions ....................................................................................................... 92

5. Killing bacteria with UCNPs ..................................................................................... 94

5.1 Materials and methods ..................................................................................... 95

5.1.1 Light irradiation ................................................................................... 95

5.1.2 Bis-GMA dental composite ................................................................. 96

5.1.3 Bacterial growth ................................................................................... 96

5.1.4 Bacterial live/dead essay ...................................................................... 96

5.1.5 Mammalian growth .............................................................................. 97

5.1.6 Mammalian live/dead analysis ............................................................. 97

5.2 Results .............................................................................................................. 97

5.2.1 Assessing the light irradiation effect on S. mutans .............................. 97

5.2.2 Assessing the light irradiation effects on NIH3T3 fibroblast survivals 99

5.3 Discussion ...................................................................................................... 101

vi

6. Summary of the thesis and future work ................................................................... 104

6.1 Radiative transfer equation based forward solver .......................................... 104

6.2 Application of upconverting nanoparticles .................................................... 105

References ...................................................................................................................... 108

vii

LIST OF TABLES

Table 2-1. RMSE with different numerical quadrature (𝜇𝑎 = 0.02 mm-1

, 𝜇𝑠 = 5 mm-1

,

𝑔 = 0.9). .................................................................................................................. 39

Table 2-2. MRE with different numerical quadrature (𝜇𝑎 = 0.02 mm-1

, 𝜇𝑠 = 5 mm-1

,

𝑔 = 0.9). .................................................................................................................. 39

Table 2-3. RMSE for different optical properties at different depth along the line 𝑥 = 0

mm (𝑔 = 0.9). ......................................................................................................... 40

Table 2-4. MRE for different optical properties at different depth along line 𝑥 = 0 mm

(𝑔 = 0.9). ................................................................................................................. 41

Table 2-5. RMSE for different optical properties to 3-D rectangle (𝑔 = 0.9). ............... 42

Table 2-6. MRE for different optical properties to 3-D rectangle (𝑔 = 0.9). ................. 43

Table 2-7. 3-D RMSE and MRE for two rectangles with Gaussian shape beam (𝑔 = 0.9).

................................................................................................................................. 43

Table 2-8. Comparison of time consumption between solving RTE and MC simulations

with Gaussian beam. ................................................................................................ 52

Table 3-1. SNR of fluorescence signal (SNR) for 1 mm depth of one tooth. ................. 62

Table 3-2. SNR of fluorescence signal (SNR) for 2 mm depth of one tooth. ................. 63

Table 3-3. Comparison of reconstructed volume by MFMT and measurement. ............. 64

Table 3-4. Comparison of reconstructed dye centroid by MFMT and micro-CT. .......... 65

Table 4-1. The volume of reconstruction with 70 detectors for different optical contrast

agent (the unit of the volume is mm3). .................................................................... 83

Table 4-2. The volume of reconstruction with 105 detectors for different optical contrast

agent (the unit of the volume is mm3). .................................................................... 83

viii

Table 4-3. The maximum difference of 3 coordinates between reconstructed centroid and

real centroid (The unit in the table is mm). ............................................................. 85

Table 4-4. The error between optical reconstructions and real value. ............................. 90

ix

LIST OF FIGURES

Figure 2-1. Distribution of discrete points on unit sphere Ω of 4 numerical quadratures

(1st octant): (a) Level symmetric quadrature ( 𝑁 = 80 ); (b) Product Gaussian

quadrature (𝑁 = 72); (c) Legendre equal-weight quadrature (𝑁 = 80); (d) Lebedev

quadrature (𝑁 = 86). ............................................................................................... 26

Figure 2-2. The 3-D rectangle used in simulations: (a) 3-D mesh of rectangle; (b) The

slice at 𝑥 = 0 mm, where the arrow indicates the position in which the photon is

launched perpendicularly to the plane 𝑧 = 0 mm. ................................................... 32

Figure 2-3. 3-D rectangle used in simulation 2.2.2: (a) Mesh of 3-D rectangle; (b)

Irradiated light intensity modeled by Gaussian shape function. .............................. 34

Figure 2-4. Anatomical structure of teeth model: (a) Outline of Tooth 1; (b) The ball

inclusion embedded into Tooth 1; (c) Different portion of Tooth 2, grey color

represents pulp, yellow color represents dentin while magenta (dark color at the

bottom) represents enamel. ...................................................................................... 36

Figure 2-5. Demonstration of Barycentric coordinate system: (a) Arbitrary triangle; (b)

Standard triangle. ..................................................................................................... 37

Figure 2-6. 3-D rectangle: (a) Mesh of rectangle; (b) Detectors located at the bottom

surface of rectangle (𝑧 = −8 mm) .[101] ................................................................ 38

Figure 2-7. The contours of logarithm of solutions to RTE and MC results, solid curve

for the solution to RTE and dashed curve for MC results: (a) The contours within

the plane 𝑥 = 0 mm; (b) The contours within the plane 𝑦 = 0 mm, the value of the

outermost curve is -1.5; (c) The contours within the plane 𝑧 = −3.0 mm.............. 41

Figure 2-8. Comparison between the solutions to RTE, DE and MC methods at different

depths, along the line 𝑥 = 0 mm: (a) 𝑧 = −0.4 mm ; (b) 𝑧 = −1.0 mm ; (c)

𝑧 = −2.0 mm; (d) 𝑧 = −3.0 mm; (e) 𝑧 = −4.0 mm; (f) 𝑧 = −5.0 mm. .............. 42

x

Figure 2-9. The contours of logarithm of photon densities of Rect. 1 at 3 planes, in which

the solid curves represent the solution to the RTE while the dashed curves represent

MC results: (a) 𝑥 = 0 mm; (b) 𝑦 = 0 mm, the value of the outermost contour is -1.5;

(c) 𝑧 = −3 mm. ....................................................................................................... 44

Figure 2-10. The contours of logarithm of photon densities of Rect. 2 at 3 planes, in

which the solid curves represent the solution to the RTE while the dashed curves

represent MC results: (a) 𝑥 = 0 mm, the value of outermost curve is -2.5; (b) 𝑦 = 0

mm, the value of outermost curve is -2.5; (c) 𝑧 = −3 mm. .................................... 44

Figure 2-11. The contours of logarithm of photon densities in Tooth 1 within 3 planes, in

which the solid curves represent solutions to the RTE and dashed curves represent

MC simulations: (a) 𝑦𝑂𝑧 plane; (b) 𝑧𝑂𝑥 plane; (c) 𝑧 = −3.0 mm. ........................ 45

Figure 2-12. Demonstrations of the position of tooth and Gaussian beam source. ......... 46

Figure 2-13. Photon distributions of Tooth 2 within 3 planes: (a) 𝑦𝑂𝑧 plane; (b) 𝑧𝑂𝑥

plane; (c) 𝑧 = −2 mm. ............................................................................................. 46

Figure 2-14. Contours of logarithm of solutions to the RTE with H-G phase function and

MC simulations within 3 planes under irradiation by Gaussian beam, in which the

solid curves represent the solutions to RTE and dashed curves represent MC

simulations: (a) 𝑦𝑂𝑧 plane; (b) 𝑧𝑂𝑥 plane; (c) 𝑧 = −4 mm. .................................. 47

Figure 2-15. Contours of logarithm of solutions to the RTE with d-E phase function and

MC simulations within 3 planes under irradiation by Gaussian beam, in which the

solid curves represent the solutions to RTE and dashed curves represent MC

simulations: (a) 𝑦𝑂𝑧 plane; (b) 𝑧𝑂𝑥 plane; (c) 𝑧 = −4 mm.[101] .......................... 48

Figure 2-16. Comparison of output flux obtained by solutions to the RTE with H-G

phase function or d-E phase function and MC simulations under irradiation of

Gaussian shape beam: (a) H-G phase function; (b) d-E phase function.[101] ........ 48

xi

Figure 2-17. Contours of logarithm of solutions to the RTE with H-G phase function and

DA solutions within 3 planes with internal source, in which the solid curves

represent the solutions to RTE and dashed curves represent DA solutions: (a) 𝑦𝑂𝑧

plane; (b) 𝑧𝑂𝑥 plane; (c) 𝑧 = −4 mm. .................................................................... 49

Figure 2-18. Contours of logarithm of solutions to the RTE with d-E phase function and

DA solutions within 3 planes with internal source, in which the solid curves

represent the solutions to RTE and dashed curves represent DA solutions: (a) 𝑦𝑂𝑧

plane; (b) 𝑧𝑂𝑥 plane; (c) 𝑧 = −4 mm. .................................................................... 50

Figure 2-19. Comparison of output flux obtained by solutions to the RTE with H-G

phase function or d-E phase function and DA solutions with internal source: (a) H-

G phase function; (b) d-E phase function. ............................................................... 51

Figure 2-20. Comparison between the logarithm of contours with and without phase

function normalization: (a) With phase function normalization (b) Without phase

function normalization. ............................................................................................ 51

Figure 3-1. Settings of the teeth phantom: (a) Red rectangle shows the FOV irradiated by

light; (b) Side view of the tooth to show the hole to hold the capillary; (c) Markers

on the surface of tooth. ............................................................................................ 56

Figure 3-2. FOV showed with different modalities and white light photo: (a) White light

photo of real teeth; (b) MFMT background image (without fluorophore); (c)

Reconstructed surface of teeth using micro-CT with ImageJ. ................................. 57

Figure 3-3. Setup of MFMT. ........................................................................................... 59

Figure 3-4. Fiducial plane shown with MFMT and micro-CT: (a) Overlap of micro-CT

image and one background image from MFMT (red color represents MFMT image);

(b) Fiducial plane shown with 3-D surface of tooth; (c) Top view from the tooth to

show fiducial plane. ................................................................................................. 61

xii

Figure 3-5. Merged image of MFMT and micro-CT images: (a,b) 1 mm depth hole with

26 M and 13 M respectively; (c,d,e) 2 mm depth hole with 26 M, 13 M and

6.5 M respectively. ................................................................................................ 64

Figure 4-1. Energy levels of different upconverting processes. Red, green, blue arrows

represent photon absorption, energy transfer and upconverting emission

respectively: (a) ESA; (b) ETU, note that ion 1 is pumped to E1 twice; (c) CSU. .. 71

Figure 4-2. Demonstrations of processing of teeth sample: (a) Photo of Tooth 1, in which

the red circle indicates one mark; (b) One slice of CT images directly acquired by

micro-CT; (c) Binary image of the same slice as (b); (d) Mesh generated by CGAL.

................................................................................................................................. 74

Figure 4-3. Different views of the simulated cylinder inside a tooth: (a) 𝑦𝑧 view; (b) 𝑧𝑥

view. ......................................................................................................................... 78

Figure 4-4. Distribution of detectors: (a) 70 detectors; (b) 105 detectors. ...................... 79

Figure 4-5. Distributions of sources: (a) 15 sources; (b) 20 sources; (c) 25 sources; (d) 27

sources; (e) 35 sources; (f) 36 sources; (g) 45 sources; (h) 63 sources. .................. 80

Figure 4-6. Sensitivity profiles at the same cross section of Tooth 1 with different optical

contrast agents: (a) UCNPs with emission of blue light (475 nm); (b) UCNPs with

emission of near infrared light (800 nm); (c) Linear fluorophore with emission of

red or near infrared light. ......................................................................................... 81

Figure 4-7. Condition number of different optical contrast and eigenvalue of sensitivity

matrix: (a) Condition number of sensitivity matrix with different number of sources

under 70 detectors; (b) Condition number of sensitivity matrix with different

number of sources under 105 detectors; (c) Eigenvalue of sensitivity matrix. ........ 82

Figure 4-8. The difference between reconstruction volume and real volume with 70

detectors for different optical contrast agents: (a) The relationship between

reconstruction volume and number of sources, the black horizontal line indicates

xiii

the real volume and the dashed lines indicate the relative error 10% off the real

volume; (b) The relationship between volume relative error and number of sources.

................................................................................................................................. 84

Figure 4-9. The difference between reconstruction volume and real volume with 105

detectors to different optical contrast agents: (a) The relationship between

reconstruction volume and number of sources, the black horizontal line indicates

the real volume and the dashed lines indicate the relative error 10% off the real

volume; (b) The relationship between volume relative error and number of sources.

................................................................................................................................. 84

Figure 4-10. Simulated measurements at the positions of detectors for different optical

contrast: (a) UCNPs emission blue light; (b) UCNPs emission NIR; (c) Linear dye.

................................................................................................................................. 87

Figure 4-11. The blue emission band of UCNPs: (a) Relation between emission intensity

and wavelength between 440 nm and 510 nm at different laser power; (b) Relation

between max emission intensity and laser power. ................................................... 88

Figure 4-12. Front merged image of UCNPs: (a) UCNPs with blue emission; (b) Bis-

GMA with UCNPs (20% w.t.) with blue emission; (c) UCNPs with NIR emission;

(d) Bis-GMA with UCNPs (20% w.t.) with NIR emission. .................................... 89

Figure 4-13. Lateral merged image of UCNPs: (a) UCNPs with blue emission; (b) Bis-

GMA with UCNPs (20% w.t.) with blue emission; (c) UCNPs with NIR emission;

(d) Bis-GMA with UCNPs (20% w.t.) with NIR emission. .................................... 89

Figure 4-14. Merged image of linear dye: (a) Front view; (b) Lateral view. .................. 90

Figure 4-15. Comparison of raw emission between linear dye and UCNPs with NIR

emission: (a) Linear dye emission with irradiation of source No. 25; (b) UCNPs

with NIR emission with irradiation of source No. 25. ............................................. 92

Figure 5-1. Merged fluorescence images for S. mutans survival: Top row: Blue light

irradiation decreases live cells (green) and increases dead cells (red) at 15, 30, and

xiv

60 minutes after irradiation; Bottom rows: Survival of S. mutans cells is not

affected by the presence of upconversion particles in Bis-GMA (second column

compared to first column). Irradiation with NIR light induces does not significantly

alter cell survival on Bis-GMA (third column compared to first column).

Irradiation with NIR light on Bis-GMA containing upconversion particles increases

the number of dead cells (last column compared to first column). Live S. mutans

cells are stained green, and dead S. mutans cells are stained red. ........................... 98

Figure 5-2. Percentage of S. mutans alive following light irradiation, indicating

irradiation with blue light or NIR irradiation of upconversion particles reduces

percentage of alive S. mutans. Asterisks denote mean survival percentages that

differ between groups (p < 0.01): (a) Blue light irradiation; (b) Near infrared light

irradiation. ................................................................................................................ 99

Figure 5-3. Merged fluorescence images for NIH3T3 survival: Top row: Blue light

irradiation decreases live cells (blue) and increases dead cells (red and green) at 15,

30, and 60 minutes after irradiation; Bottom rows: Survival of NIH3T3 fibroblasts

is not affected by the presence of upconversion particles in Bis-GMA (second

column compared to first column). Irradiation with NIR light does not significantly

alter fibroblast survival on Bis-GMA (third column compared to first column).

Irradiation with NIR light on Bis-GMA containing upconversion particles does not

increase the number of dead cells post 15 minute irradiation, but does so following

30 and 60 minute irradiations (last column compared to first column). Live NIH3T3

fibroblasts are stained blue, apoptotic NIH3T3 fibroblasts are stained green, and

necrotic NIH3T3 fibroblasts are stained red. ......................................................... 100

Figure 5-4. Live/Dead Ratios of NIH3T3 Fibroblasts, indicating irradiation with blue

light or NIR irradiation of upconversion particles for longer than 15 minute reduces

the proportion of live NIH3T3 fibroblasts, determined by the B/G+R fluorescence

ratio. Asterisks denote mean fluorescence B/G+R ratios that differ between groups (p

< 0.01): (a) Blue light irradiation; (b) Near infrared light irradiation.................... 101

xv

ACKNOWLEDGMENT

At the point I’ve finished the writing of thesis, I would like to express my gratitude to

the people who helped me during the preparation of thesis. First I would like to thank Dr.

Shiva Kotha who gave me the chance to study in Rensselaer Polytechnic Institute. Under

his guidance, I had the opportunities to be involved in so many interesting projects. Next,

I feel deeply grateful to the help from Prof. Xavier Intes. Almost all my work was under

his guidance. His rigorous and scientific requirements provided me a solid foundation in

optical imaging. During the collaboration with Prof. Fengyan Li, I learned much about

the basic skills and knowledge in numerical computing. Her careful revision of my paper

impressed me a lot. With the help of Prof. Ge Wang, I am more familiar with the

medical imaging field not only in optical imaging but also other important imaging

modalities. Prof. Juergen Hahn gave me some useful suggestions during my preparation

of thesis. I also want to point out the help of Dr. Wenxiang Cong who solved my

problem when I was in dilemma.

Besides, without the help from the lab, I could not finish the work successfully. I

collaborate the experiment with Mehmet. Qi told me how to calibrate the optical systems.

Ruoyang discussed with me solving the inverse problem with Monte Carlo simulations.

Kathleen carefully revised my manuscripts. The students from Dr. Kotha lab, Aditya,

Vaibhav, Amanda, Laura, Marissa and Liyun all made contributions in my bacteria

experiments.

Finally, I would like to thank the support from my parents as well as the support

from my wife, Sicong. She sacrificed a lot to support the family and encouraged me

every time when I was upset. Whenever I faced the difficulties, I know she was always

supporting me, helping me even from thousand miles away.

Thanks for all the people who helped me during my Ph. D. study.

xvi

ABSTRACT

Light based modalities can enable detection of physiologic functional parameters and

can be used as therapies, especially in structures close to the surface, such as teeth. The

integrity of teeth is dependent on the physiological function of its biological component:

cells in the pulp cavity. Previous studies suggest that impairment of the biological

component, as occurs in the presence of cavities (due to bacterial biofilms) or when the

tooth structure has been compromised by other means (such as cracks), affects some

measures of physiology (e.g. oxygenation). Because it may be possible to detect

impairment of pulp function through changes in its oxygenation, and since currently

used pulse oximetry based methods provide measures that are affected by other variables,

we evaluate whether it is possible to provide 3-D measurements of pulp activity. When

the tooth decay forms and surgery has to be performed to prevent further damage of

teeth, it is necessary to determine whether the dental fillings provide good coverage of

the surgical cavity. It is noted that teeth structures exhibit extremely high absorption and

scattering coefficients relative to other tissues on which optical methods have

conventionally been used (e.g. wounds, breast and brain). So it is important to determine

whether optical methods are sensitive enough to enable imaging through teeth to obtain

3-D measurements of pulp or dental fillings. In my dissertation, a finite element-based

method to solve the radiative transfer equation was proposed and verified by Monte

Carlo simulations, which can be employed as an alternative forward model to the mainly

used diffusion approximation approach. Second, experiments using mesoscopic

fluorescence molecular tomography with linear dyes placed at different sites

preliminarily proved the feasibility for accurate monitoring of pulp situations. Last,

upconverting nanoparticles (UCNPs) mixed with dental fillings employed as theranostics

agents are investigated to determine the distribution of fillings. Throughout the

experiments, the relative volume error of reconstructions of UCNPs (mixed in the

fillings) distribution is less than 10% compared with real distribution of UCNPs. The

position error (centroid error) is around 1 mm. Moreover, the blue emission light of

UCNPs was proven to be a bactericide by ex vivo experiments. Overall, this work

preliminarily investigated the possibility to employ optical-based methods to detect pulp

xvii

situations before root canal therapy, the fillings situation after surgery to ensure the

success of root canal therapy, and kill residual bacteria in mouth.

1

1. Introduction

1.1 Dental caries and current clinical practice

Dental caries, also called tooth decay, is a disease that gradually erodes the hard tissue of

teeth due to bacterial activity [1]. Bacteria on tooth surfaces form biofilms when teeth

are not cleaned regularly or thoroughly. Bacteria in the biofilms then consume simple

sugars from ingested food or drink and create acidic by-products that lower the pH and

dissolve the mineral in the hard tissues [2]. Specifically, with increasing acidity on tooth

surface, the hard tissues of teeth, such as enamel and dentin, most of which are

composed of hydroxyapatite, begin decomposition. At the very beginning of dental

caries, the decay of teeth is minor and invisible to human eye. At this stage, if no

precautions are taken, the decay will develop into a cavity. At the early stages of cavity

formation, tooth surfaces can be cleaned to get rid of bacteria and re-mineralization can

be induced. However, once a cavity has formed, the teeth cannot recover their intact

status without a dental procedure. There are several methods that a dentist uses to detect

cavities, with the primary means being visual and tactile investigation. At this stage,

depending on the extent of damage to the tooth, multiple surgical methods can be

applied to restore teeth and prevent further damage [1]. Without treatment, the cavity

will become larger and eventually reach the pulp inside the teeth, which has large

amount of blood vessels and nerves. At this step, patients will feel pain and their teeth

will be extremely sensitive to the temperature of food. Pain is the primary reason that

patients seek help from dental professionals. However, by the time pain is induced, the

pulp has already been partly damaged. It is noted that there are other mechanisms by

which tooth pain can be induced (e.g. mechanical damage in the form of cracks). It is

also noted that patients cannot identify the tooth that causes pain, and there are no

known modalities to help detect pain. Dental professionals use hot/cold exposure to

individual teeth as well as mechanical tests (poking) to identify the source of pain.*

Without any treatment, the pulp will eventually become necrosis and the tooth will lose

Portions of this chapter previously appeared as: F. Long, M. S. Ozturk, M. S. Wolff, X. Intes, and S. P. Kotha, “Dental

Imaging Using Mesoscopic Fluorescence Molecular Tomography: An ex Vivo Feasibility Study,” Photonics, vol. 1,

no. 4, pp. 488–502, Dec. 2014.

2

its vitality. Dental caries can cause oral sepsis, which can allow bacteria or other toxic

substances to enter the blood. Oral sepsis can be life threatening [3]. It is estimated that

about 2.43 billion people have dental caries in their teeth worldwide [4], making dental

caries one of the most prevalent diseases.

Dental caries also has local and systemic effects; directly from bacteria and their

components or bacterial by-products entering the blood stream, and indirectly from the

inflammatory response that these evoke. Because dental caries has local effects on the

pulp, it is necessary to create paradigms that can provide a measure of the pulp health

and reduce bacterial load.

From the descriptions of dental caries, which is a gradually developing oral

disease, two time points during disease development need to be particularly considered.

One period is the teeth can still recover to the intact status with appropriate treatment,

such as thoroughly cleaning the bacterial inside mouth; at this period, easy treatments

can make teeth healthy again. However, at this point, the minor dental caries can only be

detected through regular dental inspection since it will have no obvious symptoms to

patients. The other period is when the cavity has formed and dental surgery has to be

performed in order to prevent further decay. Conventional root canal therapy or

endodontic therapy will kill and extract the pulp outside of teeth. This will elicit a series

of problems in the pulpless teeth. It will be shown later that pulpless teeth, compared to

the teeth with pulp, are more vulnerable to be infected by bacteria. Thus detection of the

status of the teeth at different stages of dental caries or decision of whether the teeth are

in normal or diseased state is crucial for oral health. Besides, since dental caries is

induced by bacteria, how to reduce the population of the bacteria inside the mouth is also

important for reducing the incidence of dental caries or preventing secondary caries for

the same tooth.

This Chapter will be arranged as following shows. First, the structure of teeth

will be briefly stated. Then, the necessity of detecting the pulp situation and reduction of

bacteria will be discussed. Next, the specific aims will be proposed based on the

previous discussions. Then, the challenges of detecting the pulp, a review of methods

conventionally to detect dental caries and selection of optical methods to enable pulp

3

health measurements will be summarized. Finally, techniques for decreasing the bacteria

using optical methods will be also discussed.

1.2 Structure of teeth

The teeth have complex anatomical structure and as described below. From outside to

inside, the teeth are composed of enamel, dentin and pulp that reside in the root canal. In

the root of teeth, which is embedded into gum, the outermost tissue will be cementum,

not enamel. Based on the different positions in mouth and functions when chewing the

food, teeth can be categorized into incisors, canines, premolars and molars. The pulp of

teeth is protected by hard tissue such as enamel, dentin or cementum. A brief

introduction for different part of teeth is provided below.

1.2.1 Enamel

Enamel, which covers the crown of teeth, is the hardest substance in humans, with 96%

of the enamel composed of hydroxyapatite and the rest composed of water and proteins,

such as amelogenin, ameloblastin and enamelin [5]. During development of the enamel,

high protein content has been observed and this is significantly reduced in mature

enamel [6]. Thus, these proteins take part in formation of the enamel. For the function of

these proteins during enamel formation in detail, reader can refer [5].

1.2.2 Dentin

Dentin is the supportive tissue of teeth. On its outside surfaces, it is covered by the

enamel at the crown and by the cementum at the root. On its inside surfaces, dentin

encapsulates the pulp cavity. The most prominent structure of dentin is its microscopic

channels, called dentinal tubules. The odontoblasts, which secret the dentin, extend into

the dentin matrix [7]. The dentinal tubules originate from the pulp and radiate into the

dentinal-enamel junction. It is reported that the dentinal tubules has largest diameter and

density near the pulp [8]. The hardness of dentin is lower than that of enamel and

exhibits lower stiffness (which increases it deformability, enabling it to be flexible). It is

this flexibility of dentin that prevents fracture of enamel during mastication of hard

objects. Also it is the dentinal tubules that make the anisotropic photon propagation

inside teeth.

4

1.2.3 Pulp

The pulp plays an important role in supporting tooth function. Although pulpless teeth

can still function normally, it has been reported that pulpless teeth are more easily

infected by bacteria than teeth with pulp [9]. Furthermore, without the reparative

processes of the cells in the pulp, teeth are more prone to fracture. Hence, the pulp plays

a crucial role for teeth in defending the bacterial invasion and preventing mechanical

failure. Reader can refer to [10] for more details on the mechanism of the pulp

components preventing bacterial invasion. The pulp contains blood vessels, nerves,

odontoblasts and other cells like fibroblasts etc. The main role of odontoblasts is to form

and repair dentin and they may also be involved in sensory transduction when teeth are

stimulated [11]. Hence, this process needs large amounts of energy as well as oxygen

consumption. It has been shown that large amounts of oxygen consumption is essential

to maintain the vitality of odontoblasts [12]. It is also demonstrated that in the rat incisor,

the oxygen consumption rate can reach 3.2 ml/min per 100 g tissue which is comparable

with brain oxygen consumption [13].

When teeth are affected by caries, there is a reduction in the size and number of

odontoblasts. Herein, the oxygen consumption will decrease correspondingly. This can

happen even before inflammation of the pulp [3]. Thus, a reduction of oxygen

consumption or oxygen saturation in teeth may be an early sign for dental caries.

From the discussion above, the pulp microcirculation plays an important role in

maintaining the integrity of teeth. It provides the odontoblasts with oxygen and other

nutrients. The blood flow rate of pulp microcirculation is relatively high in comparison

to other tissues, e.g. muscle. It is estimated to be about 40 ~ 50 ml/min per 100 g tissue,

as reported in [14]. The high blood flow rate demonstrates that high oxygen

consumption takes place in the pulp. Based on this, one can also conclude that any

disease that affects the pulp, not just dental caries, will change the oxygen consumption

of pulp. This is shown by work reported in [15], where periodontal disease was

demonstrated to lower oxygen saturation in teeth.

5

1.2.4 Cementum

Cementum is a calcified substance that covers the dentin at the root of teeth. Cementum

is softer than dentin and is composed of 50% hydroxyapatite. The rest of the cementum

is composed of proteins, which are responsible for the formation of cementum. The main

function of cementum is to anchor the collagen fibers to the surface of teeth root while

also playing a crucial role in protecting the integrity of teeth [16].

1.3 The necessity of detecting pulp functions

As discussed in Section 1.2.3, the pulp microcirculation plays an important role in

maintaining the normal function of teeth. Therefore, detecting pulp function is expected

to be important when diagnosing or treating the dental diseases, such as dental caries. At

the early stages of dental caries, the pulp degeneration can occur without obvious

symptoms [17]. Besides the radiographical changes of teeth alone cannot be viewed as a

definitive evidence of pulp activity [18]. As discussed above, a change in oxygen

consumption can occur even before the inflammation of pulp. Thus it is reported in [15],

[19], [20] that oxygen saturation is an alternative parameter to distinguish whether a

tooth is in a normal state or not. One technique that has been employed to assess the

oxygen saturation of teeth is pulse oximetry. It is known that pulse oximetry can provide

accurate, real-time measurements of artery blood oxygen saturation at low cost.

However, the biggest disadvantage of using pulse oximetry is that only one value of

oxygen saturation can be provided at one sampling time. It cannot specify the spatial

information of oxygen saturation.

Providing a measure of spatial oxygen consumption of the pulp to dentists may

help provide a panoramic view of pulp vitality. This can enable measurements of early

stages of pulp compromise. In the content below, commonly used therapies for treating

dental caries are introduced. This will put in context the utility of using optical

tomography for assessing pulp vitality.

Root canal therapy or endodontic therapy is widely used clinically, because it has

been proven to be an effective method to maintain the integrity of teeth and to protect

against further bacterial invasion [21]. During the surgical procedure, a cavity is created

to remove the affected enamel, dentin and to extract the damaged pulp tissue. Then, the

6

hole is shaped by different size tapers, sterilized and processed for placement of gutta-

percha or eugenol-based cement [22]. Root canal therapy has a high success rate [23]–

[25] in treating dental caries, but there are still disadvantages to the procedure. Reader

can refer to [26] for the disadvantages of conventional root canal therapy. Herein only

the disadvantages that are closely related with pulp function are emphasized. As stated

earlier, pulpless teeth are more prone to infection as well as mechanical fracture. In [26],

it is noted that after all the pulp has been removed from the teeth and processed for root

canal therapy, there is no residual sensitivity. As a consequence, secondary caries does

not lead to pain (or other sensation), resulting in more severe caries. This increases the

chances of tooth loss in comparison to teeth that were not treated by the therapy [27]. As

a result, there is a greater awareness that only the affected pulp has to be removed even

during root canal therapy (i.e. it is still necessary to preserve the pulp).

It has been proven that the necrosis of pulp occurs gradually. Infection and

inflammation of pulp has been observed to be compartmentalized before necrosis of the

entire pulp occurs [28]. Thus, it is completely possible to only remove the inflamed pulp

and preserve the rest of normal pulp, which may further recover after surgery [26]. As

stated earlier, oxygen consumption may be one of the features that can determine pulp

health. It may also be possible to evaluate pulp health in the early stages, before the pulp

is severely compromised. To differentiate the inflamed pulp and normal pulp before root

canal therapy, one needs to evaluate techniques that can detect spatial oxygen

consumption. One primary means to detect oxygen consumption is optical tomography.

However, to our knowledge, the possibility of optical tomography of teeth has yet to be

investigated. Hence it is necessary to develop optical based imaging modality to provide

dentists a panoramic view of pulp situation before and after root canal therapy.

1.4 The importance of detecting the situation of dental fillings and

reducing the population of bacteria

After the cavity forms, dentists usually use root canal therapy to fill the diseased teeth to

prevent further decay of hard tissue. However, even though root canal surgery is the

main conventional therapy, it may still fail even when following the highest standards of

surgery [29]. Multiple factors can influence the success rate of root canal therapy, such

7

as broken instruments, over- and under-fillings of dental materials [29] etc. The failure

of the surgery would not restore and stabilize teeth and could cause secondary caries in

teeth. Hence, it is preferable to monitor the cavity status after surgery to ensure that the

cavity is filled appropriately. Unfortunately, conventional X-ray based radiography

cannot provide the required contrast to perform this task [29]. Hence, it is necessary to

investigate the potential use of optical imaging modalities to detect the filling situations

after surgery.

Since dental caries are induced directly by the bacteria [30], such as

Streptococcus mutans, reducing the population of bacteria through daily cleaning of the

teeth is necessary even for healthy oral situations. As discussed above, during the root

canal therapy, one of the most important steps is to sterilize the inside of teeth

thoroughly. It has been proven that bacteria can invade deep inside the dentinal tubules,

thus when performing the sterilization during surgery, more teeth dentin needs to be

wiped off in order to make sure the sterilization of teeth is complete. Even so, it is not

generally easy to fully sterilize and clean teeth. In some cases, the failure of the surgery

is due to persistent or secondary infection induced by bacteria. There are multiple factors

that can result in incomplete removal of bacteria. One factor is the irregular shape of

teeth, such as one tooth with several branched canals. Thus, this will prevent some part

of these irregular canals from being affected by the common chemo-mechanical

sterilization using standard technique and instruments [29]. The other factor may be the

leakage of coronal cover [29]. Reference [29] describes the factors that may cause

failure of root canal therapy in details and reader may refer to it for more information. In

summary, it is complete possible that sterilization of the bacterial and cleaning of debris

of pulp is not thoroughly and this will cause another infection or decay of teeth. To make

things worse, radiography of an apparently well-treated tooth cannot necessarily ensure

the thoroughly cleaning [29]. Herein, after the filling the cavity with the dental

composites, the survival bacteria can induce secondary dental caries and this is the may

reason why well-treated root canal therapy will fail.

Fortunately, upconverting nanoparticles (UCNP) can combine investigating the

filling situations after surgery and killing bacteria together. Herein, we investigate the

potential to use UCNPs as a theranostic agent in dental filling. UCNPs have the unique

8

ability to emit light in both the near-infrared (NIR) and visible regions after excitation

with NIR light. The advantages of UCNPs are insensitivity to background

autofluorescence [31], no photo bleaching and sharp emission band [32]. For dental

applications, the emission in the visible (blue light) can act as a means to kill bacteria

[33] and whereas the visible or NIR emission can be used for imaging. In phantom

studies, one advantage noted for optical tomography applications is the increase in the

resolution limit of fluorescence diffuse optical tomography (FDOT) due to its sharp

sensitivity profile [34]. Reader can refer to Chapter 4 for more details introduction of

UCNPs.

In summary of section 1.3 and 1.4, the dental pulp plays an important role in

maintaining the teeth integrity. Even the root canal therapy has to be processed, it is still

necessary to recover the pulp in order to prevent secondary decay of teeth infected by

bacteria. Whether killing bacteria completely before and after root canal therapy directly

decides the quality of treated teeth. Also monitoring the fillings situation is helpful for

reducing the failure rate of root canal therapy. Based on these discussions, the specific

aims will be proposed as follows: (1) Develop a tomography algorithm to image pulp

oxygen consumption as well as fillings situation after surgery; (2) Design and develop a

prototype imaging system to perform (1) ex vivo; and (3) Design and implement a

therapy to kill the bacteria, especially for sterilizing the residual bacteria after root canal

therapy.

1.5 The challenge of detecting pulp activity and status of fillings

Below, some of the main challenges in using optical means for optical tomography of

teeth are listed. It is first noted that considering the possibility that this technique may be

readily applicable in clinical dental applications, the medical instrument based on optical

tomography has to be ergonomic and low cost [18].

1.5.1 Lack of other suitable imaging modalities to assay pulp function (X-ray) and

status of fillings

Most dental caries detections are performed by dental X-ray radiography. Dental

radiography has proven to be effective in detecting dental caries even before they are

visible. An added advantage to using X-rays is that radiography is suitable in detecting

9

the caries between the teeth. However, dental radiography can only provide limited

information of pulp, because the pulp is composed of blood vessels, nerves and cells,

which are soft tissue with low X-ray contrast. Even for the dental caries cavity itself, the

use of conventional radiography is still challenging [35] because of its limited

information and sensitivity. The newer cone beam computed tomography (CBCT) may

provide improved sensitivity and detection of dental caries. But when concerning the

radiation dose of X-ray as well as the expense, the use of CBCT is still limited [36]. To

the fillings status, it also lacks X-ray contrast so it is not easy to observe the distribution

of dental fillings after the shaping of the root canal even with micro-CT ex vivo.

Conventionally used radiography will only make things worse.

As stated earlier, low cost and easy applicability in a dentist’s office is one of the

requirements for an instrument that is capable of detecting caries. Therefore, since

magnetic resonance imaging (MRI) based techniques are still expensive, these are not

with routine dental checks. Because of the complex structure of teeth, and the presence

of multiple interfaces with different properties, ultrasound is not a modality that can be

easily implemented, so, it is not suitable for dental imaging.

1.5.2 Optical imaging of soft tissue or fillings covered by hard tissues

It is widely known that optical based imaging modalities can penetrate deep into bio-

tissues and can provide metabolic information such as oxygen consumption etc. [37],

[38]. Therefore, optical based modalities should be a feasible technique for detecting the

pulp activity. However, the location of the pulp makes the use of optical based

modalities challenging. First, the pulp is covered by hard tissues, such as enamel and

dentin. The optical properties of enamel and dentin are relative large in comparison to

other human tissues. The absorption coefficient of dentin is about 0.3 mm-1

while the

scattering coefficient is about 28 mm-1

around 700 nm [39]. Large absorption coefficient

of dentin indicates that the transmitted light will be attenuated more, thus the sensitivity

of the detector will need to be relatively high in order to detect the significantly lower

light intensity that is emitted from the tooth. Second, strictly speaking, the hard tissue

covering the pulp is anisotropic, which means the optical properties depend on the

direction of light propagation. The micro-aligned structure of dentin and dentinal tubules,

10

is the reason for this anisotropic phenomenon [40]. The anisotropy of optical properties

will make the optical imaging tomography much more complex. This will be discussed

in detail in a separate section.

1.5.3 Limited volume of pulp and root canal

The volume of pulp is limited. It is reported that the mean volume of pulp for human

teeth is around 0.02 ml [41]. Thus, in order to discern the pulp, high resolution optical

imaging has to be considered. From our previous ex vivo experiments using mesoscopic

fluorescence molecular tomography (MFMT), 0.2 mm3

voxel size is appropriate to

detect pulp inside teeth [42].

1.5.4 The curvature and size of teeth

The size of individual teeth is quite small compared with tissues in which optical

tomography has been demonstrated successfully, namely, the breast and the head. Thus,

the curvature of teeth will influence optical tomography more than other tissues. Since

the shape of teeth will be very irregular, before assessing the accuracy of reconstruction

results, the system has to be calibrated carefully [42]. If image registration is required, it

also needs to be performed carefully.

1.5.5 Blood flow of pulp and sampling rate

As discussed above, pulse oximetry can be applied to detect the oxygen saturation of

pulp. And based on the principle of pulse oximetry, it can be concluded that the blood

inside the pulp is not in a static state. As blood flows in the vessels, the diameter of

blood vessels inside the pulp will expand and contract regularly at the rhythm of heart

rate. This requires the optical imaging with a sampling rate that can enable the optical

properties of pulp to be viewed as constant.

1.6 Optical based imaging modalities

Based on the discussion above, in order to detect the pulp vitality inside teeth, optical

based imaging modalities are feasible methods besides simple pulse oximetry. Multiple

imaging modalities have been developed to detect teeth vitality based on the change of

optical properties in teeth infected by bacteria. Optical imaging techniques have been

11

developed to detect carious lesions by exploiting the way white light passes through the

teeth (trans-illumination) and how light of specific wavelength reacts with the molecular

changes in the dental tissue (laser fluorescence). These diagnostic techniques are usually

based on optical contrast variations elicited by significant bacterial colonization and/or

byproducts of bacterial activity. The commercial system, DiagnodentTM

, is based on

changes in fluorescence of light irradiated on tooth surfaces resulting from porphyrins

that are produced by bacteria that populate carious lesions, in which the bacteria

fluorescence bands are in the region 580–600 nm, with another peak around 635 nm [43].

This system detects occlusal dentinal lesions with greater success than conventional

methods [44], but is not able to discern the depth and severity of early lesions [45].

Another quantitative light-induced fluorescence (QLF) system is based on detecting the

decrease in auto-fluorescence generated by aromatic amino acids as a result of bacterial

activity in lesions, when irradiated with different wavelengths of light [46]. When QLF

has been used to detect and longitudinally monitor enamel lesions, the sensitivity and

specificity of the technique is 64% and 55% for smooth surface and occlusal lesions,

respectively. However, QLF systems, which aim at detecting a decrease in the

autofluorescence signal, are confounded with signals associated with a loss of mineral

volume [47]. Moreover, as QLF is unable to retrieve the depth and structure of the lesion,

QLF does not meaningfully improve the diagnostic abilities of conventional methods [48].

Finally, trans-illumination, in which attenuation of visible light transmitted through the tooth

is detected, has been used to detect interproximal, smooth and occlusal lesions [49], [50].

Attenuation of light through the tooth structure, resulting primarily from light scattering, is

confounded by multiple variables in tooth structures. In addition, trans-illumination does

not provide 3-D information, and is unable to detect lesion depth [51].

All of the prevailing optical imaging based modalities to detect the dental pulp

have the disadvantages as discussed above. With the stated challenges in detecting pulp

function, several special issues have to be considered. Optical tomography is an inverse

problem based technique with multiple light projections through the sample to retrieve

the optical properties of the object [52], [53]. Generally, optical tomography needs to

solve two problems, namely, the forward and inverse problems. Roughly speaking, the

forward problem is to investigate the photon propagation and transportation in the tissue

12

[54] while the inverse problem is to reconstruct the optical properties based on

measurements at the boundary [55], [56]. There is an enormous amount of literature to

discuss every aspect of optical tomography in detail. However, to our knowledge, there

are no studies related to optical tomography of pulp. Our previous experiments based on

MFMT demonstrated the potential of optical tomography in dental applications first time

(see Chapter 3 for more details).

Briefly, our investigative experiments using MFMT have proven that it is

possible to resolve the depth of dye used to mimic the dental lesions. In the experiments,

several holes at different depth from the surface on which the light was irradiated were

drilled and a capillary filled with different concentration of dye was inserted into the

hole. Then the tooth worked as a phantom was imaged by MFMT. Our experiments

showed that up to till 2mm below the imaged surface, MFMT can reconstruct the dye

distribution with average error up to 20% in volume. Preliminarily, it can show the

spatial information of dye inside the teeth. However, there are still disadvantages of

directly using MFMT. One is the method that we used to generate the sensitivity matrix.

It was Monte Carlo voxel based method. Also it utilized the symmetry of the object to

reduce the calculation time. However, none of these assumptions can be used to simulate

the real situation of teeth strictly speaking. The other difficulty for directly using MFMT

on dental application is that the configuration of MFMT cannot be adapted for use in the

mouth without improvements. Next, I present the fundamentals of forward and inverse

problems, as it relates to detecting pulp function and fillings status after surgery with

optical tomography.

1.6.1 Forward problem

Generally, there are two large categories of methods that can be employed to solve the

forward problem; one is a deterministic method through solving the partial differential

equation (PDE), and the second is a stochastic method such as Monte Carlo (MC)

simulation.

MC simulations have not been widely used in optical tomography due to their

computational burden. Though it is the most accurate method in predicting the photon

distribution and viewed as the gold standard to assess the accuracy of other results, only

13

recently MC based tomography has been achieved due to efficient MC formulation and

algorithmic implementation. For instance, our group has already investigated MC

method in detail, form voxel based to mesh based [57]. Also, in order to get higher

calculation speed, graphics processing unit (GPU) based code have also been developed

[58]. However, in order to deal with the optical properties as a function of light

propagation, few MC methods have been developed unless simplified models neglecting

this phenomenon as employed in Chapter 3 and 4. In reference [40], the author

developed a MC method based on the magnetic wave transportation along an infinitely

long cylinder, which is used as a model of dentinal tubules.

An easier method dealing with this issue is to solve the radiative transfer

equation (RTE), from the steady state of RTE (we decided to use a continuous wave

light source, the reason will be specified later)

2

( , ) ( , ) ( ) ( , )d ( , )t s

S

f q' ' 's r s r s s s r s s r s (1-1)

In Eqn. (1-1) 𝒔 represents the directional vector on the unit sphere 𝑆2, 𝒓 is a position

vector in 3-D Euclidian space 𝑹3, 𝜑(𝒓, 𝒔) is the photon radiance (m-2

sr-1

), 𝜇𝑡 and 𝜇𝑠 are

attenuation coefficient (mm-1

) and scattering coefficient (mm-1

) respectively, 𝑞(𝒓, 𝒔)

models the light source inside the media. 𝑓(𝒔, 𝒔′) is the phase function and it describes

the probability of a photon scattered from 𝒔′ to 𝒔. The benefits of employing the RTE are

obvious. It can describe any phase function, including the widely used Henyey-

Greenstein (H-G) phase function [59] in bio-optical imaging. If necessary, it can

incorporate a simulation wherein the optical properties are dependent on the direction of

light propagation as described in Eqn. (1-1). The boundary condition of Eqn. (1-1) is

often described as no photon will travel inward the object (See Chapter 2 for more

descriptions). A detailed literature review can also be seen in Chapter 2.

Generally speaking, solving an RTE in Eqn. (1-1) will be more difficult than

solving the widely used diffusion approximation (DA) in optical imaging [60]–[62]. The

main reason why the RTE is as our first priority compared with DA is our application, to

detect the pulp activity. The size of teeth is relatively small in comparison to most

situations in which DA applies. Besides, the region of interest (ROI) is near the source.

Both of the circumstances do not meet the assumptions, where DA can apply [62]. Also,

14

DA based methods cannot handle the situation that the optical properties which depends

on the direction of photon propagation in tissue without improvement of original DA

[63]. The Dirichlet boundary condition or Robin boundary condition [64] is often

applied when solving DA.

In summary, for the forward problem, in order to enable 3-D optical imaging of

teeth, solving RTE with FEM to get the photon distribution is an attractive option.

Besides, under the simplified situations without considering the anisotropic photon

propagation in teeth, MC methods will also be employed due to their accuracy, easy

implementing and stable solution compared with RTE. Also novel MC methods

developed in our group will be used as the ground truth to assess the accuracy of

algorithm solving the RTE.

1.6.2 Inverse problem

The inverse problem is to reconstruct the optical properties inside the ROI through

measurements made at the boundary. Traditionally, multiple methods have been already

investigated. Also, several review papers offer a detailed description of these methods

[55], [65]. Basically, optical tomography inverse problems are transformed into an

optimization problem [52], [62], [66]. Below is a brief introduction for the optimization

problem.

If there exist 𝑁 nodes, assuming the optical properties assigned to the nodes are

represented by 𝑞𝑖, (𝑖 = 1, … , 𝑁 ), when there are 𝑀 sources at the boundary. So the

inverse problem is to recover 𝑞𝑖 from the 𝐿 detectors placed at the boundary of the object.

Thus, the target function is expressed as,

2 22

1 1

( )L M

l l

m m

l m

C z 0q q q (1-2)

In Eqn. (1-2), 𝐶𝑚𝑙 is an operator, which calculates the output flux at detector 𝑙 upon

irradiation with a source 𝑚, based on the optical properties 𝒒. 𝑧𝑚𝑙 is the measurement at

detector 𝑙 when the source 𝑚 is turned on. 𝜆 is Tikhonov regularization parameter, where

𝒒𝟎 is the initial guess of optical properties of the object based on some prior knowledge.

The inverse problem is to find the optical property distribution 𝒒 to minimize Eqn. (1-2).

15

It can be easily concluded using Taylor’s theorem that Eqn. (1-2) can be finally

transformed to a set of linear equations, as shown in Eqn. (1-3),

( ) ( )T T

0J J I q J Φ q q (1-3)

In Eqn. (1-3), the Jacobian matrix 𝑱 describes how the output flux changes at the position

of detector when the optical properties changes. 𝛿𝒒 is the updated optical properties at

each iteration while 𝛿𝝓 represents the difference between the measurement and

calculated output flux (theoretical value based on the optical properties 𝒒 ). In our

experiment, the depth dependent Tikhonov regularization method was employed as

discussed in Chapters 3 and 4.

It is obvious that calculating the Jacobian matrix correctly (or sensitivity matrix)

is crucial when solving equation Eqn. (1-3). There are 3 widely used methods to

generate the sensitivity matrix, namely direct method [67], perturbation method and

adjoint method [68]. The direct method and perturbation method are reported to have

more forward calculations than the adjoint method [69]. However, the direct method is

also efficient when dealing with a 1-D reconstruction problem [67]. In this work, the

adjoint method will be employed to calculate the sensitivity matrix 𝑱. Briefly speaking,

the adjoint method uses the reciprocity theorem, which is ‘the measurement of flux at

node 𝑗 that is due to a source at 𝑖 is equal to the measurement of the photon density at 𝑖

that is attributable to a source at 𝑗’ [68]. If using the adjoint method to calculate the

sensitivity matrix, the number of forward calculations will be 𝐿 + 𝑀.

In summary, for the inverse problem, adjoint method will be utilized to produce

sensitivity matrix. And the source and adjoint source (detector) Green’s functions will be

calculated using MC method in Chapters 3 and 4 due to its relative ease of

implementaion and higher accuracy.

1.6.3 Hardware settings of optical imaging system

Compared with complex imaging modalities like computed tomography (CT) and MRI,

optical based imaging systems are relatively simple and easily to be implemented.

Theoretically, optical imaging systems are composed of light source(s), detectors,

multiple lens and optical components to adjust the light path. However, the optical

16

imaging device needs to be very carefully calibrated in order to obtain good system

performances.

Considering the specific requirements in detecting the pulp, several choices have

to be made. First is the selection of temporal modulation of the light source. Generally,

there are 3 commonly used modulations, time domain, frequency domain and continuous

wave. Briefly, time domain methods uses very short pulses to drive the laser while

frequency domain modulation employs high frequency sine waves [70]. The continuous

wave uses the DC current to drive laser diode or light emitting diode (LED). There are

multiple papers to compare these different systems [71], [72]. Considering the

requirements for the dental clinical application, the devices should be ergonomic and

low cost. The time domain driven source is not appropriate because of its high cost. Thus,

it is not practical for every dentist to own a time domain imaging systems. Thus, the

feasible solutions can only be frequency domain or continuous wave system.

The frequency domain imaging system employs frequencies in the ranges

between 100 MHz to 200 MHz to drive the laser. In this frequency range, multiple

components have been commercialized for optical imaging systems. Although the

components for frequency domain imaging are compact, they are not yet inexpensive.

To make things worse, the high frequency circuits are difficult to design and debug, and

generally, it is not viable to debug these circuits without a printed circuit board (PCB).

Thus continuous wave imaging systems is our first choice (See Chapter 4 for more

details). Correspondingly, that is also the reason why we will mainly focus on the quasi-

steady state of the PDE’s mentioned above.

The other important component for the optical imaging systems is the detector.

Different groups have different choices, such as photo multiplier tube (PMT), or CCD

camera. Considering our driven mode of light source, continuous wave mode, the CCD

camera will be a good choice because they are relatively inexpensive, high speed, and

compact with abundant software packages. The only concern is that the sensitivity of the

camera may not be enough for detecting the light being transmitted through the teeth,

especially for UCNPs emission light. In the experiments, for most cases, an electron

multiplying CCD (EMCCD) was employed without EM gain.

17

1.7 Reducing bacteria after surgery and monitoring the situation of

fillings

As mentioned earlier, when a tooth cavity forms, the conventional treatment is root canal

therapy. After removing and cleaning all the soft tissue inside the pulp chamber, the next

step is to sterilize the teeth completely to ensure that the residual tissue or bacterial load

cannot induce further decay after filling the cavity with composites. However,

sometimes, the complete sterilization is not easy to obtain.

After the cavity is filled by the gutta-percha or eugenol-based cement, if the

bacteria survive in deep place of teeth, killing bacteria will be difficult. Since the

infected area is covered by the filling, dentists cannot reach that place directly. Thus, it is

impossible to sterilize again using chemical methods. Another proven effective method

that can kill the bacteria is to use ozone [73]. However, the infected area still needs to be

exposed to ozone. Thus this method may not be appropriate to kill bacteria covered by

dentin or filling composite. Laser irradiation has also been employed to kill bacteria in

root canal therapy [74]. One application of laser is that dentists employ thin optical fiber

that can enter the root canal to sterilize the part that is not easy to reach without

enlarging the cavity inside teeth. But if the root canal is covered by the filling

composites, it is impossible to put an optical fiber inside the teeth.

The most commonly used wavelength for killing bacteria or microorganism is in

ultra-violet (UV) range, approximately from 100 nm to 400 nm. UV light will induce

damage to bacteria by forming the dimer of RNA and DNA [75]. Recently, it has been

reported that high intensities of visible photons can also be employed to kill bacteria in

dental applications [74]. The reason that low energy photons (low frequency wavelength)

can function as a bactericide is based on the generation of reactive oxygen species (ROS)

that are harmful to bacteria. The ROS is generated because of the presence of

endogenous cellular photosensitizers such as cytochromes, porphyrins and NADH [76].

The generated ROS include oxygen radicals, singlet oxygen and peroxide [33]. We

propose to use red light and near infrared light to kill bacteria because it can penetrate

human tissues including bone from several mm to around 1 cm [77]. If NIR light enables

energy deposition deep in the tissue, a conversion mechanism from NIT to visible light

is still required for therapeutically effect.

18

UCNPs are a novel class of fluorophore that are ideal for our application. First,

unlike normal linear dye, UCNPs absorb low energy photon, normally in the NIR range.

Just as the discussions above, NIR light can penetrate deep for human tissues in general.

Thus it is possible to reach the deep teeth root embedded in jaw. Second, based on

different synthesis method, UCNPs have multiple emission bands. The NIR emission,

with lower wavelength than excitation, can also penetrate thick tissues and make the

detection easy. Thus NIR emissions can be employed to image the fillings situation after

surgery. Besides, for UCNPs have the nonlinear dependence of power, the sensitivity

profiles are normally sharper even with large scattering coefficient compared with

normal linear dye. This guarantees higher resolution which is appropriate for small root

canal imaging. Third, most UCNPs also have blue or UV light emission, which has been

proven to be a bactericide. More importantly, since UCNPs are mixed with dental

fillings, the irradiation region can be controlled that only the root canal and adjacent area

are influenced. Compared with wide field blue light irradiation, this is apparently safer.

1.8 Structure of thesis

In Chapter 2, I will introduce the theoretical framework and algorithm to solve RTE

efficiently and stably in detail with multiple numerical examples compared with MC

methods. In Chapter 3, the MFMT experiments to detect pulp situations (dental lesions

of soft tissue) will be introduced. In Chapter 4, the details of UCNPs as well as its

application in optical imaging will be demonstrated with in silico and real experiments.

In Chapter 5, ex vivo killing bacteria using UCNPs will be introduced to show the

feasibility of our proposed method. In Chapter 6, I will summarize the work and discuss

future work in detail.

19

2. Algorithms to solve the radiative transfer equation as a forward

problem

As discussed in Chapter 1, teeth with high optical properties and a relatively small scale

cannot employ the widely used diffusion approximation (DA). Moreover, if considering

dentinal tubes inside teeth, the light propagation model will be more complex. Here, we

preliminarily investigate the possibility to employ the radiative transfer equation (RTE)

to predict photon propagation inside tissues with complex shape and small size such as

teeth. Also of note is that, for involving the light propagation direction, RTE has the

potential to deal with anisotropic propagation conditions. Below is a brief literature

review of algorithms developed in the optical imaging field to solve the RTE. *

There are multiple challenges in solving the RTE analytically or numerically.

The RTE can only be solved analytically in very few situations [79], [80], such as when

semi-infinite media are used [81]. In such instances, the forward model can be computed

in a matter of seconds. However, for media with complex boundaries, it is extremely

difficult or even impossible to derive an analytical solution. Under these conditions, the

RTE needs to be solved using numerical methods.

Prior to applying the standard numerical methods, it is necessary to first process

the angular domain since the RTE retains the anisotropic nature of light propagation.

Over the years, a few methods have been proposed to perform this task. Tarvainen et. al,

[52], [82] formed a partition on the angular domain and employed local linear basis

functions to approximate the angular components of the solution. Thus, when forming

the weak form of the RTE by the finite element method (FEM), the high-dimensional

integral with respect to the space variables and angular variables had to be calculated.

Although this calculation could be transformed into two independent low dimensional

integrals by Fubini’s theorem, the tensor product of these two integral related matrices

still needed to be performed [52]. Besides using linear basis functions, Surya Mohan et.

Portions of this chapter previously appeared as: F. Long, F. Li, X. Intes, and S. P. Kotha, “Radiative transfer equation

modeling by streamline diffusion modified continuous Galerkin method,” J. Biomed. Opt., vol. 21, no. 3, pp. 036003-

1-036003-12, Mar. 2016. [78]

Portions of this chapter have been submitted as: W. Han, F. Long et al., “Radiative transfer with delta-Eddington-type

phase functions,” Applied Mathematics and Computation.

20

al. [66] also investigated the use of spherical harmonic functions to approximate the

angular components of the solution. However, this formulation still required

computation of the high dimensional integral and tensor product of matrices (for

example, when considering the 3-D space and 2-D unit sphere in angular space, a 5-D

integral was calculated in general). Another method to handle the angular domain of the

RTE is to employ the discrete ordinate method (DOM) or its modified version. The

standard DOM [83] consists of transferring the RTE to a system of coupled partial

differential equations (PDEs) by using numerical quadrature to calculate the integral

term (the term that contains the phase function). Klose et. al.[84], Fujii et. al. [85] and

Ren et. al. [86] employed level symmetric quadrature to directly discretize the RTE into

a number of different PDEs. The required number of PDEs was determined by the

number of discrete angles of quadrature. Employing the modified DOM, Asllanaj et. al.

[87] and Wang et. al. [88] partitioned the angular domain into finitely many sub-domains

and performed the integration with respect to angular variables on both sides of the RTE

with the assumption that solution to the RTE (photon radiance) was constant with

respect to angular variables inside each sub-domain. This would lead to easily-computed

integrals. However, a 4-dimensional integral, which could not be transformed into lower

dimensional integrals, still needs to be computed when dealing with the term of the RTE

containing the phase function [88]. In comparison to dealing with the angular domain as

mentioned above, DOM is relatively easy to implement as it does not require calculation

of the integral with respect to angular variables [89]. Consequently, the dimension of the

RTE is reduced, allowing for computation in 3 or 4 dimensions. The main disadvantage

of DOM implementations is the so-called ‘ray-effect’ which occurs due to angular

discretization and is prominent in optical imaging since the anisotropy factor is large

(𝑔 = 0.8-0.9) [90].

After dealing with the angular domain, multiple methods such as the finite

difference method (FDM), finite volume method (FVM), or FEM can be employed to

numerically solve the RTE. Klose et. al. [84]and Wang et. al. [88] employed an upwind

FDM to approximately calculate the partial derivatives with respect to spatial variables.

The FDM is relatively easy to implement compared with other numerical methods. The

main disadvantage of the FDM is its use of structural grids, which cannot easily and

21

accurately approximate complex, curved boundaries (although it is possible to deal with

this situation as proposed in [91]). For such scenarios, the FVM can be employed since it

can deal with arbitrary shapes [87], [92]. To take into account the directional

propagation of light, Asllanaj et. al. [87] proposed the inclusion of the nodal values

located upstream at each integration point. In order to reduce the number of unknowns

and have high resolution [87], the control elements were normally rebuilt to surround the

nodes. The main limitation of the FVM is that it can be very involved to achieve higher

order accuracy on un-structured grids as the numerical scheme of higher order FVM

requires the use of multiple nearby elements [93].

In terms of FEM-based approaches, it is widely known that employing

continuous Galerkin FEM (CG-FEM) directly to solve RTE will generate non-physical

oscillations [94] due to the transport effect. Tarvainen et. al. [52] and Mohan et. al. [66]

proposed solving the RTE by CG-FEM, and in their work the standard streamline

diffusion modification technique was employed. The discontinuous Galerkin FEM (DG-

FEM) was also employed by Eichholz et. al. [95] and Gao et. al. [83] to solve the RTE.

Compared with CG-FEM, DG-FEM has additional unknowns associated with the

relaxation of continuous constraints of functions at the boundary between adjacent

elements. However, DG-FEM allows for a straightforward approach to solving transport

equations such as RTE [93].

Herein, based on the considerations of the ease of implementation and

computational efficiency, an ‘intermediate’ method, founded on the modification of

DOM with CG-FEM, is proposed. In our work, several improvements were made, and

the performance of the methods was illustrated numerically. First, various numerical

quadratures were implemented to translate the RTE to a system of coupled PDEs with

DOM. A comparison was performed to understand the effect of different numerical

quadratures on the solution of the RTE with multiple combinations of optical properties.

Second, a phase function normalization strategy [96], [97] was employed in order to

efficiently lessen the instability (oscillations) or ‘ray-effect’ commonly encountered

when using DOM [83]. More precisely, during the discretization of the angular space,

the conservative properties of the phase function will generally be lost. Without

employing large numbers of discrete points in angular space, this will generate spurious

22

oscillations, as well as the ‘ray-effect’ mentioned above. The phase function

normalization techniques aim to preserve the conservative properties even after

discretization of the angular space so as to reduce the oscillations of solutions with a

fewer number of discrete points. Once the angular space is discretized, a streamlined

diffusion-modified CG-FEM was selected and implemented due to its fewer number of

unknowns compared with DG-FEM as well as its ability to robustly deal with the

transport nature of the RTE, following the work of Brooks et. al. [98].

2.1 Algorithms

2.1.1 Radiative transfer equation

The general RTE can be described as in the following equation [83],

1, , , , , , , , , d , ,t st t t f t q t

v t

' ' 'r s s r s r s s s r s s r s

(2-1)

In the equation, 𝑣 is the speed of light (m s-1

) in the media, photon radiance 𝜑(𝒓, 𝒔, 𝑡) (J

m-2

sr-1

) is defined on the domain 𝑿 × Ω × 𝑹+ , with the spatial domain 𝑿 being the

entire 3-D Euclidian space 𝑹𝟑 or its subspace, Ω being the unit sphere in 𝑹𝟑 (direction or

angular domain) and 𝑹+ is the real time domain. In particular, the angular variable is

expressed as 𝒔 = (sin 𝜃 cos 𝜙, sin 𝜃 sin 𝜙, cos 𝜃), where 𝜃 ∈ [0, 𝜋] is the polar angle and

𝜙 ∈ [0,2𝜋) is the azimuthal angle. 𝜇𝑡 represents the attenuation coefficient (mm-1

) of the

media, and 𝜇𝑡 = 𝜇𝑎 + 𝜇𝑠 , where 𝜇𝑎 and 𝜇𝑠 are the absorption coefficient (mm-1

) and

scattering coefficient (mm-1

), respectively. In the general case, they are all functions of

the spatial variable 𝒓. 𝑓(𝒔, 𝒔′) is the phase function and it describes the probability that

the photon scatters from direction 𝒔′ to 𝒔. The del operator ∇ operates on 𝜑(𝒓, 𝒔, 𝑡) with

respect to spatial variable. 𝑞(𝒓, 𝒔, 𝑡) models the source inside the object. Considering the

settings of our systems (see Chapters 3 and 4), only the quasi-steady state of RTE is

investigated, namely

, , , , d ,t s f q

' ' '

s r s r s s s r s s r s (2-2)

In the biomedical regime, the 3-dimensional (3-D) phase function is usually described by

Henyey-Greenstein (H-G) phase function [52], [59], shown below.

23

2

32 2

1 1,

4 1 2

gf

g g

'

'

s s

s s

(2-3)

In Eqn. (2-3), 𝑔 characterizes the anisotropy of the media, usually in the range of

approximately 0.8-0.9 for biological tissue, indicating the highly forward scattering

properties of the biological tissue.

Since the H-G phase function is difficult to handle in some situations, sometimes

a simplified version of the H-G phase function, the delta-Eddington (d-E) phase function

will be employed [99], [100]. The d-E phase function is shown in Eqn. (2-4)

1( , ) [(1 ) ( ) 2 (1 )]

4f g r g' ' '

s s s s s s (2-4)

In the equation, 𝑔 is the anisotropy as defined in the H-G phase function, 𝑟(𝒔 ∙ 𝒔′)

represents the slow and smooth part of the phase function and Dirac-Delta function

models the singularity when 𝒔 ∙ 𝒔′ approaches 1. In our algorithms, 𝑟(𝒔 ∙ 𝒔′) is chosen as

'( ) 1 3r g' 's s s s (2-5)

In which 𝑔′ is employed to model weakly anisotropic scattering [101].

The boundary condition of Eqn. (2-2) is defined so that no photons will travel

into the object through the boundary [52], [82], [88] except the region irradiated by the

external light, namely,

0

0 0

0, \ , 0,

, , 0

X S

S

r n sr s

r n s (2-6)

In Eqn. (2-6), 𝑆0 is the region on the boundary 𝜕𝑋 where the external light source exists

or the region that is irradiated by an external light source [64]. 𝒏 is the unit outward

normal vector at the boundary 𝜕𝑋.

2.1.2 Discrete ordinate method (DOM)

Following the DOM to discretize the angular space, a numerical quadrature is applied

first to compute the integral in Eqn. (2-2) and this gives

1

, , d , ,N

i i i

i

f w f

' ' '

s s r s s s s r s (2-7)

24

In Eqn. (2-7), 𝑤𝑖 represents the weight in a discrete direction 𝒔𝑖 . The selection of

discrete points on the unit sphere determines the accuracy of the solution to the RTE.

Various numerical quadratures have been proposed to calculate the integral accurately

[95], [102]–[104]. The following are just a brief description of several widely-used

quadratures. Readers can refer to [95], [102]–[104] for more details.

2.1.2.1 Level symmetric quadrature

The level symmetric quadrature is widely used in simulating neutron transport [102].

Assuming 𝐿 is an even number, the total number of quadrature points on a unit sphere is

𝑁 = 𝐿(𝐿 + 2) . Level symmetric quadrature preserves the symmetric property with

reflection and rotation to any coordinate axes [105]. The main disadvantage of level

symmetric quadrature is that its weight will be negative, which is not realistic for photon

transportation when 𝐿 is larger than 22 [102]. The distribution of level symmetric

quadrature points on the first octant is shown in Figure 2-1 (a).

2.1.2.2 Product Gaussian quadrature

The product Gaussian quadrature [95], [106] shown in Eqn. (2-8) is used to calculate the

integral,

2

1 1

( )d ,m m

j i j

i j

g w gm

s s (2-8)

When selecting 𝜃𝑗 on the polar axes, it is calculated as cos 𝜃𝑗 to be the 𝑗 th Gauss-

Legendre node on [−1,1]; 𝑤𝑗 are calculated correspondingly [106]. 𝜙𝑖 are selected to be

uniformly distributed on [0,2𝜋), namely 𝜙𝑖 = 𝑖𝜋 𝑚⁄ . For product Gaussian quadrature,

its degree of accuracy is 2𝑚 − 1 [106]. Note that for the 𝑗 th level of polar direction (all

discrete points lie on the intersection of the sphere and planes which are perpendicular to

the line connecting the points, (0,0,1) and (0,0, −1), with each plane being viewed as

one level), the weight is the same for all 𝜙𝑖.

2.1.2.3 Legendre equal-weight quadrature

The Legendre equal-weight quadrature [103] is similar to the product Gaussian

quadrature at the selection of discrete points along the polar direction. The points 𝜃𝑗 are

25

thus the same as product Gaussian quadrature. 𝑤𝑗 are also selected as product Gaussian

quadrature first, except they need to be further processed before actually applying them

to calculate the surface integral on a unit sphere. However, the selection of discrete

azimuth angle is different with product Gaussian quadrature. For the 𝑗 th level, it

uniformly selects 𝑗 node (equally divide [0, 𝜋 2⁄ ] to 𝑗 + 1 interval, then rotate to other

octant) in the azimuth direction (See Figure 2-1 (c)). And the weight (�̅�𝑖𝑗) at each point

is further calculated as �̃�𝑖𝑗 = 𝑤𝑗 𝑗⁄ . The formula is shown below.

,

d ,ij ij j

i j

g w gs s

(2-9)

2.1.2.4 Lebedev quadrature

From the construction of Lebedev quadrature, it is required that a selection of discrete

points on the unit sphere is invariant under octahedral rotation group with inversion

[107]. It is reported to have the highest accuracy among the numerical quadrature

mentioned above [108], [109], but is not commonly used in predicting the particle

propagation in media, especially in the optical imaging domain. For more details,

including the construction of Lebedev quadrature, the reader can refer to [110], [111].

26

Figure 2-1. Distribution of discrete points on unit sphere 𝛀 of 4 numerical

quadratures (1st octant): (a) Level symmetric quadrature (𝑵 = 𝟖𝟎); (b) Product

Gaussian quadrature (𝑵 = 𝟕𝟐); (c) Legendre equal-weight quadrature (𝑵 = 𝟖𝟎);

(d) Lebedev quadrature (𝑵 = 𝟖𝟔).

All the numerical quadratures mentioned above work well when the integrands

are continuous and smooth. However, the H-G function becomes singular when 𝒔 ∙ 𝒔′ is

around 1 and the anisotropy factor 𝑔 is close to 1. Note that most biological tissues

exhibit large 𝑔. In such case, small number of quadrature points can cause large error

during the calculation of the integral at the right hand side of Eqn. (2-2). To reduce such

error, one can increase the number of quadrature points, at the expense of using more

computational resources such as storage space and time and hence decreasing the

efficiency of the algorithm. On the other hand, none of the numerical quadrature

reviewed above preserves the key properties of the phase function, and this can lead to

large error in numerical solutions, as demonstrated in Section 2.3.5. All these motivate

us to examine the phase function normalization technique, which is to further modify the

numerical quadrature of our choice and will be discussed next.

(d) (c)

(b) (a)

27

2.1.3 Phase function normalization technique

The phase function normalization technique [96], [97] is a method that preserves the

conservative properties or laws of the phase function during the evaluation of the

integral term in Eqn. (2-2) with numerical quadrature by adjusting the quadrature

weights. The following properties hold for the H-G phase function.

, d 1, , df f g' ' ' ' 's s s s s s s s

(2-10)

The first property is from the requirement that the scattered energy is conserved [96],

while the second can be viewed as the definition of anisotropy 𝒈. Generally, for any

numerical quadrature discussed above, especially with large 𝒈, the discretized form of

the left hand side of Eqn. (2-10) is not equal to the right. A small difference between the

two sides of Eqn. (2-10) can generate large error in the solution to the RTE, as shown in

the numerical experiments in Section 2.3.5. The situation can worsen with large

absorption and scattering coefficients, since in these cases numerical methods may even

generate negative solutions which are non-physical.

Briefly, to apply the phase function normalization technique, we start with the

original discrete points 𝒔𝑖 and weights 𝑤𝑖 of a chosen numerical quadrature, mentioned

above in Section 2.1.2. Let 𝜃𝑖𝑗 (based on the original discrete points) represent the angle

between the 𝑖 th direction 𝒔𝑖 and the 𝑗 th direction 𝒔𝑗, and let 𝑓(cos 𝜃𝑖𝑗) be the value of

the phase function at 𝜃𝑖𝑗. Namely, 𝑓(cos 𝜃𝑖𝑗) = 𝑓(𝒔𝒊, 𝒔𝒋) with the specific form of the

H-G phase function. Then the normalized phase function, with its value at the same

point 𝜃𝑖𝑗 denoted as 𝑓𝑛𝑜𝑟𝑚𝑖𝑗

, satisfies the following discrete form of equation Eqn. (2-10)

1 1

1, cos , 1, ,N N

ij ij ij

i norm i norm

i i

w f w f g j N

(2-11)

Here the values of the normalized and the original phase function at 𝜃𝑖𝑗 are connected as

below,

(1 ) cosij ij ij

normf f (2-12)

Plugging Eqn. (2-12) into Eqn. (2-11) a system of linear equations will be obtained with

𝜉𝑖𝑗 being the unknowns. Notice that there are 2𝑁 conditions in Eqn. (2-11) while the

number of unknowns is 𝑁2. If further requiring that the normalized phase function is

28

symmetric such that 𝜉𝑖𝑗 = 𝜉𝑗𝑖, the number of unknowns will be reduced to (𝑁2 + 𝑁) 2⁄ .

In general, there are more unknowns than equations (when 𝑁 > 3), hence there will be

an infinite number of solutions. In order to deal with such equation, a least squares

solution is utilized, which minimizes the 2-norm of the solution vector. A Matlab built-in

function lsqr.m is employed to solve equation Eqn. (2-11). Sparse matrix techniques are

also used here for better efficiency in computation and in storage [105]. Through

normalization, the newly calculated weights are expressed as

�̅�𝑖𝑗 = 𝑤𝑖(1 + 𝜉𝑖𝑗) ∑ 𝑤𝑖(1 + 𝜉𝑖𝑗)𝑖⁄ .

2.1.4 DOM with continuous Galerkin finite element method

After discretizing the integral term in Eqn. (2-2) using a modified numerical quadrature

based on the phase function normalization technique, the RTE is turned into a system of

coupled first order PDEs. That is

1

, , , , , , 1, ,N

ijj j t j s j i i j

i

w f q j Ns r s r s s s r s r s

(2-13)

To solve Eqn. (2-13), CG-FEM with streamline diffusion modification of the test

function is implemented [94], [98]. For a given tetrahedron-based mesh 𝒯 , a finite

element space is defined as,

1{ ( ) ( ) | ( ), }|h KV v C X v P K Kr

Here 𝑃1(𝐾) is the set of linear polynomials defined in 𝐾 and 𝐶(𝑋) denotes the space of

the continuous functions in the closure of 𝑋.

Following the idea of the streamline diffusion method with a modified test

function 𝑣(𝒓) + 𝛿𝒔𝑗 ∙ ∇𝑣(𝒓), the following CG-FEM method will be formed: look for

𝜑𝑗(𝒓) ∈ 𝑉ℎ, such that for any 𝑣(𝒓) and any 𝑗 = 1, 2, … , 𝑁,

1

d d

, d d

j j j t j jX X

N

ijs j i i j j jX X

i

v v V v v V

w f v v V q v v V

s r r s r r r s r

s s r r s r r r s r

(2-14)

Note that 𝜑𝑗(𝒓) ∈ 𝑉ℎ approximates the solution 𝜑(𝒓, 𝒔𝑗) and 𝑞𝑗(𝒓) = 𝑞(𝒓, 𝒔𝑗) . The

parameter 𝛿 is related to the absorption and scattering coefficients, and is chosen to be

piecewise constant, defined as 𝛿|𝐾 = 𝐶ℎ𝐾 (1 + ℎ𝐾𝜇𝑡)⁄ , with ℎ𝐾 being the diameter of a

29

mesh element 𝐾 ∈ 𝒯 . 𝐶 (0.3~0.6) is a constant and the computed solution is not

sensitive to its value for the range of absorption and scattering coefficients examined in

this work on reasonably refined meshes. Our formula for 𝛿 satisfies the guiding principle

for choosing this parameter as outlined in Theorem 1 from [94]. Performing integration

by parts to the first term of Eqn. (2-14), we obtain

, 0

d d dj

j j j j j jX X X

v V v s v Vs n

s r r r r s n s r r

(2-15)

Our final scheme is obtained with Eqn. (2-15) plugged into Eqn. (2-14).

Let 𝒓𝑘 denote the 𝑘 th vertex of the mesh. Considering the Lagrange nodal basis

{𝜓𝑙𝑗(𝒓)}𝑙 of 𝑉ℎ associated with vertices, satisfying 𝜓𝑙

𝑗(𝒓𝑘) = 𝛿𝑙𝑘 , in which 𝛿𝑙𝑘 is the

Dirac-Delta function. Each basis function 𝜓𝑙𝑗(𝒓) is non-zero only within those elements

sharing the vertex 𝒓𝑙. The numerical solution 𝜑𝑗(𝒓) and the test function 𝑣(𝒓) can now

be expanded in terms of basis, namely,

1 1

( ) ( ), ( ) ( )j jM M

j j j j

j l l l l

l l

a v br r r r (2-16)

In Eqn. (2-16), 𝑀𝑗 represents the number of nodes (basis function) in the discrete

direction 𝒔𝑗 . Since the linear polynomial space is employed as discrete space, the

gradient of test function 𝑣(𝒓) will be constant in each element and can be calculated

analytically. With these assumptions, the numerical method can be converted into a

linear system, namely 𝑨𝒙 = 𝒒. In the linear system, matrix 𝑨 contains the terms related

with stiffness matrix, the mass matrix etc., 𝒙 = (𝑎11, 𝑎2

1, … , 𝑎𝑀1

1 , 𝑎12, … , 𝑎𝑀2

2 , … , 𝑎𝑀𝑁

𝑁 )𝑇. 𝒒

relates to the source term, which can be acquired by either employing the boundary

condition (2-6) or directly processing the source term 𝑞(𝒓, 𝒔) [85]. The linear equation

was then solved using generalized minimum residual method (GMRES) [66], which is

performed in the simulations with the Matlab built-in function gmres.m. We want to

point out that the source iteration method or improved source iteration method can also

be employed [83].

When the object and ambient media have different refractive indices, the

reflection and transmittance is modeled by Fresnel’s law [83], [85]. In the simulations,

30

the mismatched boundary condition will be applied when the refractive indices of the

object and the ambient media are different.

2.1.5 Diffusion approximation (DA)

Because of the computational difficulty of solving Eqn. (2-2) , an approximation model

of the RTE, the diffusion approximation (DA), has been widely employed to simulate

photon propagation in tissue [62], [64]. Briefly, photon radiance 𝜑(𝒓, 𝒔) is expanded in

terms of spherical harmonic functions. Then, the truncated spherical harmonic series,

with different number of terms, is used to approximate RTE. For example, the DE can be

viewed as the first order, 𝑃1 approximation [52] (in which case, one uses spherical

harmonic functions whose index is 0 and 1) . The quasi-steady DE can be expressed as

[62],

a qr r r r (2-17)

where Φ(𝒓) is photon density (J m-2

) and it is related to the photon radiance in Eqn. (2-2)

as follows

, d

r r s s (2-18)

Here 𝜅(𝒓) = 1 [3(𝜇𝑎 + 𝜇𝑠′ )]⁄ , where 𝜇𝑎 is the absorption coefficient (mm

-1) while 𝜇𝑠

′ is

the reduced scattering coefficient (mm-1

) and defined as 𝜇𝑠′ = (1 − 𝑔)𝜇𝑠 . 𝑞(𝒓)

represents the light source.

In this work, the Robin boundary condition is applied, that is

2 0,A X r n r r r (2-19)

where 𝐴 is defined as 𝐴 = (1 + 𝑅) (1 − 𝑅)⁄ , with 𝑅 expressed as

2 11.4399 0.7099 0.6681 0.0636R n n n

and 𝑛 represents the refractive index of the object when the ambient medium is air

(𝑛 = 1).

Eqn. (2-17) is a second order elliptic PDE with the Robin boundary condition.

The CG-FEM is a well-established approach for solving such equations and a standard

finite element procedure can be applied. In our simulation, the continuous linear finite

element space is employed and the corresponding matrices (stiffness matrix, mass matrix,

surface integral related matrix and load vector) are formed as in [62], [112], [113].

31

2.1.6 Monte Carlo simulation (MC)

MC is a stochastic method in which a large number of photons are transported to obtain

the photon distribution (photon density) in the medium. This procedure utilizes a

probability function to describe the stochastic events of photons, such as energy

dissipation during collisions with media particles and the new direction of propagation

when a photon hits the particles in media. It involves the absorption coefficient,

scattering coefficient, and phase function for computation. Some authors have

investigated the MC method for higher computational efficiency [114]. Mesh-based MC

methods have also been proposed to deal with complex shapes of media [57]. In our

simulation, mesh-based MC [57], [115], [116] was employed to allow comparisons of

identical meshes with solutions to RTE. Results obtained using the MC method are

viewed as the ground truth standard to evaluate the accuracy of solutions to RTE and DA.

2.2 Settings for numerical simulations

2.2.1 Ideal pencil beam simulation

In our simulations, a 3-D rectangular computational domain [−2.5, 2.5] × [−7.0, 7.0] ×

[−5.0, 0] mm3

is considered first, referred to as Rect. 1, which is similar to the setting

[42] we previously used to simulate the photon propagation in media (teeth) with the

voxel based MC method. A computational mesh was generated, as shown in Figure 2-2

(a) using the precompiled Computational Geometry Algorithms Library (CGAL). The 3-

D domain has 4,858 nodes with 27,456 elements. The position where the photon was

launched was 𝒓𝑠 = (0,0,0) with the direction 𝒔𝟎 = (0,0, −1) orthogonal to the plane

𝑧 = 0mm, see Figure 2-2 (b). This type of source was modeled by the Dirac-Delta

function 𝛿(𝒓 − 𝒓𝑠)𝛿(𝒔 − 𝒔0).

32

Figure 2-2. The 3-D rectangle used in simulations: (a) 3-D mesh of rectangle; (b)

The slice at 𝒙 = 𝟎 mm, where the arrow indicates the position in which the photon

is launched perpendicularly to the plane 𝒛 = 𝟎 mm.

The optical properties of the 3-D rectangle were selected according to the most

common values of human tissue [117]. For instance, the values for the healthy human

brain are less than 0.1 mm-1

for absorption coefficient and around 10 mm-1

for scattering

coefficient [118] at 674 nm. The anisotropy of the tissue was set to 𝑔 = 0.9 and the

refractive index 𝑛 was 1.5 for all simulations (pencil beam and Gaussian shape beam).

Various combinations of optical properties (absorption and scattering coefficients) were

tested. As mentioned above, results from the RTE were compared with the solution to

DE and MC simulations under the same settings. A tetrahedron mesh-based MC method

[115] was employed. Since using different numbers of photons to simulate photon

propagation in tissue by MC led to different absolute intensities, results were normalized

with respect to a global maximum value [88] in order to compare the solutions obtained

by RTE, DA and MC methods. Eqn. (2-18) was employed to transform the photon

radiance to photon density by numerical integration to compare MC simulations and

solutions to the DA. In order to easily process the data with Matlab, a structural grid

(49 × 139 × 50 with a resolution of 0.1 mm in each dimension) was generated. For the

structural points inside one element, a linear interpolant through the values of four nodes

of the element was employed.

Solutions to the RTE obtained by different numerical quadratures were compared

to MC simulations in order to assess their influence on the results. Since it was difficult

to use the same number of discrete angles of quadratures with different constructions,

comparable numbers of discrete angles were employed. Subsequently, solutions obtained

(a) (b)

33

with the most accurate quadrature for the RTE were compared with the results from DE

and MC simulations. Due to the limit of the storage space (RAM) of the computer used

for simulation, no more than 100 angles of DOM were employed. It is noted that the

number of angles used in the simulation was less than those in [86], [92] when the FVM

was used with similar accuracy. The simulations were run on a computer with Intel Xeon

2.8 GHz CPU, 24 GB RAM and Windows 7 professional 64-bit operating system. The

number of photons in these MC simulations was set to 106.

2.2.2 Gaussian shape beam simulation

Many light sources can be modeled by Gaussian distribution. In our next group of tests,

a Gaussian-shaped beam, in which the intensity of the points at the cross section of the

beam obeys the Gaussian distribution, was employed as the light source. The beam

intensity was modeled by

2 2

2

( )

22

1, , 0

2

x y

ddg x y z e

d

in which 𝑑 represents the beam width. Two different 3-D rectangles were employed,

Rect. 1 in Section 2.2.1 and Rect. 2, given as [−2.0, 2.0] × [−2.0, 2.0] × [−6.0, 0] mm3.

Note that Rect. 2 is smaller than Rect. 1. Only the solutions to the RTE and MC were

presented and compared, as the DA is not suitable for predicting photon propagation

when the region of interest (ROI) is near the source. When the light source is changed

from a point source to a Gaussian shape, solving the RTE only changes the boundary

conditions presented in Eqn. (2-6) or the source term. As a consequence, the matrix

assembling time and computation time is almost unchanged. Similarly, extended

illumination sources can be efficiently modeled using MC with overhead less than 5% of

total computational time [116], [119]. Here 105 photons were employed for MC

simulation for each single point of Gaussian beam for both rectangles, Rect. 1 and Rect.

2.

The anisotropy and refractive index were set to be the same values as in Section

2.2.1. The parameter 𝑑 was set to 0.5. In the first example (Rect. 1), the absorption

coefficient was set to 0.08 mm-1

and the scattering coefficient was set to 5 mm-1

. Since

the light source was modeled by a Gaussian function, it was much smoother than the

34

ideal pencil beam, thus fewer discrete angles were needed to obtain solutions with

comparable accuracy to those gathered for the ideal pencil beam. For Rect. 1, a mesh

with denser nodes was generated compared with the one in Section 2.2.1, and in total

there are 7,204 nodes with 41,559 elements.

For Rect. 2 (Figure 2-3 (a)), 10,996 nodes with 60,887 elements were generated.

Figure 2-3 (b) shows the Gaussian beam represented by a collection of single point

sources for MC in the computation. The optical properties of the simulation were set to

be 0.1 mm-1

for absorption coefficient and 5 mm-1

for scattering coefficient, respectively.

Figure 2-3. 3-D rectangle used in simulation 2.2.2: (a) Mesh of 3-D rectangle; (b)

Irradiated light intensity modeled by Gaussian shape function.

Two quantities were used to assess the difference between the solution to the

RTE and MC simulation as shown below. One is the root mean square error between the

solutions, defined as

2

RMSE

MC RTE

i i

i

u u

N

where 𝑁 represents the number of points in comparison. Note that what RMSE measures

is an absolute error. The relative error at each point between the solutions was defined as

MC RTE

i i

i MC

i

u ue

u

Based on this, the mean relative error (MRE, defined as ∑ 𝑒𝑖𝑖 𝑁⁄ ) and maximum relative

error can be easily calculated. Note that both quantities can be calculated for any

geometry, alone any line in a specific plane, or in the entire 3-D rectangle.

(b) (a)

35

2.2.3 Teeth model

Here two teeth sections were scanned by micro-CT (Medical VivaCT 40, Scanco,

Switzerland). According to [39], the absorption and scattering coefficients for dentin

were set to 0.3 mm-1

and 28 mm-1

respectively at 632 nm. In the first tooth (Tooth 1),

only the dentin was considered. In order to increase the complexity of model, a ball with

radius 1.5 mm located at (0,1, −3) mm was embedded into Tooth 1 with absorption

coefficient and scattering coefficient 0.02 mm-1

and 5 mm-1

respectively. In contrast, in

second tooth (Tooth 2), the real situation was considered with various portions of teeth

including dentin, enamel and soft tissues located in pulp. For the enamel, the absorption

coefficient and scattering coefficient was set to 0.1 mm-1

and 6 mm-1

respectively at 632

nm. For the pulp, the absorption coefficient was set to 0.01 mm-1

and 10 mm-1

respectively according to the soft tissue value shown in [90]. ImageJ (1.50b) was

employed to perform k-means to segment the teeth into three parts, pulp (soft tissues),

enamel and dentin. For all the simulations in this section, anisotropy was set to 0.9 and

refractive index was set to 1.5. The anatomical structures of the teeth model are shown in

Figure 2-4. In Tooth 1, the Gaussian beam with beam-width 0.45 mm located at (0,0,0)

was irradiated perpendicularly to the top layer 𝑧 = 0 mm. Under this situation, although

there existed mismatch of refractive index at the boundary, the direction of irradiation

light would not be changed. However, in the Tooth 2 model, the light was irradiated

towards the tooth from the lateral side and due to unevenness of the surface of the tooth,

the direction of light would be refracted. In this simulation, the source was set to

(0, −6, −2) mm and the incident direction was (0, 0.98,0.20) . Gaussian beam with

beam-width 0.45 mm was employed as the source.

36

Figure 2-4. Anatomical structure of teeth model: (a) Outline of Tooth 1; (b) The

ball inclusion embedded into Tooth 1; (c) Different portion of Tooth 2, grey color

represents pulp, yellow color represents dentin while magenta (dark color at the

bottom) represents enamel.

Under this situation, the intersection points between the incident light and surface of

teeth can be calculated by the following procedure. Since tetrahedrons were employed as

the element in the mesh, on the surface it was just one facet of a particular tetrahedron,

namely, the surface was composed of arbitrary triangles in 3-D space. Thus the problem

was transferred to find the intersection points between a triangle in 3-D space and a line.

The Barycentric coordinate system was employed to transfer the triangle in 3-D

space to a standard triangle, as shown in Figure 2-5. Then the problem will be changed

to finding an intersection point of the plane which the triangle resides in and the incident

light and determining whether the point is inside the triangle. From Figure 2-5 (b), it is

clear that the points belonging to triangle (located inside triangle) will obey the

following conditions,

1

2 1

0 1,

0 1

After the intersection point was located for each surface triangle face, Snell’s law was

employed to calculate the new direction of photon propagation. Also the energy lost due

(a) (b)

(c)

37

to reflection was calculated by Fresnel’s law as shown in [85]. Here notice that when

performing the integration on tetrahedron during FEM, the Barycentric coordinate

system was also employed to transfer an arbitrary tetrahedron to a standard one.

Figure 2-5. Demonstration of Barycentric coordinate system: (a) Arbitrary triangle;

(b) Standard triangle.

In the Tooth 1 model, 106 photons were used for MC simulation in order to

verify the accuracy of photon distribution inside teeth acquired by solutions to RTE.

22,126 nodes and 134,974 elements were generated by CGAL. Similarly, for Tooth 2,

10,213 nodes and 49,744 elements were generated.

2.2.4 Delta-Eddington phase function simulation

In this section, the d-E phase function was investigated and the comparison between d-E

phase function, H-G phase function and MC simulations were performed. Briefly, a 3-D

rectangle with volume 10 by 10 by 8 mm3 with coordinates [−5,5] × [−5,5] × [−8,0]

was employed to investigate the photon propagation. The optical properties were

assigned values as: absorption coefficient 0.1 mm-1

, scattering coefficient 8 mm-1

,

anisotropy 0.9 and refractive index 1.0. A Gaussian beam with center located at [0,0,0]

and beam width 0.5 was irradiated perpendicularly towards the plane 𝑧 = 0 mm. The

output flux was also investigated by setting 81 detectors with diameter 0.1 mm at the

bottom side of 3-D rectangle (𝑧 = −8 mm). See Figure 2-6.

(a) (b)

38

Figure 2-6. 3-D rectangle: (a) Mesh of rectangle; (b) Detectors located at the bottom

surface of rectangle (𝒛 = −𝟖 mm). [101]

13,615 nodes and 76,181 elements were generated by CGAL. Also MC was employed as

the ground truth to assess the accuracy of solution. The most advanced MC [116] in the

field of biomedical optics field was employed with 108 number of photons. In the

simulation, the photon distribution was acquired by Eqn. (2-18) and then the output flux

at detector 𝒓𝒅 can be calculated as shown in [60],

( ) ( )d d

r r n (2-20)

In the equation, all the variable definitions can be seen in Section 2.1.5. The output flux

acquired from solutions to the RTE and MC was normalized by global maximum value

respectively before comparison.

Except considering Gaussian beam, an internal isotropic source located at

(0,0, −1) was also employed. In this case, the DA was used as the ground truth and the

optical properties were adjusted to satisfy the conditions of applying the DA. As in

section 2.1.5, the FEM was employed to solve the DA numerically. The absorption and

scattering coefficients were set to 0.1 mm-1

and 15 mm-1

respectively. The mesh size was

the same as in Gaussian beam simulations.

2.3 Results

2.3.1 3-D rectangle simulations with ideal pencil beam

2.3.1.1 Comparison between different numerical quadrature of DOM

In order to evaluate the effect of the choices of numerical quadrature on the solution to

the RTE, all the numerical quadratures reviewed in Section 2.1.2 were tested first for all

39

sets of optical properties listed in Table 2-3. Table 2-1 and Table 2-2 only report the

errors for the optical settings with absorption coefficient 0.02 mm-1

and scattering

coefficient 5 mm-1

. Table 2-1 lists the 3-D RMSE (𝑥 ∈ [−2.5,2.5] mm, 𝑦 ∈ [−7.0,7.0]

mm with 𝑧 defined in the corresponding tables) in different regions with different

numerical quadratures. Note that the number in parentheses in the first column in Table

2-1 is the number of the discrete angles used for each numerical quadrature. From Table

2-1, it is clear that when the ROI is far from the source and boundary, the error between

the solutions to RTE and from the MC simulations is smaller for all numerical

quadratures. However, it can be seen from Table 2-1 that the error for Legendre equal-

weight quadrature is larger in deeper regions compared with other quadratures. In order

to find the quadrature with the least error in the simulations, the 3-D MRE was

calculated and shown in Table 2-2.

Table 2-1. RMSE with different numerical quadrature (𝝁𝒂 = 𝟎. 𝟎𝟐 mm-1

, 𝝁𝒔 = 𝟓

mm-1

, 𝒈 = 𝟎. 𝟗).

Numerical quadrature −𝟓. 𝟎 ≤ 𝒛 < 𝟎. 𝟎mm −𝟓. 𝟎 ≤ 𝒛 < −𝟏. 𝟎mm −𝟓. 𝟎 ≤ 𝒛 < −𝟐. 𝟎mm

Lebedev (86) 2.81E-2 1.07E-2 2.16E-3

Product Gaussian (72) 2.80E-2 1.07E-2 2.37E-3

Legendre equal-weight (80) 2.84E-2 1.14E-2 3.39E-3

Level symmetric (80) 2.81E-2 1.09E-2 2.31E-3

Table 2-2. MRE with different numerical quadrature (𝝁𝒂 = 𝟎. 𝟎𝟐 mm-1

, 𝝁𝒔 = 𝟓

mm-1

, 𝒈 = 𝟎. 𝟗).

Numerical quadrature −𝟓. 𝟎 ≤ 𝐳 < 𝟎. 𝟎mm −𝟓. 𝟎 ≤ 𝒛 < −𝟏. 𝟎mm −𝟓. 𝟎 ≤ 𝒛 < −𝟐. 𝟎mm

Lebedev (86) 10.05% 6.82% 5.45%

Product Gaussian (72) 10.64% 7.70% 6.39%

Legendre equal-weight (80) 20.90% 18.53% 17.64%

Level symmetric (80) 10.36% 7.29% 5.72%

It can be concluded that the MRE exhibits the same trend as the RMSE. From

Table 2-2, one can also see that product Gaussian quadrature has larger error (~1%) than

Lebedev quadrature and level symmetric quadrature although it employs fewer discrete

angles. Considering that the next order product Gaussian quadrature will have 98

discrete angles and this will increase the computation cost, product Gaussian quadrature

40

will not be our first choice. For Lebedev and level symmetric quadrature, the error for

the whole rectangle is around 10%, while for deeper regions it is around 5%. Hence we

can infer that level symmetric quadrature and Lebedev quadrature are the best among all

tested quadratures at least for the optical properties examined in this manuscript. It was

observed that level symmetric quadrature can generate negative weights, which is non-

physical when the number of discrete angles is larger than 528 [102]. Although this

number of discrete angles is quite impractical, level symmetric quadrature will not be

further investigated in the rest of this chapter. On the other hand, both numerical

quadratures provide similar relative error estimations, with a difference of less than 0.5%,

at deeper depths. Thus in this manuscript, Lebedev quadrature will be further

investigated in the following simulations.

2.3.1.2 Ideal pencil beam simulations

Figure 2-7 shows the contours of the logarithm of solutions to the RTE by the streamline

diffusion modified CG-FEM and from MC simulations within 3 planes. The optical

properties for these simulations were set as 𝜇𝑎 = 0.02 mm-1

and 𝜇𝑠 = 5 mm-1

. With this

combination of optical properties in tissue, the DA is not an accurate model to predict

photon propagation [88]. 86 angles were employed in Lebedev quadrature to obtain the

results for RTE. It is clear that at deeper depth, the difference between the solution to

RTE and MC simulations is small. The photon densities along the line 𝑥 = 0 mm with

different depths are shown in Figure 2-8. The RMSE and MRE along each line at

different depth are shown in Table 2-3 and Table 2-4.

Table 2-3. RMSE for different optical properties at different depth along the line

𝒙 = 𝟎 mm (𝒈 = 𝟎. 𝟗).

(𝝁𝒂, 𝝁𝒔) mm-1 −𝟎. 𝟒 mm −𝟏. 𝟎 mm −𝟐. 𝟎 mm −𝟑. 𝟎 mm −𝟒. 𝟎 mm −𝟓. 𝟎 mm

(0.02, 5) 6.84E-2 9.07E-2 1.03E-2 1.01E-3 4.19E-4 2.89E-4

(0.02, 8) 8.88E-2 2.58E-2 3.25E-3 1.02E-3 5.69E-4 2.23E-4

(0.08, 5) 7.08E-2 9.16E-2 1.02E-2 9.49E-4 2.80E-4 1.97E-4

(0.01, 10) 8.92E-2 2.33E-2 2.90E-3 7.40E-4 3.24E-4 1.94E-4

41

Table 2-4. MRE for different optical properties at different depth along line 𝒙 = 𝟎

mm (𝒈 = 𝟎. 𝟗).

(𝝁𝒂, 𝝁𝒔) mm-1 −𝟎. 𝟒 mm −𝟏. 𝟎 mm −𝟐. 𝟎 mm −𝟑. 𝟎 mm −𝟒. 𝟎 mm −𝟓. 𝟎 mm

(0.02, 5) 17.76% 17.24% 8.25% 4.73% 3.84% 3.59%

(0.02, 8) 11.94% 7.31% 4.10% 3.19% 3.46% 2.71%

(0.08, 5) 19.51% 18.98% 9.22% 5.05% 3.72% 3.44%

(0.01, 10) 13.80% 7.99% 5.54% 3.54% 3.01% 3.09%

In comparing MC to the DE, it is quite clear that the discrepancy between the

MC solution (blue dashed line in Figure 2-8) and the DE solution (green line in Figure

2-8) is large. In comparing MC to the RTE, the solutions to RTE and MC coincide quite

well except in shallow regions (Figure 2-8 (a) and (b)). The MRE for 𝑧 = −3.0, 𝑧 =

−4.0, 𝑧 = −5.0 mm is less than 5% (Table 2-4). At these depths, the maximum relative

error for all points is 10.04%, 12.03% and 9.06% respectively. In the shallow region, for

instance, 𝑧 = −1.0 mm and 𝑧 = −2.0 mm, the errors are larger as ROI gets closer to the

light source located at (0,0,0) . At the extreme shallow region, 𝑧 = −0.4 mm, the

difference between the solution to the RTE and results of the MC method is large, which

leads to large RMSE and MRE.

Figure 2-7. The contours of logarithm of solutions to RTE and MC results, solid

curve for the solution to RTE and dashed curve for MC results: (a) The contours

within the plane 𝒙 = 𝟎 mm; (b) The contours within the plane 𝒚 = 𝟎 mm, the value

of the outermost curve is -1.5; (c) The contours within the plane 𝒛 = −𝟑. 𝟎 mm.

(b)

(c)

(a)

42

Figure 2-8. Comparison between the solutions to RTE, DE and MC methods at

different depths, along the line 𝒙 = 𝟎 mm: (a) 𝒛 = −𝟎. 𝟒 𝐦𝐦; (b) 𝒛 = −𝟏. 𝟎 𝐦𝐦; (c)

𝒛 = −𝟐. 𝟎 𝐦𝐦; (d) 𝒛 = −𝟑. 𝟎 𝐦𝐦; (e) 𝒛 = −𝟒. 𝟎 𝐦𝐦; (f) 𝒛 = −𝟓. 𝟎 𝐦𝐦.

To further demonstrate the performance of the algorithms, three more

combinations of optical properties were tested. The RMSE and MRE between the

solutions to the RTE and results of the MC along line 𝑥 = 0 mm are shown in Table 2-3

and Table 2-4. For all examples tested and for depths smaller than -3.0 mm, the MRE is

less than or close to 5%. At a depth of 𝑧 = −2.0 mm, the MRE is less than 10%.

Table 2-3 and Table 2-4 only show the results along the line 𝑥 = 0 mm at

different depths. For a comprehensive comparison of the solutions to the RTE and

results from the MC, 3-D RMSE and MRE (𝑥 ∈ [−2.5,2.5] mm, 𝑦 ∈ [−7.0,7.0] mm)

were also calculated. The results are shown in Table 2-5 and Table 2-6.

Table 2-5. RMSE for different optical properties to 3-D rectangle (𝒈 = 𝟎. 𝟗).

(𝝁𝒂, 𝝁𝒔) mm-1

−𝟓. 𝟎 ≤ 𝒛 < 𝟎. 𝟎 mm −𝟓. 𝟎 ≤ 𝒛 < −𝟏. 𝟎 mm −𝟓. 𝟎 ≤ 𝒛 < −𝟐. 𝟎 mm

(0.02, 5) 2.81E-2 1.07E-2 2.16E-3

(0.02, 8) 2.54E-2 6.39E-3 1.81E-3

(0.08, 5) 2.88E-2 1.06E-2 2.02E-3

(0.01, 10) 2.60E-2 5.77E-3 1.58E-3

(d) (e) (f)

(a) (b) (c)

43

Table 2-6. MRE for different optical properties to 3-D rectangle (𝒈 = 𝟎. 𝟗).

(𝝁𝒂, 𝝁𝒔)mm-1

−𝟓. 𝟎 ≤ 𝒛 < 𝟎. 𝟎 mm −𝟓. 𝟎 ≤ 𝒛 < −𝟏. 𝟎 mm −𝟓. 𝟎 ≤ 𝒛 < −𝟐. 𝟎 mm

(0.02, 5) 10.05% 6.82% 5.45%

(0.02, 8) 8.33% 5.51% 4.78%

(0.08, 5) 10.90% 7.42% 5.82%

(0.01, 10) 8.87% 5.76% 4.92%

From Table 2-5 and Table 2-6, one can see that for each set of optical properties

(corresponding to each row in the table), the 3-D RMSE and MRE decrease when the

regions where the errors are measured are reduced. In our simulations, the largest error

occurs near the source. For the optical properties listed in Table 2-5 and Table 2-6, the

MRE for deeper regions is around 5%.

2.3.2 3-D rectangle simulations with Gaussian modeled intensity beam

The contours of the logarithm of photon densities within 3 planes of Rect. 1, obtained by

solving the RTE and using MC simulations, are shown in Figure 2-9. The contours of the

logarithm of photon densities of Rect. 2 are shown in Figure 2-10. For both Rect.1 and

Rect. 2, Lebedev quadrature with 50 angles was employed. 3-D RMSE and MRE for

different regions were also calculated and are shown in Table 2-7. When the volume of

the ROI is reduced, there is a subsequent decrease in both RMSE and MRE. Because the

Gaussian-shape intensity beam is smoother than the point source, the solution to the

RTE contains fewer oscillations near the light source. This can be easily seen through

comparison of the ROI near the source in Figure 2-7 and Figure 2-9.

Table 2-7. 3-D RMSE and MRE for two rectangles with Gaussian shape beam

(𝒈 = 𝟎. 𝟗).

Rect. 1 −5.0 ≤ 𝑧 < 0.0 mm −5.0 ≤ 𝑧 < −1.0 mm −5.0 ≤ 𝑧 < −2.0 mm

RMSE 2.05E-2 5.40E-3 2.04E-3

MRE 6.86% 4.48% 4.31%

Rect. 2 −6.0 ≤ 𝑧 < 0.0 mm −6.0 ≤ 𝑧 < −1.0 mm −6.0 ≤ 𝑧 < −2.0 mm

RMSE 1.59E-2 7.71E-3 3.21E-3

MRE 13.69% 10.64% 7.43%

44

Figure 2-9. The contours of logarithm of photon densities of Rect. 1 at 3 planes, in

which the solid curves represent the solution to the RTE while the dashed curves

represent MC results: (a) 𝒙 = 𝟎 mm; (b) 𝒚 = 𝟎 mm, the value of the outermost

contour is -1.5; (c) 𝒛 = −𝟑 mm.

Figure 2-10. The contours of logarithm of photon densities of Rect. 2 at 3 planes, in

which the solid curves represent the solution to the RTE while the dashed curves

represent MC results: (a) 𝒙 = 𝟎 mm, the value of outermost curve is -2.5; (b) 𝒚 = 𝟎

mm, the value of outermost curve is -2.5; (c) 𝒛 = −𝟑 mm.

(b)

(c)

(a)

(a) (b)

(c)

45

2.3.3 Teeth model simulations with Gaussian beam irradiation

The contours of logarithms of photon distributions of Tooth 1 by solving RTE and MC

simulations within 3 planes are shown in Figure 2-11. It is obvious that the difference

between solutions to the RTE and MC simulations are small even at the boundary of the

tooth. Also at least for this simulation, at the shallow region (𝑧 < 1 mm), the difference

between RTE and MC is also small. This example demonstrates the accuracy of

solutions to the RTE without considering the mismatch of refractive index between the

embedded ball and tooth (for simplicity, we assume the refractive indices of the ball and

dentin are same).

Figure 2-11. The contours of logarithm of photon densities in Tooth 1 within 3

planes, in which the solid curves represent solutions to the RTE and dashed curves

represent MC simulations: (a) 𝒚𝑶𝒛 plane; (b) 𝒛𝑶𝒙 plane; (c) 𝒛 = −𝟑. 𝟎 mm.

For Tooth 2, Figure 2-12 shows the relationship between Tooth 2 and Gaussian

beam source. The green circle indicates the location from where the Gaussian beam is

launched. Due to unevenness of the surface and refractive index mismatch, the refracted

light intensity at different points is different and is demonstrated using different heights

from the surface of Tooth 2 (shown in red color) in Figure 2-12. The photon

distributions acquired by the solutions to the RTE are shown within 3 planes in Figure

2-13.

(b) (a)

(c)

46

Figure 2-12. Demonstrations of the position of tooth and Gaussian beam source.

Figure 2-13. Photon distributions of Tooth 2 within 3 planes: (a) 𝒚𝑶𝒛 plane; (b)

𝒛𝑶𝒙 plane; (c) 𝒛 = −𝟐 mm.

2.3.4 Delta-Eddington phase function simulation

As illustrated in Section 2.2.4, first the Gaussian beam with the center located (0,0,0),

irradiated perpendicular to the top layer ( 𝑧 = 0 mm) of 3-D rectangle was performed.

Figure 2-14 demonstrates the difference between the solutions to the RTE with H-G

phase function and MC simulations. It is clear in the figure that the difference between

the solutions to RTE and MC is quite small even for the shallow region, at least visually.

(a) (b)

(c)

47

Similarly, the comparison between the solutions to RTE with dE phase function and MC

simulations is shown in Figure 2-15.

Figure 2-14. Contours of logarithm of solutions to the RTE with H-G phase

function and MC simulations within 3 planes under irradiation by Gaussian beam,

in which the solid curves represent the solutions to RTE and dashed curves

represent MC simulations: (a) 𝒚𝑶𝒛 plane; (b) 𝒛𝑶𝒙 plane; (c) 𝒛 = −𝟒 mm.

In Figure 2-15, it is clear that the solutions to RTE with the d-E phase function

also correspond with MC simulations. But the difference between the solutions to RTE

with d-E phase function and MC simulations is larger than the difference between

solutions to RTE with H-G phase function and MC simulations shown in Figure 2-14.

However this is anticipated since our MC simulations employing the H-G phase function

and d-E phase function only approximate the H-G phase function with simplified and

easily handled functions. The output flux with the comparison between solutions to the

RTE with H-G or d-E phase functions with MC simulations is shown in Figure 2-16.

Obviously, after enough scattering events, the output fluxes obtained by solutions to the

RTE with H-G phase function and d-E phase function both correspond with MC results.

(a) (b)

(c)

48

Figure 2-15. Contours of logarithm of solutions to the RTE with d-E phase function

and MC simulations within 3 planes under irradiation by Gaussian beam, in which

the solid curves represent the solutions to RTE and dashed curves represent MC

simulations: (a) 𝒚𝑶𝒛 plane; (b) 𝒛𝑶𝒙 plane; (c) 𝒛 = −𝟒 mm. [101]

Figure 2-16. Comparison of output flux obtained by solutions to the RTE with H-G

phase function or d-E phase function and MC simulations under irradiation of

Gaussian shape beam: (a) H-G phase function; (b) d-E phase function. [101]

Last in this section, we investigate the situation with an internal isotropic source

as stated in Section 2.2.4. The comparison between solutions to the RTE with H-G phase

function and DA solutions is shown in Figure 2-17.

(a) (b)

(c)

(a) (b)

49

Figure 2-17. Contours of logarithm of solutions to the RTE with H-G phase

function and DA solutions within 3 planes with internal source, in which the solid

curves represent the solutions to RTE and dashed curves represent DA solutions: (a)

𝒚𝑶𝒛 plane; (b) 𝒛𝑶𝒙 plane; (c) 𝒛 = −𝟒 mm.

In Figure 2-17, except for the region near the internal isotropic source, the

difference between the solutions to RTE with H-G phase function and MC simulations is

quite small. Since continuous global basis functions were employed to solve the RTE

with transport effect (CG-FEM), it cannot track local variations of the source as Dirac-

delta function or point source well. However, in contrast, the DA is a second order

elliptic equation without transport like term as RTE, CG-FEM can deal with the

isotropic source. This is the main reason why large error can be observed near the source.

The comparison between solutions to the RTE with the d-E phase function and the DA is

shown in Figure 2-18. Similarly in Figure 2-17, the error near the internal source is large

when at further regions, good correspondence between solutions to the RTE with d-E

phase function and DA solutions can be easily seen.

(a) (b)

(c)

50

Figure 2-18. Contours of logarithm of solutions to the RTE with d-E phase function

and DA solutions within 3 planes with internal source, in which the solid curves

represent the solutions to RTE and dashed curves represent DA solutions: (a) 𝒚𝑶𝒛

plane; (b) 𝒛𝑶𝒙 plane; (c) 𝒛 = −𝟒 mm.

Finally, the output flux with internal source is shown in Figure 2-19. Similar to

Figure 2-16, after photons experience large amount of scattering events, the output flux

between solutions to the RTE with H-G phase function and d-E phase function

corresponds well with DA solutions for all detectors.

(a) (b)

(c)

51

Figure 2-19. Comparison of output flux obtained by solutions to the RTE with H-G

phase function or d-E phase function and DA solutions with internal source: (a) H-

G phase function; (b) d-E phase function.

2.3.5 Importance of phase function normalization technique

To illustrate the importance of the normalized phase function technique to the stability of

the proposed method, especially for large anisotropy with 𝑔 around 0.8~0.9, we compare

in Figure 2-20 the numerical solutions of our methods with or without using the

normalization. Rect.1 with Gaussian beam was considered, with all the optical properties

same as in Section 2.2.2. The comparison is shown in Figure 2-20.

Figure 2-20. Comparison between the logarithm of contours with and without

phase function normalization: (a) With phase function normalization (b) Without

phase function normalization.

When processing CG-FEM without the phase function normalization, it is

observed from Figure 2-20 (b) that the algorithm is less stable and the computed solution

oscillates between positive and negative values. In the plot, the negative value is shown

as the machine epsilon in Matlab for logarithm calculation.

(a) (b)

(a) (b)

52

Table 2-8. Comparison of time consumption between solving RTE and MC

simulations with Gaussian beam.

RTE MC (105/10

6)

Rect. 1 23min 45min/402min

Rect. 2 37min 84min/772min

2.4 Discussion and future work

In this chapter, RTE was solved using our proposed algorithm with modified DOM in

the angular domain and a streamline diffusion CG-FEM in the spatial domain. The

quadrature with the highest computational efficiency, Lebedev quadrature, was chosen

to calculate the integral term, combined with the phase function normalization technique.

3-D rectangles with various common optical properties of human tissue were employed

to simulate photon propagation. The solutions to the RTE and DE were compared to

those obtained by MC methods. In addition, real sections of teeth were tested to

investigate the difference between solutions to RTE and MC simulations with complex

shape of media and large optical properties. Also, the RTE with d-E phase function was

also solved using our algorithms and the solutions to the RTE with d-E phase function

was compared with MC simulations and the DA with different types of sources

(Gaussian beam and internal isotropic source). In summary, for a 3-D rectangle

irradiated by an ideal pencil beam, the mean relative error for a region farther than 2mm

is around 5%. For a light source with Gaussian irradiation, even with a smaller

computational domain, the worst mean relative error is still less than 10% for deeper

regions. For the real teeth model and d-E phase function, the solutions to RTE also have

good correspondence with MC simulations (or DA solutions) at locations far from the

source.

One advantage of our method is that, from a mathematical perspective, it reduces

the dimension of the original RTE by using the DOM. Furthermore, a streamline

diffusion technique was employed to deal with the transport term in the RTE within a

CG-FEM framework. The difference between the test function of the standard CG-FEM

and streamline diffusion test function is the addition of a term that is the gradient of the

original test function. Hence, it is quite easy to implement the modification. In our

53

simulations of highly scattering media (a few tenths per mm of scattering coefficient),

the solution was occasionally negative when the streamline diffusion technique was not

used. In practice, with highly scattering tissue, there is no need to solve RTE to obtain

photon distribution, as the DA provides a more computationally efficient model. The

other advantage of using the streamline diffusion technique, at least with the optical

properties examined in this work, is that less iterations were needed when GMRES was

used to solve the resulting linear equations. In our work, it is important to use the

function normalization technique to calculate the integral term accurately. This

technique mitigates the instability of the DOM dramatically and further reduces the non-

physical oscillations of the solution. The oscillation of the solution to the RTE with

DOM is more prominent with ideal pencil beam. Various techniques were proposed in

literature for reducing the oscillation. For instance, in [83], the weight of numerical

quadrature is calculated based on the integration of linear interpolation of the phase

function. In comparison to the method discussed in [83], the phase function

normalization technique is more direct and easier to apply.

We would like to point out that DG-FEM is known to be a good choice for

solving problems with transport effects, such as the RTE. It is also noted that the

implementation of the DG-FEM can make the scheme parameters free and succinct.

Although DG-FEM involves more unknowns in general, it is still worth investigating in

future work.

Next, we comment on the computational time of the proposed methods for

solving the RTE directly and the MC simulations. In the case of a single point source

such as a pencil beam, solving the RTE directly is not advantageous in computational

efficiency for our algorithms. When a Gaussian beam or sources of arbitrary shape are

considered, however, solving the RTE directly is more cost effective than conventional

MC simulations. This is illustrated by Table 2-8, where computational times are reported

for solving the RTE directly with Gaussian beam and 50 discrete angles ( including time

to assemble matrices and solve linear equation), and MC simulations with 105 or 10

6

photons employed for each single point of the Gaussian beam. If using MC simulations

presented in [116], solving the RTE by the proposed algorithm will have similar

computational efficiency when a large number of photons (108) is employed in MC

54

simulations as shown in Section 2.3.4. We also would like to point out that the use of

parallel computation can reduce the time required to solve the forward photon

propagation problem. In our code, CPU-based parallel computing techniques had already

been implemented. In simulations the time would be reduced by at least 40% if parallel

computing was used (4 cores CPU) in general. Still, using GPU computing can further

decrease the time required for FEM implementation.

Finally, when considering using the RTE as a forward model, the advantages for

the inverse problem will be that the reconstruction of scattering coefficient does not need

to calculate the gradient of Green’s function (photon radiance) [52]. In contrast with the

adjoint MC method, calculating the gradient of Green’s function will be numerically

unstable, especially with a tetrahedron-based mesh. Moreover, even if employing the

reconstruction method to transfer the photon radiance to photon distribution by Eqn.

(2-18), since polynomial space is used as basis space, the gradient of Green’s function

will be calculated analytically to FEM-based framework. This will also avoid the

instability for numerical differential with arbitrary mesh, such as tetrahedron-based mesh.

2.5 Conclusion

A modified finite element method based on the streamline diffusion idea is applied to

simulate the RTE with high accuracy. It is demonstrated numerically that the method

provides an accurate and robust approach to solving the forward problem in the bio-

optical regime. Future investigation includes evaluation of the performance of this

algorithm on objects exhibiting a wider range of optical properties or objects with very

general shapes. The output flux of photons calculated by this algorithm should also be

verified by experiments.

55

3. Mesoscopic fluorescence molecular tomography applied in dental

imaging

As introduced in Chapter 1, in order to investigate whether optical imaging (tomography)

can be applied in teeth to monitor physiological variation of pulp, mesoscopic

fluorescence molecular tomography (MFMT) was first applied in human teeth ex vivo. In

this chapter, the first generation MFMT will be introduced in detail. Then the

preparation of samples and procedure of experiment will be illustrated. Finally, the

results as well as quantities to assess the accuracy of reconstructions will be provided. *

3.1 Materials and methods

3.1.1 Samples preparation and experiments procedure

The initial determination of dental application of MFMT required ex vivo experiments to

establish feasibility and performance. Extracted molar teeth were acquired from local

dentists with institutional biosafety committee (IBC) and institutional review board (IRB)

approval. A relatively flat surface in the middle part of the tooth was selected to be

imaged by MFMT. The field of view (FOV), which was irradiated by the light, was set

to 4 mm by 6 mm in the experiments (shown with red rectangle in Figure 3-1 (a)).

Further, this surface was grinded slightly to reduce the reflection of light. The teeth were

then embedded in solidified agar solution (4% w.t.) (Agarose, Sigma-Aldrich, USA) to

prevent motion during transport between two non-concurrent imaging modalities,

MFMT and micro-CT.

Portions of this chapter previously appeared as:

F. Long, M. S. Ozturk, X. Intes, and S. Kotha, “Dental imaging using laminar optical tomography and micro CT,” in

Proc. SPIE 8937, Multimodal Biomedical Imaging IX, 2014, 89370P.

F. Long, M. S. Ozturk, M. S. Wolff, X. Intes, and S. P. Kotha, “Dental Imaging Using Mesoscopic Fluorescence

Molecular Tomography: An ex Vivo Feasibility Study,” Photonics, vol. 1, no. 4, pp. 488–502, Dec. 2014.

56

Figure 3-1. Settings of the teeth phantom: (a) Red rectangle shows the FOV

irradiated by light; (b) Side view of the tooth to show the hole to hold the capillary;

(c) Markers on the surface of tooth.

To mimic the uptake of a molecular probe within the pulp of the tooth, a hole

with diameter 1.5 mm was drilled at the lateral side to hold a capillary filled with

fluorescence probe as shown in Figure 3-1 (b). To investigate the ability of MFMT to

detect and identify the fluorophore (used to simulate dental lesions or pulp situations) at

different depths, holes were drilled at 1mm, 2mm and 3mm in different teeth from the

surface which was exposed to incident light. Also note that the holes were drilled at

locations where enamel was absent and functional changes were expected. In order to

assess the accuracy of the reconstruction of fluorescence probe, micro-CT was employed

to provide rigorous information of the tooth structure, including the position of holes.

Moreover, markers (Figure 3-1 (c)) were made on the surface of the teeth to assure that

MFMT and micro-CT were sampling the same FOV. Four fiducial markers (hollow

hemispheres filled with Play-Doh) were made on the surface of teeth to demarcate the

four corners of the FOV. The Play-Doh (Hasbro, USA) in the holes was employed as

optical contrast to identify the position of holes in MFMT while the holes themselves

could be scanned clearly in micro-CT. Herein, rigorous registration between the two

modalities can be enabled. Figure 3-2 shows the markers with different modalities and a

white light photo. Note in Figure 3-2 (c), some distortions of the graph can be seen since

Figure 3-2 (c) was directly captured from 3-D reconstructions of micro-CT images.

(a)

(c)

(b)

57

Figure 3-2. FOV showed with different modalities and white light photo: (a) White

light photo of real teeth; (b) MFMT background image (without fluorophore); (c)

Reconstructed surface of teeth using micro-CT with ImageJ.

In order to mimic the dental lesions or pulp metabolism situation inside teeth,

Alexa Fluor 660 (Succinimidyl Ester, Life Tech. Inc., USA) was used as a fluorophore.

Alexa Fluor 660 has peak absorption at 660 nm and peak emission at 690 nm. When

considering using Alexa Fluor 660 to simulate the dental lesions, it is noted that the

commercial system, DiagnodentTM

, uses an excitation wavelength of 665nm and

measures the fluorescence since caries tissues induce a greater fluorescence at these

wavelengths compared with intact tissue [45]. Thus, Alexa Fluor 660 was appropriate to

be the optical contrast to simulate the lesions. Alexa Fluor 660 was then dissolved in

distilled water to make various concentration solutions such as 6.5 M, 13 and 26

to simulate different bacteria fluorescence.

Before the experiment, the teeth were first scanned using micro-CT to get the

anatomical structure as well as the accurate location of holes. The micro-CT (Medical

Viva CT 40, Scanco, Switzerland) is equipped with micro-focus X-ray cone-beam

source with 50~70 kVp and 8 W (160 Apower. In the experiment, 70 kVp voltage

setting was used. The standard resolution (38 m) was employed in the scanning,

resulting the acquisition time for one tooth of roughly less than 30 mins. This would

generate 300~400 slices based on different length of teeth. The scanned images were

then read by Matlab or ImageJ directly.

(a) (b) (c)

58

After scanning the teeth, the phantom would be scanned by MFMT. Several

series of images were captured correspondingly. For a specific tooth, first, the

background images were captured by MFMT. The definition of the background images

in MFMT is the pure sample or phantom without fluorescence inclusions and irradiation

light, but with the filter in front of detectors. Through the background images, one can

estimate the dark current of the detectors as well as system background noise. Then the

capillary with a certain concentration of dye was inserted into the teeth and the

excitation images were taken. The definition of excitation images is the sample with

fluorescence inclusions and irradiation light, but without the filter in front of detectors.

After taking the excitation images, the florescence images (emission images) were

captured. This means taking the images with fluorescence inclusions and irradiation light,

also with the filter. The division of fluorescence images and excitation images was used

as the measurement (𝒃 in Eqn. (3-2)). In order to estimate the sensitivity of MFMT, the

highest concentration of dye was tested first and, for the same tooth, lower

concentrations of dye would be tested sequentially until no efficient signal could be

detected by the system. After finishing experiments for one tooth, different teeth with

holes at different depth were examined.

3.1.2 Optical settings of MFMT

MFMT is an epi-fluorescence, non-contact imaging modality [120]. MFMT raster scans

the surface of sample. Multiple detectors laid along a line, placed radial to the irradiation

point, collect the emitted light from the surface. When the light is irradiated into the

sample, a portion of the light will be re-emitted from the same surface after multiple

scattering events with the particles of sample. The detectors far away from the

irradiation point will collect the light which penetrates deeper into the tissue [121]. The

MFMT setting is demonstrated in Figure 3-3. MFMT has been successfully applied to

preclinical imaging [122] and bio-printed tissues [123], [124]. However, these tissues

exhibit far lower attenuation than teeth.

59

Figure 3-3. Setup of MFMT.

In Figure 3-3 , 658 nm polarized excitation light (L658P040, Thorlabs, USA) is

fed to a resonant galvo-mirror and through the scanning lens focused on the surface of

sample. The emitted fluorescence light goes along the same optical path as excitation

light and is captured by the detectors. The emission signal is filtered by a long pass filter

(FF01-692/LP-25, Semrock. USA). The detectors consist of 7 avalanche photodiodes

(APD, S8550, Hamamatsu, Japan), which are sampled at 1.5 kHz (PCI-6143 DAQ, NI,

USA). Based on the demagnification of lens, 7 detectors collect light from 0.8 mm to 3.9

mm from the irradiation point. For instance, the 1st detector located the nearest to the

irradiation point will collect the light from 0.8 mm and so on. The step size of light

injection points is 50 m in the x-direction and 30 m in the y-direction. At each point,

the dwell time of the galvo-mirror is 8.3 sec. In the experiment, the field of view (FOV)

was set to 4mm by 6mm and this would lead to 80 sampling points in the x-direction and

200 sampling points in the y-direction. Herein the scanning frame rate will be

approximately 7.5 Hz. Due to the limited sensitivity of APDs, 420 frames were averaged

to increase the signal to noise ratio (SNR).

3.1.3 Optical reconstruction

Approximately, the surface of processed teeth was planar and voxel-based Monte Carlo

(MC) simulations were employed to get the sensitivity profile. The teeth are small and

have high optical properties as shown in [39]. Thus the conventional applied partial

differential equation (PDE) based method, the diffusion equation (DE), cannot be

60

applied in this situation. The consideration of applying voxel-based MC is for its

simplicity and, if considering the symmetricity of simulation region (in this case, only a

3-D rectangle was considered for simplicity), the computation time will be further

reduced to less than 10 mins if 106 photons were used in simulation. Moreover, a

forward-adjoint MC method was employed to produce Jacobians [114] for higher

computational efficiency. In general, a typical Born normalization [125] to describe the

fluorescence detected by a detector located at 𝒓𝒅 resulting from the source 𝒓𝒔 is shown

in Eqn. (3-1).

1( , ) ( ) ( , ) ( , )d

( , )

x m

B xX

U G GG

s d s d

s d

r r r r r r r rr r

(3-1)

In the equation, 𝐺𝑥 and 𝐺𝑚 are the Green’s function calculated by MC simulations at

excitation and emission (fluorescence) wavelength respectively. 𝜂(𝒓) is the distribution

of fluorophore and the quantity to be reconstructed [126]. Also notice that the index of

excitation Green’s function is 1. This will be different when employing upconverting

nanoparticles (UCNPs) in Chapter 4.

When getting the Green’s function 𝐺𝑥 and 𝐺𝑚 by MC to generate sensitivity

profile, some prior information is required, such as the geometrical information (sample

size, location of source/detector pairs and discretization level of sample) as well as

optical properties of the sample. Since the portion of the tooth irradiated by light consists

of dentin, the optical properties of dentin acquired from [39] are absorption coefficient

𝜇𝑎 = 0.3 mm-1

, scattering coefficient 𝜇𝑠 = 28 mm-1

, anisotropy 𝑔 = 0.93, and refractive

index 𝑛 = 1.54 as shown in [127]. In the experiment, to generate the sensitivity profile

with enough accuracy, 107 photons were employed. Considering the storage space of the

computer and computational speed, a 3-D rectangle (4 by 6 by 5 mm3) was discretized

into voxels with size 200 by 200 by 200 m3. Thus the number of voxels in the

experiments was 20 by 30 by 25 in total. In other words, the number of unknowns was

15,000. In practice, Eqn. (3-1) was discretized directly to obtain a linear system. Since

the reconstruction is generally ill-posed, a depth adaptive regularization method was

employed as shown in the following equation.

1( )T Tx A A D A b (3-2)

61

In Eqn. (3-2), 𝑨 is the Jacobian acquired by the forward adjoint MC (discretization of

Eqn. (3-1)), 𝝀 is Tikhonov regularization parameter acquired by L-curve analysis [128].

𝑫 is a diagonal matrix whose elements on the main diagonal are the square root of

corresponding diagonal entries of 𝑨𝑻𝑨, 𝒃 is a vector containing the measurements and

can be expressed as the emission image divided by the excitation image.

Eqn. (3-2) was then solved by Matlab built-in function cgs.m, conjugate gradient

method. The iteration method ended when either the number of maximum iterations (100

in the experiments) or the error bound 10-2

was reached. Overall, the reconstruction

procedure took less than 5 mins on a personal computer (Intel Core i7-3612 QM CPU,

2.1 GHz, 6 GB RAM, Windows 7).

3.1.4 Image registration of multimodal data sets

The critical part for rigorous registration of MFMT and micro-CT images is to find the

first layer in the 𝑧 direction between two imaging modalities. In order to get the fiducial

plane shown in Figure 3-4, the center of four markers acquired from micro-CT were

located accurately and then the fiducial plane was calculated accordingly. This plane

corresponds to the first layer of MFMT and was employed as the boundary condition in

the MFMT forward model. Aligning the top layer of MFMT reconstruction with this

plane created transformational and rotational registration between the two imaging

modalities.

Figure 3-4. Fiducial plane shown with MFMT and micro-CT: (a) Overlap of micro-

CT image and one background image from MFMT (red color represents MFMT

(a) (b) (c)

62

image); (b) Fiducial plane shown with 3-D surface of tooth; (c) Top view from the

tooth to show fiducial plane.

After the fiducial plane was located in both MFMT and micro-CT images, the

micro-CT data was down-sampled to the same discretization level as MFMT and

enabled a voxel-to-voxel match between the two modalities. Then the co-registration of

3-D images from MFMT and micro-CT were implemented and displayed by image

processing software (Amira 5.4.5, FEI visualization group, USA). Note that the inverse

problem was not cost using any CT prior information beside the shape of the sample.

3.2 Results

For all the cases investigated, inclusions of 1 mm and 2 mm yielded enough strong

fluorescence to be acquired. In order to assess the signal quality, we calculated the signal

to noise ratio (SNR). The estimation of noise was from the background image. Since the

Play-Doh used as marker had bright fluorescence also, the background images were first

processed using histogram equalization and image segmentation was employed

afterwards to differentiate the marker from real background by Matlab. The noise

estimation was processed in the areas of real background. The root mean square intensity

(RMS) was employed as the indication of power of noise or signal. Thus the SNR can be

estimated as

signal

10

noise

SNR=10logP

P

Based on the definition of SNR, the SNR of all detectors with different concentrations

and different depths were calculated as shown in Table 3-1 and Table 3-2 (two teeth with

1 mm depth hole and 2 mm depth hole respectively).

Table 3-1. SNR of fluorescence signal (SNR) for 1 mm depth of one tooth.

Dye Concentration () 1st 2nd 3rd 4th 5th 6th 7th

6.5 <0 <0 <0 <0 <0 <0 <0

13 13.5 15.4 12.9 13.7 12.2 11.5 1.9

26 21.5 23.2 23.5 23.5 20.7 12.8 7.7

63

Table 3-2. SNR of fluorescence signal (SNR) for 2 mm depth of one tooth.

Dye Concentration () 1st 2nd 3rd 4th 5th 6th 7th

6.5 0.4 1.2 <0 <0 <0 <0 <0

13 8.0 9.3 7.8 7.5 7.1 4.3 <0

26 9.9 11.3 12.8 13.1 11.4 11.9 7.8

As expected, the SNR for the 1 mm depth were generally larger than the 2 mm

depth with same concentration. However, there are some exceptions, such as for 6.5 M,

the detected signal was larger in 2 mm depth than 1 mm depth. The reason for this

phenomenon is complex. One possible reason is that the optical properties of teeth are

not homogeneous for different teeth. It is possible that the optical properties of the tooth

with 2 mm hole are smaller than the tooth with 1 mm hole and it is the difference in

optical properties that made the fluorescence at the deeper hole show larger detectable

signal intensity.

One can also observe that the mid-range detectors had larger SNR than the first

and last detectors. As described above, MFMT is a depth-resolved imaging technique

and this observed SNR behavior is due to the selective depth projection of different

source-detector pairs. High SNR values for some detectors indicate a second source,

fluorescence concentration, in the corresponding depth. This is expected as the

maximum signal intensity is dependent on the best match between the distribution of

excited photons and location of inclusions of dye. Thus, detectors close to the injection

point will receive signal from shallow inclusions whereas the detectors far from the

source will receive signal from deeper inclusions.

Reconstructions from the datasets with average SNR for seven detectors above

five were performed. Also for the special case, the 2 mm depth hole with 6.5 M was

also processed. The 3-D reconstruction of fluorescence and co-overlay with micro-CT

are shown in Figure 3-5. In the figure, the tooth surfaces were reconstructed by the

micro-CT images and the fluorescence dye distribution is shown in red. The bounding

boxes for each MFMT reconstruction are also shown.

In order to evaluate the accuracy of the reconstructions, two quantities were

proposed from the 3-D image. One is the reconstructed volume of the dye. This value

was directly compared with the real volume of dye infused into the hollow capillary.

64

Since the inner diameter of the capillary and the height of dye could be measured

directly, the volume of dye was calculated correspondingly. The other quantity is the

centroid of the reconstructed dye distribution for estimation of the spatial accuracy of

reconstruction. This parameter was compared with the hole center that was acquired by

micro-CT. Centroid error was compared in 3 directions and the maximum error was

selected, namely max (𝑒𝑥, 𝑒𝑦, 𝑒𝑧). All quantities are the relative error, which means the

difference between the values of reconstruction and real values (acquired from micro-CT

or directly measurement) divided by the real values. Table 3-3 and Table 3-4 list the

error of reconstructed dye distribution.

Figure 3-5. Merged image of MFMT and micro-CT images: (a,b) 1 mm depth hole

with 26 M and 13 M respectively; (c,d,e) 2 mm depth hole with 26 M, 13 M

and 6.5 M respectively.

Table 3-3. Comparison of reconstructed volume by MFMT and measurement.

Dye Concentration (M) MFMT Volume (mm3) Measured Volume (mm

3) Volume Err. (%)

13 (1 mm) 2.96 2.67 11

26 (1 mm) 3.18 2.86 11

6.5 (2 mm) 2.34 3.03 23

13 (2 mm) 2.04 2.67 24

(a) (b)

(c) (d) (e)

65

26 (2 mm) 2.90 2.86 1

Table 3-4. Comparison of reconstructed dye centroid by MFMT and micro-CT.

Dye Concentration (M) MFMT Centroid (mm) Micro-CT Centroid (mm) Centroid Err.(mm)

13 (1 mm) (2.1, 2.3, -1.0) (1.7, 2.0, -1.0) 0.4

26 (1 mm) (1.6, 2,2, -1.1) (1.8, 2.0, -1.0) 0.2

6.5 (2 mm) (1.9, 2.5, -1.8) (1.9, 2.4, -2.0) 0.2

13 (2 mm) (1.9, 2.5, -1.8) (1.7, 2.4, -2.0) 0.2

26 (2 mm) (2.1, 2.1, -1.8) (1.8, 2.4, -2.0) 0.3

3.3 Discussions

In this work, an ex vivo tooth model was constructed to verify preliminarily if optical

based methods were feasible in dental applications for teeth with large absorption and

scattering coefficients. Holes at different depths were employed to simulate the lesions

or pulp situations. These experiments demonstrate the potential to employ MFMT to

enable non-contact and non-invasive imaging for detecting optical changes in teeth at a

resolution of 200 m. The sensitivity of the system was designed to be established by

different concentrations of dye. Also the lesions at different depths of the teeth were

simulated by drilling holes at different places, which contained capillaries filled with dye.

The accuracy of the reconstruction was verified by rigorous registration with micro-CT

images. The error between the reconstruction and real dye locations was quantified by

comparing the volume and the centroid of reconstructed image. In summary, the overall

volume error is around 20% and the worst case is still less than 25% for all combinations

of depth and concentration of dye. As for the centroid of reconstruction, the worst case

for all the experiments is less than 0.3 mm.

The error between the reconstructed image and the experiment models is due to

several reasons. One is the method we use to generate the sensitivity matrix. In the

experiment, in order to improve the computational efficiency, voxel-based MC was

employed with simplified boundary conditions was used. This ensures that the symmetry

of the region for simulation can be employed for maximum computational efficiency.

Although, based on the voxel size, 20 × 30 × 7 = 4200 total forward computations

were needed even by the adjoint method; in fact, only 8 forward computations were

66

required. Of course, the fast computation speed sacrifices the accuracy of reconstruction

in a way. Indeed, the surface of teeth is not planar as modeled in the experiment. In order

to simulate photon propagation in arbitrary shapes of media like teeth, mesh-based MC

may be employed to increase accuracy of reconstruction [57] though this will come at an

exponential computational cost even in the case with mesh adaptive techniques [129],

[130]. Additionally, prior information is extremely important for our optical method.

This prior information not only includes the anatomical structure of teeth but also the

accurate value of optical properties, which will have great impact on reconstruction

performance. The optical properties of teeth can be found in several references, such as

[39], [131]. However, the values of optical properties are quite different in these

references. Even in the same reference, the variance is large. To make things worse,

inter- and intra-variations in optical properties may be significant. However, it was

impossible to get an accurate distribution of optical properties inside the teeth before the

experiment. This will be even more difficult in vivo. The mismatch between the model

and real samples can lead to significant error in reconstruction. Second, due to the large

absorption coefficient of teeth, less excitation light will reach the fluorophore and this

will lead to less generated fluorescence. Hence, more sensitive detectors, such as

electron multiplying CCD (EMCCD) will be more appropriate in our experiment. In

Chapter 4, an EMCCD-based transmission optical system will be introduced to

overcome the drawback of the current APD system. An optional method to increase

intensity of fluorescence is to increase the power of excitation light but in real clinical

applications, this would be limited due to safety considerations. Finally, the settings of

MFMT cannot be easily installed into the human mouth due to its limited volume.

However, micro-fabricated electromechanical systems (MEMS) based approaches to

miniaturize the light source and detector for enhanced sensitivity may enable this

application at reduced cost in the future [132]. Another optional approach is to employ a

transmission based optical system rather than the reflection optical system discussed in

this chapter. Ideally, putting the detector and light source at the opposite side of teeth

will increase the detection depth and sensitivity. This will be illustrated in detail in

Chapter 4.

67

The fluorophore used in the experiment exhibits large extinction coefficient

(132,000 cm-1

M-1

) and the quantum yield (QY) is 0.37. Thus, it will generate very strong

fluorescence. If considering employing the dye to simulate dental lesions caused by

bacteria, it will be appropriate since bacteria-induced lesions compared with intact teeth

can cause changes in fluorescence that are expected to be 10 times larger at peak

emission for various wavelengths of excitation. However, since the Alexa Fluor 660 is a

linear dye, it will generally have lower sensitivity and resolution compared with non-

linear fluorophores. This will also be discussed in detail in next chapter.

68

4. Upconverting nanoparticles and their application in dental imaging

Upconverting nanoparticles (UCNP) are a kind of particle that can absorb multiple low

energy photons to produce high energy photons. The special spectral properties of

UCNPs make them a promising theranostic agent, with potential application in root

canal therapy. When UCNPs are mixed with dental fillings, in addition to preventing

further decay of teeth, the mixture can also report the filling status of the hole through

optical imaging. Furthermore, the short wavelength visible light has been proven to be

bactericide [33]. Under the excitation of near infrared light, which is generally harmless

to human tissue, UCNPs can emit from near infrared light to visible light. The near

infrared light emission can be employed for optical tomography and the visible light for

killing residue bacteria after surgery. *

In this chapter, a brief background of UCNPs will be given. Considering the high

optical values of teeth, in order to assess the performance of the optical system, ex vivo

experiments will be investigated first. Since these preliminary experiments are the first

step for establishing the potential of clinical dental applications using optical

tomography, some simplified experiments models are employed. We emphasize here the

simplifications of our experiments. The first simplification is that during the experiment,

the incident light is irradiated at the mesiodistal surface of teeth, which in the human

mouth is not located under the gum and can be directly exposed to excitation light.

However, if we consider the real situation in human mouth then only the buccolingual

side of the teeth can be irradiated by excitation light, a problem exists in ex vivo

experiments. The buccolingual side of the tooth is smaller than the mesiodistal side. For

example, for maxillary molars, the width of the buccolingual side of the tooth (or root

thickness) is 1~2 mm [133]. In contrast, the width of the mesiodistal side for maxillary

molars is around 6~8 mm [134]. Since micro-CT is employed to verify the accuracy of

the reconstruction of optical tomography, fiducial marks have to be made at the surface

of teeth in order to rigorously register the images or are employed to indicate the

position of irradiation light. It will be easy and accurate to make marks on relatively

Portions of this chapter are to appear in: F. Long, X. Intes and S. P. Kotha, “Dental optical tomography with UCNPs,”

in Optical Tomography and Spectroscopy, 2016.

69

large surfaces compared to the smaller ones. On the other hand, theoretically one needs

to accurately pinpoint the irradiation light position when the investigated object has an

irregular outline. The curvature of buccolingual side of the tooth is larger than the

relatively planar mesiodistal side. Thus, determining accurate positions of irradiation

points by eye on the buccolingual surface of the tooth will be challenging. Moreover,

due to the limited scale of the buccolingual side, the optical model to predict photon

transportation will also be challenging considering the complex boundary conditions.

The second simplification is in the experiment is that a capillary with inner diameter 1

mm filled with optical contrast (UCNPs or linear dye) was employed to simulate the

cavity of dental caries (see Section 4.1.6 for details). The diameter of a real tooth root

canal before root canal therapy is around 0.1 ~ 0.6 mm based on different types of teeth

and position where the measurements take place [133], [135], [136]. During root canal

therapy, dentists will normally use different sizes of tapers to shape (enlarge) the root

canal [137]. This will cause enlargement of the root canal. Moreover, the diameter of the

root canal at the place of the pulp chamber will be larger compared with the root canal

near the apical foramen. On average it is reasonable to employ a 1 mm glass capillary to

approximately simulate the root canal situation after surgery.

Finally, another point to be noted is the concave shape of the root part of the

tooth. Teeth have very complexly shaped roots, such as a difference in number of roots

for individual teeth, from 1~4 roots in general [29]. For the extracted whole tooth, when

photons are emitted from one root, it is possible to re-enter another root. This will lead to

a complex photon propagation model since most numerical models for predicting photon

transport exclude this circumstance. In order to avoid this situation, the teeth will be cut

apart and only one root will be investigated in our experiments.

4.1 Materials and methods

4.1.1 Upconverting nanoparticles (UCNPs)

Lanthanide-doped UCNPs are dilute guest host systems [138], in which lanthanide ions

are distributed as a guest in the lattice of the host. Through different selections of

lanthanide ions and synthesizing methods, UCNPs can emit different colors of light,

from near infrared (NIR) to visible light and ultraviolet (UV). Unlike other multiphoton

70

absorption dyes, UCNPs do not need high power, short pulse excitation light [139]. In

contrast, continuous wave (CW) NIR laser diodes (~1 W) with low power are powerful

enough to excite UCNPs with bright emission light. Briefly, the sensitizer ions (such as,

ytterbium, Yb3+

) will capture the incident low energy photons and transfer the energy to

the activator ions (thulium, Tm3+

, erbium, Er3+

etc.), which generate the high energy

photons. Generally, the relationship between emission photon intensities and the

excitation laser power is non-linear, expressed as

n

UCNPI P (4-1)

In the equation, 𝐼𝑈𝐶𝑁𝑃 is the intensity of the emission photon, 𝑃 is the power of the laser,

and 𝑛 is the power-dependent index, normally not equal to 1 for UCNPs. For different

synthesizing methods and components, 𝑛 could be different even for the same color

emission. For instance, UCNPs (NaYF4: 30% Yb3+

, 0.5% Tm3+

) reported in [139], have

value 𝑛 equal to 5, 4, 3, 2 at upconverting emission wavelengths 345 nm, 360 nm, 475

nm and 800 nm respectively. Also notice that with different structures of UCNPs, the

illumination intensities are quite different. For example, also reported in [139], the

UCNPs with active core/active shell/inner shell structure have a higher emission

intensity compared to the UCNPs with the active core/active shell only. Another

example shows that with different components of UCNPs, the power 𝑛 will be different.

It is reported in [140], NaYF4: 18% Yb3+

/2% Er3+

when emitting the blue light, the

power 𝑛 will be 3 (2.81 in [140]). In contrast, for NaYF4: 18% Yb3+

/ 0.5% Tm3+

when

emitting the blue light, the power 𝑛 will be 2 (1.88 in [140]). From these examples

shown above, the mechanism of the upconverting process will be complex. In general,

there are 5 mechanisms employed to explain the upconverting phenomenon, which are

(1) excited-state absorption (ESA), (2) energy transfer upconversion (ETU), (3)

cooperative sensitization upconversion (CSU), (4) cross relaxation (CR) and (5) photon

avalanche (PA) [141]–[143]. Here we briefly introduce these processes. Different

mechanisms have different transfer efficiency. The three most efficient mechanisms

[144], ESA, ETU and CSU are shown in Figure 4-1.

71

4.1.1.1 ESA

ESA occurs when one ion sequentially absorbs two photons in a three level system.

When the ion absorbs one photon, its energy level will be increased from the ground

state G to the excited state E1. If the lifetime of E1 is long enough, it will increase the

possibility of another photon pumping the ion from E1 to E2. Then the upconverting

emission will occur from E2 to G. The energy system of ESA is shown in Figure 4-1 (a).

4.1.1.2 ETU

Unlike ESA, ETU involves two ions to finish one upconverting process. The sensitizer

ion will absorb one photon and transfer the energy to an adjacent activator, which

increases the energy state of the activator from G to a metastable state E1. Then the

sensitizer will relax back to ground and absorb one photon again to transfer energy to the

activator in the E1 state. The activator will be excited from E1 to E2, then emit a photon

and relax back to G.

Figure 4-1. Energy levels of different upconverting processes. Red, green, blue

arrows represent photon absorption, energy transfer and upconverting emission

respectively: (a) ESA; (b) ETU, note that ion 1 is pumped to E1 twice; (c) CSU.

4.1.1.3 CSU

Unlike ESA and ETU, CSU involves three ions during the upconverting process. The

sensitizer ion 1 and ion 2 will absorb 2 photons and be pumped to E1. Then these

sensitizer ions will cooperatively transfer the energy to the activator ion and excite ion 3

to its higher energy level E1. After ion 3 relaxes back to G, it will emit one higher energy

photon.

(a) (b) (c)

72

4.1.1.4 CR

Strictly speaking, CR is not a process to emit upconverting photons directly. However, it

is an intermediate, energy transfer process of PA. When ion 1 is in the high energy state

E2, it can interact with another ion 2 to transfer its energy to ion 2, pumping ion 2 from

G to E1. Simultaneously, the energy state of ion 1 will relax from E2 to E1.

4.1.1.5 PA

PA is a process that produce a large amount of upconverting photons above a certain

energy of excitation light [138]. In this process, ion 1 is already pumped to the E1 level

through non-resonant weak ground absorption [138]. Ion 1 will then be pumped into E2

after absorption of an excitation photon. Through CR, the energy of ion 1 will transfer to

ion 2 and make two ions at E1. Lastly, ion 2 in E1 will transfer its energy to ion 3 to

excite its energy level from G to E1. The net effect is that one E1 ion will generate two E1

ions. This process can occur continuously, generating more ions in the E1 state. After

absorption of photons, these E1 state ions will be pumped into the E2 state. Then, when

they are relaxed back to G, upconverting photons are emitted.

From the description above, the upconverting process is complex. In order to

enhance the upconverting efficiency, such as by tuning the color and processing layers

of UCNPs, multiple approaches have been reported to synthesize UCNPs. Since the

discussions of synthesizing UCNPs are beyond the scope of this thesis, readers can refer

to [138], [145], [146] for more information.

In our experiments, the UCNPs (Sodium yttrium fluoride, ytterbium and thulium

doped, NaY0.77Yb0.20Tm0.03F4, Sigma-Aldrich, USA) were employed as optical contrast

agents. The UCNPs are white powder with particle sizes less than 10 m (from

specification sheet of Sigma-Aldrich). In order to investigate the power dependence of

UCNPs employed in our experiment, a 975nm laser diode (L975P1WJ, Thorlabs, USA)

mounted by TCLDM9 (Thorlabs, USA) controlled by the current driver (LDC200C,

Thorlabs, USA) and temperature controller (TED200C, Thorlabs, USA) was used as the

excitation source of UCNPs. The power of the laser diode was recorded by the power

sensor (S310C and PM100USB, Thorlabs, USA). Also, a filter was used to remove the

excitation light. The emission light of UCNPs was coupled to an optical fiber, which was

73

connected to a spectrometer (USB 2000, Ocean optics, USA). Double logarithms plots

was employed in order to investigate the relationship between laser pump power and

intensity of UCNP emission.

Teeth phantom preparation

As discussed in Chapter 3, in order to mimic the root canal conditions after root canal

therapy, extracted teeth from local dentists were used as a model with approval from the

Institute Biosafety Committee (IBC). 3~4 fiducial marks (shallow semi-sphere dents)

determined by the size of teeth were made on the mesiodistal surface to further register

the optical reconstruction with micro-CT images (See Figure 4-2 (a)). In order to mimic

the root canal situation after root canal therapy, a simplified model as discussed above

was employed, in which a through hole was drilled from the canal foramen to the pulp

chamber with diameter 1.1 mm to contain a glass capillary with inner diameter 1.0 mm

filled with optical contrast agents. Then the teeth were cut in half to preserve only one

root in order to avoid the re-entering of photons emitted from adjacent roots. After

processing, teeth were scanned by micro-CT (Medical Viva CT 40, Scanco, Switzerland)

to get the anatomical structure of teeth (Figure 4-2 (b)). The micro-CT is equipped with

a micro-focus cone-beam X-ray source. In the experiment, the X-ray source voltage was

set to 70 kVp, and the current was 160 A with standard resolution (38 m for each side

of voxel, thus the voxel size was 38 by 38 by 38 m3). The scanning time of a single

tooth was around 20~30 mins. In total, 300~400 slices were acquired for each tooth.

Further, the image acquired by micro-CT was processed in ImageJ for k-means

segmentation or binarization (Figure 4-2 (c)).

74

Figure 4-2. Demonstrations of processing of teeth sample: (a) Photo of Tooth 1, in

which the red circle indicates one mark; (b) One slice of CT images directly

acquired by micro-CT; (c) Binary image of the same slice as (b); (d) Mesh

generated by CGAL.

4.1.3 Optical settings

Our new system is as similar features as the MFMT introduced in Chapter 3. However,

two major modifications were made. One is that the transmittance geometry, with

sources and detectors fixed at opposite side of the media, was implemented. MFMT is

based on the reflectance geometry, which means that the sources and detectors are on the

same side of the media. The disadvantage of this geometry is the limited detectable

depth, normally less than 5 mm [147]. In real situations, the mesiodistal thickness of the

tooth is less than 1 cm. Moreover, teeth are embedded into the jaw and the thickness of

the jaw is normally around 1 cm [148]. Besides, the non-contact MFMT is difficult to

place into the limited volume of the mouth. Data shown above illustrates that the

transmittance geometry is more suitable in dental applications compared with reflectance

geometry. The other modification is to replace the avalanche photo diode (APD) array

with an electron multiplying CCD (EM-CCD) camera, leading to an increased number of

measurements for improved resolution.

(a) (b)

(c) (d)

75

On the illumination side, a 975 nm excited laser (L975P1WJ, Thorlabs, USA),

controlled by current driver (LDC200C, Thorlabs, USA) and temperature controller

(TED200C, Thorlabs, USA) was fed into a 2-D galvo-scanner (GVS002, Thorlabs, USA)

and focused on the surface of the teeth by a lens. On the detection side, the upconverted

blue light (450~500 nm) or near infrared light (~800 nm) was captured by an EMCCD

(iXonEM+

885, Andor, USA) after passing through a band pass filter (FGB37M, Thorlabs,

USA or FF01-800/12-25, Semrock, USA). The field of view (FOV) was set 3 by 4 mm2

(Tooth 1); the area depends on the size of the mesiodistal surface. At each irradiation

point, the galvo-scanner would dwell 1 sec (this value can be adjusted by software run

on a PC) before moving to the next position. The number of sources was determined by

the size of the FOV and the step size between adjacent source positions (0.5 mm).

A small micro-controller unit (MCU) system was designed to control the

movements of the galvo-scanner. Briefly, the program (Visual Basic.Net) run on a PC

sends the command to MCU (MC9S08QE32, Freescale, USA) which will analyze the

protocol and control the rotation of the galvo-scanner via virtual serial port with baudrate

38,400 bps. A 16-bit digital to analog converter (DAC8565, TI, USA) is employed to

drive the movement of the galvo-scanner. After sending the signal to the DAC8565 to

make the galvo-scanner dwell at a particular position for a fixed period of time, the

MCU also sends an external trigger signal to EMCCD, which works in the external

trigger mode, to capture one image. The exposure time for the EMCCD was set to 5 ms

during experiments. Based on different optical contrast agents, the EMCCD gain was

different and the maximum value was set to 150. The MCU trigger signal is shaped by a

mono-stable circuit composed with a precision timer (NE555, TI, USA).

4.1.4 Optical reconstruction

Generally, the optical reconstruction method follows similar steps to that shown in

Chapter 3. Since the outline of the tooth is complex and the volume of the tooth is small,

only the radiative transfer equation (RTE) or mesh-based Monte Carlo can be used to

predict photon propagation. Also, as discussed in Chapter 2, our RTE algorithms are not

advantageous when dealing with ideal pencil beams. Herein, the only investigated

method will be mesh-based MC. In the experiment, after scanning the anatomical

76

structure of the tooth, the mesh was generated by pre-compiled Computational Geometry

Algorithms Library (CGAL) (See Figure 4-2 (d)). After the mesh was created,

tetrahedron mesh-based MC [119] was employed to get Green’s function for one

particular source or detector. 106 photons were employed for faster computation. A

forward-adjoint method was employed to generate Jacobians (sensitivity matrix) for high

computational efficiency. In general, the two forms employed to describe the

fluorescence or emission light from UCNPs at detector 𝒓𝒅 excited by the source 𝒓𝒔 are

shown below.

1( , ) ( )[ ( , )] ( , )d

( , )

x n m

B xX

U G GG

s d s d

s d

r r r r r r r rr r

(4-2)

( , ) ( )[ ( , )] ( , )dx n m

XU G G

s d s dr r r r r r r r (4-3)

The meaning of each expression shown in Eqn. (4-2) and (4-3) are exactly the same as

shown in Chapter 3, Eqn. (3-1). Compared with the similar reconstruction equations

shown in Chapter 3, the difference is that in Eqn. (4-2) and (4-3) the power index of the

excitation Green’s function is dependent on the optical contrast agents employed during

the experiments. For example, for linear dyes (Alexa Fluor 660 or IR Dye 800) 𝑛 = 1,

while for UCNPs 𝑛 = 2, 3, 4, … depending on the different components or synthesizing

methods of the UCNPs. In the experiments, 𝑈𝐵(𝒓𝒔, 𝒓𝒅) was calculated as 𝑈𝑒𝑚 𝑈𝑒𝑥⁄ , in

which 𝑈𝑒𝑚 represents the emission image (with optical contrast agents inside the media,

irradiated by excitation light and taken with the filter in place) and excitation image

(with optical contrast agent inside the media, irradiated by excitation light and without

the filter), respectively. In contrast, 𝑈(𝒓𝒔, 𝒓𝒅) was calculated as the difference between

the emission and background images, namely 𝑈𝑒𝑚 − 𝑈𝑏𝑔. In the expression, 𝑈𝑒𝑚 is the

same as Eqn. (4-2) while 𝑈𝑏𝑔 represents the background image (without optical contrast

agent inside the media, with irradiation by excitation light and with the filter). In a way,

the background image can be viewed as system background noise.

As discussed in Chapter 3, some prior knowledge of optical properties of dentin

must be acquired before applying forward-adjoint MC. The mismatch between the ideal

optical properties and true values may be one reason that large reconstruction errors are

generated, as shown in Chapter 3. However, the literature reports different optical

77

properties of dentin, especially for the scattering coefficient. For instance, in [39], all the

mean values of the scattering coefficients for dentin are around 28 mm-1

for a large range

of wavelengths (from visible light to infrared). However, the deviation of scattering

coefficients is large for different wavelengths of light. In contrast, the scattering

coefficient for dentin in [149], reported in Chapter 3, is around 10 mm-1

for visible light.

Thus in real experiments, different values were tested until the best performance was

achieved.

The same mesh density was employed for all teeth. The average value of volume

of elements (element volume) in the experiments was set to 0.001 mm3, leading to

different numbers of nodes and elements based on the different scales of teeth. For

instance, for the Tooth 1 employed in the in silico experiments, 32,242 nodes with

196,409 elements were generated. In other words, the number of unknowns was 32,242.

In practice, Eqn. (4-2) and (4-3) was directly discretized to get linear systems. As

discussed in Chapter 3, a depth-dependent adaptive Tikhonov regularization method was

used as shown below.

1( )T Tx A A D A b (4-4)

The equation is exactly the same as Eqn. (3-2). The equation was then solved by the

generalized minimal residual method (GMRES) using Matlab built-in function gmres.m.

The iteration method ended when either the maximum iteration number (1,000 set in the

experiment) or the error bound 10-5

was reached.

4.1.5 In silico experiment design

In order to investigate the performance of UCNPs and linear dyes in optical tomography,

an in silico experiment was designed. A cylinder with diameter 1 mm with center located

at (5, 6) was embedded in the tooth model. This was achieved by assigning the node

value 1 for the nodes located inside the cylinder. In contrast, 0 was assigned to all the

other nodes. Figure 4-3 shows the simulated cylinder inside the tooth in different views.

The sensitivity matrix was generated by the adjoint MC forward method. Then the

simulated measurements could be calculated as the multiplication of sensitivity matrix

and optical contrast agent distribution vector as discussed above. The simulated

measurements were then employed as the input to the inverse problem.

78

Figure 4-3. Different views of the simulated cylinder inside a tooth: (a) 𝒚𝒛 view; (b)

𝒛𝒙 view.

In the experiments, the number of detectors was set to 70 or 105 respectively,

with a distance of 0.5 mm between two adjacent detectors. See Figure 4-4 below. The

number of sources was varied from 15 to 63 which covered the full area of the surface

irradiated by light. See Figure 4-5 below. The field of view (FOV) was set to 4 mm by 3

mm which was the same as in the experiments for Tooth 1. The selection of optical

properties is quite cumbersome since different literature report different values as

discussed above. Thus in this section, [39] and [149] (from Chapter 3) were mainly

referenced to acquire the optical properties. Because the FOV was mainly located in

dentin of the teeth, and the absorption coefficient for dentin is quite independent from

wavelength of irradiated light. Herein, the absorption coefficient was set to 0.3 mm-1

for

all wavelengths employed in the simulation. The scattering coefficient was set to 14.2

mm-1

at 975 nm [149] and 17.1 mm-1

at 800 nm [149]. The scattering coefficient was set

to 40 mm-1

at visible light, particularly for blue light [39]. The anisotropy factor was set

to 0.93 for all the simulation and the refractive index was 1.4.

(a) (b)

79

Figure 4-4. Distribution of detectors: (a) 70 detectors; (b) 105 detectors.

When solving Eqn. (4-4), the selection of regularization parameter was

performed by L-curve analysis [128]. Moreover, the regularization parameter was

adjusted manually to find the best performance of reconstruction. In this series of

experiments, 𝜆 was in the range of 0.001 to 1. After solving Eqn. (4-4), a structured

mesh (with voxel size 0.001 mm3) was generated and a linear interpolant was employed

to transfer the node value of the tetrahedron to a structured mesh (voxel value) for data

processing by Matlab (gradient, display figure etc.). Iso-surface with iso-value 0.5 was

extracted from the normalized data with respect to the global maximum value. In order

to assess the accuracy of the reconstruction, two quantities were proposed. One quantity

is the volume under the FOV. When calculating the volume of reconstruction, all the

voxel value larger than or equal to 0.5 were counted. The other quantity is the centroid

of the reconstruction. As in the volume calculation, only the voxels with value larger

than or equal to 0.5 were counted. The relative error between the real values and

reconstructed values was calculated. For the centroid, the maximum difference along 3

dimensions was estimated, namely max{𝑒𝑥, 𝑒𝑦 , 𝑒𝑧}. Since the surface of the tooth is not

even, as stated in Chapter 2, an algorithm calculating the intersection points between

arbitrary rays and triangle surface element was employed. For the details, readers can

refer to Chapter 2, Section 2.2.3.

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Figure 4-5. Distributions of sources: (a) 15 sources; (b) 20 sources; (c) 25 sources;

(d) 27 sources; (e) 35 sources; (f) 36 sources; (g) 45 sources; (h) 63 sources.

4.1.6 Procedure of experiments

After the samples were prepared, the teeth were imaged in our transmission optical

system. Normally, background images were captured first. After that, the glass capillary

filled with pure UCNP powder was inserted into the teeth. Than the excitation images

and emission images were taken sequentially. Next, UCNPs were mixed with bisphenol

A glycol dimethacrylate (Bis-GMA), a conventional dental resin. The mixture was first

cured by blue light, and then inserted inside the teeth. The excitation images and

emission images were then taken sequentially. We repeated the same procedure for the

linear dye.

4.2 Results

4.2.1 In silico simulations

The differences in sensitivity profiles of linear dye (Alexa Fluor or IR Dye 800) and

UCNPs with different emission light within the same plane of Tooth 1 are shown in

Figure 4-6. In Figure 4-6, the source was set at the right side of Tooth 1 and the detector

was located at the left side of Tooth 1. It is clear that the sensitivity profiles for UCNPs

are sharper than those of linear dyes due to the non-linear transportation properties of

UCNPs (see Eqn. (4-2) or (4-3), the power of the excitation Green’s function is not 1).

(a) (b) (c) (d)

(e) (f) (g) (h)

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From physics, this usually represents higher resolution as suggested in [34]. In the

simulations, although the scattering coefficient of dentin at blue light was set quite larger

than the coefficient in NIR light, the sensitivity profile of UCNPs emitting blue light is

still sharper than sensitivity profile of UCNPs emitting NIR light (Figure 4-6 (a) and (b)).

Here the power of the Green’s function in Eqn. (4-2) or (4-3) was selected to be the

theoretical value, 3 for blue emission and 2 for NIR emission. Note that in real situations,

due to strong saturation effects of UCNPs, the power index may be lower than the

theoretical value.

Figure 4-6. Sensitivity profiles at the same cross section of Tooth 1 with different

optical contrast agents: (a) UCNPs with emission of blue light (475 nm); (b) UCNPs

with emission of near infrared light (800 nm); (c) Linear fluorophore with emission

of red or near infrared light.

Although the sensitivity profiles of UCNPs are normally sharper than those of

normal linear dyes, the reconstructions (inverse problem) for UCNPs are generally more

difficult than for linear dyes because the condition number of the sensitivity matrix of

high power dependent UCNPs is usually large. This can be easily seen from Figure 4-7

(a) and (b). From the graph, with an increasing number of sources under a fixed number

of detectors, the condition number increases; this means it is more difficult to solve Eqn.

(4-4). From a comparison between Figure 4-7 (a) and (b) it is shown that with a larger

(a) (b)

(c)

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number of detectors, the condition number for the sensitivity matrices is also larger for

the same number of sources, as expected. Besides, it is clear that an increasing power

dependent index will increase the condition number dramatically. That means that the

cost for higher resolution is higher difficulty in solving Eqn. (4-4). Figure 4-7 (c) shows

the positive eigenvalues of the sensitivity matrix with different optical contrasts under 70

detectors. It is obvious that the sensitivity matrix describing the procedure of UCNPs

emitting blue light has a much smaller positive eigenvalue than the other situations. This

also illustrates that the sensitivity matrix with higher power dependent UCNPs has larger

condition number, as discussed above.

Figure 4-7. Condition number of different optical contrast and eigenvalue of

sensitivity matrix: (a) Condition number of sensitivity matrix with different

number of sources under 70 detectors; (b) Condition number of sensitivity matrix

with different number of sources under 105 detectors; (c) Eigenvalue of sensitivity

matrix.

As discussed in Section 4.1.5, since the FOV was set to 4 mm by 3 mm and the

diameter of the simulated cylinder is 1 mm, the volume of optical contrast under the

FOV is 𝑉0 = 3.14 mm3. Thus the relative error, shown in Table 4-1 and Table 4-2, can

be calculated as

(a) (b)

(c)

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0

0

| |Err 100%

V V

V

Table 4-1. The volume of reconstruction with 70 detectors for different optical

contrast agent (the unit of the volume is mm3).

UCNP blue light UCNP NIR Linear dye

No. of Src. Volume Error (%) Volume Error (%) Volume Error (%)

15 2.52 19.8 4.14 31.9 2.84 9.6

20 2.06 34.4 4.24 35.0 3.29 4.8

25 2.22 29.3 3.44 9.6 3.21 2.2

27 3.80 21.0 4.20 33.8 3.15 0.3

35 2.82 10.2 2.97 5.4 3.11 1.0

36 3.50 11.4 3.24 3.2 3.10 1.2

45 3.42 8.9 3.37 7.3 3.12 0.6

63 3.04 3.2 3.04 3.2 3.09 1.6

Table 4-2. The volume of reconstruction with 105 detectors for different optical

contrast agent (the unit of the volume is mm3).

UCNP blue light UCNP NIR Linear dye

No. of Src. Volume Error (%) Volume Error (%) Volume Error (%)

15 2.31 26.4 4.26 35.7 2.86 8.9

20 2.46 21.7 4.34 38.2 2.86 8.9

25 2.29 27.1 3.52 12.1 2.95 6.1

27 3.57 13.7 3.40 8.3 2.98 5.1

35 2.85 9.2 3.47 10.5 3.01 4.1

36 2.96 5.7 3.36 7.0 3.05 2.9

45 3.35 6.7 2.95 6.1 3.09 1.6

63 3.09 1.7 3.32 5.7 3.19 1.6

From Table 4-1 and Table 4-2, for a fixed number of detectors with increasing

number of sources, the relative error tends to decrease and converge to 0 in general. As

discussed above, since the condition number of the sensitivity matrix with higher power

dependent UCNPs is larger, the relative error of reconstructions is also larger compared

with linear dye reconstructions. In practice, the optimized regularization parameters for a

sensitivity matrix with high power dependent UCNPs have to be selected very carefully

to get the best performance. The reconstruction volume also has a tendency to converge

to the real volume of cylinder 𝑉0. The behavior of the reconstructions with respect to

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different number of source-detector pairs can also be observed from Figure 4-8 and

Figure 4-9 below.

Figure 4-8. The difference between reconstruction volume and real volume with 70

detectors for different optical contrast agents: (a) The relationship between

reconstruction volume and number of sources, the black horizontal line indicates

the real volume and the dashed lines indicate the relative error 10% off the real

volume; (b) The relationship between volume relative error and number of sources.

Figure 4-9. The difference between reconstruction volume and real volume with

105 detectors to different optical contrast agents: (a) The relationship between

reconstruction volume and number of sources, the black horizontal line indicates

the real volume and the dashed lines indicate the relative error 10% off the real

volume; (b) The relationship between volume relative error and number of sources.

From Figure 4-8 and Figure 4-9, with increasing number of sources under fixed

number of detectors, the relative error decreases. Also with a large number of sources

(a) (b)

(a) (b)

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(larger than 35), all the relative errors are less than or around 10% (the reconstruction

volume will be located within the band whose width is 0.62 centered at the real volume).

Similarly, the centroid of reconstruction can be obtained as discussed in Section

4.1.5. The real centroid of the cylinder can be estimated as (5.0, 6.0, 4.2) where the 𝑧

coordinate of the center of the FOV was employed as the 𝑧 coordinate of the real

centroid. The maximum absolute error of 3 coordinates (defined in Section 4.1.5) is

shown in Table 4-3.

Table 4-3. The maximum difference of 3 coordinates between reconstructed

centroid and real centroid (The unit in the table is mm).

70 Det. 105 Det.

No. of Src. UCNP blue UCNP NIR Linear dye UCNP blue UCNP NIR Linear dye

15 0.9 0.5 0.08 1.2 0.5 0.10

20 1.4 0.5 0.07 1.5 0.5 0.09

25 1.2 0.5 0.05 1.4 0.5 0.07

27 0.8 0.5 0.06 1.0 0.5 0.09

35 1.0 0.7 0.05 1.3 0.5 0.06

36 1.0 0.6 0.04 1.2 0.7 0.06

45 1.1 0.5 0.03 1.3 0.6 0.04

63 1.0 0.6 0.03 1.2 0.6 0.03

From Table 4-3, for UCNPs, the maximum error occurs along the 𝑥 direction

which is the direction of excitation light irradiation on the teeth or 𝑧 direction. Also the

error remains almost constant with increasing number of sources under a fixed number

of detectors. In contrast, for the linear dye, the maximum error occurs along the 𝑧

direction. Also, the error for the linear dye is quite small compared with the error for

UCNPs since the condition number of the sensitivity matrix of linear dyes is much

smaller than the ones for UCNPs. From physics, the sensitivity matrix describes the

amount of change of the measurement with respect to the change of optical properties or

optical contrast agent distribution [150]. Intuitively, the areas with lower values of

sensitivity profiles will most likely have more distributions of optical contrast agent with

the same measurements. As in the sensitivity profiles with different optical contrast

agents shown in Figure 4-6, within the same plane the values of the sensitivity matrix of

UCNPs are much smaller than those of linear dyes near the position of the detectors.

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Therefore, the reconstruction will be closer to the detectors for the UCNPs compared

with linear dye.

The large deviation in centroid position of the reconstructions with UCNPs along

the 𝑧 direction is because the thickness of Tooth 1 is not uniform. This can be clearly

seen from Figure 4-3 (b), which shows that the bottom of Tooth 1 is thinner than the top

part. If directly observing the simulated measurements for different detectors shown in

Figure 4-10, we can see that for the linear dye, the measurements are quite uniform

which indicates that the measurements of the detectors located at the top of Tooth 1 are

not obviously smaller than the measurements of detectors located at the bottom although

the light travels a longer distance. However, in contrast, for the UCNPs the

measurements of the top detectors are smaller than the measurements of detectors

located at the bottom of Tooth 1 (See Figure 4-10 (a) and (b) left). This is more obvious

for UCNPs with emission of blue light (power dependent index is 3). When

reconstructing the distributions of UCNPs, the algorithm itself cannot differentiate

whether the smaller measurement is due to the longer penetration distance or less

UCNPs at corresponding positions. This may explain why larger error will occur along

the 𝑧 direction as well as why the reconstructions of UCNPs with emission of blue light

deviate from the real centroid more than UCNPs with emission of NIR.

87

Figure 4-10. Simulated measurements at the positions of detectors for different

optical contrast: (a) UCNPs emission blue light; (b) UCNPs emission NIR; (c)

Linear dye.

In summary, increasing the number of sources under a fixed number of detectors

will reduce the reconstruction error with the cost of reducing computational efficiency

(longer calculation time and more computational resources). For the UCNP, because of

its non-linear power dependent optical properties, its sensitivity profiles have a larger

gradient close to the source and are sharper near the detectors compared with those of

linear dyes. This should lead to higher resolution as reported in the references. Under the

same number of detectors, the reconstructed volume error will be below 10% with

number of sources larger than 35. The sensitivity matrices of UCNPs are more sensitive

with the thickness of teeth and have larger error in estimation of the centroid compared

with the linear dye.

4.2.2 Determination of power index of UCNPs with emission of blue light

As discussed above, since different synthesizing methods as well as different doping

materials will produce UCNPs with different power dependent indices even with the

same emission band, it is necessary to determine the value of the power dependent index.

For NIR emissions, no matter the theoretical calculation or the experiment, the power

(a) (b)

(c)

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dependent index is 2. In contrast, for 475 nm blue emission, different papers report

different values, as discussed in Section 4.1.1. Herein, an experiment was performed to

investigate the value of power index. The settings of the experiments can be seen in

Section 4.1.1.

Figure 4-11. The blue emission band of UCNPs: (a) Relation between emission

intensity and wavelength between 440 nm and 510 nm at different laser power; (b)

Relation between max emission intensity and laser power.

From Figure 4-11, it can be calculated that the power dependent index for blue

emission is 𝑛 = 1.78, which is quite similar to the value reported in [140], indicating

strong saturation effects [151], [152]. In this case, two NIR photons will produce one

blue photon emission.

4.2.3 Ex vivo experiments

From the in silico experiments, although using a large number of detectors and sources is

computationally more expensive, the relative error for the volume and centroid are

generally the smallest. Thus, in real ex vivo experiments 63 sources and 105 detectors

were employed for Tooth 1. As discussed in Section 4.1.6, several optical contrast agents

were tested, including pure UCNPs with emission of NIR and blue light, 20% w.t.

UCNPs mixed with Bis-GMA with emission of NIR and blue light, and 26 M Alexa

Fluor 660 with emission of 690 nm light.

Since mesh-based MC was used in this study, the outline of the tooth will be

preserved after reconstruction. Thus, using built in affine registration algorithms in

Amira 5.4.5 (FEI, USA), the optical reconstruction and CT images can be registered

(a) (b)

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rigorously. Figure 4-12 and Figure 4-13 show the directly merged images of UCNPs

optical reconstruction and CT images (front view and lateral view respectively). Figure

4-14 shows two views of merged images of linear dye.

Figure 4-12. Front merged image of UCNPs: (a) UCNPs with blue emission; (b) Bis-

GMA with UCNPs (20% w.t.) with blue emission; (c) UCNPs with NIR emission; (d)

Bis-GMA with UCNPs (20% w.t.) with NIR emission.

Figure 4-13. Lateral merged image of UCNPs: (a) UCNPs with blue emission; (b)

Bis-GMA with UCNPs (20% w.t.) with blue emission; (c) UCNPs with NIR

emission; (d) Bis-GMA with UCNPs (20% w.t.) with NIR emission.

(a) (b) (c) (d)

(a) (b) (c) (d)

90

Figure 4-14. Merged image of linear dye: (a) Front view; (b) Lateral view.

Based on the FOV (see Section 4.1.5), the real volume under the FOV is 3.14

mm3 and the centroid of the cylinder can be estimated from micro-CT images to be (4.5,

7.0, 4.2) mm. Thus, as performed in silico experiments, the reconstructed volume and

centroid were calculated and are shown in Table 4-4. In the table, the mixture represents

the Bis-GMA mixed with UCNPs (20% w.t.).

Table 4-4. The error between optical reconstructions and real value.

Volume (mm3) Relative Err. (%) Centroid (mm) Error (mm)

UCNPs (blue) 3.09 1.6 (4.0, 7.5, 3.3) 0.9

Mixture (blue) 3.24 3.2 (4.3, 7.3, 2.9) 1.3

UCNPs (NIR) 3.11 1.0 (4.1, 7.3, 3.4) 0.8

Mixture (NIR) 3.36 7.0 (4.4, 7.4, 3.5) 0.7

Linear dye 2.74 12.7 (4.7, 6.8, 4.1) 0.2

It is quite clear from Table 4-4 that for the UCNPs reconstruction, the volume

relative error is less than 10% for all cases. For Bis-GMA mixed with UCNPs, the error

is little bit larger than the pure UCNP powder. As demonstrated in Section 4.2.1, the

largest error occurs along the 𝑧 direction due to the thickness at the top of Tooth 1

compared to the bottom of Tooth 1 (clearly seen in Figure 4-13). This will make the

reconstructions more concentrated at the bottom of FOV. However, for the linear dye

reconstructions, the results were the worst. Additionally, there are more artifacts in the

linear dye reconstructions than for UCNPs (see Figure 4-14).

(a) (b)

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The non-optimal reconstruction of linear dye may be due to various reasons. First,

the power of excitation light for the linear dye and UCNPs were different, ~40 mW and

~100 mW respectively. However, the signal to noise ratio (SNR) of linear dye and

UCNPs images are quite similar. The calculation of SNR was the same as introduced in

Chapter 3. For instance, the average SNR for 63 sources of linear dye is 4.23 dB and in

contrast, the average SNR for 63 sources of UCNPs that emit NIR light is 4.57 dB. If

observing the raw emission images, visually the linear dye emission intensity is smaller

than UCNPs emission (NIR) intensity (see Figure 4-15). If compared with UCNP blue

emission, the light intensity for the linear dye is even larger since the attenuation

coefficient for the red or NIR light is smaller than blue light. Thus the emission light

intensity should not be the main reason for the bad reconstructions of linear dye.

In order to get better reconstruction, background noise and autofluorescence must

be minimized [31]. However, for dentin and enamel, it is reported that there exists

autofluorescence in various bands [153]. Thus, the mixture of emission fluorescence

with autofluorescence might be one reason for generation of relatively inferior

reconstruction results. Even though the wider sensitivity profiles of the linear dyes

compared with those of UCNPs reduce the condition number of the sensitivity matrix,

they will result in wider regions to reconstruct and reduction in resolution as reported.

Also, due to the small scale of teeth, the location of reconstruction is close to the

boundary and sources. Here, wider sensitivity profiles of linear dye will most likely

involve more artifacts. The other reason may be the mismatch of optical properties when

calculating the forward model and real situations. Because it will be impractical to

obtain the accurate optical properties for each tooth in clinical settings, some standard

values acquired from the references have been employed for the experiments. This could

generate relatively significant mismatch between model and experiments, compromising

the reconstructions accuracy.

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Figure 4-15. Comparison of raw emission between linear dye and UCNPs with NIR

emission: (a) Linear dye emission with irradiation of source No. 25; (b) UCNPs

with NIR emission with irradiation of source No. 25.

4.3 Discussions

In this chapter, a series of preliminary experiments using UCNPs to detect the fillings

status were implemented. Good reconstructions results were acquired for UCNPs with

two emission bands: blue and NIR. Note that these are the first experimental

reconstructions based on UCNPs to be reported to the best of our knowledge. Also, the

accuracy of the reconstruction was estimated by comparison with the real volume and

centroid. On average, less than 10% relative volume error can be obtained including

linear dye reconstructions and for the worst case of calculating the centroid of

reconstructions, the error is also less than 1.5 mm.

According to the in silico experiments, the condition number of the sensitivity

matrices of UCNPs are larger than for the linear dye, meaning higher difficulty in

solving the resulting linear equations. The situations will worsen with increasing power

dependent index. However, in real experiments, multiple advantages will overcome

these disadvantages. One advantage is that there is no autofluorescence when UCNPs are

employed due to the NIR excitation wavelength; most tissues cannot produce longer

wavelength fluorescence upon irradiation of light in NIR range. The other advantage is

sharper sensitivity profiles of UCNPs. The sharper sensitivity profiles indicate higher

(a) (b)

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resolution and smaller full width half maxima (FWHM) of the point spread functions

(PSF). Also in our case, with reconstruction area close to the boundary and sources,

better reconstructions with fewer artifacts are also acquired compared with linear dyes.

We postulate that this is due to the narrower sensitivity profiles involving less area for

reconstructions.

Because we plan to use UCNPs as a theranostic agent, in this chapter, we have

proven the feasibility of employing UCNPs as well as mixture with dental fillings to

indicate the situations of filled dental holes. In the next chapter, we will show the

therapy application of UCNPs to kill bacteria ex vivo.

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5. Killing bacteria with UCNPs

Streptococcus mutans plays an important role in the formation of dental caries due to its

anaerobic metabolism that transforms food debris into acid. The resulting acid will then

erode the hard tissues of teeth until the formation of dental caries [154]. After the caries

form, dentists often employ resin to fill the hole after carefully removing the residual

pulp, shaping the root canal and sterilizing it thoroughly. However, due to the shrinkage

of the resin being cured, the resin leads to a margin, which bacteria can infiltrate [155].

Besides, the resin itself cannot prevent the growth of bacteria. If the bacteria infiltrate

into the deep region of the tooth, a second caries may occur and lead to failure of

restoration. *

To resolve these issues, several researchers developed a resin that contains

antibacterial ions to give the resin itself endogenous antibacterial properties [157], [158].

However, these resins can only work when they have the ability to release ions. The

release of these ions can compromise the mechanical integrity of the materials due to the

presence of voids, which are regions where the teeth and fillings lack adhesion [159].

Hence it is necessary to investigate external bactericide methods to kill residual bacteria

after root canal therapy.

Based on different mechanisms, different wavelengths of light, from near

infrared to UV light have all been tested as a means to reduce bacterial activity. Among

them, blue and UV light are the most commonly employed. However, applying blue and

UV light has also been proven to be inimical to mammalian cells [160], [161]. In certain

cells and bacteria, blue light will lead to generation of reactive oxygen species (ROS)

[162]. The ROS will then interact with and disrupt the proper functions of cellular

machinery and lead to cell death. A wide variety of bacterial species have been killed in

vitro through the use of blue light [33]. Similar effects can be observed in mammalian

cells albeit with longer exposure time and a more powerful source [163]. Another

disadvantage of employing blue light is limited penetration depth [164]. As a

This chapter has been submitted as: A. Srinivasan, F. Long, et al., “Bactericidal effects of dental composites

containing upconvertion particles,” J. Dent. Res. [156]

95

consequence, direct irradiation of external blue light or UV outside the oral surface is not

an ideal method to kill bacteria located deep inside the teeth.

In contrast to employing blue and UV light to kill bacteria, irradiation with red or

near infrared (NIR) light with low power has been shown to increase the activity of

bacteria; with high power, it can also kill the bacteria. NIR light has been investigated to

treat mammalian cells over the last 40 years, including promoting wound, tissue, and

nerve healing although the mechanisms of these procedures are not entirely known [165].

These previous studies indicated that a kind of materials which could absorb NIR

photons and emit blue or UV light would be an ideal approach to kill bacteria inside the

teeth without inducing negative effects on mammalian cells.

As introduced in the previous chapters, upconverting nanoparticles (UCNPs) will

be the ideal materials for this application. They can also be employed as optical contrast

to detect the fillings’ situations. One such particle is ytterbium and thulium doped

sodium yttrium fluoride. These kinds of particles can generate relatively intense blue

light with low concentrations in cultured media [166]. These studies indicate that

UCNPs have the potential as an alternative to traditional antibiotic therapies. Here, we

postulate that near infrared light irradiated dental composites containing embedded

UCNPs may exhibit the same bactericidal properties against S. mutans as direct blue

light irradiation. We also postulate that the particle containing composites will exhibit

lower cytotoxic effects in mammalian cells relative to direct blue light irradiation.

5.1 Materials and methods

5.1.1 Light irradiation

Samples were placed below a 975 nm laser diode (L975P1WJ, Thorlabs, USA) emitting

near infrared light. The laser diode mounted by TCLDM9 (Thorlabs, USA), was

controlled by the current controller (LDC220C, Thorlabs, USA) and temperature

controller (TED200C, Thorlabs, USA). The current of laser diode was set to 1350 mA

and the work temperature was set to 25oC. The power of the laser diode was measured

by power-meter (S310C and PM100USB, Thorlabs, USA) and the power was 820 mW.

The blue light power generated from the upconversion particles was estimated to be 70

mW/cm2 at the sample. A blue light emitting source (LED, SR-01-B0040, Luxeon,

96

Canada) was also used to irradiate samples generating a power of 70 mW/cm2 at the

sample. This was used to assess the effects of direct blue light irradiation. Bacterial

samples were placed upside down under the light source (the near infrared light will

reach the resin mixed with UCNPs first), while mammalian cells were placed directly

under the light source. This simulated the conditions found in composites in a human

mouth; bacteria would be found deeper within the tooth (light needed to penetrate the

composite) while mammalian cells would be exposed to the light directly at the surface

(the gums surrounding a tooth).

5.1.2 Bis-GMA dental composite

250 L Bisphenol A glycol dimethacrylate (Bis-GMA) was placed into individual wells

of a 96 well Costar tissue culture plate. Bis-GMA was cured using blue light (470 nm) to

solidify the polymer inside the well, and then plasma treated to generate free radicals.

Serum containing media was subsequently deposited onto plasma treated wells. Half the

composite wells did not contain UCNPs (Sodium yttrium fluoride, ytterbium and

thulium doped, NaY0.77Yb0.20Tm0.03F4, Sigma-Aldrich, USA) and half the composite

wells contained the upconversion particle at 5% w/v. Streptococcus mutans and NIH3T3

fibroblasts were inoculated onto the surface of the Bis-GMA plasma treated wells

following plasma treatment and media deposition.

5.1.3 Bacterial growth

Streptococcus mutans was grown for two hours in brain heart infusion liquid media

preceding inoculation onto the dental composite. The inoculated wells were then

incubated in a 37oC room for a 14-16 hours period. Plates were irradiated following this

growth period.

5.1.4 Bacterial live/dead essay

Cells were irradiated while adherent to the composite. Directly following irradiation,

each well was washed with a .85% sodium chloride (w/v) solution. Bacteria were found

suspended in the sodium chloride solution. Percentages of live and dead bacteria of each

suspension were found through use of the Live/Dead BacLight Bacterial Viability and

Counting Kit (L7007, Molecular Probes, USA). The respective green (FITC filter) and

97

red (Texas Red filter) fluorescence of each sample was found. The green/red

fluorescence ratio was then converted to percentage of live cells using a calibration

curve from the kit. The curve obtained was G/R Ratio = .0331(% Cells Alive) - .002, R

2

= .9994.

5.1.5 Mammalian growth

NIH3T3 fibroblasts were maintained in flasks with Dulbecco’s Modified Eagle Media

(15% fetal bovine serum, 1x Penicillin and Streptomycin) and incubated at standard

conditions – 37oC, 5% carbon dioxide. Cells were allowed to reach 60-70% confluence

and then were plated on the plasma treated Bis-GMA wells. Cells were incubated for 24

hours before irradiation and subsequent staining.

5.1.6 Mammalian live/dead analysis

Cells were irradiated while adherent to the composite. Directly following irradiation,

adherent NIH3T3 fibroblasts (on Bis-GMA) were stained using the

Apoptotic/Necrotic/Healthy Cells detection kit (PK-CA707-30018, PromoKine, USA).

Stained cells were detached and suspended in trypsin. Healthy cells fluoresce blue

(DAPI filter), apoptotic cells fluoresce green (FITC filter), and necrotic cells fluoresce

red (Texas Red filter). The blue/ (green + red) fluorescence were found for each

treatment group.

5.2 Results

5.2.1 Assessing the light irradiation effect on S. mutans

In the first experiment, S. mutans was irradiated with blue light for 0 minutes, 15

minutes, 30 minutes, and 60 minutes. ANOVA analysis shows that the mean survival

rates differ from one another (p < .0001). Further post-hoc analysis using the Turkey

HSD test shows that each mean survival rate is different from the others (p < .01) as

shown in Figure 5-1 and Figure 5-2.

S. mutans was also grown on Bis-GMA composite or Bis-GMA composite with

particles. Each well was irradiated with near infrared light for 0, 15, 30, or 60 minutes.

ANOVA analysis at each stimulation time (15 minutes, 30 minutes, and 60 minutes)

98

shows that mean survival rates of S. mutans are not similar for each irradiation time (p

< .0001). Post-hoc analysis using the Turkey HSD test shows that the S. mutans mean

survival rate on irradiated Bis-GMA plates with the upconversion particle differs from

the survival rate of all other groups for each treatment time (p < .01 for all treatment

times) (see Figure 5-1 and Figure 5-2).

Figure 5-1. Merged fluorescence images for S. mutans survival: Top row: Blue light

irradiation decreases live cells (green) and increases dead cells (red) at 15, 30, and

60 minutes after irradiation; Bottom rows: Survival of S. mutans cells is not

affected by the presence of upconversion particles in Bis-GMA (second column

compared to first column). Irradiation with NIR light induces does not

significantly alter cell survival on Bis-GMA (third column compared to first

column). Irradiation with NIR light on Bis-GMA containing upconversion

particles increases the number of dead cells (last column compared to first column).

Live S. mutans cells are stained green, and dead S. mutans cells are stained red.

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Figure 5-2. Percentage of S. mutans alive following light irradiation, indicating

irradiation with blue light or NIR irradiation of upconversion particles reduces

percentage of alive S. mutans. Asterisks denote mean survival percentages that

differ between groups (p < 0.01): (a) Blue light irradiation; (b) Near infrared light

irradiation.

5.2.2 Assessing the light irradiation effects on NIH3T3 fibroblast survivals

NIH3T3 fibroblasts were plated on Bis-GMA composite wells and subsequently

irradiated with blue light for 0 minutes, 15 minutes, 30 minutes, and 60 minutes.

ANOVA analysis for the B/G+R blue/red fluorescence ratio shows that the mean

fluorescence ratios differ across treatment groups (p < .0001). Further post-hoc analysis

using the Turkey HSD test shows that each mean fluorescence ratio differs from the

others (p < .01 for all cases).

NIH3T3 fibroblasts were also plated on Bis-GMA composite and Bis-GMA

composite with the upconversion particle. ANOVA analysis for the 15 treatment time

shows that the mean B

/G+R fluorescence ratios do not differ significantly for NIH3T3

fibroblasts. For the 30 minute and 60 minute treatment times, ANOVA analysis does

show a significant difference in B/G+R ratios (p < .0001 for both treatment times). Post-

hoc analysis using the Turkey HSD test shows that the B/G+R mean fluorescence ratio on

the 30 minute irradiated Bis-GMA and particle well differs from the B/G+R mean

fluorescence ratio of the Bis-GMA control, Bis-GMA and particle control, and 30

minute NIR irradiated Bis-GMA wells (p < .01). The Turkey HSD test also shows that

the B/G+R mean fluorescence ratio on the 60 minute irradiated Bis-GMA and particle well

differs from the B/G+R mean fluorescence ratio of the Bis-GMA control, Bis-GMA and

particle control, and 60 minute NIR irradiated Bis-GMA wells (p < .01).

(a) (b)

100

Figure 5-3. Merged fluorescence images for NIH3T3 survival: Top row: Blue light

irradiation decreases live cells (blue) and increases dead cells (red and green) at 15,

30, and 60 minutes after irradiation; Bottom rows: Survival of NIH3T3 fibroblasts

is not affected by the presence of upconversion particles in Bis-GMA (second

column compared to first column). Irradiation with NIR light does not significantly

alter fibroblast survival on Bis-GMA (third column compared to first column).

Irradiation with NIR light on Bis-GMA containing upconversion particles does not

increase the number of dead cells post 15 minute irradiation, but does so following

30 and 60 minute irradiations (last column compared to first column). Live

NIH3T3 fibroblasts are stained blue, apoptotic NIH3T3 fibroblasts are stained

green, and necrotic NIH3T3 fibroblasts are stained red.

101

Figure 5-4. Live/Dead Ratios of NIH3T3 Fibroblasts, indicating irradiation with

blue light or NIR irradiation of upconversion particles for longer than 15 minute

reduces the proportion of live NIH3T3 fibroblasts, determined by the B/G+R

fluorescence ratio. Asterisks denote mean fluorescence B/G+R ratios that differ

between groups (p < 0.01): (a) Blue light irradiation; (b) Near infrared light

irradiation.

5.3 Discussion

We hypothesized that irradiating UCNPs with near infrared light would result in S.

mutans death while posing minimal risk to mammalian cells. Direct blue light irradiation

was expected to kill both mammalian and S. mutans cells. Direct blue light irradiation of

S. mutans caused death, and longer irradiation times increased the percentage of dead

bacterial cells. Direct blue light irradiation also resulted in fibroblast apoptosis and

necrosis, with increasing numbers of apoptotic and necrotic cells found as the time of

irradiation increased from 15 to 30 to 60 minutes. Blue light irradiation has been

previously shown to induce both mammalian and bacterial cell death in a variety of

species [167]–[169]. In contrast, near infrared irradiation of both mammalian and

bacterial cells caused slight increases in the number of alive cells. Previous studies have

shown beneficial effects of near infrared light [170], [171]. Near infrared irradiation of

UCNPs produced blue light which then induced bacterial death at 15, 30, and 60 minute

time frames. The blue light from the particles also induced fibroblast death at 30 and 60

minute timeframes, but not at the 15 minute timeframe. Thus, clinically, exposure times

greater than 15 minutes cannot be used.

(a) (b)

102

The blue light intensity from both direct irradiation and near infrared irradiation

of the dental material was 70 mW/cm2. The total energy generated from the irradiation

was 0, 63, 126, and 252 J/cm2 for the 0, 15, 30, and 60 minute irradiation times

respectively. The bactericidal activity observed in this study is consistent with previous

studies that have shown a significant reduction in oral bacteria after blue light irradiation,

albeit at a lower wavelength (405 nm) [172]. The previous study used a power of 50

mW/ cm2

for 300 seconds, resulting in a total energy of 15 J/cm2. The lack of negative

effects on mammalian cells upon exposure to blue light for 15 minutes is also supported

by previous studies, albeit with light at a lower wavelength (405nm) [173]. This study

indicated that mammalian cells recover only when exposed to total energies below 64.8

J/cm2 of blue light irradiation, which is consistent with the energy at 15 minutes used in

our study. Note that the previous studies used light with higher energies (405nm as

opposed to the 470nm used in this study). Because higher energy photons are expected

to be more damaging to cells (bacterial or mammalian), we expect that the use of light

with a wavelength of 470 nm can preserve the bactericidal effects while reducing the

deleterious effects on mammalian cells. One noteworthy aspect of using the UCNPs is

that blue light is expected to be generated only at locations where the composite is

present, thereby decreasing the harmful effects of blue light irradiation on mammalian

cells [174]. The rest of the region is expected to be subjected to NIR light, which has

been shown to be beneficial for tissue regeneration [175].

There are several limitations to these studies. We did not evaluate irradiation of

light with different durations, intensities, or powers. We also did not vary the content of

UCNPs in the composite or study if repeated inoculations could further reduce the

percentage of live bacteria. Furthermore, we also evaluated the bactericidal effects with

one relevant pathogen (S. mutans), but not with other pathogenic bacteria. In addition,

we used vertical irradiation, and changing the angle of irradiation may induce different

light intensities.

The results presented here do show promise that near infrared irradiation of the

UCNPs may promote bacterial death while posing minimal risk to nearby mammalian

tissue at irradiation times around 15 minutes. Therefore, near infrared irradiation of

composites containing ytterbium and thulium doped sodium yttrium fluoride may

103

provide a future alternative to antibiotics. In summary, while further studies are needed

to assess the efficacy of light penetration and bactericidal effects of near infrared light-

irradiated UCNPs in both tooth and in vivo models, there is promise that light therapy

may be able to augment and potentially replace existing treatments.

104

6. Summary of the thesis and future work

In this thesis, the potential of using optical based methods to detect the vitality of soft

tissues inside teeth was preliminarily investigated. Considering the small scale of teeth,

the widely used diffusion approximation (DA) cannot be applied to predict photon

transportation inside the teeth. An alternative radiative transfer equation (RTE)-based

model, with the continuous Galerkin finite element method (CG-FEM) as a solver, was

proposed and employed to simulate light propagation inside the teeth. Additionally,

voxel- and mesh-based Monte Carlo (MC) methods were used to generate the sensitivity

matrix in mesoscopic fluorescence molecular tomography (MFMT) and the experiments

of applying upconverting nanoparticles (UCNPs) as a theranostic agent. MFMT was

employed to investigate the pulp situation while UCNPs were employed as an optical

contrast agent to indicate the situation of fillings after root canal therapy. Moreover,

UCNPs have been preliminarily proven to be bactericidal, suggesting the potential use

for killing residual bacteria after surgery. Considering the results that were acquired,

there are still some problems need to be solved in the future work.

6.1 Radiative transfer equation based forward solver

One of several problems with our proposed algorithms is that it cannot accurately predict

photon propagation near the source. Even with streamline modification of test functions

and phase function normalization, this problem still occurs. For some optical properties,

oscillations can be seen near the source and this situation is worse when non-smooth

sources such as ideal pencil beam, are applied. For some optical properties, even if

oscillations do not exist, it is still possible to generate non-symmetrical photon

distributions near the source in uniform media. The reason for the above phenomenon is

the employment of continuous global test functions, which cannot approximate source

functions accurately with dramatic changes. Thus as discussed in Chapter 2, the

discontinuous Galerkin finite element method (DG-FEM) is a promising improvement

for solving the radiative transfer equation (RTE). However, due to the relaxation of

continuous constraints of test functions at the boundary between two adjacent elements,

a greater number of unknowns will be generated for DG-FEM. Hence, some

optimizations have to be involved, such as using the multi-grid method to perform

105

calculations on different layer (density) meshes. Also, to our knowledge, for almost all

cases when employing FEM to solve partial differential equations (PDEs) in engineering,

only the linear polynomials are used as basis functions. However, when increasing the

order of polynomials, increasing the order of accuracy is guaranteed. Of course, this will

increase the computational burden. However, it is still worth it to try quadratic

polynomials since it provides a good balance between increasing the accuracy of

solutions and appropriate computational loads.

6.2 Application of upconverting nanoparticles

In Chapter 4, upconverting nanoparticles (UCNPs) have been shown to have the

potential to indicate the filling status after root canal therapy with relatively small error.

In addition, they are successful in killing the residual bacteria after surgery, as discussed

in Chapter 5. However, there are still multiple improvements of the experiments.

First, our ex vivo experiments are preliminary for investigation of the possibilities

of applying UCNPs in detecting dental fillings after surgery. The tooth model used in the

experiments is a very simplified model. For instance, in order to reduce the position

error of the irradiation point, the mesiodistal surface of the tooth was employed. It is

noted that even for this relatively large surface, the accurate position of the irradiation

point is still difficult to locate since the human eye cannot discern near infrared (NIR)

light. Also due to the small scale of teeth, when the excitation light irradiates at the edge

of the tooth, some light will be diffracted into the camera directly and this will

‘contaminate’ the raw data to the sources located at the edge of teeth. Thus, a better

model to consider is teeth embedded into the jaw. The human jaw contains bone, soft

tissues such as gum, as well as blood vessels and nerves. Although considering the

anatomical structure of jaw is complex, it can mitigate the problems due to the small

scale of teeth as discussed above. The other benefit for using the jaw model is that it can

make the region of interest (ROI) far from the boundary of the medium (jaw). When

performing the reconstructions in Chapter 4, since the ROI is quite close to the boundary

and source, it has the possibility to involve more artifacts. Until now, almost all of the

methods used to predict photon propagation near the boundary or sources have large

error especially for deterministic method, such as the diffusion approximation (DA) or

106

the RTE. This is also our hypothesis for why the reconstruction of linear dye is worse

than those of UCNPs. In contrast, using a jaw model can make the ROI far from the

boundary and may reduce the artifacts.

Second, in order to employ a jaw model or in vivo experiments, the optical

settings need to be improved. Fortunately, the continuous wave (CW) transmission type

can be set into the human mouth relatively easily after minimizing the volume of the

source and detectors. Our device is microcontroller (MCU) based so all the circuits can

be easily minimized. Besides, even though we employed an EMCCD in our experiments,

the EM gain were employed only when the emission light intensity was too low, such as

for UCNPs blue light emission. For the other situations described in Chapter 4, the

emission light intensities were strong enough and no EM gain was used, at least in the

cases we tested. With improvements in CCD technology, the sensitivity of the camera

would not be a main concern in the future so our setup has the potential to employ a

commercial camera, which would reduce the cost of system.

Third, killing bacteria using UCNPs has only proven to be effective in well plate

settings. The next experiments would involve injecting the bacteria inside real teeth and

employing dental fillings as Bis-GMA mixed with UCNPs to fill the simulated holes.

Then the excitation NIR light would be irradiated to the teeth from an outside source.

Under this situation, with the guidance of Monte Carlo (MC) simulations or the RTE-

based forward problem, it is possible to estimate the power of the excitation NIR light

and UCNPs emission power. Considering the results of ex vivo experiments shown in

Chapter 5, further verifications should be relatively easy to complete; this application of

UCNPs has the potential for in vivo experiments in the near future.

Fourth, if the current imaging platform is based around a well-established raster-

scanning approach of the illumination spot, it may be beneficial to employ the wide-filed

structured light approach [176]. This new instrumental approach pioneered by our

laboratory has been established for preclinical application [177], [178]. Besides the

significant speedup in acquisition time [179], it enables delivering higher light doses as

well as on the fly adaptive imaging [180], [181]. Moreover, in conjunction with sparsity

solvers [182], it benefits for advanced preconditioning techniques for fast and accurate

reconstructions [183].

107

Last, if we have employed experimentally CT imaging for rigorous validation,

the use of internal structure as prior in the inverse problem is expected to greatly

improve 3-D optical imaging [184], [185]. As dental CT is becoming more widespread,

fusion of CT and optical imaging should enable fast bedside tomographic tissue

oxygenation monitoring as well as theranostics [185], [186].

108

References

[1] H. Silk, “Diseases of the Mouth,” Prim. Care Clin. Off. Pract., vol. 41, no. 1, pp.

75–90, Mar. 2014.

[2] Section on oral health, “Maintaining and Improving the Oral Health of Young

Children,” Pediatrics, vol. 134, no. 6, pp. 1224-1229, Nov. 2014.

[3] C. Yu and P. Abbott, “An overview of the dental pulp: its functions and responses

to injury,” Aust. Dent. J., vol. 52, no. 1 Suppl, pp. S4–S16, Mar. 2007.

[4] T. Vos, A. D. Flaxman, M. Naghavi, R. Lozano, et. al., “Years lived with

disability (YLDs) for 1160 sequelae of 289 diseases and injuries 1990-2010: a

systematic analysis for the Global Burden of Disease Study 2010,” Lancet, vol.

380, no. 9859, pp. 2163–2196, Dec. 2012.

[5] J. D. Bartlett, “Dental Enamel Development: Proteinases and Their Enamel

Matrix Substrates,” ISRN Dent., vol. 2013, Sep. 2013.

[6] J. E. Eastoe, “Organic Matrix of Tooth Enamel,” Nature, vol. 187, no. 4735, pp.

411–412, Jul. 1960.

[7] G. R. Holland, “The odontoblast process: form and function,” J. Dent. Res., vol.

64, pp. 499–514, Apr. 1985.

[8] A. Nanci, Ten Cate’s Oral Histology: Development, Structure, and Function, 8th

edition. St. Louis, MO, USA: Elsevier Mosby, 2012.

[9] S. Nagaoka, Y. Miyazaki, H. J. Liu, Y. Iwamoto, M. Kitano, and M. Kawagoe,

“Bacterial invasion into dentinal tubules of human vital and nonvital teeth,” J.

Endod., vol. 21, no. 2, pp. 70–73, Feb. 1995.

[10] L. A. Lindén, O. Källskog, and M. Wolgast, “Human dentine as a hydrogel,” Arch.

Oral Biol., vol. 40, no. 11, pp. 991–1004, Nov. 1995.

[11] B. Matthews, D. Andrew, TR. Amess, H. Ikeda and N. Vongsavan, “The

functional properties of intradental nerves,” in Dentin/Pulp Complex, M. Shimono,

T. Maeda, H. Suda, H. Takahashi, Eds, Tokyo, Japan: Quintessence, 1996, pp:

146-153.

[12] N. Hasegawa, “Effects of various culture conditions on matrix formative functions

of rat incisor odontoblasts in a pulp-dentin slice culture system,” Jpn. J. Oral Biol.,

vol. 31, no. 4, pp. 392–403, Aug. 1989.

[13] S. Chein, “Hemodynamics of the dental pulp,” J. Dent. Res., vol. 64 Spec, pp.

602–606, May 1985.

[14] M. W. Meyer, “Pulpal blood flow: use of radio-labelled microspheres,” Int. Endod.

J., vol. 26, no. 1, pp. 6–7, Jan. 1993.

[15] L. B. Giovanella, F. B. Barletta, W. T. Felippe, K. F. Bruno, A. H. G. de Alencar,

and C. Estrela, “Assessment of Oxygen Saturation in Dental Pulp of Permanent

Teeth with Periodontal Disease,” J. Endod., vol. 40, no. 12, pp. 1927–1931, Dec.

2014.

[16] P. F. Gonçalves, E. A. Sallum, A. W. Sallum, M. Z. Casati, S. de Toledo, and F. H.

Nociti Junior, “Dental cementum reviewed: development, structure, composition,

regeneration and potential functions,” Braz. J. Oral Sci., vol. 4, no. 12, pp. 651-

658, Dec. 2005.

[17] I. Kramer, “The relationship between pain and changes in the dental pulp

following the insertion of fillings,” Br. Dent. J., vol. 9, pp. 9–13, Jan. 1954.

109

[18] V. Gopikrishna, G. Pradeep, and N. Venkateshbabu, “Assessment of pulp vitality:

a review,” Int. J. Paediatr. Dent. Br. Paedodontic Soc. Int. Assoc. Dent. Child.,

vol. 19, no. 1, pp. 3–15, Jan. 2009.

[19] K. F. Bruno, F. B. Barletta, W. T. Felippe, J. A. Silva, A. H. Gonçalves de

Alencar, and C. Estrela, “Oxygen saturation in the dental pulp of permanent teeth:

a critical review,” J. Endod., vol. 40, no. 8, pp. 1054–1057, Aug. 2014.

[20] S. Samuel, A. Thomas, and N. Singh, “A comparative study of pulse oximetry

with the conventional pulp testing methods to assess vitality in immature and

mature permanent maxillary incisors,” CHRISMED J. Health Res., vol. 1, no. 4,

pp. 235-240, 2014.

[21] N. Mordohai, M. Reshad, S. Jivraj, and W. Chee, “Factors that affect individual

tooth prognosis and choices in contemporary treatment planning,” Br. Dent. J.,

vol. 202, no. 2, pp. 63–72, Jan. 2007.

[22] S. Patel and J. J. Barnes, The Principles of Endodontics, Second Edition. Oxford,

UK: Oxford University Press, 2013.

[23] T. Dammaschke, D. Steven, M. Kaup, and K. H. R. Ott, “Long-term survival of

root-canal-treated teeth: a retrospective study over 10 years,” J. Endod., vol. 29,

no. 10, pp. 638–643, Oct. 2003.

[24] K. Kojima, K. Inamoto, K. Nagamatsu, A. Hara, K. Nakata, I. Morita, H.

Nakagaki, and H. Nakamura, “Success rate of endodontic treatment of teeth with

vital and nonvital pulps. A meta-analysis,” Oral Surg. Oral Med. Oral Pathol.

Oral Radiol. Endod., vol. 97, no. 1, pp. 95–99, Jan. 2004.

[25] K. Balto, “Tooth survival after root canal treatment,” Evid. Based Dent., vol. 12,

no. 1, pp. 10–11, Mar. 2011.

[26] G. T.-J. Huang, “Dental Pulp and Dentin Tissue Engineering and Regeneration –

Advancement and Challenge,” Front. Biosci. Elite Ed., vol. 3, pp. 788–800, Jan.

2011.

[27] D. J. Caplan, J. Cai, G. Yin, and B. A. White, “Root canal filled versus non-root

canal filled teeth: a retrospective comparison of survival times,” J. Public Health

Dent., vol. 65, no. 2, pp. 90–96, Jun. 2005.

[28] S. Seltzer, I. B. Bender, and M. Ziontz, “The dynamics of pulp inflammation:

correlations between diagnostic data and actual histologic findings in the pulp,”

Oral Surg. Oral Med. Oral Pathol., vol. 16, pp. 846–871 contd, Jul. 1963.

[29] J. F. Siqueira, “Aetiology of root canal treatment failure: why well-treated teeth

can fail,” Int. Endod. J., vol. 34, no. 1, pp. 1–10, Jan. 2001.

[30] W. J. Loesche, “Microbiology of Dental Decay and Periodontal Disease,” in

Medical Microbiology, Fourth ed. Galveston (TX), USA: University of Texas

Medical Branch at Galveston, ch. 99, sec. 4, 1996.

[31] C. T. Xu, J. Axelsson, and S. Andersson-Engels, “Fluorescence diffuse optical

tomography using upconverting nanoparticles,” Appl. Phys. Lett., vol. 94, no. 25,

pp. 251107-1-251107-3, Jun. 2009.

[32] Q. Zhan, S. He, J. Qian, H. Cheng, and F. Cai, “Optimization of Optical

Excitation of Upconversion Nanoparticles for Rapid Microscopy and Deeper

Tissue Imaging with Higher Quantum Yield,” Theranostics, vol. 3, no. 5, pp. 306–

316, Mar. 2013.

110

[33] R. Lubart, A. Lipovski, Y. Nitzan, and H. Friedmann, “A possible mechanism for

the bactericidal effect of visible light,” Laser Ther., vol. 20, no. 1, pp. 17–22, Sep.

2011.

[34] C. T. Xu, P. Svenmarker, H. Liu, X. Wu, M. E. Messing, L. R. Wallenberg, and S.

Andersson-Engels, “High-Resolution Fluorescence Diffuse Optical Tomography

Developed with Nonlinear Upconverting Nanoparticles,” ACS Nano, vol. 6, no. 6,

pp. 4788–4795, Jun. 2012.

[35] Y.-S. Park, J.-S. Ahn, H.-B. Kwon, and S.-P. Lee, “Current status of dental caries

diagnosis using cone beam computed tomography,” Imaging Sci. Dent., vol. 41,

no. 2, pp. 43–51, Jun. 2011.

[36] D. A. Tyndall and S. Rathore, “Cone-beam CT diagnostic applications: caries,

periodontal bone assessment, and endodontic applications,” Dent. Clin. North Am.,

vol. 52, no. 4, pp. 825–841, Oct. 2008.

[37] X. Intes and B. Chance, “Non-PET functional imaging techniques: optical,”

Radiol. Clin., vol. 43, no. 1, pp. 221–234, Jan. 2005.

[38] Y. Chen, X. Intes, and B. Chance, “Development of high-sensitivity near-infrared

fluorescence imaging device for early cancer detection,” Biomed. Instrum.

Technol. Assoc. Adv. Med. Instrum., vol. 39, no. 1, pp. 75–85, Feb. 2005.

[39] D. Fried, R. E. Glena, J. D. B. Featherstone, and W. Seka, “Nature of light

scattering in dental enamel and dentin at visible and near-infrared wavelengths,”

Appl. Opt., vol. 34, no. 7, pp. 1278-1285, Mar. 1995.

[40] A. Kienle, F. K. Forster, R. Diebolder, and R. Hibst, “Light propagation in dentin:

influence of microstructure on anisotropy,” Phys. Med. Biol., vol. 48, no. 2, pp.

N7–14, Jan. 2003.

[41] K. B. Fanibunda, “A method of measuring the volume of human dental pulp

cavities,” Int. Endod. J., vol. 19, no. 4, pp. 194–197, Jul. 1986.

[42] F. Long, M. S. Ozturk, M. S. Wolff, X. Intes, and S. P. Kotha, “Dental Imaging

Using Mesoscopic Fluorescence Molecular Tomography: An ex Vivo Feasibility

Study,” Photonics, vol. 1, no. 4, pp. 488–502, Dec. 2014.

[43] K. Koenig, H. Schneckenburger, J. Hemmer, B. J. Tromberg, and R. W. Steiner,

“In-vivo fluorescence detection and imaging of porphyrin-producing bacteria in

the human skin and in the oral cavity for diagnosis of acne vulgaris, caries, and

squamous cell carcinoma,” in Proc. SPIE 2135, Advances in Laser and Light

Spectroscopy to Diagnose Cancer and Other Diseases, 1994, pp. 129–138.

[44] A. Lussi, B. Megert, C. Longbottom, E. Reich, and P. Francescut, “Clinical

performance of a laser fluorescence device for detection of occlusal caries lesions,”

Eur. J. Oral Sci., vol. 109, no. 1, pp. 14–19, Feb. 2001.

[45] X. Q. Shi, U. Welander, and B. Angmar-Månsson, “Occlusal caries detection with

KaVo DIAGNOdent and radiography: an in vitro comparison,” Caries Res., vol.

34, no. 2, pp. 151–158, Apr. 2000.

[46] D. Spitzer and J. J. ten Bosch, “Luminescence quantum yields of sound and

carious dental enamel,” Calcif. Tissue Res., vol. 24, no. 1, pp. 249–251, Dec. 1977.

[47] S. Tranaeus, S. Al-Khateeb, S. Björkman, S. Twetman, and B. Angmar-Månsson,

“Application of quantitative light-induced fluorescence to monitor incipient

lesions in caries-active children. A comparative study of remineralisation by

111

fluoride varnish and professional cleaning,” Eur. J. Oral Sci., vol. 109, no. 2, pp.

71–75, Apr. 2001.

[48] G. K. Stookey, “Quantitative light fluorescence: a technology for early monitoring

of the caries process,” Dent. Clin. North Am., vol. 49, no. 4, pp. 753–770, Oct.

2005.

[49] R. K. Chesters, N. B. Pitts, G. Matuliene, A. Kvedariene, E. Huntington, R.

Bendinskaite, I. Balciuniene, J. R. Matheson, J. A. Nicholson, A. Gendvilyte, R.

Sabalaite, J. Ramanauskiene, D. Savage, and J. Mileriene, “An abbreviated caries

clinical trial design validated over 24 months,” J. Dent. Res., vol. 81, no. 9, pp.

637–640, Sep. 2002.

[50] M. M. T. Curnow, C. M. Pine, G. Burnside, J. A. Nicholson, R. K. Chesters, and

E. Huntington, “A randomised controlled trial of the efficacy of supervised

toothbrushing in high-caries-risk children,” Caries Res., vol. 36, no. 4, pp. 294–

300, Aug. 2002.

[51] J. D. Featherstone, “The science and practice of caries prevention,” J. Am. Dent.

Assoc., vol. 131, no. 7, pp. 887–899, Jul. 2000.

[52] T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction

method for optical tomography using the finite element solution of the radiative

transfer equation,” J. Quant. Spectrosc. Radiat. Transf., vol. 109, no. 17–18, pp.

2767–2778, Nov. 2008.

[53] X. Intes, “Time-domain optical mammography SoftScan: initial results,” Acad.

Radiol., vol. 12, no. 8, pp. 934–947, Aug. 2005.

[54] J. Chen and X. Intes, “Time-gated perturbation Monte Carlo for whole body

functional imaging in small animals,” Opt. Express, vol. 17, no. 22, pp. 19566–

19579, Oct. 2009.

[55] S. R. Arridge and J. C. Schotland, “Optical tomography: forwad and inverse

problems,” Inverse Probl., vol. 25, no. 12, p. 123010, Dec. 2009.

[56] L. Zhao, H. Yang, W. Cong, G. Wang, and X. Intes, “Lp regularization for early

gate fluorescence molecular tomography,” Opt. Lett., vol. 39, no. 14, pp. 4156–

4159, Jul. 2014.

[57] J. Chen, Q. Fang, and X. Intes, “Mesh-based Monte Carlo method in time-domain

widefield fluorescence molecular tomography,” J. Biomed. Opt., vol. 17, no. 10,

pp. 1060091–1060098, Oct. 2012.

[58] E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with

graphics processing units for high-speed Monte Carlo simulation of photon

migration,” J. Biomed. Opt., vol. 13, no. 6, pp. 060504-1-060504-3, Nov. 2008.

[59] A. Kienle, F. K. Forster, and R. Hibst, “Influence of the phase function on

determination of the optical properties of biological tissue by spatially resolved

reflectance,” Opt. Lett., vol. 26, no. 20, pp. 1571–1573, Oct. 2001.

[60] S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element

approach for modeling photon transport in tissue,” Med. Phys., vol. 20, no. 2, pp.

299–309, Mar. 1993.

[61] M. Schweiger and S. Arridge, “The Toast++ software suite for forward and

inverse modeling in optical tomography,” J. Biomed. Opt., vol. 19, no. 4, pp.

040801-1-040801-15, Apr. 2014.

112

[62] H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M.

Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography

using NIRFAST: Algorithm for numerical model and image reconstruction,”

Commun. Numer. Methods Eng., vol. 25, no. 6, pp. 711–732, Aug. 2008.

[63] J. Heiskala, I. Nissilä, T. Neuvonen, S. Järvenpää, and E. Somersalo, “Modeling

anisotropic light propagation in a realistic model of the human head,” Appl. Opt.,

vol. 44, no. 11, pp. 2049–2057, Apr. 2005.

[64] M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element

method for the propagation of light in scattering media: Boundary and source

conditions,” Med. Phys., vol. 22, no. 11, pp. 1779–1792, Nov. 1995.

[65] S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl., vol. 15,

no. 2, pp. R41–R93, Apr. 1999.

[66] P. Surya Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge,

“Variable order spherical harmonic expansion scheme for the radiative transport

equation using finite elements,” J. Comput. Phys., vol. 230, no. 19, pp. 7364–7383,

Aug. 2011.

[67] Y. Epshteyn, T. Khan, and B. Rivière, “Numerical Solution of a One-dimensional

Inverse Problem by the Discontinuous Galerkin Method,” Math Comput Simul,

vol. 79, no. 7, pp. 1989–2000, Mar. 2009.

[68] S. R. Arridge and M. Schweiger, “Photon-measurement density functions Part 2:

Finite-element-method calculations,” Appl. Opt., vol. 34, no. 34, pp. 8026–8037,

Dec. 1995.

[69] H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling

and image reconstruction in diffuse optical tomography,” Philos. Trans. R. Soc.

Lond. Math. Phys. Eng. Sci., vol. 367, no. 1900, pp. 3073–3093, Aug. 2009.

[70] S. Rajoria, L. Zhao, X. Intes, and M. Barroso, “FLIM-FRET for Cancer

Applications,” Curr. Mol. Imaging, vol. 3, no. 2, pp. 144–161, 2014.

[71] S. Keren, O. Gheysens, C. S. Levin, and S. S. Gambhir, “A comparison between a

time domain and continuous wave small animal optical imaging system,” IEEE

Trans. Med. Imaging, vol. 27, no. 1, pp. 58–63, Jan. 2008.

[72] Y. Lu, C. Darne, I.-C. Tan, B. Zhu, R. Rightmer, J. Rasmussen, and E. Sevick-

Muraca, “Experimental comparison of continuous-wave and frequency-domain

fluorescence tomography in a commercial multi-modal scanner,” IEEE Trans.

Med. Imaging, vol. 34, no. 6, pp. 1197–1211, Nov. 2014.

[73] L. A. Fletcher, L. F. Gaunt, C. B. Beggs, S. J. Shepherd, P. A. Sleigh, C. J.

Noakes, and K. G. Kerr, “Bactericidal action of positive and negative ions in air,”

BMC Microbiol., vol. 7, no. 1, pp. 32–40, Apr. 2007.

[74] Y. Kimura, P. Wilder-Smith, and K. Matsumoto, “Lasers in endodontics: a

review,” Int. Endod. J., vol. 33, no. 3, pp. 173–185, May 2000.

[75] T. D. Cutler and J. J. Zimmerman, “Ultraviolet irradiation and the mechanisms

underlying its inactivation of infectious agents,” Anim. Health Res. Rev., vol. 12,

no. 1, pp. 15–23, Jun. 2011.

[76] R. Lavi, M. Sinyakov, A. Samuni, S. Shatz, H. Friedmann, A. Shainberg, H.

Breitbart, and R. Lubart, “ESR detection of 1O2 reveals enhanced redox activity

in illuminated cell cultures,” Free Radic. Res., vol. 38, no. 9, pp. 893–902, Sep.

2004.

113

[77] J. R. Jagdeo, L. E. Adams, N. I. Brody, and D. M. Siegel, “Transcranial red and

near infrared light transmission in a cadaveric model,” PloS One, vol. 7, no. 10, p.

e47460, Oct. 2012.

[78] F. Long, F. Li, X. Intes, and S. P. Kotha, “Radiative transfer equation modeling by

streamline diffusion modified continuous Galerkin method,” J. Biomed. Opt., vol.

21, no. 3, pp. 036003-1-036003-12, Mar. 2016.

[79] W. Cai, M. Xu, and R. R. Alfano, “Analytical form of the particle distribution

based on the cumulant solution of the elastic Boltzmann transport equation,” Phys.

Rev. E Stat. Nonlin. Soft Matter Phys., vol. 71, no. 4, pp. 041202-1-041202-10,

Apr. 2005.

[80] A. Kim and M. Moscoso, “Chebyshev Spectral Methods for Radiative Transfer,”

SIAM J. Sci. Comput., vol. 23, no. 6, pp. 2074–2094, Jan. 2002.

[81] A. Liemert and A. Kienle, “Exact and efficient solution of the radiative transport

equation for the semi-infinite medium,” Sci. Rep., vol. 3, Jun. 2013.

[82] T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element

model for the coupled radiative transfer equation and diffusion approximation,”

Int. J. Numer. Methods Eng., vol. 65, no. 3, pp. 383–405, 2006.

[83] H. Gao and H. Zhao, “A Fast-Forward Solver of Radiative Transfer Equation,”

Transp. Theory Stat. Phys., vol. 38, no. 3, pp. 149–192, Sep. 2009.

[84] A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The Inverse Source Problem

Based on the Radiative Transfer Equation in Optical Molecular Imaging,” J

Comput Phys, vol. 202, no. 1, pp. 323–345, Jan. 2005.

[85] H. Fujii, Y. Hoshi, S. Okawa, T. Kosuge, and S. Kohno, “Numerical modeling of

photon propagation in biological tissue based on the radiative transfer equation,”

in AIP Conf. Proc., 2013, vol. 1518, pp. 579–585.

[86] K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the

equation of radiative transferin the frequency domain,” Opt. Lett., vol. 29, no. 6,

pp. 578–580, Mar. 2004.

[87] F. Asllanaj, S. Contassot-Vivier, A. Liemert, and A. Kienle, “Radiative transfer

equation for predicting light propagation in biological media: comparison of a

modified finite volume method, the Monte Carlo technique, and an exact

analytical solution,” J. Biomed. Opt., vol. 19, no. 1, pp. 15002-1-15002-10, Jan.

2014.

[88] Y. Wang, L. Zhang, H. Zhao, F. Gao, and J. Li, “Diffuse fluorescence tomography

based on the radiative transfer equation for small animal imaging,” in Proc. SPIE

8937, Multimodal Biomedical Imaging IX, 2014, p. 89370W.

[89] H.-S. Li, G. Flamant, and J.-D. Lu, “Mitigation of Ray Effects in the Discrete

Ordinates Method,” Numer. Heat Transf. Part B Fundam., vol. 43, no. 5, pp. 445–

466, May 2003.

[90] S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med.

Biol., vol. 58, no. 11, pp. R37–R61, Jun. 2013.

[91] A. D. Klose, “Transport-theory-based stochastic image reconstruction of

bioluminescent sources,” J. Opt. Soc. Am. A Opt. Image Sci. Vis., vol. 24, no. 6,

pp. 1601–1608, Jun. 2007.

114

[92] K. Ren, G. Bal, and A. Hielscher, “Frequency Domain Optical Tomography

Based on the Equation of Radiative Transfer,” SIAM J. Sci. Comput., vol. 28, no.

4, pp. 1463–1489, Jan. 2006.

[93] J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods. New

York, NY, USA: Springer New York, 2008.

[94] G. Kanschat, “A robust finite element discretization for radiative transfer

problems with scattering,” East West J. Numer. Math., vol. 6, no. 4, pp. 265–272,

Aug. 1998.

[95] J. A. Eichholz, “Discontinuous Galerkin methods for the radiative transfer

equation and its approximations,” Ph. D. dissertation, Dept. Applied Mathematical

and Computational Sci., University of Iowa, IA, 2011.

[96] B. Hunter and Z. Guo, “A New and Simple Technique to Normalize the HG Phase

Function for Conserving Scattered Energy and Asymmetry Factor,” Numer. Heat

Transf. Part B Fundam., vol. 65, no. 3, pp. 195–217, Mar. 2014.

[97] B. Hunter and Z. Guo, “Conservation of asymmetry factor in phase function

discretization for radiative transfer analysis in anisotropic scattering media,” Int. J.

Heat Mass Transf., vol. 55, no. 5–6, pp. 1544–1552, Feb. 2012.

[98] A. N. Brooks and T. J. R. Hughes, “Streamline upwind/Petrov-Galerkin

formulations for convection dominated flows with particular emphasis on the

incompressible Navier-Stokes equations,” Comput. Methods Appl. Mech. Eng.,

vol. 32, no. 1–3, pp. 199–259, Sep. 1982.

[99] W. Cong, H. Shen, A. X. Cong, and G. Wang, “Integral equations of the photon

fluence rate and flux based on a generalized Delta-Eddington phase function,” J.

Biomed. Opt., vol. 13, no. 2, p. 024016, Mar. 2008.

[100] B. Subasilar, “Analytical approaches to the delta-Eddington model of the radiative

transfer through vertically inhomogeneous optical depths,” Appl. Math. Model.,

vol. 32, no. 4, pp. 514–534, Apr. 2008.

[101] W. Han, F. Long, W. Cong, X. Intes, and G. Wang, “Radiative transfer with delta-

Eddington-type phase functions,” Appl. Math. Comput, in press.

[102] E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport. New

York, NY, USA: Wiley, 1984.

[103] G. Longoni and A. Haghighat, “Development of new quadrature sets with the

Ordinate Splitting technique,” in Proc. of Int. Meeting on Mathematical Methods

For Nuclear Application, 2001.

[104] V. Lebedev and D. Laikov, “A quadrature formula for the sphere of the 131st

algebraic order of accuracy,” Dokl. Math., vol. 59, no. 3, pp. 477–481, Apr. 1999.

[105] B. Hunter and Z. Guo, “Comparison of Quadrature Schemes in DOM for

Anisotropic Scattering Radiative Transfer Analysis,” Numer. Heat Transf. Part B

Fundam., vol. 63, no. 6, pp. 485–507, Jun. 2013.

[106] K. Atkinson, “Numerical integration on the sphere,” ANZIAM J., vol. 23, no. 03,

pp. 332–347, Jan. 1982.

[107] V. I. Lebedev, “Values of the nodes and weights of ninth to seventeenth order

gauss-markov quadrature formulae invariant under the octahedron group with

inversion,” USSR Comput. Math. Math. Phys., vol. 15, no. 1, pp. 44–51, Oct. 1975.

115

[108] R. Koch and R. Becker, “Evaluation of quadrature schemes for the discrete

ordinates method,” J. Quant. Spectrosc. Radiat. Transf., vol. 84, no. 4, pp. 423–

435, Apr. 2004.

[109] B. A. Gregersen and D. M. York, “High-order discretization schemes for

biochemical applications of boundary element solvation and variational

electrostatic projection methods,” J. Chem. Phys., vol. 122, no. 19, p. 194110,

May 2005.

[110] V. I. Lebedev, “Quadratures on a sphere,” USSR Comput. Math. Math. Phys., vol.

16, no. 2, pp. 10–24, 1976.

[111] V. I. Lebedev, “Spherical quadrature formulas exact to orders 25–29,” Sib. Math.

J., vol. 18, no. 1, pp. 99–107, Jan. 1977.

[112] C. Johnson, Numerical solution of partial differential equations by the finite

element method. Mineola, NY, USA: Courier Dover Publications, 2012.

[113] S. C. Brenner and R. Scott, The mathematical theory of finite element methods.

New York, NY, USA: Springer, 2008.

[114] J. Chen and X. Intes, “Comparison of Monte Carlo methods for fluorescence

molecular tomography—computational efficiency,” Med. Phys., vol. 38, no. 10,

pp. 5788–5798, Oct. 2011.

[115] Q. Fang and D. A. Boas, “Tetrahedral Mesh Generation from Volumetric Binary

and Gray-scale Images,” in Proc. 6th IEEE Int. Conf. on Symp. on Biomedical

Imaging: From Nano to Macro, 2009, pp. 1142–1145.

[116] R. Yao, X. Intes, and Q. Fang, “Generalized mesh-based Monte Carlo for wide-

field illumination and detection via mesh retessellation,” Biomed. Opt. Express,

vol. 7, no. 1, pp. 171–184, Jan. 2016.

[117] J. L. Sandell and T. C. Zhu, “A review of in-vivo optical properties of human

tissues and its impact on PDT,” J. Biophotonics, vol. 4, no. 11–12, pp. 773–787,

Nov. 2011.

[118] F. P. Bevilacqua, D. Piguet, P. Marquet, J. D. Gross, B. J. Tromberg, and C. D.

Depeursinge, “In-vivo local determination of tissue optical properties:

applications to human brain,” Appl. Opt., vol. 38, no. 22, pp. 4939–4950, Aug.

1999.

[119] Q. Fang and D. R. Kaeli, “Accelerating mesh-based Monte Carlo method on

modern CPU architectures,” Biomed. Opt. Express, vol. 3, no. 12, pp. 3223–3230,

Dec. 2012.

[120] M. S. Ozturk, C.-W. Chen, R. Ji, L. Zhao, B.-N. B. Nguyen, J. P. Fisher, Y. Chen,

and X. Intes, “Mesoscopic Fluorescence Molecular Tomography for Evaluating

Engineered Tissues,” Ann. Biomed. Eng., vol. 44, no. 3, pp. 667–679, Mar. 2016.

[121] A. Dunn and D. Boas, “Transport-based image reconstruction in turbid media

with small source-detector separations,” Opt. Lett., vol. 25, no. 24, pp. 1777–1779,

Dec. 2000.

[122] M. S. Ozturk, D. Rohrbach, U. Sunar, and X. Intes, “Mesoscopic Fluorescence

Tomography of a Photosensitizer (HPPH) 3D Biodistribution in Skin Cancer,”

Acad. Radiol., vol. 21, no. 2, pp. 271–280, Feb. 2014.

[123] L. Zhao, V. K. Lee, S.-S. Yoo, G. Dai, and X. Intes, “The integration of 3-D cell

printing and mesoscopic fluorescence molecular tomography of vascular

116

constructs within thick hydrogel scaffolds,” Biomaterials, vol. 33, no. 21, pp.

5325–5332, Jul. 2012.

[124] M. S. Ozturk, V. K. Lee, L. Zhao, G. Dai, and X. Intes, “Mesoscopic fluorescence

molecular tomography of reporter genes in bioprinted thick tissue,” J. Biomed.

Opt., vol. 18, no. 10, pp. 100501-1-100501-3, Oct. 2013.

[125] V. Ntziachristos and R. Weissleder, “Experimental three-dimensional

fluorescence reconstruction of diffuse media by use of a normalized Born

approximation,” Opt. Lett., vol. 26, no. 12, pp. 893–895, Jun. 2001.

[126] J. Chen, V. Venugopal, and X. Intes, “Monte Carlo based method for fluorescence

tomographic imaging with lifetime multiplexing using time gates,” Biomed. Opt.

Express, vol. 2, no. 4, pp. 871–886, Mar. 2011.

[127] Z. Meng, X. S. Yao, H. Yao, Y. Liang, T. Liu, Y. Li, G. Wang, and S. Lan,

“Measurement of the refractive index of human teeth by optical coherence

tomography,” J. Biomed. Opt., vol. 14, no. 3, pp. 034010-1-034010-4, Jun. 2009.

[128] X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo

continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med.

Phys., vol. 30, no. 6, pp. 1039–1047, Jun. 2003.

[129] M. Guven, B. Yazici, K. Kwon, E. Giladi, and X. Intes, “Effect of discretization

error and adaptive mesh generation in diffuse optical absorption imaging: I,”

Inverse Probl., vol. 23, no. 3, pp. 1115–1133, May 2007.

[130] A. Edmans and X. Intes, “Mesh Optimization for Monte Carlo-Based Optical

Tomography,” Photonics, vol. 2, no. 2, pp. 375–391, Apr. 2015.

[131] A. Thylstrup, S. A. Leach, and C. of Europe, Dentine and Dentine Reactions in

the Oral Cavity. Oxford, UK: IRL Press, 1987.

[132] T. J. Muldoon, S. A. Burgess, B. R. Chen, D. Ratner, and E. M. C. Hillman,

“Analysis of skin lesions using laminar optical tomography,” Biomed. Opt.

Express, vol. 3, no. 7, pp. 1701–1712, Jun. 2012.

[133] J. Martos, G. H. Tatsch, A. C. Tatsch, L. F. M. Silveira, and C. M. Ferrer-Luque,

“Anatomical evaluation of the root canal diameter and root thickness on the apical

third of mesial roots of molars,” Anat. Sci. Int., vol. 86, no. 3, pp. 146–150, Sep.

2011.

[134] T. Al-Gunaid, M. Yamaki, and I. Saito, “Mesiodistal tooth width and tooth size

discrepancies of Yemeni Arabians: A pilot study,” J. Orthod. Sci., vol. 1, no. 2, pp.

40–45, Apr. 2012.

[135] H.-H. Kim, J.-B. Min, and H.-K. Hwang, “Apical foramen morphology according

to the length of merged canal at the apex,” Restor. Dent. Endod., vol. 38, no. 1, pp.

26–30, Feb. 2013.

[136] F. J. Vertucci, “Root canal morphology and its relationship to endodontic

procedures,” Endod. Top., vol. 10, no. 1, pp. 3–29, Mar. 2005.

[137] M. Hülsmann, O. A. Peters, and P. M. H. Dummer, “Mechanical preparation of

root canals: shaping goals, techniques and means,” Endod. Top., vol. 10, no. 1, pp.

30–76, Mar. 2005.

[138] G. Chen, H. Qiu, P. N. Prasad, and X. Chen, “Upconversion Nanoparticles:

Design, Nanochemistry, and Applications in Theranostics,” Chem. Rev., vol. 114,

no. 10, pp. 5161–5214, May 2014.

117

[139] H. Qiu, C. Yang, W. Shao, J. Damasco, X. Wang, H. Ågren, P. N. Prasad, and G.

Chen, “Enhanced Upconversion Luminescence in Yb3+/Tm3+-Codoped Fluoride

Active Core/Active Shell/Inert Shell Nanoparticles through Directed Energy

Migration,” Nanomaterials, vol. 4, no. 1, pp. 55–68, Jan. 2014.

[140] S. Hu, X. Wu, Z. Tang, Z. Xi, Z. Chen, P. Hu, Y. Yu, H. Yan, Y. Liu, S. Hu, X.

Wu, Z. Tang, Z. Xi, Z. Chen, P. Hu, Y. Yu, H. Yan, and Y. Liu, “Upconversion

NaYF4 Nanoparticles for Size Dependent Cell Imaging and Concentration

Dependent Detection of Rhodamine B, Upconversion NaYF4 Nanoparticles for

Size Dependent Cell Imaging and Concentration Dependent Detection of

Rhodamine B,” J. Nanomater., vol. 2015, p. e598734, Dec. 2015.

[141] G. Y. Chen, Y. Liu, Y. G. Zhang, G. Somesfalean, Z. G. Zhang, Q. Sun, and F. P.

Wang, “Bright white upconversion luminescence in rare-earth-ion-doped Y2O3

nanocrystals,” Appl. Phys. Lett., vol. 91, no. 13, pp. 133103-1-133103-3, Sep.

2007.

[142] G. Y. Chen, Y. G. Zhang, G. Somesfalean, Z. G. Zhang, Q. Sun, and F. P. Wang,

“Two-color upconversion in rare-earth-ion-doped ZrO2 nanocrystals,” Appl. Phys.

Lett., vol. 89, no. 16, pp. 163105-1-163105-3, Oct. 2006.

[143] G. Y. Chen, G. Somesfalean, Z. G. Zhang, Q. Sun, and F. P. Wang, “Ultraviolet

upconversion fluorescence in rare-earth-ion-doped Y_2O_3 induced by infrared

diode laser excitation,” Opt. Lett., vol. 32, no. 1, pp. 87–89, Jan. 2007.

[144] F. Auzel, “Upconversion and Anti-Stokes Processes with f and d Ions in Solids,”

Chem. Rev., vol. 104, no. 1, pp. 139–174, Jan. 2004.

[145] M. Haase and H. Schäfer, “Upconverting nanoparticles,” Angew. Chem. Int. Ed

Engl., vol. 50, no. 26, pp. 5808–5829, Jun. 2011.

[146] F. Wang, D. Banerjee, Y. Liu, X. Chen, and X. Liu, “Upconversion nanoparticles

in biological labeling, imaging, and therapy,” The Analyst, vol. 135, no. 8, pp.

1839–1854, Aug. 2010.

[147] E. M. C. Hillman, D. A. Boas, A. M. Dale, and A. K. Dunn, “Laminar optical

tomography: demonstration of millimeter-scale depth-resolved imaging in turbid

media,” Opt. Lett., vol. 29, no. 14, pp. 1650–1652, Jul. 2004.

[148] V. V. Kumar, N. A. Malik, C. M. Visscher, S. Ebenezer, K. Sagheb, and F.

Lobbezoo, “Comparative evaluation of thickness of jaw-closing muscles in

patients with long-standing bilateral temporomandibular joint ankylosis: a

retrospective case-controlled study,” Clin. Oral Investig., vol. 19, no. 2, pp. 421–

427, Mar. 2015.

[149] J. R. Zijp, “Optical properties of dental hard tissues,” Ph. D. dissertation, Dept.

Material Sci. and Application, Rijksuniversiteit Groningen, Netherlands, 2001.

[150] M. Chu and H. Dehghani, “Image reconstruction in diffuse optical tomography

based on simplified spherical harmonics approximation,” Opt. Express, vol. 17, no.

26, pp. 24208–24223, Dec. 2009.

[151] H. Sun, Z. Duan, G. Zhou, C. Yu, M. Liao, L. Hu, J. Zhang, and Z. Jiang,

“Structural and up-conversion luminescence properties in Tm3+/Yb3+-codoped

heavy metal oxide-halide glasses,” Spectrochim. Acta. A. Mol. Biomol. Spectrosc.,

vol. 63, no. 1, pp. 149–153, Jan. 2006.

[152] F. Pandozzi, F. Vetrone, J.-C. Boyer, R. Naccache, J. A. Capobianco, A. Speghini,

and M. Bettinelli, “A spectroscopic analysis of blue and ultraviolet upconverted

118

emissions from Gd3Ga5O12:Tm3+, Yb3+ nanocrystals,” J. Phys. Chem. B, vol.

109, no. 37, pp. 17400–17405, Sep. 2005.

[153] Y.-K. Lee, “Fluorescence properties of human teeth and dental calculus for

clinical applications,” J. Biomed. Opt., vol. 20, no. 4, pp. 040901-1-040901-8, Apr.

2015.

[154] S. D. Forssten, M. Björklund, and A. C. Ouwehand, “Streptococcus mutans,

Caries and Simulation Models,” Nutrients, vol. 2, no. 3, pp. 290–298, Mar. 2010.

[155] D. C. Sarrett, “Clinical challenges and the relevance of materials testing for

posterior composite restorations,” Dent. Mater. Off. Publ. Acad. Dent. Mater., vol.

21, no. 1, pp. 9–20, Jan. 2005.

[156] A. Srinivasan, F. Long et. al., “Bactericidal effects of dental composites

containing upconvertion particles.” J. Dent. Res. Submitted for publication.

[157] B. Aydin Sevinç and L. Hanley, “Antibacterial activity of dental composites

containing zinc oxide nanoparticles,” J. Biomed. Mater. Res. B Appl. Biomater.,

vol. 94, no. 1, pp. 22–31, Jul. 2010.

[158] S. Kasraei, L. Sami, S. Hendi, M.-Y. AliKhani, L. Rezaei-Soufi, and Z.

Khamverdi, “Antibacterial properties of composite resins incorporating silver and

zinc oxide nanoparticles on Streptococcus mutans and Lactobacillus,” Restor.

Dent. Endod., vol. 39, no. 2, pp. 109–114, May 2014.

[159] S. K. Gupta, P. Saxena, V. A. Pant, and A. B. Pant, “Release and toxicity of dental

resin composite,” Toxicol. Int., vol. 19, no. 3, pp. 225–234, 2012.

[160] G. J. Fisher, M. E. Watts, K. B. Patel, and G. E. Adams, “Sensitization of

ultraviolet radiation damage in bacteria and mammalian cells,” Br. J. Cancer.

Suppl., vol. 3, pp. 111–114, Jun. 1978.

[161] H. Ashkenazi, Z. Malik, Y. Harth, and Y. Nitzan, “Eradication of

Propionibacterium acnes by its endogenic porphyrins after illumination with high

intensity blue light,” FEMS Immunol. Med. Microbiol., vol. 35, no. 1, pp. 17–24,

Jan. 2003.

[162] D. B. Lockwood, J. C. Wataha, J. B. Lewis, W. Y. Tseng, R. L. W. Messer, and S.

D. Hsu, “Blue light generates reactive oxygen species (ROS) differentially in

tumor vs. normal epithelial cells,” Dent. Mater., vol. 21, no. 7, pp. 683–688, Jul.

2005.

[163] Y. Kuse, K. Ogawa, K. Tsuruma, M. Shimazawa, and H. Hara, “Damage of

photoreceptor-derived cells in culture induced by light emitting diode-derived

blue light,” Sci. Rep., vol. 4, Jun. 2014.

[164] J. V. Frangioni, “In vivo near-infrared fluorescence imaging,” Curr. Opin. Chem.

Biol., vol. 7, no. 5, pp. 626–634, Oct. 2003.

[165] J. Tafur and P. J. Mills, “Low-intensity light therapy: exploring the role of redox

mechanisms,” Photomed. Laser Surg., vol. 26, no. 4, pp. 323–328, Aug. 2008.

[166] A. N. Kuzmin, A. V. Kachynski, P. N. Prasad, A. A. Demidovich, L. E. Batay, A.

Bednarkiewicz, W. Strek, and A. N. Titov, “Blue up-conversion emission in Yb-

and Tm-codoped potassium yttrium tungstate,” J. Appl. Phys., vol. 95, no. 12, pp.

7862–7866, Jun. 2004.

[167] A. Lipovsky, Y. Nitzan, A. Gedanken, and R. Lubart, “Visible light-induced

killing of bacteria as a function of wavelength: implication for wound healing,”

Lasers Surg. Med., vol. 42, no. 6, pp. 467–472, Aug. 2010.

119

[168] M. Maclean, S. J. MacGregor, J. G. Anderson, and G. Woolsey, “Inactivation of

Bacterial Pathogens following Exposure to Light from a 405-Nanometer Light-

Emitting Diode Array,” Appl. Environ. Microbiol., vol. 75, no. 7, pp. 1932–1937,

Apr. 2009.

[169] J. D. Stoien and R. J. Wang, “Effect of Near-Ultraviolet and Visible Light on

Mammalian Cells in Culture II. Formation of Toxic Photoproducts in Tissue

Culture Medium by Blacklight*,” Proc. Natl. Acad. Sci. U. S. A., vol. 71, no. 10,

pp. 3961–3965, Oct. 1974.

[170] E. L. Nussbaum, L. Lilge, and T. Mazzulli, “Effects of 630-, 660-, 810-, and 905-

nm laser irradiation delivering radiant exposure of 1-50 J/cm2 on three species of

bacteria in vitro,” J. Clin. Laser Med. Surg., vol. 20, no. 6, pp. 325–333, Dec.

2002.

[171] T. Karu, “Primary and secondary mechanisms of action of visible to near-IR

radiation on cells,” J. Photochem. Photobiol. B, vol. 49, no. 1, pp. 1–17, Mar.

1999.

[172] M. Fukui, M. Yoshioka, K. Satomura, H. Nakanishi, and M. Nagayama,

“Specific-wavelength visible light irradiation inhibits bacterial growth of

Porphyromonas gingivalis,” J. Periodontal Res., vol. 43, no. 2, pp. 174–178, Apr.

2008.

[173] R. S. McDonald, S. Gupta, M. Maclean, P. Ramakrishnan, J. G. Anderson, S. J.

MacGregor, R. M. D. Meek, and M. H. Grant, “405 nm Light exposure of

osteoblasts and inactivation of bacterial isolates from arthroplasty patients:

potential for new disinfection applications?,” Eur. Cell. Mater., vol. 25, pp. 204–

214, Mar. 2013.

[174] F. Suzuki, A. Han, G. R. Lankas, H. Utsumi, and M. M. Elkind, “Spectral

dependencies of killing, mutation, and transformation in mammalian cells and

their relevance to hazards caused by solar ultraviolet radiation,” Cancer Res., vol.

41, no. 12 Pt 1, pp. 4916–4924, Dec. 1981.

[175] Y.-Y. Huang, A. C.-H. Chen, J. D. Carroll, and M. R. Hamblin, “Biphasic Dose

Response in Low Level Light Therapy,” Dose-Response, vol. 7, no. 4, pp. 358–

383, Sep. 2009.

[176] J. Chen, V. Venugopal, F. Lesage, and X. Intes, “Time-resolved diffuse optical

tomography with patterned-light illumination and detection,” Opt. Lett., vol. 35,

no. 13, pp. 2121–2123, Jul. 2010.

[177] V. Venugopal, J. Chen, and X. Intes, “Development of an optical imaging

platform for functional imaging of small animals using wide-field excitation,”

Biomed. Opt. Express, vol. 1, no. 1, pp. 143–156, Aug. 2010.

[178] V. Venugopal, J. Chen, F. Lesage, and X. Intes, “Full-field time-resolved

fluorescence tomography of small animals,” Opt. Lett., vol. 35, no. 19, pp. 3189–

3191, Oct. 2010.

[179] S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse

optical tomography based on structured illumination,” J. Biomed. Opt., vol. 15, no.

1, pp. 016006-1-016006-7, Feb. 2010.

[180] L. Zhao, K. Abe, M. Barroso, and X. Intes, “Active wide-field illumination for

high-throughput fluorescence lifetime imaging,” Opt. Lett., vol. 38, no. 19, pp.

3976–3979, Oct. 2013.

120

[181] V. Venugopal and X. Intes, “Adaptive wide-field optical tomography,” J. Biomed.

Opt., vol. 18, no. 3, pp. 036006-1-036006-14, Mar. 2013.

[182] F. Yang, M. S. Ozturk, L. Zhao, W. Cong, G. Wang, and X. Intes, “High-

Resolution Mesoscopic Fluorescence Molecular Tomography Based on

Compressive Sensing,” IEEE Trans. Biomed. Eng., vol. 62, no. 1, pp. 248–255,

Jan. 2015.

[183] R. Yao, Q. Pian, and X. Intes, “Wide-field fluorescence molecular tomography

with compressive sensing based preconditioning,” Biomed. Opt. Express, vol. 6,

no. 12, pp. 4887–4898, Nov. 2015.

[184] Q. Pian, C. Wang, X. Chen, J. Liang, L. Zhao, G. Wang, and X. Intes,

“Multimodal Biomedical Optical Imaging Review: Towards Comprehensive

Investigation of Biological Tissues,” Curr. Mol. Imaging, vol. 3, no. 2, pp. 72–87,

Feb. 2015.

[185] X. Intes, C. Maloux, M. Guven, B. Yazici, and B. Chance, “Diffuse optical

tomography with physiological and spatial a priori constraints,” Phys. Med. Biol.,

vol. 49, no. 12, pp. N155–163, Jun. 2004.

[186] Q. Pian, R. Yao, L. Zhao, and X. Intes, “Hyperspectral time-resolved wide-field

fluorescence molecular tomography based on structured light and single-pixel

detection,” Opt. Lett., vol. 40, no. 3, pp. 431–434, Feb. 2015.