monte carlo simulations for the interaction of multiple scattered light and ultrasound a thesis
TRANSCRIPT
Monte Carlo Simulations for the Interaction ofMultiple Scattered Light and Ultrasound
A Thesis Presented
by
Luis A. Nieva
to
The Department of Electrical and Computer Engineering
in partial fulfillment of the requirementsfor the degree of
Master of Science
in
Electrical and Computer Engineering
in the field of
Electromagnetics, Plasma and Optics
Northeastern UniversityBoston, Massachusetts
January 22, 2003
1
Abstract
Monte Carlo Simulations for the Interaction of
Multiple Scattered Light and Ultrasound
Acousto-Photonic Imaging is a new frequency domain optical technique for non-invasive
medical imaging. It is based on the combination of Diffuse Optical Tomography (DOT)
and focused ultrasound. Diffuse Optical Tomography, due to its diffuse nature, can not
provide good spatial resolution by itself. Therefore, the objective is to use the ultrasound
to acoustically generate optical diffuse sources at different modulation frequencies, spaced
approximately one wavelength apart in the focus of the ultrasound beam. This will improve
the spatial resolution as well as acquire the optical properties of human tissue. In addition,
the study of the physics behind this interaction is of particular interest and still is not
completely understood.
We present Monte Carlo simulations for the interaction of Near-Infrared light (NIR) and
ultrasound in dense turbid media with high albedo. The strength of the optical signals for
the continuous wave, diffuse wave, and acousto-photonic wave is computed and compared
in order to have a quantitative idea of the signals generated. Experiments based on the
speckle pattern modulation and the diffuse photon density waves modulation are described.
The experimental techniques were performed with the goal of imaging in tissue-like phan-
toms made of titanium dioxide (TiO2) suspended in polyacrylamide gel that is acoustically
impedance matched with water.
2
Acknowledgements
The completion of this thesis could not have been possible without the help, advice, and
friendship of Prof. Charles DiMarzio. I would like to thank Prof. DiMarzio for being my
friend, advisor and the motivator in my research. Through invaluable conversations not
only about optics and engineering, but also about any topic that the daily interaction had
brought up, I have learned how to be a better researcher and to be a better person. I am
deeply in debt to Chuck for his guidance.
I would also like to thank Prof. Ronald Roy. With his expertise in the field of acoustics,
he helped me have a better understanding of ultrasound wave propagation through many
meetings in which I listened to and discussed his comments with particular interest. I thank
Prof. Dana Brooks for his time, for the useful corrections in my thesis work, and for being a
member of my thesis committee. I am grateful to Dr. Gerhard Sauermann for his valuable
conversations about physics and for sharing with me his multiple experiences.
I am very thankful to the people that work and have worked at the Optical Science Labora-
tory during the last two years while completing this work. I have learned a little bit of each
of them and I hope to maintain, throughout the years, the friendship that we have built.
Finally, I wish to thank my family for all the support and the love they provide me.
Contents
1 Introduction 9
2 Background: Light, Sound and Their Interaction 11
2.1 Frequency Domain Biomedical Optics . . . . . . . . . . . . . . . . . . . . . 11
2.2 Diffuse Optical Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Biomedical Applications of Diffuse Optical Tomography . . . . . . . 16
2.3 Ultrasound as a Biomedical Tool . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Biomedical Imaging Using Ultrasound . . . . . . . . . . . . . . . . . 22
2.4 Acousto-Optic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Acousto-Photonic Effect 29
3.1 Approaches to Explain the Interaction of Multiple Scattered Light and Ul-
trasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Mathematical Models for Acousto-Photonic Imaging . . . . . . . . . . . . . 32
3.2.1 Acoustic Modulation of the Diffuse Photon Density Waves . . . . . . 32
3
CONTENTS 4
3.3 Temporal Light Correlation of Multiple Scattered Light and its Interaction
with ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Numerical Simulations: Monte Carlo Approach 40
4.1 Monte Carlo Methods for Multiple Scattered Light Simulation . . . . . . . . 41
4.2 Frequency Domain Monte Carlo Approach for Diffuse Optical Tomography 42
4.3 Monte Carlo Simulations for Acousto-Photonic Imaging . . . . . . . . . . . 45
4.3.1 First Order Approximation of the Light-Ultrasound Weight . . . . . 47
4.3.2 Acoustic-Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.3 Monte Carlo-Acoustic Simulation Ensemble . . . . . . . . . . . . . . 51
4.3.4 Discussion and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 Experimental Methods 61
5.1 Laser Speckle Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.2 Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Acoustic Modulated Diffuse Photon Density Waves . . . . . . . . . . . . . . 66
5.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.2 Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . 68
6 Conclusions and Future Work 70
A Matlab Monte Carlo-Acoustic Simulation code 72
CONTENTS 5
B Transport Theory 81
C Raman-Nath/Bragg Effect 84
References 89
List of Figures
2.1 Hemoglobin Absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Acoustic Bragg diffraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Raman-Nath regime for the Acoustic diffraction of a light beam (Z) traveling
through an acoustic beam (X). . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1 Representation of the interaction of light and sound in scattering media. . . 32
4.1 Frequency domain representation of the constitutive sidebands in the inter-
action between multiple scattered light and ultrasound. . . . . . . . . . . . 46
4.2 Basic interaction of light, ultrasound and the particles in the medium. . . . 47
4.3 Ultrasound simulation shows the displacement of the particles in the beam
and phase variations in the focus. . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Geometry for the simulation ensemble of multiple scattered light and ultra-
sound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Flow diagram of Monte Carlo-Acoustic Simulation. . . . . . . . . . . . . . . 54
6
LIST OF FIGURES 7
4.6 Amplitude and phase modulation of diffuse light interacting with a plane
ultrasound wave in scattering media. The ultrasonic wavelength (≈ 640µm),
is well defined and modulates the optical path lengths. . . . . . . . . . . . . 55
4.7 Amplitude and phase modulation of diffuse light interacting with a focussed
ultrasound wave in scattering media. . . . . . . . . . . . . . . . . . . . . . . 56
4.8 Simulation with 1 million photons. . . . . . . . . . . . . . . . . . . . . . . . 57
4.9 Simulation with 5 million photons. . . . . . . . . . . . . . . . . . . . . . . . 57
4.10 Simulation with 10 million photons. . . . . . . . . . . . . . . . . . . . . . . 58
4.11 Simulation with 20 million photons. . . . . . . . . . . . . . . . . . . . . . . 58
4.12 Signal levels of the DPDW signal with respect to the CW signal. . . . . . . 59
4.13 Signal levels of the API signal with respect to the CW signal. . . . . . . . . 59
5.1 Setup for Speckle Contrast measurements. . . . . . . . . . . . . . . . . . . . 62
5.2 Pressure at the focus of the ultrasound vs voltage supply. . . . . . . . . . . 63
5.3 Speckle pattern with and without the prescence of the ultrasound. Notice
the bluriness of the image on the right (ultrasound on) with respect to the
one on the left (ultrasound off). . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4 Speckle contrast for various sets of data. . . . . . . . . . . . . . . . . . . . . 66
5.5 Setup for DPDW experiments. . . . . . . . . . . . . . . . . . . . . . . . . . 67
List of Tables
2.1 Ultrasound intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1 Tabulated data for speckle contrast . . . . . . . . . . . . . . . . . . . . . . . 65
8
Chapter 1
Introduction
Extensive research is being conducted in the field of Medical Imaging. Qualitative and
quantitative information as well as spatial resolution are the requirements to be fulfilled to
provide the medical practitioner with useful information to help diagnose the illness.
Diffuse Optical Tomography (DOT), among the various optical imaging techniques, has
been shown to be a good way to acquire information about tissue optical properties which
in turn are related to metabolic processes through the absorption of light by hemoglobin
(Hb) [1]. The non-invasive nature of this technique as well as the quantitative information
that it can provide, makes DOT an interesting field of study and a promising tool that can
work in parallel with current medical imaging methods such us MRI, X-rays, etc. Photon
migration can be explained using radiative transport theory and has been the subject of
recent and extensive research [2, 3, 4, 5]. In particular, the use of modulated near-infrared
(NIR) light in medical imaging and dignostic applications provides us with a spectral win-
dow through which is possible to get quantitative information about the absorption and
scattering properties of human tissue. The applications range from oximetry and tissue
spectroscopy to image of brain and breast tumors and functional imaging of the brain.
On the other hand, ultrasound provides a very well stablished imaging technique with good
9
CHAPTER 1. INTRODUCTION 10
spatial resolution compared with the resolution that DOT can provide. The sound wave
does not scatter as much as the light wave inside human tissue.
With the goal of obtaining the best of both imaging modalities, we have studied the inter-
action of multiple scattered light and ultrasound, which is know with the name of Acousto-
Photonic Effect, and its primarily application, the Acousto-Photonic Imaging (API). This
study is expected to lead us to imaging the optical properties of tissue with the spatial
resolution of ultrasound.
Various approaches have attempted to explain the physics behind this process. The density
changes due to the acoustic wave and therefore, the change on the density of particles,
produce index of refraction modulation and small particle displacements at the ultrasound
frequency which in turn change the photons wave vectors producing changes in the speckle
pattern, and modulation of diffuse photon density waves (DPDW).
The focus of this thesis is to investigate the feasibility of the API method through numerical
simulations and experimental techniques that try to explain the behavior and interaction
between multiple scattered light and ultrasound inside turbid media. Chapter 2 gives a
general introduction to the medical imaging applications of the two fields of study and
also describe the theories and approaches that try to explain this combination in non-
scattering media. Chapter 3 briefly reviews the mathematical theory behind the Acousto-
Photonic Effect as a matter of understanding the work done by this author. Chapter 4
presents frequency domain Monte Carlo simulations coupled with finite-difference time-
domain (FDTD) acoustic simulations that shows the expected signal level strenghts of the
interaction between diffuse waves and ultrasound. Chapter 5 presents the experimental
work done in order to study the two most important optical phenomena, which are the
laser speckle modulation and the diffuse photon density waves modulation. Finally, chapter
6 discusses the conclusions and the future work proposed for this project.
Chapter 2
Background: Light, Sound and
Their Interaction
2.1 Frequency Domain Biomedical Optics
Biomedical optics has been topic of intensive research during the last years with the goal of
developing another tool, based on the study of the optical properties of human tissue, that
can help the medical community to diagnose disease and choose the right treatment for the
patient.
Quantitative light absorption at specific wavelengths has been used since the 1930’s for
determining the oxygen content of blood, and now the scientific community is making efforts
to use this technique in imaging. In the late 1980’s the research was directed towards imaging
the transmission of light through tissue. Light in the near infrared range (wavelengths
from 700 to 1200nm) penetrates tissue and interacts with it in a random process, with
the predominant effects being absorption and scattering. Laser optical tomograpy involves
reconstruction of the amount of transmitted laser light through an object along multiple
paths. Moreover, the modulation of the light source at radio frequencies, which is the basis
11
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 12
of diffuse optical tomography, has showed to provide information about the amplitude and
the phase of the diffuse waves, which can be used for better reconstruction of images or
more accurate measurements of optical properties of tissue. The modulation of the optical
source and its applications are usually known as frequency domain techniques.
Different applications have been developed with the use of frequency domain optical tech-
niques but we attempt to divide them in two groups. The first group would be tissue
spectroscopy and oximetry in which we treat tissue as macroscopically homogeneous and
we can use the approximations for the theory behind diffuse optical tomography that we will
review in the next subsections. Applications range from measurement of optical absorption
of tissue and Near Infrared (NIR) tissue oximetry to measurements of optical scattering in
tissues.
The second group would be related to the optical imaging of tissues. Since the objective
is to map the spatial distribution of the tissue optical properties we can not treat it as
macroscopically homogeneous anymore and then the use of the mathematical theory must
account for the spatial dependence of its constitutive parameters.
2.2 Diffuse Optical Tomography
Among the variety of optical techniques that exist to monitor and to image inside human
tissue, Diffuse Optical Tomography (DOT) has emerged as one of the most promising and
important, leading to research for different applications in the biomedical community [1, 4].
DOT has a spatial resolution of about 10mm; thus it can not provide images with the
resolution quality of X-rays, Magnetic Resonance Imaging (MRI) scans, Positron Emission
Tomography (PET) scans or ultrasound. However, this method does have a number of
practical applications even at low resolution. These include the measurement of tissue oxy-
genation for the study of muscular dystrophy [6] (which is any of a group of diseases chara-
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 13
terized by progressive wasting of muscles), tissue perfusion in the extremities for diabetic
disease [7], the detection of brain hemorrhaging [8], monitoring stroke patients [9], the study
of brain activity during specific tasks [5, 10], breast tumor detection and characterization
[5] and possibly the study of glucose concentration changes [11]. The clinical potentials of
determining the oxygenation level and functional imaging in the brain of young children has
been demonstrated [10, 12]. Beyond DOT, similar principles can be applied for fluorescense
imaging [13]: substances which play a crucial role in the body’s metabolic (energy making)
processes, such as NAD/NADH (nicotinamide adenosine diphosphate), exhibit flourescent
properties which allow detection after being excited by light. Their assessment by indirect
measurements has important potential for medical applications.
The scattering of light can be explained using standard mathematical models. In analytical
theory we start with Maxwell’s equations and introduce the scattering and absorption of
particles, which lead us to obtain the appropiate differential or integral equations for statis-
tical quantities such as variances and correlation functions [14, 15]. This is mathematically
rigorous but in practice is impossible to obtain a closed form solution that includes all the
scattering, diffraction and interference effects. Therefore, the research community has used
transport theory (radiative transfer theory) [16] in order to explain the behavior of light
in turbid media. This mathematical approach does not start with the wave equation. It
deals directly with the transport of energy through a medium containing particles. Trans-
port theory is not mathematically rigourous and does not include diffraction or polarization
effects. It is assumed in transport theory that there is no correlation between fields, and
therefore, it uses the addition of powers rather than the addition of fields [4, 14].
Transport Theory was initialy treated by Schuster in 1903 [17] and the basic differential
equation is called the equation of transfer and is equivalent to the Boltzmann’s equation
used in kinetic theory of gases and in neutron transport theory [16, 18]. This formulation is
capable of treating many physical phenomena and has been succesfully employed for prob-
lems including underwater visibility, marine biology, biomedical optics, and the propagation
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 14
of radiant energy in the atmospheres of planets, stars and galaxies.
The Boltzmann Transport Equation (BTE) is a balance relationship that describes the flow
of particles in scattering and absorbing media. This mathematical approach can be used
to model the propagation of light in optically turbid media, where the photons are treated
as the transported particles. This theory has been investigated extensively in the literature
and a brief summary can be found in Appendix B of this thesis for the convinience of the
reader. We are going to base this formulation in the work that can be found in Ref. [5, 19].
If we denote the angular photon density with u(r, Ω, t), which is defined as the number of
photons per unit volume per unit solid angle traveling in direction Ω at position r and time
t, we can write the BTE as follows:
∂u(r, Ω, t)∂t
= −v Ω · ∇u(r, Ω, t)− v(µa + µs)u(r, Ω, t)
+ vµs
∫
4πu(r, Ω′, t) f(Ω′, Ω) dΩ′ + q(r, Ω, t), (2.2.1)
where v is the speed of light in the medium, µa is the absorption coefficient in cm−1, µs is
the scattering coefficient in cm−1, f(Ω′, Ω) is the phase function or the probability density
of scattering a photon that travels along direction Ω′ into direction Ω, and q(r, Ω, t) is the
source term. q(r, Ω, t) has units of s−1 m−3 sr−1 and represents the number of photons
injected by the light source per unit volume, per unit time, per unit solid angle at position r,
time t, and direction Ω. The left hand side of Eq. (2.2.1) represents the temporal variation
of the angular photon density. Each one of the terms on the right hand side represents a
specific contribution to this variation. The first term is the net gain of photons at position
r and direction Ω due to the flow of photons. The second term is the loss of photons at
position r and direction Ω due to absorption and scattering. The third term is the gain of
photons at r and Ω due to scattering. Finally, the fourth term is the gain of photons due
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 15
to the light sources.
Some of the quantities used to describe photon transport are angular photon density
u(r, Ω, t), photon radiance L(r, Ω, t) = vu(r, Ω, t), photon density U(r, t) =∫4π u(r, Ω, t) dΩ,
photon fluence rate Φ(r, t) = v U(r, t) and photon flux J(r, t) =∫4π u(r, Ω, t) Ω dΩ.
The BTE is a difficult differential equation to solve. Therefore, a first order approximation
is commonly used to obtain closed form solutions. The validity of this approximation relies
in the assumption that scattering is much stronger than absorption (µ′s À µa), which is
true for biological tissue. The radiance L can be expressed as an isotropic photon fluence
rate Φ plus a small directional photon flux J. This approximation takes the name of the
diffusion approximation or the diffusion equation [16, 18]. See also Appendix B.
Since DOT is based on the modulation of the light source at radio frequencies we seek
for solutions of the diffusion equation in terms of the frequency ω. The frequency domain
expression for the solution of the diffusion equation for a homogeneous, infinite medium
containing a harmonically modulated point source of power P (ω) at r = 0 is
U(r, ω) =P (ω)4πD
eikr
r, (2.2.2)
where D is the diffusion coefficient given by D = v/(3µ′s + µa). The expressions for the
average photon density (UDC), and for the amplitude (UAC) and phase (φ) of the diffuse
photon density wave, derived from Eq. (2.2.2), are [5, 19]
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 16
UDC(r) =PDC
4πD
e−r(vµa/D)1/2
r, (2.2.3)
UAC(r, ω) =P (ω)4πD
e−r(vµa/D)1/2[(1+ ω2
v2µ2a
)1/2+1]1/2
r, (2.2.4)
φ(r, ω) = r(vµa/2D)1/2
[(1 +
ω2
v2µ2a
)1/2
− 1
]1/2
+ φs , (2.2.5)
where φs is the phase of the source in radians. As we can see from Eq. (2.2.4), the oscillating
photon density is proportional to the power of the point source P (ω) and will oscillate at
the same frequency. These are scalar, damped, traveling waves. The imaginary part of the
wavenumber must be greater than zero in order to satisfy the physical condition that the
amplitude is exponentially attenuated while the wave travels through the medium.
2.2.1 Biomedical Applications of Diffuse Optical Tomography
As discussed in the introduction of this chapter we can divide the applications that use
photon migration in human tissue in two groups. The first group is tissue spectroscopy and
oximetry where we deal basically with absorption and scattering in a macroscopic view.
The absorption is mainly because of oxy-hemoglobin, deoxy-hemoglobin, and water. The
absorption spectra ranging from 300 to 1100 nm is shown in Fig. 2.1 with data compiled by
Prahl [20]. We observe that the absorption around 700 to 900 nm is low compared to other
wavelengths. This is the so-called “medical spectral” window. As a result of this, light in
this spectral range penetrates deeply into tissues, thus allowing us to perform noninvasive
investigations. This is the reason why our work is based in the use of NIR light.
The scattering properties are determined mainly by the size of the scattering particles rela-
tive to the wavelength of light, and by the refractive index mismatch between the scattering
particles and the surrounding medium. In biological tissues, the scattering is mainly forward
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 17
200 300 400 500 600 700 800 900 100010
−2
10−1
100
101
102
Wavelegth (nm)
Abs
orpt
ion
coef
fcie
nt (
µ a)
Hemoglobin absorption spectra
HbHbO
2
Figure 2.1: Hemoglobin Absorption.
directed for wavelengths in the medical spectral window because the cellular organelles and
cells have dimensions comparable to the optical wavelength. Therefore, the scattering prop-
erties are described by two parameters: the scattering coeffcient µs and the average cosine
of the scattering angle g (anisotropy). Even though each scattering event is mainly forward
directed, after a number of scattering events a photon loses memory of its original direction
of propagation. It is customary to use the reduced scattering coefficient µ′s = µs(1 − g)
which represents the reciprocal average distance over which the direction of propagation of
a photon is randomized.
When we work with human tissue µ′s is typically much larger than µa, therefore, we can
assume that NIR light propagation is mainly due to scattering. This is one of the condi-
tions for the derivation of the diffusion equation (See Appendix B). The frequency-domain
solution given by Eq. (2.2.2) provides a good quantitative description of photon migration
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 18
in an infinite medium with uniform optical properties. However, for real investigation and
experimentation we have to consider real boundary conditions. In a reflectance geometry
one typically applies the semi-infinite boundary condition. This is a reasonable assumption
if the tissue depth is greater that the optical penetration (2 - 3 cm or less) and inhomo-
geneities are small [21, 22]. This assumption is not valid in transmission geometry since the
source and the detector are located in opposite sides of the tissue. In this cases it is better
to use more appropiate boundary conditions such as a slab, cylinder or sphere geometry
[23].
The objective of tissue spectroscopy is determining certain properties of the investigated
tissue volume like the oxygenation or the hemoglobin concentration of a muscle based on
the measurement of the optical properties of the tissue. Particularly, the absorption de-
pends on the presence of different chromophores like oxy-hemoglobin, deoxy-hemoglobin,
water, cytochrome oxidase, melanin, bilirubin, and lipids. Therefore, measurements at
different wavelengths have been employed to determine the relative contributions of each
chromophore according to their concentration in the tissue of study. In many cases only
three of them are sufficient to give a good description of the absorption properties of tissues.
These three chromophores are oxy-hemoglobin, deoxy-hemoglobin, and water [24].
We can also measure the scattering properties of tissue. In the past, studies were focused in
the light absorption of tissue [24], but recent research has suggested that the reduced scat-
tering coefficient itself may provide information about physiologically relevant parameters.
For instance it has been shown that mitochondria are the main source of light scatttering in
the liver, and possibly in other tissues as well [25]. Since a number of metabolic processes
related to cellular respiration occur in the mitochondria, the reduced scattering coefficient
may be related to the cellular activity and viability.
The second group of applications as we defined them is the optical imaging of tissues. These
applications are based on the sensitivity to optical properties of tissues. The contrast in
NIR imaging originates from spatial variations in the optical absorption and scattering
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 19
properties of the tissue. These spatial variations can be due to a local change in hemoglobin
concentration or oxygen saturation, a localized change in the tissue architecture, or the
concentration of cellular organelles. It is safe to point out that the promise of diffuse optical
tomography is not in achieving a high spatial resolution, but in achieving high contrast and
specificity.
The goal of the imaging technique is to generate spatial maps that display either structural
or functional properties of tissues. Therefore, we are dealing now with inhomogeneous
media, and now we have to use the diffusion equation for inhomogeneous media
−∇ ·D(r)∇U(r, t) + vµa(r)U(r, t) +∂U(r, t)
∂t= q(r, t). (2.2.6)
Notice the dependence on r of the diffusion coefficent D and the absorption coefficient µa.
Analytical solutions for this equation are available only for a few simple geometries like
spherical or cylindrical. For arbitrary inhomogeneous cases Eq. (2.2.6) can be solved using
numerical methods such as the finite difference method or finite element method. Alter-
natively, a perturbation expansion in µa and D leads to a solution of Eq. (2.2.6) in terms
of a volume integral involving the appropiate Green’s function. A similar procedure using
perturbation techniques has been developed for the modulation of DPDW by an ultrasonic
beam [26] which will be briefly reviewed in Chapter 3 of this thesis. Besides diffusion the-
ory, the case of inhomogeneous media can also be treated with stochastic methods such as
Monte Carlo simulations [27, 28] or lattice random walk models [29].
Among the most important applications of diffuse optical imaging we have noninvasive
optical mammography and optical imaging of the human brain, specifically for the detection
of intracranial hematomas and functional imaging of the brain.
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 20
2.3 Ultrasound as a Biomedical Tool
Ultrasound is a real-time tomographic imaging modality. Ultrasound is able to produce
images of the position of reflecting surfaces like internal organs and structures, but it also
can be used to produce real-time images of tissue and blood motion.
Ultrasound denotes the use of acoustical waves at frequencies greater than 20 KHz. Gener-
ally, medical ultrasound is performed at frequencies in the range of 1 MHz. The technique
is used to determine the location of surfaces within tissues by measuring the time interval
between the production of an ultrasonic pulse and the detection of its echo resulting from
the pulse being reflected from those surfaces. By measuring the time interval between the
transmitted and detected pulse, we can calculate the distance between the transmitter and
the object.
The ultrasound pulses are both produced and detected by a piezoelectric crystal transducer.
The crystal has the property of changing its physical dimensions in response to an electric
field, and can produce an electric field if its physical shape is changed mechanically. Thus,
ultrasonic compression waves (vibrations) are produced by applying an oscillating potential
across the crystal. The reflected ultrasound imposes a distortion on the crystal, which
in turn produces an oscillating voltage in the crystal. The same crystal is used for both
transmission and reception. There are a wide variety of transducers commercially available
which can produce an acoustic wave by mechanical or electronic means. The latter is used
with arrays of piezo-electric crystals, each one producing a small acoustic wave in phase
with the other crystals in the array to produce together the complete ultrasound beam.
If a structure is stationary, the frequency of the reflected wave will be identical to that of
the impinging wave. A moving structure will cause a back-scattered signal frequency shifted
higher or lower depending on the structure’s velocity toward or away from the transducer.
Imaging based on this principle is known as Doppler ultrasound.
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 21
For example, when an impinging sound pulse passes through a blood vessel, scattering and
reflection occur from the moving red cells. In this process, small amounts of sound energy
are absorbed by each red cell, and then re-radiated in all directions. If the cell is moving
with respect to the source, the back scattered energy returning to the source will be shifted
in frequency, with the magnitude and direction proportional to the velocity of the respective
blood cell. Thus, if we use ultrasound to image the cross-sectional area of the blood vessel,
the volume of blood flow can be calculated from the area of the vessel and the average
velocities of the blood cells.
The major use of Doppler ultrasound is the study of the heart and human carotid artery
disease where imaging and frequency shift are combined to produce images of artery and
ventricle lumens. The frequency shift data is used to color the image, showing direction of
flow (e.g. carotid arteries in red and veins in blue). Obstructions to blood flow are readily
evaluated by this method using hand held scanning devices. In addition to imaging heart
valves and blood vessels, ultrasound is the most convenient and inexpensive method for
medical evaluations such as fetal monitoring and gallbladder stones. Ultrasound imaging
is also being used for monitoring therapy methods such as hyperthermia, cryosurgery, drug
injections, and as a guide during biopsies and catheter placements.
The propagation of an acoustic wave through human tissue can be fully predicted and
described if we take into account the mass and stiffness of the media, and its conformance
with basic physical laws. In the very basic process a sinusoidal wave will accelerate adjacent
tissue particles and compress that part of the medium nearest to it as it moves forward from
rest. This also is going to impart a forward momentum to the particles which is going to
be transmitted to their neighbors which were at rest. These particles in turn move closer
to their neighbors, with which they collide, and so on.
When an acoustic wave is propagated in the medium, several changes occur. The particles
are accelerated and as a result are displaced from their rest positions. The particle velocity
at any point is not zero except at certain instants during a cycle. The temperature at any
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 22
point will vary above and below the ambient value. Also, the pressure at any point will
vary above and below the ambient pressure. This incremental variation of pressure is called
the acoustic pressure. A pressure variation, in turn, causes a change in the density called
the incremental density. An increase in sound pressure at a point causes an increase in the
density of the medium at that point.
Ultrasound propagation through human tissue can be explained with wave theory. The
acoustic wave equation has the form
∂2ξ
∂t2= v2
a
∂2ξ
∂x2(2.3.1)
where ξ is the displacement or the particles and va is the speed of sound in the medium.
The solution to this differential equation is well known and can be found elsewhere, but in
particular, we use the solution for a Gaussian ultrasound wave rather than a plane wave
because the transducers used in our experiments radiate a Gaussian wave. This has the
purpose of maximizing the interaction between diffuse light and sound in the focus of the
ultrasound beam. See Ref. [30] for a complete treatment of ultrasound wave propagation.
2.3.1 Biomedical Imaging Using Ultrasound
Multiple applications in the biomedical field have been developed using ultrasonic waves
[31]. The discovery of the piezo-electric effect at the end of the nineteenth century, and
the development of an ultrasonic echo-sounding device in the early 1930s by Paul Langevin
and Constantin Chilowsky, formed the basis for the development of medical pulsed-echo
SONAR.
Ultrasound can be used in therapy and as a diagnostic tool. In therapy basically the thermal
energy is used when sound is propagated through tissues. Muscle and bone have been found
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 23
to absorb more energy at interfaces with other heterogeneous tissues, because at these
surfaces, the longitudinal waves of ultrasound are reflected and transformed into transverse
waves, creating a heating effect. This happens commonly in the areas in between the muscle
and bone or between the muscle and tendon. By applying ultrasonic waves to these areas,
physical therapists can take advantage of this thermal affect to reduce inflammation and
increase mobility in the joints.
As a diagnostic tool ultrasound is used primarily in imaging. Real-time ultrasonic imaging is
possible with the use of state-of-the-art piezo-electric transducers. Since there do not appear
to be any biologically significant adverse effects of ultrasound at the levels used for diagnosis,
ultrasonic imaging has become the most frequently utilized technique in obstetrics. This
helps to diagnose multiple complications that can be present during pregnancy and also to
monitor throughout the pregnancy process. For this purpose two general types of ultrasound
scanners are available, real-time scanners for depiction of structures within the body, and
scanners that are mounted in articulated arms, which, when manually moved over the
body produce static images. Most scanning studies are performed today with real-time
scanners. Other applications are renal and urological imaging, echocardiography to examine
the structure and functioning of the heart for abnormalities and disease, and pediatric
imaging among others.
The use of ultrasound with biological tissue has to take into account potential damage and
bioeffects. There are two primary mechanisms by which ultrasound can produce biological
effects: heat and cavitation. Attenuation in tissues is, on the average, 90 to 95 percent
absorption, specifically conversion to heat. Temperature rise in a particular spot where
the acoustic wave is radiated depend upon the ultrasound intensity, frequency, specific
heat and thermal conductivity of the tissue, and vascularization. These factors combine
in complicated ways to determine the ultimate temperature rise at a given site exposed
to an ultrasound beam. A higher frequency results in a higher absorption coefficient but
greater attenuation so that a given source intensity would result in a lower intensity at a
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 24
given depth. Some concern is warranted with pulsed ultrasound Doppler and flow imaging
equipment in color where high power levels and time-averaged intensities may result in large
values in thermal index.
On the other hand, cavitation is the generation, growth, and dynamics of bubbles in a
medium. It can be generated in media that have cavitation nuclei, which are microbub-
bles. Cavitation is divided in two types: stable and transient. Stable cavitation describes
the situation in which bubbles are oscillating repeatedly in an acoustic field. Transient
cavitation refers to the situation in which cavities reach resonance size, at which violent
nonlinear dynamics occur, with the cavitiy collapsing and producing pressure shock waves
and extreme temperature gradients.
The American Institute for Ultrasound in Medicine (AIUM) has multiple publications in
which we can find information about the biological effects and safety of diagnostic ultrasound
[32]. In these references we can find that in the low megahertz frequency range there have
been no independently confirmed significant biological effects in tissue for Spatial Peak
Temporal Average (SPTA) intensites below 100mW/cm2. The above information is for
scanning imaging systems. Doppler instrument outputs can be significantly higher than
those for imaging. Typical intensities according to the National Council on Radiation
Protection and Measurements (NCRP) and the AIUM are presented in Table 2.1.
We can say that there is presently no identified risk to the use of ultrasound at intensity
levels used commonly in diagnosis. However, a prudent approach should always be employed
with the use of this technique.
2.4 Acousto-Optic Effect
The first type of interaction between light and sound was studied in non-scattering media.
Acousto-optic interaction occurs in all optical media when an acoustic wave and a ray
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 25
Instrument Type ISPTA Range (mW/cm2)Static B-scan 10 - 170
M-mode 10 - 160Dynamic B-scan
sector 6 - 30linear 0.01 - 12
DopplerCW 0.6 - 2500
pulsed 50 - 1945
Table 2.1: Ultrasound intensities
of light are present in the medium. When an acoustic wave is launched into the optical
medium, it generates a refractive-index wave that behaves like a sinusoidal grating. An
incident laser beam passing through this grating will diffract the laser beam into several
orders. Its angular position is linearly proportional to the acoustic frequency, so that the
higher the frequency, the larger the diffracted angle.
Various attempts to explain this phenomena have been developed during this century [33,
34]. We can treat this as a parametric process in which the optic and the acoustic wave
are mixed via the elasto-optic effect. This works as a oscillator system with frequency ωa
modulated by a frequency ωp which we would call the pump frequency. The typical equation
of motion would for this type of oscillator is
d2y
dt2+ γ
dy
dt+
(ω2
a + α sinωpt)y = 0.
In the case of the acousto-optic effect the pump frequency would be the light wave frequency
and the signal wave that that gets frequency shifted due the pump wave has a frequency
of ωp + ωa, where ωa is the acoustic wave frequency. This model does not provide a clear
explanation of the physical behavior of the acousto-optic effect; therefore, it has not been
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 26
further developed.
The approach that has proven to be successful in the treatment of this phenomena is the
one based on scattering theory in which the light wave is treated as photons and sound
waves as particles. This leads us to two regimes for the acousto-optic effect: The Bragg
regime [35] and the Raman-Nath regime [36, 37].
The Bragg regime was first discovered by Brillouin in the 1920’s by a set of basic experiments
using a sound column and light waves propagating with angles of incidence that were chosen
to ensure constructive interference of the light beams reflected off the crests of the sound
wave. This is also the condition for x-ray diffraction. This leads to critical angles of incidence
sinφin = pλa
2λ(2.4.1)
which are called the Bragg angles. The angle of reflection on the crest of the acoustic beam
is twice the Bragg angle. This is shown in Fig. 2.2. Brillouin makes the observation that
the sound wave is a sinusoidal grating and that therefore we should only expect two critical
angles given by the relation in Eq. (2.4.1), for p = +1 and p = −1. This is only valid for a
thick column, in contrast to the Raman-Nath regime. The validity for this assumption and
the limits will be explained in the next section.
The Raman-Nath regime owes its name to the Indian researchers C. V. Raman and N. S.
N. Nath, who based their work on the previous work done by Brillouin, Lucas and Biquard,
and Debye and Sears, which showed that the existence of numerous modes was due to the
fact that the interaction length was too small (considering a thin sound column), but in a
more direct way for the mathematical explanation of the acousto-optic interaction. This
work considers a very thin acoustic column as a phase grating which the light rays traverse
in straight lines. Because of the phase shift suffered by each ray, the total wavefront is
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 27
Figure 2.2: Acoustic Bragg diffraction.
corrugated as it leaves the sound field creating a set of upshifted and downshifted waves
with frequency ω±nωa and propagation in the direction kn such that knx = nka with angle
of propagation φn given by
sinφn =nka
k=
nλ
λa(2.4.2)
where φn is the angle of diffraction of the light beam at the output of the acoustic column,
ka is the acoustic wavenumber, k is the light wavenumber, n the order of the diffracted light
beam, λ the optical wavelength and λa the acoustic wavelength. This procces is shown in
Fig. 2.3. Later on, a generalization of the work by Raman and Nath done by Van Cittert
CHAPTER 2. BACKGROUND: LIGHT, SOUND AND THEIR INTERACTION 28
Figure 2.3: Raman-Nath regime for the Acoustic diffraction of a light beam (Z) travelingthrough an acoustic beam (X).
[38], which considers a thick film of sound as a collection of thin films, confirms the Bragg
regime and the expectation of only two critical angles. A more complete treatment of the
Acousto-optic effect can be found in Ref. [39] or in Appendix C.
Chapter 3
Acousto-Photonic Effect
Although DOT and photon migration based imaging modalities in general have proved to
be a feasible technique among the multiple imaging methods and are in the proccess of
becoming comercially available for the medical community, they lack the spatial resolution
for measurements made deep into biological tissue. This fact led our research group and a
few other optical research groups to suggest the combination of multiple scattered light and
ultrasound with the goal of enhancing the spatial resolution and provide a more complete
medical imaging tool. The combination between multiple scattered light and ultrasound
for imaging purposes was initially reported by Brooksby et.al. [40, 41] in 1993 where they
showed a basic interaction and postulate the tagging of light with ultrasound. Later on in
1995, Leutz et.al. [42], Wang et.al. [43, 44, 45] and Kempe et.al. [46] combined continuous
wave light with ultrasound reporting one and two dimensional images using single detectors.
Bocarra et.al. [47, 48] applied the same technique but used paralled detection to study the
modulation of the speckle pattern generated by the ultrasound. In 1996 Gaudette and
DiMarzio [49], postulated the modulation of diffuse photon density waves by a focused
ultrasound field which is the basis of the Acousto-Photonic Imaging (API) technique. The
subject of this thesis is the continuation of this challenging work of trying to explain the
foundations of API.
29
CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 30
Along with the experimental techniques, various mathematical models have been developed
to explain, and to find the best way to achieve, this interaction. There are mathematical
methods based on the temporal correlation of the electric fields [42, 53]. These methods
rely on the index of refraction modulation of the medium due to the ultrasound, which in
turn, modulates the laser speckle generated by the diffusing medium. There are also other
methods based purely on transport theory that try to explain the acoustic generated diffuse
photon density waves [26].
The author of this thesis has developed a numerical simulation for the API technique based
on a frequency domain Monte Carlo method [50] and finite difference time domain (FDTD)
simulations of a focused ultrasound wave [51], which will be extremely useful to corroborate
the experimental results as well as to establish the viability of this imaging technique. This
work will be presented in Chapter 4.
3.1 Approaches to Explain the Interaction of Multiple Scat-
tered Light and Ultrasound
Laser Speckle Modulation
The first idea we present to explain this phenomena is to study the behavior of the particles
as independent scatterers in the presence of an ultrasound beam. For this purpose it is
assumed that the acoustic wave does not scatter in the medium. The light source has a
carrier frequency ωc with wavelength in the medical spectral range. The ultrasound wave
has the general form of S(r)cos(ka · r − ωat) where S(r) is the amplitude of the acoustic
wave and ka is the acoustic wavevector.
The presence of this oscillating mechanical field will produce variations on the index of
refraction of the medium following the ultrasound amplitude and, therefore, modulate the
optical path lengths of the light interacting with the acoustic wave. This will cause a change
CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 31
in the speckle pattern formed by the multiple scattered light leaving the tubid media. Also
due to this mechanical wave the particles in the medium will oscillate at the frequency of
the acoustic wave. This will produce variations in the optical phase which will vary the
speckle pattern produced by the light exiting the medium. Leutz and Maret [42] as well as
Mahan [52] studied these particle displacements and developed mathematical models trying
to provide an explanation of the phenomena. Wang [53] developed a mathematical model
in which he took into account this variation of the index of refraction together with the
contributions to the modulated signal by the particle displacements. Wang’s model is based
on the temporal correlation of the multiple scattered light. This mathematical models will
be reviewed in the following sections.
Diffuse Photon Density Waves Modulation
Another approach to the combination of light and sound is the one developed by Gaudette
[26] in which he proposed that the combination of DPDW and ultrasound will provide a new
way of obtaining information using this technique. This approach is based on the creation
of virtual acoustic generated diffuse photon density waves due to a change in the density
of the medium, at the combination frequencies ω + ωa and ω − ωa where ω is the intensity
modulation frequency of the monochromatic light. This technique considers a DPDW from
source to receiver near the surface as show in Fig. 3.1. The received signal will contain
information about the optical properties of the media along a path parallel to the surface
at small depths. With the use of the ultrasound we expect to receive this virtual diffuse
sources, which due to the quantum noise and DC optical levels, will be significantly small
compared with the pure diffuse optical signal, but will provide information about the paths
that the diffuse waves have traveled, since we know the position of the ultrasound wave, we
can map these paths and provide enhanced resolution of the tissue in study.
This density modulation model is based on the diffusion equation which was stated at the
begining of the chapter, and not in Maxwell’s equations. Therefore, this does not take
CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 32
Figure 3.1: Representation of the interaction of light and sound in scattering media.
into account the coherent characteristic of light which is reasonable since human tissue
is a strongly scattering media with µ′s ≈ 10cm−1 and µa ≈ 0.1cm−1. This mathematical
approach, which is the basis of what we call Acousto-Photonic Imaging, and its implications
will be revised in the next section which is dedicated to the mathematical models dealing
with this phenomena.
3.2 Mathematical Models for Acousto-Photonic Imaging
3.2.1 Acoustic Modulation of the Diffuse Photon Density Waves
Physical considerations
This approach was developed by Gaudette [26] where he postulated that the primary source
of interaction was the modulation of DPDW due to the density modulation of the medium
by the ultrasound. This model is based on the diffusion equation and not in any type of
correlation of the electric fields. It only takes into account the fluence rate. This model
postulates that the interaction of the ultrasound beam with the DPDW is caused by the
CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 33
time-harmonic change of the number of scatterers and absorbers per unit volume. Therefore,
this change in the number of particles produces a change in density, and this implies a change
in volume, where it is assumed that the change in scattering and absorption of the media
is only due to this fact. In consequence, this will produce a change in the photon diffusion
coefficient D = v/(3µ′s + µa).
Considering that the size of the particles does not change considerably in the presence of
the ultrasound it was assumed that
µs = σsN
V(3.2.1a)
µa = σaN
V, (3.2.1b)
where σs and σa are the scattering and the absorption cross sections respectively, N is the
number of particles, and V is the volume.
Since µs and µa are assumed only to be a function of the volume, the change in these
parameters can be expressed as follows
dµs
µs= −dV
V= C (3.2.2a)
dµa
µa= −dV
V= C, (3.2.2b)
where C is the compression factor due to the ultrasound wave. This will produce a change
on the diffusion coefficient expressed by
D =v
3(µ′s + µa)(1− C), (3.2.3)
CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 34
assuming that C2 ¿ 1.
Similarly, the decay rate is affected by this change in volume and would take the form
α = (µa + µaC)v = α0 + Cα0. (3.2.4)
The compression factor C can be modeled as a Gaussian wave since the ultrasound beam
used experimentally is a Gaussian beam.
Development of the Acoustic Generated Diffusion Equation
Based on the previous analysis it is assumed that the DPDW in the presence of the ultra-
sound travels through a change in the diffusion coefficient. Assuming a small perturbation
for this model, this can be described with the following parameters:
Φ = Φ0 + Φ+1 + Φ−1
D = D0 + D1
α = α0 + α1 (3.2.5)
where the subscript 0 corresponds to the original DPDW and the subscript 1 is due to
the particle modulation. The + and − signs refer to the ultrasound modulated DPDW at
frequencies ω+ωa and ω−ωa for Φ+1 and Φ−1 respectively. If we expand the diffusion equation
using Eq. (3.2.5), considering only one sideband at frequency ω +ωa, and neglecting higher
order terms we are left only with an equation in terms of the perturbation Φ+1 and Φ0
CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 35
D0∇2Φ+1 −
∂Φ+1
∂t− α0Φ+
1 + D1∇2Φ0 +∇D1 · ∇Φ0 − α1Φ0 = 0, (3.2.6)
The first three terms of Eq. (3.2.6) look like a normal diffusion equation, therefore, the
second part is treated as a source term. Thus, the diffusion equation for the perturbed
wave is
D0∇2Φ+1 −
∂Φ+1
∂t− α0Φ+
1 =v
nQ, (3.2.7)
which is the basic equation of what we call the virtual acoustic DPWD source. Since the
term Q depends on D1 and α1, which depend on C, Q depends on the Gaussian profile of
the ultrasound beam given in the factor of compression
C = Aeiωate
− (x2+y2)
w20(1+z2/b2)
e
ika(x2+y2)
2(z+b2/z)
e(i tan−1( zb))eikaz (3.2.8)
with A in Eq. (3.2.8) given by
A = −i
√2P
πw20(1 + z2/b2)ρ0v3
a
. (3.2.9)
where P is the acoustic power, ρ0 the density of the medium, va the velocity of sound in
the medium, w0 is the waist radius of the beam and b is the Rayleigh range. The solution
for the diffusion equation for the virtual DPDW source and a complete treatment of this
mathematical model can be found in Ref. [26].
CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 36
3.3 Temporal Light Correlation of Multiple Scattered Light
and its Interaction with ultrasound
Another approach to explain this interesting physical phenomena is the use of the temporal
autocorrelation functions of the scattered light, based on the use of Maxwell’s equations
and probability distribution functions of the scatterers. A basic theoretical model was first
developed by Leutz and Maret [42], in which they took into account the autocorrelation
function of the electric fields defined by
G1(τ) = 〈E(t)E∗(t + τ)〉 =∫ ∞
tP (s)〈Es(t)E∗
s (t + τ)〉ds. (3.3.1)
where Es is the scattered electric field along a path s and P (s) is the distribution function
of the fraction of the incident intensity scattered into paths of length S. For this analysis
they consider Brownian and ultrasonic motion as the principal sources of modulation of
the laser speckle produced by the scattered light. The average field correlation for this two
cases are
〈Es(t)E∗s (t + τ)〉B = exp
(−2τs
τ0l
)(3.3.2)
〈Es(t)E∗s (t + τ)〉U =
⟨exp
−i
s/l∑
j=1
∆φj(t, τ)
⟩. (3.3.3)
Eq. (3.3.3) is the autocorrelation function due to the ultrasonic field along a path with
s/l scatterers (s À l). l is the scattering mean free path of light along succesive scatterers
and s is the length of the scattering path. ∆φj(t) is the phase variation due to the particle
displacements generated by the ultrasound. The expression for this phase variation is
CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 37
s/l∑
j=1
∆φj(t, τ) =s/l∑
j=1
kj(∆rj+1,j(t, τ)−∆rj+1,j(t)), (3.3.4)
where ∆rj+1,j(t, τ) is the distance between two succesive scatterers
∆rj+1,j(t, τ) = (rj + Asin[ka · rj − ωat])
−(rj+1 + Asin[ka · rj+1 − ωat]), (3.3.5)
and kj is the light wavevector after the jth scattering event. Assuming that the phase
change at each scattering event are independent variables with a Gaussian distribution and
transmission of light through a slab of thickness L, Leutz obtained the following expression
for the field correlation function using the procedure as in Ref. [54].
G(t) =
√6
(Ll
2) (
tτ0
+ (k0A)2(1− cos[ωat])α)
sinh
(√6
(Ll
2)(
tτ0
+ (k0A)2(1− cos[ωat])α)) . (3.3.6)
A similar approach, considering only the displacement of the particles due to the ultrasound
is the one developed by Kempe [46].
Leutz does not consider the modulation of the index of refraction, which is considered in
the mathematical model developed by Wang [53], in which he postulates that there is also
a phase change produced by the change of the index of refraction due to the acoustic wave.
This will add a second term in Eq. (3.3.3) and the expression that Wang presents as the
principal one responsible for the interaction between multiple scattered light and sound is
CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 38
the following
〈Es(t)E∗s (t + τ)〉U =
⟨exp
−i
s/l∑
j=1
∆φj(t, τ) +s/l+1∑
j=1
∆φnj(t, τ)
⟩(3.3.7)
where ∆φnj(t, τ) = φnj(t + τ) − φnj(t) and φnj is the phase variation induced by the
modulated index of refraction along the jth free path. This phase variation is
φnj(t) =∫ lj
0k0∆n(rj−1, sj , θj , t)dsj , (3.3.8)
where lj is the length of the jth free path, k0 is the optical wavevector, ∆n is the modulated
index of refraction, rj is the location of the jth scatterer, sj is the distance along the jth
free path, and θj is the angle between the optical wave vector in the scattering event j and
the acoustic wavevector ka.
Wang assumes that the modulated index of refraction is due to an acoustic plane wave
and the piezooptical properties of the medium expressed by η which is related to the
piezooptical coefficient of the material ∂n/∂p, the density ρ and the acoustic velocity va by
η = (∂n/∂p)ρv2a.
After expanding the variance of the phase variation into quadratic and cross terms he
presents the following approximation
⟨−i
s/l+1∑
j=1
∆φnj(t, τ)
2⟩≈ (s/l + 1) (2n0k0A)2δn × [1− cos(ωaτ)], (3.3.9)
CHAPTER 3. ACOUSTO-PHOTONIC EFFECT 39
where δn = (αn1 +αn2)η2, in which αn1 and αn2 are related to the averages done to simplify
the expansions into the quadratic and cross terms. A similar procedure is followed to derive
an approximate expression for the first term in Eq. (3.3.7) which is
⟨−i
s/l∑
j=1
∆φj(t, τ)
2⟩≈ s
l(2n0K0A)2δd[1− cos(ωaτ)], (3.3.10)
where δd = 1/6. Adding these two contributions to the autocorrelation function G1(τ)
Wang obtains the following expression for the autocorrelation function using the procedure
as in Ref. [54].
G1(τ) =(L/l)sinh[ε[1− cos(ωaτ)]1/2]sinh[(L/l)ε[1− cos(ωaτ)]1/2]
, (3.3.11)
where ε = 6(δn + δd)(n0k0A)2. Based on these results, Wang infers that an important
difference between δn and δd is that the average sum of the cross terms of the variations
due to the particle displacements ∆φj vanishes in the diffusion limit (s/l À 1) which means
that the contributions from the displacement by different scattering events are independent
but the contributions from the index of refraction by different free paths are correlated.
The complete derivation of this mathematical model can be found in Ref. [53].
It is important to remark that this mathematical approach assumed the use of continuous
wave light; consequently, there is no presence of a DPDW.
Chapter 4
Numerical Simulations: Monte
Carlo Approach
This chapter covers the numerical simulations performed in order to study the Acousto-
Photonic effect. The first part gives an overview of the Monte Carlo method and its appli-
cation to the simulation of scattered light. The second part studies the specific application
of the Monte Carlo algorithm in the simulation of frequency domain optical techniques. Fi-
nally, the remaining sections of this chapter are devoted to explaining the main goal of this
simulation, which is to combine this frequency domain Monte Carlo with a finite difference
time domain simulation of a focussed ultrasound beam. This will lead to an important tool
that will help us determine the feasibility of this interaction for its future use in medical
imaging.
40
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 41
4.1 Monte Carlo Methods for Multiple Scattered Light Sim-
ulation
The Monte Carlo algorithm is a very useful method for simulating random processes and
in particular, light propagation in tissue. Monte Carlo refers to a technique first proposed
by Metropolis and Ulam to simulate physical processes using a stochastic model [55]. In a
radiative transport problem, the Monte Carlo method consists of recording photon’s travel
histories as they are scattered and absorbed. Monte Carlo programs with great sophis-
tication have been developed for different applications that deal with multiple scattered
light and that take into account the different geometries and boundary conditions that the
specific problem has. The Monte Carlo method is being used by the biomedical research
community to model laser tissue interactions, evaluate scattering and absorption properties,
and in general as a tool to solve inverse problems.
The simulation is based on the random walks that photons make as they travel through
tissue, which are chosen by statistically sampling the probability distributions for step size
and angular deflection per scattering event. After propagating many photons, the net
distribution of all the photon paths yields an accurate approximation to reality.
There are a variety of ways to implement Monte Carlo simulations of light transport. One
approach is to predict steady-state light distributions. Another approach is to predict time-
resolved light distributions. A third approach is to implement Monte Carlo in the frequency
domain to predict amplitude and phase information, which is specially useful when working
with DPDW.
The basic producedure is as follows: after setting up the intial conditions and launching the
photons in the medium, a photon is moved a distance s where it may be scattered, absorbed,
propagated ballistically, internally reflected, or transmitted out of the tissue. The photon
is repeatedly moved until it either escapes from or is absorbed by the tissue. If the photon
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 42
escapes from the tissue, the reflection or transmission of the photon is recorded. If the
photon is absorbed, the position of the absorption is recorded. This process is repeated until
the desired number of photons have been propagated. The recorded reflection, transmission,
and absorption profiles will approach true values (according to the optical parameters of
the tissue being studied) as the number of photons propagated approaches infinity. The
exact formulas and procedures to implement this method are well explained in the literature
[27, 56].
4.2 Frequency Domain Monte Carlo Approach for Diffuse
Optical Tomography
The use of frequency domain techniques for near-infrared spectroscopy has led to new ways
to do numerical computations. Therefore, one of the approaches is to use the Monte Carlo
algorithm to directly predict the modulation and phase for a frequency domain measurement
without storing data related to temporal events. For this purpose, Yaroslavsky [50] reduced
the time-dependent radiative transport equations to a stationary one for the case of a
harmonically modulated radiation source.
This technique avoids the tracking of the time-histories of each individual photon and esti-
mates the quantities relevant to frequency-domain measurements. The propagation of the
photons having a complex weight is simulated in CW-regime and the resulting modula-
tion and phase, respectively, are computed directly. This method reduces the amount of
computational time and decreases the amount of information that needs to be stored.
Basically, if we work with the transport equation as in Eq. (2.2.1), rewritten here in terms
of radiance for convenience
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 43
1v
∂L(r, Ω, t)∂t
= −∇ · L(r, Ω, t)− µtL(r, Ω, t)
+µs
∫
4πL(r, Ω′, t) f(Ω′, Ω) dΩ′ + q(r, Ω, t) (4.2.1)
where the harmonically modulated source q(r, Ω, t) is given by the expression
q(r, Ω, t) = qDC(r, Ω) [1 + <(m0 exp (iωmt))] , (4.2.2)
in which m0 is the incident modulation depth, and ωm is the light source modulation angular
frequency (ωm = 2πfm where fm is the modulation frequency).
Due to the linearity of Eq. (4.2.1) with respect to the radiance L(r, Ω, t), the solution will
have the general form of a wave composed of a DC and an AC component, as was shown
in Eq. (2.2.3) and Eq. (2.2.4), given by
L(r, Ω, t) = LDC(r, Ω) + <(LAC(r, Ω, ω) exp (iωmt)). (4.2.3)
which are written in terms of radiance for the convenience of our formulation.
Once the frequency domain Monte Carlo is implemented the objective is to estimate a
functional for the oscillating radiance LAC over a certain detector area D. This functional
will have the form
JAC(ω) =∫ ∫
DLAC(r, Ω, ω) dr dΩ, (4.2.4)
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 44
Also we can define a functional for LDC of the same form as Eq. (4.2.4). Based on these
expressions we can define in our algorithm complex weights for the photons, leading us to
find directly the information for the amplitude and the phase of the DPDW. The functionals
needed are function of the weight and the number of photons launched into the system and
will have the form
JDC =1N
N∑
h=1
W hLgh, (4.2.5)
JAC =1N
N∑
h=1
ZhLgh, (4.2.6)
where h represents the h-th photon in the system. W hL is the weight for the DC migration
of photons and ZhL is the complex weight for the AC migration of photons which has the
form ZhL = mh
L exp(iϕhL), where mh
L is the intensity modulation factor of the DPDW and
ϕhL is the initial phase of the DPDW. gh is 1 if the h-th photon has reached the detector
or 0 otherwise. L is the number of scattering events that the photons had gone through
before reaching the detector. The values for the initilization of these parameters are L = 0,
W h0 = mh
0 = 1 and ϕh0 = 0.
During the simulation the values of the weights are updated at each scattering event by the
following relations:
W hl = c W h
l−1 (4.2.7)
mhl = c mh
l−1 (4.2.8)
where c = µs/(µs + µa) is the albedo which is approximately constant for the medium.
The phase ϕhl depends on the travel direction of the photon at each scattering event and
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 45
is randomly updated according to the step size and direction of the new scattering event.
In particular, for the Acousto-Photonic effect the harmonic displacements of the particles
due to the ultrasound play an important role in this change of phase. This idea and its
implementation in the Monte Carlo simulation will be addressed in detail in the next section.
4.3 Monte Carlo Simulations for Acousto-Photonic Imaging
In order to simulate the interaction between the sound and the optical fields, we have
implemented this frequency domain approach to the Monte Carlo method, taking into
account the presence of the acoustic field inside the medium. We define our statistical
weights including the effect of the ultrasound as small harmonic displacements about the
rest position of the particles in the medium. These displacements are going to produce
a small change on the optical path lengths of the optical carrier and its sidebands. The
latter are due to the diffuse waves generated at 72.4 MHz. This frequency was chosen by
convenience since it is the same frequency used in the experiments, but we could have used
a different frequency in the MHz range. See Fig. 4.1.
Fig. 4.1 shows the distribution of optical signals from a frequency domain perspective. We
find the optical carrier band ωc in the THz range which is going to produce the DC photon
migration. If we now modulate the power of the optical source with a radio-frequency signal
around the 100MHz range we get what is known as a DPDW, with frequency ωc ± ωm,
where the subscript m stands for modulated. After this, if the ultrasound is present in the
medium at the same time as the light, it is going to modulate the optical signals creating
acoustic generated sidebands at frequencies ωc ± ωa and ωc ± ωm ± ωa. The first set are
the frequencies that the acoustic modulated carrier is going to have and the latter, are the
frequencies that the acoustic modulated DPDW will have. Thus, in the frequency domain
Monte Carlo-Acoustic simulation our goal is to find the amplitude and phase of acoustic
modulated DPDW signals, and if necessary, information about the other optical sidebands
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 46
Figure 4.1: Frequency domain representation of the constitutive sidebands in the interactionbetween multiple scattered light and ultrasound.
as well.
The statistical weight for the API interaction is defined as follows:
WnhJ =J∏
j=1
[Ahj + ahj ] exp(iknShj + i~shj ·∆~knhje
iωat + i~s∗hj ·∆~knhje−iωat
)(4.3.1)
where the capital letters represent static quantities due to the optical field, and the lower
case letters represent variations at the acoustic frequency which in fact, are smaller than
the pure optical quantities. WnhJ is the statistical weight for one of the n = 9 frequency
domain sidebands that are generated in the process necessary to get the API signal. This
weight represents the value of the weight for the photon h at frequency n and scattering
event j. The value of ∆knhj is defined as the vectorial difference between the wave vector
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 47
Figure 4.2: Basic interaction of light, ultrasound and the particles in the medium.
of the incoming photon and its wave vector after the scattering event, which is going to be
random due to the diffuse nature of the process. The dot product between ∆knhj and the
acoustic wave displacement ~shj is responsible for the interaction between the ultrasound
and the scattered light. We can see a graphical representation of this process in Fig. 4.2,
where we show a photon h at frequency n and at scattering event j. The incoming photon
has certain direction kin and after it scatters is going to come out at a direction kout. The
vectorial difference between the wavevectors in these directions gives us ∆knhj , that along
with the ultrasound induced displacement of the particles, is going to give us the change in
the optical path length for the photons in the diffusive medium.
4.3.1 First Order Approximation of the Light-Ultrasound Weight
The phase change due to the ultrasound that produces the API signal is defined as ∆φnhj =
~shj ·∆~knhj . Now, if we expand in Taylor series the exponentials containing the information
about the ultrasound in Eq. (4.3.1), considering only the first order term we have:
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 48
WnhJ ≈J∏
j=1
[Ahj + ahj ] eiknShj(1 + ishj ·∆knhje
iωat) (
1 + is∗hj ·∆knhje−iωat
)(4.3.2)
In order to find a more useful expression for Eq. 4.3.2 we are going to make use of the
following mathematical derivation which has the same structure as the expression for the
statistical weights:
∏
j
[Xj + xj ] = [X1 + x1] [X2 + x2] [X3 + x3] . . .
≈ X1X2X3 · · ·+ [X2X3 . . . ]x1 + [X1X3 . . . ] x2 . . .
=∏
j
[Xj ]
1 +
J∑
j=1
xj/Xj
. (4.3.3)
The terms involving the lower case letters have been neglected since they are going to be
small compared to the mixing between optical (upper case letters) and acoustical contribu-
tions. Reordering the expression in Eq. 4.3.2 we can express the weight with the form of
Eq. (4.3.3) as:
WnhJ ≈
J∏
j=1
Ahj
1 +
J∑
j=1
ahj/Ahj
×
J∏
j=1
exp (iknShj)
1 +
J∑
j=1
(ishj ·∆knhje
iωat + is∗hj ·∆knhje−iωat
) (4.3.4)
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 49
WnhJ ≈J∏
j=1
[Ahj exp (iknShj)]×1 +
J∑
j=1
ahj/Ahj
1 +
J∑
j=1
(ishj ·∆knhje
iωat + is∗hj ·∆knhje−iωat
) (4.3.5)
Assuming that the optical amplitudes are greater that the acoustic amplitudes (Ahj À ahj)
we can drop the term[1 +
∑Jj=1 ahj/Ahj
]since it is approximately 1. Then, the final
expression for the complex weights that include the API interaction would be:
WnhJ ≈J∏
j=1
[Ahj exp (iknShj)]×1 +
J∑
j=1
(ishj ·∆knhje
iωat + is∗hj ·∆knhje−iωat
) (4.3.6)
Eq. (4.3.6) is implemented in the Matlab code listed in Appendix A as the weights defining
the expression for the functional for the DPDW.
For a preliminary model we used a plane ultrasonic wave defined as
~shj = ~sa exp (i ~ka · ~r), (4.3.7)
where ~sa is amplitude of the displacement of the particles due to the acoustic wave. The
amplitude of this displacement was considered 1µm, the ultrasound frequency was fa =
2.4MHz, and the velocity of the ultrasound in tissue was considered va = 1480m/s. The
results of this simulation showed that this basic interaction was possible and are going to
be presented in the next sections. However, in order to get more realistic results a finite-
difference time-domain acoustic simulation was implemented which gave us information of
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 50
the displacement of a focussed ultrasound wave. The details of this simulation will be
reviewed in the next section.
4.3.2 Acoustic-Simulation
The initial Monte Carlo simulations considered a plane acoustic wave to make the assump-
tions simpler. In order to get a more realistic and accurate numerical simulation of the
interaction between multiple scattered light and ultrasound, the pressure field of a focussed
circular transducer was simulated using a two-dimensional axisymmetric finite-difference
time-domain (FDTD) code developed by Manneville (see Ref. [51]). This simulation uses
as a basis the nonlinear propagation equation for an acoustic field inside an absorbing
medium, which is
∇2p− 1va
∂2p
∂t2+
2α
vaω2a
∂3p
∂t3+
β
ρv4a
∂2p
∂t2= 0 (4.3.8)
where p is the acoustic pressure field, va the acoustic velocity, α is the absorption coefficient,
ωa is the acoustic angular frequency, β the nonlinearity coefficient and ρ is the density of
the medium.
Eq. (4.3.8) was solved in cylindrical coordinates (r, z) on a grid with δr = δz = λa/13 and
δt = Ta/100 where Ta is the acoustic wave period. The fluid is supposed to be a homogeneous
medium which, for this particular simulation, used particles of Titanium Dioxide (TiO2)
suspended in water. The fact that the medium is considered homogeneous neglects the
influence of the suspension on the pressure field. The simulation is run for a time long
enough to reach steady state over the whole computational domain.
The pressure on the transducer surface was p = 100KPa. and the particle displacement is
shown in Fig. 4.3. The characteristics of the transducer are f = 2.4MHz, radius r ≈ 1.2 cm
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 51
0
1
2
3
4
5
x 10−8
X direction
Rad
ial d
irect
ion
Displacement in the direction of propagation
200 400 600 800 1000 1200 1400
100
200
300
400
500
600
−3
−2
−1
0
1
2
3
X direction
Rad
ial d
irect
ion
Phase of the acoustic pressure in the focus
50 100 150 200 250 300 350 400
20
40
60
80
100
Figure 4.3: Ultrasound simulation shows the displacement of the particles in the beam andphase variations in the focus.
and focal length fl ≈ 3.5 cm. The transducer surface is S = 5.25x10−4m2. Using the
formula Power = (Sp2)/(ρva) with rho = 998kg/m3 and va = 1480m/s, the acoustic
power is 3.55 watts.
The results of this acoustic simulation provide us with information about the axial and
radial displacement, along with the corresponding phase of the particles. This simulation
also provides information about the streaming of the particles due to the pressure level used
in the simulation.
4.3.3 Monte Carlo-Acoustic Simulation Ensemble
One of the objectives of this work is to combine these two numerical simulations into a single
one. For this purpose the information regarding the axial and radial particle displacement
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 52
(~shj) due to the ultrasound wave was taken into account in the weights defined for the optical
field in Eq. (4.3.6). These displacements are complex numbers with information about the
amplitude and phase of the displacement of the particles due to the acoustic field and were
implemented in the Matlab code together with the Monte Carlo simulation. Assuming that
the ultrasound beam is symmetric with respect to its propagation axis, we can consider the
radial information the same for the azimuthal angle which gives us the possibility to make
a three dimensional simulation. The displacements due to the streaming provided by the
acoustic simulation were not used considering that in our experimental setup we are using
Acrylamide gel phantoms to simulate human tissue. In these gels the particles have fixed
positions. Therefore, the particles of TiO2 are not subject to streaming. This fact will be
explained in Chapter 5.
Figure 4.4: Geometry for the simulation ensemble of multiple scattered light and ultrasound.
In order to explain in detail how the ensemble simulation takes into account the acoustic
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 53
information and the propagation of the light it is important to understand the geometry
of the problem. Fig. 4.4 shows the ultrasound beam in an arbitrary postion in the Monte
Carlo computational grid. The light source launches the photons in the direction specified
by the initial conditions stated at the end of section 4.2. The light source is positioned at
the origin of the coordinate system launching the photons in the +Z direction. In order to
simplify the computations, this simulation uses cartesian and cylindrical coordinates. This is
useful because the random steps of the photons are easily computed in cartesian coordinates
but the calculation of the acoustic interaction is less difficult if cylindrical coordinates are
used. Moreover, the computational time is reduced by combining both coordinates systems.
This simulation gives the possibility to explore the API signal for different positions of the
ultrasound beam with respect to the light source. For these simulations the light source was
positioned off-axis with respect to the ultrasound beam radiating in the positive Z direction.
The simulation considers each photon as an individual element in a row vector and the
information about its position and weight is computed for each scattering event. A three
dimensional array is defined with size equal to the size of the discretized volume that the
ultrasound simulation occupies. This array is initialized with weight 0, since there is no
API signal at the begining. Basically, each time that a photon randomly arrives to one
voxel of the computational grid, it contributes to the complex weight of that particular
voxel, giving information about the amplitude and the phase change which is stored in the
corresponding position of the three dimensional array. This volume array is shown in Fig.
4.4 where the red lines illustrate the trayectories that the photons will follow during the
simulation and the box in the middle illustrates the position of the ultrasound focus with
respect to the light source. It is important to mention that the photons were assumed to
travel in an infinite turbid media with defined optical characteristics, but only the weight
histories of the photons that interacted with the ultrasound were stored. This reduces the
amount of data and the computational time that we have to deal with.
Fig. 4.5 shows the flow diagram for the ensemble simulation. The optical parameters con-
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 54
sidered for this grid were: reduced scattering coeffcient µ′s = 10cm−1, absorption coefficient
µa = 1cm−1 and anisotropic coefficient g = 0.9. The wavelength λ of the light source is 690
nm with the power output modulated at 72.4MHz. N is the number of scattering events
and is increased until we get stable results as we update the statistical weights for the DC,
DPDW and API signals.
Figure 4.5: Flow diagram of Monte Carlo-Acoustic Simulation.
4.3.4 Discussion and Results
The results of this numerical simulations are presented in this section. Once the information
of the weights for the different sidebands is computed, the data is presented as a two
dimensional figure or as the amount of generated signal with respect to the continuous
wave value, captured by an optical detector.
For the case of the 2-D figure, depending on the position of the ultrasound, we averaged
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 55
all the non zero voxels of the three dimensional weight array in a direction perpendicular
to the ultrasound propagation. We can consider each voxel of the computational grid as a
single detector so that the 2-D figure will represent an snapshot of an array of detectors.
This shows how the diffuse wave gets modulated by the ultrasound. The amplitude of the
wave is presented decibels.
Fig. 4.6 presents the results for the very basic simulation of a plane ultrasound wave
and light in a 2x2x2 cm computational grid with voxel size of 100x100x100 µm. For this
simulation 106 photons were used. The optical properties of the tissue were µ′s = 10cm−1
and µa = 0.1cm−1. The number of photons in the Monte Carlo simulation is very small
for the quantity of photons needed to have results close to the real physical interaction,
however, this gave an insight that we were in the right track.
−20
−10
0
10
20
30
40
50
60
70
80
X direction (cm)
Y d
irect
ion
(cm
)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1−3
−2
−1
0
1
2
3
X direction (cm)
Y d
irect
ion
(cm
)
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 4.6: Amplitude and phase modulation of diffuse light interacting with a plane ultra-sound wave in scattering media. The ultrasonic wavelength (≈ 640µm), is well defined andmodulates the optical path lengths.
The next step was to include the information about the particle displacement provided by
the acoustic simulation which is shown in Fig. 4.3. In order to use all the information we
included the axial as well as the radial displacement of the particles with the same optical
parameters in the medium. 4x107 photons were used. The last version of the code has a
computational time of approximatelly 30 minutes for 106 photons in a Pentium IV 1.9 GHz.
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 56
−100
−90
−80
−70
−60
−50
−40
X axis (cm)
Y a
xis
(cm
)
Magnitude and Phase of Modulated Diffuse Wave − 40 million photons
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
−3
−2
−1
0
1
2
3
X axis (cm)
Y a
xis
(cm
)
Phase
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
Figure 4.7: Amplitude and phase modulation of diffuse light interacting with a focussedultrasound wave in scattering media.
Figure 4.7 presents the results for the simulation using 4x107 photons. We have found that
as we increase the number of photons in the media the generated API signal gets weaker.
One possible explanation of this fact is that the increase in the number of photons increase
the probabilty of photons being scattered in random ways, which in terms of the radiance,
increases the amount of DC light in the system. Another possibility is that the displacement
of the particles due to the ultrasound is not big enough to mantain the signal characteristics.
The process of generating the signal by increasing the number of photons in the simulation
is shown in Figs. 4.8, 4.9, 4.10 and 4.11 where 1, 5, 10 and 20 million photons were
used respectively. A decrease in specificity of the pattern is observed but the amplitude
information shows that the optical signal is increasing in that part of the medium where the
ultrasound is located. The levels of the signal are 60 dB and below this value with reference
to the DC signal which is assumed to be 0 dB. Another characteristic of this simulation is
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 57
that the figures are showing the interaction inside the ultrasound beam.
−130
−120
−110
−100
−90
−80
−70
−60
X axis (cm)Y
axi
s (c
m)
Magnitude dB
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
−3
−2
−1
0
1
2
3
X axis (cm)
Y a
xis
(cm
)
Phase
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
Figure 4.8: Simulation with 1 million photons.
−120
−110
−100
−90
−80
−70
−60
X axis (cm)
Y a
xis
(cm
)
Magnitude dB
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
−3
−2
−1
0
1
2
3
X axis (cm)
Y a
xis
(cm
)
Phase
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
Figure 4.9: Simulation with 5 million photons.
In order to get more realistic results, instead of computing and averaging the 3D com-
putational grid, we considered a semi-infinite media in which the photons traveled in the
presence of the ultrasound. We assumed a source at position 0 and a detector fiber at the
boundary with collection area 1mm2. The signal strengths were computed for the CW,
DPDW, and API signals for different distances between the source and detectors providing
similar results to those previously shown in Figs. 4.8 4.9 4.10 and 4.11. For this simulations
2x107 photons were used and the results are shown in Fig. 4.12 and Fig. 4.13.
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 58
−120
−110
−100
−90
−80
−70
−60
−50
X axis (cm)
Y a
xis
(cm
)
Magnitude dB
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
−3
−2
−1
0
1
2
3
X axis (cm)
Y a
xis
(cm
)
Phase
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
Figure 4.10: Simulation with 10 million photons.
−120
−110
−100
−90
−80
−70
−60
−50
X axis (cm)
Y a
xis
(cm
)
Magnitude dB
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
−3
−2
−1
0
1
2
3
X axis (cm)
Y a
xis
(cm
)
Phase
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2
0.4
0.6
Figure 4.11: Simulation with 20 million photons.
Fig. 4.12 shows the signal strength of the pure optical diffuse wave with respect to the
original modulated signal. The strength of the modulated optical source was normalized
to 1 in order to show the decrease in the modulation depth on the DPDW signal as we
increase the separation the source and detector optical fibers. These results were fitted
to the analytical solution of the diffusion equation considering the appropiate boundary
conditions for a semi-infinite medium [22]. Fig. 4.13 shows the signal strength of the
acoustic generated diffuse wave (API signal) with respect to the modulated optical source.
We can see that the levels of the API signals are at least 60 dB lower than the pure optical
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 59
DPDW signals which are set to 0 dB. The values of these signals were computed using
Eq. (4.2.5). Since the model developed by Gaudette [26] does not consider a semi-infinite
medium for the solution of the diffusion equation, a decaying exponential function was fitted
to the values computed by the numerical simulation in order to show its decaying behaviour.
This exponential function has the same form as the analytical solution used for the results
on Fig. 4.12.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Distance between source and detector (cm)
Mod
ulat
ion
dept
h
Figure 4.12: Signal levels of the DPDW signal with respect to the CW signal.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
−6
Distance between source and detector (cm)
Mod
ulat
ion
dept
h
Figure 4.13: Signal levels of the API signal with respect to the CW signal.
One of the disadvantages of the Monte Carlo method is the lack of efficiency in terms of
computational time. In order to improve this efficiency we worked in parallel with experts
CHAPTER 4. NUMERICAL SIMULATIONS: MONTE CARLO APPROACH 60
in the field of Computer Engineering at Northeastern University, achieving promissing re-
sults in reducing the computational times for this simulation. The simulation times were
reduced by approximately 90 %. See the work done by Ashouei in Ref. [57]. This opti-
mization was based on the parallelization of the code originally written in Matlab. This
code was converted to C and the time-consuming functions of Matlab were re-written and
optimized for the calculations that the Monte Carlo Simulation needed. The fact that the
computational optimization was done for general purpose functions within matlab and for
algorithms commonly used in computer science makes possible to use this computational
techniques in the simulation of different physical processes. A detailed explanation can be
found in Ref. [57].
Chapter 5
Experimental Methods
This chapter presents the two types of experiments performed in this project to study
the interaction between multiple scattered light and ultrasound. In the first setup the
modulation of the speckle pattern due to the ultrasound is investigated using a continuous
wave light source. Measurements of the change of the speckle contrast with and without
the ultrasound were done. In the second set of experiments the modulation of the DPDW
was studied using a power modulated laser source and lock-in measurement techniques.
5.1 Laser Speckle Measurements
One of the most important effects when diffuse light is in the presence of an ultrasound
beam is the change of the speckle pattern [47]. One parameter that can be measured is the
speckle contrast [58] which is defined as
SC =σI
〈I〉 , (5.1.1)
61
CHAPTER 5. EXPERIMENTAL METHODS 62
where σI is the standard deviation of the speckle pattern and I is the intensity. Even though
the detection of the modulation of a single speckle is extremely difficult, the calculation of
the speckle contrast, and the decrease in its value when the ultrasound is present, shows
that this speckle pattern is being changed by acoustic field.
5.1.1 Setup
The setup for these measurements is as shown in Fig. 5.1. We are using a 633 nm Helium-
Neon Melles-Griot laser with 15 mW of optical power. The detector is an 8-bit NEC
TI-23EX CCD camera, 484x512 pixels of resolution. The ultrasound transducer is manu-
factured by NTD Systems and driven at a frequency of 2.3MHz. The diameter is 1 inch
and has a spherical focus of 1.5 inches.
Figure 5.1: Setup for Speckle Contrast measurements.
The modulation for the transducer is provided by the ENI power amplifier model 325LA.
The measured pressure response vs. the input voltage in the amplifier for the transducer
used in our experiments is presented in Fig. 5.2. The usual pressure used in these experi-
CHAPTER 5. EXPERIMENTAL METHODS 63
ments is 1 MPa, and the diameter of the beam focus is approximately 1.4λa. Therefore the
power per square centimeter in the focus is 67.7W/cm2, which is higher than the maximum
safety value given in Table 2.1. Once a complete understanding of this phenomena is ob-
tained, the interaction should be optimized in order to reduce these values to the diagnostic
levels.
0 10 20 30 40 50 600
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Pre
ssur
e (M
Pa)
Voltage ENI (Vpp)
Pressure at the transducer focus
Figure 5.2: Pressure at the focus of the ultrasound vs voltage supply.
The output of the CCD camera is connected to a Cortex-1 frame grabber manufactured
by Imagenation Inc. and redirected to a TV screen. The combination between the fo-
cussed ultrasound beam and the light takes place in an acrylamide gel phantom which has
titanium dioxide particles as scatterers. The use of these phantoms is useful because its
acoustic impedance is matched with that of the water and therefore, close to human tissue
characteristics. The ultrasound transducer and the gel phantom are immersed in water for
better impedance coupling.
CHAPTER 5. EXPERIMENTAL METHODS 64
pixels
pixe
ls
Ultrasound OFF
50 100 150 200 250 300 350 400 450
50
100
150
200
250
300
350
400
pixels
pixe
ls
Ultrasound ON
50 100 150 200 250 300 350 400 450
50
100
150
200
250
300
350
400
Figure 5.3: Speckle pattern with and without the prescence of the ultrasound. Notice thebluriness of the image on the right (ultrasound on) with respect to the one on the left(ultrasound off).
5.1.2 Experiments and Results
The measurements of the speckle contrast were done taking sets of 10 images with the
frame grabber and evaluating Eq. (5.1.1). We observed a decrease on the speckle contrast
when the ultrasound was present and was somewhat noticiable in the TV screen for forward
scattering and the CCD detector camera off-axis of the laser beam.
Fig. 5.3 shows the speckle pattern as was seen on the TV screen. The picture on the
left was obtained without the presence of the ultrasound while the one on the right was
obtained with the presence of the ultrasound. This data was obtained by capturing sets of
5 images for each case with the CCD camera and the frame grabber. The change on speckle
contrast (SC) calculated for this set, averaging the corresponding images, was SCon =
0.2010 and SCoff = 0.2130 where SCon stands for speckle contrast with the ultrasound on
and SCoff stands for speckle contrast with the ultrasound off. This represents a change
of approximatelly 2%, which agrees with the percentage change observed by Li et.al. [59].
The picture on the right shows that when the ultrasound is present the speckle pattern gets
blurred. This is mainly due to the change of the optical path lengths that the photons have
CHAPTER 5. EXPERIMENTAL METHODS 65
to travel to exit the scattering media while the ultrasound is harmonically modulating the
particle positions.
The values for similar experiments are tabulated in Table 5.1 and shown in Fig. 5.4. We
can notice from Fig. 5.4 that the amount in the change of the value of the speckle contrast
is approximately constant for all the measurements. In Fig. 5.4 the vertical axis shows the
value of the speckle contrast calculated for pairs of images with and without the presence
of the ultrasound. The experiments consisted in obtaining sets of 50 pictures. An element
of each set is a pair of images of the speckle pattern with and without the ultrasound. Fig.
5.4 show 8 sets obtained in the experiments.
In Fig. 5.4 the surface on top represents the speckle contrast measurements without the
presence of the ultrasound and the surface in the bottom with the presence of the ultrasound
for 8 sets of pictures. The values obtained in Table 5.1 were obtained averaging the 50
images obtained from the frame grabber. Also for these experiments the position of the
ultrasound was optimized to get the greater change in speckle contrast obtaining a change
up to approximately 12% in some of the cases. This shows us that there is an effective
change in the optical signal due to the ultrasound.
SC UltrasoundOFF
SC UltrasoundON
Percentage ofchange %
0.4372 0.3821 12.600.3164 0.2819 10.880.5046 0.4469 11.430.4835 0.4283 11.410.4561 0.4178 08.400.4340 0.3885 10.480.3584 0.3212 10.400.3112 0.2644 15.05
Table 5.1: Tabulated data for speckle contrast
CHAPTER 5. EXPERIMENTAL METHODS 66
010
2030
4050
12
34
56
78
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Number of pictures per setNumber of sets
spec
kle
cont
rast
Figure 5.4: Speckle contrast for various sets of data.
5.2 Acoustic Modulated Diffuse Photon Density Waves
On the other hand, based on the idea developed by Gaudette [26] we conducted experiments
using a modulated diode laser to create a DPDW in the acrylamide phantoms and combine
them with the ultrasound beam. This process is complicated due to the high scattering in
the medium which attenuates the virtual diffuse waves that are created from this combina-
tion. In the next sections we present a detailed description of the experimental setup and
the experiments conducted to pursue this optical phenomena.
5.2.1 Setup
The setup is shown as a block diagram in Fig. 5.5 and uses a 690nm diode laser manufac-
tured by Melles Griot with approximately 25 mW of output power. The electronic setup is
more elaborate than the one to measure the speckle contrast.
As a detector we use the H6573 modulated photomultiplier tube module manufactured
CHAPTER 5. EXPERIMENTAL METHODS 67
Figure 5.5: Setup for DPDW experiments.
by Hamamatsu. For this experiment we need to use three frequency generators. The first
generator at a fixed frequency of 72.3 MHz drives the modulation in the diode laser to create
the DPDW and also gets combined with a fixed frequency generator at 70 MHz, from which,
after passing through a low pass filter stage, we get the 2.3 MHz frequency needed to feed
the power amplifier for the ultrasound transducer. These two fixed frequency generators
are manufactured by Wilmanco and have a power output of +17 dBm. The third generator
is a Hewlett Packard 8647A variable signal generator which is operated at 70.025 MHz and
feeds the Minicircuits power amplifier with a gain of +33 dBm which is connected to the
photomultiplier, modulating the second dynode voltage supply. This modulation of the
photomultiplier allows us to down-convert the API signal coming out from the interaction
between the scattered light and the ultrasound which is 70 MHz. Thus, this 70 MHz is
mixed with the 70.025 MHz giving out a 25KHz signal which can be handled by the final
stages of the photomultiplier since these do not have a good frequency response. Again the
CHAPTER 5. EXPERIMENTAL METHODS 68
ultrasound and the light get combined in an acrylamide gel phantom.
The output of the photomultiplier is conected to a transimpedance amplifier which is
conected to a Lock-in amplifier. The reference signal for the Lock-in is obtained from
the mixing of the 70 MHz and the 70.025MHz signals.
5.2.2 Experiments and Results
Different experiments were conducted in order to obtain the desired effect of combining
DPDW and ultrasound in highly scattering media. At this stage of the research we have
not been able to obtain the desired interaction between the DPDW and the ultrasound
but we are still making efforts to optimize the setup. We have considered a number of
improvements that may have to be done.
We have to perform changes in the position of the ultrasound beam with respect to the light
source in order to optimize the position in which we would have the maxium interaction
between the two fields. Since the expected API signals are weak compared to the original
DPDW this is a disadvantage for our purposes. The Monte Carlo simulation will help us to
solve this problem since we have the flexibility to position the light source in any position
near the ultrasound beam.
The detection system, although we are using a photomultiplier tube which gives us more
gain, also gives us more noise in the low frequencies, and since we are down-converting the
signals to the KHz range, this is a disadvantage for the detection process since the API signal
that we are expecting is very small. A solution has been suggested by our collaborators
at Boston University which involves the use of a Photorefractive Crystal (PRC) which will
work as a phase conjugator of the the signal buried into the speckle pattern. We will be
able to combine the diffuse radiance coming of from the turbid media with a reference beam
inside the PRC in order to generate phase conjugate waves that will enhance the levels of
the expected signal.
CHAPTER 5. EXPERIMENTAL METHODS 69
On the other hand, we do not have a complete characterization of the optical properties
of the gel phantom. Although it gives us good acoustical properties that resemble those of
human tissue, we can not infere that the optical properties are optimized to match those of
biological tissue.
Chapter 6
Conclusions and Future Work
In this thesis we have presented a numerical simulation based on the Monte Carlo method
with the addition of an acoustic simulation that will help us determine the feasibility of the
interaction between diffuse photon density waves and ultrasound.
This numerical simulation is very flexible and provides us with different ways to examine
the relative positions between the ultrasound and the light source. We can also use it
considering a single detector in order to find information about the amplitude and phase
of the diffuse wave in fixed points within the medium. This simulation considers a semi-
infinite medium and the reflection of the photons in tissue air interface. However, it still has
parameters that have not been considered. We have to take into account the elimination
of photons when the statistical weights are small. This is a secondary problem since the
value of these weights approach to zero as the number of scattering events increase, but it
will help to reduce the computational time of the simulation. The results show that the
interaction between ultrasound and diffuse photon density waves, although is very weak
and around 70 dB less than the pure optical signals, is possible.
Another situation that has to be studied is what happens when the acoustic modulated
diffuse wave leaves the ultrasound beam. As we can see the interaction is small and we can
70
CHAPTER 6. CONCLUSIONS AND FUTURE WORK 71
not assure that when the diffuse wave leaves the region of interaction it is going to remain
strong enough to come out from the medium since these diffuse waves are highly damped.
Also, with the use of these simulations we need to obtain signal levels for all the optical
sidebands in order to compare the strength of the API signal with the common DPDW
and DC photon migration. We expect the API signal to be small compared to the other
two, but the numerical simulations will help us to determine if the API signal is within the
range in which we would be able to detect it or it will be buried in the optical noise and
the remaining stronger signals. Also, we can specify different modulation frequencies either
for the light source and the ultrasound in order to optimize the devices that we should use
with this technique.
Current research shows that the speckle modulation phenomena is one of the most impor-
tant characteristics of the ultrasound modulated optical tomography. The development of
mathematical models with more insight in the acoustic part has to be considered in order
to have full understanding of the medium in which these diffuse waves are going to travel
and how the medium is going to affect their behavior.
A variety of experiments have been performed with the objective of modulating the DPDW,
and even though we have not yet been able to obtain optimum results there is still work
to be done. We need to optimize the optical properties of the medium in order get our
phantoms to resemble human tissue. In parallel, the numerical simulations as well as the
mathematical models will help us to determine which way to follow in order to obtain the
maximum benefits from this imaging technique.
We can implement coherent detection systems using a chopper and averaging the signals
that contain the information of the interaction between DPDW and ultrasound. This will
help us to increase the SNR. On the other hand, our collaborators at Boston University are
working using a Photorefractive Crystal to collect the coherent information of the diffuse
wave modulated at 2.3 MHz by a pulsed focused ultrasound beam.
Appendix A
Matlab Monte Carlo-AcousticSimulation code
% Monte-Carlo program for API%% by Alex Nieva - Chuck DiMarzio,% Optical Science Laboratory% Northeastern University, August 2001%%%% Coordinate axis%% ------------------> X% |% |% |% |% |% |% Z%%%% NP is the number of photons launched in the system%%
ticglobal CHANCE CRIT_ANG
72
APPENDIX A. MATLAB MONTE CARLO-ACOUSTIC SIMULATION CODE 73
CHANCE = 0.1;CRIT_ANG = 0.8509;
rand(’state’,sum(100*clock));s=load(’disp5’);acouwave = s.dis_z; acourad = s.dis_r;clear s
% setup parameterslambda = 0.69e-4; % optical wavelength (cm)fm = 72.4e6; % modulation frequencyc = 2.99e10; % cm/sg = 0.9; % Anisotropyus = 100; % Scattering coefficient cm^-1ua = 0.1; % Absorption coefficient cm^-1ut = ua + us; % Total interaction coefficientalbedo = us/ut;alph = 2.5e-3;
% Calculated parametersf0=c/lambda; % optical carrierk0=2*pi*f0/c;fm1=f0-fm; % lower sidebandkm1=2*pi*fm1/c;fp1=f0+fm; % upper sidebandkp1=2*pi*fp1/c;
% Grid sizearraystep=5e-3; % cm (50 microns)
% Definition of volume of interaction with ultrasoundminx = 0; miny = -1.5; minz = 0;maxx = 3; maxy = 1.5; maxz = 4;
% Initial position and direction - x,y,z is position,% u,v,w direction% x(i) is x coordinate of photon number i, etc.% NP is number of photons for the simulation.NP = 100000; N = NP;x = zeros(NP,1);y = x;z = x;u = zeros(size(x));
APPENDIX A. MATLAB MONTE CARLO-ACOUSTIC SIMULATION CODE 74
v = zeros(size(x));w = ones(size(x));
% Initial photon weight for 9 frequency bandsthe = zeros(size(x));dpp10 = exp(i.*the);
%weightm1m1 = zeros(size(x));%weightm10 = 0.5*ones(size(x));%weightm1p1 = zeros(size(x));
%weight0m1 = zeros(size(x));weight00 = ones(size(x));%weight0p1 = zeros(size(x));
weightp1m1 = zeros(size(x));weightp10 = ones(size(x));%weightp1p1 = zeros(size(x));
weightamp = albedo; %sqrt(albedo);%factor to reduce weight as field interactionweightampdpdw = albedo/sqrt(1+alph^2);
% weightamp = albedo; factor to reduce weight as power interaction% Power --> Wnew = Wold*albedo;% Field --> Wnew = Wold*sqrt(albedo);
% Photon step size using Prahl et. al.
s = photonstep(x,ut);
% move (first step)xold = x; yold = y; zold = z;x=x+s.*u; y=y+s.*v; z=z+s.*w;
offsetx = (maxx + minx)/2;offsety = (maxy + miny)/2;offsetz = minz;
% Loop on scattering events. Scattering events last until% all the photons had gone outside of the volume of interacion or% had died due to ruse roulette.
n = 0; %counter for # of iterations
APPENDIX A. MATLAB MONTE CARLO-ACOUSTIC SIMULATION CODE 75
modDC = 0; modAC = 0; modAPI = 0;
for n=1:800%while N >= 1,
number_of_photons = Nn = n + 1
% New directions[unew,vnew,wnew] = deflect(g,x,u,v,w);deltak = (2*pi/lambda)*[sqrt((unew-u).^2+(vnew-v).^2),(wnew-w)];u = unew; v = vnew; w = wnew;clear unew vnew wnew
% Here we are going to find the box in which we have to% put the weights% We also find out which of the photons are inside the% area of interaction% for the pure optical signalsinx = find(x<maxx & x>minx);iny = find(y<maxy & y>miny);inz = find(z<maxz & z>minz);photonin = intersect(intersect(inx,inz),iny);
clear inx iny inz
% Here we find the photons that have reached the detector.% match is the variable that has the position in the vector of photons% of the photons that have crossed the boundary.match = find(z<0);if length(match)~=0
[a,b] = check_detector(match,s(match),u(match),v(match),...z(match),xold(match),zold(match),yold(match));
modDC = modDC + sum(weight00(a));modAC = modAC + sum(weightp10(a));
modAPI = modAPI + sum(weightp1m1(a));
weight00(a) = 0;weightp10(a) = 0;
weightp1m1(a) = 0;
[w_new,ref,trans] = fresnel(b,w(b));weight00(trans) = 0;
weightp10(trans) = 0;weightp1m1(trans) = 0;w(b) = w_new;
APPENDIX A. MATLAB MONTE CARLO-ACOUSTIC SIMULATION CODE 76
new_s = (s(ref).*zold(ref))./(zold(ref) - z(ref));z(ref) = w(ref).*(s(ref)-new_s);
end
clear match new_s w_new ref trans a b
posr = ceil(sqrt((x-offsetx).^2 + (y-offsety).^2)./arraystep);posz = ceil(z./arraystep);
auxr = find(posr==0); posr(auxr) = 1; clear auxrauxr = find(posr>300); posr(auxr) = 300; clear auxrauxz = find(posz==0); posz(auxz) = 1; clear auxzauxz = find(posz>300); posz(auxz) = 300; clear auxz
% acoustic interactiondeltasa = zeros(NP,2);for h = 1:length(photonin)
deltasa(photonin(h),1) = acourad(posz(photonin(h)),...posr(photonin(h)));
deltasa(photonin(h),2) = acouwave(posz(photonin(h)),...posr(photonin(h)));
endclear posr posz
% recalculate weight%dpm1m1 = exp(i*km1*s).*(-i*dot(deltak,deltasa,2));%dpm10 = exp(i*km1*s);%dpm1p1 = exp(i*km1*s).*(i*dot(deltak,deltasa,2));
%dp0m1 = exp(i*k0*s).*(-i*dot(deltak,deltasa,2));dp00 = exp(i*k0*s);%dp0p1 = exp(i*k0*s).*(i*dot(deltak,deltasa,2));
%dpp10 = exp(i*kp1*s);the = atan((sin(the) - alph.*cos(the))./...
(cos(the) + alph.*sin(the)));dpp10 = exp(i.*the);
dpp1m1 = dpp10.*(-i*dot(deltak,deltasa,2));%dpp1p1 = exp(i*kp1*s).*(i*dot(deltak,deltasa,2));
%weightm1m1 = weightm1m1 + weightm10*weightamp.*dpm1m1;%weightm1p1 = weightm1p1 + weightm10*weightamp.*dpm1p1;%weightm10 = weightm10*weightamp.*dpm10;
%weight0m1 = weight0m1 + weight00*weightamp.*dp0m1;
APPENDIX A. MATLAB MONTE CARLO-ACOUSTIC SIMULATION CODE 77
%weight0p1 = weight0p1 + weight00*weightamp.*dp0p1;weight00 = weight00.*weightamp;%.*dp00;
weightp1m1 = weightp1m1 + weightp10.*weightamp.*dpp1m1;%weightp1p1 = weightp1p1 + weightp10*weightamp.*dpp1p1;weightp10 = weightp10.*weightampdpdw.*dpp10;
% ruse rouletteweight00 = roulette(abs(weight00));
N = length(find(weight00));
% Calculate next step size of photonss = photonstep(x,ut);
% New position of the photonsxold = x; yold = y; zold = z;
x=x+s.*u; y=y+s.*v; z=z+s.*w;
clear r h photonin aux deltasaend;
time = tocsave(’name’,’modDC’,’modAC’,’modAPI’,’time’)clear
% Calculation of the photon step at each scattering% event%%% Usage:% s = photonstep(x,ut)% where:% x is used to let the function know the number of photons,% and ut is the total interaction coefficient ut = us + ua.%% Original code in standard C by Wang-Jacques.% "Monte Carlo Modeling of Light Transport in% Multi-layered Tissues in Standard C" 1995
function s = photonstep(x,ut)
s = -log(rand(size(x)))/ut;
APPENDIX A. MATLAB MONTE CARLO-ACOUSTIC SIMULATION CODE 78
return
% Calculation of the deflection angle THETA and% PSI for photon scattering, as well as the new% direction cosines.% When g = 0, we aproximate cos(theta)=2*rand -1% When g > 0, we use the Henyey-Greenstein function.%% Usage:% [unew,vnew,wnew] = deflect(g,x,u,v,w)% where:%% unew,vnew,wnew are the new direction cosines% and u,v,w are the old ones.% g is the anisotropy and we use x just to let the% function know the size of the matrix that has the% photons.%% Original code in standard C by Wang-Jacques.% "Monte Carlo Modeling of Light Transport in% Multi-layered Tissues in Standard C" 1995
function [unew,vnew,wnew] = deflect(g,x,u,v,w)
if g==0costheta = 2*rand(size(x)) - 1;
elseaux = ((1-g^2)./(1-g+2*g*rand(size(x)))).^2;costheta = (1+g^2-aux)/(2*g);
end
psi = 2*pi*rand(size(x));
sintheta = sin(acos(costheta));sinpsi = sin(psi);cospsi = cos(psi);
if min(abs(w))>0.9999 % In the future we have to consider% each photon at a time.
unew = sintheta.*cospsi;vnew = sintheta.*sinpsi;
APPENDIX A. MATLAB MONTE CARLO-ACOUSTIC SIMULATION CODE 79
wnew = sign(w).*costheta;else
den = sqrt(1 - w.^2);coef = ((sintheta)./den);unew = coef.*(u.*w.*cospsi-v.*sinpsi) + u.*costheta;vnew = coef.*(v.*w.*cospsi+u.*sinpsi) + v.*costheta;wnew = -sintheta.*cospsi.*den + w.*costheta;
end
return;
% Function to check if the photon has reached the% detector and in what iteration.% The output is the position in the vector of photons,% of the photons that have reached the detector.
function [a,b] = check_detector(match,s,u,v,z,xold,zold,yold)
detxmin = 1.3; detxmax = 1.4;detymin = -0.05; detymax = 0.05;radius = (detxmax - detxmin)/2;
new_s = (s.*zold)./(zold - z);xzplane = xold + u.*new_s;yzplane = yold + v.*new_s;
rzplane = sqrt((xzplane - (detxmax - detxmin)/2).^2 ...+ (yzplane - (detymax - detymin)/2).^2);
aa = find(rzplane<radius);bb = find(rzplane>=radius);a = match(aa); b=match(bb);
return
% Internal reflection of photons% using Fresnel reflection coefficient
function [w_new,ref,trans] = fresnel(b,w_new)
global CRIT_ANG
APPENDIX A. MATLAB MONTE CARLO-ACOUSTIC SIMULATION CODE 80
% This part finds which photons have been internally reflected% because of the critical angle.R = zeros(size(b));phi_i = acos(abs(w_new));aux_i = find(phi_i >= CRIT_ANG);R(aux_i) = 1;
aux_t = find(phi_i < CRIT_ANG);aux_phi_i = phi_i(aux_t);phi_t = asin(1.33.*sin(aux_phi_i));
aux_R = 0.5*((sin(aux_phi_i - phi_t).^2)./(sin(aux_phi_i + phi_t).^2) + ...(tan(aux_phi_i - phi_t).^2)./(tan(aux_phi_i + phi_t).^2));
R(aux_t) = aux_R;
test = rand(size(R));aux_test = find(test>R);R(aux_test) = 0; % This means that the photon escaped the tissue.R = ceil(R); % R = 1 when the photon is internally reflected.
% 0 otherwise.reflected = find(R); transmitted = find(R==0);
w_new(reflected) = -w_new(reflected);ref = b(reflected); trans = b(transmitted);return
% This is the russian roulette algorithm to kill% the photons that have a weight lower that the% pre-defined CHANCEfunction new_weight=roulette(weight)
global CHANCE
a = find(weight<0.001 & weight~=0);rand_num = rand(size(a));b = find(rand_num>CHANCE);rand_num(b) = 0;weight(a) = weight(a).*rand_num;new_weight = weight;
return
Appendix B
Transport Theory
The angular photon density was denoted with u(r, Ω, t), and the BTE written as follows:
∂u(r, Ω, t)∂t
= −v Ω · ∇u(r, Ω, t)− v(µa + µs)u(r, Ω, t)
+ vµs
∫
4πu(r, Ω′, t) f(Ω′, Ω) dΩ′ + q(r, Ω, t), (B.0.1)
The BTE can also be written in terms of the radiance L(r, Ω, t) = vu(r, Ω, t)
1v
∂L(r, Ω, t)∂t
= −∇ · L(r, Ω, t)− µtL(r, Ω, t)
+µs
∫
4πL(r, Ω′, t) f(Ω′, Ω) dΩ′ + q(r, Ω, t), (B.0.2)
where µt = µa +µs, is the total interaction coefficient or transport coefficient, and the otherquantites remain as defined before. The photon fluence rate is given by:
Φ(r, t) =∫
L(r, Ω, t) dΩ. (B.0.3)
The photon flux, or current density, is given by
J(r, t) =∫
L(r, Ω, t) Ω dΩ. (B.0.4)
81
APPENDIX B. TRANSPORT THEORY 82
The derivation of the diffusion approximation is well documented in many other works(see Ref. [5, 19]). Basically, the method used for this approximation is know as the PN
approximation [18]. This method is based in the expansion of the radiance, phase function,and source term in spherical harmonics Yl,m, which is a mathematical tool that is based onthe solutions of the general Helmholtz’s equation. For the case of human tissue, the albedo,which is defined as W = µs/(µs + µa), is close to unity, therefore the P1 approximation isquite good and this lead us to the standard diffusion equation given by
−D∇2Φ(r, t) + vµaΦ(r, t) +∂Φ(r, t)
∂t= vq0(r, t), (B.0.5)
or in terms of the photon density
−D∇2U(r, t) + vµaU(r, t) +∂U(r, t)
∂t= q0(r, t), (B.0.6)
where D = v/(3µ′s + µa) is the photon diffusion coefficient and µ′s is the reduced scatteringcoefficient defined as µ′s = µs(1 − g), which is defined because of the usefulness of thisquantity in the mathematical derivations for the diffusion approximation. The scatteringcoefficient and the scattering anisotropy do not explicitly appear in the P1 equation northe diffusion equation but instead appear together as the reduced scattering coefficient.q0(r, t) is the monopole (isotropic) moment of the source. We can also express the diffusioncoefficient as D = v/(3µ′s) considering that µ′s À µa.
The standard photon diffusion equation is obtained assuming an isotropic source which istrue also for collimated sources displaced one mean free path into the scattering media fromthe collimated source.
We also can write the photon diffusion equation (see Eq. (B.0.6)) in the frequency domainwhich takes the form of the Helmholtz equation
(∇2 + k2)U(r) =
−q0(r)D
, (B.0.7)
where the wavenumber is complex and takes the form
k2 =−vµa + iω
D. (B.0.8)
The solution for Eq. (B.0.7) in the frequency domain for a homogeneous, infinite mediumcontaining a harmonically modulated point source of power P (ω) at r = 0 is
APPENDIX B. TRANSPORT THEORY 83
U(r, ω) =P (ω)4πD
eikr
r, (B.0.9)
with the following expressions for the average photon density (UDC), and for the amplitude(UAC) and phase (φ) [5, 19]
UDC(r) =PDC
4πD
e−r(vµa/D)1/2
r, (B.0.10)
UAC(r, ω) =P (ω)4πD
e−r(vµa/D)1/2[(1+ ω2
v2µ2a
)1/2+1]1/2
r, (B.0.11)
φ(r, ω) = r(vµa/2D)1/2
[(1 +
ω2
v2µ2a
)1/2
− 1
]1/2
+ φs , (B.0.12)
Appendix C
Raman-Nath/Bragg Effect
In this section we are going to give an overview of the Raman-Nath regime and the derivationof the Raman-Nath equations, followed by the considerations that make this effect to beconsidered in the Bragg regime.
Let us represent the pressure of the sound field in two dimensions by the expression
s(x, z, t) = <[S(x, z) exp jωat] (C.0.1)
where S(x, z) is a phasor with associated frequency ωa; and the electric field of light wavewith the expression
e(x, z, t) = <[E(x, z) exp jωt] (C.0.2)
where E(x, z) is a phasor with associated frequency ω. If we consider the acoustic wavetraveling in the postive +X direction and the light wave traveling in the positive +Zdirection the following relations hold as shown in Fig. 2.3.
ka =ωa
va(C.0.3)
k =ω
c=
n0ω
cν= n0kν , (C.0.4)
where the subscript ν represent vacuum values and we assume the the medium is nonmag-netic (µν = µ0).
84
APPENDIX C. RAMAN-NATH/BRAGG EFFECT 85
Next we assume that the polarization in the medium is related to the electric field by asound-induced time-varying electric susceptibility
χ(t) = χ0 + χ1 cos (ωat− kax + φa). (C.0.5)
where χ1 is real and may be negative. The relative dielectric constant is given by
εr0 = (1 + χ0) = n20 (C.0.6)
so that
n(t)2 = [1 + χ(t)] = [1 + χ0 + χ1 cos (ωat− kax + φa)]. (C.0.7)
Assuming that |χ1| ¿ 1, we may write n(t) expanding Eq. (C.0.7) in Taylor series
n(t) ≈ n0 ± |∆n| cos (ωat− kax + φa) (C.0.8)
|∆n| =χ1
2n0. (C.0.9)
Therefore, we can define
δn(x, z, t) = |∆n| cos (ωat− kax + φa) = b(t) cos (β(t)− kax). (C.0.10)
It is important to write δn in terms of b(t) and β(t) as in Eq. (C.0.10) because the temporalvariation of b and β is assumed to be extremely slow compared to the transit time of thelight through the sound field. Therefore, the analysis is going to consider the sound fieldstationary during which b and β are constant. Also, as for the frequencies of sound andlight, it will be assumed throughout this thesis that ωa/ω ¿ 1, and that ka/k ¿ 1.
Let us assume normal incidence of the light wave on the acoustic field (z = 0). Since we areassuming that we can treat the optical field as in geometrical optics, the total phase shiftof the light ray will be given by
θ(x, L, t) = −kν
∫ L
0δn(x, z, t)dz − kL. (C.0.11)
APPENDIX C. RAMAN-NATH/BRAGG EFFECT 86
Using Eq. (C.0.10) we find that
θ(x, L, t) = −kνLb(t) cos (β(t)− kax)− kL. (C.0.12)
Thus, the electric field at z = L is given by
E(x, L, t) = Eie−jkL e−jkνLb(t) cos (β(t)−kax). (C.0.13)
Eq. (C.0.13) represents a spatially corrugated wavefrom coming out the thin acousticfilm. Also this equation results in many sidebands due to the phase modulation, and theamplitudes are given by Bessel functions. Therefore, it is shown that
E(x, L, t) = e−jkLn=+∞∑n=−∞
(−j)nEiJn(kνLb(t)) e(jnβ(t)−jnkax) (C.0.14)
where Jn denotes the nth order Bessel function.
The physical interpretation is that the nth term causes a plane wave to propagate in thedirection kn such that knx = nka as said in the introduction to the acousto-optic effect andthe angle of propagation is that of Eq. (2.4.2). If we consider paraxial propagation thisangle may be written as
φn ≈ nka
k=
nλ
λa. (C.0.15)
Now if we use Eq. (C.0.14) and Eq. (C.0.10) we find that the nth order contribution to thetotal exit field is
En(x, L, t) = Ene−jknx−jknz+jnωat, (C.0.16)
where
knx = k sin (φn) (C.0.17a)knz = k cos (φn) (C.0.17b)En = (−j)nejnφs)Jn(ν)Ei , (C.0.17c)
APPENDIX C. RAMAN-NATH/BRAGG EFFECT 87
where ν is called the Raman-Nath parameter and is defined as
ν = kνL|∆n|. (C.0.18)
Then we can write the complete expression for the electric field plane wave in the regionz ≥ l
en(x, z, t) = <Enej(ω+nωa)t−jkx sin (φn)−jkz cos (φn). (C.0.19)
It is seen from Eq. (C.0.19) that the nth order is shifted in frequency by nωa. This is aDoppler shift. If an observer sees the sound-induced radiating dipoles moving upward withsound velocity V , the velocity component in his direction is given by V sin (φn), hence usingEq. (2.4.2) we find for the Doppler shift
∆ω =(ω
c
)V sin (φn) = k
(ωa
ka
)sin (φn) = nωa. (C.0.20)
This mathematical approach describes Raman-Nath diffraction and a similar procedure canbe done for oblique incidence of the light wave (see Ref. [39]). For oblique incidence theangle of diffraction of the optical wave is given by
φn = φ0 +nλ
λa= φ0 +
nka
k. (C.0.21)
The considerations for the Raman-Nath regime were that the sound field is thin enoughto ignore optical diffraction effects and that the sound is weak enough to ignore opticalray-bending effects. The condition for this to occur is that
L ¿ λ2a
2πλ(C.0.22)
where L is the thickness of the acoustic beam; or using the Klein-Cook parameter Q = Lk2a/k
[39].
Q ¿ 1 (C.0.23a)Qν ¿ 1 (C.0.23b)
APPENDIX C. RAMAN-NATH/BRAGG EFFECT 88
where ν is defined in Eq. (C.0.18).
If the interaction is long enough we can consider the sound column as an ensemble ofthin gratings. This will create a phase mismatch in the exiting optical field, preventingany cumulative contributions from neighboring orders and the net effect would be that nodiffraction occurs at all. However, there exist two conditions where there is phase matchingbetween light impinging at an angle φ0 and one neighboring order. This is when φ0 = φ1, inwhich the +1 order interacts sinchronously with the incident light and or when φ0 = φ−1,in which case, the interaction takes place with the −1 order. This is know as the Braggregime.
Using Eq. (C.0.21) we see that the first case for the order +1 is satisfied when
φ0 =−ka
2k= −φin (C.0.24)
and for the case of order −1 we have
φ0 =+ka
2k= +φin. (C.0.25)
This is shown in Fig. 2.2. For this regime the conditions that must hold are
Q À 1 (C.0.26a)Q/ν À 1. (C.0.26b)
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