Near Infrared Optical Imaging Of Biological

Tissue In Three Dimension Using External

Fluorescent Agent And Interfacing With

Instrumentation

By

M.Nagendra Babu

(1227003)

INDEX

Introduction

DOT

F-DOT

Results

Conclusion

Future work to be done

Introduction

Non-Invasive Imaging has become an indispensable

tool in medical diagnosis.

However most of these methods have intrinsic

drawbacks.

Diffuse Optical Tomography (DOT) is a relatively

new medical imaging modality which promises to

address some of these problems.

DOT basics

Forward Model

Reverse Model

DOT

DOT Basics

Propagation of light into tissue

Why NIR

Optical Properties of tissues

Absorption

Scattering

Propagation of light into tissue [15]

Absorption at different wavelength[15]

Forward model

Aim of forward model is to find out of the path

traveled by photon.

If the magnitude of the isotropic fluence within tissue

is significantly larger than the directional flux

magnitude, the light field is ‘diffuse’

The Radiative Transport Equation

Light propagation in tissue behaves more like

erratically moving photons migrating on average

through the medium than like a propagating wave

or a ray. Thus we use linear transport theory to

model the propagation of light.

In this approach light is treated as composed of

distinct photons, propagating through a medium

modeled as a background which has constant

scattering and absorption characteristics.

L(r, Ω, t) radiance at position ‘r’ in the direction ‘Ω’ at time ‘t’,

F (Ω, Ω′) is the scattering phase function,

Q(r, Ω, t) is the radiant source function, v- velocity in medium.

The left-hand side of accounts for photons leaving the tissue, and

the right-hand side accounts for photons entering it.

Time derivative of the radiance, which equals the net

number of photons entering the tissue.

accounts for the flux of photons along the direction Ω

The scattering and absorption of photons within the

phase element

Photons scattered from an element in phase space are

balanced by the scattering into another element in

phase space. The balance is handled by the integral

term which accounts for photons at position r being

scattered from all directions' Ω into direction Ω.

photon source.

Photon Diffusion Equation

If the scattering probability is much larger than that

of absorption within the medium we can use a

simple approximation.

The basic idea is that if the reduced scattering

coefficient is much greater than the absorption

coefficient, the radiance can be approximated as a

weighted sum of the photon fluence rate and the

photon flux.

This approximation is valid when the radiance is

almost angularly uniform, having only a relatively

small flux in any particular angular direction.

Expressing the radiance in this form, allows for the

simplification of the linear transport equation to the

variable-scattering form of what is known as the

photon diffusion equation:

FEM implementation

When the RI is homogeneous, the finite element

discretization of a volume, Ω, can be obtained by

subdividing the domain into D elements joined at V

vertex nodes.

. In finite element formalism, the fluence at a given

point, Φ(r) is approximated by the piecewise

continuous polynomial function,

where Ω, is a finite dimensional subspace

spanned by basis functions Ui,i=1…V

1(r) ( )

Vh h

i iu r

The diffusion equation in the FEM framework can be

expressed as system of linear algebraic equations:

0

1(K(k) C( ) F) q

2a

m

i

c A

(r) u (r). u (r)dn

ij i jK k r

( (r) ) u (r) u (r)d(r)

n

ij a i j

m

iC r

c

1(r) (r)dn

ij i jF u u r

0 0(r)q (r)dn

iq u r

Inverse model

The goal of the inverse problem is the recovery of

optical properties μ at each FEM node within the

domain using measurements of light fluence from

the tissue surface. This inversion can be achieved

using a modified-Tikhonov minimization

min

22 2

0

1 1

( )NM NN

M C

i i j

i j

X

We minimize this ‘objective’ function:

2 2

1

( )NM

M C

i i

i

X

2 2 2

0 0 0( ) ( ) ( ) ....X X d X

d

1

1

T Tc c c

c M

i i

1[J J I] ( )T T c MJ

c

J

This derivative is called Jacobian

Here is the regularization parameter which helps in converging the solution.

And will yield to better result, instead of this regularization parameter we can also

use priori conditions, Which can be obtained from already performed medical

operations.

Introduction

Forward model

Inversion scheme

F-DOT

Introduction

Fluorescence tomography methods aim at reconstructing the concentration of fluorophores within the imaged object

Diffuse measurement of the fluorescence emissions are obtained on the boundary of the object

Excitation is performed through external laser sources at various position

Important terms to know

Stoke shift

Quantum yield

Molar excitation

Forward model

Fluorochrome within domain Ω increases the

absorption at λ by

C is the Spatially varying Concentration

is the molar excitation of fluorochrome.

• The fluorochrome will emit at a wavelength λwith the probability of

• Assuming that only two distinct wavelength are

present

(r)c

&x m

We can write the equations as

Where 1st equation stands for excitation wavelength

and 2nd equation for emission wavelength

Under the assumption that stokes shift is small

0( (r) c(r)) (r) ( )x ax x x s sD r r

( D (r)) (r) (r) (r)m am f m x xc

x mD D D

ax am a

Solving the equations…

The final equation is diffusion equation and hence

both equations becomes independent and can be

computed completely in parallel.

0 0(r) (r) (r r ) ( D (r)) (r)x s a xc

0

1( (r))( (r) (r)) (r r )a f x s sD

1( (r) (r))f x t

0( (r)) (r r )a t s sD

0

1( (r))( (r) (r)) (r r )a f x s sD

The 1st Equation describes the propagation of excitation light with

absorption of both tissue and the inside fluorophores.

The Quantum yield is defined as the ratio between emitting fluorescence photon

numbers and the number of excitation photon absorbed by fluorophore

1(r)f

It compensates the excitation photon density absorbed by fluorescent.

Thus the 2nd Equation describes the transportation of excitation

light in tissue with assumed no fluorophore inside.

0( (r) c(r)) (r) ( )x ax x x s sD r r

Parallel inversion scheme

SIMULATION RESULTS

SIMULATION IN MATLAB

Original Mesh Reconstructed image

Error vs. Iteration Graph

Reconstruction with priors

Without Priori condition With priori condition

3 D Reconstructed image using nirfast

Conclusion

In this presentation the work towards development

and demonstration of DOT algorithms in two

dimensions progress towards f-DOT which have

certain advantages over the existing ones is

presented .

In the first set of simulations it is shown that recovery

of optical properties and the location of the

inhomogenities in two dimensions. And in three

dimension the simulations is done using the NIRFAST.

Future Progress

As the further work I will like to progress on with the

image reconstruction of biological tissue with diffuse

optical tomography and fluorescence diffuse optical

tomography in three dimension with Matlab.

I will also extend my work towards the software

development for interfacing with instrumentation of

diffuse optical tomography so that it can be

implemented in real time.

References

Kanmani Buddhi, “Studies on improvement of reconstruction methods in diffuseoptical tomography”, Department of Instrumentation, Indian Institute of Science,April (2006).

Tuchin V, ‘Tissue Optics Light scattering methods and instruments for medicaldiagnosis’, SPIE (2000).

Hamid Dehghani, Matthew E. Eames, Phaneendra K. Yalavarthy, Scott C. Davis,Subhadra Srinivasan, Colin M. Carpenter, Brian W. Pogue and Keith D. Paulsen,“Near infrared optical tomography using NIRFAST: Algorithm for numerical modeland image reconstruction”, Wiley InterScience Publications, Commun. Numer. Meth.Engng(2008).

Arridge SR, Schweiger M., “Direct calculation of the moments of the distribution ofphoton time-of-flight in tissue with a finite-element method.” Applied Optics (1995).

Arridge SR., “Optical tomography in medical imaging. Inverse Problems”, (1999).

Brooksby B, Jiang S, Kogel C, Doyley M, Dehghani H, Weaver JB, Poplack SP, PogueBW, Paulsen KD., “Magnetic resonance-guided near-infrared tomography of thebreast. Review of Scientific Instruments (2004).

Schweiger M, Arridge SR, Hiroaka M, Delpy DT. The finite element model for thepropagation of light in scattering media: boundary and source conditions. MedicalPhysics (1995).

Paulsen KD, Jiang H. “Spatially varying optical property reconstruction using a finiteelement diffusion equation approximation”. Medical Physics(1995).

Ben A. Brooks. “Combining near infrared tomography and magnetic resonanceimaging to improve breast tissue chromophore and scattering assessment”, “ThayerSchool of Engineering Dartmouth College Hanover, New Hampshire”- May 2005.

Xiaolei Song, Ji Yi, and Jing Bai. “A Parallel Reconstruction Scheme in fluorescenceTomography Based on Contrast of Independent Inversed Absorption Properties”,Department of Biomedical Engineering, Tsinghua University, Beijing 100084, China,Accepted 13 August (2006).

R. B. Schulz, J. Peter, W. Semmler, and W. Bangerth, “Indepen-dent modeling offluorescence excitation and emission with the finite element method,” inProceedingsof OSA Biomedical Topic al Meeting s, Miami, Fla, USA, April 2004.

David A. Boas, Dana H. Brooks, Eric L. Miller, Charles A. DiMarzio, Misha Kilmer,Richard J. Gaudette and Quan Zhang, ““Imaging the Body With Diffuse OpticalTomography” November 2001.