This document describes research on using near-infrared optical imaging techniques for 3D biological tissue imaging. It discusses diffuse optical tomography (DOT) and fluorescence DOT (F-DOT). For DOT, it covers the photon diffusion equation, forward and inverse models, and finite element method implementation. For F-DOT, it discusses the fluorescence transport equations and parallel inversion schemes. Simulation results using MATLAB and NIRFAST show reconstructed optical property maps and fluorescence distributions in 2D and 3D geometries. Future work aims to further develop 3D imaging software for interfacing with DOT instrumentation.

1. Near Infrared Optical Imaging Of Biological
Tissue In Three Dimension Using External
Fluorescent Agent And Interfacing With
Instrumentation
By
M.Nagendra Babu
(1227003)

2. INDEX
 Introduction
 DOT
 F-DOT
 Results
 Conclusion
 Future work to be done

3. 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.

4. DOT basics
Forward Model
Reverse Model
DOT

5. 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]

6. 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’

7. 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.

8. 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.

9. 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.

10. 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.

11.  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:

12. 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   

13.  The diffusion equation in the FEM framework can be
expressed as system of linear algebraic equations:
0
1
(K(k) C( ) F) q
2
a
m
i
c A

    
(r) u (r). u (r)d
n
ij i jK k r

  
( (r) )u (r)u (r)d
(r)
n
ij a i j
m
i
C r
c



 
1
(r) (r)dn
ij i jF u u r

 
0 0(r)q (r)dn
iq u r

 

14. 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

  
 
 
     
 
 

15.  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 
  


        
          
         

16. 1
[J J I] ( )T T c M
J  
    
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.


17. Introduction
Forward model
Inversion scheme
F-DOT

18. 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

19. 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 λ
with the probability of
• Assuming that only two distinct wavelength are
present
(r)c


&x m 

20. 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   

21. 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       

22. 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         

23. Parallel inversion scheme

24. SIMULATION RESULTS
 SIMULATION IN MATLAB
Original Mesh Reconstructed image

25. Error vs. Iteration Graph

26. Reconstruction with priors
Without Priori condition With priori condition

27. 3 D Reconstructed image using nirfast

28. 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.

29. 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.

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