A finite element approach for
modeling Diffusion equation
Subha Srinivasan
10/30/09
Forward Problem Definition
• Given a distribution of light sources qon the
boundary  of an object  and a distribution
of tissue parameters pwithin  , to find the
resulting measurement set M on 
Light Propagation in a
3-D Breast Model using BEM
Light Propagation in a
3-D Breast Model using BEM
Inverse Problem Definition
• Given a distribution of light sources qand a
distribution of measurements M on the
boundary  , to derive the distribution of
tissue parameters p within 
Diffusion equation in frequency domain
i 

. (r)(r,  )   a (r)   (r,  )  q0 (r,  )

c
•  is the isotropic fluence,  is the Diffusion
coefficient,  a is the absorption coefficient and
q 0 is the isotropic source
 (r) 
•
 s ' is
1
3a (r)  s '(r)
the reduced scattering coefficient
Solutions to Diffusion equation
• Analytical solutions exist in simple geometries
• Finite difference methods (FDM) use approximations
for differentiation and integration. Works well for 2D
problems with regular boundaries parallel to coordinate
axis, cumbersome for regions with curved or irregular
boundaries
• Finite element methods (FEM) can be easily applied to
complicated and inhomogeneous domains and
boundaries. Versatile and computationally feasible
(compared to Monte Carlo methods)
Using FEM for Modeling
• Main concept: divide a volume/area into
elements and build behavior in entire area by
characterizing each element (Mosaic)
• Uses integral formulation to generate a set of
equations
• Uses continuous piecewise smooth functions
for approximating unknown quantities
Basis Functions
For a set of basis functions, we can choose anything. For simplicity here,
shown are piecewise linear “hat functions”.
Our solution will be a linear combination of these functions.
φ1
x1=0
φ2
φ3
x2=L/2
x3=L
Derivation of FEM formulation
for Diffusion Equation
i 

. (r)(r,  )   a (r)   (r,  )  q0 (r,  )

c
N
• The approximate solution is:
• And for flux:
    j j
j 1
F  .n$,
N
F   Fj j
j 1
• Galerkin formulation gives the weighted residual to equal zero:
 w R(x)dx  0
i
R,w  0

• Galerkin weak form:
• Green’s identity:
• Substituting:
i 

., w   a   , w  q0 , w

c
µ wds
2u,w  u.w  —
 (u.n)

.w   a 


i 
.n$.wds
 ,w  q0 ,w  —

c
Matrix form of FEM Model
N


i 

 ji ds
  j   j .i   a  c   j ,i   q0 ,w   Fj —

j 1
j 1


N
• Discretizing parameters:
• Overall:


 j
j 1

N
K
L
k 1
l 1
   k k , a   ll
N

 L
i 
 ji ds
 k k  j .i    l l  c   j ,i   q0 , w   Fj —

k 1
l 1
j 1

K
• Matrix form:
 Abb

 AIb
AbI  b   Bbb
 
AII   I   0
0   Fb  Cb 
 0   C 
0    I
• For A,B detailed, refer to Paulsen et al, Med Phys, 1995
• Need to apply BCs
Boundary Conditions
• Type III BC, Robin type
( ) 

n̂.( )  0

• α incorporates reflection at the boundary due to
refractive index change

A
1
,
2A
2 / (1  R0 )  1  cos  c
1  cos  c
(n  1)2
R0 
(n  1)2
2
3
Source Modeling
• Point source: contribution of source to element in
which it falls
• Gaussian source: modeled with known FWHM
• Distributed source model:
• Hybrid monte-carlo model: Monte carlo model close
to source & diffusion model away from source
Paulsen et al, Med Phys, 1995
Forward Model:
Forward Model for Homogeneous
Domain: Multiple Sources
Forward Model
with Inclusion
Boundary
Measurements
Hybrid Source Model
Ashley Laughney summer project, 2007
Plots near the source
Ashley Laughney summer project, 2007
References
•
•
•
•
Arridge et al, Med Phys, 20(2), 1993
Schweiger et al, Med Phys, 22(11), 1995
Paulsen et al, Med Phys, 22(6), 1995
Wang et al, JOSA(A), 10(8), 1993