A finite element approach for modeling Diffusion equation

Subha Srinivasan

10/30/09

Forward Problem Definition

• Given a distribution of light sources on the boundary of an object and a distribution of tissue parameters within , to find the resulting measurement set on

∂Ω Ω

p{ } Ω

M{ } ∂Ω

q{ }

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 and a distribution of measurements on the boundary , to derive the distribution of tissue parameters within

∂Ω

Ωp{ }

M{ }

q{ }

Diffusion equation in frequency domain

• is the isotropic fluence, is the Diffusion coefficient, is the absorption coefficient and is the isotropic source

• is the reduced scattering coefficient

−∇.κ (r)∇Φ(r,ω ) + μ a (r) +iω

c⎛⎝⎜

⎞⎠⎟

Φ(r,ω ) = q0 (r,ω )

Φ κμa

q0

μs '

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

x1=0 x2=L/2 x3=L

φ1φ3

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.

φ2

Derivation of FEM formulation for Diffusion Equation

−∇.κ (r)∇Φ(r,ω ) + μ a (r) +iω

c⎛⎝⎜

⎞⎠⎟

Φ(r,ω ) = q0 (r,ω )

• The approximate solution is:

• And for flux:

• Galerkin formulation gives the weighted residual to equal zero:

• Galerkin weak form:

• Green’s identity:

• Substituting:

Φ = Φ jj=1

N

∑ φ j

wiR(x)dx =0Ω∫ R,w =0

−∇.κ∇Φ,w + μ a +iω

c⎛⎝⎜

⎞⎠⎟

Φ,w = q0 ,w

∇2u,w = −∇u.∇w + (∇u.n)µ

Γ—∫ wds

κ∇Φ.∇w + μ a +

iω

c⎛⎝⎜

⎞⎠⎟

Φ,w = q0 ,w + κ∇Φ.n$—∫ .wds

Matrix form of FEM Model

• Discretizing parameters:

• Overall:

• Matrix form:

• For A,B detailed, refer to Paulsen et al, Med Phys, 1995

• Need to apply BCs

Φ j

j=1

N

∑ κ∇φ j .∇φi + μ a +iω

c⎛⎝⎜

⎞⎠⎟φ j ,φi

⎡

⎣⎢

⎤

⎦⎥= q0 ,w + Fj

j=1

N

∑ φ jφi ds—∫

κ = κ kk=1

K

∑ ψ k ,μ a = μ ll=1

L

∑ ϕ l

Φ j

j=1

N

∑ κ kψ kk=1

K

∑ ∇φ j .∇φi + μ lϕ ll=1

L

∑ +iω

c

⎛⎝⎜

⎞⎠⎟φ j ,φi

⎡

⎣⎢

⎤

⎦⎥= q0 ,w + Fj

j=1

N

∑ φ jφi ds—∫

Abb AbI

AIb AII

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Φb

Φ I

⎧⎨⎩

⎫⎬⎭=

Bbb 0

0 0

⎛

⎝⎜

⎞

⎠⎟

Fb

0⎧⎨⎩

⎫⎬⎭+

Cb

CI

⎧⎨⎩

⎫⎬⎭

Boundary Conditions

• Type III BC, Robin type

• α incorporates reflection at the boundary due to refractive index change

Φ(γ ) +κ

αn̂.∇Φ(γ ) = 0

α =1

2A,

A =2 / (1− R0 ) −1+ cosθc

3

1− cosθc

2

R0 =(n −1)2

(n +1)2

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