OSA / BIOMED/DH 2010
BSuE1.pdf
Finite Element Solution of the Fokker-Planck Equation for Optical Tomography Ossi Lehtikangas1 , Tanja Tarvainen1,2 , Ville Kolehmainen1, Aki Pulkkinen1,3 , Simon R. Arridge2 , and Jari P. Kaipio1,4 1 Department
of Physics, University of Kuopio, P.O. Box 1627, 70211 Kuopio, Finland E-mail:
[email protected]
2 Department
of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom 3 Sunnybrook Research Institute, Sunnybrook Health Sciences Centre, 2075 Bayview Ave., Toronto, ON, M4N 3M5, Canada 4 Department of Mathematics, University of Auckland, Private Bag 92019, Auckland Mail Centre, Auckland 1142, New Zealand
Abstract: Light propagation is modeled with the Fokker-Planck equation which approximates the radiative transport equation when scattering is forward-peaked. The Fokker-Planck equation is solved with the finite element method. c 2009 Optical Society of America
OCIS codes: (170.5280) Photon migration; (100.3190) Inverse problems (170.7050) Turbid media
1
Introduction
The image reconstruction in diffuse optical tomography (DOT) is a nonlinear ill-posed inverse problem. The iterative solution of this problem requires several solutions of the forward problem. Moreover, due to ill-posedness of the reconstruction problem, even small errors in the modelling can produce large errors into the reconstructions. Therefore, an accurate and computationally feasible forward model is needed. Light propagation in biological material is governed by the transport theory [3]. This leads to describing the multiple scattering phenomenon in tissues using the radiative transport equation (RTE). Due to computational complexity of the RTE, different approximations have been developed to ease up the computation of the forward problem. The most often used model for the solution of the forward problem in DOT is the diffusion approximation (DA). The DA is computationally feasible but it fails to describe light propagation accurately in low-scattering regions as well as in the proximity of the light source and boundaries [1]. The idea of using the Fokker-Planck equation as the forward model in DOT was introduced in [4]. The Fokker-Planck equation can be used to describe light propagation accurately when scattering is strongly forward dominated which is the case in biological tissues. In this work, the Fokker-Planck equation is solved with the finite element method (FEM) [5]. The approach is tested with simulations and compared with the FE-solution of the DA and Monte Carlo (MC) simulations. 2
Methods
Let Ω ∈ Rn be the physical domain and n = 2,3 be the dimension of the domain. In addition, let sˆ ∈ S n−1 denote a unit vector in the direction of interest. The frequency domain version of the RTE is of the form iω φ(r, sˆ) + sˆ · ∇φ(r, sˆ) + µa φ(r, sˆ) = µs Lφ(r, sˆ) + q(r, sˆ), (1) c where i is the imaginary unit, c is the speed of light in medium, ω is the angular modulation frequency of the input signal, q(r, sˆ) is the source inside Ω, φ(r, sˆ) is the radiance, and µs and µa are the scattering and absorption parameters of the medium. The scattering operator L is defined as Z ′ ′ ′ (2) Lφ(r, sˆ) = −φ(r, sˆ) + Θ(ˆ s · sˆ )φ(r, sˆ )dˆ s S n−1
′
where Θ(ˆ s · sˆ ) is the scattering phase function. The Henyey-Greenstein phase function is of the form ( 1 1−g2 n=2 2π (1+g2 −2gˆ s·ˆ s′ ) ′ Θ(ˆ s · sˆ ) = 1−g2 1 n=3 4π (1+g2 −2gˆ s·ˆ s′ )3/2
(3)
OSA / BIOMED/DH 2010
BSuE1.pdf
where parameter g ∈ [−1, 1] defines the shape of the probability distribution. For forward dominated scattering g > 0. In DOT, we use the vacuum boundary condition φ0 (r, sˆ) r ∈ ∪j εj sˆ · n ˆ
