A Finite-Volume Algorithm for Modeling Light Transport with the Time-Independent Simplified Spherical Harmonics Approximation to the Equation of Radiative Transfer Ludguier D. Montejo*a , Hyun K. Kima , Andreas H. Hielscher*a,b,c a Department
of Biomedical Engineering, Columbia University, New York, NY 10027 of Radiology, Columbia University Medical Center, New York, NY 10032 c Department of Electrical Engineering, Columbia University, New York, NY 10027
b Department
ABSTRACT In this work we introduce the finite volume (FV) approximation to the simplified spherical harmonics (SPN ) equations for modeling light propagation in tissue. The SPN equations, with partly reflective boundary conditions, are discretized on unstructured grids. The resulting system of linear equations is solved with a Krylov subspace iterative method called the generalized minimal residual (GMRES) algorithm. The accuracy of the FV-SPN algorithm is validated through numerical simulations of light propagation in a numerical phantom with embedded inhomogeneities. We use a FV implementation of the equation of radiative transfer (ERT) as the benchmark algorithm. Solutions obtained using the FV-SPN (N > 1) algorithm are compared to solutions obtained with the ERT and the diffusion equation (SP1 ). Compared to the SP1 , the SP3 solutions obtained using the FV-SPN algorithm can better approximate ERT solutions near boundary sources and in the vicinity of void-like regions. Solutions using the SP3 algorithm are obtained 9.95 times faster than solutions with the ERT-based algorithm. Keywords: Diffuse optical tomography, light propagation, simplified spherical harmonics, equation of radiative transfer, finite volume method
1. INTRODUCTION The ERT has been shown to be an accurate deterministic model for the propagation of light in tissue in optical tomography (OT).1 Widespread use of the ERT in OT has been limited because numerical solutions to the equation are computationally demanding. The main reasons for high computational cost are: 1) the ERT is only first order accurate, and as such, ensuring numerical stability requires the spatial discretization of the computation domain to be small, therefore requiring the use of highly dense meshes; 2) accurate modeling of highly forward peaked light (as is often the case in tissue) requires high angular discretization to accurately capture the scattering pattern of light in tissue; 3) the presence of scattering makes intensities at different directions strongly coupled, leading to slow convergence. Altogether, the total number of equations that results from the spatial and angular discretization of the ERT is very large, making the computation time required to solve the system of equations impractical, and can often take hours.1–3 As a result of these computational considerations, approximations to the ERT have become increasingly important and common. The approximation most commonly used is the diffusion equation (DE), which assumes that light propagates through tissue in a diffuse manner. This assumption is valid only under conditions where light is highly scattered and infrequently absorbed. Thus, the DE has been shown to be a poor approximation to the ERT in tissue of small volume, with high absorption, void-like regions, or internal light sources near the boundary.1, 2 This limits the applications for which the DE approximation can be used, making it a particularly poor light propagation model for optical tomographic imaging of fingers (small geometries, high absorption, and void like regions), brains and small animals (i.e. mice and rats).
Corresponding authors:
[email protected] (L.D. Montejo) and
[email protected] (A.H. Hielscher) Optical Tomography and Spectroscopy of Tissue IX, Edited by Bruce J. Tromberg, Arjun G. Yodh, Mamoru Tamura, Eva M. Sevick-Muraca, Robert R. Alfano, Proc. of SPIE Vol. 7896, 78960J · © 2011 SPIE · CCC code: 1605-7422/11/$18 · doi: 10.1117/12.875967 Proc. of SPIE Vol. 7896 78960J-1 Downloaded from SPIE Digital Library on 21 Feb 2011 to 128.59.149.137. Terms of Use: http://spiedl.org/terms
The SPN approximation to the ERT has been shown to be superior to the DE at modeling light propagation in tissue of small volume, with higher absorption, and with lower scattering.4 The main advantage of the SPN equations is that they are a better approximation to the ERT than the DE and are far less computationally demanding than the ERT. Computational demand is reduced because the number of coupled equations resulting from the SPN approximation is substantially fewer than those resulting from the ERT. This occurs because the SPN equations are second-order accurate, and therefore, the computation mesh does not need to be as dense as the mesh required to accurately solve the ERT. The SPN approximation to the ERT has been used in the steady state domain using finite differences discretization,4 in the frequency domain using finite element discretization,5 and in the time domain using the finite element method.6 In this work we introduce the FV approximation to the SPN equations (FV-SPN ). We employ a nodecentered FV discretization in the computational domain.7 The resulting system of linear algebraic equations is solved with a Krylov subspace iterative method called GMRES.8 This new approach to solving the SPN equations is attractive because the node-centered FV method takes advantage of the beneficial properties of both the finite element and finite volume methods by combining the conservation properties of the finite volume formulation and the geometric flexibility of the finite element approach. Furthermore, the FV implementation is advantageous over the finite differences discretization of the SPN equations because it can be easily formulated on unstructured grids. This is a particularly important distinction because OT applications often involve imaging of complex geometries (e.g. mice) that can most accurately be modeled using unstructured grids. The algorithm is validated with numerical simulations. Numerical simulations on an inhomogeneous phantom are performed with the SPN equations. The relative percent difference between solutions computed using the SP1 , SP3 , and ERT equations are presented in this paper. Computation time using the SPN equations is compared to computation time using the ERT-based algorithm.
2. METHODS 2.1 SPN approximation to ERT The SPN equations of order 1, 3, 5, and 7 are presented in detail by Klose and Larsen.4 The SP3 equations are reproduced below. 2 1 μa ϕ2 ∇ϕ1 + μa ϕ1 = Q + (1) −∇ · 3μa1 3 2 5 2 4 1 μa + μa2 ϕ2 = − Q + μa ϕ1 ∇ϕ2 + (2) −∇ · 7μa3 9 9 3 3 Where μan = μa + (1 − g n ) μs are the nth order absorption coefficients, Q is a volumetric source, and ϕi are the composite moments. The Legendre moments, φi , can be written as function of the composite moments as follows: 1 (3) φ2 = ϕ2 , 3 2 φ0 = ϕ1 − ϕ2 . (4) 3 The corresponding sets of boundary conditions are, 1 + B1 1 D1 1 + A1 ϕ1 + + C1 ϕ2 + S (Ω) 2|Ω · n| dΩ, (5) n · ∇ϕ1 = n · ∇ϕ2 + 2 3μa1 8 μa3 Ω·n 1cm). Solutions to the light propagation problem were obtained 9.95 times faster with the SP3 algorithm than with the ERT-based benchmark algorithm. Ongoing work on this project involves validating the SPN algorithm by comparing numerical simulations and experimental results.
ACKNOWLEDGMENTS This work was funded in part by two grants from the National Cancer Institute (NCI) which is part of the National Institutes of Health (5U54CA126513-03: Tumor Microenvironment Network - The role of inflammation and stroma in digestive cancer and 4R33CA118666: Small animal tomography system for green fluorescent protein imaging).
REFERENCES 1. Klose, A.D., Radiative Transfer of Luminescence in Biological Tissue, in Light Scattering Reviews 4, Springer, Berlin, 293-345 (2009). 2. Ren, K., Bal, G., Hielscher, A.H., Transport- and diffusion-based optical tomography in small domains: A comparative study, Applied Optics 46(27), 6669-6679 (2007). 3. Montejo, L.D., Klose, A.D., Hielscher, A.H., Implementation of the equation of radiative transfer on blockstructured grids for modeling light propagation in tissue, Biomedical Optics Express 1(3), 861-878 (2010). 4. Klose, A.D., Larsen, E.W., Light transport in biological tissue based on the simplified spherical harmonics equations, J. Com. Phys. 220, 441-470 (2006).
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5. Chu, M., Vishwanath, K., Klose, A.D., Dehghani, H., Light transport in biological tissue using threedimensional frequency-domain simplified spherical harmonics equations, Phys. Med. Biol. 54(8), 2493-2509 (2009). 6. Dominguez, J.B., Brub-Lauzire, Y., Diffuse light propagation in biological media by a time-domain parabolic simplified spherical harmonics approximation with ray-divergence effects, Applied Optics 49(8), 1414-1429 (2010). 7. Minkowycz, S., Sparrow, E., Murthy, J., [Handbook of numerical heat transfer], J. Wiley, Hoboken NJ, (2006). 8. Saad, Y., Schultz, M., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 3, 856-869 (1989). 9. Kim, H.K, Lee, J.H., Hielscher, A.H., PDE-constrained Fluorescence Tomography with the FrequencyDomain Equation of Radiative Transfer, IEEE Journal of Selected Topics in Quantum Electronics 16(4), 793-803 (2010). 10. Kim, H.K., Hielscher, A.H., A PDE-constrained SQP algorithm for optical tomography based on the frequency-domain equation of radiative transfer, Inverse Problems 25, 015010 (2009).
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