New Approach to Solving the Radiative Transport Equation and Its Applications

1. George Y. Panasyuk Bioengineering UPenn, Philadelphia georgey@seas.upenn.edu New Approach to Solving the Radiative Transport Equationand Its Applications

2. Outline of the talk • New approach for solving the RTE in a 3D macroscopically homogeneous medium; • Application of the method to: (1) Calculation of the RTE Green’s function for the case of free boundary; (2) Generation of forward data for an inverse problem in optical tomography.

3. Spectral Method Solve system of N equations: for M different values of parameter Z; V is a matrix are vectors of length N Spectral method: • Find eigenvectors and • eigenvaluesof V “Naïve” approach: and solve 2) For every Z, for each M values of Z with Computational complexity: Computational complexity:

4. Spectral Method for the RTE S is diagonal

5. z y' z' q k y j q x x' Rotated Reference Frames To avoid k-dependence, use spherical harmonics defined in a reference frame whose z-axis is aligned with the direction of k (“rotated” frames): Wigner D-functions Euler angles = polar angles

6. - diagonal, In HG model is the spectral parameter, where is a tridiagonal real symmetric matrix with eigenfunctions Ψn and eigenvalues λn Green’s function of RTE Analytical dependences on all variables Details: J.Phys.A 39, 115 (2006)

7. Evanescent Waves and the BVP Evanescent waves: vacuum Z=0 ρ Solution of the half-space BVP: Half-space z > 0 medium z J.Phys.A 39, 115 (2006)

8. Point uni-directional (sharply-peaked) source in an infinite medium placed atr0 = (0, 0, 0) and illuminatingin the-direction. Forward and backword propagation: r = (0, 0, z),

9. Point uni-directional (sharply-peaked) source in an infinite medium placed atr0 = (0, 0, 0) and illuminatinginthe -direction. Off-axis case: r = (0, y, 0) • Two cases: • s is in the yz plane • b) s is in the xy plane: z S0= z s a y s b x

10. * * * * * *

11. 3.0 10-16 lmax = 1 lmax = 3 lmax= 34 lmax=10 lmax = 34 0 10-18 -3.0 10-20 0 πα, rad 2π 0 α0πα, rad 2π Convergence of the specific intensity with lmax (left) and the converged result at lmax= 34 (right); r = (0, 26l*, 0), φ = 0, g = 0.98, μa/μs = 0.2

12. s0 y

13. Application to optical tomography -Recover absorptioncoefficientof inhomogeneous medium from multiple measurements with different source-detector pairs, encoded in data function -G is Green’s function within DA -l* is the transport free path CCD • Based on DA when lowest order correction in l* are taken into account, • Applicable when diffusion theory breaks down (thin samples, near boundaries or sources, etc) APL, 87, 101111 (2005)

14. 1.0 corr. δα(x)/ δα(0) no corr. slab thickness = 0.5cm 0.5 Δx = 0.2l* 2Δx 0 x/l* -3.0 -1.5 0 1.5 3.0 1D profiles of reconstructed absorption coefficient α(x) = α0 +δα(x) of a point absorber using corrected (red) and uncorrected (green) DA Data function was simulated by the MRRF for the RTE APL, 87, 101111 (2005)

15. CONCLUSIONS • The method of rotated reference frames takes advantage of all symmetries of the RTE (symmetry with respect to rotations and inversions of the reference frame). • The angular and spatial dependence of the solutions is expressed in terms of analytical functions. • The analytical part of the solution is of considerable mathematical complexity. This is traded for relative simplicity of the numerical part. We believe that we have reduced the numerical part of the computations to the absolute minimum allowed by the mathematical structure of the RTE

16. Co-Authors: Vadim A. Markel John C. Schotland Publications: 1. V.A.Markel, "Modified spherical harmonics method for solving the radiative transport equation," Letter to the Editor, Waves in Random Media14(1), L13-L19 (2004).   2. G.Y.Panasyuk, J.C.Schotland, and V.A.Markel, "Radiative transport equation in rotated reference frames," Journal of Physics A, 39(1), 115-137 (2006).    3. G.Y.Panasyuk, V.A.Markel, and J.C.Schotland, Applied Physics Letters 87, 101111 (2005). Available on the web at http://whale.seas.upenn.edu/vmarkel/papers.html