New Approach to Solving the Radiative Transport Equation and Its Applications George Y. Panasyuk Bioengineering UPenn, Philadelphia

New approach for solving the RTE in a 3D macroscopically homogeneous medium; Application of the method to: Outline of the talk (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.

Solve system of N equations: for M different values of parameter Z ; V is a matrix Spectral Method are vectors of length N “Naïve” approach: and solve for each M values of Z with Computational complexity: Spectral method: 1)Find eigenvectors and eigenvalues of V 2) For every Z, Computational complexity:

Spectral Method for the RTE S is diagonal

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): x y z z' y'y' x'x'  k   Wigner D-functions Euler angles = polar angles

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

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

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

  x y z s s Two cases: a)s is in the yz plane b) s is in the xy plane: S 0 = z Point uni-directional (sharply-peaked) source in an infinite medium placed at r 0 = (0, 0, 0) and illuminating in the -direction. Off-axis case: r = (0, y, 0)

* * ** **

l max = 1 l max = 3 l max =10 0 π α, rad 2π0 α 0 π α, rad 2π Convergence of the specific intensity with l max (left) and the converged result at l max = 34 (right); r = (0, 26l*, 0), φ = 0, g = 0.98, μ a /μ s = 0.2 l max = 34

s0s0 y

-Recover absorption coefficient of inhomogeneous medium from multiple measurements with different source-detector pairs, encoded in data function Application to optical tomography CCD - G is Green’s function within DA -l * is the transport free path -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, (2005)

δα(x)/ δα(0) corr. no corr. x/l * slab thickness = 0.5cm 2Δx2Δx Δx = 0.2l* 1D profiles of reconstructed absorption coefficient α(x) = α 0 + 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, (2005)

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

1. V.A.Markel, "Modified spherical harmonics method for solving the radiative transport equation," Letter to the Editor, Waves in Random Media 14(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), (2006). 3. G.Y.Panasyuk, V.A.Markel, and J.C.Schotland, Applied Physics Letters 87, (2005). Co-Authors: Vadim A. Markel John C. Schotland Publications: Available on the web at