The Diffusion Approximation to Photon Transport

Chapter 14 in the 4th edition of Intermediate Physics for Medicine and Biology contains a section describing the diffusion approximation to photon transport.

When photons enter a substance, they may scatter many times before being absorbed or emerging from the substance. This leads to turbidity, which we see, for example, in milk or clouds. The most accurate studies of multiple scattering are done with “Monte Carlo” computer simulation, in which probabilistic calculations are used to follow a large number of photons as they repeatedly interact in the tissue being simulated. However, Monte Carlo techniques use lots of computer time. Various approximate analytical solutions also exist... One of the approximations, the diffusion approximation, is described here. It is valid when many scattering events occur for each photon absorption.

Optical Mapping of Cardiac
Excitation and Arrhythmias,
edited by Rosenbaum and Jalife.

Today, I would like to present a new homework problem about the diffusion approximation, based on a brief communication I published in the August 2008 issue of IEEE Transactions on Biomedical Engineering (Volume 55, Pages 2102–2104). I was interested in the problem because of its role in optical mapping of transmembrane potential in the heart, discussed briefly at the end of Sec 7.10 and reviewed exhaustively in the excellent book Optical Mapping of Cardiac Excitation and Arrhythmias, edited by David Rosenbaum and Jose Jalife. Enjoy the problem, which belongs at the bottom of the left column of page 394.

Section 14.5

Problem 16 ½ Consider light with fluence rate φ₀ continuously and uniformly irradiating a half-infinite slab of tissue having an absorption coefficient μ_a and a reduced scattering coefficient μ'_s. Divide the photons into two types: the incident ballistic photons that have not yet interacted with the tissue, and the diffuse photons undergoing multiple scattering. The diffuse photon fluence rate, φ, is governed by the steady state limit of the photon diffusion equation (Eq. 14.26). The source of diffuse photons is the scattering of ballistic photons, so the source term in Eq. 14.26 is s = μ'_s exp(-z/λ_unatten), where z is the depth below the tissue surface. At the surface (z=0), the diffuse photons obey the boundary condition φ = 2 D dφ/dz.

(a) Derive an analytical expression for the diffuse photon fluence rate in the tissue, phi(z). 

(b) Plot φ(z) versus z for μ_a=0.08 mm⁻¹ and μ'_s=4 mm⁻¹.  

(c) Evaluate λ_unatten and λ_diffuse for these parameters.

The most interesting aspect of this calculation is that the diffuse photon fluence rate is not maximum at the tissue surface, but rather it builds up to a peak below the surface, somewhat like the imparted energy from 10 MeV photons shown in Fig. 15.32. This has some interesting implications for optical mapping of the heart: subsurface tissue may contribute more to the optical signal than surface tissue.

If you want the solution, send me an email (roth@oakland.edu) and I will gladly supply it.