Exact analytical Green's function of the diffusion equation in a multilayered cylinder with Robin boundary conditions

摘要: The diffusion equation is the standard reduced model of radiative
transfer in biological tissue, and its Green's function---the impulse
response of the turbid medium---is the forward engine of diffuse optical
tomography and spatially resolved reflectance spectroscopy. We construct
an exact analytical Green's function for the steady-state diffusion
equation in an infinitely long multilayered concentric cylinder
modelling a human limb: bone, muscle, fat, and skin, with the point
source in the outermost (skin) layer. The exact Robin (impedance)
condition is imposed directly on the physical outer surface, avoiding
the extrapolated-boundary approximation (EBC) and its
geometry-dependent $O(z_b/R)$ systematic error on curved interfaces.
Fourier--Bessel modal expansion reduces the problem to a block upper
bidiagonal system solved by forward recursive substitution in $O(N)$
operations per mode, and a layer-wise normalised Bessel basis eliminates
the catastrophic cancellation that otherwise limits small-radius
geometries, extending reliable convergence from $m\le6$ to $m\ge20$.
Validation against Monte Carlo simulations of the radiative transfer
equation gives total reflectance agreement within $0.4\%$ and a mean
surface-profile deviation of $2.2\%$ over $z>2$~mm, a factor of about $2.7$ smaller
than for the EBC model. In a finger geometry, the EBC overestimates
every azimuthal mode of the surface reflectance, the relative error
growing monotonically from $+3\%$ at $m=0$ to $+113\%$ at $m=20$.