Steady state fluence effective coefficient for the semi-infinite medium

[alefy]

Hello Michael,

First, thank you for this great little package, with many well cited sources.

I was looking at your source [1] (Kienle and Patterson 1997) for the semi-infinite medium problem.

It seems to me that the article, for the steady state fluence (Eq. 3), defines the μ effective as μeff = sqrt(3 * μa * (μa + μsp)) which appears different in your source https://github.com/heltonmc/LightPropagation.jl/blob/968a5485a48d99abcf42a2f16beaf847686f648e/src/forwardmodels/Diffusion%20Approximation/Semi-infinite/DAsemiinf.jl#L126

Just to inform you. I might have missed something as well.

Best regards, Adrien

[heltonmc]

Hi Adrien,

Thanks for your kind note and report. The straightforward answer here is that μeff is defined as μeff=sqrt(μa/D) where μa is the absorption coefficient and D is the diffusion coefficient. Therefore, this discrepancy comes from the choice of how to define the diffusion coefficient. This topic has received a lot of attention over the past 30 years so without detailing that here the original Kienle papers use D=1 / (3 * (musp + mua)) whereas more recently it is becoming more accepted to use D=1 / (3 * (musp)) as that has matched better MC simulations for larger absorption coefficients. Essentially, when mua/musp is only like 0.1-0.3 the diffusion coefficient independent of mua has better agreement and therefore, for consistency the whole package assumes this diffusion coefficient.

For typical biological media where musp >> mua the effect is fairly negligible so it is isn't too big of a consideration but for larger absorbing media it does matter. Though, recently it is better to consider the lossless diffusion equation (no absorption) and then apply the absorption dependence from solutions to the RTE which better matches a diffusion coefficient independent of absorption. Essentially, a diffusion coefficient independent of absorption matches the physical meaning derived from RTE better. This difference is negligible for low absorbing media, matters some for moderate absorbing media, and probably shouldn't be using diffusion for high absorbing media.

Happy to expand further on this as I've left out a lot of papers and discussion on this! And I will try to make a note of this in the source files.

• Michael

[alefy]

Hello Michael,

With delays... but I am grateful for your prompt answer. I suspected something along these lines. However, I have not investigated such a point. The only bibliography that I have on this is from S. Arridge's review on Optical tomography in medical imaging in Inverse problems from 1999. He quotes Yamada [21] on this point.

Such a definition of D seems odd to me because Fick's law derived from the Radiative Transport Equation leads to D=1 / (3 * (musp + mua)).

If you have the time and willingness to write back with a bibliography and thoughts, they are very welcome.

Best wishes, Adrien

[heltonmc]

Sorry for the late reply! You are absolutely right that when the DE is derived from the RTE, the diffusion coefficients shows a dependence on absorption.

This I believe is the "classical" result but ultimately that derivation has the inherent limitations by all the assumptions used to derive the DE. The idea behind seeking alternative estimates of the diffusion coefficient is to better approximate RTE solutions with the DE that perhaps overcomes some of the limitations of diffusion.

For example, experimentally it has been shown that a better choice of D can lead to better matching with DT and MC. I think there is exploration in different approaches when approximating the DE or Fick's law from the RTE that could lead to better agreement. So perhaps it might be best to say that we are looking to replace the traditional D that includes limitation of diffusion approximations with a different D that better matches the RTE.

I listed a few papers below that explore this topic:

1. "Photon diffusion coefficient in scattering and absorbing media"
2. "Independence of the diffusion coefficient on absorption: Experimental and numerical evidence"
3. "Experimental determination of photon propagation in highly absorbing and scattering media"
4. Expression of the optical diffusion coefficient in high-absorption turbid media
5. "Photon diffusion coefficient in scattering and absorbing media"
6. Diffusion approximation for a dissipative random medium and the applications
7. Diffusion coefficient in photon diffusion theory