Efficient computation of the steady-state and time-domain solutions of the photon diffusion equation in layered turbid media

Abstract

Accurate and efficient forward models of photon migration in heterogeneous geometries are important for many applications of light in medicine because many biological tissues exhibit a layered structure of independent optical properties and thickness. However, closed form analytical solutions are not readily available for layered tissue-models, and often are modeled using computationally expensive numerical techniques or theoretical approximations that limit accuracy and real-time analysis. Here, we develop an open-source accurate, efficient, and stable numerical routine to solve the diffusion equation in the steady-state and time-domain for a layered cylinder tissue model with an arbitrary number of layers and specified thickness and optical coefficients. We show that the steady-state ($$< 0.1$$ < 0.1 ms) and time-domain ($$< 0.5$$ < 0.5 ms) fluence (for an 8-layer medium) can be calculated with absolute numerical errors approaching machine precision. The numerical implementation increased computation speed by 3 to 4 orders of magnitude compared to previously reported theoretical solutions in layered media. We verify our solutions asymptotically to homogeneous tissue geometries using closed form analytical solutions to assess convergence and numerical accuracy. Approximate solutions to compute the reflected intensity are presented which can decrease the computation time by an additional 2–3 orders of magnitude. We also compare our solutions for 2, 3, and 5 layered media to gold-standard Monte Carlo simulations in layered tissue models of high interest in biomedical optics (e.g. skin/fat/muscle and brain). The presented routine could enable more robust real-time data analysis tools in heterogeneous tissues that are important in many clinical applications such as functional brain imaging and diffuse optical spectroscopy.

Two-layered blood-lipid phantom and method to determine absorption and oxygenation employing changes in moments of DTOFs

“…Light in tissue can be modelled using the radiative transfer equation (RTE) and a commonly employed solution is the diffusion equation, which relies on the diffusion approximation [ 24 – 26 ]. Helton et al [ 27 ] presented an open-access numerical routine for solving the diffusion equation for an arbitrary number of layers, for the steady-state case and in the time domain. Geiger et al [ 28 ] modeled light propagation using spherical harmonics instead of the diffusion equation and presented the analytical solutions for the RTE.…”