Analysis of relative error in perturbation Monte Carlo simulations of radiative transport

Perturbation and differential Monte Carlo (pMC/dMC) methods, used in conjunction with nonlinear optimization methods, have been successfully applied to solve inverse problems in diffuse optics. Application of pMC to systems over a large range of optical properties requires optimal "placement" of baseline conventional Monte Carlo (cMC) simulations to minimize the pMC variance. The inability to predict the growth in pMC solution uncertainty with perturbation size limits the application of pMC, especially for multispectral datasets where the variation of optical properties can be substantial.

We aim to predict the variation of pMC variance with perturbation size without explicit computation of perturbed photon weights. Our proposed method can be used to determine the range of optical properties over which pMC predictions provide sufficient accuracy. This method can be used to specify the optical properties for the reference cMC simulations that pMC utilizes to provide accurate predictions over a desired optical property range.

We utilize a conventional error propagation methodology to calculate changes in pMC relative error for Monte Carlo simulations. We demonstrate this methodology for spatially resolved diffuse reflectance measurements with ±20% scattering perturbations. We examine the performance of our method for reference simulations spanning a broad range of optical properties relevant for diffuse optical imaging of biological tissues. Our predictions are computed using the variance, covariance, and skewness of the photon weight, path length, and collision distributions generated by the reference simulation.

We find that our methodology performs best when used in conjunction with reference cMC simulations that utilize Russian Roulette (RR) method. Specifically, we demonstrate that for a proximal detector placed immediately adjacent to the source, we can estimate the pMC relative error within 5% of the true value for scattering perturbations in the range of [-15%,+20%]. For a distal detector placed at ∼3 transport mean free paths relative to the source, our method provides relative error estimates within 20% for scattering perturbations in the range of [-8%,+15%]. Moreover, reference simulations performed at lower (μs'/μa) values showed better performance for both proximal and distal detectors.

These findings indicate that reference simulations utilizing continuous absorption weighting (CAW) with the Russian Roulette method and executed using optical properties with a low (μs'/μa) ratio spanning the desired range of μs values, are highly advantageous for the deployment of pMC to obtain radiative transport estimates over a wide range of optical properties.