Virtually increased acceptance angle for efficient estimation of spatially resolved reflectance in the subdiffusive regime: a Monte Carlo study

Abstract

Light propagation in biological tissues is frequently modeled by the Monte Carlo (MC) method, which requires processing of many photon packets to obtain adequate quality of the observed backscattered signal. The computation times further increase for detection schemes with small acceptance angles and hence small fraction of the collected backscattered photon packets. In this paper, we investigate the use of a virtually increased acceptance angle for efficient MC simulation of spatially resolved reflectance and estimation of optical properties by an inverse model. We devise a robust criterion for approximation of the maximum virtual acceptance angle and evaluate the proposed methodology for a wide range of tissue-like optical properties and various source configurations.

Keywords: (110.4234) Multispectral and hyperspectral imaging; (160.4760) Optical properties; (170.3660) Light propagation in tissues; (170.3880) Medical and biological imaging; (170.3890) Medical optics instrumentation; (170.5280) Photon migration; (170.7050) Turbid media.

Fig. 1

Example of a typical imaging system with a narrow nominal acceptance angle θ₀ (a). The nominal acceptance angle θ₀ can be virtually increased in the MC simulations to capture a larger fraction of the backscattered photon packets (b). If the quotient between the R(θ₀, r), captured at the nominal acceptance angle, and the R(θ_v, r), captured at the virtually increased acceptance angle θ_v, is approximately constant and independent of r, the virtual detection scheme can be used to reduce the computation time without introducing additional errors (c).

Fig. 2

Reflectance error arising from the virtually increased acceptance angle θ_v quantified by the spatial dependence of the absolute relative SRR error ARE (a) and the relative root mean square SRR error rRMSE (b). Results are presented for a nominal acceptance angle of 3° and for a turbid sample with absorption coefficient of 2.0 cm⁻¹, reduced scattering coefficient of 45.6 cm⁻¹, and phase function parameter γ of 1.65. A predefined 1% threshold value for E_t is marked by a horizontal black dashed line and the estimated maximum virtual acceptance angle θ_max with a vertical red dashed line.

Fig. 3

Maximum virtual acceptance angle θ_max as a function of the absorption μ_a and reduced scattering ${\mu }_s^{\prime }$ coefficients at γ = 1.75 (a) and γ = 2.15 (b). The minimum values of θ_max observed across the full range of μ_a and ${\mu }_s^{\prime }$ as a function of γ (c) and the corresponding values of μ_a (d) and ${\mu }_s^{\prime }$ (e).

Fig. 4

Scatter plots of the true and estimated values of the absorption coefficient μ_a (first column) reduced scattering coefficient ${\mu }_s^{\prime }$ (second column) and γ (third column) for virtual acceptance angles θ_v of 3° (first row), 10° (second row) and 40° (third row) obtained for a 50 μm uniform beam and nominal acceptance angle θ₀ = 3°. Ideal relation between the true and estimated values is denoted by a dashed gray line.

Fig. 5

Relative root-mean-square error (rRMSE) of the estimated μ_a (a), ${\mu }_s^{\prime }$ (b) and γ (c) as a function of the virtual acceptance angle θ_v obtained for different sources. Dashed line marks 1% rRMSE.

Fig. 6

Root-mean-square error (RMSE) of the estimated μ_a (a), ${\mu }_s^{\prime }$ (b) and γ (c) as a function of the virtual acceptance angle θ_v obtained for different sources.

Fig. 7

Normalized distribution of the number of launched photon packets over the full range of optical properties as a function of the virtual acceptance angle θ_v and for a fixed total weight of the backscattered photons packets W_tot = 10⁶. The median value of each distribution is denoted by a vertical dashed line.

Fig. 8

Required computation time and relative root-mean-square error (rRMSE) of the estimated optical properties as a function of the virtual acceptance angle θ_v.