Monte Carlo: A flexible and accurate technique for modeling light transport in food and agricultural products

Highlights

• Procedure of MC for modeling light transport and different MC models were described.

• Advanced methods for accelerating MC simulations were presented.

• Applications of MC in food and agricultural products were reviewed.

• Challenges and future perspectives for MC modeling of light transport were discussed.

Abstract

Background

Monte Carlo (MC) has been widely used in fields such as biomedicine and computer graphics owing to its unique capabilities of flexibility, high-accuracy and simplicity for modeling light transport in tissues, but the applications in food and agricultural domain are limited or hindered due to the lack of knowledge on the optical properties of food products. Thanks to major breakthroughs in optical measuring and computing technologies since the year of 2000, significant advances have been made in sensing techniques for measuring tissue optical properties. Therefore, MC has witnessed great progress in food and agricultural domain over the past two decades.

Scope and approach

The development of MC for modeling light transport in food and agricultural products, including the principle, advanced MC methods, relevant applications, and future perspectives were reviewed. The paper is aimed at helping interested researchers to gain a better understanding of the MC technique, thus stimulating quality and safety assessment of food and agricultural products.

Key findings and conclusions

This paper provides an overview of the procedure of MC modeling for light transport in food and agricultural products and commonly used MC models. Advanced methods for accelerating MC simulations are then presented. Applications of MC simulations in food and agricultural products, since the year of 2000, for optimizing the design of sensing configuration and parameter, estimating tissue optical property, and assessing quality and safety are then reviewed. Finally, challenges and future perspectives for MC technique in modeling light transport are discussed.

Introduction

Recent technological advancements in modeling light transport through turbid media have spurred great progress toward the development of property, quality and safety assessment of food and agricultural products (Li, Sun, & Cheng, 2016; Liu, Guo, Li, & Xie, 2019b; Sanchez, Hashim, Shamsudin, & Nor, 2020; Zhang et al., 2018; Zhang, Lv, & Xiong, 2018). The process of light interaction with biological or food/plant tissues is rather complex at the microscale, due to the fact that the tissue is a complex system formed by different components with different structural, chemical and optical characteristics (Lu, Van Beers, Saeys, Li, & Cen, 2020; Pathmanaban, Gnanavel, & Anandan, 2019). However, in studying light-tissue interaction, the food/plant tissues may be treated as being primarily composed of absorption particles (e.g., chromophores) and scattering particles (e.g., organelles), and thus light propagation through tissues can be simplified as mainly involving the process of photon interactions with the absorption and scattering particles. Photons mainly interact with particles in two ways: (1) absorbed by the absorption particles through which the electromagnetic energy is transformed into other forms of energy (e.g., heat, fluorescence, etc.); (2) scattered by the scattering particles through which photons change traveling direction. Absorption process can be characterized by the absorption coefficient (μ_a), while scattering process is defined by the scattering coefficient (μ_s) and anisotropy factor (g) (Hu, Fu, Wang, & Ying, 2015). Anisotropy factor (g) is an optical parameter which is defined for determining the fraction of photons that would be scattered to a specific direction; g = -1, 0 and 1 represent total backward scattering, isotropic scattering and total forward scattering, respectively. Since scattering is dominant (i.e., μ_s » μ_a) for most food and agricultural products, the anisotropy factor can be lumped into the scattering coefficient, forming a new optical parameter, called reduced scattering coefficient (μ_s^'= (1-g)μ_s). Hence, with the knowledge of μ_a and μ_s^', one can, in principle, describe light-tissue interaction or model light transport in food and plant tissues.

Light propagation in tissues is best described by radiative transfer equation (RTE), which provides a mathematical expression of photon transport through turbid media by employing the principle of energy conservation (Ishimaru, 1978). It considers the energy balance of incoming, outgoing, internal source, absorbed, and emitted photons for an infinitesimal volume in the medium. A detailed description on the derivation of the RTE can be found in Lu (2016). However, no analytical solution to the RTE is available, because it is expressed in an integro-differential form with six unknown variables. Hence, the RTE must be simplified based on some assumptions or solved numerically. Approximate analytical solutions are popular because they are computationally fast (Farrell, Patterson, & Wilson, 1992; Kienle & Patterson, 1997). Among them, diffusion equation, which is derived under the assumption that scattering is dominant over absorption, has been widely used with different optical measuring techniques (e.g., time-resolved, spatially resolved, spatial-frequency domain imaging, etc.) for estimating optical absorption and scattering coefficients of turbid media (Hu, Lu, Ying, & Fu, 2019; Huang, Lu, Hu, & Chen, 2018; Wang, Lu, & Xie, 2017). However, the diffusion equation is only applicable for describing light distributions in high scattering media (μ_s^' » μ_a), and also the radiance is only considered at a sufficiently large distance (d) from the point of illumination [d » (μ_a+μ_s^')⁻¹] (Martelli, Del Bianco, Ismaelli, & Zaccanti, 2010). These assumptions do not always hold for biological tissues due to the high absorption by chromophores, such as chlorophyll, carotenoid and water in food and agricultural products in visible and near-infrared spectral region. To address these limitations, numerical methods, such as adding-doubling (AD) and Monte Carlo (MC), have been proposed and used for modeling light transport in biological tissues. While the AD method is fast and accurate in solving the RTE, even when scattering is not dominant over absorption and source-detector distance is smaller than one mean free path [mfp^'= (μ_a+μ_s^')⁻¹], it is restricted to layered geometries with homogeneous optical properties for each layer and cannot be used to retrieve information of spatial light distribution (Aernouts et al., 2014).

Unlike the AD method, MC method is applicable in simulating the energy transfer process in arbitrary geometries with complex boundary conditions or spatial localization of inhomogeneities (Lu et al., 2020). MC method is a category of computational methods that involves the random sampling of physical quantities (Flock, Patterson, Wilson, & Wyman, 1989; Wang, Jacques, & Zheng, 1995). Incident light is simulated by multiple photon packets, which are injected orthogonally into a turbid medium, and propagate through the medium depending on tissue's optical properties (i.e., the refractive index n, absorption coefficient μ_a, scattering coefficient μ_s and anisotropy factor g). Photons are traced through a turbid medium until they exit at the tissue surface or they are absorbed. A large number of photons are simulated to estimate photon distributions in a tissue (Watte, Aernouts, & Saeys, 2015). Owing to its flexibility, high-accuracy and simplicity in modeling light propagation in turbid media, MC method is considered to be the gold standard (Bevilacqua & Depeursinge, 1999; Wang et al., 1995). Over the past tens of years, MC method has been widely used for modeling light propagation in different areas, such as biomedicine, statistics, and computer graphics (Fang & Boas, 2009; Valentine, Wood, Brown, Ibbotson, & Moseley, 2012). However, the application in food and agricultural domain has so far been limited or hindered until the year of 2000. One of the major reasons for the slower adoption of MC method in food and agricultural domain is the lack of knowledge on the optical properties of those products. Thanks to major breakthroughs in analytical solutions to the RTE as well as optical measuring and computing technologies since the year of 2000, significant advances have been made in nondestructive techniques for measuring the optical properties of food and agricultural products (Hu et al., 2015; Lu, 2016). Therefore, researchers in the food and agricultural domain have recently turned to MC methods for modeling light transport inside an intact tissue.

Therefore, the objective of this paper is to provide an overview of development in Monte Carlo technique for modeling light transport in food and agricultural products, including the principle, advanced MC methods, and relevant applications. Discussions are also given on the critical issues, challenges and future perspectives for the MC technique.