Monte Carlo: A flexible and accurate technique for modeling light transport in food and agricultural products
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 (μa), while scattering process is defined by the scattering coefficient (μs) 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., μs » μa) 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 (μs'= (1-g)μs). Hence, with the knowledge of μa and μs', 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 (μs' » μa), and also the radiance is only considered at a sufficiently large distance (d) from the point of illumination [d » (μa+μs')−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'= (μa+μs')−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 μa, scattering coefficient μs 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.
Section snippets
General procedure of MC modeling for light transport in tissues
Fig. 1(a) shows the general procedure of MC modeling for tracing the random walk steps of a single photon packet through a tissue. The procedure mainly includes launching photon packet, determining step size, hitting boundary, photon packet absorption, scattering and termination. At the launching of photon packet, the initial position, direction and weight of the photon packet should be assigned. Then the step size will be sampled randomly from the probability distribution for the photon
Methods for accelerating MC simulations
Although the MC method is considered to be the gold standard in modeling light transport in turbid media, a sufficiently large number of photon packets need to be simulated to obtain acceptable results due to the stochastic nature of MC modeling, thus making it computationally intensive. Great efforts have been made to speed up the MC simulations by using different improved methods, such as scaling MC, perturbation MC, hybrid MC, variance reduction and parallel-computed MC.
Scaling MC. A scaling
Applications of MC simulations in food and agricultural products
Due to the flexibility, high-accuracy and simplicity of MC modeling, and recent advances in computation speed, increasing attention has been received for MC simulations in the food and agricultural domain since the year of 2000. The most common application is to model light propagation in tissues by measuring physical quantities of interest (e.g., diffuse reflectance, transmittance, photon flux, photon absorption power density, etc.) for a given set of tissue's optical properties, which is
Challenges and future perspectives
Over the past two decades, we have seen significant research efforts in the development and application of Monte Carlo simulation for modeling light transport in food and agricultural products. While MC simulation offers new opportunities for optimizing the design of sensing configuration and parameter, estimating optical absorption and scattering properties, and assessing quality and safety for food and agricultural products, there still exist some issues or challenges using this technique due
Conclusions
Rapid advances in MC simulations have been taking place over the past twenty years, since MC serves as a flexible and accurate technique for simulating light propagation in food and agricultural products. In this paper, the procedure of MC modeling for light transport though tissues was first described, followed with commonly used MC models, including TR-MC, FD-MC, SR-MC and SFD-MC. Then diverse advanced methods for speeding up MC simulations (scaling MC, perturbation MC, hybrid MC, variance
Author contributions
Hu Dong: Conceptualization, Data Curation, Writing - Original Draft.
Wang Aichen: Conceptualization, Writing- Reviewing and Editing.
Sun Tong: Resources, Writing- Reviewing and Editing.
Yao Lijian: Resources, Visualization.
Yang Zidong: Resources, Visualization.
Ying Yibin: Supervision, Funding Acquisition.
Declaration of competing interest
None.
Acknowledgement
This work was supported by the Natural Science Foundation of Zhejiang Province (No. LQ20C130002) and the Project of Key Laboratory in the Ministry of Agriculture and Rural Areas (No. 2016NYZD18003).
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