Computational modeling aspects of photon transport equations in biological media

This paper presents an overview of modeling light propagation through biological media by solving the photon transport equation. Different variants of the photon transport equation (PTE) are discussed. Several methods for modeling static distributions and the transient response are presented. A discussion on how to mix and match electromagnetic problems with the PTE is also provided.

Generalized coupled photon transport equations for handling correlated photon streams with distinct frequencies

A generalized form of coupled photon transport equations that can handle correlated light beams with distinct frequencies is introduced. The derivation is based on the principle of energy conservation. For a single frequency, the current formulation reduces to a standard photon transport equation, and for fluorescence and phosphorescence, the diffusion models derived from the proposed photon transport model match for homogenous media. The generalized photon transport model is extended to handle wideband inputs in the frequency domain. © 2012 Optical Society of America.

An approximate numerical technique for characterizing optical pulse propagation in inhomogeneous biological tissue

An approximate numerical technique for modeling optical pulse propagation through weakly scattering biological tissue is developed by solving the photon transport equation in biological tissue that includes varying refractive index and varying scattering/absorption coefficients. The proposed technique involves first tracing the ray paths defined by the refractive index profile of the medium by solving the eikonal equation using a Runge-Kutta integration algorithm. The photon transport equation is solved only along these ray paths, minimizing the overall computational burden of the resulting algorithm. The main advantage of the current algorithm is that it enables to discretise the pulse propagation space adaptively by taking optical depth into account. Therefore, computational efficiency can be increased without compromising the accuracy of the algorithm.

Use of artificial neural networks as explicit finite difference operators

Results of a numerical exercise, substituting a numerical operator by an artificial neural network (ANN) are presented in this paper. The numerical operator used is the explicit form of the finite difference (FD) scheme. The FD scheme was used to discretize the one-dimensional transport equation, which included both the advection and dispersion terms. Inputs to the ANN are the FD representation of the transport equation, and the concentration was designated as the output. Concentration values used for training the ANN were obtained from analytical solutions. The numerical operator was reconstructed from a back calculation of the weights of the ANN. Linear transfer functions were used for this purpose. The ANN was able to accurately recover the velocity used in the training data, but not the dispersion coefficient. This capability was improved when numerical dispersion was taken into account; however, it is limited to the condition: C/P<0.5 , where C is the Courant number and P , the Peclet number (i.e., the restriction imposed by the Neumann stability condition).

Concepts and dimensionality in modeling unsaturated water flow and solute transport

Many environmental studies require accurate simulation of water and solute fluxes in the unsaturated zone. This paper evaluates one- and multi-dimensional approaches for soil water flow as well as different spreading mechanisms to model solute behavior at different scales. For quantification of soil water fluxes,Richards equation has become the standard. Although current numerical codes show perfect water balances, the calculated soil water fluxes in case of head boundary conditions may depend largely on the method used for spatial averaging of the hydraulic conductivity. Atmospheric boundary conditions, especially in the case of phreatic groundwater levels fluctuating above and below a soil surface, require sophisticated solutions to ensure convergence. Concepts for flow in soils with macro pores and unstable wetting fronts are still in development. One-dimensional flow models are formulated to work with lumped parameters in order to account for the soil heterogeneity and preferential flow. They can be used at temporal and spatial scales that are of interest to water managers and policymakers. Multi-dimensional flow models are hampered by data and computation requirements.Their main strength is detailed analysis of typical multi-dimensional flow problems, including soil heterogeneity and preferential flow. Three physically based solute-transport concepts have been proposed to describe solute spreading during unsaturated flow: The stochastic-convective model (SCM), the convection-dispersion equation (CDE), and the fraction aladvection-dispersion equation (FADE). A less physical concept is the continuous-time random-walk process (CTRW). Of these, the SCM and the CDE are well established, and their strengths and weaknesses are identified. The FADE and the CTRW are more recent,and only a tentative strength weakness opportunity threat (SWOT)analysis can be presented at this time. We discuss the effect of the number of dimensions in a numerical model and the spacing between model nodes on solute spreading and the values of the solute-spreading parameters. In order to meet the increasing complexity of environmental problems, two approaches of model combination are used: Model integration and model coupling. Amain drawback of model integration is the complexity of there sulting code. Model coupling requires a systematic physical domain and model communication analysis. The setup and maintenance of a hydrologic framework for model coupling requires substantial resources, but on the other hand, contributions can be made by many research groups.

Modelling solute transport in structured soils: performance evaluation of the ADR and TRM models

The movement of chemicals through the soil to the groundwater or discharged to surface waters represents a degradation of these resources. In many cases, serious human and stock health implications are associated with this form of pollution. The chemicals of interest include nutrients, pesticides, salts, and industrial wastes. Recent studies have shown that current models and methods do not adequately describe the leaching of nutrients through soil, often underestimating the risk of groundwater contamination by surface-applied chemicals and overestimating the concentration of resident solutes. This inaccuracy results primarily from ignoring soil structure and nonequilibrium between soil constituents, water, and solutes. A multiple sample percolation system (MSPS), consisting of 25 individual collection wells, was constructed to study the effects of localized soil heterogeneities on the transport of nutrients (NO−3, Cl−, PO3−4) in the vadose zone of an agricultural soil predominantly dominated by clay. Very significant variations in drainage patterns across a small spatial scale were observed (one-way ANOVA, p < 0.001 indicating considerable heterogeneity in water flow patterns and nutrient leaching. Using data collected from the multiple sample percolation experiments, this paper compares the performance of two mathematical models for predicting solute transport, the advective-dispersion model with a reaction term (ADR), and a two-region preferential flow model (TRM) suitable for modelling nonequilibrium transport. These results have implications for modelling solute transport and predicting nutrient loading on a larger scale.