A Random Walk Approach to Transport in Tissues and Complex Media: From Microscale Descriptions to Macroscale Models

Transport processes in biological systems span multiple spatial and temporal scales. From organism-level behaviors to microscopic molecular movements, transport is influenced by complex interactions of physicochemical, topological, and geometrical factors. Prior research has shown that simplified models often fail to capture the full complexity of these interactions. Experimental data frequently mask the underlying mechanisms that govern transport. Understanding how microscale properties translate into macroscale transport remains a challenge. This gap motivated researchers to explore more sophisticated modeling approaches. No prior work had resolved how to integrate spatial jumps, immobilization, and internal state changes into a unified framework. Existing models lack the ability to capture stochastic processes and heterogeneous environments. This study builds on prior knowledge of transport phenomena in biological systems.

This study aims to develop a modeling framework that connects microscale transport mechanisms with observable macroscale transport properties. The specific problem is the lack of a unified approach to model transport across spatial and temporal scales. The motivation stems from the need to understand how microscopic behaviors influence macroscopic outcomes in biological systems. The study focuses on transport in tissues and complex media, where spatial heterogeneity and stochasticity are key. The goal is to provide a generalizable framework applicable to a wide range of biological and non-biological systems. The approach uses multistate continuous-time random walks and generalized master equations. The framework allows for modeling spatial jumps, immobilization, and internal state changes. The study seeks to clarify how microscale properties determine macroscale transport coefficients.

The study employs multistate continuous-time random walks and generalized master equations to model transport processes. These models incorporate spatial jumps, immobilization at defined sites, and stochastic internal state changes. The spatial models are represented as graphs with different node classes. Walkers in the model have internal states governed by a Markov process. Fourier-Laplace transforms and asymptotic analysis are used to derive general solutions. The analysis is applied to several spatially infinite regular lattices in one and two dimensions. The framework is extended to general graphs to capture complex spatial structures. The methods allow for the derivation of transport coefficients from microscale properties.

The study derives general solutions for transport processes using Fourier-Laplace transforms and asymptotic analysis. These solutions are applied to one- and two-dimensional spatially infinite regular lattices. The framework successfully models spatial jumps, immobilization, and internal state changes. The analysis reveals how microscale properties influence macroscale transport coefficients. The models show that transport behavior depends on the interplay between spatial structure and internal state dynamics. The results demonstrate that transport coefficients can be predicted from microscale parameters. The approach is validated on general graphs representing complex spatial environments. The findings suggest that the framework can be applied to a broad range of transport problems in biological and non-biological systems.

The study concludes that microscale properties significantly influence macroscale transport coefficients in biological systems. The authors propose that their framework provides a unified approach to model transport across spatial and temporal scales. The findings suggest that spatial structure and internal state dynamics are key determinants of transport behavior. The study supports the idea that transport coefficients can be derived from microscale parameters. The authors suggest that the framework is applicable to a wide range of biological and non-biological systems. The results highlight the importance of incorporating stochasticity and heterogeneity in transport models. The study does not claim that the framework is the only approach to modeling transport. The authors emphasize that their findings are specific to the models and assumptions used in the study.