Estimation of Parameter Distributions for Reaction-Diffusion Equations with Competition using Aggregate Spatiotemporal Data

TL;DR

This paper extends a parameter distribution estimation framework to reaction-diffusion models with competition, demonstrating improved prediction accuracy and efficiency in modeling heterogeneous tumor subpopulations using simulated glioblastoma data.

Contribution

It introduces a novel approach combining random differential equations with the Prokhorov metric to estimate joint distributions in competitive reaction-diffusion models.

Findings

Random differential equation model outperforms PDE models in prediction accuracy.

The approach efficiently estimates heterogeneity in subpopulation parameters.

Clustering of subpopulations is feasible from recovered distributions.

Abstract

Reaction diffusion equations have been used to model a wide range of biological phenomenon related to population spread and proliferation from ecology to cancer. It is commonly assumed that individuals in a population have homogeneous diffusion and growth rates, however, this assumption can be inaccurate when the population is intrinsically divided into many distinct subpopulations that compete with each other. In previous work, the task of inferring the degree of phenotypic heterogeneity between subpopulations from total population density has been performed within a framework that combines parameter distribution estimation with reaction-diffusion models. Here, we extend this approach so that it is compatible with reaction-diffusion models that include competition between subpopulations. We use a reaction-diffusion model of Glioblastoma multiforme, an aggressive type of brain cancer, to test our approach on simulated data that are similar to measurements that could be collected in practice. We use Prokhorov metric framework and convert the reaction-diffusion model to a random differential equation model to estimate joint distributions of diffusion and growth rates among heterogeneous subpopulations. We then compare the new random differential equation model performance against other partial differential equation models' performance. We find that the random differential equation is more capable at predicting the cell density compared to other models while being more time efficient. Finally, we use $k$-means clustering to predict the number of subpopulations based on the recovered distributions.