# Computational framework to capture the spatiotemporal density of cells with a cumulative environmental coupling

## TL;DR

This paper introduces a new computational framework that models cell density considering environmental factors, providing a more efficient alternative to stochastic agent-based simulations for studying cellular processes.

## Contribution

The work develops a continuum PDE model coupled with environmental variables, including stability analysis and a numerical method using fundamental solutions.

## Key findings

- Validated the PDE model against agent-based simulations.
- Demonstrated the model's ability to capture cell death due to toxic exposure.
- Provided a stable and well-posed mathematical formulation.

## Abstract

Stochastic agent-based models can account for millions of cells with spatiotemporal movement that can be a function of different factors. However, these simulations can be computationally expensive. In this work, we develop a novel computational framework to describe and simulate stochastic cellular processes that are coupled to the environment. Specifically, through upscaling, we derive a continuum governing equation that considers the cell density as a function of time, space, and a cumulative variable that is coupled to the environmental conditions. For this new governing equation, we consider the stability through an energy analysis, as well as proving uniqueness and well-posedness. To solve the governing equations in free-space, we propose a numerical method using fundamental solutions. As an application, we study a cell moving in an infinite domain that contains a toxic chemical, where a cumulative exposure above a critical value results in cell death. We illustrate the validity of this new modeling framework and associated numerical methods by comparing the density of live cells to results from the corresponding agent-based model.