Numerical integration methods for large-scale biophysical simulations

TL;DR

This paper evaluates numerical integration methods for simulating large-scale biophysical systems, focusing on stiff equations and discrete events in tumor spheroid growth models, to improve robustness and stability.

Contribution

It assesses the applicability of existing integration methods to complex, event-driven biophysical simulations, providing insights for their effective implementation.

Findings

Certain integration methods are more stable for stiff equations.

Explicit methods struggle with frequent discrete events.

Implicit methods offer better robustness in the tested scenarios.

Abstract

Simulations of biophysical systems inevitably include steps that correspond to time integrations of ordinary differential equations. These equations are often related to enzyme action in the synthesis and destruction of molecular species, and in the regulation of transport of molecules into and out of the cell or cellular compartments. Enzyme action is almost invariably modeled with the quasi-steady-state Michaelis-Menten formula or its close relative, the Hill formula: this description leads to systems of equations that may be stiff and hard to integrate, and poses unusual computational challenges in simulations where a smooth evolution is interrupted by the discrete events that mark the cells' lives. This is the case of a numerical model (Virtual Biophysics Lab - VBL) that we are developing to simulate the growth of three-dimensional tumor cell aggregates (spheroids). The program must be robust and stable, and must be able to accept frequent changes in the underlying theoretical model: here we study the applicability of known integration methods to this unusual context and we describe the results of numerical tests in situations similar to those found in actual simulations.