Minimal Morphoelastic Models of Solid Tumour Spheroids: A Tutorial

TL;DR

This paper develops a hierarchy of simple, analytically tractable morphoelastic models to understand mechanical effects on tumour spheroid growth, aligning well with experimental data and offering mechanistic insights.

Contribution

It introduces a minimal, refined morphoelastic model of tumour spheroid growth that balances simplicity, analytical tractability, and biological realism, advancing the understanding of mechanical regulation.

Findings

The final model agrees with classical experimental results.

Iterative model refinement improves physical realism.

Simple models can yield mechanistic insights into tumour growth.

Abstract

Tumour spheroids have been the focus of a variety of mathematical models, ranging from Greenspan's classical study of the 1970s through to contemporary agent-based models. Of the many factors that regulate spheroid growth, mechanical effects are perhaps some of the least studied, both theoretically and experimentally, though experimental enquiry has established their significance to tumour growth dynamics. In this tutorial, we formulate a hierarchy of mathematical models of increasing complexity to explore the role of mechanics in spheroid growth, all the while seeking to retain desirable simplicity and analytical tractability. Beginning with the theory of morphoelasticity, which combines solid mechanics and growth, we successively refine our assumptions to develop a somewhat minimal model of mechanically regulated spheroid growth that is free from many unphysical and undesirable behaviours. In doing so, we will see how iterating upon simple models can provide rigorous guarantees of emergent behaviour, which are often precluded by existing, more complex modelling approaches. Perhaps surprisingly, we also demonstrate that the final model considered in this tutorial agrees favourably with classical experimental results, highlighting the potential for simple models to provide mechanistic insight whilst also serving as mathematical examples.