A one-dimensional individual-based mechanical model of cell movement in heterogeneous tissues and its coarse-grained approximation.

R. J. Murphy,R. E. Baker,M. J. Simpson,P. R. Buenzli

doi:10.1098/rspa.2018.0838

Abstract

Mechanical heterogeneity in biological tissues, in particular stiffness, can be used to distinguish between healthy and diseased states. However, it is often difficult to explore relationships between cellular-level properties and tissue-level outcomes when biological experiments are performed at a single scale only. To overcome this difficulty, we develop a multi-scale mathematical model which provides a clear framework to explore these connections across biological scales. Starting with an individual-based mechanical model of cell movement, we subsequently derive a novel coarse-grained system of partial differential equations governing the evolution of the cell density due to heterogeneous cellular properties. We demonstrate that solutions of the individual-based model converge to numerical solutions of the coarse-grained model, for both slowly-varying-in-space and rapidly-varying-in-space cellular properties. We discuss applications of the model, such as determining relative cellular-level properties and an interpretation of data from a breast cancer detection experiment.

Highlights

Biological tissues are heterogeneous and multi-scale by their very nature
We present a one-dimensional cell-based model with heterogeneous cell properties, and its coarse-grained continuum approximation
Our results shows that solutions of the discrete model approach the solution of the continuum model as the number of springs per cell increases while the spring length tends to zero, even for rapidly varying spatial cell properties

Summary

Introduction

Biological tissues are heterogeneous and multi-scale by their very nature (figure 1a). In this work, we focus on models which relate cellular-level details to tissue-level outcomes These models have been developed with a variety of coarse-graining techniques and assumptions, including the use of slowly varying and periodic assumptions on the heterogeneity in the model [29,30], correlation functions [31,32] and interaction forces from potentials [33]. The key focus of this work is to present a novel coarse-grained system of partial differential equations governing the evolution of the cell density, cell stiffness and resting cell length, from a heterogeneous cell-based model of epithelial tissue mechanics. Key algorithms used to generate results are available on GitHub

Model description

Results and discussion

Conclusion and future work