A level-set method for the evolution of cells and tissue during curvature-controlled growth

TL;DR

This paper introduces a level-set method to simulate tissue growth influenced by curvature, capturing complex phenomena like topological changes and cell density evolution with high accuracy.

Contribution

It generalizes existing models by incorporating an additional Eulerian field for cell density, enabling simulation of topological changes in 3D tissue growth.

Findings

Numerical conservation of cells indicates simulation accuracy.

The method effectively simulates tissue fragmentation and fusion.

The model captures the influence of curvature on tissue evolution.

Abstract

Most biological tissues grow by the synthesis of new material close to the tissue's interface, where spatial interactions can exert strong geometric influences on the local rate of growth. These geometric influences may be mechanistic, or cell behavioural in nature. The control of geometry on tissue growth has been evidenced in many in-vivo and in-vitro experiments, including bone remodelling, wound healing, and tissue engineering scaffolds. In this paper, we propose a generalisation of a mathematical model that captures the mechanistic influence of curvature on the joint evolution of cell density and tissue shape during tissue growth. This generalisation allows us to simulate abrupt topological changes such as tissue fragmentation and tissue fusion, as well as three dimensional cases, through a level-set-based method. The level-set method developed introduces another Eulerian field than the level-set function. This additional field represents the surface density of tissue synthesising cells, anticipated at future locations of the interface. Numerical tests performed with this level-set-based method show that numerical conservation of cells is a good indicator of simulation accuracy, particularly when cusps develop in the tissue's interface. We apply this new model to several situations of curvature-controlled tissue evolutions that include fragmentation and fusion.