# Equations describing semi-confluent cell growth (II) colony formation on a flat surface

## TL;DR

This paper presents a mathematical model to describe how cells grow in colonies on a flat surface, accounting for contact inhibition and transitions from single to multiple layers.

## Contribution

A new model using spherical cap approximation to describe non-symmetrical multilayer cell growth governed by two parameters.

## Key findings

- The model uses ordinary differential equations to describe variable contact inhibition in non-symmetrical cell colonies.
- It effectively captures transitions from monolayer to multilayer growth on a solid plate.
- The model allows quantitative analysis of colony shape, size, and contact angle dynamics.

## Abstract

Individual cell growth can be affected by the presence of adjacent cells through a complex and multi-factorial biological process known alternatively as contact inhibition or confluence sensing. In a previous paper (Hall D (2024) Equations describing semi-confluent cell growth (I) Analytical approximations. Biophys Chem 307:107173), sets of differential equations (with implicit analytical solutions) were developed to describe completely symmetrical cases of multicellular colony growth affected by variable levels of contact inhibition. Here we develop a model based on a spherical cap approximation of colony growth, that is able to describe variable contact inhibition for non-symmetrical multilayer cell formation on a solid plate. Although the model is realized as a set of interrelated ordinary differential equations, it is effectively governed by two parameters and is therefore capable for use in quantitative analysis of the kinetics of cell culture parameters such as shape, colony size and receding contact angle. The model is capable of accounting for transitions from monolayer to multilayer growth in a robust fashion.