# Lie symmetries of nonlinear parabolic-elliptic systems and their application to a tumour growth model

## TL;DR

This paper extends Lie symmetry methods to classify and solve a coupled reaction-diffusion system modeling tumor growth, providing exact solutions and biological insights, especially for radially symmetric cases.

## Contribution

It introduces a comprehensive Lie symmetry classification for nonlinear reaction-diffusion systems with arbitrary nonlinearities, including applications to tumor growth models.

## Key findings

- Complete Lie symmetry classification achieved.
- Exact solutions derived for tumor growth models.
- Graphical and biological interpretation provided.

## Abstract

A generalisation of the Lie symmetry method is applied to classify a coupled system of reaction-diffusion equations wherein the nonlinearities involve arbitrary functions in the limit case in which one equation of the pair is quasi-steady but the other not. A complete Lie symmetry classification, including a number of the cases characterised being unlikely to be identified purely by intuition, is obtained. Notably, in addition to the symmetry analysis of the PDEs themselves, the approach is extended to allow the derivation of exact solutions to specific moving-boundary problems motivated by biological applications tumour growth). Graphical representations of the solutions are provided and biological interpretation addressed briefly. The results are generalised on multi-dimensional case under assumption of radially symmetrical shape of the tumour.