Analysis and Simulation on a Model for the Evolution of Tumors under the Influence of Nutrient and Drug Application

TL;DR

This paper models tumor evolution using nonlinear PDEs, analyzing how nutrients and drugs influence tumor growth and necrosis, supported by simulations that reflect medical observations and explore various boundary and tissue conditions.

Contribution

It introduces a comprehensive nonlinear PDE model for tumor growth incorporating mechanical, nutritional, and drug effects, along with an explicit numerical scheme for simulation.

Findings

Necrotic core formation when nutrient levels are low

Tumor shrinkage or growth depending on nutrient-drug balance

Inhomogeneous conditions affect tumor shape irregularities

Abstract

We investigate the evolution of tumor growth relying on a nonlinear model of partial differential equations which incorporates mechanical laws for tissue compression combined with rules for nutrients availability and drug application. Rigorous analysis and simulations are presented which show the role of nutrient and drug application in the progression of tumors. We construct an explicit convergent numerical scheme to approximate solutions of the nonlinear system of partial differential equations. Extensive numerical tests show that solutions exhibit a necrotic core when the nutrient level falls below a critical level in accordance with medical observations. The same numerical experiment is performed in the case of drug application for the purpose of comparison. Depending on the balance between nutrient and drug both shrinkage and growth of tumors can occur. The role of inhomogeneous boundary conditions, vascularization and anisotropies in the development of tumor shape irregularities are discussed.