On a Brain Tumor Growth Model with Lactate Metabolism, Viscoelastic Effects, and Tissue Damage
Giulia Cavalleri, Pierluigi Colli, Alain Miranville, Elisabetta Rocca
TL;DR
This paper analyzes a nonlinear, time-evolving model of brain tumor growth that integrates lactate metabolism, viscoelastic tissue response, and potential tissue damage. The authors formulate a coupled system of four PDEs for $\varphi$ (tumor fraction), $\sigma$ (intracellular lactate), $\boldsymbol{u}$ (displacement), and $z$ (damage), combining a Fischer–Kolmogorov type equation, a reaction–diffusion lactate equation, a quasi-static viscoelastic balance, and a damage evolution inclusion with a maximal monotone graph. They establish global existence of weak solutions, obtain higher regularity under stronger data assumptions, and prove continuous dependence with respect to data, ensuring well-posedness of the brain-tumor model. The analysis employs a Schauder fixed-point framework with maximum principle and Moser iteration, a time-discrete scheme for the displacement, and a Yosida regularization for the damage term, together with compactness arguments to pass to the limit. These results underpin the mathematical soundness of a biologically motivated, mechanically coupled tumor-growth model and pave the way for optimal control or therapeutic optimization studies.
Abstract
In this paper, we study a nonlinearly coupled initial-boundary value problem describing the evolution of brain tumor growth including lactate metabolism. In our modeling approach, we also take into account the viscoelastic properties of the tissues as well as the reversible damage effects that could occur, possibly caused by surgery. After introducing the PDE system, coupling a Fischer-Kolmogorov type equation for the tumor phase with a reaction-diffusion equation for the lactate, a quasi-static momentum balance with nonlinear elasticity and viscosity matrices, and a nonlinear differential inclusion for the damage, we prove the existence of global in time weak solutions under reasonable assumptions on the involved functions and data. Strengthening these assumptions, we subsequently prove further regularity properties of the solutions as well as their continuous dependence with respect to the data, entailing the well-posedness of the Cauchy problem associated with the nonlinear PDE system.