Optimal control on a brain tumor growth model with lactate metabolism, viscoelastic effects, and tissue damage
Giulia Cavalleri, Alain Miranville
TL;DR
This work studies an optimal-control problem for a brain-tumor model that couples tumor density, intracellular lactate, tissue mechanics, and damage via a Fisher–Kolmogorov equation, a reaction–diffusion equation, a quasi-static elasticity balance, and a differential inclusion. The authors define a control-to-state map, establish existence of optimal controls, and prove Fréchet differentiability of the map by deriving a linearized state system with robust a priori estimates. They then formulate an adjoint system and derive a first-order necessary condition for optimality, reframing the gradient computation in terms of adjoint variables to enable efficient optimization. The results rely on strengthened regularity assumptions and a strict separation property for the damage variable, ensuring well-posedness of the state, linearized, and adjoint problems and providing a rigorous foundation for coupling chemotherapy and lactate-targeting therapy in a mechanically active brain-tumor setting.
Abstract
In this paper, we study an optimal control problem for a brain tumor growth model that incorporates lactate metabolism, viscoelastic effects, and tissue damage. The PDE system, introduced in [G. Cavalleri, P. Colli, A. Miranville, E. Rocca, On a Brain Tumor Growth Model with Lactate Metabolism, Viscoelastic Effects, and Tissue Damage (2025)], couples a Fisher-Kolmogorov type equation for tumor cell density with a reaction-diffusion equation for the lactate, a quasi-static force balance governing the displacement, and a nonlinear differential inclusion for tissue damage. The control variables, representing chemotherapy and a lactate-targeting drug, influence tumor progression and treatment response. Starting from well-posedness, regularity, and continuous dependence results already established, we define a suitable cost functional and prove the existence of optimal controls. Then, we analyze the differentiability of the control-to-state operator and establish a necessary first-order condition for treatment optimality.