A Liouville dynamical modelling and control framework for oesophageal cancer-induced immune response

A Liouville-based framework is developed for optimizing treatment strategies in oesophageal cancer (OC) while accounting for inter-patient heterogeneity. Variability within a patient cohort is represented through uncertain initial cancer and immune cell populations, whose probability distribution is propagated by a Liouville partial differential equation while the underlying individual dynamics remains deterministic. Two optimization problems are considered. First, model parameters are estimated from population-level data using an adjoint-based projected nonlinear conjugate-gradient (NCG) method. Second, optimal time-dependent combinations of chemotherapy and immunotherapy are determined using a sequential quadratic Hamiltonian (SQH) scheme based on Pontryagin's maximum principle (PMP). The Liouville equation is discretized using a finite-volume method designed to preserve positivity and conservation of the probability density. Numerical experiments with synthetic population data are used to demonstrate the parameter-estimation framework and the ability of the proposed optimal control strategy to determine treatment protocols that effectively modify the population-level cancer-immune response while limiting treatment exposure.