Nonlinear Partial Differential Equation Models for Tumour Growth and Treatment Response in Nigerian Cancer Patient Cohorts

Mathematical modelling of tumour growth and treatment response has advanced significantly in global oncology research, yet no rigorous mathematical framework has been developed and validated against clinical data from Nigerian patient cohorts, who present with distinct tumour biology, late-stage presentation characteristics, and treatment access constraints compared to populations from which existing models are calibrated. This dissertation makes original theoretical and empirical contributions to mathematical oncology by developing a system of nonlinear reaction-diffusion partial differential equations for tumour growth, angiogenesis, and chemo-radiotherapy response calibrated to retrospective clinical data from the Lagos University Teaching Hospital and University of Nigeria Teaching Hospital, Enugu, for breast and cervical cancer patients diagnosed between 2018 and 2023. Theoretical contributions include the proof of global existence, uniqueness, and non-negativity of solutions to the proposed PDE system under biologically realistic initial and boundary conditions, using a fixed-point theorem approach. A parameter estimation framework based on adjoint-state sensitivity analysis is developed to calibrate the model to longitudinal tumour measurement data from patient records. The calibrated model accurately reproduces observed patient-level treatment response trajectories with mean prediction error of 8.7 percent in tumour volume. Computational experiments explore optimal chemotherapy scheduling under the model dynamics, identifying alternating intensified and rest cycles as superior to conventional continuous dosing in the cohort context. Keywords: tumour growth model, reaction-diffusion PDE, mathematical oncology, Nigerian patients, treatment optimisation.