Abstract

The tumour microenvironment is highly heterogeneous, especially with respect to oxygen availability, nutrient supply, composition of the extracellular matrix, infiltration of immune cells, and drug penetration. The fact that they are heterogeneous has a significant impact on tumour growth, metastatic potential, drug resistance and therapeutic outcome. This work represents a mathematical model to describe spatial heterogeneity in tumour micro-environment based on partial differential equations (PDEs). The model combines reaction–diffusion processes, cellular proliferation dynamics, nutrient transport, and chemotactic interactions and is capable of simulating the complex spatial and temporal evolution of tumour tissues. The mathematical modelling uses coupled non-linear PDEs to describe the diffusion, convection, and reaction processes in the tumour microenvironment. The spatial gradients of oxygen and nutrient concentrations are described by diffusion equations, and the migration and proliferation of tumour cells are described by reaction–diffusion systems of logistic growth equations. Additionally, the model includes angiogenesis, which is a mechanism describing the formation of new blood vessels in response to hypoxia. Numerical simulations are complete to analyze the morphology of the tumour, its invasion patterns and the concentration gradients in space for different biological conditions using finite difference and finite elements methods. Results show that spatial heterogeneity has a significant influence on tumour evolution and response to therapy. Low oxygen areas form necrotic cores, and highly vascularized areas result in a rapid expansion of the tumour. The simulations show that the nutrients, cell growth and immune response are interacting in a non-linear way, which results in irregular boundaries of the tumour and in non-uniform growth patterns as seen in the clinic. The sensitivity analysis shows that diffusion coefficient, proliferation rate and vascular density are important parameters of the tumor behavior. The study emphasizes the relevance of PDE-based modeling in the study of multi-scale dynamics of tumors and in predicting the therapeutic response. The framework offers valuable insight into target drug delivery, radiation therapy optimisation, and personalised cancer treatment strategies. Moreover, coupling computational simulation with biological data is a promising avenue for enhancing predictive oncology and precision medicine efforts in cancer research.