Mathematical modelling of drug resistance in malignant tumour treatment.
Resistance to conventional chemotherapies, especially to anti-cancer agents, is rapidly becoming a global pandemic. Mutations, in combination with genetic instabilities, play an important role in the molecular heterogeneity of cancerous cells that display resistance to chemotherapeutic drugs. Currently, mechanisms involved in drug resistance phenotype resulting from the interaction of a tumour and anti-cancer agents are not fully understood. In this dissertation, we propose two new dynamical models for the interaction between a tumour and a chemotherapeutic drug. Our focus is only on resistance which is caused by random genetic point mutations. The models consist of coupled systems of ordinary and partial differential equations. Tumour cells are divided into two classes, namely; sensitive and resistant cells. We determine the equilibrium points of the model equations and investigate their stability. In the frst instance, after reviewing the basic modelling assumptions and main results found in the mathematical modelling literature on drug resistance, we present the ordinary differential equation (ODE) model. To account for spatial growth effects, we then extend the model to a partial differential equation (PDE) model that describes the local interaction of the tumour with the anti-cancer agent through convection, reaction and diffsion processes. Some analytical solutions of the PDE model that are comparable to those found in the literature are obtained. One novel outcome of the models in this dissertation is the qualitative demonstration of the possible success of the therapy for certain initial conditions, number of sensitive cells and their interaction with the chemotherapeutic drug. Parameter sensitivity analysis is carried out to determine the influence of each individual parameter in the model. For all the models, numerical solutions which showed the effct of therapeutic agents on the growth and spread of the tumour cells, subject to evolving drug resistance phenomenon, were attained and presented here.
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