Study of Tumour Growth under Immunotherapy and Chemotherapy Modelling and Simulation

Abstract

In this work, modelling of tumour growth under immunotherapy and chemotherapy has been done and solutions to the models have been obtained by various simulation techniques to understand the growth pattern of tumour and its interactions with the immune cells or with the infused drug. We analyse our model with respect to its stability and its sensitivity. We formulated our model based on predator-prey model to study the growth of a solid tumour in the presence of regular doses of lymphocytes. We further extend it to take care of the periodic behaviour of the lymphocytes, which are used for stimulating the immune system. Cell carrying capacity has been specified and a cell kill rate under immunotherapy is used to take care of how different metabolisms will react to the treatment. A simulation study of periodic drug infusion of a chemotherapeutic drug has been done by considering a set of non-linear coupled differential equations and we have been able to show that the periodic drug infusion strategy is better than the constant drug infusion technique. A theoretical model describing the growth of tumour and immunocompetent cell density is proposed incorporating an element of heterogeneity in the population. For the purpose of illustration, Stepanova s model for homogeneous population has been modified. The model is formulated in terms of non-linear differential equations by treating the growth parameters as random variables. The nonlinearity of the problem in conjunction with random parameters makes the analytical solution intractable. For gaining insight into the system, Monte-Carlo simulation framework is used to generate the tumour and immunocompetent cell growth trajectories. This framework is extended to updating various moments of interest in a recursive manner. Both the equilibrium and transient analysis of the system have been investigated. An important finding is that in contrast to deterministic model corresponding to homogeneous population, the equilibrium points show dispersion around their average values. Further, the model has been extended to study the effect of therapy for heterogeneous population by introducing constant-dose therapies. It is observed that by increasing the therapy parameter and#119862;and#119905;, there exists a critical level beyond which the complete regression of tumour size follows and the corresponding entropy also depicts a transition newline