Search

Highlights • A spatiotemporal mathematical model, in the form of a moving boundary problem, to explain cancer dormancy is proposed. • Temporal-model analysis replicates experimental observations of immune-induced tumour dormancy. • Travelling wave solutions of the spatiotemporal model are determined using the hyperbolic tangent method. • Minimum wave speeds of invasion depict that cell invasion dynamics are mainly driven by their motion and growth rates. • Stability analysis of the spatiotemporal model shows a possibility of dynamical stabilization of the tumour-free steady state. Abstract A spatio-temporal mathematical model, in the form of a moving boundary problem, to explain cancer dormancy is developed. Analysis of the model is carried out for both temporal and spatio-temporal cases. Stability analysis and numerical simulations of the temporal model replicate experimental observations of immune-induced tumour dormancy. Travelling wave solutions of the spatio-temporal model are determined using the hyperbolic tangent method and minimum wave speeds of invasion are calculated. Travelling wave analysis depicts that cell invasion dynamics are mainly driven by their motion and growth rates. A stability analysis of the spatio-temporal model shows a possibility of dynamical stabilization of the tumour-free steady state. Simulation results reveal that the tumour swells to a dormant level. [ABSTRACT FROM AUTHOR]

Cancer treatment is an inexact science despite traditional cancer therapies. The traditional cancer treatments have high levels of toxicity and relatively low efficacy. Current research and clinical trials have indicated that virotherapy, a procedure which uses replication-competent viruses to kill cancer cells, has less toxicity and a high efficacy. However, the interaction dynamics of the tumor host, the virus, and the immune response is poorly understood due to its complexity. We present a mathematical analysis of models that study tumor-immune-virus interactions in the form of differential equations with spatial effects. A stability analysis is presented and we obtained analytical traveling wave solutions. Numerical simulations were obtained using fourth order Runge–Kutta and Crank–Nicholson methods. We show that the use of viruses as a cancer treatment can reduce the tumor cell concentration to a very low cancer dormant steady state or possibly deplete all tumor cells in body tissue. The traveling waves indicated an exponential increase and decrease in the cytotoxic-T-lymphocytes (CTLs) density and tumor load in the long term respectively. [ABSTRACT FROM AUTHOR]

• A mathematical model of oncolytic virus spread using a reaction-telegraph equation is formulated. • Existence and uniqueness of model solutions is proved. • Temporal model stability results reveal several possibilities which have been discussed in detail. • Exact solutions depict that, with time, the ECM density either degenerates or exponentially grows while the viral load decays. • Spatiotemporal simulations show that the tumor and the virus densities decrease with time while the ECM density increases. A mathematical model of oncolytic virus spread using a reaction-telegraph equation is formulated. The model considers the interaction of three main components, namely, tumor cells, extra-cellular matrix (ECM), and an oncolytic virus. Analysis of the model explores several cases in which the effect of oncolytic virus spread on a tumor and the role of the extra-cellular matrix in facilitating viral spread is measured. Analysis of the model includes proving existence of positive unique solutions for the temporal and spatiotemporal cases and determining steady state solutions and investigating their stability. Under a steady state assumption of tumor cells, exact solutions for the virus and ECM densities are determined. Numerical simulations are as well carried out to supplement the analytical results. Temporal model stability results reveal several possibilities which have been discussed in detail. Temporal model simulations reveal that tumor cells would initially eat up the ECM but as the viral load increases, tumor cells are later lysed and eliminated and the ECM would grow to maximum capacity. Exact solutions depict that, with time, the ECM density either degenerates or exponentially grows while the viral load decays. Spatiotemporal simulations show that the tumor and virus densities decrease at the centre of the domain and the ECM density which is initially uniform throughout the domain becomes normally distributed and increases with time. [ABSTRACT FROM AUTHOR]

Highlights: [•] We model the response of tumour infiltrating cytotoxic lymphocytes (TICLs) to a solid tumour without any medical intervention. [•] Mathematical analysis and numerical simulations show the existence of cancer dormancy. [•] We investigate the effect of boosting CTLIs with immunotherapy by adding a class of IL2 in the initial model. [•] The analysis in the last case shows a considerable reduction of the size of the dormant tumour. [Copyright &y& Elsevier]