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2 International Journal of Applied Mathematics and Computer Science

2 2023

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Travelling waves for low-grade glioma growth and response to a chemotherapy model

Bartłomiejczyk Agnieszka, Bodnar Marek, Bogdańska Magdalena U., Piotrowska Monika J.

International Journal of Applied Mathematics and Computer Science

2023

Vol. 33, no. 4

569--581

Low-grade gliomas (LGGs) are primary brain tumours which evolve very slowly in time, but inevitably cause patient death. In this paper, we consider a PDE version of the previously proposed ODE model that describes the changes in the densities of functionally alive LGGs cells and cells that are irreversibly damaged by chemotherapy treatment. Besides the basic mathematical properties of the model, we study the possibility of the existence of travelling wave solutions in the framework of Fenichel’s invariant manifold theory. The estimates of the minimum speeds of the travelling wave solutions are provided. The obtained analytical results are illustrated by numerical simulations.

On the analysis of a mathematical model of CAR-T cell therapy for glioblastoma: Insights from a mathematical model

Bodnar Marek, Foryś Urszula, Piotrowska Monika J., Bodzioch Mariusz, Romero-Rosales Jose A., Belmonte-Beitia Juan

International Journal of Applied Mathematics and Computer Science

2023

Vol. 33, no. 3

379--394

Chimeric antigen receptor T (CAR-T) cell therapy has been proven to be successful against different leukaemias and lymphomas. Its success has led, in recent years, to its use being tested for different solid tumours, including glioblastoma, a type of primary brain tumour, characterised by aggressiveness and recurrence. This paper presents an analytical study of a mathematical model describing the competition of CAR-T and glioblastoma tumour cells, taking into account their immunosuppressive capacity. The model is formulated in a general way, and its basic properties are investigated. However, most of the analysis considers the model with exponential tumour growth, assuming this growth type for simplicity. The existence and stability of steady states are studied, and the subsequent focus is on two different types of treatment: constant and periodic. Finally, protocols for CAR-T cell therapy of glioblastoma are numerically derived; these are aimed at preventing the tumour from reaching a critical size and at prolonging the patients’ survival time as much as possible. The analytical and numerical results provide theoretical support for the treatment of glioblastoma using CAR-T cells.