Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy

Abstract

A minimally parameterized mathematical model for low-dose metronomic chemotherapy is formulated that takes into account angiogenic signaling between the tumor and its vasculature and tumor inhibiting effects of tumor-immune system interactions. The dynamical equations combine a model for tumor development under angiogenic signaling formulated by Hahnfeldt et al. with a model for tumor-immune system interactions by Stepanova. The dynamical properties of the model are analyzed. Depending on the parameter values, the system encompasses a variety of medically realistic scenarios that range from cases when (i) low-dose metronomic chemotherapy is able to eradicate the tumor (all trajectories converge to a tumor-free equilibrium point) to situations when (ii) tumor dormancy is induced (a unique, globally asymptotically stable benign equilibrium point exists) to (iii) multi-stable situations that have both persistent benign and malignant behaviors separated by the stable manifold of an unstable equilibrium point and finally to (iv) situations when tumor growth cannot be overcome by low-dose metronomic chemotherapy. The model forms a basis for a more general study of chemotherapy when the main components of a tumor's microenvironment are taken into account.

Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation

“…More recently, Urszula Ledzewicz and Heinz Schättler have tackled the problem of combining two classes of anti-cancer drugs, one hitting the tumour cell population directly, and another one indirectly, choking it by an antiangionetic effect on its vascular environment, see, e.g., [163]. Furthermore, the effects of metronomic therapy may be compared with those of the more classic maximal tolerated dose (MTD) [23] still in use in most clinical oncology departments [229]. These studies, based on the Hahnfeldt model [116] or some of its variants, that include both the tumour cell population and its environment considered as the tumour carrying capacity as a target for antiangiogenic drugs, have the remarkable feature to propose exact optimal solutions when it is possible.…”

“…The relevance of the use of metronomic therapy as an alternative to maximum tolerated dose of cytotoxic drugs in chronic forms of cancer is a question that has been actively studied for some time [23,199], and more recently with optimal control methods [229]. Its achievements were firstly attributed to angiogenic effects of cytotoxic drugs [154], then to effects of low doses of chemotherapy on stimulating the immune response [199], in particular due to the effects of tumour lysates on the participation of memory T-cells in the immune response [265].…”

Role of vascular normalization in benefit from metronomic chemotherapy

“…To date, only a few mathematical models of metronomic therapy have been published, none of which accounts for interactions among cancer cells, CSCs, immune cells, and tumor blood vessels (26)(27)(28)(29)(30)(31)(32). To this end, we have developed a mathematical model for tumor growth that accounts for three different phenotypes of cancer cells-nonstem cancer cells (CCs), CSCs (which are more resistant to drugs, hypoxia, and the immune system), and CCs that are induced by chemotherapy to acquire a more stem-like phenotype, which we refer to as treatment-induced cancer cells (ICCs) (33), as well as immune cells [natural killer (NK) cells, CD8 + T cells, and Tregs], tumor vasculature, and their interactions (Fig.…”

Optimal Control of Cancer Treatments: Mathematical Models for the Tumor Microenvironment

“…In the paper [51] we have analyzed the system (29)- (31) for a constant metronomic dosing, u.t/ D u Á const, from a dynamical systems point of view and have shown that this system exhibits a similar wide range of dynamical behaviors as the model considered in Sect. 3 and encompasses the same variety of medical scenarios.…”

“…This is the case we consider here. [51]. We again call the low equilibrium point benign and its region of attraction the benign region and call the high equilibrium point malignant and its region of attraction the malignant region.…”