Vascular tumor growth and treatment: Consequences of polyclonality, competition and dynamic vascular support

Abstract

A mathematical model is presented to describe the evolution of a vascular tumor in response to traditional chemotherapeutic treatment. Particular attention is paid to the effects of a dynamic vascular support system in a tumor comprised of competing cell populations that differ in proliferation rates and drug susceptibility. The model consists of a system of partial differential equations governing intratumoral drug concentration, cancer cell density, and blood vessel density. The balance between cell proliferation and death along with vessel production and destruction within the tumor generates a velocity field which drives the expansion or regression of the neoplasm. Radially symmetric solutions are obtained for the case when only one cell type is present and when the proportion of the tumor occupied by blood vessels remains constant. The stability of these solutions to asymmetric perturbations and to a small semi-drug resistant cell population is then investigated. The analysis shows that drug concentrations which are sufficient to insure eradication of a spherical tumor may be inadequate for the successful treatment of non-spherical tumors. When the drug is continuously infused, linear analysis predicts that whether or not a cure is possible is crucially dependent on the proliferation rate of the semi-resistant cells and on the competitive effect of the sensitive cells on the resistant population. When the blood vessel density is allowed to change dynamically, the model predicts a dramatic increase in the tumors growth and decrease in its response to therapy.