European Journal of Applied Mathematics



Modelling the interactions between tumour cells and a blood vessel in a microenvironment within a vascular tumour


C. J. W. BREWARD a1a2, H. M. BYRNE a2 and C. E. LEWIS a1
a1 Section of Oncology and Pathology, Division of Genomic Medicine, University of Sheffield Medical School, Beech Hill Road, Sheffield S10 2RX, UK
a2 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Abstract

In this paper, we develop a mathematical model to describe interactions between tumour cells and a compliant blood vessel that supplies oxygen to the region. We assume that, in addition to proliferating, the tumour cells die through apoptosis and necrosis. We also assume that pressure differences within the tumour mass, caused by spatial variations in proliferation and degradation, cause cell motion. We couple the behaviour of the blood vessel into the model for the oxygen tension. The model equations track the evolution of the densities of live and dead cells, the oxygen tension within the tumour, the live and dead cell speeds, the pressure and the width of the blood vessel. We present explicit solutions to the model for certain parameter regimes, and then solve the model numerically for more general parameter regimes. We show how the resulting steady-state behaviour varies as the key model parameters are changed. Finally, we discuss the biological implications of our work.

(Received July 13 2000)
(Revised January 26 2001)