Calculus from the past: multiple delay systems arising in cancer cell modelling


  • Graeme Charles Wake Massey University Auckland
  • Helen M Byrne University of Oxford



nonlocal calculus, cell phases, asymptotics


Nonlocal calculus is often overlooked in the mathematics curriculum. In this paper we present an interesting new class of nonlocal problems that arise from modelling the growth and division of cells, especially cancer cells, as they progress through the cell cycle. The cellular biomass is assumed to be unstructured in size or position, and its evolution governed by a time-dependent system of ordinary differential equations with multiple time delays. The system is linear and taken to be autonomous. As a result, it is possible to reduce its solution to that of a nonlinear matrix eigenvalue problem. This method is illustrated by considering case studies, including a model of the cell cycle developed recently by Simms, Bean and Koeber. The paper concludes by explaining how asymptotic expressions for the distribution of cells across the compartments can be determined and used to assess the impact of different chemotherapeutic agents. doi:10.1017/S1446181113000102

Author Biographies

Graeme Charles Wake, Massey University Auckland

Professor of Industrial Mathematics, Centre for Mathematics in Industry, Institute for Information and Mathematical Sciences.

Helen M Byrne, University of Oxford

Professor, Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute,





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