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CALCULUS FROM THE PAST: MULTIPLE DELAY SYSTEMS ARISING IN CANCER CELL MODELLING

Part of: Functional-differential and differential-difference equations Physiological, cellular and medical topics Qualitative theory

Published online by Cambridge University Press:  30 April 2013

G. C. WAKE  and
H. M. BYRNE
G. C. WAKE*
Affiliation:
Centre for Mathematics in Industry, Institute of Natural & Mathematical Sciences, Massey University, Private Bag 102-904, North Shore Mail Centre, Auckland 0932, New Zealand
H. M. BYRNE
Affiliation:
Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute, University of Oxford, 24–29 St. Giles’, Oxford OX1 3LB, UK email Helen.Byrne@maths.ox.ac.uk Computational Biology Group, Department of Computer Science, University of Oxford, South Parks Road, Oxford OX1 3QD, UK
*
✉g.c.wake@massey.ac.nz
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Abstract

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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.

Keywords

nonlocal calculuscell phasesasymptotics

MSC classification

Secondary: 34K06: Linear functional-differential equations 92C37: Cell biology 34C12: Monotone systems

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 54 , Issue 3 , January 2013 , pp. 117 - 126
Copyright
Copyright ©2013 Australian Mathematical Society 

References

Basse, B., Baguley, B. C., Marshall, E. S., Joseph, W. R., van Brunt, B., Wake, G. C. and Wall, D. J. N., “A mathematical model for analysis of the cell cycle in cell lines derived from human tumors”, J. Math. Biol. 47 (2003) 295–312; doi:10.1007/s00285-003-0203-0.CrossRefGoogle ScholarPubMed
Basse, B., Baguley, B. C., Marshall, E. S., Joseph, W. R., van Brunt, B., Wake, G. C. and Wall, D. J. N., “Modelling cell death in human tumour cell lines exposed to the anticancer drug paclitaxel”, J. Math. Biol. 49 (2004) 329–357; doi:10.1007/s00285-003-0254-2.CrossRefGoogle Scholar
Basse, B., Baguley, B. C., Marshall, E. S., Wake, G. C. and Wall, D. J. N., “Modelling the flow of cytometric data obtained from unperturbed human tumour cell lines: parameter fitting and comparison”, Bull. Math. Biol. 67 (2005) 815–830; doi:10.1016/j.bulm.2004.10.003.CrossRefGoogle ScholarPubMed
Bellman, R. and Cooke, K. L., Differential-difference equations (Academic Press, New York, 1963).Google Scholar
Hale, J. K., Infante, E. F. and Tsen, F.-S. P., “Stability in linear delay equations”, J. Math. Anal. Appl. 105 (1985) 533–555; doi:10.1016/0022-247X(85)90068-X.CrossRefGoogle Scholar
Jain, H. V. and Byrne, H. M., “Qualitative analysis of an integro-differential model of periodic chemotherapy”, Appl. Math. Lett. 25 (2012) 2132–2136; doi:10.1016/j.aml.2012.04.024.CrossRefGoogle Scholar
Johnson, L. A., Byrne, H. M., Willis, A. E. and Laughton, C. A., “An integrative biological approach to the analysis of tissue culture data: application to the anti-tumour agent RHPS4”, Integr. Biol. 3 (2011) 843–849; doi:10.1039/C1IB00025J.CrossRefGoogle Scholar
Rasmussen, H., Wake, G. C. and Donaldson, J., “Analysis of a class of distributed delay logistic differential equations”, Math. Comput. Model. 38 (2003) 123–132; doi:10.1016/S0895-7177(03)90010-0.CrossRefGoogle Scholar
Simms, K., “A mathematical model of cell cycle progression applied to breast cancer cell lines”, Ph. D. Thesis, University of Adelaide, 2011.Google Scholar
Simms, K., Bean, N. and Koerber, A., “A mathematical model of cell cycle progression applied to the MCF-7 breast cancer cell line”, Bull. Math. Biol. 74 (2012) 736–767; doi:10.1007/s11538-011-9700-2.CrossRefGoogle Scholar