×

Tumours with cancer stem cells: a PDE model. (English) Zbl 1371.92061

Summary: The role of cancer stem cells (CSC) in tumour growth has received increasing attention in the recent literature. Here we stem from an integro-differential system describing the evolution of a population of CSC and of ordinary (non-stem) tumour cells formulated and studied in a previous paper, and we investigate an approximation in which the system reduces to a pair of nonlinear coupled parabolic equation. We prove that the new system is well posed and we examine some general properties. Numerical simulations show more on the qualitative behaviour of the solutions, concerning in particular the so-called tumour paradox, according to which an increase of the mortality rate of ordinary (non-stem) tumour cells results asymptotically in a faster growth.

MSC:

92C50 Medical applications (general)
35Q92 PDEs in connection with biology, chemistry and other natural sciences

Software:

BACOLR
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Beretta, E.; Capasso, V.; Morozova, N., Mathematical modelling of cancer stem cells population behavior, Math. Model. Nat. Phenom., 7, 1, 279-305 (2012) · Zbl 1241.92035
[3] Capasso, V.; Serio, G., A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42, 1-2, 43-61 (1978) · Zbl 0398.92026
[5] Enderling, H.; Anderson, R.; Chaplain, M.; Beheshti, A.; Hlatky, L.; Hahnfeldt, P., Paradoxical dependencies of tumor dormancy and progression on basic cell kinetics, Cancer Res., 69, 22, 8814-8821 (2009)
[6] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice-Hall · Zbl 0144.34903
[7] Ganguly, R.; Puri, I. K., Mathematical model for the cancer stem cell hypothesis, Cell Prolif., 39, 1, 3-14 (2006)
[8] Hillen, T.; Enderling, H.; Hahnfeldt, P., The tumor growth paradox and immune system-mediated selection for cancer stem cells, Bull. Math. Biol., 75, 1, 161-184 (2013) · Zbl 1272.92026
[9] Ladyzhenskaia, O.; ., V. S.; Ural’tseva, N., Linear and Quasi-linear Equations of Parabolic Type. Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Translations of Mathematical Monographs (1988), American Mathematical Society
[10] Maddalena, L., Analysis of an integro-differential system modeling tumor growth, Appl. Math. Comput., 245, C, 152-157 (2015) · Zbl 1335.92042
[11] Sole, R.; Rodriguez-Caso, C.; Deisboeck, T.; J. Saldance, Cancer stem cells as the engine of unstable tumor progression, J. Theor. Biol., 253, 4, 629-637 (2008) · Zbl 1398.92132
[12] Stiehl, T.; Marciniak-Czochra, A., Mathematical modeling of leukemogenesis and cancer stem cell dynamics, Math. Model. Nat. Phenom., 7, 01, 166-202 (2012) · Zbl 1241.92045
[13] Wang, R.; Keast, P.; Muir, P., Algorithm 874: Bacolr: Spatial and temporal error control software for PDEs based on high-order adaptive collocation, ACM Trans. Math. Softw., 34, 3, 15:1-15:28 (2008) · Zbl 1291.65400
[14] Youssefpour, H.; Li, X.; Lander, A.; Lowengrub, J., Multispecies model of cell lineages and feedback control in solid tumors, J. Theor. Biol., 304, 39-59 (2012) · Zbl 1397.92381
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.