Modelling cell populations with spatial structure: Steady state and treatment-induced evolution of tumour cords

Discrete and Continuous Dynamical Systems - Series B

Volumn 4, Issue 1, 2004, Pages 161-186

  Bertuzzi, Alessandro ^a   D'Onofrio, Alberto ^b   Fasano, Antonio ^c   Gandolfi, Alberto ^a  

^a CNR   (Italy)

^b EUROPEAN INSTITUTE OF ONCOLOGY   (Italy)

^c UNIVERSITY OF FLORENCE   (Italy)

Author keywords

Cell population; Free boundary problems; Nonlinear systems of differential and integral equations; Tumour cord

Indexed keywords

EID: 1042291061     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2004.4.161     Document Type: Conference Paper

References (33)

1. J.A. Adam and S.A. Maggelakis, Diffusion regulated growth characteristics of a spherical prevascular carcinoma, Bull. Math. Biol. 52 (1990), 549-582.
2. D. Ambrosi and L. Preziosi, On the closure of mass balance models for tumour growth, Math. Models Methods Appl. Sci. 12 (2002), 737-754.
3. A. Bertuzzi and A. Gandolfi, Cell kinetics in a tumour cord, J. theor. Biol. 204 (2000), 587-599.
4. A. Bertuzzi, A. Fasano and A. Gandolfi, A mathematical model for the growth of tumor cords incorporating the dynamics of a nutrient, in "Free Boundary Problems: Theory and Applications II" (ed. N. Kenmochi), Gakkotosho, Tokyo (2000), 31-46.
5. A. Bertuzzi, A. Fasano, A. Gandolfi and D. Marangi, Cell kinetics in tumour cords studied by a model with variable cell cycle length, Math. Biosci. 177&178 (2002), 103-125.
6. A. Bertuzzi, A. Fasano and A. Gandolfi, A free boundary problem with unilateral constraints describing the evolution of a tumour cord under the influence of cell killing agents, submitted.
7. A. Bertuzzi, A. d'Onofrio, A. Fasano and A. Gandolfi, Regression and regrowth of tumour cords following single-dose anticancer treatment, Bull. Math. Biol. 65 (2003), 903-931.
8. M.I.G. Bloor and M.J. Wilson, A mathematical model of a micrometastasis, J. Theor. Med. 1 (1997), 153-168.
9. M.I.G. Bloor and M.J. Wilson, The non-uniform spatial development of a micrometastasis, J. Theor. Med. 2 (1999), 55-71.
10. C.J.W. Breward, H.M. Byrne and C.E. Lewis, The role of cell-cell interactions in a two-phase model for avascular tumour growth, J. Math. Biol. 45 (2002), 125-152.
11. R.G. Bristow and R.P. Hill, Molecular and cellular basis of radiotherapy, in "The Basic Science of Oncology" (eds. I.F. Tannock and R.P. Hill), McGraw-Hill, New York (1998), 295-321.
12. H.M. Byrne and M.A.J. Chaplain, Growth of nonnecrotic tumors in the presence and absence of inhibitors, Math. Biosci. 130 (1995), 151-181.
13. J.J. Casciari, S.V. Sotirchos and R.M. Sutherland, Variations in tumor cell growth rates and metabolism with oxygen concentration, glucose concentration, and extracellular pH, J. Cell. Physiol. 151 (1992), 386-394.
14. S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl. 255 (2001), 636-677.
15. J. Dyson, R. Villella-Bressan and G. Webb, The steady state of a maturity structured tumor cord cell population, Discrete Contin. Dynam. Systems B, this issue.
16. K.H. Falkvoll, The relationship between changes in tumour volume, tumour cell density and parenchimal cord radius in a human melanoma xenograft exposed to single dose irradiation, Acta Oncol. 29 (1990), 935-939.
17. H.P. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math. 51 (1972), 317-340.
18. K. Groebe and P. Vaupel, Evaluation of oxygen diffusion distances in human breast cancer xenografts using tumor-specific in vivo data: role of various mechanisms in the development of tumor hypoxia, J. Radiat. Oncol. Biol. Phys. 15 (1988), 691-697.
19. D.G. Hirst and J. Denekamp, Tumour cell proliferation in relation to the vasculature, Cell Tissue Kinet. 12 (1979), 31-42.
20. D.G. Hirst, V.K. Hirst, B. Joiner, V. Praise and K.M. Shaffi, Changes in tumour morphology with alterations in oxygen availability: further evidence for oxygen as a limiting substrate, Br. J. Cancer 64 (1991), 54-58.
21. J. Holash, P.C. Maisonpierre, D. Compton, P. Boland, C.R. Alexander, D. Zagzag, G.D. Yankopoulos and S.J. Wiegand, Vessel cooption, regression, and growth in tumors mediated by angiopoietins and VEGF, Science 284 (1999), 1994-1998.
22. L. Holmgren, M.S. O'Reilly and J. Folkman, Dormancy of micrometastases: balanced proliferation and apoptosis in presence of angiogenesis suppression, Nature Med. 1 (1995), 149-153.
23. S.R. Lubkin, and T. Jackson, Multiphase mechanics of capsule formation in tumors, J. Biomech. Eng. 124 (2002), 237-243.
24. D.L.S. McElwain and P.J. Ponzo, A model for the growth of a solid tumor with non-uniform oxygen consumption, Math. Biosci. 35 (1977), 267-279.
25. F. Montalenti, G. Sena, P. Cappella and P. Ubezio, Simulating cancer cell kinetics after drug treatment: application to Cisplatin on ovarian carcinoma, Phys. Rev. E 57 (1998), 5877-5887.
26. J.V. Moore, H.A. Hopkins and W.B. Looney, Dynamic histology of a rat hepatoma and the response to 5-fluorouracil, Cell Tissue Kinet. 18 (1980), 53-63.
27. J.V. Moore, H.A. Hopkins and W.B. Looney, Response of cell populations in tumor cords to a single dose of cyclophosphamide or radiation, Eur. J. Cancer Clin. Oncol. 19 (1983), 73-79.
28. J.V. Moore, H.A. Hopkins and W.B. Looney, Tumour-cord parameters in two rat hepatomas that differ in their radiobiological oxygenation status, Radiat. Environ. Biophys. 23 (1984), 213-222.
29. J.V. Moore, P.S. Hasleton and C.H. Buckley, Tumour cords in 52 human bronchial and cervical squamous cell carcinomas: Inferences for their cellular kinetics and radiobiology, Br. J. Cancer 51 (1985), 407-413.
30. S.I. Rubinow, A maturity-time representation for cell populations, Biophys. J. 8 (1968), 1055-1073.
31. I.F. Tannock, The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumour, Br. J. Cancer 22 (1968), 258-273.
32. I. Tannock, and A. Howes, The response of viable tumor cords to a single dose of radiation, Radiat. Res. 55 (1973), 477-486.
33. G.F. Webb, The steady state of a tumor cord cell population, J. Evolut. Equat. 2 (2002), 425-438.