Mathematical modelling of avascular-tumour growth

IMA Journal of Mathemathics Applied in Medicine and Biology

Volumn 14, Issue 1, 1997, Pages 39-69

  Ward, J P ^a   King, J R ^a  

^a UNIVERSITY OF NOTTINGHAM   (United Kingdom)

Author keywords

Asymptotic analysis; Avascular; Mathematical modelling; Numerical solution; Tumour growth

Indexed keywords

CELL DEATH; NONLINEAR EQUATIONS; NUMERICAL MODELS; NUTRIENTS; PARTIAL DIFFERENTIAL EQUATIONS; TUMORS;

AVASCULAR; LIVE CELL; MATHEMATICAL MODELING; NUMERICAL SOLUTION; SYSTEM OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS; TUMOR GROWTH; TUMOR SPHEROID; TWO-STATE; VELOCITY FIELD; VOLUME CHANGE;

ASYMPTOTIC ANALYSIS;

ANIMAL; BIOLOGICAL MODEL; CELL DEATH; CELL DIVISION; HUMAN; KINETICS; MATHEMATICS; MULTICELLULAR SPHEROID; PATHOLOGY; REVIEW;

ANIMALS; CELL DEATH; CELL DIVISION; HUMANS; KINETICS; MATHEMATICS; MODELS, BIOLOGICAL; SPHEROIDS, CELLULAR;

EID: 0031087466     PISSN: 02650746     EISSN: None     Source Type: Journal    
DOI: 10.1093/imammb/14.1.39     Document Type: Article

References (50)

1. ADAM, J. A., 1986. A simplified mathematical model of tumour growth. Math. Biosci. 81, 224-9.
2. ADAM, J. A., 1987a. A mathematical model of tumour growth II: Effects of geometry and symmetry. Math. Biosci. 86, 183-211.
3. ADAM, J. A., 1987b. A mathematical model of tumour growth III: Comparison with experiment. Math. Biosci. 86, 213-27.
4. ADAM, J. A., & Maggelakis, S. A., 1989. Mathematical models of tumour growth IV: Effects of necrotic core. Math. Biosci. 97, 121-36.
5. BURTON, A. C., 1966. Rate of growth of solid tumours as a problem of diffusion. Growth. 30, 157-76.
6. BYRNE, H. M., & CHAPLAIN, M. A. J., 1995. Growth of nonnccrotic tumours in the presence and absence of inhibitors. Math. Biosci. 130, 151-81.
7. BYRNE, H. M., & CHAPLAIN. M. A. J., 1996. Growth of necrotic tumours in the presence and absence of inhibitors. Math. Biosci., 135, 187-216.
8. CARLSSON, J., 1977. A proliferation gradient in three-dimensional colonies of cultured human glioma cells. Int. J. Cancer. 20, 129-36.
9. CASCIARI, J. J., SOTIRCHOS. S. V., & SUTHERLAND, R. M. 1988. Glucose diffusivity in multicellular tumour spheroids. Cancer Res. 48, 3905-9.
10. CASCIARI, J. J., SOTIRCHOS, S. V. & SUTHERLAND, R. M., 1992a. Mathematical modelling of micro-environment and growth in EMT6/RO multicellular tumour spheroids. Cell Prolif. 25, 1-22.
11. CASCIARI, J. J., SOTIRCHOS, S. V., & SUTHERLAND, R. M., 1992b. Variation in tumour cell growth rates and metabolism with oxygen-concentration, glucose-concentration and extracellular pH. J Cell. Physiol. 151, 386-94.
12. CHAPLAIN, M. A. J., & BRITTON, N. F., 1993. On the concentration profile of a growth-inhibitory factor in multicell spheroids. Math. Biosci. 115, 233-43.
13. CHAPLAIN, M. A. J., BENSON, D. L., & MAINI, P. K., 1994. Non-linear diffusion of a growth inhibitory factor in multicellular spheroids. Math. Biosci. 121, 1-13.
14. CONGER, A. D., & ZISKIN, M. C., 1983. Growth of mammalian multicellular tumour spheroids. Cancer Res. 43, 558-60.
15. COURANT, R., ISAACSON, E., & REES, M., 1952. On the solution of nonlinear hyperbolic differential equations by finite differences. Commun. Pure Appl. Math. V, 245-55.
16. DEAKIN, A. S., 1975. Model for the growth of a solid in vitro tumour. Growth, 39, 159-65.
17. FOLKMAN, J., 1971, Tumour angiogenesis: Therapeutic implications. New Engl. J. Med. 285, 1182-6.
18. FOLKMAN, J..& HOCHBERG, M., 1973. Self-regulation of growth in three dimensions. J. Exp. Med. 138, 745-53.
19. FREYER, J. P., 1988. Role of necrosis in regulating the growth saturation of multicell spheroids. Cancer Res. 48, 2432-9.
20. FREYER, J. P.. & SCHOR, P. L., 1987. Regrowth of cells from multicell tumour spheroids. Cell Tiss. Kinet. 20, 249.
21. FREYER. J. P., & SUTHERLAND, R. M., 1983. Determination of apparent diffusion constants for metabolites in multicell tumour spheroids. Adv. Exp. Med. Biol. 159, 463-75.
22. FREYER, J. P., & SUTHERLAND, R. M., 1986. Regulation of growth saturation and development of necrosis in EMT6/RO multicellular spheroids glucose and oxygen supply. Cancer Res. 46, 3504-12.
23. GLASS, L., 1973. Instability and mitotic patterns in tissue growth. Trans. ASME J. Dyn. Syst. Meas. Cont. G95, 324.
24. GREENSPAN, H. P., 1972. Models for the growth of a solid tumour by diffusion. Stud. Appl. Math. 51, 317-40.
25. HAJI-KARIM, M., & CARLSSON, J., 1978. Proliferation and viability in cellular spheroids of human origin. Cancer Res. 38, 1457-64.
26. HLATKY, L.. & ALPEN. E. L., 1985. Two-dimensional diffusion limited system for cell growth. Cell Tiss. Kinet. 18, 597-611
27. HLATKY, L., SUCHS, R. K., & ALPEN, E. L., 1988, Joint oxygen-glucose deprivation as the cause of necrosis in a tumour analogue. J. Cell Phys. 134, 167-78.
28. INCH, W. R., MCCREDIE, J. A., & SUTHERLAND, R. M., 1970. Growth of modular carcinomas in rodents compared with multicell spheroids in tissue culture. Growth 34, 271-82.
29. KOCH, C. J., KRUUV. J.. & FREY, H. E., 1973. The effect of hypoxia on the generation time of mammalian cells. Radiat. Res. 53, 43-8.
30. LANDRY, J,. FREYER. J. P.. & SUTHERLAND, R. M,. 1981. Shedding of mitotic cells from the surface of multicell spheroids during growth. J. Cell Physiol. 106, 23-32.
31. LANDRY, J., FREYER. J. P., & SUTHERLAND, R. M., 1982. A model for the growth of multicell spheriods. Cell Tiss. Kinet. 15, 585-94.
32. LEVINE, A. E., HAMILTON, D. A.. YEOMAN, L. C., BUSCH, H., & BRATTAIN, M. G., 1984. Identification of a tumour inhibitory factor in rat ascites fluid. Biochem. Biophys. Res. Comm. 119, 76-82
33. LI, C. K. N., 1982a. The glucose distribution in 9L rat brain multicell tumour spheroids and its effect on cell necrosis. Cancer, 50, 2066-73.
34. LI, C. K. N., 1982b. The role of glucose on the growth of 9L multicell tumour spheroids. Cancer, 50, 2074-8.
35. LIN, S. H., 1976. Oxygen diffusion in a spheroid cell with non-linear uptake kinetics, J. Theor. Biol. 60, 449-57.
36. MAGGELAKIS, S. A., 1992. Effects of non-uniform inhibitor production on the growth of cancer cell cultures. Appl. Math. Lett. 5, 11-4.
37. MAGGELAKIS, S. A., & ADAM, J.A., 1990. Mathematical model of prevascular growth of a spheroid carcinoma. Math. Comput. Mod. 13, 23-38.
38. MCELWAIN, D. L. S., 1978, A reexamination of oxygen diffusion in a spheroid cell with Michaelis-Mentor oxygen uptake kinetics. J. Theor. Biol. 71, 255-67.
39. MCELWAIN, D. L. S., & MORRIS, L. E., 1978. Apoptosis as a volume loss mechanism in mathematical models of solid tumour growth. Math. Biosci. 39, 147-57.
40. MCELWAIN, D. L. S., & PETTET, G. J., 1993. Cell migration in multicell spheroids: Swimming against the tide. Bull. Math. Biol. 55, 655-74.
41. MCELWAIN, D. L. S., & PONZO, P. J., 1977. A model for the growth of a solid tumour with nonuniform oxygen consumption. Math. Biosci. 35, 267-79.
42. MELICOW, M. M., 1982. The three-steps to cancer; A new concept of carcinogenesis. J. Theor. Biol. 94, 471-511.
43. MOORE. J. V., 1987. Death of cells and necrosis in tumours. In: Perspectives in Mammalian Cell Death (C. S. Potten, ed,). Oxford University Press.
44. MUELLER-KLIESER, W. F., & SUTHERLAND, R. M., 1982. Oxygen tension in multicell spheroids of two cell lines. Br. J. Cancer, 45, 256-64.
45. MUELLER-KLIESER, W. F., FREYER, J. P., & SUTHERLAND, R.M. 1983, Evidence for a major role of glucose in controlling development of necrosis in EMT6/RO multicell tumour spheroids, Adv. Exp. Med. Biol. 159, 487-95.
46. MUELLER-KLIESER, W. F., FREYER, J. P., & SUTHERLAND, R. M., 1985. Influence of glucose and oxygen supply conditions on the oxygenation of multicellular tumour spheroids. Br. J. Cancer, 53, 345-53.
47. SHYMKO, R. M., & GLASS, L., 1976. Cellular and geometric control of tissue growth and mitotic instability. J. Theor. Biol. 63, 355-74.
48. THOMLINSON, R. H., & GRAY, L. H., 1955. Histological structure of some human lung cancers and the possible implications for radiotherapy. Br. J. Cancer, 9, 539-49.
49. VAUPEL, P. W., FRINAK, S., & BICHER, H. I., 1981. Heterogenous oxygen partial pressure and pH distribution in C3H mouse mammary adenocarcinoma. Cancer Res. 41, 2008-13.
50. WEISS, L., 1978. Some mechanisms involved in cancer cell detachment by necrotic material, Int. J. Cancer, 22, 196-203.