A multiple scale model for tumor growth

Multiscale Modeling and Simulation

Volumn 3, Issue 2, 2005, Pages 440-475

  Alarcon T ^a   Byrne, Helen M ^b   Maini, P K ^c  

^a UNIVERSITY COLLEGE LONDON   (United Kingdom)

^b UNIVERSITY OF NOTTINGHAM   (United Kingdom)

^c UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Blood flow heterogeneity; Cell cycle; Growth laws; Multiple scales; p27; p53; Physiologically structured lattice model

Indexed keywords

ALGORITHMS; BIFURCATION (MATHEMATICS); BLOOD; BOUNDARY CONDITIONS; CELLS; COMPUTER SIMULATION; GROWTH (MATERIALS); ORDINARY DIFFERENTIAL EQUATIONS; TISSUE; TUMORS;

BLOOD FLOW HETEROGENEITY; CELL CYCLE; GROWTH LAWS; MULTIPLE SCALE MODELS; PHYSIOLOGICALLY STRUCTURED LATTICE MODEL; TUMOR GROWTH;

MATHEMATICAL MODELS;

EID: 21844462331     PISSN: 15403459     EISSN: None     Source Type: Journal    
DOI: 10.1137/040603760     Document Type: Article

References (72)

1. J. A. Adam, A mathematical model of tumour growth II: Effects of geometry and spatial uniformity on stability, Math. Biosci., 86 (1987), pp. 183-211.
2. J. A. Adam, A mathematical model of tumour growth III: Comparison with experiment, Math. Biosci., 86 (1987), pp. 213-227.
3. T. Alarcón, H. M. Byrne, and P. K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment, J. Theor. Biol., 225 (2003), pp. 257-274.
4. T. Alarcón, H. M. Byrne, and P. K. Maini, Mathematical model of the effect of hypoxia on the cell-cycle of normal and cancer cells, J. Theor. Biol., 229 (2004), pp. 395-411.
5. M. S. Alber, M. A. Kiskowski, J. A. Glazier, and Y. Jiang, On cellular automaton approaches to modeling biological cells, in Mathematical Systems Theory in Biology, Communications, Computation, and Finance, J. Rosenthal and D. S. Gilliam, eds., IMA Vol. Math. Appl. 134, Springer-Verlag, New York, 2003, pp. 1-39.
6. B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts, and J. D. Watson, Molecular Biology of the Cell, 3rd ed., Wiley-Interscience, New York, 1994.
7. A. R. A. Anderson and M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol., 60 (1998), pp. 857-899.
8. S. A. Barber, Soil Nutrient Bioavailability: A Mechanistic Approach, Garland, New York, 1984.
9. N. Bellomo and L. Preziosi, Modelling and mathematical problems related to tumour evolution and its interaction with the immune system, Math. Comput. Modelling, 32 (2000), pp. 413-452.
10. C. J. W. Breward, H. M. Byrne, and C. E. Lewis, A multiphase model describing vascular tumour growth, Bull. Math. Biol., 65 (2003), pp. 609-640.
11. A. Bru, J. M. Pastor, I. Fernaud, I. Bru, S. Melle, and C. Berenguer, Super-rough dynamics on tumor growth, Phys. Rev. Lett., 81 (1998), pp. 4008-4011.
12. H. M. Byrne and M. A. J. Chaplain, Free boundary value problems associated with the growth and development of multicellular spheroids European J. Appl. Math., 8 (1997), pp. 639-658.
13. H. M. Byrne, J. R. King, and D. L. S. McElwain, A two-phase model of solid tumour growth, Appl. Math. Lett., 16 (2003), pp. 567-573.
14. P. Carmeliet, Y. Dor, J.-M. Herbert, D. Fukumura, K. Brusselmans, M. Dewerchin, M. Neeman, F. Bono, R. Abramovitch, P. Maxwell, K. Koch, P. Ratcliffe, L. Moons, R. Jain, D. Collen, and E. Keshet, Role of HIF-∞ in hypoxia-mediated apoptosis, cell proliferation and tumour angiogenesis, Nature, 394 (1998), pp. 485-490.
15. M. A. J. Chaplain, Avascular growth, angiogenesis and vascular growth in solid tumours: The mathematical modelling of the stages of tumour development, Math. Comput. Modelling, 23 (1996), pp. 47-87.
16. R. Chatuverdi, J. A. Izaguirre, C. Huang, T. Cickovski, P. Virtue, G. Thomas, G. Forgacs, M. Alber, G. Hentschel, S. A. Newman, and J. A. Glazier, Multi-model simulations of chicken limb morphogenesis, in International Conference on Computational Science, Lecture Notes in Comput. Sci. 2659, Springer-Verlag, Berlin, 2003, pp. 39-49.
17. J. L. Cherry and F. R. Adler, How to make a biological switch, J. Theor. Biol., 203 (2000), pp. 117-133.
18. S. Cory, D. C. S. Huang, and J. M. Adams, The Bcl-2 family: Roles in cell survival and oncogenesis, Oncogene, 22 (2003), pp. 8590-8607.
19. J. Crown, High dose chemotherapy of metastatic breast cancer: The end of the beginning?, British J. Cancer, 75 (1997), pp. 467-469.
20. A. Deutsch and S. Dormann, Modeling of avascular tumor growth with a hybrid cellular automaton, In Silico Biol., 2 (2002), pp. 1-14.
21. D. Drasdo and S. Hohme, Individual-based approaches to birth and death in avascular tumors, Math. Comput. Modelling, 37 (2003), pp. 1163-1175.
22. J. Folkman and M. Hochberg, Self-regulation of growth in three dimensions, J. Exp. Med., 138 (1973), pp. 745-753.
23. S. J. Franks, H. M. Byrne, J. R. King, and C. E. Lewis, Modelling the growth of ductal carcinoma in situ, J. Math. Biol., 47 (2003), pp. 424-452.
24. Y. C. Fung, Biomechanics, Springer-Verlag, New York, 1993.
25. J. O. Funk, Cancer cell cycle control, Anticancer Res., 19 (1999), pp. 4772-4780.
26. D. Gammak, H. M. Byrne, and C. E. Lewis, Estimating the selective advantage of mutant p53 tumour cells to repeated rounds of hypoxia, Bull. Math. Biol., 19 (1999), pp. 4772-4780.
27. 1/S transition through regulation of p27 expression, J. Biol. Chem., 276 (2001), pp. 7919-7926.
28. R. A. Gatenby, Application of competition theory to tumour growth: Implications for tumour biology and treatment, Eur. J. Cancer., 32A (1996), pp. 722-726.
29. R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), pp. 5745-5753.
30. D. Goldman and A. S. Popel, A computational study of the effect of capillary network anastomoses and tortuosity on oxygen transport, J. Theor. Biol., 206 (2000), pp. 181-194.
31. T. G. Graeber, C. Osmanian, T. Jacks, D. E. Housman, C. J. Koch, S. W. Lowe, and A. J. Giaccia, Hypoxia-mediated selection of cells with diminished apoptotic potential in human tumours, Nature, 379 (1996), pp. 89-91.
32. F. Graner and J. A. Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model, Phys. Rev. Lett., 69 (1992), pp. 2013-2016.
33. H. P. Greenspan, Models for the growth of a solid tumour by diffusion, Stud. Math. Appl., 51 (1972), pp. 317-340.
34. H. P. Greenspan, On the growth and stability of cell cultures and solid tumours, J. Theor. Biol., 56 (1976), pp. 229-242.
35. D. Hart, E. Scochat, and Z. Agur, The growth law of primary breast cancer as inferred from mammography screening trial data, British J. Cancer, 78 (1998), pp. 382-387.
36. P. J. Hunter, P. Robbins, and D. Noble, The IUPS human physiome project, Pflügers Archiv- Eur. J. Physiol., 445 (2002), pp. 1-9.
37. P. J. Hunter, P. Kohl, and D. Noble, Integrative models of the heart: Achievements and limitations, Philos. Trans. Roy. Soc. London Ser. A, 359 (2001), pp. 1049-1054.
38. T. Jackson and H. M. Byrne, A mathematical model of tumour encapsulation, Math. Biosci., 180 (2002), pp. 307-328.
39. R. K. Jain, Delivery of molecular and cellular medicine to solid tumours, Advanced Drug Delivery Reviews, 46 (2001), pp. 149-168.
40. A. R. Kansal, S. Torquato, G. R. Harsh, IV, E. A. Chiocca, and T. S. Deisboeck, Simulated brain tumour growth dynamics using a three-dimensional cellular automaton, J. Theor. Biol., 203 (2000), pp. 367-382.
41. F. D. L. Kelpin, M. A. Kirkilionis, and B. W. Kooi, Numerical methods and parameter estimation of a structured population model with discrete events in the life history, J. Theor. Biol., 207 (2000), pp. 217-230.
42. K. W. Kinzler and B. Vogelstein, Life (and death) in a malignant tumour, Nature, 379 (1996), pp. 19-20.
43. R. M. Kirla, H. K. Haapasalo, H. Kalimo, and E. K. Salminen, Low expression of p27 indicates a poor prognosis in patients with high grade astrocytomas, Cancer, 97 (2003), pp. 644-648.
44. M. A. konerding, E. Fait, and A. Gautmann, 3D microvascular architecture of precancerous lesions and invasive carcinomas, British J. Cancer, 84 (2001), pp. 1354-1362.
45. H. A. Levine, S. Pamuk, B. D. Sleeman, and M. Nilsen-Hamilton, Mathematical modeling of capillary formation and development in tumor angiogenesis: Penetration into the stroma, Bull. Math. Biol., 63 (2001), pp. 801-863.
46. B. P. Marchant, J. Norbury, and J. A. Sherratt, Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), pp. 1653 1671.
47. D. L. S. McElwain and L. E. Morris, Apoptosis as a volume loss mechanims in mathematical models of tumour growth, Math. Biosci., 39 (1978), pp. 147-157.
48. J. Moreira and A. Deutsch, Cellular automaton models of tumour development: A critical review, Adv. Complex Sys., 5 (2002), pp. 247-267.
49. W. K. Nisioka and R. M. Welsh, Susceptibility to cytotoxic T lymphocite-induced apoptosis is a function of the proliferative status of the target, J. Exp. Med., 179 (1994), pp. 769-774.
50. P. H. Nye and P. B. Tinker, Solute Movement in the Soil-Root System, Blackwell Science, Oxford, UK, 1977.
51. M. E. Orme and M. A. J. Chaplain, A mathematical model of vascular tumour growth and invasion, Math Comput. Modelling, 23 (1996), pp. 43-60.
52. A. A. Patel, E. T. Gawlinski, S. K. Lemieux, and R. A. Gatenby, A cellular automaton model of early tumor growth and invasion: The effects of native tissue vascularity and increased anaerobic tumor metabolism J. Theor. Biol., 213 (2001), pp. 315-331.
53. G. Pettet, C. P. Please, M. J. Tindall, and D. L. S. McElwain, The migration of cells in multicell tumor spheroids, Bull. Math. Biol., 63 (2001), pp. 231-257.
54. C. P. Please, C. Pettet, and D. L. S. McElwain, A new approach to modelling the formation of necrotic regions in tumours, Appl. Math. Lett., 11 (1998), pp. 89-94.
55. M. J. Poole, A. V. Holden, and J. V. Tucker, Hierarchical reconstruction of cardiac tissue, Chaos Solitons Fractals, 13 (2002), pp. 1581-1612.
56. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, UK, 1992.
57. L. Preziosi., ED., Cancer Modelling and Simulation, CRC Press, Boca Raton, FL, 2003.
58. A. R. Pries, T. W. Secomb, T. Gessner, M. B. Sperandio, J. F. Gross, and P. Gaehtgens, Resistance to blood flow in microvessels in vivo, Circ. Res., 75 (1994), pp. 904-915.
59. A. R. Pries, T. W. Secomb, and P. Gaehtgens, Design principles of vascular beds, Circ. Res., 77 (1995), pp. 1017-1023.
60. A. R. Pries, T. W. Secomb, and P. Gaehtgens, Structural adaptation and stability of microvascular networks: Theory and simulations, Am. J. Physiol., 275 (1998), pp. H349-H360.
61. B. Ribba, K. Marron, Z. Agur, T. Alarcón, and P. K. Maini, Doxorubicin treatment efficacy on non-Hodgkin's lymphomas: Computer model and simulations, Bull. Math. Biol., submitted.
62. T. Roose, A. C. Fowler, And P. R. Darrah, A mathematical model of plant nutrient uptake, J. Math. Biol., 42 (2001), pp. 347-360.
63. D. H. Rothman and S. Zalesky, Lattice-Gas Cellular Automata: Simple Models of Complex Hydrodynamics, Cambridge University Press, Cambridge, UK, 1997.
64. J. A. Royds, S. K. Dower, E. E. Qwarstrom, and C. E. Lewis, Response of tumour cells to hypoxia: Role of p53 and NFkB, J. Clin. Path.: Mol. Pathol., 51 (1998), pp. 55-61.
65. Z. Sinik, T. Alkibay, O. Ataoglu, H. Biri, S. Sozen, N. Dinez, U. Karaoglan, and I. Bozkirli, Nuclear p53 overexpression in bladder, prostate, and renal carcinomas, Int. J. Urol., 4 (1997), pp. 546-551.
66. N. P. Smith, P. J. Mulquiney, M. P. Nash, C. P. Bradley, D. P. Nickerson, and P. J. Hunter, Mathematical modelling of the heart: Cell to organ, Chaos Solitons Fractals, 13 (2002), pp. 1613-1621.
67. M. J. Tindall, Modelling Cell Movement and the Cell Cycle in Multicellular Tumour Spheroids, Ph.D. dissertation, University of Southampton, Southampton, UK, 2002.
68. S. Turner and J. A. Sherrat, Intercellular adhesion and cancer invasion: A discrete simulation using the extended Potts model, J. Theor. Biol., 216 (2002), pp. 85-100.
69. J. J. Tyson and B. Novak, Regulation of the eukariotic cell cycle: Molecular antagonism, hysteresis, and irreversible transitions, J. Theor. Biol., 210 (2001), pp. 249-263.
70. J. Valenciano and M. A. J. Chaplain, An explicit subparametric spectral element method of lines applied to a tumour angiogenesis system of partial differential equations, Math. Models Methods Appl. Sci., 14 (2004), pp. 165-187.
71. J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth, IMA J. Math. Appl. Med., 14 (1997), pp. 39-69.
72. J. P. Ward and J. R. King, Mathematical modelling of avascular-tumour growth II: Modelling growth saturation, IMA J. Math. Appl. Med., 16 (1999), pp. 171-211.