Modelling solid tumour growth using the theory of mixtures

Mathematical Medicine and Biology

Volumn 20, Issue 4, 2003, Pages 341-366

  Byrne, Helen ^a   Preziosi, Luigi ^b  

^a UNIVERSITY OF NOTTINGHAM   (United Kingdom)

^b POLITECNICO DI TORINO   (Italy)

Author keywords

Cellular dynamics; Deformable porous media; Tumour growth

Indexed keywords

CELL DEATH; CELL PROLIFERATION; MIXTURES; POROUS MATERIALS; STRESS ANALYSIS;

CELL PROLIFERATION RATE; CELLULAR DYNAMICS; DEFORMABLE POROUS MEDIA; LIQUID PHASIS; MODEL SOLIDS; SOLID TUMORS; THEORY OF MIXTURES; TUMOR GROWTH; TWO PHASE; TWO-PHASE MODEL;

TUMORS;

ARTICLE; CANCER MODEL; CELL DEATH; CELL PROLIFERATION; CLINICAL FEATURE; HUMAN; HUMAN CELL; MATHEMATICAL ANALYSIS; MATHEMATICAL PARAMETERS; PREDICTION; SENSITIVITY ANALYSIS; SOLID TUMOR; STATISTICAL ANALYSIS; TUMOR CELL; TUMOR GROWTH; TUMOR VOLUME;

EID: 0842281982     PISSN: 14778599     EISSN: None     Source Type: Journal    
DOI: 10.1093/imammb/20.4.341     Document Type: Article

References (52)

1. Adam, J. A. (1987) A mathematical model of tumour growth. II. Effects of geometry and spatial uniformity on stability. Math. Biosci., 86, 183-211.
2. Ambrosi, D. & Preziosi, L. (2002) On the closure of mass balance models of tumour growth. Math: Mod. Meth. Appl. Sci., 12, 737-754.
3. Barker, M. K., & Seedhom, B. B. (1997) Articular cartilage deformation under physiological cycling loading. J. Biomech., 30, 377-381.
4. Bellomo, N. & Preziosi, L. (1995) Modelling, Mathematical Methods and Scientific Computation. Boca Raton, FL: CRC Press.
5. Bowen, R. M. (1976) Theory of mixtures. Continuum Physics. (A. C. Eringen, ed.). New York: Academic.
6. Bowen, R. M. (1980) Incompressible porous media model by use of the theory of mixtures. Int. J. Eng. Sci., 18, 1129-1148.
7. Breward, C. J. W., Byrne, H. M. & Lewis, C. E. (2002) The role of cell-cell interactions in a two-phase of solid tumour growth. J. Math. Biol., 45, 125-152.
8. Byrne, H. M. & Chaplain, M. A. J. (1997) Free boundary value problem associated with the growth and development of multicellular sphroids. Eur. J. Appl. Math., 8, 639-658.
9. Byrne, H. M. (2003) Modelling avascular tumour growth. Cancer Modelling and Simulation, Mathematical Biology and Medicine Series, 3. (L. Preziosi, ed.). London: Chapman and Hall/CRC, pp. 75-120.
10. Byrne, H. M., King, J. R., McElwain, D. L. S. & Preziosi, L. (2003) A two-phase model of solid tumour growth. Appl. Math. Lett., 16, 567-573.
11. Carmeliet, P. & Jain, R. K. (2000) Angiogenesis in cancer and other diseases. Nature, 407, 249-257.
12. Chen, C. Y., Byrne, H. M. & King, J. R. (2001) The influence of growth -induced stress from the surrounding medium on the development of multicell spheroids. J. Math. Biol., 43, 191-220.
13. Craft, P. S. & Harris, A. L. (1994) Clinical prognostic-significance of tumour angiogenesis. Ann. Oncol., 5, 305-311.
14. Curtis, A. S. G. & Sheehar, G. M. (1971) The control of cell division by tension or diffusion. Nature, 274, 52-53.
15. Drew, D. A. & Segel, L. A. (1971) Averaged equations for two-phase flows. Stud. Appl. Math., 50, 205-231.
16. Ehlers, W. (1993) Constitutive equations for granular materials in geomechanical context. Environmental Sciences and Geophysics, CISM Courses and Lectures, 337. (K. Hutter, ed.). Berlin: Springer.
17. Elliott, C. M. (1985) The Stefan problem with a non-monotone constitutive relation. IMA J. Appl. Math., 35, 257-264.
18. Farina, A. & Preziosi, L. (2000) Deformable porous media and composites manufacturing. Heterogenous Media: Micromechanics, Modelling, Methods and Simulations. (K. Markov & L. Preziosi, eds). Basel: Birkhäuser.
19. Farina, A. & Preziosi, L. (2001). On Darcy's law for growing porous media. Int. J. Nonlinear Mech., 37, 485-491.
20. Folkman, J. (1974) Tumor angiogenesis. Adv. Cancer Res., 19, 331-358.
21. Folkman, J. & Hochberg, M. (1973) Self-regulation of growth in three dimensions. J. Exp. Med., 138, 745-753.
22. Fowler, A. C. (1997) Mathematical Models in the Applied Sciences. Cambridge: Cambridge University Press.
23. Frijns, A. J. H., Huyghe, J. M. & Janssen, J. D. (1997) A validation of the quadriphasic mixture thory for intervertebral disc tissue. Int. J. Eng. Sci., 35, 1419-1429.
24. Franks, S. J., Byrne, H. M., Underwood, J. C. E. & Lewis, C. E. (in press). Mathematical modelling of comedo ductal carcinoma in situ of the breast. IMAMMB.
25. Gatenby, R. A. & Gawlinski, E. T. (1996) A reaction-diffusion model of cancer invasion. Cancer Res., 56, 5745-5753.
26. Greenspan, H. P. (1972) Models for the growth of a solid tumour by diffusion. Stud. Appl. Math., 52, 317-340.
27. Greenspan, H. P. (1976). On the growth and stability of cell cultures and solid tumours. J. Theor. Biol., 56, 229-242.
28. Helmlinger, G., Netti, P. A., Lichtenbeld, H. C., Melder, R. J. & Jain, R. K. (1997) Solid stress inhibits the growth of multicellular tumour spheroids. Nature Biotech., 15, 778-783.
29. Hou, J. S., Holmes, M. H., Lai, W. M. & Mow, V. C. (1989) Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. J. Biomech. Eng., 111, 78-87.
30. Kunz-Schughart, L., Doetsch, J., Mueller-Klieser, W. & Groebe, K. (2000). Proliferative activity and tumourigenic conversion: Impact on cellular metabolism in 3-D culture. Amer. J. Physiol. Cell Physiol., 278, C765-C780.
31. Lai-Fook, S. J. (1988) Pressure-flow behaviour of pulmonary interstitium. J. Appl. Physiol., 64, 2372-2380.
32. Landman, K. & Please, C. P. (2001) Tumor dynamics and necrosis: Surface tension and stability. IMA J. Maths. Appl. Med. Biol., 18, 131-158.
33. McElwain, D. L. S. & Morris, L. E. (1978) Apoptosis as a volume loss mechanism in mathematical models of solid tumour growth. Math. Biosci., 39, 147-157.
34. Mow, V. C. & Lai, W. M. (1979) Mechanics of animal joints. Ann. Rev. Fluid Mech., 11, 247-288.
35. Netti, P. A., Baxter, L. T., Boucher, Y., Skalak, R. & Jain, R. K. (1997) Macro- and microscopic fluid transport in living tissues: Application to solid tumours. AIChE J., 43, 818-834.
36. Nicholson, C. (1985) Diffusion from an injected volume of a substance in brain tissues with arbitary volume fraction and tortuosity. Brain Res., 333, 325-329.
37. Please, C. P., Pettet, G. & McElwain, D. L. S. (1998) A new approach to modelling the formation of necrotic regions in tumours. Appl. Math. Lett., 11, 89-94.
38. Preziosi, L. (1996) The theory of deformable porous media and its application to composite material manufacturing. Surv. Math. Industr., 6, 167-214.
39. Rajagopal, K. R. & Tao, L. (1995) Mechanics of Mixtures. Singapore: World Scientific.
40. Sherrat, J. A. & Nowak, M. A. (1992) Oncogenes, anti-oncogenes and the immune response to cancer. Proc. R. Soc. B, 248, 261-271.
41. Sorek, S. & Sideman, S. (1986) A porous medium approach for modelling heart mechanics: Part 1. Theory. Math. Biosci., 81, 1-14.
42. Stetler-Stevensson, W. G., Aznavoorian, S. & Liotta, L. A. (1993) Tumor cell interactions with the extracellular matrix during invasion and metastasis. Ann. Rev. Cell Biol., 9, 541-573.
43. Sutherland, R. M. (1988) Cell and environment interactions in tumour microregions: The multicell spheroid model. Science, 240, 177-184.
44. Sutherland, R. M. & Durand, R. E. (1984) Growth of cellular characteristics of multicell spheroids. Recent Results in Cancer Research, Vol. 95. Berlin: Springer, pp. 24-49.
45. Wang, C. C. (1969) On representations for isotropic functions. II. Isotropic functions of skew-symmetric tensors, symmetric tensors, and vectors. Arch. Ration. Mech. Anal., 33, 268-287.
46. Wang, C. C. (1970) A new representation theorem for isotropic functions: An answer to Professor G.F. Smith's criticism of my papers on representations for isotropic functions. II. Vector-valued isotropic functions, symmetric ten tensor-valued isotropic functions, and skew-symmetric tensor-valued isotropic functions. Arch. Ration. Mech. Anal., 36, 198-223.
47. Ward, J. P. & King, J. R. (1997) Mathematical modelling of avascular-tumour growth. IMA J. Math. Appl. Med. Biol., 14, 39-69.
48. Ward, J. P. & King, J. R. (1998) Mathematical modelling of avascular-tumour growth II: Modeling growth saturation. IMA J. Math. Appl. Med. Biol., 15, 1-42.
49. Ward, J. P. & King, J. R. (1999) Mathematical modelling of the effects of mitotic inhibitors on avascular tumour growth. J. Theor. Med., 1, 171-211.
50. Witelski, T. P. (1995) Shocks in nonlinear diffusion. Appl. Math. Lett., 8, 27-32.
51. Wu, J. Z. & Epstein, M. (1997) An improved solution for the contact of two biphasic cartilages layers. J. Biomech., 30, 371-375.
52. Yang, M., Taber, L. A. & Clark, E. B. (1994) A non-linear poroelasticity model for the trabecular embryonic heart. J. Biomech. Engng, 116, 213-223.