# A history of the study of solid tumour growth: The contribution of mathematical modelling

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## Abstract

A miscellany of new strategies, experimental techniques and theoretical approaches are emerging in the ongoing battle against cancer. Nevertheless, as new, ground-breaking discoveries relating to many and diverse areas of cancer research are made, scientists often have recourse to mathematical modelling in order to elucidate and interpret these experimental findings. Indeed, experimentalists and clinicians alike are becoming increasingly aware of the possibilities afforded by mathematical modelling, recognising that current medical techniques and experimental approaches are often unable to distinguish between various possible mechanisms underlying important aspects of tumour development.

This short treatise presents a concise history of the study of solid tumour growth, illustrating the development of mathematical approaches from the early decades of the twentieth century to the present time. Most importantly these mathematical investigations are interwoven with the associated experimental work, showing the crucial relationship between experimental and theoretical approaches, which together have moulded our understanding of tumour growth and contributed to current anti-cancer treatments.

Thus, a selection of mathematical publications, including the influential theoretical studies by Burton, Greenspan, Liotta *et al.*, McElwain and co-workers, Adam and Maggelakis, and Byrne and co-workers are juxtaposed with the seminal experimental findings of Gray *et al.* on oxygenation and radio-sensitivity, Folkman on angiogenesis, Dorie *et al.* on cell migration and a wide variety of other crucial discoveries. In this way the development of this field of research through the interactions of these different approaches is illuminated, demonstrating the origins of our current understanding of the disease.

### References

- Adam, J. A. (1986). A simplified mathematical model of tumor growth.
*Math. Biosci.***81**, 229–244.MATHCrossRefGoogle Scholar - Adam, J. A. (1987a). A mathematical model of tumor growth. ii. Effects of geometry and spatial uniformity on stability.
*Math. Biosci.***86**, 183–211.MATHCrossRefGoogle Scholar - Adam, J. A. (1987b). A mathematical model of tumor growth. iii. Comparison with experiment.
*Math. Biosci.***86**, 213–227.MATHCrossRefGoogle Scholar - Adam, J. A. (1989). Corrigendum: a mathematical model of tumor growth by diffusion.
*Math. Biosci.***94**, 155.MathSciNetCrossRefGoogle Scholar - Adam, J. A. and N. Bellomo (1997).
*A Survey of Models for Tumor-Immune System Dynamics*, Boston: Birkhauser.MATHGoogle Scholar - Adam, J. A. and S. A. Maggelakis (1989). Mathematical models of tumor growth. iv. Effects of a necrotic core.
*Math. Biosci.***97**, 121–136.CrossRefMATHGoogle Scholar - Adam, J. A. and S. A. Maggelakis (1990). Diffusion regulated characteristics of a spherical prevascular carcinoma.
*Bull. Math. Biol.***52**, 549–582.CrossRefMATHGoogle Scholar - Adam, J. A. and R. D. Noren (2002). Equilibrium model of a vascularized spherical carcinoma.
*J. Math. Biol.***31**, 735–745.MathSciNetCrossRefGoogle Scholar - Adam, J. A. and J. C. Panetta (1995). A simple mathematical model and alternative paradigm for certain chemotherapeutic regimens.
*Math. Comput. Modelling***22**, 49–60.MathSciNetCrossRefMATHGoogle Scholar - Alberts, B., A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter (2002).
*Molecular Biology of the Cell*, 4th edn, New York: Garland Science.Google Scholar - Ambrosi, D. and F. Mollica (2002). On the mechanics of a growing tumor.
*Int. J. Eng. Sci.***40**, 1297–1316.MathSciNetCrossRefGoogle Scholar - Ambrosi, D. and F. Mollica (2003). Numerical simulation of the growth of a multicellular spheroid, in
*Second M.I.T. Conference on Computational Fluid and Solid Mechanics*, UK: Elsevier Ltd, pp. 1608–1612.Google Scholar - Ambrosi, D. and L. Preziosi. On the closure of mass balance models for tumour growth.
*Math. Models Methods Appl. Sci.*(in press).Google Scholar - Anderson, A. R. A. and M. A. J. Chaplain (1998). Continuous and discrete mathematical models of tumour-induced angiogenesis.
*Bull. Math. Biol.***60**, 857–899.CrossRefMATHGoogle Scholar - Anderson, A. R. A., M. A. J. Chaplain, R. J. Steele, E. L. Newman and A. Thompson (2000). Mathematical modelling of tumour invasion and metastasis.
*J. Theor. Med.***2**, 129–154.MATHGoogle Scholar - Araujo, R. P. and D. L. S. McElwain. A linear-elastic model of anisotropic tumour growth.
*Eur. J. Appl. Math.*(in press-a).Google Scholar - Araujo, R. P. and D. L. S. McElwain. The nature of the stresses induced during tissue growth.
*Appl. Math. Lett.*(in press-b).Google Scholar - Araujo, R. P. and D. L. S. McElwain. A mixture theory for the genesis of residual stresses in growing tissues.
*SIAM J. Appl. Math.*(submitted-a).Google Scholar - Araujo, R. P. and D. L. S. McElwain. New insights into vascular collapse and growth dynamics in solid tumours.
*J. Theor. Biol.*(submitted-b).Google Scholar - Araujo, R. P. and D. L. S. McElwain (2003a). An anisotropic model of vascular tumor growth: implications for vascular collapse, in
*Second M.I.T. Conference on Computational Fluid and Solid Mechanics*, Oxford, UK: Elsevier Ltd, pp. 1613–1616.Google Scholar - Araujo, R. P. and D. L. S. McElwain (2003b). The genesis of residual stresses and vascular collapse in solid tumours, in
*Proceedings of the Sixth Engineering Mathematics and Applications Conference*, Engineering Mathematics Group, ANZIAM, pp. 1–6.Google Scholar - Arve, B. H. and A. I. Liapis (1988). Oxygen tension in tumors predicted by a diffusion with absorption model involving a moving free boundary.
*Math. Comput. Modelling***10**, 159–174.CrossRefMATHGoogle Scholar - Australian Institute of Health and Welfare (AIHM). Cancer in Australia 1999. Available from http://www.aihw.gov.au/publications/can/ca99/.
- Aznavoorian, S., M. L. Stracke, H. Krutzsch, E. Schiffman and L. A. Liotta (1990). Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumour cells.
*J. Cell Biol.***110**, 1427–1438.CrossRefGoogle Scholar - Barr, L. C. (1989). The encapsulation of tumours.
*Clin. Exp. Metastasis***7**, 1813–1816.CrossRefGoogle Scholar - Barr, L. C., R. L. Carter and A. J. S. Davies (1988). Encapsulation of tumours as a modified wound healing response.
*Lancet***ii**, 135–137.CrossRefGoogle Scholar - Baxter, L. T. and R. K. Jain (1989). Transport of fluid and macromolecules in tumors: I. Role of interstitial pressure and convection.
*Microvasc. Res.***37**, 77–104.CrossRefGoogle Scholar - Baxter, L. T. and R. K. Jain (1990). Transport of fluid and macromolecules in tumors: Ii. Role of heterogeneous perfusion and lymphatics.
*Microvasc. Res.***40**, 246–263.CrossRefGoogle Scholar - Baxter, L. T. and R. K. Jain (1991). Transport of fluid and macromolecules in tumors: Iv. A microscopic model of the perivascular distribution.
*Microvasc. Res.***41**, 252–272.CrossRefGoogle Scholar - Baxter, L. T. and R. K. Jain (1996). Pharmacokinetic analysis of microscopic distribution of enzyme-conjugated antibodies and prodrugs: comparison with experimental data.
*Br. J. Cancer***73**, 447–456.Google Scholar - Berenblum, L. (1970). The nature of tumour growth, in
*General Pathology*, 4th edn, H. E. W. Florey (Ed.), Lloyd-Luke.Google Scholar - Bertuzzi, A., A. Fasano and A. Gandolfi (2003). Cell kinetics in tumour cords studied by a model with variable cell cycle length, in
*Second M.I.T. Conference on Computational Fluid and Solid Mechanics*, pp. 1631–1633.Google Scholar - Bertuzzi, A., A. Fasano, A. Gandolfi and D. Marangi (2002). Cell kinetics in tumour cords studied by a model with variable cell cycle length.
*Math. Biosci.***177 & 178**, 103–125.MathSciNetCrossRefGoogle Scholar - Bertuzzi, A. and A. Gandolfi (2000). Cell kinetics in a tumour cord.
*J. Theor. Biol.***204**, 587–599.CrossRefGoogle Scholar - Boucher, Y. and R. K. Jain (1992). Microvascular pressure is the principal driving force for interstitial hypertension in solid tumors: implications for vascular collapse.
*Cancer Res.***52**, 5110–5114.Google Scholar - Boucher, Y., J. M. Kirkwoord, D. Opacic, M. Desantis and R. K. Jain (1991). Interstitial hypertension in superficial metastatic melanomas in humans.
*Cancer Res.***51**, 6691–6694.Google Scholar - Boucher, Y., M. Leunig and R. K. Jain (1996). Tumor angiogenesis and interstitial hypertension.
*Cancer Res.***56**, 4264–4266.Google Scholar - Bowen, R. M. (1982). Compressible porous mediamodels by use of the theory of mixtures.
*Int. J. Eng. Sci.***20**, 697–735.MATHCrossRefGoogle Scholar - Bowen, R. M. (1976). Theory of mixtures, in
*Continuum Physics*, A. C. Eringen (Ed.), vol. 3, New York: Academic Press.Google Scholar - Bowen, R. M. (1980). Incompressible porous media models by use of the theory of mixtures.
*Int. J. Eng. Sci.***18**, 1129–1148.MATHCrossRefGoogle Scholar - Bowen, R. M. and J. C. Wiese (1969). Diffusion in mixtures of elastic materials.
*Int. J. Eng. Sci.***7**, 689–722.CrossRefMATHGoogle Scholar - Breward, C. J. W., H. M. Byrne and C. E. Lewis (2002). The role of cell-cell interactions in a two-phase model for avascular tumour growth.
*J. Math. Biol.***45**, 125–152.MathSciNetCrossRefMATHGoogle Scholar - Breward, C. J. W., H. M. Byrne and C. E. Lewis (2003). A multiphase model describing vascular tumour growth.
*Bull. Math. Biol.***65**, 609–640.CrossRefGoogle Scholar - Britton, N. F. and M. A. J. Chaplain (1992). A qualitative analysis of some models of tissue growth.
*Math. Biosci.***113**, 77–89.CrossRefGoogle Scholar - Brody, S. (1945).
*Bioenergetics and Growth*, New York: Reinhold Publ. Co.Google Scholar - Brown, N. J., C. A. Staton, G. R. Rodgers, K. P. Corke, J. C. E. Underwood and C. E. Lewis (2002). Fibrinogen e fragment selectively disrupts the vasculature and inhibits the growth of tumours in a syngeneic murine model.
*Br. J. Cancer***86**, 1813–1816.CrossRefGoogle Scholar - Bullough, W. S. (1965). Mitotic and functional homeostasis: a speculative review.
*Cancer Res.***25**, 1683–1727.Google Scholar - Bullough, W. S. and J. U. R. Deol (1971). The pattern of tumour growth.
*Symp. Soc. Exp. Biol.***25**, 255–275.Google Scholar - Burton, A. C. (1966). Rate of growth of solid tumours as a problem of diffusion.
*Growth***30**, 157–176.Google Scholar - Byrne, H. B., J. R. King, D. L. S. McElwain and L. Preziosi. A two-phase model of solid tumour growth.
*Appl. Math. Lett.*(in press).Google Scholar - Byrne, H. M. (1997a). The effect of time delays on the dynamics of avascular tumour growth.
*Math. Biosci.***144**, 83–117.MATHMathSciNetCrossRefGoogle Scholar - Byrne, H. M. (1997b). The importance of intercellular adhesion in the development of carcinomas.
*IMA J. Math. Appl. Med. Biol.***14**, 305–323.MATHGoogle Scholar - Byrne, H. M. (1999a). Using mathematics to study solid tumour growth, in
*Proceedings of the 9th General Meetings of European Women in Mathematics*, pp. 81–107.Google Scholar - Byrne, H. M. (1999b). A weakly nonlinear analysis of a model of avascular solid tumour growth.
*J. Math. Biol.***39**, 59–89.MATHMathSciNetCrossRefGoogle Scholar - Byrne, H. M. and M. A. J. Chaplain (1995). Growth of nonnecrotic tumours in the presence and absence of inhibitors.
*Math. Biosci.***130**, 151–181.CrossRefMATHGoogle Scholar - Byrne, H. M. and M. A. J. Chaplain (1996a). Growth of necrotic tumours in the presence and absence of inhibitors.
*Math. Biosci.***135**, 187–216.CrossRefMATHGoogle Scholar - Byrne, H. M. and M. A. J. Chaplain (1996b). Modelling the role of cell-cell adhesion in the growth and development of carcinomas.
*Math. Comput. Modelling***24**, 1–17.CrossRefMATHGoogle Scholar - Byrne, H. M. and M. A. J. Chaplain (1997). Free boundary value problems associated with the growth and development of multicellular spheroids.
*Eur. J. Appl. Math.***8**, 639–658.MathSciNetCrossRefMATHGoogle Scholar - Byrne, H. M. and M. A. J. Chaplain (1998). Necrosis and apoptosis: distinct cell loss mechanisms in a mathematical model of avascular tumour growth.
*J. Theor. Med.***1**, 223–235.MATHCrossRefGoogle Scholar - Byrne, H. M. and S. A. Gourley (1997). The role of growth factors in avascular tumour growth.
*Math. Comput. Modelling***4**, 35–55.MathSciNetCrossRefGoogle Scholar - Byrne, H. M. and L. Preziosi. Modelling solid tumor growth using the theory of mixtures.
*IMA J. Appl. Math. Lett.*(in press).Google Scholar - Casciari, J. J., S. V. Sotirchos and R. M. Sutherland (1984). Identification of a tumour inhibitory factor in rat ascites fluid.
*Biochem. Biophys. Res. Comm.***119**, 76–82.CrossRefGoogle Scholar - Chaplain, M. A. J. (1993). The development of a spatial pattern in a model for cancer growth, in
*Experimental and Theoretical Advances in Biological Pattern Formation*, H. G. Othmer*et al.*(Eds), Plenum Press.Google Scholar - Chaplain, M. A. J. (1996). Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modeling of the stages of tumour development.
*Math. Comput. Modelling***23**, 47–87.MATHCrossRefGoogle Scholar - Chaplain, M. A. J., D. L. Benson and P. K. Maini (1994). Nonlinear diffusion of a growth inhibitory factor in multicell spheroids.
*Math. Biosci.***121**, 1–13.CrossRefMATHGoogle Scholar - Chaplain, M. A. J. and N. F. Britton (1993). On the concentration profile of a growth inhibitory factor in multicell spheroids.
*Math. Biosci.***115**, 233–245.CrossRefMATHGoogle Scholar - Chaplain, M. A. J. and L. Preziosi. Macroscopic modelling of the growth and development of tumor masses.
*Math. Models Methods Appl. Sci.*(in press).Google Scholar - Chaplain, M. A. J. and B. D. Sleeman (1993). Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory.
*J. Math. Biol.***31**, 431–473.MathSciNetCrossRefMATHGoogle Scholar - Chen, C. Y., H. M. Byrne and J. R. King (2001). The influence of growth-induced stress from the surrounding medium on the development of multicell spheroids.
*J. Math. Biol.***43**, 191–220.MathSciNetCrossRefMATHGoogle Scholar - Chen, Y.-C. and A. Hoger (2000). Constitutive functions of elastic materials in finite growth and deformation.
*J. Elasticity***59**, 175–193.CrossRefMATHGoogle Scholar - Clarke, R., R. G. Dickson and N. Brunner (1988). The process of malignant progression in human breast cancer.
*Cancer Genet. Cytogenet.***14**, 125–134.Google Scholar - Cowin, S. C. (1996). Strain or deformation rate dependent finite growth in soft tissues.
*J. Biomech.***29**, 647–649.CrossRefGoogle Scholar - Cristini, V., J. Lowengrub and Q. Nie (2003). Nonlinear simulation of tumor growth.
*J. Math. Biol.***46**, 191–224.MathSciNetCrossRefMATHGoogle Scholar - Cruveilier, (1829).
*Anatomie Pathologique du Corps Humain*, Paris: Bailliere.Google Scholar - Cui, S. (2002). Analysis of a mathematical model for the growth of tumors under the action of external inhibitors.
*J. Math. Biol.***44**, 395–426.MATHMathSciNetCrossRefGoogle Scholar - Cui, S. and A. Friedman (2000). Analysis of a mathematical model of the effect of inhibitors on the growth of tumors.
*Math. Biosci.***164**, 103–137.MathSciNetCrossRefMATHGoogle Scholar - Danova, M., A. Riccardi and G. Mazzini (1990). Cell cycle-related proteins and flow cytometry.
*Haematologica***75**, 252–264.Google Scholar - Darzynkiewicz, Z. (1995). Apoptosis in antitumour strategies: modulation of cell-cycle or differentiation.
*J. Cell. Biochem.***58**, 151–159.CrossRefGoogle Scholar - David, P. D.
*et al.*(2002). Zd6126: A novel vascular-targeting agent that causes selective destruction of tumor vasculature.*Cancer Res.***62**, 7247–7253.Google Scholar - De Angelis, E. and L. Preziosi (2000). Advection-diffusion models for solid tumour evolution in vivo and related free boundary problem.
*Math. Models Methods Appl. Sci.***10**, 379–408.MathSciNetCrossRefMATHGoogle Scholar - Deakin, A. S. (1975). Model for the growth of a solid in vitro tumour.
*Growth***39**, 159–165.Google Scholar - Dorie, M. J., R. F. Kallman, D. F. Rapacchietta, D. van Antwerp and Y. R. Huang (1982). Migration and internalization of cells and polystyrene microspheres in tumour cell spheroids.
*Exp. Cell Res.***141**, 201–209.CrossRefGoogle Scholar - Dorie, M. J., R. F. Kallman and M. A. Coyne (1986). Effect of cytochalasin b, nocodazole and irradiation on migration and internalization of cells and microspheres in tumour cell spheroids.
*Exp. Cell Res.***166**, 370–378.CrossRefGoogle Scholar - Drew, D. A. (1971). Averaged field equations for two-phase media.
*Stud. Appl. Math.***50**, 205–231.MATHGoogle Scholar - Drew, D. A. (1976). Two-phase flows: constitutive equations for lift and Brownian motion and some basic flows.
*Arch. Rat. Mech. Anal.***62**, 149–163.MATHMathSciNetCrossRefGoogle Scholar - Drew, D. A. and S. L. Passman (1999).
*Theory of Multicomponent Fluids*, New York: Springer.Google Scholar - Drew, D. A. and L. A. Segel (1971). Averaged equations for two-phase flows.
*Stud. Appl. Math.***50**, 205–231.MATHGoogle Scholar - Durand, R. E. (1976). Cell cycle kinetics in an in vitro tumor model.
*Cell Tissue Kinet.***9**, 403–412.Google Scholar - Eddy, H. A. and G. W. Casarett (1972). Development of the vascular system in the hamster malignant neurilemmoma.
*Microvasc. Res.***6**, 63–82.CrossRefGoogle Scholar - Fisher, R. A. (1937). The wave of advance of advantageous genes.
*Ann. Eugenics***7**, 355–369.MATHGoogle Scholar - Fitt, A. D., P. D. Howell, J. R. King, C. P. Please and D. W. Schwendeman (2002). Multiphase flow in a roll press nip.
*Eur. J. Appl. Math.***13**, 225–259.MathSciNetCrossRefMATHGoogle Scholar - Folkman, J., P. Cole and S. Zimmerman (1966). Tumor behavior in isolated perfused organs: in vitro growth and metastases of biopsy material in rabbit thyroid and canine intestinal segment.
*Ann. Surg.***164**, 491–502.Google Scholar - Folkman, J. and M. Hochberg (1973). Self-regulation of growth in three dimensions.
*J. Exp. Med.***138**, 745–753.CrossRefGoogle Scholar - Fowler, A. C. (1997).
*Mathematical Models in the Applied Sciences*, Cambridge University Press.Google Scholar - Franks, S. J., H. M. Byrne, J. R. King, J. C. E. Underwood and C. E. Lewis (2003). Modelling the early growth of ductal carcinoma in situ of the breast.
*J. Math. Biol.***47**, 424–452.MathSciNetCrossRefMATHGoogle Scholar - Freyer, J. P., P. L. Schor and A. G. Saponara (1988). Partial purification of a protein-growth inhibitor from multicell spheroids.
*Biochem. Biophys. Res. Comm.***152**, 463–468.CrossRefGoogle Scholar - Froese, G. (1962). The respiration of ascites tumour cells at low oxygen concentrations.
*Biochim. Biophys. Acta***57**, 509–519.CrossRefGoogle Scholar - Fung, Y. C. (1991). What are the residual stresses doing in our blood vessels?
*Ann. Biomed. Eng.***19**, 237–249.MathSciNetGoogle Scholar - Fung, Y. C. (1993).
*Biomechanics: Mechanical Properties of Living Tissues*, New York: Springer.Google Scholar - Gatenby, R. A. (1991). Population ecology issues in tumor growth.
*Cancer Res.***51**, 2542–2547.Google Scholar - Gatenby, R. A. (1995a). Models of tumor-host interactions as competing populations: implications for tumor biology and treatment.
*J. Theor. Biol.***176**, 447–455.CrossRefGoogle Scholar - Gatenby, R. A. (1995b). The potential role of transformation-induced metabolic changes in tumor-host interaction.
*Cancer Res.***55**, 4151–4156.Google Scholar - Gatenby, R. A. (1996a). Altered glucose metabolism and the invasive tumor phenotype: insights provided through mathematical models.
*Int. J. Oncol.***8**, 597–601.Google Scholar - Gatenby, R. A. (1996b). Application of competition theory to tumour growth: implications for tumour biology and treatment.
*Eur. J. Cancer***32A**, 722–726.CrossRefGoogle Scholar - Gatenby, R. A. (1998). Mathematical models of tumor-host interactions.
*Cancer J.***11**, 289–293.Google Scholar - Gatenby, R. A. and E. T. Gawlinski (1996). A reaction-diffusion model of cancer invasion.
*Cancer Res.***56**, 5745–5753.Google Scholar - Gatenby, R. A. and E. T. Gawlinski (2001). Mathematical models of tumor invasion mediated by transformation-induced alteration of microenvironmental pH, in
*The Tumour Microenvironment: Causes and Consequences of Hypoxia and Acidity*, K. A. Goode and D. J. Chadwick (Eds), Chichester, UK: John Wiley and Sons Ltd.Google Scholar - Gatenby, R. A. and E. T. Gawlinski (2003). The glycolytic phenotype in carcinogenesis and tumor invasion: insights through mathematical models.
*Cancer Res.***63**, 3847–3854.Google Scholar - Gimbrone, M. A., R. H. Aster, R. S. Cotran, J. Corkery, J. H. Jandl and J. Folkman (1969). Preservation of vascular integrity in organs perfused in vitro with a platelet-richmedium.
*Nature***221**, 33–36.Google Scholar - Glass, L. (1973). Instability and mitotic patterns in tissue growth.
*J. Dyn. Syst. Meas. Control***95**, 324–327.Google Scholar - Goldacre, R. J. and B. Sylven (1962). On the access of blood-borne dyes to various tumour regions.
*Br. J. Cancer***16**, 306–321.Google Scholar - Gompertz, G. (1825). On the nature of the function expressive of the law of human mortality, and on the new mode of determining the value of life contingencies.
*Philos. Trans. R. Soc. London***115**, 513–585.Google Scholar - Gray, L. H., A. D. Conger, M. Ebert, S. Hornsey and O. C. A. Scott (1955). The concentration of oxygen dissolved in tissues at the time of irradiation as a factor in radiotherapy.
*Br. J. Radiol.***9**, 638–648.Google Scholar - Greene, H. S. N. (1961). Heterologous transplantation of mammalian tumors.
*J. Exp. Med.***73**, 461.CrossRefGoogle Scholar - Greenspan, H. P. (1972). Models for the growth of a solid tumor by diffusion.
*Stud. Appl. Math.***52**, 317–340.Google Scholar - Greenspan, H. P. (1974). On the self-inhibited growth of cell cultures.
*Growth***38**, 81–95.Google Scholar - Greenspan, H. P. (1976). On the growth and stability of cell cultures and solid tumors.
*J. Theor. Biol.***56**, 229–242.MathSciNetGoogle Scholar - Griffiths, J. D. and A. J. Salsbury (1965).
*Circulating Cancer Cells*, Chicago: Charles C. Thomas, Publishers.Google Scholar - Griffon-Etienne, G., Y. Boucher, C. Brekken, H. D. Suit and R. K. Jain (1999). Taxane-induced apoptotis decompresses blood vessels and lowers interstitial fluid pressure in solid tumors: clinical implications.
*Cancer Res.***59**, 3776–3782.Google Scholar - Gurtin, M. E. (1973). A system of equations for age-dependent population diffusion.
*Stud. Appl. Math.***40**, 389–392.Google Scholar - Gurtin, M. E. and R. C. MacCamy (1977). On the diffusion of biological populations.
*Math. Biosci.***33**, 35–49.MathSciNetCrossRefMATHGoogle Scholar - Gutmann, R., M. Leunig, J. Feyh, A. E. Goetz, K. Messmer, E. Kastenbauer and R. K. Jain (1992). Interstitial hypertension in head and neck tumors in patients: correlation with tumor size.
*Cancer Res.***52**, 1993–1995.Google Scholar - Haddow, A. (1938). The biological characters of spontaneous tumours of the mouse, with special reference to the rate of growth.
*J. Path. Bact.***47**, 553–565.CrossRefGoogle Scholar - Hahnfeldt, P., D. Panigrahy, J. Folkman and L. Hlatky (1999). Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy.
*Cancer Res.***59**, 4770–4775.Google Scholar - Haji-Karim, M. and J. Carlsson (1978). Proliferation and viability in cellular spheroids of human origin.
*Cancer Res.***38**, 1457–1464.Google Scholar - Harel, L., G. Chatelain and A. Golde (1984). Density dependent inhibition of growth: inhibitory diffusible factors from 3t3 and rous sarcoma virus (rsv)-transformed 3t3 cells.
*J. Cell Phys.***119**, 101–106.CrossRefGoogle Scholar - Hashizume, H., P. Baluk, S. Morikawa, J. W. McLean, G. Thurston, S. Roberge, R. K. Jain and D. M. McDonald (2000). Openings between defective endothelial cells explain tumor vessel leakiness.
*Am. J. Pathol.***156**, 1363–1380.Google Scholar - Helmlinger, G., P. A. Netti, H. D. Lichtenbeld, R. J. Melder and R. K. Jain (1997). Solid stress inhibits the growth of multicellular tumour spheroids.
*Nat. Biotechnol.***15**, 778–783.CrossRefGoogle Scholar - Hickman, J. A., C. S. Potten, A. J. Merritt and T. C. Fisher (1994). Apoptosis and cancer chemotherapy.
*Philos. Trans. R. Soc. Lond. B***345**, 319–325.Google Scholar - Hill, A. V. (1928). The diffusion of oxygen and lactic acid through tissues.
*R. Soc. Proc. B***104**, 39–96.Google Scholar - Hirst, D. G. and J. Denekamp (1979). Tumour cell proliferation in relation to the vasculature.
*Cell Tissue Kinet.***12**, 31–42.Google Scholar - Hirst, D. G., J. Denekamp and B. Hobson (1982). Proliferation kinetics of endothelial and tumour cells in three mouse mammary carcinomas.
*Cell Tissue Kinet.***15**, 251–261.Google Scholar - Hirst, D. G., V. K. Hirst, B. Joiner, V. Prise and K. M. Shaffi (1991). Changes in tumour morphology with alterations in oxygen availability: further evidence for oxygen as a limiting substrate.
*Br. J. Cancer***64**, 54–58.Google Scholar - Hoger, A., T. J. Van Dyke and V. A. Lubarda (2002). On the growth part of deformation gradient for residually-stressed biological materials (submitted).Google Scholar
- Hughes, F. and C. McCulloch (1991). Quantification of chemotactic response of quiescent and proliferating fibroblasts in Boyden chambers by computer-assisted image analysis.
*J. Histochem. Cytochem.***39**, 243–246.Google Scholar - Huxley, J. S. (1932).
*Problems of Relative Growth*, New York: The Dial Press.Google Scholar - Huyghe, J. M. and J. D. Janssen (1997). Quadriphasic mechanics of swelling incompressible porous media.
*Int. J. Eng. Sci.***35**, 793–802.CrossRefMATHGoogle Scholar *Imperial Cancer Research Fund*, ICRF, 1994.Google Scholar- Iwata, K. K., C. M. Fryling, W. B. Knott and G. J. Todaro (1985). Isolation of tumour cell growth-inhibiting factors from a human rhabdomysarcoma cell line.
*Cancer Res.***45**, 2689–2694.Google Scholar - Jackson, T. L. (2002). Vascular tumor growth and treatment: consequences of polyclonality, competition and dynamic vascular support.
*J. Math. Biol.***44**, 201–226.MATHMathSciNetCrossRefGoogle Scholar - Jackson, T. L. and H. M. Byrne (2000). A mathematical model to study the effects of drug resistance and vasculature on the response of solid tumors to chemotherapy.
*Math. Biosci.***164**, 17–38.MathSciNetCrossRefMATHGoogle Scholar - Jain, R. K. (1987). Transport of molecules across tumor vasculature.
*Cancer Metastasis Rev.***6**, 559–593.CrossRefGoogle Scholar - Jain, R. K. (1994). Barriers to drug delivery in solid tumors.
*Sci. Am.***271**, 58–66.CrossRefGoogle Scholar - Jain, R. K. and L. T. Baxter (1998). Mechanisms of heterogeneous distribution of monoclonal antibodies and other macromolecules in tumour: significance of elevated interstitial pressure.
*Cancer Res.***48**, 7022–7032.Google Scholar - Jain, R. K. and J. Wei (1977). Dynamics of drug transport in solid tumors: distributed parameter model.
*J. Bioeng.***1**, 313–330.Google Scholar - Jiang, W. G. and R. E. Mansel (1996). Progress in anti-invasion and anti-metastasis research and treatment.
*Int. J. Oncol.***9**, 1013–1028.Google Scholar - Jones, A. F., H. M. Byrne, J. S. Gibson and J. W. Dold (2000). A mathematical model of the stress induced during avascular tumour growth.
*J. Math. Biol.***40**, 473–499.MathSciNetCrossRefMATHGoogle Scholar - Kastan, M. B., C. E. Canman and C. J. Leonard (1995). P53, cell cycle control and apoptosis: implications for cancer.
*Cancer Metast. Rev.***14**, 3–15.CrossRefGoogle Scholar - Kendall, D. G. (1948). On the role of variable cell generation time in the development of a stochastic birth process.
*Biometrika***35**, 316–330.MATHMathSciNetCrossRefGoogle Scholar - Kerr, J. F. R. (1971). Shrinkage necrosis: a distinct mode of cellular death.
*J. Path.***105**, 13–20.CrossRefGoogle Scholar - Kerr, J. F. R., A. H. Wyllie and A. R. Currie (1972). Apoptosis: a basic biological phenomenon with wide-ranging implications in tissue kinetics.
*Br. J. Cancer***26**, 239–257.Google Scholar - Kerr, J. F. R., J. Searle, B. V. Harmon and C. J. Bishop (1987). Apoptosis, in
*Perspectives in Mammalian Cell Death*, C. S. Potten (Ed.), UK: Oxford University Press.Google Scholar - King, W. E., D. S. Schultz and R. A. Gatenby (1986a). An analysis of systematic tumor oxygenation using multi-region models.
*Chem. Eng. Commun.***64**, 137–153.Google Scholar - King, W. E., D. S. Schultz and R. A. Gatenby (1986b). Multi-region models for describing oxygen tension profiles in human tumors.
*Chem. Eng. Commun.***47**, 73–91.Google Scholar - Kleinerman, J. and L. A. Liotta (1977). Release of tumor cells, in
*Cancer Invasion and Metastasis: Biologic Mechanisms and Therapy*, S. B. Day*et al.*(Eds), New York: Raven Press.Google Scholar - Koike, A. (1964). Mechanism of blood-borne metastases. i. Some factors affecting lodgment and growth of tumour cells in the lungs.
*Cancer***17**, 450–460.Google Scholar - Kolmogorov, A., A. Petrovsky and N. Piscounoff (1988). Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem, in
*Dynamics of Curved Fronts*, Boston: Academic Press.Google Scholar - Kunz-Schughart, L. A., M. Kreutz and R. Knuechel (1998). Multicellular spheroids: a three-dimensional in vitro culture system to study tumour biology.
*Int. J. Exp. Pathol.***79**, 1–23.CrossRefGoogle Scholar - Kuo, T. H., T. Kubota, M. Watanabe, T. Furukawa, T. Teramoto, K. Ishibiki, M. Kitajima, A. R. Moosa, S. Penman and R. M. Hoffman (1995). Liver colonization competence governs colon cancer metastasis.
*Proc. Natl. Acad. Sci. USA***92**, 12085–12089.Google Scholar - Lai, W. M., J. S. Hou and V. C. Mow (1991). A triphasic theory for the swelling and deformation behaviors of articular cartilage.
*J. Biomech. Eng.***113**, 245–258.Google Scholar - Lai, W. M., V. C. Mow and W. Zhu (1993). Constitutive modelling of articular cartilage and biomacromolecular solutions.
*Trans. ASME***115**, 474–480.Google Scholar - Laird, A. K., S. A. Tyler and A. D. Barton (1965). Dynamics of normal growth.
*Growth***21**, 233–248.Google Scholar - Landman, K. A. and C. P. Please (2001). Tumour dynamics and necrosis: surface tension and stability.
*IMA J. Math. Appl. Med. Biol.***18**, 131–158.MATHGoogle Scholar - Landry, J., J. P. Freyer and R. M. Sutherland (1981). Shedding of mitotic cells from the surface of multicell spheroids during growth.
*J. Cell. Physiol.***106**, 23–32.CrossRefGoogle Scholar - Landry, J., J. P. Freyer and R. M. Sutherland (1982). A model for the growth of multicellular spheroids.
*Cell Tissue Kinet.***15**, 585–594.Google Scholar - Leu, A. J., D. A. Berk, A. Lymboussaki, K. Alitalo and R. K. Jain (2000). Absence of functional lymphatics within a murine sarcoma: a molecular and functional evaluation.
*Cancer Res.***60**, 4324–4327.Google Scholar - Leunig, M., F. Yuan, M. D. Menger, Y. Boucher, A. E. Goetz, K. Messmer and R. K. Jain (1992). Angiogenesis, microvascular architecture, microhemodynamics, and interstitial fluid pressure during early growth of human adnocarcinoma ls174t in scid mice.
*Cancer Res.***52**, 6553–6560.Google Scholar - Levine, E. L.
*et al.*(1995). Apoptosis, intrinsic radiosensitivity and prediction of radiotherapy response in cervical carcinoma.*Radiother. Oncol.***37**, 1–9.CrossRefGoogle Scholar - Liapis, A. I., G. G. Lipscomb and O. K. Crossier (1982). A model of oxygen diffusion in absorbing tissue.
*Math. Modelling***3**, 83–92.MathSciNetCrossRefMATHGoogle Scholar - Lin, S. H. (1976). Oxygen diffusion in a spheroid cell with non-linear uptake kinetics.
*J. Theor. Biol.***60**, 449–457.CrossRefGoogle Scholar - Liotta, L. A. and C. DeLisi (1977). Method for quantitating tumor cell removal and tumor cell-invasive capacity in experimental metastases.
*Cancer Res.***37**, 4003–4008.Google Scholar - Liotta, L. A., J. Kleinerman and G. M. Saidel (1974a). Quantitative relationships of intravascular tumor cells, tumor vessels, and pulmonary metastases following tumor implantation.
*Cancer Res.***34**, 997–1004.Google Scholar - Liotta, L. A., J. Kleinerman and G. M. Saidel (1974b). Quantitative relationships of intravascular tumor cells, tumor vessels, and pulmonary metastases following tumor implantation.
*Cancer Res.***34**, 997–1004.Google Scholar - Liotta, L. A., G. M. Saidel and J. Kleinerman (1974c). Diffusion model of tumor vascularization and growth.
*Bull. Math. Biol.***34**, 117–128.Google Scholar - Liotta, L. A., J. Kleinerman and G. M. Saidel (1976a). The significance of hematogenous tumor cell clumps in the metastatic process.
*Cancer Res.***36**, 889–894.Google Scholar - Liotta, L. A., G. M. Saidel and J. Kleinerman (1976b). Stochastic model of metastases formation.
*Biometrics*535–550.Google Scholar - Liotta, L. A., G. M. Saidel, J. Kleinerman and C. DeLisi (1977). Micrometastasis therapy: theoretical concepts, in
*Cancer Invasion and Metastasis: Biologic Mechanisms and Therapy*, S. B. Day*et al.*(Eds), New York: Raven Press.Google Scholar - Loewenstein, W. R. (1981). Junctional intercellular communication: the cell-to-cell membrane channel.
*Physiol. Rev.***61**, 829–889.Google Scholar - Lomer, E. (1883). Zur frage der heilbarkeit des carcinoms.
*Z. Geburtsh. Gynaek.***9**, 277.Google Scholar - Lubarda, V. A. and A. Hoger (2002). On the mechanics of solids with a growing mass.
*Int. J. Solids Structures***39**, 4627–4664.CrossRefMATHGoogle Scholar - Lubkin, S. R. and T. Jackson (2002). Multiphase mechanics of capsule formation in tumors.
*J. Biomech. Eng.***124**, 237–243.CrossRefGoogle Scholar - Lynch, M. P., S. Nawaz and L. E. Gerschenson (1986). Evidence for soluble factors regulating cell death and cell proliferation in primary cultures of rabbit endometrial cells grown on collagen.
*Proc. Natl. Acad. Sci.***83**, 4784–4788.CrossRefGoogle Scholar - MacArthur, B. D. and C. P. Please (2003). Residual stress generation and necrosis formation in multi-cell tumour spheroids.
*J. Math. Biol.*(submitted).Google Scholar - Maggelakis, S. A. and J. A. Adam (1990). Mathematical model of prevascular growth of a spherical carcinoma.
*Math. Comput. Modelling***13**, 23–38.CrossRefMATHGoogle Scholar - Mantzaris, N., S. Webb and H. G. Othmer (2003). Mathematical modelling of tumour-induced angiogenesis.
*J. Math. Biol.*(in press).Google Scholar - Marchant, B. P., J. Norbury and A. J. Perumpanani (2000). Traveling shock waves arising in a model of malignant invasion.
*SIAM J. Appl. Math.***60**, 463–476.MathSciNetCrossRefMATHGoogle Scholar - Marchant, B. P., J. Norbury and J. A. Sherratt (2001). Travelling wave solutions to a haptotaxis-dominated model of malignant invasion.
*Nonlinearity***14**, 1653–1671.MathSciNetCrossRefMATHGoogle Scholar - Martin, G. R. and R. K. Jain (1994). Noninvasive measurement for interstitial pH profiles in normal and neoplastic tissue using fluorescent ration imaging microscopy.
*Cancer Res.***54**, 5670–5674.Google Scholar - Mayneord, W. V. (1932). On a law of growth of Jensen’s rat sarcoma.
*Am. J. Cancer***16**, 841–846.Google Scholar - McDougall, S. R., A. R. A. Anderson, M. A. J. Chaplain and J. A. Sherratt (2002). Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies.
*Bull. Math. Biol.***64**, 673–702.CrossRefGoogle Scholar - McElwain, D. L. S. (1978). A reexamination of oxygen diffusion in a spheroid cell with Michaelis-Menten oxygen uptake kinetics.
*J. Theor. Biol.***71**, 255–267.CrossRefGoogle Scholar - McElwain, D. L. S., R. Callcott and L. E. Morris (1979). A model of vascular compression in solid tumours.
*J. Theor. Biol.***78**, 405–415.CrossRefGoogle Scholar - McElwain, D. L. S. and L. E. Morris (1978). Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth.
*Math. Biosci.***39**, 147–157.CrossRefGoogle Scholar - McElwain, D. L. S. and G. J. Pettet (1993). Cell migration in multicell spheroids: swimming against the tide.
*Bull. Math. Biol.***55**, 655–674.CrossRefMATHGoogle Scholar - McElwain, D. L. S. and P. J. Ponzo (1977). A model for the growth of a solid tumor with non-uniform oxygen consumption.
*Math. Biosci.***35**, 267–279.CrossRefMATHGoogle Scholar - Miyasaka, M. (1995). Cancer metastasis and adhesion molecules.
*Clin. Orth. Rel. Res.***312**, 10–18.Google Scholar - Moore, J. V., H. A. Hopkins and W. B. Looney (1983). Response of cell populations in tumor cords to a single dose of cyclophosphamide or radiation.
*Eur. J. Cancer Clin. Oncol.***19**, 73–79.CrossRefGoogle Scholar - Moore, J. V., H. A. Hopkins and W. B. Looney (1984). Tumour-cord parameters in two rat hepatomas that differ in their radiobiological oxygenation status.
*Radiat. Envir. Biophys.***23**, 213–222.CrossRefGoogle Scholar - Moore, J. V., P. S. Haselton and C. M. Buckley (1985). Tumour cords in 52 human bronchial and cervical squamous cell carcinomas: inferences for their cellular kinetics and radiobiology.
*Br. J. Cancer***51**, 407–413.Google Scholar - Mottram, J. C. (1936). Factor of importance in radiosensitivity of tumors.
*Br. J. Radiol.***9**, 606–614.CrossRefGoogle Scholar - Mow, V. C., J. S. Hou, J. M. Owens and A. Ratcliffe (1990a). Biphasic and quasilinear viscoelastic theories for hydrated soft tissues, in
*Biomechanics of Diarthrodial Joints*, New York: Springer.Google Scholar - Mow, V. C., J. S. Hou, J. M. Owens and A. Ratcliffe (1990b). Biphasic and quasilinear viscoelastic theories for hydrated soft tissues, in
*Biomechanics of Diarthrodial Joints*, New York: Springer.Google Scholar - Mueller-Klieser, W. F. and R. M. Sutherland (1982). Oxygen tensions in multicell spheroids of two cell lines.
*Br. J. Cancer***45**, 256–263.Google Scholar - Mueller-Klieser, W. (2000). Tumor biology and experimental therapeutics.
*Crit. Rev. Oncol. Hematol.***36**, 123–139.Google Scholar - Murray, J. D. (2002).
*Mathematical Biology, I: An Introduction*, Berlin: Springer.Google Scholar - Nagle, R. B., J. D. Knox, C. Wolf, G. T. Bowden and A. E. Cress (1994). Adhesion molecules, extracellular matrix and proteases in prostrate carcinoma.
*J. Cell. Biochem. Suppl.***19**, 232–237.Google Scholar - Netti, P. A., L. T. Baxter, Y. Boucher, R. Skalak and R. K. Jain (1995). Time-dependent behavior of interstitial fluid pressure in solid tumors: implications for drug delivery.
*Cancer Res.***55**, 5451–5458.Google Scholar - Netti, P. A., L. T. Baxter, Y. Boucher, R. Skalak and R. K. Jain (1997). Macro-and microscopic fluid transport in living tissues: application to solid tumors.
*AIChE J.***43**, 818–834.CrossRefGoogle Scholar - Ng, I. O. L., E. C. S. Lai, M. M. T. Lai and S. T. Fan (1992). Tumor encapsulation in hepatocellular carcinoma: a pathological study of 198 cases.
*Cancer (N.Y.)***70**, 395–413.Google Scholar - Nor, J. E., J. Christensen, J. Liu, M. Peters, D. J. Mooney, R. M. Strieter and P. J. Polverini (2001). Up-regulation of bcl-2 in microvascular endothelial cells enhances intratumoral angiogenesis and accelerates tumor growth.
*Cancer Res.***61**, 2183–2188.Google Scholar - Olson, J. S. (2002).
*Bathsheba’s Breast: Women, Cancer and History*, Baltimore: Johns Hopkins University Press.Google Scholar - Orme, M. E. and M. A. J. Chaplain (1996). A mathematical model of vascular tumour growth and invasion.
*Math. Comput. Modelling***23**, 43–60.MathSciNetCrossRefMATHGoogle Scholar - Osgood, E. E. (1957). A unifying concept of the etiology of the leukemias, lymphomas, and cancers.
*J. Natl. Cancer Inst.***18**, 155–166.Google Scholar - Palka, J., B. Adelmann-Grill, P. Francz and K. Bayreuther (1996). Differentiation stage and cell cycle position determine the chemotactic response of fibroblasts.
*Folia Histochem. Cytobiol.***34**, 121–127.Google Scholar - Panetta, J. C. and J. A. Adam (1995). A mathematical model of cycle-specific chemotherapy.
*Math. Comput. Modelling***22**, 67–82.MathSciNetCrossRefMATHGoogle Scholar - Passman, S. L. and J. W. Nunziato (1984). A theory of multiphase mixtures, in
*Rational Thermodynamics*, C. Truesdell (Ed.), New York: Springer.Google Scholar - Perumpanani, A. J. Malignant and morphogenetic waves, PhD thesis, Oxford University, Hilary Term, 1996.Google Scholar
- Perumpanani, A. J. and H. M. Byrne (1999). Extracellular matrix concentration exerts selection pressure on invasive cells.
*Eur. J. Cancer***35**, 1274–1280.CrossRefGoogle Scholar - Perumpanani, A. J. and J. Norbury (1999). Numerical interactions of random and directed motility during cancer invasion.
*Math. Comput. Modelling***30**, 123–133.MathSciNetCrossRefMATHGoogle Scholar - Perumpanani, A. J., J. A. Sherratt, J. Norbury and H. M. Byrne (1996). Biological inferences from a mathematical model for malignant invasion.
*Invasion Metastasis***16**, 209–221.Google Scholar - Perumpanani, A. J., J. A. Sherratt, J. Norbury and H. M. Byrne (1999). A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion.
*Physica D***126**, 145–159.CrossRefGoogle Scholar - Pettet, G. J., D. L. S. McElwain and J. Norbury (2000). Lotka-Volterra equations with chemotaxis: walls, barriers and travelling waves.
*IMA J. Math. Appl. Med. Biol.***17**, 395–413.MATHGoogle Scholar - Pettet, G. J., C. P. Please, M. J. Tindall and D. L. S. McElwain (2001). The migration of cells in multicell tumor spheroids.
*Bull. Math. Biol.***63**, 231–257.CrossRefGoogle Scholar - Pioletti, D. P., L. R. Rakotomanana, J.-F. Benvenuti and P.-F. Leyvraz (1998). Viscoelastic constitutive law in large deformations: application to human knee ligaments and tendons.
*J. Biomech.***31**, 753–757.CrossRefGoogle Scholar - Pisani, P., F. Bray and D. M. Parkin (2001). Estimates of the world-wide prevalence of cancer for 25 sites in the adult population.
*Int. J. Cancer***97**, 72–81.CrossRefGoogle Scholar - Please, C. P., G. J. Pettet and D. L. S. McElwain (1998). A new approach to modelling the formation of necrotic regions in tumours.
*Appl. Math. Lett.***11**, 89–94.MathSciNetCrossRefMATHGoogle Scholar - Please, C. P., G. J. Pettet and D. L. S. McElwain (1999). Avascular tumour dynamics and necrosis.
*Math. Models Methods Appl. Sci***9**, 569–579.CrossRefMATHGoogle Scholar - Porter, R. (1997).
*The Greatest Benefit to Mankind: A Medical History of Humanity from Antiquity to the Present*, London: Harper Collins Publishers.Google Scholar - Preziosi, L. and A. Farina (2002). On Darcy’s law for growing porous media.
*Int. J. Non-Linear Mechanics***37**, 485–491.CrossRefMATHGoogle Scholar - Raff, M. C. (1992). Social controls on cell survival and cell death.
*Nature***356**, 397–400.CrossRefGoogle Scholar - Rajagopal, K. R. and A. R. Srinivasa (1998). Mechanics of the inelastic behavior of materials. i. Theoretical underpinnings.
*Int. J. Plast.***14**, 945–967.CrossRefMATHGoogle Scholar - Rajagopal, K. R. and L. Tao (1995).
*Mechanics of Mixtures*, Singapore: World Scientific.MATHGoogle Scholar - Ritchie, A. C. (1970). The classification morphology and behaviour of tumours, in
*General Pathology*, 4th edn, H. E. W. Florey (Ed.), Lloyd-Luke.Google Scholar - Robbins, S. L., R. S. Cotran and V. Kuman (1984). Neoplasia, in
*Pathologic Basis of Disease*, 3rd edn, W. B. Saunders (Ed.), Philadelphia: Saunders Co.Google Scholar - Robertson, T. B. (1923).
*The Chemical Basis of Growth and Senescence*, Philadelphia: J. B. Lippincott Co.Google Scholar - Rodriguez, E. K., A. Hoger and A. D. McCulloch (1994). Stress-dependent finite growth in soft elastic tissues.
*J. Biomech.***27**, 455–467.CrossRefGoogle Scholar - Ruoslahti, E. (1996). How cancer spreads.
*Sci. Am.*42–47.Google Scholar - Ruoslahti, E. (2002). Specialization of tumour vasculature.
*Nat. Rev. Cancer***2**, 83–90.CrossRefGoogle Scholar - Saidel, G. M., L. A. Liotta and J. Kleinerman (1976). System dynamics of a metastatic process from an implanted tumor.
*J. Theor. Biol.***56**, 417–434.Google Scholar - Schultz, D. S. and W. E. King (1987). On the analysis of oxygen diffusion in biological systems.
*Math. Biosci.***83**, 179–190.CrossRefMATHGoogle Scholar - Seftor, R. E., E. A. Seftor, K. R. Gehlsen and W. G. Stetler-Stevenson (1992). Role of alpha-v-beta-3 integrin in human melanoma cell invasion.
*Proc. Natl. Acad. Sci. USA***89**, 1557–1561.CrossRefGoogle Scholar - Shannon, M. A. and B. Rubinsky (1992). The effect of tumour growth on the stress distribution in tissue.
*Adv. Biol. Heat Mass Transfer***231**, 35–38.Google Scholar - Sherratt, J. A. (1990). Wave front propagation in a competition equation with a new motility term modelling contact inhibition between cell populations.
*Proc. R. Soc. Lond. B***241**, 29–36.Google Scholar - Sherratt, J. A. (1993). Cellular growth and travelling waves of cancer.
*SIAM Appl. Math.***53**, 1713–1730.MATHMathSciNetCrossRefGoogle Scholar - Sherratt, J. A. (2000). Wave front propagation in a competition equation with a new motility term modelling contact inhibition between cell populations.
*Proc. R. Soc. Lond. A.***53**, 2365–2386.MathSciNetGoogle Scholar - Sherratt, J. A. and M. A. J. Chaplain (2001). A new mathematical model for avascular tumour growth.
*J. Math. Biol.***43**, 291–312.MathSciNetCrossRefMATHGoogle Scholar - Shymko, R. M. and L. Glass (1976). Cellular and geometric control of tissue growth and mitotic instability.
*J. Theor. Biol.***63**, 355–374.CrossRefGoogle Scholar - Skalak, R. (1981). Growth as a finite displacement field, in
*Proceedings of the IUTAM Symposium on Finite Elasticity*, The Hague, D. E. Carlson and R. T. Shield (Eds), Martinus Nijhoff Publishers, pp. 347–355.Google Scholar - Skalak, R., G. Dasgupta and M. Moss (1982). Analytical description of growth.
*J. Theor. Biol.***94**, 555–577.MathSciNetCrossRefGoogle Scholar - Skalak, R., S. Zargaryan, R. K. Jain, P. A. Netti and A. Hoger (1996). Compatibility and the genesis of residual stress by volumetric growth.
*J. Math. Biol.***34**, 889–914.MATHGoogle Scholar - Sleeman, B. D. and H. R. Nimmo (1998). Fluid transport in vascularized tumours and metastasis.
*IMA J. Math. Appl. Med. Biol.***15**, 53–63.MATHGoogle Scholar - Snijders, H., J. Huyghe, P. Willems, M. Drost, J. Janssen and A. Huson (1992). A mixture approach to the mechanics of the human intervertebral disc, in
*Mechanics of Swelling*, T. K. Karalis (Ed.), Berlin, Heidelberg: Springer.Google Scholar - Stainsby, W. N. and A. B. Otis (1961). Blood flow, blood oxygen tension, oxygen uptake and oxygen transport in skeletal muscle.
*Am. J. Physiol.***201**, 117–122.Google Scholar - Stetler-Stevenson, W. G., S. Aznavoorian and L. A. Liotta (1993). Tumor cell interactions with the extra-cellular matrix during invasion and metastasis.
*Annu. Rev. Cell Biol.***9**, 541–573.CrossRefGoogle Scholar - Stohrer, M., Y. Boucher, M. Stangassinger and R. K. Jain (2000). Oncotic pressure in solid tumors is elevated.
*Cancer Res.***60**, 4251–4255.Google Scholar - Sutherland, R. M. (1988). Cell and environment interactions in tumor microregions: the multicell spheroid model.
*Science***240**, 177–184.Google Scholar - Sutherland, R. M. and R. E. Durand (1973). Hypoxic cells in an in vitro tumour model.
*Int. J. Radiat. Biol.***23**, 235–246.Google Scholar - Sutherland, R. M., J. A. McCredie and W. R. Inch (1971). Growth of multicell spheroids in tissue culture as a model of nodular carcinomas.
*J. Natl. Cancer Inst.***46**, 113–120.Google Scholar - Swan, G. W. (1981).
*Lecture Notes in Biomathematics*, Vol. 42, Berlin: Springer.Google Scholar - Taber, L. A. (1995). Biomechanics of growth, remodeling and morphogenesis.
*Appl. Mech. Rev.***48**, 487–545.CrossRefGoogle Scholar - Taber, L. A. and D. W. Eggers (1996). Theoretical study of stress-modulated growth in the aorta.
*J. Theor. Biol.***180**, 343–357.CrossRefGoogle Scholar - Taber, L. A. and R. Perucchio (2000). Modeling heart development.
*J. Elasticity***61**, 165–197.MathSciNetCrossRefMATHGoogle Scholar - Takahashi, M. (1966). Theoretical basis for cell cycle analysis. i. Labelled mitosis wave method.
*J. Theor. Biol.***13**, 202–211.CrossRefGoogle Scholar - Takahashi, M. (1968). Theoretical basis for cell cycle analysis. ii. Further studies on labelled mitosis wave method.
*J. Theor. Biol.***18**, 195.CrossRefGoogle Scholar - Tannock, I. F. (1968). The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumour.
*Br. J. Cancer***22**, 258–273.Google Scholar - Tannock, I. F. and A. Howes (1973). The response of viable tumor cords to a single dose of radiation.
*Radiat. Res.***55**, 477–486.Google Scholar - Thames, H. D., A. C. C. Ruifrok, L. Milas, N. Hunter, K. A. Mason, N. H. A. Terry and R. A. White (1996). Accelerated repopulation during fractionated-irradiation of a murine ovarian carcinoma: down-regulation of apoptosis as a possible mechanism.
*Int. J. Radiat. Oncol. Biol. Phys.***35**, 951–962.CrossRefGoogle Scholar - Thomlinson, R. H. and L. H. Gray (1955). The histological structure of some human lung cancers and the possible implications for radiotherapy.
*Br. J. Cancer***9**, 539–549.Google Scholar - Thompson, K. E. and H. M. Byrne (1999). Modelling the internalization of labelled cells in tumour spheroids.
*Bull. Math. Biol.***61**, 601–623.CrossRefGoogle Scholar - Tracqui, P., G. C. Cruywagen, D. E. Woodward, G. T. Bartoo, J. D. Murray and E. G. Alvord (1995). A mathematical model of glioma growth—the effect of chemotherapy on spatiotemporal growth.
*Cell Prolif.***28**, 17–31.Google Scholar - Truesdell, C. and W. Noll (1965). The non-linear field theories of mechanics, in
*Handbuch der Physik*, Flugge (Ed.), Vol. III/III, Berlin: Springer.Google Scholar - Truesdell, C. and R. Toupin (1960). The classical field theories, in
*Handbuch der Physik*, S. Flugge (Ed.), vol. III/I, Berlin: Springer.Google Scholar - Tyzzer, E. E. (1913). Factors in production and growth of tumour metastasis.
*J. Med. Res.***28**, 309–332.Google Scholar - Vaidya, V. G. and F. J. Alexandro (1982). Evaluation of some mathematical models for tumour growth.
*Int. J. Biomed. Comput.***13**, 19–35.MathSciNetCrossRefGoogle Scholar - Van Dyke, T. J. and A. Hoger (2001). Rotations in the theory of growth for soft biological materials, in
*Proceedings of the 2001 ASME Bioengineering Conference, BED-Vol. 50*, USA, pp. 647–648.Google Scholar - Van Lancker, M., C. Goor, R. Sacre, J. Lamote, S. Van Belle, N. De Coene, A. Roelstraete and G. Storme (1995). Patterns of axillary lymph node metastasis in breast cancer.
*Am. J. Clin. Oncol.***18**, 267–272.Google Scholar - Volpe, J. G. P. (1988). Genetic instability of cancer: why a metastatic tumor is unstable and a benign tumor is stable.
*Cancer Genet. Cytogenet.***14**, 125–134.CrossRefGoogle Scholar - von Bertalanffy, L. (1960).
*Fundamental Aspects of Normal and Malignant Growth*, W. W. Nowinsky (Ed.), Amsterdam: Elsevier, pp. 137–259 (Chapter 2).Google Scholar - Warburg, O. (1930).
*The Metabolism of Tumors*, London: Constable Press.Google Scholar - Ward, J. P. (1997). Mathematical modelling of avascular tumour growth, PhD thesis, Nottingham University.Google Scholar
- Ward, J. P. and J. R. King (1997). Mathematical modelling of avascular tumour growth.
*IMA J. Math. Appl. Med. Biol.***14**, 36–69.Google Scholar - Ward, J. P. and J. R. King (1999a). Mathematical modelling of avascular tumour growth, ii. Modelling growth saturation.
*IMA J. Math. Appl. Med. Biol.***16**, 171–211.MATHGoogle Scholar - Ward, J. P. and J. R. King (1999b). Mathematical modelling of the effects of mitotic inhibitors on avascular tumour growth.
*J. Theor. Med.***1**, 287–311.MATHGoogle Scholar - Ward, J. P. and J. R. King (2000). Modelling the effect of cell shedding on avascular tumour growth.
*J. Theor. Med.***2**, 155–174.MATHGoogle Scholar - Ward, J. P. and J. R. King (2003). Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures.
*Math. Biosci.***181**, 177–207.MathSciNetCrossRefMATHGoogle Scholar - Webb, S. D., J. A. Sherratt and R. G. Fish (1999a). Alterations in proteolytic activity at low pH and its association with invasion: a theoretical model.
*Clin. Exp. Metastasis***17**, 397–407.CrossRefGoogle Scholar - Webb, S. D., J. A. Sherratt and R. G. Fish (1999b). Mathematical modelling of tumour acidity: regulation of intracellular pH.
*J. Theor. Biol.***196**, 237–250.CrossRefGoogle Scholar - Wein, L. M., J. T. Wu, A. G. Ianculescu and R. K. Puri (2002). A mathematical model of the impact of infused targeted cytotoxic agents on brain tumours: implications for detection, design and delivery.
*Cell Prolif.***35**, 343–361.CrossRefGoogle Scholar - Weiss, L. (2000). The morphologic documentation of clinical progression, invasion metastasis—staging.
*Cancer Metastasis Rev.***19**, 303–313.CrossRefGoogle Scholar - Wette, R., I. N. Katz and E. Y. Rodin (1974a). Stochastic processes for solid tumor kinetics i. Surface-regulated growth.
*Math. Biosci.***19**, 231–255.CrossRefMATHGoogle Scholar - Wette, R., I. N. Katz and E. Y. Rodin (1974b). Stochastic processes for solid tumor kinetics ii. Diffusion-regulated growth.
*Math. Biosci.***21**, 311–338.CrossRefMATHGoogle Scholar - Winsor, C. P. (1932). The Gompertz curve as a growth curve.
*Proc. Natl. Acad. Sci. USA*1–7.Google Scholar - Yuhas, J. M. and A. P. Li (1978). Growth fraction as the major determinant of multicellular tumor spheroid growth rates.
*Cancer Res.***38**, 1528–1532.Google Scholar - Yuhas, J. M., A. E. Tarleton and K. B. Molzen (1978). Multicellular tumor spheroid formation by breast cancer cells isolated from different sites.
*Cancer Res.***38**, 2486–2491.Google Scholar - Znati, C. A., M. Rosenstein, Y. Boucher, M. W. Epperly, W. D. Bloomer and R. K. Jain (1996). Effect of radiation on interstitial fluid pressure and oxygenation in a human tumor xenograft.
*Cancer Res.***56**, 964–968.Google Scholar