Abstract
Medically, tumours are classified into two important classes — benign and malignant. Generally speaking, the two classes display different behaviour with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumour growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid tumour using membrane and thick-shell theory. A central feature of the model is the characterisation of the material composition of the model through the use of a strain-energy function, thus permitting a mathematical description of the degree of differentiation of the tumour explicitly in the model. Conditions are given in terms of the strain-energy function for the processes of invasion and metastasis occurring in a tumour, being interpreted as the bifurcation modes of the spherical shell which the tumour is essentially modelled as. Our results are compared with actual experimental results and with the general behaviour shown by benign and malignant tumours. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumours (in particular, the Gaussian and mean curvatures of the surface of a solid tumour) in an attempt to produce a mathematical formulation and description of the important medical processes of staging and grading cancers. We hope that this approach may form the basis of a practical application.
Similar content being viewed by others
References
Adam, J. A.: A simplified mathematical model of tumour growth. Math. Biosci. 81, 224–229 (1986)
Aroesty, J., Lincoln, T., Shapiro, N., Boccia, G.: Tumour growth and chemotherapy: mathematical methods, computer simulations, and experimental foundations. Math. Biosci. 17, 243–300 (1973)
Balding, D., McElwain, D. L. S.: A mathematical model of tumour-induced capillary growth. J. Theor. Biol. 114, 53–73 (1985)
Bogen, D. K.: Strain energy descriptions of biological swelling I: single fluid compartment models. ASME J. Biomech. Eng. 109, 252–256 (1987)
Brzakovic, D., Luo, X. M., Brzakovic, P.: An approach to automated detection of tumours in mammograms. IEEE Trans. Med. Imag. 9, 233–241 (1990)
Burton, A. C.: Rate of growth of solid tumours as a problem of diffusion. Growth 30, 157–176 (1966)
Chaplain, M. A. J., Sleeman, B. D.: An application of membrane theory to tip morphogenesis in Acetabularia. J. Theor. Biol. 146, 177–200 (1990)
Chaplain, M. A. J., Sleeman, B. D.: A mathematical model for the production and secretion of tumour angiogenesis factor in tumours. IMA J. Math. Appl. Med. Biol. 7, 93–108 (1990)
Cummings, F. W.: On surface geometry coupled to morphogen. J. Theor. Biol. 137, 215–219 (1989)
Demiray, H.: Large deformation analysis of some basic problems in biophysics. Bull. Math. Biol. 38, 701–712 (1976)
Demiray, H.: Large deformation analysis of some soft biological tissues. ASME J. Biomech. Eng. 103, 73–78 (1981)
Dhawan, A. P., Buelloni, G., Gordon, R.: Enhancement of mammographic features by optimal adaptive neighborhood image processing. IEEE Trans. Med. Imag. 5, 8–15 (1986)
Duncan, J. S., Lee, F. A., Smeulders, A. W. M., Zaret, B. L.: A bending energy model for measurement of cardiac shape deformity. IEEE Trans. Med. Imag. 10, 307–320 (1991)
Feodos'ev, V. I.: On equilibrium modes of a rubber spherical shell. Prikl. Mat. Mekh. 32, 335–341 (1968)
Folkman, J.: The vascularization of tumours. Sci. Am. 234, 58–73 (1976)
Folkman, J., Moscona, A.: The role of cell shape in growth control. Nature (London) 273, 345–349 (1978)
Fung, Y. C.: Biomechanics. Berlin Heidelberg New York: Springer 1981
Gallez, D.: Cell membranes after malignant transformation part I: dynamic stability at low surface tension. J. Theor. Biol. 111, 323–340 (1984)
Gou, P. F.: Strain energy functions for biological tissues. J. Biomech. 3, 547–550 (1970)
Greenspan, H. P.: Models for the growth of a solid tumour by diffusion. Stud. Appl. Math. 51, 317–340 (1972)
Greenspan, H. P.: On the self-inhibited growth of cell cultures. Growth 38, 81–97 (1974)
Greenspan, H. P.: On the growth and stability of cell cultures and solid tumours. J. Theor. Biol. 56, 229–242 (1976)
Greenspan, H. P.: On the dynamics of cell cleavage. J. Theor. Biol. 65, 79–99 (1977a)
Greenspan, H. P.: On the deformation of a viscous droplet caused by variable surface tension. Stud. Appl. Math. 57, 45–58 (1977b)
Greenspan, H. P.: On fluid-mechanical simulations of cell division and movement. J. Theor. Biol. 70, 125–134 (1978)
Gyllenberg, M., Webb, G. F.: Quiescence as an explanation of Gompertzian tumour growth. Growth Dev. Aging 53, 25–33 (1989)
Haralick, R. M., Watson, L. T., Laffey, T. J.: The topographic primal sketch. Int. J. Robot. Res. 2, 50–72 (1983)
Hart, T. N., Trainor, L. E. H.: Geometrical aspects of surface morphogenesis. J. Theor. Biol. 138, 271–296 (1989)
Haughton, D. M., Ogden, R. W.: On the incremental equations in nonlinear elasticity — I Membrane theory. J. Mech. Phys. Solids 26, 93–110 (1978a)
Haughton, D. M., Ogden, R. W.: On the incremental equations in nonlinear elasticity — II Bifurcation of pressurized spherical shells. J. Mech. Phys. Solids 26, 111–138 (1978b)
Hettiaratchi, D. R. P., O'Callaghan, J. R.: A membrane model of plant cell extension. J. Theor. Biol. 45, 459–465 (1974)
Hettiaratchi, D. R. P., O'Callaghan, J. R.: Structural mechanics of plant cells. J. Theor. Biol. 74, 235–257 (1978)
Humphrey, J. D., Yin, F. C. P.: On constitutive relations and finite deformations of passive cardiac tissue: I. a pseudostrain energy function. ASME J. Biomech. Eng. 109, 298–304 (1987)
Isenberg, C.: The Science of Soap Films and Soap Bubbles. Tieto: Woodspring 1978
Lai, S-M., Li, Z., Bischof, W. F.: On techniques for detecting circumscribed masses in mammograms. IEEE Trans. Med. Imag. 8, 377–386 (1989)
Landau, L. D., Lifschitz, E. M.: Theory of Elasticity. London: Pergamon 1959
McCoy, E. L.: The strain energy function in axial plant growth. J. Math. Biol. 27, 575–594 (1989)
Melicow, M. M.: The three steps to cancer: a new concept of cancerigenesis. J. Theor. Biol. 94, 471–511 (1982)
Muir, Sir Robert: In: Anderson, J. R. (ed.) Muir's Textbook of Pathology, 12 ed. London: E. Arnold 1985
Needleman, A.: Necking of pressurized spherical membranes. J. Mech. Phys. Solids 24, 339–359 (1976)
Needleman, A.: Inflation of spherical rubber membranes. Int. J. Solids Struct. 13, 409–421 (1977)
Nicolson, G.: Transmembrane control of the receptors on normal and tumour cells II Surface changes associated with transformation and malignancy. Biochim. Biophys. Acta 458, 1–72 (1976)
Norton, L., Simon, R., Brereton, H. D., Bogden, A. E.: Predicting the course of Gompertzian growth. Nature 264, 542–545 (1976)
Ogden, R. W.: Large deformation isotropic elasticity — on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc., Ser. A 326, 565–584 (1972)
Ogden, R. W.: On stress rates in solid mechanics with application to elasticity theory. Proc. Camb. Philos. Soc. 75, 303–319 (1974)
Perham, D. M., Robertson, A. J., Brown, R. A.: Morphometric analysis of breast carcinoma: association with survival. J. Clin. Pathol. 41, 173–177 (1988)
Paweletz, N., Knierim, M.: Tumour-related angiogenesis. Crit. Rev. Oncol. Hematol. 9, 197–242 (1989)
Richardson, D.: Random growth in a tessellation. Proc. Camb. Philos. Soc. 74, 515–528 (1973)
Robb, R. A., Barillot, C.: Interactive display and analysis of 3-D medical images. IEEE Trans. Med. Image. 8, 217–226 (1989)
Schwegler, H., Tarumi, K., Gerstman, B.: Physico-chemical model of a protocell. J. Math. Biol. 22, 335–348 (1985)
Sekimura, T., Hotani, H.: The morphogenesis of liposomes viewed from the aspect of bending energy. J. Theor. Biol. 149, 325–337 (1991)
Skalak, R., Tozeren, A., Zarda, R. P., Chien, S.: Strain energy function of red blood cell membranes. Biophys. J. 13, 245–264 (1973)
Shymko, R. M., Glass, L.: Cellular and geometric control of tissue growth and geometric instability. J. Theor. Biol. 63, 355–375 (1976)
Sutherland, R. M., McCredie, J. A., Inch, W. R.: Growth of multicell spheroids in tissue culture as a model of nodular carcinomas. J. Nat. Cancer Inst. 46, 113–120 (1971)
Svetina, S., Ottova-Leitmannova, A., Glaser, R.: Membrane bending energy in relation to bilayer couples concept of red blood cell shape transformations. J. Theor. Biol. 94, 13–23 (1982)
Terzopoulos, D., Platt, J., Barr, A., Fleischer, K.: Elastically deformable models. Comput. Graph. 21, 205–214 (1987)
Thomlinson, R. H., Grey, L. H.: The histological structure of some human lung cancers and the possible implications for radiotherapy. Br. J. Cancer 9, 539–549 (1955)
Thompson, D'Arcy W. (ed.): On Growth and Form, abridged. Cambridge: Cambridge University Press (1961)
Todd, P. H.: Gaussian curvature as a parameter of biological surface growth. J. Theor. Biol. 113, 63–68 (1985)
Tubiana, M.: The kinetics of tumour cell proliferation and radiotherapy. Br. J. Radiol. 44, 325–347 (1971)
Vito, R.: A note on arterial elasticity. J. Biomech. 6, 561–564 (1973)
Williams, T., Bjerknes, R.: Stochastic model for abnormal clone spread through epithelial basal layer. Nature 236, 19–21 (1972)
Willmott, N., Goldberg, J., Anderson, J., Bessent, R., McKillop, J., McArdle, C.: Abnormal vascualature of solid tumours: significance for microsphere-based targetting strategies. Int. J. Radiat. Biol. (to appear)
Wu, H., Spence, R. D., Sharpe, P. J. H.: Plant cell wall elasticity II: polymer elastic properties of the microfibrils. J. Theor. Biol. 133, 239–253 (1988)
Zinemanas, D., Nir, A.: On the viscous deformation of biological cells under anisotropic surface tension. J. Fluid Mech. 193, 217–241 (1988)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chaplain, M.A.J., Sleeman, B.D. Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory. J. Math. Biol. 31, 431–473 (1993). https://doi.org/10.1007/BF00173886
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00173886