Multiphase Models of Tumour Growth

Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


Constitutive Equation Interaction Force Mass Balance Equation Mixture Theory Growth Term 


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  1. [AM02]
    Ambrosi, D., Mollica, F.: On the mechanics of a growing tumor. Int. J. Engng. Sci.,40, 1297–1316 (2002)CrossRefMathSciNetGoogle Scholar
  2. [AM04]
    Ambrosi, D., Mollica, F.: The role of stress in the growth of a multicell spheroid. J. Math. Biol.,48, 477–499 (2004)MATHCrossRefMathSciNetGoogle Scholar
  3. [AM05]
    Araujo, R., McElwain, D.: A mixture theory for the genesis of residual stresses in growing tissues, I: A general formulation. SIAM J. Appl. Math.,65, 1261–1284 (2005)MATHCrossRefMathSciNetGoogle Scholar
  4. [AP02]
    Ambrosi, D., Preziosi, L.: On the closure of mass balance models for tumor growth. Math. Mod. Meth. Appl. Sci.,12, 737–754 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. [AP08]Ambrosi,
    Ambrosi, D., Preziosi, L.: umors as elasto-viscoplastic growing bodies. Biomechanics and Modeling in Mechanobiology (2008), submittedGoogle Scholar
  6. [AT08]Astanin,
    Astanin, S., Tosin, A.: Mathematical model of tumour cord growth along the source of nutrient. Math. Mod. Nat. Phen. (2008), in pressGoogle Scholar
  7. [BB90]
    Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic Publishers, Dordrecht (1990)Google Scholar
  8. [BBL02]
    Breward, C., Byrne, H.: Lewis, C., The role of cell-cell interactions in a two-phase model for avascular tumor growth. J. Math. Biol.,45, 125–152 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. [BBL03]
    Breward, C., Byrne, H., Lewis, C.: A multiphase model describing vascular tumor growth. Bull. Math. Biol.,65, 609–640 (2003)CrossRefGoogle Scholar
  10. [BHN+00]
    Baumgartner, W., Hinterdorfer, P., Ness, W., Raab, A., Vestweber, D., Schindler, H., Drenckhahn, D.: Cadherin interaction probed by atomic force microscopy. Proc. Nat. Acad. Sci. USA,97, 4005–4010 (2000)CrossRefGoogle Scholar
  11. [BKMP03]
    Byrne, H., King, J., McElwain, D., Preziosi, L.: A two-phase model of solid tumor growth. Appl. Math. Letters,16, 567–573 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. [Bow76]
    Bowen, R.M.: The theory of mixtures. In: A. Eringen (ed.) Continuum Physics, vol. 3, Academic Press, New York (1976)Google Scholar
  13. [BP04]
    Byrne, H., Preziosi, L.: Modeling solid tumor growth using the theory of mixtures. Math. Med. Biol.,20, 341–366 (2004)CrossRefGoogle Scholar
  14. [CDLV05]
    Canetta, E., Duperray, A., Leyrat, A., Verdier, C.: Measuring cell viscoelastic properties using a force-spectrometer: Influence of the protein-cytoplasm interactions. Biorheology,42, 298–303 (2005)Google Scholar
  15. [CGP06]
    Chaplain, M., Graziano, L., Preziosi, L.: Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development. Math. Med. Biol.,23, 197–229 (2006)MATHCrossRefGoogle Scholar
  16. [CM97]
    Chambers, A., Matrisian, L.: Changing views of the role of matrix metalloproteinases in metastasis. J. Natl. Cancer Inst.,89, 1260–1270 (1997)CrossRefGoogle Scholar
  17. [FBK+03]
    Franks, S., Byrne, H., King, J., Underwood, J., Lewis, C.: Modelling the early growth of ductal carcinoma in situ of the breast. J. Math. Biol.,47, 424–452 (2003)MATHCrossRefMathSciNetGoogle Scholar
  18. [FBM+03]
    Franks, S., Byrne, H., Mudhar, H., Underwood, J., Lewis, C.: Mathematical modelling of comedo ductal carcinoma in situ of the breast. Math. Med. Biol.,20, 277–308 (2003)MATHCrossRefGoogle Scholar
  19. [FFSS98]
    Forgacs, G., Foty, R., Shafrir, Y., Steinberg, M.: Viscoelastic properties of living embryonic tissues: A quantitative study. Biophys. J.,74, 2227– 2234 (1998)CrossRefGoogle Scholar
  20. [FK03]
    Franks, S., King, J.: Interactions between a uniformly proliferating tumor and its surroundings. Uniform material properties. Math. Med. Biol.,20, 47–89 (2003)MATHCrossRefGoogle Scholar
  21. [GG07]Gillies,
    Gillies, R., Gatenby, R.: Hypoxia and adaptive landscapes in the evolution of carcinogenesis. Cancer Metastasis Rev. (2007), e-publicationGoogle Scholar
  22. [GGG+06]
    Gatenby, R., Gawlinski, E., Gmitro, A., Kaylor, B., Gillies, R.: Acidmediated tumour invasion: A multidisciplinary study. Cancer Res.,66, 5216–5223 (2006)CrossRefGoogle Scholar
  23. [GP07]
    Graziano, L., Preziosi, L.: Mechanics in tumour growth. In: F. Mollica, L. Preziosi, K. Rajagopal (eds.) Modeling of Biological Materials, 267– 328, Birkhäuser, Boston, MA (2007)Google Scholar
  24. [HHLM89]
    Hou, J., Holmes, M., Lai, W., Mow, V.: Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. J. Biomech. Engng.,111, 78–87 (1989)Google Scholar
  25. [HR02]
    Humphrey, J., Rajagopal, K.: A constrained mixture model for growth and remodeling of soft tissues. Math. Mod. Meth. Appl. Sci.,12, 407– 430 (2002)MATHCrossRefMathSciNetGoogle Scholar
  26. [Iord08]
    Iordan, A., Duperray, A., Verdier, C.: Fractal approach to the rheology of concentrated cell suspension. Phys. Rev. E,77, 011911 (2008)CrossRefGoogle Scholar
  27. [Jay83]
    Jayaraman, G.: Water transport in the arterial wall. A theoretical study. J. Biomech.,16, 833–840 (1983)Google Scholar
  28. [Ken79]
    Kenion, D.: A mathematical model of water flux through aortic tissue. Bull. Math. Biol.,41, 79–90 (1979)MathSciNetGoogle Scholar
  29. [KLM90]
    Kwan, M., Lai, W., Mow, V.: A finite deformation theory of cartilage and other soft hydrated connective tissues: Part I – Equilibrium results. J. Biomech.,23, 145–155 (1990)CrossRefGoogle Scholar
  30. [KT87]
    Klanchar, M., Tarbell, J.: Modeling water flow through arterial tissue. Bull. Math. Biol.,49, 651–669 (1987)MATHMathSciNetGoogle Scholar
  31. [LHM90]
    Lai, W., Hou, J., Mow, V.: A triphasic theory for the swelling properties of hydrated charged soft biological tissues. In: Biomechanics of Diarthroidal Joints, vol. 1, 283–312, Springer, New York (1990)Google Scholar
  32. [Mat92]
    Matrisian, L.: The matrix-degrading metalloproteinases. Bioessays,14, 455–463 (1992)CrossRefGoogle Scholar
  33. [MHL84]
    Mow, V., Holmes, M., Lai, W.: Fluid transport and mechanical problems of articular cartilage: A review. J. Biomech.,17, 377–394 (1984)CrossRefGoogle Scholar
  34. [MKLA80]
    Mow, V., Kuei, S., Lai, W., Armstrong, C.: Biphasic creep and stress relaxation of articular cartilage: Theory and experiment. J. Biomech. Engng.,102, 73–84 (1980)Google Scholar
  35. [ML79]
    Mow, V., Lai, W.: Mechanics of animal joints. Ann. Rev. Fluid Mech.,11, 247–288 (1979)CrossRefGoogle Scholar
  36. [MPR07]
    Mollica, F., Preziosi, L., Rajagopal, K.: Modelling of Biological Materials. Birkhäuser, Boston, MA (2007)CrossRefGoogle Scholar
  37. [MRS90]
    Mow, V., Ratcliffe, A., Savio, L.Y: (eds.) Biomechanics of Diarthroidal Joints. Springer-Verlag, New York (1990)Google Scholar
  38. [NC02]Nelson,
    Nelson, D., Cox, M.: I principi di biochimica di Lehninger. Zanichelli (2002), translated by M. Averna, E. Melloni, A. SdraffaGoogle Scholar
  39. [Nic85]
    Nicholson, C.: Diffusion from an injected volume of a substance in brain tissue with arbitrary volume fraction and tortuosity. Brain Res.,333, 325–329 (1985)CrossRefGoogle Scholar
  40. [OVCG87]
    Oomens, C., Van Campen, D., Grootenboer, H.: A mixture approach to the mechanics of skin. J. Biomech.,20, 877–885 (1987)CrossRefGoogle Scholar
  41. [Pre89]
    Preziosi, L.: On an invariance property of the solution to Stokes first problem for viscoelastic fluids. J. Non-Newtonian Fluid Mech.,33, 225-228 (1989)MATHCrossRefGoogle Scholar
  42. [PF01]
    Preziosi, L., Farina, A.: On Darcy’s law for growing porous media. Int. J. Nonlinear Mech.,37, 485–491 (2001)CrossRefGoogle Scholar
  43. [PJ87]
    Preziosi, L., Joseph, D.: Stokes’ first problem for viscoelastic fluids. J. Non-Newtonian Fluid Mech.,25, 239–259 (1987)MATHCrossRefGoogle Scholar
  44. [PT08]Preziosi,
    Preziosi, L., Tosin, A.: Multiphase modeling of tumor growth and extracellular matrix interaction: Mathematical tools and applications. J. Math. Biol. (2008), in pressGoogle Scholar
  45. [PWB+97]
    Parson, S., Watson, S., Brown, P., Collins, H., Steele, R.: Matrix metalloproteinases. Brit. J. Surg.,84, 160–166 (1997)CrossRefGoogle Scholar
  46. [RHR03]
    Rao, I., Humphrey, J., Rajagopal, K.: Biological growth and remodeling: A uniaxial example with possible application to tendons and ligaments. Comp. Mod Engr. Sci.,4, 439–455 (2003)MATHGoogle Scholar
  47. [RT95]
    Rajagopal, K., Tao, L.: Mechanics of Mixtures. World Scientific, Singapore (1995)MATHGoogle Scholar
  48. [SCZM06]
    Sarntinoranont, M., Chen, X., Zhao, J., Mareci, T.: Computational model of interstitial transport in the spinal cord using diffusion tensor imaging. Ann. Biomed. Eng.,34, 1304–1321 (2006)CrossRefGoogle Scholar
  49. [SGG+07]
    Smallbone, K., Gatenby, R., Gillies, R., Maini, P., Gavaghan, D.: Metabolic changes during carcinogenesis: Potential impact on invasiveness. J. Theor. Biol.,244, 703–713 (2007)CrossRefMathSciNetGoogle Scholar
  50. [SGH+05]
    Sun, M., Graham, J., Hegedus, B., Marga, F., Zhang, Y., Forgacs, G., Grandbois, M.: Multiple membrane tethers probed by atomic force microscopy. Biophys. J.,89, 4320–4329 (2005)CrossRefGoogle Scholar
  51. [SS86]
    Sorek, S., Sideman, S.: A porous medium approach for modelling heart mechanics. Part 1: Theory. Math. Biosci.,81, 1–14 (1986)MATHCrossRefMathSciNetGoogle Scholar
  52. [SSHC96]
    Stetler-Stevenson, W., Hewitt, R., Corcoran, M.: Matrix metallo– proteinases and tumour invasion: From correlation to causality to the clinic. Cancer Biol.,7, 147–154 (1996)CrossRefGoogle Scholar
  53. [TIZB84]
    Tsaturyan, A., Izacov, V., Zhelamsky, S., Bykov, B.: Extracellular fluid filtration as the reason for the viscoelastic behaviour of the passive myocardium. J. Biomech.,17, 749–755 (1984)CrossRefGoogle Scholar
  54. [WSF05]
    Winters, B., Shepard, S., Foty, R.: Biophysical measurement of brain tumor cohesion. Int. J. Cancer,114, 371–379 (2005)CrossRefGoogle Scholar
  55. [YTC94]
    Yang, M., Taber, L., Clark, E.: A nonlinear poroelastic model for the trabecular embryonic heart. J. Biomech. Engng.,116, 213–223 (1994)CrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di TorinoItaly

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