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Multiphase Models of Tumour Growth

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Selected Topics in Cancer Modeling

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References

  1. Ambrosi, D., Mollica, F.: On the mechanics of a growing tumor. Int. J. Engng. Sci.,40, 1297–1316 (2002)

    Article  MathSciNet  Google Scholar 

  2. Ambrosi, D., Mollica, F.: The role of stress in the growth of a multicell spheroid. J. Math. Biol.,48, 477–499 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. 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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ambrosi, D., Preziosi, L.: On the closure of mass balance models for tumor growth. Math. Mod. Meth. Appl. Sci.,12, 737–754 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ambrosi, D., Preziosi, L.: umors as elasto-viscoplastic growing bodies. Biomechanics and Modeling in Mechanobiology (2008), submitted

    Google Scholar 

  6. Astanin, S., Tosin, A.: Mathematical model of tumour cord growth along the source of nutrient. Math. Mod. Nat. Phen. (2008), in press

    Google Scholar 

  7. Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic Publishers, Dordrecht (1990)

    Google Scholar 

  8. 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)

    Article  MATH  MathSciNet  Google Scholar 

  9. Breward, C., Byrne, H., Lewis, C.: A multiphase model describing vascular tumor growth. Bull. Math. Biol.,65, 609–640 (2003)

    Article  Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. Byrne, H., King, J., McElwain, D., Preziosi, L.: A two-phase model of solid tumor growth. Appl. Math. Letters,16, 567–573 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bowen, R.M.: The theory of mixtures. In: A. Eringen (ed.) Continuum Physics, vol. 3, Academic Press, New York (1976)

    Google Scholar 

  13. Byrne, H., Preziosi, L.: Modeling solid tumor growth using the theory of mixtures. Math. Med. Biol.,20, 341–366 (2004)

    Article  Google Scholar 

  14. 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. 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)

    Article  MATH  Google Scholar 

  16. Chambers, A., Matrisian, L.: Changing views of the role of matrix metalloproteinases in metastasis. J. Natl. Cancer Inst.,89, 1260–1270 (1997)

    Article  Google Scholar 

  17. 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)

    Article  MATH  MathSciNet  Google Scholar 

  18. 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)

    Article  MATH  Google Scholar 

  19. Forgacs, G., Foty, R., Shafrir, Y., Steinberg, M.: Viscoelastic properties of living embryonic tissues: A quantitative study. Biophys. J.,74, 2227– 2234 (1998)

    Article  Google Scholar 

  20. Franks, S., King, J.: Interactions between a uniformly proliferating tumor and its surroundings. Uniform material properties. Math. Med. Biol.,20, 47–89 (2003)

    Article  MATH  Google Scholar 

  21. Gillies, R., Gatenby, R.: Hypoxia and adaptive landscapes in the evolution of carcinogenesis. Cancer Metastasis Rev. (2007), e-publication

    Google Scholar 

  22. Gatenby, R., Gawlinski, E., Gmitro, A., Kaylor, B., Gillies, R.: Acidmediated tumour invasion: A multidisciplinary study. Cancer Res.,66, 5216–5223 (2006)

    Article  Google Scholar 

  23. 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. 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. Humphrey, J., Rajagopal, K.: A constrained mixture model for growth and remodeling of soft tissues. Math. Mod. Meth. Appl. Sci.,12, 407– 430 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Iordan, A., Duperray, A., Verdier, C.: Fractal approach to the rheology of concentrated cell suspension. Phys. Rev. E,77, 011911 (2008)

    Article  Google Scholar 

  27. Jayaraman, G.: Water transport in the arterial wall. A theoretical study. J. Biomech.,16, 833–840 (1983)

    Google Scholar 

  28. Kenion, D.: A mathematical model of water flux through aortic tissue. Bull. Math. Biol.,41, 79–90 (1979)

    MathSciNet  Google Scholar 

  29. 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)

    Article  Google Scholar 

  30. Klanchar, M., Tarbell, J.: Modeling water flow through arterial tissue. Bull. Math. Biol.,49, 651–669 (1987)

    MATH  MathSciNet  Google Scholar 

  31. 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. Matrisian, L.: The matrix-degrading metalloproteinases. Bioessays,14, 455–463 (1992)

    Article  Google Scholar 

  33. Mow, V., Holmes, M., Lai, W.: Fluid transport and mechanical problems of articular cartilage: A review. J. Biomech.,17, 377–394 (1984)

    Article  Google Scholar 

  34. 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. Mow, V., Lai, W.: Mechanics of animal joints. Ann. Rev. Fluid Mech.,11, 247–288 (1979)

    Article  Google Scholar 

  36. Mollica, F., Preziosi, L., Rajagopal, K.: Modelling of Biological Materials. Birkhäuser, Boston, MA (2007)

    Book  Google Scholar 

  37. Mow, V., Ratcliffe, A., Savio, L.Y: (eds.) Biomechanics of Diarthroidal Joints. Springer-Verlag, New York (1990)

    Google Scholar 

  38. Nelson, D., Cox, M.: I principi di biochimica di Lehninger. Zanichelli (2002), translated by M. Averna, E. Melloni, A. Sdraffa

    Google Scholar 

  39. 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)

    Article  Google Scholar 

  40. Oomens, C., Van Campen, D., Grootenboer, H.: A mixture approach to the mechanics of skin. J. Biomech.,20, 877–885 (1987)

    Article  Google Scholar 

  41. 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)

    Article  MATH  Google Scholar 

  42. Preziosi, L., Farina, A.: On Darcy’s law for growing porous media. Int. J. Nonlinear Mech.,37, 485–491 (2001)

    Article  Google Scholar 

  43. Preziosi, L., Joseph, D.: Stokes’ first problem for viscoelastic fluids. J. Non-Newtonian Fluid Mech.,25, 239–259 (1987)

    Article  MATH  Google Scholar 

  44. Preziosi, L., Tosin, A.: Multiphase modeling of tumor growth and extracellular matrix interaction: Mathematical tools and applications. J. Math. Biol. (2008), in press

    Google Scholar 

  45. Parson, S., Watson, S., Brown, P., Collins, H., Steele, R.: Matrix metalloproteinases. Brit. J. Surg.,84, 160–166 (1997)

    Article  Google Scholar 

  46. 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)

    MATH  Google Scholar 

  47. Rajagopal, K., Tao, L.: Mechanics of Mixtures. World Scientific, Singapore (1995)

    MATH  Google Scholar 

  48. 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)

    Article  Google Scholar 

  49. 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)

    Article  MathSciNet  Google Scholar 

  50. 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)

    Article  Google Scholar 

  51. Sorek, S., Sideman, S.: A porous medium approach for modelling heart mechanics. Part 1: Theory. Math. Biosci.,81, 1–14 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  52. 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)

    Article  Google Scholar 

  53. 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)

    Article  Google Scholar 

  54. Winters, B., Shepard, S., Foty, R.: Biophysical measurement of brain tumor cohesion. Int. J. Cancer,114, 371–379 (2005)

    Article  Google Scholar 

  55. Yang, M., Taber, L., Clark, E.: A nonlinear poroelastic model for the trabecular embryonic heart. J. Biomech. Engng.,116, 213–223 (1994)

    Article  Google Scholar 

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Correspondence to Sergey Astanin .

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Astanin, S., Preziosi, L. (2008). Multiphase Models of Tumour Growth. In: Selected Topics in Cancer Modeling. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4713-1_9

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