Skip to main content
SpringerLink
Log in

Micromechanical analysis of volumetric growth in the context of open systems thermodynamics and configurational mechanics. Application to tumor growth

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

We adopt in this paper the physically and micromechanically motivated point of view that growth (resp. resorption) occurs as the expansion (resp. contraction) of initially small tissue elements distributed within a host surrounding matrix, due to the interfacial motion of their boundary. The interface motion is controlled by the availability of nutrients and mechanical driving forces resulting from the internal stresses that built in during the growth. A general extremum principle of the zero potential for open systems witnessing a change of their mass due to the diffusion of nutrients is constructed, considering the framework of open systems thermodynamics. We postulate that the shape of the tissue element evolves in such a way as to minimize the zero potential among all possible admissible shapes of the growing tissue elements. The resulting driving force for the motion of the interface sets a surface growth models at the scale of the growing tissue elements, and is conjugated to a driving force identified as the interfacial jump of the normal component of an energy momentum tensor, in line with Hadamard’s structure theorem. The balance laws associated with volumetric growth at the mesoscopic level result as the averaging of surface growth mechanisms occurring at the microscopic scale of the growing tissue elements. The average kinematics has been formulated in terms of the effective growth velocity gradient and elastic rate of deformation tensor, both functions of time. This formalism is exemplified by the simulation of the avascular growth of multicell spheroids in the presence of diffusion of nutrients, showing the respective influence of mechanical and chemical driving forces in relation to generation of internal stresses.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adam, J.A., Maggelakis, S.A.: Diffusion regulated growth characteristics of a spherical prevascular carcinoma. Bull. Math. Biol. 52(4), 549–582 (1990)

    Article  MATH  Google Scholar 

  2. Ambrosia, D., Ateshian, G.A., Arruda, E.M., Cowin, S.C., Dumais, J., Goriely, A., Holzapfel, G.A., Humphrey, J.D., Kemkemer, R., Kuhl, E., Olberdingc, J.E., Taber, L.A., Garikipati, K.: Perspectives on biological growth and remodeling. J. Mech. Phys. Solids 59(4), 863–883 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Ambrosi, D., Guillou, A.: Growth and dissipation in biological tissues. Contin. Mech. Thermodyn. 19(5), 245–251 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Blatz, P.J., Ko, W.L.: Application of finite elasticity theory to the deformation of rubbery materials. Trans. Soc. Rheol. 6, 223–251 (1962)

    Article  Google Scholar 

  7. Brú, A., Albertos, S., Subiza, J.L., García-Asenjo, J.L., Brú, I.: The universal dynamics of tumor growth. Biophys. J. 85(5), 2948–2961 (2003)

    Article  ADS  Google Scholar 

  8. Byrne, H.: Mathematical Biomedicine and Modeling Avascular Tumor Growth. OCCAM, University of Oxford, UK. Internal report (2012)

  9. Callen, H.B.: Thermodynamics and An Introduction to Thermostatics. Wiley, New York (1985)

    MATH  Google Scholar 

  10. Cowin, S.C., Hegedus, D.H.: Bone remodeling I: theory of adaptive elasticity. J. Elast. 6(3), 313–326 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Drasdo, D., Höhme, S.: A single-cell-based model of tumor growth in vitro: monolayers and spheroids. Phys. Biol. 2(3), 133–147 (2005)

    Article  ADS  Google Scholar 

  12. De Donder, T.: Leçons de thermodynamique et de chimie physique. Gauthiers-Villars, Paris (1920)

    Google Scholar 

  13. Drozdov, A.D.: Volumetric growth of viscoelastic solids. Mech. Solids 25, 99–106 (1990)

    Google Scholar 

  14. Entov, V.M.: Mechanical model of scoliosis. Mech. Solids 18, 199–206 (1983)

    Google Scholar 

  15. Epstein, M.: Kinetics of boundary growth. Mech. Res. Commun. 37(5), 453–457 (2010)

    Article  MATH  Google Scholar 

  16. Epstein, M., Maugin, G.A.: Thermomechanics of volumetric growth in uniform bodies. Int. J. Plast. 16, 951–978 (2000)

    Article  MATH  Google Scholar 

  17. Eshelby, J.D.: The force on an elastic singularity. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 244(877), 87–112 (1951)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Folkman, J., Hochbergand, M.: Self-regulation of growth in three dimensions. J. Exp. Med. 138, 745–753 (1973)

    Article  Google Scholar 

  19. Ganghoffer, J.F.: Eshelby tensors, thermodynamics and calculus of variations. Application to volumetric growth. Int. J. Eng. Sci. 48(12), 2081–2098 (2010)

    Article  MATH  Google Scholar 

  20. Ganghoffer, J.F.: Mechanical modeling of growth considering domain variation—part II: volumetric and surface growth involving Eshelby tensors. J. Mech. Phys. Solids 58(9), 1434–1459 (2010)

    Article  ADS  MATH  Google Scholar 

  21. Ganghoffer, J.F.: On Eshelby tensors in the context of the thermodynamics of open systems: application to volumetric growth. Int. J. Eng. Sci. (2010) doi:10.1016/j.ijengsci.2010.04.003

  22. Ganghoffer, J.F.: Mechanics and Thermodynamics of surface growth viewed as moving discontinuities. Mech. Res. Commun. 38, 372–377 (2011)

    Article  MATH  Google Scholar 

  23. Ganghoffer, J.F.: A contribution to the mechanics and thermodynamics of surface growth. Application to bone external remodeling. Int. J. Eng. Sci 50(1), 166–191 (2012)

    Article  MathSciNet  Google Scholar 

  24. Ganghoffer, J.F., Sokolowski, J.: A micromechanical approach to volumetric and surface growth in the framework of shape optimization. Int. J. Eng. Sci. 74, 207–226 (2014)

    Article  MathSciNet  Google Scholar 

  25. Goodstein, J.: States of Matter (Dover Phoenix Edition). Prentice Hall, Englewood Cliffs (1975)

    Google Scholar 

  26. Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin (2000)

    MATH  Google Scholar 

  27. Helmlinger, G., Netti, P.A., Lichtenbeld, H.C., Melder, R.J., Jain, R.K.: Solid stress inhibits the growth of multicellular tumour spheroids. Nat. Biotechnol. 15, 778–783 (1997)

    Article  Google Scholar 

  28. Hsu, F.H.: The influences of mechanical loads on the form of a growing elastic body. J. Biomech. 1, 303–311 (1968)

    Article  Google Scholar 

  29. Hubbard, M.E., Byrne, H.M.: Multiphase modelling of vascular tumour growth in two spatial dimensions. J. Theor. Biol. 316, 70–89 (2013)

    Article  MathSciNet  Google Scholar 

  30. Kunz-Schughart, L.A.: Multicellular tumour spheroids: intermediates between monolayer culture and in vivo tumour. Cell Biol. Int. 23(3), 157–161 (1999)

    Article  Google Scholar 

  31. Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman et al., London (1993)

    Book  MATH  Google Scholar 

  32. Menzel, A., Kuhl, E.: Frontiers in growth and remodeling. Mech. Res. Commun. 42, 1–14 (2012)

    Article  Google Scholar 

  33. Novotny, A.A., Sokolowski, J.: Topological Derivatives in Shape Optimization. Springer, Berlin (2012)

    MATH  Google Scholar 

  34. Paszek, M.J., DuFort, C.C., Rossier, O., Bainer, R., Mouw,J.K., Godula, K., Hudak, J.E., Lakins, J.N., Wijekoon, A.C., Cassereau, L., Rubashkin, M.G., Magbanua, M.J., Thorn, K.S., Davidson, M.W., Rugo, H.S., Park, J.W., Hammer, D.A., Giannone, G., Bertozzi, C.R., Weaver, V.M.: The cancer glycocalyx mechanically primes integrin-mediated growth and survival. Nature 511(7509), 319–325 (2014)

  35. Prigogine, I.: Introduction à la thermodynamique des processus irréversibles. Dunod, Paris (1968)

    Google Scholar 

  36. Ricken, T., Bluhm, J.: Remodeling and growth of living tissue: a multiphase theory. Arch. Appl. Mech. 80, 453–465 (2010)

    Article  ADS  MATH  Google Scholar 

  37. Rodriguez, E.K., Hoger, A., McCulloch, A.D.: Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27(4), 455–467 (1994)

    Article  Google Scholar 

  38. Roose, T., Chapman, S.J., Maini, P.K.: Mathematical models of avascular tumor growth. SIAM Rev. 49(2), 179–208 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Skalak, R.: Growth as a finite displacement field. In: Carlsson, D.E., Shield, R.T. (eds.) Proceedings of the IUTAM Symposium on Finite Elasticity, pp. 347–355. Martinus Nijhoff, The Hague (1981)

    Chapter  Google Scholar 

  40. Skalak, R., Farrow, D.A., Hoger, A.: Kinematics of surface growth. J. Math. Biol. 35, 869–907 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  41. Sokolowski, J., Zolesio, J.P.: Introduction to Shape Optimization—Shape Sensitivity Analysis. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  42. Spratt, J.A., von Fournier, D., Spratt, J.S., Weber, E.E.: Decelerating growth and human breast cancer. Cancer 71(6), 2013–9 (1993)

    Article  Google Scholar 

  43. Stein, A.A.: The deformation of a rod of growing biological material under longitudinal compression. J. Appl. Math. Mech. 59, 139–146 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  44. Thompson, D.W.: On Growth and Form (Dover reprint of 1942), 2nd edn. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  45. Tung, J.C., Barnes, J.M., Desai, S.R., Sistrunk, C., Conklin, M.W., Schedin, P., Eliceiri, K.W., Keely, P.J., Seewaldt, V.L., Weaver, V.M.: Tumor mechanics and metabolic dysfunction. Free Radic. Biol. Med. 79, 269–280 (2015)

    Article  Google Scholar 

  46. Weedon-Fekjær, H., Lindqvist, B.H., Vatten, L.J., Aalen, O.O., Tretli, S.: Breast cancer tumor growth estimated through mammography screening data. Breast Cancer Res. 10(3), R41 (2008)

    Article  Google Scholar 

  47. Zolesio, J.P.: Identification de domaines par déformations. Université de Nice, Thèse d’Etat (1979)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LEMTA, Université de Lorraine, Nancy, France

    J. F. Ganghoffer & M. B. Boubaker

Authors
  1. J. F. Ganghoffer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. B. Boubaker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. F. Ganghoffer.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ganghoffer, J.F., Boubaker, M.B. Micromechanical analysis of volumetric growth in the context of open systems thermodynamics and configurational mechanics. Application to tumor growth. Continuum Mech. Thermodyn. 29, 429–455 (2017). https://doi.org/10.1007/s00161-016-0539-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-016-0539-5

Keywords

Search

Navigation