Skip to main content
SpringerLink
Log in

Remodeling and growth of living tissue: a multiphase theory

  • Special Issue
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

A continuum triphase model (i.e., a solid filled with fluid containing nutrients) based on the theory of porous media (TPM) is proposed for the phenomenological description of growth and remodeling phenomena in isotropic and transversely isotropic biological tissues. In this study, particular attention is paid on the description of the mass exchange during the stress–strain- and/or nutrient-driven phase transition of the nutrient phase to the solid phase. In order to define thermodynamically consistent constitutive relations, the entropy inequality of the mixture is evaluated in analogy to Coleman and Noll (Arch Ration Mech Anal 13:167–178, 1963). Thereby, the choose of independent process variables is motivated by the fact that the resulting phenomenological description derives both a physical interpretability and a comprehensive description of the coupled processes. Based on the developed thermodynamical restrictions constitutive relations for stress, mass supply and permeability are proposed. The resulting system of equation is implemented into a mixed finite element scheme. Thus, we obtain a coupled calculation concept to determine the solid motion, inner pressure as well as the solid, fluid and nutrient volume fractions.

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. Ball J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)

    Article  MATH  Google Scholar 

  2. Balzani, D.: Polyconvex anisotropic energies and modeling of damage applied to arterial walls. Scientific report of the Institute of Mechanics, University of Duisburg-Essen, Diss. (2006)

  3. Balzani D., Neff P., Schröder J., Holzapfel G.A.: A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int. J. Solids Struct. 43, 6052–6070 (2006)

    Article  MATH  Google Scholar 

  4. Betten, J.: Formulation of anisotropic constitutive equations, In: Boehler, J.P. (eds.) Applications of Tensor Functions in Solid Mechanics, CISM Course No. 292. Springer, Berlin (1987)

  5. Biot M.A.: Le probléme de la consolidation des matières argileuses sous une charge. Annales de la Societé scientifique de Bruxelles 55, 110–113 (1935)

    Google Scholar 

  6. Biot M.A.: General theory of three dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)

    Article  Google Scholar 

  7. Biot M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J. Acoust. Soc. Am. 28, 168–178 (1956)

    Article  MathSciNet  Google Scholar 

  8. Bluhm J.: Modelling of saturated thermo-elastic porous solids with different phase temperatures. In: Ehlers, W., Bluhm, J. (eds) Porous Media: Theory. Experiments and Numerical Applications, Springer, Berlin (2002)

    Google Scholar 

  9. Boehler, J.P.: Introduction of the invariant formulation of anisotropic constitutive equations. In: Boehler, J.P. (eds.) Applications of Tensor Functions in Solid Mechanics. CISM Courses and Lectures No. 292, pp. 13–30. Springer, Wien (1987)

  10. de Boer R.: Highlights in the historical development of the porous media theory—toward a transistent macroscopic theory. Appl. Mech. Rev. 4, 201–262 (1996)

    Article  Google Scholar 

  11. de Boer R.: Theory of porous media—highlights in the historical development and current state. Springer, Berlin (2000)

    Google Scholar 

  12. Bowen R.M.: Incompressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 18, 1129–1148 (1980)

    Article  MATH  Google Scholar 

  13. Bowen R.M.: Compressible porous media models by use of the theory of mixtures. Int. J. Eng. Sci. 20, 697–735 (1982)

    Article  MATH  Google Scholar 

  14. Carter D.R., Hayes W.C.: The compressive behavior of bone as a two-phase porous structure. J. Bone Joint Surg. 7, 954–962 (1977)

    Google Scholar 

  15. Christensen P.W., Klarbring A.: An Introduction to Structural Optimization. Springer, Berlin (2009)

    MATH  Google Scholar 

  16. Coleman B.D., Noll W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ehlers, W.: Poröse Medien – ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie/Forschungsbericht aus dem Fachbereich Bauwesen 47, Universität-GH Essen. 1989 (Forschungsbericht aus dem Fach Bauwesen, Heft 47) – Habilitation

  18. Ehlers W.: Foundations of multiphasic and porous materials. In: Ehlers, W., Bluhm, J. (eds) Porous Media: Theory, Experiments and Numerical Applications., pp. 3–86. Springer, Berlin (2002)

    Google Scholar 

  19. Ehlers W., Karajan N., Markert B., Acartürk A.: Continuum biomechanics applied to the human intervertebral disc. Proc. Appl. Math. Mech. 5, 27–30 (2005)

    Article  Google Scholar 

  20. Eipper, G.: Theorie und Numerik finiter elastischer Deformationenin fluidgesättigten Porösen Medien / Universität Stuttgart. 1998 (Bericht Nr. II-1 aus dem Institut für Mechanik (Bauwesen)) – Dissertation

  21. Fung Y.: Biomechanics—Motion, Flow, Stress, and Growth. Springer, Berlin (1990)

    MATH  Google Scholar 

  22. Fung Y.: Biomechanics—Mechanical Properties of Living Tissues. Springer, Berlin (1993)

    Google Scholar 

  23. Holzapfel G.A., Gasser T.C., Ogden R.W.: A new constitutive framework for arterial wall mechanics and a compara-tive study of material models. J. Elast. 61, 1–48 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Krstin N., Nackenhorst U., Lammering R.: Zur konstitutiven Beschreibung des anisotropen beanspruchungsadaptiven Knochenumbaus. Technische Mechanik 20(1), 31–40 (2000)

    Google Scholar 

  25. Nackenhorst U.: Numerical simulations of stress stimulated bone remodeling. Technische Mechanik 1, 31–40 (1997)

    Google Scholar 

  26. Reese S., Raible T., Wriggers P.: Finite element modelling of orthotropic material behaviour in pneumatic membranes. Int. J. Solids Struct. 38, 9525–9544 (2001)

    Article  MATH  Google Scholar 

  27. Rice J.C., Cowin S.C., Bowman J.A.: On the dependence of the elasticity and strength of cancellous bone on apparent density. J. Biomech. 21, 155–168 (1988)

    Article  Google Scholar 

  28. Ricken, T.: Kapillarität in porösen Medien - theoretische Untersuchung und numerische Simulation, Dissertation, Shaker Verlag, Aachen, Diss. (2002)

  29. Ricken T., de Boer R.: Multiphase flow in a capillary porous medium. Comput. Mater. Sci. 28, 704–713 (2003)

    Article  Google Scholar 

  30. Ricken T., Schwarz A., Bluhm J.: A triphasic model of transversely isotropic biological tissue with application to stress and biological induced growth. Comput. Mater. Sci. 39, 124–136 (2007)

    Article  Google Scholar 

  31. Schröder J., Neff P.: Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int. J. Solids Struct. 40, 401–445 (2003)

    Article  MATH  Google Scholar 

  32. Schröder J., Neff P., Balzani D.: A variational approach for materially stable anisotropic hyperelasticity. Int. J. Solids Struct. 42, 4352–4371 (2005)

    Article  MATH  Google Scholar 

  33. Spencer, A.J.M.: Theory of Invariants. In: Eringen, A.C. (eds.): Continuum Physics, Bd. 1, pp. 239–353. Academic Press, New York (1971)

  34. Taber L.A.: Biomechanics of growth, remodeling, and morphogenesis. Appl. Mech. Rev. 58, 487–545 (1995)

    Article  Google Scholar 

  35. Zheng Q.S., Spencer A.J.M.: On the canonical representations for Kronecker powers of orthogonal tensors with application to material symmetry problems. Int. J. Eng. Sci. 4, 617–635 (1993)

    Article  MathSciNet  Google Scholar 

  36. Zheng Q.S., Spencer A.J.M.: Tensors which characterize anisotropies. Int. J. Eng. Sci. 5, 679–693 (1993)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computational Mechanics, University of Duisburg-Essen, Campus Essen, Universitätsstraße 15, 45117, Essen, Germany

    Tim Ricken

  2. Institute of Mechanics, University of Duisburg-Essen, Campus Essen, Universitätsstraße 15, 45117, Essen, Germany

    Joachim Bluhm

Authors
  1. Tim Ricken
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joachim Bluhm
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tim Ricken.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ricken, T., Bluhm, J. Remodeling and growth of living tissue: a multiphase theory. Arch Appl Mech 80, 453–465 (2010). https://doi.org/10.1007/s00419-009-0383-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-009-0383-1

Keywords

Search

Navigation