Biomechanics and Modeling in Mechanobiology

, Volume 3, Issue 3, pp 147–171 | Cite as

Modelling of anisotropic growth in biological tissues

A new approach and computational aspects
Original Paper

Abstract

In this contribution, we develop a theoretical and computational framework for anisotropic growth phenomena. As a key idea of the proposed phenomenological approach, a fibre or rather structural tensor is introduced, which allows the description of transversely isotropic material behaviour. Based on this additional argument, anisotropic growth is modelled via appropriate evolution equations for the fibre while volumetric remodelling is realised by an evolution of the referential density. Both the strength of the fibre as well as the density follow Wolff-type laws. We however elaborate on two different approaches for the evolution of the fibre direction, namely an alignment with respect to strain or with respect to stress. One of the main benefits of the developed framework is therefore the opportunity to address the evolutions of the fibre strength and the fibre direction separately. It is then straightforward to set up appropriate integration algorithms such that the developed framework fits nicely into common, finite element schemes. Finally, several numerical examples underline the applicability of the proposed formulation.

References

  1. Almeida ES, Spilker RL (1998) Finite element formulations for hyperelastic transversely isotropic biphasic soft tissues. Comput Meth Appl Mech Eng 151:513–538CrossRefGoogle Scholar
  2. Ambrosi D, Mollica F (2002) On the mechanics of a growing tumor. Int J Eng Sci 40:1297–1316CrossRefGoogle Scholar
  3. Angeles J (1988) Rational kinematics. In: Springer tracts in natural philosopy, vol 34. Springer, Berlin Heidelberg New YorkGoogle Scholar
  4. Antman SS (1995) Nonlinear problems of elasticity. In: Applied mathematical sciences, vol 107. Springer, Berlin Heidelberg New YorkGoogle Scholar
  5. Ascher UM, Petzold LR (1998) Computer methods for ordinary differential equations and differential-algebraic equations. SIAM, Philadelphia, PAGoogle Scholar
  6. Ball JM (1977) Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal 63:337–403Google Scholar
  7. Beatty MF (1987) A class of universial relations in isotropic elasticity theory. J Elast 17:113–121Google Scholar
  8. Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures. Wiley, New YorkGoogle Scholar
  9. Betsch P, Steinmann P (2002) Frame-indifferent beam finite elements based upon the geometrically exact beam theory. Int J Numer Meth Eng 54:1775–1788CrossRefGoogle Scholar
  10. Betsch P, Menzel A, Stein E (1998) On the parametrization of finite rotations in computational mechanics. A classification of concepts with application to smooth shells. Comput Meth Appl Mech Eng 155:273–305CrossRefGoogle Scholar
  11. Biot MA (1965) Mechanics of incremental deformations. Wiley, New YorkGoogle Scholar
  12. Boehler JP (ed) (1987) Applications of tensor functions in solid mechanics. Number 292 in CISM courses and lectures. Springer, Berlin Heidelberg New YorkGoogle Scholar
  13. Bowen RM (1976) Theory of mixtures. In: Eringen AC (ed) Continuum physics, vol III—Mixtures and EM field theories. Academic, New York, pp 1–127Google Scholar
  14. Chadwick P (1999) Continuum mechanics. Dover, Mineola, NYGoogle Scholar
  15. Chen Y-C, Hoger A (2000) Constitutive functions of elastic materials in finite growth and deformation. J Elast 59:175–193CrossRefGoogle Scholar
  16. Ciarlet PG (1988) Mathematical elasticity—vol 1: three dimensional elasticity, vol 20 of studies in mathematics and its applications. North-Holland, AmsterdamGoogle Scholar
  17. Ciarletta M, Ieşan D (1993) Non-classical elastic solids In: Pitman research notes in mathematics series, vol 293. Longman, New YorkGoogle Scholar
  18. Conti A, DeSimone A, Dolzmann G (2002) Soft elastic response of stretched sheets of nematic elastomers: a numerical study. J Mech Phys Solids 50:1431–1451CrossRefGoogle Scholar
  19. Cowin SC (1985) The relationship between the elasticity tensor and the fabric tensor. Mech Mater 4:137–147CrossRefGoogle Scholar
  20. Cowin SC (1994) Optimization of the strain energy density in linear anisotropic elasticity. J Elast 34:45–68Google Scholar
  21. Cowin SC (1995) On the minimization and maximization of the strain energy density in cortical bone tissue. J Biomech 28(4):445–447, Technical noteCrossRefGoogle Scholar
  22. Cowin SC (1996) Strain or deformation rate dependent finite growth in soft tissues. J Biomech 29(5):647–649, Technical noteCrossRefGoogle Scholar
  23. Cowin SC (1997) Remarks on coaxiality of strain and stress in anisotropic elasticity. J Elast 47:83–84, Technical noteCrossRefGoogle Scholar
  24. Cowin SC (1998) Imposing thermodynamic restrictions on the elastic constant-fabric tensor relationship. J Biomech 31:759–762, Technical noteCrossRefGoogle Scholar
  25. Cowin SC (1999a) Bone poroelasticity. J Biomech 32:217–238CrossRefGoogle Scholar
  26. Cowin SC (1999b) Structural change in living tissues. Meccanica 34:379–398CrossRefGoogle Scholar
  27. Cowin SC (ed) (2001) Bone mechanics handbook, 2nd edn. CRC, Boca Raton, FLGoogle Scholar
  28. Cowin SC, Hegedus DH (1976) Bone remodeling I: theory of adaptive elasticity. J Elast 6(3):313–326Google Scholar
  29. Cowin SC, Humphrey JD (2000) Cardiovascular soft tissue mechanics. Kluwer, DordrechtGoogle Scholar
  30. Currey JD (2003) The many adaptions of bone. J Biomech 36:1487–1495CrossRefGoogle Scholar
  31. Dacorogna B (1989) Direct methods in the calculus of variations. In: Applied mathematical sciences, vol 78. Springer, Berlin Heidelberg New YorkGoogle Scholar
  32. De Hart J, Peters GWM, Schreurs PJG, Baaijens FPT (2004) Collagen fibers reduce stresses and stabilize motion of aortic valve leaflets during systole. J Biomech 37:303–311CrossRefPubMedGoogle Scholar
  33. Dennis JE Jr, Schnabel RB (1996) Numerical methods for unconstrained optimization and nonlinear equations. In: Classics in applied mathematics, vol 16. SIAM, Philadelphia, PAGoogle Scholar
  34. Dolzmann G (2003) Variational methods for crystalline microstructure—analysis and computation. Lecture notes in mathematics, vol 1803. Springer, Berlin Heidelberg New YorkGoogle Scholar
  35. Driessen NJB, Peters GWM, Huyghe JM, Bouten CVC, Baaijens FPT (2003) Remodelling of continuously distributed collagen fibres in soft tissues. J Biomech 36:1151–1158. Erratum 36:1235CrossRefGoogle Scholar
  36. Epstein M, Maugin GA (2000) Thermomechanics of volumetric growth in uniform bodies. Int J Plasticity 16:951–978CrossRefGoogle Scholar
  37. Fellin W, Ostermann A (2002) Consistent tangent operators for constitutive rate equations. Int J Numer Anal Meth Geomech 26:1213–1233CrossRefGoogle Scholar
  38. Fung YC (1993) Biomechanics: mechanical properties of living tissues, 2nd edn. Springer, Berlin Heidelberg New YorkGoogle Scholar
  39. Garikipati K, Narayanan H, Arruda EM, Grosh K, Calve S (2004) Material forces in the context of biotissue remodelling. In: Steinmann P, Maugin GA (eds) Mechanics of material forces. Euromech Colloquium 445 (preprint)Google Scholar
  40. Gasser TC, Holzapfel GA (2003) A rate-independent elastoplastic constitutive model for biological fiber-reinforced composites at finite strains: continuum basis, algorithmic formulation and finite element implementation. Comput Mech 4–5:340–360Google Scholar
  41. Giusti E (2003) Direct methods in the calculus of variantions. World Scientific, SingaporeGoogle Scholar
  42. Govindjee S, Mihalic A (1996) Computational methods for inverse finite elastostatics. Comput Meth Appl Mech Eng 136:47–57CrossRefGoogle Scholar
  43. Govindjee S, Mihalic A (1998) Computational methods for inverse deformations in quasi-incompressible finite elasticity. Int J Numer Meth Eng 43:821–838CrossRefGoogle Scholar
  44. Green AE, Rivlin RS, Shield RT (1952) General theory of small elastic deformations superposed on finite elastic deformations. Proc R Soc Lond Ser–A 211:128–154Google Scholar
  45. de Groot SR (1961) Thermodynamics of irreversible processes. North Holland, AmsterdamGoogle Scholar
  46. Harrigan TP, Hamilton JJ (1992) An analytical and numerical study of the stability of bone remodelling theories: dependence on microstructural stimulus. J Biomech 25(5):477–488 (corrigendum 26(3):365–366)CrossRefGoogle Scholar
  47. Harrigan TP, Hamilton JJ (1993) Finite element simulation of adaptive bone remodelling: a stability criterion and a time stepping method. Int J Numer Meth Eng 36:837–854Google Scholar
  48. Harrigan TP, Hamilton JJ (1994) Necessary and sufficient conditions for global stability and uniqueness in finite element simulations of adaptive bone remodelling. Int J Solids Struct 31:97–107CrossRefGoogle Scholar
  49. Haupt P (2000) Continuum mechanics and theory of materials. Advanced texts in physics. Springer, Berlin Heidelberg New YorkGoogle Scholar
  50. Haupt P, Pao Y-H, Hutter K (1992) Theory of incremental motion in a body with initial elasto-plastic deformation. J Elast 28:193–221Google Scholar
  51. Hoger A (1993a) The constitutive equation for finite deformations of a transversely isotropic hyperelastic material with residual stress. J Elast 33:107–118Google Scholar
  52. Hoger A (1993b) The dependence of the elasticity tensor on residual stress. J Elast 33:145–165Google Scholar
  53. Hoger A (1993c) The elasticity tensors of a residually stressed material. J Elast 31:219–237Google Scholar
  54. Hoger A (1996) The elasticity tensor of a transversely isotropic hyperelastic material with residual stress. J Elast 42:115–132Google Scholar
  55. Hoger A (1997) Virtual configurations and constitutive equations for residually stressed bodies with material symmetry. J Elast 48:125–144CrossRefGoogle Scholar
  56. Holzapfel GA (2000) Nonlinear solid mechanics, a continuum approach for engineering. Wiley, New YorkGoogle Scholar
  57. Holzapfel GA (2001) Biomechanics of soft tissue. In: Lemaitre J (ed) The handbook of materials behavior models. Multiphysics behaviors, vol 3, chapter 10, composite media, biomaterials, Academic, New York, pp 1049–1063Google Scholar
  58. Holzapfel GA, Gasser TC (2001) A viscoelastic model for fibre-reinforced composites at finite strains: continuum basis, computational aspects and applications. Comput Meth Appl Mech Eng 190:4379–4403CrossRefGoogle Scholar
  59. Holzapfel GA, Ogden RW (eds) (2003) Biomechanics of soft tissue in cardiovascular systems. Number 441 in CISM courses and lectures. Springer, Berlin Heidelberg New YorkGoogle Scholar
  60. Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48CrossRefGoogle Scholar
  61. Huiskes R, Chao EYS (1983) A survey of finite element analysis in orthopedic biomechanics: The first decade. J Biomech 16(6):385–409CrossRefGoogle Scholar
  62. Humphrey JD (2002) Cardiovascular solid mechanics. Cells, tissues, and organs. Springer, Berlin Heidelberg New YorkGoogle Scholar
  63. Husikes R, Weinans H, Grootenboer HJ, Dalastra M, Fudala B, Sloof TJ (1987) Adaptive bone-remodeling theory applied to prothetic-design analysis. J Biomech 20(11/12):1135–1150Google Scholar
  64. Ieşan D (1989) Prestressed bodies In: Pitman research notes in mathematics series, vol 195. Longman, New YorkGoogle Scholar
  65. Imatani S, Maugin GA (2002) A constitutive model for material growth and its application to three-dimensional finite element analysis. Mech Res Commun 29:477–483CrossRefGoogle Scholar
  66. Jacobs CR, Levenston ME, Beaupré GS, Simo JC, Carter DR (1995) Numerical instabilities in bone remodelling simultaneous: the advantages of a node-based finite element approach. J Biomech 28(4):449–459CrossRefGoogle Scholar
  67. Jacobs CR, Simo JC, Beaupré GS, Carter DR (1997) Adaptive bone remodelling incorporationg simultaneous density and anisotropy considerations. J Biomech 30(6):603–613CrossRefGoogle Scholar
  68. Kaliske M (2000) A formulation of elasticity and viscoelasticity for fibre reinforced material at small and finite strains. Comput Meth Appl Mech Eng 185:225–243CrossRefGoogle Scholar
  69. Katchalsky A, Curran PF (1965) Nonequilibrium thermodynamics in biophysics. In: Harvard books in biophysics, vol 1. Harvard Univ Press, Cambridge, MAGoogle Scholar
  70. Kestin J (1966) A course in thermodynamics. Blaisdell, Waltham, MAGoogle Scholar
  71. Krstin N, Nackenhorst U, Lammering R (2000) Zur konstitutiven Beschreibung des anisotropen beanspruchungsadaptiven Knochenumbaus. Techn Mech 20(1):31–40Google Scholar
  72. Kuhl E, Steinmann P (2003a) Mass- and volume specific views on thermodynamics for open systems. Proc R Soc Lond Ser–A 459:2547–2568Google Scholar
  73. Kuhl E, Steinmann P (2003b) On spatial and material settings of thermo-hyperelastodynamics for open systems. Acta Mech 160:179–217CrossRefGoogle Scholar
  74. Kuhl E, Steinmann P (2003c) Theory and numerics of geometrically nonlinear open system mechanics. Int J Numer Meth Eng 58:1593–1615CrossRefGoogle Scholar
  75. Kuhl E, Steinmann P (2004) Material forces in open system mechanics. Comput Meth Appl Mech Eng 193:2357–2381CrossRefGoogle Scholar
  76. Kuhl E, Menzel A, Steinmann P (2003) Computational modeling of growth—a critical review, a classification of concepts and two new consistent approaches. Comput Mech 32:71–88CrossRefGoogle Scholar
  77. Liu I-S (2002) Continuum mechanics. Advanced texts in physics. Springer, Berlin Heidelberg New YorkGoogle Scholar
  78. Lubarda VA, Hoger A (2002) On the mechanics of solids with a growing mass. Int J Solids Struct 39:4627–4664CrossRefGoogle Scholar
  79. Maugin GA, Imatani S (2003a) Anisotropic growth of materials. J Phys IV France 105:365–372Google Scholar
  80. Maugin GA, Imatani S (2003b) Material growth in solid–like materials. In: Miehe C (ed) Computational mechanics of solid materials at large strains, vol 108 of solid mechanics and its applications, IUTAM. Kluwer, Dordrecht, pp 221–234Google Scholar
  81. Menzel A, Steinmann P (2001a) On the comparison of two strategies to formulate orthotropic hyperelasticity. J Elast 62:171–201CrossRefGoogle Scholar
  82. Menzel A, Steinmann P (2001b) A theoretical and computational setting for anisotropic continuum damage mechanics at large strains. Int J Solids Struct 38(52):9505–9523CrossRefGoogle Scholar
  83. Menzel A, Steinmann P (2003a) Geometrically nonlinear anisotropic inelasticity based on fictitious configurations: application to the coupling of continuum damage and multiplicative elasto-plasticity. Int J Numer Meth Eng 56:2233–2266CrossRefGoogle Scholar
  84. Menzel A, Steinmann P (2003b) On the spatial formulation of anisotropic multiplicative elasto-plasticity. Comput Meth Appl Mech Eng 192:3431–3470CrossRefGoogle Scholar
  85. Menzel A, Steinmann P (2003c) A view on anisotropic finite hyper-elasticity. Euro J Mech A - Solids 22:71–87Google Scholar
  86. Menzel A, Ekh M, Runesson K, Steinmann P (2004) A framework for multiplicative elastoplasticity with kinematic hardening coupled to anisotropic damage. Int J Plasticity (in press)Google Scholar
  87. Merodio J, Ogden RW (2003) Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation. Int J Solids Struct 40:4707–4727CrossRefGoogle Scholar
  88. Miehe C (1996) Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity. Comput Meth Appl Mech Eng 134:223–240CrossRefGoogle Scholar
  89. Murray JD (2002) Mathematical biology II: spatial models and biomedical applications. In: Interdisciplinary applied mathematics, vol 18, 3rd edn. Springer, Berlin Heidelberg New YorkGoogle Scholar
  90. Nackenhorst U (1996) Numerical simulation of stress stimulated bone remodelling. Techn Mech 17(1):31–40Google Scholar
  91. Oden JT (1972) Finite elements of nonlinear continua. In: Advanced engineering series. McGraw-Hill, New YorkGoogle Scholar
  92. Ogden RW (1997) Non-linear elastic deformations. Dover, New YorkGoogle Scholar
  93. Papadopoulos P, Lu J (2001) On the formulation and numerical solution of problems in anisotropic finite plasticity. Comput Meth Appl Mech Eng 190:4889–4910CrossRefGoogle Scholar
  94. Pérez-Foguet A, Rodriguez-Ferran A, Huerta A (2000a) Numerical differentiation for local and global tangent operators in computational plasticity. Comput Meth Appl Mech Eng 189:277–296CrossRefGoogle Scholar
  95. Pérez-Foguet A, Rodriguez-Ferran A, Huerta A (2000b) Numerical differentiation for non-trivial consistent tangent matrices: an application to the MRS–lade model. Int J Numer Meth Eng 48:159–184CrossRefGoogle Scholar
  96. Petersen P (1989) On optimal orientation of orthotropic materials. Struct Optim 1:101–106Google Scholar
  97. Podio-Guidugli P (2000) A primer in elasticity. J Elast 58(1):1–103CrossRefGoogle Scholar
  98. Reese S, Raible T, Wriggers P (2001) Finite element modelling of orthotropic material behaviour in pneumatic membranes. Int J Solids Struct 38(52):9525–9544CrossRefGoogle Scholar
  99. Rodriguez EK, Hoger A, McCulloch D (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27(4):455–467CrossRefPubMedGoogle Scholar
  100. Schneck DJ (1990) Engineering principles of physiological function. New York University Biomedical Engineering Series. New York University Press, New YorkGoogle Scholar
  101. Schneck DJ (1992) Mechanics of muscle, 2nd edn. New York University Biomedical Engineering Series. New York University Press, New YorkGoogle Scholar
  102. Schröder J, Neff P (2003) Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int J Solids Struct 40:401–445CrossRefGoogle Scholar
  103. Sgarra C, Vianello M (1997) Rotations which make strain and stress coaxial. J Elast 47:217–224CrossRefGoogle Scholar
  104. Sherratt JA, Martin P, Murray JD, Lewis J (1992) Mathematical models of wound healing in embryonic and adult epidermis. IMA J Math Appl Med Biol 9:177–196PubMedGoogle Scholar
  105. Šilhavý M (1997) The mechanics and thermomechanics of continuous media. Texts and monographs in physics. Springer, Berlin Heidelberg New YorkGoogle Scholar
  106. Silver FH, Seehra GP, Freeman JW (2003) Collagen self-assemby and the development of tendon mechanical properties. J Biomech 36:1529–1553CrossRefGoogle Scholar
  107. Simo JC, Armero F (1992) Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Meth Eng 33:1413–1449Google Scholar
  108. Skalar R, Zargaryan S, Jain RK, Netti PA, Hoger A (1996) Compatibility and the genesis of residual stress by volumetic growth. J Math Biol 34:889–914CrossRefPubMedGoogle Scholar
  109. Skalar R, Farrow DA, Hoger A (1997) Kinematics of surface growth. J Math Biol 35:869–907CrossRefPubMedGoogle Scholar
  110. Spencer AJM (1984) Constitutive theory of strongly anisotropic solids. In: Spencer AJM (ed) Continuum theory of the mechanics of fibre-reinforced composites, CISM courses and lectures, vol 282. Springer, Berlin Heidelberg New York, pp 1–32Google Scholar
  111. Taber LA (1995) Biomechanics of growth, remodelling, and morphogenesis. ASME Appl Mech Rev 48(8):487–545Google Scholar
  112. Truesdell C (1966) Continuum mechanics I: the mechanical foundations of elasticity and fluid dynamics. In: International science review series, vol 8. Gordon and Beach, New YorkGoogle Scholar
  113. Truesdell C, Noll W (2004) In: Antman SS (ed) The non-linear field theories of mechanics, 3rd edn. Springer, Berlin Heidelberg New YorkGoogle Scholar
  114. Truesdell C, Toupin RA (1960) The classical field theories. In: Flügge S (ed) Encyclopedia of physics, vol III/1. Springer, Berlin Heidelberg New York, pp 226–793Google Scholar
  115. Vianello M (1996a) Coaxiality of strain and stress in anisotropic linear elasticity. J Elast 42:283–289Google Scholar
  116. Vianello M (1996b) Optimization of the stored energy and coaxiality of strain and stress in finite elasticity. J Elast 44:193–202Google Scholar
  117. Weinans H, Huiskes R, Grootenboer HJ (1992) The behavior of adaptive bone-remodelling simulation models. J Biomech 25(12):1425–1441CrossRefPubMedGoogle Scholar
  118. Weiss JA, Maker BN, Govindjee S (1996) Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput Meth Appl Mech Eng 135:107–128CrossRefGoogle Scholar
  119. Weng S (1998) Ein anisotropes Knochenumbaumodell und dessen Anwendung. Techn Mech 18(3):173–180Google Scholar
  120. Wren TAL (2003) A computational model for the adaption of muscle and tendon length to average muscle length and minimum tendon strain. J Biomech 36:1117–1124CrossRefPubMedGoogle Scholar
  121. Zheng Q-S (1993) On transversversely isotropic, orthotropic and relative isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Part I: two dimensional orthotropic and relative isotropic functions and three dimensional relative isotropic functions. Int J Eng Sci 31(10):1399–1409CrossRefGoogle Scholar
  122. Zheng Q-S, Spencer AJM (1993) Tensors which characterize anisotropies. Int J Eng Sci 31(5):679–693CrossRefGoogle Scholar
  123. Zysset PK (2003) A review of morphology—elasticity relationships in human trabecular bone: theories and experiments. J Biomech 36:1469–1485CrossRefPubMedGoogle Scholar
  124. Zysset PK, Curnier A (1995) An alternative model for anisotropic elasticity based on fabric tensors. Mech Mater 21:243–250CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Chair of Applied Mechanics, Department of Mechanical and Process EngineeringUniversity of KaiserslauternKaiserslauternGermany

Personalised recommendations