Abstract
A new remodeling theory accounting for mechanically driven collagen fiber reorientation in cardiovascular tissues is proposed. The constitutive equations for the living tissues are motivated by phenomenologically based microstructural considerations on the collagen fiber level. Homogenization from this molecular microscale to the macroscale of the cardiovascular tissue is performed via the concept of chain network models. In contrast to purely invariant-based macroscopic approaches, the present approach is thus governed by a limited set of physically motivated material parameters. Its particular feature is the underlying orthotropic unit cell which inherently incorporates transverse isotropy and standard isotropy as special cases. To account for mechanically induced remodeling, the unit cell dimensions are postulated to change gradually in response to mechanical loading. From an algorithmic point of view, rather than updating vector-valued microstructural directions, as in previously suggested models, we update the scalar-valued dimensions of this orthotropic unit cell with respect to the positive eigenvalues of a tensorial driving force. This update is straightforward, experiences no singularities and leads to a stable and robust remodeling algorithm. Embedded in a finite element framework, the algorithm is applied to simulate the uniaxial loading of a cylindrical tendon and the complex multiaxial loading situation in a model artery. After investigating different material and spatial stress and strain measures as potential driving forces, we conclude that the Cauchy stress, i.e., the true stress acting on the deformed configuration, seems to be a reasonable candidate to drive the remodeling process.
Similar content being viewed by others
References
Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P (2002) Molecular biology of the cell, 4th edn. Garland Science Taylor Francis Group, New York
Arruda EM, Boyce MC (1993) J Mech Phys Solids 41:389
Bischoff JE, Arruda EM, Grosh K (2002) J Appl Mech 69:570
Bischoff JE, Arruda EM, Grosh K (2002) J Appl Mech 69:198
Boyce MC (1996) Rubber Chem Technol 69:781
Boyce MC, Arruda EM (2000) Rubber Chem Technol 73:504
Bustamante C, Bryant Z, Smith SB (2003) Nature 421:423
Bustamante C, Smith S, Marko JF, Siggia ED (1994) Science 265:1599
Cowin SC (1984) Calc Tissue Int 36:S99
Cowin SC (1995) J Elast 34:45
Driessen NJB, Bouten CVC, Baaijens FPT (2005) J Biomech Eng 127:494
Driessen NJB, Peters GWM, Huyghe JM, Bouten CVC, Baaijens FPT (2003) J Biomech 36:1151
Driessen NJB, Wilson W, Bouten CVC, Baaijens FPT (2004) J Theor Biol 36:53
Elbischger PJ, Bischof H, Holzapfel GA, Regitnig P (2005) In: Suri JS, Yuan C, Wilson DL, Laxminarayan S (eds) Plaque imaging: pixel to molecular level, vol 113. Studies in Health Technology and Informatics, IOS Press, p 97
Elbischger PJ, Bischof H, Regitnig P, Holzapfel GA (2004) Pattern Anal Appl 7:269
Finlay HM, Mc Cullough L, Canham PB (1995) J Vasc Res 32:301
Flory PJ (1969) Statistical mechanics of chain molecules. John Wiley Sons, Chichester–New York
Freed AD, Einstein DR, Vesely I (2005) Biomech Model Mechanobiol 4:100
Garikipati K, Arruda EM, Grosh K, Narayanan H, Calve S (2004) J Mech Phys Solids 52:1595
Garikipati K, Olberding JE, Narayanan H, Arruda EM, Grosh K, Calve S (2006) J Mech Phys Solids 54:1493
Gasser TC, Ogden RW, Holzapfel GA (2006) J R Soc Interface 3:15
Gleason RL, Humphrey JD (2004) J Vasc Res 41:352
Gleason RL, Humphrey JD (2005) Math Med Biol 22:347
Hagerman PJ (1988) Annu Rev Biophys Biophys Chem 17:265
Hariton I, de Botton G, Gasser TC, Holzapfel GA (2006) Biomechanics and modeling in mechanobiology, pp available online, DOI 10.1007/s10237-006-0049-7
Himpel G, Menzel A, Kuhl E, Steinmann P (2007) Int J Num Meth Eng, in press
Holzapfel GA, Gasser TC, Stadler M (2002) European J Mech A Solids 21:441
Holzapfel GA, Ogden RW (2003) Biomechanics of soft tissue in cardiovascular systems CISM courses and lectures no 441. Springer Verlag, Wien–New York
Holzapfel GA, Ogden RW (2006) mechanics of biological tissue. Springer, Berlin–Heidelberg–New York
Holzapfel GA, Stadler M, Schulze-Bauer CAJ (2002) Ann Biomed Eng 30:753
Huang H, Kamm R, Lee RT (2003) Am J Physiol Cell Physiol 287:C1
Humphrey JD (2001) J Biomech Eng 123:638
Humphrey JD (2002) Cardiovasular solid mechanics. Springer Verlag, Berlin–Heidelberg–New York
Ingber DE (2003) Ann Intern Med 35:564
Klein-Nulend J, Bacabac RG, Mullender MG (2005) Pathol Biol (Paris) 53:576
Kratky O, Porod G (1949) Recl Trav Chim 68:1106
Kuhl E, Garikipati K, Arruda EM, Grosh K (2005) J Mech Phys Solids 53:1552
Kuhl E, Maas R, Himpel G, Menzel A (2006) Biomechanics and modeling in mechanobiology, pp available online, DOI 10.1007/s10237-006-0062-x
Kuhl E, Menzel A, Garikipati K (2006) Phil Mag 86:3241
Lehoux S, Castier Y, Tedgui A (2006) J Intern Med 259:381
Leung DYM, Glagov S, Mathews MB (1976) Science 191:475
Leung DYM, Glagov S, Mathews MB (1977) Circ Res 41:316
Lubarda VA, Hoger A (2002) Int J Solids Struct 39:4627
Marko JF, Siggia ED (1995) Macromolecules 28:8759
Menzel A (2006) In: Holzapfel GA, Ogden RW (eds) IUTAM Symposium mechanics of biological tissue. Springer Verlag, p 91
Menzel A (2005) Biomech Model Mechanobiol 3:147
Mofrad MRK, Kamm R (eds) (2006) Cytoskeletal mechanics models and measurements. Cambridge University Press
Nerem RM, Seliktar D (2001) Annu Rev Biomed Eng 3:225
Rodriguez EK, Hoger A, Mc Culloch AD (1994) J Biomech 27:455
Stopak D, Harris AK (1982) Dev Biol 90:383
Sun YL, Luo ZP, Fertala A, An KN (2002) Biochem Biophys Res Commun 295:382
Taber LA, Humphrey JD (2001) J Biomech Eng 123:528
Treloar LRG (1975) The physics of rubber elasticity. Clarendon Press, Oxford
Vianello M (1996) J Elast 42:283
Wang JH, Thampatty BP (2006) Biomech Model Mechanobio 5:1
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kuhl, E., Holzapfel, G.A. A continuum model for remodeling in living structures. J Mater Sci 42, 8811–8823 (2007). https://doi.org/10.1007/s10853-007-1917-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10853-007-1917-y