Modelling of anisotropic growth in biological tissues

Menzel, Andreas

doi:10.1007/s10237-004-0047-6

Cited by 130 publications

(117 citation statements)

References 111 publications

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“…In this section we provide a thermodynamically consistent approach to characterize the material response (Kuhl & Steinmann, 2003;Menzel, 2004). The material version of the Clasius-Planck inequality for isothermal processes, which is a reasonable assumption for biological tissue with a relatively constant temperature, can be expressed as…”

Section: Continuum Approachmentioning

confidence: 99%

Mathematical modeling of collagen turnover in biological tissue

et al. 2012

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We present a theoretical and computational model for collagen turnover in soft biological tissues. Driven by alterations in the mechanical environment, collagen fiber bundles may undergo important chronic changes, characterized primarily by alterations in collagen synthesis and degradation rates. In particular, hypertension triggers an increase in tropocollagen synthesis and a decrease in collagen degradation, which lead to the well-documented overall increase in collagen content. These changes are the result of a cascade of events, initiated mainly by the endothelial and smooth muscle cells. Here, we represent these events collectively in terms of two internal variables, the concentration of growth factor TGF-β and tissue inhibitors of metalloproteinases TIMP. While the upregulation of the former increases the collagen density, the upregulation of the latter increases the collagen density through decreasing matrix metalloproteinase MMP. We establish a mathematical theory for mechanically-induced collagen turnover and introduce a computational algorithm for its robust and efficient solution. We demonstrate that our model can accurately predict the experimentally observed collagen increase in response to hypertension reported in literature. Ultimately, the model can serve as a valuable tool to predict the chronic adaptation of collagen content to restore the homeostatic equilibrium state in vessels with arbitrary micro-structure and geometry.

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Section: Continuum Approachmentioning

confidence: 99%

Mathematical modeling of collagen turnover in biological tissue

et al. 2012

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“…More recent theories have instead represented volumetric growth by a tensor quantity (Skalak et al 1996(Skalak et al , 1997 and were first used to study the growth of vascular tissues (Rodriguez et al 1994;Taber and Eggers 1996;Taber 1998). In addition to our work on the development of cartilage growth mixture models, in recent years there has been much interest in the development of continuum growth models for single constituents (Chen and Hoger 2000;Epstein and Maugin 2000;DiCarlo and Quiligotti 2002;Lubarda and Hoger 2002;Kuhl and Steinmann 2003;Huang 2004;Volokh 2004;Lappa 2005;Menzel 2005), mixtures (Quiligotti 2002;Ganghoffer and Haussy 2005) and mixtures that employ a stress balance hypothesis (Humphrey and Rajagopal 2002;Preziosi and Farina 2002;Breward et al 2003;Garikipati et al 2004).…”

Section: Introductionmentioning

confidence: 99%

A nonlinear finite element model of cartilage growth

Davol

Bingham

Sah

et al. 2007

Biomech Model Mechanobiol

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The long range objective of this work is to develop a cartilage growth finite element model (CGFEM), based on the theories of growing mixtures that has the capability to depict the evolution of the anisotropic and inhomogeneous mechanical properties, residual stresses, and nonhomogeneities that are attained by native adult cartilage. The CGFEM developed here simulates isotropic in vitro growth of cartilage with and without mechanical stimulation. To accomplish this analysis a commercial finite element code (ABAQUS) is combined with an external program (MAT-LAB) to solve an incremental equilibrium boundary value problem representing one increment of growth. This procedure is repeated for as many increments as needed to simulate the desired growth protocol. A case study is presented utilizing a growth law dependent on the magnitude of the diffusive fluid velocity to simulate an in vitro dynamic confined compression loading protocol run for 2 weeks. The results include changes in tissue size and shape, nonhomogeneities that develop in the tissue, as well as the variation that occurs in the tissue constitutive behavior from growth.

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“…In this contribution, focus is placed on the remodelling of a single solid phase. The particular approach developed extends reorientation formulations previously proposed for the alignment of individual fibre directions as discussed in Imatani and Maugin (2002), Driessen et al (2004) and Menzel (2005Menzel ( , 2007. In particular, we will make use of an evolution equation in terms of a shear-related driving force that reorients the respective direction vectors.…”

Section: Introductionmentioning

confidence: 99%

A microsphere-based remodelling formulation for anisotropic biological tissues

Menzel

Waffenschmidt

2009

Phil. Trans. R. Soc. A.

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Biological tissues possess the ability to adapt according to the respective local loading conditions, which results in growth and remodelling phenomena. The main goal of this work is the development of a new remodelling approach that, on the one hand, reflects the alignment of fibrous soft biological tissue with respect to representative loading directions. On the other hand, the continuum approach proposed is based on a sound micro-mechanically motivated formulation. To be specific, use of a worm-like chain model is made to describe the behaviour of long-chain molecules as present in, for instance, collageneous tissues. The extension of such a one-dimensional constitutive equation to the three-dimensional macroscopic level is performed by means of a microsphere formulation. Inherent with the algorithmic treatment of this type of modelling approach, a finite number of unit vectors is considered for the numerical integration over the domain of the unit sphere. As a key aspect of this contribution, remodelling is incorporated by setting up evolution equations for the referential orientations of these integration directions. Accordingly, the unit vectors considered now allow interpretation as internal variables, which characterize the material's anisotropic properties. Several numerical studies underline the applicability of the model that, moreover, nicely fits into iterative finite element formulations so that general boundary value problems can be solved.

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Modelling of anisotropic growth in biological tissues

Cited by 130 publications

References 111 publications

Mathematical modeling of collagen turnover in biological tissue

Mathematical modeling of collagen turnover in biological tissue

A nonlinear finite element model of cartilage growth

A microsphere-based remodelling formulation for anisotropic biological tissues

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