Fibre Dispersion Law
Holger
Schmid
Bioengineering Institute, University of Auckland
This CellML model was implemented in a manner that could be used for peforming finite element model simulations on the CMISS software program developed at the Bioengineering Institute, University of Auckland. The model file presented here focuses on the functional form of the fibre dispersion law corresponding to the the combination 1,2,4,8 (FiberDispersion_law_1248), with 1 and 2 representing the isotropic components of the functional form, and 4 and 8 representing the anisotropic components of the fucntional form. For additional information on implementation of cellML files in CMISS, please refer to the following Link.
Model Structure
Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress-strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel and Gasser (Holzapfel and Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379-4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1-48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.
The original paper reference is cited below:
Hyperelastic modelling of arterial layers with distributed collagen fibre orientations, T.C. Gasser, R.W. Ogden and G.A. Holzapfel, 2006, Journal of the Royal Society, 3(6), 15-35. PubMed ID: 16849214
$\mathrm{I1}=3+2\mathrm{Emm}+2\mathrm{Enn}+2\mathrm{Ess}$
$\mathrm{I2}=3-4\mathrm{Emn}\mathrm{Emn}+4\mathrm{Enn}-4\mathrm{Ems}\mathrm{Ems}-4\mathrm{Ens}\mathrm{Ens}+4(1+\mathrm{Enn})\mathrm{Ess}+4(1+\mathrm{Enn})\mathrm{Ess}+4(1+\mathrm{Enn}+\mathrm{Ess})\mathrm{Emm}$
$\mathrm{I4}=1+2\mathrm{Emm}$
$\mathrm{I6}=1+2\mathrm{Enn}$
$\mathrm{I8}=1+2\mathrm{Ess}$
$\mathrm{Tdevmm}=2\mathrm{c11}e^{\mathrm{c12}(\mathrm{I1}-3)^{2}}(\mathrm{I1}-3)+4\mathrm{c21}e^{\mathrm{c22}(\mathrm{I2}-3)^{2}}(\mathrm{I2}-3)(1+\mathrm{Enn}+\mathrm{Ess})+2\mathrm{c41}e^{\mathrm{c42}(\mathrm{I4}(1-3\mathrm{kappa1})-1)^{2}}(\mathrm{I4}(1-3\mathrm{kappa1})-1)(1-3\mathrm{kappa1})$
$\mathrm{Tdevmn}=-4\mathrm{c21}e^{\mathrm{c22}(\mathrm{I2}-3)^{2}}(\mathrm{I2}-3)\mathrm{Emn}$
$\mathrm{Tdevms}=-4\mathrm{c21}e^{\mathrm{c22}(\mathrm{I2}-3)^{2}}(\mathrm{I2}-3)\mathrm{Ems}$
$\mathrm{Tdevnn}=2\mathrm{c11}e^{\mathrm{c12}(\mathrm{I1}-3)^{2}}(\mathrm{I1}-3)+4\mathrm{c21}e^{\mathrm{c22}(\mathrm{I2}-3)^{2}}(\mathrm{I2}-3)(1+\mathrm{Emm}+\mathrm{Ess})$
$\mathrm{Tdevss}=2\mathrm{c11}e^{\mathrm{c12}(\mathrm{I1}-3)^{2}}(\mathrm{I1}-3)+4\mathrm{c21}e^{\mathrm{c22}(\mathrm{I2}-3)^{2}}(\mathrm{I2}-3)(1+\mathrm{Emm}+\mathrm{Enn})+2\mathrm{c81}e^{\mathrm{c82}(\mathrm{I8}(1-3\mathrm{kappa2})-1)^{2}}(\mathrm{I8}(1-3\mathrm{kappa2})-1)(1-3\mathrm{kappa2})$
$\mathrm{Tdevns}=-4\mathrm{c21}e^{\mathrm{c22}(\mathrm{I2}-3)^{2}}(\mathrm{I2}-3)\mathrm{Ens}$
Vignesh Kumar
Hyperelastic modelling of arterial layers with distributed collagen fibre orientations
3(6)
15
35
Fibre Dispersion Law (Version B)
University of Auckland
Auckland Bioengineering Institute
University of Auckland
G
Holzapfel
A
This is a CellML version of the Fibre Dispersion constitutive material law,
defining the relation between the six independent strain components
and the stress components. It is assumed that the strain components
will be controlled externally by the application using this CellML
model.
R
Ogden
W
T
Gasser
C
Holger
Schmid
2006-02-22 00:00
16849214
2003-12-28
Added metadata to the model.
Vignesh Kumar
This CellML file focuses on the development and introduction of a structural continuum framework that is able to represent the dispersion of
collagen fibres orientation within arterial walls. Thus allowing the development of a new hyperelastic free-energy function
that is well suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls. This file focuses on the combination 1,2,4,8 of the fibre dispersion law. With '1' and '2' representing the isotropic components of the functional form and '4' and '8' representing the anisotropic components of the functional form.
2007-01-01T00:00:00+13:00
keyword
mechanical constitutive laws
constitutive material law
In this simple model we only have one component, which holds the
six equations.
2007-02-01
The University of Auckland
The Bioengineering Institute
Yikan
Wang
M
Journal of the Royal Society Interface
h.schmid@auckland.ac.nz
We'll use this component as the "interface" to the model, all
other components are hidden via encapsulation in this component.
Holger
Schmid
h.schmid@auckland.ac.nz