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A micro-sphere-based remodelling formulation for transversely isotropic soft biological tissue

Motivation

 

Soft biological tissues possess a pronounced composite-type multi-scale structure together with strongly anisotropic mechanical properties. A fibre-like network structure is characteristic for this kind of materials. If the tissue will be exposed to mechanical loading, an initially unstructured collagen fibre network tends to reorient with the local dominant stretch direction – it adapts according to the particular loading conditions. In general, biological tissues exhibit changes in mass, which is denoted as growth, and changes in internal structure, which is commonly referred to as remodelling. In this regard, an anisotropic micromechanically motivated model that incorporates this time-dependent remodelling effect will be introduced here.




Basic underlying concept

 

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The continuum approach used here is based on a one-dimensional micro-mechanically motivated constitutive equation – the so-called worm-like chain model. The extension of this one-dimensional constitutive law to the three-dimensional macroscopic level is performed by means of a microsphere formulation. Characteristic for this approach is a finite number of unit vectors ri to be considered for the numerical integration over the unit sphere or in other words: for every integration direction ri we set up the one-dimensional constitutive equation, integrate over the whole sphere and finally end up with the particular contribution at every Gauss-point. However, the key aspect of this contribution is that remodelling is incorporated by setting up deformation-driven evolution equations for these integration directions, which means that they are not constant any more but evolve with time. Based on this, we will show appropriate remodelling approaches for the transversely isotropic case with one preferred fibre direction.

 



Uniaxial tension test

 

We investigate the model first for a homogeneous deformation, namely uniaxial tension. In the animation below, we observe that the loading is linearly increased within a time period of 20 time steps and then we fix its value up to a time period of 400 steps. The diagram in the lower left shows the saturation behaviour of the anisotropy evolution by means of visualising the degree of anisotropy over time. Obviously it takes place in a viscous manner as the loading is fixed after 20 steps and thedegree of anisotropy continues to increase. The anisotropy evolution additionally is visualised in a more descriptive orientation-distribution-based manner in the upper right corner. We observe for the different states of deformation, that the anisotropy continuously evolves with time as the orientation-distribution-function increasingly deviates from its initially spherical distribution. The whole information can be obtained by directly taking a look at the integration directions in the lower right corner and we see that they indeed align according to the dominant principal stretch direction.


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Tissue specimen with cut

 

We next qualitatively study the inhomogeneous deformation of a tissue-related strip loaded under tension. The particular specimen considered has been discretised with 3200 finite elements. Quasi static loading conditions are applied by means of displacement boundary conditions so that the tissue is longitudinally stretched while the transverse displacements at these boundaries are clamped. By analogy with the homogeneous deformations shown above the respective load increments are linearly increased within 40 time steps, whereas the boundary conditions thereafter remain fixed for a period of another 60 steps.

The animation below displays the distribution of the maximal principal Cauchy-stresses with respect to the macroscopic longitudinal loading direction. Apparently, the stress values increase with deformation even for the loading interval within which the prescribed displacements are fixed. Practically speaking, this effect cleary underlines the viscous anisotropy evolution.

As experimentally observed, the fibre distribution aligns according to the local loading direction. This effect is recaptured by the proposed remodelling formulation and visualised by the orientation-distribution-functions (odf) calculated at four different positions of the specimen. Note that the upper left odf is related to the very upper left corner of the tissue, the lower left odf reflects the material behaviour to the left side of the cut, while the upper and lower right odfs characterise the behaviour in the middle of the specimen. As expected, we observe the most pronounced anisotropy to be present at left side of the cut (as it clearly deviates from its initally isotropic, i.e. spherical distribution), whereas at the middle of the specimen anisotropy evolution is much lower.



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