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Invariant-based constitutive models for arterial tissues

Arterial wall is composed of three concentric layers, intima, media, and adventitia, separated by elastic membranes. At microscopic level, each layer appear as a fibrous network containing mainly collagen fibers embedded in a ground matrix, consisting of elastin, proteoglycans and water. The higher quantity of water in the tissue makes arteries nearly incompressible. Elastin and collagen are the mainly components determining the mechanical behavior of the tissue. While elastin load-bearing at low stresses and at small strains, collagen fibers providing highest stiffness and tensile strength at higher stresses. During a loading process, the progressive recruitment to the resistance of collagen fibers leads to the J-shaped stress-strain curve typical of soft tissues. The collagen fibers organization confers to the material an orthotropic structure.
Neglecting temperature as well as viscous and time-dependent effects, it is possible to model soft tissue mechanics in the context of nonlinear hyper-elasticity. Consequently, the constitutive law of such a tissue is obtained by derivation of a strain-energy function per unit reference volume with respect to strain tensors. Within the framework of nonlinear hyper-elasticity [R1] and invariant theory [R2], the anisotropic behavior of arterial tissue is usually described defining additional invariants which are associated with the directions of collagen fibres and their dispersion. In the literature, two main invariant-based formulations have been developed and used for a variety of soft tissues. These approaches differ by whether the collagen orientation is modeled as a discrete distribution with fibers perfectly alligned along the preffered directions [R3, R4], or as a continuous distribution over a range of fiber orientations. Regarding continuous fibers distribution, such dispersion can be obtained either by angular integration of infinitesimal fractions of fibers aligned in a given direction [R5, R6] or by generalized structure tensors formulations [R7, R8].

 

References

  1. [R1] J. D. Humphrey, Cardiovascular solid mechanics. Cells, tissue and organs, SpringerVerlag, New York, 2002.
  2. [R2] A. J. M. Spencer, Continuum theory of the mechanics of fibre-reinforced composites, Springer-Verlag, New York, 1984.
  3. [R3] G. A. Holzapfel, T. C. Gasser, R. W. Ogden. A new constitutive framework for arterial wall mechanics and a comparative study of material models, Journal of Elasticity 61 (2000) 1–48.
  4. [R4] S. Baek, R. L. Gleason, K. R. Rajagopal, J. D. Humphrey, Theory of small on large: Potential utility in computations of fluid solid interactions in arteries, Computer Methods in Applied Mechanics and Engineering 196 (2007) 3070–3078.
  5. [R5] Y. Lanir, O. Lichtenstein, O. Imanuel. Optimal design of biaxial tests for structural material characterization of flat tissues. Journal of Biomechanical Engineering 118 (1996) 41–47.
  6. [R6] M.S. Sacks. Incorporation of experimentally-derived fiber orientation into a structural constitutive model for planar collagenous tissues. Journal of Biomechanical Engineering 125 (2003) 187–280.
  7. [R7] A.D. Freed, D.R. Einstein, I. Vesely. Invariant formulation for dispersed transverse isotropy in aortic heart valves. Biomech Model Mechanobiol 4 (2005) 100–117.
  8. [R8] T. C. Gasser, R. W. Ogden, G. A. Holzapfel, Hyperelastic modelling of arterial layers with distributed collagen fibre orientations, Journal of the Royal Society Interface 3 (2006) 15–35.