$324 Journal o f Biomechanics 2006, Vol. 39 (Suppl 1) Oral Presentations

14.13.3. Mathematical and Computational Modelling

6745 Mo, 14:00-14:15 (P11) On experimental testing methods for characterizing the mechanical properties of soft biological materials such as arterial tissues G.A. Holzapfel 1,2, R.W. Ogden 3. 1Royal Institute of Technology (KTH), School of Engineering Sciences, Stockholm, Sweden, 2 Graz University of Technology, Computational Biomechanics, Graz, Austria, 3University of Glasgow, Department of Mathematics, University Gardens, Glasgow, UK

Efficient experimental tests are crucial, in particular, for the characterization of the mechanical properties of soft biological tissues and for the development of constitutive laws that can be used for predictive purposes. Several testing methods are in common use, including uniaxial tension and planar biaxial tension, simultaneous extension, inflation, and torsion experiments of arterial segments, (multiaxial) shear tests and combinations of these protocols. For many biological specimens, such as components of (diseased) arteries, the preparation of intact separated tissue patches for suitable tests is, however, often difficult to achieve because of the small size of the specimens (see, e.g., [1]). In many cases the preparation of such patches is not feasible, and uniaxial tension tests on strips in two perpendicular directions are sometimes considered instead. They are very popular, relatively easy to perform, can provide detailed information about the nature and degree of the anisotropy by using specimens from different orientations within the tissue, and they provide appropriate data towards the material characterization. We summarize the range of information that can be gained from tests on transversely isotropic materials and orthotropic materials such as arterial tissues under various loading and boundary conditions, and we highlight, in particular, the theoretical impossibility of fully characterizing the properties of anisotropic materials on the basis of planar biaxial tests alone. In this connection, we point out some fundamental errors relating to biaxial testing that have appeared in the literature. Possible sets of independent tests required for full characterization are enumerated.

References [1] G.A. Holzapfel, G. Sommer, P. Regitnig, Anisotropic mechanical properties of

tissue components in human atherosclerotic plaques. Journal of Biomechanical Engineering 2004; 126: 657-665.

4343 Mo, 14:15-14:30 (P11) Computat ional model l ing o f fibre reorientation and growth in o r tho t rop ic biological tissues A. Menzel. Chair of Applied Mechanics, University of Kaiserslautem, Kaiserslautem, Germany

Biological tissues usually exhibit strongly anisotropic response and undergo finite deformations. Collagen fibres are often identified to determine the load capacity of such tissues. The continuum mechanical framework proposed in this contribution accounts for these characteristic properties by introducing two representative so-called fibre families. As such, orthotropic symmetry is automatically captured within the formulation. According to the observation that biological tissues tend to maximise their load capacity, these fibre families are aligned with respect to (predominant) principal stretch directions. The related algorithmic treatment is embedded into a nonlinear finite element setting whereby concepts established for viscoelasticity are adopted - in other words, the fibre-alignment is performed in a time dependent context. Apart from this remodelling formulation, anisotropic growth is incorporate. As a fundamental modelling approach, a multiplicative decomposition of the deformation gradient is applied. The underlying growth part enables the incorporation of residual stresses after so-called unloading (even though the definition of an unloaded state being non-trivial). Moreover, changes in mass and volume are directly re- lated to this growth term of the deformation gradient and appropriate evolution equations close the proposed constitutive framework.

References [1] G.A. Holzapfel and R.W. Ogden, editors. Biomechanics of Soft Tissue in

Cardiovascular Systems. Number 441 in CISM Courses and Lectures. Springer, 2003.

[2] J.D. Humphrey. Cardiovascular Solid Mechanics. Cells, Tissues, and Organs. Springer, 2002.

[3] A. Menzel. Modelling of anisotropic growth in biological tissues - A new approach and computational aspects. Biomechan. Model. Mechanobiol. 2005; 3(3): 147-171.

5339 Mo, 14:30-14:45 (P11) Micromechanical mot ivated analysis of the behavior o f arterial wall G. deBotton, I. Hariton, E.A. Socolsky. Department ef Mechanical Engineering, Ben-Gurion University, Beer-Sheva, Israel

For understanding and modeling the behavior of biological tissues an accurate model for describing their mechanical behavior is extremely important. At the

present phenomenological models which are based on a limited number of experimental measurements are commonly being used. In this work we extend this approach and introduce a micromechanical based model in which the tissue is treated as a collagen fibers reinforced connective tissue. Beginning with rigorous definitions of homogenization under finite deformations, we determine exact expressions for the overall response of a tissue with one family of unidirectional collagen fibers subjected to different boundary conditions. These solutions enable to extract a close form expression for the effective strain energy-density function (E-SEDF) that characterizes the spatial behavior of the heterogeneous tissue. The expression for the E-SEDF is given in terms of the transversely isotropic invariants, the properties of the collagen fibers, their concentration, and the properties of the ground substance in the extracellular matrix. We demonstrate that the expression we introduce correctly reduces to well-established results in the field of infinitesimal elasticity. Moreover, a comparison of the stress-strain curves derived from the proposed E-SEDF with corresponding finite element simulations of a tissue with periodic microstructure under finite deformation reveals an excellent agreement. Finally, we also demonstrate that widely accepted experimentally motivated models for characterizing the mechanical behavior of soft biological tissues are in agreement with this micromechanics motivated E-SEDE While we find that this agreement is remarkable, we emphasize that the advantage of the proposed approach is clear on grounds of the fact that it allows to accurately determine the properties of the tissue in terms of the properties of its constituents and morphology. Thus, for example, to determine local variations in the properties of the tissue due to variations in the concentration of the collagen fibers, a situation which is frequently observed in arterial walls. Corresponding devel- opments for tissues with two families of fibers are currently considered.

5210 Mo, 14:45-15:00 ( P l l ) Mechanical character izat ion of arteries: compar ison o f square and cruciform biaxial tests using inverse modeling technique

J.O.V. Delgadillo 1,2, S. Delorme 1 , S.G. Hatzikiriakos 2. 1Industrial Materials Institute, CNRC - NRC, Boucherville, QC, Canada, 2Department of Chemical & Biological Engineering, UBC, Vancouver, BC, Canada

Background: The elastic behaviour of arteries is nonlinear when subjected to large deformations. In order to measure their anisotropic behaviour, planar biaxial tests are often used. Typically, hooks are attached along the borders of a square sample of arterial tissue [1,2,3,4]. Cruciform samples clamped with grips are often used for testing industrial materials. The effect of different biaxial test boundary conditions has been evaluated with computer simulations [5]. In this study, we demonstrate the use of inverse modeling to fit the material properties, and discuss the differences between the two types of tests. Methods: Square and cruciform samples were dissected from pig aortas and tested equi-biaxially. On the square samples, forces where applied on each side and strains were measured in the center using optical tracking of ink dots. On the cruciform samples, displacements were applied on the grips and forces were measured on the grips. Each type of experiment was simulated with the finite element method. Mooney-Rivlin and Ogden constitutive models were adjusted with an optimization algorithm so that the simulation predictions fitted the experimental results. Results and Discussion: Higher stretch ratios (>1.5) were reached with the cruciform samples than with the square samples before failure. Convergence of the optimization algorithm was easier with the Mooney-Rivlin than with Ogden model. Advantages of cruciform samples over square samples include: 1) higher strain range; 2) simpler data acquisition. However, the non uniform stress distribution in cruciform samples requires the use of inverse modeling adjustment of constitutive model parameters.

References [1] Choudbury N.Z., 2005. Master's Thesis, Chem. Eng. Dept., McGill University,

Montreal, Quebec. [2] Lally C., et. al. Ann. Biomed. Eng. 2004; 32: 1355-1364. [3] Okamoto R.J., et. al. Ann. Biomed. Eng. 2002; 30: 624~35. [4] Prendergast P. J., et. al. J. Biomech. Eng. 2003; 125: 692~99. [5] Sun W., et. al. J. Biomech. Eng. 2005; 127: 709-715.

6201 Mo, 15:00-15:15 ( P l l ) The misuse of the Laplace law for the measurement of wall stiffness J. Brasseur, A. Thatte. The Pennsylvania State University, University Park, USA

The Laplace law (LL) is commonly used with concurrent measurement of intraluminal pressure and cross sectional geometry to obtain the stress within the muscle wall ob a biological vessel as a function of lumen distension. The LL stress is often plotted against strain to estimate the elastic modulus (stiffness) from the slope of the stress-strain curve, to evaluate compliance. However, to derive the LL from Newton's law, one assumes that muscle layer thickness T is very small compared to the lumen radius R. From the derivation, one