

Inverse problemThe analysis of our experiments does not always require solution of the inverse problem. For example, soft tissue characterization for identification of pathological states (diagnosis) can be based on simple scalar quantities directly extracted from the experimental data. Examples are the parameters called “stiffness”, “creep”, “softening” and “rising time” introduced for the classification of the response of the uterine cervix or liver in aspiration experiments, see [11p], [14p], [2t], [5t], [39p]. Solution of the inverse problem is required when the objective of the experiment is the determination of the constitutive model of the tissue under consideration. This is needed for simulation and prediction of the mechanical response of tissues, such as for simulation of surgery (planning or virtual reality based training, e.g. [3p]), for design and optimization of implants, or for the analysis and improvement of our experimental procedures. Conventional material testing (e.g. tensile test, torsion test, bending tests) aim at characterizing the mechanical response under simple and homogeneous deformation and stress states. In this way, constitutive model parameters can be determined directly from the experimental data or using simple analytical procedures. Our “unconventional” experiments investigate the response of soft biological tissue to large and nonhomogeneous deformations related to multiaxial and time dependent mechanical loads. Our “testpieces” are often nonhomogeneous, and made of nonlinear viscoelastic materials, thus requiring model equations with a larger number of parameters. Two main challenges arise from this complexity: 1 Finite element based inverse problem solution. We develop optimization routines and algorithms for the iterative determination of “best fit” model parameters ([2t], [1t], [17p], [6p], [4p], [2p], [1p], [3t], [7t], [29p]). We use a finite element software which allows implementation of user defined constitutive models. We have implemented a number of model equations for anisotropic hyperelastic and nonlinear viscoelastic materials (e.g., the models proposed in: Rubin, M. B., Bodner, S. R., 2002. International Journal of Solids and Structures 39 (19), 50815099; Ehret, A.E., Itskov, M. A., 2007, Journal of Materials Science 42, 8853–8863; Eberlein, R., Holzapfel, G. A., SchultzeBauer, C. A., 2001, Computer Methods in Applied Mechanics and Engineering 4, 209229; Weiss, J. A., Maker, B. N., Govindjee, S., 1996, Computer Methods in Applied Mechanics and Engineering 135 (12), 107128) considering issues related to invariance requirements and consistent determination of the tangent stiffness matrix ([4t], [5t], [21p]). 2 Nonuniqueness of constitutive model determination. Our experiments provide only limited information on the mechanical behavior of the material and lead thus to nonunique model equations. The constitutive model determined from one experiment has to be validated under different loading conditions and deformation states. Within inherent limitations related to biological tissues, we apply this approach, e.g. with comparison of results from aspiration and indentation experiments, or distinct and combined measurements on parenchyma and liver capsule ([10p], [9p], [35p], [29p]).
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