Biomechanics of Arterial Walls under Supra-Physiological Loading Conditions
Mechanics
Final Report Abstract
In the first phase of the project, the focus of the work group Balzani was on the constitutive modeling of damage in supra-physiologically loaded soft biological tissues. A construction principle for phenomenological damage formulations was derived where damage initializes at the transition of the physiological to the supra-physiological load range and the saturation behavior observed in experiments is captured. By adjustment of specified models to experimental data an excellent correlation could be obtained. Next, in order to in principle enable the measurement of the damage parameters based on microscopy, a micromechanically motivated continuum damage approach was introduced. It basically considers the integration of a statistical distribution function associated with a microstructure parameter in order to quantify the damage variable as the amount of broken proteoglycan bridges between collagen fibrils. The latter model was specified under consideration of statistically distributed proteoglycan orientations, internal length parameters and ultimate proteoglycan stretches. Adjusting the resulting formulations to uniaxial cyclic tension tests of human arteries, it was found that strain-energies with a damage function considering statistically distributed proteoglycan orientations performed best in capturing the experimental stress-strain curves among all investigated models. In the second phase of the project, a three-dimensional relaxed incremental variational approach for damage was proposed. It avoids a loss of convexity, which typically occurs at certain deformations when treating stress-softening phenomena. Thereby, mesh-independent solutions of boundary value problems computed by finite elements are obtained. The model considers a numerical integration over distributed fiber orientations, where for each fiber direction a relaxed formulation is applied. This relaxed formulation was obtained by constructing a convex hull whenever the originally (unrelaxed) energy loses convexity. For the relaxed energy a global minimization problem has to be solved at each integration point and thus, an evolutionary algorithm combined with Newton minimization was implemented. In the convexified regime elastic unloading and reloading paths were constructed in order to reflect hysteresis behavior as observed in supra-physiologically loaded arterial tissues. To more efficiently adjust the proposed model to complex experimental softening hysteresis observed in experiments, a special scheme using a surrogate model was proposed. This is required due to a lack of knowledge of the regime where convexity is lost for different parameters as they naturally arise during the adjustment procedure, which would require to solve the expensive global optimization problem to identify the convexified hull in every direction of the numerical integration scheme. Finally, a method for the assessment of rupture probabilities in damaged soft collagenous tissues based on optimal uncertainty quantification was proposed. Instead of assuming probability density functions for the tissue properties, only a limited but known set of statistical information, as the mean, variance and further statistical moments, is included. As a consequence, no precise value of the probability of failure can be determined, but a range under the constraints of the given statistical information can be identified. The capabilities of all formulations were demonstrated by implementing them in a finite element program and by simulating simplified atherosclerotic arteries under supra-physiological loadings using the parameters obtained from the adjustment to real experimental data. After all, the simulation of arterial walls could be enhanced to account for a more realistic behavior. Thus, an important improvement of the prerequisites for the simulation and assessment of therapeutic interventions such as balloon angioplasty or stent implantation could be achieved.
Publications
- Statistical approach for a continuum description of damage evolution in soft collageneous tissues, Computer Methods in Applied Mechanics and Engineering 278, 41-61 (2014)
T. Schmidt, D. Balzani, G.A. Holzapfel
(See online at https://doi.org/10.1016/j.cma.2014.04.011) - Comparative analysis of damage functions for soft tissues: properties at damage initialization, Mathematics and Mechanics of Solids 20, 480–492 (2015)
D. Balzani, T. Schmidt
(See online at https://doi.org/10.1177/1081286513504945) - Modeling the biomechanics of arterial walls under supra-physiological loading, dissertation thesis, Technische Universität Dresden (2015)
T. Schmidt
- Selective enzymatic removal of elastin and collagen from human abdominal aortas: uniaxial mechanical response and constitutive modeling, Acta Biomaterialia 17, 125–136 (2015)
A.J. Schriefl, T. Schmidt, D. Balzani, G. Sommer, G.A. Holzapfel
(See online at https://doi.org/10.1016/j.actbio.2015.01.003) - Relaxed incremental variational approach for the modeling of damage-induced stress hysteresis in arterial walls, Journal of the Mechanical Behavior of Biomedical Materials 58, 149–162 (2016)
T. Schmidt, D. Balzani
(See online at https://doi.org/10.1016/j.jmbbm.2015.08.005) - Method for the quantification of rupture probability in soft collagenous tissues, International Journal for Numerical Methods in Biomedical Engineering 33, e02781 (2017)
D. Balzani, T. Schmidt, M. Ortiz
(See online at https://doi.org/10.1002/cnm.2781)
