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An algorithmic scheme for the automated calculation of fiber orientations in arterial walls
We propose an algorithmic scheme for the numerical calculation of fiber orientations in arterial walls. The basic assumption behind the procedure is that the fiber orientations are mainly governed by the principal tensile stress directions resulting in an improved load transfer within the artery as a consequence of the redistribution of stresses. This reflects the biological motivation that soft tissues continuously adapt to their mechanical environment in order to optimize their load-bearing capacities. The algorithmic scheme proposed here enhances efficiency of the general procedure given in Hariton et al. (Biomech Model Mechanobiol 6(3):163-175, 2007), which consists of repeatedly identifying a favored fiber orientation based on the principal tensile stresses under a certain loading scenario, and then re-calculating the stresses for that loading scenario with the modified favored fiber orientation. Since the method still depends on a highly accurate stress approximation of the finite element formulation, which is not straightforward to obtain in particular for incompressible and highly anisotropic materials, furthermore, a modified model is introduced. This model defines the favored fiber orientation not only in terms of the local principal stresses, but in terms of the volume averages of the principal stresses computed over individual finite elements. Thereby, the influence of imperfect stress approximations can be weakened leading to a stabilized convergence of the reorientation procedure and a more reasonable fiber orientation with less numerical noise. The performance of the proposed fiber reorientation scheme is investigated with respect to different finite element formulations and different favored fiber orientation models, Hariton et al. (Biomech Model Mechanobiol 6(3):163-175, 2007) and Cyron and Humphrey (Math Mech Solids 1-17, 2014). In addition, it is applied to calculate the fiber orientation in a patient-specific arterial geometry.