Multiscale numerical analyses of arterial tissue with embedded elements in the finite strain regime

Multiscale models based on representative volume elements (RVEs) might help unraveling the ways in which macroscopic loadings affect the microstructure of tissues reinforced by collagen fibers, and vice versa. Tissues such as arteries, however, are characterized by a significant collagen dispersion. Therefore, many fibers have to be included in the RVE to achieve a representative geometrical model of the microstructure. For this reason, when the finite element method is employed in the numerical homogenization, fibers are commonly modeled as 1D elements, either by considering a network of trusses or the embedded elements technique. With regard to the latter, there has been little attention in previous works concerning the influence of the chosen multiscale boundary conditions and RVE size. In order to address this issue, the present work combines a sound multiscale framework with the classical embedded elements technique to simulate four increasingly larger RVEs, which resemble the microstructure of the medial layer of the arterial wall. Each RVE is modeled as a ground substance with embedded collagen fibers and subjected to a macroscopic isochoric equibiaxial stretch up to 10%, according to four classical multiscale boundary conditions: Taylor-Voigt, linear boundary displacements, periodic boundary fluctuations and minimally constrained model. Results are evaluated both at the macroscopic level (homogenized response) and the microscopic level (strains in the ground substance and fiber stretches). At the macroscopic level, the homogenized response for the periodic boundary condition seems to converge faster than the other three with increasing RVE size. At the microscopic level, the periodic model is also less prone to concentrated effects at the boundaries of the RVE. Therefore, among the four classical multiscale boundary conditions, the periodic model seems to be better suited to simulate the microstructure of fibrous tissues employing the embedded elements technique. Importantly, the resulting microscopic strain fields are characterized by a considerable degree of inhomogeneity and some values are significantly larger than the macroscopic (imposed) strain. That could help to shed more light on relevant mechanotransduction mechanisms, e.g., cell signaling, that are known to happen in the arterial media and are linked to biological processes such as growth and remodeling. Therefore, the framework herein proposed may serve as a valuable tool for the investigation of microstructural phenomena that happen in arteries, or even other fibrous tissues.