UC Berkeley

Understanding the mechanical behavior of arterial walls under various physiological loading and boundary conditions is essential for achieving the following: (1) improved therapeutics that are based on mechanical procedures (e.g. arterial segmenting and suturing), (2) study of mechanical factors that may trigger the onset of arterial aneurysms (i.e. focal blood-filled dilatation of the vessel wall caused by disease) and (3) investigations on tissue variations due to health, age, hypertension and atherosclerosis, all of which hold immense clinical relevance. In general, the physiological conditions on an any arterial segment can include axial stretch, torsional twist and transmural (internal, radial) pressure which often provoke large wall-tissue deformations that require theories of continuum hyperelasticity. Further, the presence of collagen fibers throughout the two structural layers (media, adventitia) of the arterial wall require anisotropic strain energy functions for more histological accurate models. Nonlinear computational methods are therefore essential for this class of boundary-value-problems (BVPs) which often do not contain closed-form solutions.

We begin by modeling the arterial vessel wall as a thin sheet in the form of a circular cylinder in the reference configuration. We seek to employ a bio-type strain energy function on this constitutive framework to investigate the onset of non-linear instabilities in a thin-walled, hyperelastic tube under (remote) axial stretch and internal pressure. Viscoelastic effects are also considered in this model. We then build to investigating the effects of various combinations of axial stretch and transmural pressure on the global deformation and through-thickness stress and strain fields of an arterial segment modeled as a two-layer, fiber-reinforced composite and idealized as a thick-walled cylinder in the reference configuration. We further consider (in both models) the presence of local tissue lesions, or portions of the arterial wall having either stiffer (i.e. thrombosis or scar tissue) or softer (i.e. diseased tissue) material characteristics, relative to the surrounding tissue. We account for this by appropriately scaling the elastic constants of the strain energy functions for regions with a lesion and without. For the three-dimensional model, we employ the strain energy function of Holzapfel et al. which has been modified by constraints on the principal invariants by Balzani et al. in order to ensure material polyconvexity. We choose a particular vessel, the human common carotid (HCC) artery, with appropriate geometric and material properties found from various experimentally-based studies (e.g. Fung et al.). We focus on distinct elastic constants for each layer (media, adventitia) that have been obtained through biaxial (i.e. not simply uniaxial data - reasons for this are discussed later) testing of in vitro HCC arteries. The loading conditions are combinations of axial extension and transmural pressure, in the presence and absence of material lesions. The loading is consistent with in vivo conditions on a general segment of the vessel wall.

We find that as a two-dimensional surface, the overall deformation from internal pressure (i.e. the bulge) depends on the magnitude and, more importantly, the rate of axial stretch and transmural pressure, the elastic material parameters of the bio-strain energy function, and of course local inhomogeneities in the material description of the tissue. When modeled as a three-dimensional solid undergoing pure axial stretch, the majority of the stress is in the medial tissue, which displays a significant gradient in the axial direction, whereas the stress in the adventitia is constant throughout the length of the vessel. For supra-physiological pressures (i.e. 20-30 kPa, or about 50% higher than in vivo conditions) the adventitia contributes to the load sharing and the gradient in the medial layer evens out. For narrow (2% of the length), stiff (100x stiffer than surrounding tissue), ring-like lesions under the same pressures and axial stretch, the overall vessel deformation is considerably smaller in the radial direction. The overall segment shape is stabilized by this type of material abnormality. For local spot-like stiff (100x stiffer than surrounding tissue) lesions, the deformation leads to an inward bulge (i.e. a clot) that will likely affect fluid flow characteristics, hence growth and remodeling of the tissue at the wall. For these loading conditions, when the spot-like and ring-like lesions are approximately two-times softer than the surrounding tissue, no significant differences appear in the stress and strain fields.