The degree of nonlinearity and anisotropy of blood vessel elasticity

Abstract

Blood vessel elasticity is important to physiology and clinical problems involving surgery, angioplasty, tissue remodeling, and tissue engineering. Nonlinearity in blood vessel elasticity in vivo is important to the formation of solitons in arterial pulse waves. It is well known that the stress-strain relationship of the blood vessel is nonlinear in general, but a controversy exists on how nonlinear it is in the physiological range. Another controversy is whether the vessel wall is biaxially isotropic. New data on canine aorta were obtained from a biaxial testing machine over a large range of finite strains referred to the zero-stress state. A new pseudo strain energy function is used to examine these questions critically. The stress-strain relationship derived from this function represents the sum of a linear stress-strain relationship and a definitely nonlinear relationship. This relationship fits the experimental data very well. With this strain energy function, we can define a parameter called the degree of nonlinearity, which represents the fraction of the nonlinear strain energy in the total strain energy per unit volume. We found that for the canine aorta, the degree of nonlinearity varies from 5% to 30%, depending on the magnitude of the strains in the physiological range. In solving any surgical problems of a blood vessel, physicians need to know the mechanical properties of the blood vessels. In devising any strategy to heal or remodel an artery, we must know the stress and strain in the vessel (1-8). To determine the stress and strain, we must know the constitutive equations of the materials (1,4,5,7,9). To determine the constitutive equation, we must know the zero-stress state of the material (10, 11). The constitutive equation of the blood vessel wall has been studied by Bergel, Patel, Vaishnav, Fung, Hayashi and many others (see refs. 1, 2, 5, 8, and 12 and references therein). Conventionally, tests were done on cannulated excised straight segments of blood vessels by longitudinal stretching and circumferential distension with internal pressure. Fung (13,14) summarized the experimental viscoelastic results by describing the arterial wall as pseudo-elastic, which has the characteristics that its hysteresis is essentially independent of frequency over a wide range of 5 to 6 decades. This pseudo-elasticity is explained by a stress-relaxation function that has a continuous relaxation spectrum. For pseudo-elastic bodies, the loading and unloading curves are repeatable in cyclic testing, hence each can be treated as an elastic curve. Aorta is pseudo-elastic, and its hysteresis is small (ϳ5%). Until 1983, however, it was not known that the zero-stress state of the aorta (or more precisely, the state of zero stress-resultants and stressmoments) is not an unloaded tube, but is an open sector (10, 11). Thus, all pre-1983 data and most newer ones were not referred to the correct zero-stress state. On the other hand, newer data that refer to the zero-stress state have been limited to uniaxial strain va...

Rate-Dependent Stiffening and Strain Localization in Physically Associating Solutions

“…At larger strains, the finite extensibility of the network strands and the subsequent strain-stiffening behavior is accurately described by a single fitting parameter, J*. This expression has been employed previously to describe the large strain behavior of elastic, self-assembled triblock copolymer gels deformed in uniaxial compression 1 and shear 14 and equivalent functions have been applied to describe the non-linear elasticity of biological systems 33 and recently stiff polymer networks. 34 The physically associating solution is assumed to deform affinely, so that the local extension ratios We assume that strain energy is stored in deformed 'network strands' or 'bonds', which in our case correspond to bridging midblocks that span different endblock aggregates.…”

Evaluating the mechanical behavior of arterial tissue using digital image correlation

“…However, displacements can be measured by applying artificial markers to the tissue in its relaxed state and monitoring the position of the markers during mechanical loading. For example, Zhou and Fung 11 and Humphrey et al 12 introduced a four-point grid on the surface of tissue samples using a permanent ink marker to evaluate the non-linear elastic behavior of blood vessels. An inherent assumption in this approach is that the mechanical response is uniform and homogeneous between the marker points.…”

A bilinear stress–strain relationship for arteries

“…Other behaviors including the existence of residual strains, hysteresis and active contraction are also ignored. After Fung's exponential-type pseudoelastic strain energy function was proposed [15], it has been widely used to model the highly nonlinear and anisotropic behavior of blood vessels, e.g., [16][17][18]. In this model, the second Piola-Kirchhoff stress is obtained by differentiating the strain energy with respect to the Green-Lagrange strain.…”

“…(4)) for the thoracic aorta of a mongrel dog: C = 120.2 kPa, a 1 = 0.320, a 2 = 0.451, a 4 = 0.0681, [16], we first plot the distribution of Kirchhoff stress vs. Green strain in the circumferential and axial directions in Figs. 1(a) and (b), respectively.…”

“…1(a) and (b), respectively. The range of interest is chosen to cover typical physiological values in [16] and beyond: 1.1 < λ θ < 1.8 and 1.1 < λ z < 1.6, i.e., 0.105 < E θθ < 1.12 and 0.105 < E zz < 0.78. We then consider the sample data points (11×11 grids) in Fig.…”