Comparative study of viscoelastic arterial wall models in nonlinear one-dimensional finite element simulations of blood flow

It is well known that blood vessels exhibit viscoelastic properties, which are modeled in the literature with different mathematical forms and experimental bases. The wide range of existing viscoelastic wall models may produce significantly different blood flow, pressure, and vessel deformation solutions in cardiovascular simulations. In this paper, we present a novel comparative study of two different viscoelastic wall models in nonlinear one-dimensional (1D) simulations of blood flow. The viscoelastic models are from papers by Holenstein et al. in 1980 (model V1) and Valdez-Jasso et al. in 2009 (model V2). The static elastic or zero-frequency responses of both models are chosen to be identical. The nonlinear 1D blood flow equations incorporating wall viscoelasticity are solved using a space-time finite element method and the implementation is verified with the Method of Manufactured Solutions. Simulation results using models V1, V2 and the common static elastic model are compared in three application examples: (i) wave propagation study in an idealized vessel with reflection-free outflow boundary condition; (ii) carotid artery model with nonperiodic boundary conditions; and (iii) subject-specific abdominal aorta model under rest and simulated lower limb exercise conditions. In the wave propagation study the damping and wave speed were largest for model V2 and lowest for the elastic model. In the carotid and abdominal aorta studies the most significant differences between wall models were observed in the hysteresis (pressure-area) loops, which were larger for V2 than V1, indicating that V2 is a more dissipative model. The cross-sectional area oscillations over the cardiac cycle were smaller for the viscoelastic models compared to the elastic model. In the abdominal aorta study, differences between constitutive models were more pronounced under exercise conditions than at rest. Inlet pressure pulse for model V1 was larger than the pulse for V2 and the elastic model in the exercise case. In this paper, we have successfully implemented and verified two viscoelastic wall models in a nonlinear 1D finite element blood flow solver and analyzed differences between these models in various idealized and physiological simulations, including exercise. The computational model of blood flow presented here can be utilized in further studies of the cardiovascular system incorporating viscoelastic wall properties.     

Transmural strain distribution in the blood vessel wall

The transmural distributions of stress and strain at the in vivo state have important implications for the physiology and pathology of the vessel wall. The uniform transmural strain hypothesis was proposed by Takamyzawa and Hayashi (Takamizawa K and Hayashi K. J Biomech 20: 7-17, 1987; Biorheology 25: 555-565, 1988) as describing the state of arteries in vivo. From this hypothesis, they derived the residual stress and strain at the no-load condition and the opening angle at the zero-stress state. However, the experimental evidence cited by Takamyzawa and Hayashi (J Biomech 20: 7-17, 1987; and Biorheology 25: 555-565, 1988) to support this hypothesis was limited to arteries whose opening angles (theta) are <180 degrees. It is well known, however, that theta > 180 degrees do exist in the cardiovascular system. Our hypothesis is that the transmural strain distribution cannot be uniform when theta; is >180 degrees. We present both theoretical and experimental evidence for this hypothesis. Theoretically, we show that the circumferential stretch ratio cannot physically be uniform across the vessel wall when theta; exceeds 180 degrees and the deviation from uniformity will increase with an increase in theta; beyond 180 degrees. Experimentally, we present data on the transmural strain distribution in segments of the porcine aorta and coronary arterial tree. Our data validate the theoretical prediction that the outer strain will exceed the inner strain when theta > 180 degrees. This is the converse of the gradient observed when the residual strain is not taken into account. Although the strain distribution may not be uniform when theta exceeds 180 degrees, the uniformity of stress distribution is still possible because of the composite nature of the blood vessel wall, i.e., the intima-medial layer is stiffer than the adventitial layer. Hence, the larger strain at the adventitia can result in a smaller stress because the adventitia is softer at physiological loading.     

Platelet - vessel wall interaction: role of blood clotting

Vascular damage initiates not only the adhesion and aggregation of blood platelets but also coagulation, which is of mixed (intrinsic and extrinsic) origin. Evidence is presented that thrombin, generated as a result of the injury, is a prerequisite for platelet aggregation. Platelets, after activation, in their turn promote coagulation. Prostaglandin I2 (PGI2 or prostacyclin) inhibits coagulation induced by damaged vascular tissue. This effect of PGI2 is mediated by the inhibition of platelets in their participation in the generation of factor Xa and thrombin. Dietary cod liver oil, by changing plasma coagulability, decreases the procoagulation activity of vessel walls, and arterial thrombosis. Another fish oil with similar effects on plasma coagulability and some other haemostatic parameters does not modify vessel wall-induced clotting, nor does it significantly lower arterial thrombosis tendency; this indicates the physiological relevance of vessel wall-induced clotting in arterial thrombus formation. Some evidence is also given for the importance of vessel wall-induced clotting in primary haemostasis.     

Flow in an arteriosclerotic blood vessel with rigid permeable wall

This paper concerns the fluid-mechanical study of the effects of permeability of the wall of an arteriosclerotic blood vessel by idealizing the tissue space as a porous medium bounding the blood vessel and the arteriosclerotic blood vessel as a constricted axisymetric tube of slowly but arbitrarily varying cross-secton. An analytical solution in the general case is obtained by perturbation technique and several important particular geometries of constriction are discussed as special cases. Numerical results for the effects of permeability on the wall shear stress and filtration velocity are shown graphically.     

Control of blood vessel structure: insights from theoretical models

Blood vessels are capable of continuous structural adaptation in response to changing local conditions and functional requirements. Theoretical modeling approaches have stimulated the development of new concepts in this area and have allowed investigation of the complex relations between adaptive responses to multiple stimuli and resulting functional properties of vascular networks. Early analyses based on a minimum-work principle predicted uniform wall shear stress in all segments of vascular networks and led to the concept that vessel diameter is controlled by a feedback system based on responses to wall shear stress. Vascular reactions to changes in transmural pressure suggested feedback control of circumferential wall stress. However, theoretical simulations of network adaptation showed that these two mechanisms cannot, by themselves, lead to stable and realistic network structures. Models combining reactions to fluid shear stress, circumferential stress, and metabolic status of tissue, with propagation of stimuli upstream and downstream along vascular segments, are needed to explain stable and functionally adequate adaptation of vascular structure. Such models provide a basis for predicting the response of vascular segments exposed to altered conditions, as, for example, in vascular grafts.     

Protective role of heme oxygenase in the blood vessel wall during atherogenesis.

Several lines of evidence suggest that antioxidant processes and (or) endogenous antioxidants inhibit proatherogenic events in the blood vessel wall. Heme oxygenase (HO), which catabolizes heme to biliverdin, carbon monoxide, and catalytic iron, has been shown to have such antioxidative properties. The HO-1 isoform of heme oxygenase is ubiquitous and can be increased several fold by stimuli that induce cellular oxidative stress. Products of the HO reaction have important effects: carbon monoxide is a potent vasodilator, which is thought to play a role in modulation of vascular tone; biliverdin and its by-product bilirubin are potent antioxidants. Although HO induction results in an increase in catalytic free iron release, the enhancement of intracellular ferritin protein through HO-1 has been reported to decrease the cytotoxic effects of iron. Oxidized LDL has been shown to increase HO-1 expression in endothelial and smooth muscle cell cultures, and during atherogenesis. Further evidence of HO-1 expression associated with atherogenesis has been demonstrated in human, murine and rabbit atherosclerotic lesions. Moreover, genetic models of HO deficiency suggest that the actions of HO-1 are important in modulating the severity of atherosclerosis. Recent experiments in gene therapy using the HO gene suggest that interventions aimed at HO in the vessel wall could provide a novel therapeutic approach for the treatment or prevention of atherosclerotic disease.     

Effects of hypercholesterolemia on the electrical and contractile properties of the blood vessel wall

The investigations were performed on the ring strips of rabbit aorta. The electrical activity of the vessels smooth muscle cells was registered by the "sucrose gap" method; the contractile activity of the strips was determined simultaneously. Experimental atherosclerosis was induced by keeping rabbits on a special diet enriched by cholesterol for 2 and 4 months. Strips were prepared with intact or mechanically removed endothelium. Hypercholesterolemia was shown to inhibit the reactivity of the vessel's wall to the weakening action of acetylcholine due to endothelial stimulation. The cause of these changes was the inhibition of the endothelial functional activity and inactivation of mechanisms by which endothelium influences smooth muscle cells.     

Changes in the blood vessel wall elastin of spontaneously hypertensive (SHR) rats

It has been demonstrated that both elastin and tropoelastin preparations obtained from aortae of spontaneously hypertensive rats at the stage of established hypertension differ in their amino acid composition from age-matching controls. The differences refer to an increased proportion of polar amino acids, particularly aspartic and glutamic acid (about a two-fold increase compared to the controls) and arginine and tyrosine (1.5 times the control value). On the other hand, this increase is compensated for by a decrease in the valine concentration. Furthermore, direct estimation of the number of val-pro sequence in different elastin preparations indicated a drop from 49.3 to 29.2 per 1,000 residues in normotensive controls and preparations obtained from spontaneously hypertensive rats respectively.     

Physical properties of resistance vessel wall in peripheral blood flow regulation--I. Mathematical model

A mathematical model is introduced to investigate the influence of the physical properties of the resistance vessel wall on the metabolic and myogenic mechanisms. The resistance vessel wall is assumed to have an elastic property and the elastic modulus to be a function of pressure (myogenic) and flow (metabolic). Blood is Poiseuille's flow. The resulting mathematical equations for pressure-flow, pressure-diameter, pressure-wall tension and pressure-wall elastic modulus relationships introduced obey Laplace's law. Poiseuille's law and Hooke's law. In comparison with the experimental data (pressure diameter), the mathematical model is confirmed to explain well the dynamic behavior of the resistance vessel wall in vivo.     

One-dimensional viscoelastic cell motility models

In this paper we consider a class of one-dimensional cell motility models with increasing complexities beginning with a kinematic model and ending with a model based on viscoelastic theory. In many of these models, we establish the existence of traveling cell solutions and show numerically that the solutions of the time-dependent problem converge to the traveling cell solutions as t → ∞. As a result, we are able to predict the eventual length and speed of the cell.     

Numerical analysis of pulsatile blood flow and vessel wall mechanics in different degrees of stenoses

Hemodynamics factors and biomechanical forces play key roles in atherogenesis, plaque development and final rupture. In this paper, we investigated the flow field and stress field for different degrees of stenoses under physiological conditions. Disease is modelled as axisymmetric cosine shape stenoses with varying diameter reductions of 30%, 50% and 70%, respectively. A simulation model which incorporates fluid-structure interaction, a turbulence model and realistic boundary conditions has been developed. The results show that wall motion is constrained at the throat by 60% for the 30% stenosis and 85% for the 50% stenosis; while for the 70% stenosis, wall motion at the throat is negligible through the whole cycle. Peak velocity at the throat varies from 1.47 m/s in the 30% stenosis to 3.2m/s in the 70% stenosis against a value of 0.78 m/s in healthy arteries. Peak wall shear stress values greater than 100 Pa were found for > or =50% stenoses, which in vivo could lead to endothelial stripping. Maximum circumferential stress was found at the shoulders of plaques. The results from this investigation suggest that severe stenoses inhibit wall motion, resulting in higher blood velocities and higher peak wall shear stress, and localization of hoop stress. These factors may contribute to further development and rupture of plaques.