Nonlinear buckling of blood vessels: a theoretical study

Buckling of Arteries With Noncircular Cross Sections: Theory and Finite Element Simulations

The stability of blood vessels is essential for maintaining the normal arterial function, and loss of stability may result in blood vessel tortuosity. The previous theoretical models of artery buckling were developed for circular vessel models, but arteries often demonstrate geometric variations such as elliptic and eccentric cross-sections. The objective of this study was to establish the theoretical foundation for noncircular blood vessel bent (i.e., lateral) buckling and simulate the buckling behavior of arteries with elliptic and eccentric cross-sections using finite element analysis. A generalized buckling equation for noncircular vessels was derived and finite element analysis was conducted to simulate the artery buckling behavior under lumen pressure and axial tension. The arterial wall was modeled as a thick-walled cylinder with hyper-elastic anisotropic and homogeneous material. The results demonstrated that oval or eccentric cross-section increases the critical buckling pressure of arteries and having both ovalness and eccentricity would further enhance the effect. We conclude that variations of the cross-sectional shape affect the critical pressure of arteries. These results improve the understanding of the mechanical stability of arteries.

Arterial Wall Stiffening in Caveolin-1 Deficiency-Induced Pulmonary Artery Hypertension in Mice

BACKGROUND

Pulmonary artery hypertension (PAH) is a complex disorder that can lead to right heart failure. The generation of caveolin-1 deficient mice (CAV-1-/-) has provided an alternative genetic model to study the mechanisms of pulmonary hypertension. However, the vascular adaptations in these mice have not been characterized.

OBJECTIVE

To determine the histological and functional changes in the pulmonary and carotid arteries in CAV-1-/- induced PAH.

METHODS

Pulmonary and carotid arteries of young (4-6 months old) and mature (9-12 months old) CAV-1-/- mice were tested and compared to normal wild type mice.

RESULTS

Artery stiffness increases in CAV-1-/- mice, especially the circumferential stiffness of the pulmonary arteries. Increases in stiffness were quantified by a decrease in circumferential stretch and transition strain, increases in elastic moduli, and an increase in total strain energy at physiologic strains. Changes in mechanical properties for the pulmonary artery correlated with increased collagen content while carotid artery mechanical properties correlated with decreased elastin content.

CONCLUSIONS

We demonstrated that an increase in artery stiffness is associated with CAV-1 deficiency-induced pulmonary hypertension. These results improve our understanding of artery remodeling in PAH.

Artery buckling stimulates cell proliferation and NF-κB signaling

Tortuous carotid arteries are often seen in aged populations and are associated with atherosclerosis, but the underlying mechanisms to explain this preference are unclear. Artery buckling has been suggested as one potential mechanism for the development of tortuous arteries. The objective of this study, accordingly, was to determine the effect of buckling on cell proliferation and associated NF-κB activation in arteries. We developed a technique to generate buckling in porcine carotid arteries using long artery segments in organ culture without changing the pressure, flow rate, and axial stretch ratio. Using this technique, we examined the effect of buckling on arterial wall remodeling in 4-day organ culture under normal and hypertensive pressures. Cell proliferation, NF-κB p65, IκB-α, ERK1/2, and caspase-3 were detected using immunohistochemistry staining and immunoblot analysis. Our results showed that cell proliferation was elevated 5.8-fold in the buckling group under hypertensive pressure (n = 7, P < 0.01) with higher levels of NF-κB nuclear translocation and IκB-α degradation (P < 0.05 for both). Greater numbers of proliferating cells were observed on the inner curve side of the buckled arteries compared with the outer curve side (P < 0.01). NF-κB colocalized with proliferative nuclei. Computational simulations using a fluid-structure interaction model showed reduced wall stress on the inner side of buckled arteries and elevated wall stress on the outer side. We conclude that arterial buckling promotes site-specific wall remodeling with increased cell proliferation and NF-κB activation. These findings shed light on the biomechanical and molecular mechanisms of the pathogenesis of atherosclerosis in tortuous arteries.

Stability of carotid artery under steady-state and pulsatile blood flow: a fluid-structure interaction study

It has been shown that arteries may buckle into tortuous shapes under lumen pressure, which in turn could alter blood flow. However, the mechanisms of artery instability under pulsatile flow have not been fully understood. The objective of this study was to simulate the buckling and post-buckling behaviors of the carotid artery under pulsatile flow using a fully coupled fluid-structure interaction (FSI) method. The artery wall was modeled as a nonlinear material with a two-fiber strain-energy function. FSI simulations were performed under steady-state flow and pulsatile flow conditions with a prescribed flow velocity profile at the inlet and different pressures at the outlet to determine the critical buckling pressure. Simulations were performed for normal (160 ml/min) and high (350 ml/min) flow rates and normal (1.5) and reduced (1.3) axial stretch ratios to determine the effects of flow rate and axial tension on stability. The results showed that an artery buckled when the lumen pressure exceeded a critical value. The critical mean buckling pressure at pulsatile flow was 17-23% smaller than at steady-state flow. For both steady-state and pulsatile flow, the high flow rate had very little effect (&lt;5%) on the critical buckling pressure. The fluid and wall stresses were drastically altered at the location with maximum deflection. The maximum lumen shear stress occurred at the inner side of the bend and maximum tensile wall stresses occurred at the outer side. These findings improve our understanding of artery instability in vivo.

Critical buckling pressure in mouse carotid arteries with altered elastic fibers

Arteries can buckle axially under applied critical buckling pressure due to a mechanical instability. Buckling can cause arterial tortuosity leading to flow irregularities and stroke. Genetic mutations in elastic fiber proteins are associated with arterial tortuosity in humans and mice, and may be the result of alterations in critical buckling pressure. Hence, the objective of this study is to investigate how genetic defects in elastic fibers affect buckling pressure. We use mouse models of human disease with reduced amounts of elastin (Eln+/-) and with defects in elastic fiber assembly due to the absence of fibulin-5 (Fbln5-/-). We find that Eln+/- arteries have reduced buckling pressure compared to their wild-type controls. Fbln5-/- arteries have similar buckling pressure to wild-type at low axial stretch, but increased buckling pressure at high stretch. We fit material parameters to mechanical test data for Eln+/-, Fbln5-/- and wild-type arteries using Fung and four-fiber strain energy functions. Fitted parameters are used to predict theoretical buckling pressure based on equilibrium of an inflated, buckled, thick-walled cylinder. In general, the theoretical predictions underestimate the buckling pressure at low axial stretch and overestimate the buckling pressure at high stretch. The theoretical predictions with both models replicate the increased buckling pressure at high stretch for Fbln5-/- arteries, but the four-fiber model predictions best match the experimental trends in buckling pressure changes with axial stretch. This study provides experimental and theoretical methods for further investigating the influence of genetic mutations in elastic fibers on buckling behavior and the development of arterial tortuosity.

Artery buckling analysis using a four-fiber wall model

Artery bent buckling has been suggested as a possible mechanism that leads to artery tortuosity, which is associated with aging, hypertension, atherosclerosis, and other pathological conditions. It is necessary to understand the relationship between microscopic wall structural changes and macroscopic artery buckling behavior. To this end, the objectives of this study were to develop arterial buckling equations using a microstructure-based 4-fiber reinforced wall model, and to simulate the effects of vessel wall microstructural changes on artery buckling. Our results showed that the critical pressure increased nonlinearly with the axial stretch ratio, and the 4-fiber model predicted higher critical buckling pressures than what the Fung model predicted. The buckling equation using the 4-fiber model captured the experimentally observed reduction of critical pressure induced by elastin degradation and collagen fiber orientation changes in the arterial wall. These results improve our understanding of arterial stability and its relationship to microscopic wall remodeling, and the model provides a useful tool for further studies.

An in vivo rat model of artery buckling for studying wall remodeling

Theoretical modeling and in vitro experiments have demonstrated that arterial buckling is a possible mechanism for the development of artery tortuosity. However, there has been no report of whether artery buckling develops into tortuosity, partially due to the lack of in vivo models for long-term studies. The objective of this study was to establish an in vivo buckling model in rat carotid arteries for studying arterial wall remodeling after buckling. Rat left carotid arteries were transplanted to the right carotid arteries to generate buckling under in vivo pressure and were maintained for 1 week to examine wall remodeling and adaptation. Our results showed that a significant buckling was achieved in the carotid arterial grafts with altered wall stress. Cell proliferation and matrix metalloprotinease-2 (MMP-2) expression in the buckled arteries increased significantly compared with the controls. The tortuosity level of the grafts also slightly increased 1 week post-surgery, while there was no change in vessel dimensions, blood pressure, and blood flow velocity. The artery buckling model provides a useful tool for further study of the adaptation of arteries into tortuous shapes.

Cap buckling as a potential mechanism of atherosclerotic plaque vulnerability

Plaque rupture in atherosclerosis is the primary cause of potentially deadly coronary events, yet about 40% of ruptures occur away from the plaque cap shoulders and cannot be fully explained with the current biomechanical theories. Here, cap buckling is considered as a potential destabilizing factor which increases the propensity of the atherosclerotic plaque to rupture and which may also explain plaque failure away from the cap shoulders. To investigate this phenomenon, quasistatic 2D finite element simulations are performed, considering the salient geometrical and nonlinear material properties of diverse atherosclerotic plaques over the range of physiological loads. The numerical results indicate that buckling may displace the location of the peak von Mises stresses in the deflected caps. Plaque buckling, together with its deleterious effects is further observed experimentally in plaque caps using a physical model of deformable mock coronary arteries with fibroatheroma. Moreover, an analytical approach combining quasistatic equilibrium equations with the Navier-Bresse formulas is used to demonstrate the buckling potential of a simplified arched slender cap under intraluminal pressure and supported by foundations. This analysis shows that plaque caps - calcified, fibrotic or cellular - may buckle in specific undulated shapes once submitted to critical loads. Finally, a preliminary analysis of intravascular ultrasonography recordings of patients with atherosclerotic coronary arteries corroborates the numerical, experimental and theoretical findings and shows that various plaque caps buckle in vivo. By displacing the sites of high stresses in the plaque cap, buckling may explain the atherosclerotic plaque cap rupture at various locations, including cap shoulders.

Finite Element simulation of buckling-induced vein tortuosity and influence of the wall constitutive properties

The mechanisms giving rise to vein tortuosity, which is often associated with varicosis, are poorly understood. Recent works suggest that significant biological changes in the wall of varicose veins may precede the mechanical aspects of the disease. To test the hypothesis of tortuosity being a consequence of these changes, a Finite Element model was developed based on previous experimental work on vein buckling. The model was then used to evaluate the effect of alterations of the mechanical behavior of the wall on tortuosity onset and severity. The results showed that increasing anisotropy toward the circumferential direction promotes tortuosity. An increase in wall stiffness tends to decrease the level of tortuosity but interestingly, if the vein segment is little or not pre-stretched such increase will not prevent, or it will even promote, the onset of tortuosity. These results provide additional arguments supporting the hypothesis of tortuosity being the consequence of biologically-induced changes in the varicose vein wall. Based on a 3D model of the leg and in vivo identification of the material properties of varicose veins, a clinical validation of these findings is being developed.

Mechanical buckling of arterioles in collateral development

Collateral arterioles enlarge in both diameter and length, and develop corkscrew-like tortuous patterns during remodeling. Recent studies showed that artery buckling could lead to tortuosity. The objective of this study was to determine arteriole critical buckling pressure and buckling pattern during arteriole remodeling. Arterioles were modeled as elastic cylindrical vessels with an elastic matrix support and underwent axial and radial growth. Our results demonstrated that arteriole critical buckling pressure decreased with increasing axial growth ratio and radius growth ratio, but increased with increasing wall thickness. Arteriole buckling mode number increased (wavelength decreased) with increasing axial growth ratio, but decreased with increasing radius growth ratio and wall thickness. Our study suggests that axial growth in arterioles makes them prone to buckling and that buckling leads to tortuous collaterals. These results shed light on the mechanism of collateral arteriole tortuosity.

Smooth muscle cell contraction increases the critical buckling pressure of arteries

Recent in vitro experiments demonstrated that arteries under increased internal pressure or decreased axial stretch may buckle into the tortuous pattern that is commonly observed in aging or diseased arteries in vivo. It suggests that buckling is a possible mechanism for the development of artery tortuosity. Vascular tone has significant effects on arterial mechanical properties but its effect on artery buckling is unknown. The objective of this study was to determine the effects of smooth muscle cell contraction on the critical buckling pressure of arteries. Porcine common carotid arteries were perfused in an ex vivo organ culture system overnight under physiological flow and pressure. The perfusion pressure was adjusted to determine the critical buckling pressure of these arteries at in vivo and reduced axial stretch ratios (1.5 and 1.3) at baseline and after smooth muscle contraction and relaxation stimulated by norepinephrine and sodium nitroprusside, respectively. Our results demonstrated that the critical buckling pressure was significantly higher when the smooth muscle was contracted compared with relaxed condition (97.3mmHg vs 72.9mmHg at axial stretch ratio of 1.3 and 93.7mmHg vs 58.6mmHg at 1.5, p<0.05). These results indicate that arterial smooth muscle cell contraction increased artery stability.

Artery buckling: new phenotypes, models, and applications

Arteries are under significant mechanical loads from blood pressure, flow, tissue tethering, and body movement. It is critical that arteries remain patent and stable under these loads. This review summarizes the common forms of buckling that occur in blood vessels including cross-sectional collapse, longitudinal twist buckling, and bent buckling. The phenomena, model analyses, experimental measurements, effects on blood flow, and clinical relevance are discussed. It is concluded that mechanical buckling is an important issue for vasculature, in addition to wall stiffness and strength, and requires further studies to address the challenges. Studies of vessel buckling not only enrich vascular biomechanics but also have important clinical applications.

Mechanical buckling of artery under pulsatile pressure

Tortuosity that often occurs in carotid and other arteries has been shown to be associated with high blood pressure, atherosclerosis, and other diseases. However the mechanisms of tortuosity development are not clear. Our previous studies have suggested that arteries buckling could be a possible mechanism for the initiation of tortuous shape but artery buckling under pulsatile flow condition has not been fully studied. The objectives of this study were to determine the artery critical buckling pressure under pulsatile pressure both experimentally and theoretically, and to elucidate the relationship of critical pressures under pulsatile flow, steady flow, and static pressure. We first tested the buckling pressures of porcine carotid arteries under these loading conditions, and then proposed a nonlinear elastic artery model to examine the buckling pressures under pulsatile pressure conditions. Experimental results showed that under pulsatile pressure arteries buckled when the peak pressures were approximately equal to the critical buckling pressures under static pressure. This was also confirmed by model simulations at low pulse frequencies. Our results provide an effective tool to predict artery buckling pressure under pulsatile pressure.

Effects of elastin degradation and surrounding matrix support on artery stability

Tortuous arteries are often associated with aging, hypertension, atherosclerosis, and degenerative vascular diseases, but the mechanisms are poorly understood. Our recent theoretical analysis suggested that mechanical instability (buckling) may lead to tortuous blood vessels. The objectives of this study were to determine the critical pressure of artery buckling and the effects of elastin degradation and surrounding matrix support on the mechanical stability of arteries. The mechanical properties and critical buckling pressures, at which arteries become unstable and deform into tortuous shapes, were determined for a group of five normal arteries using pressurized inflation and buckling tests. Another group of nine porcine arteries were treated with elastase (8 U/ml), and the mechanical stiffness and critical pressure were obtained before and after treatment. The effect of surrounding tissue support was simulated using a gelatin gel. The critical pressures of the five normal arteries were 9.52 kPa (SD 1.53) and 17.10 kPa (SD 5.11) at axial stretch ratios of 1.3 and 1.5, respectively, while model predicted critical pressures were 10.11 kPa (SD 3.12) and 17.86 kPa (SD 5.21), respectively. Elastase treatment significantly reduced the critical buckling pressure (P < 0.01). Arteries with surrounding matrix support buckled into multiple waves at a higher critical pressure. We concluded that artery buckling under luminal pressure can be predicted by a buckling equation. Elastin degradation weakens the arterial wall and reduces the critical pressure, which thus leads to tortuous vessels. These results shed light on the mechanisms of the development of tortuous vessels due to elastin deficiency.

Effects of Geometric Variations on the Buckling of Arteries

Arteries often demonstrate geometric variations such as elliptic and eccentric cross sections, stenosis, and tapering along the longitudinal axis. Effects of these variations on the mechanical stability of the arterial wall have not been investigated. The objective of this study was to determine the buckling behavior of arteries with elliptic, eccentric, stenotic, and tapered cross sections. The arterial wall was modeled as a homogenous anisotropic nonlinear material. Finite element analysis was used to simulate the buckling process of these arteries under lumen pressure and axial stretch. Our results demonstrated that arteries with an oval cross section buckled in the short axis direction at lower critical pressures compared to circular arteries. Eccentric cross-sections, stenosis, and tapering also decreased the critical pressure. Stenosis led to dramatic pressure variations along the vessel and reduced the buckling pressure. In addition, tapering shifted the buckling deformation profile of the artery towards the distal end. We conclude that geometric variations reduce the critical pressure of arteries and thus make the arteries more prone to mechanical instability than circular cylindrical arteries. These results improve our understanding of the mechanical behavior of arteries.

Pulmonary arterial hypertension: what the large pulmonary arteries tell us

The morphology of the large intrapulmonary arteries (PAs) in pulmonary hypertension (PH) has received limited attention. Dilation, pruning, abrupt tapering, and tortuosity of PAs occur, but whether different patients have distinct PA phenotypes is unknown. Pulmonary arteriograms from 41 pediatric patients with PH were blindly reviewed by four experts who assigned each angiogram one of three designations: straight (S), tortuous (T), or ambiguous (A). Hemodynamic variables and outcomes were compared to the phenotypes. Thirty patients were either T (19) or S (11); 11 were A. The phenotypes were not associated with age. Tortuous patients had higher PA pressure and resistance than the S group and less likely to react to inhaled nitric oxide than S patients (p < 0.05). Clinical outcomes were similar for the three groups. Thus, in PH patients two subtypes of PA morphology can often be discerned, a reflection of variability in PA tortuosity. These morphological subtypes have differing hemodynamic characteristics. The mechanism(s) underlying these differences is unknown, but neither hydrodynamic factors nor duration of PH are fully explanatory. Because PA morphology might reveal information regarding the biology of pathological remodeling, it might prove enlightening to assess the large PA phenotype in future studies of PH.

Determination of the critical buckling pressure of blood vessels using the energy approach

The stability of blood vessels under lumen blood pressure is essential to the maintenance of normal vascular function. Differential buckling equations have been established recently for linear and nonlinear elastic artery models. However, the strain energy in bent buckling and the corresponding energy method have not been investigated for blood vessels under lumen pressure. The purpose of this study was to establish the energy equation for blood vessel buckling under internal pressure. A buckling equation was established to determine the critical pressure based on the potential energy. The critical pressures of blood vessels with small tapering along their axis were estimated using the energy approach. It was demonstrated that the energy approach yields both the same differential equation and critical pressure for cylindrical blood vessel buckling as obtained previously using the adjacent equilibrium approach. Tapering reduced the critical pressure of blood vessels compared to the cylindrical ones. This energy approach provides a useful tool for studying blood vessel buckling and will be useful in dealing with various imperfections of the vessel wall.

A Nonlinear Thin-Wall Model for Vein Buckling

Tortuous or twisted veins are often seen in the retina, cerebrum, and legs (varicose veins) of one-third of the aged population, but the underlying mechanisms are poorly understood. While the collapse of veins under external pressure has been well documented, the bent buckling of long vein segments has not been studied. The objectives of this study were to develop a biomechanical model of vein buckling under internal pressure and to predict the critical pressure. Veins were modeled as thin-walled nonlinear elastic tubes with the Fung exponential strain energy function. Our results demonstrated that veins buckle due to high blood pressure or low axial tension. High axial tension stabilized veins under internal pressure. Our buckling model estimated the critical pressure accurately compared to the experimental measurements. The buckling equation provides a useful tool for studying the development of tortuous veins.

The theoretical foundation for artery buckling under internal pressure

The stability of blood vessels under the lumen blood pressure is essential to the maintenance of normal arterial function. Buckling equations have been established recently for linear and nonlinear elastic artery models with assumed sinusoidal mode shapes. However, the theoretical base for the assumption is not clear. This study established differential equations of artery buckling and then proved that straight arteries bifurcated into sinusoidal mode shapes when buckling occurs. These results set the buckling equation on a solid theoretical foundation.

Mechanical buckling of veins under internal pressure

Venous tortuosity is associated with multiple disease states and is often thought to be a consequence of venous hypertension and chronic venous disease. However, the underlying mechanisms of vein tortuosity are unclear. We hypothesized that increased pressure causes vein buckling that leads to a tortuous appearance. The specific aim of this study was to determine the critical buckling pressure of veins. We determined the buckling pressure of porcine jugular veins and measured the mechanical properties of these veins. Our results showed that the veins buckle when the transmural pressure exceeds a critical pressure that is strongly related to the axial stretch ratio in the veins. The critical pressures of the eight veins tested were 14.2 +/- 5.4 and 26.4 +/- 9.0 mmHg at axial stretch ratio 1.5 and 1.7, respectively. In conclusion, veins buckle into a tortuous shape at high lumen pressures or reduced axial stretch ratios. Our results are useful in understanding the development of venous tortuosity associated with varicose veins, venous valvular insufficiency, diabetic retinopathy, and vein grafts.