Effects of Geometric Variations on the Buckling of Arteries

Morphology of middle cerebral artery using computed tomography angiographic study in a tertiary care hospital

Increased tortuosity of vessel is associated with high incidence of plaque formation leading to atherosclerosis. Surgical procedures are done after analyzing morphology of middle cerebral artery (MCA). However, literature describing MCA morphology using computed tomography angiography (CTA) is limited, so this study was planned to determine its incidence in Indian population. Datasets of CTA from 289 patients (180 males and 109 females), average age: 49.29±16.16 years (range: 11 to 85 years), from a tertiary care hospital were systematically reviewed for morphology of MCA. Cases involving aneurysms and infarcts were excluded. Four shapes of MCA were recognized: straight, U, inverted U, and S-shaped. MCA was straight in 44% (254/578), U-shaped in 37% (215/578), S shaped in 15% (89/578) and inverted U-shaped in 3% (20/578) cases. In males, MCA was straight in 46% (166/360), U-shaped in 37% (134/360), S-shaped in 16% (58/360) and inverted U-shaped in 4% (14/360) cases. In females, MCA was straight in 42% cases (92/218), U-shaped in 37% (81/218), S-shaped in 17% (36/218) and inverted U-shaped in 4% (9/218). On comparing shape with various age groups using chi square test, U shaped (P≤0.001) and S-shaped (P=0.003) MCA were found to be statistically significant. The incidence of straight shape was higher in advanced age group (>60 years). Knowledge of MCA shape will be useful for clinicians and surgeons in successful endovascular recanalization. Also, this data would help surgeons during neurointerventional procedures.

The effects of axial twisting and material non-symmetry on arterial bent buckling

Artery buckling occurs due to hypertensive lumen pressure or reduced axial tension and other pathological conditions. Since arteries in vivo often experience axial twisting and the collagen fiber alignment in the arterial wall may become nonsymmetric, it is imperative to know how axial twisting and nonsymmetric collagen alignment would affect the buckling behavior of arteries. To this end, the objective of this study was to determine the effect of axial twisting and nonsymmetric collagen fiber distribution on the critical pressure of arterial bent buckling. The buckling model analysis was generalized to incorporate an axial twist angle and nonsymmetric fiber alignment. The effect of axial twisting on the critical pressure was simulated and experimentally tested in a group of porcine carotid arteries. Our results showed that axial twisting tends to reduce the critical pressure depending on the axial stretch ratio and twist angle. In addition, nonsymmetric fiber alignment reduces the critical pressure. Experimental results confirmed that a twist angle of 90° reduces the critical pressure significantly (p < 0.05). It was concluded that axial twisting and non-axisymmetric collagen fibers distribution could make arteries prone to bent buckling. These results enrich our understanding of artery buckling and vessel tortuosity. The model analysis and results could also be applicable to other fiber reinforced tubes under lumen pressure and axial twisting.

TORSIONAL BEHAVIOR OF STENTS: THE ROLE OF LINKER AND STENT TAPERING INVESTIGATED WITH NUMERICAL SIMULATION

The torsional performance is a major mechanical property of stent. A stent with good torsional performance is easy to deform along blood vessels without damaging the vascular wall to avoid in-stent restenosis (ISR). Therefore, this study aimed to study the effect of stent parameters on torsional performance. The effect of stent parameters on torsional performance was studied via finite element method (FEM). The twist metric (TM) and stress distribution of various stents were compared. The TM values of stents with I-, S-, M-, C-, and V-shaped linkers were 0.0190, 0.0191, 0.0184, 0.0141, and 0.0201[Formula: see text][Formula: see text], respectively. In addition, the TM value of the stent increased by 35.85 times when the number of linkers was increased from 2 to 8 and the stent was twisted at the same angular displacement in clockwise direction. The TM value of the stent with 1.13∘ tapering was 0.010 [Formula: see text], which was lower by 47.64% compared with that of cylindrical stent. Compared with the shape of the linker, the number of linkers had a more remarkable effect on torsional performance. Torsional performance was observably enhanced with the decrease in the number of linkers. Among the five stents with different linker shapes, the torsional performance of the stent with C-shaped linker was the best. Besides, the torsional performance of the tapered stent was better than that of the cylindrical stent. Moreover, the torsional performance increased by increasing the stent tapering. This work might provide insights into better stent design and clinical decisions.

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.

Stability Analysis of Arteries Under Torsion

Vascular tortuosity may impede blood flow, occlude the lumen, and ultimately lead to ischemia or even infarction. Mechanical loads like blood pressure, axial force, and also torsion are key factors participating in this complex mechanobiological process. The available studies on arterial torsion instability followed computational or experimental approaches, yet single available theoretical study had modeled the artery as isotropic linear elastic. This paper aim is to validate a theoretical model of arterial torsion instability against experimental data. The artery is modeled as a single-layered, nonlinear, hyperelastic, anisotropic solid, with parameters calibrated from experiment. Linear bifurcation analysis is then performed to predict experimentally measured stability margins. Uncertainties in geometrical parameters and in measured mechanical response were considered. Also, the type of rate (incremental) boundary conditions (RBCs) impact on the results was examined (e.g., dead load, fluid pressure). The predicted critical torque and twist angle followed the experimentally measured trends. The closest prediction errors in the critical torque and twist rate were 22% and 67%, respectively. Using the different RBCs incurred differences of up to 50% difference within the model predictions. The present results suggest that the model may require further improvements. However, it offers an approach that can be used to predict allowable twist levels in surgical procedures (like anastomosis and grafting) and in the design of stents for arteries subjected to high torsion levels (like the femoropopliteal arteries). It may also be instructive in understanding biomechanical processes like arterial tortuosity, kinking, and coiling.

Computational simulations of the helical buckling behavior of blood vessels

Tortuous vessels are often observed in vivo and could hinder or even disrupt blood flow to distal organs. Besides genetic and biological factors, the in vivo mechanical loading seems to play a role in the formation of tortuous vessels, but the mechanism for formation of helical vessel shape remains unclear. Accordingly, the aim of this study was to investigate the biomechanical loads that trigger the occurrence of helical buckling in blood vessels using finite element analysis. Porcine carotid arteries were modeled as thick-walled cylindrical tubes using generalized Fung and Holzapfel-Gasser-Ogden constitutive models. Physiological loadings, including axial tension, lumen pressure, and axial torque, were applied. Simulations of various geometric dimensions, different constitutive models and at various levels of axial stretch ratios, lumen pressures, and twist angles were performed to identify the mechanical factors that determine the helical stability. Our results demonstrated that axial torsion can cause wringing (twist buckling) that leads to kinking or helical coiling and even looping and winding. The specific buckling patterns depend on the combination of lumen pressure, axial torque, axial tension, and the dimensions of the vessels. This study elucidates the mechanism of how blood vessels buckle under various mechanical loads and how complex mechanical loads yield helical buckling.

Effect of Axial Stretch on Lumen Collapse of Arteries

The stability of the arteries under in vivo pressure and axial tension loads is essential to normal arterial function, and lumen collapse due to buckling can hinder the blood flow. The objective of this study was to develop the lumen buckling equation for nonlinear anisotropic thick-walled arteries to determine the effect of axial tension. The theoretical equation was developed using exponential Fung strain function, and the effects of axial tension and residual stress on the critical buckling pressure were illustrated for porcine coronary arteries. The buckling behavior was also simulated using finite-element analysis. Our results demonstrated that lumen collapse of arteries could occur when the transmural pressure is negative and exceeded a critical value. This value depends upon the axial stretch ratio and material properties of the arterial wall. Axial tensions show a biphasic effect on the critical buckling pressure. The lumen aspect ratio of arteries increases nonlinearly with increasing external pressure beyond the critical value as the lumen collapses. These results enhance our understanding of artery lumen collapse behavior.

Fluid-structure interaction modeling of aneurysmal arteries under steady-state and pulsatile blood flow: a stability analysis

Tortuous aneurysmal arteries are often associated with a higher risk of rupture but the mechanism remains unclear. The goal of this study was to analyze the buckling and post-buckling behaviors of aneurysmal arteries under pulsatile flow. To accomplish this goal, we analyzed the buckling behavior of model carotid and abdominal aorta with aneurysms by utilizing fluid-structure interaction (FSI) method with realistic waveforms boundary conditions. FSI simulations were done under steady-state and pulsatile flow for normal (1.5) and reduced (1.3) axial stretch ratios to investigate the influence of aneurysm, pulsatile lumen pressure and axial tension on stability. Our results indicated that aneurysmal artery buckled at the critical buckling pressure and its deflection nonlinearly increased with increasing lumen pressure. Buckling elevates the peak stress (up to 118%). The maximum aneurysm wall stress at pulsatile FSI flow was (29%) higher than under static pressure at the peak lumen pressure of 130 mmHg. Buckling results show an increase in lumen shear stress at the inner side of the maximum deflection. Vortex flow was dramatically enlarged with increasing lumen pressure and artery diameter. Aneurysmal arteries are more susceptible than normal arteries to mechanical instability which causes high stresses in the aneurysm wall that could lead to aneurysm rupture.

Effect of different expansion strategies on coronary stent deployment in a tapered artery

BACKGROUND

Vascular stenting has been widely used to treat vessel stenosis. However, long-term successes of the procedure are often compromised by late stent thrombosis (ST) and in-stent restenosis (ISR), especially in tapered arteries.

OBJECTIVE

The aim of this study was to choose a reasonable expansion strategy for tapered arteries.

METHODS

A balloon-expandable coronary stent deployment in a tapered vessel was numerically studied fol-lowing three strategies: (i) selecting the proximal diameter of the tapered vessel as the reference diameter to expand the stent, (ii) selecting the middle diameter of the tapered vessel as the reference diameter to expand the stent, and (iii) selecting the distal diameter of the tapered vessel as the reference diameter to expand the stent.

RESULTS

Computational results showed that the first expansion strategy resulted in the maximum vessel stress and the best stent apposition, while the third strategy resulted in the minimal vessel stress and the worst stent appo-sition. Meanwhile, the second expansion strategy achieved a trade-off between the first and third strategies, leading to acceptable vessel stress and stent apposition.

CONCLUSIONS

The second expansion strategy is the most reasonable choice for tapered vessels, and it should be considered when implanting a stent.

ASSESSMENT OF CORONARY STENT DEPLOYMENT IN TAPERED ARTERIES: IMPACT OF ARTERIAL TAPERING

Coronary stents are used to prop open blocked arteries in order to restore normal blood flow. A major setback in this technology is in-stent restenosis (ISR), which gravely limits the clinical success of stents, especially in tapered vessels. The present study used the finite element method to study the effects of arterial tapering on the biomechanical behavior of both stents and vessels during stent deployment inside tapered arteries. The effect of arterial tapering was demonstrated by a combination of corresponding tapered arteries with various tapering angles, including a straight artery case for comparison. Results indicated that an increase of vessel tapering led to an increase in stent radial recoil, stent tapering following expansion, and von Mises stresses on vessels. However, an increase of vessel tapering also led to a decrease in stent foreshortening. The analysis provides suggestions for clinical application in tapered vessels. The finite element method evaluated mechanical stent behavior in tapered vessels, and can help designers to optimize the design of stents for tapered vessels.

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.

Artery buckling affects the mechanical stress in atherosclerotic plaques

Background

Tortuous arteries are often seen in patients with hypertension and atherosclerosis. While the mechanical stress in atherosclerotic plaque under lumen pressure has been studied extensively, the mechanical stability of atherosclerotic arteries and subsequent effect on the plaque stress remain unknown. To this end, we investigated the buckling and post-buckling behavior of model stenotic coronary arteries with symmetric and asymmetric plaque.

Methods

Buckling analysis for a model coronary artery with symmetric and asymmetric plaque was conducted using finite element analysis based on the dimensions and nonlinear anisotropic materials properties reported in the literature.

Results

Artery with asymmetric plaque had lower critical buckling pressure compared to the artery with symmetric plaque and control artery. Buckling increased the peak stress in the plaque and led to the development of a high stress concentration in artery with asymmetric plaque. Stiffer calcified tissue and severe stenosis increased the critical buckling pressure of the artery with asymmetric plaque.

Conclusions

Arteries with atherosclerotic plaques are prone to mechanical buckling which leads to a high stress concentration in the plaques that can possibly make the plaques prone to rupture.

BALLOON-EXPANDABLE STENTS EXPANSION IN TAPERED VESSELS AND THEIR INTERACTIONS

In-stent restenosis (ISR) after stent implantation, especially in tapered vessels, remains an obstacle in the long-term benefits of stenting. In the present study, a finite element method (FEM) was employed to investigate the expansion process of balloon-expandable stents in tapered vessels (the TV model) and their interactions. For comparison, a numerical model of the same stent deployment in a straight vessel was also investigated. Results showed that in the TV model, the peak tissue stresses took place at the distal end of the tapered vessel. The node displacements of the stent's proximal and distal ends remained consistent before the stent contacted the tapered vessel, while the proximal end was larger than the distal end after the stent contacted the tapered vessel. The regions of maximum stresses in the stent after expansion were concentrated in the corners of the diamond cells of the stent's proximal end. The investigation provided some interpretations of the clinical observations in tapered vessels and also provided stent design proposals for tapered vessels. The FEM quantified the mechanical properties of stents in tapered vessels, and can help clinicians select appropriate stents, assist designers in pretests and create new stents made especially for tapered vessels.

Mechanical instability of normal and aneurysmal arteries

Tortuous arteries associated with aneurysms have been observed in aged patients with atherosclerosis and hypertension. However, the underlying mechanism is poorly understood. The objective of this study was to determine the effect of aneurysms on arterial buckling instability and the effect of buckling on aneurysm wall stress. We investigated the mechanical buckling and post-buckling behavior of normal and aneurysmal carotid arteries and aorta's using computational simulations and experimental measurements to elucidate the interrelationship between artery buckling and aneurysms. Buckling tests were done in porcine carotid arteries with small aneurysms created using elastase treatment. Parametric studies were done for model aneurysms with orthotropic nonlinear elastic walls using finite element simulations. Our results demonstrated that arteries buckled at a critical buckling pressure and the post-buckling deflection increased nonlinearly with increasing pressure. The presence of an aneurysm can reduce the critical buckling pressure of arteries, although the effect depends on the aneurysm's dimensions. Buckled aneurysms demonstrated a higher peak wall stress compared to unbuckled aneurysms under the same lumen pressure. We conclude that aneurysmal arteries are vulnerable to mechanical buckling and mechanical buckling could lead to high stresses in the aneurysm wall. Buckling could be a possible mechanism for the development of tortuous aneurysmal arteries such as in the Loeys-Dietz syndrome.

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.

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.

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.

Twist buckling behavior of arteries

Arteries are often subjected to torsion due to body movement and surgical procedures. While it is essential that arteries remain stable and patent under twisting loads, the stability of arteries under torsion is poorly understood. The goal of this work was to experimentally investigate the buckling behavior of arteries under torsion and to determine the critical buckling torque, the critical buckling twist angle, and the buckling shape. Porcine common carotid arteries were slowly twisted in vitro until buckling occurred while subjected to a constant axial stretch ratio (1.1, 1.3, 1.5 (in vivo level) and 1.7) and lumen pressure (20, 40, 70 and 100 mmHg). Upon buckling, the arteries snapped to form a kink. For a group of six arteries, the axial stretch ratio significantly affected the critical buckling torque ([Formula: see text]) and the critical buckling twist angle ([Formula: see text]). Lumen pressure also significantly affected the critical buckling torque ([Formula: see text]) but had no significant effect on the critical twist angle ([Formula: see text]). Convex material constants for a Fung strain energy function were determined and fit well with the axial force, lumen pressure, and torque data measured pre-buckling. The material constants are valid for axial stretch ratios, lumen pressures, and rotation angles of 1.3-1.5, 20-100 mmHg, and 0-270[Formula: see text], respectively. The current study elucidates the buckling behavior of arteries under torsion and provides new insight into mechanical instability of blood vessels.

Cerebrovascular geometry in the anterior circulation: an analysis of diameter, length and the vessel taper

BACKGROUND

A study was undertaken to determine the typical length, diameter and taper of vessels in the anterior cerebral circulation.

METHODS

The sample size was calculated at 100 patients based on similar measurements in the literature and divided into cohorts based on gender and side. These patients were consecutively collected from a population that had undergone CT angiography and did not have any vascular abnormality. The arterial diameter was measured at the proximal cavernous internal carotid artery (ICA), the ICA terminus, the middle cerebral artery (MCA) origin and an M2 origin. The length between these endpoints was calculated along the center line. The vessel taper was calculated for the ICA as the change in caliber per unit length.

RESULTS

The mean length of the ICA from the proximal cavernous segment to the ICA terminus was 33.1 ± 6.1 mm. The mean diameter at the cavernous ICA and the ICA terminus was 5 ± 0.6 mm and 3.6 ± 0.4 mm, respectively. The mean ICA taper was 0.04 ± 0.02 mm/1 mm. For the MCA, the diameter at the MCA and M2 origins measured 3.1 ± 0.4 mm and 2.4 ± 0.4 mm, respectively. The mean MCA length was 22.5 ± 8.1 mm. There was no significant difference based on gender or between right and left sides. Patients aged >60 years had longer ICAs (p=0.02), larger cavernous ICA (p=0.003), ICA terminus (p<0.0001) and MCA origin (p=0.01) diameters than those aged 40-60 years. The ICA vessel taper did not change with age.

CONCLUSION

ICA and MCA vessel size did not change based on gender or side. Older patients had more redundant vessels based on diameter and length. The ICA has a gentle taper from its proximal cavernous segment to the ICA terminus. This information can be important in planning interventions or designing endovascular devices.

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.

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.

[Patient education in chronic polyarthritis. 3. Intermediate results of a prospective, controlled study of the effectiveness and side effects of patient seminars for polyarthritis patients]

Efficacy and possible negative side effects of a patient education program for rheumatoid arthritis were evaluated in a controlled, prospective study over 3 months. 34 outpatients were educated over a total of 8 h in three groups within a patient-centred design. Before the program the knowledge of the disease depended only on the formal grade of education but not on disease-related variables such as disease duration or disability. Probably due to its individualizing method, the program improved the knowledge of all patients to the same extent, regardless of their intellectual and social prerequisites. The increased cognitive knowledge did not result in negative side effects like increased pain or depression. The pain score remained unchanged. Depression decreased after the education. The group sessions made us suppose that the participants may have represented a selected group of active, psychologically stable patients, who cope well with rheumatoid arthritis. In contrast, we felt that non-participation was the response of the inactive, fatalistic patients with rheumatoid arthritis, who live in social isolation and especially need our care. Therefore, future efforts must particularly focus on the problem of motivation and on an increase in the rate of participation.