A mechanical analysis of conduit arteries accounting for longitudinal residual strains

Variability in structure, morphology, and mechanical properties of the descending thoracic and infrarenal aorta around their circumference

Aortic diseases, such as aneurysms, atherosclerosis, and dissections, demonstrate a preferential development and progression around the aortic circumference, resulting in a highly heterogeneous disease state around the circumference. Differences in the aorta's structural composition and mechanical properties may be partly responsible for this phenomenon. Our goal in this study was to analyze the mechanical and structural properties of the human aorta at its lateral, anterior, posterior, and medial quadrants in two regions prone to circumferentially inhomogeneous diseases, descending Thoracic Aorta (TA) and Infrarenal Aorta (IFR). Human aortas were obtained from 10 donors (64 ± 11 years) and dissected from their loose surrounding tissue. Mechanical properties were determined in all four quadrants of TA and IFR using planar biaxial testing and fitted to three common constitutive models. The structure of tissues was assessed using Movat Pentachrome stained histology slides. We observed that the anterior quadrant exhibited the greatest thickness, followed by the lateral region, in both the TA and IFR. In TA, the posterior wall appeared as the stiffest location in most samples, while in IFR, the anterior wall was the stiffest. We observed a higher glycosaminoglycans content in the lateral and posterior regions of the IFR. We found elastin density to be similar in TA lateral, anterior, and posterior quadrants, while in IFR, the anterior region demonstrated the highest elastin density. Despite significant variations between subjects, this study highlights the distinct morphometrical, mechanical, and structural properties between the quadrants of both TA and IFR.

An inverse fitting strategy to determine the constrained mixture model parameters: application in patient-specific aorta

The Constrained Mixture Model (CMM) is a novel approach to describe arterial wall mechanics, whose formulation is based on a referential physiological state. The CMM considers the arterial wall as a mixture of load-bearing constituents, each of them with characteristic mass fraction, material properties, and deposition stretch levels from its stress-free state to the in-vivo configuration. Although some reports of this model successfully assess its capabilities, they barely explore experimental approaches to model patient-specific scenarios. In this sense, we propose an iterative fitting procedure of numerical-experimental nature to determine material parameters and deposition stretch values. To this end, the model has been implemented in a finite element framework, and it is calibrated using reported experimental data of descending thoracic aorta. The main results obtained from the proposed procedure consist of a set of material parameters for each constituent. Moreover, a relationship between deposition stretches and residual strain measurements (opening angle and axial stretch) has been numerically proved, establishing a strong consistency between the model and experimental data.

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.

Residual strains in ascending thoracic aortic aneurysms: The effect of valve type, layer, and circumferential quadrant

The stress distribution in ascending thoracic aortic aneurysms is determined by the mechanical properties, geometry, loading conditions, and zero-stress state of the aneurysmal aorta. Our objective was to fully characterize the zero-stress state of the aneurysmal aorta in twelve tricuspid aortic valve patients and eight (age/aortic diameter-matched) bicuspid aortic valve patients, for which little data are available. Opening angles and residual stretches were measured for the intact wall and individual layers according to quadrant and were similar in the two patient groups. The intact-wall and medial opening angles were comparable; their circumferential but not their axial ones peaked in the left lateral quadrant, with non-significant regional differences in the other layers. The intima's circumferential opening angles were the highest of all layers (∼300 deg) and the adventitia's the lowest (∼165 deg), with lesser layer differences in the axial opening angles. Upon radially cutting aortic rings, the released circumferential residual stretches were tensile (of large magnitude) externally and compressive (of small magnitude) internally, unlike the axial residual stretches released when cutting intact-wall strips, whose magnitude was small externally and large internally. Nevertheless, large circumferential compressive residual stretches were released in the adventitia upon layer dissection, counteracting the large circumferential tensile stretches of the intact wall externally. Moreover, the large axial tensile residual stretches of the intima counteracted the large axial compressive stretches of the intact wall internally. These layer-specific residual stretches may moderate the in-vivo stress gradients across wall thickness, serving as a protective mechanism against aortic dissection or rupture.

Stretch and stress distributions in the human artery based on two-layer model considering residual stresses

The objective is to know the stress distributions in the arterial walls under residual stresses based on two-layer model. Human common carotid arteries were analysed to show stress distributions at physiological and supraphysiological intraluminal pressures. The analyses for the loaded states were performed with stretch ratios with reference to a Riemannian stress-free configuration which is a 3D non-Euclidean manifold due to the nonzero Riemann curvature tensor. The experimental data obtained by other literature were used for the common carotid arteries to analyse the stretch and stress distributions in the arterial wall although kinematics is different from the literature. The stretches and stresses were calculated for the unloaded state, i.e. the residual stretches and stresses. And those at the axial stretch ratio 1.1 with reference to the unloaded state were calculated at the intraluminal pressures 16, 50, and 100 kPa. The stresses increased from the inner surface to the outer surface at all pressures analysed. These results suggest that in the human arteries the mechanical loads are mainly supported with the adventitia even though the media and intima play an important role to control of physiological functions.

Layer-Specific Residual Deformations and Their Variation Along the Human Aorta

This study described the regional distribution of layer-specific residual deformations in fifteen human aortas collected during autopsy. Circumferentially and axially cut strips of standardized dimensions from the anterior quadrant of nine consecutive aortic levels were photographed to obtain the zero-stress state for the intact wall. The strips were then dissected into layers that were also photographed to obtain their zero-stress state. Changes in layer-specific opening angle, residual stretches, and thickness at each aortic level and direction were determined via image analysis. The circumferential and axial opening angles of the intima were ∼240 deg and ∼30 deg, respectively, throughout the aorta; those of the adventitia were ∼150 deg and -20 deg to 70 deg. The opening angles of the intact wall and media were similar in either direction. The circumferential residual stretches of the intima and the axial residual stretches of the media showed high values in the aortic arch, decreasing in the descending thoracic aorta and increasing toward the iliac artery bifurcation, while the axial residual stretches of the adventitia increased distally. The remaining residual stretches did not vary significantly with aortic level, suggesting an intimal role in determining circumferential, as well as medial and adventitial roles in determining axial residual stretches. We conclude that the tensile residual stretches released in the intima and media upon separation, and the compressive residual stretches released in the adventitia may moderate the inverse transmural stress gradients under physiologic loads, resulting from the >180 deg circumferential opening angle of the intact wall.

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.

Three-Dimensional Contractile Mechanics of Artery Accounting for Curl of Axial Strip Sectioned from Vessel Wall

PURPOSE

It is well known that a sliced ring of arterial wall opens by a radial cut. An axial strip sectioned from arterial wall also curls into an arc. These phenomena imply that there exist residual strains in the circumferential and axial directions. How much do the axial residual strains affect the stress distributions of arterial wall? The aim of the present study is to know stress distributions of arterial wall with the residual strains under the passive and constricted conditions.

METHODS

We analyzed the stress distributions under passive and constricted conditions with considering a Riemannian stress-free configuration. In the analysis, we used strain energy functions to describe the passive and active mechanical properties of artery.

RESULTS

The present study provided distributions of stretch ratio with reference to the stress-free state (Riemannian stress-free configuration) and stress with and without the curl of axial strip of a homogenous cylindrical arterial model under the passive and constricted smooth muscle conditions. The circumferential and axial stresses with activated smooth muscle (noradrenaline 10^-5 M) at the intraluminal pressure 16 kPa and the axial stretch ratio 1.5 with reference to the unloaded vessel decreased by 3.5 and 13.8% at the inner surface with considering the axial residual strain, respectively.

CONCLUSIONS

We have shown that the Riemannian stress-free configuration is appropriate tool to analyze stress distributions of arterial wall under passive and activated conditions with the residual stresses.

Variation of Axial Residual Strains Along the Course and Circumference of Human Aorta Considering Age and Gender

Our understanding of aortic biomechanics is customarily limited by lack of information on the axial residual stretches of the vessel in both humans and experimental animals that would facilitate the identification of its actual zero-stress state. The aim of this study was thus to acquire hitherto unreported quantitative knowledge of axial opening angle and residual stretches in different segments and quadrants of the human aorta according to age and gender. Twenty-three aortas were harvested during autopsy from the aortic root to the iliac bifurcation and were divided into ≥12 segments and 4 quadrants. Morphometric measurements were taken in the excised/curled configuration of rectangular strips considered to be under zero-stress using image-analysis software to study the axial/circumferential variation of axial opening angle, internal/external residual stretch, and thickness of the aortic wall. The measured data demonstrated: (1) an axial opening angle peak at the arch branches, decreasing toward the ascending and to a near-constant value in the descending thoracic aorta, and increasing in the abdominal aorta; (2) the variation of residual stretches resembled that of opening angle, but axial differences in external residual stretch were more prominent; (3) wall thickness showed a progressive diminution along the vessel; (4) the highest opening angle/residual stretches were found in the inner quadrant and the lowest in the outer quadrant; (5) the anterior was the thinnest quadrant throughout the aorta; (6) age caused thickening but greatly reduced axial opening angle/residual stretches, without differences between males and females.

Altered Placental Chorionic Arterial Biomechanical Properties During Intrauterine Growth Restriction

Intrauterine growth restriction (IUGR) is a pregnancy complication due to placental dysfunction that prevents the fetus from obtaining enough oxygen and nutrients, leading to serious mortality and morbidity risks. There is no treatment for IUGR despite having a prevalence of 3% in developed countries, giving rise to an urgency to improve our understanding of the disease. Applying biomechanics investigation on IUGR placental tissues can give important new insights. We performed pressure-diameter mechanical testing of placental chorionic arteries and found that in severe IUGR cases (RI > 90th centile) but not in IUGR cases (RI < 90th centile), vascular distensibility was significantly increased from normal. Constitutive modeling demonstrated that a simplified Fung-type hyperelastic model was able to describe the mechanical properties well, and histology showed that severe IUGR had the lowest collagen to elastin ratio. To demonstrate that the increased distensibility in the severe IUGR group was related to their elevated umbilical resistance and pulsatility indices, we modelled the placental circulation using a Windkessel model, and demonstrated that vascular compliance (and not just vascular resistance) directly affected blood flow pulsatility, suggesting that it is an important parameter for the disease. Our study showed that biomechanics study on placenta could extend our understanding on placenta physiology.

A Novel Approach to Assess the In Situ Versus Ex Vivo Mechanical Behaviors of the Coronary Artery

Ex vivo mechanical testing has provided tremendous insight toward prediction of the in vivo mechanical behavior and local mechanical environment of the arterial wall; however, the role of perivascular support on the local mechanical behavior of arteries is not well understood. Here, we present a novel approach for quantifying the impact of the perivascular support on arterial mechanics using intravascular ultrasound (IVUS) on cadaveric porcine hearts. We performed pressure-diameter tests (n = 5) on the left anterior descending coronary arteries (LADCAs) in situ while embedded in their native perivascular environment using IVUS imaging and after removal of the perivascular support of the artery. We then performed standard cylindrical biaxial testing on these vessels ex vivo and compared the results. Removal of the perivascular support resulted in an upward shift of the pressure-diameter curve. Ex vivo testing, however, showed significantly lower circumferential compliance compared to the in situ configuration. On a second set of arteries, local axial stretch ratios were quantified (n = 5) along the length of the arteries. The average in situ axial stretch ratio was 1.28 ± 0.16; however, local axial stretch ratios showed significant variability, ranging from 1.01 to 1.70. Taken together, the data suggest that both the perivascular loading and the axial tethering have an important role in arterial mechanics. Combining nondestructive testing using IVUS with traditional ex vivo cylindrical biaxial testing yields a more comprehensive assessment of the mechanical behavior of arteries.

Residual deformations in ocular tissues

Residual deformations strongly influence the local biomechanical environment in a number of connective tissues. The sclera is known to be biomechanically important in healthy and diseased eyes, such as in glaucoma. Here, we study the residual deformations of the sclera, as well as the adjacent choroid and retina. Using freshly harvested porcine eyes, we developed two approaches of quantifying residual deformations in the spherically shaped tissues of interest. The first consisted of punching discs from the posterior wall of the eye and quantifying the changes in the area and eccentricity of these samples. The second consisted of cutting a ring from the equatorial sclera and making stress-relieving cuts in it. Measurements of curvature were made before and after the stress-relieving cuts. Using the first approach, we observed a 42% areal contraction of the choroid, but only modest contractions of the sclera and retina. The observed contractions were asymmetric. In the second approach, we observed an opening of the scleral rings (approx. 10% decrease in curvature). We conclude that residual bending deformations are present in the sclera, which we speculate may be due to radially heterogeneous growth and remodelling of the tissue during normal development. Further, residual areal deformations present in the choroid may be due to the network of elastic fibres in this tissue and residual deformations in the constituent vascular bed. Future studies of ocular biomechanics should attempt to include effects of these residual deformations into mechanical models in order to gain a better understanding of the biomechanics of the ocular wall.

Methods for using 3-D ultrasound speckle tracking in biaxial mechanical testing of biological tissue samples

Being multilayered and anisotropic, biological tissues such as cardiac and arterial walls are structurally complex, making the full assessment and understanding of their mechanical behavior challenging. Current standard mechanical testing uses surface markers to track tissue deformations and does not provide deformation data below the surface. In the study described here, we found that combining mechanical testing with 3-D ultrasound speckle tracking could overcome this limitation. Rat myocardium was tested with a biaxial tester and was concurrently scanned with high-frequency ultrasound in three dimensions. The strain energy function was computed from stresses and strains using an iterative non-linear curve-fitting algorithm. Because the strain energy function consists of terms for the base matrix and for embedded fibers, spatially varying fiber orientation was also computed by curve fitting. Using finite-element simulations, we first validated the accuracy of the non-linear curve-fitting algorithm. Next, we compared experimentally measured rat myocardium strain energy function values with those in the literature and found a matching order of magnitude. Finally, we retained samples after the experiments for fiber orientation quantification using histology and found that the results satisfactorily matched those computed in the experiments. We conclude that 3-D ultrasound speckle tracking can be a useful addition to traditional mechanical testing of biological tissues and may provide the benefit of enabling fiber orientation computation.

Noninvasive in vivo determination of residual strains and stresses

Vascular growth and remodeling during embryonic development are associated with blood flow and pressure induced stress distribution, in which residual strains and stresses play a central role. Residual strains are typically measured by performing in vitro tests on the excised vascular tissue. In this paper, we investigated the possibility of estimating residual strains and stresses using physiological pressure-radius data obtained through in vivo noninvasive measurement techniques, such as optical coherence tomography or ultrasound modalities. This analytical approach first tested with in vitro results using experimental data sets for three different arteries such as rabbit carotid artery, rabbit thoracic artery, and human carotid artery based on Fung's pseudostrain energy function and Delfino's exponential strain energy function (SEF). We also examined residual strains and stresses in the human swine iliac artery using the in vivo experimental ultrasound data sets corresponding to the systolic-to-diastolic region only. This allowed computation of the in vivo residual stress information for loading and unloading states separately. Residual strain parameters as well as the material parameters were successfully computed with high accuracy, where the relative errors are introduced in the range of 0-7.5%. Corresponding residual stress distributions demonstrated global errors all in acceptable ranges. A slight discrepancy was observed in the computed reduced axial force. Results of computations performed based on in vivo experimental data obtained from loading and unloading states of the artery exhibited alterations in material properties and residual strain parameters as well. Emerging noninvasive measurement techniques combined with the present analytical approach can be used to estimate residual strains and stresses in vascular tissues as a precursor for growth estimates. This approach is also validated with a finite element model of a general two-layered artery, where the material remodeling states and residual strain generation are investigated.

Residual shear deformations in the coronary artery

Quantifying arterial residual deformations is critical for understanding the stresses and strains within the arterial wall during physiological and pathophysiological conditions. This study presents novel findings on residual shear deformations in the left anterior descending coronary artery. Residual shear deformations are most evident when thin, long axial strips are cut from the artery. These strips deform into helical configurations when placed in isotonic solution. A residual shear angle is introduced as a parameter to quantify the residual shear deformations. Furthermore, a stress analysis is performed to study the effects of residual shear deformations on the intramural shear stress distribution of an artery subjected to pressure, axial stretch, and torsion using numerical simulation. The results from the stress analyses suggest that residual shear deformations can significantly modulate the intramural shear stress across the arterial wall.

Non-Euclidean stress-free configuration of arteries accounting for curl of axial strips sectioned from vessels

It is well known that arteries are subject to residual stress. In earlier studies, the residual stress in the arterial ring relieved by a radial cut was considered in stress analysis. However, it has been found that axial strips sectioned from arteries also curled into arcs, showing that the axial residual stresses were relieved from the arterial walls. The combined relief of circumferential and axial residual stresses must be considered to accurately analyze stress and strain distributions under physiological loading conditions. In the present study, a mathematical model of a stress-free configuration of artery was proposed using Riemannian geometry. Stress analysis for arterial walls under unloaded and physiologically loaded conditions was performed using exponential strain energy functions for porcine and human common carotid arteries. In the porcine artery, the circumferential stress distribution under physiological loading became uniform compared with that without axial residual strain, whereas a gradient of axial stress distribution increased through the wall thickness. This behavior showed almost the same pattern that was observed in a recent study in which approximate analysis accounting for circumferential and axial residual strains was performed, whereas the circumferential and axial stresses increased from the inner surface to the outer surface under a physiological condition in the human common carotid artery of a two-layer model based on data of other recent studies. In both analyses, Riemannian geometry was appropriate to define the stress-free configurations of the arterial walls with both circumferential and axial residual strains.

Comparisons of planar and tubular biaxial tensile testing protocols of the same porcine coronary arteries

To identify the orthotropic biomechanical behavior of arteries, researchers typically perform stretch-pressure-inflation tests on tube-form arteries or planar biaxial testing of splayed sections. We examined variations in finite element simulations (FESs) driven from planar or tubular testing of the same coronary arteries to determine what differences exist when picking one testing technique vs. another. Arteries were tested in tube-form first, then tested in planar-form, and fit to a Fung-type strain energy density function. Afterwards, arteries were modeled via finite element analysis looking at stress and displacement behavior in different scenarios (e.g., tube FESs with tube- or planar-driven constitutive models). When performing FESs of tube inflation from a planar-driven constitutive model, pressure-diameter results had an error of 12.3% compared to pressure-inflation data. Circumferential stresses were different between tube- and planar-driven pressure-inflation models by 50.4% with the planar-driven model having higher stresses. This reduced to 3.9% when rolling the sample to a tube first with planar-driven properties, then inflating with tubular-driven properties. Microstructure showed primarily axial orientation in the tubular and opening-angle configurations. There was a shift towards the circumferential direction upon flattening of 8.0°. There was also noticeable collagen uncrimping in the flattened tissue.

The fiber orientation in the coronary arterial wall at physiological loading evaluated with a two-fiber constitutive model

A patient-specific mechanical description of the coronary arterial wall is indispensable for individualized diagnosis and treatment of coronary artery disease. A way to determine the artery's mechanical properties is to fit the parameters of a constitutive model to patient-specific experimental data. Clinical data, however, essentially lack information about the stress-free geometry of an artery, which is necessary for constitutive modeling. In previous research, it has been shown that a way to circumvent this problem is to impose extra modeling constraints on the parameter estimation procedure. In this study, we propose a new modeling constraint concerning the in-situ fiber orientation (β (phys)). β (phys), which is a major contributor to the arterial stress-strain behavior, was determined for porcine and human coronary arteries using a mixed numerical-experimental method. The in-situ situation was mimicked using in-vitro experiments at a physiological axial pre-stretch, in which pressure-radius and pressure-axial force were measured. A single-layered, hyperelastic, thick-walled, two-fiber material model was accurately fitted to the experimental data, enabling the computation of stress, strain, and fiber orientation. β (phys) was found to be almost equal for all vessels measured (36.4 ± 0.3)°, which theoretically can be explained using netting analysis. In further research, this finding can be used as an extra modeling constraint in parameter estimation from clinical data.