Effect of residual stress and heterogeneity on circumferential stress in the arterial wall

Quantifying the stress distribution through the arterial wall is essential to studies of arterial growth and disease. Previous studies have shown that both residual stress, as measured by opening angle, and differing material properties for the media-intima and the adventitial layers affect the transmural circumferential stress (sigma theta) distribution. Because a lack of comprehensive data on a single species and artery has led to combinations from multiple sources, this study determined the sensitivity of sigma theta to published variations in both opening angle and layer thickness data. We fit material properties to previously published experimental data for pressure-diameter relations and opening angles of rabbit carotid artery, and predicted sigma theta through the arterial wall at physiologic conditions. Using a one-layer model, the ratio of sigma theta at the internal wall to the mean sigma theta decreased from 2.34 to 0.98 as the opening angle increased from 60 to 130 deg. In a two-layer model using a 95 deg opening angle, mean sigma theta in the adventitia increased (112 percent for 25 percent adventitia) and mean sigma theta in the media decreased (47 percent for 25 percent adventitia). These results suggest that both residual stress and wall layers have important effects on transmural stress distribution. Thus, experimental measurements of loading curves, opening angles, and wall composition from the same species and artery are needed to accurately predict the transmural stress distribution in the arterial wall.     

Impact of transmural heterogeneities on arterial adaptation

Recent experimental and computational studies have shown that transmurally heterogeneous material properties through the arterial wall are critical to understanding the heterogeneous expressions of constituent degrading molecules. Given that expression of such molecules is thought to be intimately linked to local magnitudes of stress, modelling the transmural stress distribution is critical to understanding arterial adaption during disease. The aim of this study was to develop an arterial growth and remodelling framework that can incorporate both transmurally heterogeneous constituent distributions and residual stresses, into a 3-D finite element model. As an illustrative example, we model the development of a fusiform aneurysm and investigate the effects of elastinous and collagenous heterogeneities on the stress distribution during evolution. It is observed that the adaptive processes of growth and remodelling exhibit transmural variations. For physiological heterogeneous constituent distributions, a stress peak appears in the media towards the intima, and a stress plateau occurs towards the adventitia. These features can be primarily attributed to the underlying heterogeneity of elastinous constituents. During arterial adaption, the collagen strain is regulated to remain in its homoeostatic level; consequently, the partial stress of collagen has less influence on the total stress than the elastin. However, following significant elastin degradation, collagen plays the dominant role for the transmural stress profile and a marked stress peak occurs towards the adventitia. We conclude that to improve our understanding of the arterial adaption and the aetiology of arterial disease, there is a need to: quantify transmural constituent distributions during histopathological examinations, understand and model the role of the evolving transmural stress distribution.     

Dependence of local left ventricular wall mechanics on myocardial fiber orientation: a model study.

The dependence of local left ventricular (LV) mechanics on myocardial muscle fiber orientation was investigated using a finite element model. In the model we have considered anisotropy of the active and passive components of myocardial tissue, dependence of active stress on time, strain and strain rate, activation sequence of the LV wall and aortic afterload. Muscle fiber orientation in the LV wall is quantified by the helix fiber angle, defined as the angle between the muscle fiber direction and the local circumferential direction. In a first simulation, a transmural variation of the helix fiber angle from +60 degrees at the endocardium through 0 degrees in the midwall layers to -60 degrees at the epicardium was assumed. In this simulation, at the equatorial level maximum active muscle fiber stress was found to vary from about 110 kPa in the subendocardial layers through about 30 kPa in the midwall layers to about 40 kPa in the subepicardial layers. Next, in a series of simulations, muscle fiber orientation was iteratively adapted until the spatial distribution of active muscle fiber stress was fairly homogeneous. Using a transmural course of the helix fiber angle of +60 degrees at the endocardium, +15 degrees in the midwall layers and -60 degrees at the epicardium, at the equatorial level maximum active muscle fiber stress varied from 52 kPa to 55 kPa, indicating a remarkable reduction of the stress range. Moreover, the change of muscle fiber strain with time was more similar in different parts of the LV wall than in the first simulation. It is concluded that (1) the distribution of active muscle fiber stress and muscle fiber strain across the LV wall is very sensitive to the transmural distribution of the helix fiber angle and (2) a physiological transmural distribution of the helix fiber angle can be found, at which active muscle fiber stress and muscle fiber strain are distributed approximately homogeneously across the LV wall.     

Branching exponent heterogeneity and wall shear stress distribution in vascular trees

A bifurcating arterial system with Poiseuille flow can function at minimum cost and with uniform wall shear stress if the branching exponent (z) = 3 [where z is defined by (D(1))(z) = (D(2))(z) + (D(3))(z); D(1) is the parent vessel diameter and D(2) and D(3) are the two daughter vessel diameters at a bifurcation]. Because wall shear stress is a physiologically transducible force, shear stress-dependent control over vessel diameter would appear to provide a means for preserving this optimal structure through maintenance of uniform shear stress. A mean z of 3 has been considered confirmation of such a control mechanism. The objective of the present study was to evaluate the consequences of a heterogeneous distribution of z values about the mean with regard to this uniform shear stress hypothesis. Simulations were carried out on model structures otherwise conforming to the criteria consistent with uniform shear stress when z = 3 but with varying distributions of z. The result was that when there was significant heterogeneity in z approaching that found in a real arterial tree, the coefficient of variation in shear stress was comparable to the coefficient of variation in z and nearly independent of the mean value of z. A systematic increase in mean shear stress with decreasing vessel diameter was one component of the variation in shear stress even when the mean z = 3. The conclusion is that the influence of shear stress in determining vessel diameters is not, per se, manifested in a mean value of z. In a vascular tree having a heterogeneous distribution in z values, a particular mean value of z (e.g., z = 3) apparently has little bearing on the uniform shear stress hypothesis.     

Stress-modulated growth, residual stress, and vascular heterogeneity.

A simple phenomenological model is used to study interrelations between material properties, growth-induced residual stresses, and opening angles in arteries. The artery is assumed to be a thick-walled tube composed of an orthotropic pseudoelastic material. In addition, the normal mature vessel is assumed to have uniform circumferential wall stress, which is achieved here via a mechanical growth law. Residual stresses are computed for three configurations: the unloaded intact artery, the artery after a single transmural cut, and the inner and outer rings of the artery created by combined radial and circumferential cuts. The results show that the magnitudes of the opening angles depend strongly on the heterogeneity of the material properties of the vessel wall and that multiple radial and circumferential cuts may be needed to relieve all residual stress. In addition, comparing computed opening angles with published experimental data for the bovine carotid artery suggests that the material properties change continuously across the vessel wall and that stress, not strain, correlates well with growth in arteries.     

Effects of myocardial constraint on the passive mechanical behaviors of the coronary vessel wall

The large epicardial coronary arteries and veins span the surface of the heart and gradually penetrate into the myocardium. It has recently been shown that remodeling of the epicardial veins in response to pressure overload strongly depends on the degree of myocardial support. The nontethered regions of the vessel wall show significant intimal hyperplasia compared with the tethered regions. Our hypothesis is that such circumferentially nonuniform structural adaptation in the vessel wall is due to nonuniform wall stress and strain. Transmural stress and strain are significantly influenced by the support of the surrounding myocardial tissue, which significantly limits distension of the vessel. In this finite-element study, we modeled the nonuniform support by embedding the left anterior descending artery into the myocardium to different depths and analyzed deformation and strain in the vessel wall. Circumferential wall strain was much higher in the untethered than tethered region at physiological pressure. On the basis of the hypothesis that elevated wall strain is the stimulus for remodeling, the simulation results suggest that large epicardial coronary vessels have a greater tendency to become thicker in the absence of myocardial constraint. This study provides a mechanical basis for understanding the local growth and remodeling of vessels subjected to various degrees of surrounding tissue.     

Uniform strain hypothesis and thin-walled theory in arterial mechanics

The authors (1987), and Chuong and Fung (1986) have shown that the strain and stress distributions in the arterial wall should be more uniform than those calculated on the basis of the conventional assumption that there is no stress over the cross-section of the arterial wall when all external force is removed (zero initial stress hypothesis). Instead of this assumption, the authors have proposed a new hypothesis that the circumferential strain uniformly distributes through the wall thickness at a physiologically normal loading, and named it 'uniform strain hypothesis'. Their results suggest the validity of the thin-walled theory in the vascular mechanics. This paper shows that if the uniform strain hypothesis is applied, the thin-walled theory can be used to accurately determine the constants included in the strain energy density function which describes the mechanical properties of the arterial wall. There were, however, significant differences in the values of the constants between the thin-walled theory and the thick-walled theory if assuming the conventional zero initial stress hypothesis.     

Strain energy density function and uniform strain hypothesis for arterial mechanics

Stress distribution through the wall thickness of the canine carotid artery was analyzed on the basis of the uniform strain hypothesis in which the wall circumferential strain was assumed to be constant over the wall cross-section under physiological loading condition. A newly proposed logarithmic type of strain energy density function was used to describe the wall properties. In contrast with other studies, this hypothesis gave almost uniform distribution of wall stresses under the physiological condition and non-zero residual stresses when all external forces were removed.     

Stress gradients in the walls of large arteries

Elastic behavior of vascular wall, assuming the vessels to be ‘thick-walled’ and utilizing finite deformation theory, was investigated. It was found that canine carotid arterial wall is neither isotropic nor transversely isotropic. Previously, stress-strain relations were obtained for carotid arteries on the basis of membrane theory (Doyle and Dobrin, 1971). Since strain gradients across the wall are fairly steep, the applicability of such expressions, for pointwise evaluation of stress, required examination. The study indicated that these relationships between mean circumferential stress and mean extension ratio in the circumferential direction could be used to relate the specific circumferential stress value to the specific extension ratio at any designated point within the wall. From this analysis it was possible to evaluate circumferential and radial wall stresses. Both of these stresses are maximal at the inner surface of the intima. At this point the radial stress is equal to the transmural pressure and is compressive, while the circumferential stress is tensile and is 1·5 to 2 times the value of the mean stress, i.e. the product of transmural pressure and the ratio of internal radius-to-wall thickness. Both stresses are lowest at the outer edge of the adventitia. These stress distributions were considered with respect to the spacing of the elastic lamellae and the absence of discernible vasa vasora in the inner third of the wall.     

Transmural flow bioreactor for vascular tissue engineering

Nutrient transport limitation remains a fundamental issue for in vitro culture of engineered tissues. In this study, perfusion bioreactor configurations were investigated to provide uniform delivery of oxygen to media equivalents (MEs) being developed as the basis for tissue‐engineered arteries. Bioreactor configurations were developed to evaluate oxygen delivery associated with complete transmural flow (through the wall of the ME), complete axial flow (through the lumen), and a combination of these flows. In addition, transport models of the different flow configurations were analyzed to determine the most uniform oxygen profile throughout the tissue, incorporating direct measurements of tissue hydraulic conductivity, cellular O₂ consumption kinetics, and cell density along with ME physical dimensions. Model results indicate that dissolved oxygen (DO) uniformity is improved when a combination of transmural and axial flow is implemented; however, detrimental effects could occur due to lumenal pressure exceeding the burst pressure or damaging interstitial shear stress imparted by excessive transmural flow rates or decreasing hydraulic conductivity due to ME compaction. The model was verified by comparing predicted with measured outlet DO concentrations. Based on these results, the combination of a controlled transmural flow coupled with axial flow presents an attractive means to increase the transport of nutrients to cells within the cultured tissue to improve growth (increased cell and extracellular matrix concentrations) as well as uniformity. Biotechnol. Bioeng. 2009; 104: 1197–1206. © 2009 Wiley Periodicals, Inc.     

Influence of left-ventricular shape on passive filling properties and end-diastolic fiber stress and strain

Passive filling is a major determinant for the pump performance of the left ventricle and is determined by the filling pressure and the ventricular compliance. In the quantification of the passive mechanical behaviour of the left ventricle and its compliance, focus has been mainly on fiber orientation and constitutive parameters. Although it has been shown that the left-ventricular shape plays an important role in cardiac (patho-)physiology, the dependency on left-ventricular shape has never been studied in detail. Therefore, we have quantified the influence of left-ventricular shape on the overall compliance and the intramyocardial distribution of passive fiber stress and strain during the passive filling period. Hereto, fiber stress and strain were calculated in a finite element analysis of passive inflation of left ventricles with different shapes, ranging from an elongated ellipsoid to a sphere, but keeping the initial cavity volume constant. For each shape, the wall volume was varied to obtain ventricles with different wall thickness. The passive myocardium was described by an incompressible hyperelastic material law with transverse isotropic symmetry along the muscle fiber directions. A realistic transmural distribution in fiber orientation was assumed. We found that compliance was not altered substantially, but the transmural distribution of both passive fiber stress and strain was highly dependent on regional wall curvature and thickness. A low curvature wall was characterized by a maximum in the transmural fiber stress and strain in the mid-wall region, while a steep subendocardial transmural gradient was present in a high curvature wall. The transmural fiber stress and strain gradients in a low and high curvature wall were, respectively, flattened and steepened by an increase in wall thickness.     

Imaging coronary artery microstructure using second-harmonic and two-photon fluorescence microscopy

The microstructural basis for the mechanical properties of blood vessels has not been directly determined because of the lack of a nondestructive method that yields a three-dimensional view of these vascular wall constituents. Here, we demonstrate that multiphoton microscopy can be used to visualize the microstructural basis of blood vessel mechanical properties, by combining mechanical testing (distension) of excised porcine coronary arteries with simultaneous two-photon excited fluorescence and second-harmonic generation microscopy. Our results show that second-harmonic generation signals derived from collagen can be spectrally isolated from elastin and smooth muscle cell two-photon fluorescence. Two-photon fluorescence signals can be further characterized by emission maxima at 495 nm and 520 nm, corresponding to elastin and cellular contributions, respectively. Two-dimensional reconstructions of spectrally fused images permit high-resolution visualization of collagen and elastin fibrils and smooth muscle cells from intima to adventitia. These structural features are confirmed by coregistration of multiphoton microscopy images with conventional histology. Significant changes in mean fibril thickness and overall wall dimension were observed when comparing no load (zero transmural pressure) and zero-stress conditions to 30 and 180 mmHg distension pressures. Overall, these data suggest that multiphoton microscopy is a highly sensitive and promising technique for studying the morphometric properties of the microstructure of the blood vessel wall.     

The Effect of Longitudinal Pre-Stretch and Radial Constraint on the Stress Distribution in the Vessel Wall: A New Hypothesis

It is well known that blood vessels shorten axially when excised. This is due to the perivascular tethering constraint by side branches and the existence of pre-stretch of blood vessels at the \textit {in situ} state. Furthermore, vessels are radially constrained to various extents by the surrounding tissues at physiological loading. Our hypothesis is that the axial pre-stretch and radial constraint by the surrounding tissue homogenizes the stress and strain distributions in the vessel wall. A finite element analysis of porcine coronary artery and rabbit thoracic aorta based on measured material properties, geometry, residual strain and physiological loading is used to compute the intramural stresses and strains. We systematically examined the effect of pre-stretch and external radial constraint in both vessels. Our results show that both stretching in the axial direction and compression in the radial direction lead to a more homogeneous strain and stress state in the blood vessel wall. A ``uniform biaxial strain' hypothesis is proposed for the blood vessel wall and the ramifications are discussed.     

A mechanism for arteriolar remodeling based on maintenance of smooth muscle cell activation

Structural adaptation in arterioles is part of normal vascular physiology but is also seen in disease states such as hypertension. Smooth muscle cell (SMC) activation has been shown to be central to microvascular remodeling. We hypothesize that, in a remodeling process driven by SMC activation, stress sensitivity of the vascular wall is a key element in the process of achieving a stable vascular structure. We address whether the adaptive changes in arterioles under different conditions can arise through a common mechanism: remodeling in a stress-sensitive wall driven by a shift in SMC activation. We present a simple dynamic model and show that structural remodeling of the vessel radius by rearrangement of the wall material around a lumen of a different diameter and driven by differences in SMC activation can lead to vascular structures similar to those observed experimentally under various conditions. The change in structure simultaneously leads to uniform levels of circumferential wall stress and wall strain, despite differences in transmural pressure. A simulated vasoconstriction caused by increased SMC activation leads to inward remodeling, whereas outward remodeling follows relaxation of the vascular wall. The results are independent of the specific myogenic properties of the vessel. The simulated results are robust in the face of parameter changes and, hence, may be generalized to vessels from different vascular beds.     

Interlayer micromechanics of the aortic heart valve leaflet

While the mechanical behaviors of the fibrosa and ventricularis layers of the aortic valve (AV) leaflet are understood, little information exists on their mechanical interactions mediated by the GAG-rich central spongiosa layer. Parametric simulations of the interlayer interactions of the AV leaflets in flexure utilized a tri-layered finite element (FE) model of circumferentially oriented tissue sections to investigate inter-layer sliding hypothesized to occur. Simulation results indicated that the leaflet tissue functions as a tightly bonded structure when the spongiosa effective modulus was at least 25 % that of the fibrosa and ventricularis layers. Novel studies that directly measured transmural strain in flexure of AV leaflet tissue specimens validated these findings. Interestingly, a smooth transmural strain distribution indicated that the layers of the leaflet indeed act as a bonded unit, consistent with our previous observations (Stella and Sacks in J Biomech Eng 129:757–766, 2007) of a large number of transverse collagen fibers interconnecting the fibrosa and ventricularis layers. Additionally, when the tri-layered FE model was refined to match the transmural deformations, a layer-specific bimodular material model (resulting in four total moduli) accurately matched the transmural strain and moment-curvature relations simultaneously. Collectively, these results provide evidence, contrary to previous assumptions, that the valve layers function as a bonded structure in the low-strain flexure deformation mode. Most likely, this results directly from the transverse collagen fibers that bind the layers together to disable physical sliding and maintain layer residual stresses. Further, the spongiosa may function as a general dampening layer while the AV leaflets deforms as a homogenous structure despite its heterogeneous architecture.     

The effect of residual strain on the diastolic function of the left ventricle as predicted by a structural model

The unloaded heart is not stress-free. It is subjected to residual stress and strain. Their extent and influence on the global performance of the left ventricle and on local phenomena in the ventricular wall are studied by model simulation. The analysis focuses on the equatorial region of the ventricle, with an approximate thick-walled cylindrical geometry. The in vivo myocardium is considered to be incompressible, consisting of fibers embedded in a fluid matrix, with transmurally varying anisotropic microstructure in accordance with morphological characteristics.

The results show that residual strain is transmurally distributed with a pattern and magnitude which agree well with measurements. The calculated residual strains are within mean ± one standard deviation of the measured ones. Their magnitude was found to increase with increasing opening angle and with increasing wall thickness. The residual strain was found to have several effects on ventricular function: At volumes higher than the reference one it gives rise to more uniform transmural distributions of stress and intramyocardial pressure; it causes about 50% increase in the ventricular compliance at high volumes and doubles the suction of atrial blood at low volumes, thus facilitating the diastolic filling. In addition, residual strains cause bias of in vivo measured strains from their true values. This may significantly affect physiological interpretation of measured ventricular deformations.

In conclusion, the present structural analysis predicts that residual strain has favorable effect on left-ventricular diastolic performance, and gives rise to more uniform ventricular stress distribution.     

Changes in the rheological properties of blood vessel tissue remodeling in the course of development of diabetes.

Rheological properties of blood vessels are expected to change in disease process if the structure of the vessel wall changes. This is illustrated in diabetes, which can be induced in rat by a single injection of Streptozocin. One of the rheological properties of the blood vessel is the stress-strain relationship. The nonlinear stress-strain relationship of arteries is best expressed as derivations of a strain-energy function. In this paper, the stress-strain relations are measured and the coefficients in the strain energy function of arteries are determined for diabetic and control rats. The meaning of these coefficients are explained. The influence of diabetes on the elastic property of the arteries is expressed by the changes of these coefficients. A point of departure of the present paper from all other blood vessel papers published so far is that all strains used here are referred to the zero-stress state of the arteries, whereas all other papers refer strains to the no-load state. The existence of a large difference between the zero-stress state and no-load state of arteries is one of our recent findings. We have explained that the use of zero-stress state as a basis of strain measurements reveals that the in vivo circumferential stress distribution is quite uniform in the vessel wall at the homeostatic condition. It also makes the strain energy function much more accurate than those in which the residual stress is ignored. Using these new results, the stress and strain distribution in normal and diabetic arteries are presented.     

Transmural sheet strains in the lateral wall of the ovine left ventricle

In an attempt to provide a better understanding of our finding that regions with contracting left ventricular myofibers need not develop a significant transmural systolic wall thickening gradient, the analytic approach of Costa et al. was applied to the four-dimensional dynamic data obtained 1 and 8 wk after surgical implantation of transmural radiopaque beads in the lateral equatorial left ventricular wall in seven ovine hearts. Quantitative histology of tissue blocks demonstrated that fiber angles varied linearly across the wall in this region from -37 degrees in the subepicardium to +18 degrees in the subendocardium. Sheet angles exhibited a pleated-sheet behavior, alternating sign from subepicardium to subendocardium. From end diastole (reference configuration) to end systole (deformed configuration), fiber strain was uniformly negative, sheet extension and sheet thickening were uniformly positive, and sheet-normal shear contributed to wall thickening at all wall depths. Subepicardial radial wall thickening increased significantly from week 1 to week 8, with significant increases in the contributions from subepicardial sheet extension and sheet-normal shear. At 1 and 8 wk, the contribution of sheet-normal shear to wall thickening was substantial at all transmural depths; the contribution of sheet extension to wall thickening was greatest in the subepicardium and least in the subendocardium, and the contribution of sheet thickening to wall thickening was greatest in the subendocardium and least in the subepicardium. A mechanistic model is proposed that provides a working hypothesis that a selective decrease in subepicardial intercellular matrix stiffness is responsible for elimination of the transmural wall thickening gradient 1-8 wk after marker implantation surgery.     

Mechanical mechanisms of thrombosis in intact bent microvessels of rat mesentery

The hypothesis that thrombus can be induced by localized shear stresses/rates, such as in the bent/stretched microvessels, was tested both experimentally and computationally. Our newly designed in vivo experiments were performed on the microvessels (post-capillary venules, 20-50mum diameter) of rat mesentery. These microvessels were bent/stretched with no/minimum injuries. In less than 60min after the microvessels were bent/stretched, thrombi were formed in 19 out of 61 bent locations (31.1%). Interestingly, thrombi were found to be initiated at the inner wall of the curvature in these bent/stretched vessels. To investigate the mechanical mechanisms of thrombus induction, we performed a 3-D computational simulation using commercial software, FLUENT. To simulate the bending and stretching, we considered the vessels with different curvatures (0 degrees , 90 degrees and 180 degrees ) as well as different shaped cross-sections (circular and elliptic). Computational results demonstrated that the highest shear stress/rate and shear stress/rate gradient are located at the inner wall of the curved circular-shaped vessels. They are located at the two apexes of the wall with shorter axis for the 0 degrees (straight) elliptic-shaped vessel and towards the inner side when the vessels are bent. The differences of the shear stresses/rates and of the shear stress/rate gradients between the inner and outer walls become larger in more bent and elliptic-shaped microvessels. Comparison of our experimental and numerical simulation results suggests that the higher shear stress/rate and the higher shear stress/rate gradient at the inner wall are responsible for initiating the thrombosis in bent post-capillary venules.     

Phage like it HOT: solution structure of the bacteriophage P1-encoded HOT protein, a homolog of the theta subunit of E. coli DNA polymerase III

DNA polymerase III, the main replicative polymerase of E. coli, contains a small subunit, theta, that binds to the epsilon proofreading subunit and appears to enhance the enzyme's proofreading function--especially under extreme conditions. It was recently discovered that E. coli bacteriophage P1 encodes a theta homolog, named HOT. The (1)H-(15)N HSQC spectrum of HOT exhibits more uniform intensities and less evidence of conformational exchange than that of theta; this uniformity facilitates a determination of the HOT solution structure by NMR. The structure contains three alpha helices, as reported previously for theta; however, the folding topology of the two proteins is very different. Residual dipolar coupling measurements on labeled theta support the conclusion that it is structurally homologous with HOT. As judged by CD measurements, the melting temperature of HOT was 62 degrees C, compared to 56 degrees C for theta, consistent with other data suggesting greater thermal stability of the HOT protein.