Remodeling of conduit arteries in hypertension and flow-overload obeys a minimum energy principle

Arterial remodeling is an important process in physiology and pathophysiology. Based on an energy minimization method, Murray's law predicts the optimal inner radius. Application of Darcy's law in the wall results in an optimal outer radius. The average wall stress is computed by the Laplace's law. Using these formulas, a large porcine coronary artery in hypertension is studied. The results reveal how wall thickness and average circumferential stress change after increasing blood pressure and volume flow rate. The theoretical predictions are in good qualitative agreement with experimental observations. The advantage and limitation of the current approach are discussed.     

Effects of stent design parameters on normal artery wall mechanics

A stent is a device designed to restore flow through constricted arteries. These tubular scaffold devices are delivered to the afflicted region and deployed using minimally invasive techniques. Stents must have sufficient radial strength to prop the diseased artery open. The presence of a stent can subject the artery to abnormally high stresses that can trigger adverse biologic responses culminating in restenosis. The primary aim of this investigation was to investigate the effects of varying stent "design parameters" on the stress field induced in the normal artery wall and the radial displacement achieved by the stent. The generic stent models were designed to represent a sample of the attributes incorporated in present commercially available stents. Each stent was deployed in a homogeneous, nonlinear hyperelastic artery model and evaluated using commercially available finite element analysis software. Of the designs investigated herein, those employing large axial strut spacing, blunted corners, and higher amplitudes in the ring segments induced high circumferential stresses over smaller areas of the artery's inner surface than all other configurations. Axial strut spacing was the dominant parameter in this study, i.e., all designs employing a small stent strut spacing induced higher stresses over larger areas than designs employing the large strut spacing. Increasing either radius of curvature or strut amplitude generally resulted in smaller areas exposed to high stresses. At larger strut spacing, sensitivity to radius of curvature was increased in comparison to the small strut spacing. With the larger strut spacing designs, the effects of varying amplitude could be offset by varying the radius of curvature and vice versa. The range of minimum radial displacements from the unstented diastolic radius observed among all designs was less than 90 microm. Evidence presented herein suggests that stent designs incorporating large axial strut spacing, blunted corners at bends, and higher amplitudes exposed smaller regions of the artery to high stresses, while maintaining a radial displacement that should be sufficient to restore adequate flow.     

Regional epicardial strain in the embryonic chick heart during the early looping stages

Epicardial strains were measured in Hamburger-Hamilton stage 11 and 12 embryonic chick hearts (1.6-2.0 days of incubation). These stages include part of the early phase of cardiac looping, as the initially straight heart tube bends and twists to form a curved c-shaped tube. By analyzing the motion of microbeads placed on the myocardial surface, we measured strains near the outer curvature, in the central region, and near the inner curvature of the primitive ventricle. No significant differences in strain were found between stages. Relative to end diastole, all three regions shortened by about 10% during systole in the circumferential direction, and the outer curvature shortened longitudinally by about 5%. In contrast, and unlike strains in older hearts, the inner curvature and central regions elongated by approximately 5-10% in the longitudinal direction during systole. These results are consistent with microstructural data and suggest that the material properties of the outer curvature are relatively isotropic, whereas the properties of the central and inner curvature regions are orthotropic, with contractile stress exerted primarily in the circumferential direction.     

Healthy and diseased coronary bifurcation geometries influence near-wall and intravascular flow: A computational exploration of the hemodynamic risk

Local hemodynamics has been identified as one main determinant in the onset and progression of atherosclerotic lesions at coronary bifurcations. Starting from the observation that atherosensitive hemodynamic conditions in arterial bifurcation are majorly determined by the underlying anatomy, the aim of the present study is to investigate how peculiar coronary bifurcation anatomical features influence near-wall and intravascular flow patterns. Different bifurcation angles and cardiac curvatures were varied in population-based, idealized models of both stenosed and unstenosed bifurcations, representing the left anterior descending (LAD) coronary artery with its diagonal branch. Local hemodynamics was analyzed in terms of helical flow and exposure to low/oscillatory shear stress by performing computational fluid dynamics simulations.Results show that bifurcation angle impacts lowly hemodynamics in both stenosed and unstenosed cases. Instead, curvature radius influences the generation and transport of helical flow structures, with smaller cardiac curvature radius associated to higher helicity intensity. Stenosed bifurcation models exhibit helicity intensity values one order of magnitude higher than the corresponding unstenosed cases. Cardiac curvature radius moderately affects near-wall hemodynamics of the stenosed cases, with smaller curvature radius leading to higher exposure to low shear stress and lower exposure to oscillatory shear stress. In conclusion, the proposed controlled benchmark allows investigating the effect of various geometrical features on local hemodynamics at the LAD/diagonal bifurcation, highlighting that cardiac curvature influences near wall and intravascular hemodynamics, while bifurcation angle has a minor effect.     

Three-dimensional stress and strain distribution in a two-layer model of a coronary artery

For a right coronary artery, three-dimensional stress and strain distributions at a physiological intraluminal pressure and an axial extension ratio were computed on the basis of a two-layer elastic model. To validate the model, curves of external radius versus pressure and of axial force versus pressure were computed for three axial extension ratios. To analyze mechanical properties, stress-free configurations of media and adventitia, and the constitutive law of each layer in literature, were used. The present study showed that the peak circumferential stress and the peak axial stress appear in the media at the boundary between the media and adventitia. This result is due to the opening angle of the media being larger than π (rad) and the larger value of a material constant of the strain energy function for the media than for the adventitia. The circumferential stress and strain were discontinuous at the boundary. On the other hand, the radial stress was continuous at the boundary because of the boundary condition for stress. The circumferential stress and axial stress in the adventitia were almost uniformly distributed, and smaller than in the media. The residual stress and strain were also computed. The circumferential residual stress and strain were almost linearly distributed in each layer, although discontinuity appeared at the boundary between the two layers.     

Measured effects of diacylglycerol on structural and elastic properties of phospholipid membranes.

Diacylglycerol, a biological membrane second messenger, is a strong perturber of phospholipid planar bilayers. It converts multibilayers to the reverse hexagonal phase (HII), composed of highly curved monolayers. We have used x-ray diffraction and osmotic stress of the HII phase to measure structural dimensions, spontaneous curvature, and bending moduli of dioleoylphosphatidylethanolamine (DOPE) monolayers doped with increasing amounts of dioleoylglycerol (DOG). The diameter of the HII phase cylinders equilibrated in excess water decreases significantly with increasing DOG content. Remarkably, however, all structural dimensions at any specific water/lipid ratio that is less than full hydration are insensitive to DOG. By plotting structural parameters of the HII phase with changing water content in a newly defined coordinate system, we show that the elastic deformation of the lipid monolayers can be described as bending around a pivotal plane of constant area. This dividing surface includes 30% of the lipid volume independent of the DOG content (polar heads and a small fraction of hydrocarbon chains). As the mole fraction of DOG increases to 0.3, the radius of spontaneous curvature defined for the pivotal surface decreases from 29 A to 19 A, and the bending modulus increases from approximately 11 to 14 (+/-0.5) kT. We derive the conversion factors and estimate the spontaneous curvatures and bending moduli for the neutral surface which, unlike the pivotal plane parameters, are intrinsic properties that apply to other deformations and geometries. The spontaneous curvature of the neutral surface differs from that of the pivotal plane by less than 10%, but the difference in the bending moduli is up to 40%. Our estimate shows that the neutral surface bending modulus is approximately 9kT and practically does not depend on the DOG content.     

3-D MRI assessment of regional left ventricular systolic wall stress in patients with reperfused MI

The goal of this study was to assess the regional variations of end-systolic wall stress in patients with reperfused Q wave acute myocardial infarction (AMI), with the use of a three-dimensional (3-D) approach. Fifteen normal volunteers and fifty patients with reperfused AMI underwent cardiac MRI that used a short-axis fast-gradient-echo sequence. The end-systolic wall stress was calculated with the use of the Grossman formula with the radius and the wall thickness defined with a 3-D approach using the tridimensional curvature. The mean wall stress was significantly increased at each level of the short-axis plane only in patients with anterior AMI. When calculated at a regional level in patients with anterior AMI, wall stress significantly increased in anterior sector as well as normal sector. In patients with inferior AMI, wall stress significantly increased only in inferior and lateral sectors. In conclusion, the quantification of regional wall stress by cardiac MRI is better with the 3D approach than other methods for precise evaluation in patients with AMI. Despite early reperfusion, the wall stress remained high in patients with anterior AMI.     

A 2D finite element model of the interventricular septum under normal and abnormal loading

The interventricular septum is the structure that separates the left and right ventricles of the heart. Under normal loading conditions, it is concave to the left ventricle, but under abnormal loading the septum flattens and occasionally inverts. In the past, the septum has frequently been modelled as integral to the left ventricle with the effects of pressure from the right ventricle being ignored. Under abnormal loading, the septum has been described as behaving equivalent to a "flapping sail". There has been no consideration of structural behaviour under these conditions. A 2-D plane stress FE model of the septum was used to investigate the difference in structural behaviour of the septum during diastole between normal and abnormal loading. The biaxial stress patterns that develop are distinctively disparate. Under normal loading, the septum behaves much like a thick-walled cylinder subject to internal and external pressure, with the resulting stresses being circumferential tension and radial compression, both varying with radius. These stresses are very low throughout most of diastole. However, under abnormal loading, the septum behaves in an arch-like fashion, with high compressive stresses almost circumferential in direction, combined with radial compression. We conclude that right ventricular pressures cause bending effects in the wall of the heart, and that under abnormal loading, the compressive stresses that develop in the septum may lead to an understanding of certain, previously unexplained, pathological conditions.     

Stress Distribution in the Canine Left Ventricle during Diastole and Systole

A model is proposed for stress analysis of the left ventricular wall (LV wall) based on the realistic assumption that the myocardium is essentially composed of fiber elements which carry only axial tension and vary in orientation through the wall. Stress analysis based on such a model requires an extensive study of muscle fiber orientation and curvature through the myocardium. Accordingly, the principal curvatures were studied at a local site near the equator in ten dog hearts rapidly fixed in situ at end diastole and end systole; the fiber orientation for these hearts had already been established in a previous study. The principal radii of curvature were (a) measured by fitting templates to the endocardial and epicardial wall surfaces in the circumferential and longitudinal directions and (b) computed from measured lengths of semiaxes of ellipsoids of revolution representing the LV wall (“ellipsoid” data). The wall was regarded as a tethered set of nested shells, each having a unique fiber orientation. Results indicate the following. (a) Fiber curvature, k, is maximum at midwall at end systole; this peak shifts towards endocardium at end diastole. (b) The pressure or radial stress through the wall decreases more rapidly near the endocardium than near the epicardium at end diastole and at end systole when a constant tension is assumed for each fiber through the wall. (c) At end diastole the curve for the circumferential stress vs. wall thickness is convex with a maximum at midwall. In the longitudinal direction the stress distribution curve is concave with a minimum at midwall. Similar distributions are obtained at end systole when a constant tension is assumed for each fiber through the wall. (d) The curvature and stress distributions obtained by direct measurements at a selected local site agree well with those computed from “ellipsoid” data.     

A 2D Finite Element Model of the Interventricular Septum Under Normal and Abnormal Loading

Abstract

The interventricular septum is the structure that separates the left and right ventricles of the heart. Under normal loading conditions, it is concave to the left ventricle, but under abnormal loading the septum flattens and occasionally inverts. In the past, the septum has frequently been modelled as integral to the left ventricle with the effects of pressure from the right ventricle being ignored. Under abnormal loading, the septum has been described as behaving equivalent to a “flapping sail”. There has been no consideration of structural behaviour under these conditions. A 2-D plane stress FE model of the septum was used to investigate the difference in structural behaviour of the septum during diastole between normal and abnormal loading. The biaxial stress patterns that develop are distinctively disparate. Under normal loading, the septum behaves much like a thick-walled cylinder subject to internal and external pressure, with the resulting stresses being circumferential tension and radial compression, both varying with radius. These stresses are very low throughout most of diastole. However, under abnormal loading, the septum behaves in an arch-like fashion, with high compressive stresses almost circumferential in direction, combined with radial compression. We conclude that right ventricular pressures cause bending effects in the wall of the heart, and that under abnormal loading, the compressive stresses that develop in the septum may lead to an understanding of certain, previously unexplained, pathological conditions.     

Calculation and application of the anterior surface area of a model human cornea

The macroscopic anterior surface area was calculated for three models of the average human cornea. Two models, a general ellipsoid and a rotational ellipse (rotationally symmetric ellipsoid) gave a surface area of 132 mm2, while a spherical model gave 126 mm2. A general ellipsoidal model having the maximum radius horizontal (with-the-rule corneal astigmatism) has less surface area than a rotational ellipse with the same horizontal radius. For a corneal sagittal height of 2.59 mm, the surface area of an ellipsoidal cornea equals -19.2Q + 16.3R -0.476 which specifies a rotational ellipse (radius R, asphericity Q) of equal surface area. In a cornea with the maximum radius vertical (against-the-rule corneal astigmatism), the ellipsoid has slightly more surface area than a rotational ellipse with the same horizontal radius of curvature. For a given horizontal radius of curvature, the sphere has the lowest surface area. For a corneal sagittal height s of 2.59 mm, the sphere underestimates by 8% the surface area of a rotational ellipse with asphericity -0.5. The anterior corneal surface area of a rotational ellipse model, radius R, asphericity Q is given by 2 pi Rs- 19.2Q. In all three models, the surface area increases with horizontal radius of curvature. In the rotational ellipse model, the rate of increase (slope) is independent of asphericity, and the slope found in with-the-rule astigmatism is less than the slope found with against-the-rule astigmatism. The calculated surface area predicts a precorneal tear volume of 0.86 microliter for a 6.5 micron tear thickness. The apparent, or plane projected are of an epithelial lesion underestimates the curved surface area with a percentage error that increases rapidly with lesion diameter. For a 12 mm diameter lesion on a rotational ellipse model, the apparent area underestimates the surface area by 18%. The average posterior corneal surface in human is not spherical but imitates the anterior surface, and has an area of 137 mm2 or 3.8% greater than the anterior area.     

The semicircular canals of the hagfish Myxine glutinosa

The internal radius (r) and radius of curvature (R) of the single semicircular canals of Myxine glutinosa have unusual dimensions. In mammals and fish the increase in dimension of r and R with respect to body weight is small; in fish r is larger than in mammals of equivalent weight in order to increase the sensitivity of the canals to angular rotation and R increases correspondingly (Jones & Spells, 1963). In Myxine r is larger than in fish or mammals yet R is smaller. It is argued that the large internal radius is the result of the need to increase the sensitivity of a single canal which has to signal rotation in three planes while the small radius of curvature follows from the absence of a cupula. In order to verify that the cristae of the canals do respond to rotational velocity, recordings were obtained from the nerves serving the canals during rotation in the horizontal plane. The frequency response of several afferents recorded simultaneously at sinusoidal rotations between 0.25 and 2.0 Hz was in the form of a sine wave 90 in advance of head position, as would be expected of a velocity transducer. The gain of single afferents was an order of magnitude less than those reported for other vertebrates.     

Curvature of the forelimb bones of anthropoid primates: Overall allometric patterns and specializations in suspensory species

It has previously been reported that brachiating primates, particularly gibbons, are characterized by distinctively straight forelimb long bones, yet no hypotheses have been proposed to explain why straight limb bones may be adaptive to suspensory locomotion. This study explores quantitatively the curvature of the long bones in 13 species of anthropoid primates and analyzes the functional consequences of curvature in biomechanical terms. These analyses demonstrate that, whereas the humeri of gibbons and spider monkeys are functionally less curved than those of other taxa, the ulnae of brachiators are neither more nor less curved than those of other anthropoids, and the gibbon radius is far more curved than would be predicted from body size alone. The humerus is likely significantly less curved in brachiators because of its torsion-dominated loading regime and the greatly increased stress magnitude developed in torsionally loaded curved beams. The large curvature of the radius is localized in the region of attachment of the supinator muscle. Analysis presented here of muscle mass allometry in catarrhines demonstrates that gibbons are characterized by an extremely massive supinator, and the large radial curvature is therefore most likely due to forearm muscle mechanics. This study also demonstrates that the overall pattern of limb bone curvature for anthropoids is distinct from the pattern reported for mammals as a whole. This distinctive scaling relationship may be related to the increased length of the limb bones of primates in comparison to other mammals.     

Frequency dependence of dynamic curvature effects on flow through coronary arteries

The flow through a curved tube model of a coronary artery was investigated computationally to determine the importance of time-varying curvature on flow patterns that have been associated with the development of atherosclerosis. The entry to the tube was fixed while the radius of curvature varied sinusoidally in time at a frequency of 1 or 5 Hz. Angiographic data from other studies suggest that the radius of curvature waveform contains significant spectral content up to 6 Hz. The overall flow patterns were similar to those observed in stationary curved tubes; velocity profile skewed toward the outer wall, secondary flow patterns, etc. The effects of time-varying curvature on the changes in wall shear rate were expressed by normalizing the wall shear rate amplitude with the shear rate calculated at the static mean radius of curvature. It was found that the wall shear rate varied as much as 94 percent of the mean wall shear rate at the mid wall of curvature for a mean curvature ratio of 0.08 and a 50 percent change in radius of curvature. The effects of 5 Hz deformation were not well predicted by a quasi-static approach. The maximum values of the normalized inner wall shear rate amplitude were found to scale well with a dimensionless parameter equivalent to the product of the mean curvature ratio (delta), normalized change in radius of curvature (epsilon), and a Womersley parameter (alpha). This parameter was less successful at predicting the amplitudes elsewhere in the tube, thus additional studies are necessary. The mean wall shear rate was well predicted with a static geometry. These results indicate that dynamic curvature plays an important role in determining the inner wall shear rates in coronary arteries that are subjected to deformation levels of epsilon delta alpha > 0.05. The effects were not always predictable with a quasi-static approach. These results provide guidelines for constructing more realistic models of coronary artery flow for atherogenesis research.     

Temporal trends in femoral curvature and length in medieval and modern Scotland

We measured how much the radius of the anterior curvature and the length of the femoral shaft of cadaveric bones have changed from medieval to recent times. Around 20 (x, y) coordinates of a virtual coordinate system were measured at intervals of 1.5 cm along the shaft of the femur to calculate one single radius of a virtual circle in the (x, y) plane. The median radii of curvature were 119, 141, and 158 cm for medieval, early, and late 20th century femora, respectively. Early and late 20th century femora were of similar length (45 cm), but medieval femora were shorter (43.5 cm). Femora have become not only longer but also straighter since the Middle Ages. These findings account in part for the increase in height of modern generations. Size and shape changes may have significant implications for the biomechanical response of the femur to the forces to which it is subjected in everyday life, in trauma, and following surgical intervention.     

Human complement subcomponent C2: purification and proteolytic cleavage in fluid phase by C1s, C1r2-C1s2 and C1

Photosynthetic membranes contain considerable regions of high surface curvature, notably at their margins, where the average radius of curvature is about 10 nm. The proportion of total membrane lipid in the outer and inner thylakoid margin monolayers is estimated at 21% and 13%, respectively. The major thylakoid lipid, monogalactosyldiacylglycerol, is roughly cone-shaped and will not form complete lamellar bilayer phases, even in combination with other thylakoid lipids. It is proposed that this galactolipid plays a role in: (a) stabilising regions of concave curvature in thylakoids; and (b) packaging hydrophobic proteins in planar bilayer regions by means of inverted micelles. This model predicts substantial asymmetries in the distribution of lipids both across and along the thylakoid bilayer plane.     

Semicircular canal radii of curvature (R) in cat,guinea pig and man

The radii of curvature (R) of the horizontal (Rh), anterior (Ra) and posterior (Rp) semicircular canals were measured by a new technique (called ROTA) for cat, guinea pig and man. For each canal, data points from the ossecus canal were rotated and plotted by computer such that the plane of the sheet of computer plot corresponded to the plane best fitting that canal. The radius of each osseous canal was determined and where necessary, the radius of the are of data points was corrected for thickness of the absent tissue. For cat, guinea pig and man there are differences in R between canals within a labyrinth suggesting that if other things are equal there could be differences in the average mechanical sensitivity of the canals, which is consistent with physiological recordings from primary vestibular neurons in the cat, The Rs determined by ROTA are compared with Rs determined by conventional histological means.     

Regional nonuniformity of normal adult human left ventricle

Regional nonuniformity is a feature of both diseased and normal left ventricles (LV). With the use of magnetic resonance (MR) myocardial tagging, we performed three-dimensional strain analysis on 87 healthy adults in local cardiac and fiber coordinate systems (radial, circumferential, longitudinal, and fiber strains) to characterize normal nonuniformities and to test the validity of wall thickening as a parameter of regional function. Regional morphology included wall thickness and radii of curvature measurements. With respect to transmural nonuniformity, subendocardial strains exceeded subepicardial strains. Going from base to apex, wall thickness and circumferential radii of curvature decreased, whereas longitudinal radii of curvature increased. All of the strains increased from LV base to apex, resulting in a higher ejection fraction (EF) at the apex than at the base (70.9 +/- 0.4 vs. 62.4 +/- 0.4%; means +/- SE, P < 0.0001). When we looked around the circumference of the ventricle, the anterior part of the LV was the flattest and thinnest and showed the largest wall thickening (46.6 +/- 1.2%) but the lowest EF (64.7 +/- 0.5%). The posterior LV wall was thicker, more curved, and showed a lower wall thickening (32.8 +/- 1.0%) but a higher EF (71.3 +/- 0.5%). The regional contribution of the LV wall to the ejection of blood is thus highly variable and is not fully characterized by wall thickening alone. Differences in regional LV architecture and probably local stress are possible explanations for this marked functional nonuniformity.