Wall shear in pulsatile flow

This paper deals with a mathematical attempt to determine the wall shear during normal flow of blood in the ascending and the descending thoracic aorta. A simple model is used, but the results obtained are in agreement with published experimental results for the descending thoracic aorta. It is suggested that the degree of fluctuation in the pressure gradient at a given station is the major factor in determining the level of wall shear at that point.     

Maximal wall shear stress in arterial stenoses: application to the internal carotid arteries

Maximal wall shear stress (MWSS) in the convergent part of a stenosis is calculated by the interactive boundary-layer theory. A dimensional analysis of the problem shows that MWSS depends only on a few measurable parameters. A simple relationship between MWSS and these parameters is obtained, validated, and used to calculate the magnitude of MWSS in a carotid stenosis, as a function of the patency of the circle of Willis and the stenotic pattern. This demonstrates the huge effect of collateral pathways. Elevated MWSS are observed even in moderate stenoses, provided they are associated with a contralateral occlusion, a large anterior, and narrow posterior communicating arteries, suggesting a potential risk of embolus release in this configuration.     

Wall shear stress during pulsatile flow distal to a normal porcine aortic valve

The shear stress at the wall has been of interest as one of the possible fluid dynamic factors that may be damaging in the region of prosthetic valves. The purpose of this study was to measure the axial wall shear stresses in the region of a 29 mm tissue annulus diameter porcine stent mounted prosthetic aortic valve (Hancock, Model 242). Studies were performed in an in vitro pulse duplicating system. The axial wall shear stress was calculated from velocities obtained near the wall with a laser Doppler anemometer. The largest axial wall shear stress was 29 dyn cm-2 and it occurred at the highest stroke volume used (80 ml). At a stroke volume of 50 ml, the largest axial wall shear stress was 17 dyn cm-2 and at a stroke volume of 35 ml, it was 15 dyn cm-2. Stresses of these magnitudes are far below those reported to be damaging to the endothelial surface. These stresses may be high enough, however, to affect platelet function.     

Response of a concentrated monoclonal antibody formulation to high shear

There is concern that shear could cause protein unfolding or aggregation during commercial biopharmaceutical production. In this work we exposed two concentrated immunoglobulin‐G1 (IgG1) monoclonal antibody (mAb, at >100 mg/mL) formulations to shear rates between 20,000 and 250,000 s^?1 for between 5 min and 30 ms using a parallel‐plate and capillary rheometer, respectively. The maximum shear and force exposures were far in excess of those expected during normal processing operations (20,000 s^?1 and 0.06 pN, respectively). We used multiple characterization techniques to determine if there was any detectable aggregation. We found that shear alone did not cause aggregation, but that prolonged exposure to shear in the stainless steel parallel‐plate rheometer caused a very minor reversible aggregation (<0.3%). Additionally, shear did not alter aggregate populations in formulations containing 17% preformed heat‐induced aggregates of a mAb. We calculate that the forces applied to a protein by production shear exposures (<0.06 pN) are small when compared with the 140 pN force expected at the air–water interface or the 20–150 pN forces required to mechanically unfold proteins described in the atomic force microscope (AFM) literature. Therefore, we suggest that in many cases, air‐bubble entrainment, adsorption to solid surfaces (with possible shear synergy), contamination by particulates, or pump cavitation stresses could be much more important causes of aggregation than shear exposure during production. Biotechnol. Bioeng. 2009;103: 936–943. © 2009 Wiley Periodicals, Inc.     

Wall shear stress estimates in coronary artery constrictions.

Wall shear stress estimates from laminar boundary layer theory were found to agree fairly well with the magnitude of shear stress levels along coronary artery constrictions obtained from solutions of the Navier Stokes equations for both steady and pulsatile flow. The relatively simple method can be used for in vivo estimates of wall shear stress in constrictions by using a vessel shape function determined from a coronary angiogram, along with a knowledge of the flow rate.     

Wall shear stress fixed points in cardiovascular fluid mechanics

Complex blood flow in large arteries creates rich wall shear stress (WSS) vectorial features. WSS acts as a link between blood flow dynamics and the biology of various cardiovascular diseases. WSS has been of great interest in a wide range of studies and has been the most popular measure to correlate blood flow to cardiovascular disease. Recent studies have emphasized different vectorial features of WSS. However, fixed points in the WSS vector field have not received much attention. A WSS fixed point is a point on the vessel wall where the WSS vector vanishes. In this article, WSS fixed points are classified and the aspects by which they could influence cardiovascular disease are reviewed. First, the connection between WSS fixed points and the flow topology away from the vessel wall is discussed. Second, the potential role of time-averaged WSS fixed points in biochemical mass transport is demonstrated using the recent concept of Lagrangian WSS structures. Finally, simple measures are proposed to quantify the exposure of the endothelial cells to WSS fixed points. Examples from various arterial flow applications are demonstrated.     

Wall shear stress in normal left coronary artery tree

Despite the fact that the role of wall shear stress (WSS) as a local mechanical factor in atherogenesis is well established, its distribution over the entire normal human left coronary artery (LCA) tree has not yet been studied. A three-dimensional computer generated model of the epicardial LCA tree, based on averaged human data set extracted from angiographies, was adopted for finite-element analysis of the Navier-Stokes flow equations treating blood as non-Newtonian fluid. The LCA tree includes the left main coronary artery (LMCA), the left anterior descending (LAD), the left circumflex artery (LCxA) and their major branches. In proximal LCA tree regions where atherosclerosis frequently occurs, low WSS appears. Low WSS regions occur at bifurcations in regions opposite the flow dividers, which are anatomic sites predisposed for atherosclerotic development. On the LMCA bifurcation, at regions opposite to the flow divider, dominant low WSS values occur ranging from 0.75 to 2.25 N/m2. High WSS values are encountered at all flow dividers. This work determines, probably for the first time, the topography of the WSS in the entire normal human LCA epicardial tree. It is also the first work determining the spatial WSS differentiation between proximal and distal normal human LCA parts. The haemodynamic analysis of the entire epicardial LCA tree further verifies the implications of the WSS in atherosclerosis mechanisms.     

Strain distribution over plaques in human coronary arteries relates to shear stress

Once plaques intrude into the lumen, the shear stress they are exposed to alters with hitherto unknown consequences for plaque composition. We investigated the relationship between shear stress and strain, a marker for plaque composition, in human coronary arteries. We imaged 31 plaques in coronary arteries with angiography and intravascular ultrasound. Computational fluid dynamics was used to obtain shear stress. Palpography was applied to measure strain. Each plaque was divided into four regions: upstream, throat, shoulder, and downstream. Average shear stress and strain were determined in each region. Shear stress in the upstream, shoulder, throat, and downstream region was 2.55+/-0.89, 2.07+/-0.98, 2.32+/-1.11, and 0.67+/-0.35 Pa, respectively. Shear stress in the downstream region was significantly lower. Strain in the downstream region was also significantly lower than the values in the other regions (0.23+/-0.08% vs. 0.48+/-0.15%, 0.43+/-0.17%, and 0.47+/-0.12%, for the upstream, shoulder, and throat regions, respectively). Pooling all regions, dividing shear stress per plaque into tertiles, and computing average strain showed a positive correlation; for low, medium, and high shear stress, strain was 0.23+/-0.10%, 0.40+/-0.15%, and 0.60+/-0.18%, respectively. Low strain colocalizes with low shear stress downstream of plaques. Higher strain can be found in all other plaque regions, with the highest strain found in regions exposed to the highest shear stresses. This indicates that high shear stress might destabilize plaques, which could lead to plaque rupture.     

Effect of high shear on proteins

Shear is present in almost all bioprocesses and high shear is associated with processes involving agitation and emulsification. The purpose of this study is to investigate the effect of high shear and high shear rate on proteins. Two concentric cylinder-based shear systems were used. One was a closed concentric-cylinder shear device (CCSD) and the other was a homogenizer with a rotor/stator assembly. Mathematical modeling of these systems allowed calculation of the shear rate and shear. The CCSD generated low shear rates (a few hundred s(-1)), whereas the homogenizer could generate very high shear rates (> 10(5) s(-1)). High shear could be achieved in both systems by increasing the processing time. Recombinant human growth hormone (rhGH) and recombinant human deoxyribonuclease (rhDNase) were used as the model proteins in this study. It was found that neither high shear nor high shear rate had a significant effect on protein aggregation. However, a lower melting temperature and enthalpy were detected for highly sheared rhGH by using scanning microcalorimetry, presumably due to some changes in protein's conformation. Also, SDS-PAGE indicated the presence of low molecular-weight fragments, suggesting that peptide bond breakage occurred due to high shear. rhDNase was relatively more stable than rhGH under high shear. No conformational changes and protein fragments were observed. (c) 1996 John Wiley & Sons, Inc.     

Wall shear rate measurements in an elastic curved artery model

Since atherosclerotic lesions tend to be localized at bends and branching points, knowledge of wall shear rate patterns in models of these geometries may help elucidate the mechanism of atherogenesis. This study uses the photochromic method of flow visualization to determine both the mean and amplitude of the wall shear rate waveform in straight and curved elastic arterial models to demonstrate the effects of curvature, elasticity, and the phase angle between the flow and pressure waveforms (impedance phase angle). Under sinusoidal flow conditions characteristic of large arteries, the mean shear rate at the inner wall of the curved tube is reduced 40–56% from its steady flow value, depending on the phase angle. Wall shear rate amplitudes in the curved tube are significantly reduced by wall motion (36–55% of the Womersley amplitude for a straight rigid tube). The shear rate amplitude at the outer wall decreases 30% as the phase angle is reduced from −20° to −66°, while the shear rate amplitude at the inner wall increases 45%. As a result, the oscillatory nature of flow at the outer wall decreases with decreasing negative phase angle, but flow at the inner wall becomes much more oscillatory. At large negative phase angles, characteristic of hypertension or vasoactive agents, the shear rate at the inner wall has a small mean and cycles through positive and negative values; the shear rate at the outer wall remains positive throughout the flow cycle. Thus, the impedance phase angle could affect atherogenesis along the inner wall if temporal and directional changes in wall shear rate play a role.     

Wall shear stress at the initiation site of cerebral aneurysms

Hemodynamics are believed to play an important role in the initiation of cerebral aneurysms. In particular, studies have focused on wall shear stress (WSS), which is a key regulator of vascular biology and pathology. In line with the observation that aneurysms predominantly occur at regions of high WSS, such as bifurcation apices or outer walls of vascular bends, correlations have been found between the aneurysm initiation site and high WSS. The aim of our study was to analyze the WSS field at an aneurysm initiation site that was neither a bifurcation apex nor the outer wall of a vascular bend. Ten cases with aneurysms on the A1 segment of the anterior cerebral artery were analyzed and compared with ten controls. Aneurysms were virtually removed from the vascular models of the cases to mimic the pre-aneurysm geometry. Computational fluid dynamics (CFD) simulations were created to assess the magnitude, gradient, multidirectionality, and pulsatility of the WSS. To aid the inter-subject comparison of hemodynamic variables, we mapped the branch surfaces onto a two-dimensional parametric space. This approach made it possible to view the whole branch at once for qualitative evaluation. It also allowed us to empirically define a patch for quantitative analysis, which was consistent among subjects and encapsulated the aneurysm initiation sites in our dataset. To test the sensitivity of our results, CFD simulations were repeated with a second independent observer virtually removing the aneurysms and with a 20 % higher flow rate at the inlet. We found that branches harboring aneurysms were characterized by high WSS and high WSS gradients. Among all assessed variables, the aneurysm initiation site most consistently coincided with peaks of temporal variation in the WSS magnitude.     

Geometry of human ribs pertinent to orthopedic chest-wall reconstruction

Orthopedic reconstruction of blunt chest trauma can aid restoration of pulmonary function to reduce the mortality associated with serial rib fractures and flail chest injuries. Contemporary chest wall reconstruction requires contouring of generic plates to the complex surface geometry of ribs. This study established a biometric foundation to generate specialized, anatomically contoured osteosynthesis hardware for rib fracture fixation. On human cadaveric ribs three through nine, the surface geometry pertinent to anatomically conforming osteosynthesis plates was characterized by quantifying the apparent rib curvature C(A), the longitudinal twist alpha(LT) along the diaphysis, and the unrolled curvature C(U). In addition, the rib cross-sectional geometry pertinent to intramedullary fixation strategies was characterized in terms of cross-section height, width, area, and cortex thickness. The rib surface exhibited a curvature C(A) ranging from 3.8 m(-1) in the anteromedial section of rib seven to 17.3 m(-1) in the posterior section of rib three. All ribs had in common a longitudinal twist alpha(LT), ranging from 41-60 degrees. The unrolled curvature C(U) decreased gradually from ribs three to five, and increased gradually with reversed orientation from rib six to nine. The cross-sectional area remained constant along the rib diaphysis. However, the medullary canal increased in size from 29.9 mm(2) posteriorly to 41.2 mm(2) in anterior rib segments. Results of this biometric rib characterization describe a novel strategy for intraoperative plate contouring and provide a foundation for the development of specialized rib osteosynthesis strategies.     

A geometry-based model for non-invasive estimation of pressure gradients over iliac artery stenoses

The aim of this study was to develop and verify a model that provides an accurate estimation of the trans-lesion hyperemic pressure gradient in iliac artery stenoses in seconds by only using patient-specific geometric properties obtained from 3-dimensional rotational angiography (3DRA).Twenty-one patients with symptomatic peripheral arterial disease (PAD), iliac artery stenoses and an ultrasound based peak systolic velocity ratio between 2.5 and 5.0 underwent 3DRA and intra-arterial pressure measurements under hyperemic conditions. For each lesion, geometric properties were extracted from the 3DRA images using quantitative vascular analysis software. Hyperemic blood flow was estimated based on stenosis geometry using an empirical relation. The geometrical properties and hyperemic flow were used to estimate the pressure gradient by means of the geometry-based model. The predicted pressure gradients were compared with in vivo measured intra-arterial pressure measurements performed under hyperemic conditions.The developed geometry-based model showed good agreement with the measured hyperemic pressure gradients resulting in a concordance correlation coefficient of 0.86. The mean bias ± 2SD between the geometry-based model and in vivo measurements was comparable to results found by evaluating the actual computational fluid dynamics model (−1.0 ± 14.7 mmHg vs −0.9 ± 12.7 mmHg).The developed model estimates the trans-lesional pressure gradient in seconds without the need for an additional computational fluid dynamics software package. The results justify further study to assess the potential use of a geometry-based model approach to estimate pressure gradient on non-invasive CTA or MRA, thereby reducing the need for diagnostic angiography in patients suffering from PAD.     

Biaxial mechanical response of bioprosthetic heart valve biomaterials to high in-plane shear

Utilization of novel biologically-derived biomaterials in bioprosthetic heart valves (BHV) requires robust constitutive models to predict the mechanical behavior under generalized loading states. Thus, it is necessary to perform rigorous experimentation involving all functional deformations to obtain both the form and material constants of a strain-energy density function. In this study, we generated a comprehensive experimental biaxial mechanical dataset that included high in-plane shear stresses using glutaraldehyde treated bovine pericardium (GLBP) as the representative BHV biomaterial. Compared to our previous study (Sacks, JBME, v.121, pp. 551-555, 1999), GLBP demonstrated a substantially different response under high shear strains. This finding was underscored by the inability of the standard Fung model, applied successfully in our previous GLBP study, to fit the high-shear data. To develop an appropriate constitutive model, we utilized an interpolation technique for the pseudo-elastic response to guide modification of the final model form. An eight parameter modified Fung model utilizing additional quartic terms was developed, which fitted the complete dataset well. Model parameters were also constrained to satisfy physical plausibility of the strain energy function. The results of this study underscore the limited predictive ability of current soft tissue models, and the need to collect experimental data for soft tissue simulations over the complete functional range.     

Wall shear stress and near-wall flows in the stenosed femoral artery

Stenotic artery hemodynamics are often characertised by metrics including oscillatory shear index (OSI) and residence time (RT). This analysis was conducted to clarify the link between the near-wall flow behaviour and these resultant flow metrics. A computational simulation was conducted of a stenosed femoral artery, with an idealised representative geometry and a physiologically realistic inlet profile. The overall flow behaviour was characterised through consideration of the axial flow, which was non-dimensionalised against mean flow velocity. The OSI and RT metrics, which are a useful indicator of likely atherosclerotic sites, were explained through a discussion of the WSS values at different time points, the velocity behaviour and velocity profiles, with a particular focus on the near-wall behaviour which influences wall shear stress and the transient evolution of the wall shear stress. While, the stenosis throat experiences high values of wall shear stress, the smooth flow through this contracted region results in low variation in wall shear stress vectors and limited opportunity for any particle stasis. However, regions were noted distal and proximal (though to a lesser extent), where the change in recirculation zones over the cycle created highly elevated regions of both OSI and RT.     

Heterozygote advantage fails to explain the high degree of polymorphism of the MHC

Major histocompatibility (MHC) molecules are encoded by extremely polymorphic genes and play a crucial role in vertebrate immunity. Natural selection favors MHC heterozygous hosts because individuals heterozygous at the MHC can present a larger diversity of peptides from infectious pathogens than homozygous individuals. Whether or not heterozygote advantage is sufficient to account for a high degree of polymorphism is controversial, however. Using mathematical models we studied the degree of MHC polymorphism arising when heterozygote advantage is the only selection pressure. We argue that existing models are misleading in that the fitness of heterozygotes is not related to the MHC alleles they harbor. To correct for this, we have developed novel models in which the genotypic fitness of a host directly reflects the fitness contributions of its MHC alleles. The mathematical analysis suggests that a high degree of polymorphism can only be accounted for if the different MHC alleles confer unrealistically similar fitnesses. This conclusion was confirmed by stochastic simulations, including mutation, genetic drift, and a finite population size. Heterozygote advantage on its own is insufficient to explain the high population diversity of the MHC.Electronic Supplementary Material Supplementary material is available in the online version of this article at