Wall shear stress and related hemodynamic parameters in the fetal descending aorta derived from color Doppler velocity profiles

Measurement of Wall Shear Rate Across the Entire Vascular Wall Using Ultrasound Speckle Decorrelation

OBJECTIVE

The accurate measurement of the wall shear rate (WSR) plays a crucial role in the early diagnosis of cardiovascular disease progression and acute events such as aneurysms and atherosclerotic plaque ruptures. To address this need, the speckle decorrelation (SDC) technique has been used to measure WSR based on the 2-D out-of-plane blood flow speed. This technique is particularly advantageous because it enables the use of a 1-D array transducer to measure WSR over the entire luminal area. This study aims to develop a region-based singular value decomposition (SVD) filtering technique that selectively suppresses clutter noise in the vascular region to measure WSR using SDC.

METHOD

Ultrasound simulations, in-vitro flow experiments, and an in-vivo human study were conducted to evaluate the feasibility of this method's clinical application.

RESULTS

The results demonstrated that WSR can be effectively measured across entire vascular walls using a conventional 1-D array transducer along with the proposed methodology.

CONCLUSION

This study successfully demonstrates a noninvasive and accurate SDC-based method for measuring vital vascular WSR. This approach holds significant promise for assessing vascular WSR in both healthy individuals and high-risk cardiovascular disease patients.

Shear stress regulation in cylindrical arteries through medial growth and nitric oxide release

The mechanisms employed by blood vessels in order to adapt to their hemodynamic environment are important for our general understanding of disease and development. Changes in arterial geometry are generally induced by two effects: vasodilation and/or constriction; and growth and remodeling ("G &R"). The first can occur over short periods of a few minutes, while the second usually occurs over timescales of weeks or months. The free radical Nitric oxide (NO) is one of the few biological signaling molecules that is gaseous. When smooth muscle cells internalize NO, they lengthen and ultimately induce a relaxation of the artery. Platelet-Derived Growth Factor (PDGF) is a growth factor released by smooth muscle cells and platelets that regulates cell growth and division. In this paper we present a single-layered, axisymmetric hyperelastic model for a deforming, growing artery in which the opening angle is regulated by NO and growth is induced by PDGF. Our model describes vasodilation and G &R in a long cylindrical artery regulated by a steady-state Poiseuille flow. The transport of NO released by the endothelium is governed by a diffusion equation with a shear-stress dependent flux boundary condition. Arterial opening angle is assumed to be a Hill function of the wall-averaged NO concentration. We find that both growth and NO help to regulate shear stress with respect to the flow rate, but regulation through growth occurs only at large times. In contrast, regulation through NO is immediate but can only occur as long as the opening angle is able to continually decrease as a function of flow rate. Our model is calibrated using experimental data from ligated, control, and anastomosed carotid arteries of adult and weanling rabbits. Our results generate shear stress/flow rate and lumen radius/flow rate curves that agree with experimental data from control and NO-inhibited rabbit carotid arteries.

Recent advances in quantifying the mechanobiology of cardiac development via computational modeling

Mechanical forces are essential for coordinating cardiac morphogenesis, but much remains to be discovered about the interactions between mechanical forces and the mechanotransduction pathways they activate. Due to the elaborate and fundamentally multi-physics and multi-scale nature of cardiac mechanobiology, a complete understanding requires multiple experimental and analytical techniques. We identify three fundamental tools used in the field to probe these interactions: high resolution imaging, genetic and molecular analysis, and computational modeling. In this review, we focus on computational modeling and present recent studies employing this tool to investigate the mechanobiological pathways involved with cardiac development. These works demonstrate that understanding the detailed spatial and temporal patterns of biomechanical forces is crucial to building a comprehensive understanding of mechanobiology during cardiac development, and that computational modeling is an effective and efficient tool for obtaining such detail. In this context, multidisciplinary studies combining all three tools present the most compelling results.

Time-Resolved Wall Shear Rate Mapping Using High-Frame-Rate Ultrasound Imaging

In atherosclerosis, low wall shear stress (WSS) is known to favor plaque development, while high WSS increases plaque rupture risk. To improve plaque diagnostics, WSS monitoring is crucial. Here, we propose wall shear imaging (WASHI), a noninvasive contrast-free framework that leverages high-frame-rate ultrasound (HiFRUS) to map the wall shear rate (WSR) that relates to WSS by the blood viscosity coefficient. Our method measures WSR as the tangential flow velocity gradient along the arterial wall from the flow vector field derived using a multi-angle vector Doppler technique. To improve the WSR estimation performance, WASHI semiautomatically tracks the wall position throughout the cardiac cycle. WASHI was first evaluated with an in vitro linear WSR gradient model; the estimated WSR was consistent with theoretical values (an average error of 4.6% ± 12.4 %). The framework was then tested on healthy and diseased carotid bifurcation models. In both scenarios, key spatiotemporal dynamics of WSR were noted: 1) oscillating shear patterns were present in the carotid bulb and downstream to the internal carotid artery (ICA) where retrograde flow occurs; and 2) high WSR was observed particularly in the diseased model where the measured WSR peaked at 810 [Formula: see text] due to flow jetting. We also showed that WASHI could consistently track arterial wall motion to map its WSR. Overall, WASHI enables high temporal resolution mapping of WSR that could facilitate investigations on causal effects between WSS and atherosclerosis.

Numerical Investigation of the Fetal Left Heart Hemodynamics During Gestational Stages

Flow-driven hemodynamic forces on the cardiac tissues have critical importance, and have a significant role in the proper development of the heart. These mechanobiological mechanisms govern the cellular responses for the growth and remodeling of the heart, where the altered hemodynamic environment is believed to be a major factor that is leading to congenital heart defects (CHDs). In order to investigate the mechanobiological development of the normal and diseased hearts, identification of the blood flow patterns and wall shear stresses (WSS) on these tissues are required for an accurate hemodynamic assessment. In this study, we focus on the left heart hemodynamics of the human fetuses throughout the gestational stages. Computational fetal left heart models are created for the healthy fetuses using the ultrasound images at various gestational weeks. Realistic inflow boundary conditions are implemented in the models using the Doppler ultrasound measurements for resolving the specific blood flow waveforms in the mitral valve. Obtained results indicate that WSS and vorticity levels in the fetal left heart decrease with the development of the fetus. The maximum WSS around the mitral valve is determined around 36 Pa at the gestational week of 16. This maximum WSS decreases to 11 Pa at the gestational week of 27, indicating nearly three-times reduction in the peak shear stress. These findings reveal the highly dynamic nature of the left heart hemodynamics throughout the development of the human fetus and shed light into the relevance of hemodynamic environment and development of CHDs.

A Review of Biomechanics Analysis of the Umbilical-Placenta System With Regards to Diseases

Placenta is an important organ that is crucial for both fetal and maternal health. Abnormalities of the placenta, such as during intrauterine growth restriction (IUGR) and pre-eclampsia (PE) are common, and an improved understanding of these diseases is needed to improve medical care. Biomechanics analysis of the placenta is an under-explored area of investigation, which has demonstrated usefulness in contributing to our understanding of the placenta physiology. In this review, we introduce fundamental biomechanics concepts and discuss the findings of biomechanical analysis of the placenta and umbilical cord, including both tissue biomechanics and biofluid mechanics. The biomechanics of placenta ultrasound elastography and its potential in improving clinical detection of placenta diseases are also discussed. Finally, potential future work is listed.

Pressure and Flow Relations in the Systemic Arterial Tree Throughout Development From Newborn to Adult

Objective: Distributed models of the arterial tree allow studying the effect of physiological and pathophysiological changes in the vasculature on hemodynamics. For the adult, several models exist; however, a model encompassing the full age range from newborn to adult was until now lacking. Our goal is to describe a complete distributed hemodynamic model for normal development from newborn to adult. Methods: The arterial system was modeled by 121 segments characterized by length, radius, wall thickness, wall stiffness, and wall viscosity. The final segments ended in three-element Windkessels. All parameters were adapted based on body height and weight as a function of age as described in the literature. Results: Pressures and flows are calculated as a function of age at sites along the arterial tree. Central to peripheral transfer functions are given. Our results indicate that peripheral pressure in younger children resembles central pressure. Furthermore, total arterial compliance, inertance and impedance are calculated. Findings indicate that the arterial tree can be simulated by using a three-element Windkessel system. Pulse wave velocity in the aorta was found to increase during development. Conclusions: The arterial system, modeled from newborn to adult bears clinical significance, both for the interpretation of peripheral measured pressure in younger and older children, and for using a Windkessel model to determine flow from pressure measurements.

Characterization of the hemodynamic wall shear stresses in human umbilical vessels from normal and intrauterine growth restricted pregnancies

Significant reductions in blood flow and umbilical diameters were reported in pregnancies affected by intrauterine growth restriction (IUGR) from placental insufficiency. However, it is not known if IUGR umbilical blood vessels experience different hemodynamic wall shear stresses (WSS) compared to normal umbilical vessels. As WSS is known to influence vasoactivity and vascular growth and remodeling, which can regulate flow rates, it is important to study this parameter. In this study, we aim to characterize umbilical vascular WSS environment in normal and IUGR pregnancies, and evaluate correlation between WSS and vascular diameter, and gestational age. Twenty-two normal and 21 IUGR pregnancies were assessed via ultrasound between the 27th and 39th gestational week. IUGR was defined as estimated fetal weight and/or abdominal circumference below the 10th centile, with no improvement during the remainder of the pregnancy. Vascular diameter was determined by 3D ultrasound scans and image segmentation. Umbilical artery (UA) WSS was computed via computational flow simulations, while umbilical vein (UV) WSS was computed via the Poiseuille equation. Univariate multiple regression analysis was used to test for the differences between normal and IUGR cohort. UV volumetric flow rate, UA and UV diameters were significantly lower in IUGR fetuses, but flow velocities and WSS trends in UA and UV were very similar between normal and IUGR groups. In both groups, UV WSS showed a significant negative correlation with diameter, but UA WSS had no correlation with diameter, suggesting a constancy of WSS environment and the existence of WSS homeostasis in UA, but not in UV. Despite having reduced flow rate and vascular sizes, IUGR UAs had hemodynamic mechanical stress environments and trends that were similar to those in normal pregnancies. This suggested that endothelial dysfunction or abnormal mechanosensing was unlikely to be the cause of small vessels in IUGR umbilical cords.

Common Carotid Artery Flow Measured by 3-D Ultrasonic Vector Flow Imaging and Validated with Magnetic Resonance Imaging

Ultrasound (US) examination of the common carotid artery was compared with a through-plane magnetic resonance imaging (MRI) sequence to validate a recently proposed technique for 3-D US vector flow imaging. Data from the first volunteer examined were used as the training set, before volume flow and peak velocities were calculated for the remaining eight volunteers. Peak systolic velocities (PSVs) and volume flow obtained with 3-D US were, on average, 34% higher and 24% lower than those obtained with MRI, respectively. A high correlation was observed for PSV (r = 0.79), whereas a lower correlation was observed for volume flow (r = 0.43). The overall standard deviations were ±5.7% and ±5.7% for volume flow and PSV with 3-D US, compared with ±2.7% and ±3.2% for MRI. Finally, the data were re-processed with a change in the parameter settings for the echo-canceling filter to investigate its influence on overall performance. PSV was less affected by the re-processing, whereas the difference in volume flow between 3-D vector flow imaging and MRI was reduced to -9%, and with an improved overall standard deviation of ±4.7%. The results illustrate the feasibility of using 3-D US for precise and angle-independent volume flow and PSV estimation in vivo.

Wall Shear Rate Measurement: Validation of a New Method Through Multiphysics Simulations

Wall shear stress is known to affect the vessel endothelial function and to be related to important pathologies like the development of atherosclerosis. It is defined as the product of the blood viscosity by the blood velocity gradient at the wall position, i.e., the wall shear rate (WSR). The WSR measurement is particularly challenging in important cardiovascular sites, like the carotid bifurcation, because of the related complex flow configurations characterized by high spatial and temporal gradients, wall movement, and clutter noise. Moreover, accuracy of any method for WSR measurement can be effectively tested only if reliable gold standard WSR values, considering all the aforementioned disturbing effects, are available. Unfortunately, these requirements are difficult to achieve in a physical phantom, so that the accuracy test of the novel WSR measurement methods was so far limited to straight pipes and/or similar idealistic configurations. In this paper, we propose a new method for WSR measurement and its validation based on a mathematical model of the carotid bifurcation, which, exploiting fluid-structure simulations, is capable of reproducing realistic flow configuration, wall movement, and clutter noise. In particular, the profile near the wall, not directly measurable because affected by clutter, is estimated through a power-law fitting and compared with the gold standard provided by the model. In this condition, the WSR measurements featured an accuracy of ±20 %. A preliminary test on a volunteer confirmed the feasibility of the WSR method for in vivo application.

Fluid mechanics of human fetal right ventricles from image-based computational fluid dynamics using 4D clinical ultrasound scans

There are 0.6–1.9% of US children who were born with congenital heart malformations. Clinical and animal studies suggest that abnormal blood flow forces might play a role in causing these malformation, highlighting the importance of understanding the fetal cardiovascular fluid mechanics. We performed computational fluid dynamics simulations of the right ventricles, based on four-dimensional ultrasound scans of three 20-wk-old normal human fetuses, to characterize their flow and energy dynamics. Peak intraventricular pressure gradients were found to be 0.2–0.9 mmHg during systole, and 0.1–0.2 mmHg during diastole. Diastolic wall shear stresses were found to be around 1 Pa, which could elevate to 2–4 Pa during systole in the outflow tract. Fetal right ventricles have complex flow patterns featuring two interacting diastolic vortex rings, formed during diastolic E wave and A wave. These rings persisted through the end of systole and elevated wall shear stresses in their proximity. They were observed to conserve ∼25.0% of peak diastolic kinetic energy to be carried over into the subsequent systole. However, this carried-over kinetic energy did not significantly alter the work done by the heart for ejection. Thus, while diastolic vortexes played a significant role in determining spatial patterns and magnitudes of diastolic wall shear stresses, they did not have significant influence on systolic ejection. Our results can serve as a baseline for future comparison with diseased hearts.

Non-invasive vascular radial/circumferential strain imaging and wall shear rate estimation using video images of diagnostic ultrasound

The aim of this work was to develop a convenient method for radial/circumferential strain imaging and shear rate estimation that could be used as a supplement to the current routine screening for carotid atherosclerosis using video images of diagnostic ultrasound. A reflection model-based correction for gray-scale non-uniform distribution was applied to B-mode video images before strain estimation to improve the accuracy of radial/circumferential strain imaging when applied to vessel transverse cross sections. The incremental and cumulative radial/circumferential strain images can then be calculated based on the displacement field between consecutive B-mode images. Finally, the transverse Doppler spectra acquired at different depths along the vessel diameter were used to construct the spatially matched instantaneous wall shear values in a cardiac cycle. Vessel phantom simulation results revealed that the signal-to-noise ratio and contrast-to-noise ratio of the radial and circumferential strain images were increased by 2.8 and 5.9 dB and by 2.3 and 4.4 dB, respectively, after non-uniform correction. Preliminary results for 17 patients indicated that the accuracy of radial/circumferential strain images was improved in the lateral direction after non-uniform correction. The peak-to-peak value of incremental strain and the maximum cumulative strain for calcified plaques are evidently lower than those for other plaque types, and the echolucent plaques had higher values, on average, than the mixed plaques. Moreover, low oscillating wall shear rate values, found near the plaque and stenosis regions, are closely related to plaque formation. In conclusion, the method described can provide additional valuable results as a supplement to the current routine ultrasound examination for carotid atherosclerosis and, therefore, has significant potential as a feasible screening method for atherosclerosis diagnosis in the future.

Clinical benefits of integrating cardiac and vascular models

INTRODUCTION

Recent advances in computational methods and medical imaging techniques have enabled non-invasive exploration of cardiovascular pathologies, from cardiac level to complex arterial networks. The potential of cardiac and vascular modeling in guiding and monitoring therapies could be further extended through the integration of the two systems.

AREAS COVERED

This review includes advances in methods for cardiac electromechanics and vascular flow simulations. The results of a literature search depicting the state of the art in cardiac and vascular modeling are reviewed. The paper goes on to address the benefits and challenges of combined cardiovascular modeling, highlighting the relevance of specific cardiovascular features and implementation. Various alternative approaches and insights on future directions are presented and analyzed with respect to their applicability to clinical practice.

EXPERT OPINION

The article has emerged from the exploration of currently available cardiac and vascular mathematical tools and their corresponding clinical application. The summarized analysis suggests that future efforts should be aimed at developing more accurate and patient-specific mathematical models integrating cardiac and vascular functions to enhance the knowledge of cardiovascular pathologies.

Estimation of blood velocity, volumetric flow and wall shear rate using Doppler ultrasound

Commercial ultrasound systems can make a number of measurements related to haemodynamics which are relevant to clinical practice and to clinical research. These include maximum velocity, volumetric flow and wall shear rate. Using appropriate protocols, measurements can be made averaged over the cardiac cycle, or as a function of time through the cardiac cycle. Maximum velocity underpins most of these measurements. Maximum velocity is overestimated as a result of geometric spectral broadening, by typically up to 30%, but by much larger amounts as the angle approaches 90°. Though not used in clinical practice, a simple correction technique using a string phantom can substantially reduce these errors. For volumetric flow and wall shear rate, methods such as specialist multi-gate ultrasound systems, magnetic resonance imaging and image guided modelling are available. Before resorting to these more complex methods users might consider that, with care and attention to procedure, high quality information may be obtained using commercial ultrasound systems. Manufacturers could make more use of the colour flow image for quantification of velocity, and adopt vector Doppler techniques.

Development of a compliant and cytocompatible micro-fibrous polyethylene terephthalate vascular scaffold

Bioengineering approaches have been intensively applied to create small diameter vascular grafts using artificial materials. However, a fully successful, high performing and anti-thrombogenic structure has not been achieved yet. In this study, we have designed and fabricated a novel non-woven fibrous vascular graft with biomechanical properties closely resembling those of native vessels. Vascular cell growth, preservation of cell phenotype, retention of vasoactive properties, as well as the effect of gelatin coating on the cellular interaction with the scaffolds under static and shear stress conditions were investigated. The non-woven fibrous scaffolds were made from melt blown polyethylene terephthalate fiber webs stacked by means of a consolidation technique. The scaffold variables were fiber diameter distribution and the number of consolidated web stacks. SEM analysis confirmed various fiber diameter and pore size ranges corresponding to the different conditions. The scaffolds showed burst pressure values of ∼1500 mmHg and compliance (8.4 ± 1.0 × 10(-2) % mmHg(-1) ) very similar to those of native arteries (∼8 × 10(-2) % mmHg(-1) ). The structure with the smallest fiber diameter range (1-5 μm) and pore size range (1-20 μm) was the most suitable for the growth of human brain endothelial cells and aortic smooth muscle cells. The cells maintained their specific cell phenotype, expressed collagen and elastin and produced cAMP in response to α-calcitonin gene-related peptide. However, under shear stress conditions (0.9 dyne cm(-2) ), only 30% of the cells were retained in both uncoated and gelatin-coated scaffolds indicating the need for improving the cell retention capacity of these structures, which is our future research direction. This study indicates that the biomechanical and biocompatible properties of this novel vascular scaffold are promising for the development of a vascular graft with similar characteristics to those of native vessels.

Assessment of blood volume flow in slightly curved arteries from a single velocity profile

Non-invasive estimation of arterial blood volume flow (BVF) has become a central issue in assessment of cardiovascular risk. Poiseuille and Womersley approaches are commonly used to assess the BVF from centerline velocity, but both methods neglect the influence of curvature. Based on the assumption that the velocity in curved tubes as function of the circumferential position for a given radial position can be approximated by a cosine, the BVF can also be estimated by averaging velocities at opposite radial positions, referred to as the cosine theta model (CTM). This study investigates the accuracy of BVF estimation in slightly curved arteries for BVF waveforms obtained in the brachial artery of 6 volunteers. Computational fluid dynamics simulations were used to compute the influence of curvature on velocity profiles. The BVF was then estimated from the simulation results with the CTM and methods based on Poiseuille, Womersley and using the center stream velocity and the velocity waveform at the position where the maximum velocity is observed, and compared to the prescribed BVF. The simulations show that the influence of curvature is strongest when the flow decelerates. For Poiseuille and Womersley, the time average BVF was underestimated by maximally 10.4% and 7.8% for a radius of curvature of 50 and 100 mm, respectively. The estimation error is lower for the CTM and equals 4.2% and 1.2% for a radius of curvature of 50 and 100mm, respectively. From this study, we can conclude that the velocity waveform at the position of the maximum rather than the center stream velocity waveform combined with the Womersley method should be chosen. The CTM improves current estimation techniques if in-vivo velocity distributions are available.

Haemodynamics and blood flow measured using ultrasound imaging

Visualization of, and measurements related to, haemodynamic phenomena in arteries may be made using ultrasound systems. Most ultrasound technology relies on simple measurements of blood velocity taken from a single site, such as the peak systolic velocity for assessment of the degree of lumen reduction caused by an arterial stenosis. Real-time two-dimensional (2D) flow field visualization is possible using several methods, such as colour flow, blood flow imaging, and echo particle image velocimetry; these have applications in the examination of the flow field in diseased arteries and in heart chambers. Three-dimensional (3D) and four-dimensional ultrasound systems have been described. These have been used to provide 2D velocity profile data for the estimation of volumetric flow. However, they are limited for haemodynamic evaluation in that they provide only one component of the velocity. The provision of all seven components (three space, three velocity, and one time) is possible using image-guided modelling, in which 3D ultrasound is combined with computational fluid dynamics. This method also allows estimation of turbulence data and of relevant quantities such as the wall shear stress.

Blood pressure estimation in the human fetal descending aorta

OBJECTIVES

The objectives of this study were to estimate fetal blood pressure non-invasively from two-dimensional color Doppler-derived aortic blood flow and diameter waveforms, and to compare the results with invasively derived human fetal blood pressures available from the literature.

METHODS

Aortic pressures were calculated from digitally recorded color Doppler cineloops of the fetal descending aorta by applying the Womersley model in combination with the two-element Windkessel model, assuming constant pulse wave velocity during the second half of pregnancy. The results were compared with invasively derived human fetal blood pressures obtained from the literature.

RESULTS

In 21 normal pregnancies the estimated mean aortic pressure regression line increased linearly from 28 mmHg at 20 weeks of gestation to 45 mmHg at 40 weeks of gestation. The pulse pressure based on the regression line increased linearly from 21 mmHg at 20 weeks of gestation to 29 mmHg at 40 weeks of gestation. The aortic compliance exhibited a log linear relationship with the gestational age and a statistically significant eightfold increase was observed between 20 and 40 weeks. The aortic downstream peripheral resistance exhibited an exponentially decaying relationship across the same gestational age range. Non-invasively derived aortic systolic and diastolic aortic pressures were comparable with previously reported invasively derived systolic and diastolic umbilical arterial pressures; however, the mean pressures differed significantly from those reported in the umbilical artery in a separate study. The aortic systolic pressures calculated in this study were significantly higher than invasively derived left ventricular systolic pressures that have been previously reported in the literature.

CONCLUSIONS

This study demonstrates the feasibility of estimating arterial blood pressure in the human fetus. The method described is of potential use in assessing fetal blood pressure non-invasively, particularly for studying relative changes with time.

A method to estimate wall shear rate with a clinical ultrasound scanner

A simple technique to estimate the wall shear rate in healthy arteries using a clinical ultrasound scanner has been developed. This method uses the theory of fully developed oscillatory flow together with a spectral Doppler trace and an estimate of mean arterial diameter. A method using color flow imaging was compared with the spectral Doppler method in vascular phantoms and found to have errors that were on average 35% greater. Differences from the theoretic value for the time averaged wall shear rate using the spectral Doppler method varied by artery: brachial -9 (1) %; carotid -7 (1) %; femoral -22 (4) %; and fetal aorta -17 (10) %. Test measurements obtained from one healthy volunteer demonstrated the feasibility of the technique in vivo.

Large variations in absolute wall shear stress levels within one species and between species

Wall shear stress (WSS), the frictional force between blood and endothelium, is an important determinant of vascular function. It is generally assumed that WSS remains constant at a reference value of 15 dyn/cm(2). In a study of small rodents, we realized that this assumption could not be valid. This review presents an overview of recent studies in large and small animals where shear stress was measured, derived from velocity measurements or otherwise, in large vessels. The data show that large variations exist within a single species (human: variation of 2-16 N/m(2)). Moreover, when we compared different species at the same location within the arterial tree, an inverse relationship between animal size and wall shear stress was noted. When we related WSS to diameter, a unique relationship was derived for all species studied. This relationship could not be described by the well-known r(3) law of Murray, but by the r(2) law introduced by Zamir et al. in 1972. In summary, by comparing data from the literature, we have shown that: (i) the assumption of a physiological WSS level of approximately 15 dyn/cm(2) for all straight vessels in the arterial tree is incorrect; (ii) WSS is not constant throughout the vascular tree; (iii) WSS varies between species; (iv) WSS is inversely related to the vessel diameter. These data support an "r(2) law" rather than Murray's r(3) law for the larger vessels in the arterial tree.

Abnormal arterial flows by a distributed model of the fetal circulation

Modeling the propagation of blood pressure and flow along the fetoplacental arterial tree may improve interpretation of abnormal flow velocity waveforms in fetuses. The current models, however, either do not include a wide range of gestational ages or do not account for variation in anatomical, vascular, or rheological parameters. We developed a mathematical model of the pulsating fetoumbilical arterial circulation using Womersley's oscillatory flow theory and viscoelastic arterial wall properties. Arterial flow waves are calculated at different arterial locations from which the pulsatility index (PI) can be determined. We varied blood viscosity, placental and brain resistances, placental compliance, heart rate, stiffness of the arterial wall, and length of the umbilical arteries. The PI increases in the umbilical artery and decreases in the cerebral arteries, as a result of increasing placental resistance or decreasing brain resistance. Both changes in resistance decrease the flow through the placenta. An increased arterial stiffness increases the PIs in the entire fetoplacental circulation. Blood viscosity and peripheral bed compliance have limited influence on the flow profiles. Bradycardia and tachycardia increase and decrease the PI in all arteries, respectively. Umbilical arterial length has limited influence on the PI but affects the mean arterial pressure at the placental cord insertion. The model may improve the interpretation of arterial flow pulsations and thus may advance both the understanding of pathophysiological processes and clinical management.