Periodic flow of a viscous liquid in a thick-walled elastic tube

Arterial blood flow is analyzed on the basis of a realistic model consisting of a viscous liquid contained in a thick-walled viscoelastic tube. Approximate forms of the Navier-Stokes and continuity equations are derived for this model and solved in conjunction with the equations of motion of an elastic solid. Expressions are found for the displacement of the tube wall, velocity distribution, volume flow rate and phase velocity of the pressure wave. Changes in the shape of the pressure wave caused by damping and dispersion are determined, and the effect of viscoelasticity is assessed. Numerical results are presented which correspond to observed parameters of the circulatory systems of living animals.     

Exercise reduces arterial pressure augmentation through vasodilation of muscular arteries in humans

Exercise markedly influences pulse wave morphology, but the mechanism is unknown. We investigated whether effects of exercise on the arterial pulse result from alterations in stroke volume or pulse wave velocity (PWV)/large artery stiffness or reduction of pressure wave reflection. Healthy subjects (n = 25) performed bicycle ergometry. with workload increasing from 25 to 150 W for 12 min. Digital arterial pressure waveforms were recorded using a servo-controlled finger cuff. Radial arterial pressure waveforms and carotid-femoral PWV were determined by applanation tonometry. Stroke volume was measured by echocardiography, and brachial and femoral artery blood flows and diameters were measured by ultrasound. Digital waveforms were recorded continuously. Other measurements were made before and after exercise. Exercise markedly reduced late systolic and diastolic augmentation of the peripheral pressure pulse. At 15 min into recovery, stroke volume and PWV were similar to baseline values, but changes in pulse wave morphology persisted. Late systolic augmentation index (radial pulse) was reduced from 54 +/- 3.9% at baseline to 42 +/- 3.7% (P < 0.01), and diastolic augmentation index (radial pulse) was reduced from 37 +/- 1.8% to 25 +/- 2.9% (P < 0.001). These changes were accompanied by an increase in femoral blood flow (from 409 +/- 44 to 773 +/- 48 ml/min, P < 0.05) and an increase in femoral artery diameter (from 8.2 +/- 0.4 to 8.6 +/- 0.4 mm, P < 0.05). In conclusion, exercise dilates muscular arteries and reduces arterial pressure augmentation, an effect that will enhance ventricular-vascular coupling and reduce load on the left ventricle.     

The respiratory change in preejection period: a new method to predict fluid responsiveness.

The accuracy and clinical utility of preload indexes as bedside indicators of fluid responsiveness in patients after cardiac surgery is controversial. This study evaluates whether respiratory changes (Delta) in the preejection period (PEP; DeltaPEP) predict fluid responsiveness in mechanically ventilated patients. Sixteen postcoronary artery bypass surgery patients, deeply sedated under mechanical ventilation, were enrolled. PEP was defined as the time interval between the beginning of the Q wave on the electrocardiogram and the upstroke of the radial arterial pressure. DeltaPEP (%) was defined as the difference between expiratory and inspiratory PEP measured over one respiratory cycle. We also measured cardiac output, stroke volume index, right atrial pressure, pulmonary arterial occlusion pressure, respiratory change in pulse pressure, systolic pressure variation, and the Deltadown component of SPV. Data were measured without positive end-expiratory pressure (PEEP) and after application of a PEEP of 10 cmH2O (PEEP10). When PEEP10 induced a decrease of >15% in mean arterial pressure value, then measurements were re-performed before and after volume expansion. Volume loading was done in eight patients. Right atrial pressure and pulmonary arterial occlusion pressure before volume expansion did not correlate with the change in stroke volume index after the fluid challenge. Systolic pressure variation, DeltaPEP, Deltadown, and change in pulse pressure before volume expansion correlated with stroke volume index change after fluid challenge (r2 = 0.52, 0.57, 0.68, and 0.83, respectively). In deeply sedated, mechanically ventilated patients after cardiac surgery, DeltaPEP, a new method, can be used to predict fluid responsiveness and hemodynamic response to PEEP10.     

Periodic flow of a viscous liquid in a thick-walled elastic tube

Arterial blood flow is analyzed on the basis of a realistic model consisting of a viscous liquid contained in a thick-walled viscoelastic tube. Approximate forms of the Navier-Stokes and continuity equations are derived for this model and solved in conjunction with the equations of motion of an elastic solid. Expressions are found for the displacement of the tube wall, velocity distribution, volume flow rate and phase velocity of the pressure wave. Changes in the shape of the pressure wave caused by damping and dispersion are determined, and the effect of viscoelasticity is assessed. Numerical results are presented which correspond to observed parameters of the circulatory systems of living animals. This research was partially supported by the National Science Foundation; it was done in part by D. K. Whirlow in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Carnegie Institute of Technology.     

Wave propagation in a model of the arterial circulation

The propagation of the arterial pulse wave in the large systemic arteries has been calculated using a linearised method of characteristics analysis to follow the waves generated by the heart. The model includes anatomical and physiological data for the 55 largest arteries adjusted so that the bifurcating tree of arteries is well matched for forward travelling waves. The peripheral arteries in the model are terminated by resistance elements which are adjusted to produce a physiologically reasonable distribution of mean blood flow. In the model, the pressure and velocity wave generated by the contraction of the left ventricle propagates to the periphery where it is reflected. These reflected waves are re-reflected by each of the bifurcations that they encounter and a very complex pattern of waves is generated. The results of the calculations exhibit many of the features of the systemic arteries, including the increase of the pulse pressure with distance away from the heart as well as the initial decrease and then the large increase in the magnitude of back flow during late systole going from the ascending aorta to the abdominal aorta to the arteries of the leg. The model is then used to study the effects of the reflection or absorption of waves by the heart and the mechanisms leading to the incisura are investigated. Calculations are carried out with the total occlusion of different arterial segments in order to model experiments in which the effects of the occlusion of different arteries on pressure and flow in the ascending aorta were measured. Finally, the effects of changes in peripheral resistance on pressure and velocity waveforms are also studied. We conclude from these calculations that the complex pattern of wave propagation in the large arteries may be the most important determinant of arterial haemodynamics.     

Measurement of total pulmonary arterial compliance using invasive pressure monitoring and MR flow quantification during MR-guided cardiac catheterization

Pulmonary hypertensive disease is assessed by quantification of pulmonary vascular resistance. Pulmonary total arterial compliance is also an indicator of pulmonary hypertensive disease. However, because of difficulties in measuring compliance, it is rarely used. We describe a method of measuring pulmonary arterial compliance utilizing magnetic resonance (MR) flow data and invasive pressure measurements. Seventeen patients with suspected pulmonary hypertension or congenital heart disease requiring preoperative assessment underwent MR-guided cardiac catheterization. Invasive manometry was used to measure pulmonary arterial pressure, and phase-contrast MR was used to measure flow at baseline and at 20 ppm nitric oxide (NO). Total arterial compliance was calculated using the pulse pressure method (parameter optimization of the 2-element windkessel model) and the ratio of stroke volume to pulse pressure. There was good agreement between the two estimates of compliance (r = 0.98, P < 0.001). However, there was a systematic bias between the ratio of stroke volume to pulse pressure and the pulse pressure method (bias = 61%, upper level of agreement = 84%, lower level of agreement = 38%). In response to 20 ppm NO, there was a statistically significant fall in resistance, systolic pressure, and pulse pressure. In seven patients, total arterial compliance increased >10% in response to 20 ppm NO. As a population, the increase did not reach statistical significance. There was an inverse relation between compliance and resistance (r = 0.89, P < 0.001) and between compliance and mean pulmonary arterial pressure (r = 0.72, P < 0.001). We have demonstrated the feasibility of quantifying total arterial compliance using an MR method.     

Cardiac limitation to maximal oxygen transport and changes in components after jogging across the US.

Observations were made before and 3-5 days after prolonged endurance jogging an average of 42 miles/day, 6 days/wk for 2.5 mo by a young male adult who voluntarily initiated a run across the United States. Both arterial PO2 and lactic acid increased. In each instance, the first limitation in circulatory delivery of oxygen was a plateau in stroke volume and cardiac output. Afterward, pulse deficit and systemic arterial pressure fell with exercise and heart rate accelerated. Although there was no change in oxygen transport (Q X CAO2), a reduction in stroke volume was exactly balanced by a rise in arterial oxygen content. Vital capacity, residual volume, and total lung capacity and diffusion capacity for carbon monoxide, hematocrit, and red cell mass increased, while plasma volume diminished and heart size and total blood volume were unchanged.     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

A fast computational model for circulatory dynamics: effects of left ventricle–aorta coupling

The course of diseases such as hypertension, systolic heart failure and heart failure with a preserved ejection fraction is affected by interactions between the left ventricle (LV) and the vasculature. To study these interactions, a computationally efficient, biophysically based mathematical model for the circulatory system is presented. In a four-chamber model of the heart, the LV is represented by a previously described low-order, wall volume-preserving model that includes torsion and base-to-apex and circumferential wall shortening and lengthening, and the other chambers are represented using spherical geometries. Active and passive myocardial mechanics of all four chambers are included. The cardiac model is coupled with a wave propagation model for the aorta and a closed lumped-parameter circulation model. Parameters for the normal heart and aorta are determined by fitting to experimental data. Changes in the timing and magnitude of pulse wave reflections by the aorta are demonstrated with changes in compliance and taper of the aorta as seen in aging (decreased compliance, increased diameter and length), and resulting effects on LV pressure–volume loops and LV fiber stress and sarcomere shortening are predicted. Effects of aging of the aorta combined with reduced LV contractile force (failing heart) are examined. In the failing heart, changes in aortic properties with aging affect stroke volume and sarcomere shortening without appreciable augmentation of aortic pressure, and the reflected pressure wave contributes an increased proportion of aortic pressure.

     

Acute effects of 17beta-estradiol on ventricular and vascular hemodynamics in postmenopausal women

Because premenopausal women have lower cardiovascular morbidity than postmenopausal women, it has been proposed that estrogen may have a protective role. Estrogen is involved in smooth muscle relaxation both through its specific receptor as well as through calcium channel blockade. This study examined the acute effect of estradiol on invasive cardiovascular hemodynamics in 18 postmenopausal women (age 62.6 +/- 7.6 years, means +/- SD). The effect of estradiol on left ventricular chamber performance was studied in 9 women using simultaneous left ventricular pressure-volume recordings. In a further group of 9 women, the acute effect of estradiol on arterial function was assessed using input impedance (derived from simultaneous aortic pressure and flow recordings), pressure waveform analysis, and pulse wave velocity. After 2 mg micronized 17beta-estradiol was administered, serum estradiol levels increased from 50.9 +/- 21.9 to 3,190 +/- 2,216 pmol/l, P < 0.0001. There was no effect of estradiol on either left ventricular inotropic or lusitropic function. There was no acute effect of estradiol on arterial impedance, reflection coefficient, augmentation index, or pulse wave velocity. There was a trend to decreased heart rate and cardiac output in both groups of 9 women. Because heart rate and cardiac output were common to both hemodynamic data sets, results for these parameters were pooled. Across all 18 women, there was a small but significant decrease in heart rate (69.2 +/- 10.4 vs. 67.2 +/- 9.9 beats/min, P = 0.02), as well as a significant decrease in cardiac output (4.82 +/- 1.77 vs. 4.17 +/- 1.56 l/min, P = 0.002). Despite achieving supraphysiological serum levels, this study found no significant effect of acute 17beta-estradiol on ventricular or large artery function.     

Normal hemodynamic parameters in conscious Wistar rats

The evolution of different hemodynamic parameters with ponderal growth has been studied in conscious Wistar rats. The thermodilution method has been used to determine cardiac output and related variables. The results suggest that, between animal weight and the different hemodynamic parameters, there is a direct proportional relationship to blood volume, mean arterial pressure, cardiac output, stroke volume and total peripheral resistance, and an indirect proportional relationship to heart rate, cardiac index and stroke volume index. Body weight, therefore, plays a major role in hemodynamic determination, this having to be kept in mind when designing the experiment.     

Parameters describing the pulse wave

Pulse wave analysis permits non-invasive assessment of arterial elasticity indices. The contour varies in different parts of the circulation. It depends on physiological or pathophysiological conditions of the organism. The pathological events like arteriosclerosis or diabetes have a primary effect to the artery elasticity. Hypertension or some heart diseases also influence the pulse wave velocity and resulted in earlier wave reflections. There are several methods of pulse wave measurements based on different principles and depending on the type of measured pulse wave. The evaluation parameters can be assessed from the time domain, derivations, velocity or frequency domain. The main aim of this review article is to offer a recent overview of pulse wave measurement parameters and main results obtained. The principles of pulse wave measurement and current experience in clinical practice are shortly discussed too.     

Mathematical modelling of the dependence of the performance of the left ventrical of the heart on preload and afterload

Results of the numerical simulation of the end-diastolic, end-systolic, and stroke volumes of the left ventricle of the heart are presented. The simulation was based on a published simple kinetic model of cardiac muscle and approximation of the ventricle geometry with thick-wall cylinder where the fiber orientation varied linearly from sub-epicardium towards sub-endocardium. Blood flow was modeled with a liner compartment model. This simplified approach provides correct dependencies of the stroke volume on the preand afterload, namely end-diastolic pressure and peripheral resistance. The calculations show that the stroke volume is independent of arterial compliance and blood inertia.     

Wave Propagation in a Viscous Fluid Contained in an Orthotropic Elastic Tube

The problem of pressure wave propagation through a viscous fluid contained in an orthotropic elastic tube is considered in connection with arterial blood flow. Solutions to the fluid flow and elasticity equations are obtained for the presence of a reflected wave. Numerical results are presented for both isotropic and orthotropic elastic tubes. In particular, the pressure pulse, flow rate, axial fluid velocity, and wall displacements are plotted vs. time at various stations along the ascending aorta of man. The results indicate an increase in the peak value of the pressure pulse and a decrease in the flow rate as the pulse propagates away from the heart. Finally, the velocity of wave propagation depends mainly on the tangential modulus of elasticity of the arterial wall, and anisotropy of the wall accounts in part for the reduction of longitudinal movements and an increase in the hydraulic resistance.     

Hemodynamic effects of anti-G suit inflation in a 1-G environment

This study evaluated effects of various anti-G inflation pressures on cardiac volumes and the relationship of these volume changes to mean arterial pressure changes. Ventricular volumes were calculated using two-dimensional echocardiography. An anti-G suit was inflated to 2, 4, and 6 psi in the standing and supine positions for 10 male subjects. In the supine position, mean arterial pressure increased from base line for all three inflation pressures (P = 0.05). The end-diastolic volume increased after 2-psi inflation (P = 0.03). Cardiac output or stroke volume did not change. After standing, mean arterial pressure (P = 0.002), end-diastolic volume (P = 0.002), and stroke volume (P = 0.05) fell after suit deflation. Peripheral vascular resistance fell in the 2- and 4-psi inflation profiles. In the standing protocol, mean arterial pressure, end-diastolic volume, stroke volume, and cardiac output rose with all three inflation pressures (P less than 0.05). After reclining, heart rate increased (P = 0.02) and mean arterial pressure fell (P less than 0.05) in the 4- and 6-psi inflation profiles after suit deflation. Increases in mean arterial pressure are caused by increases in cardiac preload and cardiac output after inflation of the anti-G suit while subjects were standing. Increased cardiac preload was not consistently seen after inflation while subjects were supine. Changes in end-diastolic volume and mean arterial pressure were dependent on the pressure used to inflate the anti-G suit.     

Assessment of Model Based (Input) Impedance,Pulse Wave Velocity,and Wave Reflection in the Asklepios Cohort

Objectives

Arterial stiffness and wave reflection parameters assessed from both invasive and non-invasive pressure and flow readings are used as surrogates for ventricular and vascular load. They have been reported to predict adverse cardiovascular events, but clinical assessment is laborious and may limit widespread use. This study aims to investigate measures of arterial stiffness and central hemodynamics provided by arterial tonometry alone and in combination with aortic root flows derived by echocardiography against surrogates derived by a mathematical pressure and flow model in a healthy middle-aged cohort.

Methods

Measurements of carotid artery tonometry and echocardiography were performed on 2226 ASKLEPIOS study participants and parameters of systemic hemodynamics, arterial stiffness and wave reflection based on pressure and flow were measured. In a second step, the analysis was repeated but echocardiography derived flows were substituted by flows provided by a novel mathematical model. This was followed by a quantitative method comparison.

Results

All investigated parameters showed a significant association between the methods. Overall agreement was acceptable for all parameters (mean differences: -0.0102 (0.033 SD) mmHg*s/ml for characteristic impedance, 0.36 (4.21 SD) mmHg for forward pressure amplitude, 2.26 (3.51 SD) mmHg for backward pressure amplitude and 0.717 (1.25 SD) m/s for pulse wave velocity).

Conclusion

The results indicate that the use of model-based surrogates in a healthy middle aged cohort is feasible and deserves further attention.     

Noninvasive cardiovascular phenotyping in mice

With the growth of genetic engineering, mice have become common as models of human diseases, which in turn has stimulated the development of techniques to monitor and image the murine cardiovascular system. Invasive methods are often more quantitative, but noninvasive methods are preferred when measurements must be repeated serially on living animals during development or in response to pharmacological or surgical interventions. Because of the small size and high heart rates in mice, high spatial and temporal resolutions are required to preserve signal fidelity. Monitoring of body temperature and the electrocardiogram is essential when animals must be anesthetized for a measurement or other procedure. Several other groups have developed cardiovascular imaging modalities suitable for murine applications, and ultrasound is the most widely used. Our group has developed and applied high-resolution Doppler probes and signal processing for measuring blood velocity in the heart and peripheral vessels of anesthetized mice noninvasively. We can measure cardiac filling and ejection velocities as indices of systolic and diastolic ventricular function and for timing of cardiac events; velocity pulse arrival times for determining pulse-wave velocity and arterial stiffness; peripheral velocity waveforms as indices of arterial resistance, compliance, and wave reflections; stenotic velocities for estimation of pressure drop and detection of vorticity; and tail artery velocity for determining systolic and diastolic blood pressure using a pressure cuff. These noninvasive methods are convenient and easy to apply and have been used to detect and evaluate numerous cardiovascular phenotypes in mutant mice.     

Effect of induced hypothermia on the systemic circulation

Studies of the systemic circulation in dogs (n = 5) during hypothermia showed that cardiac output, mean arterial pressure, total peripheral resistance, pulse rate, work L. V. is reduced and the stroke volume is increased. The authors think that these effects are probably due to metabolic alterations during hypothermia.     

Predicting systolic and diastolic aortic blood pressure and stroke volume in the intact sheep.

We developed a mathematical model describing the interaction between the heart and the arterial system. The model was constructed and tested on basis of invasive hemodynamic data in six sheep. Data from a first group of three animals (49 cardiac cycles) were used to assess a template time-varying elastance curve for the left ventricle, while the baseline steady-state data of a second group of three animals were used to assess reference cardiac and arterial parameters in sheep. The model is fully characterized by nine parameters, which were converted into 6 dimensionless numbers using the Buckingham pi theorem. The model was then used to generate LV pressure and volume and aortic pressure and flow for 86 conditions obtained by varying parameters 50 to 200% of their reference value. Systolic (SBP) and diastolic (DBP) blood pressure and stroke volume (SV) were determined from these model-generated curves and multiple linear regression analysis yielded the following expressions: SBP = Pisovolumic [0.638 - 0.0773 Emax C + 0.0507 RC/T] (r2 = 0.89); DBP = Pisovolumic [0.438-0.0712 Emax C + 0.0655RC/T] (r2 = 0.88) and SV = LVEDV [1.265-1.040 LVEDV/(LVEDV - Vd) + 0.125 Emax C-0.0777RC/T] (r2 = 0.93) with Pisovolumic = Emax (LVEDV - Vd), Emax and Vd being the slope and intercept of the end-systolic pressure-volume relation, R and C the total peripheral resistance and compliance, LVEDV the left ventricular end-diastolic volume, and T the cardiac cycle length. These expressions were validated using data from the second group of three animals obtained during vena cava occlusion at baseline and during administration of dobutamine (61 cycles). The correlation between measured and predicted values was 0.98, 0.97 and 0.92 for SBP, DBP and SV, respectively. Compared to the measured values, SBP and DBP were, on average, underestimated by 5 and 6mmHg, respectively, and SV overestimated by 1.4 ml. We conclude that the derived expressions for blood pressure and stroke volume remain valid in the intact sheep for various hemodynamic conditions, and, taking into account their dimensionless form, may hold in other species and in humans.     

A new similarity principle for cardiac energetics

Similarity criteria of the functional design of the mammalian cardiovascular system are scant. For the analysis of mammalian cardiac energetics physiological parameters such as mean arterial blood pressure, stroke volume, heart rate, metabolic rate and heart and body weights are considered pertinent. Based on these parameters, a new similarity principle is established via allometric equations, dimensional analysis and Buckingham's pi-theorem. The principle states that the ratio of left ventricular external work to metabolic rate is inversely proportional to resting heart rates of mammals. The proportionality constant is dimensionless and is invariant of mammalian body weights.