Estimation of arterial compliance in aortic regurgitation: three methods evaluated in pigs

Three methods for measuring arterial compliance when aortic regurgitation is present are examined. The first two methods are based on a Windkessel model composed of two elements, compliance C and resistance R. Arterial compliance was estimated from diastolic pressure waveforms and diastolic regurgitant flow for one method, and from systolic aortic pressure waveforms and systolic flow for the other method. The third method was based on a three-element Windkessel model, composed of characteristic resistance r, compliance C and resistance R. In this method arterial compliance was calculated by adjusting the model to the modulus and phase of the first harmonic term of the aortic input impedance. The three methods were compared and validated in six anaesthetised pigs over a broad range of aortic pressures. The three methods were found to give quantitatively similar estimates of arterial compliance at mean aortic pressures above 60 mm Hg. Below 60 mm Hg, estimates of arterial compliance varied widely, probably because of poor validity of the Windkessel models in the low pressure range.     

Analytical relationship between arterial input impedance and the three-element Windkessel series resistance

The recently proposed energy-balance method for estimating the series resistance of the three-element Windkessel model is reformulated in the frequency domain. New mathematical expressions are analytically derived, involving Fourier harmonics of pulsatile arterial pressure and flow. It is shown that the series resistance of the arterial three-element Windkessel model can be expressed as a weighted sum of the arterial input impedance moduli.     

The arterial Windkessel

Frank’s Windkessel model described the hemodynamics of the arterial system in terms of resistance and compliance. It explained aortic pressure decay in diastole, but fell short in systole. Therefore characteristic impedance was introduced as a third element of the Windkessel model. Characteristic impedance links the lumped Windkessel to transmission phenomena (e.g., wave travel). Windkessels are used as hydraulic load for isolated hearts and in studies of the entire circulation. Furthermore, they are used to estimate total arterial compliance from pressure and flow; several of these methods are reviewed. Windkessels describe the general features of the input impedance, with physiologically interpretable parameters. Since it is a lumped model it is not suitable for the assessment of spatially distributed phenomena and aspects of wave travel, but it is a simple and fairly accurate approximation of ventricular afterload. J.-W. Lankhaar is supported by a grant from the Netherlands Heart Foundation, the Hague, the Netherlands (NHS2003B274).     

Patient-Specific Modeling of Blood Flow and Pressure in Human Coronary Arteries

Coronary flow is different from the flow in other parts of the arterial system because it is influenced by the contraction and relaxation of the heart. To model coronary flow realistically, the compressive force of the heart acting on the coronary vessels needs to be included. In this study, we developed a method that predicts coronary flow and pressure of three-dimensional epicardial coronary arteries by considering models of the heart and arterial system and the interactions between the two models. For each coronary outlet, a lumped parameter coronary vascular bed model was assigned to represent the impedance of the downstream coronary vascular networks absent in the computational domain. The intramyocardial pressure was represented with either the left or right ventricular pressure depending on the location of the coronary arteries. The left and right ventricular pressure were solved from the lumped parameter heart models coupled to a closed loop system comprising a three-dimensional model of the aorta, three-element Windkessel models of the rest of the systemic circulation and the pulmonary circulation, and lumped parameter models for the left and right sides of the heart. The computed coronary flow and pressure and the aortic flow and pressure waveforms were realistic as compared to literature data.     

Non-invasive assessment of cardiac output during exercise in healthy young humans: comparison between Modelflow method and Doppler echocardiography method

AIMS: The Modelflow method can estimate cardiac output from arterial blood pressure waveforms using a three-element model of aortic input impedance (aortic characteristic impedance, arterial compliance, and systemic vascular resistance). We tested the reliability of a non-invasive cardiac output estimation during submaximal exercise using the Modelflow method from finger arterial pressure waveforms collected by Portapres in healthy young humans. METHODS: The Doppler echocardiography method was used as a reference method. Sixteen healthy young subjects (nine males and seven females) performed a multi-stage cycle ergometer exercise at an intensity corresponding to 70, 90, 110 and 130% of their individual ventilatory threshold for 2 min each. The simultaneous estimation of cardiac output (15 s averaged data) using the Modelflow and Doppler echocardiography methods was performed at rest and during exercise. RESULTS AND CONCLUSION: The Modelflow-estimated cardiac output correlated significantly with the simultaneous estimates by the Doppler method in all subjects (r = 0.87, P < 0.0001) and the SE of estimation was 1.93 L min-1. Correlation coefficients in each subject ranged from 0.91 to 0.98. Although the Modelflow method overestimated cardiac output, the errors between two estimates were not significantly different among the exercise levels. These results suggest that the Modelflow method using Portapres could provide a reliable estimation of the relative change in cardiac output non-invasively and continuously during submaximal exercise in healthy young humans, at least in terms of the relative changes in cardiac output.     

Incorporating Autoregulatory Mechanisms of the Cardiovascular System in Three-Dimensional Finite Element Models of Arterial Blood Flow

The cardiovascular system is a closed-loop system in which billions of vessels interact with each other, and it enables the control of the systemic arterial pressure and varying organ flow through autoregulatory mechanisms. In this study, we describe the development of mathematical models of autoregulatory mechanisms for systemic arterial pressure and coronary flow and discuss the connection of these models to a hybrid numerical/analytic closed-loop model of the cardiovascular system. The closed-loop model consists of two lumped parameter heart models representing the left and right sides of the heart, a three-dimensional finite element model of the aorta with coronary arteries, three-element Windkessel models and lumped parameter coronary vascular models that represent the systemic circulation, and a three-element Windkessel model to approximate the pulmonary circulation. Using the connection between the systemic arterial pressure and coronary flow regulation systems, and the hybrid closed-loop model, we studied how the heart, coronary vascular beds, and arterial system respond to physiologic changes during light exercise and showed that these models can realistically simulate temporal behaviors of the heart, coronary vascular beds, and arterial system during exercise of healthy subjects. These models can be used to study temporal changes occurring in the heart, coronary vascular beds, and arterial system during cardiovascular intervention or changes in physiological states.     

Influence of central and peripheral changes on the hydraulic input impedance of the systemic arterial tree

The input impedance of the systemic arterial tree of the dog has been computed by Fourier analysis. It was shown that a distance between pressure and flow transducers of less than 2 cm results in appreciable errors which manifest themselves mainly in the phase of the input impedance. The input impedance for controls, occlusions at various locations in the aorta, and an increase and decrease of peripheral resistance were studied. For the same experiments, the total arterial compliance was calculated from the peripheral resistance of the diastolic aortic-pressure curve. The characterstic impedance of the ascending aorta was also estimated. The impedance in the control situation may be modelled by means of a 3-element Windkessel consisting of a peripheral resistance and (total) arterial compliance, together with a resistance equal to the characteristic impedance of the aorta. The occlusions of the aorta show that blockage at (and beyond) the trifurcation do not result in a detectable change in input impedance, except for a slight increase of the peripheral resistance. The more proximal an aortic occlusion, the more effect it has on the pattern of the input impedance. When the aorta is occluded at the diphragm, or higher, the single (uniform) tube appears to be a much better model than the Windkessel. Occlusion of one or both carotid arteries increases the mean pressure; consequently not only the peripheral resistance increases but also the total arterial compliance decreases. The Windkessel with increased peripheral resitance and decreased compliance is again a good model. After a sudden release of occlusion of the aorta, the arterial system has a low peripheral resistance and may also be modelled by the Windkessel.     

Identification of the three-element windkessel model incorporating a pressure-dependent compliance

A new one-step computational procedure is presented for estimating the parameters of the nonlinear three-element windkessel model of the arterial system incorporating a pressure-dependent compliance. The data required are pulsatile aortic pressure and flow. The basic assumptions are a steadystate periodic regime and a purely elastic compliant element. By stating two conditions, zero mean flow and zero mean power in the compliant element, peripheral and characteristic resistances are determined through simple closed form formulas as functions of mean values of the square of aortic pressure, the square of aortic flow, and the product of aortic pressure with aortic flow. The pressure across as well as the flow through the compliant element can be then obtained so allowing the calculation of volume variation and compliance as functions of pressure. The feasibility of this method is studied by applying it to both simulated and experimental data relative to different circulatory conditions and comparing the results with those obtained by an iterative parameter optimization algorithm and with the actual values when available. The conclusion is that the proposed method appears to be effective in identifying the three-element windkessel even in the case of nonlinear compliance.     

Can a linear electrical analog model of a mechanical valve predict flow by using a pressure gradient?

The objective was to determine whether a previously developed technique for biological aortic valves could predict flow through a mechanical valve. An electrical analog model of the aortic valve that includes compliance, resistance, and inertance parameters, and corresponding second order differential equations was used to predict flow given a pressure gradient, as previously reported. Simulated pressures and flow were recorded by using a pulse duplicator system. The heart rate was varied from 60 to 180 bpm, and the stroke volume was varied from 22 to 67 cc. Resistance, inertance, and compliance parameters of the governing differential equation were estimated by using a least-squares fit to the measured flow at 120 bpm and 50 cc stroke volume. By using these parameter estimates, flow was calculated for other heart rates and stroke volumes. To achieve a better flow prediction, a nonlinear filter (third order polynomial range calibration equation) was applied to the output of the linear model (flow). The mean error, full-scale error, and spectral error in magnitude and phase between measured and predicted flow were compared. Error in mean flow ranged from 3% at medium flow rates to 90% at low flow rates. The maximum and minimum full scale errors were 12% and 5%, respectively. Error in the harmonics of measured and calculated flow ranged from 0% to 55%. Larger errors were usually present at the higher harmonics. The agreement between measured and calculated flow was better at normal and high flows but rather poor at low flows. The nonlinear filter (range calibration equation) was unable to account for the discrepancies between the measured and calculated flow over all flow ranges. It seems that this linear model and nonlinear filter have limited application, and an alternate nonlinear approach may produce better results.     

Vascular characteristics influence the aortic ultrasound Doppler signal: computer and hydraulic model simulations

There is an increasing demand for non-invasive methods for the assessment of left ventricular function. Ultrasound Doppler methods are promising, and the early systolic flow velocity signal immediately distal to the aortic valve has been used clinically for this purpose. However, the signal is influenced not only by left ventricular ejection but also by systemic vascular characteristics. Their relative contribution to the timevelocity signal has not been analysed in depth previously. A theoretical analysis, based on a three-element Windkessel model, neglecting peripheral outflow in early systole and assuming linear pressure rise, was therefore tested in computer and hydraulic model simulations where peripheral outflow was included. Significant changes in early aortic flow velocity parameters were found when vascular characteristics were altered. As predicted by the theory, with a standardized aortic valve area and aortic pressure change, the simulations confirmed that maximal flow velocity is related to compliance of the aorta and the large arteries, and that maximal acceleration is inversely related to the characteristic impedance of the aorta. Therefore, maximal velocity and acceleration can be used for assessment of left ventricular function only in situations where vascular characteristics can be considered relatively constant or where they can be estimated.     

Pulse wave attenuation measurement by linear and nonlinear methods in nonlinearly elastic tubes.

Reasons for the continuing difficulty in making definitive measurements of pulse wave attenuation in elastic tubes and arteries in the presence of reflections are sought. The measurement techniques available were re-examined in elastic tubes mimicking the arterial compliance nonlinearity, under conditions of strong reflection. The pulse was of physiological shape, and two different pulse amplitudes in the physiological range were used. Measurements of pressure, flow-rate and diameter pulsation allowed the deployment of four of the classical linear methods of analysis. In addition, a method of separating the forward- and backward-travelling waves that does not require linearising assumptions was used, and the attenuation in the forward and reverse directions was calculated from the resulting waveforms. Overall, the results obtained here suggest that a fully satisfactory way of measuring arterial attenuation has yet to be devised. The classical linear methods all provided comparable attenuation estimates in terms of average value and degree of scatter across frequency. Increased scatter was generally found at the higher pulse amplitude. When the forward waveforms from the separation were similarly compared in terms of frequency components, the average value at energetic harmonics was similar to both the value indicated by the linear methods and the values predicted from linear theory on the basis of estimated viscous and viscoelastic parameter data. The backward waveforms indicated a physically unreasonable result, attributed as the expression for this technique of the same difficulties that normally manifest in scatter. Data in the literature suggesting that one of the classical methods, the three-point, systematically over-estimates attenuation were not supported, but it was confirmed that this method becomes prone to negative attenuation estimates at low harmonics as pulse amplitude increases. Although the goal of definitive attenuation measurement remains elusive, the task provides a sensitive tool for the examination of the effect of nonlinearities in the arterial system.     

A new model-based method of reconstructing central aortic pressure from peripheral arterial pressure

We have shown in our previous study that the transfer function between central aortic pressure and tonometric radial arterial pressure can be modeled as a pure elastic lossless tube terminated with a modified Windkessel. We hypothesized, using the model-derived radial arterial flow, that central pressure could be reconstructed by adding the time-shifted forward and backward pressure components (Stergiopulos et al.: Am J Physiol 274: H1386---H1392, 1998). In eight patients (age 16--75), central micromanometric and radial arterial tonometric pressure were measured simultaneously. We imposed measured tonometric pressure to the terminal modified Windkessel to estimate radial arterial flow, with which tonometric pressure was separated into forward and backward components. These components were then appropriately time shifted, and summed to central pressure. We used average parameter values for the terminal impedance, but individualized the transmission delay. The poor correlation (r(2)) between tonometric and central pressure (0.264--0.765) was improved by both central pressure reconstruction methods (generalized transfer function: 0.887--0.974, model-based method: 0.849--0.979). The sensitivity analysis indicated that the key model parameter in reconstructing central pressure was the transmission delay. We conclude that our model-based method was capable of reconstructing central pressure as precisely as the generalized transfer function method, and also capable of individualizing the transfer function by changing the transmission delay.     

Isolated aorta setup for hemodynamic studies

A setup consisting of a high-performance hydraulic pump connected to the ascending part of an isolated aorta, including all major distal branches, each loaded with calibrated artificial resistors, was developed. The system was used to study total aortic compliance of the baboon as a function of mean aortic pressure (n=5). The aorta loaded with the resistors was mounted in a custom-designed sink table, such that it was submersed in physiological saline maintained at 37°C. Mean distending pressure in the entire aortic compliance from pressure and flow waves generated by the pump. Total aortic compliance as a function of mean pressure was fitted with a logarithmic function: Ln (Compliance)=A+B * P. The value of A(±SE) was: 1.565±0.319 and B: −0.020±0.003 (P<0.001). The results were compared with previously published results (also using the same three-element Windkessel fit) obtained in three of the same animalsin vivo. Thein vivo data were A: 1.095±0.235 and B: −0.019±0.003.In vitro data had a significantly higher value of A thanin vivo (P=0.017), implying a significantly higher aortic compliancein vitro thanin vivo. Occlusion of the proximal descending aorta was performed at a low distending pressure (55 mm Hg) to determine the proximal complicance. It was found (n=4) that 46±11% (SD) of the total arterial compliance is to be attributed to the ascending and proximal descending aorta. This work was supported in part by Grant RG 86/0066 from the scientific affairs division of Nato.     

天麻素对动脉血管顺应性以及血流动力学的影响

本文旨在研究天麻素对动脉血管顺应性和血流动力学等的作用。采用改良风箱模型来计算动脉管的顺应性和血管中血流惯性。在静脉注射天麻素前后，分别记录和计算出狗的血压，心输出量，外周阻力，血流惯性以及中央和外周动脉血管的顺应性。结果表明天麻素具有降低血压和外周务管阻力，增加动脉血管中血流惯性，以及中央和外周动脉血管的顺应性等作用。因此，天麻素是一种有效的能够改善由血管顺应性下降所致的高血压－老年性高血压的中     

Finite element solution of steady state temperature distribution in the human torso

A finite element model of the steady state temperature distribution in the human torso is developed. The torso is approximated by a circular cylinder of core surrounded by a layer of muscle and insulating layers of fat and skin. The model is simplified by neglecting longitudinal heat flow. The region occupied by a circular cross-section of the torso is discretized into a mesh of triangles and the boundary of the torso, that is, the skin surface, is consequently approximated by a polygon. The elliptic partial differential equation governing the steady state temperature distribution, together with the associated boundary conditions, are expressed in equivalent variational form. Linear basis functions are used and the resulting integral is minimized over the region bounded by the approximating polygon. Results for two numerical experiments are determined by solving systems of linear equations.     

The influence of a nonlinear resistance element upon in vitro aortic pressure tracings and aortic valve motions

In vitro testing of biological heart valves requires pressure and flow waveforms closely simulating natural conditions, which are mainly influenced by the characteristics of the vascular system. Simulation of the arterial function in artificial circulations was mostly performed by the useful Windkessel model but sometimes failed by generating inadequate systolic pressures. The integration of a novel nonlinear resistance element may improve the Windkessel function. Native porcine aortic valves were studied in a mock circulation with a novel nonlinear resistance element combined with the Windkessel compared with an aperture plate resistance. Pressure and flow measurements were performed at varying heart rates and stroke volumes and analyzed in the time and frequency domain. Aortic valve motions were evaluated using high speed video recording. With the classical afterload configuration including an aperture plate resistance, the pressure tracings showed a nonphysiologic decrease of pressure during systole after early peak pressure. By integration of the novel nonlinear resistance, peak systolic pressure occured later, peak pressure was higher, and the pressure waveform was more physiologically shaped. Leaflet motions of the aortic valves were less oscillatory and compared well with in vivo characteristics. In conclusion, a novel nonlinear resistance element in a mock circulation has the potential to provide more physiologic aortic pressure waveforms as influencing aortic valve dynamics and thus may be a helpful tool for investigation of biological heart valves.     

基于LabVIEW的心血管动力学参数分析与研究

目的:根据脉搏波理论设计出一套基于虚拟仪器的心血管动力学参数分析系统。方法:本文以LabVIEW7.0作为软件开发平台,根据示波法原理和弹性腔理论总结提炼出收缩压、平均压、舒张压、心率、每搏心输出量、每分心输出量、外周阻力等参数。结果:通过心血管动力学参数的分析能判断心血管病患者与正常人的差异性,对心血管病的预防及疾病早期发现都能发挥很大的作用。大量的实测结果表明系统有较好的准确性和稳定性。结论:此采集分析系统有操作便捷,灵活性高,可扩展性好的特点,可以直观地观测病人的心血管状况以便及时的对心血管疾病采取措施和进行预防,为心血管动力学参数的分析研究提供了一种简便有效的途径。     

Ultrasound image-based computer model of a common carotid artery with a plaque

Ultrasound scans were acquired from a common carotid artery in a patient with an early atherosclerotic plaque forming a mild asymmetrical stenosis. The 3D vascular geometry of the diseased arterial segment was reconstructed from a series of 2D cross-sectional images, and computational meshes for the flow and wall domains were developed. Numerical flow simulations incorporating coupled fluid–solid interaction were implemented using flow and pressure waveforms measured in vivo. The effects of wall distensibility were investigated by comparing the predictions obtained with different wall compliance, one with ‘natural’ compliance and another with a stiffer wall. Limited flow separation was predicted in the post-stenotic zone. The non-uniform thickness of the diseased wall restricted the wall motion locally and re-distributed the stress, giving raised concentrations at the plaque shoulders.     

Non-invasive pulsatile arterial pressure and stroke volume changes from the human finger

In this paper we review recent developments in the methodology of non-invasive finger arterial pressure measurement and the information about arterial flow that can be obtained from it. Continuous measurement of finger pressure based on the volume-clamp method was introduced in the early 1980s both for research purposes and for clinical medicine. Finger pressure tracks intra-arterial pressure but the pressure waves may differ systematically both in shape and magnitude. Such bias can, at least partly, be circumvented by reconstruction of brachial pressure from finger pressure by using a general inverse anti-resonance model correcting for the difference in pressure waveforms and an individual forearm cuff calibration. The Modelflow method as implemented in the Finometer computes an aortic flow waveform from peripheral arterial pressure by simulating a non-linear three-element model of the aortic input impedance. The methodology tracks fast changes in stroke volume (SV) during various experimental protocols including postural stress and exercise. If absolute values are required, calibration against a gold standard is needed. Otherwise, Modelflow-measured SV is expressed as change from control with the same precision in tracking. Beat-to-beat information on arterial flow offers important and clinically relevant information on the circulation beyond what can be detected by arterial pressure.     

Estimating respiratory mechanical parameters of ventilated patients: a critical study in the routine intensive-care unit

Airflow and pressure were measured post-operatively in eight mechanically ventilated patients in the routine intensive care unit. Analysis of the input impedance spectra versus frequency suggested that respiratory data cannot be adequately reproduced using the classic two-element R-C model, as the real part of input impedance decreases with frequency. To fit in with this behaviour, we adopted a three-element model with an additional parallel compliance. The three parameters of this model were estimated separately in the frequency and time domains by minimising suitable least-square criterion functions. The results demonstrate a good agreement between the parameter estimates in the frequency and time domains, and show that the three-element model reproduces the input impedance frequency pattern in the range 0.2–8 Hz. Comparison of different linear models in the time domain demonstrated that the precision of parameter estimates and the quality of best fitting sharply increase from the two-element to the three-element model. The addition of a fourth resistive parameter, like in the Mead model, does not lead to appreciable improvement and makes the model almost unidentifiable. The possible contribution of a ventilator-patient circuit of the upper airway shunting and of the peripheral airway obstruction are also discussed.