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.     

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).     

Arterial windkessel parameter estimation: A new time-domain method

We developed and validated a new, more accurate, and easily applied method for calculating the parameters of the three-element Windkessel to quantitate arterial properties and to investigate ventriculoarterial coupling. This method is based on integrating the governing differential equation of the three-element Windkessel and solving for arterial compliance. It accounts for the interaction between characteristic impedance and compliance, an important phenomenon that has been ignored by previously implemented methods. The new integral method was compared with four previously published methods as well as a new independent linear least-squares analysis, using ascending aortic micromanometric and volumetric flow measurements from eight dogs. The parameters calculated by the new integral method were found to be significantly different from those obtained by the previous methods but did not differ significantly from maximum likelihood estimators obtained by a linear leastsquares approach. To assess the accuracy of parameter estimation, pressure and flow waveforms were reconstructed in the time domain by numerically solving the governing differential equation of the three-element Windkessel model. Standard deviations of reconstructed waveforms from the experimental ensemble-averaged waveforms, which solely reflect the relative accuracy of the Windkessel parameters given by the various methods, were calculated. The new integral method invariably yielded the smallest error. These results demonstrate the improved accuracy of our new integral method in estimating arterial parameters of the three-element Windkessel.     

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.     

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.     

Fast Estimation of Arterial Vascular Parameters for Transient and Steady Beats with Application to Hemodynamic State under Variant Gravitational Conditions

Numerous parameter estimation techniques exist for characterizing the arterial system using electrical circuit analogs. These techniques are often limited by requiring steady-state beat conditions and can be computationally expensive. Therefore, a new method was developed to estimate arterial parameters during steady and transient beat conditions. A four-element electrical analog circuit was used to model the arterial system. The input impedance equations for this model were derived and reduced to their real and imaginary components. Next, the physiological input impedance was calculated by computing fast Fourier transforms of physiological aortic pressure (AoP) and aortic flow. The approach was to reduce the error between the calculated model impedance and the physiological arterial impedance using a Jacobian matrix technique which iteratively adjusted arterial parameter values. This technique also included algorithms for estimating physiological arterial parameters for nonsteady physiological AoP beats. The method was insensitive to initial parameter estimates and to small errors in the physiological impedance coefficients. When the estimation technique was applied to in vivo data containing steady and transient beats it reliably estimated Windkessel arterial parameters under a wide range of physiological conditions. Further, this method appears to be more computationally efficient compared to time-domain approaches. © 1999 Biomedical Engineering Society. PAC99: 8719Uv, 8710+e, 0230Qy     

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.     

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.     

硝普钠对动物血管力学参数的影响

有研究表明硝普钠具有增加动脉顺应性的趋势和降低外周阻力的作用,但是硝普钠对血管中血流惯性的影响却不统一。本文以Coldwyn等建立的动脉系改良风箱力学模型为主要研究方法对动物动脉血管顺应性(包括中央和外周顺应性),血流惯性和外周阻力等进行了研究,并建立了动脉系总阻抗的公式。然后评价硝普钠对总阻抗的影响以及总阻抗做为一种血管力学参数的灵敏性。实验结果表明:硝普钠具有统计学意义地增加外周血管的顺应性(P<0.05);有增加中央动脉顺应性和血流惯性的趋性;并能降低外周阻力和动脉系总阻抗(P<0.05)。结果还表明动脉系总阻抗是一种灵敏度高的血管力学参数。因此,硝普钠能影响血管参数,对外周血管顺应性和血管总阻抗的作用尤其明显。     

Outflow Boundary Conditions for Arterial Networks with Multiple Outlets

Simulation of blood flow in three-dimensional geometrically complex arterial networks involves many inlets and outlets and requires large-scale parallel computing. It should be based on physiologically correct boundary conditions, which are accurate, robust, and simple to implement in the parallel framework. While a secondary closure problem can be solved to provide approximate outflow conditions, it is preferable, when possible, to impose the clinically measured flow rates. We have developed a new method to incorporate such measurements at multiple outlets, based on a time-dependent resistance boundary condition for the pressure in conjunction with a Neumann boundary condition for the velocity. Convergence of the numerical solution for the specified outlet flow rates is achieved very fast at a computational complexity comparable to the widely used Resistance or Windkessel boundary conditions. The method is verified using a patient-specific cranial vascular network involving 20 arteries and 10 outlets.     

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     

Validity of the transfer function model of the human arterial system of the lower limb in man

A transfer function model of the human arterial system has been examined and tested on a group of 21 patients undergoing reconstructive arterial surgery in the leg. It was found that use of the model to estimate parameters relating to proximal (above femoral artery) arterial wall stiffness and radius is not reliable, the model being affected by the peripheral resistance. The value of the model parameters relating to the peripheral arterial radius is more consistent with the theory although there are still contradictions. It is concluded that the application of transfer function modelling to the diagnosis of arterial disease should be approached with caution.     

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.     

Multi-branched model of the human arterial system

A model of the human arterial system was constructed based on the anatomical branching structure of the arterial tree. Arteries were divided into segments represented by uniform thin-walled elastic tubes with realistic arterial dimensions and wall properties. The configuration contains 128 segments accounting for all the central vessels and major peripheral arteries supplying the extremities including vessels of the order of 2·0 mm diameter. Vascular impedance and pressure and flow waveforms were determined at various locations in the system and good agreement was found with experimental measurements. Use of the model is illustrated in investigating wave propagation in the arterial system and in simulation of arterial dynamics in such pathological conditions as arteriosclerosis and presence of a stenosis in the femoral artery.     

Effects of oscillation on the arteries measured by arterial impedance during left ventricular assistance

Oscillating blood flow has effects on the arteries similar to the effects of pulsatile blood flow at lower frequencies. Alternating-current theory is useful to study the pulse in the circulatory system. Arterial impedance is a good index to estimate the frequency characteristics of the artery. In this study, total vascular resistance and arterial impedance were studied in animal experiments during left ventricular assistance. A centrifugal pump was used for comparison with a VFP (vibrating-flow pump). Left ventricular assistance was performed in animal experiments using goats. Total vascular resistance and arterial impedance were studied to estimate the frequency characteristics of the artery. Total vascular resistance during steady flow assistance decreased compared with that during nonassistance. Arterial walls were extended by the blood flow assistance at steady flow. The resistance during oscillating blood flow was different at each driving frequency, showing the frequency dependency, or pulse effect, of the arterial system under nonsteady flow. Arterial impedance was also studied during oscillating blood flow and showed a slight increase at a driving frequency of 25 or 30 Hz. These fluctuations in impedance are influenced by the reflection of the pulse. Arterial impedance should be taken into consideration when analyzing pulsatile blood flow because the pulse reflection may have some effects on the arterial wall. Some variation of blood pressure and blood flow might be necessary for stable support with artificial circulatory assistance.     

Computing total arterial compliance of the arterial system from its input impedance

The total arterial compliance of the arterial system was computed from its input impedance by expressing the impedance in terms of its frequency-response vector diagram (f.r.v.) The f.r.v. plot of a 3-element windkessel subjected to random pacing follows, theoretically, a circular path. Since the windkessel model serves as a good approximation for the arterial system, we have used the simple properties of its f.r.v. plot to obtain the compliance, which is otherwise normally determined from the peripheral resistance and the time constant of the diastolic pressure decay. The arterial compliance can also be determined from the impulse response function of the arterial system. Data obtained from dog experiments during no intervention, aortic occlusion and during occlusion of both carotid arteries have been analysed.     

Wave reflection and hydraulic impedance in the healthy arterial system: a controversial subject

According to the prevailing view today, wave reflections play an important part in determining the pressure and flow waveforms in the ascending aorta, and therefore the arterial input impedance. This review surveys the literature and shows that it is not clear whether reflections are diffuse, arising along a continuous length of the system or localised with discrete reflecting sites. Moreover, if they are localised, there is no general agreement as to the exact location of the reflecting site or sites. According to another view, however, significant wave reflections are absent in the central arterial system. This view, if accepted calls for an explanation of the pressure and flow waveforms as observed in the arterial system.     

Norwood procedure with non-valved right ventricle to pulmonary artery shunt improves ventricular energetics despite the presence of diastolic regurgitation: a theoretical analysis

When the Norwood procedure is conducted for the hypoplastic left heart syndrome using a non-valved right ventricle (RV) to pulmonary artery (PA) shunt, diastolic regurgitation from PA to RV may have an adverse effect on postoperative hemodynamics. In this study, we examined the impact of the diastolic regurgitation on ventricular energetics by computational analysis using a combination of a time-varying elastance chamber model and a modified three-element Windkessel vascular model. This study revealed that use of the valved or non-valved RV-PA shunt eliminated pulmonary over-circulation which was observed when using the systemic to pulmonary artery shunt (modified Blalock–Taussig shunt). Although the valved RV-PA shunt improved pulmonary blood supply and consequently increased pulmonary artery flow and oxygen saturation compared to the non-valved RV-PA shunt, the non-valved RV-PA shunt improved ventricular energetics in spite of the presence of PA to RV regurgitation.     

外周阻力的测量与临床研究

目的：研究外周阻力与脉搏波之间的关系及其外周阻力在临床中的应用。方法：利用脉搏波理论和弹性腔模型提出一种便捷易行的外周阻力算法。采用临床中测试高血压患者125人和健康者125人对外周阻力与血压进行相关性分析。结果：通过大量病例分析和临床测试证实了算法的有效性和可靠性，而且此算法已经应用LabVIEW7．0开发成软件并在医院进行临床应用。结论：外周阻力能帮助人们了解自己的动脉硬化情况，随时提醒人们注意身体状况，减少疾病的发生，同时也能够为医生判断病人的心血管系统疾病提供重要的参考指标。