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.     

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.     

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.     

Dynamic model of the short-term regulation of arterial pressure in the cat

The purpose of this study was to characterise the dynamics of the short-term control of arterial pressure in the cat with the aid of a model consisting of a nonlinear negative-feedback control system. The arterial system was described by a three element windkessel model (peripheral resistance, R, aortic characteristic impedance, R_c, and total arterial compliance, C). The resistance regulation was represented by a second-order system with static gain G_R, a damping factor σ and an undamped natural frequency ω_n. The resistance gain, G_R, and the windkessel parameters were obtained from measurements of aortic and venous pressures and cardiac output in two steady states. The parameters σ and ω_n were estimated from mean pressure and mean flow during the transient from control to the new steady state. Pressure reductions averaged 10 per cent and resistance changes averaged 12 per cent. Average windkessel model parameters in the control condition were: C=(25·9±6·1) 10⁻⁶ g⁻¹ cm⁴ s², R_c=(2·51±0·53) 10³ g cm⁻⁴ s⁻¹, R=(40·9±9·8) 10³ g cm⁻⁴ s⁻¹. Average estimates of parameters of the resistance regulator were: G_R=(4·14±2·38) 10⁻³ min ml⁻¹, ω_n = 1·0 ± 1·0 rad s⁻¹, σ=0·41±0·19. A satisfactory fit was found between model predicted and measured pressure. The results suggest that the dynamic short-term control of pressure is underdamped and oscillatory. The amplitude of these oscillations is affected by arterial compliance, suggesting an interaction between the arterial system and short-term resistance regulation.     

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.     

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.     

Quantification of aortic regurgitation by Doppler echocardiography: a new method evaluated in pigs

We have developed a method to quantify aortic regurgitant orifice and volume, based on measurements of the velocity of the regurgitant jet, aortic systolic flow, the systolic and diastolic arterial pressures, a Windkessel arterial model, and a parameter estimation technique. In six pigs we produced aortic regurgitant flows between 2·1 and 17·8 ml per beat, i.e. regurgitant fractions from 0·06 to 0·58. Pulmonary and aortic flows were measured with electromagnetic flow probes, aortic pressure was measured invasively, and the regurgitant jet velocity was obtained with continuous-wave Doppler. The parameter estimation procedure was based on the Kalman filter principle, resulting primarily in an estimate of the regurgitant orifice area. The area was multiplied by the velocity integral of the regurgitant jet to estimate regurgitant volume. A strong correlation was found between the regurgitant volumes obtained by parameter estimation and the electromagnetic flow measurement. These results from our study in pigs suggest that it may be possible to quantify regurgitant orifice and volume in patients completely noninvasively from Doppler and blood pressure measurements.     

Wave transmission and input impedance of a model of skeletal muscle microvasculature

We analyzed wave transmission properties and input impedance of a microvascular network model. The model, derived from rat spinotrapezius muscle and previously described and validated by other investigators for steady pressure-flow relations, was expanded to include pulsatile phenomena. Microvessels are considered purely elastic, with compliances a function of vessel type; viscous dissipation follows Poiseuille's law. Linear and nonlinear results are presented. In the nonlinear case, shear rate-dependent viscosity of blood and transmural pressure-dependent vascular diameters were calculated and small signal perturbations were imposed around several working points. We investigated effects on input impedance of physiological variability of network parameters and structure: distribution of capillary diameters, capillary segment length, and presence or absence of cross-connecting capillaries. Results show that although wave transmission properties are complex, input impedance is simple. Apparent wave speeds differ substantially from phase velocities and change markedly from branch to branch; pressure and flow waves appear to travel at different speeds. These features result from the mesh-like structure of the network and the prominence of reflection at branchpoints. Input impedance displays a similar form under all conditions: Magnitude is a monotonically decreasing function of frequency, and phase decreases from 0 to approximately −45°. Consideration of the characteristic impedance of a microvessel leads to modification of the three-element Windkessel as a reduced model of the observed input impedance.     

Continuous measurement of aortic radius changein vivo with an intra-aortic ultrasonic catheter

The radii of the inner and outer walls of the aorta and the intravascular blood pressure were recorded simultaneously in the descending thoracic aorta of intact, living dogs using 7·5 MHz ultrasound. Blood pressure and the A-mode signals containing wall echoes were also recorded on videotape which was later replayed for processing. Thein vivo data were compared with data obtained on the same vessels post mortem. The change in radius due to a pressure change from 80 to 125 mmHg was calculated from thein vivo andin vitro data. After normalising the radius changes with respect to the radius at 80 mm Hg, the ratio of thein vivo andin vitro values ranged from 0·66 to 1·36 with a mean of 0·94. The changes in radius were comparable with previously reported values obtained using various techniques.     

Linear-lumped-parameter modeling of pulmonary impedance in monkeys

Pulmonary impedance, Z_L, measured from 2 to 32 Hz in anesthetized, intubated and paralyzed bonnet monkeys (Macaca radiata) was fitted to a variety of linearlumped parameter mechanical networks. Parameter values for each network were obtained by minimizing the average of the percent distance, D_r, between the computed network impedance and measured Z_L at all frequencies. Measured resistance, R_L, decreased from 2 to 8 Hz and increased from 8 to 32 Hz indicating that a single series resistance-inertance-compliance (RIC) network was not optimal (D_r∼19%). Networks consisting of two series RIC pathways in parallel resulted in a lower D_r (∼14%), but parameter values were difficult to interpret. Despite not modeling the decrease in R_L with frequency below 8 Hz, an airway wall compliance, C _aw , network in which the airways were separated into central and peripheral components resulted in an even lower D_r (∼11%). In addition, parameter values were easy to interpret, consistent among our “normal” monkeys and changed consistently and explainably with change in lung mechanics induced by decrease in lung volume. We conclude that (1) networks containing both parallel pathways and C _aw are necessary to model Z_L over the entire frequency range (2–32 Hz), (2) the effect of C _aw is an important determinant of Z_L above 8 Hz, and (3) a six-parameter C _aw network with the ratio of C _aw to parenchymal compliance, C_p, fixed may prove useful in interpreting changes in Z_L induced by alterations in lung mechanics in monkeys.     

In Vitro and finite-element model investigation of the conductance technique for measurement of aortic segmental volume

This investigation examined the feasibility of applying the conductance catheter technique for measurement of absolute aortic segmental volume. Aortic segment volume was estimated simultaneouslyin vitro by using the conductance catheter technique and sonomicrometer crystals. Experiments were performed in five isolated canine aortas. Vessel diameter and pressure were altered, as were the conductive properties of the surrounding medium. In addition, a three-dimensional finite-element model of the vessel and apparatus was developed to examine the electric field and parallel conductance volume under different experimental conditions. The results indicated that in the absence of parallel conductance volume, the conductance catheter technique predicted absolute changes in segmental volumes and segmental pressure-volume relationships that agreed closely with those determined by sonomicrometry. The introduction of parallel conductance volume added a significant offset error to measurements of volume made with the conductance catheter that were nonlinearly related to the conductive properties of the surrounding medium. The finite-element model was able to predict measured resistance and parallel conductance volume, which correlated strongly with those measuredin vitro. The results imply that absolute segmental volume and distensibility may be determined only if the parallel conductance volume is known. If the offset volume is not known precisely, the conductance catheter technique may still be applied to measure absolute changes in aortic segmental volume and compliance.     

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     

Short-term regulation of arterial pressure and the calculation of open-loop gain in the intact anesthetized dog

Open-loop gain of the short-term systemic pressure regulation was determined under closed-loop conditions in the closed chest anesthetized dog (n=5). For this purpose, cardiac output and mean systemic pressure were varied by ventricular pacing after the production of complete heart block. From the pressure-flow data resistance gain (the ratio of peripheral resistance change to pressure change in the steady state) was obtained by means of a simple model. The value of this gain was automatically estimated by fitting the pressure-flow relation described by the model to the experimental data. The model allows the pressure-flow relation to be straight or curved with or without a zero-flow pressure intercept. The best fit was obtained when the pressure-flow curve was convex to the pressure axis and had no intercept. When the model was linearized about the control values of pressure and flow (operating point), open-loop gain could be calculated from resistance gain. Its averaged value in the control condition, 1.63±0.45, is in agreement with values found by other investigators in open-loop conditions. During vasoconstriction open-loop gain, at the (new) operating point, increased to 2.51±0.51; during vasodilation it decreased to 1.17±0.27. Open-loop gain about an operating point thus can be determined in the intact animal from measurements of mean pressure and mean flow in the steady state.     

Comparison of arterial waves derived by classical wave separation and wave intensity analysis in a model of aortic coarctation

Coarctation of the aorta may develop during fetal life and impair quality of life in the adult because upper body hypertension and aneurysm formation in the descending aorta may develop. We used our computational model of the young adult arterial circulation, incorporated aorta coarctation over a range from 0 to 80% and evaluated the effects in terms of forward pressure (P ⁺ ) and backward pressure (P ⁻ ). Predictions at several sites proximal and distal to the coarctation using an impedance-based waveform separation method (WSA) and the time-domain technique of wave intensity analysis (WIA) yielded comparable outcomes. A large reflected backward compression wave was seen proximal to the coarctation. Both techniques, WSA and WIA, gave the same results in terms of P ⁺ and P ⁻ . A descending index (DI) was formulated as the difference between peak systolic pressure and valve closure pressure, divided by the pulse pressure. DI increased with stenosis severity for mild to moderate aortic coarctations that did not yet cause evident hypertension. This index may allow for early diagnosis by noninvasive estimation of coarctation severity.     

Macromolecular Transport in the Arterial Wall: Alternative Models for Estimating Barriers

Early atherosclerosis, or atherogenesis, is characterized by the abnormal accumulation of plasma-borne macromolecules (e.g., LDL) in the arterial intima. The change of barrier characteristics of tissue in the arterial wall requires evaluation of macromolecular transport across the endothelial cell layer (ECL) and internal elastic lamina (IEL), the luminal and abluminal boundaries of the arterial intima, respectively. In this study, alternative mathematical models are derived from dynamic mass balances to describe macromolecular transport across the arterial wall. One model considers each medial layer as a spatially lumped compartment, whereas another model consists of a spatially lumped intima and spatially distributed media. Model simulations of a tracer concentration distribution in the arterial wall are compared with concentration distributions of horseradish peroxidase (HRP) after i.v. injection in mice. For each model, optimal parameter values are obtained that yield model outputs matching the data well for two different HRP circulation times. The model parameter estimates show that the ECL is the major barrier for macromolecular transport across the normal arterial wall. Sensitivity analysis indicates that the parameter estimates of the transport coefficients of the ECL and IEL are well determined. Optimal circulation times are determined and expected to yield improved precision of parameter estimates in future experiments to reflect disease progression.     

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.     

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.     

Analytic solution of the Variable-Volume Double-Pool urea kinetics model applied to parameter estimation in hemodialysis

An analytic solution of the Variable-Volume Double-Pool urea kinetics model and its application to the estimation of clinically relevant parameters of the patient-machine system, are presented. These include the urea distribution volume and generation rate and the mean dialyzer clearance. The estimation of these parameters is based on the assumption of constant values for the diffusion coefficient between the two pools and the intra-extracellular volume ratio. Results obtained by computer simulations show that the effect of a ± 50% variation of these parameters influences the estimates less than standard measurement errors.

Starting from these results, four methods to in vivo estimate the urea distribution volume and generation rate from blood samples are compared. Two methods are based on the analytic solution of the double-pool model using seven samples (reference method) or three samples (new clinical method). The remaining methods are based on urea mass-balance and are largely used in the clinical practice. These last techniques differ from each other for the blood sample taken at the end of the treatment or 30 min later.

The results obtained from hemofiltration sessions show that the urea generation rate is accurately estimated by all methods. The total distribution volume is still accurately estimated by the new clinical method while it is systematically underestimated by the urea mass-balance when the blood sample at the end of dialysis is used. Instead, a high overcompensation results using the blood sample taken 30 min after the end of dialysis. Finally, the new clinical method also provides reliable estimates for the dialyzer clearance starting from only three blood samples all taken during dialysis.     

Computer modeling of the effects of aortic valve stenosis and arterial system afterload on left ventricular hypertrophy

The degree of left ventricular hypertrophy is generally thought to reflect the severity of aortic stenosis. However, the compounded influence of arterial system load is poorly understood. We developed a computer model to investigate the effects of aortic valve stenosis in combination with various systemic arterial parameters in the development of left ventricular hypertrophy. Data show that an increased peripheral resistance and/or aortic valve resistance, results in an increase in left ventricular wall thickness and mass, while peak systolic wall stress remains constant. Changing arterial compliance to above normal level would not induce significant changes in wall thickness, while reduction in arterial compliance below normal would cause an increase in ventricular wall thickness. When a double load is imposed on the left ventricle by way of a stenotic valve and an increased arterial afterload, a greater and an aggregated increase in wall thickness results, hastening the hypertrophic process.     

Assessment of aortic pressure power components and their link to overall elastic and resistive arterial properties

This paper reviews the analytical expressions for in-phase and quadrature aortic power components associated with the real and imaginary parts of aortic input admittance, respectively. It is shown that active power Wact, and its steady, Wstdy, and pulsatile, Wpuls, components logically follow from in-phase power. Reactive power follows from quadrature power only for sinusoidal signals. The definition of reactive power indexes for real aortic pressures and flows requires extreme care. The link between overall arterial properties and pressure power components (and indexes) is investigated, making use of a three-element windkessel model and ascending aortic pressure and flow data taken from eight anaesthetised dogs, under basal state and after treatment with a vasoconstrictor (methoxamine). Seven dogs are normotensive in the baseline state (NBA cases, n = 7), the average (+/- SE) of mean pressure being 86.5 +/- 5.2 mmHg. The eighth dog has a baseline mean pressure of 134 mmHg and is considered to be hypertensive. The two experimental cases from this dog are grouped with those from the other seven dogs after vasoconstriction, to form the NVC + H group (n = 9). On average, fitting the model to the experimental data yields a 100% increase (p < 0.05) in total peripheral resistance, a 63% decrease (p < 0.01) in total arterial compliance and a 10% decrease (p > 0.05) in aortic characteristic impedance, from the NBA group to the NVC + H. Correspondingly, the peak-to-peak amplitude of quadrature power shows a 69% increase (p < 0.02). Wact, Wstdy, and Wpuls show a 28% increase (p > 0.05), a 40% increase (p < 0.02) and a 43% decrease (p > 0.05), respectively. Energetic efficiency of the arterial system, Eart = 1 - (Wpuls/Wact), increases by 8% (p < 0.02). From analysis of the estimates of power components and arterial parameters in relation to low-frequency phase angles of aortic impedance, it is concluded that the decrease in total arterial compliance with increasing pressure reduces the power lost in pulsation. This happens at the expense of an increase in quadrature power and absolute values of related reactive power indexes.