Four-element Windkessel Model Research Articles

Temporal complexity in photoplethysmography and its influence on blood pressure.

Objective: The temporal complexity of photoplethysmography (PPG) provides valuable information about blood pressure (BP). In this study, we aim to interpret the stochastic PPG patterns with a model-based simulation, which may help optimize the BP estimation algorithms. Methods: The classic four-element Windkessel model is adapted in this study to incorporate BP-dependent compliance profiles. Simulations are performed to generate PPG responses to pulse and continuous stimuli at various timescales, aiming to mimic sudden or gradual hemodynamic changes observed in real-life scenarios. To quantify the temporal complexity of PPG, we utilize the Higuchi fractal dimension (HFD) and autocorrelation function (ACF). These measures provide insights into the intricate temporal patterns exhibited by PPG. To validate the simulation results, continuous recordings of BP, PPG, and stroke volume from 40 healthy subjects were used. Results: Pulse simulations showed that central vascular compliance variation during a cardiac cycle, peripheral resistance, and cardiac output (CO) collectively contributed to the time delay, amplitude overshoot, and phase shift of PPG responses. Continuous simulations showed that the PPG complexity could be generated by random stimuli, which were subsequently influenced by the autocorrelation patterns of the stimuli. Importantly, the relationship between complexity and hemodynamics as predicted by our model aligned well with the experimental analysis. HFD and ACF had significant contributions to BP, displaying stability even in the presence of high CO fluctuations. In contrast, morphological features exhibited reduced contribution in unstable hemodynamic conditions. Conclusion: Temporal complexity patterns are essential to single-site PPG-based BP estimation. Understanding the physiological implications of these patterns can aid in the development of algorithms with clear interpretability and optimal structures.

Analytical solution to Windkessel models using piecewise linear aortic flow waveform

Objective. Deriving time-domain analytical solutions to two- three- and four-element Windkessel models, which are commonly used in teaching and research to analyse the behaviour of the arterial pressure-flow relationship. Approach. The governing (first-order, non-homogeneous, linear) differential equations are solved analytically, based on a piecewise linear function that can accurately approximate typical aortic flow waveforms. Main results. Closed-form expressions for arterial pressure are obtained both in transient conditions and in steady-state periodic regime. Significance. In most cases Windkessel models are studied in the frequency domain and when studied in the time domain, numerical methods are used. The main advantage of the proposed expressions is that they are an explicit, exact, and easily understood mathematical description of the model behaviour. Moreover, they avoid the use of Fourier analysis or numerical solvers to integrate the differential equations.

Blood pressure assessment with in-ear photoplethysmography

Objective. In this study, we aimed to estimate blood pressure (BP) from in-ear photoplethysmography (PPG). This novel implementation provided an unobtrusive and steady way of recording PPG, whereas previous PPG measurements were mostly performed at the wrist, finger, or earlobe. Methods. The time between forward and reflected PPG waves was very short at the ear site. To minimize errors introduced by feature extraction, a multi-Gaussian decomposition of in-ear PPG was performed. Both hand-crafted and whole-based features were extracted and the best combination of features was selected using a backward-search wrapper method and evaluated by the Akaike information criteria. Hemodynamic parameters such as compliance and inertance were estimated from a four-element Windkessel (WK4) model, which was used to pre-classify PPG signals and generate different BP estimation algorithms. Calibration was done by using previous measurements from the same class. To validate this novel approach, 53 subjects were recruited for a one-month follow-up study, and 17 subjects were recruited for a two-month follow-up study. Calibrated systolic BP estimation accuracy was significantly improved with inertance-based pre-classification, while diastolic BP showed less improvement. Results. With proper feature selection, pre-classification and calibration, we have achieved a mean absolute error of 5.35 mmHg for SBP estimation, compared to 6.16 mmHg if no pre-classification was carried out. The performance did not deteriorate in two months, showing a decent BP trend-tracking ability. Conclusion. The study demonstrated the feasibility of in-ear PPG to reliably measure BP, which represents an important technological advancement in terms of unobtrusiveness and steadiness.

Study of the Flow Rate in Blood Vessels by Using a Four-element Windkessel Model

Study of the Flow Rate in Blood Vessels by Using a Four-element Windkessel Model

A novel electrical impedance function to estimate central aortic blood pressure waveforms

Background and objectiveCentral aortic blood pressure waveforms convey imperative information about cardiovascular risk, but these measurements are obtained invasively. The study aims to develop an electrical impedance function (EIF) to estimate the central (aortic) blood pressure waveform from the radial blood pressure waveform without the need of invasive procedures. MethodsThis paper shows a method of using the four-element Windkessel model to derive an EIF using circuit analysis to estimate the central aortic blood pressure. The four elements were identified from radial waveforms by using the Nelder-Mead simplex algorithm and then used to construct the EIF to obtain the aortic pressure waveform, given the radial waveform. ResultsWaveforms generated by EIF gave the lowest values for Root Mean Square Error (RMSE) (5.69) and Mean Average Percentage Error (MAPE) (0.0402) compared to the generalized transfer function (GTF) and N-point moving average (NPMA), and the lowest MAPE and a comparable RMSE when compared to the Adaptive Transfer Function (ATF). Moreover, EIF has a significantly lower computing time (0.008 ms) when compared with GTF, NPMA and ATF. ConclusionsThe proposed function had comparable or better performance to existing methods when tested on both simulated and actual patient data, and was able to give a better signal shape retention at the dicrotic notch (99.94 % correlation), which is a crucial feature for cardiovascular disease detection. SignificanceOverall the EIF performance validates its use in estimating central aortic pressure waveforms, making it potentially viable for better embedded systems for non-invasive central blood pressure monitoring due to its low computation time and high accuracy.

Observer-Based Monitoring of the Cardiovascular System

An observer-based state and fault estimation method for the cardiovascular system is presented in this brief. The approach is based on the use of a four-element windkessel model which is written in a singular state-space form. This particular representation allows to design an observer to estimate the systemic pressure, the aortic blood flow, and possible problems in the flow through the aorta for monitoring purposes. Numerical simulations are performed in order to demonstrate the pertinence of this representation to conceive an adequate observer to monitor the cardiovascular activity.

MRI-based aortic blood flow model in 3D ballistocardiography.

Ballistocardiography (BCG) is a non-invasive technique which measures the acceleration of a body induced by cardiovascular activity, namely the force exerted by the beating heart. A one dimensional aortic flow model based on the transmission lines theory is developped and applied to the simulation of three dimensional BCG. A four-element Windkessel model is used to generate the pressure-wave. Using transverse MRI slices of a human subject, a reconstruction of the aorta allows the extraction of parameters used to relate the local change in mass of the 1D flow model to 3D acceleration BCG. Simulated BCG curves are then compared qualitatively with the ensemble average curves of the same subject recorded in sustained microgravity. Confirming previous studies, the main features of the y-axis are well simulated. The simulated z-axis, never attempted before, shows important similarities. The simulated x-axis is less faithful and suggests the presence of reflections.

Measurements and Modeling of Transient Blood Flow Perturbations Induced by Brief Somatosensory Stimulation

Proper interpretation of BOLD fMRI and other common functional imaging methods requires an understanding of neurovascular coupling. We used laser speckle-contrast optical imaging to measure blood-flow responses in rat somatosensory cortex elicited by brief (2 s) forepaw stimulation. Results show a large increase in local blood flow speed followed by an undershoot and possible late-time oscillations. The blood flow measurements were modeled using the impulse response of a simple linear network, a four-element windkessel. This model yielded excellent fits to the detailed time courses of activated regions. The four-element windkessel model thus provides a simple explanation and interpretation of the transient blood-flow response, both its initial peak and its late-time behavior.

In Vitro Identification of Four-Element Windkessel Models Based on Iterated Unscented Kalman Filter

Mock circulatory loops (MCLs) have been widely used to test left ventricular assist devices. The hydraulic properties of the mock systemic arterial system are usually described by two alternative four-element windkessel (W4) models. Compared with three-element windkessel model, their parameters, especially the inertial term, are much more difficult to estimate. In this paper, an estimator based on the iterated unscented Kalman filter (IUKF) algorithm is proposed to identify model parameters. Identifiability of these parameters for different measurements is described. Performance of the estimator for different model structures is first evaluated using numerical simulation data contaminated with artificial noise. An MCL is developed to test the proposed algorithm. Parameter estimates for different models are compared with the calculated values derived from the mechanical and hydraulic properties of the MCL to validate model structures. In conclusion, the W4 model with an inertance and an aortic characteristic resistance arranged in series is proposed to represent the mock systemic arterial system. Once model structure is appropriately selected, IUKF can provide reasonable estimation accuracy in a limited time and may be helpful for future clinical applications.

In Vitro Comparison of Support Capabilities of Intra‐Aortic Balloon Pump and Impella 2.5 Left Percutaneous

The Impella 2.5 left percutaneous (LP), a relatively new transvalvular assist device, challenges the position of the intra-aortic balloon pump (IABP), which has a long record in supporting patients after myocardial infarction and cardiac surgery. However, while more costly and more demanding in management, the advantages of the Impella 2.5 LP are yet to be established. The aim of this study was to evaluate the benefits of the 40 cc IABP and the Impella 2.5 LP operating at 47,000 rpm in vitro, and compare their circulatory support capabilities in terms of cardiac output, coronary flow, cardiac stroke work, and arterial blood pressure. Clinical scenarios of cardiogenic preshock and cardiogenic shock (CS), with blood pressure depression, lowered cardiac output, and constant heart rate of 80 bpm, were modeled in a model-controlled mock circulation, featuring a systemic, pulmonary, and coronary vascular bed. The ventricles, represented by servomotor-operated piston pumps, included the Frank-Starling mechanism. The systemic circulation was modeled with a flexible tube having close-to-human aortic dimensions and compliance properties. Proximally, it featured a branch mimicking the brachiocephalic arteries and a physiological correct coronary flow model. The rest of the systemic and pulmonary impedance was modeled by four-element Windkessel models. In this system, the enhancement of coronary flow and blood pressure was tested with both support systems under healthy and pathological conditions. Hemodynamic differences between the IABP and the Impella 2.5 LP were small. In our laboratory model, both systems approximately yielded a 10% cardiac output increase and a 10% coronary flow increase. However, since the Impella 2.5 LP provided significantly better left ventricular unloading, the circulatory support capabilities were slightly in favor of the Impella 2.5 LP. On the other hand, pulsatility was enhanced with the IABP and lowered with the Impella 2.5 LP. The support capabilities of both the IABP and the Impella 2.5 LP strongly depended on the simulated hemodynamic conditions. Maximum hemodynamic benefits were achieved when mechanical circulatory support was applied on a simulated scenario of deep CS.

Physiological interpretation of inductance and low-resistance terms in four-element windkessel models: Assessment by generalized sensitivity function analysis

Physiological relevance of parameters of three arterial models, denominated W4P, W4S and IVW, was assessed by computation of parameter-related generalized sensitivity functions (GSFs), which allow the definition of heart-cycle time intervals where the information content of experimental data, useful for estimation of each model parameter, is concentrated. The W4P and W4S are derived from the three-element windkessel by connecting an inductance, L, in parallel or in series, respectively, with aortic characteristic impedance, R c . In the IVW, L is placed in series at the input of a viscoelastic windkessel, incorporating a Voigt cell (a resistor, R d , in series with a capacitor, C). Pressure and flow measured in the ascending aorta of five ferrets and five dogs were used to estimate all model parameters, by fitting to pressure. For each model structure, parameter-related GSFs were generated. Focusing on controversial L, R c and R d physical meaning, our GSF analysis yielded the conclusion that, in both the W4S and the IVW, but not in the W4P, the L-term is suitable to represent the inertial properties of blood motion. Moreover, the meaning of aortic characteristic impedance ascribed to R c is questionable; while R d is likely to account for viscous losses of arterial wall motion.

Estimation of Three- and Four-Element Windkessel Parameters Using Subspace Model Identification

A windkessel model is widely used to operationalize vascular characteristics. In this paper, we employ a noniterative subspace model identification (SMI) algorithm to estimate parameters in a three- and four-element windkessel model by application of physical foreknowledge. Simulation data of the systemic circulation were used to investigate systematic and random errors in the parameter estimations. Results were compared with different methods as proposed in the literature: one closed-loop and two iterative methods for the three-element model, and one iterative method for the four-element model. For the three-element model, no significant systematic errors were observed using SMI. Concerning random errors, SMI appeared more robust in parameter estimations compared with the other methods (P < 0.05 for a signal-to-noise ratio of 18 dB). For the four-element model, a significant systematic error in the estimate of the arterial inertance L was observed (P = 0.011). However, for all methods, an increasing number of outliers in parameter estimates were observed at increased noise levels. These outliers were almost exclusive due to errors in estimates of L. In conclusion, with SMI physical parameters can mathematically be derived by application of physiological foreknowledge. For a three-element windkessel model, SMI appeared a very robust method to estimate parameters. However, application to a four-element windkessel model was less accurate because of low identifiability of L. Therefore, based on the simulation results, the use of the four-element windkessel model is questionable.

A lumped model for blood flow and pressure in the systemic arteries based on an approximate velocity profile function

Previously, by assuming a viscous dominated flow in the boundary layer and an inertia dominated flow in the vessel core, a velocity profile function for a 1D-wave propagation model was derived. Because the time dependent shape of the velocity profile in this boundary layer model depends on the size of the inviscid core and the boundary layer, and thus on the Womersley number, it differs along the arterial tree. In this study we evaluated a lumped model for a vessel segment in which the element configuration is based on physical phenomena described by the boundary layer model and for which all parameters have a physically based quantitative value dependent on the Womersley numbers. The proposed electrical analog consists of a Womersley number dependent resistor and an inductor arranged in parallel, representing the flow impedence in respectively the vessel core and the boundary layer, in series with a second resistor. After incorporating a capacitor representing the vessel compliance in this rigid tube model, the element configuration resembles the configuration of the four-element windkessel model. For arbitrary Womersley numbers the relative impedence of Womersley theory is approximated with high accuracy. In the limits for small and large Womersley numbers the relative impedences of the proposed lumped model correspond exactly to Womersley theory.

Three- and four-element Windkessel models: Assessment of their fitting performance in a large cohort of healthy middle-aged individuals

Lumped-parameter models are used to estimate the global arterial properties by fitting the model to measured (aortic) pressure and flow. Different model configurations coexist, and it is still an open question as to which model optimally reflects the arterial tree and leads to correct estimates of arterial properties. An assessment was made of the performance of (a) the three-element Windkessel model (WK3) consisting of vascular resistance R, total arterial compliance C, and characteristic impedance Zc; (b) a four-element model with an inertance element L placed in parallel with Zc (WK4-p); and (c) a four-element model with L placed in series with Zc (WK4-s). Models were fitted to data measured non-invasively in 2404 healthy subjects, aged between 35 and 55 years. It was found that model performance segregated into two groups. In a group containing 20 per cent of the dataset (characterized by low blood pressure and wave reflection) the WK4-p model outperformed the other models, with model behaviour as envisioned by its promoters. In these cases, the WK3 and WK4-s models led to increased overestimation of total arterial compliance and underestimation of characteristic impedance. However, in about 80 per cent of the cases, the WK4-p model showed a behaviour that was very similar to that of the WK3 and WK4-s models. Here, the WK4-s model yielded the best quality of fit, although model parameters reached physically impossible values for L in about 12 per cent of all cases. The debate about which lumped-parameter model is the better approximation of the arterial tree is therefore still not fully resolved.

ALTERATION OF RIGHT VENTRICULAR-PULMONARY VASCULAR COUPLING IN A PORCINE MODEL OF PROGRESSIVE PRESSURE OVERLOADING

In acute pulmonary embolism, right ventricular (RV) failure may result from exceeding myocardial contractile resources with respect to the state of vascular afterload. We investigated the adaptation of RV performance in a porcine model of progressive pulmonary embolism. Twelve anesthetized pigs were randomly divided into two groups: gradual pulmonary arterial pressure increases by three injections of autologous blood clot (n=6) or sham-operated controls (n=6). Right ventricular pressure-volume (PV) loops were recorded using a conductance catheter. Right ventricular contractility was estimated by the slope of the end-systolic PV relationship (Ees). After load was referred to as pulmonary arterial elastance (Ea) and assessed using a four-element Windkessel model. Right ventricular-arterial coupling (Ees/Ea) and efficiency of energy transfer (from PV area to external mechanical work [stroke work]) were assessed at baseline and every 30 min for 4 h. Ea increased progressively after embolization, from 0.26+/-0.04 to 2.2+/-0.7 mmHg mL(-1) (P<0.05). Ees increased from 1.01+/-0.07 to 2.35+/-0.27 mmHg mL(-1) (P<0.05) after the first two injections but failed to increase any further. As a result, Ees/Ea initially decreased to values associated with optimal SW, but the last injection was responsible for Ees/Ea values less than 1, decreased stroke volume, and RV dilation. Stroke work/PV area consistently decreased with each injection from 79%+/-3% to 39%+/-11% (P<0.05). In response to gradual increases in afterload, RV contractility reserve was recruited to a point of optimal coupling but submaximal efficiency. Further afterload increases led to RV-vascular uncoupling and failure.

Development of systemic arterial mechanical properties from infancy to adulthood interpreted by four-element windkessel models

Aortic impedance data of infants, children and adults (age range 0.8-54 yr), previously reported by others, were interpreted by means of three alternative four-element windkessel models: W4P, W4S, and IVW. The W4P and W4S are derived from the three-element windkessel (W3) by connecting an inertance (L) in parallel or in series, respectively, with the aortic characteristic resistance (Rc). In the IVW, L is connected in series with a viscoelastic windkessel (VW). The W4S and IVW (same input impedance) fit the data best. The W4S, however, suffers from the assumption that Rc is part of total peripheral resistance (Rp). The IVW model offers a new paradigm for interpretation of resistive properties in terms of viscous (Rd) properties of vessel wall motion, distinguished from Rp. Results indicated that rapid reduction of Rd/Rp during early development is functional to modulation of decay time constant (taud) of pressure in diastole, such that normalization over heart period (taud/T) is independent of body size. Estimates of total arterial compliance (C) vs. age were fitted by a bell-shaped curve with a maximum at 33 yr. With body weight (BW) factored out by normalization, the C/BW data scattered about a bell-shaped curve centered at 66 mmHg. Inertance was significantly higher in pediatric patients than in adults, in accordance with a lower cross-sectional area of the vasculature, commensurate to a lower aortic flow. Changes of arterial properties appear functional to control the ratio of pulsatile power to active power and keep arterial efficiency as high as 97% in infants and children.

Estimation of arterial compliance in small size animal by four-element windkessel models of aortic input impedance

Estimation of arterial compliance in small size animal by four-element windkessel models of aortic input impedance

Conductance catheter-based assessment of arterial input impedance, arterial function, and ventricular-vascular interaction in mice

Global assessment of both cardiac and arterial function is important for a meaningful interpretation of pathophysiological changes in animal models of cardiovascular disease. We simultaneously acquired left ventricular (LV) and aortic pressure and LV volume (V(LV)) in 17 open-chest anesthetized mice (26.7 +/- 3.2g) during steady-state (BL) and caval vein occlusion (VCO) using a 1.4-Fr dual-pressure conductance catheter and in a subgroup of eight animals during aortic occlusion (AOO). Aortic flow was obtained from numerical differentiation of V(LV). AOO increased input impedance (Z(in)) for the first two harmonics, increased characteristic impedance (0.025 +/- 0.007 to 0.040 +/- 0.011 mmHg x microl(-1) x s, P < 0.05), and shifted the minimum in Z(in) from the third to the sixth harmonic. For all conditions, the Z(in) could be well represented by a four-element windkessel model. The augmentation index increased from 116.7 +/- 7.8% to 145.9 +/- 19.5% (P < 0.01) as well as estimated pulse-wave velocity (3.50 +/- 0.94 to 5.95 +/- 1.62 m/s, P < 0.05) and arterial elastance (E(a), 4.46 +/- 1.62 to 6.02 +/- 1.43 mmHg/microl, P < 0.01). AOO altered the maximal slope (E(max), 3.23 +/- 1.02 to 5.53 +/- 1.53 mmHg/microl, P < 0.05) and intercept (-19.9 +/- 8.6 to 1.62 +/- 13.51 microl, P < 0.01) of the end-systolic pressure-volume relation but not E(a)/E(max) (1.44 +/- 0.43 to 1.21 +/- 0.37, not significant). We conclude that simultaneous acquisition of Z(in) and arterial function parameters in the mouse, based solely on conductance catheter measurements, is feasible. We obtained an anticipated response of Z(in) and arterial function parameters following VCO and AOO, demonstrating the sensitivity of the measuring technique to induced physiological alterations in murine hemodynamics.

Effect of hemodiafiltration on pulmonary hemodynamics in endotoxic shock.

Hemofiltration can improve pulmonary hemodynamics during septic shock. The main objective of the study was to determine whether hemodiafiltration (HDF) would also have beneficial effects on pulmonary hemodynamics during septic shock. In the Endo group, six anesthetized pigs received a 0.5 mg/kg endotoxin infusion over 30 min. In the HDF group (n = 6), HDF was started 30 min after the end of the endotoxin infusion, while in the Control group (n = 4) they received HDF but no endotoxin infusion. Pulmonary hemodynamics were analyzed in detail with a four-element windkessel model. Although in the Control group, HDF did not alter pulmonary hemodynamic parameters, in the HDF group, it was responsible for an amplification of the deleterious pulmonary vascular response to endotoxin insult. Our results show that HDF must be used cautiously in septic shock since it can precipitate right heart failure by increasing pulmonary vascular resistance.

Systemic and pulmonary hemodynamics assessed with a lumped-parameter heart-arterial interaction model

Arterial pressure and flow result from the interaction between the (actively) ejecting ventricle and the (passive) arterial circulation. The main objective was to construct a model, accounting for this interaction, that is simple enough so that (i) model parameters can be derived from data measured in experimental and/or clinical conditions, and (ii) the model can be applied to support the analysis and interpretation of these data. It is demonstrated how an established conceptual model of ventricular function (the time-varying elastance) can be coupled to a four-element windkessel model of the arterial system to yield an elegant model of heart-arterial interaction. The coupling leads to a set of three ordinary differential equations. The model allows the study of the effect of changes in cardiac and/or arterial properties on arterial pressure and flow. As an illustration, cardiac and arterial model parameters are derived from measured experimental data in the systemic circulation of a pig and in the pulmonary circulation of a dog. It is evaluated how well measured cardiac and arterial function actually adhere to their assumed theoretical models (time-varying elastance and four-element windkessel model). It is further assessed how well the simple model of heart-arterial interaction describes systemic and pulmonary hemodynamics by comparing simulated and measured experimental data. The limitations and pitfalls of the model, as well as possible applications in the clinical field, are discussed.