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

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.

Twenty-four hour blood pressure variability and the prevalence and the progression of cerebral white matter hyperintensities

Blood pressure variability (BPV) is related to cerebral white matter hyperintensities (WMH), but longitudinal studies assessing WMH progression are scarce. Patients with cardiovascular disease and control participants of the Heart-Brain Connection Study underwent 24-hour ambulatory blood pressure monitoring and repeated brain MRI at baseline and after 2 years. Using linear regression, we determined whether different measures of BPV (standard deviation, coefficient of variation, average real variability (ARV), variability independent of the mean) and nocturnal dipping were associated with WMH and whether this association was mediated or moderated by baseline cerebral perfusion. Among 177 participants (mean age: 65.9 ± 8.1 years, 33.9% female), the absence of diastolic nocturnal dipping was associated with higher WMH volume at baseline (β = 0.208, 95%CI: 0.025-0.392), but not with WMH progression among 91 participants with follow-up imaging. None of the BPV measures were associated with baseline WMH. Only 24-hour diastolic ARV was significantly associated with WMH progression (β = 0.144, 95%CI: 0.030-0.258), most profound in participants with low cerebral perfusion at baseline (p-interaction = 0.042). In conclusion, absent diastolic nocturnal dipping and 24-hour diastolic ARV were associated with higher WMH volume. Whilst requiring replication, these findings suggest that blood pressure patterns and variability may be a target for prevention of small vessel disease.

Imaging-photoplethysmography-guided optical microangiography

We report a method to image facial cutaneous microvascular perfusion using wide-field imaging photoplethysmography (iPPG) and handheld swept-source optical coherence tomography (OCT). The iPPG system employs a 16-bit-depth camera to provide a 2D wide-field blood pulsation map that is then used as a positioning guidance for OCT imaging of cutaneous microvasculature. We show the results from iPPG and OCT to demonstrate the ability of guided imaging of cutaneous microvasculature, which is potentially useful for the assessment of skin conditions in dermatology and cosmetology.

Design, Development, and Temporal Evaluation of a Magnetic Resonance Imaging-Compatible In Vitro Circulation Model Using a Compliant Abdominal Aortic Aneurysm Phantom

Biomechanical characterization of abdominal aortic aneurysms (AAAs) has become commonplace in rupture risk assessment studies. However, its translation to the clinic has been greatly limited due to the complexity associated with its tools and their implementation. The unattainability of patient-specific tissue properties leads to the use of generalized population-averaged material models in finite element analyses, which adds a degree of uncertainty to the wall mechanics quantification. In addition, computational fluid dynamics modeling of AAA typically lacks the patient-specific inflow and outflow boundary conditions that should be obtained by nonstandard of care clinical imaging. An alternative approach for analyzing AAA flow and sac volume changes is to conduct in vitro experiments in a controlled laboratory environment. In this study, we designed, built, and characterized quantitatively a benchtop flow loop using a deformable AAA silicone phantom representative of a patient-specific geometry. The impedance modules, which are essential components of the flow loop, were fine-tuned to ensure typical intraluminal pressure conditions within the AAA sac. The phantom was imaged with a magnetic resonance imaging (MRI) scanner to acquire time-resolved images of the moving wall and the velocity field inside the sac. Temporal AAA sac volume changes lead to a corresponding variation in compliance throughout the cardiac cycle. The primary outcome of this work was the design optimization of the impedance elements, the quantitative characterization of the resistive and capacitive attributes of a compliant AAA phantom, and the exemplary use of MRI for flow visualization and quantification of the deformed AAA geometry.

A coupled one dimension and transmission line model for arterial flow simulation

A broad choice of numerical schemes and methods currently exists for blood flow simulations. The results rely critically on the prescription of boundary conditions. The outflow boundary condition for a one-dimensional (1D) flow solver is usually prescribed via a Windkessel or lumped parameter model. The weakness of such an approach is the determination of the parameters. In the present work, we use an alternative approach, that is, a reflection coefficient (RC), to lumped parameter models for distal boundary conditions. With such a RC, the number of parameters required is reduced to one. We derive the theoretical foundation for the RC. Specifically, we couple a transmission line theory for peripheral resistance with a 1D arterial flow solver. We apply this method to a healthy and a stenosed virtual aorta, and show this method can reproduce some subtle features in arterial pressure propagation, such as the steepened pressure waveform and the reflection from the stenosed site. In summary, the RC parameter has strong physical implications in the theory of wave propagation and may be used in flow simulations where reflections need to be explicitly modeled. NOVELTY STATEMENT: A novel coupled one-dimensional-transimission line model has been developed in this work with detailed implementations. Only one outflow boundary condition, that is, the refection coefficient is required in the model. Reflections for a pulse wave from aortic terminals as well as from a stenotic site are numerically simulated.

Cardiovascular system identification: Simulation study using arterial and central venous pressures

The paper presents a study of the identifiability of a lumped model of the cardiovascular system. The significance of this work from the existing literature is in the potential advantage of using both arterial and central venous (CVP) pressures, two signals that are frequently monitored in the critical care unit. The analysis is done on the system's state-space representation via control theory and system identification techniques. Non-parametric state-space identification is preferred over other identification techniques as it optimally assesses the order of a model, which best describes the input-output data, without any prior knowledge about the system. In particular, a recent system identification algorithm, namely Observer Kalman Filter Identification with Deterministic Projection, is used to identify a simplified version of an existing cardiopulmonary model. The outcome of the study highlights the following two facts. In the deterministic (noiseless) case, the theoretical indicators report that the model is fully identifiable, whereas the stochastic case reveals the difficulty in determining the complete system's dynamics. This suggests that even with the use of CVP as an additional pressure signal, the identification of a more detailed (high order) model of the circulatory system remains a challenging task.

Simulation of motor current waveforms in monitoring aortic valve state during ventricular assist device support

Monitoring of aortic valve (AV) opening and closure during left ventricular assist device (LVAD) heart pump support is crucial in preventing AV abnormalities and remodeling caused by anomalous resirculation. In this study, simulations of LVAD motor current waveforms were undertaken to investigate AV response to rotary blood pump assistance, as well as to detect AV open and close status under heart failure conditions. A two-dimensional fluid-structure interaction finite-element model is presented to predict AV state during LVAD outflow. The data will be useful in the development of a pump speed controller for optimal management of pump outflow.

WAVE PROPAGATION IN A 1D FLUID DYNAMICS MODEL USING PRESSURE-AREA MEASUREMENTS FROM OVINE ARTERIES

This study considers a 1D fluid dynamics arterial network model with 14 vessels developed to assimilate ex vivo 0D temporal data for pressure-area dynamics in individual vessel segments from 11 male Merino sheep. A 0D model was used to estimate vessel wall parameters in a two-parameter elastic model and a four-parameter Kelvin viscoelastic model. This was done using nonlinear optimization minimizing the least squares error between model predictions and measured cross-sectional areas. Subsequently, estimated values for elastic stiffness and unstressed area were related to construct a nonlinear relationship. This relation was used in the network model. A 1D single vessel model of the aorta was then developed and used to estimate the inflow profile and parameters for total resistance and compliance for the downstream network and to demonstrate effects of incorporating viscoelasticity in the arterial wall. Lastly, the extent to which vessel wall parameters estimated from ex vivo data can be used to realistically simulate pressure and area in a vessel network was evaluated. Elastic wall parameters in the network simulations were found to yield pressure-area relationships across all vessel locations and sheep that were in ranges comparable to those in the ex vivo data.

Simulation of motor current waveform as an index for aortic valve open-close condition during ventricular support

Monitoring of aortic valve (AV) opening and closure during heart pump support by a left ventricular assist device (LVAD) is crucial in preventing adverse events such as thrombus formation near the AV. In preventing adverse events such as thrombus formation near the AV. In this paper, simulations of LVAD motor current waveform were undertaken to evaluate its suitability for ascertaining aortic valve status. A two-dimensional fluid-structure interaction finite-element model is presented to predict AV closure during LVAD outflow, useful in the development of a pump speed controller.

Review of zero-D and 1-D models of blood flow in the cardiovascular system

Background

Zero-dimensional (lumped parameter) and one dimensional models, based on simplified representations of the components of the cardiovascular system, can contribute strongly to our understanding of circulatory physiology. Zero-D models provide a concise way to evaluate the haemodynamic interactions among the cardiovascular organs, whilst one-D (distributed parameter) models add the facility to represent efficiently the effects of pulse wave transmission in the arterial network at greatly reduced computational expense compared to higher dimensional computational fluid dynamics studies. There is extensive literature on both types of models.

Method and Results

The purpose of this review article is to summarise published 0D and 1D models of the cardiovascular system, to explore their limitations and range of application, and to provide an indication of the physiological phenomena that can be included in these representations. The review on 0D models collects together in one place a description of the range of models that have been used to describe the various characteristics of cardiovascular response, together with the factors that influence it. Such models generally feature the major components of the system, such as the heart, the heart valves and the vasculature. The models are categorised in terms of the features of the system that they are able to represent, their complexity and range of application: representations of effects including pressure-dependent vessel properties, interaction between the heart chambers, neuro-regulation and auto-regulation are explored. The examination on 1D models covers various methods for the assembly, discretisation and solution of the governing equations, in conjunction with a report of the definition and treatment of boundary conditions. Increasingly, 0D and 1D models are used in multi-scale models, in which their primary role is to provide boundary conditions for sophisticate, and often patient-specific, 2D and 3D models, and this application is also addressed. As an example of 0D cardiovascular modelling, a small selection of simple models have been represented in the CellML mark-up language and uploaded to the CellML model repository http://models.cellml.org/. They are freely available to the research and education communities.

Conclusion

Each published cardiovascular model has merit for particular applications. This review categorises 0D and 1D models, highlights their advantages and disadvantages, and thus provides guidance on the selection of models to assist various cardiovascular modelling studies. It also identifies directions for further development, as well as current challenges in the wider use of these models including service to represent boundary conditions for local 3D models and translation to clinical application.

Quantification of right ventricular afterload in patients with and without pulmonary hypertension

Right ventricular (RV) afterload is commonly defined as pulmonary vascular resistance, but this does not reflect the afterload to pulsatile flow. The purpose of this study was to quantify RV afterload more completely in patients with and without pulmonary hypertension (PH) using a three-element windkessel model. The model consists of peripheral resistance (R), pulmonary arterial compliance (C), and characteristic impedance (Z). Using pulmonary artery pressure from right-heart catheterization and pulmonary artery flow from MRI velocity quantification, we estimated the windkessel parameters in patients with chronic thromboembolic PH (CTEPH; n = 10) and idiopathic pulmonary arterial hypertension (IPAH; n = 9). Patients suspected of PH but in whom PH was not found served as controls (NONPH; n = 10). R and Z were significantly lower and C significantly higher in the NONPH group than in both the CTEPH and IPAH groups (P < 0.001). R and Z were significantly lower in the CTEPH group than in the IPAH group (P < 0.05). The parameters R and C of all patients obeyed the relationship C = 0.75/R (R(2) = 0.77), equivalent to a similar RC time in all patients. Mean pulmonary artery pressure P and C fitted well to C = 69.7/P (i.e., similar pressure dependence in all patients). Our results show that differences in RV afterload among groups with different forms of PH can be quantified with a windkessel model. Furthermore, the data suggest that the RC time and the elastic properties of the large pulmonary arteries remain unchanged in PH.

Reduction of compartment compliance increases venous flow pulsatility and lowers apparent vascular compliance: Implications for cerebral blood flow hemodynamics

The global compliance of a fixed-volume, incompressible compartment may play a significant role in determining the inherent vascular compliance. For the intracranial compartment, we propose that the free-displacement of the cerebral spinal fluid (CSF) directly relates to cerebral vascular compliance. To test this hypothesis, an in vivo surrogate intracranial compartment was made by enclosing a rabbit's kidney within a rigid, fluid-filled container. Opening/closing a port atop the box modulated the free flow of box fluid (open-box state). We observed that the pulsatility of the renal venous outflow increased in response to hampering the free flow of fluid in-and-out of the container (closed-box state). To associate the observed pulsatility changes with the compliance changes, a parametric method was proposed for the computation of the apparent compliance (C(app)) of the whole renal vascular system. The calculated C(app) for each experiment's closed-box state was favorably compared to a time-domain compliance assessment method at the mean heart rate. In addition, it was revealed that C(app) in the open-box state was greater than that in the closed-box state only when the calculations were performed at frequencies lower than the heart rate and closer to the ventilation rate. These experimental results support the concept that the vessel compliance of vascular systems enclosed within a rigid compartment is a function of the global compartment compliance.

Echocardiographic assessment of aortic elastic properties with automated border detection in an ICU: in vivo application of the arctangent Langewouters model

We studied whether combined pressure and transesophageal ultrasound monitoring is feasible in the intensive care unit (ICU) setting for global cardiovascular hemodynamic monitoring [systemic vascular resistance (SVR) and total arterial compliance (CPPM)] and direct estimation of local ascending and descending aortic mechanical properties, i.e., distensibility and compliance coefficients (DC and CC). Pressure-area data were fitted to the arctangent Langewouters model, with aortic cross-sectional area obtained via automated border detection. Data were measured in 19 subjects at baseline, during infusion of sodium nitroprusside (SNP), and after washout. SNP infusion lowered SVR from 1.15 ± 0.40 to 0.80 ± 0.32 mmHg·ml−1·s ( P < 0.05), whereas CPPM increased from 0.87 ± 0.46 to 1.02 ± 0.42 ml/mmHg ( P < 0.05). DC and CC increased from 0.0018 ± 0.0007 to 0.0025 ± 0.0009 l/mmHg ( P < 0.05) and from 0.0066 ± 0.0028 to 0.0083 ± 0.0026 cm2/mmHg ( P < 0.05), respectively, at the descending, but not ascending, aorta. The Langewouters model fitted the descending aorta data reasonably well. Assessment of local mechanical properties of the human ascending aorta in a clinical setting by automated border detection remains technically challenging.

Lumped parameter estimation for the embryonic chick vascular system: a time-domain approach using MLAB

We have evaluated several lumped parameter analog models for the early chick embryonic vascular system that may be used to infer loading characteristics of the developing heart. We measured dorsal aortic pressure and flow simultaneously with a servo-null pressure system and a pulsed Doppler velocimeter. Four different analog circuit models were chosen for comparisons. We formulated the time-domain differential equations specifying the relations between pressure and flow in the models, and then estimated the lumped parameters that produced the best fit. The MLAB mathematical modeling software was used for solving differential equations, and for minimizing the difference between model-predicted values and experimental data. The traditional three-element Windkessel model with an added inductance term was most often the best-fitting model. This is compatible with the previous study using a frequency-domain approach. The procedures developed for the current study are adaptable for the study of a variety of nonlinear models, and distributed parameter models for mammalian cardiovascular development with mechanically, pharmacologically, or genetically altered conditions.

Simulation of arterial hemodynamics after partial prosthetic replacement of the aorta

BACKGROUND

Replacing parts of the aorta with a non-compliant vascular prosthesis results in marked alterations of the aortic input impedance and influences arterial hemodynamics. We propose a mathematical model of circulation that can predict hemodynamic changes after simulation of vascular grafting.

METHODS

A new mathematical model of the human arterial system was developed on a 75-MHz Pentium personal computer using Matlab software. The human arterial tree was delineated according to a 128-branch design encompassing bifurcations and physical properties of the arterial wall. A digitized aortic flow wave was chosen as the input signal to the system. After determination of the modules of elasticity of native vascular tissue and standard prostheses in technical experiments, replacement of any part of the aorta with a prosthesis was simulated by increasing the elasticity in the parts desired.

RESULTS

During control conditions, the model displayed a physiologic distribution of flow and pressure waves throughout the arterial system. Simulated replacement of the aorta resulted in an increase in pressure amplitude and a partial loss of the aortic "Windkessel" function. Calculation of the aortic input impedance showed an increase in the characteristic impedance, whereas the peripheral resistance remained unaltered.

CONCLUSIONS

This mathematical model of the arterial circulation is useful for simulating hemodynamic changes after implantation of vascular grafts. The results of the model analysis are consistent with those in previous experimental work.

Time domain method to identify simultaneously parameters of the windkessel model applied to the pulmonary circulation

Lumped models are frequently used to provide a satisfactory description of the hemodynamic properties of the pulmonary vasculature. The purpose of this study is to describe a method to identify simultaneously the parameters values of windkessel models components. The following equation was used to obtain R1 (characteristic resistance), R2 (peripheral resistance), C (total compliance) and L (inertance): [formula: see text] where ki are the following functions of L, R1, R2 and C: [formula: see text] To assess the accuracy of the method, estimates of R1, R2, and C were compared to characteristic impedance Rc, vascular resistance PVR and pulmonary arterial compliance Cd respectively computed from referenced methods. Comparison between R1 and Rc, PVR and R1 + R2, C and Cd were obtained in 5 anaesthetised pigs during basal conditions and after endotoxin-shock. The results indicate that in both conditions, comparisons evidenced highly significant correlations between values computed by the different approaches (p < 0.0001). Although our method yielded to consistently lower values than values provided by referenced methods, the results were concordant with respect to the expected response of pulmonary vasculature to endotoxin insult. We conclude that our method of identification is suitable for the assessment of lumped parameters windkessel model estimates. The main interest is that actual resistance and compliance values can be obtained easily and simultaneously by a global method approach.

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.

A new description of arterial function: the compliance-pressure loop

Hypertension and atherosclerosis are associated with reduced arterial compliance, which is the principal component that reflects the dynamic behavior of the arterial system. Hence, change in arterial compliance has been used as a compass of arterial wall properties, as well as an effective parameter for assessing therapeutic treatment efficacy. The arterial compliance-blood pressure loop concept is introduced here for assessment of arterial function. Aortic pressure and flow were measured in experimental dogs during normal and acute hypertension. The compliance-pressure loops were constructed from pulsatile blood pressure waveforms and the corresponding compliances. The features of the loop are that, for any given heart beat, arterial compliance is seen to be maximal in early systole to facilitate ventricular ejection, compliance decreasing during the remainder of systole owing to increased blood pressure and reduced aortic flow, compliance in diastole increasing as pressure declines. The arteries are stiffer with reduced compliance in hypertension. Thus, the compliance-pressure loop can provide an effective characterization of the dynamic behavior of the arterial system in terms of pressure-flow relation and blood vessel properties.

New closed-form expressions for the estimation of arterial windkessel compliance

New closed-form mathematical expressions, in the time- and frequency-domain, are derived for estimating the arterial windkessel compliance. The proposed expressions assume the three-element windkessel to model the arterial system and require the measurements of the entire waveforms of arterial pressure and flow. The resistance parameters are estimated using the recently proposed energy-balance method, then compliance is analytically calculated in order to minimize the pressure error in the compliant element. The derived expressions remain valid even when the windkessel compliance is assumed to be pressure-dependent. Also, it is shown that the method, either time- or frequency-domain formulation, provides parameter estimates, which minimize the arterial pressure square error. The method has been applied to simulated data as well as to pressure and flow data measured in the ascending aorta of three anaesthetized dogs under different circulatory conditions.