Characterization of pulmonary arterial input impedance with lumped parameter models

Current Understanding of the Right Ventricle Structure and Function in Pulmonary Arterial Hypertension

Pulmonary arterial hypertension (PAH) is a disease resulting in increased right ventricular (RV) afterload and RV remodeling. PAH results in altered RV structure and function at different scales from organ-level hemodynamics to tissue-level biomechanical properties, fiber-level architecture, and cardiomyocyte-level contractility. Biomechanical analysis of RV pathophysiology has drawn significant attention over the past years and recent work has found a close link between RV biomechanics and physiological function. Building upon previously developed techniques, biomechanical studies have employed multi-scale analysis frameworks to investigate the underlying mechanisms of RV remodeling in PAH and effects of potential therapeutic interventions on these mechanisms. In this review, we discuss the current understanding of RV structure and function in PAH, highlighting the findings from recent studies on the biomechanics of RV remodeling at organ, tissue, fiber, and cellular levels. Recent progress in understanding the underlying mechanisms of RV remodeling in PAH, and effects of potential therapeutics, will be highlighted from a biomechanical perspective. The clinical relevance of RV biomechanics in PAH will be discussed, followed by addressing the current knowledge gaps and providing suggested directions for future research.

Particle swarm optimizer for arterial blood flow models

BACKGROUND AND OBJECTIVE

Mathematical modeling and computational simulations of arterial blood flow network can offer an insilico platform for both diagnostics and therapeutic phases of patients that suffer from cardiac diseases. These models are normally complex and involve many unknown parameters. For physiological relevance, these parameters should be optimized using in-vivo human/animal data sets. The main goal of this work is to develop an efficient, yet an accurate optimization algorithm to compute parameters in the arterial blood flow models.

METHODS

The particle swarm optimization (PSO) method is proposed herein for the first time, as an accurate algorithm that applies to computing parameters in the Windkessel type model of blood flow in the arterial system. We begin by defining a 6-element Windkessel (WK6) arterial flow model, which is then implemented and validated using multiple flow rate and aortic pressure measurements obtained from different subjects including dogs, pigs and humans. The parameters in the model are obtained using the PSO technique which minimizes the pressure root mean square (P-RMS) error between the computed and the measured aortic pressure waveform.

RESULTS

Model parameters obtained using the proposed PSO method were able to recover the pressure waveform in the aorta during the cardiac cycle for both healthy and diseased species (animals/humans). The PSO method provides an accurate approach to solve this challenging multi-dimensional parameter identification problem. The results obtained by PSO algorithm was compared with the classical gradient-based, namely the non-linear square fit (NLSF) algorithm.

CONCLUSIONS

The results indicate that the PSO method offers alternative and accurate method to find optimal physiological parameters involved in the Windkessel model for the study of arterial blood flow network. The PSO method has performed better than the NLSF approach as depicted from the P-RMS calculations. Finally, we believe that the PSO method offers a great potential and could be used for many other biomedicine optimization problems.

A hybrid Windkessel-Womersley model for blood flow in arteries

A hybrid Windkessel-Womersley (WK-W) coupled mathematical model for the study of pulsatile blood flow in the arterial system is proposed in this article. The model consists of the Windkessel-type proximal and distal compartments connected by a tube to represent the aorta. The blood flow in the aorta is described by the Womersley solution of the simplified Navier-Stokes equations. In addition, we defined a 6-elements Windkessel model (WK6) in which the blood flow in the connecting tube is modeled by the one-dimensional unsteady Bernoulli equation. Both models have been applied and validated using several aortic pressure and flow rate data acquired from different species such as, humans, dogs and pigs. The results have shown that, both models were able to accurately reconstruct arterial input impedance, however, only the WK-W model was able to calculate the radial distribution of the axial velocity in the aorta and consequently the model predicts the time-varying wall shear stress, and frictional pressure drop during the cardiac cycle more accurately. Additionally, the hybrid WK-W model has the capability to predict the pulsed wave velocity, which is also not possible to obtain when using the classical Windkessel models. Moreover, the values of WK-W model parameters have found to fall in the physiologically realistic range of values, therefore it seems that this hybrid model shows a great potential to be used in clinical practice, as well as in the basic cardiovascular mechanics research.

Pulsatile Perfusion Bioreactor for Biomimetic Vascular Impedances

Pulsatile waves of blood pressure and flow are continuously augmented by the resistance, compliance, and inertance properties of the vasculature, resulting in unique wave characteristics at distinct anatomical locations. Hemodynamically generated loads, transduced as physical signals into resident vascular cells, are crucial to the maintenance and preservation of a healthy vascular physiology; thus, failure to recreate biomimetic loading in vitro can lead to pathological gene expression and aberrant remodeling. As a generalized approach to improve native and engineered blood vessels, we have designed, built, and tested a pulsatile perfusion bioreactor based on biomimetic impedances and a novel five-element electrohydraulic analog. Here, the elements of an incubator-based culture system were formulaically designed to match the vascular impedance of a brachial artery by incorporating both the inherent (systemic) and added elements of the physical system into the theoretical approach. Freshly harvested porcine saphenous veins were perfused within a physiological culture chamber for 6 h and the relative expression of seven known mechanically sensitive remodeling genes analyzed using the quantitative polymerase chain reaction (qPCR) method. Of these, we found plasminogen activator inhibitor-1 (SERPINE1) and fibronectin-1 (FN1) to be highly sensitive to differences between arterial- and venous-like culture conditions. The analytical approach and biological confirmation provide a framework toward the general design of long-term hemodynamic-mimetic vascular culture systems.

Diagnosis of pulmonary hypertension from magnetic resonance imaging-based computational models and decision tree analysis

Accurately identifying patients with pulmonary hypertension (PH) using noninvasive methods is challenging, and right heart catheterization (RHC) is the gold standard. Magnetic resonance imaging (MRI) has been proposed as an alternative to echocardiography and RHC in the assessment of cardiac function and pulmonary hemodynamics in patients with suspected PH. The aim of this study was to assess whether machine learning using computational modeling techniques and image-based metrics of PH can improve the diagnostic accuracy of MRI in PH. Seventy-two patients with suspected PH attending a referral center underwent RHC and MRI within 48 hours. Fifty-seven patients were diagnosed with PH, and 15 had no PH. A number of functional and structural cardiac and cardiovascular markers derived from 2 mathematical models and also solely from MRI of the main pulmonary artery and heart were integrated into a classification algorithm to investigate the diagnostic utility of the combination of the individual markers. A physiological marker based on the quantification of wave reflection in the pulmonary artery was shown to perform best individually, but optimal diagnostic performance was found by the combination of several image-based markers. Classifier results, validated using leave-one-out cross validation, demonstrated that combining computation-derived metrics reflecting hemodynamic changes in the pulmonary vasculature with measurement of right ventricular morphology and function, in a decision support algorithm, provides a method to noninvasively diagnose PH with high accuracy (92%). The high diagnostic accuracy of these MRI-based model parameters may reduce the need for RHC in patients with suspected PH.

Determinants of right ventricular afterload (2013 Grover Conference series)

Right ventricular (RV) afterload consists of both resistive and capacitive (pulsatile) components. Total afterload can be measured directly with pulmonary artery input impedance spectra or estimated, either with lumped-parameter modeling or by pressure-volume analysis. However, the inverse, hyperbolic relationship between resistance and compliance in the lung would suggest that the pulsatile components are a predictable and constant proportion of the resistive load in most situations, meaning that total RV load can be estimated from mean resistive load alone. Exceptions include elevations in left atrial pressures and, to a lesser extent, chronic thromboembolic disease. The pulsatile components may also play a more significant role at normal or near-normal pulmonary artery pressures. Measures of coupling between RV afterload and RV contractility may provide important information not apparent by other clinical and hemodynamic measures. Future research should be aimed at development of noninvasive measures of coupling.

In vitro validation of finite-element model of AAA hemodynamics incorporating realistic outlet boundary conditions

The purpose of this study is to validate numerical simulations of flow and pressure in an abdominal aortic aneurysm (AAA) using phase-contrast magnetic resonance imaging (PCMRI) and an in vitro phantom under physiological flow and pressure conditions. We constructed a two-outlet physical flow phantom based on patient imaging data of an AAA and developed a physical Windkessel model to use as outlet boundary conditions. We then acquired PCMRI data in the phantom while it operated under conditions mimicking a resting and a light exercise physiological state. Next, we performed in silico numerical simulations and compared experimentally measured velocities, flows, and pressures in the in vitro phantom to those computed in the in silico simulations. There was a high degree of agreement in all of the pressure and flow waveform shapes and magnitudes between the experimental measurements and simulated results. The average pressures and flow split difference between experiment and simulation were all within 2%. Velocity patterns showed good agreement between experimental measurements and simulated results, especially in the case of whole-cycle averaged comparisons. We demonstrated methods to perform in vitro phantom experiments with physiological flows and pressures, showing good agreement between numerically simulated and experimentally measured velocity fields and pressure waveforms in a complex patient-specific AAA geometry.

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.

In vitro validation of finite element analysis of blood flow in deformable models

The purpose of this article is to validate numerical simulations of flow and pressure incorporating deformable walls using in vitro flow phantoms under physiological flow and pressure conditions. We constructed two deformable flow phantoms mimicking a normal and a restricted thoracic aorta, and used a Windkessel model at the outlet boundary. We acquired flow and pressure data in the phantom while it operated under physiological conditions. Next, in silico numerical simulations were performed, and velocities, flows, and pressures in the in silico simulations were compared to those measured in the in vitro phantoms. The experimental measurements and simulated results of pressure and flow waveform shapes and magnitudes compared favorably at all of the different measurement locations in the two deformable phantoms. The average difference between measured and simulated flow and pressure was approximately 3.5 cc/s (13% of mean) and 1.5 mmHg (1.8% of mean), respectively. Velocity patterns also showed good qualitative agreement between experiment and simulation especially in regions with less complex flow patterns. We demonstrated the capabilities of numerical simulations incorporating deformable walls to capture both the vessel wall motion and wave propagation by accurately predicting the changes in the flow and pressure waveforms at various locations down the length of the deformable flow phantoms.

Development of A Physical Windkessel Module to Re-Create In-Vivo Vascular Flow Impedance for In-Vitro Experiments

PURPOSE

To create and characterize a physical Windkessel module that can provide realistic and predictable vascular impedances for in-vitro flow experiments used for computational fluid dynamics validation, and other investigations of the cardiovascular system and medical devices.

METHODS

We developed practical design and manufacturing methods for constructing flow resistance and capacitance units. Using these units we assembled a Windkessel impedance module and defined its corresponding analytical model incorporating an inductance to account for fluid momentum. We tested various resistance units and Windkessel modules using a flow system, and compared experimental measurements to analytical predictions of pressure, flow, and impedance.

RESULTS

The resistance modules exhibited stable resistance values over wide ranges of flow rates. The resistance value variations of any particular resistor are typically within 5% across the range of flow that it is expected to accommodate under physiologic flow conditions. In the Windkessel impedance modules, the measured flow and pressure waveforms agreed very favorably with the analytical calculations for four different flow conditions used to test each module. The shapes and magnitudes of the impedance modulus and phase agree well between experiment and theoretical values, and also with those measured in-vivo in previous studies.

CONCLUSIONS

The Windkessel impedance module we developed can be used as a practical tool to provide realistic vascular impedance for in-vitro cardiovascular studies. Upon proper characterization of the impedance module, its analytical model can accurately predict its measured behavior under different flow conditions.

How to measure pulmonary vascular and right ventricular function

Long-standing pulmonary hypertension causes significant peripheral and proximal arterial remodeling and right ventricular dysfunction. The clinical metric most often used to assess the progression of PH is the pulmonary vascular resistance (PVR). However, even when measured from multipoint pressure-flow curves, PVR provides information only on the peripheral arterial function, not the proximal arterial function and gives only an incomplete description of all the forces that oppose right ventricular (RV) flow output. Pulmonary vascular impedance spectra (PVZ) capture the impact of proximal and peripheral arterial structure and function on RV function. Analyses of ventricular-vascular coupling give insight into the efficiency of mechanical and metabolic interactions between the right ventricle and the pulmonary vasculature. Here we review techniques for measuring PVZ in humans and animal models and for determining RV function.

Cardiovascular haemodynamics and ventriculo-arterial coupling in an acute pig model of coronary ischaemia-reperfusion

Although reperfusion after coronary occlusion is mandatory for myocardial salvage, reperfusion may trigger a cascade of harmful events (reperfusion injury) adding to myocardial injury. We investigated effects of reperfusion on left ventricular (LV) haemodynamics and ventriculo-arterial (VA) coupling in pigs following acute myocardial ischaemia induced by coronary artery occlusion. Experiments were performed in six animals, with measurements of cardiac and arterial function at baseline, after 60 min of ischaemia (T60) and after 2 (T180) and 4 h of reperfusion (T300). Ventriculo-arterial coupling was assessed using the ventriculo-arterial elastance ratio of paper, as well as using a 'stiffness coupling' and 'temporal coupling' index. Reperfusion following ischaemia (T180 versus T60) induced a progressive decline in cardiovascular function, evidenced by a decrease in mean arterial blood pressure, cardiac output and ejection fraction which was not restored at T300. Although reperfusion also induced an increase in slope of the end-systolic pressure-volume relationship (ESPVR), the ESPVR curve shifted to the right, associated with a depression of contractile function. Histology demonstrated irreversible myocardial damage at T300. The ventriculo-arterial elastance ratio and the 'stiffness coupling' index were unaffected throughout the protocol, but the 'temporal coupling' parameter indicated a relative shift between heart period and the time constant of the arterial system. It is unlikely that these alterations are attributable to ischaemic injury alone. The combination of both the stiffness and temporal coupling index may provide more information when studying ventriculo-arterial coupling than the more commonly used ventricular end-systolic stiffness/effection arterial elastance (E(es)/E(a)) ratio.

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.