Beat-to-beat estimation of windkessel model parameters in conscious rats

Work-loop contractions reveal that the afterload-dependent time course of cardiac Ca2+ transients is modulated by preload

Preload and afterload dictate the dynamics of the cyclical work-loop contraction that the heart undergoes in vivo. Cellular Ca²⁺ dynamics drive contraction, but the effects of afterload alone on the Ca²⁺ transient are inconclusive. To our knowledge, no study has investigated whether the putative afterload dependence of the Ca²⁺ transient is preload dependent. This study is designed to provide the first insight into the Ca²⁺ handling of cardiac trabeculae undergoing work-loop contractions, with the aim to examine whether the conflicting afterload dependency of the Ca²⁺ transient can be accounted for by considering preload under isometric and physiological work-loop contractions. Thus, we subjected ex vivo rat right-ventricular trabeculae, loaded with the fluorescent dye Fura-2, to work-loop contractions over a wide range of afterloads at two preloads while measuring stress, length changes, and Ca²⁺ transients. Work-loop control was implemented with a real-time Windkessel model to mimic the contraction patterns of the heart in vivo. We extracted a range of metrics from the measured steady-state twitch stress and Ca²⁺ transients, including the amplitudes, time courses, rates of rise, and integrals. Results show that parameters of stress were afterload and preload dependent. In contrast, the parameters associated with Ca²⁺ transients displayed a mixed dependence on afterload and preload. Most notably, its time course was afterload dependent, an effect augmented at the greater preload. This study reveals that the afterload dependence of cardiac Ca²⁺ transients is modulated by preload, which brings the study of Ca²⁺ transients during isometric contractions into question when aiming to understand physiological Ca²⁺ handling.NEW & NOTEWORTHY This study is the first examination of Ca²⁺ handling in trabeculae undergoing work-loop contractions. These data reveal that reducing preload diminishes the influence of afterload on the decay phase of the cardiac Ca²⁺ transient. This is significant as it reconciles inconsistencies in the literature regarding the influence of external loads on cardiac Ca²⁺ handling. Furthermore, these findings highlight discrepancies between Ca²⁺ handling during isometric and work-loop contractions in cardiac trabeculae operating at their optimal length.

A method of parameter estimation for cardiovascular hemodynamics based on deep learning and its application to personalize a reduced-order model

Precise model personalization is a key step towards the application of cardiovascular physical models. In this manuscript, we propose to use deep learning (DL) to solve the parameter estimation problem in cardiovascular hemodynamics. Based on the convolutional neural network (CNN) and fully connected neural network (FCNN), a multi-input deep neural network (DNN) model is developed to map the nonlinear relationship between measurements and the parameters to be estimated. In this model, two separate network structures are designed to extract the features of two types of measurement data, including pressure waveforms and a vector composed of heart rate (HR) and pulse transit time (PTT), and a shared structure is used to extract their combined dependencies on the parameters. Besides, we try to use the transfer learning (TL) technology to further strengthen the personalized characteristics of a trained-well network. For assessing the proposed method, we conducted the parameter estimation using synthetic data and in vitro data respectively, and in the test with synthetic data, we evaluated the performance of the TL algorithm through two individuals with different characteristics. A series of estimation results show that the estimated parameters are in good agreement with the true values. Furthermore, it is also found that the estimation accuracy can be significantly improved by a multicycle combination strategy. Therefore, we think that the proposed method has the potential to be used for parameter estimation in cardiovascular hemodynamics, which can provide an immediate, accurate, and sustainable personalization process, and deserves more attention in the future.

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.

Arterial Hypertension and Unusual Ascending Aortic Dilatation in a Neonate With Acute Kidney Injury: Mechanistic Computer Modeling

Background

Neonatal asphyxia caused kidney injury and severe hypertension in a newborn. An unusually dilatated ascending aorta developed. Dialysis and pharmacological treatment led to partial recovery of the ascending aortic diameters. It was hypothesized that the aortic dilatation may be associated with aortic stiffening, peripheral resistance, and cardiovascular changes. Mathematical modeling was used to better understand the potential causes of the hypertension, and to confirm our clinical treatment within the confines of the model's capabilities.

Methods

The patient's systolic arterial blood pressure showed hypertension. Echocardiographic exams showed ascending aorta dilatation during hypertension, which partially normalized upon antihypertensive treatment. To explore the underlying mechanisms of the aortic dilatation and hypertension, an existing lumped parameter hemodynamics model was deployed. Hypertension was simulated using realistic literature informed parameter values. It was also simulated using large parameter perturbations to demonstrate effects. Simulations were designed to permit examination of causal mechanisms. The hypertension inducing effects of aortic stiffnesses, vascular resistances, and cardiac hypertrophy on blood flow and pressure were simulated. Sensitivity analysis was used to stratify causes.

Results

In agreement with our clinical diagnosis, the model showed that an increase of aortic stiffness followed by augmentation of peripheral resistance are the prime causes of realistic hypertension. Increased left ventricular elastance may also cause hypertension. Ascending aortic pressure and flow increased in the simultaneous presence of left ventricle hypertrophy and augmented small vessel resistance, which indicate a plausible condition for ascending aorta dilatation. In case of realistic hypertension, sensitivity analysis showed that the treatment of both the large vessel stiffness and small vessel resistance are more important in comparison to cardiac hypertrophy.

Conclusion and Discussion

Large vessel stiffness was found to be the prime factor in arterial hypertension, which confirmed the clinical treatment. Treatment of cardiac hypertrophy appears to provide significant benefit but may be secondary to treatment of large vessel stiffness. The quantitative grading of pathophysiological mechanisms provided by the modeling may contribute to treatment recommendations. The model was limited due to a lack of data suitable to permit model identification.

Mechanical loading of isolated cardiac muscle with a real-time computed Windkessel model of the vasculature impedance

To date, the mechanical loads imposed on isolated cardiac muscle tissue in vitro have been oversimplified. Researchers typically applied loads that are time-invariant, resulting in either isometric and auxotonic contractions, or flat-topped (isotonic shortening) work-loops. These contraction types do not fully capture the dynamic response of contracting tissues adapting to a variable load, such as is experienced by ventricular tissue in vivo. In this study, we have successfully developed a loading system that presents a model-based, time-varying, continuously updated, load to cardiac tissue preparations. We combined a Windkessel model of vascular fluid impedance together with Laplace's Law and encoded it in a real-time hardware-based force-length control system. Experiments were carried out on isolated rat left ventricular trabeculae; we directly compare the work-loops arising from this protocol with those of a typical simplified isotonic shortening work-loop system. We found that, under body conditions, cardiac trabeculae achieved greater mechanical work output against our new loading system, than with the simplified isotonic work-loop protocol. We further tested whether loading the tissue with a mechanical impedance defined by "diseased" Windkessel model parameters had an effect on the performance of healthy trabeculae. We found that trabecula shortening decreased when applying the set of Windkessel parameters describing the hypertensive condition, and increased in the hypotensive state. Our implementation of a real-time model of arterial characteristics provides an improved, physiologically derived, instantly calculated load for use in studying isolated cardiac muscle, and is readily applicable to study various disease conditions.

The Failing Heart Under Stress: Echocardiography is an Essential Monitoring Tool in the Intensive Care Unit

Echocardiography has been evolving to play a pivotal role in hemodynamic management, both intraoperatively and at the bedside. A full assessment of hemodynamics necessitates the use of all of the options available on modern echocardiographs. This introductory review provides insight into three important issues of hemodynamic monitoring by echocardiography: evaluation of preloading conditions, assessment of systolic function, and contractility and estimation of afterload. Mastering these three features will help in a minimally invasive approach of hemodynamic instability.

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.

A time-invariant visco-elastic windkessel model relating blood flow and blood volume

The difference between the rate of change of cerebral blood volume (CBV) and cerebral blood flow (CBF) following stimulation is thought to be due to circumferential stress relaxation in veins (Mandeville, J.B., Marota, J.J.A., Ayata, C., Zaharchuk, G., Moskowitz, M.A., Rosen, B.R., Weisskoff, R.M., 1999. Evidence of a cerebrovascular postarteriole windkessel with delayed compliance. J. Cereb. Blood Flow Metab. 19, 679-689). In this paper we explore the visco-elastic properties of blood vessels, and present a dynamic model relating changes in CBF to changes in CBV. We refer to this model as the visco-elastic windkessel (VW) model. A novel feature of this model is that the parameter characterising the pressure-volume relationship of blood vessels is treated as a state variable dependent on the rate of change of CBV, producing hysteresis in the pressure-volume space during vessel dilation and contraction. The VW model is nonlinear time-invariant, and is able to predict the observed differences between the time series of CBV and that of CBF measurements following changes in neural activity. Like the windkessel model derived by Mandeville, J.B., Marota, J.J.A., Ayata, C., Zaharchuk, G., Moskowitz, M.A., Rosen, B.R., Weisskoff, R.M., 1999. Evidence of a cerebrovascular postarteriole windkessel with delayed compliance. J. Cereb. Blood Flow Metab. 19, 679-689, the VW model is primarily a model of haemodynamic changes in the venous compartment. The VW model is demonstrated to have the following characteristics typical of visco-elastic materials: (1) hysteresis, (2) creep, and (3) stress relaxation, hence it provides a unified model of the visco-elastic properties of the vasculature. The model will not only contribute to the interpretation of the Blood Oxygen Level Dependent (BOLD) signals from functional Magnetic Resonance Imaging (fMRI) experiments, but also find applications in the study and modelling of the brain vasculature and the haemodynamics of circulatory and cardiovascular systems.

Increase in pulse wavelength causes the systemic arterial tree to degenerate into a classical windkessel

Two competing schools of thought ascribe vascular disease states such as isolated systolic hypertension to fundamentally different arterial system properties. The "windkessel school" describes the arterial system as a compliant chamber that distends and stores blood and relates pulse pressure to total peripheral resistance (R(tot)) and total arterial compliance (C(tot)). Inherent in this description is the assumption that arterial pulse wavelengths are infinite. The "transmission school," assuming a finite pulse wavelength, describes the arterial system as a network of vessels that transmits pulses and relates pulse pressure to the magnitude, timing, and sites of pulse-wave reflection. We hypothesized that the systemic arterial system, described by the transmission school, degenerates into a windkessel when pulse wavelengths increase sufficiently. Parameters affecting pulse wavelength (i.e., heart rate, arterial compliances, and radii) were systematically altered in a realistic, large-scale, human arterial system model, and the resulting pressures were compared with those assuming a classical (2-element) windkessel with the same R(tot) and C(tot). Increasing pulse wavelength as little as 50% (by changing heart rate -33.3%, compliances -55.5%, or radii +50%) caused the distributed arterial system model to degenerate into a classical windkessel (r(2) = 0.99). Model results were validated with analysis of representative human aortic pressure and flow waveforms. Because reported changes in arterial properties with age can markedly increase pulse wavelength, results suggest that isolated systolic hypertension is a manifestation of an arterial system that has degenerated into a windkessel, and thus arterial pressure is a function only of aortic flow, R(tot), and C(tot).

Characterization of circulatory disorders in beta-thalassemic mice by noninvasive ultrasound biomicroscopy

Beta-thalassemia is an inherited hematological disease caused by a decrease or absence of production of beta-globin that requires chronic therapeutic interventions. This condition leads to important arterial and venous thromboembolic events, transitory ischemic attacks, and microcirculatory obstructions, indicative of circulatory disturbances. To investigate the presence of microcirculatory disorders without the confounding effect of treatments, we used beta-thalassemic mice with typical clinical characteristics of human beta-thalassemia major. One impediment to the understanding of microcirculatory physiology, in particular for beta-thalassemic mice, has been the lack of an appropriate noninvasive imaging approach. We thus developed a novel noninvasive high-frequency ultrasound imaging method to evaluate murine vascular hemodynamic properties. In our beta-thalassemic mice, total peripheral vascular resistance was significantly increased (P < 0.01) compared with wildtype littermates, whereas mean blood pressure, heart rate, and cardiac output were similar (P = nonsignificant). Importantly, the vascular hemodynamics in beta-thalassemic mice were significantly affected according to the Pourcelot indexes measured in the common carotid artery and abdominal aorta (P < 0.01 and P < 0.05, respectively). Hence, our beta-thalassemia characterization of vascular hemodynamics by noninvasive ultrasonic approaches proves the existence and provides unique quantitative assessment of microcirculatory flow disturbances in those mice.

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.

Spontaneous beat-by-beat fluctuations of total peripheral and cerebrovascular resistance in response to tilt

Beat-by-beat estimates of total peripheral resistance (TPR) can be obtained from continuous measurements of cardiac output by using Doppler ultrasound and noninvasive mean arterial blood pressure (MAP). We employed transfer function analysis to study the heart rate (HR) and vascular response to spontaneous changes in blood pressure from the relationships of systolic blood pressure (SBP) to HR (SBP→HR), MAP to total peripheral resistance (TPR) and cerebrovascular resistance index (CVRi) (MAP→TPR and MAP→CVRi), as well as stroke volume (SV) to TPR in nine healthy subjects in supine and 45° head-up tilt positions. The gain of the SBP→HR transfer function was reduced with tilt in both the low- (0.03–0.15 Hz) and high-frequency (0.15–0.35 Hz) regions. In contrast, MAP→TPR transfer function gain was not affected by head-up tilt, but it did increase from low- to high-frequency regions. The phase relationships between MAP→TPR were unaffected by head-up tilt, but, consistent with an autoregulatory system, changes in MAP were followed by directionally similar changes in TPR, just as observed for the MAP→CVRi. The SV→TPR had high coherence with a constant phase of 150–160°. Together, these data that showed changes in MAP preceded changes in TPR, as well as a possible link between SV and TPR, are consistent with complex interactions between the vascular component of the arterial and cardiopulmonary baroreflexes and intrinsic properties such as the myogenic response of the resistance arteries.

Beat-to-beat stroke volume estimation from aortic pressure waveform in conscious rats: comparison of models

Several methods for estimating stroke volume (SV) were tested in conscious, freely moving rats in which ascending aortic pressure and cardiac flow were simultaneously (beat-to-beat) recorded. We compared two pulse-contour models to two new statistical models including eight parameters extracted from the pressure waveform in a multiple linear regression. Global as well as individual statistical models gave higher correlation coefficients between estimated and measured SV (model 1, r = 0.97; model 2, r = 0.96) than pulse-contour models (model 1, r = 0.83; model 2, r = 0.91). The latter models as well as statistical model 1 used the pulsatile systolic area and thus could be applied to only 47 +/- 17% of the cardiac beats. In contrast, statistical model 2 used the pressure-increase characteristics and was therefore established for all of the cardiac beats. The global statistical model 2 applied to data sets independent of those used to establish the model gave reliable SV estimates: r = 0.54 +/- 0.07, a small bias between -8% to +10%, and a mean precision of 7%. This work demonstrated the limits of pulse-contour models to estimate SV in conscious, unrestrained rats. A multivariate statistical model using eight parameters easily extracted from the aortic waveform could be applied to all cardiac beats with good precision.