A mathematical model of the carotid baroregulation in pulsating conditions

Trans-aortic Valvular Ejection Fraction for Monitoring Recovery of Patients with Ventricular Systolic Heart Failure

Durable mechanical circulatory support in the form of left ventricular (LV) assist device (LVAD) therapy is increasingly considered in the context of the recovery of native cardiac function. Progressive improvement in LV function may facilitate LVAD explantation and a resultant reduction in device-related risk. However, ascertaining LV recovery remains a challenge. In this study, we investigated the use of trans-aortic valvular flow rate and trans-LVAD flow rate to assess native LV systolic function using a well-established lumped parameter model of the mechanically assisted LV with pre-existing systolic dysfunction. Trans-aortic valvular ejection fraction (TAVEF) was specifically found to characterize the preload-independent contractility of the LV. It demonstrated excellent sensitivity to simulated pharmacodynamic stress tests and volume infusion tests. TAVEF may prove to be useful in the ascertainment of LV recovery in LVAD-supported LVs with pre-existing LV systolic dysfunction.

Seven Mathematical Models of Hemorrhagic Shock

Although mathematical modelling of pressure-flow dynamics in the cardiocirculatory system has a lengthy history, readily finding the appropriate model for the experimental situation at hand is often a challenge in and of itself. An ideal model would be relatively easy to use and reliable, besides being ethically acceptable. Furthermore, it would address the pathogenic features of the cardiovascular disease that one seeks to investigate. No universally valid model has been identified, even though a host of models have been developed. The object of this review is to describe several of the most relevant mathematical models of the cardiovascular system: the physiological features of circulatory dynamics are explained, and their mathematical formulations are compared. The focus is on the whole-body scale mathematical models that portray the subject's responses to hypovolemic shock. The models contained in this review differ from one another, both in the mathematical methodology adopted and in the physiological or pathological aspects described. Each model, in fact, mimics different aspects of cardiocirculatory physiology and pathophysiology to varying degrees: some of these models are geared to better understand the mechanisms of vascular hemodynamics, whereas others focus more on disease states so as to develop therapeutic standards of care or to test novel approaches. We will elucidate key issues involved in the modeling of cardiovascular system and its control by reviewing seven of these models developed to address these specific purposes.

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.

Cardiovascular dynamics during head-up tilt assessed via pulsatile and non-pulsatile models

This study develops non-pulsatile and pulsatile models for the prediction of blood flow and pressure during head-up tilt. This test is used to diagnose potential pathologies within the autonomic control system, which acts to keep the cardiovascular system at homeostasis. We show that mathematical modeling can be used to predict changes in cardiac contractility, vascular resistance, and arterial compliance, quantities that cannot be measured but are useful to assess the system's state. These quantities are predicted as time-varying parameters modeled using piecewise linear splines. Having models with various levels of complexity formulated with a common set of parameters, allows us to combine long-term non-pulsatile simulations with pulsatile simulations on a shorter time-scale. We illustrate results for a representative subject tilted head-up from a supine position to a [Formula: see text] angle. The tilt is maintained for 5 min before the subject is tilted back down. Results show that if volume data is available for all vascular compartments three parameters can be identified, cardiovascular resistance, vascular compliance, and ventricular contractility, whereas if model predictions are made against arterial pressure and cardiac output data alone, only two parameters can be estimated either resistance and contractility or resistance and compliance.

Parameter estimation in a minimal model of cardio-pulmonary interactions

Mechanical ventilation is a widely used breathing support for patients in intensive care. Its effects on the respiratory and cardiovascular systems are complex and difficult to predict. This work first presents a minimal mathematical model representing the mechanics of both systems and their interaction, in terms of flows, pressures and volumes. The aim of this model is to get insight on the two systems' status when mechanical ventilation settings, such as positive end-expiratory pressure, are changing. The parameters of the model represent cardiac elastances and vessel compliances and resistances. As a second step, these parameters are estimated from 16 experimental datasets. The data come from three pig experiments reproducing intensive care conditions, where a large range of positive end-expiratory pressures was imposed by the mechanical ventilator. The data used for parameter estimation is limited to information available in the intensive care unit, such as stroke volume, central venous pressure and systemic arterial pressure. The model is able to satisfactorily reproduce this experimental data, with mean relative errors ranging from 1 to 26%. The model also reproduces the dynamics of the cardio-vascular and respiratory systems, and their interaction. By looking at the estimated parameter values, one can quantitatively track how the two coupled systems mechanically react to changes in external conditions imposed by the ventilator. This work thus allows real-time, model-based management of ventilator settings.

Practical identifiability analysis of a minimal cardiovascular system model

BACKGROUND AND OBJECTIVE

Parameters of mathematical models of the cardiovascular system can be used to monitor cardiovascular state, such as total stressed blood volume status, vessel elastance and resistance. To do so, the model parameters have to be estimated from data collected at the patient's bedside. This work considers a seven-parameter model of the cardiovascular system and investigates whether these parameters can be uniquely determined using indices derived from measurements of arterial and venous pressures, and stroke volume.

METHODS

An error vector defined the residuals between the simulated and reference values of the seven clinically available haemodynamic indices. The sensitivity of this error vector to each model parameter was analysed, as well as the collinearity between parameters. To assess practical identifiability of the model parameters, profile-likelihood curves were constructed for each parameter.

RESULTS

Four of the seven model parameters were found to be practically identifiable from the selected data. The remaining three parameters were practically non-identifiable. Among these non-identifiable parameters, one could be decreased as much as possible. The other two non-identifiable parameters were inversely correlated, which prevented their precise estimation.

CONCLUSIONS

This work presented the practical identifiability analysis of a seven-parameter cardiovascular system model, from limited clinical data. The analysis showed that three of the seven parameters were practically non-identifiable, thus limiting the use of the model as a monitoring tool. Slight changes in the time-varying function modeling cardiac contraction and use of larger values for the reference range of venous pressure made the model fully practically identifiable.

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.

Impeller-pump model derived from conservation laws applied to the simulation of the cardiovascular system coupled to heart-assist pumps

Previous numerical models of impeller pumps for ventricular assist devices utilize curve-fitted polynomials to simulate experimentally-obtained pressure difference versus flow rate characteristics of the pumps, with pump rotational speed as a parameter. In this paper the numerical model for the pump pressure difference versus flow rate characteristics is obtained by analytic derivation. The mass, energy and angular momentum conservation laws are applied to the working fluid passing through the impeller geometry and coupled with the turbomachine's velocity diagram. This results in the construction of a pressure difference versus flow rate characteristic for the specific pump geometry, with pump rotational speed as parameter. Overall this model allows modifications of the pump geometry, so that the pump avoids undesirable operating conditions, such as regurgitant flow. The HeartMate III centrifugal pump is used as an example to demonstrate the application of the technique. The parameterised numerical model for HeartMate III derived by this technique is coupled with a numerical model for the human cardiovascular system, and the combination is used to investigate the cardiovascular response under different conditions of impeller pump support. Conditions resulting in regurgitant pump flow, the pump resulting in aortic valve closure and taking over completely the pumping action from the diseased heart, and inner ventricular wall suction at pump inlet are predicted by the model. The simulation results suggest that for normal HeartMate III operation the pump speed should be maintained between 3,100 and 4,500 rpm to avoid regurgitant pump flow and ventricular suction. To obtain optimal overall cardiovascular system plus pump response, the pump operating speed should be 3,800 rpm.

Modeling the afferent dynamics of the baroreflex control system

In this study we develop a modeling framework for predicting baroreceptor firing rate as a function of blood pressure. We test models within this framework both quantitatively and qualitatively using data from rats. The models describe three components: arterial wall deformation, stimulation of mechanoreceptors located in the BR nerve-endings, and modulation of the action potential frequency. The three sub-systems are modeled individually following well-established biological principles. The first submodel, predicting arterial wall deformation, uses blood pressure as an input and outputs circumferential strain. The mechanoreceptor stimulation model, uses circumferential strain as an input, predicting receptor deformation as an output. Finally, the neural model takes receptor deformation as an input predicting the BR firing rate as an output. Our results show that nonlinear dependence of firing rate on pressure can be accounted for by taking into account the nonlinear elastic properties of the artery wall. This was observed when testing the models using multiple experiments with a single set of parameters. We find that to model the response to a square pressure stimulus, giving rise to post-excitatory depression, it is necessary to include an integrate-and-fire model, which allows the firing rate to cease when the stimulus falls below a given threshold. We show that our modeling framework in combination with sensitivity analysis and parameter estimation can be used to test and compare models. Finally, we demonstrate that our preferred model can exhibit all known dynamics and that it is advantageous to combine qualitative and quantitative analysis methods.

Verification of a computational cardiovascular system model comparing the hemodynamics of a continuous flow to a synchronous valveless pulsatile flow left ventricular assist device

The purpose of this investigation is to use a computational model to compare a synchronized valveless pulsatile left ventricular assist device with continuous flow left ventricular assist devices at the same level of device flow, and to verify the model with in vivo porcine data. A dynamic system model of the human cardiovascular system was developed to simulate the support of a healthy or failing native heart from a continuous flow left ventricular assist device or a synchronous pulsatile valveless dual-piston positive displacement pump. These results were compared with measurements made during in vivo porcine experiments. Results from the simulation model and from the in vivo counterpart show that the pulsatile pump provides higher cardiac output, left ventricular unloading, cardiac pulsatility, and aortic valve flow as compared with the continuous flow model at the same level of support. The dynamic system model developed for this investigation can effectively simulate human cardiovascular support by a synchronous pulsatile or continuous flow ventricular assist device.

EQUILIBRIUM MULTIPLICITY IN A CARDIOVASCULAR SYSTEM MODEL

The aim of this paper is to show that, for some parameters, a two-dimensional cardiovascular system model can exhibit intrinsic equilibrium multiplicity generated by a backward bifurcation, regardless of the baroreflex effect. The model considers the dynamics of arterial and venous compartments and a feedback effect in the stroke volume induced by venous pressure changes. The results of the mathematical analysis indicate that multiple non-trivial equilibrium points exist when the stroke volume function is convex around the origin. Interestingly, this equilibrium point structure would imply that under certain stroke volume functions, the baroreflex system would have to stabilize and regulate an unstable operating condition produced by certain values of the stroke volume. The paper ends with the discussion of some implications for the reliability and robustness of the baroreflex-feedback mechanism.

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.

Numerical simulation of global hydro-dynamics in a pulsatile bioreactor for cardiovascular tissue engineering

Previous numerical simulations of the hydro-dynamic response in the various bioreactor designs were mostly concentrated on the local flow field analysis using computational fluid dynamics, which cannot provide the global hydro-dynamics information to assist the bioreactor design. In this research, a mathematical model is developed to simulate the global hydro-dynamic changes in a pulsatile bioreactor design by considering the flow resistance, the elasticity of the vessel and the inertial effect of the media fluid in different parts of the system. The developed model is used to study the system dynamic response in a typical pulsatile bioreactor design for the culturing of cardiovascular tissues. Simulation results reveal the detailed pressure and flow-rate changes in the different positions of the bioreactor, which are very useful for the evaluation of hydro-dynamic performance in the bioreactor designed. Typical pressure and flow-rate changes simulated agree well with the published experimental data, thus validates the mathematical model developed. The proposed mathematical model can be used for design optimization of other pulsatile bioreactors that work under different experimental conditions and have different system configurations.

Numerical Comparison of Hemodynamics With Atrium to Aorta and Ventricular Apex to Aorta VAD Support

We report the first attempt to study with numerical methods ventricular assist device (VAD) models and the effects of various inlet VAD cannulations, coupling physical explanations and numerical investigation conclusions with clinical research results. We compared the hemodynamic response with VAD support by using two distinct VAD-inlet cannulation configurations: left atrium to aorta and left ventricular apex to aorta. Impeller pump and displacement pump VADs are considered. Constant VAD flow rate and counterpulsation motion models are simulated. The native cardiovascular system is modeled using the concentrated-parameter method by considering the flow resistance, vessel elasticity, and inertial effect of blood flow in cardiovascular system individual segments. Impeller and displacement pump dynamic models are represented by corresponding inlet and outlet flow rate changes in the VADs. Results show that the two VAD inlet cannulation configurations produce similar cardiac response (flows, pressures, volumes), except that when the VAD flow approaches the 100% assisting condition, the peak left ventricular systolic pressure and diastolic volume increase slightly in the left atrial cannulation, whereas they drop markedly in the left ventricular apex cannulation, suggesting increased ventricular wall tension and ventricular dilatation in the left atrial cannulation and that hemodynamically the left ventricular apex cannulation is more advantageous.

Model-based cardiac diagnosis of pulmonary embolism

A minimal cardiac model has been shown to accurately capture a wide range of cardiovascular system dynamics commonly seen in the intensive care unit (ICU). However, standard parameter identification methods for this model are highly non-linear and non-convex, hindering real-time clinical application. An integral-based identification method that transforms the problem into a linear, convex problem, has been previously developed, but was only applied on continuous simulated data with random noise. This paper extends the method to handle discrete sets of clinical data, unmodelled dynamics, a significantly reduced data set theta requires only the minimum and maximum values of the pressure in the aorta, pulmonary artery and the volumes in the ventricles. The importance of integrals in the formulation for noise reduction is illustrated by demonstrating instability in the identification using simple derivative-based approaches. The cardiovascular system (CVS) model and parameter identification method are then clinically validated on porcine data for pulmonary embolism. Errors for the identified model are within 10% when re-simulated and compared to clinical data. All identified parameter trends match clinically expected changes. This work represents the first clinical validation of these models, methods and approach to cardiovascular diagnosis in critical care.

Simulation of cardiovascular system diseases by including the autonomic nervous system into a minimal model

Diagnosing cardiovascular system (CVS) diseases from clinically measured data is difficult, due to the complexity of the hemodynamic and autonomic nervous system (ANS) interactions. Physiological models could describe these interactions to enable simulation of a variety of diseases, and could be combined with parameter estimation algorithms to help clinicians diagnose CVS dysfunctions. This paper presents modifications to an existing CVS model to include a minimal physiological model of ANS activation. A minimal model is used so as to minimise the number of parameters required to specify ANS activation, enabling the effects of each parameter on hemodynamics to be easily understood. The combined CVS and ANS model is verified by simulating a variety of CVS diseases, and comparing simulation results with common physiological understanding of ANS function and the characteristic hemodynamics seen in these diseases. The model of ANS activation is required to simulate hemodynamic effects such as increased cardiac output in septic shock, elevated pulmonary artery pressure in left ventricular infarction, and elevated filling pressures in pericardial tamponade. This is the first known example of a minimal CVS model that includes a generic model of ANS activation and is shown to simulate diseases from throughout the CVS.

Numerical Simulation of Cardiovascular Dynamics With Left Heart Failure and In-series Pulsatile Ventricular Assist Device

This article presents a numerical model for investigations of the human cardiovascular circulation system response, where the function of the impaired left ventricle is augmented by the pumping action of a pulsatile ventricular assist device (VAD) connected in series to the native heart. The numerical model includes a module for detailed heart valve dynamics, which helps to improve the accuracy of simulation in studying the pulsatile type VAD designs. Simulation results show that, for the case with left ventricular (LV) failure, the VAD support successfully compensates the impaired cardiovascular response, and greatly reduces the after-load of the diseased ventricle, thus assisting possible recovery of the ventricle from the diseased condition. The effects of these conditions on pulmonary circulation are also shown. To investigate the effect of different pumping-activation functions (VAD motion profiles) on the cardiovascular response, three different VAD motion profiles are investigated. The numerical results suggest that Hermitian type motion profiles (smooth curves skewed toward early systole) have the advantage of requiring minimum power to the VAD, and producing the minimum after-load to the left ventricle, minimum ventricular wall stress, and minimum ventricular work to the diseased ventricle; while sawtooth type motions need slightly more power input, and induce slightly increased aortic pressure in diastole, thus improving coronary perfusion.

Effects of atrial contraction, atrioventricular interaction and heart valve dynamics on human cardiovascular system response

Various simulation models of different complexity have been proposed to model the dynamic response of the human cardiovascular system. In a related paper we proposed an improved numerical model to study the dynamic response of the cardiovascular system, and the pressures, volumes and flow-rates in the four chambers of the heart, which included the effects of atrial contraction, atrioventricular interaction, and heart valve dynamics. This paper investigates the effects of each one of these aspects of the model on the overall dynamic system response. The dynamic response is studied under different situations, with and without including the effect of various features of the model, and these situations are studied and compared among themselves and to detailed aspects of expected healthy-system response. As an important contribution with potential clinical applications, this paper examines the corresponding effects of atrioventricular interaction, and heart valve opening and closing dynamics to the general system dynamic response. This isolation of physical cause-effect relationships is difficult to study with purely experimental methods. The simulation results agree well with results in the open literature. Comparison shows that introduction of these new features greatly improves the simulation accuracy of the effects of a, v and c waves, and in predicting regurgitant valve flow, the dichrotic notch, and E/A velocity ratio.

A concentrated parameter model for the human cardiovascular system including heart valve dynamics and atrioventricular interaction

Numerical modeling of the human cardiovascular system has always been an active research direction since the 19th century. In the past, various simulation models of different complexities were proposed for different research purposes. In this paper, an improved numerical model to study the dynamic function of the human circulation system is proposed. In the development of the mathematical model, the heart chambers are described with a variable elastance model. The systemic and pulmonary loops are described based on the resistance-compliance-inertia concept by considering local effects of flow friction, elasticity of blood vessels and inertia of blood in different segments of the blood vessels. As an advancement from previous models, heart valve dynamics and atrioventricular interaction, including atrial contraction and motion of the annulus fibrosus, are specifically modeled. With these improvements the developed model can predict several important features that were missing in previous numerical models, including regurgitant flow on heart valve closure, the value of E/A velocity ratio in mitral flow, the motion of the annulus fibrosus (called the KG diaphragm pumping action), etc. These features have important clinical meaning and their changes are often related to cardiovascular diseases. Successful simulation of these features enhances the accuracy of simulations of cardiovascular dynamics, and helps in clinical studies of cardiac function.

Efficient implementation of non-linear valve law and ventricular interaction dynamics in the minimal cardiac model

A minimal model of the cardio-vascular system (CVS) with ventricular interaction and inertial effects that accurately captures the physiological trends of a variety of disease states has been developed. However, the physiologically accurate open on pressure, close on flow valve law is computationally heavy to implement, reducing the model's potential clinical benefit. A significantly simpler representation of the valve law using Heaviside functions is derived and the ventricular interaction equations are reformulated to obtain a unique closed form analytical solution. The new formulation is tested and compared with the previous formulation for a healthy human and four clinically significant disease states: mitral and aortic stenosis, pulmonary embolism and septic shock. The new model formulation matches the previous model definition, differing by a mean model response error of no more than 0.2%. Computationally, it is 24 x faster than the previous method. More specifically, a short 20-beat simulation that took 102 s now requires 4.3 s, significantly improving the model's potential for practical use in a diagnostic and/or decision support role in the intensive care unit.

Numerical simulation of cardiovascular dynamics with healthy and diseased heart valves

This paper presents a new concentrated parameter model for cardiovascular dynamics that includes an innovative model of heart valve dynamics, which is embedded in the overall model of the four chambers of the heart and the systemic and pulmonary circulation loops. The heart chambers are described with a variable elastance model, and the systemic and pulmonary loops are described with modified Windkessel models. In modelling the heart valve dynamics, the various factors that influence the valve motion are examined, and the governing differential equation for valve motion is derived. The heart valve model includes the influence of the blood pressure effect, the friction effect from the tissue, and from blood motion. As improvement from previous works, the contribution of the blood vortex effect in the vicinity of the valve leaflets to valve motion is specially considered. The proposed model is then used in simulation of healthy and certain pathological conditions such as mitral valve stenosis and aortic regurgitation. The predicted results agree well with results illustrated in cardiology textbooks.

Mathematical modeling of cardiovascular system dynamics using a lumped parameter method

This work reviews the main aspects of cardiovascular system dynamics with an emphasis on modeling hemodynamic characteristics by the use of a lumped parameter approach. The methodological and physiological aspects of the circulation dynamics are summarized with the help of existing mathematical models. The main characteristics of the hemodynamic elements, such as the heart and arterial and venous systems, are first described. Distributed models of an arterial network are introduced, and their characteristics are compared with those of lumped parameter models. We also discuss the nonlinear characteristics of the pressure-volume relationship in veins. Then the control pathways that participate in feedback mechanisms (baroreceptors and cardiopulmonary receptors) are described to explain the interaction between hemodynamics and autonomic nerve control in the circulation. Based on a set-point model, the computational aspects of reflex control are explained.

Neural activity of the cardiac baroreflex decreases with age in normotensive and hypertensive subjects

OBJECTIVES

Baroreflex control of the heart rate is impaired in hypertensive subjects and decreases with age. The decrease in cardiac baroreflex sensitivity (BRS) is often ascribed to decreased distension of the pressure-sensing arterial wall segments. However, alterations in the sensing and processing of neural signals may be involved as well.

DESIGN

Conventionally, both vessel wall stiffness and the sensing and neural processing of the baroreflex are incorporated in the measure of pressure-derived BRS. We introduce stretch-derived BRS, which only considers the sensing and neural components of the baroreflex.

METHODS

To determine stretch-derived BRS in a non-invasive manner, we measured the spontaneously occurring low-frequency variations (range, 0.06-0.12 Hz) in the carotid artery diameter and the corresponding R-R interval fluctuations, and determined the associated transfer function. The stretch-derived BRS in a group of age-matched (age range, 25-72 years) normotensive (n = 20) and hypertensive (n = 21) subjects was compared.

RESULTS

In both subject groups the stretch-derived BRS decreased significantly with age. Moreover, the stretch-derived BRS of both groups was only different below 50 years of age.

CONCLUSIONS

The analysis of low-frequency fluctuations in the carotid artery diameter demonstrates that aging as well as hypertension are associated with impaired neural control of the baroreflex. Beyond 50 years of age the effect of hypertension cannot be distinguished from the effect of aging.

Minimal haemodynamic system model including ventricular interaction and valve dynamics

Characterising circulatory dysfunction and choosing a suitable treatment is often difficult and time consuming, and can result in a deterioration in patient condition, or unsuitable therapy choices. A stable minimal model of the human cardiovascular system (CVS) is developed with the ultimate specific aim of assisting medical staff for rapid, on site modelling to assist in diagnosis and treatment. Models found in the literature simulate specific areas of the CVS with limited direct usefulness to medical staff. Others model the full CVS as a closed loop system, but they were found to be very complex, difficult to solve, or unstable. This paper develops a model that uses a minimal number of governing equations with the primary goal of accurately capturing trends in the CVS dynamics in a simple, easily solved, robust model. The model is shown to have long term stability and consistency with non-specific initial conditions as a result. An "open on pressure close on flow" valve law is created to capture the effects of inertia and the resulting dynamics of blood flow through the cardiac valves. An accurate, stable solution is performed using a method that varies the number of states in the model depending on the specific phase of the cardiac cycle, better matching the real physiological conditions. Examples of results include a 9% drop in cardiac output when increasing the thoracic pressure from -4 to 0 mmHg, and an increase in blood pressure from 120/80 to 165/130 mmHg when the systemic resistance is doubled. These results show that the model adequately provides appropriate magnitudes and trends that are in agreement with existing data for a variety of physiologically verified test cases simulating human CVS function.

SHORT-TERM AUTONOMIC CONTROL OF CARDIOVASCULAR FUNCTION: A MINI-REVIEW WITH THE HELP OF MATHEMATICAL MODELS

In this work the main aspects of the short-term regulation of the cardiovascular system are reviewed and critically discussed, laying special emphasis on the role of the autonomic neural mechanisms involved, on their mutual interrelationships and complex integration. All these aspects are summarized with the help of mathematical models developed by the authors in past years. The main characteristics of the uncontrolled system (i.e., the heart and vessels) and of the efferent neural branches (sympathetic and vagal) working on it are first described. Then, the afferent pathways which participate in feedback mechanisms (baroreceptors, chemoreceptors, lung-stretch receptors, direct CNS response), and the feedforward mechanisms anticipating cardiovascular requirements are introduced, and their role discussed with reference to various cardiovascular perturbations (hemorrhage or posture changes, hypoxia, asphyxia, dynamic exercise). Analysis of physiological data via mathematical equations, and results of computer simulations, emphasize the great complexity, richness and variability of the autonomic cardiovascular control, including redundant mechanisms and antagonistic requirements. The use of mathematical models is essential to capture this richness, and to summarize apparent contradictory data into a coherent and comprehensive theoretical setting.

Velocity profile method for time varying resistance in minimal cardiovascular system models

This paper investigates the fluid dynamics governing arterial flow used in lumped parameter cardiovascular system (CVS) models, particularly near the heart where arteries are large. Assumptions made in applying equations conventionally used in lumped parameter models are investigated, specifically that of constant resistance to flow. The Womersley number is used to show that the effects of time varying resistance must be modelled in the pulsatile flow through the large arteries near the heart. It is shown that the equation commonly used to include inertial effects in fluid flow calculations is inappropriate for including time varying resistance. A method of incorporating time varying resistance into a lumped parameter model is developed that uses the Navier-Stokes equations to track the velocity profile. Tests on a single-chamber model show a 17.5% difference in cardiac output for a single-chamber ventricle model when comparing constant resistance models with the velocity profile tracking method modelling time varying resistance. This increase in precision can be achieved using 20 nodes with only twice the computational time required. The method offers a fluid dynamically and physiologically accurate method of calculating large Womersley number pulsatile fluid flows in large arteries around the heart and valves. The proposed velocity profile tracking method can be easily incorporated into existing lumped parameter CVS models, improving their clinical application by increasing their accuracy.

Simulation study of the cardiovascular functional status in hypertensive situation

An extended cardiovascular model was established based on our previous work to study the consequences of physiological or pathological changes to the homeostatic functions of the cardiovascular system. To study hemodynamic changes in hypertensive situations, the impacts of cardiovascular parameter variations (peripheral vascular resistance, arterial vessel wall stiffness and baroreflex gain) upon hemodynamics and the short-term regulation of the cardiovascular system were investigated. For the purpose of analyzing baroregulation function, the short-term regulation of arterial pressure in response to moderate dynamic exercise for normotensive and hypertensive cases was studied through computer simulation and clinical experiments. The simulation results agree well with clinical data. The results of this work suggest that the model presented in this paper provides a useful tool to investigate the functional status of cardiovascular system in normal or pathological conditions.

Influence of baroreflex on volume elasticity of heart and aorta in the rabbit

Optimal ventriculoaortic coupling includes tuning of elastic properties. The ratio of effective arterial elastance and left ventricular endsystolic elastance is often taken as a measure for mechanical and energetical efficiency. The present study determined the time course of ventricular and aortic volume elasticity (VE = dp/dV) throughout a complete heartbeat. This was achieved by using changes of eigenfrequency of two catheter-transducer systems under closed chest conditions in rabbits. Short-term VE modulation was studied by a baroreflex response, as induced by pressure changes applied to the carotid sinus. Long-term changes were studied in atherosclerotic rabbits (12 wk of high-cholesterol feeding). The time course and mean values of ventricular and aortic VE were changed by the baroreflex stimulus. Cholesterol feeding diminished the response. The degree of ventriculoaortic coupling, as quantified by VE(Aorta)/VE(Ventricle) ratio, varied during a single ejection period. The large span allows either maximal energetical efficiency or maximal stroke work. Although normal rabbits adjusted their ventriculoaortic coupling during baroreflex input, the cholesterol-fed rabbits failed to do so.

Dynamics of cerebral blood flow regulation explained using a lumped parameter model

The dynamic cerebral blood flow response to sudden hypotension during posture change is poorly understood. To better understand the cardiovascular response to hypotension, we used a windkessel model with two resistors and a capacitor to reproduce beat-to-beat changes in middle cerebral artery blood flow velocity (transcranial Doppler measurements) in response to arterial pressure changes measured in the finger (Finapres). The resistors represent lumped systemic and peripheral resistances in the cerebral vasculature, whereas the capacitor represents a lumped systemic compliance. Ten healthy young subjects were studied during posture change from sitting to standing. Dynamic variations of the peripheral and systemic resistances were extracted from the data on a beat-to-beat basis. The model shows an initial increase, followed approximately 10 s later by a decline in cerebrovascular resistance. The model also suggests that the initial increase in cerebrovascular resistance can explain the widening of the cerebral blood flow pulse observed in young subjects. This biphasic change in cerebrovascular resistance is consistent with an initial vasoconstriction, followed by cerebral autoregulatory vasodilation.

Computer simulation of the baroregulation in response to moderate dynamic exercise

A baroregulation model, based on a previous pulsatile non-linear multielement cardiovascular model, is extended and used to study short-term regulation mechanisms. Using this model, the responses of several cardiovascular variables to different exercise levels are simulated and compared with the experimental data reported in the literature. The impact of physiological or pathological changes on the short-term regulation of arterial pressure under the stimulus of moderate dynamic exercise is then studied. The simulation results indicate that baroreflex feedback plays a critical role in the short-term regulation of arterial pressure. When the baroreflex gain decreases to one-third of the normal value, the response of the mean arterial pressure to moderate dynamic exercise and post-exercise recovery time increases by factors of 1.7 and 2.3, respectively. Clinical data from 36 subjects (two groups: normal and hypertensive) are collected to validate the model. Computer simulations for the hypertensive group show that the elastic modulus of the arterial vessel wall is increased by 1.5 times, and peripheral resistance is increased by 1.3 times the normal value, and the baroreflex gain decreases from 0.55 (for the normal group) to 0.40. The simulation results for normal and hypertensive groups agree well with the clinical data.