Mathematical model of cardiovascular mechanics for diagnostic analysis and treatment of heart failure: Part 1. Model description and theoretical analysis

Model-based assessment of cardiopulmonary autonomic regulation in paced deep breathing

Autonomic dysfunction can lead to many physical and psychological diseases. The assessment of autonomic regulation plays an important role in the prevention, diagnosis, and treatment of these diseases. A physiopathological mathematical model for cardiopulmonary autonomic regulation, namely Respiratory-Autonomic-Sinus (RSA) regulation Model, is proposed in this study. A series of differential equations are used to simulate the whole process of RSA phenomenon. Based on this model, with respiration signal and ECG signal simultaneously acquired in paced deep breathing scenario, we manage to obtain the cardiopulmonary autonomic regulation parameters (CARP), including the sensitivity of respiratory-sympathetic nerves and respiratory-parasympathetic nerves, the time delay of sympathetic, the sensitivity of norepinephrine and acetylcholine receptor, as well as cardiac remodeling factor by optimization algorithm. An experimental study has been conducted in healthy subjects, along with subjects with hypertension and coronary heart disease. CARP obtained in the experiment have shown their clinical significance.

Cardiorenal Systems Modeling: Left Ventricular Hypertrophy and Differential Effects of Antihypertensive Therapies on Hypertrophy Regression

Cardiac and renal function are inextricably connected through both hemodynamic and neurohormonal mechanisms, and the interaction between these organ systems plays an important role in adaptive and pathophysiologic remodeling of the heart, as well as in the response to renally acting therapies. Insufficient understanding of the integrative function or dysfunction of these physiological systems has led to many examples of unexpected or incompletely understood clinical trial results. Mathematical models of heart and kidney physiology have long been used to better understand the function of these organs, but an integrated model of renal function and cardiac function and cardiac remodeling has not yet been published. Here we describe an integrated cardiorenal model that couples existing cardiac and renal models, and expands them to simulate cardiac remodeling in response to pressure and volume overload, as well as hypertrophy regression in response to angiotensin receptor blockers and beta-blockers. The model is able to reproduce different patterns of hypertrophy in response to pressure and volume overload. We show that increases in myocyte diameter are adaptive in pressure overload not only because it normalizes wall shear stress, as others have shown before, but also because it limits excess volume accumulation and further elevation of cardiac stresses by maintaining cardiac output and renal sodium and water balance. The model also reproduces the clinically observed larger LV mass reduction with angiotensin receptor blockers than with beta blockers. We further provide a mechanistic explanation for this difference by showing that heart rate lowering with beta blockers limits the reduction in peak systolic wall stress (a key signal for myocyte hypertrophy) relative to ARBs.

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.

Algebraic formulas characterizing an alternative to Guyton's graphical analysis relevant for heart failure

Although Guyton's graphical analysis of cardiac output-venous return has become a ubiquitous tool for explaining how circulatory equilibrium emerges from heart-vascular interactions, this classical model relies on a formula for venous return that contains unphysiological assumptions. Furthermore, Guyton's graphical analysis does not predict pulmonary venous pressure, which is a critical variable for evaluating heart failure patients' risk of pulmonary edema. Therefore, the purpose of the present work was to use a minimal closed-loop mathematical model to develop an alternative to Guyton's analysis. Limitations inherent in Guyton's model were addressed by 1) partitioning the cardiovascular system differently to isolate left ventricular function and lump all blood volumes together, 2) linearizing end-diastolic pressure-volume relationships to obtain algebraic solutions, and 3) treating arterial pressures as constants. This approach yielded three advances. First, variables related to morbidities associated with left ventricular failure were predicted. Second, an algebraic formula predicting left ventricular function was derived in terms of ventricular properties. Third, an algebraic formula predicting flow through the portion of the system isolated from the left ventricle was derived in terms of mechanical properties without neglecting redistribution of blood between systemic and pulmonary circulations. Although complexities were neglected, approximations necessary to obtain algebraic formulas resulted in minimal error, and predicted variables were consistent with reported values.

A quantitative systems physiology model of renal function and blood pressure regulation: Model description

Renal function plays a central role in cardiovascular, kidney, and multiple other diseases, and many existing and novel therapies act through renal mechanisms. Even with decades of accumulated knowledge of renal physiology, pathophysiology, and pharmacology, the dynamics of renal function remain difficult to understand and predict, often resulting in unexpected or counterintuitive therapy responses. Quantitative systems pharmacology modeling of renal function integrates this accumulated knowledge into a quantitative framework, allowing evaluation of competing hypotheses, identification of knowledge gaps, and generation of new experimentally testable hypotheses. Here we present a model of renal physiology and control mechanisms involved in maintaining sodium and water homeostasis. This model represents the core renal physiological processes involved in many research questions in drug development. The model runs in R and the code is made available. In a companion article, we present a case study using the model to explore mechanisms and pharmacology of salt‐sensitive hypertension.

Numerical modeling of vessel geometry to measure hemodynamics parameters non-invasively in cerebral arteriovenous malformation

Cerebral arteriovenous malformations (CAVM) are congenital lesions that contain a cluster of multiple arteriovenous shunts (NIDUS). Cardiac arrhythmia in CAVM patients causes irregular changes in blood flow and pressure in the NIDUS area. This paper proposes the framework for creating the lumped model of tortuous vessel structure near NIDUS based on radiological images. This lumped model is to analyze flow variations, with various combinations of the transient electrical signals, which simulate similar conditions of cardiac arrhythmia in CAVM patients. This results in flow variation at different nodes of the lumped model. Here we present two AVM patients with evaluation of 150 vessels locations as node points, with an accuracy of 93%, the sensitivity of 95%, and specificity of 94%. The calculated p-value is smaller than the significance level of 0.05.

Closing the loop: modelling of heart failure progression from health to end-stage using a meta-analysis of left ventricular pressure-volume loops

Introduction

The American Heart Association (AHA)/American College of Cardiology (ACC) guidelines for the classification of heart failure (HF) are descriptive but lack precise and objective measures which would assist in categorising such patients. Our aim was two fold, firstly to demonstrate quantitatively the progression of HF through each stage using a meta-analysis of existing left ventricular (LV) pressure-volume (PV) loop data and secondly use the LV PV loop data to create stage specific HF models.

Methods and Results

A literature search yielded 31 papers with PV data, representing over 200 patients in different stages of HF. The raw pressure and volume data were extracted from the papers using a digitising software package and the means were calculated. The data demonstrated that, as HF progressed, stroke volume (SV), ejection fraction (EF%) decreased while LV volumes increased. A 2-element lumped parameter model was employed to model the mean loops and the error was calculated between the loops, demonstrating close fit between the loops. The only parameter that was consistently and statistically different across all the stages was the elastance (Emax).

Conclusions

For the first time, the authors have created a visual and quantitative representation of the AHA/ACC stages of LVSD-HF, from normal to end-stage. The study demonstrates that robust, load-independent and reproducible parameters, such as elastance, can be used to categorise and model HF, complementing the existing classification. The modelled PV loops establish previously unknown physiological parameters for each AHA/ACC stage of LVSD-HF, such as LV elastance and highlight that it this parameter alone, in lumped parameter models, that determines the severity of HF. Such information will enable cardiovascular modellers with an interest in HF, to create more accurate models of the heart as it fails.

Adaptive multi-infusion decision support for the multivariable circulatory management of critically ill patients

We have developed a novel adaptive multi-infusion advisory system for circulatory management of critically ill patients which co-ordinates infusion adjustments to ensure safe trajectories. This system should reduce patient hospital stay and improve patient outcome by enhancing the quality of patient circulatory control; alleviating the clinical cognitive load, giving staff more time for direct patient care, while also reducing infusion adjustment errors. We have applied three derived circulatory variables which relate to the three main types of cardiovascular infusions (inotropic, vasoactive and fluid). A lumped parameter steady flow model of the human circulatory system and the effects of cardiovascular infusions was constructed for algorithm development, clinical experts providing feedback on a representative test set of simulated patients in circulatory shock. Independent self-learning fuzzy logic controllers (SLFLC) were found to give good adaptation to variable patient infusion sensitivities. A supervisory, rule-based module co-ordinates infusion adjustments to ensure safe circulatory trajectories. Monitoring of manual infusion adjustments allows timely advice and also a critiquing capability which can train junior staff and reduce infusion adjustment errors. A physical mock circulatory loop was used to construct and test our physical advisory system. Preliminary clinical results show good clinical utility of our adaptive multi-infusion advisory system.

The role of the autonomic nervous system in hypertension: a bond graph model study

A bond graph model of the cardiovascular system with embedded autonomic nervous regulation was developed for a better understanding of the role of the autonomic nervous system (ANS) in hypertension. The model is described by a pump model of the heart and a detailed representation of the head and neck, pulmonary, coronary, abdomen and extremity circulation. It responds to sympathetic and parasympathetic activities by modifying systemic peripheral vascular resistance, heart rate, ventricular end-systolic elastance and venous unstressed volumes. The impairment of ANS is represented by an elevation of the baroreflex set point. The simulation results show that, compared with normotensive, in hypertension the systolic and diastolic blood pressure (SBP/DBP) rose from 112/77 mmHg to 144/94 mmHg and the left ventricular wall thickness (LVWT) increased from 10 mm to 12.74 mm. In the case that ANS regulation was absent, both the SBP and DBP further increased by 8 mmHg and the LVWT increased to 13.22 mm. The results also demonstrate that when ANS regulation is not severely damaged, e.g. the baroreflex set point is 97 mmHg, it still has an effect in preventing the rapid rise of blood pressure in hypertension; however, with the worsening of ANS regulation, its protective role weakens. The results agree with human physiological and pathological features in hemodynamic parameters and carotid baroreflex function curves, and indicate the role of ANS in blood pressure regulation and heart protection. In conclusion, the present model may provide a valid approach to study the pathophysiological conditions of the cardiovascular system and the mechanism of ANS regulation.

Modelling in the study of interaction of Hemopump device and artificial ventilation

The aim of this work is to evaluate in different ventricular conditions the influence of joint mechanical ventilation (MV) and Hemopump assistance. To perform this study, we used a computer simulator of human cardiovascular system where the influence of MV was introduced changing thoracic pressure to positive values. The simulation confirmed that haemodynamic variables are highly sensitive to thoracic pressure changes. On the other hand, Hemopump assistance raises, among the others, mean aortic pressure, total cardiac output (left ventricular output flow plus Hemopump flow) and coronary flow. The simulation showed that the joint action of Hemopump and positive thoracic pressure diminishes these effects.

A cardiovascular-respiratory control system model including state delay with application to congestive heart failure in humans

This paper considers a model of the human cardiovascular-respiratory control system with one and two transport delays in the state equations describing the respiratory system. The effectiveness of the control of the ventilation rate is influenced by such transport delays because blood gases must be transported a physical distance from the lungs to the sensory sites where these gases are measured. The short term cardiovascular control system does not involve such transport delays although delays do arise in other contexts such as the baroreflex loop (see [46]) for example. This baroreflex delay is not considered here. The interaction between heart rate, blood pressure, cardiac output, and blood vessel resistance is quite complex and given the limited knowledge available of this interaction, we will model the cardiovascular control mechanism via an optimal control derived from control theory. This control will be stabilizing and is a reasonable approach based on mathematical considerations as well as being further motivated by the observation that many physiologists cite optimization as a potential influence in the evolution of biological systems (see, e.g., Kenner [29] or Swan [62]). In this paper we adapt a model, previously considered (Timischl [63] and Timischl et al. [64]), to include the effects of one and two transport delays. We will first implement an optimal control for the combined cardiovascular-respiratory model with one state space delay. We will then consider the effects of a second delay in the state space by modeling the respiratory control via an empirical formula with delay while the the complex relationships in the cardiovascular control will still be modeled by optimal control. This second transport delay associated with the sensory system of the respiratory control plays an important role in respiratory stability. As an application of this model we will consider congestive heart failure where this transport delay is larger than normal and the transition from the quiet awake state to stage 4 (NREM) sleep. The model can be used to study the interaction between cardiovascular and respiratory function in various situations as well as to consider the influence of optimal function in physiological control system performance.

Numerical Simulation of a Systemic Flow Test Rig

In this study, numerical simulation was carried out to investigate the dynamic response of a systemic flow test rig that is widely used for in vitro study of prosthesis in the cardiovascular system. In the system the physiological impedance of systemic circulation was modeled as a resistance-capacitance-resistance type. The system analysis was directly based on differential equations describing the system dynamics, and numerically solved using the fourth-order Runge-Kutta method. Results showed that pressure in the systemic circulation test rig could be successfully simulated with the developed model. From the numerical experiment, it was found that the maximum stroke of the driving mechanism, the flow coefficients and opening of the control valves, and the initial volume of air in the compliance strongly affect the dynamic performance of the test rig. The numerical method developed is a useful tool in the design and optimization of the system configuration.

Simulation and prediction of cardiotherapeutical phenomena from a pulsatile model coupled to the Guyton circulatory model

In order to use simulation prediction for cardiotherapeutical purposes, the well-documented and physiologically validated circulatory Guyton model was coupled to a cardiac pulsatile model which comprises the hemodynamics of the four chambers including valvular effects, as well as the Hill, Frank-Starling, Laplace, and autonomic nervous system (ANS) effects. The program is written in the "C" language and available for everybody. The program system was submitted to validation and plausibility tests both as to the steady-state and the dynamic properties. Pressures, volumes and flows and other variables turned out to be compatible with published experimental and clinical recordings both under physiological and pathophysiological conditions. The results from the application to cardiac electrotherapy emphasize the importance of atrial contraction to ventricular filling, the adequate atrio-ventricular delay, the effect of impaired ventricular relaxation, and the significance of the choice of the adequate cardiac pacemaker, both with respect to the stimulation site and the adequate sensor controlling pacing rate. The simulation will be further developed, tested and applied for cardiological purposes.

Computer simulation of haemodynamic parameters changes with left ventricle assist device and mechanical ventilation

Left Ventricular Assist Device is used for recovery in patients with heart failure and is supposed to increase total cardiac output, systemic arterial pressure and to decrease left atrial pressure. Aim of our computer simulation was to assess the influence of Left Ventricular Assist Device (LVAD) on chosen haemodynamic parameters in the presence of ventilatory support. The software package used for this simulation reproduces, in stationary conditions, the heart and the circulatory system in terms of pressure and volume relationships. Different circulatory sections (left and right heart, systemic and pulmonary arterial circulation, systemic and pulmonary venous circulation) are described by lumped parameter models. Mechanical properties of each section are modelled by RLC elements. The model chosen for the representation of the Starling's law of the heart for each ventricle is based on the variable elastance model. The LVAD model is inserted between the left atrium and the aorta. The contractility of the heart and systemic arterial resistance were adjusted to model pathological states. Our simulation showed that positive thoracic pressure generated by mechanical ventilation of the lungs dramatically changes left atrial and pulmonary arterial pressures and should be considered when assessing LVAD effectiveness. Pathological changes of systemic arterial resistance may have a considerable effect on these parameters, especially when LVAD is applied simultaneously with mechanical ventilation. Cardiac output, systemic arterial and right atrial pressures are less affected by changes of thoracic pressure in cases of heart pathology.

The influence of left ventricle assist device and ventilatory support on energy-related cardiovascular variables

One of the main purposes in using Left Ventricle Assist Devices (LVAD) to assist recovery in patients with heart failure, is to reduce the external work (EW) of the left natural ventricle. The simultaneous presence of mechanical ventilatory support can affect the value of this variable. The aim of our computer simulation was to trace the influence of LVAD on EW, cardiac mechanical efficiency (CME) and pressure volume area (PVA) in the presence of ventilatory support. Pathological conditions of the heart were reproduced. Peripheral systemic arterial resistance (Ras) was also changed to model physiological and pathological states. The influence of mechanical ventilation was introduced by changing levels of mean thoracic pressure. In this way we were able to predict changes of EW, CME and PVA in both ventricles, during ventilatory (mechanical) and cardiovascular (LVAD) support. Our simulation showed that positive thoracic pressure seems to affect the energy-related cardiovascular variables and should be taken into account during the assessment of LVAD effectiveness. Pathological changes of systemic peripheral resistance have a considerable effect on EW, CME and PVA of left ventricle. On the other hand energy-related parameters of the right ventricle are not especially affected by changes in systemic peripheral resistance.

Correlation of end-diastolic pressure and myocardial elasticity with the transit time of the left atrial pressure wave (A-Ar interval)

Contraction of the left atrium in diastole generates a pressure wave that moves along the postero-lateral wall of the left ventricle (LV), rebounds off the LV apex, and is then directed toward the outflow tract. The movement of this atrial pressure wave may be detected with pulsed Doppler echocardiography by placing a sample volume in the LV outflow tract. The resulting spectral profile shows the initial. A velocity wave and also the Ar velocity wave, which is caused by the atrial pressure wave rebounding off the LV apex. The transit time from the inflow tract to the outflow tract of the atrial pressure wave (A-Ar interval) may be determined from the time axis of the spectral profile by measuring the peak-to-peak separation of the A and Ar, velocity waves. It occurs in the range 25 to 80 milliseconds. The primary determinant of the A-Ar interval is the elasticity of the LV myocardium. We correlated ventricular elasticity with the A-Ar interval in 47 patients and found a significant negative linear correlation (r = -0.782, p < 0.001). Because the pressure in a viscoelastic conduit such as the LV is determined by the elasticity of the ventricular wall, we correlated end-diastolic pressure with the A-Ar interval and again showed a significant negative linear correlation (r = -0.701, p < 0.001). The A-Ar interval is an easily measured noninvasive index of the diastolic function of the LV that reflects its intrinsic elasticity and end-diastolic pressure. It is therefore a quantitative measurement of LV wall stiffness and end-diastolic pressure.

Mathematical model of cardiovascular mechanics for diagnostic analysis and treatment of heart failure: Part 2. Analysis of vasodilator therapy and planning of optimal drug therapy

Using a mathematical model of cardiovascular mechanics, various complicated responses to vasodilator therapy for heart failure have been well accounted for through common logic: (i) the differential effects of various vasodilators on cardiac output; (ii) the opposite response of cardiac output to sodium nitroprusside in a normal state and heart failure state; (iii) the different responses of cardiac index, arterial pressure and left ventricular end-diastolic pressure to hydralazine in different types of heart failure. The response to combined vasodilator-inotropic agent therapy was simulated well by the model. The optimal therapeutic regimen was then formulated to simultaneously control the cardiac output, systemic and pulmonary arterial and venous pressures, and the degree of coronary ischaemia by multiple drug delivery, and the problem was solved using the model. We conclude that the model provides a useful basis for obtaining a guidance for more appropriate therapeutic regimen in heart failure.