Incorporating Autoregulatory Mechanisms of the Cardiovascular System in Three-Dimensional Finite Element Models of Arterial Blood Flow

The cardiovascular system is a closed-loop system in which billions of vessels interact with each other, and it enables the control of the systemic arterial pressure and varying organ flow through autoregulatory mechanisms. In this study, we describe the development of mathematical models of autoregulatory mechanisms for systemic arterial pressure and coronary flow and discuss the connection of these models to a hybrid numerical/analytic closed-loop model of the cardiovascular system. The closed-loop model consists of two lumped parameter heart models representing the left and right sides of the heart, a three-dimensional finite element model of the aorta with coronary arteries, three-element Windkessel models and lumped parameter coronary vascular models that represent the systemic circulation, and a three-element Windkessel model to approximate the pulmonary circulation. Using the connection between the systemic arterial pressure and coronary flow regulation systems, and the hybrid closed-loop model, we studied how the heart, coronary vascular beds, and arterial system respond to physiologic changes during light exercise and showed that these models can realistically simulate temporal behaviors of the heart, coronary vascular beds, and arterial system during exercise of healthy subjects. These models can be used to study temporal changes occurring in the heart, coronary vascular beds, and arterial system during cardiovascular intervention or changes in physiological states.     

On Coupling a Lumped Parameter Heart Model and a Three-Dimensional Finite Element Aorta Model

Aortic flow and pressure result from the interactions between the heart and arterial system. In this work, we considered these interactions by utilizing a lumped parameter heart model as an inflow boundary condition for three-dimensional finite element simulations of aortic blood flow and vessel wall dynamics. The ventricular pressure–volume behavior of the lumped parameter heart model is approximated using a time varying elastance function scaled from a normalized elastance function. When the aortic valve is open, the coupled multidomain method is used to strongly couple the lumped parameter heart model and three-dimensional arterial models and compute ventricular volume, ventricular pressure, aortic flow, and aortic pressure. The shape of the velocity profiles of the inlet boundary and the outlet boundaries that experience retrograde flow are constrained to achieve a robust algorithm. When the aortic valve is closed, the inflow boundary condition is switched to a zero velocity Dirichlet condition. With this method, we obtain physiologically realistic aortic flow and pressure waveforms. We demonstrate this method in a patient-specific model of a normal human thoracic aorta under rest and exercise conditions and an aortic coarctation model under pre- and post-interventions.     

A LabVIEW™ Model Incorporating an Open-Loop Arterial Impedance and a Closed-Loop Circulatory System

While numerous computer models exist for the circulatory system, many are limited in scope, contain unwanted features or incorporate complex components specific to unique experimental situations. Our purpose was to develop a basic, yet multifaceted, computer model of the left heart and systemic circulation in LabVIEW™ having universal appeal without sacrificing crucial physiologic features. The program we developed employs Windkessel-type impedance models in several open-loop configurations and a closed-loop model coupling a lumped impedance and ventricular pressure source. The open-loop impedance models demonstrate afterload effects on arbitrary aortic pressure/flow inputs. The closed-loop model catalogs the major circulatory waveforms with changes in afterload, preload, and left heart properties. Our model provides an avenue for expanding the use of the ventricular equations through closed-loop coupling that includes a basic coronary circuit. Tested values used for the afterload components and the effects of afterload parameter changes on various waveforms are consistent with published data. We conclude that this model offers the ability to alter several circulatory factors and digitally catalog the most salient features of the pressure/flow waveforms employing a user-friendly platform. These features make the model a useful instructional tool for students as well as a simple experimental tool for cardiovascular research.     

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.     

Physiological and pathophysiological implications of ventricular/vascular coupling

The purpose of this paper is to consider “ideal” ventricular/vascular coupling, and how this may be manifest in the time domain and in the frequency domain. The paper will also consider how such “ideal” coupling is achieved, and how it might be disturbed. The arterial system plays a crucial role in ventricular/vascular coupling since it separates the smallest vessels where flow is almost perfectly continuous from the ventricle, whose output is intermittent. Ventricular/vascular coupling can be assessed from measurements of pressure and flow in the ascending aorta (AA) (for left ventricle/systemic circulation), and in the main, pulmonary artery (MPA) (for right ventricle/pulmonary circulation). Ideal coupling is manifest as low pressure fluctuation in AA and MPA. Low pressure fluctuation results in pressure during systole being only slightly greater than pressure throughout the whole cardiac cycle, and pressure during diastole being only slightly less. This is desirable because pressure during systole determines ventricular output (when inotropic state and ventricular filling are constant), and ventricular metabolic requirement, while pressure during diastole in AA is a major determinant of coronary blood flow. In the frequency domain, “ideal” coupling is manifest as a correspondence between minimal values of impedance modulus in AA and MPA with maximal values of flow harmonics in AA and MPA, respectively. Factors responsible for “ideal” coupling have been identified as high distensibility of proximal arteries (with decreasing distensibility in peripheral arteries), wave reflection at arterial terminations, and a “match” between heart rate on the one hand and arterial length and wave velocity on the orther. This favourable “match” results in the heart operating for both systemic and pulmonary circulations close to a node of pressure and antinode of flow; this match is improved under conditions which simulate flight and fight. While ventricular/vascular coupling appears to be close to ideal in most large mammals, it appears to be less than ideal in adult humans and some small mammals including guinea pigs, rats, and mice. The cause for mismatch in small mammals is unclear. In humans however, finding are attributable to progressive arterial degeneration which is known to commence in childhood and is apparent in the elderly as dilated tortuous arteries, high pulse pressure, and high likelihood of developing ventricular failure.     

Large-Scale 3-D Geometric Reconstruction of the Porcine Coronary Arterial Vasculature Based on Detailed Anatomical Data

The temporal and spatial distribution of coronary blood flow, pressure, and volume are determined by the branching pattern and three-dimensional (3-D) geometry of the coronary vasculature, and by the mechanics of heart wall and vascular tone. Consequently, a realistic simulation of coronary blood flow requires, as a first step, an accurate representation of the coronary vasculature in a 3-D model of the beating heart. In the present study, a large-scale stochastic reconstruction of the asymmetric coronary arterial trees (right coronary artery, RCA; left anterior descending, LAD; and left circumflex, LCx) of the porcine heart has been carried out to set the stage for future hemodynamic analysis. The model spans the entire coronary arterial tree down to the capillary vessels. The 3-D tree structure was reconstructed initially in rectangular slab geometry by means of global geometrical optimization using parallel simulated annealing (SA) algorithm. The SA optimization was subject to constraints prescribed by previously measured morphometric features of the coronary arterial trees. Subsequently, the reconstructed trees were mapped onto a prolate spheroid geometry of the heart. The transformed geometry was determined through least squares minimization of the related changes in both segments lengths and their angular characteristics. Vessel diameters were assigned based on a novel representation of diameter asymmetry along bifurcations. The reconstructed RCA, LAD and LCx arterial trees show qualitative resemblance to native coronary networks, and their morphological statistics are consistent with the measured data. The present model constitutes the first most extensive reconstruction of the entire coronary arterial system which will serve as a geometric foundation for future studies of flow in an anatomically accurate 3-D coronary vascular model.     

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.     

Computerized left ventricular mechanics and control system analyses models relevant for cardiac diagnosis

Two left ventricular models, relevant to cardiac diagnosis, are presented. The first is a Computer-based Finite Element Stress Analysis model of the left ventricle (LV) aimed to account for the irregular left ventricular geometry obtainable from single plane cineangiocardiography. The in vivo data for the model consists of instantaneous left ventricular chamber pressure and cine. The calculated wall stresses are compared with idealized geometry models. The finite element model is better able to delineate the stress concentrations at sites of large curvatures.The second model consists of a control system model of the LV, which formulates the interaction of the mechanics of the LV and the lumped parameter circulatory system by the central nervous system's monitoring of the mean arterial pressure, regulation of the heart rate, left ventricular contractility and the perepheral impedance. The model is parametrically simulated for a subject by means of the continuous system modelling program (CSMP); then the simulated model's response to a physiological stress simulating pressure perturbation is determined; the model enables an assessment of the physiological stress sustaining capacity of a subject.     

Computer simulation of the mechanically-assisted failing canine circulation

A model of the cardiovascular system is presented. The model includes representations of the left and right ventricles, a nonlinear multielement model of the aorta and its main branches, and lumped models of the systemic veins and the pulmonary circulation. A simulation of the intra-aortic balloon pump and representations of physiological compensatory mechanisms are also incorporated in the model. Parameters of the left ventricular model were set to simulate either the normal or failing canine circulation. Pressure and flow waveforms throughout the circulation as well as ventricular pressure and volume were calculated for the normal, failing, and assisted failing circulation. Cardiac oxygen supply and consumption were calculated from the model. They were used as direct indices of cardiac energy supply and utilization to assess the effects of cardiac assistance. This work was supported in part by Grant No. EET-8620120 from the National Science Foundation.     

Anatomy of the ferret heart: An animal model for cardiac research

The hearts of 38 black-footed ferrets (Mustela nigripes) were studied with the use of physiologic, microdissection, vascular injection and histologic methods. These animals had a mean heart rate of 265 per minute, a heart weight of 3.7–5.2 gm, and a mean aortic pressure of 139.5 mm Hg. The predominant left coronary artery supplied usually both the SA and AV nodes, as well as the AV bundle, bundle branches and most of the ventricular myocardium. The cells of a well differentiated cardiac conduction system increase in cytoplasmic diameter from the SA node to the distal bundle branches. A cartilaginous right fibrous trigone and thick anulus fibrosus form useful landmarks for delineating AV node and AV bundle relationships. Small size, discrete nodal masses and a unique coronary arterial pattern make this heart an ideal model for histochemical, ultrastructural, electrophysiologic and pathologic circulation research.     

脉搏波传播和血管顺应性对体循环和肺循环系统血流特性和心室功率的影响

本文采用耦合模型研究脉搏波传播和血管顺应性对体循环和肺循环血流特性和心室功率的影响。左心室和右心室均采用Ｅ－Ｒ模型,后负荷系统对于体动脉和肺动脉分别采用Ｔ－Ｙ管模型和稍微不对称Ｔ管模型,应用脉冲响应法将二者耦合起来。选取生理范围的参数,计算了两个系统的每搏输出量（ＳＶ）、每搏输出功（ＳＷ）、定常功率（Ｗs）、脉动功率（Ｗo)和总功率（Ｗt）等。详细分析了血管顺应性、脉搏波波速等对这些参量的影响。得到的主要结果是脉动功率对参数的变化比定常功率更敏感。因此,脉动功率百分比或许是评估心室效率的一个较好参数之一。     

Dependence of Intramyocardial Pressure and Coronary Flow on Ventricular Loading and Contractility: A Model Study

The phasic coronary arterial inflow during the normal cardiac cycle has been explained with simple (waterfall, intramyocardial pump) models, emphasizing the role of ventricular pressure. To explain changes in isovolumic and low afterload beats, these models were extended with the effect of three-dimensional wall stress, nonlinear characteristics of the coronary bed, and extravascular fluid exchange. With the associated increase in the number of model parameters, a detailed parameter sensitivity analysis has become difficult. Therefore we investigated the primary relations between ventricular pressure and volume, wall stress, intramyocardial pressure and coronary blood flow, with a mathematical model with a limited number of parameters. The model replicates several experimental observations: the phasic character of coronary inflow is virtually independent of maximum ventricular pressure, the amplitude of the coronary flow signal varies about proportionally with cardiac contractility, and intramyocardial pressure in the ventricular wall may exceed ventricular pressure. A parameter sensitivity analysis shows that the normalized amplitude of coronary inflow is mainly determined by contractility, reflected in ventricular pressure and, at low ventricular volumes, radial wall stress. Normalized flow amplitude is less sensitive to myocardial coronary compliance and resistance, and to the relation between active fiber stress, time, and sarcomere shortening velocity.     

Arterial pressure-flow relations in the awake standing dog.

The transient circulatory changes following paced heart rate increase are reported from 133 trials with 6 unanesthetized dogs with chronically implanted monitoring devices for heart rate, cardiac output, aortic blood pressure, and mean right atrial pressure. In 62 trials with 2 of the dogs, pulmonary artery, and left ventricular end-diastolic pressure, as well as left ventricular dP/dt were also studied. The sequence of changes in pressures and flows is analyzed in terms of probable underlying mechanisms, particularly with respect to the nature of vascular resistances. The rise in aortic pressure and flow during the first 3 s of paced heart rate increase, before arterial stretch receptor reflexes become active, is more consistent with an effective downstream pressure of about 49 mmHg, presumably at the arteriolar level, than with an effective downstream pressure close to 0 mmHg at the right atrial level. In the pulmonary circulation where vascular reflex effects are less prominent, the pattern of pulmonary arterial pressure and flow for the entire 30 s of observation is consistent with an effective downstream pressure of 9 mmHg, presumably at the alveolar or pulmonary arteriolar level, rather than at the level of the left ventricular end-diastolic pressure.     

应用于零维左心血液循环的二尖瓣模型的研究

提出一个可以准确合理地模拟二尖瓣动力学特性的瓣叶运动流阻模型。考虑影响二尖瓣瓣叶运动的跨瓣压差和血流推力,建立二尖瓣运动的控制方程,提出依赖于瓣叶打开角度θ的瓣叶运动流阻模型,把该模型应用于零维左心血液循环系统,得到血液动力学特性。在保持心输出量和反流分数一致的条件下,比较该模型、瞬态关闭的阶梯流阻模型和经验指定的时变流阻模型。结果发现,瓣叶运动流阻模型能反映瓣膜关闭过程中的血液动力学,如压差和流量的滞后性以及关闭流量,同时该模型可以通过调整单位转动惯量跨瓣压差影响系数K_p和血流影响系数K_b的大小,改变瓣膜打开过程和关闭过程所需时间,瓣膜打开和关闭时间分别为50.0和40.2 ms。该模型可弥补阶梯流阻模型中忽略瓣膜运动过程的瞬态关闭的缺点,同时也能避免时变流阻模型中关闭起始时间的不合理性。此模型较为合理准确地模拟二尖瓣关闭过程的动力学特性,且简单易控制。     

Comparison of two mathematical models for the study of vascular reactivity

This work is based on the premise that fingertip temperature variation during arterial occlusion and subsequent reperfusion can be used as an indirect measurement of vascular reactivity, commonly assessed by directly measuring flow and its temporal alterations in response to arterial occlusion. Temperature of the fingers depends on blood perfusion and environmental factors. The temperature change experienced during hyperemia or high blood flow after occlusion depend on the capacity of the occluded arteries to restore normal circulation or vascular reactivity.This work uses two mathematical models of heat transfer to show the relationship between blood flow and changes in fingertip temperature experienced during vascular occlusion and reperfusion. The models consider different levels of complexity and anatomical detail. One model is a lumped or zero-order model that neglects tissue composition; the second model (first-order model) considers a simplified anatomy of the finger and allows the analysis of tissue composition which cannot be addressed with the lumped system.Thermal models provide a way of estimating the influence of different factors on the dynamic temperature response recorded during the reactivity tests. The models intend to increase the interest of the clinical community in the thermal study of vascular reactivity compared to other techniques that focus on the analysis of flow.The differences of the calculated dynamic temperature during arterial occlusion and reperfusion using both models were analyzed. Such comparison indicated that the zero-order model suffices to analyze the temperature variation during the reactivity test, as long as the proper variation between initial condition and environmental parameters affecting the response is used.     

左心室压力-容积关系的仿真

目的　仿真前、后负荷改变及各种瓣膜病情况下 ,左心室压力 容积关系的改变。方法　应用Ursino所建立的全循环系统集中参数模型 ,对跨瓣流动方程作了适当修改。结果　首先 ,讨论了心室前负荷、后负荷和心肌收缩力对左心室压力 容积关系的影响 ;在此基础上仿真了二尖瓣狭窄、二尖瓣关闭不全 ,主动脉瓣狭窄和主动脉瓣双病变的左心室压力 容积环。结论　仿真结果显示了心脏瓣膜疾病对左心室功能的影响 ,同时说明集中参数模型不但能描述血管系统 ,也能较好地模拟心脏功能     

Cardiac assist by intrathoracic and abdominal pressure variations: A mathematical study

A lumped parameter mathematical model of the complete cardiovascular system simulating cardiac assist by external pressure variations is presented. The various vascular compartments are assumed to have appropriate resistance, capacitance and inductance values. The right and left ventricles are represented by time varying capacitances reciprocal to the time varying elastances. The effects of external pressure variations on the circulation are evaluated for different modes of operation. Systolic thoracic pressure waves, diastolic abdominal pressure waves and a combination of both waves are shown to have a favourable effect on the circulation in cardiogenic shock. The different mechanism of cardiocirculatory assist involved are presented and discussed below     

The Effect of Progesterone on Coronary Blood Flow in Anaesthetized Pigs

The present study was designed to investigate the effect of progesterone on the coronary circulation and to determine the mechanisms involved. In pigs anaesthetized with sodium pentobarbitone, changes in left circumflex or anterior descending coronary blood flow caused by intravenous infusion of progesterone at constant heart rate and arterial blood pressure were assessed using an electromagnetic flowmeter. In 14 pigs, infusion of 1 mg h(-1) of progesterone caused an increase in coronary blood flow without affecting left ventricular dP/dtmax (rate of change of left ventricular systolic pressure) and filling pressures of the heart. In a further four pigs, this vasodilatory coronary effect was enhanced by graded increases in the dose of the hormone of between 1, 2 and 3 mg h(-1). The mechanisms of the above response were studied in the 14 pigs by repeating the experiment after haemodynamic variables had returned to the control values observed before infusion. In six pigs, blockade of muscarinic cholinoceptors and adrenoceptors with atropine, propranolol and phentolamine did not affect the coronary vasodilatation caused by progesterone. In the remaining eight pigs, this response was abolished by intracoronary injection of N(omega)-nitro-L-arginine methyl ester (L-NAME) even when performed after reversing the increase in arterial blood pressure and coronary vascular resistance caused by L-NAME with continuous intravenous infusion of papaverine. The present study showed that intravenous infusion of progesterone primarily caused coronary vasodilatation. The mechanism of this response was shown to involve the endothelial release of nitric oxide.     

Three-Wall Segment (TriSeg) Model Describing Mechanics and Hemodynamics of Ventricular Interaction

A mathematical model (TriSeg model) of ventricular mechanics incorporating mechanical interaction of the left and right ventricular free walls and the interventricular septum is presented. Global left and right ventricular pump mechanics were related to representative myofiber mechanics in the three ventricular walls, satisfying the principle of conservation of energy. The walls were mechanically coupled satisfying tensile force equilibrium in the junction. Wall sizes and masses were rendered by adaptation to normalize mechanical myofiber load to physiological standard levels. The TriSeg model was implemented in the previously published lumped closed-loop CircAdapt model of heart and circulation. Simulation results of cardiac mechanics and hemodynamics during normal ventricular loading, acute pulmonary hypertension, and chronic pulmonary hypertension (including load adaptation) agreed with clinical data as obtained in healthy volunteers and pulmonary hypertension patients. In chronic pulmonary hypertension, the model predicted right ventricular free wall hypertrophy, increased systolic pulmonary flow acceleration, and increased right ventricular isovolumic contraction and relaxation times. Furthermore, septal curvature decreased linearly with its transmural pressure difference. In conclusion, the TriSeg model enables realistic simulation of ventricular mechanics including interaction between left and right ventricular pump mechanics, dynamics of septal geometry, and myofiber mechanics in the three ventricular walls.