The aim of cardiovascular fluid dynamics is to predict pressure, velocity and tension as a function of time and space based on a realistic model of vascular geometry, fluid-structure interaction, blood’s behaviour and vessel wall response.
Modelling of cardiovascular fluid mechanics is challenging because of the heterogeneity of the vascular system in terms of geometrical and mechanical properties. This aspect requires the development and use of different models according to the need and the assumptions made.
Three-dimensional (3-D) models are appropriate when detailed information on the flow field is needed in a specific region of the vascular system. The high computational power required and the challenge related to fluid-structure interaction remain the main limitations to their practical application, which is confined to defined regions such as arterial bifurcation, aortic arch and aneurysm formation.
One-dimensional (1-D) models overcome this limitation with the assumption of axial symmetry and the restriction of spatial variation to the axial direction. Since the wavelength range of the pressure waves generated by the heart is much longer than the diameter of the blood vessels, 1-D modelling is well suited for the analysis of arterial wave propagation and highlights the resistance, elastic and inertial effects of blood motion and arterial wall.
Zero-dimensional (0-D) or lumped-parameter models eliminate the variation in space and allow the description of pressure and flow as a function of time in a specific compartment of the circulatory system. 0-D modelling is widely used particularly for the analysis of average values and the interaction between an assist device and the cardiovascular system. Although 0-D modelling gives less detailed predictions of pressure and flow waveforms, it has shown great potential and flexibility for clinical application with particular reference to the patho-physiology of heart failure, guidance for patient selection and the haemodynamic impact of device intervention. A combined approach of lumped-parameter modelling, pressure-volume analysis and modified time-varying elastance has a significant potential for daily use within the constraints of a clinical setting.
Numerical models are helpful for the analysis of variables which cannot be measured directly with conventional methods; they can test hypotheses or predict the behaviour of the system under investigation. Numerical simulation is one of the methods used to study the behaviour of the cardiovascular and respiratory system in different physiological and pathological conditions.
CARDIOSIM© is a software simulator of the human cardiocirculatory and respiratory system, developed in the Cardiovascular Numerical/Hybrid Modelling Lab of the CNR Institute of Clinical Physiology (Rome). The main components of the cardiovascular system are two pumps in series, the peripheral or systemic circulation and the pulmonary circulation.
The cardiovascular simulator runs on PC with Microsoft Windows operating system; it is implemented in Visual Basic language.
This software simulator platform has a modular structure that consisting of seven different general sections (see figure), which can be assembled to reproduce different patho-physiological conditions. The complexity of the assembled model depends on the context in which it must be used.
Figure shows the seven sections implemented in CARDIOSIM©: left and right heart (atrium and ventricle) sections, systemic and pulmonary arterial compartments, systemic and pulmonary venous sections and finally the coronary circulation.
This numerical simulator can reproduce the most important circulatory phenomena in terms of pressure and volume relationships. It represents the whole circulation using a lumped-parameter model and enables the simulation of different cardiovascular conditions on the basis of the Starling’s law of the heart and a modified time-varying elastance model.
CARDIOSIM© is currently being used in Italy for the training of medical, bioengineering and clinical engineering students. This software is also recommended for Continuing Medical Education (CME) promoted by Italian Training Institutions in cooperation with the Ministry of Health to ensure high quality teaching among health care professionals. These courses are developed in cooperation with the Institute of Clinical Physiology (CNR - Rome), the Department of Cardiovascular, Respiratory, Nephrology, Anaesthetic and Geriatric Sciences (Sapienza University in Rome) and with the Laboratory for Extracorporeal Circulation Research (ECCLab - RWTH Aachen University). This educational activity is addressed to medical students and a wide range of healthcare professionals (e.g. cardiologists, cardiac surgeons, general practitioners, anaesthetists, nurses, researchers, paramedics, etc.).
The Frank-Starling law of the heart is the fundamental principle of cardiac performance which states that the force of contraction of the cardiac muscle is proportional to its initial length. The energy set free at each contraction is a simple function of cardiac filling (preload). When the diastolic filling of the heart is increased or decreased with a given volume, the displacement of the heart increases or decreases with this volume. As described elsewhere, cardiac output increases or decreases in response to changes in heart rate or stroke volume. When a person stands up, for example, cardiac output falls because of a fall in central venous pressure, which leads to a decrease in stroke volume. Another example is limb movement (muscle pump) during exercise, which enhances venous return to the heart and causes an increase in stroke volume. The ability of the heart to change its force of contraction and therefore stroke volume in response to changes in venous return is called the Frank-Starling mechanism. Increased venous return increases the ventricular filling (end-diastolic volume) and therefore preload, which is the initial stretching of the cardiac myocytes prior to contraction. Increasing preload increases the active tension developed by the muscle fibre and increases the velocity of fibre shortening at a given afterload and inotropic state.
Frank-Starling law describes the ability of the native ventricle to change the stroke volume (beat-to-beat) as a function of the atrial pressure (preload) in terms of end-diastolic and end-systolic volumes.
Ventricular filling depends on the amount of venous return in relation to potential loss through the atrioventricular valve and ventricular ability to change its volume in relation to a given change in pressure or ventricular compliance.
Ventricular emptying depends on its ability to contract and on the state of the vascular bed.
The circulatory network can be rearranged in CARDIOSIM© to reproduce Frank-Starling law of the heart as follows:
Figure shows the electric analogue of the left circulatory network with the left ventricle implemented using the ventricular numerical model described in the Numerical Heart Model (1) section. In this configuration the preload and the afterload can be set (manually) to the desired value.
Numerical Heart Models
CARDIOSIM © reproduces the heart’s behaviour using a modified time-varying elastance model. Two different numerical models have been implemented into the software simulator, each with specific features which are described below.
- In the first one the left and right ventricular filling and ejection phases are described separately. The contraction and ejection phases are implemented using a modified time-varying elastance model. The left (right) ventricular loop, the End-Systolic Pressure-Volume Relationship (ESPVR) and the End-Diastolic Pressure-Volume Relationship (EDPVR) can be plotted on the pressure-volume plane. The behaviour of left (right) atrium is described as a linear capacity with a constant value of compliance and unstressed volume neglecting the contractile atrial activity.
- In the second one a modified time-varying elastance model is used to describe left and right ventricular function and applied for the analysis of ventricular interdependence through the inter-ventricular septum, where the properties of one ventricle are a function of the properties of the contra-lateral one. Left and right atrial function is described in a similar manner. This setting is useful for the analysis and simulation of inter-ventricular and/or intra-ventricular conduction delay (dyssynchrony). Ventricular dyssynchrony may cause a number of deleterious effects on cardiac function such as reduced diastolic filling time, weakened contractility, severe mitral regurgitation and regional wall motion abnormalities. The atrioventricular delay that occurs in atrioventricular block and atrial fibrillation can also be simulated.
Systemic Circulation Models
The systemic circulation is a high pressure, high resistance system with pulsatile behaviour where blood pressure ranges from 120 mmHg in systole and 80 mmHg in diastole.
Pressure and flow in the systemic circulation fluctuate in view of the pumping action of the left ventricle. During early systole, pressure and flow rate rise in phase with each other emphasizing the fact that a pressure gradient drives the flow. Later in systole, the synchrony between pressure and flow rate breaks down due to the arrival of reflected components of the pressure wave.
Aortic pressure and flow result from the interaction of the heart, the pump, and the arterial system, the load. The quantitative contribution of the heart and the arterial system to pressure and flow plays a key role in the understanding of the mechanism of hypertension, heart failure and other cardiovascular disease. If the heart is modelled according to the time-varying elastance theory and the arterial system is modelled with a three-element Windkessel, the contribution of each parameter to pressure and flow can be quantified.
CARDIOSIM© reproduces the systemic circulation based on different numerical models with variable complexity. The numerical analysis is based on lumped-parameter models (0-D) using RLC circuit elements (Windkessel model). The Windkessel model (WM) was designed in the late 1800’s by the German physiologist Otto Frank and consists of:
- Compliance (C): refers to the elasticity and extensibility of the major artery during the cardiac cycle;
- Resistance (R): refers to the flow resistance encountered by the blood as it flows through the arterial/venous/capillary system;
- Inertance (L): simulates the inertia of the blood as it is cycled through the heart.
In the electrical analogue, the compliance (C) is represented as a capacitor with electric charge storage properties; the resistance (R) is represented as an energy dissipating resistor. The effect of blood inertia can be introduced (incorporated to construct a 4-element WM (see Systemic Arterial Tree in figure reported in Frank-Starling Law section) in the form of the inductor (L) equivalent electrical component.
The flow of blood is analogous to that of a current flowing in the circuit and the blood pressure P(t) is modelled as a time-varying electric potential.
Windkessel models are used as hydraulic load for isolated hearts and in studies of the whole circulation. They are used to estimate total arterial compliance from pressure and flow and describe the general features of the input impedance with physiologically interpretable parameters.
Three types of Windkessel models are implemented in the software simulator to reproduce the behaviour of different districts of the systemic circulation:
- 2-WM consists of one resitance (R) and one compliance (C).
- 3-WM consists of one compliance (C) and two resistances.
- 4-WM consists of one compliance (C), two resistances and one inertance (L).
Since WM is a lumped model, it is not suitable for the assessment of spatially distributed phenomena and aspects of wave travel. Nevertheless, it remains a simple and fairly accurate approximation of ventricular afterload.
The systemic circulation is reproduced in CARDIOSIM© choosing one of seven different network:
- Systemic Network 1 reproducing the the arterial section with one 4-WM and venous circulation with one variable resistances and one compliance.
- Systemic Network 2 reproducing the the arterial section with two RLC modules and venous circulation with two variable resistances and one compliance.
- Systemic Network 3 consisting of aortic, thoracic and abdominal sections implemented with RLC elements (three-cells model) and venous section reproduced with two variable resistances and one compliance.
- Systemic Network 4 consisting of systemic arterial section, splanchnic and extra-splanchnic peripheral and venous sections, peripheral and venous circulation in active muscle compartment and systemic thoracic veins section.
- Systemic Network 5 is reproduced with ascending and descending aorta sections, carotid and peripheral arteries networks, systemic veins section and vena cava compartment.
- Systemic Network 6 reproducing ascending and discending aorta and aortic arch sections, thoracic section, two abdominal sections, superior vena cava compartment, arms and head sections, renal and hepatic sections, inferior and abdominal vena cava compartments, splanchnic section and legs section.
- Systemic Network 7 consisting of ascending aorta compartment, thoracic and abdominal aorta sections, upper and lower body sections and superior and inferior vena cava compartments.
Pulmonary Circulation Models
The pulmonary circulation is a low pressure, low resistance system with pulsatile behaviour where blood pressure ranges from 25 mmHg in systole and 10 mmHg in diastole. This system conveys the output of the right ventricle via the pulmonary arteries to the alveolar capillaries and returns the blood to the left atrium via the pulmonary veins. The mechanics of the pulmonary circulation is affected to a major extent by the mechanics of the lungs, whose main function is the exchange of oxygen and carbon dioxide between the air and the blood. The pulmonary blood vessels are effective in attenuating pressure and flow oscillations generated by opposing contraction of the right and left side of the heart. The pulmonary circulation acts also as a sieve to remove abnormal particulate material circulating in the blood. Finally, the pulmonary circulation appears to have an important role in influencing the chemical composition of the blood by either removing or adding substances to it.
Six different numerical models describing the pulmonary circulation have been implemented in CARDIOSIM© library based on RLC electrical representation.
The six electrical analogue network are:
- Pulmonary Network 1 reproducing the arterial section with a 4-WM and the venous circulation with one resistance and one compliance.
- Pulmonary Network 2 reproducing the the arterial section with a 4-WM and venous circulation with 2-WM.
- Pulmonary Network 3 consisting of arterial, peripheral and venous sections implemented with 3-WM and two 2-WM elements respectively.
- Pulmonary Network 4 consisting of main and small arterial compartments, arteriole and capillary sections and venous circulation.
- Pulmonary Network 5 (three-cell model) describing main and small arterial peripheral compartments with 3-WM elements. The pulmonary veins circulation is reproduced with a 2-WM.
- Pulmonary Network 6 modelling the pulmonary circulation with the following compartments: arteries, capillary and veins pulmonary districts. The different circulatory networks have been implemented using 4-WM and 2-WM.
Coronary Circulation Models
The coronary circulation supplies the myocardium. The left coronary artery supplies approximately 75% of the myocardium; the right coronary artery supplies the remaining 25%. The left coronary artery consists of the left main stem (LMS), which divides into the left anterior descending (LAD) and circumflex (CX) arteries. It supplies the anterior and lateral wall of the left ventricle and the anterior two thirds of the inter-ventricular septum. The right coronary artery divides into the posterior descending artery (PDA) and the left ventricular branch (LV-Br). It supplies the right ventricle, the posterior wall of the left ventricle and posterior third of the septum.
The coronary circulation delivers 5% of the total cardiac output at rest, which is equivalent to 250 ml/min or 0.8 ml/min/g of heart muscle, to maintain myocardial function. The myocardium has the highest oxygen consumption with a rate of 10 ml O_2/min/100 g at rest. Coronary perfusion pressure, perfusion time and vessel diameter are determinants of coronary blood flow, which is the highest in diastole and the lowest in systole in view of cardiac muscle contraction. The coronary perfusion pressure is the difference between the aortic diastolic pressure and the left ventricular end-diastolic pressure (LVEDP) and it plays a key role in the physiology of the coronary circulation.
The diastolic pressure time index (DPTI) is the product of coronary perfusion pressure and diastolic time and is a measure of coronary blood supply. The tension time index (TTI) is the product of systolic pressure and systolic time and is a measure of oxygen demand. When DPTI and TTI are combined together, the endocardial viability ratio is obtained (EVR = DPTI/TTI), which represents the myocardial oxygen supply/demand balance.
Several mathematical/numerical models have been proposed to reproduce the behaviour of the coronary circulation and the relation between intra-myocardial pressure, volume and resistance to flow.
The earliest mathematical intra-myocardial pump models (IMPM) proposed the mechanism of systolic vascular compression leading to displacement of fluid. They highlighted the importance of the vascular capacitance to store fluid during diastole with subsequent active pumping during myocardial contraction. Flow variations between systole and diastole are not directly coupled to a variable resistance but to intra-myocardial volume variations in terms of capacitance of the coronary vasculature, later adjusted to resistance and capacitance dependent on intra-myocardial pressure.
The “Vascular waterfall mechanism” proposed by Downey and Kirk (1975) considers the intramural coronary artery as a collapsible tube dependent on the intra-myocardial pressure surrounding the vessel. If the intra-myocardial pressure exceeds coronary perfusion pressure, the vessel collapses, otherwise it remains patent with a fixed resistance. A lumped-parameter model can reproduce the “Vascular waterfall mechanism”. 0-D models are useful for the interpretation of haemodynamic data measured following cardiac catheterisation where coronary input impedance is defined as the ratio of (sinusoid) perfusion pressure to (sinusoid) coronary flow and affected by the distribution of the resistive and capacitive component within the coronary tree.
According to Spaan (1981), the electric circuit presented in figure cannot describe all the oscillatory pressure-flow relations observed experimentally. For instance, systolic arterial back-flow related to low pressure cannot be explained with the waterfall model.
The following coronary numerical models based on IMPM have been implemented in the software library:
- Coronary Network 1 reproducing the the the “Vascular waterfall mechanism” proposed by Downey and Kirk with resistor, diode and battery.
- Coronary Network 2 modelling the coronary circulation within the wall of the left ventricle with 3-WM.
- Coronary Network 3 representing the coronary bed with a single arterial and venous path, both modelled with 3-WM.
- Coronary Network 4 where the coronary circulation is modelled using three parallel vascular branches equivalent to the sub-endocardial, middle and sub-epicardial layers of the left ventricular wall.
Ventricular pressure-volume analysis enables to understand cardiac mechanics in animal experiments and humans. A pressure-volume (PV) loop (green line) in figure is generated by real-time measurement (or by simulation) of pressure and volume within the ventricle during a cardiac cycle. The following haemodynamic and energetic information is obtained:
- End Diastolic Pressure-Volume Relationship (EDPVR);
- End Systolic Pressure-Volume Relationship (ESPVR) whose slope represents the End-systolic elastance (Ees), which is an index of myocardial contractility;
- End Systolic Volume (ESV);
- End Diastolic Volume (EDV);
- Isovolumic relaxation phase;
- Isovolumic contraction phase;
- Ventricular filling phase;
- Ventricular empting phase.
Figure shows the ventricular loop plotted in the pressure-volume plain.
The difference between the End Diastolic Volume (EDV) and the End Systolic Volume (ESV) defines the Stroke Volume (SV):
SV = EDV - SV
The Stroke Volume is affected by changes in preload, afterload, and contractility (see the Starling's Law).
Ejection Fraction (EF) during each ventricular contraction is defined as:
EF = SV / EDV
In figure the white area enclosed within the green loop is the ventricular External Work (EW), which is an energetic parameter representing the work performed by the left (right) ventricle to eject the SV into the aorta (pulmonary artery).
In figure the area under the triangle comprised between the green, red and blue lines represents the ventricular elastic Potential Energy (PE), which is the energy stored in the ventricular wall at the end of systole.
The ventricular Pressure Volume Area (PVA) is defined by:
PVA = EW + PE
PVA correlates linearly with myocardial oxygen consumption per beat (MVo2):
MVo2 = a * PVA + b
and represents the total mechanical energy of contraction.
Mechanical Circulatory Support Systems
Patients with advanced heart or pulmonary failure may need mechanical assistance to keep their heart pumping enough blood.
Mechanical Circulatory Support Systems (MCSS) are designed to replace or assist heart function in patients with advanced heart failure. Short- and mid-term ventricular assist devices are frequently used to bridge patients with severe heart failure either to recovery or heart transplant. Long-term ventricular assist devices (VADs) and Total Artificial Hearts (TAHs) are increasingly used as a bridge to heart transplant or as a permanent solution in patients with end-stage heart failure who are not eligible for heart transplant.
Bridging a patient from temporary MCSS to a long-term device and / or heart transplant is challenging and requires a multidisciplinary approach. In the absence of myocardial recovery, MCSS remains an exit strategy either as a bridge to transplant or long-term support in selected patients. Preoperative assessment, planning and timing for intervention play a key role for a successful outcome. Numerical modelling and simulation may be used as an additional tool for device selection and optimisation with potential for outcome prediction.
Different MCSS have been implemented in CARDIOSIM© such as:
Pulsatile and Continuous Flow Pumps are mechanical assist devices that can be used as:
The connection between the native heart and an assist device is either "in series" or "in parallel". The pump is connected "in parallel" with the native ventricle when aspirates blood from the atrium and pumps it into the aorta (or pulmonary arterial tree). An "in series" connection is obtained when the device aspirates blood from the native ventricle and ejects it into the aorta (or pulmonary arterial tree).
In the presence of cardiogenic shock, the IABP can be considered as a bridge between conventional medical therapy and MCSS. The IABP is widely available, less invasive, easy to insert with relatively low risk and ideal in an acute setting. Current data suggest that IABP may assist aortic valve opening in patients requiring peripheral Veno-arterial Extra-Corporeal Membrane Oxygenation (VA‐ECMO) and should not necessarily be removed prior to VA‐ECMO support.
CARDIOSIM© software simulator may become an aid for a more quantitative approach for patient assessment, selection and device treatment suitability where the clinician has the final say based on the available data.
Thoracic Artificial Lung
Patients assisted with MCSS (e.g. LVAD, BVAD, PUCA pump, Hemopump, IABP etc.) may require prolonged mechanical ventilation due to postoperative respiratory failure. During the initial period after assist device insertion, patients are managed in the intensive care unit (ICU) where they are mechanically ventilated (MV). MV is the standard treatment for acute lung disease. In patients affected by acute and/or chronic pulmonary dysfunction, thoracic artificial lung (TAL) may be used as an alternative to conventional mechanical ventilation. The TAL device is designed to take over or to supplement lung function. Progress has been recently made towards a clinical trial for the TAL device to test its potential.
The TAL device is implemented in CARDIOSIM© using a lumped-parameter model. It is connected between the right ventricle and the pulmonary circulation either in series, in parallel or in hybrid mode (see TAL model section).