Eur J Cardiothorac Surg 2004;26:239-247
© 2004 Elsevier Science NL
A new computer model of mitral valve hemodynamics during ventricular filling
Gábor Szabóa*,
Davis Soansb,
Andy Grafa,
Carsten J. Bellera,
Lee Waiteb,
Siegfried Hagla
a Department of Cardiac Surgery, University of Heidelberg, Im Neuenheimer Feld 110, 69120 Heidelberg, Germany
b Rose-Hulman Institue of Technology, Terre Haute, IN, USA
Received 28 August 2003;
received in revised form 16 January 2004;
accepted 11 March 2004.
* Corresponding author. Tel.: +49-6221-566-111; fax: +49-6221-5655-85
e-mail: dzsi{at}hotmail.com
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Abstract
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Objective: Quantitative description of left ventricular diastolic filling and mitral valve function remains difficult despite advances in echocardiography. The purpose of the present study was to develop a lumped parameter model of left ventricular filling and validate it in porcine trials under physiological conditions and after valve replacement. Methods: Six animals were instrumented with aortic flow meter, left atrial pressure catheter and combined left ventricular pressure–conductance catheter. The model simulates ventricular and arterial pressures and flows during diastolic filling. Input parameters include maximum mitral valve area, blood viscosity and density, atrial compliance, left ventricular active relaxation characteristics and initial pressure and flow values. The outputs of the model are atrial and ventricular pressure as well as transmitral flow as a function of time. The model primarily consists of a system of four first-order, non-linear ordinary differential equations which were solved with MATLAB software. Results: Left atrial and ventricular pressure data and model flow curves were nearly identical under baseline conditions, during rapid preload reduction by vena caval occlusion and after prosthetic valve replacement. Measured and model based calculation of early diastolic filling volume (E-wave), showed an excellent correlation under all three conditions (r=0.998, P<0.0001; r=0.997, P<0.0001; r=0.974, P<0.0001, respectively) with a mean difference less then two percent. Conclusion: The new lumped parameter model of left ventricular filling allows for the first time a detailed simulation of pressure and flow curves in the left heart including transmitral hemodynamics.
Key Words: Mitral valve • Pressure • Flow • Computer model • Valve replacement
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1. Introduction
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Evaluation of diastolic ventricular function and quantification of valve stenosis and mitral regurgitation are important in clinical practice as well as in physiological research [1–5]. Measurement of transmitral flow, atrio-ventricular pressure gradient and mitral valve area are cornerstones of severity assessment and decision making for therapy. Although the early and accurate diagnosis would be essential in the treatment of these patients, none of the so far available diagnostic methods allows a complete description of transmitral pressure–flow relationships [6–9] and each method has specific limitations. Invasive cardiac catheterization provides accurate pressure measurements but little information about intracardiac flow and mitral valve area. Non-invasive assessment of Doppler transmitral velocity curve is often used to assess left ventricular diastolic function providing transmitral velocity and flow values, however, the accurate calculation of pressure gradients and mitral valve area with different methods [6–9] remains a topic of future research. Many non-invasive Doppler studies have qualitatively related left ventricular relaxation and left ventricular and atrial compliance derived from transmitral flow velocity parameters, to ventricular function but a quantitative relationship has in general not been demonstrated yet [6,10]. A number of models have been developed to describe flow and pressure relationships in the human heart, to help investigators better understand the relationship between Doppler velocity profiles and physiologic parameters [1–5,10–14].
The aim of this research was the development and validation of a new computer model of pressure and flow through the mitral valve during early diastolic ventricular filling (E-wave) which allows an accurate simulation of transmitral flow curves from pressure signals or inversely, the estimation of pressure values from transmitral flow signals.
Furthermore, the second aim was to test the model after valve replacement in vivo because quality and accuracy of non-invasive estimates of valve and diastolic function after valve replacement are reduced due to artifacts of Doppler signals and thereby the assessment of transmitral flow in the presence of a prosthetic valve is difficult [15,16].
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2. Material and methods
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2.1. Mathematical model of left ventricular filling
This model describes the flow and pressure relationships between the left atrium and the left ventricle during early ventricular filling prior to atrial systole. The model begins at the time when pressures are equal in the atrium and the ventricle, i.e. begins from the instant of mitral valve opening. The model then describes flow and pressures during ventricular filling up to the point of atrial systole. The model consists of a system of three first-order, non-linear ordinary differential equations. The equations are solved in MATLAB 5.3.0 using the ode45 command which solves systems of non-stiff differential equations using the medium order method. A more complete description of the fluid mechanics described by this system of equations can also be found in the work published by Thomas and Weyman (1989) [1]. The first of the three equations is written as follows:
 | (1) |
where q represents flow rate, dq/dt represents the time rate of change of flow rate, Pa represents left atrial pressure, Pv represents left ventricular pressure, and M represents the inertance term, which is analogous to inductance in the electrical circuit analogy, Rv represents viscous resistance, which is analogous to resistance in the electrical circuit analogy and Rc represents convective resistance, which, in an electrical circuit model, would be analogous to a resistor whose variable resistance is dependent on current through the resistor. M, Rv and Rc were calculated from blood density and viscosity, blood column length (inertial length) through the mitral valve (comparable to valve thickness) and mitral valve effective area (see Appendix B). The second differential equation is written as follows:
 | (2) |
where Ca represents atrial compliance. The calculation of compliance is based on the chamber pressure–volume relationships (see Appendix C). The third equation is written as follows:
 | (3) |
where 
represents the ratio of pressure at the instant of mitral valve opening (the point where ventricular pressure falls below atrial pressure) to the minimum diastolic ventricular pressure, T is the time constant in seconds, t represents the time variable, 
v represents chamber stiffness and V ventricular volume.
2.2. Variable area mitral valve model
A portion of the entire lumped parameter model of flow through the mitral valve is a model of the intrinsic and extrinsic characteristics of the valve itself, disregarding the heart chambers. This portion of the model is referred to as the variable-area mitral valve model. The model proposed for the behavior of the mitral valve during diastolic filling consists two parts—the term describing the dynamics of the valve and a term modelling the forces acting on the valve. The dynamics term is a simple linear second-order differential equation, but the forces term assumes that any kind of pressure—static, dynamic or the pressure caused by the acceleration or deceleration of the fluid—only acts on the occluded valve area. This effectively limits the valve aperture to the maximum aperture, which is a parameter of the model. This is the complete differential equation for the proposed model:
 | (4) |
A represents the aperture of the mitral valve, 
0 represents natural frequency of the mitral valve. It is influenced by the mass of the valve cusps and the modulus of elasticity of the tissue. D represents the damping coefficient of the mitral valve. The valve cusps have comparatively little mass, but have a large area, and they are moving in blood, a viscous fluid. Therefore, the system will possess significant damping (D>1). The viscosity of the blood and the size of the valve cusps govern the magnitude of the damping coefficient. Amax is the maximum aperture of the mitral valve; Qv represents flow across the mitral valve. Ks, KD and KT are gain coefficients for static and dynamic pressure and for pressure caused by acceleration and deceleration of the fluid, respectively. These ‘gain factors’ represent the sensitivity of the valve to static, dynamic and acceleration-induced pressure.
2.3. Animal studies
All animals used in this study received care in compliance with the ‘Guide for the Care and Use of Laboratory Animals’ prepared by the Institute of Laboratory Animals Resources, National Research Council, and published by the National Academy Press, Revised 1996. The investigation was approved by the Institutional Animal Care and Use Committee of the state of Baden-Württemberg, located in Karlsruhe, Germany.
2.4. General preparation
Six domestic pigs (20–24 kg) were studied. The anesthesia was initiated using propofol (10 mg i.v.). The animals were intratracheally intubated and the internal jugular vein was cannulated for intravenous drug delivery. The anesthesia was maintained by continuous infusion of piritramid (15 µg/kg/min i.v.) The animals were paralyzed with pancuronium bromide (0.1 mg/kg as a bolus and then 4 µg/kg/min i.v.) The pigs were ventilated with a mixture of N2O and O2 (40:60%) at a frequency of 12–15/min and a tidal volume starting at 15 ml/kg/min. The settings were adjusted by maintaining arterial partial CO2 pressure levels between 35–40 mmHg. The femoral artery and vein were cannulated for recording aortic pressure and for taking blood samples for the analysis of blood gases, electrolytes and pH. Basic intravenous volume substitution was carried out with Ringer's solution at rate of 1 ml/min/kg. If necessary, the rate of volume substitution was modified according to the continuously controlled input–output balance in order to maintain cardiac output at baseline levels. According to the values of K+, HCO3– and base excess, substitution included administration of potassium chloride and sodium bicarbonate (8.4%). No catecholamines or other hormonal or pressor substances were administered. Rectal temperature and standard peripheral electrocardiogram were monitored continuously.
After median sternotomy the pericardium was incised. The great vessels were dissected and the swine was instrumented as follows: a perivascular ultrasonic flow probe was positioned at the proximal ascending aorta to measure aortic flow and cardiac output. Stroke volume was calculated from the integrated flow signal. Left ventricular pressure–volume data were recorded by a 6F dual-field combined volume–conductance micromanometer tipped catheter (Millar Instruments, Inc, Houston, Tex). Parallel conductance was estimated by rapid injection of one ml of hypertonic saline into the pulmonary artery. The volume signal provided by the conductance catheter was registered continuously (Sigma F5, Leycom, Leiden, The Netherlands) and computed by the Conduct PC software (Leycom, Leiden, The Netherlands). Left atrial pressure was measured by 5F micromanometer tipped catheter (Millar Instruments, Inc, Houston, Tex). Left atrial pressure, left ventricular pressure (LVP) and left ventricular volume, aortic pressure, aortic flow, and ECG were registered continuously. All pressure data were amplified by a custom made analog-digital amplifier (Schubart, Wiesbaden, Germany). All data were collected and stored with a sampling frequency of 512 Hz on a PC for further off-line analysis.
2.5. Mitral valve replacement on the beating heart
To avoid the influence of ischemia/reperfusion injury mitral valve replacement was performed on the beating heart. After baseline measurements and systemic anticoagulation with sodium heparin (300 U/kg) the right atrium and the distal ascendent aorta were cannulated for cardiopulmonary bypass. Additionally, a pulmonary vent was inserted into the main pulmonary artery. The extracorporeal circuit consisted of a heat exchanger, a venous reservoir, a roller pump and a membrane oxygenator primed with Ringer lactate solution (700 ml) supplemented with heparin (150 U/kg) and 20 ml sodium bicarbonate (8.4%). On normothermic cardiopulmonary bypass, a 3 cm longitudinal incision was made on the left atrium and the mitral leaflets were cut. A 23 mm St Jude Medical prosthetic mitral valve was implanted with single Teflon-armed stitches. The whole procedure lasted 35–40 min during which the heart was beating regularly. The left atrium was closed with a continuous suture. To avoid air embolism the heart were electrically fibrillated during this time. This period lasted less than 2 min, and thereby the effects on metabolic demand were negligible. After deairing and defibrillation, the animals were weaned from the cardiopulmonary bypass. One hour after weaning, the hemodynamic measurements were repeated.
2.6. Data analysis and protocol
Basically, the present model is able to generate pressure, flow and atrial and ventricular stiffness values. In the present study, primary input parameters were the measured diastolic atrial and ventricular pressure curves and left ventricular active relaxation time constant as well as mitral valve geometry, blood density, and blood viscosity. Atrial and ventricular compliance can be considered as inputs to the model; or by matching waveforms, the model can be used to predict values for these parameters. The primary output of the model was calculated flow across the mitral valve. Since data of the mitral valve aperture are not available, the waveforms of experimental and calculated flow across the mitral valve were compared to validate the model. The mitral flow was determined by differentiating the ventricular volume data with respect to time.
Data regarding the ventricular and atrial pressures, ECG, and ventricular volume measurements were filtered in MATLAB and averaged over 20 heartbeats to obtain mean files. The data in the files were used to set the initial pressures and flows for the model. The values for each parameter were then obtained by the use of an iterative program written in MATLAB. This program runs the model 200 times over the data to obtain values for the parameters of interest. In order to facilitate the estimation of the parameters, a simple iterative algorithm was developed that tries to match the experimental and simulated pressure waveforms. Starting with initial values given by the user, the value of each parameter is incremented or decremented by the product of the criterion and a so-called learn factor during each iteration. The learn factor can be used to control the rate of convergence and to prevent instability. The algorithm to determine the atrial and ventricular stiffness as well as ventricular pressure decay time constant t tries to minimize the difference between the slopes of the simulated and the experimental pressure waveforms toward the end of the simulation time.
Simulations were performed under three different conditions: (1) under baseline conditions the average of 20 beats was used to calculate flow across the mitral valve. (2) A series of beats was registered during transient vena caval occlusion to achieve different preload conditions and thereby filling states. Here, simulation of pressures and flows were based on the datasets of single beats. (3) After mitral valve replacement, simulations were repeated on the average of 20 beats.
If mean values are given, these values are given as mean±SD. In order to validate the model, linear regression was used to compare measured and calculated flow or volume values. To assess for error and bias, the Altman and Bland analysis method was used. Mean hemodynamic values before and after valve replacement were compared by paired t-test, intergroup analysis was performed by one-way analysis of variance. Statististical significance was defined with P<0.05.
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3. Results
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Table 1 shows steady state hemodynamic data before and after valve replacement. They were in the physiologic range. Fig. 1 illustrates typical results of the computer simulation in comparison with measured pressure and flow curves. The top panel of Fig. 1 shows mean left ventricular and left atrial curves for the diastolic filling phase in pigs with a natural heart valve. The data from porcine trials were filtered using the Butterworth filter tool in MATLAB and the mean of 20 consecutive diastolic filling events was taken to obtain the mean curve for that data set. The computer model predictions are also shown. There was a very good agreement between measured and subsequently simulated pressure curves. The primary output of the model simulated transmitral flow is depicted on the bottom panel of Fig. 1, which was nearly identical with the measured values.
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Fig. 1. Original registration of measured and model pressure (top panel) and transmitral flow curves (bottom panel) under baseline conditions. LVP, left ventricular pressure; LAP, left atrial pressure.
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After valve replacement, the measured and simulated curves were also nearly identical (Fig. 2)
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Fig. 2. Original registration of measured and model pressure (top panel) and transmitral flow curves (bottom panel) after valve replacement. LVP, left ventricular pressure; LAP, left atrial pressure.
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Rapid occlusion of the vena cava led to a marked reduction of preload and thereby left ventricular pressures and volumes. Model simulations were performed from the dataset of each single beat throughout the obtained series during vena caval occlusion. There was a very good agreement between measured and simulated curves. Calculated and simulated early diastolic filling volumes (E-wave) are depicted in Fig. 3
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Fig. 3. Linear correlation between model and measured early diastolic filling volumes (E-wave) under baseline conditions (left panel), during vena caval occlusion (mid panel) and after valve replacement (right panel). Dashed lines indicate 95% confidence intervals dotted lines indicate 95% prediction intervals.
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To further validate the model, we compared measured and calculated filling volumes during passive diastole (E-wave). We found an excellent correlation between calculated and measured values under physiologic conditions with natural heart valve (r=0.998, P<0.0001), during vena caval occlusion (r=0.997, P<0.0001) as well as after prosthetic valve replacement (r=0.974, P<0.0001) (Fig. 3). Quality control plots using the Altman and Bland method with conductance catheter derived E-waves as a reference technique showed that there was no overall underestimation or overestimation of the calculated E-wave (Fig. 4)
. The mean signed differences were small under baseline (0.060±0.605 ml), during vena cava occlusion (0.155±0.551 ml) and after valve replacement (0.266±0.766 ml). The slope and intercept for the regression of the difference onto the mean E waves were not different from zero emphasizing the absence of systemic bias in the model.
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Fig. 4. Altman and Bland analysis of measured and model early diastolic filling volumes (E-wave) under baseline conditions (left panel), during vena caval occlusion (mid panel) and after valve replacement (right panel). The mean of measured and model E-wave (X-axes) is plotted against the difference of measured and model E-wave (Y-axes).
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4. Discussion
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In the present study, we developed and validated a new lumped parameter computer model of left ventricular filling and transmitral hemodynamics. To our best knowledge, this is the first study in which (1) a variable mitral valve area was included, (2) the model was validated in vivo in porcine trials, and (3) the model was used to simulate transmitral flow and after prosthetic valve replacement. We found that based on measured pressure curves as input parameters, the model is able to generate transmitral flow curves nearly identical to the measured ones. Furthermore, corresponding measured and calculated diastolic filling volumes (E-waves) were also nearly the same.
4.1. Mathematical models of transmitral flow dynamics
The study of flow through heart valves has long been a topic of interest to engineers and physicians. In 1951 Gorlin [17] published a formula, which predicts flow through a heart valve based on its cross-sectional area, the pressure gradient across the valve and an empirical constant (discharge coefficient). Thomas and Weyman developed a lumped parameter model of mitral flow which they validated in vitro [1,3]. They found that early mitral flow (E-wave) is governed mainly by ventricular and atrial compliance properties and by impedance characteristics of the valve. The model could be used to calculate Doppler and hemodynamic indices which are of clinical importance. Thomas and Weyman later developed a lumped parameter model and computer simulation to further study transmitral velocity [10]. They studied the effects of isolated changes in atrial compliance, ventricular compliance, ventricular relaxation and valvular morphology. In this study they showed that transmitral blood velocity is affected mainly by, (1) the transmitral pressure difference and (2) the net atrial and ventricular compliance. Isaaz [18] developed a theoretical method for non-invasive assessment of transmitral pressure-flow relationships using Doppler ultrasound. He performed an ultrasound study using Doppler echocardiography for eight cardiac cycles in one human patient to validate the lumped parameter mathematical model. She et al. [19] developed a lumped parameter model of flow versus pressure drop. The study discusses the use of an electrical analogue to model the heart valve and the relationship between static and dynamic characteristics. In 1995, Nudelman et al. [12] published a study comparing a number of lumped parameter models of transmitral flow. They developed, tested and verified a method of acquisition and reproduction of Doppler velocity profiles and then performed a numerical determination of model-to-model and model-to-data goodness-of-fit. They concluded that automated, quantitative characterization of clinical Doppler E-wave contours is feasible.
Latest models focused on three-dimensional flow geometry and intraventricular flow patterns. Lemmon and Yoganathan [13,14] developed a three-dimensional computational model of left heart diastolic function using the immersed boundary method. They reported agreement between model predicted velocity waveforms and published clinical data. A very recent intraventricular flow model study [20] showed that mitral valve opening during left ventricular filling has an influence on left ventricular flow patterns.
As mentioned above the present model includes for the first time a variable mitral valve area and provides in contrast to the above described studies an in vivo validation under physiologic conditions and for the first time after valve replacement. In a previous work [21], we tested the above described model with a fixed mitral valve area. Even if measured and calculated E-waves then showed a good correlation the time course of transmitral flow showed a considerable difference. Therefore, we developed this new model with a variable mitral valve area.
4.2. Validation of the model
We used the pressure measurement as primary input of the model. If atrial and ventricular stiffnesses are given, the matching of simulated and measured pressure waveforms may be sufficient to validate the mathematical model. However, the comparison between predicted and measured early diastolic filling volumes allows a more complete validation. As arterial and ventricular stiffness were not given in the present study we predicted atrial and ventricular stiffness by matching measured and simulated pressure curves. Therefore, the validation of the model is based on transmitral flow curves. Mitral flow occurs in two stages that correspond to the valve openings in early and late diastole. We modeled only early diastolic filling (E-wave). The model was effective over a wide range of filling volumes. Values of early diastolic filling ranged from 2.2 to 35 ml in the pig. Predicted volumes from the computer model agreed within 5% of the measured values. Moreover, not only measured volumes but also the time course of measured and simulated transmitral flow curves were nearly identical. We observed a significant reduction of both simulated and measured transmitral flow and effective orifice area during preload reduction. This is in accordance with the Doppler study of Triulzi et al. [22]. They demonstrated in humans that preload reduction results in decreased peak transmitral flow velocity and the time-velocity integral of early diastolic inflow.
4.3. Limitations
While we could compare pressure and flow curves, the measurement of the variable mitral valve area was not possible. To our best knowledge, no such data have been published for humans or pigs until now. Using the newest developments in magnetic resonance technology and four-dimensional echocardiography the authors are currently involved in a study to determine the time course of mitral valve opening. Even if we were not able to measure variable mitral valve area, the very good agreement between measured and calculated pressure and flow curves indicates that the simulated curves of mitral valve opening should be also close to the natural time course.
In this model atrial contraction is not included. Further work will be needed to extend this model to include atrial systole and late diastolic filling.
4.4. Applications of the model
The present model may have numerous applications. During cardiac catheterization high resolution left atrial and ventricular pressure measurements are easily available, whereas continuous volume measurements or intracardiac flow measurements are difficult and the accuracy of these measurements is rather questionable. Using the present variable mitral valve area model it is easy to determine transmitral flow and effective orifice area as well as atrial and ventricular stiffness. This would contribute to a more accurate and possibly earlier diagnosis of different types of mitral valve disease and diastolic heart failure.
The application of the model can be even more useful in the field of non-invasive diagnosis. The ability to determine model parameters from clinical transmitral flow waveforms is equivalent to solve the inverse problem: calculation of pressure gradients or absolute values or indexes of chamber stiffness. Doppler transmitral velocity curve is commonly used to assess left ventricular diastolic function. Previous investigations, however, relating Doppler mitral indexes to ventricular compliance, relaxation and preload have been inconclusive and at times contradictory [3,6,9,22]. Comparisons between model outputs and in-vivo porcine experiments suggest that the model may be used in the future to assess atrial and ventricular compliance as well as effective orifice area based on non-invasive Doppler waveforms more accurately. If the model based on measured Doppler velocity curves can estimate parameters like atrial compliance, ventricular compliance, and effective area as well as absolute pressures and/or pressure gradients in the left heart, then the model could prove clinically useful.
This the first study that describes transmitral flow and mitral valve opening kinetics after valve replacement. We could show that the present mathematical model provides an accurate description not only under physiologic conditions but also at the presence of a prosthetic heart valve, which may be useful in the clinical follow up of patients with mitral valve replacement. Using the beating heart technique, we tried to minimize operative injury, however, we cannot exclude that altered filling patterns may occur related, at least partly, to a momentary alteration of myocardial relaxation or chamber stiffness in association with the operative procedure. To clarify this issue, chronic valve replacement experiments would be necessary with complete recovery of the animals. Apart from this, the model can be useful in development and in vivo evaluation of new types of mitral valve prosthesis. The model also can be used to predict the hemodynamic effects of mitral valve surgery in individual patients (‘virtual valve surgery’). Using preoperative cardiac catheterization data the effects of a changed effective area after reconstructive surgery or the effects of a replaced valve can be easily calculated. Here further experimental and clinical studies are warranted to elucidate the predictive value of this model.
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Acknowledgments
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This work was supported by the German Research Foundation SFB414/H2 to G.S. and in part by the Eli Lilly and Company Life Sciences Research Center at Rose-Hulman Institute of Technology, Terre Haute, Indiana, USA to L.W.
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Footnotes
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Presented at the joint 17th Annual Meeting of the European Association for Cardio-thoracic Surgery and the 11th Annual Meeting of the European Society of Thoracic Surgeons, Vienna, Austria, October 12–15, 2003.
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Appendix A. Conference discussion
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Dr M. Antunes (Coimbra, Portugal): This is a purely experimental work, I have difficulty in dealing with, but I appreciate that it has a very high intrinsic scientific value. Do you think that whatever technology and hardware comes behind this concept will be easily applicable in the clinical situation?
Dr Szabo: I think it will be relatively easy to use this model engine in cardiac diagnostics. There is a software program which was developed to make the calculations, and this software program can use the registered left ventricle and arterial pressures during cardiac catheterization, and it will give us output transmitral flows over time with high resolution. The other possible applications have to be validated in future settings. If we know the mechanical behavior of a valve, a natural valve or a prosthetic valve, we can test the hemodynamic performance or simulate the hemodynamic performance of the valve. So that is a second possible application also.
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appendix b
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Here we represent the calculation of the resistive elements from the first equitation:
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| (B.1) |
where q represent flow rate in m3/s, dq/dt represents the time rate of change of flow rate in m3/s2, Pa represents left atrial pressure in N/m2, Pv represents left ventricular pressure in N/m2, and M represents the inertance term, which is analogous to inductance in the electrical circuit analogy and has the units of kg/m4.
M can be calculated as:
 | (B.2) |
where 
represents blood density in kg/m3, l represents the blood column length (inertial length) through the mitral valve (comparable to valve thickness) in m, and A represents the mitral valve effective area in m2.
Rv represents viscous resistance, which is analogous to resistance in the electrical circuit analogy and has the units kg/m4 s. Rv can be calculated as:
 | (B.3) |
where µ represents viscosity with units of N s/m2
Rc represents convective resistance, which, in an electrical circuit model, would be analogous to a resistor whose variable resistance is dependent on current through the resistor, and has the units kg/m7. Rc can be calculated as:
 | (B.4) |
It should be noted, that the calculation of all three resistive component includes mitral valve area. Instead of taking a fixed effective orifice, we used a variable valve area (see Section 2, Eq. (4)). As a consequence, resistive elements are changing over the time during diastolic filling.
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appendix c
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Here we show the calculation of atrial compliance in the second equation:
 | (C.1) |
where Ca represents atrial compliance in m5/N (m3/N/m2).
In the electrical circuit analogy, Ca is analogous to capacitance. In this case the value of capacitance is not constant but depends on voltage (pressure) across the capacitor, as in a semiconductor. Compliance can be realistically modeled with the pressure volume relationship:
 | (C.2) |
where V represents chamber volume in m3 and represents chamber stiffness in units of m–3.
Atrial stiffness, dP/dV=1/Ca, is therefore PoeV or P and the second equation can be rewritten as:
 | (C.3) |
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appendix d
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The third equation is written as follows:
 | (D.1) |
Note that the first term on the right hand side of the equation follows the same form as Eq. (C.3).
In order to model exponential pressure decay over time to represent active ventricular relaxation, the second term on the right hand side of Eq. (D.1) is introduced. The equation is modified with an offset term, which can effectively shift the baseline of the ventricular pressure curve for the cases where Pv is less than zero. The equation given below, which reflects this exponential pressure decay, is given in Weyman [1] and modified here to include negative ventricular pressures:
 | (D.2) |
Solving Eq. (D.2) for Po(exp(vV))
 | (D.3) |
and now we can rewrite Eq. (D.1) as:
 | (D.4) |
where represents the ratio of pressure at the instant of mitral valve opening (the point where ventricular pressure falls below atrial pressure) to the minimum diastolic ventricular pressure, T is the time constant in seconds, and t represents the time variable.
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D. Tanne, L. Kadem, R. Rieu, and P. Pibarot
Hemodynamic impact of mitral prosthesis-patient mismatch on pulmonary hypertension: an in silico study
J Appl Physiol,
December 1, 2008;
105(6):
1916 - 1926.
[Abstract]
[Full Text]
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Abstract
1. Introduction
