Disease-Based Models of Cardiac Electromechanics
2. Simulation of the normal heart
The normal heart is simulated first in order to validate the model and to calibrate the parameters to the real values. The resulting simulation protocol will then be considered as reference when tailoring the algorithm to the pathologies under study.
2.1 Parameters of the normal heart
To simulate the normal heart, the geometrical parameters are defined as follows:
- The degree of the super-ellipsoid that models the right-ventricle is chosen equal to 2.5 in order to have a simulated anatomy consistent with the clinical observations (see next table). The right ventricle is therefore defined by the equation:
- The orientation of the cardiac muscle fibres varies from +90° on the endocardium to -90° on the epicardium (Sermesant et al., 2006)
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INTERACTIVE 3D MESH |
Left panel: simulated geometry, the black lines represent the cardiac muscle fibres.
Right panel: interactive 3-D mesh, left click: rotation, right click: zoom.
The electrical parameters are defined according to (Sermesant et al., 2006) and to (Aliev and Panfilov, 1996): e = 0.01, k = 8, a = 0.15 (more information). The actual transmembrane potential E in mV is obtained by using the conversion formula E = 100 * u - 80 and the actual time is computed by scaling the normalised time used in the electro-physiological equations (where the duration of the simulated action potential is equal to 1.0) to get a more realistic value of 300 ms.
Finally, the conductivity d0 in the fibre direction is equal to 1.0 and the conductivity in the transverse plane r is equal to 0.25.
The biomechanical parameters are tailored to the new geometry and are chosen in order to simulate a normal cardiac function in children. In this way, the contraction rate αc is now equal to 30, the relaxation rate αr to 20 and the maximum contraction σ0 to 0.025. A delay between the electrical signal and the biomechanical answer is also considered: Td = 0.01 and Tr = 0.3.
In this model the pressure in the atria and in the arteries are constant all along the cardiac cycle. We therefore set these parameters as close as possible to their mean values during the cardiac cycle. Nevertheless, a compromise between numerical stability and respect of the actual values had to be found. Thus, pressure in the left atrium is set equal to 15 mmHg, in the aorta to 60 mmHg, in the right atrium to 7.5 mmHg and in the pulmonary artery to 30 mmHg.
The following table provides a unified view of the parameters of the simulation.
| Normal heart |
| Geometrical parameters |
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Degree of the super-ellipsoid: 2.5 RV passive volume = 65 mL; LV passive volume = 72 mL; RV/LV passive ratio = 0.90 Fibre orientations: +90° to -90° |
| Electrical parameters |
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Transmembrane propagation parameters: e = 0.01, k =
8, a = 0.15 Anisotropic conductivity: d0 = 1, r = 0.25 |
| Biomechanical parameters |
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Contraction parameters: αc = 30;
αr = 20;
σ0=0.025 Electromechanical coupling: Td = 0.01; Tr = 0.3 |
| Boundary conditions |
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Right atrium pressure: 7.5 mmHg; left atrium pressure: 15
mmHg Pulmonary artery pressure: 30 mmHg; Aorta pressure: 60 mmHg |
2.2 Results and discussion
Four cardiac cycles are simulated but only the last three are considered for the evaluation. The first cycle is a transitional state of the system and may thus bias the results.
At each cardiac cycle and for each ventricle we calculate four clinical measurements:
- end-diastolic volume (EDV): volume of the cavity when the heart quits the filling phase and starts the isovolumetric contraction,
- end-systolic volume (ESV): volume of the cavity when the heart quits the ejection phase and starts the isovolumetric relaxation,
- stroke volume (SV): amount of blood ejected during a cardiac cycle, defined by SV = EDV - ESV,
- ejection fraction (EF): indicator of global contraction, defined by EF = SV / EDV.
Since the system converges after two cardiac cycles, only the fourth and last simulated cycle is reported in next table. The diagram of the volume variations over the entire simulation is shown in the figure below.
| Right Ventricle (RV) | Left Ventricle (LV) |
| RV EDV = 75.82 mL | LV EDV = 83.48 mL |
| RV ESV = 38.85 mL | LV ESV = 30.69 mL |
| RV SV = 36.97 mL | LV SV = 52.79 mL |
| RV EF = 48.76 % | LV EF = 63.24 % |
| End-diastolic Volume Ratio: 0.90 | |
Left ventricle. According to (Helbing et al., 1995), the simulated end-diastolic and end-systolic volumes of the left ventricle are consistent with clinical observations in healthy children (LVEDV = 89 ± 26 mL, RVEDV = 26 ± 9 mL). Left ejection fraction can be seen as normal or slightly subnormal. In (Helbing et al., 1995), LVEF = 70% ± 6.
Right ventricle. If the simulated end-diastolic volume is consistent with clinical observations (RVEDV = 92 ± 25 mL according to (Helbing et al., 1995)), the simulated end-systolic volume is moderately high and cannot be considered as a normal value in children (39.77 mL versus 27 ± 9 mL (Helbing et al., 1995)). Consequently, the simulated ejection fraction is lower than the observed ejection fraction in healthy children (47.26% versus 70% ± 4, (Helbing et al., 1995)).
The ratio between the end-diastolic volumes of the right and left ventricles (equal to 0.9) is consistent with the observed values (between 0.9 and 1.1 depending on the studies (Graham et al., 1973; Helbing et LL., 1995; Lorenz 2000)).
Finally, the simulation has been assessed visually. Both ventricles stay synchronised during all the cardiac phases, and no abnormal septal movement is apparent. The simulated normal heart is shown in the following video.
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Simulation of the normal heart. Click to play the video.
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2.3 Conclusion and future works
The disease-based model previously presented has been calibrated by simulating the normal heart. The parameters of the simulation have been set according to clinical information and, for those where no clinical data was available, to get simulated clinical parameters consistent with actual measurements.
The resulting simulations are promising. The values of the simulated cardiac parameters related to the left ventricle fall in the range of the observed values in healthy children, whereas those related to the right ventricle are slightly different but close to the clinical references.
The inconsistencies in the right-ventricle parameters are mainly due to a limitation of the anatomical model. Indeed, to build the synthetic anatomy, the geometrical primitives are cut at the basal plane. Part of the right ventricle, in particular the outflow tract, is therefore ignored by our 3-D model. The volume of the right cavity is thus under-estimated and the right-ventricular wall truncated.
A way to solve this issue would consist in modelling the heart anatomy by cutting the primitives by the valvar plane instead of the basal plane. The use of advanced techniques such as those presented in the last section of this document constitutes another solution currently under study in Health-e-Child.
Another limitation raised by this first simulation is the use of constant pressures as boundary conditions. Indeed, in the real heart these pressures vary significantly during the cardiac cycle. The reliability of the simulations would therefore be improved if such phenomena were implemented. Ejection fraction for example may be more accurate. To this aim, we plan to implement advanced techniques that simulate the pressure variations along the cardiac cycle, such as the Windkessel model (Sermesant et al., 2006b).
Yet, promising results have been obtained despite these limitations. They show that it is possible to simulate paediatric cardiac diseases by using 3-D electromechanical models of the heart. This drove us to perform preliminary simulations of right-ventricle overload and dilated and hypertrophic cardiomyopathy. The first results are presented in the following sections.
2.4 References
- Aliev, R., Panfilov, A., 1996. A simple two-variable model of cardiac excitation. Chaos, Solitons, Fractals, vol. 7, no. 3, pp. 293-301.
- Graham, T. P., Jarmakani, J. M., Atwood, G. F., Canent R. V., 1973. Right Ventricular Volume Determinations in Children: Normal Values and Observations with Volume or Pressure Overload. Circulation, Vol. 47, No. 1., pp. 144-153.
- Helbing, W. A., Rebergen, S. A., Maliepaard, C., Hansen, B., Ottenkamp, J., Reiber, J. H., de Roos A., 1995. Quantification of right ventricular function with magnetic resonance imaging in children with normal hearts and with congenital heart disease. Am Heart J, Vol. 130, No. 4., pp. 828-837.
- Lang, R. M., Bierig, M., Devereux, R. B., Flachskampf, F., A., Foster, E., Pellikka, P., A., Picard, M., H., Roman, M., J., Seward, J., Shanewise, J., S., Solomon, S., D., Spencer, K., T., Sutton, M., St J., Stewart, W., J., 2005, Recommendations for Chamber Quantification: A Report fromt the American Society of Echovardiography's Guidelings end Standards Committee and the Chamber Quantification Writing Group, Developed in Conjunction with the European Association of Echocardiography, a Brandch of the European Society of Cardiology. J. Am. Soc. Echocardiogr, no 18, pp. 1440-1463
- Lorenz, C. H., 2000. The Range of Normal Values of Cardiovascular Structures in Infants, Children, and Adolescents Measured by Magnetic Resonance Imaging. Pediatric Cardiology, Vol. 21, No. 1., pp. 37-46.
- Pfisterer, M. E., Battler, A., Zaret, B. L., 1985. ange of normal values for left and right ventricular ejection fraction at rest and during exercise assessed by radionuclide angiocardiography. Eur Heart J, vol. 6, no. 8, pp. 647-655
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