August  2020, 13(8): 2231-2258. doi: 10.3934/dcdss.2020126

Modified bidomain model with passive periodic heterogeneities

1. 

Institut de Mathématiques de Bordeaux, CNRS UMR 5251, L'Institut de rythmologie et modélisation cardiaque LIRYC, Université de Bordeaux, Carmen, Inria Bordeaux–Sud-Ouest, Bordeaux, France

2. 

Carmen, Inria Bordeaux–Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251, L'Institut de rythmologie et modélisation cardiaque LIRYC, Université de Bordeaux, Département de biologie computationnelle, Institut Pasteur, USR 3756 CNRS, Paris, France

3. 

Monc, Inria Bordeaux–Sud-Ouest, Institut de Mathématiques de Bordeaux, CNRS UMR 5251, Université de Bordeaux, Bordeaux, France

* Corresponding author: Anđela Davidović

Received  April 2018 Revised  January 2019 Published  August 2020 Early access  October 2019

Fund Project: Y.C. and A.D. have been partially funded by ANR projects, references: ANR-13-MONU-0004 and ANR-10-IAHU-04. C.P. has been partially funded by the Plan Cancer projects DYNAMO (PC201515) and NUMEP (PC201615), and Inria associate team NUM4SEP

In this paper we study how mesoscopic heterogeneities affect electrical signal propagation in cardiac tissue. The standard model used in cardiac electrophysiology is a bidomain model - a system of degenerate parabolic PDEs, coupled with a set of ODEs, representing the ionic behviour of the cardiac cells. We assume that the heterogeneities in the tissue are periodically distributed diffusive regions, that are significantly larger than a cardiac cell. These regions represent the fibrotic tissue, collagen or fat, that is electrically passive. We give a mathematical setting of the model. Using semigroup theory we prove that such model has a uniformly bounded solution. Finally, we use two–scale convergence to find the limit problem that represents the average behviour of the electrical signal in this setting.

Citation: Yves Coudière, Anđela Davidović, Clair Poignard. Modified bidomain model with passive periodic heterogeneities. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2231-2258. doi: 10.3934/dcdss.2020126
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.  doi: 10.1137/0523084.

[2]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 374, American Mathematical Soc., 2011. doi: 10.1090/chel/374.

[3]

M. Boulakia, M. A. Fernandez, J.-F. Gerbeau and N. Zemzemi, A coupled system of PDEs and ODEs arising in electrocardiograms modeling, Applied Mathematics Research eXpress, 2 (2008), Art. ID abn002, 28 pp, https://dx.doi.org/10.1093/amrx/abn002. doi: 10.1093/amrx/abn002.

[4]

Y. BourgaultY. Coudiere and C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology, Nonlinear Analysis: Real World Applications, 10 (2009), 458-482.  doi: 10.1016/j.nonrwa.2007.10.007.

[5]

P. CamellitiT. K. Borg and P. Kohl, Structural and functional characterisation of cardiac fibroblasts, Cardiovascular Research, 65 (2005), 40-51.  doi: 10.1016/j.cardiores.2004.08.020.

[6]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13, Oxford University Press on Demand, 1998.

[7]

R. ClaytonO. BernusE. CherryH. DierckxF. FentonL. MirabellaA. PanfilovF. SachseG. Seemann and H. Zhang, Models of cardiac tissue electrophysiology: progress, challenges and open questions, Progress in Biophysics and Molecular Biology, 104 (2011), 22-48.  doi: 10.1016/j.pbiomolbio.2010.05.008.

[8]

A. Collin and S. Imperiale, Mathematical analysis and 2-scale convergence of a heterogeneous microscopic bidomain model, Mathematical Models and Methods in Applied Sciences, 28 (2018), 979-1035.  doi: 10.1142/S0218202518500264.

[9]

Y. Coudière, A. Davidović and C. Poignard, The modified bidomain model with periodic diffusive inclusions, in Computing in Cardiology Conference (CinC), 2014 (ed. Alan Murray), IEEE, 2014, 1033–1036, https://ieeexplore.ieee.org/abstract/document/7043222.

[10]

A. Davidović, Multiscale Mathematical Modelling of Structural Heterogeneities in Cardiac Electrophysiology, PhD thesis, Université de Bordeaux, 2016, https://hal-univ-tlse3.archives-ouvertes.fr/U-BORDEAUX/tel-01478145v1.

[11]

A. Davidović, Y. Coudière and Y. Bourgault, Image-based modeling of the heterogeneity of propagation of the cardiac action potential. example of rat heart high resolution mri, in International Conference on Functional Imaging and Modeling of the Heart, Springer, 2017,260–270. doi: 10.1007/978-3-319-59448-4_25.

[12]

M. Ethier and Y. Bourgault, Semi-implicit time-discretization schemes for the bidomain model, SIAM Journal on Numerical Analysis, 46 (2008), 2443-2468.  doi: 10.1137/070680503.

[13]

L. C. Evans, Partial Differential Equations, , in Graduate Studies in Mathematics, vol. 19, Am. Math. Soc., 1998.

[14]

P. C. Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro-and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis, Springer, 50 (2002), 49–78. doi: 10.1007/978-3-0348-8221-7_4.

[15]

D. B. Geselowitz and W. Miller III, A bidomain model for anisotropic cardiac muscle, Annals of Biomedical Engineering, 11 (1983), 191-206.  doi: 10.1007/BF02363286.

[16]

F. Hecht, New development in freefem++, Journal of Numerical Mathematics, 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[17]

O. KavianM. LeguèbeC. Poignard and L. Weynans, "Classical" electropermeabilization modeling at the cell scale, Journal of Mathematical Biology, 68 (2014), 235-265.  doi: 10.1007/s00285-012-0629-3.

[18]

P. KohlA. KamkinI. Kiseleva and D. Noble, Mechanosensitive fibroblasts in the sino-atrial node region of rat heart: Interaction with cardiomyocytes and possible role, Experimental Physiology, 79 (1994), 943-956.  doi: 10.1113/expphysiol.1994.sp003819.

[19]

P. KohlP. CamellitiF. L. Burton and G. L. Smith, Electrical coupling of fibroblasts and myocytes: Relevance for cardiac propagation, Journal of Electrocardiology, 38 (2005), 45-50.  doi: 10.1016/j.jelectrocard.2005.06.096.

[20]

W. Krassowska and J. Neu, Effective boundary conditions for syncytial tissues, Biomedical Engineering, IEEE Transactions on, 41 (1994), 143-150.  doi: 10.1109/10.284925.

[21]

M. LeguebeA. SilveL. M. Mir and C. Poignard, Conducting and permeable states of cell membrane submitted to high voltage pulses: mathematical and numerical studies validated by the experiments, Journal of Theoretical Biology, 360 (2014), 83-94.  doi: 10.1016/j.jtbi.2014.06.027.

[22]

K. A. MacCannellH. BazzaziL. ChiltonY. ShibukawaR. B. Clark and W. R. Giles, A mathematical model of electrotonic interactions between ventricular myocytes and fibroblasts, Biophysical Journal, 92 (2007), 4121-4132.  doi: 10.1529/biophysj.106.101410.

[23]

W. T. Miller and D. B. Geselowitz, Simulation studies of the electrocardiogram. I. the normal heart, Circulation Research, 43 (1978), 301-315.  doi: 10.1161/01.RES.43.2.301.

[24]

C. C. Mitchell and D. G. Schaeffer, A two-current model for the dynamics of cardiac membrane, Bulletin of mathematical biology, 65 (2003), 767-793.  doi: 10.1016/S0092-8240(03)00041-7.

[25]

A. Muler and V. Markin, Electrical properties of anisotropic neuromuscular syncytia. I. distribution of the electrotonic potentia, Biofizika, 22 (1977), 307-12. 

[26]

J. Neu and W. Krassowska, Homogenization of syncytial tissues, Critical Reviews in Biomedical Engineering, 21 (1992), 137-199. 

[27]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623.  doi: 10.1137/0520043.

[28]

M. PennacchioG. Savaré and P. C. Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM Journal on Mathematical Analysis, 37 (2005), 1333-1370.  doi: 10.1137/040615249.

[29]

M. Rioux and Y. Bourgault, A predictive method allowing the use of a single ionic model in numerical cardiac electrophysiology, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 987-1016.  doi: 10.1051/m2an/2012054.

[30]

F. B. SachseA. P. MorenoG. Seemann and J. Abildskov, A model of electrical conduction in cardiac tissue including fibroblasts, Annals of Biomedical Engineering, 37 (2009), 874-889.  doi: 10.1007/s10439-009-9667-4.

[31]

M. Veneroni, Reaction–diffusion systems for the macroscopic bidomain model of the cardiac electric field, Nonlinear Analysis: Real World Applications, 10 (2009), 849-868.  doi: 10.1016/j.nonrwa.2007.11.008.

[32]

J. C. Weaver and Y. A. Chizmadzhev, Theory of electroporation: A review, Bioelectrochemistry and Bioenergetics, 41 (1996), 135-160.  doi: 10.1016/S0302-4598(96)05062-3.

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.  doi: 10.1137/0523084.

[2]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 374, American Mathematical Soc., 2011. doi: 10.1090/chel/374.

[3]

M. Boulakia, M. A. Fernandez, J.-F. Gerbeau and N. Zemzemi, A coupled system of PDEs and ODEs arising in electrocardiograms modeling, Applied Mathematics Research eXpress, 2 (2008), Art. ID abn002, 28 pp, https://dx.doi.org/10.1093/amrx/abn002. doi: 10.1093/amrx/abn002.

[4]

Y. BourgaultY. Coudiere and C. Pierre, Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology, Nonlinear Analysis: Real World Applications, 10 (2009), 458-482.  doi: 10.1016/j.nonrwa.2007.10.007.

[5]

P. CamellitiT. K. Borg and P. Kohl, Structural and functional characterisation of cardiac fibroblasts, Cardiovascular Research, 65 (2005), 40-51.  doi: 10.1016/j.cardiores.2004.08.020.

[6]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13, Oxford University Press on Demand, 1998.

[7]

R. ClaytonO. BernusE. CherryH. DierckxF. FentonL. MirabellaA. PanfilovF. SachseG. Seemann and H. Zhang, Models of cardiac tissue electrophysiology: progress, challenges and open questions, Progress in Biophysics and Molecular Biology, 104 (2011), 22-48.  doi: 10.1016/j.pbiomolbio.2010.05.008.

[8]

A. Collin and S. Imperiale, Mathematical analysis and 2-scale convergence of a heterogeneous microscopic bidomain model, Mathematical Models and Methods in Applied Sciences, 28 (2018), 979-1035.  doi: 10.1142/S0218202518500264.

[9]

Y. Coudière, A. Davidović and C. Poignard, The modified bidomain model with periodic diffusive inclusions, in Computing in Cardiology Conference (CinC), 2014 (ed. Alan Murray), IEEE, 2014, 1033–1036, https://ieeexplore.ieee.org/abstract/document/7043222.

[10]

A. Davidović, Multiscale Mathematical Modelling of Structural Heterogeneities in Cardiac Electrophysiology, PhD thesis, Université de Bordeaux, 2016, https://hal-univ-tlse3.archives-ouvertes.fr/U-BORDEAUX/tel-01478145v1.

[11]

A. Davidović, Y. Coudière and Y. Bourgault, Image-based modeling of the heterogeneity of propagation of the cardiac action potential. example of rat heart high resolution mri, in International Conference on Functional Imaging and Modeling of the Heart, Springer, 2017,260–270. doi: 10.1007/978-3-319-59448-4_25.

[12]

M. Ethier and Y. Bourgault, Semi-implicit time-discretization schemes for the bidomain model, SIAM Journal on Numerical Analysis, 46 (2008), 2443-2468.  doi: 10.1137/070680503.

[13]

L. C. Evans, Partial Differential Equations, , in Graduate Studies in Mathematics, vol. 19, Am. Math. Soc., 1998.

[14]

P. C. Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro-and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis, Springer, 50 (2002), 49–78. doi: 10.1007/978-3-0348-8221-7_4.

[15]

D. B. Geselowitz and W. Miller III, A bidomain model for anisotropic cardiac muscle, Annals of Biomedical Engineering, 11 (1983), 191-206.  doi: 10.1007/BF02363286.

[16]

F. Hecht, New development in freefem++, Journal of Numerical Mathematics, 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[17]

O. KavianM. LeguèbeC. Poignard and L. Weynans, "Classical" electropermeabilization modeling at the cell scale, Journal of Mathematical Biology, 68 (2014), 235-265.  doi: 10.1007/s00285-012-0629-3.

[18]

P. KohlA. KamkinI. Kiseleva and D. Noble, Mechanosensitive fibroblasts in the sino-atrial node region of rat heart: Interaction with cardiomyocytes and possible role, Experimental Physiology, 79 (1994), 943-956.  doi: 10.1113/expphysiol.1994.sp003819.

[19]

P. KohlP. CamellitiF. L. Burton and G. L. Smith, Electrical coupling of fibroblasts and myocytes: Relevance for cardiac propagation, Journal of Electrocardiology, 38 (2005), 45-50.  doi: 10.1016/j.jelectrocard.2005.06.096.

[20]

W. Krassowska and J. Neu, Effective boundary conditions for syncytial tissues, Biomedical Engineering, IEEE Transactions on, 41 (1994), 143-150.  doi: 10.1109/10.284925.

[21]

M. LeguebeA. SilveL. M. Mir and C. Poignard, Conducting and permeable states of cell membrane submitted to high voltage pulses: mathematical and numerical studies validated by the experiments, Journal of Theoretical Biology, 360 (2014), 83-94.  doi: 10.1016/j.jtbi.2014.06.027.

[22]

K. A. MacCannellH. BazzaziL. ChiltonY. ShibukawaR. B. Clark and W. R. Giles, A mathematical model of electrotonic interactions between ventricular myocytes and fibroblasts, Biophysical Journal, 92 (2007), 4121-4132.  doi: 10.1529/biophysj.106.101410.

[23]

W. T. Miller and D. B. Geselowitz, Simulation studies of the electrocardiogram. I. the normal heart, Circulation Research, 43 (1978), 301-315.  doi: 10.1161/01.RES.43.2.301.

[24]

C. C. Mitchell and D. G. Schaeffer, A two-current model for the dynamics of cardiac membrane, Bulletin of mathematical biology, 65 (2003), 767-793.  doi: 10.1016/S0092-8240(03)00041-7.

[25]

A. Muler and V. Markin, Electrical properties of anisotropic neuromuscular syncytia. I. distribution of the electrotonic potentia, Biofizika, 22 (1977), 307-12. 

[26]

J. Neu and W. Krassowska, Homogenization of syncytial tissues, Critical Reviews in Biomedical Engineering, 21 (1992), 137-199. 

[27]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623.  doi: 10.1137/0520043.

[28]

M. PennacchioG. Savaré and P. C. Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM Journal on Mathematical Analysis, 37 (2005), 1333-1370.  doi: 10.1137/040615249.

[29]

M. Rioux and Y. Bourgault, A predictive method allowing the use of a single ionic model in numerical cardiac electrophysiology, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 987-1016.  doi: 10.1051/m2an/2012054.

[30]

F. B. SachseA. P. MorenoG. Seemann and J. Abildskov, A model of electrical conduction in cardiac tissue including fibroblasts, Annals of Biomedical Engineering, 37 (2009), 874-889.  doi: 10.1007/s10439-009-9667-4.

[31]

M. Veneroni, Reaction–diffusion systems for the macroscopic bidomain model of the cardiac electric field, Nonlinear Analysis: Real World Applications, 10 (2009), 849-868.  doi: 10.1016/j.nonrwa.2007.11.008.

[32]

J. C. Weaver and Y. A. Chizmadzhev, Theory of electroporation: A review, Bioelectrochemistry and Bioenergetics, 41 (1996), 135-160.  doi: 10.1016/S0302-4598(96)05062-3.

Figure 1.  On the left: the idealised full 2D domain, $ \Omega $. On the right: the periodic cell, $ Y $
Figure 2.  The convergence study for $ L^2 $ errors of $ v_{\varepsilon} $ and $ h_{\varepsilon} $ in log-log scale. Observed convergence rates are $ 1.39 $ for $ v_{\varepsilon} $, and $ 0.63 $ for $ h_{\varepsilon} $
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