March 2019, 14(1): 173-204. doi: 10.3934/nhm.2019009

The cardiac bidomain model and homogenization

Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

* Corresponding author: Erik Grandelius

Received  August 2018 Revised  November 2018 Published  January 2019

Fund Project: This work was supported by the Research Council of Norway (project 250674/F20)

We provide a rather simple proof of a homogenization result for the bidomain model of cardiac electrophysiology. Departing from a microscopic cellular model, we apply the theory of two-scale convergence to derive the bidomain model. To allow for some relevant nonlinear membrane models, we make essential use of the boundary unfolding operator. There are several complications preventing the application of standard homogenization results, including the degenerate temporal structure of the bidomain equations and a nonlinear dynamic boundary condition on an oscillating surface.

Citation: Erik Grandelius, Kenneth H. Karlsen. The cardiac bidomain model and homogenization. Networks & Heterogeneous Media, 2019, 14 (1) : 173-204. doi: 10.3934/nhm.2019009
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.

[2]

G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, in Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media (May 1995) (ed. A. Bourgeat et al.), World Scientific Pub., Singapore, 1996, 15–25.

[3]

H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glas. Mat. Ser. III, 35 (2000), 161-177.

[4]

M. AmarD. AndreucciP. Bisegna and R. Gianni, A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case, Differential Integral Equations, 26 (2013), 885-912.

[5]

B. AndreianovM. BendahmaneK. H. Karlsen and C. Pierre, Convergence of discrete duality finite volume schemes for the cardiac bidomain model, Netw. Heterog. Media, 6 (2011), 195-240. doi: 10.3934/nhm.2011.6.195.

[6]

M. Bendahmane and K. H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185-218. doi: 10.3934/nhm.2006.1.185.

[7]

M. Boulakia, M. A. Fernández, J.-F. Gerbeau and N. Zemzemi, A coupled system of PDEs and ODEs arising in electrocardiograms modeling, Appl. Math. Res. Express. AMRX, (2008), Art. ID abn002, 28pp.

[8]

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

[9]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[10]

D. CioranescuA. DamlamianP. DonatoG. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718-760. doi: 10.1137/100817942.

[11]

D. CioranescuA. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620. doi: 10.1137/080713148.

[12]

D. Cioranescu and P. Donato, An Introduction to Homogenization, vol. 17 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1999.

[13]

P. Colli Franzone, L. F. Pavarino and S. Scacchi, Mathematical Cardiac Electrophysiology, vol. 13 of MS & A. Modeling, Simulation and Applications, Springer, Cham, 2014. doi: 10.1007/978-3-319-04801-7.

[14]

P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), vol. 50 of Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 2002, 49–78.

[15]

P. DonatoK. H. Le Nguyen and R. Tardieu, The periodic unfolding method for a class of imperfect transmission problems, J. Math. Sci. (N.Y.), 176 (2011), 891-927. doi: 10.1007/s10958-011-0443-2.

[16]

P. Donato and K. H. Le Nguyen, Homogenization of diffusion problems with a nonlinear interfacial resistance, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1345-1380. doi: 10.1007/s00030-015-0325-2.

[17]

R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bulletin of Mathematical Biology, 17 (1955), 257-278. doi: 10.1007/BF02477753.

[18]

M. GahnM. Neuss-Radu and P. Knabner, Homogenization of reaction–diffusion processes in a two-component porous medium with nonlinear flux conditions at the interface, SIAM Journal on Applied Mathematics, 76 (2016), 1819-1843. doi: 10.1137/15M1018484.

[19]

M. Gahn and M. Neuss-Radu, A characterization of relatively compact sets in $L^p(\Omega, B)$, Stud. Univ. Babeş-Bolyai Math., 61 (2016), 279–290.

[20]

I. Graf and M. A. Peter, Diffusion on surfaces and the boundary periodic unfolding operator with an application to carcinogenesis in human cells, SIAM J. Math. Anal., 46 (2014), 3025-3049. doi: 10.1137/130921015.

[21]

E. Grandelius, The Bidomain Equations of Cardiac Electrophysiology, Master's thesis, University of Oslo, 2017.

[22]

C. S. Henriquez and W. Ying, The bidomain model of cardiac tissue: From microscale to macroscale, Springer US, Boston, MA, 2009, 401–421.

[23]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544.

[24]

U. Hornung and W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media, J. Differential Equations, 92 (1991), 199-225. doi: 10.1016/0022-0396(91)90047-D.

[25]

J. P. Keener and A. V. Panfilov, A biophysical model for defibrillation of cardiac tissue, Biophysical Journal, 71 (1996), 1335-1345. doi: 10.1016/S0006-3495(96)79333-5.

[26]

J. P. Keener, The effect of gap junctional distribution on defibrillation, Chaos, 8 (1998), 175-187. doi: 10.1063/1.166296.

[27]

J. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, no. v. 3 in Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag, 1972.

[28]

D. LukkassenG. Nguetseng and P. Wall, Two-scale convergence., Int. J. Pure Appl. Math., 2 (2002), 35-86.

[29]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques, SIAM J. Math. Anal., 40 (2008), 215-237. doi: 10.1137/050645269.

[30] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.
[31]

J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Crit. Rev. Biomed. Eng., 21 (1993), 137-199.

[32]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720. doi: 10.1137/060665452.

[33]

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

[34]

M. PennacchioG. Savaré and P. Colli Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM J. Math. Anal., 37 (2005), 1333-1370. doi: 10.1137/040615249.

[35]

G. Richardson, A multiscale approach to modelling electrochemical processes occurring across the cell membrane with application to transmission of action potentials, Mathematical Medicine and Biology: A Journal of the IMA, 26 (2009), 201-224. doi: 10.1093/imammb/dqn027.

[36]

G. Richardson and S. J. Chapman, Derivation of the bidomain equations for a beating heart with a general microstructure, SIAM J. Appl. Math., 71 (2011), 657-675. doi: 10.1137/090777165.

[37]

J. Simon, Compact sets in the space $L^ p(0, T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[38]

J. Sundnes, G. T. Lines, X. Cai, B. F. Nielsen, K.-A. Mardal and A. Tveito, Computing the Electrical Activity in the Heart, Springer, 2006.

[39]

L. Tung, A bi-domain model for describing ischemic myocardial D-C potentials, PhD thesis, MIT, Cambridge, MA, 1978.

[40]

A. Tveito, K. H. Jæger, M. Kuchta, K.-A. Mardal and M. E. Rognes, A cell-based framework for numerical modeling of electrical conduction in cardiac tissue, Frontiers in Physics, 5 (2017), 48. doi: 10.3389/fphy.2017.00048.

[41]

M. Veneroni, Reaction-diffusion systems for the microscopic cellular model of the cardiac electric field, Math. Methods Appl. Sci., 29 (2006), 1631-1661. doi: 10.1002/mma.740.

[42]

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

[43]

Z. Yang, The periodic unfolding method for a class of parabolic problems with imperfect interfaces, ESAIM Math. Model. Numer. Anal., 48 (2014), 1279-1302. doi: 10.1051/m2an/2013139.

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.

[2]

G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, in Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media (May 1995) (ed. A. Bourgeat et al.), World Scientific Pub., Singapore, 1996, 15–25.

[3]

H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glas. Mat. Ser. III, 35 (2000), 161-177.

[4]

M. AmarD. AndreucciP. Bisegna and R. Gianni, A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case, Differential Integral Equations, 26 (2013), 885-912.

[5]

B. AndreianovM. BendahmaneK. H. Karlsen and C. Pierre, Convergence of discrete duality finite volume schemes for the cardiac bidomain model, Netw. Heterog. Media, 6 (2011), 195-240. doi: 10.3934/nhm.2011.6.195.

[6]

M. Bendahmane and K. H. Karlsen, Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue, Netw. Heterog. Media, 1 (2006), 185-218. doi: 10.3934/nhm.2006.1.185.

[7]

M. Boulakia, M. A. Fernández, J.-F. Gerbeau and N. Zemzemi, A coupled system of PDEs and ODEs arising in electrocardiograms modeling, Appl. Math. Res. Express. AMRX, (2008), Art. ID abn002, 28pp.

[8]

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

[9]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.

[10]

D. CioranescuA. DamlamianP. DonatoG. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718-760. doi: 10.1137/100817942.

[11]

D. CioranescuA. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620. doi: 10.1137/080713148.

[12]

D. Cioranescu and P. Donato, An Introduction to Homogenization, vol. 17 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1999.

[13]

P. Colli Franzone, L. F. Pavarino and S. Scacchi, Mathematical Cardiac Electrophysiology, vol. 13 of MS & A. Modeling, Simulation and Applications, Springer, Cham, 2014. doi: 10.1007/978-3-319-04801-7.

[14]

P. Colli Franzone and G. Savaré, Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level, in Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), vol. 50 of Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 2002, 49–78.

[15]

P. DonatoK. H. Le Nguyen and R. Tardieu, The periodic unfolding method for a class of imperfect transmission problems, J. Math. Sci. (N.Y.), 176 (2011), 891-927. doi: 10.1007/s10958-011-0443-2.

[16]

P. Donato and K. H. Le Nguyen, Homogenization of diffusion problems with a nonlinear interfacial resistance, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1345-1380. doi: 10.1007/s00030-015-0325-2.

[17]

R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bulletin of Mathematical Biology, 17 (1955), 257-278. doi: 10.1007/BF02477753.

[18]

M. GahnM. Neuss-Radu and P. Knabner, Homogenization of reaction–diffusion processes in a two-component porous medium with nonlinear flux conditions at the interface, SIAM Journal on Applied Mathematics, 76 (2016), 1819-1843. doi: 10.1137/15M1018484.

[19]

M. Gahn and M. Neuss-Radu, A characterization of relatively compact sets in $L^p(\Omega, B)$, Stud. Univ. Babeş-Bolyai Math., 61 (2016), 279–290.

[20]

I. Graf and M. A. Peter, Diffusion on surfaces and the boundary periodic unfolding operator with an application to carcinogenesis in human cells, SIAM J. Math. Anal., 46 (2014), 3025-3049. doi: 10.1137/130921015.

[21]

E. Grandelius, The Bidomain Equations of Cardiac Electrophysiology, Master's thesis, University of Oslo, 2017.

[22]

C. S. Henriquez and W. Ying, The bidomain model of cardiac tissue: From microscale to macroscale, Springer US, Boston, MA, 2009, 401–421.

[23]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500-544.

[24]

U. Hornung and W. Jäger, Diffusion, convection, adsorption, and reaction of chemicals in porous media, J. Differential Equations, 92 (1991), 199-225. doi: 10.1016/0022-0396(91)90047-D.

[25]

J. P. Keener and A. V. Panfilov, A biophysical model for defibrillation of cardiac tissue, Biophysical Journal, 71 (1996), 1335-1345. doi: 10.1016/S0006-3495(96)79333-5.

[26]

J. P. Keener, The effect of gap junctional distribution on defibrillation, Chaos, 8 (1998), 175-187. doi: 10.1063/1.166296.

[27]

J. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, no. v. 3 in Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag, 1972.

[28]

D. LukkassenG. Nguetseng and P. Wall, Two-scale convergence., Int. J. Pure Appl. Math., 2 (2002), 35-86.

[29]

A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques, SIAM J. Math. Anal., 40 (2008), 215-237. doi: 10.1137/050645269.

[30] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.
[31]

J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Crit. Rev. Biomed. Eng., 21 (1993), 137-199.

[32]

M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39 (2007), 687-720. doi: 10.1137/060665452.

[33]

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

[34]

M. PennacchioG. Savaré and P. Colli Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM J. Math. Anal., 37 (2005), 1333-1370. doi: 10.1137/040615249.

[35]

G. Richardson, A multiscale approach to modelling electrochemical processes occurring across the cell membrane with application to transmission of action potentials, Mathematical Medicine and Biology: A Journal of the IMA, 26 (2009), 201-224. doi: 10.1093/imammb/dqn027.

[36]

G. Richardson and S. J. Chapman, Derivation of the bidomain equations for a beating heart with a general microstructure, SIAM J. Appl. Math., 71 (2011), 657-675. doi: 10.1137/090777165.

[37]

J. Simon, Compact sets in the space $L^ p(0, T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[38]

J. Sundnes, G. T. Lines, X. Cai, B. F. Nielsen, K.-A. Mardal and A. Tveito, Computing the Electrical Activity in the Heart, Springer, 2006.

[39]

L. Tung, A bi-domain model for describing ischemic myocardial D-C potentials, PhD thesis, MIT, Cambridge, MA, 1978.

[40]

A. Tveito, K. H. Jæger, M. Kuchta, K.-A. Mardal and M. E. Rognes, A cell-based framework for numerical modeling of electrical conduction in cardiac tissue, Frontiers in Physics, 5 (2017), 48. doi: 10.3389/fphy.2017.00048.

[41]

M. Veneroni, Reaction-diffusion systems for the microscopic cellular model of the cardiac electric field, Math. Methods Appl. Sci., 29 (2006), 1631-1661. doi: 10.1002/mma.740.

[42]

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

[43]

Z. Yang, The periodic unfolding method for a class of parabolic problems with imperfect interfaces, ESAIM Math. Model. Numer. Anal., 48 (2014), 1279-1302. doi: 10.1051/m2an/2013139.

Figure 1.  The rescaled sets $ \Omega_i^{\varepsilon} $, $ \Omega_e^{\varepsilon} $, $ \Gamma^{\varepsilon} $ (left) and the unit cell $ Y $ (right)
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