4601:
4057:
4596:{\displaystyle {\begin{cases}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=\chi \left(C_{m}{\dfrac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right)-I_{s_{1}}&\mathbf {x} \in \mathbb {H} \\\nabla \cdot \left(\left(\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e}\right)\nabla v_{e}\right)=-\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+I_{s_{2}}&\mathbf {x} \in \mathbb {H} \\\mathbf {\Sigma } _{i}(\nabla v+\nabla v_{e})\cdot \mathbf {n} =0&\mathbf {x} \in \partial \mathbb {H} \\\left\cdot \mathbf {n} =0&\mathbf {x} \in \partial \mathbb {H} \end{cases}}}
843:
526:
838:{\displaystyle {\begin{alignedat}{2}&\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right)-I_{s_{1}}\\&\nabla \cdot \left(\left(\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e}\right)\nabla v_{e}\right)=-\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+I_{s_{2}}\end{alignedat}}}
112:
2383:
3765:
3062:
1176:
3559:
2027:
3414:
2211:
1774:
The second assumption is that the heart is isolated so that the current that leaves one region need to flow into the other. Then, the current density in each of the intracellular and extracellular domain must be equal in magnitude but opposite in sign, and can be defined as the product of the surface
119:
The bidomain domain is principally represented by two main regions: the cardiac cells, called intracellular domain, and the space surrounding them, called extracellular domain. Moreover, usually another region is considered, called extramyocardial region. The intracellular and extracellular domains,
3627:
5433:
Niederer, S. A.; Kerfoot, E.; Benson, A. P.; Bernabeu, M. O.; Bernus, O.; Bradley, C.; Cherry, E. M.; Clayton, R.; Fenton, F. H.; Garny, A.; Heidenreich, E.; Land, S.; Maleckar, M.; Pathmanathan, P.; Plank, G.; Rodriguez, J. F.; Roy, I.; Sachse, F. B.; Seemann, G.; Skavhaug, O.; Smith, N. P. (3
3255:
is the vector that represents the outwardly unit normal to the myocardial surface of the heart. Since the intracellular potential is not explicitily presented in the bidomain formulation, this condition is usually described in terms of the transmembrane and extracellular potential, knowing that
2902:
1037:
1729:
1478:
The first assumption is that the intracellular current can flow only between the intracellular and extracellular regions, while the intracellular and extramyocardial regions can comunicate between them, so that the current can flow into and from the extramyocardial regions but only in the
66:
in anisotropic tissues is not unique in all directions, but it is different in parallel and perpendicular direction with respect to the fiber one. Moreover, in tissues with unequal anisotropy ratios, the ratio of conductivities parallel and perpendicular to the fibers are different in the
3421:
2797:
3962:
2542:
50:
Since it is a continuum model, rather than describing each cell individually, it represents the average properties and behaviour of group of cells organized in complex structure. Thus, the model results to be a complex one and can be seen as a generalization of the
1426:
2146:
3231:
1886:
3305:
75:
it is about 5:2. Mathematically, unequal anisotropy ratios means that the effect of anisotropy cannot be removed by a change in the distance scale in one direction. Instead, the anisotropy has a more profound influence on the electrical behavior.
1278:
2643:
1612:
2676:
3847:
2433:
2378:{\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{i})-\nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})=-\nabla \cdot (\mathbf {\Sigma } _{e}\nabla v_{e})-\nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})}
3561:
Here the normal vectors from the perspective of both domains are considered, thus the negative sign are necessary. Moreover, a perfect transmission of the potential on the cardiac boundary is necessary, which gives
3620:
3760:{\displaystyle \left(\mathbf {\Sigma } _{i}\nabla v\right)\cdot \mathbf {n} =-\left((\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e})\nabla v_{e}\right)\cdot \mathbf {n} \quad \mathbf {x} \in \partial \mathbb {H} .}
3819:
3057:{\displaystyle \nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right),}
1874:
1171:{\displaystyle {\begin{cases}{\dfrac {\partial \mathbf {w} }{\partial t}}=\mathbf {F} (v,\mathbf {w} )&{\text{in }}\mathbb {H} \\\mathbf {w} (t=0)=\mathbf {w} _{0}&{\text{in }}\mathbb {H} \end{cases}}}
2851:
2206:
1342:
984:
2051:
3157:
3554:{\displaystyle \left(\mathbf {\Sigma } _{e}\nabla v_{e}\right)\cdot \mathbf {n} _{e}=-\left(\mathbf {\Sigma } _{0}\nabla v_{0}\right)\cdot \mathbf {n} _{0}\quad \mathbf {x} \in \partial \mathbb {H} .}
115:
Bidomain model domain, considering the intracellular and extracellular region as a unique physical region representing the heart, and an extramyocardial region representing the torso or a fluid bath.
1196:
In some cases, an extramyocardial region is considered. This implies the addition to the bidomain model of an equation describing the potential propagation inside the extramyocardial domain.
4656:
Lines, G.T.; Buist, M.L.; Grottum, P.; Pullan, A.J.; Sundnes, J.; Tveito, A. (1 July 2002). "Mathematical models and numerical methods for the forward problem in cardiac electrophysiology".
1563:
1206:
2022:{\displaystyle {\begin{alignedat}{2}\nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{i})&=\chi I_{m}\\\nabla \cdot (\mathbf {\Sigma } _{e}\nabla v_{e})&=-\chi I_{m}\end{alignedat}}}
4045:
4013:
1334:
418:
386:
223:
195:
422:
The transmembrane current flows between the intracellular and extracellular regions and it is in part described by the corresponding ionic current over the membrane per unit area
172:
to simulate physiological conditions. The boundary of the two principal physical domains defined are important to solve the bidomain model. Here the heart boundary is denoted as
3984:
1454:
447:
3300:
2897:
2429:
353:
3253:
1607:
1526:
1006:
166:
144:
3137:
3103:
911:
877:
3839:
1504:
3418:
For the extracellular potential, if the myocardial region is presented, a balance in the flow between the extracellular and the extramyocardial regions is considered
3409:{\displaystyle (\mathbf {\Sigma } _{i}\nabla v)\cdot \mathbf {n} =-(\mathbf {\Sigma } _{i}\nabla v_{e})\cdot \mathbf {n} \quad \mathbf {x} \in \partial \mathbb {H} .}
28:. It consists in a continuum (volume-average) approach in which the cardiac microstructure is defined in terms of muscle fibers grouped in sheets, creating a complex
1032:
3624:
Instead, if the heart is considered as isolated, which means that no myocardial region is presented, a possible boundary condition for the extracellular problem is
1800:
1305:
494:
474:
284:
257:
2559:
1187:
physiological models, which take into account both macroscopic behaviour and cell physiology with a quite detailed description of the most important ionic current.
4622:. Special considerations can be made for the numerical solution of these equations, due to the high time and space resolution needed for numerical convergence.
1769:
1749:
1585:
307:
5491:
Pathmanathan, Pras; Bernabeu, Miguel O.; Bordas, Rafel; Cooper, Jonathan; Garny, Alan; Pitt-Francis, Joe M.; Whiteley, Jonathan P.; Gavaghan, David J. (2010).
4054:
If the heart is considered as an isolated tissue, which means that no current can flow outside of it, the final formulation with boundary conditions reads
3565:
5337:
5192:
1805:
5335:
Trayanova N, Roth BJ, Malden LJ (1993). "The response of a spherical heart to a uniform electric field: A bidomain analysis of cardiac stimulation".
4693:
Information processing in the nervous system; proceedings of a symposium held at the State
University of New York at Buffalo, 21st-24th October, 1968
1724:{\displaystyle {\begin{alignedat}{2}J_{i}&=-\mathbf {\Sigma } _{i}\nabla v_{i}\\J_{e}&=-\mathbf {\Sigma } _{e}\nabla v_{e}.\end{alignedat}}}
5563:
3147:
In order to solve the model, boundary conditions are needed. The more classical boundary conditions are the following ones, formulated by Tung.
3154:
section, there ca not been any flow of current between the intracellular and extramyocardial domains. This can be mathematically described as
3071:
section is obtained through a generalization, considering possible external stimulus which can be given through the external applied currents
4700:
2792:{\displaystyle \nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{i}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{ion}\right).}
32:
structure with anisotropical properties. Then, to define the electrical activity, two interpenetrating domains are considered, which are the
4636:
3777:
1460:
169:
3957:{\displaystyle \nabla \cdot (\mathbf {\Sigma } \nabla v)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right)-I_{s}}
2537:{\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v)=-\nabla \cdot ((\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e})\nabla v_{e}).}
4716:
Muler AL, Markin VS (1977). "Electrical properties of anisotropic nerve-muscle syncytia-I. Distribution of the electrotonic potential".
168:). The extramyocardial region can be considered as a fluid bath, especially when one wants to simulate experimental conditions, or as a
2802:
2157:
5548:
5086:
5145:
Roth BJ (1992). "How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle".
1531:
4770:
Muler AL, Markin VS (1977). "Electrical properties of anisotropic nerve-muscle syncytia-III. Steady form of the excitation front".
932:
531:
4893:
4743:
Muler AL, Markin VS (1977). "Electrical properties of anisotropic nerve-muscle syncytia-II. Spread of flat front of excitation".
5558:
5147:
926:
5568:
509:
4066:
1891:
1617:
358:
Moreover, some important parameters need to be taken in account, especially the intracellular conductivity tensor matrix
1421:{\displaystyle (\mathbf {\Sigma } _{0}\nabla v_{0})\cdot \mathbf {n} _{0}=0\quad \mathbf {x} \in \partial \mathbb {T} ,}
520:, while the second one computes the extracellular potential starting from a given transmembran potential distribution.
5573:
513:
2141:{\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{i})=-\nabla \cdot (\mathbf {\Sigma } _{e}\nabla v_{e}).}
1339:
Moreover, an isolated domain assumption is considered, which means that the following boundary conditions are added
62:
Many of the interesting properties of the bidomain model arise from the condition of unequal anisotropy ratios. The
5553:
3226:{\displaystyle (\mathbf {\Sigma } _{i}\nabla v_{i})\cdot \mathbf {n} =0\quad \mathbf {x} \in \partial \mathbb {H} }
5383:
Boulakia, Muriel; Cazeau, Serge; Fernández, Miguel A.; Gerbeau, Jean-Frédéric; Zemzemi, Nejib (24 December 2009).
922:
4611:
5079:
Mathematically modelling the electrical activity of the heart : from cell to body surface and back again
4018:
3989:
1310:
1184:
phenomenological models, which are the simplest ones and used to reproduce macroscopic behavior of the cell.
517:
394:
362:
287:
200:
84:
63:
175:
1472:
5492:
4615:
3967:
5190:
Henriquez CS (1993). "Simulating the electrical behavior of cardiac tissue using the bidomain model".
1430:
1273:{\displaystyle -\nabla \cdot (\mathbf {\Sigma } _{0}\nabla v_{0})=0\quad \mathbf {x} \in \mathbb {T} }
5447:
5291:
5234:
4941:
4619:
67:
intracellular and extracellular spaces. For instance, in cardiac tissue, the anisotropy ratio in the
1046:
425:
5225:
3259:
2856:
2388:
312:
97:
the effect of fiber curvature on the transmembrane potential distribution during an electric shock.
72:
68:
5440:
Philosophical
Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
4610:
There are various possible techniques to solve the bidomain equations. Between them, one can find
3236:
1590:
1509:
989:
149:
127:
5415:
5362:
5172:
5127:
5102:
Roth BJ (1997). "Electrical conductivity values used with the bidomain model of cardiac tissue".
4673:
1878:
By combining the previous assumptions, the conservation of current densities is obtained, namely
21:
3108:
3074:
882:
848:
496:
need to be considered to derive the bidomain model formulation, which is done in the following
5512:
5473:
5407:
5354:
5317:
5260:
5201:
5164:
5119:
5082:
4994:
4967:
4910:
4873:
4838:
4779:
4752:
4725:
4696:
3824:
1489:
121:
29:
2638:{\displaystyle I_{m}=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\text{ion}}\right),}
5504:
5463:
5455:
5399:
5346:
5307:
5299:
5250:
5242:
5156:
5111:
4957:
4949:
4902:
4865:
4856:
Peskoff A (1979). "Electric potential in three-dimensional electrically syncytial tissues".
4828:
4665:
4631:
3842:
1200:
1011:
1778:
1283:
479:
452:
262:
235:
40:
domains, representing respectively the space inside the cells and the region between them.
5493:"A numerical guide to the solution of the bidomain equations of cardiac electrophysiology"
5436:"Verification of cardiac tissue electrophysiology simulators using an N-version benchmark"
5533:
5451:
5295:
5238:
4945:
3964:
where the only variable is now the transmembrane potential, and the conductivity tensor
2151:
This equation states exactly that all currents exiting one domain must enter the other.
5468:
5435:
5384:
5312:
5279:
5255:
5220:
4962:
4929:
2544:
Then, knowing the transmembral potential, one can recover the extracellular potential.
1754:
1734:
1570:
292:
91:
5303:
5246:
4953:
4906:
4869:
4800:
Tung L (1978). "A bi-domain model for describing ischemic myocardial d-c potentials".
5542:
5508:
4677:
37:
33:
5419:
5131:
2154:
From here, it is easy to find the second equation of the bidomain model subtracting
5366:
3774:
By assuming equal anisotropy ratios for the intra- and extracellular domains, i.e.
2548:
52:
44:
5176:
4891:
Peskoff A (1979). "Electric potential in cylindrical syncytia and muscle fibers".
1483:
1775:
to volume ratio of the cell membrane and the transmembrane ionic current density
146:), while the extramyocardial domain is a unique physical space adjacent of them (
2547:
Then, the current that flows across the cell membrane can be modelled with the
1459:
If the extramyocardial region is the human torso, this model gives rise to the
309:, which is defined as the difference of the potential across the cell membrane
5403:
4669:
111:
94:
produced by an action potential wave front propagating through cardiac tissue,
4833:
4816:
5516:
5477:
5459:
5411:
5358:
5321:
5264:
5205:
5168:
5123:
4998:
4971:
232:
The unknowns in the bidomain model are three, the intracellular potential
4914:
4877:
4842:
4783:
4756:
4729:
1486:
and a quasi-static assumption, the gradient of a scalar potential field
5160:
3615:{\displaystyle v_{e}=v_{0}\quad \mathbf {x} \in \partial \mathbb {H} .}
1771:
represent the intracellular and extracellular quantities respectively.
124:, are considered to be a unique physical space representing the heart (
5350:
5115:
3064:
which describes the evolution of the transmembrane potential in time.
1456:
being the unit normal directed outside of the extramyocardial domain.
3814:{\displaystyle \mathbf {\Sigma } _{i}=\lambda \mathbf {\Sigma } _{e}}
4985:
Neu JC, Krassowska W (1993). "Homogenization of syncytial tissues".
4817:"Simulation studies of the electrocardiogram, I. The normal heart"
1869:{\displaystyle -\nabla \cdot J_{i}=\nabla \cdot J_{e}=\chi I_{m}.}
110:
47:
in 1969 before being formulated mathematically in the late 1970s.
25:
2846:{\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})}
2201:{\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})}
5385:"Mathematical Modeling of Electrocardiograms: A Numerical Study"
5280:"Electric and magnetic fields from two-dimensional bisyncytia"
79:
Three examples of the impact of unequal anisotropy ratios are
979:{\displaystyle I_{\text{ion}}=I_{\text{ion}}(v,\mathbf {w} )}
5077:
Pullan, Andrew J.; Buist, Martin L.; Cheng, Leo K. (2005).
4589:
1475:
of the electromagnetism, considering some simplifications.
1164:
4050:
Formulation with boundary conditions in an isolated domain
3841:, the model can be reduced to one single equation, called
2385:
and knowing that the transmembral potential is defined as
87:
during unipolar stimulation of a sheet of cardiac tissue,
55:
to higher dimensions and, going to define the so-called
2899:, one can get the first equation of the bidomain model
523:
Thus, the bidomain model can be formulated as follows:
913:
can be defined as applied external stimulus currents.
4169:
4060:
4021:
3992:
3970:
3850:
3827:
3780:
3630:
3568:
3424:
3308:
3262:
3239:
3160:
3111:
3077:
2905:
2859:
2805:
2679:
2562:
2436:
2391:
2214:
2160:
2054:
1889:
1808:
1781:
1757:
1737:
1615:
1593:
1573:
1534:
1512:
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1433:
1345:
1313:
1286:
1209:
1050:
1040:
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992:
935:
885:
851:
529:
482:
476:
and the surface to volume ratio of the cell membrane
455:
428:
397:
365:
315:
295:
265:
238:
203:
178:
152:
130:
1587:
represent the current density of the electric field
1008:
is called ionic variable. Then, in general, for all
449:. Moreover, the membrane capacitance per unit area
5221:"Current injection into a two-dimensional bidomain"
4695:. Springer-Science and Business. pp. 325–331.
1307:is the potential in the extramyocardial region and
4595:
4039:
4007:
3978:
3956:
3833:
3813:
3759:
3614:
3553:
3408:
3294:
3247:
3225:
3131:
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3056:
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2791:
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2423:
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2200:
2140:
2021:
1868:
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1199:Usually, this equation is a simple generalized
921:The ionic current is usually represented by an
5026:
5024:
5022:
5020:
5018:
5016:
5014:
5012:
5010:
5008:
1558:{\displaystyle \mathbf {E} =-\nabla \varphi .}
5534:Scholarpedia article about the bidomain model
4930:"Electrical properties of spherical syncytia"
4928:Eisenberg RS, Barcilon V, Mathias RT (1979).
4795:
4793:
390:the extracellular conductivity tensor matrix
8:
5497:Progress in Biophysics and Molecular Biology
1471:The bidomain equations are derived from the
5338:IEEE Transactions on Biomedical Engineering
5104:IEEE Transactions on Biomedical Engineering
1180:Different ionic models have been proposed:
5193:Critical Reviews in Biomedical Engineering
4987:Critical Reviews in Biomedical Engineering
1336:is the corresponding conductivity tensor.
508:The bidomain model is defined through two
5467:
5311:
5254:
5219:Sepulveda NG, Roth BJ, Wikswo JP (1989).
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154:
153:
151:
132:
131:
129:
43:The bidomain model was first proposed by
24:to define the electrical activity of the
4648:
4040:{\displaystyle \mathbf {\Sigma } _{e}.}
3067:The final formulation described in the
4802:PhD Dissertation, MIT, Cambridge, Mass
4658:Computing and Visualization in Science
4008:{\displaystyle \mathbf {\Sigma } _{i}}
2045:from which, summing the two equations
1329:{\displaystyle \mathbf {\Sigma } _{0}}
929:(ODEs). Mathematically, one can write
413:{\displaystyle \mathbf {\Sigma } _{e}}
381:{\displaystyle \mathbf {\Sigma } _{i}}
218:{\displaystyle \partial \mathbb {T} .}
4815:Miller WT III; Geselowitz DB (1978).
3150:First of all, as state before in the
190:{\displaystyle \partial \mathbb {H} }
7:
4637:Forward problem of electrocardiology
2553:
1880:
1461:forward problem of electrocardiology
197:while the torso domain boundary is
4578:
4536:
4505:
4496:
4464:
4424:
4415:
4350:
4327:
4303:
4255:
4205:
4202:
4199:
4180:
4172:
4129:
4106:
4092:
4069:
3979:{\displaystyle \mathbf {\Sigma } }
3930:
3927:
3924:
3906:
3898:
3865:
3851:
3746:
3711:
3648:
3601:
3540:
3498:
3442:
3395:
3362:
3324:
3215:
3176:
3040:
3037:
3034:
3016:
3008:
2966:
2943:
2929:
2906:
2827:
2806:
2753:
2745:
2703:
2680:
2605:
2597:
2515:
2473:
2458:
2437:
2359:
2338:
2319:
2298:
2276:
2255:
2236:
2215:
2182:
2161:
2119:
2098:
2076:
2055:
1976:
1955:
1915:
1894:
1831:
1812:
1701:
1652:
1546:
1407:
1361:
1234:
1213:
1192:Model of an extramyocardial region
1063:
1053:
800:
777:
753:
705:
669:
666:
663:
645:
637:
595:
572:
558:
535:
204:
179:
14:
1506:can describe an electrical field
5509:10.1016/j.pbiomolbio.2010.05.006
5392:Annals of Biomedical Engineering
5278:Sepulveda NG, Wikswo JP (1987).
4894:Bulletin of Mathematical Biology
4858:Bulletin of Mathematical Biology
4571:
4558:
4526:
4483:
4457:
4444:
4402:
4384:
4340:
4288:
4273:
4239:
4119:
4082:
4024:
3995:
3972:
3861:
3801:
3783:
3739:
3733:
3698:
3683:
3663:
3638:
3594:
3533:
3521:
3488:
3465:
3432:
3388:
3382:
3352:
3337:
3314:
3241:
3208:
3196:
3166:
2956:
2919:
2817:
2799:Finally, adding and subtracting
2693:
2502:
2487:
2448:
2349:
2309:
2266:
2226:
2172:
2109:
2066:
1966:
1905:
1802:per unit area, which means that
1691:
1642:
1609:, two equations can be obtained
1595:
1536:
1514:
1449:{\displaystyle \mathbf {n} _{0}}
1436:
1400:
1382:
1351:
1316:
1258:
1224:
1139:
1115:
1091:
1077:
1057:
994:
969:
790:
738:
723:
585:
548:
400:
368:
5148:Journal of Mathematical Biology
3737:
3592:
3531:
3386:
3206:
1398:
1256:
927:ordinary differential equations
497:
5564:Partial differential equations
4518:
4493:
4437:
4412:
3871:
3857:
3708:
3678:
3375:
3347:
3330:
3309:
3189:
3161:
2840:
2812:
2528:
2512:
2482:
2479:
2464:
2443:
2372:
2344:
2332:
2304:
2289:
2261:
2249:
2221:
2195:
2167:
2132:
2104:
2089:
2061:
1989:
1961:
1928:
1900:
1374:
1346:
1247:
1219:
1131:
1119:
1095:
1081:
973:
959:
512:(PDE) the first of which is a
510:partial differential equations
442:{\displaystyle I_{\text{ion}}}
259:, the extracellular potential
1:
5304:10.1016/S0006-3495(87)83381-7
5247:10.1016/S0006-3495(89)82897-8
4954:10.1016/S0006-3495(85)83800-5
4907:10.1016/s0092-8240(79)80032-4
4870:10.1016/s0092-8240(79)80031-2
3770:Reduction to monodomain model
3295:{\displaystyle v=v_{i}-v_{e}}
2892:{\displaystyle v=v_{i}-v_{e}}
2424:{\displaystyle v=v_{i}-v_{e}}
348:{\displaystyle v=v_{i}-v_{e}}
3248:{\displaystyle \mathbf {n} }
2853:on the left and rearranging
1602:{\displaystyle \mathbf {E} }
1521:{\displaystyle \mathbf {E} }
1001:{\displaystyle \mathbf {w} }
161:{\displaystyle \mathbb {T} }
139:{\displaystyle \mathbb {H} }
71:is about 10:1, while in the
3151:
2669:
2663:
514:reaction diffusion equation
5590:
2208:from both sides. In fact,
120:which are separate by the
5549:Cardiac electrophysiology
5404:10.1007/s10439-009-9873-0
4670:10.1007/s00791-003-0101-4
4612:finite difference schemes
3132:{\displaystyle I_{s_{2}}}
3098:{\displaystyle I_{s_{1}}}
906:{\displaystyle I_{s_{2}}}
872:{\displaystyle I_{s_{1}}}
3834:{\displaystyle \lambda }
1499:{\displaystyle \varphi }
4834:10.1161/01.res.43.2.301
4691:Schmitt, O. H. (1969).
518:transmembrane potential
288:transmembrane potential
228:Unknowns and parameters
85:transmembrane potential
64:electrical conductivity
5559:Differential equations
5460:10.1098/rsta.2011.0139
4616:finite element schemes
4597:
4041:
4009:
3980:
3958:
3835:
3815:
3761:
3616:
3555:
3410:
3296:
3249:
3227:
3133:
3099:
3058:
2893:
2847:
2793:
2639:
2538:
2425:
2379:
2202:
2142:
2023:
1870:
1796:
1765:
1745:
1725:
1603:
1581:
1559:
1522:
1500:
1450:
1422:
1330:
1301:
1274:
1172:
1028:
1027:{\displaystyle t>0}
1002:
980:
917:Ionic current equation
907:
873:
839:
490:
470:
443:
414:
382:
349:
303:
280:
253:
219:
191:
162:
140:
116:
5569:Mathematical modeling
4620:finite volume schemes
4598:
4042:
4010:
3981:
3959:
3836:
3816:
3762:
3617:
3556:
3411:
3297:
3250:
3228:
3134:
3100:
3059:
2894:
2848:
2794:
2661:Combining equations (
2640:
2539:
2426:
2380:
2203:
2143:
2024:
1871:
1797:
1795:{\displaystyle I_{m}}
1766:
1746:
1726:
1604:
1582:
1560:
1523:
1501:
1479:extracellular space.
1451:
1423:
1331:
1302:
1300:{\displaystyle v_{0}}
1275:
1173:
1029:
1003:
981:
908:
874:
840:
491:
489:{\displaystyle \chi }
471:
469:{\displaystyle C_{m}}
444:
415:
383:
350:
304:
281:
279:{\displaystyle v_{e}}
254:
252:{\displaystyle v_{i}}
220:
192:
163:
141:
114:
5081:. World Scientific.
4821:Circulation Research
4058:
4019:
3990:
3986:is a combination of
3968:
3848:
3825:
3778:
3628:
3566:
3422:
3306:
3260:
3237:
3158:
3109:
3075:
3069:standard formulation
2903:
2857:
2803:
2677:
2560:
2434:
2389:
2212:
2158:
2052:
1887:
1806:
1779:
1755:
1735:
1731:where the subscript
1613:
1591:
1571:
1532:
1510:
1490:
1431:
1343:
1311:
1284:
1207:
1038:
1012:
990:
933:
925:through a system of
883:
849:
527:
504:Standard formulation
480:
453:
426:
395:
363:
313:
293:
263:
236:
201:
176:
150:
128:
83:the distribution of
5452:2011RSPTA.369.4331N
5446:(1954): 4331–4351.
5296:1987BpJ....51..557S
5284:Biophysical Journal
5239:1989BpJ....55..987S
5226:Biophysical Journal
4946:1985BpJ....48..449E
4934:Biophysical Journal
3843:monodomain equation
3143:Boundary conditions
1528:, which means that
1473:Maxwell's equations
1034:, the system reads
73:extracellular space
69:intracellular space
5574:Numerical analysis
5161:10.1007/BF00948895
4606:Numerical solution
4593:
4588:
4188:
4037:
4005:
3976:
3954:
3831:
3811:
3757:
3612:
3551:
3406:
3292:
3245:
3223:
3129:
3095:
3054:
2889:
2843:
2789:
2635:
2534:
2421:
2375:
2198:
2138:
2019:
2017:
1866:
1792:
1761:
1741:
1721:
1719:
1599:
1577:
1555:
1518:
1496:
1446:
1418:
1326:
1297:
1270:
1168:
1163:
1071:
1024:
998:
976:
903:
869:
835:
833:
486:
466:
439:
410:
378:
345:
299:
276:
249:
215:
187:
158:
136:
117:
57:bidomain equations
22:mathematical model
5554:Electrophysiology
5351:10.1109/10.245611
5116:10.1109/10.563303
4702:978-3-642-87086-6
4187:
3913:
3023:
2760:
2659:
2658:
2624:
2612:
2043:
2042:
1764:{\displaystyle e}
1744:{\displaystyle i}
1580:{\displaystyle J}
1154:
1103:
1070:
956:
943:
652:
436:
302:{\displaystyle v}
122:cellular membrane
30:three-dimensional
5581:
5521:
5520:
5503:(2–3): 136–155.
5488:
5482:
5481:
5471:
5430:
5424:
5423:
5398:(3): 1071–1097.
5389:
5380:
5371:
5370:
5332:
5326:
5325:
5315:
5275:
5269:
5268:
5258:
5216:
5210:
5209:
5187:
5181:
5180:
5142:
5136:
5135:
5099:
5093:
5092:
5074:
5003:
5002:
4982:
4976:
4975:
4965:
4925:
4919:
4918:
4888:
4882:
4881:
4853:
4847:
4846:
4836:
4812:
4806:
4805:
4797:
4788:
4787:
4767:
4761:
4760:
4740:
4734:
4733:
4713:
4707:
4706:
4688:
4682:
4681:
4653:
4632:Monodomain model
4602:
4600:
4599:
4594:
4592:
4591:
4585:
4574:
4561:
4553:
4549:
4548:
4547:
4535:
4534:
4529:
4517:
4516:
4492:
4491:
4486:
4471:
4460:
4447:
4436:
4435:
4411:
4410:
4405:
4395:
4387:
4380:
4379:
4378:
4377:
4360:
4356:
4349:
4348:
4343:
4320:
4316:
4315:
4314:
4302:
4298:
4297:
4296:
4291:
4282:
4281:
4276:
4250:
4242:
4235:
4234:
4233:
4232:
4215:
4211:
4210:
4209:
4208:
4189:
4186:
4178:
4170:
4167:
4166:
4146:
4142:
4141:
4140:
4128:
4127:
4122:
4102:
4098:
4091:
4090:
4085:
4046:
4044:
4043:
4038:
4033:
4032:
4027:
4014:
4012:
4011:
4006:
4004:
4003:
3998:
3985:
3983:
3982:
3977:
3975:
3963:
3961:
3960:
3955:
3953:
3952:
3940:
3936:
3935:
3934:
3933:
3914:
3912:
3904:
3896:
3894:
3893:
3864:
3840:
3838:
3837:
3832:
3821:for some scalar
3820:
3818:
3817:
3812:
3810:
3809:
3804:
3792:
3791:
3786:
3766:
3764:
3763:
3758:
3753:
3742:
3736:
3728:
3724:
3723:
3722:
3707:
3706:
3701:
3692:
3691:
3686:
3666:
3658:
3654:
3647:
3646:
3641:
3621:
3619:
3618:
3613:
3608:
3597:
3591:
3590:
3578:
3577:
3560:
3558:
3557:
3552:
3547:
3536:
3530:
3529:
3524:
3515:
3511:
3510:
3509:
3497:
3496:
3491:
3474:
3473:
3468:
3459:
3455:
3454:
3453:
3441:
3440:
3435:
3415:
3413:
3412:
3407:
3402:
3391:
3385:
3374:
3373:
3361:
3360:
3355:
3340:
3323:
3322:
3317:
3301:
3299:
3298:
3293:
3291:
3290:
3278:
3277:
3254:
3252:
3251:
3246:
3244:
3232:
3230:
3229:
3224:
3222:
3211:
3199:
3188:
3187:
3175:
3174:
3169:
3138:
3136:
3135:
3130:
3128:
3127:
3126:
3125:
3104:
3102:
3101:
3096:
3094:
3093:
3092:
3091:
3063:
3061:
3060:
3055:
3050:
3046:
3045:
3044:
3043:
3024:
3022:
3014:
3006:
3004:
3003:
2983:
2979:
2978:
2977:
2965:
2964:
2959:
2939:
2935:
2928:
2927:
2922:
2898:
2896:
2895:
2890:
2888:
2887:
2875:
2874:
2852:
2850:
2849:
2844:
2839:
2838:
2826:
2825:
2820:
2798:
2796:
2795:
2790:
2785:
2781:
2780:
2779:
2761:
2759:
2751:
2743:
2741:
2740:
2720:
2716:
2715:
2714:
2702:
2701:
2696:
2653:
2644:
2642:
2641:
2636:
2631:
2627:
2626:
2625:
2622:
2613:
2611:
2603:
2595:
2593:
2592:
2572:
2571:
2554:
2543:
2541:
2540:
2535:
2527:
2526:
2511:
2510:
2505:
2496:
2495:
2490:
2457:
2456:
2451:
2430:
2428:
2427:
2422:
2420:
2419:
2407:
2406:
2384:
2382:
2381:
2376:
2371:
2370:
2358:
2357:
2352:
2331:
2330:
2318:
2317:
2312:
2288:
2287:
2275:
2274:
2269:
2248:
2247:
2235:
2234:
2229:
2207:
2205:
2204:
2199:
2194:
2193:
2181:
2180:
2175:
2147:
2145:
2144:
2139:
2131:
2130:
2118:
2117:
2112:
2088:
2087:
2075:
2074:
2069:
2037:
2028:
2026:
2025:
2020:
2018:
2014:
2013:
1988:
1987:
1975:
1974:
1969:
1950:
1949:
1927:
1926:
1914:
1913:
1908:
1881:
1875:
1873:
1872:
1867:
1862:
1861:
1846:
1845:
1827:
1826:
1801:
1799:
1798:
1793:
1791:
1790:
1770:
1768:
1767:
1762:
1750:
1748:
1747:
1742:
1730:
1728:
1727:
1722:
1720:
1713:
1712:
1700:
1699:
1694:
1678:
1677:
1664:
1663:
1651:
1650:
1645:
1629:
1628:
1608:
1606:
1605:
1600:
1598:
1586:
1584:
1583:
1578:
1564:
1562:
1561:
1556:
1539:
1527:
1525:
1524:
1519:
1517:
1505:
1503:
1502:
1497:
1455:
1453:
1452:
1447:
1445:
1444:
1439:
1427:
1425:
1424:
1419:
1414:
1403:
1391:
1390:
1385:
1373:
1372:
1360:
1359:
1354:
1335:
1333:
1332:
1327:
1325:
1324:
1319:
1306:
1304:
1303:
1298:
1296:
1295:
1279:
1277:
1276:
1271:
1269:
1261:
1246:
1245:
1233:
1232:
1227:
1201:Laplace equation
1177:
1175:
1174:
1169:
1167:
1166:
1160:
1155:
1152:
1148:
1147:
1142:
1118:
1109:
1104:
1101:
1094:
1080:
1072:
1069:
1061:
1060:
1051:
1033:
1031:
1030:
1025:
1007:
1005:
1004:
999:
997:
985:
983:
982:
977:
972:
958:
957:
954:
945:
944:
941:
912:
910:
909:
904:
902:
901:
900:
899:
878:
876:
875:
870:
868:
867:
866:
865:
844:
842:
841:
836:
834:
830:
829:
828:
827:
810:
806:
799:
798:
793:
770:
766:
765:
764:
752:
748:
747:
746:
741:
732:
731:
726:
703:
699:
698:
697:
696:
679:
675:
674:
673:
672:
653:
651:
643:
635:
633:
632:
612:
608:
607:
606:
594:
593:
588:
568:
564:
557:
556:
551:
533:
516:in terms of the
495:
493:
492:
487:
475:
473:
472:
467:
465:
464:
448:
446:
445:
440:
438:
437:
434:
421:
419:
417:
416:
411:
409:
408:
403:
389:
387:
385:
384:
379:
377:
376:
371:
354:
352:
351:
346:
344:
343:
331:
330:
308:
306:
305:
300:
285:
283:
282:
277:
275:
274:
258:
256:
255:
250:
248:
247:
224:
222:
221:
216:
211:
196:
194:
193:
188:
186:
167:
165:
164:
159:
157:
145:
143:
142:
137:
135:
5589:
5588:
5584:
5583:
5582:
5580:
5579:
5578:
5539:
5538:
5530:
5525:
5524:
5490:
5489:
5485:
5434:October 2011).
5432:
5431:
5427:
5387:
5382:
5381:
5374:
5334:
5333:
5329:
5277:
5276:
5272:
5218:
5217:
5213:
5189:
5188:
5184:
5144:
5143:
5139:
5101:
5100:
5096:
5089:
5076:
5075:
5006:
4984:
4983:
4979:
4927:
4926:
4922:
4890:
4889:
4885:
4855:
4854:
4850:
4814:
4813:
4809:
4799:
4798:
4791:
4769:
4768:
4764:
4742:
4741:
4737:
4715:
4714:
4710:
4703:
4690:
4689:
4685:
4655:
4654:
4650:
4645:
4628:
4608:
4587:
4586:
4568:
4539:
4524:
4508:
4481:
4480:
4476:
4473:
4472:
4454:
4427:
4400:
4397:
4396:
4381:
4369:
4364:
4338:
4337:
4333:
4306:
4286:
4271:
4270:
4266:
4265:
4261:
4252:
4251:
4236:
4224:
4219:
4193:
4179:
4171:
4158:
4157:
4153:
4132:
4117:
4116:
4112:
4080:
4079:
4075:
4062:
4056:
4055:
4052:
4022:
4017:
4016:
3993:
3988:
3987:
3966:
3965:
3944:
3918:
3905:
3897:
3885:
3884:
3880:
3846:
3845:
3823:
3822:
3799:
3781:
3776:
3775:
3772:
3714:
3696:
3681:
3677:
3673:
3636:
3635:
3631:
3626:
3625:
3582:
3569:
3564:
3563:
3519:
3501:
3486:
3485:
3481:
3463:
3445:
3430:
3429:
3425:
3420:
3419:
3365:
3350:
3312:
3304:
3303:
3282:
3269:
3258:
3257:
3235:
3234:
3179:
3164:
3156:
3155:
3145:
3117:
3112:
3107:
3106:
3083:
3078:
3073:
3072:
3028:
3015:
3007:
2995:
2994:
2990:
2969:
2954:
2953:
2949:
2917:
2916:
2912:
2901:
2900:
2879:
2866:
2855:
2854:
2830:
2815:
2801:
2800:
2765:
2752:
2744:
2732:
2731:
2727:
2706:
2691:
2690:
2686:
2675:
2674:
2651:
2617:
2604:
2596:
2584:
2583:
2579:
2563:
2558:
2557:
2518:
2500:
2485:
2446:
2432:
2431:
2411:
2398:
2387:
2386:
2362:
2347:
2322:
2307:
2279:
2264:
2239:
2224:
2210:
2209:
2185:
2170:
2156:
2155:
2122:
2107:
2079:
2064:
2050:
2049:
2035:
2016:
2015:
2005:
1992:
1979:
1964:
1952:
1951:
1941:
1931:
1918:
1903:
1885:
1884:
1853:
1837:
1818:
1804:
1803:
1782:
1777:
1776:
1753:
1752:
1733:
1732:
1718:
1717:
1704:
1689:
1679:
1669:
1666:
1665:
1655:
1640:
1630:
1620:
1611:
1610:
1589:
1588:
1569:
1568:
1530:
1529:
1508:
1507:
1488:
1487:
1469:
1434:
1429:
1428:
1380:
1364:
1349:
1341:
1340:
1314:
1309:
1308:
1287:
1282:
1281:
1237:
1222:
1205:
1204:
1194:
1162:
1161:
1149:
1137:
1111:
1110:
1098:
1062:
1052:
1042:
1036:
1035:
1010:
1009:
988:
987:
949:
936:
931:
930:
919:
891:
886:
881:
880:
857:
852:
847:
846:
832:
831:
819:
814:
788:
787:
783:
756:
736:
721:
720:
716:
715:
711:
701:
700:
688:
683:
657:
644:
636:
624:
623:
619:
598:
583:
582:
578:
546:
545:
541:
525:
524:
506:
478:
477:
456:
451:
450:
429:
424:
423:
398:
393:
392:
391:
366:
361:
360:
359:
335:
322:
311:
310:
291:
290:
266:
261:
260:
239:
234:
233:
230:
199:
198:
174:
173:
148:
147:
126:
125:
109:
107:Bidomain domain
104:
12:
11:
5:
5587:
5585:
5577:
5576:
5571:
5566:
5561:
5556:
5551:
5541:
5540:
5537:
5536:
5529:
5528:External links
5526:
5523:
5522:
5483:
5425:
5372:
5345:(9): 899–908.
5327:
5290:(4): 557–568.
5270:
5233:(5): 987–999.
5211:
5182:
5155:(6): 633–646.
5137:
5110:(4): 326–328.
5094:
5088:978-9812563736
5087:
5004:
4993:(2): 137–199.
4977:
4940:(3): 449–460.
4920:
4901:(2): 183–192.
4883:
4864:(2): 163–181.
4848:
4827:(2): 301–315.
4807:
4789:
4778:(4): 671–675.
4762:
4751:(3): 518–522.
4735:
4724:(2): 307–312.
4708:
4701:
4683:
4664:(4): 215–239.
4647:
4646:
4644:
4641:
4640:
4639:
4634:
4627:
4624:
4607:
4604:
4590:
4584:
4580:
4577:
4573:
4569:
4567:
4564:
4560:
4556:
4552:
4546:
4542:
4538:
4533:
4528:
4523:
4520:
4515:
4511:
4507:
4504:
4501:
4498:
4495:
4490:
4485:
4479:
4475:
4474:
4470:
4466:
4463:
4459:
4455:
4453:
4450:
4446:
4442:
4439:
4434:
4430:
4426:
4423:
4420:
4417:
4414:
4409:
4404:
4399:
4398:
4394:
4390:
4386:
4382:
4376:
4372:
4367:
4363:
4359:
4355:
4352:
4347:
4342:
4336:
4332:
4329:
4326:
4323:
4319:
4313:
4309:
4305:
4301:
4295:
4290:
4285:
4280:
4275:
4269:
4264:
4260:
4257:
4254:
4253:
4249:
4245:
4241:
4237:
4231:
4227:
4222:
4218:
4214:
4207:
4204:
4201:
4196:
4192:
4185:
4182:
4177:
4174:
4165:
4161:
4156:
4152:
4149:
4145:
4139:
4135:
4131:
4126:
4121:
4115:
4111:
4108:
4105:
4101:
4097:
4094:
4089:
4084:
4078:
4074:
4071:
4068:
4067:
4065:
4051:
4048:
4036:
4031:
4026:
4002:
3997:
3974:
3951:
3947:
3943:
3939:
3932:
3929:
3926:
3921:
3917:
3911:
3908:
3903:
3900:
3892:
3888:
3883:
3879:
3876:
3873:
3870:
3867:
3863:
3859:
3856:
3853:
3830:
3808:
3803:
3798:
3795:
3790:
3785:
3771:
3768:
3756:
3752:
3748:
3745:
3741:
3735:
3731:
3727:
3721:
3717:
3713:
3710:
3705:
3700:
3695:
3690:
3685:
3680:
3676:
3672:
3669:
3665:
3661:
3657:
3653:
3650:
3645:
3640:
3634:
3611:
3607:
3603:
3600:
3596:
3589:
3585:
3581:
3576:
3572:
3550:
3546:
3542:
3539:
3535:
3528:
3523:
3518:
3514:
3508:
3504:
3500:
3495:
3490:
3484:
3480:
3477:
3472:
3467:
3462:
3458:
3452:
3448:
3444:
3439:
3434:
3428:
3405:
3401:
3397:
3394:
3390:
3384:
3380:
3377:
3372:
3368:
3364:
3359:
3354:
3349:
3346:
3343:
3339:
3335:
3332:
3329:
3326:
3321:
3316:
3311:
3289:
3285:
3281:
3276:
3272:
3268:
3265:
3243:
3221:
3217:
3214:
3210:
3205:
3202:
3198:
3194:
3191:
3186:
3182:
3178:
3173:
3168:
3163:
3144:
3141:
3124:
3120:
3115:
3090:
3086:
3081:
3053:
3049:
3042:
3039:
3036:
3031:
3027:
3021:
3018:
3013:
3010:
3002:
2998:
2993:
2989:
2986:
2982:
2976:
2972:
2968:
2963:
2958:
2952:
2948:
2945:
2942:
2938:
2934:
2931:
2926:
2921:
2915:
2911:
2908:
2886:
2882:
2878:
2873:
2869:
2865:
2862:
2842:
2837:
2833:
2829:
2824:
2819:
2814:
2811:
2808:
2788:
2784:
2778:
2775:
2772:
2768:
2764:
2758:
2755:
2750:
2747:
2739:
2735:
2730:
2726:
2723:
2719:
2713:
2709:
2705:
2700:
2695:
2689:
2685:
2682:
2657:
2656:
2647:
2645:
2634:
2630:
2620:
2616:
2610:
2607:
2602:
2599:
2591:
2587:
2582:
2578:
2575:
2570:
2566:
2549:cable equation
2533:
2530:
2525:
2521:
2517:
2514:
2509:
2504:
2499:
2494:
2489:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2455:
2450:
2445:
2442:
2439:
2418:
2414:
2410:
2405:
2401:
2397:
2394:
2374:
2369:
2365:
2361:
2356:
2351:
2346:
2343:
2340:
2337:
2334:
2329:
2325:
2321:
2316:
2311:
2306:
2303:
2300:
2297:
2294:
2291:
2286:
2282:
2278:
2273:
2268:
2263:
2260:
2257:
2254:
2251:
2246:
2242:
2238:
2233:
2228:
2223:
2220:
2217:
2197:
2192:
2188:
2184:
2179:
2174:
2169:
2166:
2163:
2149:
2148:
2137:
2134:
2129:
2125:
2121:
2116:
2111:
2106:
2103:
2100:
2097:
2094:
2091:
2086:
2082:
2078:
2073:
2068:
2063:
2060:
2057:
2041:
2040:
2031:
2029:
2012:
2008:
2004:
2001:
1998:
1995:
1993:
1991:
1986:
1982:
1978:
1973:
1968:
1963:
1960:
1957:
1954:
1953:
1948:
1944:
1940:
1937:
1934:
1932:
1930:
1925:
1921:
1917:
1912:
1907:
1902:
1899:
1896:
1893:
1892:
1865:
1860:
1856:
1852:
1849:
1844:
1840:
1836:
1833:
1830:
1825:
1821:
1817:
1814:
1811:
1789:
1785:
1760:
1740:
1716:
1711:
1707:
1703:
1698:
1693:
1688:
1685:
1682:
1680:
1676:
1672:
1668:
1667:
1662:
1658:
1654:
1649:
1644:
1639:
1636:
1633:
1631:
1627:
1623:
1619:
1618:
1597:
1576:
1554:
1551:
1548:
1545:
1542:
1538:
1516:
1495:
1468:
1465:
1443:
1438:
1417:
1413:
1409:
1406:
1402:
1397:
1394:
1389:
1384:
1379:
1376:
1371:
1367:
1363:
1358:
1353:
1348:
1323:
1318:
1294:
1290:
1268:
1264:
1260:
1255:
1252:
1249:
1244:
1240:
1236:
1231:
1226:
1221:
1218:
1215:
1212:
1193:
1190:
1189:
1188:
1185:
1165:
1159:
1150:
1146:
1141:
1136:
1133:
1130:
1127:
1124:
1121:
1117:
1113:
1112:
1108:
1099:
1097:
1093:
1089:
1086:
1083:
1079:
1075:
1068:
1065:
1059:
1055:
1048:
1047:
1045:
1023:
1020:
1017:
996:
975:
971:
967:
964:
961:
952:
948:
939:
918:
915:
898:
894:
889:
864:
860:
855:
826:
822:
817:
813:
809:
805:
802:
797:
792:
786:
782:
779:
776:
773:
769:
763:
759:
755:
751:
745:
740:
735:
730:
725:
719:
714:
710:
707:
704:
702:
695:
691:
686:
682:
678:
671:
668:
665:
660:
656:
650:
647:
642:
639:
631:
627:
622:
618:
615:
611:
605:
601:
597:
592:
587:
581:
577:
574:
571:
567:
563:
560:
555:
550:
544:
540:
537:
534:
532:
505:
502:
485:
463:
459:
432:
407:
402:
375:
370:
342:
338:
334:
329:
325:
321:
318:
298:
273:
269:
246:
242:
229:
226:
214:
210:
206:
185:
181:
156:
134:
108:
105:
103:
100:
99:
98:
95:
92:magnetic field
88:
18:bidomain model
13:
10:
9:
6:
4:
3:
2:
5586:
5575:
5572:
5570:
5567:
5565:
5562:
5560:
5557:
5555:
5552:
5550:
5547:
5546:
5544:
5535:
5532:
5531:
5527:
5518:
5514:
5510:
5506:
5502:
5498:
5494:
5487:
5484:
5479:
5475:
5470:
5465:
5461:
5457:
5453:
5449:
5445:
5441:
5437:
5429:
5426:
5421:
5417:
5413:
5409:
5405:
5401:
5397:
5393:
5386:
5379:
5377:
5373:
5368:
5364:
5360:
5356:
5352:
5348:
5344:
5340:
5339:
5331:
5328:
5323:
5319:
5314:
5309:
5305:
5301:
5297:
5293:
5289:
5285:
5281:
5274:
5271:
5266:
5262:
5257:
5252:
5248:
5244:
5240:
5236:
5232:
5228:
5227:
5222:
5215:
5212:
5207:
5203:
5199:
5195:
5194:
5186:
5183:
5178:
5174:
5170:
5166:
5162:
5158:
5154:
5150:
5149:
5141:
5138:
5133:
5129:
5125:
5121:
5117:
5113:
5109:
5105:
5098:
5095:
5090:
5084:
5080:
5073:
5071:
5069:
5067:
5065:
5063:
5061:
5059:
5057:
5055:
5053:
5051:
5049:
5047:
5045:
5043:
5041:
5039:
5037:
5035:
5033:
5031:
5029:
5027:
5025:
5023:
5021:
5019:
5017:
5015:
5013:
5011:
5009:
5005:
5000:
4996:
4992:
4988:
4981:
4978:
4973:
4969:
4964:
4959:
4955:
4951:
4947:
4943:
4939:
4935:
4931:
4924:
4921:
4916:
4912:
4908:
4904:
4900:
4896:
4895:
4887:
4884:
4879:
4875:
4871:
4867:
4863:
4859:
4852:
4849:
4844:
4840:
4835:
4830:
4826:
4822:
4818:
4811:
4808:
4803:
4796:
4794:
4790:
4785:
4781:
4777:
4773:
4766:
4763:
4758:
4754:
4750:
4746:
4739:
4736:
4731:
4727:
4723:
4719:
4712:
4709:
4704:
4698:
4694:
4687:
4684:
4679:
4675:
4671:
4667:
4663:
4659:
4652:
4649:
4642:
4638:
4635:
4633:
4630:
4629:
4625:
4623:
4621:
4617:
4613:
4605:
4603:
4575:
4565:
4562:
4554:
4550:
4544:
4540:
4531:
4521:
4513:
4509:
4502:
4499:
4488:
4477:
4461:
4451:
4448:
4440:
4432:
4428:
4421:
4418:
4407:
4388:
4374:
4370:
4365:
4361:
4357:
4353:
4345:
4334:
4330:
4324:
4321:
4317:
4311:
4307:
4299:
4293:
4283:
4278:
4267:
4262:
4258:
4243:
4229:
4225:
4220:
4216:
4212:
4194:
4190:
4183:
4175:
4163:
4159:
4154:
4150:
4147:
4143:
4137:
4133:
4124:
4113:
4109:
4103:
4099:
4095:
4087:
4076:
4072:
4063:
4049:
4047:
4034:
4029:
4000:
3949:
3945:
3941:
3937:
3919:
3915:
3909:
3901:
3890:
3886:
3881:
3877:
3874:
3868:
3854:
3844:
3828:
3806:
3796:
3793:
3788:
3769:
3767:
3754:
3743:
3729:
3725:
3719:
3715:
3703:
3693:
3688:
3674:
3670:
3667:
3659:
3655:
3651:
3643:
3632:
3622:
3609:
3598:
3587:
3583:
3579:
3574:
3570:
3548:
3537:
3526:
3516:
3512:
3506:
3502:
3493:
3482:
3478:
3475:
3470:
3460:
3456:
3450:
3446:
3437:
3426:
3416:
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296:
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227:
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171:
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101:
96:
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54:
48:
46:
41:
39:
38:extracellular
35:
34:intracellular
31:
27:
23:
19:
5500:
5496:
5486:
5443:
5439:
5428:
5395:
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5224:
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231:
118:
78:
61:
56:
53:cable theory
49:
42:
17:
15:
5200:(1): 1–77.
923:ionic model
170:human torso
102:Formulation
5543:Categories
4643:References
1467:Derivation
4772:Biofizika
4745:Biofizika
4718:Biofizika
4678:123211416
4618:and also
4579:∂
4576:∈
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180:∂
5517:20553747
5478:21969679
5420:10114284
5412:20033779
5132:24225323
4626:See also
2673:) gives
1203:of type
1153:in
1102:in
286:and the
5469:3263775
5448:Bibcode
5367:7593406
5359:8288281
5322:3580484
5313:1329928
5292:Bibcode
5265:2720084
5256:1330535
5235:Bibcode
5206:8365198
5169:1640183
5124:9125816
4999:8243090
4972:4041538
4963:1329358
4942:Bibcode
2667:) and (
498:section
45:Schmitt
5515:
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4676:
3233:where
3152:derive
1482:Using
1280:where
986:where
845:where
5416:S2CID
5388:(PDF)
5363:S2CID
5173:S2CID
5128:S2CID
4674:S2CID
26:heart
20:is a
5513:PMID
5474:PMID
5408:PMID
5355:PMID
5318:PMID
5261:PMID
5202:PMID
5165:PMID
5120:PMID
5083:ISBN
4995:PMID
4968:PMID
4911:PMID
4874:PMID
4839:PMID
4780:PMID
4753:PMID
4726:PMID
4697:ISBN
4015:and
3105:and
1751:and
1019:>
879:and
90:the
36:and
16:The
5505:doi
5501:102
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5308:PMC
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5251:PMC
5243:doi
5157:doi
5112:doi
4958:PMC
4950:doi
4903:doi
4866:doi
4829:doi
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2623:ion
955:ion
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320:=
317:v
297:v
272:e
268:v
245:i
241:v
213:.
209:T
184:H
155:T
133:H
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