Bidomain model

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The bidomain model is a mathematical model for the electrical properties of cardiac muscle that takes into account the anisotropy of both the intracellular and extracellular spaces. It is formed of the bidomain equations.

The bidomain model was developed in the late 1970s. [1] [2] [3] [4] [5] [6] [7] [8] It is a generalization of one-dimensional cable theory. The bidomain model is a continuum model, meaning that it represents the average properties of many cells, rather than describing each cell individually. [9]

Many of the interesting properties of the bidomain model arise from the condition of unequal anisotropy ratios. The electrical conductivity in anisotropic tissue is different parallel and perpendicular to the fiber direction. In a tissue with unequal anisotropy ratios, the ratio of conductivities parallel and perpendicular to the fibers is different in the intracellular and extracellular spaces. For instance, in cardiac tissue, the anisotropy ratio in the intracellular space is about 10:1, while in the extracellular space it is about 5:2. [10] Mathematically, unequal anisotropy ratios means that the effect of anisotropy cannot be removed by a change in the distance scale in one direction. [11] Instead, the anisotropy has a more profound influence on the electrical behavior. [12]

Three examples of the impact of unequal anisotropy ratios are

  • the distribution of transmembrane potential during unipolar stimulation of a sheet of cardiac tissue,[13]
  • the magnetic field produced by an action potential wave front propagating through cardiac tissue,[14]
  • the effect of fiber curvature on the transmembrane potential distribution during an electric shock.[15]

The bidomain model is now widely used to model defibrillation of the heart.

Formulation[edit]

Standard formulation[edit]

The bidomain model can be formulated as follows:

where is the membrane surface area per unit volume (of tissue), is the electrical capacitance of the membrane per unit area, where is the interstitial voltage and is the extracellular voltage, and is the ionic current over the membrane per unit area.

Formulation with boundary conditions and surrounding tissue[edit]

The surrounding tissue can be included to give reasonable boundary conditions to make the system solvable:

Derivation[edit]

Let with boundary be the set of all points in the heart. In each point in there is an intra- and extracellular voltage and current, denoted by , , and respectively. Let and be the intra- end extracellular conductivity tensor matrices respectively.

We assume Ohmic current-voltage relationship and get

We require that there is no accumulation of charge anywhere in , and therefore that

giving one of the model equations:

 

 

 

 

(1)

This equation states that all current exiting one domain must enter the other.

The transmembrane current is given by

 

 

 

 

(2)

We model the membrane similarly to that of the cable equation,

 

 

 

 

(3)

where is the membrane surface area per unit volume (of tissue), is the electrical capacitance of the membrane per unit area, and is the ionic current over the membrane per unit area.

Combining equations (2) and (3) gives

which can be rearranged using to get another model equation:

 

 

 

 

(4)

Boundary conditions[edit]

In order to solve the model, boundary conditions are needed. One way to define the boundary condition is to extend the model with a volume with perimeter that surrounds the heart and represent the body tissue.

As was the case for , we assume no accumulation of charge in , i.e.

 

 

 

 

(5)

where is the conductance tensor of the body tissue and is the voltage in .

Assuming that the body is electrically surrounded from the environment, there can be no current component on the surface in the normal direction, hence:

 

 

 

 

(6)

On the surface of the heart, a common assumption is that there is a direct connection between the surrounding tissue and the extracellular domain. This means that the potentials and must be equal on the heart surface, i.e.

 

 

 

 

(7)

This direct connection also require that all ionic current exiting on the heart surface, must enter the extracellular domain, and vica versa. This gives another boundary condition:

 

 

 

 

(8)

Finally, we assume that there is a complete isolation of the intracellular domain and the surrounding tissue. Similarly to equation (2), we get

which can be rewritten using to

 

 

 

 

(9)

Extending the model to include equations (5)-(9) gives a solvable system of equations.

Reduction to monodomain model[edit]

By assuming equal anisotropy ratios for the intra- and extracellular domains, i.e. for some scalar , the model can be reduced to the monodomain model.

Numerical solution[edit]

There are some special considerations for numerical solution of these equations, due to high time and space resolution needed for numerical convergence.[16] [17]

References[edit]

  1. ^ Muler AL, Markin VS (1977). "Electrical properties of anisotropic nerve-muscle syncytia-I. Distribution of the electrotonic potential.". Biofizika. 22 (2): 307–312. PMID 861269. 
  2. ^ Muler AL, Markin VS (1977). "Electrical properties of anisotropic nerve-muscle syncytia-II. Spread of flat front of excitation.". Biofizika. 22 (3): 518–522. PMID 889914. 
  3. ^ Muler AL, Markin VS (1977). "Electrical properties of anisotropic nerve-muscle syncytia-III. Steady form of the excitation front.". Biofizika. 22 (4): 671–675. PMID 901827. 
  4. ^ Tung L (1978). "A bi-domain model for describing ischemic myocardial d-c potentials.". PhD dissertation, MIT, Cambridge, Mass. 
  5. ^ Miller WT III; Geselowitz DB (1978). "Simulation studies of the electrocardiogram, I. The normal heart.". Circulation Research. 43 (2): 301–315. doi:10.1161/01.res.43.2.301. PMID 668061. 
  6. ^ Peskoff A (1979). "Electric potential in three-dimensional electrically syncytial tissues.". Bulletin of Mathematical Biology. 41 (2): 163–181. doi:10.1016/s0092-8240(79)80031-2. PMID 760880. 
  7. ^ Peskoff A (1979). "Electric potential in cylindrical syncytia and muscle fibers.". Bulletin of Mathematical Biology. 41 (2): 183–192. doi:10.1016/s0092-8240(79)80032-4. PMID 760881. 
  8. ^ Eisenberg RS, Barcilon V, Mathias RT (1979). "Electrical properties of spherical syncytia.". Biophysical Journal. 48 (3): 449–460. Bibcode:1985BpJ....48..449E. doi:10.1016/S0006-3495(85)83800-5. PMC 1329358Freely accessible. PMID 4041538. 
  9. ^ Neu JC, Krassowska W (1993). "Homogenization of syncytial tissues.". Critical Reviews of Biomedical Engineering. 21: 137–199. 
  10. ^ Roth BJ (1997). "Electrical conductivity values used with the bidomain model of cardiac tissue.". IEEE Transactions on Biomedical Engineering. 44 (4): 326–328. doi:10.1109/10.563303. PMID 9125816. 
  11. ^ Roth BJ (1992). "How the anisotropy of the intracellular and extracellular conductivities influences stimulation of cardiac muscle.". Journal of Mathematical Biology. 30 (6): 633–646. doi:10.1007/BF00948895. PMID 1640183. 
  12. ^ Henriquez CS (1993). "Simulating the electrical behavior of cardiac tissue using the bidomain model.". Critical Reviews of Biomedical Engineering. 21: 1–77. 
  13. ^ Sepulveda NG, Roth BJ, Wikswo JP (1989). "Current injection into a two-dimensional bidomain.". Biophysical Journal. 55 (5): 987–999. Bibcode:1989BpJ....55..987S. doi:10.1016/S0006-3495(89)82897-8. PMC 1330535Freely accessible. PMID 2720084. 
  14. ^ Sepulveda NG, Wikswo JP (1987). "Electric and magnetic fields from two-dimensional bisyncytia.". Biophysical Journal. 51 (4): 557–568. Bibcode:1987BpJ....51..557S. doi:10.1016/S0006-3495(87)83381-7. PMC 1329928Freely accessible. PMID 3580484. 
  15. ^ Trayanova N, Roth BJ, Malden LJ (1993). "The response of a spherical heart to a uniform electric field: A bidomain analysis of cardiac stimulation.". IEEE Transactions on Biomedical Engineering. 40 (9): 899–908. doi:10.1109/10.245611. PMID 8288281. 
  16. ^ 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 October 2011). "Verification of cardiac tissue electrophysiology simulators using an N-version benchmark". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 369 (1954): 4331–4351. Bibcode:2011RSPTA.369.4331N. doi:10.1098/rsta.2011.0139. 
  17. ^ Pathmanathan, Pras; Bernabeu, Miguel O.; Bordas, Rafel; Cooper, Jonathan; Garny, Alan; Pitt-Francis, Joe M.; Whiteley, Jonathan P.; Gavaghan, David J. "A numerical guide to the solution of the bidomain equations of cardiac electrophysiology". Progress in Biophysics and Molecular Biology. 102 (2-3): 136–155. doi:10.1016/j.pbiomolbio.2010.05.006.