Bidomain model

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. ∇ ⋅ ( Σ i ∇ v i ) + ∇ ⋅ ( Σ e ∇ v e ) = 0. {displaystyle abla cdot left(mathbf {Sigma } _{i} abla v_{i} ight)+ abla cdot left(mathbf {Sigma } _{e} abla v_{e} ight)=0.}     (1) J t = ∇ ⋅ ( Σ i ∇ v i ) = − ∇ ⋅ ( Σ e ∇ v e ) . {displaystyle J_{t}= abla cdot left(mathbf {Sigma } _{i} abla v_{i} ight)=- abla cdot left(mathbf {Sigma } _{e} abla v_{e} ight).}     (2) J t = χ ( C m ∂ v ∂ t + I i o n ) , {displaystyle J_{t}=chi left(C_{m}{frac {partial v}{partial t}}+I_{mathrm {ion} } ight),}     (3) ∇ ⋅ ( Σ i ∇ v ) + ∇ ⋅ ( Σ i ∇ v e ) = χ ( C m ∂ v ∂ t + I i o n ) . {displaystyle abla cdot left(mathbf {Sigma } _{i} abla v ight)+ abla cdot left(mathbf {Sigma } _{i} abla v_{e} ight)=chi left(C_{m}{frac {partial v}{partial t}}+I_{mathrm {ion} } ight).}     (4) ∇ ⋅ ( Σ 0 ∇ v 0 ) = 0 x ∈ T , {displaystyle abla cdot left(mathbf {Sigma } _{0} abla v_{0} ight)=0,,,,,,,mathbf {x} in mathbb {T} ,}     (5) n → ⋅ ( Σ 0 ∇ v 0 ) = 0 x ∈ ∂ T . {displaystyle {vec {n}}cdot left(mathbf {Sigma } _{0} abla v_{0} ight)=0,,,,,,,mathbf {x} in partial mathbb {T} .}     (6) v e = v 0 x ∈ ∂ H . {displaystyle v_{e}=v_{0},,,,,,,mathbf {x} in partial mathbb {H} .}     (7) n → ⋅ ( Σ 0 ∇ v 0 ) = n → ⋅ ( Σ e ∇ v e ) x ∈ ∂ H . {displaystyle {vec {n}}cdot left(mathbf {Sigma } _{0} abla v_{0} ight)={vec {n}}cdot left(mathbf {Sigma } _{e} abla v_{e} ight),,,,,,,mathbf {x} in partial mathbb {H} .}     (8) n → ⋅ ( Σ i ∇ v ) + n → ⋅ ( Σ i ∇ v e ) = 0 x ∈ ∂ H . {displaystyle {vec {n}}cdot left(mathbf {Sigma } _{i} abla v ight)+{vec {n}}cdot left(mathbf {Sigma } _{i} abla v_{e} ight)=0,,,,,,,mathbf {x} in partial mathbb {H} .}     (9) 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.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. 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.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.