Forward problem of electrocardiology

Stylized representation of a human torso that describes the domain and the notation considered for the forward problem of electrocardiography. Two separated areas and their boundary are considered, which represent the heart and human torso around it.

The basic assumption of the bidomain model is that the heart tissue can be divided in two ohmic conducting continuous media, connected but separated through the cell membrane. This media are called intracellular and extracellular regions, the former representing the cellular tissues, and the latter representing the space between cells.[2][1]

The standard formulation of the bidomain model, including a dynamical model for the ionic current, is the following[2]

$${\displaystyle {\begin{cases}\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla V_{m})+\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla u_{e})=A_{m}\left(C_{m}{\dfrac {\partial V_{m}}{\partial t}}+I_{\text{ion}}(V_{m},w)\right)+I_{\text{app}}&{\text{in }}\Omega _{H}\\\nabla \cdot \left(({\boldsymbol {\sigma }}_{i}+{\boldsymbol {\sigma }}_{e})\nabla u_{e}\right)+\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla V_{m})=0&{\text{in }}\Omega _{H}\\{\frac {\partial w}{\partial {t}}}+g(V_{m},w)=0&{\text{in }}\Omega _{H}\end{cases}}}$$

where ${\displaystyle V_{m}}$ and ${\displaystyle u_{e}}$ are the transmembrane and extracellular potentials respectively, ${\displaystyle I_{\text{ion}}}$ is the ionic current, which depends also from a so-called gating variable ${\displaystyle w}$ (accounting for cellular-level ionic behavior), and ${\displaystyle I_{\text{app}}}$ is an external current applied to the domain. Moreover, ${\displaystyle {\boldsymbol {\sigma }}_{i}}$ and ${\displaystyle {\boldsymbol {\sigma }}_{e}}$ are the intracellular and extracellular conductivity tensors, ${\displaystyle A_{m}}$ is the surface to volume ratio of the cell membrane and ${\displaystyle C_{m}}$ is the membrane capacitance per unit area. Here the domain ${\displaystyle \Omega _{H}}$ represents the heart muscle.[2]

The boundary conditions for this version of the bidomain model are obtained through the assumption that there is no flow of intracellular potential outside of the heart, which means that

$${\displaystyle {\boldsymbol {\sigma }}_{i}\nabla V_{m}\cdot \mathbf {n} +{\boldsymbol {\sigma }}_{i}\nabla u_{e}\cdot \mathbf {n} =0\quad \quad {\text{on }}\Sigma }$$

where ${\displaystyle \Sigma =\partial \Omega _{H}}$ denotes the boundary of the heart domain and ${\displaystyle \mathbf {n} }$ is the outward unit normal to ${\displaystyle \Sigma }$.[2]

The monodomain model is a simplification of the bidomain model that, in spite of some unphysiological assumptions, is able to represent realistic electrophysiological phenomena at least for what concerns the transmembrane potential ${\displaystyle V_{m}}$.[2][1]

The standard formulation is the following partial differential equation, whose only unknown ${\displaystyle V_{m}}$ is the transmembrane potential:

$${\displaystyle \chi C_{m}{\frac {\partial V_{m}}{\partial t}}-\nabla \cdot \left({\boldsymbol {\sigma }}_{i}{\frac {\lambda }{1+\lambda }}\nabla V_{m}\right)+\chi I_{\text{ion}}=I_{\text{app}}\quad \quad {\text{in }}\Omega _{H}}$$

where ${\displaystyle \lambda }$ is a parameter that relates the intracellular and extracellular conductivity tensors.[2]

The boundary condition used for this model is[9]

$${\displaystyle \left({\boldsymbol {\sigma }}_{i}\nabla V_{m}\right)\cdot \mathbf {n} =0\quad \quad {\text{on }}\partial \Omega _{H}.}$$

As for the bidomain model, the torso model can be derived from the Maxwell's equations and the continuity equation after some assumptions. First of all, since the electrical and magnetic activity inside the body is generated at low level, a quasi-static assumption can be considered. Thus, the body can be viewed as a passive conductor, which means that its capacitive, inductive and propagative effect can be ignored.[1]

Under quasi-static assumption, the Maxwell's equations are[1]

$${\displaystyle {\begin{cases}\nabla \cdot \mathbf {E} &={\frac {\rho }{\epsilon _{0}}}\\\nabla \times \mathbf {E} &=0\\\nabla \cdot \mathbf {B} &=0\\\nabla \times \mathbf {B} &=\mu _{0}\mathbf {J} \end{cases}}}$$

and the continuity equation is[1]

$${\displaystyle \nabla \cdot \mathbf {J} =0.}$$

Since its curl is zero, the electrical field can be represented by the gradient of a scalar potential field, the torso potential

${\displaystyle \mathbf {E} =-\nabla u_{T}}$

(1)

where the negative sign means that the current flows from higher to lower potential regions.[1]

Then, the total current density can be expressed in terms of the conduction current and other different applied currents so that, from continuity equation,[1]

${\displaystyle \nabla \cdot \mathbf {J} =\nabla \cdot (\mathbf {J} _{\text{app}}+{\boldsymbol {\sigma }}_{T}\mathbf {E} )=0}$

(2)

Then, substituting (1) in (2)

$${\displaystyle \nabla \cdot ({\boldsymbol {\sigma }}_{T}\nabla u_{T})=\nabla \cdot \mathbf {J} _{\text{app}}=I_{v}}$$

in which ${\displaystyle I_{v}}$ is the current per unit volume.[1]

Finally, since aside from the heart there is no current source inside the torso, the current per unit volume can be set to zero, giving the generalized Laplace equation, which represents the standard formulation of the diffusive problem inside the torso[1]

$${\displaystyle \nabla \cdot ({\boldsymbol {\sigma }}_{T}\nabla u_{T})=0\quad \quad {\text{in }}\Omega _{T}.}$$

The boundary conditions accounts for the properties of the media surrounding the torso, i.e. of the air around the body. Generally, air has null conductivity which means that the current cannot flow outside the torso. This is translated in the following equation[1]

$${\displaystyle {\boldsymbol {\sigma }}_{T}\nabla u_{T}\cdot \mathbf {n} _{T}=0\quad \quad {\text{on }}\Gamma _{T}}$$

where ${\displaystyle \mathbf {n} _{T}}$ is the unit outward normal to the torso and ${\displaystyle \Gamma _{T}}$ is the torso boundary, which means the torso surface.[1][2]

Usually, the torso is considered to have isotropic conductivity, which means that the current flows in the same way in all directions. However, the torso is not an empty or homogeneous envelope, but contains different organs characterized by different conductivity coefficients, which can be experimentally obtained. A simple example of conductivity parameters in a torso that considers the bones and the lungs is reported in the following table.[2]

Values of the torso conductivity.[2]

${\displaystyle \sigma _{T}^{\text{lungs}}}$	${\displaystyle \sigma _{T}^{\text{bones}}}$	${\displaystyle \sigma _{T}^{\text{remaining regions}}}$
(S/cm)	(S/cm)	(S/cm)
${\displaystyle 2.4\times 10^{-4}}$	${\displaystyle 4\times 10^{-5}}$	${\displaystyle 6\times 10^{-4}}$

The complete coupling between the heart and the torso is obtained imposing a perfect electrical transmission condition between the heart and the torso. This is done considering the following two equations, that establish a relationship between the extracellular potential and the torso potential[2]

$${\displaystyle {\begin{aligned}u_{e}&=u_{T}&{\text{on }}\Sigma \\({\boldsymbol {\sigma }}_{e}\nabla u_{e})\cdot \mathbf {n} _{e}&=-({\boldsymbol {\sigma }}_{T}\nabla u_{T})\cdot \mathbf {n} _{T}\quad &{\text{on }}\Sigma .\end{aligned}}}$$

This equations ensure the continuity of both the potential and the current across the epicardium.[2]

Using these boundary conditions, it is possible to obtain two different fully coupled heart-torso models, considering either the bidomain or the monodomain model for the heart electrical activity. From the numerical viewpoint, the two models are computationally very expensive and have similar computational costs.[2]

Boundary conditions that represent a perfect electrical coupling between the heart and the torso are the most used and the classical ones. However, between the heart and the torso there is the pericardium, a sac with a double wall that contains a serous fluid which has a specific effect on the electrical transmission. Considering the capacitance ${\displaystyle C_{p}}$ and the resistive${\displaystyle R_{p}}$ effect that the pericardium has, alternative boundary conditions that take into account this effect can be formulated as follows[10]

$${\displaystyle {\begin{aligned}R_{p}{\boldsymbol {\sigma }}_{e}\nabla u_{e}\cdot \mathbf {n} &=R_{p}C_{p}{\frac {\partial (u_{T}-u_{e})}{\partial t}}+(u_{T}-u_{e})\quad \quad &{\text{on }}\Sigma \\{\boldsymbol {\sigma }}_{e}\nabla u_{e}\cdot \mathbf {n} &={\boldsymbol {\sigma }}_{T}\nabla u_{T}\cdot \mathbf {n} \quad \quad &{\text{on }}\Sigma \end{aligned}}}$$

The fully coupled heart-torso model, considering the bidomain model for the heart electrical activity, in its complete form is[2]

$${\displaystyle {\begin{cases}\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla V_{m})+\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla u_{e})=A_{m}\left(C_{m}{\dfrac {\partial V_{m}}{\partial t}}+I_{\text{ion}}(V_{m},w)\right)+I_{\text{app}}&{\text{in }}\Omega _{H}\\[1ex]\nabla \cdot \left(({\boldsymbol {\sigma }}_{i}+{\boldsymbol {\sigma }}_{e})\nabla u_{e}\right)+\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla V_{m})=0&{\text{in }}\Omega _{H}\\[1ex]{\dfrac {\partial w}{\partial {t}}}+g(V_{m},w)=0&{\text{in }}\Omega _{H}\\[1ex]\nabla \cdot ({\boldsymbol {\sigma }}_{T}\nabla u_{T})=0&{\text{in }}\Omega _{T}\\[1ex]({\boldsymbol {\sigma }}_{i}\nabla V_{m})\cdot \mathbf {n} +({\boldsymbol {\sigma }}_{i}\nabla u_{e})\cdot \mathbf {n} =0&{\text{on }}\Sigma \\[1ex]u_{e}=u_{T}&{\text{on }}\Sigma \\[1ex]({\boldsymbol {\sigma }}_{e}\nabla u_{e})\cdot \mathbf {n} =({\boldsymbol {\sigma }}_{T}\nabla u_{T})\cdot \mathbf {n} &{\text{on }}\Sigma \\[1ex]({\boldsymbol {\sigma }}_{T}\nabla u_{T})\cdot \mathbf {n} =0&{\text{on }}\Gamma _{T}\end{cases}}}$$

where the first four equations are the partial differential equations representing the bidomain model, the ionic model and the torso model, while the remaining ones represent the boundary conditions for the bidomain and torso models and the coupling conditions between them.[2]

The fully coupled heart-torso model considering the monodomain model for the electrical activity of the heart is more complicated that the bidomain problem. Indeed, the coupling conditions relate the torso potential with the extracellular potential, which is not computed by the monodomain model. Thus, it is necessary to use also the second equation of the bidomain model (under the same assumptions under which the monodomain model is derived), yielding:[2]

$${\displaystyle \nabla \cdot \left(({\boldsymbol {\sigma }}_{i}+{\boldsymbol {\sigma }}_{e})\nabla u_{e}\right)+\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla V_{m})=0\quad \quad {\text{in }}\Omega _{H}.}$$

This way, the coupling conditions do not need to be changed, and the complete heart-torso model is composed of two different blocks:[2]

• First the monodomain model with its usual boundary condition must be solved:
  $${\displaystyle {\begin{cases}\chi C_{m}{\dfrac {\partial V_{m}}{\partial t}}-\nabla \cdot \left({\boldsymbol {\sigma }}_{i}{\dfrac {\lambda }{1+\lambda }}\nabla V_{m}\right)+\chi I_{\text{ion}}=I_{\text{app}}&{\text{in }}\Omega _{H}\\[1ex]\left({\boldsymbol {\sigma }}_{i}\nabla V_{m}\right)\cdot \mathbf {n} =0&{\text{on }}\partial \Omega _{H}.\end{cases}}}$$

  
• Then, the coupled model that includes the computation of the extracellular potential, the torso model and the coupling conditions must be solved:
  $${\displaystyle {\begin{cases}\nabla \cdot \left(({\boldsymbol {\sigma }}_{i}+{\boldsymbol {\sigma }}_{e})\nabla u_{e}\right)+\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla V_{m})=0&{\text{in }}\Omega _{H}\\\nabla \cdot ({\boldsymbol {\sigma }}_{T}\nabla u_{T})=0&{\text{in }}\Omega _{T}\\{\boldsymbol {\sigma }}_{T}\nabla u_{T}\cdot \mathbf {n} =0&{\text{on }}\Gamma _{T}\\u_{e}=u_{T}&{\text{on }}\Sigma \\({\boldsymbol {\sigma }}_{e}\nabla u_{e})\cdot \mathbf {n} =({\boldsymbol {\sigma }}_{T}\nabla u_{T})\cdot \mathbf {n} &{\text{on }}\Sigma \end{cases}}}$$

  

The fully coupled heart-torso models are very detailed models but they are also computationally expensive to solve.[2] A possible simplification is provided by the so-called uncoupled assumption in which the heart is considered completely electrically isolated from the heart.[2] Mathematically, this is done imposing that the current cannot flow across the epicardium, from the heart to the torso, namely[2]

$${\displaystyle {\boldsymbol {\sigma }}_{e}\nabla u_{e}\cdot \mathbf {n} =0\quad \quad {\text{on }}\Sigma .}$$

Applying this equation to the boundary conditions of the fully coupled models, it is possible to obtained two uncoupled heart-torso models, in which the electrical models can be solved separately from the torso model reducing the computational costs.[2]

The uncoupled version of the fully coupled heart-torso model that uses the bidomain to represent the electrical activity of the heart is composed of two separated parts:[2]

• The bidomain model in its isolated form
  $${\displaystyle {\begin{cases}\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla V_{m})+\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla u_{e})=A_{m}\left(C_{m}{\dfrac {\partial V_{m}}{\partial t}}+I_{\text{ion}}(V_{m},w)\right)+I_{\text{app}}&{\text{in }}\Omega _{H}\\\nabla \cdot \left(({\boldsymbol {\sigma }}_{i}+{\boldsymbol {\sigma }}_{e})\nabla u_{e}\right)+\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla V_{m})=0&{\text{in }}\Omega _{H}\\{\dfrac {\partial w}{\partial {t}}}+g(V_{m},w)=0&{\text{in }}\Omega _{H}\\({\boldsymbol {\sigma }}_{i}\nabla V_{m})\cdot \mathbf {n} +({\boldsymbol {\sigma }}_{i}\nabla u_{e})\cdot \mathbf {n} =0&{\text{on }}\Sigma \\{\boldsymbol {\sigma }}_{e}\nabla u_{e}\cdot \mathbf {n} =0&{\text{on }}\Sigma .\end{cases}}}$$

  
• The torso diffusive model in its standard formulation, with the potential continuity condition
  $${\displaystyle {\begin{cases}\nabla \cdot ({\boldsymbol {\sigma }}_{T}\nabla u_{T})=0&{\text{in }}\Omega _{T}\\{\boldsymbol {\sigma }}_{T}\nabla u_{T}\cdot \mathbf {n} =0&{\text{on }}\Gamma _{T}\\u_{T}=u_{e}&{\text{on }}\Sigma \\\end{cases}}}$$

  

As in the case of the fully coupled heart-torso model which uses the monodomain model, also in the corresponding uncoupled model extracellular potential needs to be computed. In this case, three different and independent problems must be solved:[2]

• The monodomain model with its usual boundary condition:
  $${\displaystyle {\begin{cases}\chi C_{m}{\dfrac {\partial V_{m}}{\partial t}}-\nabla \cdot \left({\boldsymbol {\sigma }}_{i}{\dfrac {\lambda }{1+\lambda }}\nabla V_{m}\right)+\chi I_{\text{ion}}=I_{\text{app}}&{\text{in }}\Omega _{H}\\[1ex]\left({\boldsymbol {\sigma }}_{i}\nabla V_{m}\right)\cdot \mathbf {n} =0&{\text{on }}\partial \Omega _{H}.\end{cases}}}$$

  
• The problem to compute the extracellular potential with a boundary condition on the epicardium prescribing no intracellular current flow:
  $${\displaystyle {\begin{cases}\nabla \cdot \left(({\boldsymbol {\sigma }}_{i}+{\boldsymbol {\sigma }}_{e})\nabla u_{e}\right)+\nabla \cdot ({\boldsymbol {\sigma }}_{i}\nabla V_{m})=0&{\text{in }}\Omega _{H}\\[1ex]({\boldsymbol {\sigma }}_{i}+{\boldsymbol {\sigma }}_{e})\nabla u_{e}\cdot \mathbf {n} =-({\boldsymbol {\sigma }}_{i}\nabla V_{m})\cdot \mathbf {n} &{\text{on }}\Sigma \end{cases}}}$$

  
• The torso diffusive model with the potential continuity boundary condition at the epicardium:
  $${\displaystyle {\begin{cases}\nabla \cdot ({\boldsymbol {\sigma }}_{T}\nabla u_{T})=0&{\text{in }}\Omega _{T}\\{\boldsymbol {\sigma }}_{T}\nabla u_{T}\cdot \mathbf {n} =0&{\text{on }}\Gamma _{T}\\u_{T}=u_{e}&{\text{on }}\Sigma \\\end{cases}}}$$