Deriving the Bidomain Model of Cardiac Electrophysiology From a Cell-Based Model; Properties and Comparisons

Figure 1. (A) Illustration of an EMI model domain for four connected cells. The intracellular space (orange) is denoted by Ωi and the extracellular space (red) is denoted by Ωe. The cell membrane is defined as the interface between the intracellular and extracellular spaces and is denoted by Γ. Similarly, the intercalated discs (purple) are denoted by Γg and are defined as the interface between neighboring cells. Each cell is shaped as a cylinder with a diameter increasing slightly toward the center of the cell. (B) Illustration of the finite element mesh used to represent a single cell in the EMI model simulations.

Figure 2. Illustration of a tissue block, Δ, containing a number of cells (orange) and a surrounding extracellular space (red). In Step 1 of the derivation we approximate the discontinuous intracellular space (A) consisting of individual cells connected by gap junctions by a continuous intracellular space (B).

Figure 3. Intracellular potential, ui, at time t = 20 ms in EMI model and bidomain model simulations using a passive membrane model (Equation 74) and the default parameter values specified in Table 1, except that the value of Rg is increased by the factor indicated by the column titles. In addition, the cell length (Lcell) is varied as described for each row of plots.

Figure 4. Conduction velocity as Rg is increased in EMI model and bidomain model simulations of a strand of 20 connected cells with an active membrane model (Jæger et al., 2021b). The values on the x-axis represent the factor with which the default Rg value in Table 1 is multiplied. The remaining parameter values are as specified in Table 1. Note that the plot is separated into three panels in order to improve the visibility of the data. Note also that we consider two different discretization resolutions in the simulations of each model. In the default case, Δt = 0.001 ms and Δx in the bidomain model and the typical edge length in the EMI model is 10 μm, whereas in the refined case (dashed lines), both the spatial and temporal discretization steps are reduced to half of the default values.

Figure 5. Membrane potential, v, and extracellular potential, ue, at time t = 5 ms in EMI model and bidomain model simulations using the default parameter values specified in Table 1 and an active membrane model (Jæger et al., 2021b). This simulation required a CPU time of 132 min for the EMI model and 2 min for the bidomain model.

Figure 6. Membrane potential, v, and extracellular potential, ue, at time t = 20 ms in EMI model and bidomain model simulations using and active membrane model (Jæger et al., 2021b) and the default parameter values specified in Table 1, except that the value of Rg is increased by a factor of 200.