Eikonal models describe the myocardial state transition in terms of the depolarization wavefront arrival time. The wavefront contour Γ(*t*) is formed by all positions *x* having a first arrival time *τ* equal to *t*:

$$\mathrm{\Gamma}\left(t\right):=\left\{\mathbf{x}\right|\tau \left(\mathbf{x}\right)=t\}$$(1)

Neglecting the curvature of the front the simplest eikonal model formulation is

$$\sqrt{\u3008\nabla \hspace{0.17em}\tau ,{\mathrm{ACV}}^{\ast}{\mathrm{CV}}^{\ast}{\mathbf{A}}^{T}\nabla \hspace{0.17em}\tau \u3009}=1,$$(2)

where **A** denotes the local fiber coordinate system. **CV**^{*} is the local conduction velocity tensor. It contains the scalar planar conduction velocities CV^{l}, CV^{t} and CV^{n} for the respective propagation in longitudinal, tangential and normal direction of the local fiber coordinate system. All variables given in local coordinates will be marked by an asterisk in the following.

While the distribution of local conduction velocities just represents the model parameters in terms of an eikonal model formulation it depends on many parameters within the bidomain model equations. Among these are the entries of the intra- and extracellular conductivity tensors **M**_{i}^{*} and **M**_{e}^{*}, parameters of the single cell model, the stimulation current density and the basic cycle length of the stimulation protocol as illustrated in Figure 1. The conduction velocity further depends on the local front curvature.

Figure 1 Left: Atrial action potential waveforms for different basic cycle lengths. Right: Change caused by diffusive coupling within the bidomain model in comparison to the uncoupled single cell response.

A known result from eikonal model derivations is

$$\text{CV}\propto \sqrt{\overrightarrow{\mathbf{n}}\cdot {r}_{m}\mathbf{M}\overrightarrow{\mathbf{n}}},$$(3)

where **M** is the conductivity tensor of the monodomain model. CV denotes the conduction velocity along the depolarization front normal $\overrightarrow{\mathbf{n}}$. *r*_{m} is defined by the passive conductance of the cell membrane per unit volume at resting membrane potential *v*_{m, rest}. Assuming just coincident basis vectors for the intracellular and extracellular space within the bidomain model formulation **M**_{e}^{*} can be expressed as

$${\mathbf{M}}_{e}^{\ast}=\mathrm{\Lambda}{\mathbf{M}}_{i}^{\ast}\mathit{\hspace{1em}}\iff $$(4)

$$=\left(\begin{array}{ccc}\hfill \frac{{\sigma}_{e}^{l}}{{\sigma}_{i}^{l}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\sigma}_{e}^{t}}{{\sigma}_{i}^{t}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{{\sigma}_{e}^{n}}{{\sigma}_{i}^{n}}\hfill \end{array}\right){\mathbf{M}}_{i}^{\ast}.$$(5)

Using the notation

$$\begin{array}{ccccc}{\mathrm{\lambda}}_{l}:=\frac{{\sigma}_{e}^{l}}{{\sigma}_{i}^{l}}\hfill & & {\mathrm{\lambda}}_{t}:=\frac{{\sigma}_{e}^{t}}{{\sigma}_{i}^{t}}\hfill & & {\mathrm{\lambda}}_{n}:=\frac{{\sigma}_{e}^{n}}{{\sigma}_{i}^{n}}\hfill \end{array}$$(6)

and neglecting the stimulation currents the monodomain model equation can then be restated as

$$\nabla \cdot \left(\mathbf{M}\nabla \hspace{0.17em}{v}_{m}\right)=\chi \left({c}_{m}\frac{\partial \hspace{0.17em}{v}_{m}}{\partial \hspace{0.17em}t}+{i}_{\text{ion}}\right)$$(7)

where *χ* is the surface-to-volume ratio. The local conductivity tensor **M**^{*} = **A**^{T} **MA** is given by

$${\mathbf{M}}^{\ast}=\left(I+{\left(I+\mathrm{\Lambda}\right)}^{-1}\right){\mathbf{M}}_{i}^{\ast}$$(8)

$$\begin{array}{cccc}& =\left(\begin{array}{ccc}\hfill \frac{{\mathrm{\lambda}}_{l}}{1+{\mathrm{\lambda}}_{l}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\mathrm{\lambda}}_{t}}{1+{\mathrm{\lambda}}_{t}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{{\mathrm{\lambda}}_{n}}{1+{\mathrm{\lambda}}_{n}}\hfill \end{array}\right){\mathbf{M}}_{i}^{\ast}.\hfill & & \end{array}$$(9)

This formulation was introduced as augmented monodomain model by Bishop and Plank [2], [3].

The trace of ventricular action potentials is known to vary in dependence on the heart chamber and the location of the cell in transmural and apico-basal direction [1], [12], [13]. These effects are called repolarization gradients or dispersion of action potential duration. The question arise, if repolarization gradients influence as well the conduction velocity distribution.

The study was performed in the following steps: Initially the entries of the conduction velocity tensor **CV**^{*} were derived based on planar wavefront propagation. The 1D bidomain model simulations comprised atrial and ventricular single cell models to facilitate the parametrization of whole heart models. In a second step the sensitivity of **CV**^{*} in response to changes in the intra- and extracellular conductivity was investigated. In a third step the error

$$e=\frac{\overrightarrow{\mathbf{n}}\cdot \mathbf{cv}\overrightarrow{\mathbf{n}}-{\text{CV}}_{\text{exact}}}{{\text{CV}}_{\text{exact}}}$$(10)

relying on **CV**^{*} in the presence of nonnegligible front curvature was determined. CV_{exact} denotes the conduction velocity of the bidomain model in direction of the respective front normal. The bidomain model equations were computed with Galerkin finite element methods (FEM) and operator splitting techniques as described by Sundness et al. [11]. Two different human cell models of ventricular myocardium were taken into account. The first ventricular cell model integrated is a model with minimum complexity presented by Bueno-Orovio et al. in 2008 [1]. It will be referred as BCF model in the following. The second ventricular single cell model is the ten Tusscher-Noble-Noble-Panfilov (TNNP) model [12], [13]. For action potential propagation in atrial myocardium the Courtemanche-Ramirez-Nattel (CRN) model [5] was applied.

Assumptions on the electrical parameters differ among published simulation studies. Parameter values for the capacitance per unit cell membrane and the amount of cell area per unit volume of ventricular myocardium were chosen according to Li et al. [7]. The entries of the conductivity tensors applied by Sundnes et al. were scaled to achieve an unchanged diffusion constant with respect to Bueno-Orovio et al. The resulting values for the entries of the conductivity tensors in longitudinal direction ${\sigma}_{i}^{l}=7.12\text{mS/cm}$ and ${\sigma}_{e}^{l}=4.74\text{mS/cm}$ were well in accordance with the values applied by Weiss et al. [15]. In the last-mentioned work the anisotropic ratios between longitudinal and tangential directions were considerably larger than those taken into account by Sundnes et al. The resulting entries of **M**^{*}_{e} and **M**^{*}_{i} are far from being linearly dependent.

Atrial myocardium was modeled as isotropic. Parameter values were set according to Mainardi et al. [8].

Spatial discretization for all simulations was $\mathrm{\Delta}x=25\mathrm{\mu}\text{m}$ in accordance with [10]. A discretization in time $\mathrm{\Delta}t=1\mathrm{\mu}\text{s}$ was chosen based on an evaluation of the single cell model convergence.

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