Both minimum and maximum values of the electrical potential were calculated. The generated signals were smooth and densely sampled such that no pre-filtering was required.

Following clinical practice, we also report the peak-to-peak difference, Φ_pp = ϕ _max − ϕ _min.

The width of the EGM or EGM duration should reflect the time interval over which the peak is formed but is tedious to define in practice due to a vast variation in the signal morphology. Here, we defined the fraction of completion for both QRS and T waves, given as follows: S t = ∫ − ∞ t | Φ t | d t ∫ − ∞ ∞ | Φ t | d t . (7) Then, in analogy to the measurement of the action potential duration, we define the start and stop time of the signal as follows: S (t ₅) = 0.05 and S (t ₉₅) = 0.95. Then, the time interval, including 90% of absolute surface variations under the signal curve, was as follows: W 90 = t 95 − t 5 . (8) This quantity was used in the following as the width of the EGM.

To quantify the resemblance between two signals Φ₁ and Φ₂, we used the root-mean-squared error (RMSE) on the interval of interest t ∈ [t ₀, t ₁]: RMSE = 1 t 1 − t 0 ∫ t 0 t 1 | Φ 1 − Φ 2 | 2 d t , (9) which yielded a value in mV.

The set of coupled Eq. 2a has not yet been solved analytically. However, if the distribution of the transmembrane potential V m ( r ⃗ , t ) is known, Eq. 2a can be regarded as a Poisson problem: − ∇ ⃗ ⋅ G i + G e ∇ ⃗ Φ = ρ x , t , (10) with charge density ρ = ∇ ⃗ ⋅ G i ∇ ⃗ V m . (11) A similar reasoning was used to approximate electrical signals from monodomain simulations in the pseudo-bidomain method or the lead-field method (Bishop and Plank, 2011a).

The shape of the wave front in a slab with uniform fibers is in general non-planar, due to bath-loading effects; the presence of a conducting layer around the myocardium tends to accelerate the wave front near the tissue–bath interface (Roth, 1991; Bishop and Plank, 2011b; Bishop et al., 2011). In our simulation results with conductivity values taken from the literature, we found that the wave front was approximately planar, see Figure 2C. Therefore, within this study, we neglected bath loading effects and assumed that membrane potential V _m changed rapidly from resting state ϕ _rest = −86 mV to ϕ _max = +10 mV, over a plane parallel to the YZ plane. For a wave traveling to the positive X-direction, it was found that V m = ϕ max − ϕ max − ϕ rest H x − v t , (12) with H(x) as the Heaviside step function.

By inserting this Ansatz (Eq. 12) into the bidomain of Eq. 2a, we obtained a Poisson problem (Eq. 10), with the following charge density: ρ x , t = − ϕ m a x − ϕ r e s t g i , x x δ ′ x − v t . (13)

Here, δ(x) is the Dirac distribution that represents a localized charge (Arfken and Weber, 1995); its spatial derivative, as appearing in (Eq. 13), is the mathematical representation of an electrical dipole layer.

The general solution to this case could be obtained by linear superposition, in the form of Green’s functions. If G ( r ⃗ , r ⃗ 0 ) was the electrical potential Φ created by a unit point charge at r ⃗ 0 − ∇ ⃗ ⋅ G i + G e ∇ ⃗ Φ = δ r ⃗ − r ⃗ 0 , (14) respecting the boundary conditions, then the solution to (Eq. 10) is as follows: Φ r ⃗ , t = ∭ G r ⃗ , r ⃗ 0 ρ r ⃗ 0 d 3 r 0 . (15) So, the search for an analytical solution came down to 1) finding the Green’s function and 2) integrating it over the source configuration. In our present scope, we only considered wave fronts parallel to the YZ plane, whence Φ r ⃗ , t = ∫ − L 0 G r ⃗ , z 0 ρ z 0 d z 0 . (16) For general anisotropic and inhomogeneous media, the Green’s function has no closed form and should be computed numerically. In this paragraph, we continue the simplest case possible, where the conductivity in the Poisson’s Eq. 10 in the tissue is the same as in the blood and torso, in addition to being constant and isotropic. This implies that G i + G e = g B I = g T I = g ̃ I , where we used g ̃ to denote the simplified, lumped conductivity. We refer to this assumption as the “homogenized” case, since the differences between the conductivities of the myocardium, blood, and torso are neglected and have been replaced by a homogeneous conductivity g ̃ . Our use of the term “homogenized” should not be confounded with the homogenization of the cellular structure within the tissue, which takes place during the derivation of bidomain equations (Neu and Krassowska, 1993).

In the analytical derivation, the bath was taken to be unbounded as opposed to the simulations. To denote that the potential too is an approximation, we indicated it as Φ ̃ .

Under this condition, the problem is reduced to a classical electrostatic problem: g ̃ Δ Φ ̃ = ϕ m a x − ϕ r e s t g i , x x δ ′ x − x 0 H z + L H − z , (17) with the dipole sheet position, x ₀ = vt.

In the first step, we replaced the source by a line charge of unit strength parallel to Y that is placed at x = x ₀ and z = z ₀, i.e., ρ = δ(x − x ₀)δ(z − z ₀). Then, it was found from Gauss’ law that (Jackson, 1975) g ̃ Δ Φ ̃ line = − δ x − x 0 δ z − z 0 ⇒ Φ ̃ line = − 1 2 π g ̃ ln x − x 0 2 + z − z 0 2 . (18) Differentiating with respect to x ₀ gave the potential distribution of a dipole line, with the dipole oriented in the X-direction: g ̃ Δ Φ ̃ dipole line = ϕ m a x − ϕ r e s t g i , x x δ ′ x − x 0 δ z − z 0 ⇒ Φ ̃ dipole line = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ x − x 0 x − x 0 2 + z − z 0 2 . (19) To obtain the potential generated by the wave front, the dipole line charges are needed to be stacked on top of each other in the Z-direction, z ₀ ∈ [−L, 0], where z = 0 corresponds to the endocardium. Φ ̃ dipole layer x , x 0 , z = ∫ − L 0 Φ ̃ dipole line d z 0 , = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ ∫ − L 0 x − x 0 x − x 0 2 + z − z 0 2 d z 0 , = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ arctan z + L x − x 0 − arctan z x − x 0 , (20a) = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ arctan x − x 0 z − arctan x − x 0 z + L . (20b)

Here, arctan is the inverse of the tangent function on the interval (−π/2, π/2), see Figure 3. In the last step, we used that arctan ( 1 / x ) = π 2 sgn ( x ) − arctan ( x ) . It should further be noted that arctan (−x) = − arctan(x). ¹

Since the wave profile propagated at constant speed v to the right, the wave front was located at x ₀(t) = vt, leading to the following spatiotemporal potential distribution: Φ ̃ x , z , t = Φ ̃ dipole layer x , v t , z = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ × arctan x − v t z − arctan x − v t z + L . (21)

The potential registered by an electrode at position (x, z) = (0, h) is then the measured unipolar signal in the blood pool: Φ ̃ unipolar t = Φ ̃ dipole layer 0 , v t , h = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ × − arctan v t h + arctan v t h + L . (22)

Snapshots of spatial profiles at different times and different distances h to the endocardial surface are shown in Figure 4. When the signal is recorded on the endocardial surface, one can set h → > 0 in (Eq. 22), such that for an electrode at x = 0: Φ ̃ unipolar t = − ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ arctan L v t , (23) which first entails an upward deflection, followed by a finite downward jump at t = 0 and a restoring phase, see Figure 4A.

Figure 5 shows how this result can be interpreted geometrically. Since tan ( θ 0 ) = v t h and tan ( θ 1 ) = v t h + L , the result can be written in terms of the angle Θ₁ = θ ₁ − θ ₀ under which the wave front is seen as follows: Φ ̃ unipolar t = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ Θ 1 . (24)

This result is in agreement with the classical solid angle theory for the electrogram (Bayley and Berry, 1964; Holland and Arnsdorf, 1977; Van Oosterom, 2002; Macfarlane et al., 2010). The angle θ from Figure 5 can be extended with the angle ξ that a point on the wave front makes with respect to the Y-axis. Then, ξ and θ are spherical coordinates centered on the Y-axis, which obey dΩ = dθdξ sin ξ. Since we worked with a slab geometry, the angle ξ under which the wave front is seen always extends from 0 to π, whence Ω θ 0 , θ 1 = ∬ d Ω = ∫ 0 π d ξ ∫ θ 0 θ 1 d θ ⁡ sin ⁡ ξ = 2 θ 1 − θ 0 = 2 Θ 1 . (25) Hence, doubling the planar angles Θ in this work will give the corresponding solid angle in 3D. It should be noted that angles Θ are expressed in radians, while the other angles used in this work are reported in degrees.

Thus, in the homogenized approximation, the unipolar EGM is proportional to the angle Θ₁ subtended by the wave front when viewed from the electrode at any given time. The potential difference ϕ _max − ϕ _rest can be measured in experiments and has a value of around 120 mV (Plonsey, 1965; Plonsey, 1974; Holland and Arnsdorf, 1977). The value of g _i,xx can also be measured.

However, in the homogenized theory, there is no fixed rule to estimate the lumped conductivity g ̃ as a weighted average of the conductivities in the problem: g _i,l, g _i,t, g _e,l, g _e,t, g _B, and g _T. In the following, we showed the relation between them from the exact analytical solution in the case with and without anisotropy.

Analytical calculations of potentials generated by dipole charges in inhomogeneous media representing the heart have been carried out before (Bayley and Berry, 1964; Kempner and Grayzel, 1970; Plonsey, 1974). However, these are focused on ECG generation and, therefore, adopted a circular geometry representing a cross-section through the heart.

In this paragraph, we incorporated the different conductivities of the blood (g _B), myocardium (g _M, given by g _i,xx + g _e,xx ), and torso (g _T), for now assuming that the myocardium is an isotropic layer of thickness L. The mathematical problem is similar to finding the electrostatic potential in a three-layered medium with different electrical permittivities. For two layers, the result was detailed in the study by Jackson (1975) using the method of mirror charges, and it was used by Plonsey and Barr (1987) to obtain potentials inside the myocardium. The case where the two outer layers have equal properties and the recording is made in the middle layer has also been described in a study of quantum dots (Escribano et al., 2017).

In case of a three-layered medium, the solution method came down to reflecting the position of the sources within the myocardium on the other side of myocardium–torso and myocardium–blood interfaces and solving the interface conditions to recursively find all the strengths of the mirror sources.

This procedure is outlined in our Supplementary Appendix A1. We chose Cartesian coordinates, such as z > 0, which represents the blood pool, −L < z < 0 the (isotropic) myocardium and z< −L the torso domain (assumed homogeneous).

The factors relating to the strengths of different mirror sources are, with * equal to B (blood) or T (torso): h * = g M − g * g M + g * , v * = 2 g * g M + g * . (26)

Solving the recursion relation then led to the following explicit series solution: Φ B x , z , t = v B g M ϕ m a x − ϕ r e s t 2 π g B ∑ j = 0 ∞ h B ⌊ j 2 ⌋ h T ⌈ j 2 ⌉ × arctan x − v t z + j L − arctan x − v t z + j + 1 L , (27a) = v B g M ϕ m a x − ϕ r e s t 2 π g B ∑ j = 0 ∞ h B ⌊ j 2 ⌋ h T ⌈ j 2 ⌉ Θ j + 1 . (27b)

We observed that including inhomogeneous conductivities led to focusing and defocusing of electrical field lines [see (Jackson, 1975)], which can alternatively be interpreted as reflections of the source at the endo- and epicardial boundaries.

Although the result was formulated using the angle Θ, Eq. 27 go beyond the solid angle theory, as the solid angles subtended by the reflection are also included. Considering only the first term of the expansion led to Φ B ( x , z , t ) ≈ − v B g M ( ϕ m a x − ϕ r e s t ) 2 π g B Θ 1 . A comparison with the homogenized result (Eq. 24) suggested that g ̃ = g B v B = g M + g B 2 . Thus, in the homogenized approach, the chosen effective conductivity should be equal to the arithmetic mean of myocardial and blood conductivities. This result was refined in the following paragraph to also include anisotropy.

To add anisotropy of the myocardium to the derivation, we denoted g _M = g _i,xx + g _e,xx and g _zz = g _i,zz + g _e,zz . To reuse the isotropic solution from the previous paragraph, we restored isotropy in the myocardial domain by re-scaling the Z-axis according to the square root of the conductivity ratio (see Figure 8), inspired by previous work on cardiac anisotropy (Wellner et al., 2002; Verschelde et al., 2007; Young and Panfilov, 2010). Thus, we used a (dimensionless) re-scaling factor, which also appeared in the study by Plonsey and Barr (1987): η = g M g z z = g i , x x + g e , x x g i , z z + g e , z z . (28) For η = 1, the configuration is that of three parallel layers described previously.

For η ≠ 1, we defined a re-scaled Z-coordinate as follows: Z = z + L − η L , i f z < − L η z , i f − L ≤ z ≤ 0 z . i f 0 < z (29)

In the following, we took the re-scaled tissue thickness to be ℓ = ηL. In Supplementary Appendix A2, it was verified that this reduced the problem to the one with three homogeneous layers, but at the interface conditions, g _M needed to be replaced by g _M/η. Anisotropy, thus, affects three elements. First, the transfer coefficients are changed to the following: H * = g M − η g * g M + η g * , V * = 2 g * g M + η g * . (30)

Second, the effective tissue thickness becomes ℓ = ηL, meaning that the myocardium appears to be thicker, for wave propagation along the myofiber direction (as then, both g _xx and η increase). The geometric re-scaling also affects the solid angles, which we will denote as Θ ̃ here. For unipolar signal scales in the linear leading order with Θ₁, we expect that maximal signal amplitude is reached for the case ψ = 0, i.e., wave propagation parallel to the myofibers. Third, wave speed v also depends on the myofiber orientation.

At the end of our derivation, the electrical potential measured in the blood is as follows: Φ B x , z , t = V B g i , x x ϕ m a x − ϕ r e s t 2 π g B ∑ j = 0 ∞ H B ⌊ j 2 ⌋ H T ⌈ j 2 ⌉ × arctan x − v t z + j η L − arctan x − v t z + j + 1 η L , (31a) = V B g i , x x ϕ m a x − ϕ r e s t 2 π g B ∑ j = 0 ∞ H B ⌊ j 2 ⌋ H T ⌈ j 2 ⌉ Θ ̃ j + 1 . (31b)

Equation 31 is the main analytical result of this paper. It goes beyond the solid angle theory, as it uses re-scaled angles due to anisotropy and takes into account several mirror layer reflections, due to the inhomogeneity of the layers, see Figure 8. Keeping only the first term in (Eq. 31), we get the following: ϕ B one X , Z ≈ g i , x x ϕ m a x − ϕ r e s t V B 2 π g B Θ ̃ 1 , = ϕ m a x − ϕ r e s t π g i , x x g M + η g B , Θ ̃ 1 = A Θ ̃ 1 . (32)

Here, we introduced the notation A as the proportionality factor between the electrical potential measurement and the angle. This prefactor depends on the conductivities and fiber orientation angle ψ. If one would furthermore neglect the re-scaling in the angles (i.e., Θ j ≈ Θ ̃ j , if η ≈ 1), one recovers (Eq. 24) with the homogenized conductivity being equal to an anisotropy-weighted average: g ̃ = g B V B = g M + η g B 2 , (33) with anisotropy ratio η in the direction of wave propagation given by (Eq. 28).

Bipolar signals are the difference of unipolar signals, see (Eq. 4).

For the homogenized case, from (Eq. 23), we found, with p used for the proximal and d for the distal electrode positions: V bipolar t = Φ unipolar x d , z d , t − Φ unipolar x p , z p , t = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ arctan x d z d − arctan x d z d + L + arctan x p z p + L − arctan x p z p . (34)

Inserting the electrode coordinates x _d = x ₀ − vt, z _d = h, and x _p = x ₀ + d cos β cos α − vt, z _p = h + d sin α yielded the following: V bipolar t = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ arctan x 0 − v t h − arctan x 0 − v t h + L + arctan x 0 + d ⁡ cos ⁡ β ⁡ cos ⁡ α − v t h + d ⁡ sin ⁡ α + L − arctan x 0 + d ⁡ cos ⁡ β ⁡ cos ⁡ α − v t h + d ⁡ sin ⁡ α . (35)

If the catheter was held parallel to the wave propagation direction or perpendicular to the endocardium, we get the following expressions: V bip , ‖ t = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ arctan x 0 − v t h − arctan x 0 − v t h + L + arctan x 0 + d − v t h + L − arctan x 0 + d − v t h , V bip , ⊥ t = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ arctan x 0 − v t h − arctan x 0 − v t h + L + arctan x 0 − v t h + d + L − arctan x 0 − v t h + d . (36)

The bipolar signals in the main directions are shown as a difference of unipolar signals in Figure 6.

From the analytical result in case of anisotropy (Eq. 31), an exact solution for the bipolar electrogram can be found, see Supplementary Appendix A3.

The only remaining unknown parameter in our analytical solution is the wave speed v. This speed depends on the local cell kinetics, i.e., the term i _ion in bidomain equations. Within this study, we neglected the bath-loading effect, such that the wave became planar. Its propagation velocity will then depend on the conductivities in the X-direction and was, therefore, affected by g _{i, xx} , g _{e, xx} , g _B, and g _T. In one spatial dimension, the intra- and extracellular conductivity tensors are numbers and are, hence, proportional to each other, under which condition the bidomain model could be simplified to a monodomain description (Bishop et al., 2011; Clayton et al., 2011). The resulting diffusion constant equals D = 1 β m C m g i , x x g e , x x g i , x x + g e , x x . (37)

Through the dimensional analysis of the monodomain equation, it was found that the conduction velocity is proportional to D , whence v ψ = v ‖ g i,l + g e,l g i,l g e,l 1 / 2 g i,l ⁡ cos 2 ⁡ ψ + g i,t ⁡ sin 2 ⁡ ψ g e,l ⁡ cos 2 ⁡ ψ + g e,t ⁡ sin 2 ⁡ ψ g e,l + g i,l cos 2 ⁡ ψ + g e,t + g i,t sin 2 ⁡ ψ 1 / 2 . (38)

This reasoning is held for the bulk of a bidomain region, neglecting the bath-loading effects.

From simulations with ψ = 0° or 90°, we measured that v _‖ = 0.42 m/s. This value was inserted into the following analytical solutions to have no more free parameters in the theory.

In Figure 7, the analytical approximations (homogenized, green solid line and anisotropic, dotted orange line) are compared to the simulated unipolar and bipolar signal in the two main directions. The repolarization (T-wave) was not included in the theoretical framework here and is, therefore, absent in theory graphs.

Both homogenized and anisotropic solutions follow the same qualitative behavior, and the amplitudes are in the correct range. When looked at more closely, it can be seen that the amplitudes of the anisotropic solution agree slightly better than the homogenized approximation. The difference, however, is small. The RMSE (Eq. 9) of the anisotropic solution with the simulated signal is shown on each plot.

To understand why keeping the first terms in the expansion works, let us consider the relative importance of the first mirrored layer (Θ₂) to the myocardium (Θ₁). We found that Θ₂ ≪Θ₁ during wave passing if the electrode was close to the tissue (h ≪ ηL), see Figure 8. Thus, we expect the best convergence and a good approximation when the electrode is closer to the tissue, where the tissue is thick and η is large, i.e., for wave propagation along the myofiber direction. The latter can be seen in the top row of Figure 9, where the unipolar voltage (Φ_B or Φ_uni) is shown. The higher the ψ value, the more the analytical solution differs from the simulated curve.

In Figure 9, the homogenized isotropic theory is plotted together with the inhomogeneous anisotropic theory (using only one or three terms). Strikingly, the amplitude of the EGM decreases significantly if the wave propagates perpendicular to the myofibers. For example, for unipolar potential Φ _B , the peak-to-peak amplitude changes from 4.64 mV at ψ = 0° to 1.27 mV at ψ = 90°, implying a reduction factor of 3.6.

Figure 10 shows simulated and theoretical bipolar signals for different catheter orientation angles α and β, as defined in Figure 1. The case α = β = 0° (top left panel) corresponds to a bipolar electrode directed parallel to the wave propagation and shows a signal equal to V _bip,‖(t), see (Eq. 36). This signal is even and has two zeros (see Figure 6B) and three extrema. Similarly, a bipolar electrode directed normally toward the endocardium (α = 90° for any β value) yields V _bip,⊥(t) (Eq. 36), a signal with uneven symmetry (see Figure 6C and bottom row of Figure 10). This signal has only one zero crossing and vanishes for x − vt = ±∞. For intermediate orientations, the result gradually evolves from V _bip,‖ to V _bip,⊥. In all cases, it is well represented by the analytical solution.

The effect of measuring further away from the cardiac wall (higher h) is shown in Figure 11A. At a larger distance, the maximal solid angle subtended Θ₀ or Θ ̃ 0 becomes smaller, such that the amplitude of the signal is reduced.

Bipolar signals for different interelectrode distances d are shown in Figure 11B. For both orientations, the signal amplitude grows with the increased interelectrode distance. This can be understood by a first-order Taylor approximation around the distal electrode: V bip t = Φ r ⃗ dist − Φ r ⃗ dist + d ⃗ ≈ − ∇ ⃗ Φ r ⃗ dist ⋅ d ⃗ , (39) which grows linearly with the interelectrode distance d = ‖ d ⃗ ‖ .

Figure 12 shows simulated bipolar signals for the different catheter orientations for a slab of thickness L = 5 mm. The minimum, maximum, peak-to-peak amplitude, and signal width were calculated as detailed in Section 2.4, in the same manner for simulated and analytical signals. In simulated signals, both the QRS complex and T-wave can be distinguished and the amplitude and width were also extracted for the T-wave.

Electrogram amplitude and width are shown as a function of myocardial wall thickness in Figure 13. For the QRS complex, there is a good agreement between (full anisotropic) the theory and simulations.

We first observe that thicker walls yield larger iEGM amplitudes, both for the QRS complex and T-wave. This is in line with the solid angle theory and the exact solution in this paper; each unipolar signal arises as a difference between an endocardial and epicardial contribution. For thicker walls, the epicardial contribution decreases, such that the recorded amplitude will saturate at the amplitude of the unipolar signal originating from the endocardium.

Second, the width of the QRS complex also increases with the wall thickness. The reason is the same as for the amplitude: in the limit of L → ∞, only endocardial contributions to (Eq. 36) matter, and this profile has the largest width.

The effect of the myofiber direction on the amplitude and width of bipolar iEGMs is shown in Figure 14.

The amplitude is the highest for waves propagating along fibers and decreases for non-zero fiber angle ψ. This effect was already visible in Figure 9. From (Eq. 31), we can see that anisotropy affects the EGM amplitude in two different ways: via changing the effective tissue thickness and, hence, the solid angle and via the prefactor A of the leading order term (Eq. 32). By plotting hypothetical EGMs with one factor being left out, we found that the change in the amplitude is mostly caused by prefactor A(ψ) from Eq. 32 and only in a limited manner by re-scaled solid angles.

The myofiber orientation also affects the width of the measured EGMs. From (Eq. 32), we learn that two effects occur. First, increasing ψ decreases effective wall thickness ℓ, which leads to a less wide EGM, see paragraph Section 3.3.2. Second, the wave speed will decrease for larger ψ, as the wave does not propagate along the myofiber direction anymore. This effect will increase the EGM width. From Figure 14, we conclude that the reduction of the propagation velocity is the dominant effect, increasing the EGM width in case of a propagation transverse to the myofiber direction.

It can be further observed in Figure 9 that the theoretically predicted width of the QRS complex for ψ = 90° shows an outlier. The corresponding EGM was shown in Figure 9 (middle row, rightmost column). Due to the small signal amplitude, the width is determined by wide positive lobes, which makes the calculation of the width sensitive to small deviations.

In this work, we derived the iEGM-shape from bidomain model equations, with increasing accuracy in the approximations: a homogeneous medium, a three-layered medium, and a three-layered medium with anisotropy. In earlier works, mirror sources have been applied in cylindrical or spherical heart and torso geometries (Bayley and Berry, 1964; Gelernter and Swihart, 1964; Selvester et al., 1967) to obtain insights on the body-surface ECG and on a two-layered myocardium with parallel fibers (Plonsey and Barr, 1987), to mimic recordings of electrodes placed within the tissue. Here, we focused on intracardiac EGMs measured in the blood pool as used in the clinic, requiring the inclusion of anisotropy. Figures 9, 14 show that the effect of anisotropy can indeed cause a 3.6-fold change in the amplitude of a bipolar signal.

The classical solid angle theory (Geselowitz, 1989; Van Oosterom, 2002) can be applied here for the case of equal conductivities (without myofibers), but fails for the anisotropic slab model. Figure 5 shows how the arctan function can be seen as the angle (in 2D and as a solid angle in 3D) from the electrode position toward the dipole sheet. In three-layered theories, reflections (mirror sources) also need to be taken into account.

Our analytical results also confirm that there are no points of infinite potential (singularities) when measuring close to a cardiac depolarization wave. This fact has also been recognized by others [e.g., Holland and Arnsdorf (1977)] using the solid angle theory, but we, nevertheless, prefer to repeat the arguments here.

A classical result in electromagnetism states that the potential near an electrical dipole scales as 1/R ². This reasoning seems to suggest that there is a singularity (infinite value) of the potential on the myocardial surface (i.e., for R → 0). Such a result would clearly be non-physical. The flaw in the argumentation is that if the measurement electrode is put next to the endocardium (where R = 0), the dipole charge is distributed along the wave front; if the depolarization front has constant surface density charge σ = p/A (with p as the local dipole moment and A as the surface), then in a region of radius R around the electrode, a total dipole moment P = σ π R 2 2 is present, leading to a potential proportional to P / R 2 = σ π 2 , which is finite. The absence of singularity in the potential is reflected in our analytical result, both in the homogenized case (Eq. 22) and the full anisotropic series solution (Eq. 31), see Figures 2, 4. The multi-valuedness of the arctan function accounts for both the discontinuity of Φ within the myocardial wall and the continuity of Φ outside it. The underlying mathematical object is a branch cut (Arfken and Weber, 1995), see also a recent application of this mathematical concept in cardiology as phase discontinuity and phase defects in the studies by Arno et al., 2021 and Tomii et al., 2021.

At large distances, we can find by which power law unipolar and bipolar voltages decay, using the Taylor series arctan(ξ) ≈ ξ − ξ ³/3 + ⋯ for small ξ. From the homogenized solution (Eq. 22), we have the following for an electrode at a large distance h = R from the endocardium: Φ ̃ unipolar t = ( ϕ m a x − ϕ r e s t ) g i , x x 2 π g ̃ − v t R − − v t R + L + O R − 3 ∝ L R 2 + O R − 3 . (40)

A similar argument is held for each term in the full anisotropic series solution. Thus, the unipolar signal decays as R ⁻² with a distance from the myocardium for large R. Similarly, we find the following for parallel and perpendicular bipolar recordings: V bip , ‖ ∝ d R 2 + O R − 3 , V bip , ⊥ ∝ d R 3 + O R − 4 . (41)

The different decay law for both cases should be noted. This is rooted in the fact that holding the catheter perpendicular to the myocardial wall effectively takes the derivative with respect to the distance from myocardium (R), according to (Eq. 39). As the derivative of R ⁻² is −2R ⁻³, this explains the cubic power decay of V _bip,⊥. Both bipolar recordings are also proportional in the leading order to interelectrode distance d.

The analytical solution presented here was designed for a planar wave front that remains perpendicular to endo- and endocardial boundaries. This is a simplification, as the correct boundary condition to V _m , i.e., (Eq. 2j), will cause a V-shaped wave front due to bath loading (Roth, 1991). This effect is small in the studied regime, see Figure 2C. Future works should also address the bath-loading effect, which requires the inclusion of the wave front shape into the full PDE solution, which may be possible using our present results. Other extensions could, e.g., address rotational anisotropy and spatially varying wall parameters, such as anisotropy, conduction velocity, thickness, and repolarization.

A clinical study by Glashan et al. (2018) suggests that it is important to account for the wall thickness when analyzing electro-anatomical voltage mapping data. A linear dependence of bipolar and unipolar voltages on the wall thickness was proposed. Our results agree with the positive dependence of voltage on thickness; however, the exact dependence is more complicated than being linear, see (Eq. 35). In Figure 13, we see that the amplitude increases with increasing L both for the QRS complex and T-wave. For large L, the epicardium has a minimal influence and the difference becomes low; therefore, we see a saturation in V _bip(L) curves, i.e., a horizontal asymptote. The bipolar electrograms observed at a position are in leading orders a difference of two contributions determined by the intersection of the propagation wave front with the boundaries of the medium. In summary, the addition of anisotropy and a three-layered medium does not qualitatively change the explanation given by the solid angle theory (Holland and Arnsdorf, 1977).

From the analytical solution for the unipolar voltage, a difference can be taken into account to obtain bipolar voltages. These signals vary in polarity between the parallel measurements to the wave front and are perpendicular to the wall. The bipolar signals measured at intermediate angles interpolate between these extremes and exhibit neither an even nor odd symmetry in time.

Anisotropy of the cardiac wall acts in different manners on iEGMs: by changing the conduction velocity and by deflecting electrical field lines at the myocardial boundary, differently from the isotropic case. The latter effect causes a different apparent tissue thickness ℓ and also affects the prefactor in our formula (Eq. 31). In Figure 9, it is clear that ψ has a pronounced effect and the amplitude decreases approximately threefold if the wave is not propagating parallel but is rather perpendicular to the myofibers. We conclude that if clinicians want to use a threshold on the unipolar or bipolar amplitude to delineate viable tissue regions, not only the wall thickness, but also the local myofiber orientation (relative to the wave propagation) should be taken into account.