## Abstract

Through a detailed mathematical analysis we seek to advance our understanding of how cardiac tissue conductances govern pivoting (spiral, scroll, rotor, functional reentry) wave dynamics. This is an important problem in cardiology since pivoting waves likely underlie most reentrant tachycardias. The problem is complex, and to advance our methods of analysis we introduce two new tools: a ray tracing method and a moving-interface model. When used in combination with an ionic model, they permit us to elucidate the role played by tissue conductances on pivoting wave dynamics. Specifically we simulate traveling electrical waves with an ionic model that can reproduce the characteristics of plane and pivoting waves in small patches of cardiac tissue. Then ray tracing is applied to the simulated pivoting waves in a manner to expose their real displacement. In this exercise we find loci with special characteristics, as well as zones where a part of a pivoting wave quickly transitions from a regenerative to a non-regenerative propagation mode. The loci themselves and the monitoring of the ionic model state variables in this zone permit to elucidate several aspects of pivoting wave dynamics. We then formulate the moving-interface model based on the information gathered with the above-mentioned analysis. Equipped with a velocity profile *v*(*s*), *s*: distance along of the pivoting wave contour and the steady- state action potential duration (APD) of a plane wave during entrainment, *APDss*(*T*), at period *T*, this simple model can predict: shape, orbit of revolution, rotation period, whether a pivoting wave will break up or not, and whether the tissue will admit pivoting waves or not. Because *v*(*s*) and *APDss*(*T*) are linked to the ionic model, dynamical analysis with the moving-interface model conveys information on the role played by tissue conductances on pivoting wave dynamics. The analysis conducted here enables us to better understand previous results on the termination of pivoting waves. We surmise the method put forth here could become a means to discover how to alter tissue conductances in a manner to terminate pivoting waves at the origin of reentrant tachycardias.

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## Acknowledgements

Funding from the Whitaker Foundation. Support provided by the Center for Computational Research from the University at Buffalo. Computational resources from the Texas Advanced Computer Center, Grant: TG-BCS110013. Funding from Complex Biosystems Inc. We thank an anonymous reviewer to: help us improve our model formulation, and make recommendations to clearly introduce our findings to analysts in the field.

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## Appendices

### Cell Model

### 1.1 Ionic Currents of the Cell Model Shina (Total of 6)

Below is the detailed formulation of each current in shina

\(J_b\): Background potassium, Luo and Rudy (1991)

(a) \(J_{Na}\): Sodium current, Ebihara and Johnson (1980), j-gate, Beeler and Reuter (1977). See Fig. 10.

\((b) J_{Si}\): Slow inward current, Beeler and Reuter (1977). See Fig. 11.

(c) \(J_{K}\): Delayed rectifier, Matsuura et al. (1987). See Fig. 12.

\(J_{Kp}\): Plateau potassium current, Yue and Marban (1988). See Fig. 13

(d) \(J_{K1}\): Inward rectifier, Luo and Rudy (1991). See Fig. 13a.

### 1.2 Note on the Sell Model:

The cell model employed here is the same that generated stationary pivoting waves in Beaumont et al. (1998). The model is termed shina which stands for a shift of \(J_{Na}\) kinetics. It emphasizes the important role played by \(J_{Na}\) kinetics. Current kinetics and APD have been adjusted to generate stationary pivoting waves as experimentally recorded in thin sheet of cardiac tissue (Beaumont et al. 1998). The model has 2 versions Shina\(\langle g_K10\rangle \) and Shina\(\langle g_K75\rangle \) where \(\bar{g}_K\) conductance been adjusted to generate realistic (Shina\(\langle g_K10\rangle \)) and abnormally brief (Shina\(\langle g_K75\rangle \)) APDs. Version Shina\(\langle g_K10\rangle \) generates stationary pivoting waves with break up and Shina\(\langle g_K75\rangle \) without break up. Historically, Shina\(\langle g_K75\rangle \) was employed in an analysis to find the conditions required to generate stationary pivoting waves in an ionic model. See (c) below for the changes in \(\bar{g}_K\) conductances. (a): As in Luo and Rudy model (Luo and Rudy 1991), the j-gate of the sodium current was added to the Ebihara and Johnson formulation (Ebihara and Johnson 1980). (b): The Beeler and Reuter (1977) formulation for \(J_{Si}\) was modified by dividing the maximal conductance by 2, and by fixing \(E_{Si}\) to 80 mV. (c): The delayed outward rectifier \(J_K\) has a voltage-dependent conductance exhibiting an inward rectification. The parameters of this current were obtained fitting experimental data published by Matsuura et al. (1987). In this study the maximal current \(\bar{g}_K(u)\, (u-E_K)\) is the origin of tail currents produced by voltage clamp pulse elicited at various activating potentials. In that study \(E_K=-75\, \mathrm{mV}\). The maximal current was divided by \(u-E_k\) and fitted with a B-Spline expansion. The current is in *nA* and for a cell. Approximating the cell surface by a cylinder 100 \(\mu m\) long and 10 \(\mu m\) in diameter, the current should be multiplied by a factor of 30 to convert it in \(\mu \mathrm{A}/\mathrm{cm}^2\). This factor is replaced by either 10 or 75 in the models Shina\(\langle g_K10\rangle \) and Shina\(\langle g_K75\rangle \) (cell size can vary up to a factor 4 (Raba et al. 2013)). The time constant was obtained fitting \(1/(\alpha (u)+\beta (u))\) with a B-Spline expansion. Figure 12 shows the steady state, time constant and maximal current (\(\bar{g}_K\, (u-E_K)\)) of this current. In Shina \(E_K=-78\). Table 5 gives the coefficients of the B-Spline expansions for the conductance and time constant. The note in appendix explains to a reader unfamiliar with such representation how to use it. (d): We used the Luo and Rudy formulation of \(J_{K1}\) (Luo and Rudy 1991) with \(E_{K1}=-87.9\) mV and \(K_o=5.4\)*mM*. But we multiplied the conductance by 2 to make it more representative of the guinea pig current (Beaumont et al. 1998; Samie et al. 2001). The *background current* for shina and LR-I, i.e., sum of all time-independent currents, is compared in Fig. 13a.

Compared to LR-I, Shina has a lower plateau, slower cell upstroke velocity, 131 *V* / *s*, compared to 421 *V* / *s* for LR-I, lower traveling wave upstroke velocity 60 *V* / *s* compared to 264 *V* / *s* for LR-I, and lower propagation velocity, 33 \(\mathrm{cm}{/}\mathrm{s}\) compared to 50 cm/s for LR-I, all of which can be appreciated in Fig. 13a, b. Most important due to the shift in \(J_{Na}\) kinetics, shina has a higher take-off potential (potential at which positive feedback at the origin of the upstroke starts), which makes this model more sensitive to a change of load. With Shina, a load increase can reduce the upstroke velocity to an extent where the sodium current is no longer activated. This does not occur with LR-I. That fact was noticed in a previous study (Beaumont et al. 1998). With LR-I no matter how one reduces APD, pivoting waves never exhibit an excitable gap. Wave front and wave tail are always adjacent to one another. This is unrealistic because excitable gaps have been measured experimentally (Beaumont et al. 1998).

### 1.3 Note on Approximations with B-spline Basis:

Figure 14 shows a B-Spline function and a basis formed with it. The \(u_n\) are knots on which B-Splines of the basis are centered. They define intervals \(e_n\). Each B-Spline is generated with elementary splines \(b_i(u)\)\(i\in [0,3]\) as follows:

with \(\bar{u}\) the normalized parametric coordinate, and where \(u_a, u_b\) are the end points of an element. Thus, on a given segment, lets say \(e_4\) delimited by \(u_4, u_5\) a function *f*(*u*) approximated in that basis is expressed by

\(\zeta _n\) is the coefficients of the expansion. Each B-Spline spans 4 intervals; thus, a given interval is crossed by the support of 4 adjacent B-splines. To insure this relation is preserved at each end interval a ghost knot is introduced at both ends. These are the first and last coefficients of the expansion given in Table 5 which are not assigned a \(u_n\).

### Curvature Along the Pivoting Wave Arm

Curvature \(\kappa (s)\) along the pivoting wave arm is by definition

Taking the partial derivative of \(\varvec{\rho }(s,t)\) with respect to *s*,

where the superscript prime denotes partial derivative with respect to “*s*.” We notice that

a key equation in the determination of the interface shape. Taking the derivative with respect to “*s*” of \(\varvec{\rho }(s)\) a second time we have

From the above and (B.1),

Recalling that

where \(\bar{s}\): coordinate of for which \(v(\bar{s})=0\), and the last term comes from taking the derivative with respect to “*s*” of (24), i.e.,

We have that

where

Consequently \(\kappa (\bar{s})\) may not be null, but will always be small for the cases that interest us. To see this consider the rate of change of the \(\kappa ^2(s)\) along the pivoting wave arm, i.e.,

where

Assuming *v*(*s*) can be approximated by,

around \(\bar{s}\), then for such velocity profile

which is deduced taking the derivative of (22) with respect to “*s*,” i.e.,

Consequently, for such interface \(a^{\prime }(\bar{s},t) = 0\), and

Thus, the curvature has an extremum at \(s=\bar{s}\). That extremum is also a minimum because it is between two maxima. Indeed as \(s \rightarrow \pm \infty \), \(\kappa (s) \rightarrow 1/\rho (s)\, \rightarrow 0\). Clearly from the wave shape \(\kappa (s)\) has only 2 maxima, one along the wave front (near the thin rim) and one along the wave back (distal to the thin rim where the tail velocity picks up). The extremum at \(s=\bar{s}\) is between these two maxima and can only be a minimum. Furthermore, due to the diffusive effect of the Laplacian operator, \(\kappa (\bar{s})\) of traveling waves in the heart will always be small. If the pivoting wave arm is nearly flat in the vicinity \(s=\bar{s}\), \(\rho (s)\) in that vicinity is expressed by

The curvature formula (B.3,B.5) holds for this case and gives \(\kappa (\bar{s})=0\). The close up on a typical pivoting wave tip generated by a ionic model in Fig. 4 supports this view.

The coordinate \(s=\bar{s}\) is also an inflexion point. Consider \(\varvec{\rho }^{\prime }(s,t)\) expressed with components along and perpendicular to \(\varvec{\rho }(s,t)\). Then since

the \(\varvec{\rho }^{\prime }(s,t)\) component perpendicular to \(\varvec{\rho }(s,t)\) is against and along \(\varvec{\rho }(s,t)\) for \(s<\bar{s}\) and \(s>\bar{s}\), respectively. Thus, \(\varvec{\rho }(s,t)\) approaches \(\varvec{\rho }(\bar{s},t)\) tangentially from below (for \(s<\bar{s}\)) and above (for \(s>\bar{s}\)) which is the signature of an inflexion point.

The coordinate \(\bar{s}\) is also the closest to the center of rotation because \(\rho ^{\prime }(\bar{s},t) \,=\, 0\) and \(\rho ^{\prime }(s,t) \,>\, 0\) for \(|s-\bar{s}|>0\). Consequently this coordinate demarcates a transition where the wave progresses toward and recedes from the center of rotation. In the ionic model such transition can only occur if the sum of all ionic currents transitions from inward to outward.

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Beaumont, J. Dynamics of Pivoting Electrical Waves in a Cardiac Tissue Model.
*Bull Math Biol* **81**, 2649–2690 (2019). https://doi.org/10.1007/s11538-019-00623-y

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DOI: https://doi.org/10.1007/s11538-019-00623-y