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Complex Structure and Dynamics of the Heart

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

As indicated in the introduction, in this section, the generic view of cardiac tissue as an excitable medium is adopted to investigate how activation patterns in these systems can be characterized. While the approach chosen here is purely theoretical, in the long run, such characterization might aid the quantitative assessment of the turbulence and complexity of cardiac arrhythmias in an experimental situation and thus contribute to the development of adaptive control strategies which make use of this information.

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Notes

  1. 1.

    If diffusion only acts on the variable \(u\) as in the original model, \(\frac{\partial ^{2} }{\partial y^{2}}\) would correspondingly only act on the \(\delta u\)-component of \(\delta \mathbf {U}\), which would lead to a different \(\mathbf {L}\) and would invalidate the following argument.

  2. 2.

    This protection is caused by the waves that are periodically emitted outwards and prevent any interaction in the opposite direction. Indeed, there is a very weak electrotonic coupling even in retrograde direction which decays exponentially with distance from the boundary, thus quickly vanishing from the accessible range of numerical accuracy.

  3. 3.

    Linear stability analysis can therefore be used to determine the modes that are responsible for instabilities, e.g. meandering and alternans modes for a rigidly rotating spiral in the unstable regime [14, 16].

  4. 4.

    One could also imagine that the spiral wave is just attracted initially by the heterogeneity but its exact position is still arbitrary, which would result in a preservation of the local symmetry.

  5. 5.

    It should be emphasized that the number of phase singularities which are detected depends on the chosen origin \((u^*,v^*)\) necessary to define the phase. Although visual inspection of the dynamics for the parameters considered in this section confirms that the detected phase singularities are reasonable, the numbers can deviate from those reported in Ref. [25].

  6. 6.

    Infinitesimal deviation \(\delta \mathbf {u}\) from the resting state and infinitesimal electric fields are equivalent assumptions, as the resting state is known to be stable.

  7. 7.

    Equation (3.16) is identical to the ad-hoc ansatz used in Refs. [28, 31] to assess electric-field induced membrane potential changes for circular obstacles. However, those publications use a different sign in Eq. (3.16b). Private communication with the main author, Alain Pumir, led to the finding that the latter is a misprint.

  8. 8.

    The size and curvature of the tissue domains can be normalized by \(\lambda \), as explained in Sect. 3.2.3. Therefore, the actual parameters used for the Fenton-Karma model do not alter the results. The only condition is that the field strength used is small enough to preserve the validity of Eq. (3.16a, 3.16b).

  9. 9.

    As \(\mathbf {E}\) has units of a potential divided by a length, applying the same transformations as for \(\mathbf {x}\) and \(e\) automatically leads to a unit \(\mathbf {E}^*\), thus being identical to \({\hat{\mathbf {E}}}\) defined above.

  10. 10.

    In fact, for the infinite stack, i.e. \(I=\mathbb R\), the electric field can also have a \(z\)-component, which does, however, not change the solution, because all normal vectors have vanishing \(z\)-components according to Eq. (3.22)

  11. 11.

    As very small domains are equivalent to very large \(\lambda \) (cf. Sect. 3.2.4), this is also clear directly from Eq. (3.16a), which turns into a homogeneous Laplace equation for large \(\lambda \).

  12. 12.

    As indicated in Sect. 2.1.3, the assumption of a constant electric field vector determining the boundary condition of Eq. (3.16b) and ultimately the \(\cos \)-term on the right hand side of Eq. (3.29) is only an approximation. In case a more realistic boundary condition is determined in the future and the right hand side of Eq. (3.29) is replaced with some arbitrary function \(f(\varphi )\), the mathematical strategy as presented here could be applied by writing \(f\) as a sum of its Fourier modes \(a_n\cos (n\varphi )\), \(b_n\sin (n\varphi )\). With the ansatz \(e^*_{a,b}=cg(r^*)\cos (n\varphi )\) (or similarly with \(\sin \)), following the derivation would lead to solutions involving \(K_n\) and \(I_n\), where \(n=1\) is only the special case considered above. As the Neumann boundary condition of Eq. (3.29) is additive, the solution could then be written as a sum of these partial solutions.

  13. 13.

    Note that, despite the adaptive space step \(h\) of the simulations, the numerical realization of the boundaries using the phase-field method slightly underestimates the electric field-induced effect in the asymptotic case (see Fig. 3.27). Assuming this trend holds for finite pulse duration, the critical field strength in the corresponding field strength range shown in Fig. 3.35 could be slightly too high.

  14. 14.

    See also Ref. [46] and references therein.

  15. 15.

    One bin containing all sizes of blood vessels beyond the upper limit of the plotting range in Figs. 3.44a and 3.43a was included in the prediction. This corresponds to blood that are large enough to be activated already at the lowest predicted field strengths.

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  1. Research Group Biomedical Physics, Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany

    Philip Bittihn

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Bittihn, P. (2015). Results. In: Complex Structure and Dynamics of the Heart. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-12232-8_3

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