A mathematical model of the unidirectional block caused by the pulmonary veins for anatomically induced atrial reentry

Abstract

It is widely believed that the pulmonary veins (PVs) of the left atrium play the central role in the generation of anatomically induced atrial reentry but its mechanism has not been analytically explained. To understand this mechanism, a new analytic approach is proposed by adapting the geometric relative acceleration analysis from spacetime physics based on the hypothesis that a large relative acceleration can translate to a dramatic increase in the curvature of a wavefront and subsequently to conduction failure. By verifying the strong dependency of the propagational direction and the magnitude of anisotropy for conduction failure, this analytic method reveals that a unidirectional block can be generated by asymmetric propagation toward the PVs. This model is validated by computational tests in a T-shaped domain, computational simulations for three-dimensional atrial reentry and previous in-silico reports for anatomically induced atrial reentry.

Appendices

Appendix A: Trajectory of the cardiac action potential on the surface ℳ

To obtain the trajectory, we use the wavefront S _j at time t _j . Let n be the parameter of S _j such that S _j =S _j (n). Let us call n the selector parameter. Let S ₀=S ₀(n) be the wavefront at time t ₀. At t ₁=t ₀+Δt, we will have another wavefront S ₁=S ₁(n), which is also parameterized by the same selector parameter n. See the left plot of Fig. 3. Suppose the wavefronts S ₀ and S ₁ are both continuous and differentiable. For an arbitrary point S ₀(n) on the wavefront S ₀, let the propagational velocity be v ⁰(n). Then, the trajectory will be defined similarly to that in classical mechanics [70]:

Definition

For an interval I ∈R, a trajectory P ₀(t) is a differentiable mapping P ₀: $I\to \mathcal {M}^{e}$ to satisfy:

$$\left. {\frac{\partial P_{0} }{\partial t}} \right|_{S^{0}\left(n \right)} =\mathrm{v}^{0}\left(n \right)= {\lim_{\Delta t\to 0}} \frac{S_{1} \left(n \right)-S_{0} \left(n \right)}{\Delta t}.$$

For example, ${P_{0}^{h}} \left (t \right )$ let be the trajectory for the selector parameter 0 in k discrete time steps. For the final time T, let Δt be the time interval Δt = T/k. In isotropic and homogeneous media, the propagational direction is normal to the wavefront but in the presence of anisotropy, it may be aligned in the direction of anisotropy. For sufficiently small Δt, the trajectory at time t ₀+Δt passing S ₀(n) is:

$${P_{0}^{h}} \left({t_{0} +\Delta t} \right)=S_{0} \left(n \right)+\mathbf{v}^{0}\left(n \right)\Delta t=S_{1} \left(n \right).$$

Note that the trajectory ${P_{n}^{h}}$ meets the wavefront S _j with the same parameter n, which is why n is called the selector of a trajectory. By repeating this procedure for t ₀−Δt, we can construct a continuous and differentiable curve P _k in a neighborhood of S ⁰(n) such that for a sufficiently small δ, we have:

$${~}_{\Delta t \to 0}^{\lim}|{P_{k}^{h}} \left({t_{0} +n\Delta t} \right)\to P_{k} \left. {\left({t_{o} +n\Delta t} \right)} \right\| < \delta , \quad \text{for}\, \, \text{all}\, \, n$$

With the affine parameter $\lambda = at + b, a, b \in \mathbb {R}^{+}$, or the time that is measured by the moving body’s clock, the trajectory is represented as P $_{k} =P_{k}(\lambda ) \in \mathcal {M}$ as well as the wavefront S _j =S _j (n) $\in \mathcal {M}$. What remains is to prove the existence and uniqueness of a trajectory for each point on the curved surface $\mathcal {M}$.

Proposition A

Consider a curved element $\mathcal {M}^{e}$ that is locally Euclidean and a two-dimensional manifold. For any point p on the wavefront S _j that is piece-wise continuous and differentiable in $\mathcal {M}^{e}$, there exists a unique and piece-wisely differentiable trajectory P _k in the neighborhood of p.

Proof

According to the above definition of the trajectory, it is sufficient to prove that, for any point p, there exists a unique vector v that is orthogonal in the sense of the metric g _ij to the tangent vector of the wavefront S _j . The consideration of general orthogonality is because of the anisotropy. Let v _Sj be the tangent vector of the wavefront S _j . The uniqueness of v _Sj is provided by the assumption that the wavefront is continuous and differentiable. Since $\mathcal {M}^{e}$ is a two-dimensional manifold, there is a function $\mathcal {C}$ that maps an Euclidean element $\Omega ^{2}\in \mathbb {R}^{2}$ into the curved element $\mathcal {M}^{e}$ such that $\mathcal {C}:\Omega ^{2}\to \mathcal {M}^{e}$. Without loss of generality, let q be the point in Ω such that $\mathcal {C}(q) =p$. Let s _k be the Euclidean axis of Ω² and let ∂/∂s _k ≡v _sk be the tangent vector of the Euclidean axis for k = 1, 2. Considering the tangent vector is mapped by $\mathcal {C}$ such that [58]:

$$\mathcal{C}\left({\mathrm{v}_{s1} } \right)\equiv \mathcal{C}\left({\frac{\partial }{\partial_{s_{k} } }} \right)=\frac{\partial \mathcal{C}}{\partial_{s_{k} } }\equiv \mathrm{v}_{k}^{\prime} , \quad \text{for}\, \, \text{each}\, \, k, $$

we see that each $\partial \mathcal {C}/\partial _{s_{k} }$ becomes the tangent vector at$^{}p\in \mathcal {M}^{e}$. Then, $\left ({\mathrm {v}_{1}^{\prime } \mathrm {, v}_{2}^{\prime } } \right )$ constitute a linear basis of the tangent plane at p where the tangent vector $\mathbf {v}_{S_{j} }$ lies, by definition. Thus, by an orthogonal procedure, for example by the Gram–Schmidt process in the tangent plane [71], there exists a unique v such that v is independent of $\mathbf {v}_{S_{j} }$ and:

$$\left({\mathrm{v}, \mathrm{v}_{S_{j} } } \right)=g_{ij} .$$

It is clear to see that the uniqueness of the trajectory in the neighborhood of a point p is the direct consequence of the uniqueness of the propagational vector and the smoothness of the wavefront. For example, for an isotropic sphere with a point initialization from the pole, the wavefront is aligned along the azimuthal angle and the trajectory along the polar angle for every point. It should be pointed out that the trajectory does not have to be orthogonal to the wavefront everywhere because of the feasible presence of anisotropy on the surface.

Appendix B: Relative acceleration from a discrete model

To explain qualitatively how a large relative acceleration can be interpreted as conduction failure for the justification of the following hypothesis, we first consider stopping conditions in a discrete model consisting in individual myocardial units that represent cells or tissues by a similar procedure as cellular automata [50]. Motivated by the widely known facts on myocardial cells or tissues, we give the following properties to myocardial units:

• (i) One unit faces multiple units such as hexagonal packing.

• (ii) One unit can excite a limited number (N _max ) of neighboring units, and when they try to excite more than N _max , none of the neighboring units reach their threshold potential (TP) for excitation.

• (iii) After a unit is excited, it undergoes a recovery process, called the refractory period (RP), and cannot be excited during RP.

For the structural property (i) of myocardial tissues as reported in ref. [72, 73], the property (ii) provides the unique characteristics of excitation propagation to restrict the maximum number of excited cells that one cell can excite, known as impedance mismatch or sink-source mismatch. Property (iii) is also well known. Depending on the ionic currents of Ca ²⁺ and K ⁺, the minimum time for the recovery of excitability is expressed as a function of RP that is also correlated with action potential duration (APD) [3].

Figure 14a shows myocardial tissues modeled as hexagonal packing. A geometric-free discrete representation is shown in Fig. 14b, where a dark color indicates the cell is excited and the white color indicates the cell is in the excitable state. Figure 14c shows the wavefront to represent the line of excited tissues at each time.

For the hexagonal packing of myocardial units as shown in Fig. 14a, we can also represent it with a geometric-free discrete model as shown in Fig. 14b in which the number of units in each layer reflects the shape of the hexagonal packing. Starting from (A1, A2) units, the excitation propagates to (B1, B2, B3) units, (C1, C2, C3, C4) units and so forth. Note that the first letter of the unit indicates that the same layer must be excited in all of the units and the excitation occurs only in alphabetical order, i.e., a C-unit cell cannot excite a B unit or a C unit. The goal of this discrete model is to expand the concept of the unit from cell to tissue, because when a cell satisfies the aforementioned properties, a tissue being a group of cells also satisfies the same properties if we neglect the time difference of excitation within a tissue.

No matter what this one unit in the discrete model represents, if we connect excited units at every time interval, we obtain the wavefront of cardiac excitation propagation. This wavefront represents a series of myocardial units that are in the process of depolarization. Consequently, the trajectory can be defined as if the cardiac wave is a physical wave, that is, a collection of moving particles. Upon these representations and notions, we describe the stopping conditions of the excitation propagation as the following: Without a loss of generality,

Let three be the maximum number of units one can excite in the model of hexagonal packing, i.e., N _max = 3.

The first stopping condition is obtained directly from property (ii). In Fig. 14d, if the A1 and A2 units attempt to excite the B-units, then each unit needs to excite an average of 3.5 units. Consequently, the electric potential of all B-units reaches below the TP and, consequently, fails to be excited. In brief, the excitation propagation stops or conduction failure occurs if:

• [SG-i] The average number of units to be excited exceeds the maximum number of excitable units.

This stopping condition has often been cited when excitation propagation faces abrupt tissue expansion, such as a narrow cell strand connected to a large rectangular cell [52]. In the same context, we can explain why a small gap between ablation lesions, burned and dead myocardial tissues by catheter, does not allow excitation leakage. It is important to note that the excessive number of B-units represents the geometric distribution of myocardial units, which can be interpreted as geometry in physics.

On the contrary, less attention has been given to the second condition. In Fig. 14f, the geometry of the hexagonal packing is the same as that of normal propagation in Fig. 14b, although the A1 unit propagates rapidly only along the B1 unit and finally to the C1 unit. This type of propagation is common in myocardial fibers or myocardial sheets, which we generally call anisotropy [59]. Because the C1 unit is the only excited cell in C-units, the C1 unit attempts to excite all the five D-units. Let T1 be the time when the C1 unit uses all of the electric potential for the D-units. At the same time, the A2 unit proceeds to excite the (B2, B3) units and subsequently the (C2, C3, C4) units. Let T2 be the time when the (C2, C3, C4) units are all excited. There is a critical difference concerning the times T1 and T2: if T2 is earlier than T1, that is, if the (C2, C3, C4) units can also contribute to exciting D-units along with the C1 unit, then all the D-units can be excited, and the propagation continues. However, if T1 is earlier than T2, that is, if the (C2, C3, C4) units are excited, the D-units are already in RP so that the (C2, C3, C4) units have no units to excite and propagation stops at the D-units. In other words, excitation propagation also stops, or conduction failure occurs if

• [SG-ii] The excitation time of other C-units exceeds the propagation time from the C1 unit to the D-units consisting in more than the maximum number of units one unit can excite.

To simplify the above condition, we introduce another property of myocardial units:

• (iv) The excitation time by a myocardial unit is roughly proportional to the number of cells that one cell has to excite.

With the property (iv), the time T1 only depends on the propagational time from the A1 unit to the C1 unit, because the time required to propagate from the C1 unit to the D-units is fixed. Moreover, the ratio of the velocity is at maximum four times higher on fibers than the standard excitable media [59]. Thus, it is reasonable to say that the time T1 is relatively fixed. On the other hand, the excitation time of other C-units depends on the geometry of the B-units and the C-units other than the B1 unit and the C1 unit. Thus, the time T2 varies widely and is dependent upon the geometry of the C-units and the D-units.

These discrete models can be naturally interpreted in continuous wave forms. This procedure can be performed by drawing a line for excited units at each time, which represents the front of the excitation propagation, called a wavefront in light of the waves. For example, Fig. 14c displays the wavefront (solid line) and trajectories (dotted line) of the excitation propagation for the discrete model of Fig. 14b. In waveforms, the shape of the wavefront and trajectories now reflect the geometry of myocardial tissues.

Appendix C: proof of proposition 1

To derive the eikonal equation of Eq. (9), we use the unique property of cardiac excitation propagation as a traveling wave solution. The traveling wave property of the membrane potential can be in vitro observable from the almost constant shape of the traveling membrane potential in homogeneous media; it has, therefore, been widely accepted and used for various analyses for cardiac action potential propagation [74, 75]. With this traveling wave of function ψ and a wave-speed c = c(y ¹, y ²), the membrane potential u can be expressed as:

$$\begin{array}{@{}rcl@{}} u\left({y^{1}, y^{2}, \mathcal{T}} \right)=\psi\left({y^{1}-c\left({y^{1}, y^{2}} \right)\mathcal{T}, y^{2}} \right), \end{array}$$

(21)

where y ¹ is the path of the propagation and y ² is the isochrone where the membrane potential uis constant with respect to y ² such that ∂u/∂y ² = 0 as shown in the right plot of Fig. 3. Also, τ indicates the elapsed time from the wavefront with respect to the axis y ¹ and y ². Consequently, τ = 0 indicates the transition layer or just the wavefront for each λ. The definition of the trajectory for y ¹and the wavefront for y ² are also confirmed by differentiating Eq. (21) with respect to y ² to yield −(∂ψ/∂y ¹)(∂c/∂y ²)τ + ∂ψ/∂y ² = 0, which shows that ∂ψ/∂y ² = 0 at the moving frame where τ = 0. In Eq. (9), the use of the traveling wave assumption (21) yields the following equality at the transition layer where τ = 0:

$$d_{y}^{11} \frac{\partial^{2}\psi }{\partial y^{1^{2}}}+\left[ {\frac{1}{\sqrt{g}}\sum\limits_{\alpha =1}^{2} {\frac{\partial }{\partial y^{\alpha} }\left({\sqrt{g}d_{y}^{\alpha 1} \frac{\partial \psi }{\partial y^{1}}} \right)+c} } \right]\frac{\partial \psi }{\partial y^{1}}+F=0, $$

(22)

where the subscript is used to indicate that the corresponding quantity is expressed with respect to the generally non-orthogonal y ^j axis. This implies that $d_{y}^{^{21}}$ may not be zero in spite of our assumption of the orthogonal diffusivity tensor (8). Since the axis y ^j depends on the shape of the wavefront, the above eikonal equation is written in a time-dependent moving axis y ^j = y ^j(t) for τ = 0 on the curved surface. Remember that this eikonal equation is similar to the three-dimensional eikonal curvature equation by Keener [61] with one main difference Due to the difference between the orthogonal curved axis and the Euclidean axis, the component $\left ({1/\sqrt {gy}} \right )\left ({\partial \sqrt {gy}/\partial y^{\alpha } } \right )d_{y}^{\alpha 1}$ in the above equation replaces the following term in the three-dimensional eikonal curvature equation, $\left ({\partial x^{k}/\partial y^{1}} \right )\left ({\partial /\partial y^{\alpha } } \right )\left ({\partial y^{1}/\partial x^{k}} \right )d_{y}^{\alpha 1}$. Equation (22) is well defined in π but is not convenient for further analysis because the axis y ^j depends on the behavior of the propagation such as the initialization of the propagation and subsequently the shape of the wavefront and the time variable, thus it is neither orthogonal nor time-independent. Because of this, it is inconvenient to express all the tensors with the axis y ^α. Instead, we express every tensor on a fixed and orthogonal curved coordinate axis x ^α, which is independent of the direction of propagation or the direction of the wavefront.

Consider that the diffusivity tensor $d_{y}^{\alpha \beta }$ is expanded with respect to the curved axis x ^k such that [57, 58]:

$$d_{y}^{\alpha 1} \left({\equiv \sum\limits_{k, l=1}^{2} {d^{kl}} \frac{\partial y^{\alpha} }{\partial x^{k}}\frac{\partial y^{1}}{\partial x^{l}}} \right)=\sum\limits_{k=1}^{2} {d^{kk}} \frac{\partial y^{\alpha} }{\partial x^{k}}\frac{\partial y^{1}}{\partial x^{k}}, $$

(23)

where the diffusivity coefficient d ^kk is expressed with respect to x ^j and the second equality is obtained from our choice of anisotropy that is only aligned along one of the orthogonal curved axes (equation (9)). As for$\sqrt {g_{y} }$, the transformation rule by multiplying the determinant of the Jacobian J ≡ [∂x ^j/∂y ⁱ] applies as $\sqrt {g_{y} }=J\sqrt {g}$ [76]. As a result, by substituting Eq. (23) into Eq. (22), we obtain:

$$\left({\sum\limits_{k=1}^{2} {E^{k}} } \right)\frac{\partial^{2}\psi }{\partial y^{1^{2}}}+\left\{ {\sum\limits_{k=1}^{2} {\frac{1}{\sqrt{g}}\frac{\partial U^{k}}{\partial x^{k}}+c_{r} } } \right\}\frac{\partial \psi }{\partial y^{1}}+F=0, $$

(24)

where we introduced the new variables $\Lambda ^{k}\equiv \partial y^{1}/\partial x^{k}, U^{k}\equiv \Lambda ^{k}\sqrt {g}d^{kk}$ and $E^{k}=\Lambda ^{k}U^{k}/\sqrt {g}$ with the speed variable c _r that is defined as:

$$c_{r} \equiv c+\sum\limits_{k=1}^{2} {\frac{U^{k}}{\sqrt{g}}} \left\{ {\frac{1}{J}\frac{\partial J}{\partial x^{k}}+\sum\limits_{\alpha =1}^{2} {\frac{\partial }{\partial y^{\alpha} }\left({\frac{\partial y^{\alpha} }{\partial x^{k}}} \right)} } \right\}.$$

Moreover, according to the supposition (10), for sufficiently small positive constants ε ₁ and ε ₂ and for k, α = 1, 2, the following quantities are sufficiently small;

$$\left\| {\frac{1}{J}\frac{\partial J}{\partial x^{k}}} \right\| < \varepsilon_{1} \ll 1,\quad \mathrm{and}\quad \left\| {\frac{\partial }{\partial y^{\alpha} }\left( {\frac{\partial y^{\alpha} }{\partial x^{k}}} \right)} \right\| < \varepsilon_{2} \ll 1.$$

Consequently, the speed function c _r is approximately the same as c, i.e., c _r ≈ c, independent of the time variable and the geometrical factors such as d ^kk, Λ^k and g _kk .

With the proposed hypothesis, what remains is to obtain the critical factors for the relative acceleration in the action potential propagation. In the ambient space of π ^e, i.e., in R³, we regard the y ¹-axis as the trajectory and project it into the ambient space of a higher dimension [47, 51]. In eikonal Eq. (22), being originally derived in π ^e, we extend the curved surface π ^e in the neighborhood of each point in the following way: consider a curved surface as a two-dimensional submanifold embedded in three-dimensional Euclidean space z ⁱ. The function ψ(y ¹, y ²) can be extended to a function ψ(z ¹, z ², z ³) in a tubular neighborhood of the surface where the directional derivative of ψ along the surface normal is zero. Define ψ on an open subset around π ^e. Let Y ₁ : $\mathbb {R}^{2}\to \mathbb {R}^{3}$ be the coordinate map for the y ¹-axis in the ambient space$\mathbb {R}^{3}$. With the affine parameter λ and the selector parameter n, we introduce the following notation,

$$P_{n} \left(\lambda \right)=\mathbf{Y}_{1} \left({\lambda , n} \right)\equiv \left({{\gamma_{n}^{1}} , {\gamma_{n}^{2}} , {\gamma_{n}^{3}} } \right)\in \mathbb{R}^{3}, $$

where ${\gamma _{n}^{i}}$ is the component of Y ₁ for the Cartesian coordinate axis z ⁱ. Moreover, since ψ(x ¹, x ²) was extended to ψ(z ¹, z ², z ³), we can express the differentiation with respect to y ¹ by the chain rule:

$$\frac{\partial \psi }{\partial y^{1}}=\sum\limits_{i=1}^{3} {\frac{\partial \psi }{\partial z^{i}}} \frac{\partial z^{i}}{\partial y^{1}}=\sum\limits_{i=1}^{3} {\frac{\partial \psi }{\partial z^{i}}\frac{\partial {\gamma_{n}^{i}} }{\partial \lambda }, }$$

(25)

where the second equality is obtained by using the fact that ∂z ⁱ/∂y ¹ is the ith component of the tangential vector of the path Y _1, which is the same as $\partial {\gamma _{n}^{i}} /\partial \lambda$. Note that the above equality is nothing but the expression of the derivative in terms of moving frames as ∇ψ⋅v ¹ where v ¹ is the tangent vector of the y ¹-axis [46]. Moreover, differentiating again with respect to y ¹ yields:

$$\frac{\partial^{2}\psi }{\partial y^{1^{2}}}=\sum\limits_{i=1}^{3} {\left({\frac{\partial \psi }{\partial z^{i}}\frac{\partial^{2}{\gamma_{n}^{i}} }{\partial \lambda^{2}}+\sum\limits_{j=1}^{3} {\frac{\partial^{2}\psi }{\partial z^{i}\partial z^{j}}} \frac{\partial {\gamma_{n}^{i}} }{\partial \lambda }\frac{\partial {\gamma_{n}^{i}} }{\partial \lambda }} \right)} .$$

(26)

By substituting the Eqs. (25) and (26) into Eq. (24), we obtain:

$$\frac{\partial \psi }{\partial z^{i}}\left\{ {E^{k}\frac{\partial^{2}{\gamma_{n}^{i}} }{\partial \lambda^{2}}+\left({\frac{1}{\sqrt{g}}\frac{\partial U^{k}}{\partial x^{k}}+c_{r} } \right)\frac{\partial^{2}{\gamma_{n}^{i}} }{\partial \lambda }} \right\}+E^{k}\frac{\partial \psi }{\partial z^{i}\partial z^{j}}\frac{\partial {\gamma_{n}^{i}} }{\partial \lambda }\frac{\partial {\gamma_{n}^{i}} }{\partial \lambda }+F=0, $$

(27)

where the summation notation is used for easier reading and the indexes i and j are summed up to three and the index k is summed up to two. To obtain the relative acceleration equation, we differentiate the above equation with respect to the selector parameter n along the wavefront. Considering that the reaction function F is constant along the wavefront, leading to ∂F/∂n = 0, we obtain:

$$\frac{\partial \psi }{\partial z^{i}}\left\{ {E^{k}\frac{\partial^{2}n^{i}}{\partial \lambda^{2}}+\frac{\partial E^{k}}{\partial n}\left({\frac{\partial v^{i}}{\partial \lambda }+\frac{\partial^{2}\psi }{\partial z^{i}\partial z^{j}}v^{j}v^{i}} \right)+\frac{\partial }{\partial n}} \right\}\left[ {\left({\frac{1}{\sqrt{g}}\frac{\partial U^{k}}{\partial x^{k}}+c_{r} } \right)v^{i}} \right]=0, $$

(28)

where we used the variable n ⁱ for the separation vector as defined in Eq. (1) and v ⁱ for the tangent vector of the trajectory Y ₁, which are defined as $n^{i}\equiv {\partial {\gamma _{n}^{i}} }\left /{\partial n}\right .$ and $v^{i}\equiv {\partial {\gamma _{n}^{i}} }\left /{\partial \lambda }\right .$ for 1 ≤ i ≤ 3. Each upper index indicates the component of a vector in the Cartesian coordinate z ⁱ, for example, n = (n ¹, n ², n ³). In addition, we used the interchangeability of the differentiation [51] in the direction of n and λ for the above equation such that:

$$\frac{\partial }{\partial n}\left({\frac{\partial^{2}{\gamma_{n}^{i}} }{\partial \lambda^{2}}} \right)=\frac{\partial }{\partial n}\left({\frac{\partial }{\partial \lambda }} \right)\left({\frac{\partial }{\partial \lambda }} \right){\gamma_{n}^{i}} =\left({\frac{\partial }{\partial \lambda }} \right)\left({\frac{\partial }{\partial \lambda }} \right)\frac{\partial {\gamma_{n}^{i}} }{\partial n}=\frac{\partial^{2}n^{i}}{\partial \lambda^{2}}.$$

Without loss of generality, we may consider the case only when the quantity inside the bracket of Eq. (28) is zero, independent of ∂ψ/∂z ⁱ. Then we obtain Eq. (11).

Appendix D: Examples and validations of the relative acceleration analysis

In the following examples on various curved surfaces, we will display how mathematical analysis from the relative acceleration Eq. (19) coincides with computational results and how the curvature of the geometry changes the impact of anisotropy for conduction failure. For easier reading, each example is organized such that Problem means the setting of cardiac excitation propagation on a curved surface, RA analysis means the analysis and predictions from Eq. (19), Computational modeling and validation means the validation by computation simulations in the above computational scheme and some references, respectively.

1.1 D.1 Anisotropic plane

Problem 1

Consider a block of anisotropy in the middle of a plane. For a plane with −100 ≤ x ≤ 100 and −100 ≤ y ≤ 100, let the block be located in −50 ≤ x ≤ 50 and −50 ≤ y ≤ 50. Let the anisotropy be aligned with the direction of the x-axis, denoted as x-anisotropy, which expresses the diffusivity tensor d as d = ς ^xx x in the block of anisotropy and ς ^yy = 1.0 everywhere. See Fig. 15a. Also, allow the excitation to propagate in the form of a plane wave in the -x direction front of the right wall and, consequently, the wavefront is in the direction of the y-axis.

Fig. 14

Representation of stopping conditions for conduction failure in individual myocardial cells and continuous waves

Fig. 15

For ς ^xx = 4.0. Anisotropy block is located in the middle (a). After initiating from the right wall, the membrane potential (u) at T = 300.0 (b), T = 500.0 (c) from the right wall

RA analysis

Since these conditions mean that g ^xx = g ^yy = 1 and Λ^x = 1, Λ^y = 0, the relative acceleration Eq. (19) boils down to:

$$-\frac{\partial^{2}n^{i}}{\partial \lambda^{2}}=\frac{\partial \left({\log \varsigma^{xx}} \right)}{\partial y}\frac{\partial v^{i}}{\partial \lambda }+\frac{1}{\varsigma^{xx}}\frac{\partial^{2}\varsigma^{xx}}{\partial x\partial y}v^{i}, $$

(29)

where we used n = y and ∂v ⁱ/∂n = 0. From the above equation, it is obvious that a large relative acceleration can be achieved only by increasing the magnitude of anisotropy in the plane, especially at the interfaces of anisotropy where ∂ ² ς ^xx/(∂x∂y) is large. This result is compatible with the break-up conditions drawn from the kinematics approach as shown by Morozov et al. [38].

Computational modeling

In computational simulation, we similarly observe that a large relative acceleration can only occur at the interfaces of the anisotropy block, i.e., in the line of y = ± 50 for the first component and two points (50, 50) and (50, −50) for the second component. See Fig. 16a for the distribution of ς ^xx and ∂(logς ^xx)/∂y along the y-axis. With ς ^xx = 4.0, conduction failure does not happen as shown in Fig. 15b and c. However, increasing the magnitude of anisotropy up to ς ^xx = 10.0, though this large magnitude seems to be unrealistic in a biological system, leads to conduction failure in the block of anisotropy because the relative acceleration has increased significantly at the interface. Note that if the anisotropy changes slowly such that ∂ ² ς ^xx/(∂x∂y) remains bounded, then no conduction failure would be induced by the anisotropy.

Fig. 16

For ς ^xx = 10.0. The distribution of ς ^xx (solid line) and ∂(logς ^xx)/∂y (dashed line) along they-axis (a). After initiating from the right wall, the membrane potenti∂xal (u) at T = 200.0 (b), T = 300.0 (c)

Problem 2 (RA analysis and validation)

Conduction failure can also occur even inside an anisotropic block. If the direction of anisotropy is oblique to the propagational direction, then conduction failure can happen inside anisotropy. Let ς be the magnitude of anisotropy and θ be the angle with respect to the direction of the excitation propagation. For the sake of simplicity, let the magnitude of anisotropy (ς) be constant in π. For the same planar propagation as before, the relative acceleration equation is expressed as:

$$-\frac{\partial^{2}n^{i}}{\partial \lambda^{2}}=-\varsigma \tan \theta \frac{\partial \theta }{\partial y}\frac{\partial v^{i}}{\partial \lambda }+\left[ {-\frac{\partial \theta }{\partial x}\frac{\partial \theta }{\partial y}-\tan \theta \frac{\partial^{2}\theta }{\partial x\partial y}} \right]v^{i}.$$

(30)

Note that this equation implies that the action potential propagation can also be blocked by the significant variable angle θ of the direction of anisotropy as well as the magnitude. This result coincides with Davydov et al. [77], who displayed that the chiral anisotropy, which is aligned along the circumferential direction of circles, can break up the excitation propagation, but Eq. (30) suggests a host of possibilities for conduction failure by variations of the direction of anisotropy.

1.2 D.2 Anisotropic sphere

Problem 3

The relative acceleration analysis can be similarly applied to the propagation on an anisotropic sphere. However, there is a difference. Contrary to the propagation in the plane, the planar propagation is impossible. Instead, the propagation should be point-wisely initiated, for example, an initiation from the north pole. Geometry tells us that point-initialized excitation follows the line of the polar angle (θ) and the wavefront is aligned along the azimuthal angle (ϕ). For a sphere of radius r = 50.0, consider a block of anisotropy located in −π/4 ≤ ϕ ≤ π/4 and −0.6 ≤ θ ≤ 0.6. See Fig. 17a. Let the direction of anisotropy be both along the θ- and ϕ axis in general but because of the propagational direction, only θ-anisotropy will contribute to the changes of the propagation.

Fig. 17

For ς ^𝜃𝜃 = 10.0. The block of anisotropy is located in the middle π/4 < θ < 3π/4 on a sphere (a). After point-initialized from the north pole, the membrane potential (u) at T = 300.0 (b), T = 400.0 (c). 10.3 Anisotropic torus

RA analysis

Geometrically, these conditions imply that g ^𝜃𝜃 = 1/r ² and Λ^θ = 1, Λ^ϕ = 0, thus relative acceleration Eq. (19) is expressed as:

$$-\frac{\partial^{2}n^{i}}{\partial \lambda^{2}}=\frac{\partial \left({\log \varsigma^{\theta \theta }} \right)}{\partial \varphi }\frac{\partial v^{i}}{\partial \lambda }+\left({\frac{1}{\varsigma^{\theta \theta }}\frac{\partial^{2}\varsigma^{\theta \theta }}{\partial \varphi \partial \theta }+\frac{\partial \left({\log \varsigma^{\theta \theta }} \right)}{\partial \varphi }\cot \theta } \right)v^{i}, $$

(31)

where we used $\left ({{1}\left /{\sqrt {g}}\right .} \right )\left ({{\partial \left ({\sqrt {g}g^{kk}} \right )}\left /{\partial x_{k} }\right .} \right )=g^{kk}\Gamma _{mk}^{m} , m\ne k$ as proved in Appendix E. Equation (31) is similar to Eq. (29) for the plane, but the last additional component, i.e., ∂(logς ^𝜃𝜃)/∂φcotθ, appears. The above component indicates the dependency of the relative acceleration on the location of anisotropy with respect to the point of initialization. For example, for the point of initialization at the pole, let θ be the azimuthal angle of the sphere from the pole. Then, the magnitude and sign of the above component change as the location of anisotropy changes. Figure 18a displays that the above component changes signs at θ = π/2 and the magnitude of it increases as θ gets closer to 0 or π. Consequently, since the change of signs of the above component can increase or decrease the total magnitude of the relative acceleration, Eq. (31) implies that conduction failure also depends on the location of anisotropy with respect to the location of point initialization. In the perspective of the trajectory, this analysis is no surprise. With point initialization at the north pole, the trajectories diverge in the northern hemisphere up to the equator and from the equator the trajectories converge in the southern hemisphere. Thus, the use of anisotropy to increase the divergence of trajectories will add its relative acceleration only for the trajectories in the northern hemisphere and will decrease it for the trajectories in the southern hemisphere.

Fig. 18

For ς ^𝜃𝜃 = 10.0. Magnitude of the last component in Eq. (31) with respect to θ (a). The block of anisotropy is located in the bottom 3 π/4 < θ < π. After point-initialized from the north pole, the membrane potential (u) at T = 600.0 (b), T = 700.0 (c)

Computational modeling

To confirm the above analysis, a block of anisotropy is placed on the sphere such that the first interface that the propagation meets is located at θ = π/4 distance from the point of initialization, i.e., in the northern hemisphere. As shown in Fig. 17b and c, the block of anisotropy with a sufficiently large magnitude causes conduction failure. On the other hand, move the point of initialization away from the block of anisotropy such that the first interface that the propagation meets is located at θ = 3π/4 distance from the point of initialization, i.e., in the southern hemisphere. Then, the block of the same anisotropy, with the same magnitude of anisotropy as before, does not cause conduction failure as shown in Fig. 18b and c. This phenomenon seems to be well explained by the change of signs of the last component in Eq. (31) representing the convergence and divergence of the trajectories.

D.3 Anisotropic torus

Problem 4

The excitation propagation in a torus is strikingly different from the propagation in a plane or sphere in the sense that we cannot find one axis to represent the direction of propagation. For example, reminder that Λ^k≡∂y ₁/∂x _k , Λ^x = 1 and Λ^y = 0 for planar propagation in the plane and Λ^θ = 1 and Λ^ϕ = 0 for a sphere with point-initialization. However, Λ^θ and Λ^φ for a torus is not always constant. With the radius of the meridian r and the radius of the great circle R, let θ be the axis around the meridian and ϕ be the axis around the great circle. Then, a torus can be parameterized such that x = ((R + rcosθ)sinϕ, (R + rcosθ)cosϕ, rcosθ)φ, 0 ≤ θ ≤ π, −π ≤ φ ≤ π. Then, no matter where or how the excitation starts, Λ^ϕ and Λ^θ are always neither constant nor zero. This fact makes the analysis complicated but to make the problem simpler, we first consider that there is no anisotropy.

RA analysis and validation

We first consider an isotropic torus such that ς ^ϕϕ = ς ^𝜃𝜃 = 1.0. To make the problem even simpler, we suppose that the wavefront is at the angle of η with respect to θ such that Λ^θ = cos η and Λ^ϕ = sin η. Let Λ^θ and Λ^ϕ be constant along the wavefront. Since g ^𝜃𝜃 = 1/r ² and g ^ϕϕ = 1/(R + rcosθ)², relative acceleration Eq. (19) is expressed as:

$$-\frac{\partial^{2}n^{i}}{\partial \lambda^{2}}=\frac{R+r\cos \theta }{\Lambda^{\theta^{2}}\left({R+r\cos \theta } \right)^{2}+\Lambda^{\phi^{2}}r^{2}}\left[ {\cos \alpha \frac{R\cos \theta +r}{R+r\cos }v^{i}-\cos \alpha \sin \theta \frac{\partial v^{i}}{\partial \lambda }} \right].$$

(32)

Note that each component shows strong dependency on the meridian r for the fixed R. In other words, the above relative acceleration equation for an isotropic torus implies that the relative acceleration can be significantly changed by resizing the meridian r. This result coincides with the kinematic analysis by Davydov et al. [35], which demonstrated that the critical curvature for breaking-up the wave can be generated by adjusting the ratio R/r of the torus.

Problem 5

Consider anisotropy on a torus. However, for the sake of simplicity, we suppose the propagation follows locally the ϕ axis only in a small region of the torus. Let R = 100.0 and r = 50.0. Let the center of the torus be located at (0, 0, 0). The block of anisotropy is placed in the area of −50.0 ≤ x ≤ 50.0 and 0.0 ≤ y ≤ 150. If the propagation is point-initialized at (−150.0, 0, 0), then the propagation approximately follows the ϕ-axis in the area of the anisotropy block.

RA analysis

With the anisotropy of both directions and Λ^θ = 0 and Λ^ϕ = 1, we obtain:

$$-\frac{\partial^{2}n^{i}}{\partial \lambda^{2}}=\frac{\partial \left({\log \varsigma^{\phi \phi}} \right)}{\partial n}\frac{\partial v^{i}}{\partial \lambda }+\frac{\partial^{2}\varsigma^{\phi \phi}}{\partial n\partial \phi}v^{i}+\frac{2r\sin \theta }{R+r\cos \theta}\left({\frac{\partial \theta }{\partial n}} \right)\left[ {\varsigma^{\phi \phi}\frac{\partial v^{i}}{\partial \lambda }+\frac{\partial \varsigma^{\phi \phi }}{\partial^{\phi} }v^{i}} \right], $$

(33)

where we used ∂ϕ/∂n = 0 according to the assumption that the propagation follows the ϕ-axis in the area, thus n = θ. The first two components are the same as the relative acceleration generated by anisotropy in the plane but the last component is an additional component for an anisotropic torus. Because of this last component, the behavior of cardiac excitation propagation displays the following unique phenomena: the first phenomenon is that the relative acceleration depends on the θ angle. Considering 2rsinθ/(R + rcosθ) increases as θ approaches π/2, it can be predicted that there is a larger relative acceleration in the area where θ is closer to π/2. The second phenomenon is as follows; because the last term adds additional relative acceleration, the critical magnitude of ς ^ϕϕ for conduction failure is slightly less than that for anisotropy in the plane. This is possible because all the components in Eq. (33) have the same sign when ς ^ϕϕ is larger than 1.0, which is the choice of our anisotropy. This result seems to support the conjecture that the curvature of geometry can increase the effects of anisotropy on curved surfaces.

Computational modeling

The predictions of these unique phenomena in a torus can be confirmed in computational simulations. Consider a torus of R = 100.0 and r = 50.0 centered at (0, 0, 0). Let the block of anisotropy be located in −50.0 ≤ x ≤ 50.0 and 0.0 ≤ y ≤ 150.0 as shown in Fig. 19a. In Fig. 19b and c, when the propagation is approximately in the direction of the ϕ axis, the anisotropy ς ^ϕϕ = 8.0, which is less than ς ^ϕϕ = 10.0 for the previous cases, generates the largest acceleration in the area where θ = π/2. But, the block of anisotropy fails to generate a sufficiently large relative acceleration for the area where θ is small. This agrees with the firstly predicted phenomenon from Eq. (33). However, the ς ^ϕϕ anisotropy cannot block the propagation in the result. This could be just a matter of direction of the propagation. Figure 20b and c displays that additional anisotropy in another direction ς ^𝜃𝜃 can add additional relative acceleration in the area where θ is small and consequently can stop the propagation. Note that in the plane, whatever the direction of the propagation, anisotropy with a magnitude of 8.0 cannot block the propagation for conduction failures. This confirms the second predicted phenomenon by relative acceleration Eq. (33).

Fig. 19

Location of the anisotropy block (darkened) with ς ^ϕϕ = 8.0 (a). After point-initialized at the rightmost point of the above torus, the membrane potential (u) at T = 500.0 (b) and at T = 1000.0 (c)

Fig. 20

Anisotropy block with ς ^ϕϕ = 8.0 and ς ^𝜃𝜃 = 8.0. After point-initialized at the rightmost point of the above torus, the membrane potential (u) at T = 800.0 (a), T = 900.0 (b) and T = 1000.0 (c)

Appendix E: differentiation of $\sqrt{g}g^{kk}$

Lemma D

Consider any index k and the index m different from k and suppose that x ^m is in the Killing direction. Then, the following equation holds:

$$\frac{1}{\sqrt{g}}\frac{\partial \left({\sqrt{g}g^{kk}} \right)}{\partial x^{k}}=g^{kk}\Gamma_{mk}^{m} .$$

(34)

Proof

In the following calculations, the sum is taken over all the indices repeated above and below except k. Let us differentiate $\sqrt {g}g^{kk}$ with respect to the axis x _k . Using the chain rule and using $\frac {\partial \sqrt {g}}{\partial x_{k} =}\sqrt {g}\Gamma _{\mu k}^{\mu }$, we obtain:

$$\frac{\partial \left({\sqrt{g}g^{kk}} \right)}{\partial x^{k}}=\sqrt{g}\left({g^{kk}\Gamma_{\mu k}^{\mu} +\frac{\partial g^{kk}}{\partial x^{k}}} \right).$$

(35)

To replace the differentiation of g ^kk with the differentiation of the metric tensor g _kk , we use the identity of $g^{i\upmu }g_{\upmu j} ={\delta _{j}^{i}}$ to obtain:

$$\frac{\partial \left({\sqrt{g}g^{kk}} \right)}{\partial x^{k}}=\sqrt{g}g^{kk}\Gamma_{\mu k}^{\mu} +\sqrt{g}g^{kv}\left({g_{v\mu } \frac{\partial g^{k\mu }}{\partial x_{k} }} \right), $$

(36)

and the equation obtained by differentiating $g^{kk}g_{k\upmu } =\delta _{\mu }^{k}$:

$$\frac{\partial g_{v\mu } }{\partial x^{k}}g^{k\mu }+g_{v\mu } \frac{\partial g^{k\mu }}{\partial x^{k}}=0.$$

Note that this reduces Eq. (36) to the following:

$$\frac{\partial \left({\sqrt{g}g^{kk}} \right)}{\partial x^{k}}=\sqrt{g}g^{kk}\Gamma_{\mu k}^{\mu} -\sqrt{g}g^{kv}g^{k\mu }\frac{\partial g_{v\mu } }{\partial x_{k} }.$$

(37)

Replace the differentiation of g ^kk with the equalities in Γ _ijk and $\Gamma _{jk}^{i}$, which are Christofel symbols of the first kind and the second kind respectively, such that:

$$\frac{\partial g_{\mathit{v}\mu } }{\partial x^{k}}=\Gamma_{\mathit{v}k\mu } +\Gamma_{\mu k\mathit{v}} , \quad \Gamma_{\mu k}^{k} =g^{k\mathit{v}}\Gamma_{\mathit{v}\mu k} .$$

With these equalities and the condition of orthogonality of the curved axis x _k , Eq. (37) becomes:

$${\begin{array}{*{20}c} {\frac{1}{\sqrt{g}}=\frac{\partial \left({\sqrt{g}g^{kk}} \right)}{\partial x^{k}}=g^{kk}\left({\Gamma_{mk}^{m} -\Gamma_{kk}^{k} } \right), } \hfill & {m\ne k} \hfill \\ \end{array} }.$$

(38)

For surfaces of revolution such as spherical shells and toruses, the curved axis x ^m representing the rotational direction is in the direction of the Killing vector, or the Killing direction, which satisfies the condition ∂g _μv /∂x ^m = 0 [51] and consequently Eq. (34).