Tony’s Take, November 2024

This month’s topics:

Topology and atrial tachycardia.

According to a November 5 article in the journal Circulation: Arrhythmia and Electrophysiology, topology can help understand atrial flutter, a condition where the heart beats too fast, and can help plan remedial surgery.

Atrial flutter is often caused by a disorder of the heart conduction network called macroreentrant atrial tachycardia. The electrical signals that govern contractions of the atria (the two upper chambers of the heart) short-circuit, pathologically accelerating the heartbeat.

To understand where topology comes in, we need some information about the heart and the heart conduction network.

Schematic diagram of the heart conduction network. The four chambers of the heart are labeled LA, RA (left atrium, right atrium) and LV, RV (left and right ventricles). The heart’s autonomous nervous system (orange) is controlled by the sinoatrial SA node and the atrioventricular AV node. The time (in seconds) to activate the different regions of the heart is indicated by green arrows and numbers. Image by Dr. Richard E. Klabunde, used with permission.

The two atria receive blood through veins from the lungs (LA) and the rest of the body (RA). The sinoatrial node SA is a cluster of cells that functions as the pacemaker. Once a cycle it initiates a depolarization wave that spreads through the atria, making them contract and expel blood into the ventricles. The atrioventricular node AV is a segment of tissue that delays this signal, giving the blood time to move, and then sends it on through the labeled pathways to the ventricles, which contract and expel blood through arteries towards the lungs (RV) and the rest of the body (LV).

Propagation of depolarization and repolarization, shown by colors superimposed on a frontal view of the heart. In each cycle, a depolarization wave (red) sweeps through the heart. Cells go from resting potential (unhighlighted) to depolarized (blue), and then repolarize (yellow). Note initiation at the SA node, and delay at the AV node. Image credit: Patrick J. Lynch, medical illustrator, used under CC BY 2.5, via Wikimedia Commons.

One way this scenario can go askew is for part of the depolarization wave to recycle around one of the atria, and to cause additional rapid contractions of that atrium. This is the “reentry” in reentrant atrial tachycardia (literally, speeding up of the heart), a potentially dangerous condition. Lead author Mattias Duytschaever and colleagues explain in their article how a topological analysis can give a useful description of this pathology and of steps to cure it.

As suggested by the depolarization wave image above, heart conduction is a 2-dimensional phenomenon. From this point of view the chambers of the heart function as surfaces. For example the left atrium, as a surface, is topologically a 2-dimensional sphere with three holes (the entry points for the two branches of each pulmonary vein are presumably close enough so that, in terms of the conduction on the surface, they constitute a single hole).

Left: Schematic of the left atrium, a sphere-like surface with one large hole (MV) at the bottom, and two pairs of small holes (RPV and LPV) at the top. Right: Sphere with three holes.
For the study of conduction, the left atrium is a surface with boundary: a sphere with three holes. The holes correspond to the right and left pulmonary veins (RPV, LPV) and to the mitral valve (MV) separating the atrium from the left ventricle. Image by Tony Phillips.

The first step in the analysis (concentrating on left atrium tachycardia) is to visualize the conduction pattern as a field of phases, assigning to each point on the surface an angle between $0$ and $2\pi$ representing when in the cycle that point is first hit by the depolarization wave. Explicitly, if LAT is the local activation time at that point, and TCL is the tachycardia cycle length, the authors define the phase at that point as $\phi = {2\pi}\frac{\rm LAT}{\rm TCL}$. Next they define the topological charge of an area on the surface as the total change in phase going counterclockwise around a curve enclosing that area, and the topological charge of a boundary component as the change in phase going counterclockwise around that component.

Three ellipses with various topological charges. Left: TC = -1, phases propagate clockwise as you move counterclockwise around the ellipse. Center: TC = 0, all points on the ellipse have the same phase. Right: TC = 1, phases propagate counterclockwise as you move counterclockwise.
Examples of areas A with topological charge
TC(A)$=-1, 0, +1$. We associate to each point a circle of possible phases, and picture a field of phases by assigning to each point a line segment in its circle, at the appropriate angle. TC(A)$=0$ happens, for example, when all the points on the boundary of A experience depolarization at the same time. Image by Tony Phillips.

The presence of left atrium tachycardia means that the depolarization wave circles around somewhere on the surface. This means that there must be a curve surrounding an area with topological charge $+1$ or $-1$. (The area within the curve must contain a hole or a point that does not conduct the wave, such as a scar.) The authors apply what they call the index theorem to assert that there must also exist another area with the opposite topological charge.

    • The index theorem. Suppose a surface with boundary is cut up by curves into regions A1, A2, $\dots$ , An. If a field of phases is defined on the entire surface, then the topological charges of A1, A2, $\dots$ , An, plus the topological charges of the boundary components, must add up to zero.

      The polygon ${\rm A}_1$ has vertices ${\bf x}_1, {\bf x}_2, {\bf x}_3, {\bf x}_4, {\bf x}_5$; the adjacent polygon ${\rm A}_2$ has vertices ${\bf x}_1, {\bf x}_2, {\bf x}_6, {\bf x}_7$. Image by Tony Phillips.

Proof. Suppose the surface includes the adjacent curvilinear polygons ${\rm A}_1$ and ${\rm A}_2$ as in the diagram. The topological charge of ${\rm A}_1$, i.e. the total rotation of the phase field around the boundary of ${\rm A}_1$, is the sum of its rotations along the edge ${\bf x}_1 {\bf x}_2$, along the edge ${\bf x}_2 {\bf x}_3$, etc., always moving counterclockwise. Similarly the topological charge of ${\rm A}_2$ is the sum of rotations along the edges ${\bf x}_2 {\bf x}_1$, ${\bf x}_1 {\bf x}_7$, ${\bf x}_7 {\bf x}_6$ and ${\bf x}_6 {\bf x}_2$, again moving counterclockwise. Note that the rotation along ${\bf x}_2 {\bf x}_1$ is the negative of the rotation along ${\bf x}_1 {\bf x}_2$. Each edge on the surface, including those along the boundary, will contribute twice to the sum of the topological charges, with opposite signs. So the total sum must be zero.

The authors call “critical boundary” (CB) a curve, usually a boundary component, around which the phase field has non-zero rotation. As a consequence of the index theorem, reentrant atrial tachycardia implies that there will be two CBs; connecting one to the other by ablation (destroying a ribbon of surface between them) reestablishes, in a majority of cases, normal heart rhythm. The article includes the results of a clinical study involving 131 patients (106 with left atrium reentrant atrial tachycardia), at the St. Jan Hospital Bruges, whose diagnosis and treatment was informed by this research.

“Thermodynamic Linear Algebra”

A system of oscillating springs can help solve linear equations. That’s the conclusion of a November 5 article in Nature‘s new journal, npj Unconventional Computing. The article’s eight authors argue that linear algebra is central to many scientific algorithms and accelerating its implementation would be very valuable. The new paper, titled “Thermodynamic Linear Algebra,” offers a “physics-based computing paradigm based on classical thermodynamics.”

Let’s look at how the thermodynamic paradigm works. We focus on the classic linear algebra problem of solving a system of linear equations. The one-dimensional case is extremely simple: We have a single equation $ax=b$, where the solution is $x=\frac{b}{a}$. To apply our paradigm to this case, we would construct a mechanical system with potential function $U=\frac{1}{2}ax^2-bx$. For example, it might be a unit mass suspended by a spring with spring constant $a$ in a gravitational field $b$ (illustrated below). We give the system an initial kick, and wait. There is no damping in this system, so it will keep bouncing forever. But the system will evolve to an equilibrium state where its average configuration corresponds to the minimum value of the potential energy function, irrespective of the particular initial conditions chosen. (This is the thermodynamic minimum energy principle.) The minimum of $U$ occurs where $\frac{dU}{dx}= ax-b =0$, i.e. at $x=\frac{b}{a}$, the solution to our algebraic problem.

The mechanical system consists of a unit weight hanging on a spring with spring constant $a$, attracted downward by a gravitational field with gravitational constant $b$. The potential function for this system, $U=\frac{1}{2}ax^2-bx$, is graphed on the right. When the system reaches thermodynamic equilibrium the measured values of $x$ will fall in an interval (grey shading) centered about $x=\frac{b}{a}$. Image by Tony Phillips.

In practice the paradigm is applied to huge systems. The $N$ unknowns $x_1, \dots, x_N$ are amalgamated into a vector ${\bf x}=(x_1,\dots,x_N)$, the coefficient $a$ becomes an $N\times N$ matrix $A$ of coefficients $(a_{ij})$, the constant $b$ becomes an $N$-vector ${\bf b}=(b_1,\dots,b_N)$, and $ax=b$ becomes the matrix equation $A{\bf x}={\bf b}$. The authors explain how we can restrict ourselves to the case where $a_{ij} = a_{ji}$, i.e., where the matrix is symmetric. This allows a mechanical implementation in which the coefficient $a_{ij}$ represents the spring coupling between unit masses at $x_i$ and $x_j~$, while $a_{ii}$ is the spring constant associated to the mass at $x_i$, and the vector ${\bf b}$ represents an $n$-tuple of “gravitational fields,” one for each mass.

This device has potential energy function $$U(x)= \frac{1}{2}{\bf x}^{\top}A{\bf x}-{\bf b}^{\top}{\bf x}$$ generalizing $\frac{1}{2}ax^2-bx$, where ${\bf x}$ is written as a column vector and ${\bf x}^{\top}$ is its row-vector transpose; similarly for ${\bf b}$ and ${\bf b}^{\top}$. Just as in the 1-dimensional case, the system is given an initial kick and again, when it reaches thermodynamic equilibrium, the average value (over time) of the coordinates will give the location of the minimum of the potential energy function. Locating this minimum involves a similar calculation to the one giving $x=\frac{b}{a}$, except with partial derivatives, and as before the coordinates $(x_1,\dots x_N)$ of the minimum are exactly the solution of the linear system we started with.

The authors include a diagram of their algorithm.

Flow chart depicting the following process: Encode system as oscillating springs, wait for equilibrium time t0, extract trajectory of the system, integrate the dynamics for a time interval starting at t0, and then estimate the solution based on the average of the dynamics.
A diagram of the algorithm for solving the system of linear equations $A{\bf x}={\bf b}$ (along with a companion algorithm for estimating the inverse matrix $A^{-1}$). The “extract trajectory” and “integrate dynamics” boxes represent the averaging process at the end of the algorithm. Image from the Open Access article licensed under a Creative Commons Attribution 4.0 International License.

—Tony Phillips, Stony Brook University