Research on Hans Dierckx' Research Page

Theory of rotors and arrhythmias

Thu, 01 Jun 2023 00:00:00 +0000

Fundamental building blocks

The contraction of our hearts is coordinated by a traveling non-linear wave of electrical depolarization, which locally triggers mechanical contraction of the cells. Hence, abnormal patterns lead to inefficient pumping of blood. Depending on the precise emergent pattern and where it takes place, this may lead to chronic fatigue, blood clot formation and stroke, or sudden cardiac death.

Remarkably, many of the precise patterns are still incompletely understood. The more complex patterns are a complicated interplay between wave fronts, wave backs and conduction blocks (when a front hits a wave back). Such conduction block may result in the formation of a spiral-shaped rotating pattern (also called scroll wave in 3D, or rotor by medical doctors) that sustains itself and was seen in tachycardia. However, experimentally observed rotors have shorter lifespan that those in simulations. In case of unstable rotors, they further break-up into an irregular pattern (fibrillation), of which chaotic behaviour is expected but not yet proven.

Previous fundamental achievements include: a geometric theory for wave fronts and rotor filaments; determining the minimal thickness below which no 3D instability will happen; extending the notion of filament tension to quasi-periodic cores as observed in experiments.

Ongoing research entails the elaboration of a new topological description that unifies the concepts of conduction block, quasi-periodic rotors and filaments via topological phase defects. Furthermore, these findings are combined with experimental data, for physics-based inversion and source reconstruction of cardiac signals.

Curved-space viewpoint on cardiac anisotropy

The cardiac muscle cells are organised in such a way that the conduction of the electrical waves through the heart go faster in one direction, called the fiber direction, than the other ones. This is comparable to the gps system that tells you, it will take 30 minutes to go from Kortrijk to Ghent when you take the highway instead of 50 minutes only using small roads. So we can redefine distance in terms of travelling time, instead of using the Euclidean distance.

In mathematics or physics terms, this comes down to endowing cardiac tissue with a metric tensor, and from geometric considerations, the heart then becomes a Riemannian manifold.

Using tensor calculus, geodesics and covariant derivatives, it is thereby possible to obtain general theoretical results on the time-evolution (drift and stability) of wave fronts and rotors in the heart. These efforts are laying the foundation for the field of “cardiac geometrodynamics”.

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Application of mathematical physics concepts to cardiac excitation

Here is a non-exhaustive list of concepts from mathematical physics that are being used in our research:

Geodesics, metric tensors, curved space
Anisotropy of wave propagation can be handled elegantly using a curved-space formalism. A glimpse thereof was added to a famous cardiology textbook. \
Symmetry breaking
Wave fronts and rotors have less Euclidean symmetries than the reaction-diffusion equation, leading to critical eigenmodes of the linearized operator (Goldstone modes).
Bra-ket notation
In perturbation theory, we typically project onto the response functions, which can be written in Dirac’s notation: e.g. $\left<Y|PV\right>$. The use of quantum mechanical notation in biological context is sometimes confusing referees.
Particle-wave duality
In contrast to quantum mechanics, our operators are non-selfadjoint. As a result, the right-hand eigenfunctions are waves (spirals) while left-hand eigenfunctions are localized, like particles. This localization explains why it is so difficult to restore chaotic activity in the heart. See this great video.
Curved-space coordinate systems
In general relativity theory, it is customary to use nearly Euclidean coordinate systems, e.g. Gauss coordinates, Fermi coordinates or Riemann normal coordinates. We apply all of these in a biological context (and sometimes need to further extend them still).
Action principle
Part of the emerging rotor dynamics can be derived from an action principle.
Topological charge & defects
Cardiac rotors revolve around a rotor filament, which is a topological defect. We recently showed that the defect in 3D should be a phase defect surface.
String-like and brane-like dynamics
We previously showed that rotor filaments act as strings in a background space that is curved due to anisotropy. In the recent phase defect interpretation, filaments become brane-like objects that are phase defect surfaces. More topological constraints apply to the edges of those phase defect surfaces.
Green’s functions
Certain aspects can be dealt with classical superposition, e.g. forward calculation of electrograms and quantifying mechano-electrical feedback on rotor drift.
Branch cuts, complex analysis
Recent work shows that at the heart of a linear-core rotor, there is a phase discontinuity or phase defect.
Feynman-Hellman theorem
We use this theorem to calculate filament rigidity, which explains why in thin tissue slabs, full-fledged 3D instability cannot occur.
Pauli matrices and commutators
Even in three dimensions, rotation of scroll waves occurs in a plane, and the set of Pauli matrices is a suitable basis shape to calculate the shape of circular scroll wave cores, as well as the isotropic invariants of higher-order corrections.
Higher-dimensional embedding
We extended Wellner’s minimal principle for rotor filaments to inhomogeneous media by adding a fourth spatial dimension which restores homogeneity.
Schrödinger’s equation
The link between the diffusion and Schrödinger’s equation goes back a long time and has inspired the path integral formalism for quantum mechanics. Here, we explained paradoxical onset of ectopic (additional) heart beats using the analogy - in the other direction.

Building theory from experimental observations

To make the connection between theory and practice, we need to test our ideas on observations of excitation patterns in real hearts. However, these data are scarse, since it is not yet possible to view inside individual patients’ hearts.

An experimental method that can be used on explanted hearts is optical voltage mapping. Here, a voltage-sensitive dye is administered to cardiac tissue to visualize excitation patterns with high resolution. Our first analysis within the group of arrhythmia patterns provided by Prof. E. Tolkacheva (Minneapolis, USA) demonstrated that cardiac rotors in rabbit hearts are organised around extended phase defect lines, rather than point singularities, forcing us to rethink the classical topological approach to cardiac arrhythmia organisation. In the next paper, we also identified the phase defects in cell cultures of human immortalized atrial myocytes, grown in the Pijnappels lab (University of Leiden, the Netherlands).

In ongoing work, we are performing pattern analysis and reconstruction on intracardiac electrograms, as well as ultrasound recordings.

Miscellaneous research topics

Tue, 21 Jun 2022 00:00:00 +0000

Geometry-driven dynamics in reaction-diffusion systems

The cardiac monodomain equations can be formulated as a set of parabolic differential equations of the reaction-diffusion type. Most of our theoretical results apply to the wide class of reaction-diffusion equations, only the existence of stable traveling waves or vortices needs to be assumed.

Therefore, our findings also apply to other reaction-diffusion systems, such as active diffusion of substances across the cell membrane, pattern formation on animal furs, signaling waves at the cellular of multicellular level, oxidation waves, and oscillating chemical reactions (e.g. the Belousov-Zhabotinsky (BZ) reaction).

One of our theoretical predictions on the drift of scroll waves in a medium of stepwise thickness, inspired by a cardiac application, was shortly thereafter experimentally verified in the BZ reaction by Steinbock et al:

H. Ke, Z. Zhang, and O. Steinbock, “Scroll Wave Drift Along Steps, Troughs, and Corners”, Chaos 25, 064303, 1-7, 2015 pdf

Spiral wave chimeras

Spiral wave chimeras are emergent structures in oscillatory media, with spatial coexistance of synchronized and asynchronized regions. The first chimeras were found in non-locally coupled media (i.e. with action-at-a-distance), but in collaboration with B.W. Li, we showed that also a classical reaction-diffusion system can generate spiral wave chimeras.

Cover image generated by Dall-E 3.

Data analysis and inversion

Fri, 17 Jun 2022 00:00:00 +0000

Ultrasound imaging of arrhythmias

Even though various heart rhythm disorders can be recognized by electrocardiograms, the exact three-dimensional spatio-temporal pattern of the cardiac activation sequence throughout the cardiac wall is not well understood during rhythm disorders. Imaging these complex wave patterns can be done directly by plunging needle electrodes or needle catheters into the heart, however in human patients this is impossible to do and indirect inverse electrocardiographic techniques are being developed in order to infer the whole image from surface measurements. The best known example is the electrocardiogram (ECG), but to find the precise activation pattern from body-surface measurements is an incompletely solved inverse problem.

A pilot study from 2018 [1] showed that mechanical deformation of the heart, which can be estimated by ultrasound data, is closely linked to the electrical phenomena during cardiac arrhythmias. This idea opens up new ways of gaining insight into the complex, inherently 3D, electrical patterns in a fast and non-invasive manner. Recent studies have shown that imaging the electromechanical activation sequence with ultrasound data can be helpfull in certain situations (see for example in silico [2], in vivo experiments [3]).

In the ICARUS project, we set out to expand upon this technique and bring it to a clinically feasible tool, working in close collaboration the University Hospital UZ Leuven (Gasthuisberg). This collaboration involves the Cardiovascular Imaging and Dynamics group of Prof. Jan Dhooge and the group of Prof. Joris Ector, head of ablation therapies at the hospital. By combining expertise in mathematical modelling, echocardiography and clinical experience, we will advance our understanding of the three-dimensional electrical patterns and improve diagnosis and localization of cardiac arrhythmias.

[1] Christoph, J., Chebbok, M., Richter, C., Schröder-Schetelig, J., Bittihn, P., Stein, S., … Luther, S. (2018). Electromechanical vortex filaments during cardiac fibrillation, Nature, 555(7698), 667–672. https://doi.org/10.1038/nature26001

[2] Lebert, J., & Christoph, J. (2019). Synchronization-based reconstruction of electromechanical wave dynamics in elastic excitable media. Chaos, 29(9), https://doi.org/10.1063/1.5101041

[3] Grubb, C. S., Melki, L., Wang, D. Y., Peacock, J., Dizon, J., Iyer, V., … Wan, E. Y. (2020). Noninvasive localization of cardiac arrhythmias using electromechanical wave imaging. Science Translational Medicine, 12(536). https://doi.org/10.1126/scitranslmed.aax6111

Inversion of cardiac electrograms

When cardiac myocytes activate or deactivate, they act as electrical dipole sources, generating a potential field in the torso. This extracellular potential is recorded on the cardiac surface during surgery, or on the body surface, where it is known as the electrocardiogram (ECG). While the ECG is routinely used to diagnose and classify arrhythmias, it is still not possible to accurately reconstruct the arrhythmia sources in the heart from body-surface recordings.

As a step-up to ECG reconstruction, we aim to first solve the inverse problem for intracardiac electrograms (iEGM). From measurements with electrodes on the inner cardiac surface (endocardium), we seek to infer the 4D wave pattern inside the myocardial wall using physics-based inversion methods.

Cover image source: Griffin Health

Towards a digital twin of the heart

Fri, 17 Jun 2022 00:00:00 +0000

Numerical modeling of arrhythmias

We have our own written C++-OpenMPI software, called Ithildin, to perform cardiac simulations using different geometries for the heart tissue. For electrogram simulations, we use the open cardiac electrophysiology simulator for in-silico experiments openCARP[1].

With these tools, we can perform a large variety of simulations to study different aspects of cardiac arrhythmias. In forward modeling, we seek quantitative predictions of pattern evolution and electrogram shapes.

[1] Plank, G., Loewe A., Neic. A et al. (2021). The openCARP simulation environment for cardiac electrophysiology. Computer Methods and Programs in Biomedicine 2021;208:106223. doi:10.1016/j.cmpb.2021.106223 *[2] Kabus D, Cloet M, Zemlin C, Bernus O, Dierckx H (2024). The Ithildin library for efficient numerical solution of anisotropic reaction-diffusion problems in excitable media. PLoS ONE 19(9): e0303674. [https://doi.org/10.1371/journal.pone.0303674] (https://doi.org/10.1371/journal.pone.0303674)

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Modeling the cardiac electrogram

Despite its widespread use, there are still fundamental insights lacking on how substrate parameters affect intracardiac electrograms, and which information can be inferred from electrogram recordings. For this, we are collaborating with A.P. Panfilov (Ghent University) and K. Zeppenfeld (University of Leiden) and P. Claus (KU Leuven).

Creation of individual models from machine learning

Mathematical models of heart function have been historically derived from detailed measurements of currents across the cell membrane. When applying the resulting model to a patient, it is silently assumed that the original model describes also the excitation properties of that person. As an alternative, we use machine learning methods to mimic this entire process and directly learn from recordings taken at the tissue scale in a specific heart.

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