Hans Dierckx

Models of cardiac tissue electrophysiology: Progress, challenges and open questions

Models of cardiac tissue electrophysiology are an important component of the Cardiac Physiome Project, which is an international effort to build biophysically based multi-scale mathematical models of the heart. Models of tissue electrophysiology can provide a bridge between electrophysiological cell models at smaller scales, and tissue mechanics, metabolism and blood flow at larger scales. This paper is a critical review of cardiac tissue electrophysiology models, focussing on the micro-structure of cardiac tissue, generic behaviours of action potential propagation, different models of cardiac tissue electrophysiology, the choice of parameter values and tissue geometry, emergent properties in tissue models, numerical techniques and computational issues. We propose a tentative list of information that could be included in published descriptions of tissue electrophysiology models, and used to support interpretation and evaluation of simulation results. We conclude with a discussion of challenges and open questions.

Construction and validation of anisotropic and orthotropic ventricular geometries for quantitative predictive cardiac electrophysiology

Reaction–diffusion computational models of cardiac electrophysiology require both dynamic excitation models that reconstruct the action potentials of myocytes as well as datasets of cardiac geometry and architecture that provide the electrical diffusion tensor D, which determines how excitation spreads through the tissue. We illustrate an experimental pipeline we have developed in our laboratories for constructing and validating such datasets. The tensor D changes with location in the myocardium, and is determined by tissue architecture. Diffusion tensor magnetic resonance imaging (DT-MRI) provides three eigenvectors ei and eigenvalues λi at each voxel throughout the tissue that can be used to reconstruct this architecture. The primary eigenvector e1 is a histologically validated measure of myocyte orientation (responsible for anisotropic propagation). The secondary and tertiary eigenvectors (e2 and e3) specify the directions of any orthotropic structure if λ2 is significantly greater than λ3—this orthotropy has been identified with sheets or cleavage planes. For simulations, the components of D are scaled in the fibre and cross-fibre directions for anisotropic simulations (or fibre, sheet and sheet normal directions for orthotropic tissues) so that simulated conduction velocities match values from optical imaging or plunge electrode experiments. The simulated pattern of propagation of action potentials in the models is partially validated by optical recordings of spatio-temporal activity on the surfaces of hearts. We also describe several techniques that enhance components of the pipeline, or that allow the pipeline to be applied to different areas of research: Q ball imaging provides evidence for multi-modal orientation distributions within a fraction of voxels, infarcts can be identified by changes in the anisotropic structure—irregularity in myocyte orientation and a decrease in fractional anisotropy, clinical imaging provides human ventricular geometry and can identify ischaemic and infarcted regions, and simulations in human geometries examine the roles of anisotropic and orthotropic architecture in the initiation of arrhythmias.

Electrical Wave Propagation in an Anisotropic Model of the Left Ventricle Based on Analytical Description of Cardiac Architecture

We develop a numerical approach based on our recent analytical model of fiber structure in the left ventricle of the human heart. A special curvilinear coordinate system is proposed to analytically include realistic ventricular shape and myofiber directions. With this anatomical model, electrophysiological simulations can be performed on a rectangular coordinate grid. We apply our method to study the effect of fiber rotation and electrical anisotropy of cardiac tissue (i.e., the ratio of the conductivity coefficients along and across the myocardial fibers) on wave propagation using the ten Tusscher–Panfilov (2006) ionic model for human ventricular cells. We show that fiber rotation increases the speed of cardiac activation and attenuates the effects of anisotropy. Our results show that the fiber rotation in the heart is an important factor underlying cardiac excitation. We also study scroll wave dynamics in our model and show the drift of a scroll wave filament whose velocity depends non-monotonically on the fiber rotation angle; the period of scroll wave rotation decreases with an increase of the fiber rotation angle; an increase in anisotropy may cause the breakup of a scroll wave, similar to the mother rotor mechanism of ventricular fibrillation.

Drift of Scroll Waves in Thin Layers Caused by Thickness Features: Asymptotic Theory and Numerical Simulations

A scroll wave in a very thin layer of excitable medium is similar to a spiral wave, but its behavior is affected by the layer geometry. We identify the effect of sharp variations of the layer thickness, which is separate from filament tension and curvature-induced drifts described earlier. We outline a two-step asymptotic theory describing this effect, including asymptotics in the layer thickness and calculation of the drift of so-perturbed spiral waves using response functions. As specific examples, we consider drift of scrolls along thickness steps, ridges, ditches, and disk-shaped thickness variations. Asymptotic predictions agree with numerical simulations.

Spiral wave chimeras in locally coupled oscillator systems

The recently discovered chimera state involves the coexistence of synchronized and desynchronized states for a group of identical oscillators. In this work, we show the existence of (inwardly) rotating spiral wave chimeras in the three-component reaction-diffusion systems where each element is locally coupled by diffusion. A transition from spiral waves with the smooth core to spiral wave chimeras is found as we change the local dynamics of the system or as we gradually increase the diffusion coefficient of the activator. Our findings on the spiral wave chimera in the reaction-diffusion systems suggest that spiral chimera states may be found in chemical and biological systems that can be modeled by a large population of oscillators indirectly coupled via a diffusive environment.

Drift laws for spiral waves on curved anisotropic surfaces.

Rotating spiral waves organize spatial patterns in chemical, physical, and biological excitable systems. Factors affecting their dynamics, such as spatiotemporal drift, are of great interest for particular applications. Here, we propose a quantitative description for spiral wave dynamics on curved surfaces which shows that for a wide class of systems, including the Belousov-Zhabotinsky reaction and anisotropic cardiac tissue, the Ricci curvature scalar of the surface is the main determinant of spiral wave drift. The theory provides explicit equations for spiral wave drift direction, drift velocity, and the period of rotation. Depending on the parameters, the drift can be directed to the regions of either maximal or minimal Ricci scalar curvature, which was verified by direct numerical simulations.

Buckling of Scroll Waves

A scroll wave in a sufficiently thin layer of an excitable medium with negative filament tension can be stable nevertheless due to filament rigidity. Above a certain critical thickness of the medium, such a scroll wave will have a tendency to deform into a buckled, precessing state. Experimentally this will be seen as meandering of the spiral wave on the surface, the amplitude of which grows with the thickness of the layer, until a breakup to scroll wave turbulence happens. We present a simplified theory for this phenomenon and illustrate it with numerical examples.

Action Potential Duration Heterogeneity of Cardiac Tissue Can Be Evaluated from Cell Properties Using Gaussian Green's Function Approach

Action potential duration (APD) heterogeneity of cardiac tissue is one of the most important factors underlying initiation of deadly cardiac arrhythmias. In many cases such heterogeneity can be measured at tissue level only, while it originates from differences between the individual cardiac cells. The extent of heterogeneity at tissue and single cell level can differ substantially and in many cases it is important to know the relation between them. Here we study effects from cell coupling on APD heterogeneity in cardiac tissue in numerical simulations using the ionic TP06 model for human cardiac tissue. We show that the effect of cell coupling on APD heterogeneity can be described mathematically using a Gaussian Greens function approach. This relates the problem of electrotonic interactions to a wide range of classical problems in physics, chemistry and biology, for which robust methods exist. We show that, both for determining effects of tissue heterogeneity from cell heterogeneity (forward problem) as well as for determining cell properties from tissue level measurements (inverse problem), this approach is promising. We illustrate the solution of the forward and inverse problem on several examples of 1D and 2D systems.

Effective dynamics of twisted and curved scroll waves using virtual filaments

Scroll waves are three-dimensional excitation patterns that rotate around a central filament curve; they occur in many physical, biological, and chemical systems. We explicitly derive the equations of motion for scroll wave filaments in reaction-diffusion systems with isotropic diffusion up to third order in the filaments twist and curvature. The net drift components define at every instance of time a virtual filament which lies close to the instantaneous filament. Importantly, virtual filaments obey simpler, time-independent laws of motion which we analytically derive here and illustrate with numerical examples. Stability analysis of scroll waves is performed using virtual filaments, showing that filament curvature and twist add as quadratic terms to the nominal filament tension. Applications to oscillating chemical reactions and cardiac tissue are discussed.

Intravoxel Fibre Structure of the Left Ventricular Free Wall and Posterior Left-Right Ventricular Insertion Site in Canine Myocardium Using Q-Ball Imaging

Q-ball imaging (QBI) is an established high-angular resolution diffusion MRI technique enabling to resolve intravoxel fibre structure. Here, we present a detailed study of myocardial fibre orientation using QBI. We compare standard diffusion tensor MRI (DTI) versus QBI in the canine left ventricular free wall (LVFW) and posterior left-right ventricular insertion site. Most voxels within the LVFW show high fractional anisotropy (FA), Gaussian diffusion profiles, and a single population of aligned fibres. In these, the difference between fibre helix angles estimated by DTI and QBI is below 5 degrees. However, we show that reduced FA near the anterior papillary muscle in the LVFW and in most of the left-right ventricular fusion site correlates with non-Gaussian diffusion. The QBI orientation distribution functions (ODF) in these regions reveal complex intravoxel fibrous structure, which cannot be inferred using DTI. Extensive ODF maps of myocardial fibre orientation are presented and discussed for the first time to our knowledge.