Intramural wave propagation in cardiac tissue: asymptotic solutions and cusp waves

Conduction in the Heart Wall: Helicoidal Fibers Minimize Diffusion Bias

The mammalian heart must function as an efficient pump while simultaneously conducting electrical signals to drive the contraction process. In the ventricles, electrical activation begins at the insertion points of the Purkinje network in the endocardium. How does the diffusion component of the subsequent excitation wave propagate from the endocardium in a healthy heart wall without creating directional biases? We show that this is a consequence of the particular geometric organization of myocytes in the heart wall. Using a generalized helicoid to model fiber orientation, we treat the myocardium as a curved space via Riemannian geometry, and then use stochastic calculus to model local signal diffusion. Our analysis shows that the helicoidal arrangement of myocytes minimizes the directional biases that could lead to aberrant propagation, thereby explaining how electrophysiological principles are consistent with local measurements of cardiac fiber geometry. We discuss our results in the context of the need to balance electrical and mechanical requirements for heart function.

Asymptotic wave propagation in excitable media

Wave shape and velocity are important issues in reaction-diffusion systems, and are often the result of competition in media with heterogeneous conduction properties. Asymptotic wave front propagation at maximal conduction velocity has been previously reported in the context of anisotropic cardiac tissue, but it is unknown whether this is a universal property of excitable tissues where conduction velocity can be locally modulated by mechanisms other than anisotropy. Here, we investigate the impact of conduction heterogeneities and boundary effects on wave propagation in excitable media. Following a theoretical analysis, we find that wave-front cusps occur where local velocity is reduced and that asymptotic wave fronts propagate at the maximal translational conduction velocity. Simulations performed in different reaction-diffusion systems, including cardiac tissue, confirm our theoretical findings. We conclude that this property can be found in a wide range of reaction-diffusion systems with excitable dynamics and that asymptotic wave-front shapes can be predicted.

Generalized minimal principle for rotor filaments

To a reaction-diffusion medium with an inhomogeneous anisotropic diffusion tensor D, we add a fourth spatial dimension such that the determinant of the diffusion tensor is constant in four dimensions. We propose a generalized minimal principle for rotor filaments, stating that the scroll wave filament strives to minimize its surface area in the higher-dimensional space. As a consequence, stationary scroll wave filaments in the original 3D medium are geodesic curves with respect to the metric tensor G=det(D)D(-1). The theory is confirmed by numerical simulations for positive and negative filament tension and a model with a non-stationary spiral core. We conclude that filaments in cardiac tissue with positive tension preferentially reside or anchor in regions where cardiac cells are less interconnected, such as portions of the cardiac wall with a large number of cleavage planes.

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

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.

Extracting surface activation time from the optically recorded action potential in three-dimensional myocardium

Optical mapping has become an indispensible tool for studying cardiac electrical activity. However, due to the three-dimensional nature of the optical signal, the optical upstroke is significantly longer than the electrical upstroke. This raises the issue of how to accurately determine the activation time on the epicardial surface. The purpose of this study was to establish a link between the optical upstroke and exact surface activation time using computer simulations, with subsequent validation by a combination of microelectrode recordings and optical mapping experiments. To simulate wave propagation and associated optical signals, we used a hybrid electro-optical model. We found that the time of the surface electrical activation (t(E)) within the accuracy of our simulations coincided with the maximal slope of the optical upstroke (t(F)*) for a broad range of optical attenuation lengths. This was not the case when the activation time was determined at 50% amplitude (t(F50)) of the optical upstroke. The validation experiments were conducted in isolated Langendorff-perfused rat hearts and coronary-perfused pig left ventricles stained with either di-4-ANEPPS or the near-infrared dye di-4-ANBDQBS. We found that t(F)* was a more accurate measure of t(E) than was t(F50) in all experimental settings tested (P = 0.0002). Using t(F)* instead of t(F50) produced the most significant improvement in measurements of the conduction anisotropy and the transmural conduction time in pig ventricles.

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 e(i) and eigenvalues λ(i) at each voxel throughout the tissue that can be used to reconstruct this architecture. The primary eigenvector e(1) is a histologically validated measure of myocyte orientation (responsible for anisotropic propagation). The secondary and tertiary eigenvectors (e(2) and e(3)) specify the directions of any orthotropic structure if λ(2) is significantly greater than λ(3)-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.

Dual excitation wavelength epifluorescence imaging of transmural electrophysiological properties in intact hearts

BACKGROUND

Epifluorescence imaging using voltage-sensitive dyes has provided unique insights into cardiac electrical activity and arrhythmias. However, conventional dyes use blue-green excitation light, which has limited depth penetration.

OBJECTIVE

The aim of this study was to demonstrate that combining a short and a long excitation wavelength using near-infrared (NIR) dyes allows for epifluorescence imaging of transmural electrophysiological properties in intact hearts.

METHODS

Epifluorescence imaging was performed in rat hearts (N = 11) using DI-4-ANEPPS and the NIR dye DI-4-ANBDQBS. Activation and action potential duration (APD) patterns were investigated at 2 excitation wavelengths (530 and 660 nm) after epicardial stimulation at various cycle lengths (160 to 70 ms).

RESULTS

Optical action potential upstrokes acquired with 660-nm excitation of DI-4-ANBDQBS were significantly longer than upstrokes obtained with 530-nm excitation of DI-4-ANEPPS (P < .001). Comparison of activation maps showed counterclockwise rotation of isochrones consistent with a transmural rotation of myofibers. Pronounced APD modulation by the activation sequence was observed at both excitation wavelengths. Significantly prolonged APDs (P = .016) and steeper APD restitution curves were found with DI-4-ANBDQBS (660-nm excitation) when compared with DI-4-ANEPPS (530-nm excitation). Dual excitation wavelength experiments using solely DI-4-ANBDQBS yielded similar results. Monophasic action potential recordings showed prolonged APD and steeper APD restitution curves in the endocardium, indicating that 660-nm excitation provides a significant endocardial contribution to the signal. Three-dimensional computer simulations confirmed our findings.

CONCLUSION

Dual excitation wavelength epifluorescence allows detecting transmural heterogeneity in intact hearts. It therefore has the potential to become an important tool in experimental cardiac electrophysiology.

Anisotropy of wave propagation in the heart can be modeled by a Riemannian electrophysiological metric

It is well established that wave propagation in the heart is anisotropic and that the ratio of velocities in the three principal directions may be as large as v(f)v(s)v(n) approximately 4(fibers)2(sheets)1(normal). We develop an alternative view of the heart based on this fact by considering it as a non-Euclidean manifold with an electrophysiological(el-) metric based on wave velocity. This metric is more natural than the Euclidean metric for some applications, because el-distances directly encode wave propagation. We develop a model of wave propagation based on this metric; this model ignores higher-order effects like the curvature of wavefronts and the effect of the boundary, but still gives good predictions of local activation times and replicates many of the observed features of isochrones. We characterize this model for the important case of the rotational orthotropic anisotropy seen in cardiac tissue and perform numerical simulations for a slab of cardiac tissue with rotational orthotropic anisotropy and for a model of the ventricles based on diffusion tensor MRI scans of the canine heart. Even though the metric has many slow directions, we show that the rotation of the fibers leads to fast global activation. In the diffusion tensor MRI-based model, with principal velocities 0.25051 m/s, we find examples of wavefronts that eventually reach speeds up to 0.9 m/s and average velocities of 0.7 m/s. We believe that development of this non-Euclidean approach to cardiac anatomy and electrophysiology could become an important tool for the characterization of the normal and abnormal electrophysiological activity of the heart.

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.

Reconstructing subsurface electrical wave orientation from cardiac epi-fluorescence recordings: Monte Carlo versus diffusion approximation

The development of voltage-sensitive dyes has revolutionized cardiac electrophysiology and made optical imaging of cardiac electrical activity possible. Photon diffusion models coupled to electrical excitation models have been successful in qualitatively predicting the shape of the optical action potential and its dependence on subsurface electrical wave orientation. However, the accuracy of the diffusion equation in the visible range, especially for thin tissue preparations, remains unclear. Here, we compare diffusion and Monte Carlo (MC) based models and we investigate the role of tissue thickness. All computational results are compared to experimental data obtained from intact guinea pig hearts. We show that the subsurface volume contributing to the epi-fluorescence signal extends deeper in the tissue when using MC models, resulting in longer optical upstroke durations which are in better agreement with experiments. The optical upstroke morphology, however, strongly correlates to the subsurface propagation direction independent of the model and is consistent with our experimental observations.

Extracting intramural wavefront orientation from optical upstroke shapes in whole hearts

Information about intramural propagation of electrical excitation is crucial to understanding arrhythmia mechanisms in thick ventricular muscle. There is currently a controversy over whether it is possible to extract such information from the shape of the upstroke in optical mapping recordings. We show that even in the complex geometry of a whole guinea pig heart, optical upstroke morphology reveals the 3D wavefront orientation near the surface. To characterize the upstroke morphology, we use V(F)(*), the fractional level at which voltage-sensitive fluorescence, V(F), has maximal time derivative. Low values of V(F)(*)( approximately 0.2) indicate a wavefront moving away from the surface, high values of V(F)(*) ( approximately 0.6) a wavefront moving toward the surface, and intermediate values of V(F)(*) ( approximately 0.4) a wavefront moving parallel to the surface. We further performed computer simulations using Luo-Rudy II electrophysiology and a simplified 3D geometry. The simulated V(F)(*) maps for free wall and apical stimulations as well as for sinus rhythm are in good quantitative agreement with the averaged experimental results. Furthermore, computer simulations show that the effect of the curvature of the heart on wave propagation is negligible.

Depth-resolved optical imaging of transmural electrical propagation in perfused heart

We present a study of the 3-dimensional (3D) propagation of electrical waves in the heart wall using Laminar Optical Tomography (LOT). Optical imaging contrast is provided by a voltage sensitive dye whose fluorescence reports changes in membrane potential. We examined the transmural propagation dynamics of electrical waves in the right ventricle of Langendorf perfused rat hearts, initiated either by endo-cardial or epi-cardial pacing. 3D images were acquired at an effective frame rate of 667Hz. We compare our experimental results to a mathematical model of electrical transmural propagation. We demonstrate that LOT can clearly resolve the direction of propagation of electrical waves within the cardiac wall, and that the dynamics observed agree well with the model of electrical propagation in rat ventricular tissue.

Detection of intramyocardial scroll waves using absorptive transillumination imaging

Optical imaging using voltage-sensitive dyes has become an important tool for studying vortex-like electrical waves in the heart. Such waves, known as spiral or scroll waves, can spontaneously form in pathological ventricular myocardium, causing ventricular fibrillation and sudden death. Until recently, observations of scroll waves were limited to their surface manifestations, thus providing little information about the shape and location of their organizing center, the filament. We use computer modeling to assess the feasibility of visualizing filaments using dynamic transillumination imaging in conjunction with near-IR voltage-sensitive absorptive dyes (absorptive transillumination). We simulate transillumination signals produced by the intramural scroll waves in a realistic slab of ventricular tissue with trabeculated endocardial surface. The computations use a detailed ionic model of electrical excitation (LRd) coupled to a photon transport model for cardiac tissue. Our simulations show that dynamic absorptive transillumination data, with subsequent processing involving either amplitude maps, time-space plots, or power-of-the-dominant-frequency maps, can be used to reliably detect intramural scroll waves through the whole thickness (approximately 10 mm) of the ventricular wall. Neither variations in the thickness of the myocardial wall nor noise impeded the detection of intramural filaments.

What can we learn from the optically recorded epicardial action potential?

Optical mapping using voltage-sensitive fluorescent dyes has become a major tool for studying excitation propagation in the heart. Computational and experimental studies have indicated that the optical upstroke morphology reflects the orientation of the subsurface excitation front. In a recent whole heart computational study performed by Bishop et al. (Bishop, M. J., B. Rodriguez, J. Eason, J. P. Whiteley, N. Trayanova, and D. J. Gavaghan. 2006. Synthesis of voltage-sensitive optical signals: application to panoramic optical mapping. Biophys. J. 90:2938-2945), an example was provided of two different directions of propagation having nevertheless very similar epicardial optical upstrokes. The goal of this comment is to clarify the interpretation of optical upstroke morphologies and reconcile the results obtained by Bishop et al. with previous computational and experimental studies.