Dynamics of Pivoting Electrical Waves in a Cardiac Tissue Model.

Through a detailed mathematical analysis we seek to advance our understanding of how cardiac tissue conductances govern pivoting (spiral, scroll, rotor, functional reentry) wave dynamics. This is an important problem in cardiology since pivoting waves likely underlie most reentrant tachycardias. The problem is complex, and to advance our methods of analysis we introduce two new tools: a ray tracing method and a moving-interface model. When used in combination with an ionic model, they permit us to elucidate the role played by tissue conductances on pivoting wave dynamics. Specifically we simulate traveling electrical waves with an ionic model that can reproduce the characteristics of plane and pivoting waves in small patches of cardiac tissue. Then ray tracing is applied to the simulated pivoting waves in a manner to expose their real displacement. In this exercise we find loci with special characteristics, as well as zones where a part of a pivoting wave quickly transitions from a regenerative to a non-regenerative propagation mode. The loci themselves and the monitoring of the ionic model state variables in this zone permit to elucidate several aspects of pivoting wave dynamics. We then formulate the moving-interface model based on the information gathered with the above-mentioned analysis. Equipped with a velocity profile v(s), s: distance along of the pivoting wave contour and the steady- state action potential duration (APD) of a plane wave during entrainment, APDss(T), at period T, this simple model can predict: shape, orbit of revolution, rotation period, whether a pivoting wave will break up or not, and whether the tissue will admit pivoting waves or not. Because v(s) and APDss(T) are linked to the ionic model, dynamical analysis with the moving-interface model conveys information on the role played by tissue conductances on pivoting wave dynamics. The analysis conducted here enables us to better understand previous results on the termination of pivoting waves. We surmise the method put forth here could become a means to discover how to alter tissue conductances in a manner to terminate pivoting waves at the origin of reentrant tachycardias.

Determining the light scattering and absorption parameters from forward-directed flux measurements in cardiac tissue.

We describe a method to accurately measure the light scattering model parameters from forward-directed flux (FDF) measurements carried out with a fiber-optic probe (optrode). Improved determination of light scattering parameters will, in turn, permit better modeling and interpretation of optical mapping in the heart using voltage-sensitive dyes. Using our optrode-based system, we carried out high spatial resolution measurements of FDF in intact and homogenized cardiac tissue, as well as in intralipid-based tissue phantoms. The samples were illuminated with a broad collimated beam at 660 and 532 nm. Measurements were performed with a plunge fiber-optic probe (NA=0.22) at a spatial resolution of up to 10 µm. In the vicinity of the illuminated surface, the FDF consistently manifested a fast decaying exponent with a space constant comparable with the decay rate of ballistic photons. Using a Monte Carlo model, we obtained a simple empirical formula linking the rate of the fast exponent to the scattering coefficient, the anisotropy parameter g, and the numerical aperture of the probe. The estimates of scattering coefficient based on this formula were validated in tissue phantoms. Potential applications of optical fiber-based FDF measurements for the evaluation of optical parameters in turbid media are discussed.

Fitting C² continuous parametric surfaces to frontiers delimiting physiologic structures.

We present a technique to fit C(2) continuous parametric surfaces to scattered geometric data points forming frontiers delimiting physiologic structures in segmented images. Such mathematical representation is interesting because it facilitates a large number of operations in modeling. While the fitting of C(2) continuous parametric curves to scattered geometric data points is quite trivial, the fitting of C(2) continuous parametric surfaces is not. The difficulty comes from the fact that each scattered data point should be assigned a unique parametric coordinate, and the fit is quite sensitive to their distribution on the parametric plane. We present a new approach where a polygonal (quadrilateral or triangular) surface is extracted from the segmented image. This surface is subsequently projected onto a parametric plane in a manner to ensure a one-to-one mapping. The resulting polygonal mesh is then regularized for area and edge length. Finally, from this point, surface fitting is relatively trivial. The novelty of our approach lies in the regularization of the polygonal mesh. Process performance is assessed with the reconstruction of a geometric model of mouse heart ventricles from a computerized tomography scan. Our results show an excellent reproduction of the geometric data with surfaces that are C(2) continuous.

Extending the conditions of application of an inversion of the Hodgkin-Huxley gating model.

We present an inversion of the Hodgkin-Huxley formalism to estimate initial conditions and model parameters, including functions of voltage, from the solutions of the underlying ordinary differential equation (ODE) subjected to multiple voltage step stimulations. As such, the procedure constitutes a means to estimate the parameters including functions of voltage of an Hodgkin-Huxley formalism from experimental data.The basic idea was developed in a previous communication (SIAM J. Appl. Math. 64:1264-1274, 2009). The inversion in question applies to currents exhibiting activation and inactivation, but the version, as published previously, cannot estimate the unknowns for channels that rapidly inactivate just after a brief opening. In such cases, the amplitude of the current, in a given voltage range, is too small to be detectable by the instrumentation using previously applied experimental protocols. This is, for example, the case for the sodium channels in a number of excitable tissue for potential in the vicinity of the cell resting potential. The current communication extends the inversion procedure in a manner to overcome this limitation.Furthermore, within the inversion framework, we can determine whether the data at the basis of the estimation sufficiently constrains the estimation problem, i.e., whether it is complete. We exploit this element of our method to document a set of stimulation protocols that constitute a complete data set for the purpose of inverting the Hodgkin-Huxley formalism.

Application of phase correlation to the montage synthesis and three-dimensional reconstruction of large tissue volumes from confocal laser scanning microscopy.

We have implemented and tested a new automatic method for the montage synthesis and three-dimensional (3D) reconstruction of large tissue volumes from confocal laser scanning microscopy data (CLSM). This method relies on maximization of the phase correlation between adjacent images. It was tested on a large specimen (a murine heart) that was cut into a number of individual sections with thickness appropriate for CLSM. The sections were scanned horizontally (in-plane) and vertically (perpendicular to the optical planes) to produce "tiles" of a 3D volume. Phase correlation maximization was applied to the montage synthesis of in-plane tiles and 3D alignment of optical slices within a given physical section. The performance of the new method is evaluated.

Laplace-Dirichlet energy field specification for deformable models. an FEM approach to active contour fitting.

The construction of large scale computer models for complex biological systems requires the fitting of curves or surfaces to anatomical data sets. Algorithms recently developed to perform this task are based on the displacement of an initial model contour. There are several problems associated with this approach. Here we present improvements which eliminate the (i) sensitivity to the initial model position and shape; (ii) existence of local minima or maxima in the field used to displace the model; and (iii) presence of multiple solutions in the rules governing model displacement. Key elements of our algorithm are first that both the energy field used to displace the model and the model displacement itself are governed by partial differential equations. Secondly, we approximate the model with a polygonal contour which facilitates accurate displacement. Tests performed against cases that are known to be problematic show that our algorithm can fit complex data sets entirely automatically.