Estimation of Cardiac Bidomain Parameters from Extracellular Measurement: Two Dimensional Study

Approaches for determining cardiac bidomain conductivity values: progress and challenges

Modelling the electrical activity of the heart is an important tool for understanding electrical function in various diseases and conduction disorders. Clearly, for model results to be useful, it is necessary to have accurate inputs for the models, in particular the commonly used bidomain model. However, there are only three sets of four experimentally determined conductivity values for cardiac ventricular tissue and these are inconsistent, were measured around 40 years ago, often produce different results in simulations and do not fully represent the three-dimensional anisotropic nature of cardiac tissue. Despite efforts in the intervening years, difficulties associated with making the measurements and also determining the conductivities from the experimental data have not yet been overcome. In this review, we summarise what is known about the conductivity values, as well as progress to date in meeting the challenges associated with both the mathematical modelling and the experimental techniques. Graphical abstract Epicardial potential distributions, arising from a subendocardial ischaemic region, modelled using conductivity data from the indicated studies.

Numerical sensitivity analysis of a variational data assimilation procedure for cardiac conductivities

An accurate estimation of cardiac conductivities is critical in computational electro-cardiology, yet experimental results in the literature significantly disagree on the values and ratios between longitudinal and tangential coefficients. These are known to have a strong impact on the propagation of potential particularly during defibrillation shocks. Data assimilation is a procedure for merging experimental data and numerical simulations in a rigorous way. In particular, variational data assimilation relies on the least-square minimization of the misfit between simulations and experiments, constrained by the underlying mathematical model, which in this study is represented by the classical Bidomain system, or its common simplification given by the Monodomain problem. Operating on the conductivity tensors as control variables of the minimization, we obtain a parameter estimation procedure. As the theory of this approach currently provides only an existence proof and it is not informative for practical experiments, we present here an extensive numerical simulation campaign to assess practical critical issues such as the size and the location of the measurement sites needed for in silico test cases of potential experimental and realistic settings. This will be finalized with a real validation of the variational data assimilation procedure. Results indicate the presence of lower and upper bounds for the number of sites which guarantee an accurate and minimally redundant parameter estimation, the location of sites being generally non critical for properly designed experiments. An effective combination of parameter estimation based on the Monodomain and Bidomain models is tested for the sake of computational efficiency. Parameter estimation based on the Monodomain equation potentially leads to the accurate computation of the transmembrane potential in real settings.

A brief history of tissue models for cardiac electrophysiology

The last four decades have produced a number of significant advances in the developments of computer models to simulate and investigate the electrical activity of cardiac tissue. The tissue descriptions that underlie these simulations have been built from a combination of clever insight and careful comparison with measured data at multiple scales. Tissue models have not only led to greater insights into the mechanisms of life-threatening arrhythmias but have been used to engineer new therapies to treat the consequences of cardiac disease. This paper is a look back at the early years in the cardiac modeling and the challenges facing the field as models move toward the clinic.

Axial anisotropic conductivity imaging based on projected current density in MREIT

An aim of magnetic resonance electrical impedance tomography (MREIT) is to visualize the internal current density and/or the conductivity of an imaging object. In MREIT, it is desirable to use just one component of the internal magnetic flux density vector B=(B(x),B(y),B(z)) caused by the injected current, measured without rotating the object. We present a method of visualizing the axial anisotropic conductivity tensor by use of the measured magnetic flux density B(z) data. The method involves the use of a projected current density, which is a uniquely and stably determined component of the internal current generated by the injected current, derived from the measured B(z) data. Each component of the axial anisotropic conductivity is recovered by matching the measured B(z) data with a determined intermediate isotropic conductivity and the projected currents. Results from numerical simulations demonstrate that the proposed algorithm is robust to noise and stably determines the anisotropic conductivity tensor on the imaging slice. For a practical implementation, we studied a postmortem canine brain case to visualize each component of the anisotropic conductivity. We observed that the reconstructed anisotropic conductivity images clearly reflects the anisotropic property of the white matter in the direction parallel to its fibers.

A controllably anisotropic conductivity or diffusion phantom constructed from isotropic layers

Phantoms with controllable and well-defined anisotropy are needed to test methods for imaging electrical anisotropy. We developed and tested a phantom that had properties similar to a homogeneous anisotropic conductive medium. The phantom was constructed with alternate slices of isotropic gel having different conductivities. The degree of anisotropy in the phantom could be varied easily by changing the relative conductivity of the two gels. We tested the stability of several phantoms and found their properties were maintained for approximately 8 h following construction. The phantom has application to electrical impedance tomography, magnetic resonance electrical impedance tomography, EEG and ECG source imaging and diffusion tensor imaging.

Construction and validation of a plunge electrode array for three-dimensional determination of conductivity in the heart

The heart's response to electrical shock, electrical propagation in sinus rhythm, and the spatiotemporal dynamics of ventricular fibrillation all depend critically on the electrical anisotropy of cardiac tissue. Analysis of the microstructure of the heart predicts that three unique intracellular electrical conductances can be defined at any point in the ventricular wall; however, to date, there has been no experimental confirmation of this concept. We report the design, fabrication, and validation of a novel plunge electrode array capable of addressing this issue. A new technique involving nylon coating of 24G hypodermic needles is performed to achieve nonconductive electrodes that can be combined to give moderate-density multisite intramural measurement of extracellular potential in the heart. Each needle houses 13 silver wires within a total diameter of 0.7 mm, and the combined electrode array gives 137 sites of recording. The ability of the electrode array to accurately assess conductances is validated by mapping the potential field induced by a point current source within baths of saline of varying concentration. A bidomain model of current injection in the heart is then used to test an approximate relationship between the monodomain conductivities measured by the array, and the full set of bidomain conductivities that describe cardiac tissue.

Cardiac electrophysiology model adjustment using the fusion of MR and optical imaging

Despite important recent efforts in cardiac electrophysiology modelling, there is still a strong need for validating macroscopic models, that are well suited for diagnosis and treatment planning. In this paper we present a method to adjust the parameters of a macroscopic electrophysiology model on depolarisation and repolarisation maps obtained ex-vivo from optical imaging. With this imaging technique, optical fluorescence data are recorded with high spatial and temporal resolution on a large healthy porcine heart. A model of the myocardium is built from the MR images of the same heart, which also integrates the myocardial fibre orientation measured with DTI. We then present the first quantitative adjustment of a personalised volumetric model of the myocardium.

Laminar Arrangement of Ventricular Myocytes Influences Electrical Behavior of the Heart

The response of the heart to electrical shock, electrical propagation in sinus rhythm, and the spatiotemporal dynamics of ventricular fibrillation all depend critically on the electrical anisotropy of cardiac tissue. A long-held view of cardiac electrical anisotropy is that electrical conductivity is greatest along the myocyte axis allowing most rapid propagation of electrical activation in this direction, and that conductivity is isotropic transverse to the myocyte axis supporting a slower uniform spread of activation in this plane. In this context, knowledge of conductivity in two directions, parallel and transverse to the myofiber axis, is sufficient to characterize the electrical action of the heart. Here we present new experimental data that challenge this view. We have used a novel combination of intramural electrical mapping, and experiment-specific computer modeling, to demonstrate that left ventricular myocardium has unique bulk conductivities associated with three microstructurally-defined axes. We show that voltage fields induced by intramural current injection are influenced by not only myofiber direction, but also the transmural arrangement of muscle layers or myolaminae. Computer models of these experiments, in which measured 3D tissue structure was reconstructed in-silico, best matched recorded voltages with conductivities in the myofiber direction, and parallel and normal to myolaminae, set in the ratio 4:2:1, respectively. These findings redefine cardiac tissue as an electrically orthotropic substrate and enhance our understanding of how external shocks may act to successfully reset the fibrillating heart into a uniform electrical state. More generally, the mechanisms governing the destabilization of coordinated electrical propagation into ventricular arrhythmia need to be evaluated in the light of this discovery.