On the electrophysiology of the atrial fast conduction system: an uncertain quantification study

Abstract

Cardiac modeling entails the epistemic uncertainty of the input parameters, such as bundles and chambers geometry, electrical conductivities and cell parameters, thus calling for an uncertainty quantification (UQ) analysis. Since the cardiac activation and the subsequent muscular contraction is provided by a complex electrophysiology system made of interconnected conductive media, we focus here on the fast conductivity structures of the atria (internodal pathways) with the aim of identifying which of the uncertain inputs mostly influence the propagation of the depolarization front. Firstly, the distributions of the input parameters are calibrated using data available from the literature taking into account gender differences. The output quantities of interest (QoIs) of medical relevance are defined and a set of metamodels (one for each QoI) is then trained according to a polynomial chaos expansion (PCE) in order to run a global sensitivity analysis with non-linear variance-based Sobol’ indices with confidence intervals evaluated through the bootstrap method. The most sensitive parameters on each QoI are then identified for both genders showing the same order of importance of the model inputs on the electrical activation. Lastly, the probability distributions of the QoIs are obtained through a forward sensitivity analysis using the same trained metamodels. It results that several input parameters—including the position of the internodal pathways and the electrical impulse applied at the sinoatrial node—have a little influence on the QoIs studied. Vice-versa the electrical activation of the atrial fast conduction system is sensitive on the bundles geometry and electrical conductivities that need to be carefully measured or calibrated in order for the electrophysiology model to be accurate and predictive.

Graphic Abstract

Appendix A: Convergence of the electrophysiology model

Fig. 7

Time behaviour of the average transmembrane potential in the atrial fast conduction network by refining a the spatial and b the temporal resolution. The insets show the same quantity within the initial fast depolarization phase

The transmembrane potential averaged over the ventricular domain as a function of time is shown in Fig. 7. Each solid curve corresponds to a different simulation of the monodomain equations with the ten Tusscher–Panfilov model on a different grid with elements number varying from 600 to 4000 and different time step sizes [12]. The averaged transmembrane potential becomes basically grid independent for mesh resolution exceeding 2000 elements and, based on this result, a mesh with 2397 elements (corresponding to a $\Delta _{X} = 0.25$ mm) and $\Delta t = 0.005$ ms is used for the UQ analysis. The corresponding computational cost to run a single simulation is of about 30 CPU-minutes and, consequently, the cost to build the UQ datasets is of about 42 CPU-days. The numerical simulation have been run on an Intel Xeon Processors (E5-2620 v3 - 15M Cache, 2.40 GHz), with 16 CPUs.

1.1 B: Convergence of the PCE analysis

Figure 8a shows the coefficient $R^{2}$ and $Q^{2}$ introduced in Sect. 2 for the case of $t_{AV}$ in the female population as a function of the training dataset size. The optimal metamodel (see Table 2) is trained each time using a different training dataset with size ranging from 500 to 2000 and tested against the same testing dataset made of 200 samples. The $R^{2}$ index is stable with respect to the training data set size as the number of samples is larger than 500, which means that the variety of the metamodel is sufficient to describe the physical phenomenon at study. Conversely, lower size of the training dataset lead to a suboptimal value of the index $Q^{2}$, which corresponds to a reduced ability of the metamodel to predict values outside the training sample. Hence, the convergence of the difference $R^{2}$–$Q^{2}$ indicates that the metamodel can be considered stable when the size of the training metamodel exceeds 1000 cases. Similar results are obtained for the other QoIs and the males population (not reported here for the sake of brevity).

Fig. 8

a $R^{2}$ and $Q^{2}$ indices for $t_{AV}$ of the female dataset. Indices are reported for an increasing size of the training dataset and a fixed testing dataset of size 200 is used to compute the corresponding $Q^{2}$ index. b Stability analysis of Sobol’ indices for $t_{AV}$ on the female dataset with 5-percentile confidence intervals calculated using a bootstrap methodology

Another approach for testing the metamodel performance consists of evaluating its stability to a perturbation of the training dataset. Specifically, the results of the UQ analysis are shown as a function of the dataset size thus determining for what size they become stable, as shown in Fig. 8b where the Sobol’ indices are seen to be stable for dataset size larger than 1000. The figure also reports the 5-percentile confidence intervals calculated using a bootstrap method on the training dataset.

1.2 C: Electrical conductivity vs conduction velocity

Fig. 9

Conduction velocity as a function of the electrical conductivity (black dots) with superimposed a square root interpolation function (blue line). The straight red lines indicate the ranges of the electrical conductivity and of the conduction velocity studied in the UQ analysis

The electrical conductivity M is an important input parameter of the electrophysiology model. However, most of the available measurements in the literature refer to the conduction velocity rather than to the electrical conductivity [1, 2, 21, 22]. For this reason, an inverse calibration has to be preformed to determine the electrical conductivities corresponding to the conduction velocities measured experimentally. In order to determine such relation between the electrical conductivity and the conduction velocity, the monodomain equations have been solved over a one-dimensional straight domain of length 100 mm for several electrical conductivity values. The corresponding conduction velocity is measured by selecting two points 50 mm apart each other inside the domain and monitoring their activation time (defined as the instant when the transmembrane potential exceeds $-70$ mV). The conduction velocity is thus measured as the ratio between the distance between the monitoring points and the time interval among their activation and is reported in Fig. 9 for spatial and temporal discretization of $\Delta x = 0.25$ mm and $\Delta t = 1\cdot 10^{-3}$ ms and current stimulus applied at one tip of the domain as defined in equation (2) with $S_{d} =$2.5 ms, $S_{a} =$1 mA/$\text {mm}^{2}$ and $SA_l = 6.85$ mm.