Gaussian process manifold interpolation for probabilistic atrial activation maps and uncertain conduction velocity

Abstract

In patients with atrial fibrillation, local activation time (LAT) maps are routinely used for characterizing patient pathophysiology. The gradient of LAT maps can be used to calculate conduction velocity (CV), which directly relates to material conductivity and may provide an important measure of atrial substrate properties. Including uncertainty in CV calculations would help with interpreting the reliability of these measurements. Here, we build upon a recent insight into reduced-rank Gaussian processes (GPs) to perform probabilistic interpolation of uncertain LAT directly on human atrial manifolds. Our Gaussian process manifold interpolation (GPMI) method accounts for the topology of the atrium, and allows for calculation of statistics for predicted CV. We demonstrate our method on two clinical cases, and perform validation against a simulated ground truth. CV uncertainty depends on data density, wave propagation direction and CV magnitude. GPMI is suitable for probabilistic interpolation of other uncertain quantities on non-Euclidean manifolds. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Keywords: Gaussian process; atrial fibrillation; cardiac conduction velocity; local activation time; manifold; probabilistic interpolation.

Figure 1.

(a) Representation of the terms in equation (2.8), where the shaded area represents the Voronoi cell with area A_i and the angles opposing the edge between vertex i and j are a_ij and b_ij. (b) Subdivision of a mesh triangle face into nine triangles by addition of vertices, including a vertex at the mesh face centroid. (Online version in colour.)

Figure 2.

Posterior LAT distribution for Patient 1, shown as CV vectors at face centroids coloured by LAT posterior mean (a) and standard deviation (b). Spheres represent LAT observations y_i (a) and observation noise σ_i (b). (Online version in colour.)

Figure 3.

Posterior LAT distribution for Patient 2, shown as CV vectors at face centroids coloured by LAT posterior mean (a) and standard deviation (b). Spheres represent LAT observations y_i (a) and observation noise σ_i (b). (Online version in colour.)

Figure 4.

CV vectors at face centroids coloured by CV magnitude. CV from simulation (a), and CV predictions using 1000 observations randomly selected from the simulation: noiseless observations (b) and noisy observations (c). (Online version in colour.)

Figure 5.

CV vectors at face centroids coloured by CV magnitude interquartile range, for the predictions in figure 4. Note that the colour scales are different for each case, due to differences in prediction uncertainty for CV magnitude. (Online version in colour.)

Figure 6.

Validation plots for the magnitude of the gradient of LAT. Simulation (ground truth) values were obtained with the ‘wave’ method for 1000 well-spaced centroid locations. For prediction, 1000 noisy LAT observations were used. The error bars on the prediction versus truth plot (left) represent the 9th and 91st percentiles. (Online version in colour.)

Figure 7.

CV predictions for Patient 1, shown as CV vectors at face centroids coloured by CV magnitude (a) and interquartile range (b). Spheres show these quantities at observation vertices (interpolated from neighbouring faces). (Online version in colour.)