Uncertainty Visualization in Forward and Inverse Cardiac Models

Abstract

Quantification and visualization of uncertainty in cardiac forward and inverse problems with complex geometries is subject to various challenges. Specific to visualization is the observation that occlusion and clutter obscure important regions of interest, making visual assessment difficult. In order to overcome these limitations in uncertainty visualization, we have developed and implemented a collection of novel approaches. To highlight the utility of these techniques, we evaluated the uncertainty associated with two examples of modeling myocardial activity. In one case we studied cardiac potentials during the repolarization phase as a function of variability in tissue conductivities of the ischemic heart (forward case). In a second case, we evaluated uncertainty in reconstructed activation times on the epicardium resulting from variation in the control parameter of Tikhonov regularization (inverse case). To overcome difficulties associated with uncertainty visualization, we implemented linked-view windows and interactive animation to the two respective cases. Through dimensionality reduction and superimposed mean and standard deviation measures over time, we were able to display key features in large ensembles of data and highlight regions of interest where larger uncertainties exist.

1. Introduction

Forward and inverse problems in electrocardiography provide a means for studying the electrical properties and patterns that arise within the heart and/or torso from the integrated electrical activity of cardiac myocytes. In the forward case, electrical inferences are made based on given source models and associated conducting media that contain these sources. Inverse solutions, in contrast, use remote observations to deduce the electrical function of the cardiac sources. Solutions to forward and inverse problems, however, require several assumptions that, due to such characteristics as the complex nature of cardiac structure or the inherent need to regularize the inverse, generate uncertainty in the results.

To understand the uncertainty associated with cardiac forward and inverse problems, visualization techniques are often applied; however, visualization of 3-dimensional data presents its own set of complexities. Obstruction and clutter can obscure or even hide regions of interest. To overcome these difficulties in visualization and to thus better understand variations in the uncertainty of electrocardiographic models, we have developed new 3-dimensional techniques to visually analyze simulation uncertainty.

In order to explore uncertainty visualization, we considered two electrocardiographic simulation cases—one for forward and one for inverse simulation. We first investigated the uncertainty associated with bidomain conductivities in the static forward model of an ischemic heart. In the second case, we explored variability in reconstructed activation times, computed by way of Tikhonov inverse, as a function of an unknown λ value. To more thoroughly investigate the uncertainties in our models, we have developed new visualization systems that overcome many of the difficulties of 3-dimensional rendering and provide a means of interactive exploration of high-dimensional data.

2. Methods

While both forward and inverse problems in electrocardiography seek to capture electric activity from the heart, they are different in formulation and solution approach. Each, therefore, required unique and specific uncertainty quantification and visualization approaches.

2.1. Static ischemic forward model

To generate data for the static ischemia model, we extracted cardiac and ischemic zone geometries from experimentally induced ischemic studies using approaches described previously [1, 2]. Fiber directions, acquired by diffusion tensor imaging (DTI), were also assigned.

In order to simulate ischemia, anisotropic conductivity values within the ischemic region were decreased (see Table 1) and a potential difference of 30 mV was applied to the transmembrane potential, V_m, to represent the weakened activity during the plateau phase of the ischemic action potential, which corresponds to the ST segment of the ECG [3]. A linear transition region (border zone) from the diseased to healthy tissue was also applied [4].

Table 1.

Ratio applied to tensor conductivity values within the ischemic region to simulate the diseased state.

Conductivity	Ischemic Ratio
σil	1/10
σit	1/1000
σel	1/2
σet	1/4

Cardiac activity under these conditions was represented by the bidomain passive current flow equation [3]

$$\nabla ·({\overline{\sigma }}_i+{\overline{\sigma }}_e)\nabla V_e=-\nabla ·{\overline{\sigma }}_i\nabla V_m,$$ (1)

where σ̄_i and σ̄_e represent the intracellular and extracellular conductivity tensors, respectively, and V_e represents the unknown extracellular potentials.

There have been many studies attempting to document the conductivity values of the heart [5, 6]; however, these values remain elusive and, therefore, provide a source of uncertainty for the model. We selected a range of conductivity values from the literature [7] onto which we applied generalized polynomial chaos-stochastic collocation methods (gPC-SC) [8]. Conductivity values were treated as uniformly distributed, stochastic processes within the conductivity range provided. gPC-SC was used to reduce the stochastic governing equations into a finite set of deterministic simulation parameters, reducing computational complexity and allowing us to extract the mean and probability density function of the resulting solution ensemble.

2.2. Activation time inverse solutions

The activation-based inverse problem aims to estimate the spread of electrical activation over the epicardium from measurements of electric potentials on the body surface during the QRS complex. Prior to computing the inverse solution, body surface potentials were generated by solving a distinct forward model from the one described in the previous section. This forward model was composed of a nonlinear function, x(τ), which mapped a set of activation times to transmembrane potentials (TMPs) from a discrete set of electrical sources along the heart surface, and a linear mapping, A, a matrix that computes body surface potentials from TMPs. This problem is not only non-linear but also ill-conditioned and hence requires regularization. A simple, yet effective approach to regularization uses the Tikhonov [9] method, resulting in the following non-linear, least squares (NLLS) optimization problem, which is non-convex [10]:

$${\tau }_{\lambda }=\mathit{argmin}{\Vert {Ax}_{\tau }-b\Vert }_{\mathit{Frob}}^2+\lambda {\Vert L{x}_{\tau }\Vert }_{\mathit{Frob}}^2,$$ (2)

where b are the body surface potentials, L is a regularization matrix, and λ is a regularization parameter that controls the trade-off between data fidelity and solution regularity.

We have previously used a convex relaxation of the above problem and discovered uncertainty in the solutions possibly due, in part, to the inherent sensitivity of the problem to signal noise, geometric errors, forward model assumptions (e.g. source amplitudes), and inverse problem parameters (e.g. L and λ) [10, 11]. In this study, we focused on the uncertainty generated by the selection of λ. We chose 50 evenly-spaced values within the interval [0.09, 0.11] on which we performed a perturbation analysis. For each λ, a solution was found for the convex relaxation, from which activation times were computed; from these values, we calculated statistics of the activation time (mean and standard deviation) for each source on the heart surface. This spatial distribution of mean and standard deviation of the activation time gave a measure of the uncertainty of the solution associated with λ selection.

2.3. μView: myocardial uncertainty viewer

To enable uncertainty visualization for the volumetric ischemic simulations we created a five-way, linked-view tool (μView), examples of which are shown in Figure 1. Simulation results, acquired in Section 2.1, were combined and dimensionality reduction was applied to provide a means of displaying large ensembles (high dimensional) data. Figures 1A and 1B show standard deviations and isovalue separation, respectively as two forms of dimensionality reduction used to visualize these data. Other methods were also employed to explore the data including: mean, minimum and maximum isosurfaces, and clustering based on several correlation metrics. Mean and standard deviation visualizations applied simple statistical measures at each node of the simulation with increased intensity representing increased values. Isovalue separation scanned each node of the simulation set and determined how many nodes were above (red) and below (blue) the assigned isovalue. The spectrum of color between these two extrema reflected nodes that had values, within the solution ensemble, that were both above and below the threshold. Minimum and maximum isosurfaces use the isovalue results to highlight only the surfaces produced by the isovalue extrema. Clustering applied k-means methods to bin similar nodes together. Similarities were determined by L2 norm, Pearson’s correlation, or histogram difference.

Figure 1.

Uncertainty Visualization of the Cardiac Ischemia Forward Model. (A) Standard deviation and (B) isolvalue rendering of cardiac forward model show regions of interest near the ischemic zone (outlined in pane ). The five-way linked view contains the following viewing modalities: volumetric, 2D parameter space, – 2D slicing planes.

To overcome difficulties associated with 3D rendering, the following visual techniques were applied: volume rendering (Figure 1A ), a 2D view of parameter space ( ) and three orthogonal 2D slicing planes (panes – ). Volume rendering and 2D slicing planes are standard visualization techniques used to display data values at each point within the myocardium. The parameter space view, in order to reduce the high dimensional data while preserving important features, uses principle component analysis to contract the solution space into a 2D representation – allowing the user to explore features not otherwise visible in the spatial domain. The shape of the resulting image is arbitrary and is a result of the selected principle components. For more information on the technical aspects of how these methods are applied, please see our prior work [12].

2.4. Uncertainty animation

Static isochronal maps of activation times are the most common visualization method for activation-based simulations of the spread of excitation in the heart. Such compression of an entire heart beat into a single image is one of the advantages of activation mapping that supports its utility in research and clinical applications. However, in the setting of visualizing uncertainty, the need to view additional parameters and their variation over both time and parameter space provides new challenges. To address this challenge, we have expanded the static, single image of activation to become an interactive animation over time. Similar to visualizing an animation of the moving activation wavefront, each frame of our animation shows spatial regions that are within one standard deviation of the current mean activation time and allows the user to move forward and backward in time as well as adjust view parameters. Figure 2 shows a single time instant from such an animation in which regions in yellow correspond to areas with low standard deviation values, i.e., represent the mean activation time values. Other regions, spanning from red to purple, display larger values of the standard deviation and indicate the spatial distributions of uncertainty for this particular time instant. Examples of this uncertainty animation can be found at http://www.sci.utah.edu/~kpotter/research/heartActivation/

Figure 2.

Spatial Distribution of Activation Times and Standard Deviations. Animations of uncertainty in the spatial distribution of activation times on heart surfaces at 30 ms. Color maps show the spatial location and the difference between the present time and the mean activation time (within one standard deviation).

3. Results

Visualizing the results of varied conductivity parameters in the forward problem and the regularization parameter of the inverse problem allowed us to identify regions of interest within simulation results. In the ischemic forward model, isosurfacing, min/max isosurfacing and clustering allowed us identify regions near the ischemic zone by scanning the solution space of the simulation. Though independent of the physical space, these methods were able to scan the solution space and provide the viewer with a sense of uncertainty with respect to a specified isovalue or number of bins. By projecting these findings onto the physical space we were able to observe at what location, the chosen solution parameters differed. Standard deviation values were able to illustrate regions of uncertainty, within the areas of interest, as a sense of a range of variance within the solution ensemble.

By animating the activation based inverse solution, we were able to identify regions on the surface of the heart that exhibited higher temporal and spatial uncertainty. During a single frame of the animation (such as is seen in Figure 2) the standard deviation in some regions span a small distance (1/10 of the LV circumference) where other regions show spatial variation as large as 1/4 of the LV circumference. The temporal discrepancy in activation times within these spatial regions is as high as 36.4 ms.

3.1. Discussion

By using several uncertainty visualization techniques, we were able to highlight regions of interest in ischemic forward simulations and activation-based inverse solutions. Questions stemming from discrepancies in experimentally measured conductivity values for cardiac tissue generated small uncertainties in the forward solution. Regions in or near the ischemic zone exhibited greater standard deviations, while regions near the epicardium (where DTI-defined fiber directions exhibited more random directional behavior) had wider discrepancies between min and max isosurfaces (not shown). These results remain inconclusive and will require further investigation. Likewise, for the inverse case, some spatial regions were shown to exhibit higher standard deviations than others based on different, user selected regularization parameters, λ, that merits additional study.