Sensitivity of Noninvasive Cardiac Electrophysiological Imaging to Variations in Personalized Anatomical Modeling

Abstract

Objective

Noninvasive cardiac electrophysiological (EP) imaging techniques rely on anatomically-detailed heart-torso models derived from high-quality tomographic images of individual subjects. However, anatomical modeling involves variations that lead to unresolved uncertainties in the outcome of EP imaging, bringing questions to the robustness of these methods in clinical practice. In this study, we design a systematic statistical approach to assess the sensitivity of EP imaging methods to the variations in personalized anatomical modeling.

Methods

We first quantify the variations in personalized anatomical models by a novel application of statistical shape modeling. Given the statistical distribution of the variation in personalized anatomical models, we then employ unscented transform to determine the sensitivity of EP imaging outputs to the variation in input personalized anatomical modeling.

Results

We test the feasibility of our proposed approach using two of the existing EP imaging methods: epicardial-based electrocardiographic imaging and transmural electrophysiological imaging. Both phantom and real-data experiments show that variations in personalized anatomical models have negligible impact on the outcome of EP imaging.

Conclusion

This study verifies the robustness of EP imaging methods to the errors in personalized anatomical modeling and suggests the possibility to simplify the process of anatomical modeling in future clinical practice.

Significance

This study proposes a systematic statistical approach to quantify anatomical modeling variations and assess their impact on EP imaging, which can be extended to find a balance between the quality of personalized anatomical models and the accuracy of EP imaging that may improve the clinical feasibility of EP imaging.

I. Introduction

Noninvasive cardiac electrophysiological (EP) imaging, also known as inverse electrocardiography, mathematically reconstructs subject-specific cardiac electrical dynamics from noninvasive body-surface potential (BSP) measurements. Different EP imaging systems have been developed, delivering the reconstruction of potential dynamics on the epicardium [1], [2], activation front on both the epicardium and the endocardium [3], and intramural action potential dynamics [4]–[8] or current density/activation front [9], [10] throughout the depth of the ventricular wall. Developments in the past three decades have led to applications of EP imaging in important clinical problems, such as ventricular arrhythmia [11], [12], dyssynchronous heart failure [13], atrial fibrillation [14], and myocardial ischemia [15]. To move toward future clinical translation, several components within current EP imaging systems remain to be examined regarding both their clinical practicability and their influence on the robustness of these systems.

One such component is personalized anatomical models that are needed to establish the mathematical relationship between the cardiac electrical source v and body-surface measurements Φ (Φ = Hv), where the constructed transfer matrix H is specific to each subject’s heart-torso geometry. The role of anatomical modeling in the accuracy of EP imaging has been previously studied. It was shown that using a generic instead of personalized anatomical model in EP imaging would fail to accurately estimate cardiac electrical activity of individual subjects [16]. Further studies also confirmed that certain global anatomical parameters, such as size, position and orientation of the heart with respect to electrode placement on the torso, must be subject-specific in order for noninvasive EP imaging to be accurate [17], [18]. Therefore, it has been a standard practice in current EP imaging systems to utilize a high-quality, anatomically-detailed heart-torso model generated from individual subjects’ tomographic scans.

Personalized anatomically-detailed heart-torso model can be constructed either directly from patient’s tomographic data or by customizing a template model to patient’s anatomy [19]. However, taking either approaches, construction of detailed anatomical models puts high demands on the quality of tomographic images needed. It also involves a time-consuming, expert-dependent image-analysis process, the complexity of which reduces the cost-effectiveness of the otherwise promising technique of EP imaging in clinical practice. Furthermore, it introduces model variations stemming from a variety of factors such as image quality, segmentation expertise, segmentation methods, and/or registration techniques. This in turn leads to unresolved uncertainties in the outcome of EP imaging systems that bring questions to their robustness in clinical practice.

In this study, we will investigate the sensitivity of EP imaging outcomes to the variation in detailed personalized anatomical modeling. This will help us understand and verify the robustness of current EP imaging systems (to modeling variations inherent in these systems). Furthermore, it will shed lights on the quality of anatomical models that is actually needed for reliable EP imaging, which in the long term will provide guidance for establishing a clinically practicable procedure of anatomical modeling for EP imaging. Considering the solution of the inverse EP problem y as a function of input anatomical model x, y = g(x), where g represents the complex process of the inverse estimation of cardiac electrical activity, we approach our problem in two steps: 1) modeling the probabilistic/statistical distribution of the input variable x, the personalized anatomical models, and 2) quantifying the uncertainty of the outcome y in relation to the uncertainty in input x. In this study, we focus on the variation in modeling the ventricles of individual subjects.

To model the variations in input variable x, we propose a novel application of statistical shape modeling (SSM) [20]. SSM provides a parametric shape model that captures the pattern of variability among a set of shapes. It is conventionally used to represent the shape variation among a population of subjects. However, in the context of EP imaging, the importance of subject-specific global anatomical parameters have been established [17], [18]. Therefore, in this study, the variation to be modeled is that of the personalized shapes, assuming that its global parameters have been correctly personalized to the individual’s images. To do so, we build an SSM for each specific subject rather than a population of subjects. Training SSM over a set of subject-specific anatomical models derived from the same subject’s anatomical images with different image quality, different inter/intra-individual segmentation, or different segmentation methods, we derive a parametric description of the variation in personalized anatomical model in terms of their mean and variance.

To assess the sensitivity of the output variable y to the uncertainties in input variable x, there are two general groups of techniques: analytical versus simulation-based approaches. Analytical approach is based on the differentiation of system equations with respect to the parameters of interest [21]. In our problem, the system g represents a complex process including the construction of personalized anatomical models, the calculation of transfer matrix H, and the inverse estimation of cardiac electrical activity. Therefore, it cannot be explicitly formulated and analytically assessed.

Alternatively, simulation-based approaches provide sensitivity evaluation based on multiple simulations of the system g for different input values. The most well-known simulation-based approach is Monte Carlo (MC) analysis that generates a large number of random samples from the input parameter space. These samples are then propagated through the system to obtain the corresponding outputs, using which the statistics of the output variable can be calculated. However, given the large dimensionality of the input anatomical model x (more than 1000 points), application of MC analysis to our problem will require a large number of simulations that is computationally expensive if not impossible.

Deterministic sampling is another class of simulation-based techniques that balances between the analytical and random sampling techniques. An example of such techniques is generalized polynomial chaos-stochastic collocation method that approximates the system function by a linear combination of orthogonal polynomials using collocation points (samples). These collocation points are derived with the aid of collocation methods originally used in numerical differentiation/integration [22]. This method is previously used to study the sensitivity of the forward electrocardiographic simulation to the position of the heart [18]. However, the number of required collocation points increases exponentially with the dimension of the input variable. Alternatively, unscented transform (UT) is a deterministic sampling method that preserves the system equation g(x) intact and approximates the distributions of input/output variables instead. Particularly, it uses optimization methods to find a minimal number of samples that can preserve the statistical moments of a given distribution up to a desired order [23]. Compared to SC, in the presence of a high-dimensional input variable x, UT requires substantially less number of samples to accurately estimate statistics of output variable up to the second moment [24].

Therefore, we will apply the UT to the SSM, trained for each subject, to draw a set of samples from the statistical distribution of input personalized anatomical models. EP imaging will be carried out on each sample to obtain the corresponding outcome. The statistics of the EP imaging outcome will be calculated to study its uncertainty subject to anatomical modeling variations. The feasibility of our approach is demonstrated on a set of phantom and real-data experiments. In these experiments, we employ two of the existing EP imaging methods as testbeds: epicardial-based electrocardiographic imaging (ECGI) [1] and transmural electrophysiological imaging (TEPI) [6]. Our results report that, as verified by statistical test of equivalence, uncertainties of personalized anatomical models have negligible impact on the accuracy of EP imaging.

The implication of this finding is two-fold. On one hand, it shows that the current practice of EP imaging—reliant on detailed personalized anatomical model—is robust to modeling errors at the level of shape variations inherently associated with image quality, segmentation users, and segmentation methods. On the other hand, it indicates the possibility that high-level complexity in personalized anatomical models might not be necessary for reliable EP imaging and, as a result, the process of anatomical modeling in current EP imaging systems may be simplified for improved clinical feasibility in future practice.

A preliminary version of this study has been reported in [25], [26].

II. Cardiac EP Imaging

Cardiac EP imaging approaches essentially aim to recover cardiac bioelectrical sources from measurements of the bioelectrical field in the torso volume conductor, in particular on the body surface. The quasi-static electromagnetism [27] explains the relation between the field measurements within the torso volume and cardiac bioelectrical sources as

$$\nabla ·(({\mathbf{D}}_{int}+{\mathbf{D}}_{\text{ext}})\nabla {\varphi }_e(\mathbf{r}))=\nabla ·(-{\mathbf{D}}_{int}\nabla u(\mathbf{r}))\forall \mathbf{r}\in {\mathrm{\Omega }}_h$$ (1)

$$\nabla ·({\mathbf{D}}_i\nabla {\varphi }_i(r))=0\forall \mathbf{r}\in {\mathrm{\Omega }}_i,\cup {\mathrm{\Omega }}_i={\mathrm{\Omega }}_{t/h}.$$ (2)

On a bidomain heart model, (1) describes how the extracellular potential ϕ_e within the heart volume Ω_h originates from the gradient of action potential u. r stands for the spatial coordinate, D_int and D_ext are the effective intracellular and extracellular conductivity tensors, and their summation D_k = D_int + D_ext is bulk conductivity tensor.

On the monodomain torso model, (2) describes how the potential ϕ_i distributes within the volume conductor Ω_i external to the heart with conductivity tensor D_i, assuming that no other active electrical source exists within the torso.

Within the myocardium (1), the anisotropic ratio of D_k is a magnitude smaller than that of D_int [28], [29]. Therefore, we only retain the intracellular anisotropy and assume isotropic bulk and torso conductivity, which converts D_k and D_i to scalars σ_blk and σ_i, respectively. This is similar to an oblique dipole layer model, where the anisotropy of primary current is considered whereas that of passive/secondary current is neglected [29]. Intracellular conductivity tensor D_int is obtained by mapping a 3-D experimentally-derived mathematical fibrous model to the personalized ventricular geometry of the subject [30]. Its conductivity is assumed to be 0.24 Sm⁻¹ in longitudinal direction and 0.024 Sm⁻¹ in transversal direction [31]. Isotropic bulk conductivity σ_blk is calculated as an intermediate value between longitudinal and transversal conductivities (0.48 Sm⁻¹ and 0.12 Sm⁻¹, respectively) [31]. Torso conductivity value σ_t is assumed to be 0.2 Sm⁻¹ [31]. Based on these assumptions, the forward relationship between cardiac action potential u and body-surface voltage data ϕ can be described in the following Poisson’s (3) within the heart and Laplace’s (4) external to the heart:

$${\sigma }_{\text{blk}}{\nabla }^2{\varphi }_e(\mathbf{r})=\nabla ·(-{\mathbf{D}}_{int}\nabla u(\mathbf{r}))\forall \mathbf{r}\in {\mathrm{\Omega }}_h$$ (3)

$${\sigma }_t{\nabla }^2\varphi (\mathbf{r})=0\forall \mathbf{r}\in {\mathrm{\Omega }}_{t/h}.$$ (4)

To preserve the continuity of potentials and currents on the interface, a boundary condition on the heart surface Γ_h is defined as

$$\begin{array}{c}{\varphi }_e(\mathbf{r})=\varphi (\mathbf{r})\\ {\sigma }_{\text{blk}}\frac{\partial {\varphi }_e(\mathbf{r})}{\partial n}+{\mathbf{D}}_{int}\frac{\partial u(\mathbf{r})}{\partial n}={\sigma }_t\frac{\partial \varphi (\mathbf{r})}{\partial n}.\end{array}\forall \mathbf{r}\in {\mathrm{\Gamma }}_h$$ (5)

Similarly, a boundary condition is defined on the torso surface that assumes no current (flux) leaves the torso surface Γ_t

$$\frac{\partial \varphi (\mathbf{r})}{\partial n}=0\forall \mathbf{r}\in {\mathrm{\Gamma }}_t$$ (6)

where n stands for the outward normal of a surface.

Existing EP imaging techniques can be generally divided into two categories, according to how the aforementioned bidomain equations are solved (also known as forward modeling). In earlier approaches, the forward model is constructed on the domain external to the heart surface by solving the Laplace (4) and (6). In these approaches, the equivalent source model takes the form of epicardium potentials [1], [2] or AT on the ventricular surface [3], [32], [33]. By formulating the source model on the heart surface, these approaches avoid the nonunique solutions induced by the underlying biophysics. Recent efforts start to examine the forward model that includes the domain of 3-D myocardium, solving both the Poison’ s (3), (5) and the Laplace (4), (6). The source model in these approaches represent the real physiological sources in the heart, in the form of 3-D transmural action potential [4]–[8] or current density/activation wavefront [9], [10] throughout the myocardial wall. Because of the 3-D source model beneath the heart surface, this type of approaches faces additional challenges of nonunique solutions induced by the underlying biophysics of the problem.

A. Surface-Based EP Imaging Methods

Surface-based EP imaging considers equivalent source models on the ventricular surface in forms of epicardial potential [1], [2], [12], or activation time (AT) [3], [32], [33]. These surface-based EP imaging methods often overcome the ill-posedness of the inverse problem by employing regularization with various constraints on spatial and/or temporal properties of the solution, such as Tikhonov regularization (zero-order, first-order, second-order) [34], truncated-SVD [35], [36], state-space filtering framework (Kalman filter) [37], and recently L1-norm-based sparsity models [11], [38], [39].

In this paper, we consider ECGI technique, as presented in [1]. ECGI solves the inverse problem for the equivalent current source models located on the epicardium of the heart. Using boundary element method, (4) and (6) can be numerically solved on a subject heart-torso model to obtain linear relationship between body-surface measurements Φ and epicardial potential v_e : Φ = Hv_e. ECGI then solves the inverse problem using zero-order Tikhonov regularization treating each time instant separately

$${\widehat{\mathbf{v}}}_e=arg{min}_{{\mathbf{v}}_e}{\Vert \mathrm{\Phi }-\mathbf{H}{\mathbf{v}}_e\Vert }^2+\beta {\Vert {\mathbf{v}}_e\Vert }^2.$$ (7)

The first term in (7) provides the least square solution while the second term imposes constraint on the spatial smoothness of the solution. The regularization parameter β controls the relative weight of two terms. For details of the ECGI algorithm, please refer to [1].

B. Transmural EP Imaging Methods

In the past decade, transmural EP imaging has been emerged that considers transmural source models in forms of transmural action potential [4]–[8], or current density/activation wavefront [9], [10]. In the reconstruction of current density or activation wavefront, different orders of smoothness constraints have been imposed on the spatial or temporal properties of the solution, through methods such as Laplacian-weighted minimum norm [4], or lead-field-normalized-weighted minimum norm in combination with truncated-SVD [9]. In the reconstruction of transmural action potential, physiological constraints are often used [6]–[8]. For instance, Ohyu et al. proposed a maximum a posteriori (MAP) estimation of transmural action potential (AP) by modeling AP as a simplified template step function [8]. Another example is the method proposed by He et al. that presents a parameterized cellular automaton heart model, where deterministic regularization is used to optimize the model parameters [7]. Wang et al. also proposed another transmural EP imaging approach that guides MAP estimation of the trans-membrane potential distribution using a 3-D whole-ventricle electrical excitation model [6].

In this study, we consider TEPI method, as presented in [6]. TEPI solves the inverse problem for the transmural action potential within the myocardium. Using a proper combination of numerical methods, such as mesh-free and boundary element methods [6], [40], one can numerically solve (3)–(6) on a subject heart-torso model, and obtain linear relationship between body-surface measurements Φ and transmural action potential u: Φ = H̃u. For detailed specification of the heart-torso models and calculation of the forward problem using mesh-free and boundary element methods, please refer to [40].

TEPI combines a physiological spatiotemporal prior of action potential with BSP data through Bayesian statistical regularization to overcome the ill-posedness of the problem. Assuming action potential u and BSP Φ as random variables, TEPI seeks the MAP estimate of u_k at time instant k that maximizes the posterior density function of u_k given all body-surface measurements, Φ_1:k, up to the time instant k, where k = 1, …, t and t is the total number of measurements over time.

$$\widehat{{\mathbf{u}}_k}=arg{max}_{{\mathbf{u}}_k}\{f({\mathbf{u}}_k\mid {\mathbf{\Phi }}_{1:k})\}.$$ (8)

At each time step k, a prediction of the distribution of u_k is first obtained by a probabilistic simulation of a macroscopic Aliev–Panfilov model [41] based on its estimation from the previous iteration u_k−1. An MAP estimation of u_k is then calculated using the predicted prior and the likelihood according to the Bayes rules. For details of TEPI, please refer to [6].

III. Sensitivity Assessment of EP Imaging to Variations in Personalized Ventricular Shapes

Anatomical modeling involves variations and uncertainties that are caused by different factors such as different segmentation techniques, inter/intra-individual difference in segmentation, and image quality. As an example, Fig. 1 presents two different segmentation results (red and blue contours) of one cardiac MRI slice at the short axis view performed by two experts. The visible difference in local details of endocardium and epicardium contours would eventually lead to two different heart models built for the same subject. In contrast with global anatomical parameters (heart size, position and orientation) whose importance on EP imaging is well established, the influence of these local shape variations on the accuracy of EP imaging has not been investigated or well understood.

Fig. 1.

Two different segmentations (red and blue contours) of one MRI slice at the short axis view.

To determine the sensitivity of the EP imaging outcome y to the local anatomical modeling variations x, we formulate the problem as y = g(x), where x contains the 3-D coordinates of all surface nodes, y is the corresponding EP imaging outcome, and g represents a complex process including the construction of personalized anatomical models, the calculation of transfer matrix H, and the inverse estimation of cardiac electrical activity. As summarized in Fig. 2, first, we will model the statistical distribution accounting for variations in personalized anatomical modeling. Second, we will quantify the output uncertainty resulting from the input variation. Detailed descriptions of these two steps are provided in the following sections.

Fig. 2.

Outline of the proposed approach to investigate the uncertainty of EP imaging subject to variations in personalized anatomical models.

A. Uncertainty Quantification in Personalized Anatomical Models

As a first step, we need to quantify the variations/uncertainties in personalized anatomical models. Because of the importance of subject-specific global anatomical parameters (heart size, position, and orientation with respect to torso) on the accuracy of EP imaging [17], [18], we train a personalized SSM for each subject on a set of anatomical models that have identical global parameters correctly customized to the subject. This set of training models will include shape variations resulted from two major causes: different image resolution and interindividual difference in segmentation. For the former, cardiac MR/CT images of each subject are downsampled to generate different image resolutions. For the latter, different experts are recruited to perform manual segmentations for each subject’ s images. These segmentation results by different users on images with different resolutions provide the basis for constructing the training set of anatomical models to build the subject-specific SSM.

For each segmentation result, a surface mesh of the heart will be built with nodal positions represented in Cartesian coordinates (x_i, y_i, z_i i = 1, 2, …, n). For each subject, all surface meshes in the training set have the same number of nodes with known correspondence. Concatenating the coordinates of all n nodes to one vector, each surface mesh (anatomical model) can be described as x = (x₁, y₁, z₁, …, x_n, y_n, z_n)^T. It must be noted that, as mentioned earlier, the training anatomical models for each subject are aligned and share identical global anatomical parameters tailored to that subject.

Constructing the statistical shape model consists of two steps: 1) extracting the mean shape, 2) calculating the covariance matrix of the training shapes [20]. The mean shape x̄ is calculated by averaging over all training samples as

$$\overline{\mathbf{x}}=\frac{1}{s}\sum _{i=1}^s{\mathbf{x}}_i.$$ (9)

The covariance matrix P_x of the training samples is calculated as

$${\mathbf{P}}_{\mathbf{x}}=\frac{1}{s-1}\sum _{i=1}^s({\mathbf{x}}_i-\overline{\mathbf{x}}){({\mathbf{x}}_i-\overline{\mathbf{x}})}^T.$$ (10)

The variation of personalized anatomical models for a specific subject is thus modeled as a high-dimensional Gaussian random variable with mean x̄ and covariance P_x : (x̄, P_x).

B. Simulation-Based Sensitivity Evaluation

Given the statistical description of personalized anatomical modeling, we are interested in quantifying the resulting statistical distribution of the EP imaging output. As described earlier, the complex relationship g between EP imaging output y and the input anatomical model x cannot be analytically formulated. Therefore, simulation-based techniques should be used to obtain output statistics based on multiple simulations of y = g(x) using samples drawn from the input distribution x. Because of the high-dimensionality of x, we employ the UT to reduce the number of samples and accordingly the number of EP imaging processes needed to be carried out.

UT is established based on the principle that, given a nonlinear function y = g(x), it is easier to approximate a probability distribution than it is to approximate an arbitrary nonlinear function [23]. Accordingly, a set of weighted points (sigma points), S = {i = 1, 2, …, p : , }, is selected deterministically so as to minimize the error in approximating the statistical moments of x up to a desired order. Applying the nonlinear function g to each point yields a set of transformed points, statistics of which gives an estimate of the output statistics that otherwise could not be derived analytically.

Given the Gaussian shape model (x̄, P_x), derived in the previous section, different sets of sigma points can be used to capture the output statistics up to the second order. An example that we employed in our study is a set of 2L sigma points (samples) symmetrically-distributed around the mean along the direction of second-order variation

$$\begin{array}{c}{\mathcal{X}}_{\mathrm{i}}=\overline{\mathbf{x}}+{(\sqrt{L{\mathbf{P}}_{\mathbf{x}}})}_i,i=1,\dots ,L\\ {\mathcal{X}}_{i+L}=\overline{\mathbf{x}}-{(\sqrt{L{\mathbf{P}}_{\mathbf{x}}})}_i,i=1,\dots ,L\\ {\mathcal{W}}_i=1/2L,i,\dots ,2L\end{array}$$ (11)

where L represents dimension of the input variable x (3n in this case, n being the number of nodes in the shape model) and ${(\sqrt{L{\mathbf{P}}_{\mathbf{x}}})}_i$ denotes the ith row or column of the matrix $\sqrt{L{\mathbf{P}}_{\mathbf{x}}}$. is the weight assigned to the ith sigma point.

Because L = 3n is large in our study (n is at the order of 10³), 2L sigma points still requires a large amount of EP imaging processes. To further reduce the number of sigma points needed, we propose to sample the sigma points only along the predominant (instead of all) directions of variance in x. To extract the principal directions of variance in x, principle component analysis is applied to P_x to decompose the covariance matrix into the principle modes of variations Ψ (eigenvectors) and the corresponding value of variances λ (eigenvalue)

$$\sqrt{{\mathbf{P}}_{\mathbf{x}}}=\sqrt{\lambda }\mathrm{\Psi }.$$ (12)

The modes of variations are sorted in an order with descending variances such that λ₁ ≥ λ₂ ≥ ⋯ ≥ λ_3n. Therefore, the covariance matrix P_x can be approximated by the c highest modes of variations as

$$\sqrt{{\stackrel{\sim }{\mathbf{P}}}_{\mathbf{x}}}=[\sqrt{{\lambda }_1}{\psi }_1;\sqrt{{\lambda }_2}{\psi }_2;\cdots ;\sqrt{{\lambda }_c}{\psi }_c].$$ (13)

Replacing the full size covariance matrix P_x with the reduced covariance matrix P̃_x in (11), we can reduce the number of required sigma points to 2c as

$$\begin{array}{c}{\mathcal{X}}_i=\overline{\mathbf{x}}+\sqrt{c{\lambda }_i}{\psi }_i,i=1,\dots ,c\\ {\mathcal{X}}_{i+c}=\overline{\mathbf{x}}-\sqrt{c{\lambda }_i}{\psi }_i,i=1,\dots ,c\\ {\mathcal{W}}_i=1/2c,i=1,\dots ,2c.\end{array}$$ (14)

To guarantee plausible shapes, $\sqrt{c}$ should be restrained to a certain interval. A common approach is to limit $\sqrt{c}$ to lie within [−3, +3], treating all modes as independent distributions [20]. This is approximately equivalent to taking the first ten eigenvectors (c = 10) in sampling the sigma points, which accounts for 98% of the total variation ( ${\sum }_{i=1}^L{\lambda }_i$). As an example, Fig. 3 presents anatomical variations around mean geometry x̄ (shown in red color) along the (a) first, (b) second, and (c) third principle directions (white and blue contours show $-3\sqrt{{\lambda }_i}$ and $+3\sqrt{{\lambda }_i}i\in \{1,2,3\}$, respectively).

Fig. 3.

Examples of anatomical variations around mean geometry model along (a) first, (b) second, and (c) third principle directions. Mean geometry model is shown in red color while the two sample geometry models are shown in white and blue.

Each sigma point represents a personalized anatomical model following the statistical distribution modeled for a specific subject using his or her SSM.

C. Statistical Analysis

For each subject, on each anatomical model sampled from his/her personalized SSM, EP imaging can be carried out using identical BSP data acquired for that specific subject

$${\mathcal{Y}}_i=g({\mathcal{X}}_i).$$ (15)

According to the UT theory, the first two moments of the output distribution can be obtained as

$$\overline{\mathbf{y}}=\sum _{i=0}^{2c}{\mathcal{W}}_i{\mathcal{Y}}_i$$ (16)

$${\mathbf{P}}_{\mathbf{y}}=\sum _{i=0}^{2c}{\mathcal{W}}_i{({\mathcal{Y}}_i-\overline{\mathbf{y}})}^2.$$ (17)

In this way, the mean and variance of EP imaging output (16) and (17) are calculated, which provide information about the distribution/uncertainty of the EP imaging solutions.

In addition, pooled from results across multiple subjects, we conduct a statistical test of equivalence to investigate the statistical equivalency of the EP imaging outputs obtained on different anatomical models drawn from subject-specific SSM models. To do so, a specific accuracy measure θ (explained in Section IV) is extracted from the EP imaging outputs. θs obtained on the same subject yet with different anatomical models (drawn from the SSM) are randomly paired up and the difference of each pair (dθ) is calculated. The hypothesis test of equivalence based on paired t-test is then performed on the dθs of multiple subjects. Assuming parameter dθ ~ (δ, σ²) to be the measure of the intrasubject difference of the paired observations of θs, statistical equivalency is obtained if δ/σ lies within an established range [−ε, ε],ε > 0 [42]. The alternative and null hypotheses are defined as

$$\begin{array}{l}\text{Alternative}\text{hypothesis}(H_1):-\epsilon \le \delta /\sigma \le \epsilon \\ \text{Null}\text{hypothesis}(H_0):\delta /\sigma <-\epsilon \text{or}\delta /\sigma >\epsilon .\end{array}$$

IV. Experiments and Results

As mentioned before, we consider two EP imaging approach (ECGI and TEPI, described in Section II) as testbeds for our experiments. Solution of ECGI is in the form of epicardial potential dynamics while TEPI solution is in the form of transmural action potential dynamics. We conduct two sets of synthetic and real-data experiments.

A. Synthetic Experiments

Synthetic experiments are conducted on four subjects including one canine heart and three human hearts. A realistic human torso model with 370 vertices on the triangulated body surface is coupled with all the heart models to set up the phantom experiments [43]. For each subject, MR/CT images are down-sampled to generate three sets of images with different resolutions. A set of seven ventricular models are developed using manual segmentation of MR/CT images with specific resolution performed by different experts (total 21 segmentations for each subject). This set serves the training set for constructing subject-specific SSM. From each subject-specific SSM, 20 ventricular models are generated using the ten highest Eigen values of the covariance matrix. BSP data, including 370 electrodes located on the vertices of the torso surface mesh, are simulated on the mean shape provided by the trained SSM for each subject and are corrupted with 20-dB white Gaussian noise. For each subject, the identical BSP data are coupled with the SSM-sampled ventricular models for EP imaging to ensure that the only factor affecting EP imaging outputs is the variation in the input ventricular shape.

In the synthetic experiments the ground truth is available in form of epicardial/action potential dynamics. Therefore, quantitative accuracy of the solution is evaluated in terms of average relative error (RE) and average correlation coefficients (CC) between estimated and true potentials over a cardiac cycle [1].

Experiments are conducted on both healthy and postinfarction settings; for the latter, the infarct region extends from basal to midanteroseptal and inferoseptal in each subject’s ventricle.

1) ECGI Results

Fig. 4 gives an example of the epicardial potential reconstructed on two ventricular models for the same subject in the healthy setting at two time instants during depolarization and repolarization stages. Variation in two personalized anatomical models is also shown in short-axis views in Fig. 5(a). Despite visible difference in anatomical details, similar epicardial potential patterns can be observed for two anatomical models during cardiac cycle. Estimated epicardial potential dynamics for two corresponding nodes on the surface meshes of two ventricular models in the healthy setting are also very similar [see Fig. 5(b)]. An example of epicardial potential reconstruction for the same anatomical models in the postinfarction setting is shown in Fig. 6. Likewise, potential dynamics are consistent for the two ventricular models during depolarization and repolarization.

Fig. 4.

ECGI solution (epicardial potential) obtained on two ventricular models for the same subject in the healthy setting, at two different time instants of cardiac depolarization and repolarization phases. (a) Venticular model 1. (b) Venticular model 2.

Fig. 5.

(a) Overlap of two ventricular models of one subject, shown in Fig. 4, in short-axis view from base to apex. (b) ECGI solution at two corresponding nodes on the two ventricular models in healthy setting.

Fig. 6.

ECGI solution (epicardial potential) obtained on two ventricular models for the same subject in the postinfarction setting, at two different time instants of cardiac depolarization and repolarization phases. (a) Venticular model 1. (b) Venticular model 2.

The mean and variance of the ECGI outputs on a mean anatomical model are calculated according to (16) and (17). Standard deviation (STD) map of ECGI outputs for healthy and postinfarction settings during the ST segment of an ECG cycle is presented in Fig. 7. STD ranges from 2.63e – 9 to 1.4e – 4 in the healthy setting while in the postinfarction setting there is a slight increase in the STD range (6.43e – 8, 0.007). Low STD in both settings indicates robustness of ECGI outcomes to the variations in personalized anatomical models.

Fig. 7.

STD map of ECGI output on a mean anatomical model during ST segment of an ECG cycle for (a) healthy, and (b) postinfarction settings.

On ECGI outputs obtained on the 80 ventricular models for the four subjects, error measures RE and CC for the same subject are randomly paired up (40 pairs for the paired t-test) and RE difference and CC difference are calculated for each pair. Paired results of the four subjects have RE difference (dθ) with mean 0.03 and variance 0.19. Assuming the commonly used equivalence limits ε = 0.5 [42], with 40 number of pairs, dθ resides in the rejection region at α level = 5%. The CC difference for the paired population has normal distribution (0.04, 0.032), also reporting the rejection of null hypothesis for the tolerance ε = 0.5 at α = 0.05.

2) TEPI Results

Fig. 8 presents four snapshots of transmural action potential estimation on two ventricular models of one subject in the healthy setting during depolarization and recovery stages. Transmural action potential propagates similarly through two models during cardiac cycle despite visible anatomical detail differences [shown in Fig. 9(a)]. Estimated transmural action potential dynamics for two corresponding nodes on the two ventricular models during one cardiac cycle are presented in Fig. 9(b). It can be seen that slight difference in the position of the corresponding nodes does not affect the estimated action potential dynamics. Fig. 10 shows snapshots of transmural action potentials obtained on two ventricular models of the same subject in the postinfarction setting, where late activation (left) and early repolarization (right) at infarct region are similar in two anatomical models.

Fig. 8.

TEPI solution (transmural action potential) obtained on two ventricular models for the same subject in the healthy setting, at two different time instants of cardiac depolarization and repolarization phases. Color encodes the value of action potential. (a) Venticular model 1. (b) Venticular model 2.

Fig. 9.

(a) Overlap of two ventricular models of one subject, shown in Fig. 8, in short-axis view from base to apex. (b) TEPI solution at two corresponding nodes on the two ventricular models in healthy setting.

Fig. 10.

TEPI solution (transmural action potential) obtained on two ventricular models for the same subject in the postinfarction setting, at two different time instants of cardiac depolarization and repolarization phases. Color encodes the value of action potential. (a) Venticular model 1. (b) Venticular model 2.

Similarly, mean and variance of TEPI outputs are determined [see (16) and (17)]. STD map of TEPI outputs on a mean anatomical model during the ST segment of an ECG cycle for both healthy and postinfarction settings are shown in Fig. 11. TEPI output STD ranges between 8.25e – 6 to 0.03 for different ventricular models during a cardiac cycle in healthy setting. In postinfarction settings, STD range increases (9.25e – 6, 0.05) compared to healthy setting. The small range of variation in TEPI output in both settings confirms its low sensitivity to the variations in input anatomical models.

Fig. 11.

STD map of TEPI output on a mean anatomical model during ST segment of an ECG cycle for (a) healthy, and (b) postinfarction settings.

Likewise, TEPI outputs on the entire population are collected and its errors in RE and CC on ventricular models for the same subject are randomly paired up (40 pairs in total for the four subjects). The difference of RE (dθ) for the paired results of all four subjects has mean 0.01 and variance 0.03 that belongs to the rejection region at α =0.05, choosing equivalence limits to be ±ε for ε = 0.5 and a number of 40. The null hypothesis is thus rejected and the alternative hypothesis can be accepted. Similarly, the difference of CC for the paired population has a normal distribution (0.02, 0.016) that reports the rejection of the null hypothesis for the tolerance ε = 0.5 at α =0.05. Therefore, in our phantom experiments, the null hypothesis is successfully rejected, and we can conclude that local variations in personalized ventricular models do not affect the accuracy of either EP imaging methods being considered. In another words, variations in the shape level does not impact the accuracy of EP imaging systems.

B. Real Data Experiments

Real data experiments are performed on four postinfarction human subjects [44]. The datasets include end-diastolic MR images of heart/torso geometries and 123-lead BSP recordings for each subject. MRI data of each subject’s heart includes ten slices from base to apex with 8-mm interslice spacing and 1.33-mm pixel spacing. The torso surface is described by boundary elements with 370 vertices. BSP measurements were recorded by Dalhousie University standards [20] at 123 known anatomical sites and interpolated to 370 nodes of the Dalhousie torso model; each BSP recording consists of a single averaged PQRST complex sampled at 2 kHz. Gold standards of the infarct are provided in terms of the location and size of the infarct with regard to the 17-segment division of the LV according to AHA standards [45].

Similar to the phantom experiments, seven ventricular models are built for each subject based on manual segmentations of three different images resolutions by different experts in order to create the SSM training set (in total 21 segmentations for each subject). Using trained SSM for each subject, 16 ventricular models are generated, giving 64 heart models in total among the four subjects. For each subject, 16 ventricular models are coupled with identical measured BSP data as inputs to the two EP imaging methods under study.

In real-data experiments, because the ground truth of epicardial/action potential dynamics (e.g., intracardiac electrogram recording) is not available, infarct quantification is used to assess the solution accuracy. Infarct region can be identified based on the characteristic features of the estimated potential dynamics. In this study, we extract two most representative features of the infarct region: AT and action potential duration (APD) which is defined as the difference between AT and repolarization time (RT). In ECGI, late AT is used as a main characteristic of the infarct region. AT is set at the maximum negative derivative of the QRS segment of epicardial potential trace [46]. In TEPI, regions associated with late AT and/or short APD are identified as infarct regions [47]. AT is set at the maximum first derivative of action potential upstroke and RT is set at the maximum second derivative of action potential downstroke [48]. Infarct region is finally quantified in terms of infarct size (IS) and center that are calculated based on the 17 segments of the left ventricle (LV) [45]. IS describes the ratio of estimated infarcted segments to the total number of segments. Infarct center (IC) represents the center of the infarct scar segments.

1) ECGI Results

Fig. 12(a) shows ECGI solutions for two ventricular models of case 3 and case 4 during depolarization. For each subject, despite the visible difference in anatomical details of the two ventricular models, similar potential dynamics can be observed. Infarct regions detected by ECGI methods for two ventricular models of the same subject are also identical, as highlighted with black contour with good match with the ground truth. Detected infarct region for case 1 corresponds to segments 1, 3, 4, 7, 9, 10, 13 and 15 with center located at segment 9. In case 2, segments 4, 5, 9, 10, 11 are associated with infarct region (center at segment 10). In case 3, detected infarct region is centered at segment 12 and extends to segments 3, 4, 5, 11 and 16. In case 4 the infarct region is detected to be at segments 1, 3, 4, 5, 7, 10, 15, and 17 with center at segment 15. The STD map of ECGI solution for these two cases at the corresponding time instants are also shown in Fig. 12(b). Low peak value on the STD map (0.004 mV) indicates robustness of ECGI solution to the variations in personalized anatomical modeling.

Fig. 12.

(a) ECGI solution for two ventricular models constructed for case 3 and case 4 during depolarization. Black and red contours show the ECGI estimated and the reference infarct region, respectively. (b) STD map of ECGI output on a mean anatomical model for case 3 and case 4 at the corresponding time instants. Color encodes the value of potential magnitude.

Hypothesis test of equivalence is conducted on randomly paired up ECGI outputs of the same subject (32 pairs in total for four subjects) using IS and IC as accuracy measures. IS difference and IC difference for paired ECGI observations have normal distribution (0.1, 0.17) and (0.6, 1.17), respectively, that falls within the rejection region of the null hypothesis. In another words, the alternative hypothesis is accepted meaning that variations in shape level have negligible impact on accuracy of ECGI output.

2) TEPI Results

Fig. 13(a) presents TEPI solutions for two ventricular models of case 3 and case 4. Despite variations in anatomical details, similar potential dynamics can be observed on two anatomical models of the same subject. The infarct regions detected for the two anatomical models (highlighted with black contours) are consistent and match the ground truth (red contour) as well. In case 1, detected infarct region (segments 1, 2, 7, 8, 9, 13, 14, and 15) centers at segment 8. The infarct region in case 2 extends to segments 2, 3, 8, 9, 14, and 15 with center 9. In case 3, segments 3, 4, 9, 10, and 15 are detected to be associated with infarct (center at segment 10). In case 4, infarct region is estimated to be at segments 9, 10, 11, and 15 (center at 15). The STD map of TEPI solution for these two cases during repolarization are also shown in Fig. 13(b). Maximum STD corresponds to low value of 0.001 mV that shows low sensitivity of TEPI method to the variations in anatomical modeling.

Fig. 13.

(a) TEPI solution for two ventricular models constructed for case 3 and case 4. Black and red contours show the TEPI estimated and the reference infarct region, respectively. (b) STD map of TEPI output on a mean anatomical model for case 3 and case 4 at the corresponding time instants. Color encodes the value of potential magnitude.

TEPI outputs on ventricular models for the same subject are randomly paired up, and hypothesis tests of equivalence are performed on the 32 pairs for the four subjects. IS difference and IC difference of paired observations for TEPI outputs have normal distribution (0.01, 0.06) and (0.4, 1.8), respectively. For a sample of 32 pairs and the equivalence limit ε = 0.5, it reports the rejection of null hypothesis and acceptance of alternative hypothesis at 5% level. Therefore, the real-data experiments results confirm the findings of synthetic experiments.

V. Discussions

A. Bound of Ventricular Shape Variations for Accurate EP Imaging

As the current study demonstrates that error in modeling anatomical details of patient-specific geometry does not significantly impact accuracy of EP imaging methods, the next natural question is what is the bound of such modeling errors in order to guarantee accurate EP imaging results. However, it is difficult to statistically model the complete range of modeling errors that could be involved in a geometrical model. The SSM built in this paper represents only the error in modeling an anatomically-detailed model. In the proposed approach of sensitivity study, sampling using the method of UT has exhausted the distribution space as defined by the SSM, i.e., by scaling the eigenvalues up to ±3λ [20]. To go beyond the distribution defined by the SSM (i.e., to add even larger scaling values), we will obtain in-valid geometrical models [20]. Therefore, an alternative approach need to be taken to investigate the “bound” of shape variations for valid EP imaging.

As a first step, we propose a minimal model that is based on analytic descriptions of regular 3-D geometrical shapes (see Fig. 14). In this minimal ventricular model, LV epicardium and endocardium are described by two concentric circles at short axis and an ellipsoid at long axis. Right ventricle (RV) surface is represented using two concentric circles intersecting with LV circles at short-axis and a parabola at long axis [49]. Subject-specific global anatomical parameters (heart size, position and orientation) are incorporate into this minimal model through parameters of the geometrical shapes such as center and radii of endocardium and epicardium circles as well as length of the ellipsoid. This minimal ventricular model (see Fig. 14) can be seen as an anatomically-plausible model that is personalized to a subject’s heart anatomy without including realistic shape details. The construction details of this model and its use in phantom experiments were previously described [50].

Fig. 14.

Cardiac minimal anatomical model. (a) LV model. (b) RV model.

Here, on the four human subjects in real-data experiments, we construct corresponding minimal anatomical model for each subject. Coupling minimal model of each subject with its BSP measurements, we reconstruct EP outcomes for each subject using ECGI and TEPI methods. Similarly, infarct quantification is performed as explained in the real-data experiments section. The results of ECGI method for case 3 and 4 are presented in Fig. 15. As can be seen, the reconstructed epicardial potential maps for the minimal model of both cases are similar to that of realistic anatomical models of Fig. 12. TEPI results on minimal models of case 3 and case 4 are presented in Fig. 16. As highlighted in the Figs. 13 and 16, there is a good agreement between the infarct region detected using detailed anatomical details (see Fig. 13) and the one detected minimal model (see Fig. 13) in both cases.

Fig. 15.

ECGI solution for minimal ventricular models of case 3 and case 4 during depolarization. Black and red contours show the ECGI estimated and the reference infarct region, respectively.

Fig. 16.

TEPI solution for minimal ventricular models of case 3 and case 4. Black and red contours show the TEPI estimated and the reference infarct region, respectively. Color encodes the value of potential magnitude.

This preliminary study reports negligible role of anatomical details in accurate EP imaging as well as potentials of the proposed minimal model to be used as a generic model that can be quickly personalized to a patient’s global geometrical parameters. In the future, it would be interesting to build a SSM that includes variations both in the global parameters and the shape details, which will then allow us to investigate the “bound” of modeling error toward nonpatient-specific models. This will require a comprehensive set of training data and substantial modeling effort in order to build an SSM like this, which will constitute the next step of our study.

B. Validation Against Intracardiac EP Data

In this study, the findings are limited to healthy and postinfarction settings. Furthermore, in real-data postinfarction setting, EP imaging methods are only validated against gold standard in terms of 17 segments of AHA standard due to the challenge of obtaining intracardiac validation data in humans. To date, transmural EP imaging validation against intracardiac measurements is only reported in animal studies [51], [52]. In future study, more analysis will be conducted on real cases with intracardiac EP measurements as gold standard to obtain sensitivity study in terms of RE and CC. The sensitivity of EP imaging methods to personalized anatomical models variations in other cardiac pathologies is remained to be verified.

C. Impact of Heart Motion

It is a common practice in cardiac EP imaging to use static heart due to model complexity, computational cost, and more importantly, the challenge to acquire temporally-aligned body-surface mapping and tomographic imaging data. However, neglecting heart motion in cardiac EP imaging will affect the solutions. In the depolarization stage, this effect can be assumed to be minimal because, as a result of the delay between electrical activation and mechanical contraction, the depolarization of the ventricles can be expected to be close to completion before it starts to contract. The repolarization of the ventricles, however, will be accompanied by the deformation of the heart and it remains unclear how the neglect of this deformation would affect EP imaging. Because the current study focuses on the effect of shape variations instead of heart motion on cardiac EP imaging, we follow the common practice in the field to adopt a static heart model to isolate the factors of interest (shape variations).

D. Impact of Fiber Architecture

In this study, fiber structures on sample ventricular models are obtained by mapping a detailed, experimentally-derived mathematical model [30] to the patient-specific geometry of our ventricular models. The 3-D fiber architecture is fixed for each patient among the multiple ventricular models with shape variations, again to isolate the factor of shape variation and its effect on cardiac EP imaging. It therefore does not affect the observations of our sensitivity study.

VI. Conclusion

Noninvasive cardiac EP imaging focuses on noninvasive and mathematical reconstruction of cardiac electrical activity using body surface measurements data. It has been shown diagnostic potential in current practice of clinical EP study for different cardiac pathologies [11]–[15]. To move toward future clinical translation, several components within current EP imaging systems remain to be examined regarding both their clinical practicability and their influence on the robustness of these systems.

One such component is personalized anatomically-detailed ventricular modeling, preparation of which involves inevitable variations and uncertainties that may impact reliability of EP imaging outcome. While global anatomical parameters (heart size, position and orientation) have been proven to be important, shape variation caused by personalized anatomical modeling errors has not been investigated. This paper, for the first time, we demonstrate that, when global anatomical parameters are correctly captured, modeling errors introduced during subject-specific modeling may not impact accuracy of EP imaging outcome. This study not only verifies the robustness of the investigated EP imaging methods to the errors in personalized geometrical modeling, but also suggests the possibility to simplify the process of anatomical modeling in future practice of EP imaging.

While this study only considers two sources of variations in personalized ventricular models, the presented approach can be used to study a wider variety of anatomical variations such as that in modeling the torso model. Furthermore, the presented approach can be extended to study different levels of model complexity starting from global anatomical parameters (heart size, location, orientation with respect to surface electrodes) to variations in shape details. This can help us find the balance between the complexity/quality of anatomical models and the accuracy of EP imaging for improved cost-effectiveness of these techniques in future clinical practice.