Physiological-Model-Constrained Noninvasive ... - IEEE Xplore

296

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 57, NO. 2, FEBRUARY 2010

Physiological-Model-Constrained Noninvasive Reconstruction of Volumetric Myocardial Transmembrane Potentials Linwei Wang∗ , Member, IEEE, Heye Zhang, Ken C. L. Wong, Huafeng Liu, Member, IEEE, and Pengcheng Shi, Member, IEEE

Abstract—Personalized noninvasive imaging of subject-specific cardiac electrical activity can guide and improve preventive diagnosis and treatment of cardiac arrhythmia. Compared to body surface potential (BSP) recordings and electrophysiological information reconstructed on heart surfaces, volumetric myocardial transmembrane potential (TMP) dynamics is of greater clinical importance in exhibiting arrhythmic details and arrythmogenic substrates inside the myocardium. This paper presents a physiologicalmodel-constrained statistical framework to reconstruct volumetric TMP dynamics inside the 3-D myocardium from noninvasive BSP recordings. General knowledge of volumetric TMP activity is incorporated through the modeling of cardiac electrophysiological system, and is used to constrain TMP reconstruction. This physiological system is reformulated into a stochastic state-space representation to take into account model and data uncertainties, and nonlinear data assimilation is developed to estimate volumetric myocardial TMP dynamics from personal BSP data. Robustness of the presented framework to practical model and data errors is evaluated. Comparison of epicardial potential reconstructions with classical regularization-based approaches is performed on computational phantom regarding right bundle branch blocks. Further, phantom experiments on intramural focal activities and an initial real-data study on postmyocardial infarction demonstrate the potential of the framework in reconstructing local arrhythmic details and identifying arrhythmogenic substrates inside the myocardium. Index Terms—Body surface potential (BSP), cardiac electrophysiological imaging, data assimilation, inverse problem of ECG (IECG), myocardial transmembrane potential (TMP).

I. INTRODUCTION ODY surface potential (BSP) recordings provide noninvasive observations of the underlying cardiac electrical activity, among which the ECG has been a standard tool for diagnosing cardiac arrhythmia in clinical practice. These observa-

B

Manuscript received April 11, 2008; revised January 10, 2009 and February 25, 2009. First published June 16, 2009; current version published January 20, 2010. This work was supported by the Rochester Institute of Technology (RIT) doctoral fellowships, by the China 973 Project, and by the Hong Kong Research Grants Council (HKRGC). Asterisk indicates corresponding author. ∗ L. Wang is with the College of Computing and Information Sciences, Rochester Institute of Technology, Rochester, NY 14623 USA (e-mail: [email protected]). H. Zhang is with the Bioengineering Institute, University of Auckland, Auckland 92019, New Zealand. K. C. L. Wong and P. Shi are with the College of Computing and Information Sciences, Rochester Institute of Technology, Rochester, NY 14623 USA. H. Liu is with the State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2009.2024531

tions, however, are limited in spatial resolution and remote from the heart. Moreover, as described by the quasi-static electromagnetic theory [1], the potential difference recorded on each pair of electrode is produced by the integration of electrical sources throughout the myocardial mass; therefore, the internal 3-Ddistributed details are absent. To improve preventive diagnosis and treatment of cardiac arrhythmia, it is of direct clinical interest to reconstruct more detailed cardiac electrophysiological information from noninvasive BSP recordings. Early solutions of this inverse problem of ECG (IECG) are formulated in terms of physical simplifications of cardiac sources [2]–[4], such as potential distribution on epicardium [3]. Compared to BSP, these solutions characterize cardiac electrical activity from closer distances to the heart with higher resolutions. Nevertheless, secondary tasks are needed to deduce diagnostically useful parameters of real cardiac sources [5]. Efforts have also been made to directly reconstruct cardiac electrophysiological parameters, such as transmembrane potential (TMP) and activation front, on heart surfaces [5]–[9]. These solutions do not provide electrophysiological details inside the myocardial volume. Because cardiac arrhythmia usually involve intramural arrhythmogenic substrates and local arrhythmic details, it is of greater clinical importance to reconstruct TMP dynamics throughout the myocardium. According to the quasi-static electromagnetism [1], unfortunately, BSP alone is not sufficient for uniquely determining the 3-D distribution of electrical sources inside the myocardium [10]. Prior physiological knowledge about volumetric TMP activity is, therefore, needed to constrain the solution space. These knowledge are usually incorporated through physiological models, under the constrains of which volumetric cardiac electrical activity can be obtained by deterministic optimization [11]. Nevertheless, physiological models are developed for the general population, and therefore, differ from subject-specific conditions. Furthermore, personal BSP data are always corrupted by noise in practice. It leads to the risk of unrealistic TMP estimates if general physiological models overly constrain subject-specific information, especially when the reconstruction is incorrectly initialized. A feasible solution is to couple the prior knowledge and observational data in a statistical perspective, so that both model and data errors are explicitly taken into account. In this paper, we develop a physiological-model-constrained statistical framework for noninvasive volumetric TMP imaging, based on: 1) utilizing a priori knowledge of general

0018-9294/$26.00 © 2009 IEEE

WANG et al.: PHYSIOLOGICAL-MODEL-CONSTRAINED NONINVASIVE RECONSTRUCTION

297

Fig. 1. (a) Three-dimensional ventricular myocardium of a specific subject, represented by mesh-free points (yellow dot) with the associated 3-D fiber structure (red line). (b) Heterogeneity of TMP shapes in epi-, endo-, midmyocardial layers. (c) Example of combined heart–torso model on the coupled mesh-free BE platform. The ventricles are represented with a cloud of mesh-free points and the torso is described by triangulated body surface.

electrophysiological activity inside the myocardium to constrain the TMP imaging and 2) coupling this general knowledge with personal data in a statistical manner to allow the existence of uncertainties in both information. To incorporate sophisticated a priori physiological knowledge, the cardiac electrophysiological system is modeled to describe the volumetric myocardial TMP activity model for system dynamics and TMP-to-BSP model for system observations. It is constructed on personalized heart–torso structures derived from tomographic images of individual subjects, and reformulated into a stochastic state-space representation to take into account model and data errors. Nonlinear data assimilation is then developed to estimate volumetric myocardial TMP dynamics from input BSP sequence. Robustness of the presented framework is validated by phantom experiments regarding various data and model errors in practical environments. Comparison with classical regularization-based IECG approaches is then performed regarding epicardial potential (EP) reconstruction for bundle branch block (BBB) conditions. Further, phantom experiments on intramural focal activities and an initial real-data study on a patient with myocardial infarction (MI) demonstrate the potential of the presented framework in reconstructing local conduction abnormality and identifying arrhythmic substrates inside the myocardium for individual subjects. II. METHODS A. Cardiac Electrophysiological System Modeling of the cardiac electrophysiological system aims at: 1) incorporating sophisticated physiological knowledge of general TMP activity inside the myocardium, and establishing its relationship with BSP observations and 2) personalizing heart and torso structures from tomographic images. We develop a coupled mesh-free boundary element (BE) platform to represent the combined heart–torso structures, based on which we derive and validate the associated volumetric myocardial TMP activity model [12], [13] and TMP-to-BSP model [14], [15]. In what follows, these models are briefly reviewed for a sufficient

background for this framework. For more details refer to related publications and technical reports [12]–[15]. 1) Personalized Heart–Torso Model: Given a set of shortaxis cardiac structural images such as MRI, we hand-trace 2-D contours of epicardium and endocardia slice by slice, and build triangulated meshes for heart surfaces. A two-stage Gaussian algorithm [16] is performed to smooth the faceted surfaces while keeping their overall size and shape features. A cloud of mesh-free points is then placed within the heart surfaces to represent the 3-D myocardium. To describe myocardial conductive anisotropy, detailed ventricular fiber structures are mapped from the mathematical model of myocardial fibrous structure established in [17]. First, fiber orientations on the epicardium and endocardia of personalized heart model are mapped from the reference model after registering corresponding surfaces of personalized heart model and reference heart model. Because ventricular fibers are spirally arranged and fiber orientations change from epi- to endocardium in a counterclockwise manner [18], fiber orientations inside the myocardium are interpolated from those on the surfaces. Fig. 1(a) exemplifies the 3-D mesh-free representation and fiber structure constructed for a canine heart. Meanwhile, transmural heterogeneity of TMP shapes across the epi-, mid-, and endomyocardial regions [19] [see Fig. 1(b)] is also taken into account. Regarding the impact of torso modeling on forward/inverse electrocardiographic problems, the consensus has been reached that geometry is the primary factor that is compared to the material property [20], [21]. To reduce model complexity, it is reasonable to emphasize the accuracy of geometrical modeling and relax excessive restrictions on tissue anisotropy or inhomogeneity. In this study, the torso is assumed as an isotropic and homogeneous volume conductor, and is described by triangulated body surface. Torso structures are effectively personalized only with a few parameters by deforming a reference model to match subject-specific structural image data [22]. Fig. 1(c) illustrates a combined heart–torso model on the coupled mesh-free BE platform, where the torso is represented with triangulated body surface, and the 3-D ventricular mass

298


is represented with a cloud of mesh-free points. In practice, the number of mesh-free points usually ranges from 2000 to 5000 for a physiological plausible yet computationally feasible representation of the 3-D myocardium. The triangulated body surface, in comparison, only possesses around 500 vertices, as we have demonstrated that the resolution of body surface mesh has less impact on the accuracy of TMP-to-BSP modeling [14], [15]. 2) Volumetric Myocardial TMP Activity Model: Complexity of cardiac electrophysiological models ranges from macroscopic-level two-variable equations [23] to the cellularlevel Luo–Rudy model [24] with an excess of 15 variables. Among them, two-variable diffusion-reaction systems [23], [25], [26] are favorable in IECG studies because of their ability to balance model plausibility with computational feasibility, which is given as  ∂u    ∂t = ∇(D∇u) + f1 (u, v)    ∂v = f2 (u, v) ∂t

(1)

where u stands for TMP, v for recovery current, D is the diffusion tensor, and ∇(D∇u) accounts for intercellular electrical propagation. Variations of f1 (u, v) and f2 (u, v) produce different TMP shapes. The mesh-free method [27] is utilized to spatially discretize (1) on personalized heart model, giving the volumetric TMP activity model over personalized 3-D myocardium [12], [13] as  ∂U −1    ∂t = −M KU + f1 (U, V)    ∂V = f2 (U, V) ∂t

(2)

where vectors U and V consist of u and v from all M mesh-free points inside the myocardium. Matrices M and K encode the 3-D myocardial structure as well as its conductive anisotropy. The variables f1 (U, V) and f2 (U, V) reflect the heterogeneity of TMP shapes across the myocardial wall. This knowledge about general spatiotemporal TMP dynamics is used to constrain the solution space for obtaining unique TMP estimates from given BSP recordings. 3) TMP-to-BSP Model: Because the torso can be assumed as a passive volume conductor, the quasi-static approximation of Maxwell’s equations effectively describes how active cardiac sources produce potential differences within the torso [1]. Common formulations of TMP-to-BSP mapping are based on biodomain theory, where any point in the myocardium is considered to be in either the intra- or extracellular space [28]. Within the myocardium volume Ωh , the bidomain theory defines the distribution of extracellular potentials φe as a result of the gradients of TMP, which is given as ∇((Di (r) + De (r))∇φe (r)) = ∇(Dk (r)∇φe (r)) = ∇(−Di (r)∇u(r))

∀r ∈ Ωh

(3)

where r stands for the spatial coordinate, Di and De are effective intracellular and extracellular conductivity tensors, and their summation Dk is termed as bulk conductivity tensor. In the region Ωt/h bounded between heart surfaces and the body surface, potentials φt are calculated assuming that no other active electrical source exists within the torso as ∇(Dt (r)∇φt (r)) = 0

∀r ∈ Ωt/h

(4)

where Dt is torso conductivity tensor. Usually, (3) is reduced to a BE formulation by ignoring the conductive anisotropy and inhomogeneity in both intra- and extracellular spaces [29]. As a result, it only considers the TMP activity on heart surfaces. By considering the conductive anisotropy in both spaces, the finite-element (FE) formulation is able to investigate volumetric electrical conduction within the myocardial mass [30]. At the same time, it increases the problem size by considering extracellular potentials in the 3-D myocardium. Because the anisotropic ratio of Dk is a magnitude smaller than that of Di , we only retain the anisotropy of Di to reduce model complexity. Within the isotropic and homogeneous volume conductor of the torso Ωt , [see (3) and (4)] is unified into a Poisson’s equation describing the distribution of potential φ as σ∇2 φ(r) = ∇(−Di (r)∇u(r))

∀r ∈ Ωt

(5)

where Ωt includes Ωt/h and Ωh , but without interface in between. This approach combines the advantages of previous BEand FE-based efforts by emphasizing the electrical anisotropy for active current conduction, but neglecting that for passive currents. It preserves the primary factors to TMP-to-BSP modeling, including 3-D anisotropic heart structures and relative heart– torso positions. Furthermore, it avoids excessive variables, and focus only on TMP and BSP, the two vectors of direct interest to TMP reconstruction. By the direct method solution in BEM [31], (5) is reformulated into ∂φ(r) φ(r)q ∗ (ξ, r) dΓt − φ∗ (ξ, r) dΓt c(ξ)φ(ξ) + ∂ n Γt Γt (∇(Di ∇u(r)))φ∗ (ξ, r) dΩt = (6) σ Ωt where c(ξ) is determined by surface smoothness of any point ξ on Γt , n is the outward normal vector of boundary surfaces, and φ∗ (ξ, r) and q ∗ (ξ, r) are the so-called fundamental solution and its normal derivative [31]. The volume integral on the right-hand side (RHS) of (6) is commonly approximated as a summarization of several current dipoles [32]. Instead, we introduce the mesh-free strategy [27] into (6) for a simpler yet more direct calculation of the volume integral. With the Green’s theorem and integration by parts, the volume integral is written as ∗ ∗ φ (ξ, r) ∇φ (ξ, r) ∂u(r) dΓt − Di Di ∇u(r) dΩt . σ ∂n σ Γt Ωt (7) The boundary condition on Γt assumes that neither active nor passive current leaves the body surface. It removes the third


term on the left-hand side (LHS) of (6) and the first term on the RHS of (7), thus simplifying (6) into φ(r)q ∗ (ξ, r) dΓt (8) c(ξ)φ(ξ) + Γt

=− Ωt

∇φ∗ (ξ, r) σ

Accordingly, with the measurement vector Y(t) = Φ(t), the TMP-to-BSP model (12) becomes the measurement equation with another additive noise ν(t) representing model and data errors as ˜ Y(t) = ( H 0 ) X(t) + ν(t) = HX(t) + ν(t).

Di ∇u(r) dΩt .

(9)

Applying the BEM [31] to the boundary integral and the mesh-free strategy [27] to the volume integral, (8) is transformed to a compact matrix formulation given as LΦ = BU

(10)

where Φ consist of φ from N vertices on Γt (M N ). Matrices L (N × N ) and B (N × M ) encode the geometrical and conductivity information in personalized heart–torso structures. Since the solution of Φ given U in (10) is underdetermined, an additional constraint AΦ = 0 (A is a 1 × N vector) is imposed to define the potential integral over Γt as zero [33]. Augmenting L to La with A and B to Ba with the corresponding all-zero (N + 1)th row, we rearrange (10) into La Φ = Ba U.

(11)

To avoid the additional computation of solving the linear system of (11) during TMP reconstruction, the minimal norm (MN) method is used to get the transfer matrix H for the linear TMP-to-BSP model as Φ = (LTa La )−1 LTa Ba U = HU.

(12)

The condition number of matrix H (cond(H)) measures how sensitive the solution of U is to perturbations of Φ. If cond(H) is small (close to 1), the relative error in U is not much larger than the relative error in Φ. If cond(H) is large, small disturbances in Φ could cause wild fluctuation in U. Because cond(H) is at a magnitude of 1016 , the inverse problem defined by (12) is severely ill-posed. To obtain unique and physiologically meaningful estimates of U from Φ, sufficient constraints are needed from prior knowledge. B. Stochastic State-Space Interpretation In constraining subject-specific TMP imaging, general physiological models [see (2) and (12)] introduce errors such as parameter variations among population and imprecise model structures in pathological conditions. Additional modeling errors also arise from the personalization of heart–torso structures from tomographic images. On the other hand, data errors always exists in structural images and BSP recordings. To explicitly allow the existence of these uncertainties, the physiological system is reformulated into a state-space representation. By letting the state vector X(t) = (U(t)T V(t)T )T , the volumetric myocardial TMP activity model (2) is rearranged into the state equation with an additive stochastic component ω(t) accounting for model uncertainty as −M−1 K 0 f1 (X(t)) ∂X(t) = + ω(t) X(t) + ∂t 0 0 f2 (X(t)) = F (X(t)) + ω(t).

299

(13)

(14)

Because BSP data are collected by discrete sampling, the continuous state-space system [see (13) and (14)] is discretized in the time domain as Xk = Fd (Xk −1 ) + ωk

(15)

˜ k + νk Yk = HX

(16)

where the ordinary differential equation of (13) is discretized by a fourth-order Runge–Kutta solver [34]. It requires high temporal resolution with automatically and adaptively selected time steps. This discretization process is embedded in TMP reconstruction, and no explicit formulation is derived for Fd . C. Volumetric Myocardial TMP Estimation by Data Assimilation Data assimilation interprets the latent state of a system by combining the information in imprecise system models and observations. Sequential data assimilation, such as the filtering technique, performs data assimilation in an iterative manner in the time domain. In each iteration, the previous estimates and system models are used to calculate the predicts of the unknowns, which are then updated to the final estimates given new data [35]. The primary challenges for developing a proper filtering algorithm for volumetric myocardial TMP estimation arise from two aspects. First, the nonlinear TMP activity model (2) does not allow easy local linearization or temporal discretization. It makes the widely used Kalman filter (KF) or extended KF [35] unsuitable because they are designed for linear or locally linearized models. At the same time, the problem domain is of large scale, huge dimensionality, and intricate structure. The alternative Monte Carlo (MC) methods [36], though excellent in dealing with model nonlinearities, become impractical in this study because they normally require intensive computation. 1) Volumetric Myocardial TMP Estimation Algorithm: The TMP estimation algorithm is developed, which is based on the unscented KF (UKF) [38], and combines the advantages of MC methods and KF updates, so as to preserve the intact model nonlinearity and maintain computational feasibility. In each iteration from time step k − 1 to k, the prediction of Xk is obtained in an MC manner to preserve the intact nonlinearity of the state model (15). First, a set of l sigma points {Xk −1,i }li=1 is generated to approximate the probabilistic disˆ k −1 tribution of Xk −1 . It is drawn from the latest estimates of X ˆ and Px k −1 in a deterministic scheme defined by the unscented transform (UT) [37]. This deterministic sampling technique alleviates the computational burden caused by random sampling and slow convergence in MC methods. Each sigma point Xk −1,i is then transformed through the deterministic state model into ¯ − , is prea new sample Xk |k −1,i . Mean of Xk , which is X k dicted as the weighted sample mean of this new ensemble set

300


Fig. 2. TMP estimation algorithm at iteration from instant k − 1 to k. The 2n + 1 sigma point set for n-D vector [37] is used as an example of selecting sigma points. Notations: M is the dimension of the state vector X; W is the weights for each sigma points; the superscript m and c representing weights for calculating mean and covariance, respectively; λ, α, and β are the parameters used to tune the distribution of sigma points; {·}− is the prior predictions of ˆ is the posterior estimates of {·}; and Q ω and R ν are the prespecified {·}; {·} k k covariance matrices for ω k and νk .

{Xk |k −1,i }li=1 , and the covariance P− x k as the weighted sample covariance added by the covariance of model errors Qω k . To maintain computational feasibility, the prediction of the obser¯ − and P− , given new vation vector Yk , and the correction of X xk k observations, are performed in the context of ordinary KF upˆ 0 and P ˆ x ), such dates. Given initial approximations on X0 (X 0 iteration (see Fig. 2) continues at each BSP sampling instant. 2) Implementations: The most expensive computation in the presented algorithm is the l model simulations, which is required in propagating all sigma points through the state model (as shown in the example in Fig. 2, l is usually proportional to the dimension of the state vector). Since V is not of interest to the TMP imaging and its change is not directly reflected in BSP observations, we skip the estimation of V to reduce the dimension of the unknowns. At the kth iteration, a set of lu sigma u is generated where lu < l. This set goes points {Uk −1,i }li=1 ¯ k −1 to genthrough the state model (15) with a single vector V lu u . erate two new sigma point sets {Uk |k −1,i }i=1 and {Vk |k −1,i }li=1 lu − − {Uk |k −1,i }i=1 is used to predict Uk and Pu k , and estimates of ˆ u are obtained from KF update rules. In the mean ˆ k and P U k u ¯ for the next iteration time, Vk is calculated from {Vk |k −1,i }li=1 (see Fig. 3). Regarding the substantially reduced computational cost and the indirect relations of V to the TMP estimation (for detailed mathematics refer to Appendix), this modified algo-

Fig. 3. Modified TMP estimation algorithm at iteration from instant k − 1 to k. Since the dimension of U is M/2, the number of sigma points becomes (2 × M/2) + 1 = M + 1. Q uω k : error covariance for errors in the state equation. Note Q ω k is an M × M matrix while Q uω k is an (M/2) × (M 2) matrix.

rithm improves the practicability of the presented TMP imaging approach. The discretized TMP activity model [see (15)] requires small time step [25], [26], and therefore, a high model resolution, with adaptive time steps determined by the Runge–Kutta solver. On the other hand, in practice, BSP data is usually sampled at an equidistant time step with lower resolution. To allow the coupling of the TMP model of fixed high resolution with BSP data of various lower resolutions, we modify the filtering update ˆ u estimated ˆ k −1 and P strategy as follows: from the current U k −1 u is from BSP data Yk −1 , a set of lu sigma points {Uk −1,i }li=1 generated and propagated through the state model [see (15)] for s model steps until the next BSP data Yk is available for KF update. s is determined by the time interval between the consecutive BSP data Yk −1 and Yk . In this way, BSP data resolution determines the number of KF updates in the whole filtering process, and therefore, the resolution of TMP estimates. It affects the frequency of interaction between prior model and input data, thereby having a direct impact on the accuracy of TMP estimates. Furthermore, it also influences the computational efficiency of the filtering process. This subject will be discussed in Section IV-C. ˆu ˆ 0 and P 3) Filtering Parameters: Initial conditions U 0 are determined by anatomical locations of earliest excitation in normal ventricles [39]. Because the uncertainty in the state model is usually unknown, in practice, Qω is constantly set as a constant diagonal matrix 1e − 004I, where I is the M × M identity matrix. This gives a signal-to-noise


ratio (SNR) [SNR = 10 log10 (power(signal))/(cov(noise)) = 20 log10 (mean(signal))/(std(noise))] averaged around 30 dB over time. In phantom experiments where SNR of input BSP data is predefined, Rν is assumed as a temporally invariant but spatially variant diagonal matrix, each diagonal component of which equals the noise power calculated from known SNR and time-average power of BSP signal at the corresponding mesh-free point. In human studies, Rν is set as a spatially homogeneous but temporally variant matrix γk I, where γk is set as one-hundredth of the mean of the input BSP data at each time instant (representing 20-dB SNR) and I is the N × N identity matrix. In this way, TMP estimation assumes no favorable prior knowledge; it, therefore, avoids overly optimistic specification of filtering parameters, and ensures unbiased TMP estimates and comparisons to related works. III. EXPERIMENTS In the following experiments, the TMP activity model is developed from the two-variable diffusion-reaction system presented in [26] because it preserves the distinct TMP features that are closely related to BSP morphology, and is given as  ∂u    ∂t = ∇(D∇u) + ku(u − a)(1 − u) − uv (17)  ∂v   = −e(v + ku(u − a − 1)). ∂t The longitudinal and transverse component of D are set as 4.0 and 1.0 according to [25]. Parameters e, k, and a are defined as follows [26]: e = 0.01, k = 8, and a = 0.15, 0.14, and 0.17 for endo-, mid-, and epicardial regions, respectively. Values for different conductivities are adopted from [40]: di l = 0.24 Sm−1 , di t = 0.48 Sm−1 , and σ = 0.2 Sm−1 , where di l and di t are the longitudinal and transverse components of Di ; values of u and v are normalized between 0 and 1. For validation of this novel TMP imaging approach, most experiments are performed on a computational phantom of realistic heart–torso structures [see Fig. 1(c)], the geometry and fiber information of which are provided in [17] and [41]. In addition, initial real-data experiments are carried out on a patient with MI. TMP and TMP-to-BSP models used in TMP estimation are constructed on the phantom or real patient’s heart–torso structure and parameterized, as described previously. Initial conditions and error covariance Qω k and Rν k are generated, as described in Section II-C, unless otherwise stated. In the following phantom experiments, the ventricles are represented by mesh-free points with detailed 3-D fiber structure [see Fig. 1(b)], and the torso is represented by triangulated body surface with 700 electrodes. The effect of the number of electrodes (spatial resolution of input data) on TMP estimation accuracy will be discussed in Section IV-C. True TMP and BSP are generated with models modified according to the specific conditions under study. At each electrode, simulated BSP signal is added with zero-mean noises with covariance calculated from predefined SNR and time-averaged BSP signal power (diagonal component in Rν ). These noise-corrupted high-resolution BSP signals (≈24 kHz as model resolution) are directly input to

301

the framework for TMP estimation without any downsampling, namely, model predicts are corrected by BSP data at each model time step. Accuracy of the TMP estimates is quantitatively measured by the relative root mean squared errors (RRMSEs), correlation coefficients (CCs), and maximum nodal errors between ˆ k and simulated TMP Uk as estimated TMP U RRMSEk =

ˆ k − Uk ) (U Uk

(18)

CCk =

ˆ k · Uk ) (U ˆ k Uk ) (U

(19)

ˆ k − Uk |) Max.errork = max(|U

(20)

where operator · denotes dot production between two vectors, the Euclidean form of a vector, | | the absolute value, and ¯ the mean of {·}. Depending on the circumstances, (18)–(20) {·} ˆ k and have two different definitions: 1) at each time step k, U Uk represent the instant TMP map, and (18)–(20) measure the accuracy of the estimated spatial distribution of TMP at that step. Their time courses exhibit the convergence of estimation errors during the filtering process and 2) for each spatial node ˆ k and Uk represent the time course of TMP dynamics, and k, U (18)–(20) measure the accuracy of the estimated temporal dynamics of TMP at that node. Their spatial variations display the different local performance of TMP estimation throughout the 3-D myocardium. Future studies would benefit from more sophisticated metrics for evaluating the quality of TMP estimates. A. Robustness Assessment Even without regard to any pathological condition, general physiological models introduce errors in subject-specific TMP imaging because of simplified model structures and variations of model parameters among the general population. Likewise, data errors always arise in medical imaging and BSP recording process. This study evaluates the robustness of the presented framework to various modeling and data errors in practical environments. 1) Data Errors: White Gaussian noise (WGN), though widely used as the default assumption for data errors in IECG studies [11], [42], might fall short to represent realistic data errors in certain situations. Because of the shortage of objective information about error sources, two common types of nonGaussian noises (Poisson and uniform) are considered in this study. Modeling of realistic data errors is out of the scope of this paper. Table I lists the accuracy of TMP estimates under different data noises, in terms of time-averaged errors (mean ± standard deviation) of estimated TMP distribution after convergence. As explained in the UKF theory [38], the presented TMP estimation algorithm is based on the Gaussian approximation of involved variables. When data noises are not Gaussian, the filtering algorithm is not optimal. But compared to standard KF methods, it has higher tolerance of such noises. As shown in Table I, the presented framework produces less accurate results under non-Gaussian noises compared to Gaussian noises. With the same noise type, higher noise levels (lower SNR) lead to lower

302


TABLE I RRMSE, CC, AND MAXIMUM LOCAL NODAL ERRORS OF TMP ESTIMATES AGAINST SIMULATED TMPS

Fig. 4. (a) Representative BSP time course on a selected electrode. Because TMP value is normalized in the TMP activity model, BSP value does not represent real absolute potential values. (b) Representative TMP time course on the mesh-free node with maximum estimation error (RRMSE = 0.1988). (c) Representative TMP time course on the mesh-free node with average estimation error (RRMSE = 0.0070). (d) Time course of RRMSE of estimated spatial TMP distribution, the initial value of which is 0.2126.

accuracy. Nevertheless, the quality of TMP estimates remains consistently high. As an example for elaboration, Fig. 4(a) illustrates a representative time course of 10-dB-WGN-corrupted BSP (versus simulated BSP, RRMSE = 34%). RRMSE of the corresponding estimates of TMP dynamics at each node is 0.0070 ± 0.0150 over the space, among which, the time courses of TMP estimates with maximum (RRMSE = 0.1988) and average errors (RRMSE = 0.0070) are shown in Fig. 4(b) and (c) versus the corresponding ground truth. Fig 4(d) displays the convergence of RRMSE of spatial TMP distribution over time, where RRMSE of the initial TMP map is 0.2126. 2) Parameter Variations: Values of TMP and TMP-to-BSP model parameters vary among general population, and the exact parameter values for specific patients are rarely available a priori. This study investigates how the variations of parameter a and di l affect the TMP estimates. True TMP and BSP are simulated with a or di l deviated from standard values by WGN, the means of which increase from 5% of the standard parameter values up to the thresholds leading to cardiac pathological conditions. Fig. 5(a) and (b) lists the change of RRMSE of spatial TMP distribution (mean ± standard deviation over time after con-

vergence) with increasing errors in a and di l , respectively. As shown in both the results, TMP estimates remain highly accurate (RRMSE < 10%) in the presence of population variations in different parameters.Note that as parameters deviate farther away from their standard values, and thus, decreasing of the estimation accuracy becomes more rapid. When the deviation is beyond certain limits, pathological conditions are involved. Applications of the presented framework in pathological conditions are investigated in the later sections. 3) Electrode Misplacement: Misplacement of electrodes is common during BSP recording practice. In the following experiments, BSPs are simulated by shifting the 700 electrodes away from the original position in the direction of each Cartesian axis with WGN of different means. As shown in Fig. 5(c), compared with other modeling errors, electrode misplacement gives rise to distinctly larger estimation errors. It has been reported that, for reconstructions of EPs, small electrode misplacement could result in immediate degradation of IECG solutions [22]. With much higher degree of freedom, the estimation of volumetric TMP is expected to have similar or even less stability in the presence of electrode misplacement. On the other hand, the presented approach might alleviate such a problem because of the constraints of prior physiological


303

Fig. 5. (a) Impact of variations in parameter a on RRMSE of the TMP estimates. (b) Impact of variations in longitudinal electrical conductivity di l on RRMSE of the TMP estimates. The x-axis represents the mean deviation of the parameters from the standard values. (c) Impact of electrode misplacement on RRMSE of the TMP estimates. The x-axis represents the mean shift of electrodes away from standard locations in the direction of each Cartesian direction. The y-axis represents time-averaged RRMSE of estimated TMP distribution, in terms of mean ± standard deviation.

Fig. 6. (a) Impact of torso size errors on RRMSE of the TMP estimates. The x-axis represents mean relative volume changes V /V t , where V denotes changed torso size used in simulations and V t denotes the standard one used in TMP estimation. (b) Impact of heart size errors on RRMSE of the TMP estimates. The x-axis represents mean relative volume changes V /V h , where V denotes changed heart size used in simulations and V h denotes the standard one used in TMP estimation. (c) Impact of heart position errors on RRMSE of the TMP estimates. The x-axis represents the mean shift of the heart away from standard position in each direction of the Cartesian axis. The y-axis represents time-averaged RRMSE of estimated TMP distribution, in terms of mean ± standard deviation.

knowledge and the statistical model–data coupling. As demonstrated in Fig. 5(c), when electrode misplacement is within a certain range, the estimates are insensitive to the corresponding errors and retain sufficient accuracy (RRMSE < 22%); when electrode misplacement exceeds certain limits, the accuracy of TMP estimates start to degrade rapidly. It verifies that the presented framework is able to deal with moderate electrode misplacements in practical environments; it also identifies the electrode position as one of the leading factors for TMP imaging, and therefore, emphasizes the importance of correct electrode positioning during BSP recording. 4) Geometrical Modeling Errors: Modeling of personalized heart–torso structures usually introduces different errors, such as errors in torso size, heart size, and relative heart–torso position. In simulating conditions with incorrect heart or torso size, the original heart or torso model is scaled to a new one. Regarding the relative heart–torso position, BSP is generated by shifting the heart away from its original location in each direction of three Cartesian axis by WGN of different means. Fig. 6 lists changes of RRMSE of spatial TMP distribution (mean ± standard deviation over time after convergence) with increasing errors in these situations. As shown, heart and torso size have less impact on the estimation accuracy. In comparison,

the relative heart–torso position has more notable influences, yet the framework still produces robust estimates (RRMSE < 20%) [see Fig. 6(c)]. Moreover, when the errors are within a certain range, the TMP estimates are insensitive to them and have high accuracy (RRMSE < 5%). It is interesting to note that, when input BSP is simulated with reduced heart sizes, estimation errors are notably larger than the situations where enlarged heart sizes are used for BSP simulation [see Fig. 6(b)]. Similar results could be observed in [21]. The possible reason is that with smaller heart size, simulated BSP is farther away from cardiac electrical sources. It smears more details in BSPs and brings more challenges to TMP estimation. B. Comparisons of EP Imaging: BBBs In normal conditions, electrical activation progresses from the endocardium to the epicardium simultaneously in both ventricles. When electrical activation arrives at the epicardium, an epicardial breakthrough occurs and generates a local potential minimum on the surface [43]. If parts of the bundle branches are blocked, the activation propagates sequentially rather than simultaneously in the ventricles. The epicardial breakthrough originally corresponding to the BBB, accordingly, will be

304


Fig. 7. Volumetric myocardial TMP dynamics at the beginning of ventricular activation in RBBB conditions. (a) Simulated true TMP dynamics. (b) Volumetric TMP estimates from 10-dB-WGN-corrupted BSPs. (a) and (b) (Top to down): Depolarization at 0, 6.4, 12.8, and 19.2 ms after electrical pulses arrive at the ventricles. The color encodes normalized TMP values, and black contours represent TMP isochrones. The estimation is initialized with erroneous knowledge, which is rapidly corrected by the BSP data (before 10 ms after the onset of ventricular activation) and TMP imaging results are close to the simulated ground truth.

Fig. 8. (a) RRMSE of TMP estimates in RBBB conditions, which drops rapidly at the beginning of the ventricular activation and remains consistently low thereafter (