Physiological-Model-Constrained Noninvasive Reconstruction of Volumetric Myocardial Transmembrane...

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

Physiological-Model-Constrained NoninvasiveReconstruction of Volumetric Myocardial

Transmembrane PotentialsLinwei Wang∗, Member, IEEE, Heye Zhang, Ken C. L. Wong,

Huafeng Liu, Member, IEEE, and Pengcheng Shi, Member, IEEE

Abstract—Personalized noninvasive imaging of subject-specificcardiac electrical activity can guide and improve preventive di-agnosis and treatment of cardiac arrhythmia. Compared to bodysurface potential (BSP) recordings and electrophysiological infor-mation reconstructed on heart surfaces, volumetric myocardialtransmembrane potential (TMP) dynamics is of greater clinical im-portance in exhibiting arrhythmic details and arrythmogenic sub-strates inside the myocardium. This paper presents a physiological-model-constrained statistical framework to reconstruct volumetricTMP dynamics inside the 3-D myocardium from noninvasive BSPrecordings. General knowledge of volumetric TMP activity is in-corporated through the modeling of cardiac electrophysiologicalsystem, and is used to constrain TMP reconstruction. This physi-ological system is reformulated into a stochastic state-space repre-sentation to take into account model and data uncertainties, andnonlinear data assimilation is developed to estimate volumetricmyocardial TMP dynamics from personal BSP data. Robustnessof the presented framework to practical model and data errors isevaluated. Comparison of epicardial potential reconstructions withclassical regularization-based approaches is performed on compu-tational phantom regarding right bundle branch blocks. Further,phantom experiments on intramural focal activities and an initialreal-data study on postmyocardial infarction demonstrate the po-tential of the framework in reconstructing local arrhythmic detailsand identifying arrhythmogenic substrates inside the myocardium.

Index Terms—Body surface potential (BSP), cardiac electro-physiological imaging, data assimilation, inverse problem of ECG(IECG), myocardial transmembrane potential (TMP).

I. INTRODUCTION

BODY surface potential (BSP) recordings provide nonin-vasive observations of the underlying cardiac electrical

activity, among which the ECG has been a standard tool for di-agnosing cardiac arrhythmia in clinical practice. These observa-

Manuscript received April 11, 2008; revised January 10, 2009 and February25, 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 ResearchGrants 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 InformationSciences, 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 onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TBME.2009.2024531

tions, however, are limited in spatial resolution and remote fromthe heart. Moreover, as described by the quasi-static electromag-netic theory [1], the potential difference recorded on each pairof electrode is produced by the integration of electrical sourcesthroughout the myocardial mass; therefore, the internal 3-D-distributed details are absent. To improve preventive diagnosisand treatment of cardiac arrhythmia, it is of direct clinical in-terest to reconstruct more detailed cardiac electrophysiologicalinformation from noninvasive BSP recordings.

Early solutions of this inverse problem of ECG (IECG)are formulated in terms of physical simplifications of cardiacsources [2]–[4], such as potential distribution on epicardium [3].Compared to BSP, these solutions characterize cardiac electricalactivity from closer distances to the heart with higher resolu-tions. Nevertheless, secondary tasks are needed to deduce diag-nostically useful parameters of real cardiac sources [5]. Effortshave also been made to directly reconstruct cardiac electrophys-iological parameters, such as transmembrane potential (TMP)and activation front, on heart surfaces [5]–[9]. These solutionsdo not provide electrophysiological details inside the myocar-dial volume.

Because cardiac arrhythmia usually involve intramural ar-rhythmogenic substrates and local arrhythmic details, it isof greater clinical importance to reconstruct TMP dynamicsthroughout the myocardium. According to the quasi-static elec-tromagnetism [1], unfortunately, BSP alone is not sufficient foruniquely determining the 3-D distribution of electrical sourcesinside the myocardium [10]. Prior physiological knowledgeabout volumetric TMP activity is, therefore, needed to con-strain the solution space. These knowledge are usually incor-porated through physiological models, under the constrains ofwhich volumetric cardiac electrical activity can be obtainedby deterministic optimization [11]. Nevertheless, physiologicalmodels are developed for the general population, and therefore,differ from subject-specific conditions. Furthermore, personalBSP data are always corrupted by noise in practice. It leads tothe risk of unrealistic TMP estimates if general physiologicalmodels overly constrain subject-specific information, especiallywhen the reconstruction is incorrectly initialized. A feasible so-lution is to couple the prior knowledge and observational datain a statistical perspective, so that both model and data errorsare explicitly taken into account.

In this paper, we develop a physiological-model-constrainedstatistical framework for noninvasive volumetric TMP imag-ing, 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 BEplatform. 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 constrainthe TMP imaging and 2) coupling this general knowledge withpersonal data in a statistical manner to allow the existence ofuncertainties in both information. To incorporate sophisticateda priori physiological knowledge, the cardiac electrophysio-logical system is modeled to describe the volumetric myocar-dial TMP activity model for system dynamics and TMP-to-BSPmodel for system observations. It is constructed on personalizedheart–torso structures derived from tomographic images of in-dividual subjects, and reformulated into a stochastic state-spacerepresentation to take into account model and data errors. Non-linear data assimilation is then developed to estimate volumetricmyocardial TMP dynamics from input BSP sequence.

Robustness of the presented framework is validated byphantom experiments regarding various data and model er-rors in practical environments. Comparison with classicalregularization-based IECG approaches is then performed re-garding epicardial potential (EP) reconstruction for bundlebranch block (BBB) conditions. Further, phantom experimentson intramural focal activities and an initial real-data study on apatient with myocardial infarction (MI) demonstrate the poten-tial of the presented framework in reconstructing local conduc-tion abnormality and identifying arrhythmic substrates insidethe myocardium for individual subjects.

II. METHODS

A. Cardiac Electrophysiological System

Modeling of the cardiac electrophysiological system aims at:1) incorporating sophisticated physiological knowledge of gen-eral TMP activity inside the myocardium, and establishing itsrelationship with BSP observations and 2) personalizing heartand torso structures from tomographic images. We develop acoupled mesh-free boundary element (BE) platform to repre-sent the combined heart–torso structures, based on which wederive and validate the associated volumetric myocardial TMPactivity model [12], [13] and TMP-to-BSP model [14], [15]. Inwhat follows, these models are briefly reviewed for a sufficient

background for this framework. For more details refer to relatedpublications and technical reports [12]–[15].

1) Personalized Heart–Torso Model: Given a set of short-axis cardiac structural images such as MRI, we hand-trace2-D contours of epicardium and endocardia slice by slice,and build triangulated meshes for heart surfaces. A two-stageGaussian algorithm [16] is performed to smooth the faceted sur-faces while keeping their overall size and shape features. A cloudof mesh-free points is then placed within the heart surfaces torepresent the 3-D myocardium. To describe myocardial conduc-tive anisotropy, detailed ventricular fiber structures are mappedfrom the mathematical model of myocardial fibrous structure es-tablished in [17]. First, fiber orientations on the epicardium andendocardia of personalized heart model are mapped from thereference model after registering corresponding surfaces of per-sonalized heart model and reference heart model. Because ven-tricular fibers are spirally arranged and fiber orientations changefrom epi- to endocardium in a counterclockwise manner [18],fiber orientations inside the myocardium are interpolated fromthose on the surfaces. Fig. 1(a) exemplifies the 3-D mesh-freerepresentation and fiber structure constructed for a canine heart.Meanwhile, transmural heterogeneity of TMP shapes across theepi-, mid-, and endomyocardial regions [19] [see Fig. 1(b)] isalso taken into account.

Regarding the impact of torso modeling on forward/inverseelectrocardiographic problems, the consensus has been reachedthat geometry is the primary factor that is compared to the ma-terial property [20], [21]. To reduce model complexity, it isreasonable to emphasize the accuracy of geometrical modelingand relax excessive restrictions on tissue anisotropy or inhomo-geneity. In this study, the torso is assumed as an isotropic andhomogeneous volume conductor, and is described by triangu-lated body surface. Torso structures are effectively personalizedonly with a few parameters by deforming a reference model tomatch subject-specific structural image data [22].

Fig. 1(c) illustrates a combined heart–torso model on thecoupled mesh-free BE platform, where the torso is representedwith triangulated body surface, and the 3-D ventricular mass


is represented with a cloud of mesh-free points. In practice,the number of mesh-free points usually ranges from 2000 to5000 for a physiological plausible yet computationally feasiblerepresentation of the 3-D myocardium. The triangulated bodysurface, in comparison, only possesses around 500 vertices, aswe have demonstrated that the resolution of body surface meshhas less impact on the accuracy of TMP-to-BSP modeling [14],[15].

2) Volumetric Myocardial TMP Activity Model: Com-plexity of cardiac electrophysiological models ranges frommacroscopic-level two-variable equations [23] to the cellular-level 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 abil-ity to balance model plausibility with computational feasibility,which is given as

∂u

∂t= ∇(D∇u) + f1(u, v)

∂v

∂t= f2(u, v)

(1)

where u stands for TMP, v for recovery current, D is the dif-fusion tensor, and ∇(D∇u) accounts for intercellular electri-cal propagation. Variations of f1(u, v) and f2(u, v) producedifferent TMP shapes. The mesh-free method [27] is utilizedto spatially discretize (1) on personalized heart model, givingthe volumetric TMP activity model over personalized 3-D my-ocardium [12], [13] as

∂U∂t

= −M−1KU + f1(U,V)

∂V∂t

= f2(U,V)

(2)

where vectors U and V consist of u and v from all M mesh-freepoints inside the myocardium. Matrices M and K encode the3-D myocardial structure as well as its conductive anisotropy.The variables f1(U,V) and f2(U,V) reflect the heterogeneityof TMP shapes across the myocardial wall. This knowledgeabout general spatiotemporal TMP dynamics is used to constrainthe solution space for obtaining unique TMP estimates fromgiven BSP recordings.

3) TMP-to-BSP Model: Because the torso can be assumedas a passive volume conductor, the quasi-static approximationof Maxwell’s equations effectively describes how active car-diac sources produce potential differences within the torso [1].Common formulations of TMP-to-BSP mapping are based onbiodomain theory, where any point in the myocardium is con-sidered to be in either the intra- or extracellular space [28].Within the myocardium volume Ωh , the bidomain theory de-fines the distribution of extracellular potentials φe as a result ofthe 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 andDe are effectiveintracellular and extracellular conductivity tensors, and theirsummation Dk is termed as bulk conductivity tensor.

In the region Ωt/h bounded between heart surfaces and thebody surface, potentials φt are calculated assuming that no otheractive 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- andextracellular spaces [29]. As a result, it only considers theTMP activity on heart surfaces. By considering the conductiveanisotropy in both spaces, the finite-element (FE) formulationis able to investigate volumetric electrical conduction withinthe myocardial mass [30]. At the same time, it increases theproblem size by considering extracellular potentials in the 3-Dmyocardium.

Because the anisotropic ratio of Dk is a magnitude smallerthan that of Di , we only retain the anisotropy of Di to reducemodel complexity. Within the isotropic and homogeneous vol-ume conductor of the torso Ωt , [see (3) and (4)] is unified into aPoisson’s equation describing the distribution of potential φ as

σ∇2φ(r) = ∇(−Di(r)∇u(r)) ∀r ∈ Ωt (5)

where Ωt includes Ωt/h and Ωh , but without interface in be-tween. This approach combines the advantages of previous BE-and FE-based efforts by emphasizing the electrical anisotropyfor active current conduction, but neglecting that for passive cur-rents. 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, andfocus only on TMP and BSP, the two vectors of direct interestto TMP reconstruction.

By the direct method solution in BEM [31], (5) is reformu-lated into

c(ξ)φ(ξ) +∫

Γ t

φ(r)q∗(ξ, r) dΓt −∫

Γ t

(∂φ(r)∂n

)

φ∗(ξ, r) dΓt

=∫

Ω t

(∇(Di∇u(r)))φ∗(ξ, r)σ

dΩt (6)

where c(ξ) is determined by surface smoothness of any point ξon Γt , n is the outward normal vector of boundary surfaces, andφ∗(ξ, r) and q∗(ξ, r) are the so-called fundamental solution andits normal derivative [31].

The volume integral on the right-hand side (RHS) of (6) iscommonly approximated as a summarization of several currentdipoles [32]. Instead, we introduce the mesh-free strategy [27]into (6) for a simpler yet more direct calculation of the volumeintegral. With the Green’s theorem and integration by parts, thevolume integral is written as∫

Γ t

(φ∗(ξ, r)

σ

)

Di∂u(r)∂n

dΓt−∫

Ω t

(∇φ∗(ξ, r)

σ

)

Di∇u(r) dΩt .

(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 theRHS of (7), thus simplifying (6) into

c(ξ)φ(ξ) +∫

Γ t

φ(r)q∗(ξ, r) dΓt (8)

= −∫

Ω t

(∇φ∗(ξ, r)

σ

)

Di∇u(r) dΩt . (9)

Applying the BEM [31] to the boundary integral and themesh-free strategy [27] to the volume integral, (8) is transformedto a compact matrix formulation given as

LΦ = BU (10)

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

LaΦ = BaU. (11)

To avoid the additional computation of solving the linearsystem of (11) during TMP reconstruction, the minimal norm(MN) method is used to get the transfer matrix H for the linearTMP-to-BSP model as

Φ = (LTa La)−1LT

a BaU = HU. (12)

The condition number of matrix H (cond(H)) measures howsensitive the solution of U is to perturbations of Φ. If cond(H) issmall (close to 1), the relative error in U is not much larger thanthe relative error in Φ. If cond(H) is large, small disturbancesin Φ could cause wild fluctuation in U. Because cond(H) isat a magnitude of 1016 , the inverse problem defined by (12) isseverely ill-posed. To obtain unique and physiologically mean-ingful estimates of U from Φ, sufficient constraints are neededfrom prior knowledge.

B. Stochastic State-Space Interpretation

In constraining subject-specific TMP imaging, general phys-iological models [see (2) and (12)] introduce errors such asparameter variations among population and imprecise modelstructures in pathological conditions. Additional modeling er-rors also arise from the personalization of heart–torso structuresfrom tomographic images. On the other hand, data errors al-ways exists in structural images and BSP recordings. To explic-itly allow the existence of these uncertainties, the physiologicalsystem is reformulated into a state-space representation. By let-ting the state vector X(t) = (U(t)T V(t)T )T , the volumetricmyocardial TMP activity model (2) is rearranged into the stateequation with an additive stochastic component ω(t) accountingfor model uncertainty as

∂X(t)∂t

=(−M−1K 0

0 0

)

X(t) +(

f1(X(t))

f2(X(t))

)

+ ω(t)

= F (X(t)) + ω(t). (13)

Accordingly, with the measurement vector Y(t) = Φ(t), theTMP-to-BSP model (12) becomes the measurement equationwith another additive noise ν(t) representing model and dataerrors as

Y(t) = (H 0 )X(t) + ν(t) = HX(t) + ν(t). (14)

Because BSP data are collected by discrete sampling, thecontinuous state-space system [see (13) and (14)] is discretizedin the time domain as

Xk = Fd(Xk−1) + ωk (15)

Yk = HXk + νk (16)

where the ordinary differential equation of (13) is discretizedby a fourth-order Runge–Kutta solver [34]. It requires hightemporal resolution with automatically and adaptively selectedtime steps. This discretization process is embedded in TMPreconstruction, and no explicit formulation is derived for Fd .

C. Volumetric Myocardial TMP Estimationby Data Assimilation

Data assimilation interprets the latent state of a system bycombining the information in imprecise system models and ob-servations. Sequential data assimilation, such as the filteringtechnique, performs data assimilation in an iterative mannerin the time domain. In each iteration, the previous estimatesand system models are used to calculate the predicts of the un-knowns, which are then updated to the final estimates given newdata [35].

The primary challenges for developing a proper filtering al-gorithm for volumetric myocardial TMP estimation arise fromtwo aspects. First, the nonlinear TMP activity model (2) doesnot allow easy local linearization or temporal discretization. Itmakes the widely used Kalman filter (KF) or extended KF [35]unsuitable because they are designed for linear or locally lin-earized models. At the same time, the problem domain is oflarge scale, huge dimensionality, and intricate structure. The al-ternative Monte Carlo (MC) methods [36], though excellent indealing with model nonlinearities, become impractical in thisstudy because they normally require intensive computation.

1) Volumetric Myocardial TMP Estimation Algorithm: TheTMP estimation algorithm is developed, which is based on theunscented KF (UKF) [38], and combines the advantages of MCmethods and KF updates, so as to preserve the intact modelnonlinearity and maintain computational feasibility.

In each iteration from time step k − 1 to k, the prediction ofXk is obtained in an MC manner to preserve the intact non-linearity of the state model (15). First, a set of l sigma pointsXk−1,il

i=1 is generated to approximate the probabilistic dis-tribution of Xk−1 . It is drawn from the latest estimates of Xk−1and Pxk −1 in a deterministic scheme defined by the unscentedtransform (UT) [37]. This deterministic sampling technique al-leviates the computational burden caused by random samplingand slow convergence in MC methods. Each sigma point Xk−1,i

is then transformed through the deterministic state model intoa new sample Xk |k−1,i . Mean of Xk , which is X−

k , is pre-dicted as the weighted sample mean of this new ensemble set


Fig. 2. TMP estimation algorithm at iteration from instant k − 1 to k. The2n + 1 sigma point set for n-D vector [37] is used as an example of selectingsigma points. Notations: M is the dimension of the state vector X; W is theweights for each sigma points; the superscript m and c representing weights forcalculating mean and covariance, respectively; λ, α, and β are the parametersused to tune the distribution of sigma points; ·− is the prior predictions of·; · is the posterior estimates of ·; and Qω k

and Rνkare the prespecified

covariance matrices for ωk and νk .

Xk |k−1,ili=1 , and the covariance P−

xkas the weighted sample

covariance added by the covariance of model errors Qωk. To

maintain computational feasibility, the prediction of the obser-vation vector Yk , and the correction of X−

k and P−xk

, given newobservations, are performed in the context of ordinary KF up-dates. Given initial approximations on X0 (X0 and Px0 ), suchiteration (see Fig. 2) continues at each BSP sampling instant.

2) Implementations: The most expensive computation in thepresented algorithm is the l model simulations, which is re-quired in propagating all sigma points through the state model(as shown in the example in Fig. 2, l is usually proportional tothe dimension of the state vector). Since V is not of interest tothe TMP imaging and its change is not directly reflected in BSPobservations, we skip the estimation of V to reduce the dimen-sion of the unknowns. At the kth iteration, a set of lu sigmapoints Uk−1,ilu

i=1 is generated where lu < l. This set goesthrough the state model (15) with a single vector Vk−1 to gen-erate two new sigma point sets Uk |k−1,ilu

i=1 and Vk |k−1,ilui=1 .

Uk |k−1,ilui=1 is used to predict U−

k and P−uk

, and estimates of

Uk and Pukare obtained from KF update rules. In the mean

time, Vk is calculated from Vk |k−1,ilui=1 for the next iteration

(see Fig. 3). Regarding the substantially reduced computationalcost and the indirect relations of V to the TMP estimation (fordetailed mathematics refer to Appendix), this modified algo-

Fig. 3. Modified TMP estimation algorithm at iteration from instant k − 1 tok. Since the dimension of U is M/2, the number of sigma points becomes (2 ×M/2) + 1 = M + 1. Qu

ω k: error covariance for errors in the state equation.

Note Qω kis an M × M matrix while Qu

ω kis an (M/2) × (M 2) matrix.

rithm improves the practicability of the presented TMP imagingapproach.

The discretized TMP activity model [see (15)] requires smalltime 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 sampledat an equidistant time step with lower resolution. To allow thecoupling of the TMP model of fixed high resolution with BSPdata of various lower resolutions, we modify the filtering updatestrategy as follows: from the current Uk−1 and Puk −1 estimatedfrom BSP data Yk−1 , a set of lu sigma points Uk−1,ilu

i=1 isgenerated and propagated through the state model [see (15)] fors model steps until the next BSP data Yk is available for KFupdate. s is determined by the time interval between the consec-utive BSP data Yk−1 and Yk . In this way, BSP data resolutiondetermines the number of KF updates in the whole filteringprocess, and therefore, the resolution of TMP estimates. It af-fects the frequency of interaction between prior model and inputdata, thereby having a direct impact on the accuracy of TMPestimates. Furthermore, it also influences the computational ef-ficiency of the filtering process. This subject will be discussedin Section IV-C.

3) Filtering Parameters: Initial conditions U0 and Pu0

are determined by anatomical locations of earliest excita-tion in normal ventricles [39]. Because the uncertainty inthe state model is usually unknown, in practice, Qω is con-stantly set as a constant diagonal matrix 1e − 004I, where Iis 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 dBover time. In phantom experiments where SNR of input BSPdata is predefined, Rν is assumed as a temporally invariantbut spatially variant diagonal matrix, each diagonal componentof which equals the noise power calculated from known SNRand time-average power of BSP signal at the correspondingmesh-free point. In human studies, Rν is set as a spatially ho-mogeneous but temporally variant matrix γkI, where γk is setas one-hundredth of the mean of the input BSP data at each timeinstant (representing 20-dB SNR) and I is the N × N identitymatrix. In this way, TMP estimation assumes no favorable priorknowledge; it, therefore, avoids overly optimistic specificationof filtering parameters, and ensures unbiased TMP estimatesand comparisons to related works.

III. EXPERIMENTS

In the following experiments, the TMP activity model is de-veloped from the two-variable diffusion-reaction system pre-sented in [26] because it preserves the distinct TMP featuresthat are closely related to BSP morphology, and is given as

∂u

∂t= ∇(D∇u) + ku(u − a)(1 − u) − uv

∂v

∂t= −e(v + ku(u − a − 1)).

(17)

The longitudinal and transverse component of D are set as4.0 and 1.0 according to [25]. Parameters e, k, and a are de-fined as follows [26]: e = 0.01, k = 8, and a = 0.15, 0.14, and0.17 for endo-, mid-, and epicardial regions, respectively. Val-ues for different conductivities are adopted from [40]: dil

=0.24 Sm−1 , dit

= 0.48 Sm−1 , and σ = 0.2 Sm−1 , where diland

ditare 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 real-istic heart–torso structures [see Fig. 1(c)], the geometry and fiberinformation 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 con-structed on the phantom or real patient’s heart–torso structureand parameterized, as described previously. Initial conditionsand error covariance Qωk

and Rνkare generated, as described

in Section II-C, unless otherwise stated.In the following phantom experiments, the ventricles are rep-

resented by mesh-free points with detailed 3-D fiber structure[see Fig. 1(b)], and the torso is represented by triangulatedbody surface with 700 electrodes. The effect of the number ofelectrodes (spatial resolution of input data) on TMP estimationaccuracy will be discussed in Section IV-C. True TMP and BSPare generated with models modified according to the specificconditions under study. At each electrode, simulated BSP signalis added with zero-mean noises with covariance calculated frompredefined SNR and time-averaged BSP signal power (diago-nal component in Rν ). These noise-corrupted high-resolutionBSP signals (≈24 kHz as model resolution) are directly input to

the framework for TMP estimation without any downsampling,namely, model predicts are corrected by BSP data at each modeltime step. Accuracy of the TMP estimates is quantitatively mea-sured by the relative root mean squared errors (RRMSEs), cor-relation coefficients (CCs), and maximum nodal errors betweenestimated TMP Uk and simulated TMP Uk as

RRMSEk =(‖Uk − Uk‖)

‖Uk‖(18)

CCk =(Uk · Uk )

(‖Uk‖‖Uk‖)(19)

Max.errork = max(|Uk − Uk |) (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)have two different definitions: 1) at each time step k, Uk andUk represent the instant TMP map, and (18)–(20) measure theaccuracy of the estimated spatial distribution of TMP at thatstep. Their time courses exhibit the convergence of estimationerrors during the filtering process and 2) for each spatial nodek, Uk and Uk represent the time course of TMP dynamics, and(18)–(20) measure the accuracy of the estimated temporal dy-namics of TMP at that node. Their spatial variations display thedifferent local performance of TMP estimation throughout the3-D myocardium. Future studies would benefit from more so-phisticated metrics for evaluating the quality of TMP estimates.

A. Robustness Assessment

Even without regard to any pathological condition, generalphysiological models introduce errors in subject-specific TMPimaging because of simplified model structures and variationsof model parameters among the general population. Likewise,data errors always arise in medical imaging and BSP recordingprocess. This study evaluates the robustness of the presentedframework to various modeling and data errors in practicalenvironments.

1) Data Errors: White Gaussian noise (WGN), thoughwidely used as the default assumption for data errors in IECGstudies [11], [42], might fall short to represent realistic data er-rors in certain situations. Because of the shortage of objectiveinformation about error sources, two common types of non-Gaussian noises (Poisson and uniform) are considered in thisstudy. Modeling of realistic data errors is out of the scope of thispaper.

Table I lists the accuracy of TMP estimates under differentdata noises, in terms of time-averaged errors (mean ± standarddeviation) of estimated TMP distribution after convergence. Asexplained in the UKF theory [38], the presented TMP estimationalgorithm is based on the Gaussian approximation of involvedvariables. When data noises are not Gaussian, the filtering al-gorithm 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 undernon-Gaussian noises compared to Gaussian noises. With thesame noise type, higher noise levels (lower SNR) lead to lower


TABLE IRRMSE, 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 representreal absolute potential values. (b) Representative TMP time course on the mesh-free node with maximum estimation error (RRMSE = 0.1988). (c) RepresentativeTMP time course on the mesh-free node with average estimation error (RRMSE = 0.0070). (d) Time course of RRMSE of estimated spatial TMP distribution, theinitial value of which is 0.2126.

accuracy. Nevertheless, the quality of TMP estimates remainsconsistently high.

As an example for elaboration, Fig. 4(a) illustrates a rep-resentative time course of 10-dB-WGN-corrupted BSP (versussimulated BSP, RRMSE = 34%). RRMSE of the correspondingestimates of TMP dynamics at each node is 0.0070 ± 0.0150over the space, among which, the time courses of TMP es-timates with maximum (RRMSE = 0.1988) and average er-rors (RRMSE = 0.0070) are shown in Fig. 4(b) and (c) versusthe corresponding ground truth. Fig 4(d) displays the conver-gence of RRMSE of spatial TMP distribution over time, whereRRMSE of the initial TMP map is 0.2126.

2) Parameter Variations: Values of TMP and TMP-to-BSPmodel parameters vary among general population, and the ex-act parameter values for specific patients are rarely available apriori. This study investigates how the variations of parametera and dil

affect the TMP estimates. True TMP and BSP aresimulated with a or dil

deviated from standard values by WGN,the means of which increase from 5% of the standard parame-ter values up to the thresholds leading to cardiac pathologicalconditions.

Fig. 5(a) and (b) lists the change of RRMSE of spatial TMPdistribution (mean ± standard deviation over time after con-

vergence) with increasing errors in a and dil, respectively. As

shown in both the results, TMP estimates remain highly accu-rate (RRMSE < 10%) in the presence of population variationsin different parameters.Note that as parameters deviate fartheraway from their standard values, and thus, decreasing of theestimation accuracy becomes more rapid. When the deviation isbeyond certain limits, pathological conditions are involved. Ap-plications of the presented framework in pathological conditionsare investigated in the later sections.

3) Electrode Misplacement: Misplacement of electrodes iscommon during BSP recording practice. In the following exper-iments, BSPs are simulated by shifting the 700 electrodes awayfrom the original position in the direction of each Cartesian axiswith WGN of different means.

As shown in Fig. 5(c), compared with other modeling errors,electrode misplacement gives rise to distinctly larger estimationerrors. It has been reported that, for reconstructions of EPs, smallelectrode misplacement could result in immediate degradationof IECG solutions [22]. With much higher degree of freedom,the estimation of volumetric TMP is expected to have similaror even less stability in the presence of electrode misplace-ment. On the other hand, the presented approach might alleviatesuch a problem because of the constraints of prior physiological


Fig. 5. (a) Impact of variations in parameter a on RRMSE of the TMP estimates. (b) Impact of variations in longitudinal electrical conductivity dilon 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 RRMSEof 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-axisrepresents 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/Vt , where V denotes changedtorso size used in simulations and Vt denotes the standard one used in TMP estimation. (b) Impact of heart size errors on RRMSE of the TMP estimates. Thex-axis represents mean relative volume changes V/Vh , where V denotes changed heart size used in simulations and Vh denotes the standard one used in TMPestimation. (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 ineach 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 demon-strated in Fig. 5(c), when electrode misplacement is within acertain range, the estimates are insensitive to the correspondingerrors and retain sufficient accuracy (RRMSE < 22%); whenelectrode misplacement exceeds certain limits, the accuracy ofTMP estimates start to degrade rapidly. It verifies that the pre-sented framework is able to deal with moderate electrode mis-placements in practical environments; it also identifies the elec-trode position as one of the leading factors for TMP imaging,and therefore, emphasizes the importance of correct electrodepositioning during BSP recording.

4) Geometrical Modeling Errors: Modeling of personalizedheart–torso structures usually introduces different errors, such aserrors in torso size, heart size, and relative heart–torso position.In simulating conditions with incorrect heart or torso size, theoriginal heart or torso model is scaled to a new one. Regardingthe relative heart–torso position, BSP is generated by shiftingthe heart away from its original location in each direction ofthree Cartesian axis by WGN of different means.

Fig. 6 lists changes of RRMSE of spatial TMP distribution(mean ± standard deviation over time after convergence) withincreasing errors in these situations. As shown, heart and torsosize have less impact on the estimation accuracy. In comparison,

the relative heart–torso position has more notable influences, yetthe framework still produces robust estimates (RRMSE < 20%)[see Fig. 6(c)]. Moreover, when the errors are within a certainrange, the TMP estimates are insensitive to them and have highaccuracy (RRMSE < 5%). It is interesting to note that, wheninput BSP is simulated with reduced heart sizes, estimationerrors are notably larger than the situations where enlarged heartsizes are used for BSP simulation [see Fig. 6(b)]. Similar resultscould be observed in [21]. The possible reason is that withsmaller heart size, simulated BSP is farther away from cardiacelectrical sources. It smears more details in BSPs and bringsmore challenges to TMP estimation.

B. Comparisons of EP Imaging: BBBs

In normal conditions, electrical activation progresses fromthe endocardium to the epicardium simultaneously in both ven-tricles. When electrical activation arrives at the epicardium, anepicardial breakthrough occurs and generates a local potentialminimum on the surface [43]. If parts of the bundle branchesare blocked, the activation propagates sequentially rather thansimultaneously in the ventricles. The epicardial breakthroughoriginally corresponding to the BBB, accordingly, will be


Fig. 7. Volumetric myocardial TMP dynamics at the beginning of ventricular activation in RBBB conditions. (a) Simulated true TMP dynamics. (b) VolumetricTMP 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 theventricles. 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 lowthereafter (<10% since 10 ms after the onset of ventricular activation). (b) Anterior view of simulated RBBB EP maps at 35 ms after QRS onset. The color encodesmagnitudes of EP and black contours represent potential isochrones. The intact LV epicardial breakthroughs (site 2) and the absence of RV breakthrough (site 1)is consistent with the results in [44].

absent. In the phantom experiments, true abnormal BBB TMPdynamics is simulated by removing the earliest-excited ventric-ular sites in the BBBs.

In the following, we compare EP estimates obtained fromour approach and classical regularization-based EP reconstruc-tions for right BBB (RBBB) on the computational phantom [seeFig. 1(c)]. An additional epicardial surface is introduced into thisheart–torso structure, and the coupled mesh-free BEM method(see Section II-A3) is used to construct forward TMP-to-EP(Tte), EP-to-BSP (Teb ), and TMP-to-BSP (Ttb = Teb × Tte)models (for details refer to [15]), based on which simulated TMPare mapped to EP and BSP as ground truth. Typical regulariza-tion methods, zero-order Tikhonov regularization and truncatedsingular value decomposition (TSVD) are used to reconstruct EPfrom 10-dB-WGN-corrupted BSP with Teb (denoted as EPTikand EPTsvd ). Regularization parameter for Tikhonov regulariza-tion is determined by the L-curve method, and singular valuesto be truncated in TSVD are determined by an empirical thresh-

old [45]. The same regularization parameters are used at eachseparate time instant, in other words, no temporal regularizationis applied in these two methods. Volumetric TMP is estimatedfrom the same input BSP by the presented approach with Ttb ,and projected to EP (denoted as EPTmp ) by Teb for comparisonswith EPTik and EPTsvd .

Fig. 7(a) lists the simulated RBBB TMP dynamics, whereinstead of normal simultaneous propagations in both ventri-cles, electrical activation propagates sequentially from the leftventricle (LV) to the right ventricle (RV). As shown in the re-sults of volumetric myocardial TMP imaging [see Fig. 7(b)],TMP estimation starts with an erroneous initial condition be-cause no prior knowledge about RBBB is utilized. This error israpidly corrected by BSP data (before 10 ms after the onset ofventricular activation), and the following intraventricular con-duction abnormality is faithfully reconstructed. As shown in thetime course of RRMSE of the estimated spatial TMP distribu-tions [see Fig 8(a)], the estimation error drops rapidly at the


Fig. 9. Anterior view of estimated EP map at 35 ms after QRS onset. (a) EPTm p : estimates projected from the TMP estimates from the presented approach.(b) EPTik : reconstruction using zero-order Tikhonov regularization, with regularization parameter determined by the L-curve method. (c) EPTsvd : reconstructionusing zero-order TSVD method. Note that the absence of RV breakthrough (site 1) and the presence of LV breakthrough (site 2) is faithfully reconstructed inEPTm p .

Fig. 10. Representative time courses of estimated EP versus simulated EP. (a) RRMSEEPT m p = 0.1829, RRMSEEPT ik = 0.4398, and RRMSEEPT sv d =0.4589. (b) RRMSEEPT m p = 0.0552, RRMSEEPT ik = 0.3446, and RRMSEEPT sv d = 0.3288.

beginning of the ventricular activation and remains consistentlylow thereafter (RRMSE < 10% since 10 ms after the onset ofventricular activation).

Fig. 8(b) displays the anterior view of simulated RBBBEP map at 35 ms after QRS onset. As shown, because ofthe block of RBB, no evidence of RV breakthrough is ob-served (absence of potential minimum at site 1). Left fascic-ular breakthrough (site 2) is similar to the normal conditionbecause LBB is intact. Fig. 9 lists EPTmp (a), EPTik (b), andEPTsvd (c) at 35 ms after QRS onset, RRMSE of which are0.1490, 0.8072, and 0.8325, respectively. Note that the ab-sence of RV breakthrough and presence of LV breakthroughis correctly reconstructed in EPTmp but not in EPTik or EPTsvd .Fig. 10(a) and (b) compares the examples of EP time coursesfrom EPTmp , EPTik , and EPTsvd versus the ground truth, wherein (a) RRMSEEPT m p = 0.1829, RRMSEEPT ik = 0.4398, andRRMSEEPT sv d = 0.4589, and in (b) RRMSEEPT m p = 0.0552,RRMSEEPT ik = 0.3446, and RRMSEEPT sv d = 0.3288. Theseresults are representative of averaged mean accuracy in the re-construction. Nevertheless, both EPTik or EPTsvd show evidentinconsistency of accuracy among different nodes in the heart.This gives a relatively high global RRMSE and large devi-ation in the accuracy of temporal TMP estimates for all the

Fig. 11. Representative time courses of input noisy BSP versus BSP projectedfrom estimated EP. RRMSEBSPT m p = 0.1697, RRMSEBSPT ik = 0.3018,and RRMSEBSPT sv d = 0.3019. Note that EP estimated from zero-orderTikhonov regularization and TSVD give similar BSP projections, which aremuch noisier than that projected from the TMP estimates obtained by the pre-sented approach.

mesh-free nodes over the space (0.1196 ± 0.0587 for EPTmp ,0.7685 ± 0.4343 for EPTik , and 0.9127 ± 0.5968 for EPTsvd ,respectively).

Fig. 11 shows representative time courses of BSP projectedfrom EPTmp , EPTik , and EPTsvd versus clean BSP signal. It


TABLE IIRRMSE, CC, AND MAXIMUM LOCAL ERRORS OF THE TMP ESTIMATES AGAINST SIMULATED TMPS IN VARIOUS BBB CONDITIONS

Fig. 12. TMP dynamics of cross section containing the ectopic foci. (a) Simulated normal TMP sequence. (b) Simulated arrhythmic TMP sequence.(c) Reconstructed arrhythmic TMP sequence. The color encodes normalized TMP values. From top to down: 15–105 ms with 30 ms interval in between.The TMP imaging results closely reconstruct the arrhythmic details originated from the ectopic foci and depicts the locations of these two focal points.

is apparent that Tikhonov and TSVD regularizations providecloser fitting of clean BSP data. RRMSE of these reprojectedBSP versus clean BSP signal are 0.0689, 0.0306, and 0.0361 forour presented method, Tikhonov, and TSVD regularizations, re-spectively. On the other hand, our framework delivers smoother(less noisy) reprojected BSP; compared with noise-corruptedinput BSP, our framework provides lowest RRMSE in the re-projected BSP (0.1697, 0.3018, and 0.3019 for our presentedmethod, Tikhonov, and TSVD regularizations, respectively).

The aforementioned comparisons of EP estimates demon-strate the improvement of estimation accuracy of the presentedapproach over traditional regularization-based approaches, andtherefore, verify the benefits of the incorporation of physio-logically meaningful constraints, and the statistical coupling ofimperfect prior knowledge and input data. In addition, Table IIsummarizes the errors of TMP estimates from 20-dB-WGN-corrupted BSPs in various BBB conditions. Consistently highaccuracy is obtained because false knowledge in the initial con-dition is timely corrected and does not have significant impacton the results.

C. Imaging of Intramural Focal Activity

Ectopic foci can lead to ventricular tachycardia, the treatmentof choice for which is radio frequency ablation. To destroy theorigin of abnormal activation, accurate localization of ectopicfoci inside the myocardium is needed. Phantom experiments arecarried out to investigate the ability of the presented frameworkin imaging the intramural foci and the arrhythmic activationoriginating from it.

The first experiment considers focal arrhythmia caused by twofocal points [see Fig. 13(a)] that spontaneously activate at thebeginning of the ventricular excitation. Fig. 12(a) and (b) liststhe sequence of simulated normal and arrhythmic TMP activityon the cross section containing the ectopic foci, where Fig. 12(b)exhibits distinct abnormal activation originating from the twofocal points. Fig. 12(c) illustrates the corresponding TMP esti-mates on the same cross section, where the arrhythmic detailsare closely reconstructed. The quantitative accuracy of the vol-umetric TMP estimates is RRMSE = 0.0227 and CC = 0.9997.According to the electrical activation time calculated from theTMP estimates and normal TMP simulations, the ectopic foci


Fig. 13. Cross section containing the two focal points, represented by mesh-free points and ectopic foci are highlighted by red color. (a) Ground truth.(b) Localization using the difference in activation time map calculated fromTMP estimates and simulated normal TMPs. The focal point in the lateralLV endocardial layer is correctly identified, while the one in the septal LVendocardial layer is 5.8 mm away from its true location.

Fig. 14. Locations of the ectopic foci inside the myocardium, highlightedby light blue. (a) Ground truth. (b) Localized using the activation time mapcalculated from the TMP estimates. The real locations of the ectopic foci aremostly captured, with the mean deviance as 4.5 mm.

are localized as the origins of abnormal activation. Note thateven though local TMP estimates are not able to immediatelycapture the foci at the very beginning of ventricular activationbecause of the erroneous initial conditions, the difference ofactivation time between TMP estimates and normal TMP sim-ulations could reflect local TMP abnormality and its origins.Fig. 13(b) depicts the estimated ectopic foci inside the my-ocardium, where one of the focal point is accurately localizedand the other one is localized at 5.8 mm away from the groundtruth. Similar to the investigation on BBB conditions, this ex-periment shows that for pathological conditions that introduceerroneous initial conditions in TMP estimation, the presentedframework is able to closely reconstruct the arrhythmic detailsand localize the origins responsible for the abnormality withoutany prior knowledge about their existence.

In the second experiment, a group of ectopic foci is pickedfrom the subendocardial layer [see Fig. 14(a)] and stimulatedat around 7 ms after excitation pulses arrive at ventricles.Asshown in Fig. 15(a) and (b), the premature excitations partiallysuppress normal sinus rhythm and disrupt the normal activationsequence. In general, quality of the TMP estimates falls intothree groups. In regions with premature excitations, consistentaccuracy is achieved with CC = 0.97 ± 0.0053 and RRMSE =0.19 ± 0.0707 [see Fig. 16(a)]. In regions with delayed excita-tions but far away from the sites initialized as earliest excited, theframework gives satisfactory results with CC = 0.97 ± 0.0070and RRMSE = 0.16 ± 0.0813 [see Fig. 16(b)]. However, forregions around sites assumed as earliest excited, the estimationfails [see Fig. 16(c)]. This is because of the strong false priorknowledge contained in the initial conditions and model struc-tures. As a result, timely and accurate capture of the foci is not

ensured. Similar to the former experiment, nevertheless, differ-ence of the activation time between TMP estimates and normalTMP simulations is able to reflect the origins of local TMP ab-normality. Fig 14(b) pictures the estimated ectopic foci insidethe myocardium, where the mean localization error is 4.5 mmusing the electrical activation time map. As demonstrated, inpathological conditions that involve not only erroneous initialconditions but also incorrect model structures, the presentedframework is able to capture the arrhythmic pattern and the in-tramural origins of the arrhythmia, yet provides limited accuracyof local TMP estimates in certain areas inside the myocardium.

D. Human Studies on MI

An initial read-data study is performed on a set of MRI andBSP data collected from a post-MI patient [46]. The cardiacMRI set contains ten slices, with 8-mm interslice spacing and1.33 mm/pixel in-plane resolution. Each BSP map is recordedfrom 123 electrodes and interpolated to 370 nodes on the torsosurface. The complete BSP sequence consists of a single aver-aged PQRST complex sampled at 2000 Hz. Anatomical loca-tions of all electrodes and nodes are available. Fig. 17(a) illus-trates the combined heart–torso model of the patient, where thetorso is represented by triangulated body surface with 370 ver-tices and the 3-D ventricular mass by a cloud of 1230 mesh-freepoints with detailed fiber structure. Fig. 17(b) exemplifies an in-put BSPM at 85 ms of ventricular activation. Reference interpre-tation of the infarct substrate is provided by cardiologists whoexamined gadolinium-enhanced images of the patient’s heart.Using the standard 17-segment division of LV [47], they identi-fied the percentage of infarcted myocardial mass (EP = 52%),the segment containing the centroid of infarcted substrates(CE = 10 or 11), and all infarcted segments (3–5, 9–12, and15 and 16). In other words, the infarct substrate is distributed inthe inferior-lateral LV with its center in the middle layer.

Fig. 18(b) lists the final imaging results of the volumetricTMP activity in patient’s heart. Compared to the TMP activityobtained by simulating normal conditions in patient’s heart [seeFig. 18(a)], the TMP imaging results exhibit distinct conduc-tion delay, and block in the inferior part of basal-middle LVand lateral part of middle-apical LV. Fig. 19(a) illustrates theventricular activation map calculated from the TMP estimates,where evident late activation time is observed in the same area.Imaging of the infarct substrate is also obtained from the TMPestimates. In brief, activation time and TMP duration of eachmesh-free point is calculated from its temporal TMP shape.Their weighted combination is used to produce the infarct met-ric Mi (0 ≤ Mi ≤ 1) in a way that mesh-free points with localTMP abnormality would have Mi closer to 1. Fig. 19(b) de-scribes the distribution of infarct substrate based on the value ofMi . As shown, location of the infarct substrate (highlighted withevidently higher value of Mi) is again consistent with referenceinterpretation.

In this study, CE of the infarct substrate is located as thesegment containing the center of all infarcted mesh-free pointsweighted by Mi . EP equals the number of infarcted mesh-freepoints divided by the total number of mesh-free points. Segment


Fig. 15. Anterior view of volumetric TMP dynamics. (a) Simulated normal TMP sequence. (b) Simulated arrhythmic TMP sequence of premature excitation.(c) Reconstructed arrhythmic TMP sequence. The color encodes normalized TMP values. From top to down: 28, 31, and 35 ms. TMP imaging results capture thearrhythmic patterns, yet in certain local areas inside the myocardium, consistent high accuracy is not ensured (see Fig. 16).

Fig. 16. Time course of TMP representative of different pathophysiological phenomena. (a) Premature excitation. (b) Delayed excitation in regions away fromareas of normal earliest excitation. (c) Delayed excitation in regions around areas of normal earliest excitation. Comparison of TMP shapes are among simulatednormal TMP (green), simulated abnormal TMP (red), and reconstructed TMP (blue). The presented framework is able to closely reconstruct the abnormality in(a) and (b), yet fails in (c).

overlap (SO) between the results and reference interpretationmeasures the percentage of correct identification. Table III com-pares these results with the reference interpretation and threeother existent results on the same dataset. In brief, results fromDawoud et al. were based on the EP imaging from BSP [48],Farina estimated the size and site of spherical infarct modelsinside the myocardium using BSP [49], and Mneimneh and

Povinelli produced the best existent results using simple ECGsignal analysis [50]. As shown, our results are close to referenceinterpretation, comparable to the best results, and substantiallyimproved over the other two IECG-based approaches. Becausereference interpretation identifies the substrate by segments,its EP measures the percentage of affected segments in all 17segments (EP = 9/17 52%). Instead, our approach improves


Fig. 17. (a) Combined heart–torso model customized from patient-specificMRIs. The ventricles are represented by a cloud of 1230 mesh-free points, andthe torso by triangulated body surface with 370 vertices. (b) Exemplary BSPMat 85 ms of ventricular activation. The color encodes BSP amplitudes and blackcontours represent potential isochrones.

the precision by differentiating between normal and infarctedtissues within the same segment. It might explain why our ap-proach produces smaller EP than reference interpretation.

IV. DISCUSSIONS

A. Physiological Modeling

Modeling of the cardiac electrophysiological system incorpo-rates a priori general physiological knowledge on personalizedheart–torso structures into noninvasive TMP imaging. With aview toward the IECG problem, this modeling aims at ade-quate physiological constraints on TMP reconstruction ratherthan realistic descriptions of electrophysiological phenomena.First of all, the level of modeling details has to maintain thereconstructibility of the unknowns and computational feasibil-ity under the restrictions of available computational resources.Meanwhile, modeling scales are limited by the spatial resolutionof patient’s structural imaging and BSP data.

With increased information and resolution in patient’s data,as well as more computational resources, this modeling couldgo into finer details and scales in future studies. For instance,the two-variable TMP activity model [see (1)] can be replacedby more sophisticated models of cardiac electrophysiologicaldynamics. Conductive inhomogeneity among different torso tis-sues could be included, and the associated TMP-to-BSP mod-els could be derived following the same line in Section II-A3.Briefly, this can be done by first solving the Poison equationor Laplace equations governing each homogeneous tissue, andthen coupling the results on the interfaces of different tissues.Besides, by providing diffusion tensor MRI of patient’s heartin the future, more personalized fiber structure can be recon-structed. In a word, models contained in this physiological sys-tem can be adjusted to varying complexity and differing spe-cialization. Moreover, with increasing computational resources,hierarchical modeling of the system could be introduced so thatthe same activity is described by multiple models from differentperspectives. These modeling improvements are gained at theexpense of increased complexity, and their feasibility deservesfurther investigations.

B. State-Space System and Uncertainty Modeling

By reformulating the physiological system into the stochasticstate-space system, model and data uncertainties are allowed toexist. The current study adopts the widely used assumptions suchas additive noises and Gaussian distributions. In the following,we present some potential solutions for relaxing these specificassumptions.

1) Non-Gaussian Distributions and UT: The (scaled) UTprovides a rather flexible and general scheme to approximatedifferent probabilistic distributions up to different orders [37].In this study, sigma points are selected based on the assump-tion of Gaussian distribution and second-order accuracy. It isa plausible simplification since, in practice, the second-orderinformation is generally sufficient, and there is always a lackof objective knowledge on the probabilistic distribution of in-terest. Nevertheless, when the second-order approximation isnot adequate for operational purposes or the probabilistic dis-tributions of interest are known, different schemes for selectingsigma points could be derived. In general, it determines howmany sigma points are needed, where they are located, andwhat weights are assigned, respectively. It can be solved by anoptimization problem regarding specific type of probabilisticdistributions and desired orders of accuracy under study [37].

2) Nonlinear Uncertainty Modeling and UKF: The pre-sented TMP estimation algorithm (see Figs. 2 and 3) is specif-ically customized to the linear measurement model [see (16)]and additive noises [see (15) and (16)]. Intricate pathologicalconditions, nevertheless, might introduce model errors that arenonlinearly injected into the physiological system. The originalformulation of UKF [38] is able to address these issues.

First, in situations with nonlinear state and measurement mod-els, the UKF passes sigma points through the nonlinear mea-surement model after passing them through the state model.Statistics of Y,Py , and Pxy are calculated using these sigmapoints. Kalman gain is computed by PxyP−1

x and the correctionis performed following KF update rules (for details refer to [38]).Compared with the presented algorithm for linear measurementmodel (see Fig. 2), generalization to nonlinear models is ful-filled at the expense of increased computational complexity.Furthermore, when uncertainties are nonlinearly injected intothe physiological models, the UKF augments the state vectorto include all such noises and carries out the standard UKF onthe reformulated state-space system (for details refer to [38]).Because of the increase in the number of the unknowns, thecomputational requirement is significantly increased.

3) Generalized State-Space System: According to the afore-mentioned potential solutions, a generalized state-space systemcould be formulated to allow for more flexible and sophisticateduncertainty modeling as

∂X(t)∂t

= F (X(t), ω(t)) (21)

Y(t) = G(X(t), ν(t)) (22)

where uncertainties ω(t) and ν(t) are related to the physiologicalsystem in linear or nonlinear manners by F (·) and G(·). Thisgeneralized formulation [see (21) and (22)] shows the potential


Fig. 18. Volumetric myocardial TMP imaging during ventricular activation of the patient with MI. (a) Simulated normal TMP activation. (b) TMP imagingresults. Top to down: 35, 64, 78, and 89 ms after the onset of ventricular activation. TMP imaging results exhibit distinct conduction delay, and block in the inferiorpart of basal-middle LV and lateral part of middle-apical LV.

Fig. 19. (a) Electrical activation time maps derived from TMP estimates. The color encodes values of activation time, and black contours represent activationisochrones. Abnormal late activation exhibits at the inferior part of basal-middle LV and lateral part of middle-apical LV. (b) Distribution maps of the infarctsubstrate. The color bar encodes the values of Mi (higher value of Mi indicates more severe infarction). The substrate is again located in the inferior part ofbasal-middle LV and lateral part of middle-apical LV.

TABLE IIICOMPARISON OF MI EVALUATION RESULTS WITH REFERENCE INTERPRETATION

(THE SECOND COLUMN) AND EXISTENT BEST RESULTS (FOURTH–SIXTH

COLUMNS)

of the presented approach in more sophisticated applications.Long-term study is needed to achieve such a generalization stepby step, addressing issues such as the computational feasibilityof augmenting the state vector in this large-scale problem andthe impact of the nonlinear uncertainty on TMP estimates.

C. Major Factors in TMP Estimation

Using phantom experiments of normal and RBBB conditions,we investigate the change of TMP accuracy as a result of sep-arate changes in several major factors of input data and priorknowledge. Data noises are fixed as 10-dB WGN, unless other-wise stated.

1) Data: Spatial and temporal resolution are two importanttraits of input data that affect TMP estimation. Reduced spatialresolution of input BSP is expected to decrease the accuracy ofTMP estimates. When the number of electrodes for input BSP iscut down to a more realistic amount of 330, RRMSE of spatialTMP distribution over time increases up to 0.0476 ± 0.0183(versus 0.0129 ± 0.0022, see Table I) in normal conditions and0.0683 ± 0.0213 in RBBB.

Regarding the temporal resolution of input BSP, simulatedhigh-resolution BSP (≈24 kHz) is downsampled to 6 and 2 kHz(≈BSP sampling frequency in practice) for inputs. Table IV liststhe time-averaged errors of estimated TMP distribution obtainedfrom 6- and 2-kHz BSP data in normal and RBBB conditions,where evident decreases of estimation accuracy can be observed.Fig. 20 displays the EP map at 35 ms after the QRS onset inRBBB, projected from the TMP estimates obtained under 2-kHzBSP input. Compared to the simulated EP map [see Fig. 8(b)]and the reconstruction from 24-kHz BSP [see Fig. 9(a)], thereconstruction from lower resolution input BSP exhibits a falsepotential minimum on RV surface and a less evident LV break-through (RRMSE = 0.4441). It indicates that 2-kHz BSP is notsufficient for accurately capturing the RBBB characteristics onepicardial breakthrough, yet the reconstruction is still improvedover those from classical regularization-based techniques [seeFig. 9(b) and (c)].


TABLE IVCOMPARISON OF TIME AVERAGE (MEAN ± STANDARD DEVIATION) OF RRMSE, CC, AND MAXIMUM LOCAL NODAL ERRORS

OF ESTIMATED TMP DISTRIBUTION UNDER DIFFERENT DATA RESOLUTION

Fig. 20. Anterior view of estimated EP map at 35 ms after QRS onset, pro-jected from TMP estimates obtained from 2-kHz input BSP (10 dB WGN-corrupted). Compared to the results obtained from 24-kHz input BSP, it isevident that reconstruction from 2-kHz BSP deviates further away from theground truth (RRMSE = 0.4441). Instead of the absence of RV breakthroughand distinct LV breakthrough, the current reconstruction exhibits a false strongpotential minimum on RV surface and a less evident LV breakthrough. Neverthe-less, it is still improved over reconstructions from classical zero-order Tikhonovregularization or TSVD methods.

Because spectrum analysis of BSP signal shows that allof its significant components reside in the low-frequency end(<1 kHz), the cause of such changes is not straightforward.As described in Section II-C2, decreased data resolution corre-sponds to reduced number of KF update, namely less interactionbetween model and data, within the same filtering process. Toexplore its impact, we examine the UT-based TMP predicts ob-tained without any interaction with data. As explained in theTMP estimation algorithm (see Figs. 2 and 3), the UT assumesthe distribution of unknown U as Gaussian, and therefore, focusonly on its mean and covariance when U undergoes the non-linear transform defined by the TMP activity model [see (2)].This approximation introduces accumulative errors as the non-linear transform goes on. Furthermore, for each prediction, Pu

is expanded by model uncertainty Qω , and therefore, the uncer-tainty of U keeps increasing. As a result, TMP predicts deviateaway from the model simulations, as shown in Fig. 21 (RRMSE= 0.0430 ± 0.0576). Because data information during KF up-date prevents the expansion of Pu , the increased frequency ofmodel–data interaction, namely increased BSP data resolution,might be able to improve the accuracy of TMP estimates. Inaddition, it is interesting to note that, in both normal and RBBBconditions, estimation accuracy degrades slightly when inputBSP resolution changes from 24 to 6 kHz, but drop notablywhen it goes further down to 2 kHz. As aforementioned, thedecreasing of input data resolution brings computational reduc-tion. Therefore, a proper choice of input BSP resolution could

Fig. 21. Time course of RRMSE between model-simulated TMP and UT-predicted TMP using the same model. Errors accumulate when the UT continuesto predict TMP without any interaction with input data (RRMSE = 0.0430 ±0.0576).

provide a desirable compromise between estimation accuracyand computational efficiency. This topic will be investigated infuture studies.

2) Model: The specification of initial conditions and errorstatistics is known to have important influences on the inversesolutions. An initial condition close to the real situation is ex-pected to deliver estimates with faster convergence and higheraccuracy. Similarly, appropriate adjustments of Qω and Rν

(confidence levels of physiological models and input data) couldlead to high filtering performance. Because of insufficient in-formation, in practice, however, the quality of TMP estimatesshould not overly depend on these filtering parameters. To putthe presented algorithm into rigorous tests, this study utilizesonly well-known general physiological knowledge for specify-ing initial conditions and error statistics. In the following, wefurther assess the quantitative impact of these filtering parame-ters on TMP estimates.

In all the presented phantom experiments, Rν is assumedas time-invariant with each diagonal component specified byknown SNR. It gives a relatively plausible approximation ofdata error statistics. Fig. 22(a) lists the change of RRMSE ofTMP estimates while Rν is defined as bPΦk

I, where PΦkis

the mean power of input BSP at time instant k and b rangesfrom 1e − 004 to 10−0.5 . As we observed, Rν generated fromb = 1e − 004 has components at the same magnitude with thosein Rν for 10-dB WGN. As a result, when the specification ofRν deviates from true data error statistics and less confidencelevel is put on the input data (b increases), RRMSE of TMPestimates grows.

Fig. 22(b) lists the changes of RRMSE of TMP estimates ob-tained under different Qω , where Qω = aI and a changes from1e − 004 to 0.01 (corresponding to model uncertainty averaged


Fig. 22. (a) Impact of increasing bk (Rνk= bk I) on RRMSE of TMP estimates in normal and RBBB conditions. The variable bk equals bPΦ k

, where PΦ kis

the mean power of input BSP at time instant k. When the specification of Rν deviates from true data error statistics and less confidence level is put on the inputdata (b increases), RRMSE of TMP estimates grows but remains low. (b) Impact of increasing a (Qω k

= aI) on RRMSE of TMP estimates in normal and RBBBconditions. As shown, increasing model uncertainties degrade the accuracy of TMP estimates, yet the impact becomes smaller when a increases and estimationaccuracy remains high with relatively large values of a (model uncertainty is ≈10 dB, when a = 0.01).

at 33.3 − 13.3 dB SNR). As shown, accuracy of TMP estimatesdecreases when a higher level of model errors is specified, yet itstill remains high in comparison to the existent IECG approachesin the presence of relatively large model errors (≈10-dB SNR).It is also expected that, in situations involving intricate patholo-gies, and therefore, complicated model errors, larger value of awould allow less trust on prior model, but more dependence oninput data, and therefore, more accurate TMP estimates. This isto be investigated in forthcoming studies.

In this study, prior knowledge on ventricular locations ofearliest excitation determines the initial condition. In theory, itcould also be used as model constraints on TMP reconstruction.However, the earliest excitation areas usually vary among indi-viduals, especially in pathological conditions. Meanwhile, suchpatient-specific knowledge is usually hard to obtain a priori non-invasively. General knowledge is likely to impose false modelconstraints and lead to incorrect TMP estimates. In comparison,despite slower convergence and lower accuracy, an inaccurateinitial condition could be corrected by input data as the filter-ing continues. This is validated by the phantom experiments onBBB and simple focal arrhythmia. On one hand, compared tonormal conditions, erroneous initial conditions in these exper-iments lead to evident decreases in TMP estimation accuracy(from ≈0.01 to ≈0.10). On the other hand, the estimation accu-racy is still improved over existent IECG approaches.

3) Summary: Both the plausibility of prior knowledge(model) and the quality of input data have important influenceson the accuracy of TMP estimates. Particularly, as shown in thepresented experiments from Section III-A–C, correctness of theprior knowledge largely affect the estimation accuracy: RRMSEof TMP estimates rises from ≈0.01 in normal conditions withcorrect model to ≈0.1 in BBB with models of incorrect initialconditions, and further up to≈0.2 in focal arrhythmia with mod-els of erroneous parameters and/or structures. Such change isexpectable. Experiments in Table I consider no model errors butonly various data errors, and therefore, prior knowledge plays acritical role in leading to very high accuracy in TMP estimates.Experiments in pathological conditions introduce different lev-els of errors into prior knowledge, leading to different level ofaccuracy degrading in the TMP estimates. Nevertheless, input

data prevent the dominance of prior knowledge and ensure thespecificity of the estimates, and therefore, satisfactory resultsare still obtained.

In summary, this study provides a complete quantitative eval-uation of the impact of major factors on the accuracy of TMPestimates. It highlights that, on one hand, the performance of thepresented framework is affected by several factors; on the other,it remains promising in rigid experiment setups (realistic spatialand temporal frequency, unbiased specification of initial condi-tion and error statistics, and the use of general normal model inpathological conditions).

Future efforts would benefit from personalizing general priorknowledge to take advantage of more subject-specific informa-tion, such as standard ECG diagnostic criteria or structural im-ages, or using simple techniques to obtain preliminary estimates.It is also of interest to conduct further studies on the optimalplan of spatial and temporal collection of BSP data in practice.

D. Implementation Issues

To reduce the dimension of the state vector, we skip the esti-mation of V during TMP reconstruction. Our experiments showhigh accuracy in TMP estimates, further proving that V has fewcontribution to the KF update (see Appendix). Nevertheless, theaccumulation of errors in Vk−1 is unavoidable as the iterationgoes on, which, in turn, might introduce errors to the estimationof Uk . To provide a quantitative evaluation of this effect, wecompare the accuracy of TMP estimates and computational timebetween experiments implemented with (see Fig. 2) and without(see Fig. 3) V estimation. This set of experiments is performedwith 2-kHz 10-dB-WGN-corrupted BSP from 330 electrodesas inputs. For a heart represented by 836 mesh-free points, av-erage computational time for each iteration is cut down fromaround 112 to 55 s. This 50% reduction of computational timeis expected as the 2 × (2 × 836) + 1 model simulation is re-duced to 2 × 836 + 1 simulation. Interestingly, RRMSE of TMPestimates with V update is 0.1432 ± 0.0994, while without Vupdate is 0.1184 ± 0.0936. It shows that, though the original al-gorithm (see Fig. 2) avoids the error caused by the skipping of Vestimation, its larger number of unknowns and higher numerical


difficulty actually lead to lower accuracy in TMP estimates. Theuse of the modified TMP estimation algorithm without V up-date is, therefore, justified for its reduced computational costand higher numerical accuracy.

E. Applications in Intricate Pathological Conditions

Regarding the robustness to model and data errors in prac-tice, [5] demonstrated that the introduction of physiological con-straints is able to improve the stability of IECG solutions overstandard regularization. In comparison, IECG solutions in thisstudy show higher accuracy and stability even with higher levelsof noises. In applications to pathological conditions involvingerroneous initial conditions, such as blocks in bundle branchesor focal excitations at the beginning of ventricular activation, thepresented framework is able to quickly correct erroneous initialconditions, accurately reconstruct arrhythmic details, and iden-tify origins responsible for the abnormality (such as blockedbundle branch and focal points). These results, on one hand,further validate the benefits of physiological constraints; on theother, they suggest the advantage of statistical model–data cou-pling in IECG problem.

When more intricate pathological conditions introduce mod-eling structure errors, such as the excitation of ectopic foci in themiddle of ventricular activation or MI, the presented frameworkis able to capture the feature of the conduction abnormality androughly approximate the arrhythmogenic substrate. Neverthe-less, accuracy of local TMP estimates inside the myocardiumneed to be improved. For instance, in focal activity originatedin the middle of the ventricular activation, TMP estimates showlow accuracy in regions close to the sites initialized as the earli-est excitation areas. It implies that BSP data are not sufficient tocorrect the false information contained in the model structure.We propose a progressive iteration scheme to solve this prob-lem by several passes of TMP estimation. The first pass of TMPestimation starts without any prior knowledge, constrained withstandard models of normal-valued parameters. Latest TMP es-timates are used to modify the constraining models for the nextpass of TMP estimation. Such iteration of feedback and esti-mation continues until no significant changes are observed inthe TMP estimates. We applied this scheme to the real patientdata, and obtained initial positive results. Experiments on an-other three sets of patient data are ongoing, aiming for a morecomplete validation of the presented approach.

Another potential solution to reduce the constraining effectsof prior knowledge is the simultaneous estimation of TMP dy-namics and electrophysiological properties of the myocardium.By using unknown parameters, this method can modify theconstraining models in accordance to patient’s data during thefiltering. Moreover, we are seeking the fusion of complemen-tary patient’s data as a further fundamental strategy to enhancethe patient specificity of the TMP estimates. Structural imagesequences, for instance, contain temporally sparse but spatiallydense volumetric myocardial kinematic measures [51], [52]. Bycardiac electromechanical coupling [53], they become obviousand ideal sources to complement BSPs with local volumetriccardiac electrophysiological information.

Meanwhile, the current study identifies arrhythmogenic sub-strates using physiological parameters calculated from volu-metric TMP estimates. For example, the late activation timeand reduced TMP duration constitute the metric for identify-ing infarcted tissues, while the ectopic foci is localized as theorigin of abnormal activation. Future studies on more real-dataexperiments are needed to investigate the robustness of thesemetrics and develop more sophisticated criterion for substrateidentification.

V. CONCLUSION

In this paper, we present a physiological-model-constrainedstatistical approach to noninvasive functional imaging of volu-metric TMP activity inside patient’s myocardium. Given BSPsand structural images of specific patients, the presented frame-work estimates subject-specific volumetric myocardial TMP ac-tivity by integrating input data and a priori physiological knowl-edge regarding the respective data and modeling errors. Theprimary value of this framework is its ability to directly depictlocal details of arrhythmic TMP activities, and identify arrhyth-mogenic substrates inside the 3-D myocardium. This advantageis verified by phantom experiments concerning intraventricularconduction abnormality and intramural focal activity, as well asan initial real-data study on transmural MI.

The key novelty of the presented approach lies in the incor-poration of volumetric TMP activity model as constraints tothe TMP imaging, and the utilization of statistical (Bayesian)approach to take into account model and data uncertainties. Itshares the idea with existent state-space-based approaches [9],[54], [55], which sophisticated that prior knowledge could beeffectively incorporated through the state models. Attempts onincorporating physiological knowledge in the 3-D myocardiumand reconstructing volumetric TMP activity via state-space for-mulation are presented here for the first time according to ourknowledge.

As discussed in Section IV, future improvements of the frame-work will focus on the following aspects. First, physiologicalmodeling will explore models of varying complexity levels anddiffering specializations. Meanwhile, more sophisticated uncer-tainty modeling will be investigated for the potential of gener-alized state-space formulations. Simultaneous imaging of TMPdynamics and tissue electrophysiological properties in the heartis ongoing. Furthermore, we are now working on the integrationof complementary patient’s data, such as BSPs and structuralimage sequences, as the fundamental development to the pre-sented framework.

Thorough validation of the presented framework is very chal-lenging since it requires not only the collection of BSP record-ings and structural images of specific subjects, but also com-plete electrical mapping on the 3-D myocardium. The presentedcomparisons with existent IECG methods on phantom experi-ments and validations with gold standards in real-data studiesprovide initial verifications and understandings about itsstrengths and limitations. More detailed comparisons may not befeasible since the presented volumetric imaging approach pro-vides spacial details and resolution that cannot be approached by


most existent techniques. Future studies will look into the pos-sibility of collecting volumetric TMP mapping data, as well asreal data in a variety of pathology conditions, for more completehuman studies.

APPENDIX I

PROOF OF FIRST ZONKLAR EQUATION

To observe the role of V in the TMP estimation, we decom-pose the relevant matrices as

H = (H 0 ) P−x =

(P−

u P−uv

P−vu P−

v

)

(23)

where P−u and P−

v are the predicted covariances of U and V,respectively, and P−

uv and P−vu are the cross covariances be-

tween U and V. As a result, Kalman gain G is equal to

G =(

P−u HT

P−vuH

T

)

(HP−u HT + Rν )−1 . (24)

In general, G is computed as PxyP−1y . In the aforementioned

equation,

Pxy =(

Puy

Pvy

)

=(

P−u HT

P−vuH

T

)

.

Namely, V does not affect the relations between U and Y(Puy = P−

u HT ), and it relates to Y indirectly through U(Pvy = P−

vuHT ). Meanwhile, Py = HP−

u HT + Rν , whichdoes not have the contribution from V. Replacing (24) intothe calculation of X, we observe that the prediction of V− doesnot contribute to the estimation of U.

REFERENCES

[1] R. Plonsey, Bioelectric Phenomena. New York: McGraw-Hill, 1969.[2] R. M. Gulrajani, H. Pham-Huy, R. A. Nadeau, P. Savard, J. de Guise,

R. E. Primeau, and F. A. Roberge, “Application of the single moving dipoleinverse solution to the study of the Wolff–Parkinson-white syndrome inman,” J. Electrocardiol., vol. 17, pp. 271–288, 1984.

[3] Y. Rudy and B. Messinger-Rapport, “The inverse problem of electrocar-diography: Solutions in terms of epicardial potentials,” CRC Crit. Rev.Biomed. Eng., vol. 16, pp. 215–268, 1988.

[4] J. J. M. Cuppen and A. van Oosterom, “Model studies with the inverselycalculated isochrones of ventricular depolarization,” IEEE Trans. Biomed.Eng., vol. BME-31, no. 10, pp. 652–659, Oct. 1984.

[5] G. Huiskamp and F. Greensite, “A new method for myocardial activationimaging,” IEEE Trans. Biomed. Eng., vol. 44, no. 6, pp. 433–446, Jun.1997.

[6] A. J. Pullan, L. K. Cheng, M. P. Nash, C. P. Bradley, and D. J. Paterson,“Noninvasive electrical imaging of the heart: Theory and model develop-ment,” Ann. Biomed. Eng., vol. 29, pp. 817–836, 2001.

[7] R. Modre, B. Tilg, G. Fischer, and P. Wach, “Noninvasive myocardialactivation time imaging: A novel inverse algorithm applied to clinical ECGmapping data,” IEEE Trans. Biomed. Eng., vol. 49, no. 10, pp. 1153–1161,Oct. 2002.

[8] B. Messnarz, B. Tilg, R. Modre, G. Fischer, and F. Hanser, “A new spa-tiotemporal regularization approach for reconstruction of cardiac trans-membrane potential patterns,” IEEE Trans. Biomed. Eng., vol. 51, no. 2,pp. 273–281, Feb. 2004.

[9] A. Ghodrati, D. H. Brooks, G. Tadmor, and R. S. MacLeod, “Wavefront-based models for inverse electrocardiography,” IEEE Trans. Biomed.Eng., vol. 53, no. 9, pp. 1821–1831, Sep. 2006.

[10] Y. Yamashita, “Theoretical studies on the inverse problem of electrocar-diography and the uniqueness of the solution,” IEEE Trans. Biomed. Eng.,vol. BME-29, no. 11, pp. 719–725, Nov. 1982.

[11] B. He, G. Li, and X. Zhang, “Noninvasive imaging of cardiac transmem-brane potentials within three-dimensional myocardium by means of a

realistic geometry aniostropic heart model,” IEEE Trans. Biomed. Eng.,vol. 50, no. 10, pp. 1190–1202, Oct. 2003.

[12] H. Y. Zhang and P. C. Shi, “A meshfree method for solving cardiacelectrical propagation,” in Proc. Int. Conf. IEEE Eng. Med. Biol. Sci.,2005, pp. 349–352.

[13] H. Zhang, P. Shi, and P. J. Hunter, “A meshfree method for simulating my-ocardial electrical activity,” Bioeng. Inst. Univ. Auckland, New Zealand,Tech. Rep., Jul. 2008.

[14] L. Wang, H. Zhang, C. L. Wong, and P. Shi, “Coupled meshfree-BEMplatform for electrocardiographic simulation: Modeling and validations,”in Proc. Int. Workshop Med. Imag. Augmented Reality, 2008, vol. 1, pp. 98–107.

[15] L. Wang, H. Zhang, K. Wong, H. Liu, and P. Shi, “Electrocardiographicsimulation on personalized heart–torso structures using coupled meshfree-BEM platform,” Int. J. Funct. Inf. Pers. Med., 2009, Under review.

[16] G. Taubin, “Curve and surface smoothing without shrinkage,” in Proc.Int. Conf. Comput. Vis., 1995, pp. 825–857.

[17] M. Nash. (1998, May). Mechanics and material properties of theheart using an anatomically accurate mathematical model. Ph.D.dissertation, Univ. Auckland, New Zealand [Online]. Available:http://www.esc.auckland.ac.nz/Nash/thesis/

[18] P. M. Nielsen, I. J. L. Grice, B. H. Smaill, and P. J. Hunter, “Mathematicalmodel of geometry and fibrous structure of the heart,” Amer. J. Physiol.,vol. 260, pp. H1365–H1378, 1991.

[19] G. X. Yan, W. Shimizu, and C. Antzelevitch, “Charactertistics and dis-tribution of M cells in arterially perfused canine left ventricular wedgepreparations,” Circulation, vol. 98, pp. 1921–1927, 1998.

[20] C. P. Brandley, A. J. Pullan, and P. J. Hunter, “Effects of material prop-erties and geometry on electrocardiographic forward simulations,” Ann.Biomed. Eng., vol. 28, pp. 721–741, 2000.

[21] L. K. Cheng, J. M. Bodley, and A. J. Pullan, “The effect of experimentaland modeling errors on the electrocardiographic inverse problem,” IEEETrans. Biomed. Eng., vol. 50, no. 1, pp. 23–32, Jan. 2003.

[22] L. Cheng, “Non-invasive electrical imaging of the heart,” Ph.D. disserta-tion, Univ. Auckland, Auckland, New Zealand, 2001.

[23] R. Fitzhugh, “Impulses and physiological states in theoretical models ofnerve membrane,” Biophys. J., vol. 1, pp. 445–466, 1961.

[24] C. Luo and Y. Rudy, “A dynamic model of the cardiac ventricular actionpotential: Simulations of ionic currents and concentration changes,” Circ.Res., vol. 74, pp. 1071–1096, 1994.

[25] J. M. Rogers and A. D. McCulloch, “A collocation-Galerkin finite elementmodel of cardiac action potential propagation,” IEEE Trans. Biomed.Eng., vol. 41, no. 8, pp. 743–757, Aug. 1994.

[26] R. R. Aliev and A. V. Panfilov, “A simple two-variable model of cardiacexcitation,” Chaos, Solitions Fractals, vol. 7, no. 3, pp. 293–301, 1996.

[27] G. Liu, Meshfree Methods. Boca Raton, FL: CRC Press, 2003.[28] C. S. Henriquez, “Simulating the electrical behaviour of cardiac tissue

using the bidomain model,” Crit. Rev. Biomed. Eng., vol. 21, pp. 1–77,1993.

[29] G. Fischer, B. Tilg, P. Wach, G. Lafter, and W. Rucker, “Analytical val-idation of the BEM-application of the BEM to the electrocardiographicforward and inverse problem,” Comput. Methods Programs Biomed.,vol. 55, pp. 99–106, 1998.

[30] G. Fischer, B. Tilg, R. Modre, G. J. M. Huiskamp, J. Fetzer, W. Rucker, andP. Wach, “A bidomain model based BEM–FEM coupling formulation foranisotropic cardiac tissue,” Ann. Biomed. Eng., vol. 28, pp. 1229–1243,2000.

[31] C. A. Brebbia, J. C. F. Telles, and L. C. Wrobel, Boundary ElementTechniques: Theory and Applications in Engineering. Berlin, Germany:Springer-Verlag, 1984.

[32] A. C. L. Barnard, I. M. Duck, and M. L. Lynn, “The application ofelectromagnetic theory to electrocardiology,” Biophys. J., vol. 7, pp. 443–462, 1967.

[33] M. Aoki, Y. Okamoto, T. Musha, and K. I. Harumi, “Three dimensionalsimulation of the ventricular depolarization and repolarization processesand body surface potentials: Normal heart and bundle branch block,”IEEE Trans. Biomed. Eng., vol. BME-34, no. 6, pp. 454–461, Jun. 1987.

[34] J. Butcher, Numerical Methods for Ordinary Differential Equations.New York: Wiley, 2008.

[35] B. D. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs,NJ: Prentice-Hall, 1979.

[36] B. Ristic, Beyond the Kalman Filter: Particle Filters for Tracking Appli-cations. Boston, MA: Artech House, 2004.

[37] S. Julier, “The scaled unscented transform,” Int. J. Numer. Methods Eng.,vol. 47, pp. 1445–1462, 2000.


[38] S. J. Julier and J. K. Uhlmann, “A new extension of the Kalman filterto nonlinear systems,” in Proc. Int. Symp. Aerosp./Defense Sens., Simul.Controls, 1997, pp. 182–193.

[39] D. Durrur, R. Dam, G. Freud, M. Janse, F. Meijler, and R. Arzbaecher,“Total excitation of the isolated human heart,” Comput. Methods Appl.Mech. Eng., vol. 41, no. 6, pp. 899–912, 1970.

[40] B. J. Roth, “Electrical conductivity values used with the bidomain modelof cardiac tissue,” IEEE Trans. Biomed. Eng., vol. 44, no. 4, pp. 326–328,Apr. 1997.

[41] R. S. MacLeod and D. H. Brooks, “Recent progress in inverse problemsin electrocardiology,” in Proc. Int. Conf. IEEE Eng. Med. Biol. Sci., 1998,vol. 17, pp. 73–83.

[42] R. N. Ghanem, J. E. Burnes, A. L. Waldo, and Y. Rudy, “Imaging dis-persion of myocardial repolarization, II: Noninvasive reconstruction ofepicardial measures,” Circulation, vol. 104, pp. 1306–1312, 2001.

[43] G. Arisi, E. Macchi, S. Baruffi, S. Spaggiari, and B. Taccardi, “Potentialfields on the ventricular surface of the exposed dog heart during normalexcitation,” Circ. Res., vol. 52, pp. 706–715, 1983.

[44] C. Ramanathan, R. N. Ghanem, P. Jia, K. Ryu, and Y. Rudy, “Noninvasiveelectrocardiographic imaging for cardiac electrophysiology and arrhyth-mia,” Nat. Med., vol. 10, no. 4, pp. 422–428, 2004.

[45] F. Greensite, “Heart surface electrocardiographic inverse solutions,” inModeling and Imaging of Bioelectrical Activity, B. He, Ed. Norwell,MA/New York: Kluwer/Plenum, 2004, pp. 119–156.

[46] A. L. Goldberger, L. A. N. Amaral, L. Glass, J. M. Hausdorff, P. C.Ivanov, R. G. Mark, J. E. Mietus, G. B. Moody, C. K. Peng, andE. Stanley, “Physiobank, physiotoolkit, and physionet components of anew research resource for complex physiological signals,” Circulation,vol. 101, pp. e215–e220, 2000.

[47] M. D. Cerqueira, N. J. Weissman, V. Dilsizian, A. K. Jacobs, S. Kaul,W. K. Laskey, D. J. Pennell, J. A. Rumberger, T. Ryan, and M. S.Verani, “Standardized myocardial segmentation and nomenclature for to-mographic imaging of the heart,” Circulation, vol. 105, pp. 539–542,2002.

[48] F. D. Dawoud, “Using inverse electrocardiography to image myocardialinfarction,” in Proc. Comput. Cardiol., 2007, pp. 177–180.

[49] D. Farina, “Model-based approach to the localization of infarction,” inProc. Comput. Cardiol., 2007, pp. 173–176.

[50] M. A. Mneimneh and R. J. Povinelli, “RPS/GMM approach toward thelocalization of myocardial infarction,” in Proc. Comput. Cardiol., 2007,pp. 185–188.

[51] Z. H. Hu, D. Metaxas, and L. Axel, “In vivo strain and stress estimation ofthe heart left and right ventricles from MRI images,” Med. Image Anal.,vol. 7, pp. 435–444, 2003.

[52] C. L. Wong, H. Y. Zhang, H. F. Liu, and P. C. Shi, “Physiome-model-basedstate-space framework for cardiac deformation recovery,” Acad. Radiol.,vol. 14, pp. 1341–1349, 2007.

[53] D. Nickerson, S. Niederer, C. Stevens, M. Nash, and P. Hunter, “A com-putational model of cardiac electromechanics,” in Proc. Int. Conf. IEEEEng. Med. Biol. Sci., 2006, pp. 5311–5314.

[54] D. Joly, Y. Goussard, and P. Savard, “Time-recursive solution to the inverseproblem of electrocardiography: A model-based approach,” in Proc. Int.Conf. IEEE Eng. Med. Biol. Sci., 1993, pp. 767–768.

[55] J. El-Jakl, F. Champagnat, and Y. Goussard, “Time-space regularizationof the inverse problem of electrocardiography,” in Proc. Int. Conf. IEEEEng. Med. Biol. Sci., 1995, pp. 213–214.

Linwei Wang (M’08) received the B.E. degree ininformation engineering from Zhejiang University,Hangzhou, China, in 2005, the M.Phil. degree in elec-trical and electronic engineering from the Hong KongUniversity of Science and Technology, Kowloon,Hong Kong, in 2007, and the Ph.D. degree in comput-ing and information science from Rochester Instituteof Technology, Rochester, NY, in 2009.

She is currently an Assistant Professor of thePh.D. program in computing and information sci-ences, Rochester Institute of Technology. Her current

research interests include personalized computational biomedicine, medical im-age computing, computational cardiac electrophysiology, and statistical inverseproblems.

Heye Zhang received the B.S. and M.E. degreesfrom Tsinghua University, Beijing, China, in 2001and 2003, respectively, and the Ph.D. degree fromthe Hong Kong University of Science and Technol-ogy, Kowloon, Hong Kong, in 2007.

He is currently a Postdoctoral Fellow at theBioengineering Institute, University of Auckland,Auckland, New Zealand. His current research inter-ests include cardiac electrophysiology and cardiacimage analysis.

Ken C. L. Wong received the B.E. degree (withfirst class honors) in electronic engineering and theM.Phil. degree in electrical and electronic engineer-ing from the Hong Kong University of Science andTechnology, Kowloon, Hong Kong, in 2002 and 2004,respectively. He is currently working toward thePh.D. degree in computing and information sciencesat Rochester Institute of Technology, Rochester, NY.

His current research interests include medi-cal image analysis, physiome modeling, and signalprocessing.

Huafeng Liu (M’03) received the B.S. degree in op-tical engineering, the M.S. degree in measurementtechniques and instrumentation, and the Ph.D. de-gree in positron emission tomography from ZhejiangUniversity, Hangzhou, China.

From 2001 to 2003, he was a Postdoctoral Fellowat the Hong Kong University of Science and Technol-ogy, where he was engaged in conducting researchon statistical filtering and inverse mechanics strate-gies for cardiac image analysis. He is currently a FullProfessor at the State Key Laboratory of Modern Op-

tical Instrumentation, Zhejiang University. His current research interests includebiomedical imaging instrumentation, image reconstruction, and medical imageanalysis.

Pengcheng Shi (M’93) received the B.S. degree inbiomedical engineering from Shanghai Jiao TongUniversity, Shanghai, China, in 1989, and the M.S.,M.Phil., and Ph.D. degrees in electrical engineeringfrom Yale University, New Haven, CT, in 1992, 1993,and 1996, respectively. He is currently working to-ward the Ph.D. degree in computing and informa-tion sciences at Rochester Institute of Technology,Rochester, NY.

He was with the Yale University School ofMedicine as a National Research Service Award

(NRSA) Fellow in diagnostic radiology. He was an Assistant/Associate Pro-fessor in computer and information sciences at the New Jersey Instituteof Technology. He was an Assistant Professor in electrical and computerengineering at the Hong Kong University of Science and Technology. Hewas the Chair Professor in biomedical engineering at the Southern Medi-cal University. His current research interests include integrative system ap-proaches to biomedical imaging, image computing and intervention, and inverseelectrophysiology.

Physiological-Model-Constrained Noninvasive Reconstruction of Volumetric Myocardial Transmembrane...

Documents

Transcript of Physiological-Model-Constrained Noninvasive Reconstruction of Volumetric Myocardial Transmembrane...