Inverse Reconstruction of Epicardial Potentials Improved byVectorcardiography and Realistic Potentials

Matthijs JM Cluitmans1,2, Pietro Bonizzi2, Joel MH Karel2,Paul GA Volders1, Ralf LM Peeters2, Ronald L Westra2

1 Cardiovascular Research Institute Maastricht, Maastricht, The Netherlands2 Department of Knowledge Engineering, Maastricht University, Maastricht, The Netherlands

Abstract

The inverse problem of electrocardiography is to recon-struct electrical activity at the level of the epicardium frombody-surface electrograms and a patient-specific torso-heart geometry. This is complicated by the ill-posednessof the inverse problem, making the reconstruction imper-fect. Previously, we have shown that the use of realisticepicardial training electrograms can improve reconstruc-tion quality in silico. Here, we apply this method in apatient and compare the resulting computed electrogramswith intracardiac measurements. Additionally, we uti-lize a new method that yields further improvements by in-corporating characteristics of vectorcardiographic infor-mation. Patient-specific vectorcardiographic optimizationcombined with training data created on a patient-specificheart improves morphologies of the reconstructed epicar-dial potentials. This underlines the need for constraintsbased on real patient-specific information in the regular-ization of the electrocardiographic inverse problem.

1. Introduction

Body-surface electrocardiograms (ECGs) are widelyused to assess cardiac arrhythmias. However, these onlyreflect the attenuated and dispersed electromagnetic propa-gation of the heart’s electrical activity to the body-surface.Direct, noninvasive assessment of electrical processes atthe level of the heart muscle would be of great benefit toclinical practice. This can be achieved by solving the in-verse problem of electrocardiography, which would yieldelectrical heart activity in terms of body-surface ECGs andthe patient-specific torso-heart geometry, see Fig. 1.

During the last decades, much progress has been madein tackling the inverse problem of electrocardiography [1]and applications in humans appear with increasing fre-quency [2]. However, reconstruction of cardiac electri-cal activity remains imperfect. This is largely due to theill-posedness of the inverse problem, meaning that small

Figure 1. Inverse electrical heart activity reconstructionas applied in this paper. (A) Body-surface potentials areobtained with an extensive set of electrodes attached to thetorso of the subject. (B) The location of the electrodes andof the outer heart-surface is determined from a CT scan.(C) The heart-torso geometry can be coupled with the mea-sured body-surface potentials. (D) By applying inverse al-gorithms, the corresponding electrical heart activity can bereconstructed.

variations (e.g. noise) in the input data will yield largeand unrealistic variations in the reconstructions. To copewith this problem, regularization is applied by incorporat-ing additional constraints to arrive at more realistic solu-tions. Previously, we described the use of realistic train-ing data to improve reconstructions [3]. Here, we applythis method in a patient and compare the resulting com-puted electrograms with intracardiac measurements. Ad-ditionally, we utilize a new method that yields further im-provements by incorporating characteristics of vectorcar-diographic patient-specific information.

2. Methods

2.1. The inverse problem of electrocardiog-raphy

In this study, we used a potential-based formulation toreconstruct heart-surface potentials (epicardial potentials).

ISSN 2325-8861 Computing in Cardiology 2013; 40:369-372.369

This is described by the following forward problem:

YB = AYH (1)

in which YB represents the vectorized body-surface poten-tials (BSP), YH the vectorized heart-surface (epicardial)potentials, and A is the transfer matrix that contains theelectromagnetic relation between those potential vectors.The transfer matrix is based solely on geometrical and con-ductivity properties of the torso and is usually determinedfrom a patient-specific Computed Tomography (CT) scan.

In the inverse problem, the body-surface potentials YBand the transfer matrixA are assumed to be known. Due tothe ill-posedness of the problem, direct solutions are verysensitive to noise. Hence, the solution will not dependcontinuously on the data [4]. By applying regularizationschemes, the ill-posed nature of this problem can be re-stricted. In this study, we compare existing regularizationschemes to our new proposed setup. This new approach isbased on exploiting realistic training data and a vectorcar-diographic representation to obtain improved reconstruc-tions of heart-surface potentials.

2.2. Realistic training data as reconstruc-tion basis

Traditional regularization methods incorporate mathe-matical or physical information to arrive at a better inversesolution. A novel approach is to use electrophysiologicalinformation as well. We previously investigated this ideain a small simulation study, where details of this methodcan be found [3]. In short, this method incorporates thefollowing steps:1 Epicardial training potentials Y #

H are simulated on thepatient’s digitized heart surface as determined from CTscan. To create a diverse training set, we used theFitzHugh-Nagumo action potential model [5] to simu-late multiple beats, originating from different, randomlychosen locations on the heart surface.

2 Singular Value Decomposition (SVD) of the training po-tentials yields Y #

H = USV T, with U and V containingthe left and right singular vectors, respectively, and Scontaining the singular values σi [6]. In this case, thecolumns of U represent a spatial basis for the set of re-alistic heart-surface potentials. Due to the descendingordering of singular values, the first columns of U aremore important for representing the training data, andtruncation may be applied to arrive at a smaller spatialbasis. A condensed basis will be beneficial as it leavesfewer possibilities for ill-posed influences that could re-sult in unrealistic solutions. Therefore, we truncate Uto a suitably small basis Ut consisting of only the first tcomponents.

3 An existing regularization method (such as the Gener-alized Minimal Residual (GMRes) method [7]) is used

to obtain reconstructed epicardial potentials YH frombody-surface potentials YB and the transfer matrix A.

4 Next, these epicardial potentials are projected onto therealistic basis Ut, yielding the final reconstruction ofepicardial potentials YH .As a novel approach, we propose a new step between

steps 3 and 4, in which the reconstructed epicardial po-tentials are optimized in a patient-specific way before pro-jection onto the realistic basis. To achieve this optimiza-tion, we aim to match the vectorcardiographic characteris-tics of the reconstructed epicardial potentials with those ofthe measured body-surface potentials.

2.3. Vectorcardiographic optimization ofepicardial potentials

Starting from YH in step 3, this set is beat-by-beat ana-lyzed with SVD to select the primary orthogonal compo-nents that contain the most information, (Z`H ), with super-script (·)` indicating the `th cardiac beat:

Y `H = UHSHVTH ⇒ Z`H =

√n(V (p)

H )T

with n the number of observations and p the numberof orthogonal components preserved. This yields a car-diac dipole in a virtual orthogonal space. A similar pro-cedure is applied to the 12-lead ECG (Y12, which wasrecorded simultaneously with YB) to obtain an orthogo-nal p-dimensional approximation of the surface potentials(Z`B):

Y `12 = U12S12VT12 ⇒ Z`B =

√n(V (p)

12 )T

We choose to apply this procedure to Y12 instead of YBto exploit a different ensemble of surface information, butit would also be possible to apply this procedure to YBdirectly.

Both sets of dipoles represent the same electrical heartactivity. However, the initial heart-surface potentials YHwere influenced by ill-posedness and their dipole mightnot match the dipole of the body-surface potentials. Tocompensate for this mismatch, a beat-to-beat regulariza-tion of the reconstructed heart-surface electrograms YH isobtained by aligning the corresponding intracardiac dipoleZ`H with the body-surface dipole Z`B , exploited as a ref-erence. This is achieved by means of a statistical signalmodel which accounts for morphological differences be-tween the intracardiac dipole loop and body-surface dipoleloop in terms of: scaling, modeled by the positive scalarα; an orthogonal transformation, modeled by the (p × p)orthonormal matrix R; time synchronization (due to pos-sible differences between Y12 and YB), modeled by a shiftmatrix Jτ , with τ being an integer time shift, such thatτ = ∆ × Fs, with ∆ the maximum time synchroniza-tion error allowed and Fs the sampling frequency [8]. This

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model optimizes the reconstructed heart-surface potentialsto match the information in the measured body-surfacepotentials that might have been lost due to ill-posedness.Thus, the regularized epicardial dipole is modeled by:

Z`H = αRZ`HJτ

Parameters α, R, and τ are estimated by solving the fol-lowing minimization problem:

ε2min = minα,R,τ

||Z`B − αRZ`HJτ ||2F (2)

where || · ||F denotes the Frobenius norm, which for ageneric matrix C is ||C||2F = tr(CTC); note that R, αand τ also depend on `. [8] This minimization cannot beachieved in a closed-form solution [9] and it is then per-formed by first finding the estimates of α and R by fixingτ . The minimization with respect to τ is then solved by agrid search in the interval [−∆; ∆]. Estimates of α and Rare obviously computed for all values of τ . In order to in-clude the time synchronization step, it is necessary to add2∆ samples to the intracardiad dipole loop Z`H , such thatZ`H = (p × n + 2∆). The time shift matrix Jτ whichcorrects for misalignment in time between the intracardiacdipole loop and the body-surface dipole loop is then de-fined by:

Jτ = [0n×∆−τ In×n 0n×∆+τ ]T

In this framework, it can be shown that the estimate of Rcan be found in analogy to the context of rotation of sub-spaces [10], and is given by:

Rτ = UτVTτ

where Uτ and V Tτ are found by calculating the SVD of

Z`BJTτ (Z`H)T. The estimate of α is then:

ατ =tr(JT

τ (Z`H)TRTτ Z`B)

tr(JTτ (Z`H)TRT

τ Z`HJτ )

Finally, the regularized epicardial beat is obtained as fol-lows:

Y `H =1√NU

(p)H S

(p)H Z`H (3)

Note that this represents the global optimum of Eq. (2).Differently from [8, 9], no average beat is defined in thisprocedure, neither for Z`H nor for Z`B , to preserve a de-tailed representation of each cardiac loop.

Following the vectorcardiographic optimization of re-constructed heart-surface potentials, we project these ontothe realistic basis as descriped in step 4 to obtain the vec-torcardiographic optimized and projected final reconstruc-tion of epicardial potentials Y `H for the `th beat.

2.4. Clinical validation data

Test and validation data were obtained from a patientin the Maastricht University Medical Center (MUMC+,Maastricht, The Netherlands). Data collection consistedof three recordings: 1) Extensive body-surface potentialrecordings; 2) A CT-scan; 3) Intracardiac lead recordings.Body-surface potential recordings YB were obtained with256 electrodes attached to the torso of the patient, at asampling frequency Fs of 2048Hz. Recordings include na-tive rhythm (with a left bundle branch block morphology,LBBB), and pacing by implanted pacemaker (pacing bothright and left ventricles). A CT scan was obtained with theelectrodes still attached to the patient’s torso. A geometrywas created from the electrode positions (representing thebody-surface) and the outer heart-surface. The conductorvolume was assumed to be homogeneous. A transfer ma-trix A relating the electrical activity at the heart-surfaceto the body-surface was computed with methods availablefrom the SCIrun software repository [11] (see Fig. 1).

A few months after this procedure, pseudo-unipolarelectrograms were recorded with the implanted pacemakerfrom an epicardial lead in the left ventricle (LV) and anendocardial lead in the right ventricle (RV). These record-ings were obtained for a paced beat and a native beat. Al-though the recordings were not obtained simultaneouslywith the other data sets, the corresponding 12-lead ECGwas comparable for those beats. Therefore, these record-ings were used for validation purposes, although theirpseudo-unipolar characteristics prevent exact morphologi-cal comparison to reconstructed unipolar electrograms.

3. Results

Figure 2 shows the results for two different types ofbeats on two different locations (right ventricle and leftventricle), where we decided by experimenting to fix regu-larization parameters p and t to 7. As can be seen, electro-grams reconstructed with the GMRes method are able toreconstruct the first part of the QRS complex correctly, butseem to miss the second deflection in that complex. Whenvectorcardiographic optimization is applied as well, thisdeflection shows up more pronounced. Subsequent projec-tion on the realistic basis gives further improvements andstabilizes the solution.

Also when looking at the ability to determine the pac-ing location from the reconstructed electrograms, the re-sults are improved by the proposed technique. For thepaced beat, GMRes reconstructs a location of first activa-tion 92 mm from the known pacing location. With vector-cardiographic optimization, this improves to 37 mm, andfurther projection on the realistic basis yields a mismatchof only 22 mm.

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Figure 2. Electrograms on a patient’s heart, measuredvia pacemaker leads (left column) and non-invasively re-constructed from body-surface potentials (other columns).The reconstructed electrograms are regularized with theGeneralized Minimal Residual method only (GMRes),with GMRes and subsequent vectorcardiographic opti-mization (GMRes+VCG), and with subsequent projectionon a realistic basis (GMRes+VCG + Real Basis).

4. Conclusions

In this study, we investigated electrocardiographic in-verse problem regularization by patient-specific vectorcar-diographic optimization and subsequent projection onto abasis obtained from realistic simulated training data. Com-parison with actual intracardiac beats from a patient showsthat epicardial beat reconstruction is improved with respectto GMRes only, notably for recovering previously misseddeflections and decreasing the mismatch when detectingthe pacing location. The physiological meaningfulness ofthese reconstructions was also confirmed by experts.

One limitation is that some of the deflections appear-ing in the GMRes regularized electrogram seem to becomeless pronounced when applying the proposed method.Therefore, care should be taken when applying our methodtoo rigorously; this can be alleviated by convenient tuningof the vectorcardiographic optimization dimension p and

the realistic basis dimension t. How to automatically de-termine these parameters is still an open question.

In conclusion, this study underlines the need for con-straints based on patient-specific information to achievea physiologically meaningful regularization of the elec-trocardiographic inverse problem. We have shown thata patient-specific vectorcardiographic approach combinedwith training data created on a patient-specific heart showspotential to fulfill this need.

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Address for correspondence:

Matthijs JM CluitmansDepartment of Knowledge Engineering, Maastricht University,P.O. Box 616, 6200 MD Maastricht, The Netherlandsm.cluitmans@maastrichtuniversity.nl

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