Analysis of the Criteria of Activation-Based InverseElectrocardiography using Convex Optimization

Burak EremCDSP Center, Dept. of ECE

Northeastern Univ., Boston, MA USAberem@ece.neu.edu

Peter M. van DamDonders Institute, Radboud Univ.

Nijmegen, The Netherlandspeter.van.dam@peacs.nl

Dana H. BrooksCDSP Center, Dept. of ECE

Northeastern Univ., Boston, MA USAbrooks@ece.neu.edu

Abstract—In inverse electrocardiography (ECG), the problemof finding activation times on the heart noninvasively from bodysurface potentials is typically formulated as a nonlinear leastsquares optimization problem. Current solutions rely on iterativealgorithms which are sensitive to the presence of local minima. Asa result, improved initialization approaches for this problem havebeen of considerable interest. However, in experiments conductedon a subject with Wolff-Parkinson-White syndrome, we haveobserved that there may be a mismatch between favorablesolutions of the optimization problem and solutions with thedesired physiological characteristics. In this work, we use amethod based on a convex optimization framework to explorethe solution space and analyze whether the optimization criteriatarget their intended objective.

I. INTRODUCTION

The inverse problem of electrocardiography (ECG) is toestimate source parameters on or in the heart given a geometricand conductivity model of the torso volume and observedelectric potentials on the body surface. Activation-based in-verse ECG models the functional sources of the heart atany location on the heart surface (synthesized by closing theventricular endocardium to its epicardium, for example) ashaving two sequential states: off, then on [1]. One can modelthe on/off waveform behavior of each source during the QRScomplex of a heart beat as a phase-shifted step function, aparameterization that reduces the temporal behavior of eachsource to a single unknown variable: the activation time [2].Because this waveform parameterization is nonlinear in theunknown activation time, the inverse problem is typicallyformulated as a nonlinear least squares (NLLS) minimizationproblem. It turns out that this problem is not convex and thatits objective function tends to have many suboptimal localminima.

This has led to several attempts to incorporate the observeddata and prior physiological knowledge into initializationmethods, with the belief that local minima found near theseinitializations are likely to be close to being optimal. Animportant example is the Fastest Route Algorithm (FRA),an initialization method that employs a simplified wavefrontpropagation model based on finding the shortest path on agraph. In this method, each edge that connects two nodesof the graph that represents the heart surface is weightedby an assumed propagation velocity [3], [4]. Each node onthe surface is considered as a candidate for the site of first

activation. An initial earliest node is selected by choosing theresulting wavefront, as determined by the estimated propa-gation pattern which would follow from a signal triggeringactivation at that node, whose predicted body surface potentials(BSPs) have the highest correlation with the data. Wavefrontsarising from later breakthroughs are then also considered andcombined node-wise by retaining the activation time of thefirst wavefront to arrive [5], [6]. The emphasis of this methodis on choosing an initialization with physiologically-plausiblepropagation behavior that is consistent with the measuredBSPs.

More recently, we introduced a method of initialization thatreformulates the NLLS problem as an optimization problemwith non-convex constraints, relaxes the constraints to beconvex, and then solves the new problem for a globally optimalsolution. This solution is typically infeasible for the NLLSproblem because it does not satisfy the original non-convexconstraints, but we suggested in [7] one possible method thatsolved for the nearest neighbor to the convex relaxation solu-tion in the original non-convex constraint set and subsequentlyused this point as initialization for the NLLS problem. Wefound that this method of initialization yielded solutions thatwere close to ground truth in experiments conducted on datasimulated from known activation times and perturbed withpseudorandom “measurement” noise.

However, when conducting experiments with clinical datarecorded from a subject with Wolff-Parkinson-White (WPW)syndrome (published previously in [8]–[10]), we observedwhat appeared to be a mismatch between the optimization cri-teria and our intended objective. The measured BSPs exhibitedtypical WPW behavior during the QRS complex, with extraactivations initiated from the Kent bundle (whose locationwas determined invasively). It was previously reported thatFRA-initialized candidate solutions to the NLLS problem wereable to localize the initial activations [10]. These candidatesolutions had activation patterns exhibiting propagation con-sistent with cardiac electrophysiology. Surprisingly, we wereable to find physiologically-inadequate candidate solutionsthat outperformed the FRA-initialized solutions in terms ofobjective value. Furthermore, we found that the duration ofthe upstroke in this optimization setting can have a significanteffect on the objective value. In this paper, we use the convexrelaxation to explore the solution space and examine whether

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Fig. 1: Isochronal activation time maps of candidate solutions (left to right: FRA-initialized, “last”, “best”, and convex relaxation)for the WPW fusion heart beat, rows showing alternate views. Plot of objective values (black=convex relaxation, blue=“last”,magenta=“best”, red=smooth FRA-initialized (upstroke=10ms), green=FRA-initialized) for a range of regularization parameters.

the NLLS objective function targets the solutions we wish tofind.

We begin the rest of the paper by presenting the NLLS prob-lem, its convex relaxation, and briefly explaining the nearestneighbor initialization method in the Background section. Inthe Methods section, we introduce an optimization procedurethat, together with the convex relaxation, we use to explorethe solution space of the NLLS problem. In the Experimentssection, we report the results we obtained by applying FRAand our newer methods to two clinical WPW datasets. Weanalyze the results and their implications in the Discussionsection below.

II. BACKGROUND

In this section, we review a framework for the activation-based problem as an optimization problem and use it to showhow the original NLLS problem can be equivalently expressedas a non-convex constrained optimization problem. We explainhow a convex relaxation can be obtained and one method ofconverting its global solution into a feasible initialization forthe NLLS problem.

For the remainder of the paper we assume that the data isregularly sampled in time and we only consider those samplesthat correspond to the QRS complex of a single heartbeat.At any given time, the linear relationship between a vectorof body surface potentials, y ∈ RM , and a vector of on/offsources on the heart, x ∈ RN , is y = Ax, where A is theforward matrix that results from solving the forward problemon spatially discretized heart and body surface domains. Fur-thermore, we assume that the waveforms for the sources areunit step functions whose true amplitudes are known (a vectorv) and have been multiplicatively absorbed into the linearforward model (A← A diag(v)) for notational simplicity.

In order to reformulate the NLLS problem as a constrainedoptimization problem, we define an alternative set of con-straints to describe the nonlinearly parameterized waveforms.The original nonlinear parameterization is that every source,xn, has the waveform

xn(t) = u(t− τn) ={

0 , t− τn < 01 , t− τn ≥ 0

where τn is the activation time. Using this parameterization,the original NLLS problem is

minimize∑

t ‖y(t)−Ax(t)‖22 + λ‖Lx(t)‖22

where L is a Tikhonov regularization matrix, λ is the regu-larization parameter, and the optimization variables are theactivation times. The Gauss-Newton algorithm and similarnonlinear least squares solvers require that the objective func-tion is differentiable, so a smoothed step function (with a spec-ified upstroke duration) is typically used and the subsequentapproximate version of the original NLLS problem is solvedinstead [2], [11].

If we let QRS correspond to the sample times t = 1, . . . , Tthen we can define a source matrix X that contains all ofthe temporal samples of each spatial source such that Xn,t =xn(t). Key characteristics of this matrix are that its values areeither 0 or 1, are nondecreasing as the column index increases,and that they always increase from 0 to 1 between columnindices 1 and T .

Let D be a first-order temporal differencing matrix (i.e. Dis T × T with 1 on the diagonal and −1 on the subdiagonal).If we define the sets R and E as

R = {X ∈ R(N×T ) | 0 ≤ X ≤ 1, XDT ≥ 0,

XDT1(T×1) = 1(N×1)}

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E = {X ∈ R(N×T ) | tr(XTX) = 1T(N×1)X1(T×1)}

(where 1(i×j) denotes a i×j matrix of ones) then X ∈ R∩E .Thus we can express the original NLLS problem as a con-strained optimization problem

minimize ‖Y −AX‖2F + λ‖LX‖2Fsubject to X ∈ R ∩ E

where the optimization variable is the matrix X and ‖·‖Fdenotes the Frobenius norm. We showed this problem wasnon-convex in [7] and formulated a convex relaxation byrelaxing the domain from R ∩ E to R. Theoretically, if thesolution to the convex relaxation satisfies X ∈ E , it is theglobal solution to the non-convex problem as well. In general,the convex relaxation does not solve the original problembecause of infeasibility. In this case, the objective value f(X)for the convex relaxation is simply a lower bound on theobjective value of the original constrained problem (whichfollows from R ∩ E ⊂ R and convex f ). If the solution isinfeasible, one can find its nearest neighbor in the feasible setand use that as initialization for the NLLS problem [7].

III. METHODS

In this section we describe an optimization procedure thatsolves a sequence of alternate NLLS problems that approachesthe original NLLS problem. This is done by using a sequenceof parameter values that effectively transforms a subset of theconvex domain R into R∩ E in its limit.

We start by modeling the values of a source matrix Xas samples taken at regular intervals from an underlyingcontinuous-time function. Rather than limit the function foreach row to phase-shifted unit step functions, we use an arbi-trary function hn that is non-decreasing with values between0 and 1. Therefore we have

Xn,t = xn(t) = hn(β(t− τn))

where β is a time scaling parameter and τn is the phaseshift. Given these conditions, as β goes from 1 → ∞, Xapproaches the set R∩E . Thus, as an alternate NLLS problem,we minimize the least squares objective function from theoriginal NLLS with this new parameterization (for fixed β andvariable τn). We note that this alternate NLLS problem andthe original NLLS problem have the same convex relaxation.

We initialize the sequence of optimization problems with theconvex relaxation, Xc, such that β = β0 = 1, the initial phaseshifts τn are the activation times from the nearest neighborinitialization method, and the waveforms hn are chosen suchthat X = Xc. We use a Gauss-Newton solver to search fora local minimizer, using this solution to initialize the nextproblem in the sequence with β = β1 > β0. The procedurecontinues in this manner until X is within some specifiedprecision of its nearest neighbor in R∩ E .

At the conclusion of every iteration, we store the minimiz-ing activation times and source matrix. When the sequenceterminates, we find the nearest feasible neighbor to all of thestored source matrices and keep the one with the best objective

value as the “best” feasible candidate solution. We also keepthe final source matrix of the sequence as the “last” candidatesolution (not necessarily feasible).

IV. EXPERIMENTS

For our experiments, we used measured body surface po-tentials during the QRS complex of beats from a subject di-agnosed with Wolff-Parkinson-White (WPW) syndrome. Sub-jects with WPW have an additional conduction system calledthe Kent bundle that leads to additional initial activations tothose caused by the regular conduction system. In this case,the location of the subject’s Kent bundle was determinedinvasively. We use two different beats for our experiments:first a fusion beat (i.e. ventricular activation initiated by boththe Kent bundle and regular conduction system), and then aKent-bundle-only beat. The fusion beat is considered a typicalWPW beat, whereas the Kent-bundle-only beat was inducedby administration of adenosine to block the AV-node. Thus,although exact activation patterns may not be known, much isknown about the beats that can be used to evaluate whethercandidate solutions to the inverse problem capture the correctbehavior.

For each beat, we computed candidate solutions by solvingthe NLLS problem initialized by FRA, applying the opti-mization procedure described in Methods for the “last” and“best” candidate solutions, and solving the convex relaxation.Optimization problems were solved for a fixed value of theregularization parameter (λ = 1) in each case, but the objectivevalues of resulting candidate solutions were compared over awider range of regularization parameters.

V. DISCUSSION

In this paper, we described the results of a procedurefor activation-based inverse electrocardiography as applied todata measured from a patient with WPW syndrome. In theseexperiments, we explored the solution space of the NLLSminimization problem of activation-based inverse ECG witha number of optimization procedures: FRA-initialized NLLS,a convex relaxation of the NLLS problem, and a sequence ofNLLS problems.

The first experiment was conducted on a fusion beat thatcombined both the Kent bundle and the normal conductionsystem. The results of the optimization procedures can beseen as isochrone maps in Figure 1. The aforementioned sitesof initial activation can be seen in the candidate solutionfound by FRA initialization. This candidate solution has itsearliest activation near the base of the heart where the Kentbundle is known to be located, and a second activation on theepicardium of the other ventricle. Furthermore, the spacing ofthe isochronal contours for this activation pattern are consistentwith the behavior of propagating wavefronts. The secondexperiment was conducted on an induced Kent-bundle-onlybeat andits results can be seen in Figure 2. Again, in this casethe FRA-initialized candidate solution accurately localizes thesite of initial activation and captures the expected behavior ofpropagating wavefronts.

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Fig. 2: Isochronal activation time maps of candidate solutions (left to right: FRA-initialized, “last”, “best”, and convexrelaxation) for the WPW Kent-bundle-only heart beat, with rows showing alternate views. Plot of objective values for arange of regularization parameters (colors of lines same as in Fig. 1).

For the fusion beat experiment, the other three candidatesolutions that were obtained using the optimization proce-dures described in the Background and Methods sectionsloosely resemble the FRA-initialized activation pattern buthave isochronal contours that are inconsistent with propagatingwavefronts. In the case of the Kent-bundle-only beat, thecandidate solutions for the same three methods find erroneousendocardial sites of initial activation in addition to those of theKent bundle. In terms of their qualitative physiological char-acteristics and the degree to which they adhere to the knownbehavior of the beats in these experiments, these three sets ofcandidate solutions are clearly unfavorable in comparison tothe comparable FRA-initialized candidate solutions.

On the other hand, in Figures 1 & 2, we also plot theobjective values for these candidate solutions evaluated fora range of regularization parameters. The green and redcurves in this plot correspond to the FRA-initialized candidatesolution and its smoothed approximation with a 10ms upstrokeduration, respectively. The other three curves correspond tothe unfavorable candidate solutions described in the previousparagraph. As the plot shows, the unfavorable candidate so-lutions have a lower objective value than FRA for all of theregularization parameter values.

In theory, the objective function of an optimization prob-lem serves as a way of ranking the suitability of candidatesolutions. The implication of the results we have presentedhere is that there is a mismatch between the objective ofactivation-based inverse ECG and the objective function ofthe corresponding optimization problem. This suggests thatthe optimization criteria for this problem need to be carefullyreviewed and modified, if necessary, to achieve the intendedgoal. Another conclusion we draw from this work is that theconvex relaxation method, which compared favorably to FRAon simulated data, does not compare favorably on this clinicalcase. We are currently investigating the possible causes of thisdiscrepancy with an eye to improving both methods.

ACKNOWLEDGMENT

Support for the work of BE and DHB was provided inpart by the NIH/NCRR Center for Integrative BiomedicalComputing (CIBC), 2P41 RR0112553-12. The authors wouldlike to thank Dr. Gerald Fischer for use of the data.

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