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Application of Stochastic Finite ElementMethods to Study the Sensitivity of ECGForward Modeling to Organ Conductivity

Sarah E. Geneser, Robert M. Kirby, and Robert S. MacLeod

Abstract— Because numerical simulation parametersmay significantly influence the accuracy of the results,evaluating the sensitivity of simulation results to variationsin parameters is essential. Although the field of sensitivityanalysis is well developed, systematic application of suchmethods to complex biological models is limited due tothe associated high computational costs and the substantialtechnical challenges for implementation. In the specific caseof the forward problem in electrocardiography, the lackof robust, feasible, and comprehensive sensitivity analysishas left many aspects of the problem unresolved andsubject to empirical and intuitive evaluation rather thansound, quantitative investigation. In this study, we havedeveloped a systematic, stochastic approach to the analysisof sensitivity of the forward problem of electrocardiographyto the parameter of inhomogeneous tissue conductivity. Weapply this approach to a two-dimensional, inhomogeneous,geometric model of a slice through the human thorax. Weassigned probability density functions for various organconductivities and applied stochastic finite elements basedon the generalized Polynomial Chaos-stochastic Galerkin(gPC-SG) method to obtain the standard deviation of theresulting stochastic torso potentials. This method utilizes aspectral representation of the stochastic process to obtainnumerically accurate stochastic solutions in a fraction ofthe time required when employing classic Monte Carlomethods.

We have shown that a systematic study of sensitivityis not only easily feasible with the gPC-SG approach butcan also provide valuable insight into characteristics of thespecific simulation.

Index Terms— electrocardiographic forward problem;stochastic finite elements; polynomial chaos; stochasticGalerkin; stochastic processes; uncertainty quantification

I. INTRODUCTION

The forward problem of electrocardiography providesvaluable information for both basic science and clinicalmedicine and has a well established set of solution

Manuscript submitted November 22, 2006; Revised March 2, 2007Geneser and Kirby are with the School of Computing and Scientific

Computing and Imaging Institute, University of Utah, Salt Lake City,UT, 84112, USA. E-mail: geneser,kirby@cs.utah.edu

MacLeod is with the Department of Bioengineering and theNora Eccles Harrison Cardiovascular Research and Training Insti-tute, University of Utah, Sale Lake City, UT, 84112, USA. E-mail:macleod@cvrti.utah.edu

approaches [1]–[3]. Computational electrocardiographicmodels based on realistic human physiology includemany input parameters, such as the geometric discretiza-tion of the torso organs, the conductivities of the tissues,and the representation of cardiac sources. Inherent ineach model parameter are “errors” associated with quan-tifying and simplifying the physiology so that numericalimplementations remain tractable. These errors can re-sult, for example, from simplifications or abstractionsof the physics or geometry (modeling errors), lack ofinstrumentation precision in obtaining model parameters(parameter errors) as well as from natural variations overpatient populations. Quantification and control of sucherrors are critical to the simulation process; only then canscientists judiciously evaluate which components of themodel are critically sensitive to variations and hence mayrequire additional refinement or more accurate measure-ments. In the specific case of the electrocardiographicforward problem, tissue conductivity is an example ofan input parameter that is very difficult to accuratelyobtain; establishing conductivity in realistic conditionsrequires careful experiments and complex interpretationwith often widely varying results [4]–[6]. The role ofsuch variations in conductivity on simulations is the topicof extensive ongoing research and discussion, with noobvious means of resolution available [7]–[13].

The goal of this study was to present and show theutility of a novel methodology of sensitivity analysis firstpresented in [14]–[16] that is applicable to a wide rangeof problem in bioelectric fields. To demonstrate its fea-sibility, we utilized the approach to evaluate the effectsof variations and uncertainty in the conductivity valuesassigned to organs in a two-dimensional model of thehuman thorax. The mathematical framework presentedherein extends naturally for use on three-dimensionalmodels; we focus on a two-dimensional model for thesake of clarity of presentation.

Sensitivity analysis in computational modeling pro-vides a quantitative description of the dependence of thesolution on various model parameters and input values.Such analysis can provide insight into the relative impactof parameter variation on solution accuracy, which, in

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turn, has implications regarding the underlying mecha-nisms of the model system. Sensitivity analysis methodswere first developed for the assessment of mathematicalmodels of control systems [17]–[19] and have a strongtradition in the analysis of mechanical systems [20].However, the value of the sensitivity approach extendsbeyond these initial applications. The approach is es-pecially useful for biological systems as they typicallyrequire a high level of simplification and abstraction toencapsulate physiological mechanisms in tractable sim-ulations. Many parameters in biological simulations areapproximations based on experiments and in many cases,the interplay between such parameter approximationsand other modeling assumptions is not clear. Such anapproach can identify aspects of the model that may besimplified or reduced without significantly impacting thesolution accuracy.

Several methods for carrying out sensitivity analy-sis exist, but they are not widely applied to complexsystems due to their computational cost and difficultyof implementation. For example, sampling-based sensi-tivity methods such as Monte Carlo [21], [22], LatinHypercube [23], [24], response surface [25], reliabilitybased methods [26], [27], and the Fourier AmplitudeSensitivity Test [28] are often impractical due to the largenumber of repetitions necessary to obtain statisticallyvalid results. The time required to compute sufficientsamples for accurate solutions often forces researchersto revert to highly under-sampled “brute-force” methods[29], [30]. Sometimes referred to as “sensitivity testing”,this technique investigates the model response to anextremely reduced set of model parameter combinations,and thus fails to provide robust quantification of themodel sensitivity.

Several methods exist to directly calculate sensitivitycoefficients rather than to extract them from large num-bers of test simulations. One scheme is based on thecoupled direct method in which the model equations aredifferentiated with respect to the parameters of interestand the resulting set of sensitivity equations is solvedsimultaneously with the model equations [31]. Anotherclass of techniques relies on the Green’s function oradjoint method, also based on differentiation of themodel equations. In this case, one constructs an auxiliaryset of Green’s functions thus minimizing the numberof differential equations that must then be solved toobtain sensitivity coefficients [32], [33]. Examples ofdifferential analysis methods include the Neumann [34]and Taylor series [35], [36] expansions, and perturbationanalysis [37]–[39]. These methods are computationallyefficient, but require that the perturbation terms be smalli.e., only limited variations in parameters are possible,and can be very cumbersome to apply to complexsimulations. Moreover, for nonlinear systems, the re-

sulting equations for these analytical methods can bemathematically intractable.

In summary, most sensitivity analysis techniques todate have suffered from one of the following limitations:(1) they are expensive to compute (in comparison withsolving the single deterministic model problem); (2) theyrequire extensive mathematical adaptation or manipula-tion of the original model (such as computing Frechetderivatives of an already complex system); or (3) theyrestrict the range of parameter or model perturbations ina way that hampers one’s ability to answer the sensitivityquestions of interest.

In contrast to all the approaches described above, thegeneralized polynomial chaos-stochastic Galerkin (gPC-SG) technique attempts to alleviate these limitationsby treating the input data and model parameters asstochastic processes and solving the resulting stochasticcomputational system of equations in order to obtaininformation about the sensitivity of the system. Toaccomplish this, it is necessary to assume a particularprobability density function (PDF) for the parameters ofinterest, which arguably allows for even more subtlety inthe sensitivity analysis. Once the stochastic solution isobtained through an orchestrated combination of deter-ministic model solutions, one can compute and examinethe standard deviation and higher statistical momentsof the result. Such information provides a quantitativedescription of the sensitivity and uncertainty of thesystem to the parameters.

Based on the Wiener-Hermite polynomial chaos ex-pansion [40], gPC has been applied to a range ofproblems in computational mechanics [41]–[49]. Thistechnique has also recently been introduced into otherdisciplines such as thermodynamics [50]–[52] and phys-ical chemistry [53], [54], in part because it leads to effi-cient solutions to stochastic problems of interest, i.e., notonly parameter sensitivity and uncertainty quantification.

To improve the computational efficiency of thismethod, we employ the stochastic Galerkin method tosolve a generalized Polynomial Chaos representation ofthe stochastic system. The stochastic Galerkin methodrepresents stochastic processes via orthogonal polyno-mials of random variables and utilizes specific sets oforthogonal polynomials to achieve computationally effi-cient representation of random processes with arbitraryprobability density functions. Such expansions exhibitfast convergence rates when the stochastic responseof the system is sufficiently smooth in the randomspace, e.g., bifurcation behavior is absent. The stochasticGalerkin method is an efficient means of reducing thestochastic governing equations to a system of determin-istic equations that can then be solved via conventionalnumerical techniques.

We present here a development of the stochastic

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Galerkin method adapted to the forward problem of elec-trocardiography. With this approach it is possible to carryout a Galerkin projection of the governing equations ontothe polynomials basis functions defined by gPC. Sucha gPC-SG approach is capable of both applying rela-tively large perturbations in the model parameters andtolerating large variations in the responses. This methodhas been successfully applied to model uncertainty incomplex stochastic solid and fluid problems [47], [48],[55] and is especially well suited to electrocardiographicforward problems because they are linear with relativelywell behaved solutions. In this study we demonstratedthe feasibility of the gPC-SG approach by successfullyapplying it to a realistic two-dimensional model of theelectrocardiographic forward problem. The results of thisevaluation support some previous findings using other,less complete sensitivity studies but also suggest somenew criteria for the creation of geometric models forbioelectric field problems.

II. METHODS

The methods section is partitioned into two subsec-tions, Section II-A outlines the standard approach weadopted for the electrocardiographic forward problemwhile Section II-B describes application of the gener-alized polynomial chaos-stochastic Galerkin (gPC-SG)technique to the standard ECG forward problem.

A. Electrocardiographic Forward Problem

The electrocardiographic forward problem is a quasi-static approximation of Maxwell’s equations expressedas follows:

∇ · (σ(x)∇u(x)) = 0, x ∈ Ω (1)u(x) = u0(x), x ∈ ΓD (2)

~n · σ(x)∇u(x) = 0, x ∈ ΓN , (3)

where Ω denotes the torso domain, ΓD and ΓN de-note the epicardial (Dirichlet) and torso (Neumann)boundaries respectively, u(x) is the potential field onthe domain Ω, u0(x) is the known epicardial poten-tial boundary function, σ(x) is the symmetric positivedefinite conductivity tensor, and ~n denotes the outwardfacing normal with respect to the torso.

1) Numerical Methodology: Solving Equations 1–3for any but the simplest geometries requires a numeri-cal approximation, typically based on methods such asboundary elements or finite elements [1]. In order tofacilitate the subsequent application of the stochasticGalerkin method, we employed a high-order finite el-ement method [56]–[58].

High-order elements refer to the polynomial order ofthe basis functions used to describe spatial variation ofthe variable(s) of interest. Basis functions with greater

than first/linear degree provide a means of capturingmore rapid variations in the system—and potentiallyachieving higher accuracy—without the need for refiningthe geometric mesh used to describe the domain understudy. For this study, we first compared results usingbasis functions of first, second, and third degree anddetermined second degree to be adequate for all experi-mental results presented in this study.

We utilized a triangular tessellation T (Ω) of thedomain with the set N denoting indices of the meshnodes as the geometric basis for the finite elementcomputations. For the case of linear finite elements, thisset consisted of the indices for the triangle vertices, andfor the case of nodal high-order finite elements, the setcontained indices for nodes at the triangle vertices aswell as the edge and internal discretization nodes. Wethen decomposed the set N into two non-intersectingsets, B and I, representing nodal indices that lie on theDirichlet boundary (and hence denote positions at whichthe potentials are known) and nodal indices for which thesolution is sought (i.e., the Ndof degrees of freedom ofthe problem), respectively.

Let φi(x) denote the global finite element interpo-lating basis functions, which have the property thatφi(xj) = δij where xj denotes a node of the meshfor j ∈ N . Solutions are then of the form:

u(x) =∑k∈N

ukφk(x) (4)

=∑k∈I

ukφk(x) +∑k∈B

ukφk(x), (5)

where the first term of Equation 5 denotes the sum overthe degrees of freedom of the problem of the values at theunknown vertices weighted by the basis functions andthe second term denotes the same sum for the (known)Dirichlet boundary conditions of the solution.

Substituting the expansion (Equation 5) into the differ-ential equation (1), multiplying by a function from thetest space φj(x); j ∈ I, taking inner products, andintegrating by parts yields a linear system of the form:

S~u = ~f, (6)

where ~u denotes a vector containing the solution of thesystem (i.e., potential values at the nodal positions inI) and S and ~f denote the stiffness matrix and right-hand-side function, respectively, given by the followingexpressions:

S = Sjk = (∇φj , σ∇φk) (7)~f = fj =

∑i∈B

ui(∇φj , σ∇φi). (8)

In the expressions above, j, k ∈ I, and (·, ·) denotesthe inner product taken over the entire spatial domainΩ. Both the stiffness matrix and right-hand side vector

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are functions of the conductivity. The stiffness matrix inEquation 6 is positive definite and hence solution of thelinear system is amenable to iterative methods such asthe preconditioned conjugate gradient method [59].

2) Geometric Model: The geometric model for thisstudy consisted of a single two-dimensional slice of theUtah Torso Model [9], [60]. We adaptively refined theoriginal mesh to achieve higher spatial density near theepicardial surface in order to improve numerical solutionaccuracy. Figure 1 contains a diagram of the modelshowing regions representing lungs, skeletal muscle,subcutaneous fat, and a miscellaneous category denoted“torso cavity”. The segmentation of the tissues ensuredthat boundaries always resided at the edges of elementsand we assumed constant conductivity values over eachorgan region and hence within every element of thatregion. Although we did not consider anisotropy in thisstudy, the model would easily support the assignment ofconductivity tensors for each element.

Fig. 1. Two-dimensional geometric model of the torso: Conductiv-ity values were assigned according to the different regions of the torsoslice which consisted of lungs, skeletal muscle, subcutaneous fat, anda miscellaneous category denoted torso cavity. Orientation of the sliceis according to a standard radiological view, looking towards the headof a supine subject.

3) Simulation Input Data: Boundary Conditions andConductivity Parameters: Solving a particular case ofthe forward problem requires tissue conductivity values,which we obtained from the literature [61] and sum-marized in Table I. Dirichlet boundary conditions tookthe form of potential values along the epicardial surfaceof the geometric model. For this we utilized a set ofpotentials interpolated from 64-channel measurementsperformed on a human subject during open-chest abla-tion therapy for severe cardiac arrhythmia [61].

TABLE ICONDUCTIVITY VALUES CORRESPONDING TO ORGANS IN THE

MODEL AND THE PERCENT AREA OF THE DOMAIN CONTRIBUTED

organ category conductivity (S/m) percent area of domain Ωlungs 0.096 36.37%

skeletal muscle 0.200 22.93%subcutaneous fat 0.045 21.61%

torso cavity 0.239 19.09%

B. Sensitivity quantification through the stochasticGalerkin method

We now present the gPC-SG approach as applied tothe ECG forward problem starting with a brief gen-eral overview of the gPC-SG formulation, followed bya description of our specific implementation of thismethodology.

1) General Formulation: The stochastic Galerkinmethod represents any stochastic process, a(ξ), by aweighted sum of orthogonal polynomials [47], [48],[55], [62] which are functions of a vector of randomvariables, ξ, of known PDF. In the case of this study,the random processes of interest are the (stochastic)conductivity values attributed to different organs withinthe torso. The random variables of interest will be chosento represent the distributions from which conductivityvalues are sampled. We denote the stochastic orthogonalpolynomial set as ψi(ξ) and can write

a(ξ) =∞∑

i=0

aiψi(ξ). (9)

For any reasonable random process, this infinite summa-tion can be truncated to P +1 terms and the polynomialweights ai are projections of the random input onto therandom polynomials

ai =∫

Γ

a(ξ)ψi(ξ)dµ = 〈a(ξ), ψi(ξ)〉µ (10)

in the stochastic domain, Γ, where the random measurespace, µ, is appropriate for the probability density func-tion of the random variable vector, ξ. The stochasticcoefficients of the trial expansion (as in the finite elementmethod) are found by solving the resulting system forthe random solution coefficients.

To apply this general approach to the specific case ofan elliptical forward problem such as that in Equation1, we consider a stochastic conductivity tensor σ(x; ξ)expressed in terms of an n-dimensional random variablevector ξ = (ξ1, ξ2, . . . , ξn)T , where the PDFs of therandom vector components are known or assumed. Be-cause the conductivity tensors are physically constrainedto be non-negative and non-zero, the PDF must besimilarly constrained. The components of the randomvariable vector denote the different random variablesnecessary to describe the process of interest. Whenexamining a stochastic process whose PDF is known,the smallest number of independent and uncorrelatedrandom variables necessary to represent the process aretypically obtained by such methods as Karhunen-Loeve(KL) expansion [63]. For example, if one were to modeltwo tissue classes as stochastic and independent (in thestatistical sense), a two-dimensional random variablevector would be utilized with the two components as-signed to each tissue class respectively. The resulting

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stochastic versions of Equations 1–3 are then

(∇ · σ(x; ξ)∇)u(x; ξ) = 0, x ∈ Ω (11)u(x; ξ) = u0(x; ξ), x ∈ ΓD (12)

~n · σ(x; ξ)∇u(x; ξ) = 0, x ∈ ΓN , (13)

where the solution u(x; ξ) is now also a function ofthe stochastic variable vector ξ and thus has a randomdistribution with mean, variance, and higher stochasticmoments. The conductivity tensor and solution field, asrandom processes, can be represented via the generalizedstochastic Galerkin expansions as

u(x; ξ) =P∑

i=0

ui(x)ψi(ξ) (14)

σ(x; ξ) =P∑

i=0

σi(x)ψi(ξ), (15)

where the functions ψi(ξ) are orthogonal polynomialsranging up to P th degree and P must be chosen to belarge enough so that the solutions will meet the accu-racy requirements for the particular system of interest.Convergence rates of the system depend on the choiceof orthogonal polynomials for the underlying probabilitydensity functions of the random model parameter. Eachprobability distribution has a corresponding optimal setof orthogonal polynomials [62]; e.g., for Gaussian ran-dom functions, Hermite polynomials provide the bestconvergence, whereas the Legendre polynomials are bestutilized for functions of uniform distributions, etc. Whenthe stochastic response contains a discontinuity, e.g., neara bifurcation point, piecewise polynomials [64], [65] canbe employed to circumvent the difficulty.

Substituting the above expressions into the stochasticelliptic system given by Equations 11–13 and projecting(in the Galerkin sense) the resulting system into therandom space spanned by the basis polynomials, ψk,leads to the following linear system:

For k = 0, . . . , P

P∑i=0

P∑j=0

Cijk∇ · (σi(x)∇uj(x)) = 0, x ∈ Ω (16)

where Cijk = 〈ψi(ξ)ψj(ξ), ψk(ξ)〉µ denotes the innerproduct over the appropriate probability measure space.The probability measure, µ, within the integration isdetermined by the PDF of the random variables used. Byusing numerical quadrature to evaluate the inner productCijk [47], [55], [62], with the appropriate boundary con-ditions (discussed in section II-B.2), the system reducesto a large linear combination of elliptic equations for thecoefficients uj(x). Since the unknowns are now onlyfunctions of space, standard finite element techniquescan be employed to transform this into a large linear

system of deterministic stiffness equations and right-hand-side vectors.

2) Implementation Details: In this study, we system-atically investigated the effect of conductivity variabilityby individually perturbing the conductivity values of theorgan groups within a two-dimensional cross section ofthe human torso, assuming uniform distributions for theconductivity variability. For the polynomial set ψi, weemployed the Legendre polynomials defined on [−1, 1],as these are most efficient for expressing random pro-cesses with uniform (or nearly uniform) distributions.

Under these assumptions, we express the conductivityas follows:

σ(x; ξ) = σ0(x)ψ0(ξ) + σ1(x)ψ1(ξ), (17)

where σ(x; ξ) is uniformly distributed on the interval[a(x), b(x)] for each point x ∈ Ω. We have then σ0(x) =0.5(a(x)+ b(x)) and σ1(x) = 0.5(b(x)−a(x)). Underour assumptions, only two terms are needed to representthe randomness of the conductivity. This does not implythat two polynomial chaos terms are sufficient to repre-sent the random (and possibly non-uniformly distributed)process denoting the potential in the torso; the numberof terms utilized to represent the stochastic processesis denoted P . Because we were interested in perturbingorgan groups individually, we set the first mode σ0(x)to the prescribed (deterministic) value for each organover all computational experiments. We then set σ1(x)to a non-zero value (corresponding to the half lengthof the uniform interval) for the organ group of interestand set σ1(x) = 0 for each of the remaining organs.For example, with a 50% uniform interval, conductivityfor that organ varied from 0.5 to 1.5 times that nominal(deterministic) value with equal probability of any valuein that range.

Under the assumption that the boundary conditionswere known and had no stochastic variability (i.e., weredeterministic), the sytem of equations reduces to thefollowing (for i = 0, 1, j = 0, . . . , P ):

u0(x) = u0(x), x ∈ ΓD (18)uj(x) = 0, x ∈ ΓD (19)

~n · σi(x)∇uj(x) = 0, x ∈ ΓN , (20)

with the appropriate boundary conditions (given in sec-tion II-B.2).

Equations 16 and 18–20 comprise a large deterministicsystem. To make concrete the form of this system underour assumptions, let S0 and S1 denote the stiffness ma-trices generated with σ0(x) and σ1(x) respectively, andlet ~f0j and ~f1j denote the right-hand-side vectors basedupon boundary conditions for uj(x) corresponding toσ0(x) and σ1(x), respectively. The linear system of

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Fig. 2. Diagram of the large linear system resulting from the linearcombination of stiffness matrices and right-hand-sides. The α valuescorresponds to inner products of the stochastic basis polynomials, withα0 = C000, α1 = C101, α2 = C211, α3 = C220, α4 = C321 andα5 = C330. The matrices, S0 and S1 are as described in the text.

equations which results is

P∑j=0

[Cij0S0 + Cij1S1] ~uj =P∑

j=0

[Cij0

~f0j + Cij1~f1j

],

(21)for i = 0, . . . , P where ~uj denotes the vector of finiteelement degrees of freedom expressing the jth stochasticmode. Given Ndof degrees of freedom for each finiteelement spatial discretization (i.e., the dimensions ofthe stiffness matrices S0 and S1 are Ndof ), then thetotal dimension of the new stochastic linear system is(P +1)Ndof . The extension of this methodology to threespatial dimensions consists of populating the equationabove with stiffness matrices and right-hand-side vectorsfrom a three-dimensional finite element model.

Figure 2 shows a schematic of the type of systemwe solved via iterative methods to obtain the stochasticmoments of the solution given a third-degree stochasticGalerkin discretization (i.e., P = 3). Note that the choiceof P = 3 has been made to help simplify the presentationof the figure.

3) Post-Processing: Given the stochastic modes, wecalculated the mean and standard deviation of the solu-tion as

mean(u(x; ξ)) = u0(x)

stdev(u(x; ξ)) =

[P∑

i=1

(ui(x))2〈ψi(ξ), ψi(ξ)〉µ

] 12

.

These modes are fields over the problem domain andhence we could visualize them over the entire two-dimensional slice through the torso surface.

In all the simulations presented here, we assumeduniform distributions for the stochastic conductances andfound that using stochastic polynomials of degree seven(i.e.,P = 7) were sufficient to capture the stochasticbehavior of the potentials in the torso. To verify thischoice, we accomplished a comparison between MonteCarlo and the proposed methodology for one of theexperiments presented in our work. We found that mean

and standard deviations calculated from the simulationof 6, 000 Monte Carlo trials were the same to foursignificant figures as those computed using degree sevenstochastic polynomials. However, the Monte Carlo trialsrequired more than 2, 700 times the CPU time requiredto compute the same result using the proposed method-ology. This performance discrepancy would be furtherexacerbated in the case of an even larger number ofMonte Carlo trials.

III. RESULTS

We present here results of a sensitivity analysis ofvariations in organ conductivity in a two-dimensionalmodel of the human thorax. The epicardial potentialsapplied to the inner boundary are taken from measure-ments on an exposed human heart during surgery forresolution of persistent ventricular arrhythmias. For thisreport, we selected a time instant late (85 ms after onset)in the QRS complex as representative and present plotsof mean and standard deviation of computed potential onthe geometric model oriented in the standard radiologicalview (looking from the feet toward the head with thesubject facing upward). For all of the results, we examineconductivity intervals of ± 50% or smaller, as theseranges fall well within the distributions of experimentallyobtained values [4]–[6].

Figure 3 shows results for variations of 50% in theconductivity of lung, muscle and fat. Panel (a) showsthe mean potentials resulting from variations in lungconductivity. As expected, the mean behavior was vir-tually identical for all organ conductivity experiments.The roughly dipolar nature of the epicardial boundaryconditions and the similarly dipolar but considerablysmoothed potential distribution on the torso surface arealso evident in this figure. Such attenuation and smooth-ing is evident when comparing the two distinct maximaof the right ventricular potentials that fuse into a singlemaximum as the current travels out to the torso surface.Panels (b)–(d) show the spatial distribution of standarddeviation over the thorax for variations in each of thethree major organs. For variations in lung conductivity,there is a large region of high standard deviation (0.21–0.23 mV)—the highest standard deviation values for anysimulation experiments at the 50% level—over the leftposterolateral region of the thorax. The same regionshowed maximal standard deviation for variations inmuscle conductivity but at a value less than half thatseen for variations in lung conductivities. Variationsin lung conductivity also produced elevated but sub-maximal standard deviations in the region on the rightposterolateral aspects of the thorax, a region once againshared (albeit at relatively lower amplitudes) in theresults for variation in muscle conductivity. In contrast,for variations in subcutaneous fat conductivity, only one

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(a) Mean for ± 50% lung conductivity (b) Standard deviation for ± 50% lung conductivity

(c) Standard deviation for ± 50% muscle conduc-tivity

(d) Standard deviation for ± 50% fat conductivity

Fig. 3. Effects of stochastic variation in conductivities of three organs on torso potentials: These contour plots correspond to the stochasticbehavior of the electrical potential across a slice through the torso resulting from stochastic organ conductivities. Figure (a) shows the meanfor a stochastic interval of ± 50% from the reference lung conductivity, while the (b) shows the associated standard deviation. The bottom twopanels show the standard deviation for 50% stochastic variation in muscle (c) and fat (d), respectively. In each figure depicting the standarddeviation of torso potentials, the region of varying conductivity is indicated by the dotted contours. All units in the plot are mV.

region located in left anterolateral area showed elevatedstandard deviation values and even there, the amplitudewas an order of magnitude smaller (0.03 mV) than themaximum for variations in lung conductivity.

We performed experiments with artificially rotatedepicardial potentials to determine the effect of epicardialpotential orientation on variations in torso potential andstandard deviation. Figure 4 shows one such example inwhich Panel (a) contains mean results for a 90 degreerotation in epicardial potentials relative to those shown inFigure 3. The associated patterns of standard deviationshifted in response to the altered epicardial potentialsbut not in a way that reflected a simple rotation. Forexample, a comparison of Panel (d) in both figures, bothshowing the standard deviation from a 50% variation infat conductivity, shows that for the original orientation, asingle area of elevated standard deviation exists (Figure3), while rotating the same potentials resulted in threewidely separate regions of elevated standard deviation(Figure 4).

Figure 4 also illustrates the effect of scaling on sensi-tivity of torso potentials to variations in conductivity.Panels (b) and (c) appear almost identical in patternbut amplitudes vary by a factor of approximately 5,equal to the ratio of the respective range of variationin conductivity (50% vs. 10%).

Another finding shown in Figure 4 is the nature ofinteractions between variations of conductivity in multi-

ple organs. Panels (b)–(d) show standard deviations forvariations in a single organ while Panels (e) and (f) showresults from allowing simultaneous variation of bothlung and fat conductivity. We selected a level of vari-ation for each component that generated approximatelyequal individual variance in the forward solutions; thecombined effects are certainly a function of the relativelevels of variation in the individual parameters. In onecombined case, Panel (e), conductivities are coupledthrough a single random variable while in the other case,Panel (f), variations are functions of two independentvariables. In both instances, there are features in themaps of cumulative standard deviation that appear tooriginate from one or the other of the associated single-parameter maps. However, the cumulative distributionsare not simple algebraic sums of the single-parameterdistributions, indicating complex interactions betweenthe conductivities of different organs. Another findingillustrated in the figure was the larger standard devia-tions of potential resulting from independent variation ofmultiple conductivities as compared to variations linkedto a single random variable.

IV. DISCUSSION

The main purpose of this study was to present anovel approach to parameter sensitivity analysis thatcould have application to a wide range of bioelectricand biomedical simulation problems.

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(a) Mean for ± 50% lung conductivity (b) Standard deviation for ± 50% lung conductivity

(c) Standard deviation ± 10% lung conductivity (d) Standard deviation for ± 50% fat conductivity

(e) Standard deviation for lung and fat conductivityas functions of the same random variable

(f) Standard deviation for lung and fat conductivityas functions of two random variables

Fig. 4. Effects of multiple stochastic dimensions on the variance of the torso potentials: These contour plots correspond to the stochasticbehavior of the electrical potential across the torso surface resulting from stochastic conductivity with uniform distribution where the epicardialpotentials have been rotated clockwise 180 degrees. Figure (a) shows the mean torso potentials for stochastic lung conductivity with valuesvarying within ± 50% from the reference value, while (b) depicts the standard deviation. Figure (c) depicts the standard deviation for stochasticlung conductivity of ± 10% as a function of a single random variable. Figure (d) depicts the standard deviation for stochastic fat conductivityof ± 50% as a function of a single random variable. Figure (e) depicts the standard deviation for stochastic lung conductivity of ± 10% andstochastic fat conductivity of ± 50% where both stochastic lung and fat are a function of the same single random variable. Figure (f) depicts theresults of stochastic lung and fat conductivity variation from the same components as in Figure (e), but each is a function of two independentand uncorrelated random variables. In each figure depicting the standard deviation of torso potentials, the region of varying conductivity isindicated by the dotted contours. The numerical value of the maximum is included in each plot and all units in the plot are mV.

A thorough sensitivity analysis as described here hasdistinct advantages over the approaches used in mostpreviously reported studies that have sought to establishthe effects of conductivity variation on forward ECGand EEG problems [7], [10]. Most notably, rather thaninvestigating only a small subset of model parametervalue combinations, in a full statistical analysis one isable to explore the impact of a continuous range of organconductivity values on torso potentials. Additionally, thisapproach provides a complete spatial distribution of theeffects of parameter variation on the forward solutionrather than a single value (typically the root meansquared of the difference between solutions) as with mostother investigations of this type. Thus it is possible to

observe that variations in organ regions result in stan-dard deviations with widely different spatial distributionsas illustrated by Figures 3 and 4. Analysis of thesespatial patterns, in turn, can suggest a correspondencebetween the effect of source potentials, problem domaingeometry, and the location and value of the conductivityvariations upon the resulting potentials throughout theproblem domain. It is these relationships that revealfeatures of, in this case, the electrocardiographic forwardproblem and its implementation under realistic condi-tions.

Although we do not report on a complete sensitivityanalysis of the forward problem, there are some observa-tions that illustrate the utility of the stochastic approach –

9

observations that are not possible using the more typicalanalyses from past studies.

As an example of the insights stochastic analysis canprovide, we noted that the mean potentials across thetorso cross section were extremely similar for multipledistributions of organ conductivities. This finding isexpected due to the symmetry of the conductivity prob-ability distributions examined and the relatively linearbehavior of the elliptic problem with respect to theconductivities. In contrast, we noted distinct differencesin the spatial distribution of the standard deviations forvariations in conductivity of the various organs. Bygenerating spatial distribution of mean potential andstandard deviation over the entire cross section, we wereable to observe the relationship between these parametersand the geometry of the organ boundaries and thus visu-alize the effects of changing conductivities. For example,Figure 3 shows that the structure of mean potential(shown in Panel (a)) did not obviously reflect underlyingorgan boundaries whereas maps of standard deviation inPanels (b) and (c) show breaks in the iso-value linesthat align with the boundary of the lungs and musclerespectively. The standard deviation from variation insubcutaneous fat (depicted in Panel (d)), shows a weakerassociation with internal organ boundaries.

Our findings appear to contradict those of Klepfer etal., who determined that both skeletal muscle and fatconductivities had greater affect than lung conductivityon the three-dimensional forward problem of electrocar-diography [7]. These findings, however, were based onthe assumption of skeletal muscle anisotropy; indeed inthe case of isotropic skeletal muscle conductivities (aswere used in our study), the effects of muscle conduc-tivity were significantly less than for lung conductivities.The fat conductivities have elevated impact in the studyby Klepfer et al., perhaps because the difference betweenthe two fat conductivity values investigated was signif-icantly larger than the intervals examined here. A fullcomparison of results will require the use of equivalent,three-dimensional models.

The accuracy of a geometric model depends not onlyon conductvity values but also—and perhaps even moreso—on the proximity of the various organs to the bound-ing surfaces of the problem and the geometric accuracyof the regions of different conductivity. The stochasticapproach we describe here is also suited to an analysis ofsuch geometric variation, a project beyond the scope ofthis report. Ultimately, one would like to allow variationin both conductivity and geometry and determine theinfluence of these variations on the forward solution. Toaddress the question of the role of anisotropy of theskeletal muscle described by Klepfer et al. [7] wouldthen require an additional parameter in the variationalanalysis. Conceptually, all this is possible using our

approach—conversely, the computational cost of any ofthe more traditional approaches of sensitivity analyseswould become all the more insurmountable if one wereto vary all the relevant parameters simultaneously.

Orientation of the epicardial potentials greatly affectedthe spatial characteristics of both the mean potentialsand especially their standard deviation. Figures 3 and4 illustrate these effects for epicardial potentials ro-tated by approximately 90, as shown in Panel (a) ofboth figures. The effect of rotation on mean potentialwas predictably a rotation of the same pattern, slightlydistorted by the asymmetric shape of the torso crosssection. The effect on standard deviation was, however,markedly more complex, indicating a more complexrelationship between potential distribution and placementof the regions of different conductivity. At this point, it isdifficult to draw many general conclusions about this re-lationship but it does appear that the organ conductivitiesimpact the potentials most significantly when they arelocated in close proximity to regions of large amplitudesof epicardial potentials. Such regions of large sourcestrength induce large current flow and one can expectthis flow to change most in regions of large conductivitychange. In addition, we noted that the regions of greateststandard deviation did not necessarily occur at the organboundaries or even within the large organs, but rathernear the torso boundary itself. Such findings suggest aform of remote effect, i.e., changes in conductivity inone region alter most strongly the potentials in anotherregion, perhaps because in these regions, the currenthas already passed through large regions of changingconductivity and the cumulative effects of deflection andattenuation have grown.

We also observed that the combined effect of con-ductivity variations in two different organs is not ad-ditive, and thus cannot be predicted by the individualstochastic behavior of single organ regions. We foundthat standard deviation for combinations of stochasticorgans was typically larger than for the individual organs,suggesting that in concert, errors are compounded. More-over, although some features of the standard deviationdistribution from combined variation were present in oneor the other of the equivalent maps of the individualvariations, there were also unique features only visiblein the combined distribution. These findings emphasizethe importance of quantifying the collective (rather thanindividual) impact of organ conductivities upon the fullmodel and further underscore the value of the gPC-SGmethod in enabling such forms of analysis.

The studies presented here were based on a two-dimensional simplification of a three-dimensional prob-lem. The gPC-SG approach is not intrinsically limitedto two dimensions and for this forward problem extendsnaturally to three, as outlined briefly in Section II. The

10

simplification from 3 to 2 dimensions also necessitatescare in interpreting the findings as they apply to thisproblem and especially the importance of extendingthem to a full, three-dimensional model. One obviouseffect of the simple model was that computed torsosurface potentials were larger than what one wouldexpect from a three-dimensional model. This is clearly aresult of the reduced torso volume relative to the extentof the epicardial sources, which decreases the electricalload on the source potentials and thus increases thepotential amplitude throughout the cross section. Onecannot reliably speculate upon the equivalent change inamplitude of standard deviations in a three-dimensionalmodel, although the linearity of the problem suggests thelikelihood of similar scaling. We do note that the patternsvisible in the cross section generally projected more orless radially to the outer boundary (the torso surfacecontour). However, the effect of a three-dimensionalmodel on spatial distributions of standard deviation isdifficult to anticipate and awaits the results from on-going studies in our laboratory with three-dimensionalimplementations of this approach.

While we present a gPC-SG analysis of the influenceof conductivity on a specific problem in electrocardiogra-phy, this approach can be extended to a number of otherbiological problems, including that of source localizationin the brain from extracranial measurements of electricpotentials or magnetic fields. Technical hurdles havepreviously made this sort of analysis prohibitive due tothe large number of samples necessary or the complexityof implementation. The gPC-SG approach provides aframework capable of overcoming these hurdles. Thecomputational tractability of gPC-SG also allows forinvestigating the effects of multiple stochastic parametersor even multiple random-dimensional stochastic param-eters in a reasonable amount of time.

ACKNOWLEDGMENT

The authors would like to thank Dr. Dongbin Xiu ofPurdue University (USA) for his generous help intro-ducing us to generalized polynomial chaos, Dr. TobiasPreusser of University of Bremen (Germany) for hiscomments concerning presentation of the material, andMr. Seungkeol Choe for providing the two-dimensionalfinite element solver used in this work. The authors alsoacknowledge the computational support and resourcesprovided by the Scientific Computing and Imaging In-stitute. This work was funded by a University of UtahSeed Grant Award, NSF Career Award (Kirby) NSF-CCF0347791, and the NIH NCRR Center for Integrative

Biomedical Computing (www.sci.utah.edu/cibc), NIHNCRR Grant No. 5P41RR012553-02. Support for theacquisition of the data used within the simulation studiescame from the Nora Eccles Treadwell Foundation.

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Sarah E. Geneser Sarah E. Geneser receivedher MA degree in mathematics and is cur-rently working toward her PhD degree incomputer science at the University of Utah.She is a member of the Scientific Comput-ing and Imaging Institute at Utah and herresearch interests lie in scientific computing.

Robert M. Kirby Robert M. Kirby receivedhis ScM degrees in computer science and inapplied mathematics and his PhD degree inapplied mathematics from Brown University.He is an assistant professor of computerscience at the University of Utah’s Schoolof Computing, an adjunct assistant professorin the Department of Bioengineering, andis a member of the Scientific Computingand Imaging Institute at Utah. His researchinterests lie in scientific computing and visu-

alization.

Robert S. MacLeod (S87M87) received boththe B.S. (’79) degree in engineering physicsand the Ph.D. (’90) degree in physiology andbiophysics from Dalhousie University, Hali-fax, N.S. Canada. He received the M.S. de-gree in electrical engineering from the Tech-nische Universitat, Graz, Austria, in 1985.He is an Associate Professor in the Bioengi-neering Department and the Department ofInternal Medicine (Division of Cardiology)at the University of Utah, Salt Lake City, and

an associate director of the Scientific Computing and Imaging Insti-tute, the Nora Eccles Harrison Cardiovascular Research and TrainingInstitute, and the Department of Bioengineering. His research inter-ests include computational electrocardiography (forward and inverseproblems), experimental investigation and clinical detection of cardiacischemia and repolarization abnormalities, cardiac imaging in atrialfibrillation, and scientific computing and visualization.