pre-printQuantification and visualization of uncertainty in cardiac forward and inverse problems with complex geometries is subject to various challenges. Specific to visualization is the observation that occlusion and clutter obscure important regions of interest, making visual assessment difficult. In order to overcome these limitations in uncertainty visualization, we have developed and implemented a collection of novel approaches. To highlight the utility of these techniques, we evaluated the uncertainty associated with two examples of modeling myocardial activity. In one case we studied cardiac potentials during the repolarization phase as a function of variability in tissue conductivities of the ischemic heart (forward case). In a second case, we evaluated uncertainty in reconstructed activation times on the epicardium resulting from variation in the control parameter of Tikhonov regularization (inverse case). To overcome difficulties associated with uncertainty visualization, we implemented linked-view windows and interactive animation to the two respective cases. Through dimensionality reduction and superimposed mean and standard deviation measures over time, we were able to display key features in large ensembles of data and highlight regions of interest where larger uncertainties exist

Burton, Brett M.

Potter, Kristin Carrie

English

The University of Utah: J. Willard Marriott Digital Library

Uncertainty Visualization in Forward and Inverse Cardiac ModelsBrett M Burton1,2, Burak Erem4, Kristin Potter1, Paul Rosen1,Chris R Johnson1, Dana H Brooks3, Rob S Macleod1,21Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, Utah, USA2Bioengineering Department, University of Utah, Salt Lake City, Utah, USA3Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, USA4Computational Radiology Laboratory, Children’s Hospital, Boston, MA, USAAbstractQuantification and visualization of uncertainty in car-diac forward and inverse problems with complex geome-tries is subject to various challenges. Specific to visualiza-tion is the observation that occlusion and clutter obscureimportant regions of interest, making visual assessmentdifficult. In order to overcome these limitations in uncer-tainty visualization, we have developed and implementeda collection of novel approaches. To highlight the utility ofthese techniques, we evaluated the uncertainty associatedwith two examples of modeling myocardial activity. In onecase we studied cardiac potentials during the repolariza-tion phase as a function of variability in tissue conductivi-ties of the ischemic heart (forward case). In a second case,we evaluated uncertainty in reconstructed activation timeson the epicardium resulting from variation in the controlparameter of Tikhonov regularization (inverse case). Toovercome difficulties associated with uncertainty visual-ization, we implemented linked-view windows and interac-tive animation to the two respective cases. Through dimen-sionality reduction and superimposed mean and standarddeviation measures over time, we were able to display keyfeatures in large ensembles of data and highlight regionsof interest where larger uncertainties exist.1. IntroductionForward and inverse problems in electrocardiographyprovide a means for studying the electrical properties andpatterns that arise within the heart and/or torso from the in-tegrated electrical activity of cardiac myocytes. In the for-ward case, electrical inferences are made based on givensource models and associated conducting media that con-tain these sources. Inverse solutions, in contrast, use re-mote observations to deduce the electrical function of thecardiac sources. Solutions to forward and inverse prob-lems, however, require several assumptions that, due tosuch characteristics as the complex nature of cardiac struc-ture or the inherent need to regularize the inverse, generateuncertainty in the results.To understand the uncertainty associated with cardiacforward and inverse problems, visualization techniquesare often applied; however, visualization of 3-dimensionaldata presents its own set of complexities. Obstruction andclutter can obscure or even hide regions of interest. Toovercome these difficulties in visualization and to thus bet-ter understand variations in the uncertainty of electrocar-diographic models, we have developed new 3-dimensionaltechniques to visually analyze simulation uncertainty.In order to explore uncertainty visualization, we consid-ered two electrocardiographic simulation cases—one forforward and one for inverse simulation. We first investi-gated the uncertainty associated with bidomain conductiv-ities in the static forward model of an ischemic heart. Inthe second case, we explored variability in reconstructedactivation times, computed by way of Tikhonov inverse, asa function of an unknown λ value. To more thoroughlyinvestigate the uncertainties in our models, we have de-veloped new visualization systems that overcome many ofthe difficulties of 3-dimensional rendering and provide ameans of interactive exploration of high-dimensional data.2. MethodsWhile both forward and inverse problems in electrocar-diography seek to capture electric activity from the heart,they are different in formulation and solution approach.Each, therefore, required unique and specific uncertaintyquantification and visualization approaches.2.1. Static ischemic forward modelTo generate data for the static ischemia model, we ex-tracted cardiac and ischemic zone geometries from exper-imentally induced ischemic studies using approaches de-scribed previously [1, 2]. Fiber directions, acquired byISSN 2325-8861 Computing in Cardiology 2013; 40:57-60. 57Table 1. Ratio applied to tensor conductivity values withinthe ischemic region to simulate the diseased state.Conductivity Ischemic Ratioσil 1/10σit 1/1000σel 1/2σet 1/4diffusion tensor imaging (DTI), were also assigned.In order to simulate ischemia, anisotropic conductivityvalues within the ischemic region were decreased (see Ta-ble 1) and a potential difference of 30 mV was applied tothe transmembrane potential, Vm, to represent the weak-ened activity during the plateau phase of the ischemic ac-tion potential, which corresponds to the ST segment of theECG [3]. A linear transition region (border zone) from thediseased to healthy tissue was also applied [4].Cardiac activity under these conditions was representedby the bidomain passive current flow equation [3]∇ · (σi + σe)∇Ve = −∇ · σi∇Vm, (1)where σi and σe represent the intracellular and extracel-lular conductivity tensors, respectively, and Ve representsthe unknown extracellular potentials.There have been many studies attempting to documentthe conductivity values of the heart [5, 6]; however, thesevalues remain elusive and, therefore, provide a source ofuncertainty for the model. We selected a range of con-ductivity values from the literature [7] onto which we ap-plied generalized polynomial chaos-stochastic collocationmethods (gPC-SC) [8]. Conductivity values were treatedas uniformly distributed, stochastic processes within theconductivity range provided. gPC-SC was used to reducethe stochastic governing equations into a finite set of de-terministic simulation parameters, reducing computationalcomplexity and allowing us to extract the mean and prob-ability density function of the resulting solution ensemble.2.2. Activation time inverse solutionsThe activation-based inverse problem aims to estimatethe spread of electrical activation over the epicardium frommeasurements of electric potentials on the body surfaceduring the QRS complex. Prior to computing the inversesolution, body surface potentials were generated by solv-ing a distinct forward model from the one described in theprevious section. This forward model was composed of anonlinear function, x(τ), which mapped a set of activationtimes to transmembrane potentials (TMPs) from a discreteset of electrical sources along the heart surface, and a lin-ear mapping, A, a matrix that computes body surface po-tentials from TMPs. This problem is not only non-linearbut also ill-conditioned and hence requires regularization.A simple, yet effective approach to regularization uses theTikhonov [9] method, resulting in the following non-linear,least squares (NLLS) optimization problem, which is non-convex [10]:τλ = argmin‖Axτ − b‖2Frob + λ‖Lxτ‖2Frob, (2)where b are the body surface potentials, L is a regulariza-tion matrix, and λ is a regularization parameter that con-trols the trade-off between data fidelity and solution regu-larity.We have previously used a convex relaxation of theabove problem and discovered uncertainty in the solutionspossibly due, in part, to the inherent sensitivity of the prob-lem to signal noise, geometric errors, forward model as-sumptions (e.g. source amplitudes), and inverse problemparameters (e.g. L and λ) [10, 11]. In this study, wefocused on the uncertainty generated by the selection ofλ. We chose 50 evenly-spaced values within the interval[0.09, 0.11] on which we performed a perturbation analy-sis. For each λ, a solution was found for the convex relax-ation, from which activation times were computed; fromthese values, we calculated statistics of the activation time(mean and standard deviation) for each source on the heartsurface. This spatial distribution of mean and standard de-viation of the activation time gave a measure of the uncer-tainty of the solution associated with λ selection.2.3. µView: myocardial uncertainty viewerTo enable uncertainty visualization for the volumetricischemic simulations we created a five-way, linked-viewtool (µView), examples of which are shown in Figure 1.Simulation results, acquired in Section 2.1, were com-bined and dimensionality reduction was applied to providea means of displaying large ensembles (high dimensional)data. Figures 1A and 1B show standard deviations and iso-value separation, respectively as two forms of dimension-ality reduction used to visualize these data. Other methodswere also employed to explore the data including: mean,minimum and maximum isosurfaces, and clustering basedon several correlation metrics. Mean and standard devi-ation visualizations applied simple statistical measures ateach node of the simulation with increased intensity rep-resenting increased values. Isovalue separation scannedeach node of the simulation set and determined how manynodes were above (red) and below (blue) the assigned iso-value. The spectrum of color between these two extremareflected nodes that had values, within the solution ensem-ble, that were both above and below the threshold. Mini-mum and maximum isosurfaces use the isovalue results tohighlight only the surfaces produced by the isovalue ex-trema. Clustering applied k-means methods to bin similar 58Figure 1. Uncertainty Visualization of the Cardiac Ischemia Forward Model. (A) Standard deviation and (B) isolvaluerendering of cardiac forward model show regions of interest near the ischemic zone (outlined in pane 1 ). The five-waylinked view contains the following viewing modalities: 1 volumetric, 2 2D parameter space, 3 - 5 2D slicing planes.nodes together. Similarities were determined by L2 norm,Pearson’s correlation, or histogram difference.To overcome difficulties associated with 3D rendering,the following visual techniques were applied: volume ren-dering (Figure 1A 1 ), a 2D view of parameter space ( 2) and three orthogonal 2D slicing planes (panes 3 - 5 ).Volume rendering and 2D slicing planes are standard vi-sualization techniques used to display data values at eachpoint within the myocardium. The parameter space view,in order to reduce the high dimensional data while preserv-ing important features, uses principle component analysisto contract the solution space into a 2D representation – al-lowing the user to explore features not otherwise visible inthe spatial domain. The shape of the resulting image is ar-bitrary and is a result of the selected principle components.For more information on the technical aspects of how thesemethods are applied, please see our prior work [12].2.4. Uncertainty animationStatic isochronal maps of activation times are the mostcommon visualization method for activation-based simu-lations of the spread of excitation in the heart. Such com-pression of an entire heart beat into a single image is oneof the advantages of activation mapping that supports itsutility in research and clinical applications. However, inthe setting of visualizing uncertainty, the need to view ad-ditional parameters and their variation over both time andparameter space provides new challenges. To address thischallenge, we have expanded the static, single image ofactivation to become an interactive animation over time.Similar to visualizing an animation of the moving activa-tion wavefront, each frame of our animation shows spa-tial regions that are within one standard deviation of thecurrent mean activation time and allows the user to moveforward and backward in time as well as adjust view pa-rameters. Figure 2 shows a single time instant from suchan animation in which regions in yellow correspond to ar-eas with low standard deviation values, i.e., represent themean activation time values. Other regions, spanning fromred to purple, display larger values of the standard devia-tion and indicate the spatial distributions of uncertainty forthis particular time instant. Examples of this uncertaintyanimation can be found at http://www.sci.utah.edu/˜kpotter/research/heartActivation/3. ResultsVisualizing the results of varied conductivity parametersin the forward problem and the regularization parameter ofthe inverse problem allowed us to identify regions of in-terest within simulation results. In the ischemic forwardmodel, isosurfacing, min/max isosurfacing and clusteringFigure 2. Spatial Distribution of Activation Times andStandard Deviations. Animations of uncertainty in the spa-tial distribution of activation times on heart surfaces at 30ms. Color maps show the spatial location and the differ-ence between the present time and the mean activation time(within one standard deviation). 59allowed us identify regions near the ischemic zone by scan-ning the solution space of the simulation. Though inde-pendent of the physical space, these methods were able toscan the solution space and provide the viewer with a senseof uncertainty with respect to a specified isovalue or num-ber of bins. By projecting these findings onto the physicalspace we were able to observe at what location, the cho-sen solution parameters differed. Standard deviation val-ues were able to illustrate regions of uncertainty, within theareas of interest, as a sense of a range of variance withinthe solution ensemble.By animating the activation based inverse solution, wewere able to identify regions on the surface of the heart thatexhibited higher temporal and spatial uncertainty. Duringa single frame of the animation (such as is seen in Figure2) the standard deviation in some regions span a small dis-tance (1/10 of the LV circumference) where other regionsshow spatial variation as large as 1/4 of the LV circumfer-ence. The temporal discrepancy in activation times withinthese spatial regions is as high as 36.4 ms.3.1. DiscussionBy using several uncertainty visualization techniques,we were able to highlight regions of interest in ischemicforward simulations and activation-based inverse solu-tions. Questions stemming from discrepancies in experi-mentally measured conductivity values for cardiac tissuegenerated small uncertainties in the forward solution. Re-gions in or near the ischemic zone exhibited greater stan-dard deviations, while regions near the epicardium (whereDTI-defined fiber directions exhibited more random direc-tional behavior) had wider discrepancies between min andmax isosurfaces (not shown). These results remain incon-clusive and will require further investigation. Likewise,for the inverse case, some spatial regions were shown toexhibit higher standard deviations than others based on dif-ferent, user selected regularization parameters, λ, that mer-its additional study.AcknowledgmentsThis project was supported by grants from the NationalCenter for Research Resources (5P41RR012553-14) andthe National Institute of General Medical Sciences (8 P41GM103545-14) from the National Institutes of Health andby Award No. KUS-C1-016-04, made by King AbdullahUniversity of Science and Technology (KAUST). Experi-ment data collection was funded by the Nora Eccles Tread-well Foundation at the Cardiovascular Research and Train-ing Institute.References[1] Shome S, MacLeod R. Simultaneous high-resolution elec-trical imaging of endocardial, epicardial and torso-tank sur-faces under varying cardiac metabolic load and coronaryflow. In Functional Imaging and Modeling of the Heart,Lecture Notes in Computer Science 4466. Springer-Verlag,2007; 320–329.[2] Aras K, Shome S, Swenson D, Stinstra J, MacLeod R.Electrocardiographic response of the heart to myocardial is-chemia. In Computers in Cardiology 2009. 2009; 105–108.[3] Hopenfeld B, Stinstra J, MacLeod R. Mechanism for STdepression associated with contiguous subendocardial is-chemia 2004;15(10):1200–1206.[4] Swenson D, Stinstra J, Burton B, Aras K, Healy L, ,MacLeod R. Evaluating the effects of border zone approx-imations with subject specific ischemia models. In DoesselO, Schlegel WC (eds.), World Congress on Med. Phys. andBiomed. Eng., volume 25/IV. Heidelberg: Springer, 2009;1680–1683.[5] Plonsey R, Barr R. A critique of impedance measurementsin cardiac tissue 1986;14:307–322.[6] Pollard A, Barr R. A biophysical model for cardiacmicroimpedance measurements. Jun 2010;298(6):H1699–H1709.[7] Johnston P, Kilpatrick D. The effect of conductivity val-ues on ST segment shift in subendocardial ischaemia 2003;50(2):150–158.[8] Xiu D. Efficient collocational approach for parametric un-certainty analysis. Comm Comp Phys 2007;2:293–309.[9] Tikhonov A, Arsenin V. Solution of Ill-posed Problems.Washington, DC: Winston, 1977.[10] Erem B, van Dam P, Brooks D. A convex relaxation frame-work for initialization of activation-based inverse electro-cardiography. In NFSI & ICBEM 8th International Sympo-sium on IEEE, 2011. 2011; .[11] Erem B, van Dam P, Brooks D. Analysis of the criteria ofactivation-based inverse electrocardiography using convexoptimization. In Conf Proc IEEE Eng Med Biol Soc. 2011;.[12] Rosen P, Burton B, Potter K, Johnson CR. Visualization forunderstanding uncertainty in the simulation of myocardialischemia. In The 3rd International Workshop on Visualiza-tion in Medicine and Life Sciences. 2013; .Address for correspondence:Brett M BurtonPO Box 171344Salt Lake City, UT 84117 bburton@sci.utah.edu 60

Uncertainty visualization in forward and inverse cardiac models

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Uncertainty visualization in forward and inverse cardiac models

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