deriving and visualizing uncertainty in kinetic pet modeling

Eurographics Workshop on Visual Computing for Biology and Medicine (2012)T. Ropinski, A. Ynnerman, C. Botha, and J. Roerdink (Editors)

Deriving and Visualizing Uncertaintyin Kinetic PET Modeling

Khoa Tan Nguyen1 and Alexander Bock1 and Anders Ynnerman1 and Timo Ropinski1

1Linköping University, Sweden

AbstractKinetic modeling is the tool of choice when developing new positron emission tomography (PET) tracers forquantitative functional analysis. Several approaches are widely used to facilitate this process. While all theseapproaches are inherently different, they are still subject to uncertainty arising from various stages of the mod-eling process. In this paper we propose a novel approach for deriving and visualizing uncertainty in kinetic PETmodeling. We distinguish between intra- and inter-model uncertainties. While intra-model uncertainty allows usto derive uncertainty based on a single modeling approach, inter-model uncertainty arises from the differencesof the results of different approaches. To derive intra-model uncertainty we exploit the covariance matrix anal-ysis. The inter-model uncertainty is derived by comparing the outcome of three standard kinetic PET modelingapproaches. We derive and visualize this uncertainty to exploit it as a basis for changing model input parameterswith the ultimate goal to reduce the modeling uncertainty and thus obtain a more realistic model of the tracerunder investigation. To support this uncertainty reduction process, we visually link abstract and spatial data byintroducing a novel visualization approach based on the ThemeRiver metaphor, which has been modified to sup-port the uncertainty-aware visualization of parameter changes between spatial locations. We have investigatedthe benefits of the presented concepts by conducting an evaluation with domain experts.

1. Introduction

Positron emission tomography (PET) is one of the mostwidely used functional imaging modalities and it is appliedin medical research as well as during everyday diagnosis.Depending on the chosen tracer injected into a subject, PETcan be used, for instance, to visualize the brain activity or toassess the impact of a cardiovascular event by enabling theanalysis of the myocardium vitality. In a typical PET study,PET data is obtained over a period of time and is composedof various signals that represent the tracer concentration foreach time step. Kinetic PET modeling is based on mathe-matical models that enable isolating those components of thesignal which are of interest. By applying this model analy-sis it becomes possible to understand the physiology and thepathophysiology of metabolism. Thus kinetic PET model-ing paves the way for quantified PET analysis which enablescomparative studies and researching the correlation betweenPET uptake and signals obtained from other modalities. Un-fortunately the modeling process itself is tedious and time-consuming. Although the costs for developing a new tracerare in general lower than the costs for developing a newdrug, the fact that 39% of new drug candidates fail due to

inadequate pharmacokinetics [Wal04] suggests that effectivemodeling approaches are of great interest.

Like many other disciplines of medical research, kineticPET modeling nowadays is performed with dedicated ap-plication packages [CHP81, BBC∗98, BB97, MC01]. Thecurrent practice is, however, to chose one modeling ap-proach early on in the analysis process as the different ki-netic modeling approaches are supported as separate fea-tures. In this paper, we propose to assess the standard ki-netic PET modeling approaches in an integrated manner inorder to improve the modeling result. This allows us to de-rive inter-model uncertainty which is based on the differ-ent results of the standard modeling approaches. We fur-ther derive intra-model uncertainty, based on a single mod-eling approach, which is visualized in an integrated man-ner together with the inter-model uncertainty. To further fa-cilitate the user in reducing the modeling uncertainties, wepropose a visualization metaphor that is based on the The-meRiver metaphor [HHN00], and takes into account thespatial and temporal properties of the kinetic PET mod-eling process. By enabling to relate the kinetic model-ing parameters with each other this metaphor bridges the

© The Eurographics Association 2012.

K. T. Nguyen & A. Bock & A. Ynnerman & T. Ropinski / Deriving and Visualizing Uncertaintyin Kinetic PET Modeling

gap between the different modeling approaches. Thus themain contributions of this paper are:

• Intra- and inter-model uncertainty derivation based onstandard kinetic PET modeling approaches.

• Integrated visualization of micro and macro parameter un-certainty to combine and compare the results of standardkinetic PET modeling approaches.

The paper is structured as follows. In Section 2 we re-view the related work covering modeling systems and vi-sualization techniques. In Section 3 we provide necessarybackground for readers, who are not familiar with kineticPET modeling. The derivation of uncertainty is presented inSection 4. In Section 5 we describe in detail the proposeduncertainty-aware visualization technique as well as the de-sign decisions for the integrated visualization of micro andmacro parameter uncertainty. Section 6 presents the evalu-ation and the feedback of domain experts. The paper con-cludes in Section 7.

2. Related Work2.1. Visualization TechniquesIn the last years several general visualization techniques al-lowing representation of time-varying data sets have beenproposed. A thorough review of the field would be beyondthe scope of this paper, we refer to the survey presentedby Aigner et al. [AMM∗08] and focus on those techniquesdirectly related to our approach. In addition, we also referto [PWL97, JS03] for more detail information on differentuncertainty visualization techniques for volumetric data sets.The most basic techniques for time-varying data, i. e., linegraphs or sequence charts, are in fact several hundred yearsold [MMHE11]. While most of the views incorporated inour visualization can be considered as standard visual rep-resentations, we also propose a novel view linking abstractand spatial representations. This view is based on the The-meRiver visualization metaphor [HHN00] which has beenoriginally proposed to visualize variations over time in doc-ument collections. The metaphor is similar in spirit to linecharts but allows a better correlation with external events.In contrast with the extension we propose in this paper theoriginal metaphor does not allow to be linked to spatial en-tities. The same is true for the widely used parallel coordi-nate display [Ins85]. Huaiqing et al. [HLX11] showed that itis possible to include an uncertainty visualization into bothparallel coordinates and star glyphs by introducing an ad-ditional dimension to the respective plot. Recently Malik etal. [MMHE11] proposed a visual analytics system that al-lows exploration of time correlations by taking into accounta spatial component. While this system puts more emphasison abstract data, related work more targeted towards scien-tific visualization also exists. Oeltze et al. [ODH∗07] pre-sented an interactive visual analysis approach for the eval-uation of perfusion data. The authors employed multipleviews for high dimensional data visualization and combinedstatistics with brushing and linking for the analysis of dif-

fusion data. A more general approach for integrating ab-stract and spatial data has been proposed by Balabanian etal. [BYV∗08, BVG10]. They integrate renderings of sub-volumes of interest into a hierarchy visualization derivedfrom the same data. Thus, they are able to show the compo-nents of a complex system in its hierarchical context. Whilethis approach also targets towards the integration of abstractand spatial representations, it differs from our approach in asense that the temporal component was not considered.

2.2. Kinetic Modeling SystemsThere are several software packages supporting the kineticPET modeling process, e. g., KMZ [BB97], BLD [CHP81],SAAM/SAAM II [BBC∗98], and PMOD (PMOD Technolo-gies Ltd., Zurich, Switzerland). The KMZ system separatesthe graphical user interface and the numerical calculationunit for extensibility. This system is, however, limited tothe application of a rather low number of pre-defined ki-netic model structures, and does not support model analy-sis and parameter modification as an integral part. PMODis a flexible successor of KMZ and commercially avail-able on various platforms. This software package allowsdifferent types of modeling, e. g., general kinetic model-ing or pixel-wise modeling. It also supports image fusionand surface-based volumetric data visualization. Muzic etal. [MC01] have proposed another flexible PET compart-mental modeling framework named COMKAT. It providesthe ability to implement arbitrary user-specified modelsthrough graphical and command-line interaction. KIS sys-tem [HTW∗05] is an Internet-based kinetic imaging sys-tem for microPET. In comparison with other existing soft-ware systems, KIS incorporates an educational aspect aid-ing users in better understanding the modeling process.Even though this package was initially designed for mi-croPET only its design allows handling general imagingstudies in tracer kinetics and pharmacokinetics in other an-imal models as well as in humans. While all these systemsare widely accepted and have thus often been used fortracer modeling, they differ from the approach presented inthis paper in mainly two ways. First, although the graph-ical modeling approaches are inherently visual, visualiza-tion does not play a major role in these systems. Second,the concurrent integration of multiple models and the abil-ity to compare uncertainty of the output arisen during theprocess through visualization is not an integral part.

3. Kinetic PET ModelingKinetic PET modeling helps to isolate the components ofinterest from the obtained PET data. As a result, the analysisof the underlying mathematical model provides insight intothe kinetic behavior of the tracer under investigation.

Kinetic PET modeling techniques may be divided intotwo groups: model-driven and data-driven methods. Themodel-driven techniques require a model structure as a-priori information (see Figure 1). In contrast data-driven al-gorithms obtain this information directly from the kinetic



Figure 1: Three-tissue (or four-compartment) compartmentmodel. This model consists of components of arterialplasma, Cp(t), free ligand in tissue, C1(t), specific binding,C2(t), non-specific binding, C3(t), and six kinetic micro pa-rameters, K1,k2 . . . , k6, describing the rate constants.

data, e. g., a dynamic PET (dPET) scan. Figure 1 shows athree-tissue (or four-compartment) compartment model. Thefirst compartment, Cp, is the arterial blood. This compart-ment is also known as the input function. From the arte-rial blood, the radioligand passes into the second compart-ment, C2, known as the free compartment. The third com-partment, C3, is the region of nonspecific-binding which weare usually interested to observe. The fourth compartment,C4, is a nonspecific binding compartment that exchangeswith the free compartment, C1. K1,k2 . . . , k6 are the kineticmicro parameters describing the constant transfer rates be-tween compartments. It is worth noting that K1 is writtenin uppercase since it differs, with respect to several proper-ties, from the other micro parameters, e. g., it is subject toa different measuring unit.

The following system of ordinary differential equations(ODEs) is the mathematical representation of the compart-ment model in Figure 1.

dC1/dt = K1Cp(t)+k5C3(t)+k4C2(t)−(k2+k3+k6)C1(t)

dC2/dt = k3C1(t)− k4C2(t) (1)

dC3/dt = k6C1(t)− k5C3(t)

where Cp(t), C1(t), C2(t), and C3(t) are radioactivity con-centrations at time t for each compartment. It should benoted that a compartment model is always analyzed with re-spect to a region of interest (ROI) which could, in the mostextreme cases, be the whole body or a single voxel insidean organ but is more often a single organ. When several or-gans are considered we refer to the model as a multi-levelcompartment model. Quantitative analysis of a compartmentmodel is done through solving its mathematical represen-tation. Depending on the given inputs different type of in-formation can be obtained by solving the system of ODEs.For instance the mathematical prediction of the tissue ac-tivity (time-activity curve - TAC) can be obtained from agiven input function, Cp(t), and micro parameters. This iscalled the forward problem. On the other hand kinetic mi-cro parameters can be estimated from a given input func-tion, Cp(t), and measured TACs. This is called the back-ward problem and can be solved by using different leastsquares fitting techniques such as linear least squares, non-linear least squares, weighted integration [CHG86], general-ized linear least squares [FHWH96] or basis function tech-niques [KHI85, CJ93, GLHC97]. In both cases the arterial

Figure 2: Dependency of the computed kinetic parameters.In a given ROI, based on the input function and the mea-sured TACs, the macro parameter can be obtained directlyfrom the Logan or the Patlak plot. The compartment model-ing allows us to obtain micro parameters from which againmacro parameters can be derived.

radioactivity concentration, Cp(t), known as the input func-tion must be provided. There are two approaches to obtainthe input function: invasive and non-invasive methods. Al-though the non-invasive methods provide less accurate mea-surements, they are usually preferred [ZFCL∗11]. Recent de-velopment of reference tissue models, e. g., [WCI00], how-ever, help to avoid the blood sampling requirement.

Logan [Log00] plots and Patlak [PB85] plots are the twostate-of-the-art data-driven modeling methods. These tech-niques employ a transformation of the data such that a linearregression of the transformed data yields the macro param-eter of interest. Despite their inherent simplicity they canreliably provide good enough output. As a result, they arefrequently used in practice. Nevertheless these techniquesare biased due to statistical noise [SL00] and requires spe-cial handling when the resulting plot becomes non-linear. Itshould be noted that these macro parameters can also be de-rived from the micro parameters described above. For in-stance the net influx of the tracer obtained from the Pat-lak plot, Ki, can be derived from the micro parameters inthe compartment model in Figure 1 as follows: Ki =

K1×k3k2+k3

.In comparison with the micro parameters, macro parame-ters represent the observed data rather than the individualparameter. Spectral analysis [CJ93] is another data-drivenmodeling approach which characterizes the system impulseresponse function (IRF). In this approach the non-negativeleast square fitting is used, and the macro parameters arecalculated as functions of the IRF. Although these tech-niques do not provide information about the underlyingmodel structure, spectral analysis provides the number of tis-sue compartments evident in the data.

4. Deriving Intra- and Inter-Model UncertaintyIn this section we explain the variety of kinetic parameterscomputed and analyzed by our system, as well as the deriva-tion of the associated intra- and inter-model uncertainty.

The dependency of modeling parameters is shown in Fig-ure 2. Micro parameters can be estimated from the com-partment model. Macro parameters can be obtained eitherby using the Logan or Patlak method or through the deriva-tion from the micro parameters. These different sources of



macro parameters enable us to compute the inter-model un-certainty, while the covariance analysis can be used to deriveintra-model uncertainty from the compartment model.

In the Patlak method the concentration of the tracer at timet after injection is described as follows

C(t) = λ ·Cp(t)+Ki

∫ t

0Cp(τ)dτ (2)

where C(t) represents activity in the tissue as measured bythe PET scanner, Cp(t) is the plasma activity, Ki is the netrate of the tracer influx into the tissue, and λ is the distri-bution volume of the tracer. By dividing both sides of Equa-tion 2 by Cp(t), the macro parameter, Ki, can be estimated asthe slope in the Patlak plot through the use of linear regres-sion analysis. The Logan technique works in a similar fash-ion. For instance, the macro parameter which is the volumedistribution of the tracer is obtained through a regression ap-proach on the transformed data.

In compartmental modeling, micro parameters can be ob-tained by solving the system of ODEs. This is an optimiza-tion problem with the goal to minimize the weighted residualsum of squares (WRSS) defined as

WRSS(λ ) = ∑Nj=1 wi

(Cobs(t j)−C(t j)

)2

= ∑Nj=1 wi

(Cobs(t j)− e−β t j

)2 (3)

where β represents the kinetic parameters to be estimated,wi is the weighting scheme, N is the number of measure-ments, and Cobs(t j) and C(t j) represent the measured dataand the model prediction of the tracer activity at time t j re-spectively. It is worth noting that the concentration of thetracer is a nonlinear function of the parameter β over time:C(t j) = e−β t j . In order to calculate the associated uncer-tainty of the estimates of the kinetic parameters the bootstraptechnique [TSB∗98] is commonly used. This technique is,however, computational expensive. In this work we proposethe analysis of the covariance matrix of the estimates to cal-culate the certainty level of the estimated kinetic parameters.In general the covariance matrix of the estimates of the ki-netic parameters is symmetric. Particularly the diagonal ele-ments are the variances of the parameter estimates which arepositive by definition and the off-diagonal elements are thecovariances which may be either positive or negative. Find-ing the optimal result from the minimization of the Equa-tion 3 is an iterative process. During this process the infor-mation matrix constructed by the weighting scheme wi, andthe derivative of the function used to fit the data representsthe minimum achievable covariance matrix as in Equation 4

Cov(p̂)≥ M−1(p̂) (4)

where p̂ is the estimates of the kinetic parameters, Cov(p̂) isthe covariance matrix of the estimates, and M is the informa-tion matrix. The information matrix approaches the covari-ance matrix of the estimates for a large sample size of tissue

Figure 3: Workflow to derive the kinetic parameters and theirassociated uncertainties. Input data is fed into three differentkinetic modeling techniques which are used to derive kineticmicro and macro parameters. These parameters are analyzedand the modeling uncertainty is derived based on the pa-rameter variation. Thus the uncertain output data serves asa basis for changing model input parameters which can beinteractively changed as depicted by the red arrows.

measurements and/or decreasing variance of the measure-ment error. Moreover the estimates also approach Gaussiandistributions. It is worth noting that despite the fact that thereare several sources of noise in PET imaging, the additionof these sources of errors tends to form a Gaussian distribu-tion [ITY∗98]. Thus, for an optimal weighted nonlinear leastsquares fitting, the covariance matrix of the estimates needsto reach the lower bound fined by the information matrix.As a result, the analysis of the covariance matrix of the esti-mates reveals the precision of the estimates which representsthe intra-model uncertainty.

Since the macro parameters, obtained from either thePatlak plot or the Logan plot, represent the observed datarather than the individual parameters, the relative compar-ison between these values and the corresponding deriva-tion from the micro parameters can provide useful in-formation. We refer to the differences of macro param-eters based on these three sources as inter-model uncer-tainty. For a thorough understanding of intra- and inter-model uncertainty, an appropriate visualization techniquesis essential, which also allows for relating the intra-modeluncertainty based on the micro parameters to the inter-model uncertainty based on the macro parameters.

5. Uncertainty-Aware Kinetic Parameter VisualizationWe motivate the proposed visualization techniques by giv-ing an overview of the underlying modeling workflowas illustrated in Figure 3. The three state-of-the-art ki-netic PET modeling approaches, Logan plotting, Patlakplotting, and compartment modeling as are employed toprovide the estimates of the kinetic parameters. By vi-sually depicting the resulting intra- and inter-model un-certainty ranges we enable the user to modify the mod-ifiable model input parameters (red arrows in Figure 3)with the goal to minimize the uncertainty and thus al-low more realistic modeling results. It is worth mention-ing that depending on the underlying problem solving ap-proach, the parameters being modifiable may vary.



Figure 4: The uncertainty-aware visualization technique de-picts differences in the micro and macro parameters. Thecolor of the bands connecting the kinetic parameters be-tween different ROIs is selected based on the pre-attentiveprocess. Uncertainty is depicted by using blurring while cer-tain values are emphasized through edge detections.

5.1. Kinetic Parameters Uncertainty VisualizationTo be able to obtain more realistic modeling results it isessential to depict the modeling parameters as well as theintra- and the inter-model uncertainties. Therefore we pro-pose an uncertainty-aware visualization technique as illus-trated in Figure 4. The proposed visualization technique isbased on the ThemeRiver metaphor [HHN00] which hasbeen extended to visualize the parameter changes and theassociated uncertainties dependent on spatial locations, i. e.,micro and macro parameters together with the intra- andinter-model uncertainties computed for different ROIs. EachROI is represented by a box that is subdivided based on theassociated micro and macro parameters. Based on the con-nectivity in the underlying compartment model the boxes areconnected through bands which depict changes in kinetic pa-rameters as well as the associated uncertainty. The colorsof these bands have been chosen to comply with the rulesof pre-attentive perception [The92]. Initially macro param-eters are in focus so they are colored in red, while the mi-cro parameters, which are of interest in a later processingstage, are colored in shades of blue. To allow a better com-parison of the micro and the macro parameters the bandsare drawn such that the width of the curves is not affectedby its curvature.

Besides the parameter changes the bands are also used todepict the uncertainties of the currently selected ROIs. Wehave chosen to keep the uncertainty depiction simple in or-der to reduce visual clutter. As the silhouette perception isan essential part of the shape perception process, silhouettesare a powerful mechanism in visual communication. There-fore in our case, to depict the underlying uncertainty we havedecided to use parameter uncertainty as a modulation factorfor dissolving or emphasizing the segments’ shapes by blur-ring or edge enhancement. It should be noted that dependingon the connected ROI representations, the uncertainty alonga segment may vary. To realize various degrees of dissolu-tion and emphasize, we combine blurring and edge detec-

Figure 5: The compartment model (left) for three differentorgans can be considered as a state diagram, where the statechanges over time. The TACs (right) are plotted over timeafter injection.

tion techniques realized as post processing of the connec-tion segments. When the uncertainty to be depicted is be-low the uncertainty threshold umin, the regions of the seg-ment are emphasized by adding a silhouette edge indicatingcertain values. All regions subject to an uncertainty exceed-ing umin are vertically blurred based on the degree of uncer-tainty. We have to consider two things. First, the degree ofblur needs to be based on the uncertainty, and second, theblurring should not affect the neighboring segment. To reachthe first goal, we allow the user to define a maximum degreeof blur specified through a maximal blur kernel size gmax aswell as an upper uncertainty threshold umax. Based on thesevalues and the current uncertainty u, the kernel size gdim forthe Gaussian kernel used to achieve the desired effect canbe derived by gdim = ((u−umin)/(umax −umin))×gmax. Toreach the second goal, and avoid bleeding effects betweenadjacent segments, the Gaussian filter is adapted in such away that for samples lying outside the current segment a de-saturated version of the color of the current segment is used.Thus we are able to achieve effects as shown in Figure 6.It is worth noting that since the uncertainty information wasnot taken into account in the image on the left in Figure 6,the color is more saturated.

To depict the spatial position of the associated ROIs, theevents originally connected along the time axis are now lo-cated in the image plane in a way that enables an easy asso-ciation with the ROIs shown in the 3D view. Thus the layoutof the boxes is associated with different ROIs in an associ-ated 3D view (see Figure 7 (top left)). This association is im-portant, as it shows how a tracer behaves in different organsand allows the domain expert to gain further insight into thekinetic behavior of the tracer under investigation.



Figure 6: A high degree of uncertainty is depicted by using vertical blurring: no uncertainty depiction applied (left), uncertaintydepiction without edge detection (middle) and uncertainty depiction with edge detection (right).

The proposed layout calculation algorithm contains twomain steps and is executed whenever the ROI positions inthe 3D view are changed, e. g., when the camera is rotatedor zoomed. In the first step the ROI representations are posi-tioned such that overlap is avoided and a direct mental link-ing with the original ROI positions in the 3D view can beestablished. In the second step we perform a relaxation ofthe position of the boxes in the horizontal and vertical direc-tion, which maximizes the distances between these objects,and thus allows us to have sufficiently large connection be-tween them. Thus, it improves the connectivity in the initiallayout. The last step of the layout algorithm is performed inan animated manner and only when the user is not changingthe positions of the ROIs in the 3D view. Thus we can en-sure a coherent association with the ROI positions in the 3Dview after the first step and a coherent transition to a layoutoptimized for the presented representation.

5.2. Linking Kinetic Parameters with Spatial andTemporal Attributes

While our uncertainty derivation and visualization meth-ods provide further insights, they must be seen as an ad-dition to previous visualization techniques and should beapplied in a combined manner with these. As most of thedata in the kinetic PET modeling process have both, spatialand temporal attributes, these properties need to be consid-ered during the combination.

The TACs are, for instance, an abstract representationshowing the tracer activity over time (see Figure 5 (right)).They are associated with a spatial position but are usuallyrepresented by a 2D plot, which has no spatial components.While the structure of the compartment model is usually pre-sented in a 2D space, the compartment model itself inher-ently represents time through which the tracer transits be-tween the compartments (see Figure 5 (left)). Moreover acompartment model is always associated with a spatial po-sition given by the analyzed ROI. In contrast to the modelstructure the measured kinetic data given by the 4D PETis inherently spatial. Since PET is a rather low resolutionmodality, it suffers from partial volume effects. Thus PETis commonly fused with a co-registered CT scan to allowbetter spatial relations communication. As a result, the pro-posed integrated visualization combines these conventional

2D and 3D views to help users to relate the uncertainty, theTACs, and the model structure to their spatial anchor.

From the requirement analysis above we make use of themental linking approach, which has shown to be a very use-ful when dealing with multiple visual representations in sev-eral application scenarios [Tor03, GRW∗00, Dol07, BBP10,KVDG∗10]. It is worth noting that linking in our case ismore difficult since the analyzed data, e. g., 4D PET scans, isinherently spatial while the analysis techniques are primarilybased on abstract representations, e. g., model structures andregression analysis plots.

Figure 7 illustrates the organization of the different viewsin an integrated visualization. The top left view is a multi-volume rendering view which has been enriched by depict-ing the ROIs through silhouette enhancement techniques.The view operates on three volumetric data sets: the PETdata, the co-registered CT data used to provide the spa-tial context, and a segmentation volume which encodesthe ROIs. Generation of this segmentation volume is notpart of our system. The 3D view is linked to standard 2D

Figure 7: Our system employs multiple views supporting thekinetic modeling process. A multi-volume rendering view(top left) allows us to assess the spatial location of the inputfunction sampling positions as well as of the ROIs. 2D sliceviews (bottom left) enable to assess tracer uptake in moredetail and support repositioning of the image-derived inputfunction sample locations. The plot (bottom right) showsthe output of the Patlak graphical analysis. An uncertainty-aware widget (top right) reveals the modeling uncertainty.



slice representations arranged in the bottom left view. Whenthe user scrolls through the shown slices, the slice indica-tors (green and red rectangle) in the 3D view are updatedand indicate the current slice position. This allows an in-detail inspection of the relevant structures, while the 3Dview provides an overview of the spatial context. These 2Dviews allow a better interaction when performing selectiontasks when changing the image-derived input function lo-cations is necessary. The output of the Logan/Patlak plotis shown on the bottom right.

6. EvaluationTo investigate the benefits of the proposed uncertainty-awaremodeling concepts we have conducted an evaluation withthree domain experts from the Turku PET Centre, which areworking in nuclear medicine and conduct research related tokinetic PET modeling. The objective was to assess the users’ability to quickly perceive the uncertainty in the estimatesof the kinetic parameters and derive meaningful informationfrom the proposed visualizations.

In order to derive the uncertainty of the estimates we im-plemented a prototype software that supports compartmentalmodeling as shown in Figure 5. Spectral analysis is used toidentify the initial number of compartments that are evidentin the input data. The system also solves both the forwardand the backward problems. In addition, the Patlak plottingand the Logan plotting techniques are also integrated.

We prepared four data sets containing different levels ofsynthetically introduced uncertainty into the ground-truthdata, which lead to the different uncertainty in the estimatesof the kinetic parameters. Figure 8 illustrates the uncertainty-aware visualization of two sets of kinetic parameters withdifferent levels of uncertainty, whereby the uncertainty dis-played in the ROI on the right is higher for all parameters.As it can be seen, color saturation and silhouette thicknessvary dependent on the degree of uncertainty. We presentedthe visualizations of these four data sets to the three domainexperts. While the ROIs on the left are subject to the samedegree of uncertainty for each individual kinetic parameter,the ROIs on the right depict different sets of kinetic param-eters with varying uncertainty. The participants were askedto perform the following tasks:1. Organizing the visualizations in an increasing order of

overall level of uncertainty in ROIs on the right.2. Identifying the image in which the macro parameter Ki

has the highest level of uncertainty.3. Identifying the micro parameters that have the strongest

influence on the uncertainty of the macro parameter Ki.By using the proposed uncertainty-aware visualization tech-nique all three participants could quickly perceive the un-derlying uncertainty of the kinetic parameters. In task 1, allthe participants provided the correct increasing order of un-certainty level in ROIs on the right. We received one incor-rect answer for the task 2. Particularly, the participant se-lected the image containing the second highest level of un-

Figure 8: The uncertainty-aware visualization supports therelative uncertainty comparison between two different setsof kinetic parameters.

certainty. In task 3, the participants identified correctly themicro parameters that contribute most uncertainty to the de-rived macro parameter Ki. As we consider these results asvery positive feedback, the proposed uncertainty-aware vi-sualization technique seems to help to depict the uncertaintyin the estimates of the kinetic parameters, while at the sametime providing insights into the uncertainty contribution ofthe micro parameters to the derived macro parameter.

Apart from the task results, we also received very posi-tive comments from the domain experts. Among them thestatement, that the proposed visualization technique enablesto identify the uncertainty contribution of the micro param-eters to the derived macro parameters, which is not possiblewith the standard bar plot used to depict the kinetic param-eters. One domain expert also requested more informationabout the software and its availability.

When querying feedback regarding the uncertaintyderivation approach the domain experts stated that the pro-posed uncertainty derivation approach is comparable to thecommonly used bootstrap technique. As the bootstrap tech-nique is a statistical approach, the analysis of the covariancematrix is a viable alternative. Since the analysis of the co-variance matrix is done with an iterative weighted nonlin-ear least squares fitting, it requires less computation than thebootstrap approach.

7. Conclusions and Future WorkIn this paper we have presented another approach to thederivation of uncertainty in the standard kinetic PET mod-eling. By combining the results of different state-of-the-artPET modeling techniques, e. g., Logan plots, Patlak plotsand compartment modeling, together with sample-based anddata-derived input functions, we are able to compute intra-and inter-model uncertainties for kinetic modeling parame-ters. To communicate the parameters and their associated un-certainty a novel uncertainty-aware visualization techniquewas proposed and incorporated into an integrated visualiza-tion. While the uncertainty-aware visualization allows theuser to estimate the differences in the model output, the inter-active visual feedback in the integrated visualization enablesto adapt the input parameters in order to minimize the uncer-tainty. Furthermore the linked views in the integrated visu-alization also depict both the spatial and temporal nature ofthe kinetic data to assist the kinetic PET modeling process.

There are several interesting research opportunities for the



future. While the uncertainty-aware visualization techniqueenables users to compare the uncertainty in the estimatesof the kinetic parameters, the combination with a machinelearning techniques to assist the uncertainty reduction pro-cess is an interesting research direction. It could be possi-ble to allow a semi-automatic uncertainty reduction basedon the interplay between human and computer. Furthermorethe integration of histological findings into the proposed in-tegrated visualization approach would be an interesting nextstep towards more effective kinetic PET modeling.

AcknowledgmentsWe gratefully acknowledge the feedback and the evaluationfrom Vesa Oikonen and his colleagues from Turku PET Cen-ter, University of Turku.

References[AMM∗08] AIGNER W., MIKSCH S., MULLER W., SCHUMANN

H., TOMINSKI C.: Visual methods for analyzing time-oriented data.IEEE Transactions on Visualization and Computer Graphics 14, 1(2008), 47 –60. 2

[BB97] BURGER C., BUCK A.: Requirements and implementationof a flexible kinetic modeling tool. Journal of Nuclear Medicine 38,11 (1997), 1818. 1, 2

[BBC∗98] BARRETT P., BELL B., COBELLI C., GOLDE H., SCHU-MITZKY A., VICINI P., FOSTER D.: SAAM II: simulation, analy-sis, and modeling software for tracer and pharmacokinetic studies.Metabolism Clinical and Experimental 47, 4 (1998), 484–492. 1, 2

[BBP10] BUSKING S., BOTHA C., POST F.: Dynamic multi-viewexploration of shape spaces. In Computer Graphics Forum (2010),vol. 29, Wiley Online Library, pp. 973–982. 6

[BVG10] BALABANIAN J., VIOLA I., GRÖLLER E.: Interactive il-lustrative visualization of hierarchical volume data. In Proceedingsof Graphics Interface (2010), Canadian Information Processing So-ciety, pp. 137–144. 2

[BYV∗08] BALABANIAN J.-P., YSTAD M., VIOLA I., LUNDER-VOLD A., HAUSER H., GRÖLLER M. E.: Hierarchical volume vi-sualization of brain anatomy. In Proceeding of Vision, Modeling andVisualization (VMV 2008) (2008), pp. 313–322. 2

[CHG86] CARSON R., HUANG S., GREEN M.: Weighted inte-gration method for local cerebral blood flow measurements withpositron emission tomography. Journal of Cerebral Blood Flow &Metabolism 6, 2 (1986), 245–258. 3

[CHP81] CARSON R., HUANG S., PHELPS M.: BLD: a softwaresystem for physiological data handling and model analysis. In Pro-ceedings of the Annual Symposium on Computer Application inMedical Care (1981), American Medical Informatics Association,p. 562. 1, 2

[CJ93] CUNNINGHAM V., JONES T.: Spectral analysis of dynamicpet studies. Journal of Cerebral Blood Flow & Metabolism 13, 1(1993), 15–23. 3

[Dol07] DOLEISCH H.: SimVis: Interactive visual analysis of largeand time-dependent 3D simulation data. In Simulation Conference,2007 Winter (2007), IEEE, pp. 712–720. 6

[FHWH96] FENG D., HUANG S., WANG Z., HO D.: An unbiasedparametric imaging algorithm for nonuniformly sampled biomedicalsystem parameter estimation. IEEE Transactions on Medical Imag-ing 15, 4 (1996), 512–518. 3

[GLHC97] GUNN R., LAMMERTSMA A., HUME S., CUNNINGHAMV.: Parametric imaging of ligand-receptor binding in pet using asimplified reference region model. Neuroimage 6, 4 (1997), 279–287. 3

[GRW∗00] GRESH D., ROGOWITZ B., WINSLOW R., SCOLLAND., YUNG C.: WEAVE: A system for visually linking 3-D andstatistical visualizations, applied to cardiac simulation and measure-ment data. In Proceedings of the conference on Visualization (2000),IEEE, pp. 489–492. 6

[HHN00] HAVRE S., HETZLER B., NOWELL L.: Themeriver: visu-alizing theme changes over time. In IEEE Symposium on Informa-tion Visualization (2000), pp. 115–123. 1, 2, 5

[HLX11] HUAIQING H., LEI Y., XU Q.: Multidimensional uncer-tainty visualization with parallel coordinate and star glyph. Interna-tional Journal of Digital Content Technology and its Applications 5,6 (2011), 412–420. 2

[HTW∗05] HUANG S., TRUONG D., WU H., CHATZIIOANNOU A.,SHAO W., WU A., PHELPS M.: An internet-based kinetic imagingsystem (KIS) for MicroPET. Molecular Imaging and Biology 7, 5(2005), 330–341. 2

[Ins85] INSELBERG A.: The plane with parallel coordinates. TheVisual Computer 1, 2 (1985), 69–91. 2

[ITY∗98] IKOMA Y., TOYAMA H., YAMADA T., UEMURA K.,KIMURA Y., SENDA M., UCHIYAMA A.: Creation of a dynamicdigital phantom and its application to a kinetic analysis]. Kaku igaku.The Japanese journal of nuclear medicine 35, 5 (1998), 293. 4

[JS03] JOHNSON C., SANDERSON A.: A next step: Visualizing er-rors and uncertainty. IEEE Computer Graphics and Applications 23,5 (2003), 6–10. 2

[KHI85] KOEPPE R., HOLDEN J., IP W.: Performance comparisonof parameter estimation techniques for the quantitation of local cere-bral blood flow by dynamic positron computed tomography. Journalof Cerebral Blood Flow & Metabolism 5, 2 (1985), 224–234. 3

[KVDG∗10] KREKEL P., VALSTAR E., DE GROOT J., POST F.,NELISSEN R., BOTHA C.: Visual analysis of multi-joint kinematicdata. In Computer Graphics Forum (2010), vol. 29, Wiley OnlineLibrary, pp. 1123–1132. 6

[Log00] LOGAN J.: Graphical analysis of pet data applied to re-versible and irreversible tracers. Nuclear medicine and biology 27,7 (2000), 661–670. 3

[MC01] MUZIC R., CORNELIUS S.: Comkat: compartment modelkinetic analysis tool. Journal of Nuclear Medicine 42, 4 (2001),636. 1, 2

[MMHE11] MALIK A., MACIEJEWSKI R., HODGESS E., EBERTD.: Describing temporal correlation spatially in a visual analyticsenvironment. In Hawaii International Conference on System Sci-ences (2011), pp. 1 –8. 2

[ODH∗07] OELTZE S., DOLEISCH H., HAUSER H., MUIGG P.,PREIM B.: Interactive visual analysis of perfusion data. IEEE Trans-actions on Visualization and Computer Graphics 13, 6 (2007), 1392–1399. 2

[PB85] PATLAK C., BLASBERG R.: Graphical evaluation of blood-to-brain transfer constants from multiple-time uptake data. general-izations. Journal of Cerebral Blood Flow & Metabolism 5, 4 (1985),584–590. 3

[PWL97] PANG A., WITTENBRINK C., LODHA S.: Approaches touncertainty visualization. The Visual Computer 13, 8 (1997), 370–390. 2

[SL00] SLIFSTEIN M., LARUELLE M.: Effects of statistical noiseon graphic analysis of pet neuroreceptor studies. Journal of NuclearMedicine 41, 12 (2000), 2083. 3

[The92] THEEUWES J.: Perceptual selectivity for color and form.Attention, Perception, & Psychophysics 51, 6 (1992), 599–606. 5

[Tor03] TORY M.: Mental registration of 2D and 3D visualizations(an empirical study). In IEEE Visualization (2003), pp. 371–378. 6

[TSB∗98] TURKHEIMER F., SOKOLOFF L., BERTOLDO A., LUCIG-NANI G., REIVICH M., JAGGI J., SCHMIDT K.: Estimation of com-ponent and parameter distributions in spectral analysis. Journal ofCerebral Blood Flow & Metabolism 18, 11 (1998), 1211–1222. 4

[Wal04] WALKER D.: The use of pharmacokinetic and pharmacody-namic data in the assessment of drug safety in early drug develop-ment. British journal of clinical pharmacology 58, 6 (2004), 601–608. 1

[WCI00] WATABE H., CARSON R., IIDA H.: The reference tissuemodel: Three compartments for the reference region. NeuroImage11, 6 (2000), S12. 3

[ZFCL∗11] ZANOTTI-FREGONARA P., CHEN K., LIOW J., FUJITAM., INNIS R.: Image-derived input function for brain pet studies:many challenges and few opportunities. Journal of Cerebral BloodFlow & Metabolism (2011). 3


deriving and visualizing uncertainty in kinetic pet modeling

Documents