1182 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 5, MAY 2010

A Spatio-Temporal Deconvolution Methodto Improve Perfusion CT Quantification

Lili He*, Burkay Orten, Synho Do, W. Clem Karl, Avinish Kambadakone, Dushyant V. Sahani, and Homer Pien

Abstract—Perfusion imaging is a useful adjunct to anatomicimaging in numerous diagnostic and therapy-monitoring settings.One approach to perfusion imaging is to assume a convolutionrelationship between a local arterial input function and thetissue enhancement profile of the region of interest via a “residuefunction” and subsequently solve for this residue function. Thisill-posed problem is generally solved using singular-value decom-position based approaches, and the hemodynamic parameters aresolved for each voxel independently. In this paper, we present aformulation which incorporates both spatial and temporal corre-lations, and show through simulations that this new formulationyields higher accuracy and greater robustness with respect toimage noise. We also show using rectal cancer tumor images thatthis new formulation results in better segregation of normal andcancerous voxels.

Index Terms—Deconvolution, perfusion computed tomography(CT), singular value decomposition, spatial and temporal correla-tions.

I. INTRODUCTION

C ONVENTIONAL medical imaging, such as computed to-mography (CT), magnetic resonance (MR), and X-ray

radiography, provides anatomical structure information. Sincemany pathologies or injuries impact not only structure but alsofunction, researchers have sought to expand the types of infor-mation that can be obtained using these modalities. One suchapproach is perfusion imaging—where hemodynamic parame-ters are obtained as the mixture of blood (endogenous or exoge-nous) and contrast agent is “perfused” throughout an anatomical

Manuscript received January 11, 2010; revised February 09, 2010; acceptedFebruary 10, 2010. First published April 08, 2010; current version publishedApril 30, 2010. This work was supported in part by grants from the Center forIntegration of Medicine and Innovative Technology (CIMIT), U.S. Army Med-ical Research, and also in part by the National Science Foundation (NSF) En-gineering Research Centers (ERC) grant to the Center for Subsurface Sensingand Imaging Systems (CenSSIS). Asterisk indicates corresponding author.

*L. He is with the Laboratory for Medical Imaging and Computations, De-partment of Radiology, Massachusetts General Hospital and Harvard MedicalSchool, Boston, MA 02114 USA (e-mail: lhe2@partners.org).B. Orten is with the Department of Electrical and Computer Engineering, BostonUniversity, Boston, MA 02215 USA (e-mail: borten@bu.edu).

S. Do and H. Pien are with the Laboratory for Medical Imaging andComputations, Department of Radiology, Massachusetts General Hospital andHarvard Medical School, Boston, MA 02114 USA(e-mail: sdo@partners.org;hpien@partners.org).

W. C. Karl is with the Department of Electrical and Computer Engineeringand Biomedical Engineering, Boston University, Boston, MA 02215 USA(e-mail: wckarl@bu.edu).

A. Kambadakone and D. V. Sahani are with the Department of Radiology,Massachusetts General Hospital and Harvard Medical School, Boston, MA02114 USA (e-mail: akambadakone@partners.org; dsahani@partners.org).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMI.2010.2043536

region. Both MR and CT perfusion (MRp and CTp) have beenfound to be useful in a number of settings, including diagnosis,risk stratification, and therapeutic monitoring [1]–[5].

There are currently several different perfusion models in use,depending on the phase of contrast enhancement and the hemo-dynamic parameter to be estimated. In this paper, we focus oncharacterizing tissue perfusion by examining the initial passageof contrast (i.e., first-pass enhancement). Even with first-pass,several perfusion models have been developed; the two mostwidely studied classes of models are those that examine the rateof enhancement (maximum slope models), and those that seek toestimate a “residue” function convolution kernel that relates theinput (arterial enhancement) and response (tissue enhancement)[1], [4], [6]. This second model forms the basis of the analysisof dynamic susceptibility contrast (DSC) MRI, is widely usedin CTp analysis, and is the focus of our study.

More specifically, the deconvolution method is based on thecentral volume principle and the assumption that the arterialinput function (AIF) and tissue time-enhancement curves (TEC)are related through a convolution integral by the residue func-tion, where the residue function describes the tracer remainingin the tissue at any instant of time [7]–[12]. As such, signifi-cant past research have focused on how best to deal with theinverse problem of estimating the residue function from theAIF and TEC. In this context, the most commonly used de-convolution method is truncated singular value decomposition(TSVD) and its extensions, such as circulant TSVD (cTSVD)[8], [9], [13]–[16]. Others, however, have noted the tendency ofTSVD-based methods to introduce unwanted oscillations [17],[18]. This has led researchers to examine different regulariza-tion methods to stabilize the deconvolution, and have shownvarying degrees of success in recovering the residue function[17], [19]–[21]. It is worth noting that these past studies havefocused exclusively on imposing temporal regularization.

Since perfusion images tend to be noisy, our aims are todevelop a method to perform deconvolution-based first-passhemodynamic parameter estimation that is more robust tonoisy inputs, and to produce perfusion parameter maps withbetter signal-to-noise characteristics. To that end, we havedeveloped a formulation which utilizes a Mumford–Shah-likefunctional to enforce both spatial and temporal correlations,and additionally terminate these correlations when a discon-tinuity between regions appears likely [22]–[24]. BecauseTSVD-based approaches estimate the residue function (andhence the perfusion parameter) for each voxel independentlyof its neighbors, our spatially- and temporally-correlated de-convolution approach mitigates the noise issues associatedwith traditional approaches. Furthermore, it is frequently the

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HE et al.: A SPATIO-TEMPORAL DECONVOLUTION METHOD TO IMPROVE PERFUSION CT QUANTIFICATION 1183

case that an ad hoc moving average filter is used to postpro-cess the noisy TSVD-derived perfusion parameter maps; ourformulation eliminates the need for such a step, potentiallypreserving the sharpness between adjacent regions. Althoughspatially-correlated image models have been used in severalcontexts (cf. [25]–[28]), to date we are not aware of any suchwork in the context of perfusion parameter estimation.

In this paper, we utilize our spatio-temporal deconvolutionmethod to estimate blood flow (BF). We utilize simulations bothto select parameters in the algorithm, and to show that our ap-proach leads to a more accurate estimate of the residue func-tion. We further show that our approach generates perfusion pa-rameter maps which are less noisy. Lastly, we use in vivo rectaltumor CTp data to demonstrate that our estimated BF valueslead to better separation between cancerous tissue—which by itsangiogenic nature tends to have greater blood flow—and normaltissue. This paper is organized as follows. In Section II, we pro-vide the theoretical background for perfusion parameter estima-tion, and introduce our spatio-temporal deconvolution method.In Section III, we define our experiments and results from usingboth simulations and clinical datasets. Finally, we provide a dis-cussion of our conclusions in Section IV.

II. THEORY

A. Perfusion Parameters Estimation

Using the theoretical treatment provided by [13], we considera bolus of nondiffusible contrast agent injected at time ina tissue volume of interest (VOI). The individual particles ofthe contrast agent follow different paths through the vascularstructure of the tissue. The distribution of their transit time isdefined by the transport function , and

(1)

where is the concentration of tracer at the venous output,is the concentration at the arterial input (i.e., the AIF), and

denotes convolution. The corresponding mean transit time(MTT) is defined as

(2)

and the blood volume (BV), represented by the amount of in-travascular contrast agent in the tissue of interest, can be calcu-lated by

(3)

The central volume theorem [29] then states that

(4)

The residue function , which measures the mass of contrastmedia remaining in a given vascular network over time, is de-fined by

(5)

The concentration of contrast agent (or tissue time-enhancementfunction), , within the given VOI can be written as

(6)

Under this model, at time , a unit of contrast agent isinjected as a bolus, and indicating that the en-tire mass of contrast agent is within the vascular network. Aftersome time when all contrast have left the vascular network,

. Note that in practice, contrast agents can re-circulate back into the vascular network; therefore this model isvalid only during the initial passage of contrast, not during thesubsequent recirculation or equilibrium phases.

Let us assume that and are measured withequally spaced time points , with time increment

. The convolution can be discretized

(7)

where

......

and

......

. . ....

Note that if can be estimated, then can be computedsince at , . As such, much of the remainder ofthis paper deals with the estimation of in order to compute

.

B. Circulant Truncated Singular Value Decomposition(cTSVD)

In singular value decomposition, matrix is factored intotwo orthogonal matrices and and a diagonal matrix ,with singular values, , in descending orderalong the diagonal

(8)

Equation (7) can then be rewritten as

(9)

Since smaller singular values are associated with higher fre-quency singular vectors, the reciprocal of these small singularvalues become large weighting coefficients of oscillatory sin-gular vectors. TSVD regularizes the solution by truncating smallsingular values to zero according to a certain threshold andthus remove the corresponding terms from the solution.

It has been noted that perfusion parameter estimations aresensitive to delays and dispersions between the AIF and the

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1184 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 5, MAY 2010

Fig. 1. Performance of cTSVD with different thresholds � at PCNR � ��.

tissue VOI. For example, in major vasculopathy cases, contrastagents may arrive earlier in the tissue than in the chosen AIF,leading errors in the estimation. Following previous studies[15], [16], [30], we use circulant (by means of a block-circulantversion of matrix) instead of linear deconvolution to avoidsuch causality problems.

The accuracy of the estimation of residue function by cTSVDis sensitive to the choice of the regularization parameter . Otherresearchers have variously set to 20% of the maximum ele-ment in at signal-to-noise ratio [13], to 4% ofthe maximum element in at [15], or adaptivelydetermined based on SNR [31], although questions about theconditions under which the adaptive technique may be usefulhas been raised [20]. In this paper, we utilize simulations to se-lect a that yields accurate BF values across a range of noiselevels. Specifically, we simulate two regions with different BFvalues—one at 70 ml/100g/min for abnormal tissue, the other at30 ml/100g/min for normal tissue. Different levels of Gaussiannoise are added to generate various levels of peak contrast-to-noise ratios (PCNRs, which are calculated by dividing the peakcontrast difference between the two regions by the Gaussiannoise standard deviation). A number of PCNRs are examinedbetween the range of 20–70, with highly consistent results. Figs.1 and 2 shows the performance of cTSVD with PCNRs of 20and 60, where the dashed lines indicate the true BF values. Theerror bars in Figs. 1 and 2 are calculated from 100 random MonteCarlo realizations. According to our simulations, when is ap-proximate 0.06 (6% of the maximum element in ), the mostaccurate BF estimates are consistently achieved.

C. Proposed Spatio-Temporal Deconvolution

Spatial correlations rest on the assumption that physiolog-ical changes to perfusion are regional effects, rather than single-voxel effects. However, since different regions may be under-going different perfusion changes, it is important to break thatspatial continuity when the data indicates edges may be present.As such, we have chosen to implement a continuity constraintin the spirit of Mumford–Shah [22]. In our model, we estimateperfusion parameters by considering both spatial and temporalcorrelations.

Fig. 2. Performance of cTSVD with different thresholds � at PCNR � ��.

1) Spatio-Temporal Regularization: To incorporate spatialcontinuity, we modify the terms in (7) such that there is bothspatial and temporal dependency. Specifically, letdenote contrast agent concentration in the tissue of the voxelat spatial location and time point . Similarly, let

be the remaining contrast agent concentrationof the voxel at spatial location and time point . Theleast-squares form of (7) is

(10)

However, noise in the data is usually magnified to such extentthat the solution may have no practical value or meaning. In thespirit of Mumford–Shah [22], we incorporate a prior of not onlytemporal but also spatial correlations through the inclusion oftwo constraints to the original least-squares cost function. Thisresults in the new cost function

(11)

where introduces a penalty if the solution behaves undesirablyaccording to the priori information, (i.e., we seek nonoscillatingand smooth solutions). Various models have been proposed (cf[17]) and a first order 4-D derivative operator suffices for ourpurpose. denotes an edge field identifying discontinuities inthe perfusion estimates, which is defined separately for the ,

, and components and its value ranges from 0 (no edgebetween voxels) to 1 (strong edge between voxels). We use thenotation , and , where

. The regularization terms and controlthe regularization strength, which must be optimized and arediscussed in the following section.

The minimizer of (11) requires both and tobe 0, then we have

(12)

and

(13)

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HE et al.: A SPATIO-TEMPORAL DECONVOLUTION METHOD TO IMPROVE PERFUSION CT QUANTIFICATION 1185

The minimizer of (11) requires that (12) and (13) are simultane-ously satisfied. The minimization is obtained iteratively by firstfixing to solve for , and then fixing to solve for . Theedge field is initialized to zero, and at each voxel is initial-ized to be a vector of zeros. Each iteration is solved by conjugategradient method.

2) Method for Choosing Regularization Parameter: To ex-amine the trade-off between data fidelity and regularization, welook at the L-curve characterizing versus

[32]. Note in particular that we focus first on op-timizing the selection of since (12) involves only . Sub-sequently, according to the (13), the ratio of serves as athreshold of whether the smoothing effect should turn on or not.We utilize the simulation of two regions with different BF valuesdescribed previously. When the gradient across the boundarybetween these two regions is much less than the ratio of ,

, which means the smoothing effect should be turnedon; on the other hand, when the gradient is much greater than

, , the smoothing effect should be turned off. Usingthis approach, is calculated based on simulation data.

III. EXPERIMENTS AND RESULTS

In this section, we first show our results from simulation-based experiments, followed by results from our clinical evalu-ations.

A. Simulations

1) AIF Generation: The AIF is simulated using a gamma-variate function [13], [33]; the analytic expression for is

(14)

where is bolus arrival time to any given region. Generally,, s are used to generate curves with much wider

peaks for adults than for children , sincethe injected contrast agent dose is usually proportional to bodyweight and the amount of injected contrast agent is much largerin adults than in children. Theses settings of and with ,

can render an AIF with a shape and size that wouldtypically be obtained using a standard injection scheme [34].

2) Residue Function Generation: The transport function, where is a vector of model parameters, is simulated

using the family of gamma distributions, as [18]

(15)

To ensure that the central volume theorem [(4)] is satisfied, weset . The residue function can then be ob-tained using (5).

3) Tissue Time-Enhancement Curve and Gaussian NoiseGeneration: We set , , , to generatethe AIF. Simulations are performed for ,and BF values varied from 20 to in

increment. Correspondingly, MTT rangesfrom 3 to 12 s. We set to generate theresidue function. Noise-free tissue time-enhancement curve canbe generated according to (7). Gaussian noise isthen added to the noise-free tissue time-enhancement signals. In

Fig. 3. Example of the L-curve (�� ��� versus ���� ��� as a function of� ) for the synthetic data at PSNR � ��. Values of the regularization parameter� are listed along the curve and the optimum � � � is chosen as the cornerof the curve.

the following, peak signal-to-noise ratio (PSNR) is calculatedby dividing the peak value of tissue time-enhancement signalby the noise standard deviation .

B. Simulation Results

The L-curve used for optimizing the choice of is shown inFig. 3. The vertical part of the curve corresponds to under-regu-larized estimations, where the solution is dominated by pertur-bation errors. On the other hand, the horizontal part of the curvecorresponds to over-smoothed estimations, where the solutionis dominated by residual fit errors. The optimum regularizationparameter is chosen as the corner between the horizontaland vertical portions of the curve, which defines the transitionbetween over- and under-regularization. It should be noted that

is a relatively insensitive parameter, and that values near the“corner” of the L-curve yield very similar results.

1) Residue Function Recovery: The residue functions recov-ered by cTSVD and spatio-temporal deconvolution methods areshown in Fig. 4. Under both circumstances of and

, the residue functions in Fig. 4(1c) and (2c) are inagreement with the true residual. On the other hand, the curvesin Fig. 4(1b) and (2b) show relatively large baseline oscillationsand less accurate peak values.

2) BF Estimation: From the recovered residue function, BFcan be estimated at . We now examine the accuracy of ourBF estimates at various noise levels. To do so, we generate asmall region containing 40 40 voxels with the same perfusioncharacteristics and compute the mean and standard deviationover this region. The comparison is carried for different valuesof BFs and at both low and high noise levels. The simulationsare repeated 25 times with random noise realizations.

Shown in Fig. 5 are the BF values as estimated by cTSVD andspatio-temporal deconvolution, for two different noise levels.Spatio-temporal estimates can be visually observed to be morerobust and accurate. To further summarize Fig. 5, we computethe mean squared error (MSE) as defined by

(16)

where and , , are the true and estimated BFvalues, respectively. The MSEs of cTSVD and spatio-temporal

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1186 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 5, MAY 2010

Fig. 4. Recovery of residue functions by cTSVD and spatio-temporal methods. The parameters used for the simulation is �� � � ������ . In the first row,�� � �� and �� � �� ������ ����; in the second row, �� � �� and �� � �� ������ ����. Column (a): True residue function. Column (b):Restored residue function by cTSVD. Column (c): Restored residue function by spatio-temporal deconvolution.

Fig. 5. Comparison of the accuracy in estimating BF by cTSVD and spatio-temporal deconvolution methods for different BF values. (a) �� � ��,��� � ���� for spatio-temporal deconvolution and ��� � ������ forcTSVD; (b) �� � ��, ��� � ���� for spatio-temporal deconvolutionand ��� � ���� for cTSVD.

deconvolution are 134.52 and 2.65, respectively, at, and 13.1 and 0.72 at .In addition, we are also interested in exploring how noise

levels affect the performance in estimating BF of these twomethods. For this purpose, we fix the BFand impose different levels of noise (PSNR varies from 5 to60); the estimation results are shown in Fig. 6. It is shown thatspatio-temporal deconvolution appears to provide more accu-rate estimates of BF than cTSVD across a broad range of noiselevels. In addition, the accuracy of cTSVD in estimating de-grades dramatically as a function of noise level, whereas thespatio-temporal deconvolution approach appears more stable.

Furthermore, for the simulated uniform region, the ideal BFvariability should be 0. Some variations are expected in the re-covered BF maps and a better method should produce smallervariation. Examples of recovered BF maps are shown in Fig. 7.When PSNR is extremely small , cTSVD behavespoorly in recovering true object from background and defining

Fig. 6. Comparison of the accuracy in estimating BF via cTSVDand spatio-temporal deconvolution methods at different PSNRs. True�� � �� ������ ����.

edges, while in the BF map produced by spatio-temporal decon-volution method, the object can be easily distinguished fromthe background and the edge is well defined. When PSNR ishigh , both methods can estimate BF values ac-curately, however, Fig. 7 (1a) contains more fluctuations than(1b). It is apparent that the estimated BF map produced by thespatio-temporal method is smoother than that by cTSVD. Andfurther quantitative comparisons are shown in Fig. 8 (where BFis varied) and Fig. 9 (where PSNR is varied). In both figures,spatio-temporal deconvolution method produces lower BF vari-ation than cTSVD does.

3) Contrast Preservation: Contrast is frequently considereda metric indicating how well different regions can be distin-guished. For the problem at hand, one would expect that thecontrast of the normal and abnormal tissue time-enhancementsignals calculated using the recovered residue function should

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HE et al.: A SPATIO-TEMPORAL DECONVOLUTION METHOD TO IMPROVE PERFUSION CT QUANTIFICATION 1187

Fig. 7. Examples of recovered BF maps via (a) cTSVD and (b) spatio-tem-poral deconvolution methods. The leftmost image is true BF map with�� � �� ������ ���. The PSNRs in first and second rows are 80 and 7,respectively.

Fig. 8. Comparison of cTSVD and spatio-temporal deconvolution methods inreducing variations over homogeneous region for different BF values. The idealvariation is 0. (a) � �� � ��. (b) � �� � ��.

Fig. 9. Comparison of cTSVD and spatio-temporal deconvolution methods inreducing variations over homogeneous region at different PSNRs. The idealvariation is 0. True �� � �� ������ ���.

comparable to that of the noise-free residue function. To com-pare the performance of cTSVD and spatio-temporal decon-volution methods in preserving contrast, we generate synthetic

Fig. 10. Simulated CTp data. First row and second row are 1-D time series(residue function) of a voxel which is spatially located in the regions I and II,respectively. The parameters used to generate residue function: � � �, � � �,� � ��� and � � � ; Voxels in region I have same perfusion characteristic,�� � �������� ���, and �� � ������� . Similarly, voxels in regionII have same perfusion characteristic, �� � �� ������ ��� and �� �� ������ .

Fig. 11. Comparison of cTSVD and spatio-temporal deconvolution methods inpreserving contrast over different levels of noise.

CTp data spatially containing two 40 40 uniform regions withdifferent perfusion characteristic ( ,

), temporal resolution of 1 s, and totalduration of 60 s, as shown in Fig. 10.

A plot of true PCNRs against estimated PCNRs is shownin Fig. 11, from which PCNRs estimated by spatio-temporaldeconvolution method can be observed to be consistently higher

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1188 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 5, MAY 2010

Fig. 12. Comparison of (a) cTSVD and (b) spatio-temporal deconvolutionmethods in preserving edges between two adjacent regions over different levelsof noise. ���� � �, 5, 10, and 20 at the first, second, third, and fourth row,respectively.

and more accurate than those by cTSVD, when .At , cTSVD produces marginally higher PCNRsthan spatio-temporal deconvolution. However, from Fig. 12,it clearly shows that spatio-temporal deconvolution performsbetter than cTSVD in preserving edges bestween two adjacentregions consistently over different levels of noise, especiallywhen PCNRs are low .

C. Clinical Studies

In this section we describe the results from comparing ourapproach with cTSVD on four subjects—two with rectal tumorsand two normal rectums. The presence and precise location oftumors were identified by board-certified radiologists.

1) Data Acquisition: Perfusion CT was performed with afour slice MDCT scanner (LightSpeed QX/I, GE Medical Sys-tems, Milwaukee, WI). 500 ml of saline was instilled rectally foradequate distension of the rectum and proper delineation of thetumor. An unenhanced scan of the pelvis (5 mm thickness) wasperformed first to localize the tumor. A 2 cm region of the tumorat its maximum dimension was selected for perfusion imaging.Dynamic scanning was then performed at a static table posi-tion in the region of the tumor following intravenous injectionof 125 mL of iopamidol (Isovue 300, Bracco) at an injectionrate of 7 cc/s through a 18 guage IV cannula. The following

scan parameters were used for the cine acquisition: 1 s gantryrotation time, 100–120 kVp, 200–240 mA, 10 s scanning delayfrom initiation of contrast media injection and the total durationof scanning was 45 s (four sections per gantry rotation of 5 mmthickness). Given the slice thickness, we processed our data inthree dimensions instead of four, since significantlydifferent anatomy can be observed in consecutive -slices.

D. Clinical Experiments Results

For cTSVD, a threshold of 6% of the maximum singularvalue is used, in accordance with our simulation studies(Section II-B). Similarly, the regularization parameters forthe spatio-temporal deconvolution method are chosen usingthe L-curve method discussed in Section II-C-2. Processingis performed on a computer with Intel Core 2 Quad CPU,Q6600@2.40 GHz and 2.39 GHz, 3 GB of RAM. It takes ap-proximately 700 s to process a clinical data set ofby spatio-temporal deconvolution method with 5 iterations, andapproximately 50 s by the cTSVD method.

1) BF Estimation: BF maps were calculated for each data setusing cTSVD and spatio-temporal deconvolution. We first com-pare two methods by visually observing the estimated BF maps.As shown in Figs. 13 and 14, variations in estimated blood flowmaps are reduced greatly by our method. The tumor region aremoreevidentlydefined,while thenormal regionaremoresmooth.

2) Cancerous Voxels Clustering: By aggregating all voxels(within the ROI) from the two normal patient data sets intoa “normal” cluster, and two tumor patient data sets into a“abnormal” cluster, we have two clusters of voxels—one con-taining samples of BF values from normal rectal voxels, theother containing samples of BF values from cancerous rectalvoxels. In our case, and . To quantify theseparability between normal and tumor BF values, we definethe distance between these two clusters as

(17)

where , are the means, and , are the standard devi-ations of BF in the normal and tumor clusters, respectively. Wehypothesized that our spatio-temporal deconvolution algorithmto produce larger distance as defined in (17), that is, to moredefinitively differentiate between normal and tumor patients.Figs. 15 and 16 show scatter plots of normal versus abnormalcluster. The coordinate value of each point is “the number ofpixels”—increasing pixel number moves from the top-left to thebottom-right of the region of interest as delineated by a radiol-ogist. It is apparent that the two clusters are more separatable indata processed via spatio-temporal deconvolutionthan cTSVD .

3) Diagnostic Test: Let us define sensitivity as the propor-tion of samples with abnormal BF values which test positive,and specificity as the proportion of samples with normal BFvalues which test negative, at a specific threshold. Fig. 17 is thereceiver operating characteristic (ROC) curve drawn based on1247 normal samples and 2269 abnormal samples in which weexamine a spectrum of thresholds. The plot shows the tradeoffbetween true positive rate (sensitivity) and false positive rate

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HE et al.: A SPATIO-TEMPORAL DECONVOLUTION METHOD TO IMPROVE PERFUSION CT QUANTIFICATION 1189

Fig. 13. BF maps and zoomed-in regions of two rectal tumor patients estimated by (a) cTSVD and (b) spatio-temporal deconvolution. Tumor regions are delineatedin red.

Fig. 14. BF maps and zoomed-in regions of two patients with normal rectums estimated by (a) cTSVD and (b) spatio-temporal deconvolution.

Fig. 15. Two clusters of normal vs. abnormal generated by cTSVD method.The distance � between two clusters is 31.37.

(1-specificity). The closer the curve is to the upper left corner,the more accurate the test. Fig. 17 shows that spatio-temporal

Fig. 16. Two clusters of normal versus abnormal generated by our spatial-tem-poral deconvolution method. The distance � between two clusters is 81.23.

deconvolution appears to be considerably more accurate thancTSVD, leading to more efficient diagnostics.

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1190 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 5, MAY 2010

Fig. 17. ROC curves generated by spatial-temporal deconvolution and TSVD.Area under curve (AUC) of spatio-temporal deconvolution is 0.9800 and theAUC of cTSVD is 0.7987.

IV. DISCUSSION AND CONCLUSION

In this paper, a spatio-temporal deconvolution method for im-proving characterization of residue function and for quantifyingperfusion parameters was introduced. The proposed method wascompared to the cTSVD, and numerical simulations with knownAIF and residue function were conducted to validate the pro-posed methodology.

For our simulation studies, a gamma-variate probabilitydensity function was used, since the corresponding residuefunctions presented a wide range of shapes, from exponen-tial to box-car (with increasing ), representative of normalto ischemic tissues. Our experiments were conducted usingdifferent shapes of residue functions and results were highlyconsistent.

The L-curve method was chosen for determining the regular-ization parameter in our work as noted in Section II-C-2. Itwould be interesting to compare L-curve with other approaches,such as discrepancy principle, generalized cross-validation, andother statistical approaches [32].

It is also noted that the first-order derivative operator wasused in the presented work, although other higher order deriva-tive operators (e.g., , ) may also be effective [17].

It has been found that the presence of bolus delay anddispersion between the AIF and the tissue of interest can intro-duce considerable errors in the quantification of BF [34]. Theproblem can generally be minimized by either correcting thedispersion or using a local AIF [35]. The proposed spatio-tem-poral deconvolution method made an effort to avoid suchproblem via the usage of a block-circulant version of , as wedid with the cTSVD method as well.

In summary, by exploiting contextual information fromneighboring voxels both spatially and temporally, a spatio-tem-poral regularizer has been introduced. It has been demonstratedby both simulated and clinical data that the proposed methodappears to be more robust than cTSVD with respect to noise,

especially high noise levels. The proposed spatio-temporaldeconvolution method improves the characterization of residuefunction by both producing a more accurate representation(which results in improved estimation of BF) and reducingthe baseline oscillations. Spatio-temporal deconvolution alsoreduces unexpected variations within a local uniform regionof interests; preserves the contrast of tissue-enhancement sig-nals between different types (i.e., between normal and tumor)tissues, and appears to facilitate clinical diagnosis.

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