Magnetic Resonance in Medicine 63:1366–1375 (2010)

An Automated Method for Nonparametric KineticAnalysis of Clinical DCE-MRI Data: Application toGlioblastoma Treated with Bevacizumab

Gregory Z. Ferl,1* Lu Xu,1 Michel Friesenhahn,2 Lisa J. Bernstein,2 Daniel P. Barboriak,3

and Ruediger E. Port1

Here, we describe an automated nonparametric method forevaluating gadolinium-diethylene triamine pentaacetic acid (Gd-DTPA) kinetics, based on dynamic contrast-enhanced–MRIscans of glioblastoma patients taken before and after treatmentwith bevacizumab; no specific model or equation structure isassumed or used. Tumor and venous blood concentration-timeprofiles are smoothed, using a robust algorithm that removesartifacts due to patient motion, and then deconvolved, yield-ing an impulse response function. In addition to smoothing,robustness of the deconvolution operation is assured by exclud-ing data that occur prior to the plasma peak; an exhaustiveanalysis was performed to demonstrate that exclusion of theprepeak plasma data does not significantly affect results. Allanalysis steps are executed by a single R script that requiresblood and tumor curves as the sole input. Statistical momentanalysis of the Impulse response function yields the area underthe curve (AUC) and mean residence time (MRT). Compari-son of deconvolution results to fitted Tofts model parameterssuggests that AUC

MRT and AUC of the Impulse response func-tion closely approximate fractional clearance from plasma totissue (K trans) and fractional interstitial volume (ve). Intervisitvariability is shown to be comparable when using the deconvolu-tion method (11%[ AUC

MRT ] and 13%[AUC]) compared to the Toftsmodel (14%[K trans] and 24%[ve]). AUC and AUC

MRT both exhibit astatistically significant decrease (P < 0.005) 1 day after adminis-tration of bevacizumab. Magn Reson Med 63:1366–1375, 2010.© 2010 Wiley-Liss, Inc.

Key words: nonparametric model; pharmacokinetics; MRI;bevacizumab; impulse response function

Dynamic Contrast Enhanced-MRI in the Development ofAntiangiogenic Therapies

Dynamic contrast-enhanced MRI (DCE-MRI) is an impor-tant tool for examining vascular function in malignanttumors (1–3). It can be used to noninvasively monitorkinetic distribution of contrast agents, such as gadolinium-diethylene triamine pentaacetic acid (Gd-DTPA), in vivo.Gd-DTPA does not cross cell membranes or bind to

1Early Development PKPD, Genentech, Inc., South San Francisco, California,USA.2Nonclinical Biostatistics, Genentech, Inc., South San Francisco, California,USA.3Department of Radiology, Duke University Medical Center, Durham, NorthCarolina, USA.*Correspondence to: Gregory Z. Ferl, Ph.D., Genentech, Inc., 1 DNA Way, MailStop 463A, South San Francisco, CA, 94080. E-mail: gregoryf@gene.comReceived 13 May 2009; revised 19 November 2009; accepted 23 November2009.DOI 10.1002/mrm.22335Published online in Wiley InterScience (www.interscience.wiley.com).

tissue components because of its high polarity and onlydistributes within blood plasma and interstitial space;metabolism of contrast agent within tissues is not observed.Thus, Gd-DTPA distribution is a function of blood flowand capillary density/permeability (2); insights into thesephysiological properties can be gained from kinetic anal-ysis of the contrast enhancement profiles over timeand are of particular interest in preclinical and clinicaldevelopment of antiangiogenic therapies, such as beva-cizumab (Avastin®) (4) and the tyrosine kinase inhibitorPTK787/ZK2 22584 (5).

Compartmental Models for DCE-MRI Data

Compartmental modeling methodology (6–8), applied toDCE-MRI data facilitates estimation of fractional masstransfer rates between tissue and plasma, volumes of distri-bution, and mean residence times (MRTs), thus providinga clearer picture of tracer kinetics within the tissue of inter-est. Typically, a single ordinary differential equation withforcing (input) function equal to the plasma concentration-time curve is fitted to Gd-DTPA contrast agent data (3,9).However, a single ordinary differential equation does notalways describe the tumor concentration-time profile opti-mally when compared to more complex tissue models (10),necessitating evaluation of multiple model structures foreach set of data. Choosing the optimal model structureto represent a particular data set can be challenging, andnumerous statistical methods have been developed to aidin this task (11).

Nonparametric Kinetic Analysis

For certain data analysis scenarios, it may be advanta-geous to augment standard compartmental analysis withnonparametric methods, such as the initial area under theGd-DTPA uptake versus time curve (IAUC) (12), whichavoid the need to specify a particular model structure. Forlinear, time-invariant systems, the impulse response func-tion (IRF ) describes a measured output following a unitstrength input that can be understood as a very brief pulseof high magnitude, often modeled mathematically by theDirac delta function or unit impulse (13). For this study,we consider tracer blood concentration as the input to thesystem (tumor) and the observed tumor concentration-timeprofile as system output; assuming linearity, the tumor IRFcan be calculated via deconvolution. Here, we utilize ourpreviously described method (14) for retrieving the tumorIRF from DCE-MRI data, which retains kinetic informa-tion without having to assume a particular compartmental

© 2010 Wiley-Liss, Inc. 1366

Nonparametric Kinetic Analysis of DCE-MRI Data 1367

structure. An alternative deconvolution method for DCE-MRI was developed independently by Fan and Karczmar(15,16) and applied to simulated data and rat tumor data.Both approaches utilize previously described numericaldeconvolution techniques, outlined in early tracer kineticmodeling (17) and control systems engineering literature(18), for use with paired input/output data points equallyspaced with respect to time; the IRF is referred to as theweighting function in these texts.

THEORY

Calculation of IRF

The convolution operation for two continuous functions,u(t) and h(t), can be written as

y (t) = u(t) ⊗ h(t) =∫ ∞

−∞u(τ )h(t − τ )dτ . [1]

A general time-invariant, linear system can be described byEq. 1, where h(t) represents the response of the system to aunit impulse, u(t) is the actual system input, and y (t) is thesystem output. For this study, u(t) is replaced by the bloodinput function, y (t) is replaced by the tumor concentration-time curve, and h(t) is the tumor IRF.

The appropriate finite-dimensional, discrete-time analogof Eq. 1 for these data is:

�y = U�h, [2]

where U is an n × n triangular matrix based on the inputfunction, �y is an n × 1 vector corresponding to the tumorconcentration-time curve, and �h is an n × 1 vector corre-sponding to the IRF (Eqs. A-1 and A-2). Measured datapoints must be equally spaced w.r.t. t; here, smoothed bloodand tumor curves are discretized into n equally spaced datapoints with a time resolution of approximately 1 sec. SinceU is triangular and can be extracted from the dynamicimages along with �y , �h is easily calculated via forwardsubstitution (Eqs. A-3 and A-4) (17).

Calculation of Area Under-the Curve, MRT and Variance ofthe Residence Time

The zeroth, first normal, and second central moments (19)of the IRF correspond to the area under the curve (AUC),MRT and variance of the residence time (VRT ) of con-trast agent in tissue as described by the IRF and arewritten as

AUC =∫ ∞

0h(t)dt [3]

and

MRT =∫ ∞

0 th(t)dtAUC

= AUMCAUC

[4]

and

VRT =∫ ∞

0 t2h(t)dtAUC

, [5]

where h(t) is the tumor IRF and AUMC is the area under thefirst moment curve. When applied to our data, each integralis evaluated from 0 to tf , the duration of the MRI scan, andis only an approximation to the true value. Discussion ofcutoff errors can be found in the Appendix. Integrals arecalculated using the trapezoidal rule.

Analysis of a single ordinary differential equation tis-sue model (Eq. A-5) reveals that the ratio of AUC to MRTand the initial value of the IRF(0) reflect flux of contrastagent from plasma into tissue (K trans) (Eqs. A-14 and A-8); likewise, the AUC and MRT of the IRF reflect ve and

1kep

, respectively (Eqs. A-12 and A-13). These quantities arecalculated and analyzed.

MATERIALS AND METHODS

Clinical Imaging Study

The DCE-MRI data analyzed here were collected in a phaseII clinical trial of bevacizumab in adults with histologicallyproven grade III–IV glioma that was progressive or recur-rent after radiation therapy (20). Twenty consecutivelyadmitted patients were enrolled in the DCE-MRI study,which was approved by the Duke Institutional ReviewBoard; informed consent was obtained from every patient.Three DCE-MRI scans were obtained for each patient: 3days and 1 day prior to treatment and 1 day after thefirst dose of bevacizumab (10 mg/kg intravenously). Twopatients missed one of the pretreatment visits. Contrast-enhancing tumor was too small for follow-up in onepatient; one patient missed the visit after bevacizumab, andin two patients DCE-MRI data had to be discarded becauseof technical errors (timing of contrast agent injection, prob-lem with T1 mapping). Thus, data from 16 patients (36–63years old [mean 50.3 years], 10 male) remain for analysis.

A 1.5-T Siemens Avanto scanner (Malvern, PA) wasused with a head coil. Multiflip images using flip anglesof 5, 10, 15, 20, 25, and 30◦ were acquired for T1 map-ping. Dynamic images were acquired over a 5-min periodusing a three-dimensional Fast Low Angle SHot (FLASH)sequence, resulting in a temporal resolution of 4.8 sec, aspatial resolution of 1×1×5 mm (16 slices), and a 256×256voxel field of view. Contrast agent, 0.1 mmol/kg Gd-DTPA,was injected at 3 mL/sec after the fifth dynamic imageacquisition.

Image Processing

For each patient visit, all T1-estimation, dynamic, andstatic images were coregistered with 1 × 1 × 5 mm res-olution, using the dynamic image at the last time pointas reference. Tumor regions of interest (ROIs) were drawnmanually on the coregistered static images, with the assis-tance of an in-house cluster analysis–based autosegmen-tation program. On selected slices, ROIs were manuallydrawn over the superior sagittal sinus to estimate the vas-cular input function for subsequent tumor kinetic analysis.Pixelwise signal enhancement to concentration conversionwas conducted based on estimated relaxation time (T1),equilibrium magnetization (S0), and known relaxivity ofGd-DTPA (21–23). Median contrast agent concentrationsover all pixels within tumor and vein ROIs were extracted

1368 Ferl et al.

FIG. 1. Postcontrast static T1-weighted DCE-MRI image. Tumorregion (ROI) outlined in red.

at each data acquisition time point, resulting in tumor andblood concentration-time data.

Figure 1 shows an MRI image with the ROI drawn onthe area corresponding to tumor. ROIs were drawn toinclude the entire contrast-enhancing tumor because theaim was to compare our novel method for kinetic analysisagainst a currently accepted standard method (24). Alter-nate segmentation criteria would be required to investigatetumor heterogeneity and regions of rapid contrast agentuptake (25).

Data Management and Processing

DCE-MRI data for all patients are stored on a central serverand accessed via a routine written with the statistical pro-gramming language R (26). In addition to importing patientdata (venous blood [Cv ] and tumor [CT ] concentration-timeprofiles), the routine performs all analyses, including sub-traction of blood fraction from CT , lag time correction, datasmoothing, calculation of the IRF, and plotting of results,which are saved on the local workstation, as described insubsequent sections. The major advantage to handling thedata in this manner is that every step of the data analy-sis process is executed in an automated fashion, beginningwith the blood and tissue concentration-time profiles.

Definition of Input and Output Data

Whole blood concentration, instead of plasma, is usedas the input function since a wide range of capillaryhematocrit values has been reported; normal hematocrit inhumans is reported as 42–52% for males and 37–47% forfemales (27). Brix et al. (28) cite a reference that lists cap-illary hematocrit at 25%, while Schreiber et al. (29) cite areference that places capillary hematocrit at 25% in whitematter and 26–37% in normal gray matter (30). Conver-sion from blood to plasma concentrations using a nominalhematocrit value would simply scale all calculated IRFs by

a constant factor and would not affect the results presentedhere.

The tumor ROI includes a small blood component so that:

C(t) = CT (t) + vbCv (t), [6]

where C(t) is the sum of tissue vascular (Cv ) and extravascu-lar (CT ) concentrations; the contribution of Cv is weightedby vb, the fractional blood volume of tissue. We subtractvbCv (t) from the measured tumor concentration-time datain order to reduce the complexity of the IRF. Althoughhigher values have been reported (31), we assume a nom-inal value of 1.5% for tumor blood fraction, based on ourprevious modeling studies and use CT (t) as model output:

CT (t) = C(t) − 0.015 · Cv (t) [7]

Cv and CT Time Shifting and Truncation

As previously described, data acquisition via DCE-MRIbegins prior to injection of the Gd-DTPA tracer, resultingin a lag time of approximately 45 sec before a rise of signalintensity in blood and tumor is observed. These data areremoved in order to ensure success of the deconvolutionalgorithm, which is highly sensitive to initial values of Cv

and CT . Based on our analysis of the data described here,the most robust method for dealing with this issue is toremove all Cv and CT data that occur prior to the time cor-responding to the Cv peak. Additionally, CT is shifted oneframe to the right (approximately 5 sec) to account for thefact that the ROI is drawn on a major vein, thus reflectingcontrast agent concentration after passing through tumor;the initial rise in CT should occur after the initial rise inCv since we are assuming an input/output relationshipbetween Cv and CT . These transformations are written as

C̃v (t) = Cv (t + tp) [8]

and

C̃T (t) = CT (t + tp − 5), [9]

where tp is the time at which Gd-DTPA is at maximum con-centration in blood; t has units of seconds. C̃v (t) and C̃T (t)are the concentration-time profiles used for all subsequentanalysis. Note that drawing the ROI on a major vein (orartery) provides only an approximation of the true bloodconcentration of contrast agent within tumor due to delayand dispersion effects as the bolus passes through the tissueand a microcapillary environment that differs from vein orartery, e.g., lower hematocrit.

Data Smoothing

The deconvolution operation performs poorly on noisydata, so it is necessary to smooth C̃v (t) and C̃T (t) priorto calculating the IRF. Using the Locfit package withinR, we applied a robust locally adaptive smoothing algo-rithm (32) to C̃v (t) and C̃T (t). The robust adaptive smootherimplemented here down-weights apparent outlier datapoints while capturing the peaks and troughs of the bloodconcentration-time profile, when applied to data occurringafter the plasma peak. This has the desirable effect of cor-recting the data for artifacts (e.g., data scatter caused by

Nonparametric Kinetic Analysis of DCE-MRI Data 1369

patient movement during data acquisition) in an objectivemanner.

Analysis of Prepeak Blood Concentration-Time Profile

The possibility of excessive information loss, regardingexclusion of Cv data occurring prior to the blood con-centration peak (Eq. 8), prompted an exhaustive analysisof the effect, on results, of varying venous and tumorconcentration-time profile truncation and time shift oftumor versus vein (Eqs. 8 and 9). Venous concentration-time profiles (Cv ) were truncated by removing early-timesmoothed data points with 1-sec time resolution one ata time, beginning with the first nonzero value, definedas data from two consecutive time points with slope>5 µmol/L·sec−1. For each truncated Cv curve, the decon-volution operation was performed 30 times, with tumortime shifts ranging from −10 to +20 sec, relative to theoriginal position of the tumor concentration-time curve; thetail ends of the tumor or vein data were always truncatedso that the lengths of the two curves were equal. Approxi-mately 10 truncated Cv curves (Eq. 8) were considered foreach scan, with 30 tumor time shifts (Eq. 9) evaluated foreach truncated Cv curve, for a total of 300 calculated IRFs.AUC, MRT , AUC

MRT , and IRF(0) were calculated and plottedfor each IRF. This analysis effectively evaluates the sensi-tivity of the deconvolution operation to distance betweenthe plasma and tissue concentration peaks.

Comparison to Extended Tofts Model

Three comparisons are made between the AUCMRT ratio and

AUC of the IRF as calculated via our deconvolution methodversus values of K trans and ve calculated by fitting theextended Tofts model to the data discussed here. Absolutevalues of these quantities, percentage reduction betweenpost- and pretreatment, and reproducibility of quantitiesbased on the two pretreatment scans are compared. TheAUCMRT ratio and AUC of the IRF, as calculated here, are anal-ogous to K trans and ve, as described in the Appendix (Eqs.A-15 and A-16).

Software

All steps in the numerical deconvolution analysis methoddescribed here have been automated using the statisticalprogramming language R (26). As summarized in Table 1,once the path to data files from the image processing stageof analysis is specified (files containing Cv and CT data), thedata of interest are retrieved and all aforementioned anal-ysis steps are performed. The method is appropriate forany linear, time-invariant system; no assumptions regard-ing a model structure are required, nor is parametric curvefitting included in any analysis steps. Plots are automati-cally generated and include a statistical analysis summary,thus providing the core code for a relatively user-friendlydeconvolution package that can be readily applied to anylinear tracer kinetic analysis problem where arterial inputfunction and tissue concentration-time data are available.

Statistical Analysis

The Wilcoxon signed rank test was used to evaluate statisti-cal significance of changes in parameters AUC, MRT , AUC

MRT

Table 1Analysis Flow Chart. Summary of Steps Executed by the R Script

Retrieve DCE-MRI data from server⇓

Extract Cv & CT curves⇓

Subtract blood fraction from CT⇓

Time shift, truncate Cv and CT⇓

Smooth Cv and CT data⇓

Calculate IRF⇓

Calculate AUC, MRT , and IRF(0)⇓

Statistical analysis⇓

Plot results and save to file

and IRF(0) after treatment, compared to baseline values.Ratio of posttreatment to mean pretreatment is comparedto a null hypothesis of ratio = 1.

To assess the performance of our method versus the cur-rently accepted gold standard, we calculate the intrapatientpretreatment percentage coefficient of variation (%CV ) asfollows:

%CV = σp

p̄√

2× 100%, [10]

where

σp =√∑n

i=1(pi,1 − pi,2)2

n[11]

and

p̄ =∑2

j=1∑n

i=1 pi,j

2n. [12]

pi,j is the calculated parameter value for patient i basedon data from pretreatment scan j. In the case where cor-relation exists between pi,1 − pi,2 and (pi,1 + pi,2)/2, %CVis calculated by applying the following equation to loge-transformed data:

%CV =√√√√exp

(σ 2

p

2

)− 1 × 100%. [13]

Kendall’s tau was calculated to test for correlation; dis-tribution of pi,1 − pi,2 was tested for normality using theShapiro-Wilk test.

RESULTS

Tumor Blood Fraction/Lag Time Correction

The open circles within the top two panels of Fig. 2 showvein and tumor DCE-MRI data that have been transformedaccording to Eqs. 8 and 9. The gray circles represent the pre-treatment MRI scans and the black represent posttreatment.For this particular patient, pretreatment blood concen-tration is lower than posttreatment, while pretreatmenttumor concentration is higher than posttreatment tumorconcentration.

1370 Ferl et al.

FIG. 2. Analysis of data from a single patient before (gray) and after(black) administration of bevacizumab. Top panel: Blood concentra-tion. Center panel: Tumor concentration. Bottom panel: IRF. Circlesrepresent DCE-MRI data; solid curves are smoothed DCE-MRI dataused for the deconvolution operation. AUCs are unitless.

Data Smoothing

The solid lines in the top two panels of Fig. 2 representthe smoothed data. The smoothed venous concentration-time curve captures the peak while eliminating the majorityof noise in the remainder of each concentration-time pro-file. The smoothed tumor curve attenuates noise and outlierdata from pre- and posttreatment scans.

Calculation of IRF

The bottom panel of Fig. 2 plots the theoretical response oftumor to a unit impulse, calculated by solving for �h in Eq.2 (See Eqs. A-1 through A-4 for details). For this particularpatient, the AUC of the posttreatment IRF, shown in black,exhibits an approximately 2-fold reduction compared tothe AUC of the pretreatment IRF, shown in gray, despite

the more modest decrease in tumor uptake illustrated inthe central panel. These IRFs suggest that the rate of con-trast agent extravasation and fractional interstitial volumeof tumor has been reduced as a result of treatment.

Calculation of AUC, MRT , and VRT

The upper-left panel of Fig. 3 compares the average IRFAUC of the two pretreatment scans to the +1 day posttreat-ment scan for each patient. All data points are normalized(divided by) to the median of all pretreatment scans. Anal-ysis of intrapatient post- to pretreatment ratios reveals astatistically significant (P < 0.005) decrease of 35% in AUCacross all patients 1 day after administration of the first doseof bevacizumab (Table 2). The mean ± standard deviationpretreatment AUC (unitless) across all patients is 0.16 ±0.07, while the average posttreatment value is 0.11 ± 0.05(Table 3).

The upper-right panel of Fig. 3 plots the MRT, as calcu-lated by Eq. 4, for each patient visit. There is no statisticallysignificant (P > 0.6) change in MRT across all patientswhen comparing pre- and posttreatment scans (Table 2).For these 5-min DCE-MRI scans, the average MRT acrossall patients and visits is 93 ± 9 sec.

The lower-left panel of Fig. 3 plots the ratio AUCMRT of the IRF

for each patient visit. Given the relatively constant MRT ,the trend observed here is similar to that of the AUC, witha 35% decrease (P < 0.005) observed in this quantity 1day after treatment (Table 2). The average pretreatment AUC

MRTacross all patients is 0.11 ± 0.05 min−1, while the averageposttreatment value is 0.069 ± 0.033 min−1 (Table 3).

The lower-right panel of Fig. 3 plots IRF(0) for eachpatient visit. A statistically significant (P < 0.005) decreaseof 36% is observed for this quantity. The average pretreat-ment IRF(0) across all patients is 103±69 min−1, while theaverage posttreatment value is 62±40 min−1. The absolutevalues of IRF(0) are highly sensitive to the Cv and CT timeshifting described in Eqs. 8 and 9 (data not shown).

The variance of contrast agent residence (VRT ) (Eq. 5)was also calculated, but no statistically significant changewas observed (P > 0.5) in the posttreatment data. Theaverage

√VRT across all patients and visits is 120 ± 8 sec.

We assessed the sensitivity of these values to vb byestimating AUC, MRT , AUC

MRT , and IRF(0), with vb rangingfrom 0–3%, yielding estimates of fractional change frompre- to posttreatment ranging from 0.62–0.68, 0.98–1.06,0.60–0.70, and 0.55–0.73, respectively.

Analysis of Prepeak Blood Concentration-Time Profile

Figure 4 illustrates the effect of Cv and Cb time-shiftingand truncation on the calculated AUC of the IRF, usingdata from a single representative patient; two pretreat-ment (gray) and the single post-treatment (black) scans areshown. The left panel shows all AUCs, calculated using thepreviously described method. In this patient, little changeis seen in the AUCs once the first two nonzero data pointsare removed from Cv (upper horizontal axis > 2). The rightpanel shows only those AUCs derived from IRFs that arepositive in magnitude over the course of the scan (smallnegative IRF values, less than 5% of the maximum IRFare allowed) since applying the numerical deconvolution

Nonparametric Kinetic Analysis of DCE-MRI Data 1371

FIG. 3. Each of the four panels shows pairs of box plots depicting summary measure values from pre- and posttreatment DCE-MRI scanssuperimposed upon individual patient data (n = 17); each data point is normalized by the median pretreatment value across all patients.The two pretreatment scans for each patient are averaged and then compared to the follow-up scan taken 1 day after receiving the firstdose of bevacizumab. Pre- and posttreatment AUC, MRT , AUC

MRT , and IRF(0) values for each patient are connected by a solid gray line andcorrespond to percentage changes in ve, 1

kep, K trans, and K trans, respectively (see Appendix).

method to the entire Cv curve often results in oscillationsand large negative values in the IRF. Prepeak analysis wasperformed on data from all scans; this result is typical forthe remaining patients and similar results are obtained for

Table 2Comparison of DCE-MRI Scans Taken Before and 1 Day AfterBevacizumab Treatment*

+1dbase Tofts

(n = 17) Standard deviation P value analog

AUC 0.65 0.17 <0.005 ve

MRT 1.01 0.10 >0.6 1kep

AUCMRT 0.65 0.19 <0.005 K trans

IRF(0) 0.64 0.25 <0.005 K trans

*The +1 day scan (+1d) is divided by the mean of the two pretreat-ment baseline scans (base) for each patient; we expect this ratio tobe unity if there is no change in the calculated metric (AUC, MRT ,AUCMRT , IRF(0)) between pre- and posttreatment scans. The Wilcoxonsigned rank test is used to assign a P value to +1d/base (n = 17),with a null hypothesis of +1d/base = 1. Tofts model analogs, derivedin the Appendix, are also shown.

MRT and AUCMRT ; however, IRF(0) is highly dependent on data

truncation and timeshifting.

Comparison to Tofts Model

Reproducibility of K trans values, calculated via the Toftsmodel, for each of the two pretreatment scans was assessedand compared to AUC

MRT calculated via our numerical

Table 3Comparison of Values (mean ± standard deviation) EstimatedUsing the Extended Tofts Model (K trans and ve) to CorrespondingValues Estimated via Numerical Deconvolution ( AUC

MRT and AUC)*

Pretreatment 1 Pretreatment 2 Posttreatment

ToftsK trans 0.049 ± 0.023 0.053 ± 0.025 0.032 ± 0.014ve 0.16 ± 0.06 0.21 ± 0.07 0.12 ± 0.05

DeconvAUCMRT 0.10 ± 0.05 0.11 ± 0.05 0.07 ± 0.03AUC 0.15 ± 0.06 0.17 ± 0.07 0.11 ± 0.05

*Fractional blood volume (vb) was estimated for each scan when fit-ting the Tofts model (values not shown), while we assume a constantnominal value of vb = 0.015 when using the deconvolution method.

1372 Ferl et al.

FIG. 4. Effect of CT and Cv time shifting and truncation of early data points on pre- (gray) and posttreatment (black) AUCs from a singlepatient. For both panels, the graph region between the two leftmost vertical dotted lines illustrates the effect on the IRF AUC of shifting CTto the left by 30 sec (starting at +20 sec relative to the original position of CT ), 1 sec at a time, where the entire Cv curve is utilized. Thegraph region between the second and third leftmost vertical dotted lines takes CT through the same time shifts, except the first nonzeroCv data point has been removed, indicated by the “1” on the upper horizontal axis, which denotes the number of excluded Cv data points.The graph region between the rightmost two dotted lines utilizes Cv data occurring after (and including) the plasma peak. Left panel: All IRFAUCs calculated by this method are shown. Right panel: The AUC of IRFs > 0 for all t are shown (small negative IRF values, less than 5%of the maximum IRF are allowed); negative IRF values are not physiologically possible and thus may be excluded. The AUC of the IRF isunitless, as seen in Fig. 2.

deconvolution method. The pretreatment %CV for K trans

is 13.6% compared to 10.7% for AUCMRT . Likewise, ve (Tofts

model) and AUC (deconvolution) were assessed for repro-ducibility and have %CVs of 23.6% and 12.7%, respec-tively. Galbraith et al. (33) report %CVs of 24% and 8.5%for K trans and ve in tumor. Mean values of K trans and ve

calculated via the Tofts model are listed in Table 3.No correlation was observed across patients between

pi,1 − pi,2 and (pi,1 + pi,2)/2 for ve or AUC (P > 0.05), whilecorrelation was observed for both K trans and AUC

MRT (P < 0.05);hence, %CVs were calculated using Eq. 10 for the formerand Eq. 13 for the latter.

The distribution of pi,1 − pi,2 exhibited no significantdeviation from normality (P > 0.05) for ve and AUC orloge-transformed values of K trans and AUC

MRT .

DISCUSSION

Automated Deconvolution

Technical problems related to calculation of the IRF fromclinical DCE-MRI data were solved by smoothing the rawCT and Cv data with a robust, adaptive smoother (32) and bytruncation of prepeak Cv data which minimized instabilityof the deconvolution operation and subsequent oscillationswithin the IRF, thus producing a robust method for obtain-ing the IRF from the 51 clinical DCE-MRI scans underconsideration here.

An exhaustive analysis of prepeak blood concentrationdata was performed and shows no major changes in cal-culated values of AUC and MRT as a result of removingdata occurring prior to the time associated with the plasmapeak (Fig. 4). Truncation of the data in this manner sta-bilizes the numerical deconvolution operation, resultingin relatively smooth IRFs for all scans analyzed. However,removal of prepeak data may result in loss of informa-tion from high-blood-flow tumor regions, limit the ability

to detect small foci of neovascularization, and may resultin an input/output mismatch between early time bloodand tissue data. Interestingly, this analysis also reveals thatcalculated values of the IRF, AUC and MRT are not substan-tially affected by mild to moderate oscillations that run theentire length of the IRF (not shown). Extreme oscillations,however, substantially affect results.

Compartmental models are the standard tool for kineticanalysis of imaging data and, when using an acceptedmodel structure, are the gold standard method for quan-tification of imaging data. However, investigation of theperformance of various model structures, each fitted to thedata sets discussed here, reveals that a single model maynot be optimal for all data in a given study. As shown here,deconvolution analysis can be used to augment or replacecompartmental analysis of kinetic tracer data, thus avoid-ing potential pitfalls related to model discrimination andnonlinear regression. Numerical deconvolution is elegantin its simplicity in that it generates the information-richIRF by simple algebraic manipulation of observed bloodand tissue concentration-time profiles. Statistical momentanalysis, i.e., calculation of the AUC , MRT , and VRT , isreadily applied to the IRF, thus facilitating comparison oftissue time activity curves across studies, independent ofactual plasma tracer kinetics.

A caveat to this approach is that the integrals used to cal-culate the AUC , MRT , and VRT only produce exact resultswhen evaluated from zero to infinity. Since the DCE-MRIscans performed for this study are 5 min in length, Eqs. 3–5are approximations to the true AUC , MRT , and VRT values(Eqs. A-17 and A-18).

Nonparametric Results vs. Tofts Model

If a Tofts model structure is assumed, it can be shown thatthe AUC of the IRF evaluated from t = 0 to t = ∞ (AUC∞)is exactly equal to ve (Eq. A-12) and MRT∞ is equal to 1

kep

Nonparametric Kinetic Analysis of DCE-MRI Data 1373

(Eq. A-13); hence, the ratio of AUC∞ to MRT∞ of the IRF isequal to the Tofts model K trans (Eq. A-14). It is also appar-ent that in order for the MRT of contrast agent to remainunchanged between scans of finite time, AUCtf , ve, andK trans must all change by the same fractional amount (Eq.A-11), where AUCtf is the AUC evaluated from t = 0 tot = tf , the duration of the DCE-MRI scan. The IRF sum-mary parameters shown in Table 2 follow this constraintalmost exactly. Based on our nonparametric analysis, wesee that the MRT does not change as a result of treatment,but the AUC and two mathematically independent mea-sures of K trans all drop by about 35%, suggesting that AUCtf ,AUCtfMRTtf

, and IRF(0) provide insight into absolute and frac-

tional changes in interstitial volume and tracer flux ratesfrom plasma to tissue.

Reproducibility analysis based on the two pretreatmentscans shows that intrapatient %CVs are similar when com-paring the Tofts model, deconvolution, and Galbraith et al.(33) results. Also, our analysis and that of Galbraith et al.(33) show there is no correlation between pretreatmentintervisit differences of ve and corresponding mean inter-visit values of ve, while a significant correlation exists forK trans values.

K trans and ve values obtained using both the Tofts anddeconvolution methods will be affected by cutoff errorsince both require tissue and plasma concentration-timedata from t = 0 to t = ∞ in order to perfectly estimatethese parameters. When using our numerical deconvolu-tion approach, cutoff error can be estimated (Eqs. A-17 andA-18); this analysis reveals that ve will always be under-estimated and K trans always overestimated, with the latterbeing more susceptible to cutoff error. Deconvolution val-ues may be corrected for cutoff error if a nominal kep valuefor the tissue of interest is known a priori. Note that if theIRF MRT is constant between scans, cutoff error cancelsout when taking the ratio of parameters measured beforeand after treatment, as shown in Table 2; thus, percentagechanges in K trans and ve from pre- to posttreatment shouldbe quite accurate when calculated using numerical decon-volution. Cutoff error is particularly important in DCE-MRIstudies due to short acquisition times and is a topic worthyof further consideration.

CONCLUSION

We have developed a highly automated nonparametricmethod of correcting and analyzing dynamic DCE-MRI datathat retains kinetic information, thus providing a robust,model-free approach to evaluation of changes in contrastagent uptake subsequent to antitumor therapy. Addition-ally, when analyzing DCE-MRI data of short duration (scantime �10 min), our method appears to provide a measure ofK trans and ve, via AUC

MRT and AUC , that is at least as consistentbetween studies as those obtained using the Tofts model.

Although unable to provide an absolute measurement ofK trans, IRF(0) can be used to quantify percentage changesin K trans when comparing pre- and posttreatment data; thatis, the tumor–blood concentration ratio at the time cor-responding to peak plasma concentration appears to beproportional to K trans.

APPENDIX

CONVOLUTION MATRIX

For two n × 1 vectors in discrete-time space, �u and �h,the convolution operation can be expressed as a matrixoperation, as shown here:

y1

y2...

yn−2

yn−1

yn

= U

h1

h2...

hn−2

hn−1

hn

, [A-1]

where U is written as

U =

u1 0 . . . 0 0 0u2 u1 0 . . . 0 0... u2

. . .. . .

... 0

un−2...

. . .. . . 0

...un−1 un−2 . . . u2 u1 0un un−1 un−2 . . . u2 u1

[A-2]

Since this n × n matrix is always triangular, �h can be com-puted via forward substitution, given that �u and �y areknown. The following equations illustrate the first twoiterations of the deconvolution operation in discrete-timespace:

h1 = y1

u1[A-3]

h2 = y2 − u2h1

u1. [A-4]

THE TOFTS-LARSSON MODEL

The classic Tofts-Larsson model (34–37), referred to as the“generalized kinetic model” in Tofts et al. (9), imposes aone-state variable, i.e., one ordinary differential equation,tissue model with plasma forcing function upon observedMRI contrast agent kinetics, written as

dCT

dt= K transCv − K trans

veCT . [A-5]

Since contrast agent is not bound or metabolized withintissue and does not cross the cell membrane, K trans can beinterpreted as fractional volume clearance (time−1) fromplasma to tissue interstitial space and ve as fractionalinterstitial volume (unitless) of the tissue of interest (9).

The IRF can be calculated analytically (7,38) and for theTofts model is written as

h(t) = K transexp(−kept), [A-6]

where

kep = K trans

ve. [A-7]

Since the first DCE-MRI data point must occur shortlyafter t = 0, K trans is written as

K trans ≈ h(0 + �t), [A-8]

1374 Ferl et al.

where �t is the data acquisition time for a single DCE-MRIframe. Thus, the first calculated value of the IRF (Eq. A-3) provides a measure of K trans, with magnitude of errordetermined in part by the time resolution of the MRI dataacquisition process.

Eqs. 3 and 4 define the AUC and MRT of the IRF ; appliedto Eq. A-6 and evaluated from t = 0 to t = tf yields:

AUCtf =∫ tf

0h(t)dt = ve(1 − exp(−keptf )), [A-9]

The MRT is equal to the AUMC divided by the AUC andis written as:

MRTtf =ve

K trans [AUCtf − tf K transexp(−keptf )]AUCtf

. [A-10]

Substituting Eq. A-9 into Eq. A-10 and rearranging yields:

MRTtf = ve

K trans + tf

[1 − ve

AUCtf

]. [A-11]

Taking the limit of Eqs. A-9 and A-11 as tf → ∞ shows that

ve = AUC∞, [A-12]

1kep

= MRT∞ [A-13]

and

K trans = AUC∞MRT∞

. [A-14]

Since we are collecting data for a finite length of time andare not assuming any particular model structure, Eqs. A-12 and A-14 are written as follows for our nonparametricanalysis method

ve ≈ AUCtf [A-15]

and

K trans ≈ AUCtf

MRTtf= [AUCtf ]2

AUMCtf, [A-16]

thus providing an approximation of ve and K trans based onthe AUC and AUMC of the IRF, with magnitude of errorprimarily determined by the product of MRI scan length(tf ) and the true value of kep (cutoff error) (39); likewise,MRTtf provides an approximation of 1

kep.

Cutoff error for AUC (ve) and AUCMRT (K trans) estimates can

be approximated via Eqs. A-9 and A-10 if a nominal valueof kep for the tissue of interest is known a priori, where

AUCtf

AUC∞= 1 − exp(−keptf ) [A-17]

and

AUCtf

MRTtf

(AUC∞MRT∞

)−1

= (1 − exp(−keptf ))2

1 − (1 + keptf )exp(−keptf ). [A-18]

Equations A-17 and A-18 suggest that cutoff error willlead to underestimation of AUC (ve) and overestimation

of AUCMRT (K trans) when using the nonparametric approach

described here. Specifically, cutoff error for ve and K trans

will be greater than 10% if keptf � 2 and keptf � 3,respectively.

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