Dual time point based quantification of metabolic uptake rates in 18F-FDG PET
© van den Hoff et al.; licensee Springer. 2013
Received: 10 January 2013
Accepted: 12 February 2013
Published: 13 March 2013
Assessment of dual time point (DTP) positron emission tomography was carried out with the aim of a quantitative determination of K m , the metabolic uptake rate of [18F]fluorodeoxyglucose as a measure of glucose consumption.
Starting from the Patlak equation, it is shown that , where m t is the secant slope of the tissue response function between the dual time point measurements centered at t = t 0. denotes arterial tracer concentration, is an estimate of the Patlak intercept, and τ a is the time constant of the c a (t) decrease. We compared the theoretical predictions with the observed relation between and K m in a group of nine patients with liver metastases of colorectal cancer for which dynamic scans were available, and K m was derived from conventional Patlak analysis. Twenty-two lesion regions of interest (ROIs) were evaluated. c a (t) was determined from a three-dimensional ROI in the aorta. Furthermore, the correlation between K m and late standard uptake value (SUV) as well as retention index was investigated. Additionally, feasibility of the approach was demonstrated in a whole-body investigation.
Patlak analysis yielded a mean V r of ml/ml. The patient averaged τ a was 99 ± 23 min. Linear regression between Patlak-derived K m and DTP-derived K s according to K s = b · K m + a yielded b = 0.98 ± 0.05 and a = -0.0054 ± 0.0013 ml/min/ml (r = 0.98) in full accordance with the theoretical predictions b = 1 and . K s exhibits better correlation with K m than late SUV and retention index, respectively. is proposed as a quantitative estimator of K m which is independent of patient weight, scan time, and scanner calibration.
Quantification of K m from dual time point measurements compatible with clinical routine is feasible. The proposed approach eliminates the issues of static SUV and conventional DTP imaging regarding influence of chosen scanning times and inter-study variability of the input function. K s and exhibit improved stability and better correlation with the true K m . These properties might prove especially relevant in the context of radiation treatment planning and therapy response control.
For many years, quantification of the metabolic rate of glucose consumption with dynamic [18F]fluorodeoxyglucose (FDG) positron emission tomography (PET) using the so-called Patlak plot, a procedure most clearly described by Patlak in his seminal papers [1, 2], has proven valuable in PET research and clinical routine.
However, in the clinical oncological setting, quantification is mostly restricted to the ubiquitously used standard uptake value (SUV). The reason is twofold: (1) no need (or even inability) to determine the arterial input function (AIF) and (2) inability to perform dynamic whole-body investigations.
Without question, the SUV (defined as the tracer uptake at a certain time point normalized to injected dose per unit body weight) has proven a valuable means of achieving a certain level of quantitative description, thus allowing, e.g., definition of standardized evaluation schemes (see  for an overview).
The approach, however, has known shortcomings [4–6]. SUVs do not directly provide information about the tracer kinetics but, by their very nature, only a static snapshot somewhere on the tissue response function (TRF). Naturally, SUVs are varying along the given TRF and are thus prone to variability when not determined at a strictly standardized time. Since SUVs do not contain any information of the actual rate of tracer accumulation (related to the slope of the tissue response function), TRFs from different tissues might in extreme cases even intersect at a certain time (thus exhibiting identical SUVs and zero image contrast at this moment) while having completely different kinetic properties. Tissue SUV stability is further compromised by not accounting for the sizable inter-study variability of arterial blood SUV which directly influences the actually obtained tissue uptake.
One quite extensively investigated way around the ‘snapshot problem’ is dual time point (DTP) investigations [7, 8] in which two successive whole-body scans are performed to obtain information regarding the rate of tracer accumulation. While being undoubtly valuable in discriminating between tumor and inflammation, quantitative evaluation of DTP measurements is usually restricted to computation of a so-called retention index, RI, representing the percentage change of SUVmax or SUVmean between early and late images (see, e.g., [9, 10]). However, the retention index, too, depends on the acquisition time of (and time difference between) early and late PET scan and, therefore, requires the same strict standardization as the SUV approach to provide useful quantitative measures. The retention index, too, is affected by the mentioned AIF variability at late times due to the evoked changes of the TRF slope.
There also have been attempts to directly use the TRF slope obtained in dynamic scans as a substitute for actual kinetic modeling [11, 12] while avoiding measurements of tracer concentration in blood. However, a convincing physiological interpretation of the slope parameter is missing. Furthermore, the approach suffers from the same problems as SUVs and retention index regarding the uncontrolled influence of the inter- and intra-subject variability of the AIF.
In this study, we propose a new assessment of DTP (and, more generally, TRF slope)-based methods with the aim of a quantitative determination of K m , the metabolic uptake rate of FDG. We demonstrate that starting from the Patlak model, one can derive an analytical relation between K m and the TRF slope m t , which only requires the image-based determination of the AIF during the respective late PET scans. The derived relation is especially compatible with dual time point whole-body investigations.
In this retrospective investigation, we evaluate the new approach in a group of patients with liver metastases of colorectal cancer for which K m was determined, both, by conventional Patlak analysis of the fully dynamic PET scans as well as by the newly developed approach.
where m t is the secant (or average) TRF slope in the chosen time interval centered at t 0 and (see Figure 1).
that approximates K m quite accurately (see Appendix and Figure 1).
We have compared these theoretical predictions with the actually observed relation between K s and K m in a group of patients with liver metastases for which fully dynamic scans were performed.
The investigated patient group included nine male subjects with liver metastases of colorectal cancer (mean age 62.8 years, range 48 to 76). For each patient, one to three dynamic PET scans of 60 min duration were performed (altogether 15 scans). Scans started immediately after injection of 346 to 430 MBq FDG. The scans were performed with an ECAT EXACT HR + (Siemens/CTI, Knoxville, TN, USA). The acquired data were sorted into 23 to 31 frames with 10 to 20 s duration during bolus passage, 30 to 150 s duration until 10 min post-injection (p.i.), and 300 s duration afterwards. Tomographic images were reconstructed using attenuation-weighted OSEM reconstruction (6 iterations, 16 subsets, 6 mm FWHM Gaussian filter).
Additionally, feasibility of the generation of parametric maps was demonstrated in a whole-body FDG investigation of a 63-year-old woman with bronchial carcinoma of the right lung and lymph node metastases of the right hilar region and the mediastinum (Philips Ingenuity TF PET/MR (Philips, Cleveland, OH, USA), injected dose 273 MBq, first scan 67 min p.i. (2 min per bed position), second scan 117 min p.i. (1.5 min per bed position)).
Region of interest (ROI) definition was performed using ROVER (ABX, Radeberg, Germany) [13, 14]. The AIF was determined from a roughly cylindrical three-dimensional (3D) ROI centered in the aorta using a concentric safety margin of at least 1 cm to exclude partial volume effects. 3D lesion ROIs were defined in 22 lesions, and the respective TRFs were computed. Further data analysis was performed using the R software for statistical computation .
For all 22 lesions, K m and V r were derived from the conventional Patlak analysis of the full dynamic data later than 20 min p.i. (at which time all Patlak plots already were linear). For comparison with the corresponding result of the subsequent DTP evaluation, τ a was determined from a mono-exponential fit to the complete AIF data in the time window used for the Patlak analysis. Variability of τ a and V r was expressed as mean ± standard deviation (SD).
where was calculated from the exponential connecting the two points and which yields .
Additionally, the retention index was computed as . Linear regression analysis was performed between K m and K s , , RI, and (the SUV of the lesions in the late image), respectively. Parametric images of K s , , and K m were generated for visual comparison after filtering of the DTP image data with a bilateral filter  (spatial filter width 9 mm, intensity filter width 2.5 SUV).
Influence of image noise
where the final ratio represents the relative SUV error of the second dual time point measurement. For Δ t ≈ 30 min and typical tumor accumulation rates of ≈ 2% to 4% per minute, one can thus estimate that the relative errors of K s are about 2.5 to 4 times higher than the corresponding SUV errors (the statistical error of is quite similar since the small correction term cannot contribute much to the total statistical uncertainty of . Although the noise in the parametric maps can thus be expected to be distinctly higher than that in the uptake images, the resulting visual quality is still quite satisfactory for reasonable choices of Δ t min) as will be demonstrated in the following.
Summary of parameters entering the K s and determination
Mean ± SD
V r (ml/ml)
0.53 ± 0.08
0.39 - 0.68
Dynamic τ a (min)
104 ± 20
79 - 156
DTP τ a (min)
99 ± 23
81 - 172
3.1 ± 0.7
2.4 - 4.5
Linear regression results: Pearson correlation coefficient r and the obtained regression parameters are shown
Slope (mean ± SD)
Intercept (mean ± SD)
K m vs. K s
0.98 ± 0.05
-0.54 ± 0.13
K m vs.
0.98 ± 0.04
0.04 ± 0.10
K m vs.
1.62 ± 0.48
2.81 ± 1.33
K m vs. RI
11.38 ± 1.34
5.44 ± 3.79
Figure 2B presents the correlation between K m and according to Equation 3. The correction term was computed using the average V r derived from the Patlak analysis of all 22 lesions, ml/ml, and individual (investigation-specific) time constants τ a derived from the DTP data (the independent determination of τ a from the full dynamic data in the Patlak time window - performed as a consistency check - yielded essentially the same result (104 ± 20 min (dynamic) vs. 99 ± 23 min (DTP)) but was not used further). As can be seen, the degree of linear correlation is distinctly improved in comparison to Figure 2A. Furthermore, the fitted straight line now essentially coincides with the line of identity. Consequently, the average difference between K m and amounts to only 1.4 ± 4.1% and exceeds 10% only in a single lesion.
Our main result is that in the investigated patient group, there is a very pronounced linear correlation K s = a + b · K m , where b is very nearly equal to one (see Figure 2A). This behavior is in complete agreement with the formalism presented in the Appendix, notably Equation 13: the variations of the (small) term V r / τ a should be essentially uncorrelated to K m so that a high (but slightly “noisy”) linear correlation between K m and K s with a slope near one is predicted. Furthermore, according to Equation 13, the modulus of the intercept, a=-0.0054 ml/min/ml, should be approximately equal to the average of V r /τ a in the investigated patient group. This prediction, too, is in complete agreement with the actual values of V r (determined from Patlak analysis) and τ a , namely ml/ml and min.
The second important finding is the fact that the degree of correlation as well as quantitative agreement between K m and K s can be further improved by assuming a reasonable constant value for (since V r is inaccessible in DTP measurements) and determining individually the rate of decrease, τ a , of the AIF (which can be estimated from the DTP measurement). The corrected K s , , exhibits an improved correlation to K m (due to compensation of the τ a variability) and also improved quantitative concordance as long as is roughly in accord with the individual true V r . Both phenomena are illustrated clearly in Figure 2B. The distinctly improved correlation (compared to Figure 2A) is achieved by the individual correction of the τ a influence. The residual deviations from the perfect correlation in Figure 2B are mainly due to the variability of V r . A nearly perfect quantitative agreement with K m is observed since was set to the mean of the actual V r values derived from the Patlak analysis. This obviously would not be possible when considering realistic DTP measurements (without a preceding complete dynamic study), and a less-than-perfect quantitative agreement should be expected in this case. Nevertheless, as the comparison of Figure 2A,B suggests, performing the correction with some roughly correct value for will always decrease the bias between K s and K m .
The rather small variability of tumor V r observed in the present investigation might seem surprising. However, the square of k 2 / (k 2 + k 3) appearing in Equation 6 will never deviate very much from unity since for FDG, k 2 quite generally is distinctly larger than k 3. The variability of V r is thus mostly controlled by the first term, K 1 / k 2. Since both K 1 and k 2 are usually identified as being associated with the facilitated diffusion across the cell membrane, it might very well be expected that the ratio K 1 / k 2 is essentially constant, independent of the actual K 1. This might be the underlying reason for the low variability of V r observed in this study. Whether V r variability is higher in other tumors remains to be investigated, but we believe this to be unlikely. V r should never be much larger than about 0.6 to 0.7 ml/ml which appears to be a rough upper bound for the K 1 / k 2 ratio. According to our own data, this is true, e.g., in the human brain (K 1 / k 2 ≈ 0.1 / 0.15 = 0.67 ml/ml) as well as the myocardium (K 1 / k 2 ≈ 0.6 / 1.4 = 0.43 ml/ml). V r in these organs is rather low (≈ 0.3 ml/ml) due to the large k 3 in both tissues.
The very high correlation between (or K s ) and K m is to be compared with the markedly inferior correlation between K m and late SUV () and retention index RI, respectively (Figure 3). Since all these parameters are ultimately intended as surrogate parameters of K m , the superiority of K s seems obvious. Since in the present study the retention index is computed from exactly the same DTP tissue data as K s , it is worth to point out that the sole factor responsible for the much better K s (K m ) correlation is adequate consideration of the substantial inter-subject variability (see Table 1). Indeed, one could write , where Δ t is just a constant in the present context. To some extent, K s might thus be considered just a more sensible definition of a retention index where the uptake difference Δ c t is normalized to (as well as Δ t) instead of .
The observed very low correlation between late SUV and K m is caused by the six data points with SUV >8 in Figure 3A. Leaving these six points out increases the correlation coefficient to 0.94 which is in good agreement with published data . Closer inspection revealed exceptionally high values (see inset graphic in Figure 3A) for the affected data points which might have physiological reasons but could also hint at erroneous SUV calibration (for which, however, a retrospective inspection did not find any evidence). In any case, the data demonstrate the high sensitivity of SUV evaluations to variations of the AIF level and incorrect SUV calibration.
The comparison of uptake and parametric images in Figure 4 demonstrates that K s as well as reproduces the essential features of the Patlak K m image, notably the increased contrast between metastasis and liver background. Regarding the targeted lesions, the image is, moreover, in good quantitative agreement with the K m image and could thus serve as a basis for regional quantitative evaluation. We, therefore, believe it is worthwhile to investigate the potential suitability of as a quantitative estimator (and not just a surrogate) of K m more thoroughly in future studies. Figure 5 demonstrates that reasonable statistical quality of the map can in fact be achieved in whole-body DTP investigations as well.
Compared to more conventional approaches, our approach has several relevant benefits. The most important one in our view is the potential to perform fully quantitative whole-body investigations based on a DTP acquisition. The only additional prerequisite is identification of the aorta or left ventricle in the DTP data. One gains the ability to directly identify regions of elevated irreversible FDG metabolism and to put the established DTP approach on a quantitative basis. A further advantage is the implied correction for the sizable inter-subject variation of the blood tracer concentration (SUV range, 2.4-4.5 in this study). The latter correction alone clearly improves the correlation between the derived parameter (K s ) and the targeted one (K m ). Another important aspect is elimination of the dependence of SUV uptake and retention index on the time of measurement(s). To the extent that the Patlak model can be considered valid (negligible k 4), the proposed procedure yields a time-independent result, namely a direct estimate of the invariant rate K m which prospectively should allow definition of improved, objective reference values. A further implication is elimination of any intra-scan time dependence in whole-body/multi-bed studies. Last but not least, the issue of ensuring correct SUV calibration is eliminated since all calibration factors cancel out when performing an image-based determination of both TRF slope and . This observation seems especially relevant for multi-center studies.
We have demonstrated that it is possible to derive a quantitative estimate of K m , the metabolic trapping rate of FDG, solely from a dual time point measurement. We believe this approach to be of potential relevance especially in the context of oncological whole-body investigations where the required AIF information is available in the field of view (aorta or left ventricle). In this case, the approach eliminates most if not all issues of static SUV and conventional dual time point imaging regarding the influence of the chosen scan times relative to the time of injection and the substantial influence of inter-study variability of the AIF. Consequently, the derived parameters K s and exhibit a much improved stability and much better correlation with the true K m . These properties might prove especially relevant in the context of radiation treatment planning and therapy response control. Whether this is indeed the case has to be investigated in appropriate future studies.
where τ 0 is always equal to one. The parameters τ n > 0 are constructed in such a way that for a mono-exponential decrease of c a (t) near t 0, we obtain τ n > 0 = τ a , where τ a is the time constant of the exponential. Actually, it is known that starting rather early after bolus injection (t > 20 min), c a (t) can be reasonably well described by a slow mono-exponential decrease with a time constant τ a ≈ 100 min (in the present study, we found an average value of τ a = 99 min, while a value of τ a = 80 min was reported in ).
The detailed derivation of Equation 12 is presented in an additional file (see Additional file 1). The factors in square brackets deviate only minimally from one up to even quite large values of Δ t. For the sake of simplicity, we will demonstrate this only for the well-established approximately mono-exponential behavior of c a (t) at later times but emphasize that the conclusions remain the same when using other reasonable parametrizations of the observed shape of the AIF at later times (e.g., by an inverse power law).
In other words, the secant slope is to a very good approximation equal to the instantaneous slope at t 0 and thus can be used instead. This in turn implies that the average slope of the TRF (derivable, e.g., by a least squares fit of a straight line in the considered time window), too, is very nearly identical to m t . Note that these conclusions are valid even if varies considerably over the considered time interval (see Figure 1). Formally, this result is identical to stating that a second-order Taylor expansion of c t (t) around t 0 turns out to be sufficiently accurate within t 0 ± Δ t / 2.
The quantitative relation between K s and K m is investigated in Figure 6. For this figure, we computed K m and V r over a range of sensible choices for the transport constants K 1, k 2, and k 3. The resulting K m and V r (top row of Figure 6) are used to compute K s from Equation 13 for a realistic value of τ a (we chose τ a = 99 min). The bottom row in Figure 6 compares the true K m to K s .
As can be seen (bottom right), the fractional deviation of K s from K m becomes large only when k 3 is very small (i.e., when there is virtually no trapping). Overall K s is a negatively biased estimator of K m , but an approximate correction of the bias is possible considering the following.
Determine the secant TRF slope m t in the time interval from a dual time point measurement of c t (t) starting at sufficiently late times after injection, typically t > (20-30) min.
Estimate and τ a from the exponential connecting the two time points t - , t +.
Compute a correction term using the individually determined τ a and a fixed value for the distribution volume. In the absence of any specific information regarding V r in the investigated tumor entity, we propose to use the average V r determined in this study, i.e., ml/ml.
- 5.Finally, compute the corrected K s , i.e.,(15)
as a quantitative estimate of the true K m .
According to Equations 14 and 15, is equal to K m if (irrespective of the values of K 1 - k 3 yielding this V r value). Therefore, remains a very good approximation of K m as long as V r does not deviate too much from the assumed value. This behavior is illustrated in Figure 7.
aFor completeness, we mention that in the presence of sizable fractional blood volume (fbv), the substitutions and would have to be performed in Equation 5 where the ‘asterisked’ quantities would be the experimentally accessible ones.
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