Abstract
Background
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 [^{18}F]fluorodeoxyglucose as a measure of glucose consumption.
Methods
Starting from the Patlak equation, it is shown that
Results
Patlak analysis yielded a mean V_{r} of
Conclusion
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 interstudy variability of the input
function. K_{s} and
Keywords:
Wholebody PET; Dual time point; Metabolic rate of FDG; PET quantification; Tracer kinetic modelingBackground
For many years, quantification of the metabolic rate of glucose consumption with dynamic [^{18}F]fluorodeoxyglucose (FDG) positron emission tomography (PET) using the socalled 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 wholebody 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 [3] for an overview).
The approach, however, has known shortcomings [46]. 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 interstudy 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 wholebody 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 socalled retention index, RI, representing the percentage change of SUV_{max} or SUV_{mean} 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 intrasubject 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 imagebased determination of the AIF during the respective late PET scans. The derived relation is especially compatible with dual time point wholebody 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.
Methods
Theory
It is well known that the TRF after a bolus injection of FDG appears to be approximately linear at later times. Closer inspection, however, reveals, that the curve exhibits a finite curvature: the slope decreases with time due to the continuously decreasing AIF (see Figure 1). In the Appendix, we demonstrate that for times t when the Patlak equation is valid (usually for t > 2030 min), the ratio between the instantaneous values of TRF slope and AIF level can be expressed in terms of the parameters K_{m} and V_{r} of the Patlak model and the time constant τ_{a} describing the essentially monoexponential decrease of the AIF in the considered time window. It is shown in the Appendix that the TRF slope at t = t_{0} is very nearly identical to the slope of the secant connecting the boundary points of a finite symmetric time interval around t_{0} (and also to the average slope in this interval).
Figure 1. AIF plus TRF calculated for K_{1 }= 0.3 ml/min/ml, k_{2 }= 0.5 /min, k_{3 }= 0.08 /min. The TRF does not become linear at later times but exhibits a visible curvature. However, the slope at some time point t_{0} (t_{0} = 60 min in this example) is nearly identical to the slope of the secant connecting the boundary points of a finite time interval centered at t_{0}. The AIF is scaled such that c_{a}(t_{0}) = 1. The data accessible in a DTP measurement are indicated by the square plotting symbols. For further details, see the main text.
One finally arrives at the relation
with
where m_{t} is the secant (or average) TRF slope in the chosen time interval centered at t_{0} and
The rate K_{s} defined by Equation 2 (i.e., the ratio between the TRF slope and AIF level at time
t_{0}) can be determined from measurements during the late phase alone. Contrary to the
Patlak method, knowledge of the full AIF is not required. To the extent that K_{s} ≫ V_{r} / τ_{a}, K_{s} might directly serve as an (negatively biased) approximation of K_{m}. Moreover, to the extent that V_{r} can be replaced by a suitable constant value
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.
Study sample
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 postinjection (p.i.), and 300 s duration afterwards. Tomographic images were reconstructed using attenuationweighted OSEM reconstruction (6 iterations, 16 subsets, 6 mm FWHM Gaussian filter).
Additionally, feasibility of the generation of parametric
Data evaluation
Region of interest (ROI) definition was performed using ROVER (ABX, Radeberg, Germany) [13,14]. The AIF was determined from a roughly cylindrical threedimensional (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 [15].
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 monoexponential 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).
Dual time point data were generated from the data 20 to 30 min and 50 to 60 min p.i.,
yielding two pairs of c_{a} and c_{t} values which were assigned to the respective frame centers t^{  / + } = 25 / 55 min (which corresponds to t_{0} = 0.5 · (t^{} + t^{+}) = 40 min and Δt = t^{+}  t^{} = 30 min). Using the abbreviations
where
For
Additionally, the retention index was computed as
Influence of image noise
Considering a single voxel and neglecting the (much smaller) statistical error of
the ROIbased
which decreases with increasing concentration difference Δc_{t}. Taking into account that measurement times of both dual time point measurements
might be adjusted in such a way that
where the final ratio represents the relative SUV error of the second dual time point
measurement. For Δt ≈ 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
Results
The obtained results are summarized in Tables 1 and 2. Figure 2A shows the correlation between Patlakderived K_{m} and K_{s}. The solid line is the line of identity, and the dashed line is the linear regression result. The linear correlation is very good, and the fitted slope is identical to one within the given error limits of about 5% (Table 2). The fitted intercept of 0.54 ml/min/100 ml thus represents the experimentally observed average underestimate of the true K_{m} by K_{s}.
Table 1. Summary of parameters entering the K_{s }and
Table 2. Linear regression results: Pearson correlation coefficient r and the obtained regression parameters are shown
Figure 2. Correlation between the metabolic rate K_{m }and the DTPderived rate constants. (A) K_{m} and K_{s} and (B) K_{m} and
Figure 2B presents the correlation between K_{m} and
For comparison, Figure 3A,B presents the correlations between K_{m} and the late SUV uptake
Figure 3. Correlation between the metabolic rate K_{m }and Standard Uptake Value and retention index, respectively. (A) K_{m} and
Figure 4A provides one example of a lesion uptake image, and the corresponding parametric
images of K_{s},
Figure 4. Liver metastasis of a colorectal carcinoma exhibiting a central necrosis. (A) A representative sagittal slice of uptake and the corresponding parametric images
of (B) K_{s}, (C)
Finally, Figure 5 demonstrates the feasibility of generating parametric
Figure 5. Comparison of parametric images of SUV and
Discussion
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
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
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.
We surmise, therefore, that V_{r} in tumors (and healthy tissue) will never deviate too much from the value of 0.53,
ml/ml used for K_{s} correction in this study. The corrected rate,
Figure 6. Visualization of the difference between K_{m }and K_{s }. The plots cover a substantial range of the parameters k_{2} and k_{3}, assuming a fixed ratio K_{1} / k_{2} = 0.7 ml/ml (a rationale for fixing this ratio is given in the discussion above). Top left, K_{m}; top right, V_{r}; bottom left, absolute difference (K_{m}  K_{s}); and bottom right, fractional difference ((K_{m}  K_{s}) / K_{m}). Parameters and their respective units: k_{2}, k_{3} (1/min); V_{r} (ml/ml); and K_{m}, K_{s} (ml/min/100 ml). Moving along the line, V_{r} ≈ 0.55 ml/ml between K_{m} = 1 and 4 ml/min/100 ml corresponds approximately to the experimental data of Figure 2A.
Figure 7. Visualization of the difference between K_{m }and
The very high correlation between
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 [17]. Closer inspection revealed exceptionally high
The comparison of uptake and parametric images in Figure 4 demonstrates that K_{s} as well as
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
wholebody 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 intersubject variation of the blood tracer concentration
(SUV range, 2.44.5 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 timeindependent 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 intrascan time dependence in wholebody/multibed
studies. Last but not least, the issue of ensuring correct SUV calibration is eliminated
since all calibration factors cancel out when performing an imagebased determination
of both TRF slope and
Conclusion
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
wholebody 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 interstudy variability of the AIF. Consequently, the derived parameters K_{s} and
Appendix
We start with the standard Patlak formula but avoid division by c_{a}(t):^{a}
where K_{m} is the metabolic trapping rate, defined by
and V_{r} is the apparent volume of distribution defined by
Equation 5 is valid for times t > T^{∗} where T^{∗} ≈ 20 to 30 min p.i.. Utilization of this equation for K_{m} determination requires measurements of the TRF only for t > T^{∗} but measurement of the complete AIF starting at time zero. We now want to eliminate the dependency on measurements prior to T^{∗}. By taking the time derivative at some time point t > T^{∗}, it follows directly from Equation 5 that
or after division by c_{a}(t) (suppressing the t argument)
Focusing on some specific time point t = t_{0}, we use the Taylor expansion of c_{a}(t) around
Introducing the parameters τ_{n} defined by
where τ_{0} is always equal to one. The parameters τ_{n > 0} are constructed in such a way that for a monoexponential 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 monoexponential 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 [18]).
Inserting the Taylor expansion from Equation 9 into Equation 7, we get (
In order to derive K_{m} from this equation, we need to reliably estimate
with
Replacing all occurrences of c_{a}(t) in Equation 11 by the Taylor series in Equation 9 (neglecting fourth and higher order terms) and executing the integration separately for each term of the series yield after some straightforward but lengthy calculations the following equation:
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 Δt. For the sake of simplicity, we will demonstrate this only for the wellestablished approximately monoexponential 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).
Additional file 1. Derivation of Equation 12. A pdf file showing the complete derivation of Equation 12 using Taylor expansion.
Format: PDF Size: 111KB Download file
This file can be viewed with: Adobe Acrobat Reader
As already pointed out, for a monoexponential decrease of c_{a}(t), all τ_{n > 0} coincide with the time constant τ_{a} of the exponential. Consider, then, choosing Δt = 60 min in Equation 12. Since τ_{a} ≈ 100 min, we have for both square brackets 1 + 1 / 24 · 0.6^{2} = 1.015. It is, therefore, permissible to replace both square brackets by one. This yields
Thus, Δc_{t} is to a very good approximation proportional to Δt. Δt can become quite large, e.g., Δt = 1 h, as long as the lower bound
and introducing the rate constant K_{s}
for the ratio of the secant slope and the blood concentration at t_{0}, we get
or
Comparison of Equation 13 with Equation 10 yields the important result
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
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.
According to Equation 14, the V_{r} and K_{m}  K_{s} maps in Figure 6 differ only by a constant factor τ_{a} (and a conversion factor of 100 due to the chosen units of ml/min/100 ml for K_{m} and K_{s}). Moreover, V_{r} does vary only modestly in comparison to the individual rate constants and to K_{m} (except when k_{3} becomes distinctly larger than k_{2}, but this is not observed in real data). Therefore, K_{m}  K_{s} does not vary much across the relevant part of the k_{2} / k_{3} plane. We, therefore, hypothesize that the difference K_{m}  K_{s} can be actually treated to be approximately constant. Consequently, we propose to estimate K_{m} using only late time measurements of c_{a}(t) and c_{t}(t) as follows:
1. Determine the secant TRF slope m_{t} in the time interval
2. Estimate
3. Compute
4. Compute a correction term
5. Finally, compute the corrected K_{s}, i.e.,
as a quantitative estimate of the true K_{m}.
According to Equations 14 and 15,
Endnote
^{a}For completeness, we mention that in the presence of sizable fractional blood volume
(fbv), the substitutions
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
JVDH derived the theoretical background, performed part of the data analysis, and is the main author of the manuscript. FH performed part of the data analysis and wrote part of the manuscript. LO and GS contributed to the derivation of the theoretical background. JL and BBB performed the PET measurements. JS and JK provided intellectual input and reviewed the manuscript. All authors read and approved the final manuscript.
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