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 , 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. Twentytwo lesion regions of interest (ROIs) were evaluated. c_{a}(t) was determined from a threedimensional 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 wholebody investigation.
Results
Patlak analysis yielded a mean V_{r} of ml/ml. The patient averaged τ_{a} was 99 ± 23 min. Linear regression between Patlakderived K_{m} and DTPderived 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.
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 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.
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 (see Figure 1).
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 , K_{s} differs from K_{m} only by a τ_{a} dependent offset that can be added to K_{s} to obtain a corrected 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 maps was demonstrated in a wholebody FDG investigation of a 63yearold 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)).
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 and Equation 2, K_{s} is given by
where was calculated from the exponential connecting the two points and which yields .
For computation according to Equation 3, we fixed V_{r} to the mean of the Patlak evaluation for all lesions ( ml/ml), while τ_{a} was estimated individually for each study from the exponential connecting and as
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 [16] (spatial filter width 9 mm, intensity filter width 2.5 SUV).
Influence of image noise
Considering a single voxel and neglecting the (much smaller) statistical error of the ROIbased value, it follows from Equation 4 that the relative statistical error of K_{s} is equal to that of Δc_{t} and thus, by Gaussian error propagation,
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 , one can get a rough estimate of the error according to
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 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 Δt min) as will be demonstrated in the following.
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 determination
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 . Solid black lines represent the line of identity; dashed lines represent the least squares straight line fits to the data.
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 (investigationspecific) 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.
For comparison, Figure 3A,B presents the correlations between K_{m} and the late SUV uptake , and K_{m} and the retention index RI, respectively. Obviously, the correlation between K_{m} and is rather poor. The correlation between K_{m} and the RI is substantially higher but still clearly below the degree of correlation between K_{m} and K_{s} or . The correlation in Figure 3A is clearly distorted by the group of the six highest observed SUV values which correspond to only moderately high K_{m} values. This phenomenon might be explained by the exceptionally high values observed in the respective patients (see inset graphic in Figure 3A).
Figure 3. Correlation between the metabolic rate K_{m }and Standard Uptake Value and retention index, respectively. (A) K_{m} and (in SUV units) and (B) K_{m} and retention index (in percent). is the tissue uptake at t = 55 min. Dashed lines represent the least squares straight line fits to the data. The inset graph in (A) additionally provides the correlation between and , which helps to explain the group of data points above the regression line with SUV > 8 (see the ‘Discussion’ section).
Figure 4A provides one example of a lesion uptake image, and the corresponding parametric images of K_{s}, , and K_{m} are shown in Figure 4B,C,D, respectively, all of which are displayed in a common scale. The comparable, enhanced targettobackground contrast of the three parametric images relative to the uptake image is obvious. In agreement with the theoretical expectation and the ROI data in Figure 2, there is good quantitative concordance between K_{m} and , while K_{s} exhibits a constant negative bias of about K_{0} ≈ 0.5 ml/min/100 ml in comparison to K_{m}.
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) , and (D) K_{m}, respectively, are shown. The means of the rate constants over the lesion are as follows: K_{s} = 2.23 ml/min/100 ml, ml/min/100 ml, and K_{m} = 2.90 ml/min/100 ml, demonstrating a quite satisfactory quantitative agreement between and K_{m}.
Finally, Figure 5 demonstrates the feasibility of generating parametric maps of reasonable statistical quality for a typical DTP wholebody FDG study.
Figure 5. Comparison of parametric images of SUV and.(A) FDG SUV 67 min. p.i. and (B) parametric map in a 63yearold woman. The patient has a bronchial carcinoma of the right lung and lymph node metastases of the right hilar region and the mediastinum (coronal maximum projection). Note the different regional contrast and the used units in both images.
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 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 lessthanperfect 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.
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, , can then be expected to be a less biased estimate of K_{m} than K_{s} over a substantial range of actually realized V_{r} values between about 0.2 and 0.7 ml/ml (see Figures 6 and 7). Whether does offer any advantages over K_{s} in terms of clinical relevance remains to be seen, but the improved correlation with K_{m} seems justification enough to perform the correction.
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 }andfor τ_{a }= 99 min and an assumed distribution volumeml/ml. In comparison with Figure 6, the systematic difference relative to the true K_{m} is mostly removed even when V_{r} deviates distinctly from the assumed value. Moving along the line, V_{r} ≈ 0.55 ml/ml between K_{m} = 1 and 4 ml/min/100ml corresponds approximately to the experimental data of Figure 2B.
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 intersubject variability (see Table 1). Indeed, one could write , where Δt 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 Δc_{t} is normalized to (as well as Δt) 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 [17]. 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 wholebody 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 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 . This observation seems especially relevant for multicenter studies.
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 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.
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 : nth derivative at t = t_{0}):
Introducing the parameters τ_{n} defined by , Equation 8 can be rewritten as
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 as well as to have knowledge of 1 / τ_{1} (the fractional rate of decrease of the AIF at t = t_{0}). Obviously, direct determination of the time derivative at t = t_{0} is not feasible in real (noisy) data. On the other hand, it is not clear whether the average slope over a necessarily rather large neighborhood (required for reasons of limited time resolution and count rate statistics) is an acceptable approximation of (since the slope changes over time). For investigation of this question, we compute from Equation 5 the difference for two time points lying symmetrically around t_{0} at a finite (possibly large) distance Δt
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.
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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 remains larger than T^{∗}. Defining the secant slope m_{t} between and
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 varies considerably over the considered time interval (see Figure 1). Formally, this result is identical to stating that a secondorder Taylor expansion of c_{t}(t) around t_{0} turns out to be sufficiently accurate within t_{0} ± Δt / 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.
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 from a dual time point measurement of c_{t}(t) starting at sufficiently late times after injection, typically t > (2030) min.
2. Estimate and τ_{a} from the exponential connecting the two time points t^{  }, t^{+}.
4. 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.,
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.
Endnote
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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