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Abstract
The distribution volume ratio (DVR), which is a linear function of receptor availability, is widely used as a model parameter in imaging studies. The DVR corresponds to the ratio of the DV of a receptor-containing region to a nonreceptor region and generally requires the measurement of an arterial input function. Here we propose a graphical method for determining the DVR that does not require blood sampling. This method uses data from a nonreceptor region with an average tissue-to-plasma efflux constant k2 to approximate the plasma integral. Data from positron emission tomography studies with [15C]raclopride (n = 20) and [11C]d-threo-methylphenidate ([11C]dMP) (n = 8) in which plasma data were taken and used to compare results from two graphical methods, one that uses plasma data and one that does not. k2 was 0.163 and 0.051 min−1 for [11C]raclopride and [11C]dMP, respectively. Results from both methods were very similar, and the average percentage difference between the methods was −0.11% for [11C]raclopride and 0.46% for [11C]dMP for DVR of basal ganglia (BG) to cerebellum (CB). Good agreement between the two methods was also achieved for DVR images created by both methods. This technique provides an alternative method of analysis not requiring blood sampling that gives equivalent results for the two ligands studied. It requires initial studies with blood sampling to determine the average kinetic constant and to test applicability. In some cases, it may be possible to neglect the b̅2 term if the BG/CB ratio becomes reasonably constant for a sufficiently long period of time over the course of the experiment.
Most techniques currently in use to quantify positron emission tomography (PET) images involve arterial blood sampling to construct an input function for the determination of model parameters. This is an invasive procedure that places stress on certain classes of patients, particularly the elderly, and can also present an added risk to health workers when studying subjects in high-risk categories for HIV and other blood-borne infections. For these reasons, we have investigated the possibility of eliminating blood sampling from some studies with receptor ligands for which we routinely report a distribution volume ratio (DVR) that is the ratio of the DV in a receptor region to the DV in a non-receptor-containing region. The DV is a linear function of free receptor concentration and is therefore frequently used as a parameter of comparison for studies with reversibly binding radiotracers (Dewey et al., 1993; Holtoff et al., 1991; Koeppe et al., 1991; Carson et al., 1993). The DV is also proportional to the ratio of transport constants (plasma to tissue and tissue to plasma), which is a function of plasma protein binding (Carson et al., 1993; Logan et al., 1994). The dependence upon plasma protein binding is eliminated by taking the ratio of DVs from a receptor-containing region to a nonreceptor region (Logan et al., 1994). There are several approaches to the determination of the DV, but they all require some amount of blood sampling and the determination of a metabolite correction for the total radioactivity to obtain a measure of the amount of unchanged radiotracer in plasma (Logan et al., 1990; Koeppe et al., 1991; Carson et al., 1993). In this study, we compute directly the DVR, which is the parameter we use in assessing receptor availability with a modification of the graphical method (requiring the plasma integral for unchanged tracer at each scan time) (Logan et al., 1990). In this modified method, PET data from a non-receptor-containing region and an average effective tissue-to-plasma efflux constant (k2) for that region are used to approximate the plasma integral. The use of the average k2 (b̅2) minimizes the small time dependence of the region-of-interest (ROI) ratio of receptor to nonreceptor regions. A similar approach in which plasma data were replaced with data from a nonreceptor region was suggested by Patlak and Blasberg (1985) in the graphical analysis of irreversible ligands. Recently Ichise et al. (1995) have proposed methods for obtaining the DVR without blood sampling using [123I]iodobenzofuran with single photon emission computed tomography, although no comparison with results using plasma data was given. Lammertsma et al. (1996) have also proposed a method without blood sampling applied to PET studies with [11C]raclopride. This method uses the cerebellum (CB) as a reference region but also requires a nonlinear regression analysis to determine four parameters including Bmax/Kd.
We have applied this graphical method for reversible ligands to [11C]raclopride (Farde et al., 1985), which binds to dopamine (DA) D2 receptors, and to [11C]d-threo-methylphenidate ([11C]dMP) (Ding et al., 1994), which binds to the DA transporter. The kinetics of these ligands differ due to differences in rates of washout from the non-receptor regions (a result of differences on the order of a factor of 3 in b̅2).
The calculation of the DVR is compared using the measured arterial plasma input function and the method based on the non–receptor region. In these studies an ROI from the CB was used as the non–receptor region since there is negligible concentration of DA receptors or transporters in this region.
THEORY AND METHODS
The equation used in the graphical analysis of reversible systems (Logan et al., 1990) is given by
| (1) |
where A(t) is the radioactivity measured by the PET camera at time t in a specified ROI (averaged over some number of pixels) or in a single pixel and Cp(t) is the radioactivity in plasma due to unmetabolized radioligand. If a time t* can be found such that the time dependence of the intercept is sufficiently small, then for times T > t*, the DV can be found as the slope of a plot of ∫ToA(t)dt/A(T) vs. ∫CpA(t)dt/A(T). The time t* can be determined as the time after which no further significant increases in slope are observed.
To distinguish the non–receptor region, the graphical analysis equation can be written as
| (2) |
in which CB is used to designate the radiotracer concentration in the non-receptor region. For the ligands used in these studies, CB is an ROI from the cerebellum. The DV and intercept are λ and −1/b̅2, respectively. Although Eq. 2 does not specify a particular model, the uptake and loss of ligand in the absence of receptors can generally be described by a two-compartment model (Wong et al., 1984):
| (3) |
where K1 is the plasma-to-tissue influx constant, which is a function of blood flow, the permeability-surface area, and plasma protein binding, and k2 is the efflux constant, also a function of non-specific binding. For the two-compartment model, λ = K1/k2. Rearranging Eq. 2 and representing the intercept in Eq. 2 as −1/b̅2 in place of −1k2:
| (4) |
Substituting for ∫ToCpdt in Eq. 1 gives
| (5) |
The DVR is DV/λ = (B′max/K′d + 1) where B′max is the free receptor concentration (receptors not occupied by endogenous neurotransmitter) and K′d is the effective equilibrium dissociation constant (K′d = Kd/fNS, where fNS is the free fraction of tracer) (Logan et al., 1994). Replacing k2 in Eq. 5 with its average value k̅2
| (6) |
where 8 is the error term given by δ = D1/k̅2)CB(T)/A(T) and CB(T)/A(T) → 1/DVR so that the time dependence of the error term should be small after some time t*. The slope determined from the linear portion of a plot of (for T > t*) ∫ToA(t)dt/A(T) vs. [∫To CBT(t)dt + CB(T/k̅2)]/A(T) gives directly the DVR of the ROI to that of the CB. The DVR can also be obtained without the use of k̅2 if the ratio A(T)/CB(T) is reasonably constant, that is,
| (7) |
so that the intercept is now
| (8) |
Although it would be simpler to use the ratio A(T)/CB(T) as the DVR, it does not, in fact, represent the true DVR even when constant except in special cases. This is because the tissue-to-plasma ratio A(T)/Cp(T) is a function of the exponent describing the decline in plasma radioactivity with time. For a three-compartment model, this can be expressed (see Table 5, Logan et al., 1990) as
| (9) |
where the eigenvalues α± are given by
| (10) |
and K = (k2 + k3 + k4) (B′max/K′d = k3/k4, where k4 is the receptor-ligand dissociation constant). Unless ‖α+ ‖ ≫ βq where βq is the smallest plasma exponent, A(T)/Cp(T) gives a value larger than the DV. For non-receptor regions, the smallest eigenvalue should be greater than for the receptor region so that the ratio CB(T)/Cp(T) may be closer to the true DV. In any case, the ratio A(T)/CB(T), even when it becomes constant, will generally overestimate the DVR. These problems can be avoided by using a controlled infusion rate of the tracer (Patlak and Pettigrew, 1976; Carson et al., 1993), but this is experimentally more complex than the bolus injection method upon which this work is based.
We have investigated the limits of applicability of this graphical method using two reversible ligands with different kinetics in the rate of loss from non-receptor regions, which is controlled by k2. All data used in these calculations were from human studies with [11C]raclopride (Volkow et al., 1994) and [11C]dMP(Volkow et al., 1995) and carried out on a whole-body, high-resolution positron emission tomograph (CTI 931; 15 planes, transaxial full width at half-maximum 6.5 mm, interslice distance 6.7 mm). Transmission scans were obtained to correct for attenuation. Arterial blood sampling was done for all studies. The amount of unchanged tracer present in plasma was determined by solid phase extraction using a laboratory robot (Alexoff et al., 1995). Experimental details for these studies can be found in Volkow et al. (1994, 1995).
Since the greatest uptake of these two ligands is in the basal ganglia (BG), which has a large concentration of both DA D2 receptors and transporters, ROI calculations were limited to this structure. Regions with less uptake would be subject to smaller error since the ratio CB(T)/A(T) for these regions would approach a constant value (steady state) in a shorter time. For use in the DVR calculation, average values of k2 for [11C]raclopride (n = 20) and [11C]dMP (n = 8) were determined graphically with measured plasma input functions and used in Eq. 7. (Although the designation “k2” is used to be consistent with the terminology of the two-compartment model, the assumption of a two-compartment model for CB is not necessary since k2 was calculated graphically from the intercept.) To assess the sensitivity of the value of k2 used, we calculated the DVR without the correction using Eq. 7 and for a range of k2 values for [11C]dMP.
A more general comparison was made by constructing parametric images of trie DVR. In this case, instead of an ROI, calculations were done for each pixel in the reconstructed image falling within the brain. That part of the field of view falling within the brain was determined by masking the reconstructed images with a binary representation of the brain shape for each plane. These binary images were obtained by averaging scans up to 6 min and applying a cutoff of 0.25 of the maximum value in the plane to separate brain from nonbrain and to create a binary image. The brain area was filled in using morphological transformations. Image calculations were done using pixels falling within this binary image. For comparison, DVR images were generated by first constructing DV images using the measured plasma input function and Eq. 1. These were then divided by the DV of the CB ROI to create DVR images. For both the [11C]raclopride BG data and the DVR images, the slopes of the graphical analysis were determined as the average of slopes for times 22–60, 22–55, 27–60, and 27–55 (min). For [11C]dMP the times used in the calculation of DVRs were 29–75 and 29–60 (min). These times were chosen by observing the times at which the plots became linear and the calculated slope appeared to be independent of the initial time. Averaging over the different end time points was done to minimize any possible effect of statistical fluctuations due to lower count rates at the last time point. This was a potential problem mainly in the parametric image construction, which is done on a pixel-by-pixel basis. No approximate blood volume correction was made in the data prior to the DVR calculations using plasma data since it was not possible to make a blood volume correction using the graphical technique presented here. Comparison with data for which a blood volume correction has been made indicates a systematic difference between the methods.
RESULTS AND DISCUSSION
[11C]Raclopride exhibits rapid clearance from non-receptor regions with b̅2 in CB of 0.163 ± 0.036 min−1, which corresponds to a half-time of 4 min, while [11C]dMP exhibits a slower loss from nonreceptor regions b̅2 = 0.051 ± 0.007 min−1, with a half-time of 13 min. It is this clearance constant that controls how rapidly the non-receptor region reaches a steady state, although as discussed previously, a constant tissue-to-plasma ratio does not necessarily yield the true DV particularly for receptor regions. For example, for raclopride the BG/CB ratio at the last scan time (60 min) exceeds the DVR determined using plasma and tissue uptake data by an average of 12 ± 6%. Although it would be expected that the BG/CB ratios for dMP would exceed the DVR to an even greater extent, they were found to be 5 ± 10% at the last time point, probably indicating that the BG/plasma ratio had not quite reached a constant value. In any case when using bolus injection protocols, some modeling technique such as the one described here is required to determine the DVR.
A summary of the comparison of DVRs using both graphical techniques for ROIs from the BG is presented in Table 1. The average percentage change between DVRs calculated using the CB with the average k2 and the DVR calculated from the ratio of DVs determined using the measured plasma integral is close to 0 for both [11C] raclopride and [11C]dMP. The absolute percentage difference is slightly higher for [11C]dMP although still small. The small percentage differences are indicative of random errors with little or no systematic error introduced with the use of an average k2. The greater sensitivity of dMP to the value of b̅2 used in the DVR analysis is apparent in the difference between results calculated using Eq. 6 compared with Eq. 7. The average percentage change for dMP between the DVR calculated using plasma data and that calculated from CB data without b̅2 is — 12% compared with — 1.8% for raclopride. The same difference (12 and 2% for dMP and raclopride, respectively) is observed when comparing absolute values of percentage change, indicating the tendency to underestimate the DVR when the k2 term is not included in Eq. 7. The difference between the result with and without the term CB(T)/b̅2 is minimized by extending to somewhat later times the initial time for the slope calculation so that the ratio A(T)/CB(T) (which is included in int′ in Eq. 7) is changing slowly with time. The intercept (int in Eqs. 1, 5, 6) becomes effectively constant before A(T)/Cp(T) [or A(T)/CB(T)] becomes constant (see Logan et al., 1990) so that earlier time points can be used when calculating the DV or the DVR using Eq. 6.
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TABLE 1. Comparison of differences between DVRs calculated graphically using a plasma input function and calculated using an average k2 and a region from the CB for n subjects
The variation of the DVR with deviations of k2 from the average value was investigated with simulations. Simulated CB data representative of dMP were generated using Eq. 3 with λ = K1/k2 = 9.2 and varying k2 from a baseline value of 0.051 min−1 by ±25 and +50%, maintaining λ constant. The DVR for a BG ROI was then determined using the simulated CB data but with the average k2 value (0.051 min−1) instead of the value actually used to generate the data. The variations in the DVR from the “true” value (2.96) are presented in Table 2. Variations of ±25% from the original k2 value of 0.051 min−1 resulted in changes in the DVR of 3–4%. Most values of k2 would be expected to fall within this range given that the coefficient of variation (SD/avg × 100) for dMP is on the order of 15. Therefore, only small random errors would be expected to result from the use of the CB ROI with average k2 in place of the measured plasma input function. By extending the times for which the slope was calculated to 39–75 min, the DVR for [11C]dMP was found to be within 2% of the “true” value with even a 50% increase in k2. Similar calculations were not done for [11C]raclopride since the difference in DVR with and without b̅2 was small.
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TABLE 2. Comparison of error in DVR for BG ROIs with variation in k2 for dMP
The graphical technique described here depends upon the substitution of the plasma integral with equivalent data from a non–receptor ROI. Figures 1 and 2 compare the integrated plasma radioactivity (filled circles) from studies with [11C]raclopride and [11C]dMP, respectively, with (1/λ)[∫T0 CB(t)dt + CB(T)/k̅2] (open circles) and with (1/λ)[∫T0 CB(t)dt (triangles). It would appear that using (1/λ)[∫T0 CB(t)dt in place of (1/λ)[∫T0 CB(t)dt + CB(T)/k2] or the integrated plasma radioactivity should introduce significant error into the calculation of the DVR. However, from Figs. 3 and 4, which compare the graphical analysis using (1/λ)[∫T0 CB(t)dt + CB(T)/k̅2] with that using (1/λ)[∫T0 CB(t)dt, the slope (DVR) is not greatly affected. The difference in the DVRs for [11C]raclopride (the slopes of the plots illustrated in Fig. 3) is −2% [3.24 with CB(T)/k2 and 3.16 without this term]. For [11C]dMP (Fig. 4), the difference in slope is larger, 2.94 including the CB(T)/k2 term and 2.73 without it for a 7% decrease. From Eq. 9, the difference appears in the intercept, which for dMP is −18.5 min [without CB(T)/k2] compared with −43.7 min. For later times, CB(T)/A(T) (which appears in δ in Eq. 7) is slowly varying function of time, which is why leaving out the term CB(T)/b̅2 does not have as great an effect on the calculation of the DVR as might be expected. By including the term CB(T)/b̅2 explicitly in the slope calculation, the time dependence of the intercept is minimized since the error term δ is proportional to (1/k2 – 1/b̅2) instead of 1/k2. While the inclusion of CB(T)/b̅2 has only a small effect on the calculation of DVRs for [11C]raclopride, it does improve agreement of the DVR calculations for [11C]dMP, although improvement is also observed by increasing the initial time for the slope calculation.
FIG. 3. Comparison of DVR for BG/CB of a [11C]raclopride study, b̅2 = 0.163 (filled circles, DVR = 3.24), and DVR calculated using Eq. 7 so that CB(T)/b̅2 was not included (open circles, DVR = 3.17). For comparison, DVR with plasma is 3.25. See text for abbreviations.
FIG. 4. Comparison of DVR for BG/CB of a [11C]dMP study, k̅2 = 0.051 (filled circles, DVR = 2.94), and DVR using Eq. 7, so that CB(T)/k̅2 was not included (open circles, DVR = 2.73). DVR calculated with plasma is 2.87. See text for abbreviations.
DVR images of three planes at the level of the basal ganglia are illustrated for [11C]raclopride in Fig. 5. The upper images were generated using the standard graphical technique and divided by the DV of the CB. The lower images were generated by the technique described here. The DVRs obtained by the two methods were compared by linear regression using Y = A + BX where Y is the pixel value from the DVR image generated with plasma data and X is the pixel value from the method using the CB ROI. The results are given in Table 3 for [11C]-raclopride and Table 4 for [11C]dMP (images for [11C]dMP are not shown). The agreement between the two methods was excellent. [11C]dMP showed slightly more variability than raclopride.
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TABLE 3. Comparison of DVR images of [11C]raclopride generated by the two graphical methods
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TABLE 4. Comparison of DVR images of[11C]dMP generated by the two graphical methods
FIG. 5. Distribution volume ratio images of three planes at the level of the basal ganglia for [11C]raclopride. The upper images were generated using the standard graphical technique and divided by the distribution volume of the cerebellum. The lower images were generated by the technique described here.
CONCLUSIONS
This method of using a non-receptor ROI in place of a measured plasma integral has been shown to give the same results for the DVRs as the graphical method using plasma data for the ligands [11C]-raclopride and [11C]dMP. Since the DVR is the parameter we use for the comparison of changes in receptor availability, we can obtain the same information without blood sampling. Before implementing this method for other ligands, it is necessary to do a set of baseline studies to establish an average k2 and determine the initial time at which the slope should be calculated. In some cases, the use of k̅2 may not be necessary if the experimental times at which the DVR is calculated are such that the time dependence of CB(T)/A(T) is sufficiently small. Also, adjusting the initial time at which the slope is calculated minimizes the time dependence of the intercept as was found for [11C]dMP, but this will also depend upon the half-life of the radioactive isotope and amount of radioactivity injected. This graphical technique provides a simple method for obtaining DVRs and is not sensitive to fluctuations of k2 from the average value used in its calculations. In this respect, it is similar to the use of average kinetic constants in the determination of glucose metabolic rates in deoxyglucose models (Sokoloff et al., 1977).
Acknowledgment:
This research was supported in part by the U.S. Department of Energy under contract DE-AC02–76CH0016, NINDS grant no. NS15380, and NIDA grant no. 5R01-DA06891. We thank Robert Carciello and David Schlyer for cyclotron operations; Alex Levy and Donald Warner for PET operations; Robert MacGregor, Colleen Shea, Richard Ferrieri, and Payton King for radiotracer preparation and analysis; Kathy Piscani for subject recruitment; Noelwah Netusil for patient care; and Carol Redvanly for scheduling and organization.
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| BG | basal ganglia |
| CB | cerebellum |
| DA | dopamine |
| dMP | d-tzreo-methylphenidate |
| DVR | distribution volume ratio |
| PET | positron emission tomography |
| ROI | region of interest |
