Correction for the Effect of Rising Plasma Glucose Levels on Quantification of MR glc with FDG-PET

Abstract

Positron emission tomography (PET) using the tracer [18F]-fluorodeoxyglucose (FDG) is commonly used for measuring metabolic rate of glucose (MR_glc) in the human brain. Conventional PET methods (e.g., the Patlak method) for quantifying MR_glc assume the tissue transport and phosphorylation mechanisms to be in steady state during FDG uptake. As FDG and glucose use the same transporters and phosphorylation enzymes, changing blood glucose levels can change the rates of FDG transport and phosphorylation. Compartmental models were used to simulate the effect of rising arterial glucose, from normal to hyperglycemic levels on FDG uptake for a typical PET protocol. The subsequent errors on the values of MR_glc calculated using the Patlak method were investigated, and a correction scheme based on measured arterial glucose concentration (G_p) was evaluated. Typically, with a 40% rise in G_p over the duration of the PET study, the true MR_glc varied by only 1%; however, the Patlak method overestimated MR_glc by 15%. The application of the correction reduced this error to −2%. In general, the application of the correction resulted in values of MR_glc consistently significantly closer to the true steady state calculation of MR_glc independently of changes to the parameters defining the model.

Introduction

Measurements of cerebral metabolic rate of glucose (MR_glc) can be made in vivo using the fluorine-18 labeled tracer, fluorodeoxyglucose (FDG) in positron emission tomography (PET) as an indicator of brain metabolism. A standard method of quantifying MR_glc is the graphical analysis technique by Patlak et al (1983), commonly known as the Patlak plot. Briefly, this method requires the measurement of regional FDG uptake over time, a measure of FDG activity in the blood plasma (e.g., from continuous arterial sampling) and a measurement of the plasma glucose concentration. The plasma glucose concentration is assumed constant throughout the duration of the scan.

In experimental protocols investigating brain metabolism it may be difficult or impossible to maintain a constant plasma glucose level for the duration of the PET acquisition. For example, studies published by the authors have used methods in which the subject's glucose level is maintained at a constant level through the uptake period using a steady rate insulin infusion and a variable rate glucose infusion (e.g., Cranston et al, 2001; Bingham et al, 2005). However, Anthony et al (2006) used a study protocol in which an insulin infusion was not available in half of the studies, and 6 of 14 subjects showed changes in plasma glucose levels over 1 mmol/L during the 90-min PET scan. The largest change was in a subject whose glucose levels rose from 6 mmol/L to a maximum of 8 mmol/L.

The purpose of this study was to estimate the effect of changing plasma glucose levels on values of MR_glc calculated using the Patlak method, and to establish a method of correcting for any such errors. FDG-PET is considered the most robust noninvasive method of evaluating MR_glc and no parallel in vivo techniques exist for measuring regional MR_glc, which could accurately compare FDG-PET methods to actual glucose metabolism. For this reason, a simulation of FDG uptake was developed based on standard models of glucose transport and phosphorylation.

The calculations of glucose transport rates across the blood-brain barrier and phosphorylation rates in the tissue are derived from the three compartment model of FDG uptake developed by Sokoloff et al (1977) and Huang et al (1980). This model requires the steady state uptake of tracer into the tissue, in which the transport and phosphorylation rates of FDG are constant for the duration of the measurements.

One way this condition can be violated without direct neuronal stimulation is if the blood glucose level changes. Both glucose transport across cell membranes and phosphorylation are saturable processes (Lund-Andersen, 1979), therefore, a change in glucose level can change the number of available glucose transporters or hexokinase binding sites for FDG and consequently the observed uptake.

The main assumption underlying this study is that a rise in plasma glucose does not trigger an increase in MR_glc. It is well known that hypoglycemia can trigger counterregulatory responses that are associated with changes in cerebral neuronal activation (Rizza et al, 1979 and Mitrakou et al, 1991). Cerebral glucose oxidation in healthy subjects, calculated using [1-¹¹C]-D-glucose has been shown to increase significantly during hyperglycemia; however, these were at plasma glucose levels of 13.8 mmol/L (Blomqvist et al, 1998). There are no reported studies of the effects of mild hyperglycemia on cerebral glucose metabolism in healthy subjects, with plasma glucose levels similar to those found in the study by Anthony et al (2006).

Materials and methods

Models for Simulation of Glucose Kinetics

The compartmental model used to simulate glucose kinetics is shown in Figure 1. Both glucose transport and phosphorylation can be described by Michaelis-Menten kinetics (Sokoloff et al, 1977) and so, in general terms, the rate of glucose transfer from compartments A to B is given by the equation,

$$K{G}_{\mathrm{A}}=\frac{V_{max}{G}_{\mathrm{A}}}{K_t+G_{\mathrm{A}}}$$

(1)

where K is the transport rate ‘constant’ between compartments A and B (e.g., K₁, k₂, or k₃). V_max is a constant defining the maximal transport rate, K_t is the half-saturation constant (equal to the concentration of glucose at which the rate is half V_max), and G_A is the glucose concentration in compartment A. As typical plasma FDG concentrations are negligible compared with plasma glucose, the rate constants of both glucose and FDG kinetics can be considered independent of FDG concentrations.

Figure 1

Three-compartment model by Sokoloff et al (1977) for glucose/FDG kinetics in the brain. The compartments, C_p, C_i, and C_m, are the concentrations of glucose/FDG in the plasma, in the cell, and as phosphorylated substrate (i.e., Glucose-6-P or FDG-6-P), respectively. Arrows with solid lines represent routes of transfer between the compartments available to both glucose and FDG. The dotted line shows the route of further glycolysis available to glucose only.

On the basis of equation 1 we developed a computer simulation to describe the evolution of the rate constants K₁, k₂, and k₃, and the intracellular glucose concentration, G_i, over time in response to a continuous change in the plasma glucose concentration, G_p. If G_p is changing over time then K_1-3 and G_i are also time dependent, and the rate ‘constants’ K_1-3 in Figure 1 are now given by

$$\begin{array}{rl}{K}_1(t)=& \frac{V_{max}^1}{K_t+G_{\mathrm{p}}(t)}\\ k_2(t)=& \frac{V_{max}^2}{K_t+G_{\mathrm{i}}(t)}\\ k_3(t)=& \frac{V_{max}^3}{K_{\mathrm{m}}+G_{\mathrm{i}}(t)}\end{array}$$

(2)

The superscripts in V ^j _max (j = 1, 2, 3) refer to the maximal rates of transfer relating to the different rate constants. The rate of change of G_i(t) is given by

$$\frac{\mathrm{d}{G}_i(t)}{\mathrm{d}t}=K_1(t)G_{\mathrm{p}}(t)-G_{\mathrm{i}}(k_2(t)+k_3(t))$$

(3)

The actual metabolic rate of the model, MR_act, is given by k₃ x G_i.

Fluorodeoxyglucose Kinetics

The same three-compartment model shown in Figure 1 was used to simulate FDG uptake, in which the time-dependent G_i, G_p, and K_1-3 are provided by the glucose simulations above. A typical uptake curve, F_T, from a dynamic FDG-PET brain study in a volume of interest (VOI) in the brain comprises unphosphorylated FDG in tissue, F_i, FDG-6-P in tissue, F_m, and a small fraction, b, of FDG in blood, F_b, assumed to be equal to the arterial FDG sample. The total FDG uptake is then given by

$$F_{\mathrm{T}}(t)=(1-b)(F_{\mathrm{i}}(t)+F_{\mathrm{m}}(t))+b{F}_{\mathrm{b}}(t)$$

(4)

where F_i and F_m are the FDG concentrations in the compartments and are denoted as C_i and C_m, respectively, in Figure 1.

Patlak for the Steady State Model

The Patlak method assumes that, under steady state conditions in which the transfer rate parameters K₁, k₂, and k₃ are constant, after a time, t', F_i equilibrates with the concentration of FDG in the plasma, F_P. Therefore, when t > t', MR_glc is related to a ‘grouped’ transfer rate constant, K_i, (see Figure 2) and the equation used to derive it is given by

$$\frac{F_{\mathrm{T}}(t)}{F_{\mathrm{p}}(t)}=K_{\mathrm{i}}\frac{{\int }_0^t{F}_{\mathrm{p}}(T)\mathrm{d}T}{F_{\mathrm{p}}(t)}+\eta$$

(5)

Figure 2

Two-compartment model by Patlak et al (1983) representing free FDG concentration in the plasma, F_p, and FDG-6-P concentration, F_m.

A plot of F_T(t)/F_p(t) against F_p(t)dt/F_p(t) should reveal a straight line with gradient K_i and intercept η for t > t'. So MR_glc is given by

$$M{R}_{\mathrm{g}\mathrm{l}\mathrm{c}}=\frac{K_{\mathrm{i}}{G}_{\mathrm{p}}}{LC}\approx k_3{G}_{\mathrm{i}}$$

(6)

where G_p is the plasma glucose concentration and LC is the lumped constant derived by Sokoloff et al (1977). As the values of LC are not of primary concern for this study, the value will be assumed to be 1 except where stated otherwise.

Glucose Corrected Patlak Method

According to the published values of K_m, K_t, and G_i (e.g., Lund-Andersen 1979 and Barros et al, 2007) in healthy human brain cells under typical conditions, hexokinase is highly saturated, whereas values of K_t are of the same order of G_p and G_i. To avoid using highly variable published values of K_m and K_t in any correction method, this modified equation assumes that the transfer system as shown in Figure 2 is completely saturated. Therefore, with changing G_p a grouped transfer rate constant, K'_i that is equivalent to K had G_p not changed (i.e., from its value at t = 0) can be calculated from

$$\frac{F_{\mathrm{T}}(t)}{F_{\mathrm{p}}(t)}=K_{\mathrm{i}}^{\prime }\frac{{\int }_0^t(G_{\mathrm{p}}(t=0)/G_{\mathrm{p}}(t))F_{\mathrm{p}}(T)\mathrm{d}T}{F_{\mathrm{p}}(t)}+{\eta }^{\prime }$$

(7)

and MR_glc is given by

$$M{R}_{\mathrm{g}\mathrm{l}\mathrm{c}}=\frac{K_{\mathrm{i}}^{\prime }{G}_{\mathrm{p}}(t=0)}{LC}=M{R}_{\mathrm{c}\mathrm{o}\mathrm{r}}$$

(8)

We investigated how well this corrected method can calculate MR_glc compared with no correction in the face of changing plasma glucose levels. MR_glc calculated from the uncorrected and corrected methods will be denoted as MR_unc and MR_cor, respectively. For situations in which G_p was changing, MR_unc was calculated using the mean value of G_p between t_start and t_end, that is,

$$M{R}_{\mathrm{u}\mathrm{n}\mathrm{c}}=\frac{K_{\mathrm{i}}\overline{G_{\mathrm{p}}}}{LC}$$

(9)

Under steady state conditions, MR_unc and MR_cor are equal and will be denoted as MR_ss.

Simulations

Plasma Glucose and Initial Kinetic Parameters

Six plasma glucose profiles were simulated as a linear rises from 5 mmol/L up to 5 (i.e., steady state), 7, 9, 11, 13, and 15 mmol/L over 2h. These profiles were denoted as GP1, GP2, GP3, GP4, GP5, and GP6, respectively.

Time dependent K_1-3, G _i , and MR_act were calculated for each plasma glucose profile with a range of initial values. K₁(t=0) was set to 0.001 ml g⁻¹ s⁻¹ and a range of initial values of k₂ and k₃ were chosen; 0.0006 to 0.002 s⁻¹ and 0.0001 to 0.0016 s⁻¹, respectively. Initial values of G_i were calculated from the initial values of G_p and K_1-3. This assumes that the system is in steady state before the rise.

The half-saturation constants used for transport and phosphorylation, K_t and K_m, respectively, were fixed at 4 and 0.05 mmol/L (typical values are presented in Barros et al, 2007 and Lund-Andersen 1979, respectively). The constants V ^j _max (j = 1 to 3) were calculated from K _t , K_m, and initial values of K_1-3 and G_p.

Fluorodeoxyglucose Kinetics and Patlak

For each plasma glucose profile and set of time-dependent kinetic parameters, a 2-h FDG time activity curve with 1-sec frames was simulated with b = 0.05 (equation 4) and a simulated arterial FDG curve, F_p, as described by Feng et al (1993). To investigate the effect of time on the Patlak calculation, a range of t_end (15 to 120 mins) was used to calculate MR_unc and MR_cor, where t_end - t_start = 300 s.

The effect of the kinetics on the efficacy of both Patlak methods was assessed using the ratios MR_unc/MR_act and MR_cor/MR_act. The effect solely of the correction term was assessed using the ratios MR_cor/MR_ss, MR_unc/MR_ss, or MR_unc/MR_cor.

Kinetic Constants

The published values of K _t and K _m are variable so a range of values either side of published values (K _t = 1 to 32 mmol/L and K_m = 0.001 to 1.0 mmol/L) were also used in the simulation to assess the effect on the Patlak correction (i.e., MR_cor/MR_unc).

The model of the LC presented in Crane et al (1983) is dependent on the values of k₂, k₃, and ‘transport coefficient’ and ‘phosphorylation coefficient’ denoted as TC and PC, respectively. TC is equal to the ratio of the transport rate constants of glucose and FDG and PC, the ratio of the phosphorylation rate constants of glucose and FDG. The effect of these parameters on the Patlak correction (i.e., MR_cor/MR_unc) was investigated using a range of TC and PC (TC = 0.2 to 2.5 and PC = 0.1 to 1.0) around the published values (Kuwabara et al, 1990) of 1.1 and 0.3, respectively.

Fluctuating Gp and Fluorodeoxyglucose Noise

Typical plasma glucose profiles are not linear so the effect of random fluctuations of G_p was also investigated. Plasma glucose samples were simulated at 5-min intervals with fluctuations of the order of those observed in real studies. Profiles were simulated with normally distributed values around the mean calculated from the base linear glucose profiles (GP1-6) and a standard deviation (s.d.) of 0.15 mmol/L (derived from profiles by Anthony et al, 2006, which showed no significant rising trend). The G_p input to the glucose model was resampled to 1-sec intervals using a spline interpolation fitting the assumption that actual plasma glucose profiles vary smoothly. About 100 randomly fluctuating glucose profiles were generated to estimate the population mean and s.d. of MR_cor and MR_unc.

Noise arising in typical FDG-PET scans was also simulated using noise levels representing different dynamic frame durations and different size VOIs. Noise was calculated from the simulated uptake and variance parameters derived from a phantom study on a GE Discovery ST in 3D mode with CT attenuation correction and scatter correction. 100 noisy FDG uptake curves were simulated for a 2-h uptake period with 5 mins frame durations, for each glucose profile and for each noise function, representing a 1 voxel (2 × 2 × 3.27 mm) VOI and a 100 voxel VOI.

To simulate the effects of randomly fluctuating glucose profiles or noisy FDG uptake curves, the following initial parameters were used, K₁ = 0.001 ml g⁻¹ s⁻¹ k₂ = 0.0012 s⁻¹, k₃ = 0.0008 s⁻¹, b = 0.05. Patlak calculations were made with t_start = 30 mins and t_end = 60 mins. These parameters are in the ranges of a typical FDG scan.

Table 1 summarizes the abbreviations and their descriptions used in the text, Figures and Table 2.

Table 1

Description of abbreviations used

Abbreviation	Description
G p > (F p )	Arterial plasma glucose (FDG) concentration
Gi (Fi)	Intercellular glucose (FDG) concentration
GP1, GP2, GP3, &	Simulated rising plasma glucose profiles. GP1 simulates euglycemia at 5 mmol/L, and GP2, 3, 4, 5, and 6 simulate rises from 5 mmol/L to 7, 9, 11, 13, and 15 mmol/L over a 2-h period, respectively
MR act	Simulated actual metabolic rate (k3 x Gi)
MR unc	Metabolic rate calculated using Patlak method, with no correction for rising plasma glucose (see equations (5) and (6))
MR cor	Metabolic rate calculated using Patlak method, with correction for rising plasma glucose (see equations (7) and (8))
MR ss	Metabolic rate calculated using Patlak method under steady state conditions, that is, correction is not necessary and MRunc = MRcor
LC	Lumped constant (from Sokoloff et al, 1977)
TC, PC	Transport and phosphorylation coefficients (from Crane et al, 1983)
DP	Glucose-6-dephosphorylase

Table 2

Estimated population means and standard deviations of MR_unc/MR_ss and MR_unc/MR_ss ratios

	1 voxel		100 voxels
	MRcor/MRss Mean (s.d.)	MRunc/MRss Mean (s.d.)	MRcor/MRss Mean (s.d.)	MRunc/MRss Mean (s.d.)
GP1	0.990 (0.171)	0.990 (0.171)	0.992 (0.019)	0.992 (0.019)
GP2	1.010 (0.166)	1.073 (0.177)	1.013 (0.018)	1.077 (0.019)
GP3	1.017 (0.193)	1.142 (0.218)	1.018 (0.019)	1.144 (0.022)
GP4	1.009 (0.209)	1.191 (0.248)	1.029 (0.022)	1.215 (0.027)
GP5	1.015 (0.187)	1.253 (0.232)	1.039 (0.021)	1.284 (0.027)
GP6	1.028 (0.200)	1.323 (0.258)	1.040 (0.022)	1.339 (0.030)

Calculated from Monte Carlo simulations of noisy FDG-PET data for VOIs of 1 voxel and 100 voxels (voxel size, 2 × 2 × 3.27 mm) from a 3D scan with 5-min frames and linearly varying plasma glucose profiles (GP1-6). The initial parameters for the FDG uptake model were K₁ = 0.001 ml g⁻¹ s⁻¹, k₂ = 0.0012 s⁻¹, k₃ = 0.0008 s⁻¹, and b = 0.05. MR_ss is the metabolic rate calculated by the Patlak method at steady state with no noise added.

Results

Plasma Glucose and Initial Kinetic Parameters

Example plots of simulated G_p, G_i, k₃, and MR_act are shown in the top two panels of Figure 3. In nonsteady state simulations in which G_p rose linearly, G_i was approximately proportional to G_p after a small period of equilibration between the two compartments. Values of k₃ fell almost inversely proportionally to the rise in G_i resulting in MR_act remaining approximately constant, showing only a very small rise with G_p.

Figure 3

Simulation of glucose kinetics in response to a rising glucose profile. Top left: G_P (solid line) and G_i (dotted line). Top right: k₃ (solid line, left y-axis, s⁻¹), and MR_act (dotted line, right y-axis, μmol g⁻¹ s⁻¹). Bottom plots show the same parameters with a fluctuating glucose profile.

MR_act rose by less than 2% over 120 mins in the most extreme cases, with the steepest rise in plasma glucose levels (GP6). The rise in MR_act was mostly dependent on the G_p profile and the initial value of k₃. Small initial values of k₃ and large rises in G_p provoked the largest change in MR_act. Substituting typical values found in our insulin study (Anthony et al, 2006) into this simulation revealed MR_act increases of around 0.5% over 120 mins. As values for MR_act are very stable over time, a single value equal to the mean over 120 mins is used to compare with MR_ss, MR_unc, and MR_cor.

Fluorodeoxyglucose Kinetics and Patlak

Under the steady state condition, the theoretical maximum ratio, from equation (4), of MR_ss to MR_act is 1-b, which would occur when the intracellular FDG concentration was in equilibrium with plasma FDG. Simulated uptake used b = 0.05, so the theoretical maximum of MR_ss/MR_act under steady state conditions was 0.95.

Steady state simulations showed MR_ss/MR_act ratio depended on values of k₂ and k₃. The value of t_start also affected this ratio. The higher values of k₂ and k₃ gave ratios closer to 0.95 as the time taken for F_i and F_p (equation 4) to equilibrate, t‘, is lower under these circumstances (see Figure 4, top panel).

Figure 4

Plots showing ratio of Patlak calculated MR to simulated actual MR (y-axis) against t_end (x-axis), where t_end - t_start = 300s. For all graphs K₁ = 0.001 ml g⁻¹ s⁻¹, k₂ = 0.0012 s⁻¹, b = 0.05 and the different line series represent results using six different initial values of k₃ (see legend in top graph). Top graph shows results with a euglycemic glucose profile (GPl), where the y-axis is MR_ss/MR_act; middle graph shows results with rising glucose profile, GP3, where the y-axis is MR_unc/MR_act; bottom graph shows results with rising glucose profile, GP3, where the y-axis is MR_cor/MR_act.

Simulations with linearly increasing G_p revealed the effect of the correction. Figure 4, middle panel, shows MR_unc/MR_act highly dependent on t_end, whereas MR_cor/MR_act (Figure 4, bottom panel) is much more stable for all except very low initial values (t = 0) of k₃ or t_end and then not much different to the dependence to either of these parameters under steady state conditions (i.e., compared with Figure 4, top panel).

Figure 5 shows the effect of the correction on rising glucose profiles for parameters in the range of a typical FDG scan protocol (K₁(t = 0) = 0.001 ml g⁻¹ s⁻¹ k₂ (t=0) = 0.0012 s⁻¹, b = 0.05, t_start = 30 mins and t_end = 60 mins). Across the range of k₃ (t = 0), MR_unc/MR_act values show a wide spread of values between GP1 and GP6. The correction brings these values closer to those from GP1. For example, with an initial value of k₃ = 0.0008 s⁻¹ with GP3, MR_unc/MR_ss = 1.149 and MR_cor/MR_ss = 1.023.

Figure 5

Plots showing ratio of Patlak calculated MR to simulated actual MR (y-axis) against initial values of k₃ (x-axis, s⁻¹), where K₁ = 0.001 ml g⁻¹ s⁻¹, k₂ = 0.0012 s⁻¹, and b = 0.05. Left plot shows MR_unc/MR_act, and the right plot shows MR_cor/MR_act. The different line series show results from different glucose profiles (filled black circles represent results from GPl, and white squares represent results using GP6). Patlak slope is calculated between t_start = 1,800 and t_end = 3,600s.

Kinetic Constants

For rising glucose profiles, MR_cor/MR_act changed with respect to K _t and K _m ; however, the effect of the correction was virtually constant within each glucose profile. For example, the ratio MR_cor/MR_unc was 0.775 (3 d.p.) for all values of K _t and K_m with GP6 (the most extreme case of rising plasma glucose) with a difference of 0.00022 between the maximum and minimum.

For varying TC and PC, across the range of K_1-3, the MR_cor/MR_unc ratio for GP6 had negligible variance (minimum, 0.7769; maximum, 0.7784).

Fluctuating Gp and Fluorodeoxyglucose Noise

The MR_cor/MR_unc ratios for linearly rising glucose for these parameters were 1.0 (exactly), 0.941, 0.890, 0.847, 0.810, and 0.777 for GP1-6, respectively.

Fluctuating glucose profiles also had negligible effect on the accuracy of the correction factor. For profiles based on GP1, the mean ± s.d. of the ratio MR_cor/MR_unc was 1.000 ± 0.008; GP2, 0.940 ± 0.008; GP3, 0.890 ± 0.007; GP4, 0.848 ± 0.006; GP5, 0.809 ± 0.006; and GP6, 0.778 ± 0.005. That is, all estimated means are comparable with the values calculated without these fluctuations, with s.d. < 1%.

Adding noise to the simulated FDG uptake also had negligible effect on the MR_cor/MR_unc ratio. In all cases MR_cor < MR_unc. For a VOI of 1 voxel the mean ± s.d. of the ratio MR_cor/MR_unc for GP2 was 0.940 ± 0.001; GP3, 0.891 ± 0.002; GP4, 0.847 ± 0.002; GP5, 0.810 ± 0.003; and GP6, 0.777 ± 0.003. For a VOI of 100 voxels: GP2, 0.940 ± 0.0004; GP3, 0.891 ± 0.0008; GP4, 0.847 ± 0.0010; GP5, 0.810 ± 0.0014; and GP6, 0.777 ± 0.0015. That is, all estimated means are comparable with the values calculated without PET noise. Table 2 shows similar data but of the ratios of MR_cor or MR_unc to the steady state Patlak calculation with the same parameters but no noise (equal to 0.927 x MR_act). The differences in the standard deviations indicate that for large VOIs, the correction is important but might not be so important for smaller VOIs.

Discussion

FDG uptake in brain was simulated during nonsteady state conditions to investigate the validity of a proposed correction to the calculated MR_glc using the Patlak method. Rising arterial glucose concentrations were simulated from normal to hyperglycemic levels (5 mmol/1 and higher).

In all cases, the correction method improved on the standard Patlak method. It is important to establish the meaning of ‘improve’ in this context. The Patlak method is used in FDG-PET studies to provide a simplified calculation of MR_glc. However, the assumptions on which this method is based mean that incomplete equilibration of FDG concentration between plasma and tissue compartments, poorly defined values for the ratios of transfer rates between FDG and glucose and the assumption of negligible blood volume in the region of interest can result in deviations of the Patlak calculations from the actual metabolic rate, even under steady state conditions. Nevertheless, it is of interest to note that, in conditions where plasma glucose levels are changing but metabolic rate changes are negligible, how well the correction to the Patlak method recreates values that would be obtained under euglycemic conditions using the standard method. Where values of MR_cor are closer to MR_ss than MR_unc, then this is considered an improvement.

MR_cor and MR_unc less accurately matched MR_act in cases where initial values of k₂ and/or k₃ were small compared with K₁, as FDG equilibrium between plasma and tissue would take longer to achieve. These values also affected the impact of t_end on the Patlak calculations in all cases. For small initial values of k₂ and/or k₃, the accuracy of the Patlak calculation in reproducing actual metabolic rates was greatly affected by the t_end chosen. This effect was greater at small values of t_end. As can be seen in Figure 4, calculations at small t_end overestimate MR_act and calculation made at later times under estimated it. The early values of t_end for which the Patlak calculations equalled MR_act differed depending on the values of k₂ and k₃, so it would not be appropriate to use an early t_end. It should be noted that the small initial values of k₂ and k₃ discussed here do not produce the characteristic FDG uptake curves observed in typical PET scans.

The values of parameters such as K_t, K_m, TC, and PC have been taken from earlier studies or assumed. The quoted values are either uncertain or cover a wide range so a single simulation may not accurately reflect this. However, a wide range of values used in the simulation did not change the effect of the Patlak correction significantly.

In studies where the Patlak method for determining MR_glc is used, the LC is usually assumed to be constant or 1.0. We have addressed here a correction for changing plasma glucose levels that we have shown to be true if the assumption of constant LC holds. In practice, LC will change during the course of the study if G_p diverges from euglycemia during this period. In this case, inaccuracies in Patlak calculation of MR_glc will be due to both changes in G_p directly and through changes in the LC. In this work, we only deal with the first of these effects.

There is no simple relationship between plasma glucose and LC and, in general, the value of LC depends on the values of the individual rate constants (K₁, k₂, k₃), and these are not available when data are acquired with a view to Patlak analysis. Changes in K_i and LC due to increasing plasma glucose both result in an underestimation of MR_glc if constant G_p and LC are assumed in the analysis. Correcting for G_p but neglecting LC changes will, therefore, still lead to an improved estimate of MR_glc. A detailed evaluation of the residual error resulting from neglecting changes in LC warrants a more detailed analysis but is beyond the scope of this paper.

The model used to simulate glucose and FDG kinetics relied on the assumption that the enzyme, dephosphorylase, is present in negligible amounts. If this assumption was invalid then another parameter (usually denoted as k₄), which would allow transfer back from compartments C_m to C_i (see Figure 1) would need to be included to represent the dephosphorylation of glucose-6-phosphate. However, this would complicate the model, as the assumption allows C_m to be ignored. Any model including k₄ would need a more sophisticated model of glucose kinetics including further metabolism of glucose from phosphorylation. Glucose-6-dephosphoryalse is reported to be present in small amounts in the human brain and studies of FDG kinetics have shown that including k₄ significantly improves the fit of the model to the data (e.g., Hasselbalch et al, 1999, Schmidt et al, 1992). These studies have shown the value of k₄ to be approximately 1/10th of k₃, yielding a 30% difference in calculated metabolic rate. A more sophisticated glucose model would allow a more rigorous simulation of glucose metabolic rate and FDG kinetics under changing plasma glucose. The Patlak method assumes that k₄ is negligible and so the presence of dephosphorylase would mean the Patlak method would further underestimate glucose metabolism; however, this would apply to both the uncorrected Patlak and corrected Patlak calculations. This study has shown that although physiological parameters defining the model may be poorly defined (e.g., K_t, TC, etc), the ratio of MR_cor/MR_unc is insensitive to these values.

The validity of this method is dependent on whether the change in the actual MR_glc due to a rise in plasma glucose is negligible. This method assumes that the change in FDG uptake is due solely to the effect of competition for glucose transporters and hexokinase. Although this assumption was not implicit in the model, the low published values of K_m suggest that phosphorylation of glucose is a highly saturated process. Even with a three-fold increase in plasma glucose (GP6), our model showed a very small increase in MR_act over the 2-h period (~2%). It is unclear whether this assumption is valid for such high plasma glucose changes in the real world but for more moderate rises it may be more so.

In conclusion, the Patlak method has been shown to yield incorrect results with plasma glucose levels rising from normoglycemia to hyperglycemia during FDG-PET acquisition. Assuming that glucose metabolic rate does not change with moderate changes in plasma glucose levels, a corrected Patlak equation may be used to calculate more accurate rates. The method described above is straightforward to implement in practice. If there is a reason to suspect that plasma glucose levels are likely to change throughout an acquisition, then in addition to the standard protocol, plasma glucose levels are required throughout the scan (typically these would be measured at 5-min intervals). These values are then substituted into equations 7 and 8 to calculate the corrected glucose metabolic rate.