Quantitation of glucose uptake in tumors by dynamic FDG-PET has less glucose bias and lower variability when adjusted for partial saturation of glucose transport
© Williams et al; licensee Springer. 2012
Received: 21 September 2011
Accepted: 1 February 2012
Published: 1 February 2012
A retrospective analysis of estimates of tumor glucose uptake from 1,192 dynamic 2-deoxy-2-(18F)fluoro-D-glucose-positron-emission tomography [ FDG-PET] scans showed strong correlations between blood glucose and both the uptake rate constant [K i] and the metabolic rate of glucose [MRGluc], hindering the interpretation of PET scans acquired under conditions of altered blood glucose. We sought a method to reduce this glucose bias without increasing the between-subject or test-retest variability and did this by considering that tissue glucose transport is a saturable yet unsaturated process best described as a nonlinear function of glucose levels.
Patlak-Gjedde analysis was used to compute K i from 30-min dynamic PET scans in tumor-bearing mice. MRGluc was calculated by factoring in the blood glucose level and a lumped constant equal to unity. Alternatively, we assumed that glucose consumption is saturable according to Michaelis-Menten kinetics and estimated a hypothetical maximum rate of glucose consumption [MRGlucMAX] by multiplying K i and (K M + [glucose]), where K M is a half-saturation Michaelis constant for glucose uptake. Results were computed for 112 separate studies of 8 to 12 scans each; test-retest statistics were measured in a suitable subset of 201 mice.
A K M value of 130 mg/dL was determined from the data based on minimizing the average correlation between blood glucose and the uptake metric. Using MRGlucMAX resulted in the following benefits compared to using MRGluc: (1) the median correlation with blood glucose was practically zero, and yet (2) the test-retest coefficient of variation [COV] was reduced by 13.4%, and (3) the between-animal COVs were reduced by15.5%. In statistically equivalent terms, achieving the same reduction in between-animal COV while using the traditional MRGluc would require a 40% increase in sample size.
MRGluc appeared to overcorrect tumor FDG data for changing glucose levels. Applying partial saturation correction using MRGlucMAX offered reduced bias, reduced variability, and potentially increased statistical power. We recommend further investigation of MRGlucMAX in quantitative studies of tumor FDG uptake.
Keywordsvariability glucose correction MRGlucMAX blood glucose partial saturation correction dynamic FDG-PET
We undertook a large series of tumor imaging studies in mice using the metabolic rate of glucose [MRGluc] from Patlak analysis as our preferred estimate of the tumor glucose uptake rate, expecting it to be relatively unbiased with respect to blood glucose. When we undertook a retrospective review of 1,192 such scans performed in study groups of 8 to 12 mice, we observed that our MRGluc data were, in fact, strongly correlated with blood glucose even though individual studies were often underpowered to convincingly show this (see Figure 2B).
The original formulations of quantitative glucose uptake measurements using radioactive uptake assays were described comprehensively 35 years ago in the seminal work of Sokoloff et al. . The importance of glucose transport processes based on saturable Michaelis-Menten kinetics has been demonstrated in biochemical studies of glucose transporter 1 [GLUT-1], the dominant glucose transporter in tumors (and erythrocytes and the blood-brain barrier), which have shown that glucose transport into cells can be characterized as a saturable process with a half-maximal-rate Michaelis constant [K M] of approximately 40 mg/dL ; it is the transport step that dominates the overall uptake and trapping rate in many situations [10, 11]. Studies in intact animals suggested that the apparent half-saturation constant, K M, for the GLUT-1-dominated blood-to-brain tissue transport was approximately 5  to 7.3 mM , equivalent to 100 to 130 mg/dL.
We reasoned that tissue glucose levels are often neither far below K M (where the glucose transport rate would be approximately proportional to blood glucose level) nor far above K M (where the glucose transport rate would be saturated and independent of blood glucose level). Consequently, tissue glucose uptake rates are likely to show an intermediate, nonlinear dependence on blood glucose levels.
We tested this hypothesis using a form of the MRGluc calculation that employs the Michaelis-Menten relationship to compute the hypothetical maximal uptake rate [MRGlucMAX] based on an empirical half-saturation K M of 130 mg/dL (see 'Results' section). This approach should reflect the relatively constant glucose uptake capacity of the tissue rather than the instantaneous uptake rate, more or less independent of variations in blood glucose. We refer to this glucose correction method as partial saturation correction.
In this paper, we review 112 separate tumor studies of 8 to 12 dynamic FDG scans each, all analyzed with the Patlak-Gjedde simplified tracer kinetic modeling methods [14–17] yielding the uptake rate constant [K i] (per second) and the MRGluc (in micromoles per minute per 100 cm3). We compared the glucose bias, test-retest, and between-animal variability of K i, MRGluc, and MRGlucMAX.
Materials and methods
Data listed by mouse strains and tumor types
Tumor cell line/mouse strain
Tissue of origin
Number of mice
Number of scans
BT474MI in beige-scid nude
HCT116 in athymic nude
PC3 in athymic nude
FaDu in C.B.-17 scid
H292 in C.B.-17 scid
H596 in huHGF transgenics
537-MEL in athymic nude
A2058 in athymic nude
A375 in athymic nude
Colo205 in athymic nude
H2122 in athymic nude
All studies were conducted with the approval of Genentech's AALAC-accredited institutional animal care and use committee. Briefly, animals were fasted overnight with free access to water prior to PET imaging. Sevoflurane in air  was used to induce and maintain anesthesia sufficient to restrain the animals while they were scanned prone on the bed of an Inveon MM scanner (Siemens Preclinical Solutions, Knoxville, TN, USA). PET scans lasted 30 min. X-ray CT scans provided attenuation correction. List mode data were typically reconstructed into images with 128 × 128 in-plane voxels of 0.4 × 0.4 mm and 0.8 mm through-plane voxel thickness using vendor-provided iterative OP-MAP implementation with the beta hyperparameter set to 0.05 . The resolution (approximately 1.5 mm), sensitivity, and other performance characteristics of this scanner have been described previously . Body temperature was maintained at 37°C by warm air flows under feedback control. When animals were re-scanned on the second or subsequent days, they were imaged on the same scanner and at the same time of the day as for their first scan. The mice received an FDG tracer dose of approximately 200 μCi by infusion through a tail vein catheter.
Blood glucose measurements
At every scan, blood glucose measurements were taken twice: once approximately 5 min before and once shortly after the scan approximately 35 min later. The glucose value used in calculations is the mean of the pre- and post-scan measurements. Data were collected with the commercially available Contour glucometer (Bayer Healthcare, Tarrytown, NY, USA). Test-retest reproducibility measurements according to Equation 5 were conducted using this instrument in 20 mice and showed a coefficient of variation of 3.7%.
Regions of interest [ROIs] were drawn using the image analysis software IRW from Siemens. For any given tumor model, all scans for all animals were analyzed by a single observer following a standard procedure: Tumor ROIs were defined as voxels exceeding a threshold percentage of the maximal tumor signal measured in the last 10 min of the scan; this excluded necrotic or otherwise hypointense regions from the analysis. Mean signal values from the ROIs were used for analysis. Image-derived signal from an ROI in the liver was used as an input function reference region in the Patlak analysis, a technique described in mice by Green et al. . This method is well suited to high-resolution whole-body scans that minimize partial volume artifacts [20, 22], such as those used here.
Time-activity curves and Patlak plots
The Patlak and subsequent statistical analyses were performed with the statistical programming language R . For each tumor model described in Table 1, examples of the time-activity curves and the resultant Patlak plots are presented in Additional file 1 to 11. K i was measured from Patlak plots of dynamic FDG-PET data [15, 16, 24]. The linear portion of the plot (beginning approximately 5 min into the time-activity curves) was used for fitting and visually verified: the correlation coefficient r 2 in each case was at least 0.99.
Kinetic modeling and partial saturation correction
MRGluc was estimated as K i × [glucose] × LC, where LC is the lumped kinetic constant (set to unity) and [glucose] is the blood glucose measurement. Some literature denotes this form of MRGluc where LC = 1 as 'MRFDG' [25, 26]; we will use 'MRGluc' for its semantic emphasis on glucose (rather than FDG or glutamate) uptake. Although the LC scales the absolute value of the K i data, it is important to note that the choice of LC has no bearing on the subsequent analysis of glucose correlation, between-animal coefficient of variation [COV], or test-retest reproducibility.
MRGlucMAX is the hypothetical maximal value of glucose uptake rate, approached asymptotically as the glucose concentration increases to saturating levels. If it were physically possible, the glucose uptake rate measured along the horizontal asymptote would be expected to show zero correlation with the glucose concentration. The curvature parameter K M is the Michaelis constant that represents the blood glucose concentration at which the glucose uptake rate is half the maximal (glucose-saturated) rate.
Computation of MRGlucMAX
All the data presented in this paper were computed using Equation 4, and group mean data were calculated by sample averaging the results for individual animals within a given study.
Estimation of K Mby minimizing the correlation between blood glucose and MRGlucMAX
We computed estimates of MRGlucMAX with a range of K M values from 40 to 200 mM and selected the K M that gave the smallest nonnegative value of the median Pearson's correlation coefficient between MRGlucMAX and [glucose] across all 112 studies. As an exploratory analysis, we also separately estimated K M for each of the 11 tumor models.
Variability and reproducibility
Both K iand MRGluc are correlated with blood glucose levels
In some studies, correlations between blood glucose levels and the FDG-PET estimates of glucose uptake rate were readily apparent. Two of these are illustrated in Figure 1: panel A for K i and panel B for MRGluc. As expected, many individual cohorts of 8 to 12 mice were statistically underpowered to show such a relationship.
More important, and remarkable, was the consistent presence and strength of this relationship between blood glucose and tissue uptake rates when seen in the meta-analysis of our large sample of studies. Figure 2 illustrates this using a box plot of Pearson's correlation coefficients between blood glucose and (A) K i, and (B) MRGluc for all 1,192 scans from the 112 studies. The data are grouped into one box for each of the 11 tumor models, with the median for each box shown as a horizontal line. Data for each tumor model comprised 4 to 30 studies; the open circles within a box show individual studies for full disclosure.
Blood glucose levels were negatively correlated with K i in 90 studies (Figure 2A). The median correlation coefficient (dashed line) was -0.4. With MRGluc as the metric of tumor glucose uptake rate, 104 studies now showed a positive correlation with a median correlation coefficient of 0.55 (Figure 2B), indicating that factoring in the glucose did not eliminate the bias, but rather changed it from negative to positive.
In this meta-analysis of 112 studies, it is possible to compute for each tumor model confidence interval around the correlation coefficients reported in Figure 2. The statistical methodology and results are presented for the interested reader in Additional file 2.
A preliminary analysis of our data simply looked for positive correlations between MRGluc and blood glucose levels in the double-reciprocal Lineweaver-Burk plots that are characteristic of a Michaelis-Menten relationship [27, 28]. Thirteen of the first 20 tumor studies we examined had some correlation, judging by eye, encouraging further consideration of the Michaelis-Menten model in our data. It was also apparent that the data were inherently noisy such that individual studies were perhaps underpowered to demonstrate a relationship. No quantitative inferences were drawn from these analyses, however. Four such studies are shown in Figure 4.
When K M= 130 mg/dL, MRGlucMAXshows zero glucose bias, on average
Figure 2C shows the correlation between MRGlucMAX and blood glucose when K M = 130 mg/dL. At this value of K M, the median correlation coefficient for all 112 studies was practically equal to zero, < 0.0004 (dashed line in Figure 2C). Of the 112 studies, 55 showed a positive correlation, 55 showed a negative correlation, and 2 had practically zero correlation (< 0.003). Increasing values of K M beyond 130 mg/dL resulted in progressively more negative median correlation coefficients.
Individual blood glucose often varies between scans
Blood glucose levels recorded at scan time for individual mice on multiple measurement days are presented in Figure 3. Each box contains a different cohort of mice studied on multiple days. Differences in group means and fluctuations over time are apparent despite consistency of handling.
Our hypothesis was that we could reduce variability by extrapolating the tumor glucose uptake rate measurement to a hypothetical asymptote where glucose is under saturating conditions, and our data seem to support this. Mathematically, the improvement in COV appears to come from the fact that MRGlucMAX is greater than MRGluc by definition. Specifically, because K M = 130 mg/dL and the glucose measurements are near 100 mg/dL, we observe that, on the average, MRGlucMAX values approximately double those of MRGluc. The standard error in MRGlucMAX is also greater than that in MRGluc, but proportionately less so, and so we get an overall reduction in COV of 15.5%.
Power estimation and sample size calculation
An overall reduction in COV of 15.5% could translate (statistically equivalently) into either a need for fewer subjects per study or into the ability to detect smaller effect sizes. Generally speaking, the standard error of sample means (and of maximum likelihood estimators in general) is inversely proportional to the square root of the sample size: this implies that to achieve the same between-animal COV when using MRGluc would require an increased sample size of 40% (i.e., 100 × 1/(1-0.155)2) compared when using MRGlucMAX.
Sensitivity analysis for the between-animal COV as a function of K M
Varying the value of K M in the range of 40 to 200 mg/dL did not change the nature of the results: MRGlucMAX gave lower between-animal variability than MRGluc. The reduction in the average COV/K M correspondence was 10% (40 mg/dL), 15% (100 mg/dL), 15.5% (130 mg/dL), and 16% (200 mg/dL).
Correlations between blood glucose levels and MRGluc
Figure 2B shows that, in our setting with anesthetized mice, there is undoubtedly a strong and persistent positive correlation between blood glucose and MRGluc across a variety of tumor models and mouse strains. It is possible to calculate confidence intervals for the correlation coefficients; these reinforce our conclusions since only one of eleven models had a 95% confidence interval that included zero (-0.01 to 0.42). These calculations and results are presented in Additional file 2 for the interested reader.
Use and applicability of MRGluc
Rigorous methods for estimating the MRGluc utilization were developed over 30 years ago and continue to be successfully applied [7, 12, 26, 30, 31], not least in tumors [2, 4, 17, 32–36]. However, capturing the rate of glucose uptake in the instant of the scan leads to MRGluc reflecting changes in blood glucose whether or not they are functionally significant to the tissue. For malignant tumors, which are highly glucose-addicted, glucose uptake capacity may well be a more important tissue characteristic to consider than the glucose uptake rate. MRGlucMAX reduces glucose bias by emphasizing capacity rather than rate.
Fundamental problem with nonlinear regression estimates of K Mand MRGlucMAX
It may be surprising to some readers that we do not employ a nonlinear regression model to simultaneously estimate MRGlucMAX and K M from measurements of K i and [glucose]. Although considerable care must be taken, this approach is known to work [37–39] for enzymatic data collected in vitro with minimal statistical noise in the measurements. However, it proved to be impossible with our data from living subjects: the objective function was difficult to optimize and subject to very large estimation errors. Mathematically, this is due to maximum likelihood estimates of K M and MRGlucMAX being highly linearly codependent, and it requires a wide range of glucose values to confidently distinguish the effects of changing K M and changing MRGlucMAX, at least when faced with relatively noisy real-world K i measurements. This argument is presented in Appendix 1 for the interested reader along with simulations.
Use of a fixed value of K M
The use of an apparent K M value derived in separate experiments and used within a physiologically reasonable range has the mathematical advantage that it reduces the number of parameters we need to estimate from the scan data and thus avoids the use of underpowered determinations made on a case-by-case basis. We used a large sample of studies to determine the K M at which there was, on average, no net correlation between MRGlucMAX and blood glucose levels (K M = 130 mg/dL). We regard this as an upper limit; the lower limit might be set by studies on isolated cells where the measurement can be made with full knowledge of the extracellular glucose concentration (K M = 40 mg/dL, ).
A biological advantage is that we are better able to fix the K M as a constant property of a certain tissue or tumor type under given conditions which are largely dictated by the discrete nature of the molecular determinants, such as the isotype of the glucose transporter, GLUT-1 versus GLUT-3, for example.
The importance of tissue glucose and blood glucose
Of particular importance should be how the (mechanistically relevant) tissue glucose relates to the (conveniently measured) blood glucose levels . Ideally, we should know the interstitial glucose concentration in the tumor microenvironment, and while the relationship between blood and tissue glucose is of intense interest and active study, it is still not trivial to measure [41–43]. For normal tissues, interstitial glucose may be modeled, but in tumors with all their heterogeneity and variability, this is likely to remain challenging, and this will likely continue to present a significant source of variability in data that depend on tissue glucose but measure blood glucose.
In some tumors, the glucose utilization is so great and the perfusion so poor that the true tissue glucose may be close to zero , leaving tissue glucose transport far from being saturated yet also decoupled from blood glucose levels. We have seen from our data is possible, but not typical. More common are cases where the FDG uptake rate does correlate with blood glucose levels, implying some degree of saturation and thus nonzero tissue glucose levels.
Mathematical expectation of a correlation between K i× [glucose] and [glucose]
Here, we report an empirical correlation between glucose as it is commonly measured (in the blood) and MRGluc as it is commonly defined and described in the literature (K i × [glucose], based on blood glucose measurements). Although it is not widely remarked upon in the literature, this correlation appears to be almost inevitable, a natural consequence of the relationship between K i, MRGluc, and [glucose]. Given the widely described result that tissue FDG uptake rates (K i) and FDG uptake levels (standardized uptake value [SUV]) are affected by [glucose] [2–6, 45–48], truly incredible circumstances must prevail to have zero correlation between MRGlu and [glucose] under all circumstances. A more extensive mathematical analysis of this problem is presented for the interested reader in the Appendix 2.
Applicability of K Mvalues across multiple tumor types
Model-specific K M values might be expected to have some benefit and were tested as an exploratory measure. They made it possible to bring the blood glucose correlation with MRGlucMAX close to zero for each tumor model independently. However, there was no additional improvement in the between-animal variability. Employing a global value of 130 mg/dL seemed adequate for these exploratory studies given that the benefits of using MRGlucMAX are not critically dependent on using a precise value of K M.
Alternative linear regression method for estimation of MRGlucMAXwith a fixed K M
Having adopted the use of a fixed value for K M, we note that a least-squares linear regression method to compute MRGlucMAX is readily apparent from Equation 3 by plotting K i as a function of (1/(K M + [glucose]), giving a straight line with MRGlucMAX as the slope when the regression line is forced through the origin. Estimation of the group mean by linear regression may perform better than averaging individual values (c.f., Section II.5 in the book by Christensen ). However, when we tested this alternative calculation, we found that it made no appreciable difference to the results. We note that linear regression should offer the greatest benefit where the data contain a wide spread of glucose concentrations; as we noted above, this is not the case for our living-subject data with its relatively narrow range of physiological blood glucose values and relatively high noise level.
Variability and statistical power of MRGlucMAX compared to MRGluc
As noted, a 40% increase in sample size would be required to achieve a 15.5% reduction in COV. However, translating a reduction in COV to improve statistical power requires additional assumptions, e.g., regarding the potential treatment effect . To make a preliminary estimate, we assume that the relative treatment effect is the same for MRGlucMAX and MRGluc when expressed as a percentage change from baseline (a conservative assumption since MRGlucMAX is an asymptote). In this case, the reduction in the required sample size while maintaining the same error rates (i.e., the same statistical power) is 28.6% (equal to 100 × (1-(1-0.155)2)) (see Equation 2 in van Belle and Martin ). The actual sample size savings achieved in practice are likely to be smaller than this because assumptions will not hold exactly. In particular, glucose uptake is only approximated by Michaelis-Menten kinetics; K M is not known exactly; and the error distribution may be neither Gaussian nor perfectly homoscedastic.
No doubt there are many sources of physiological noise contributing to the total observed variability in K i , and blood glucose may be only a small part of that. Nevertheless, a 15.5% reduction in between-animal COV is not trivial and could well become important over the course of many studies or in marginal cases. Also, this improvement should not be considered in isolation, but seen as one step in the evolution of PET methodology over the years.
Glucose normalization and bias
Although biologically appealing, mixed results have come from previous studies of linear glucose normalizations applied to FDG-PET data [3–5]. Multiplying K i by blood glucose (or normalized glucose, i.e., [glucose]/100 mg/dL) did not eliminate bias in our data. Some have found that this normalization actually increased variability and was unhelpful [8, 52, 53], possibly because of the noise introduced by the glucose assay. However, glucose bias was significantly reduced with the nonlinear MRGlucMAX function, while simultaneously achieving reductions in between-animal and test-retest COVs compared to both K i and MRGluc. This is very encouraging and warrants further investigation.
Other nonlinear glucose corrections
Given the significant biological noise that remains in FDG-PET data even after various corrections are applied, other line equations that approximate the Michaelis-Menten equation should fit the data and give broadly similar bias reductions and improvements in glucose-derived variability. For example, Wong et al. have demonstrated that using a square-root function of the glucose concentration allowed their clinical SUV data to better classify indolent and aggressive lymphomas . They also suggested, referring to Langen et al. , that this correction would not be necessary with dynamic scans quantified with MRGluc.
Applicability to SUV data
Our preliminary observations confirm that tumor SUV values correlate highly with our K i data, showing a negative correlation with blood glucose across hundreds of mice and dozens of studies. However, our SUV values were derived from time-activity curves at no more than 30 min after injection of FDG, and with only 5 min of acquisition time, making them statistically noisy compared to purposeful SUV data. We expect that partial saturation correction will have similar benefits with SUV data, but more appropriate experimental data will be required before this can be properly explored. However, applying the square root of glucose SUV correction of Wong et al.  to our tumor data did reduce glucose bias and variability compared to MRGluc, almost as much as MRGlucMAX. The converse should also be true, suggesting that multiplying SUV by (K M + [glucose]) would be effective in the clinical lymphoma setting, while the mechanistic foundation of this correction may make it possible to rationally optimize K M in different tissues or tumor types.
Outliers and blood glucose changes during the scan
Individual outliers often exhibited large differences between their pre-scan and post-scan blood glucose levels. We tested some exclusion criteria which censored out data from scans where there had been a 75% or greater change in blood glucose level during the course of the scan. This helped reduce the between-animal variability in some studies. However, a more attractive alternative may be to track and account for the changes in blood glucose occurring during a scan as proposed by Dunn et al. . It would be interesting to evaluate a combination of partial saturation correction and the method of Dunn et al.  to better account for both between-scan and intra-scan blood glucose changes.
Measured in a very large sample of 1,192 nonclinical dynamic FDG-PET scans, it was clear that the rate of tumor glucose uptake estimated by MRGluc was, in most studies, positively correlated with blood glucose levels. This gave an unwanted bias and additional variability in our estimates of tumor glucose uptake rates.
By assuming a Michaelis-Menten relationship, the simple use of K M + [glucose] in place of [glucose] as the glucose correction factor had several benefits: the hypothetical glucose-saturated MRGlucMAX was less correlated with blood glucose, had lower between-animal variability, and had lower test-retest variability compared to MRGluc.
This reduced bias and reduced variability may translate into a significant reduction in sample size (up to 28%) for nonclinical treatment studies. Further performance comparisons of MRGluc and MRGlucMAX applied to detect confirmed treatment responses in our nonclinical tumor models have been completed and will be described separately.
It will be very interesting, and straightforward, to see if these findings can be translated to studies of clinical trial data where saturation-corrected SUV data could be calculated by multiplying SUV by (100 mg/dL + [glucose]), rather than the more commonly reported glucose-normalized SUV employing ([glucose]/100 mg/dL). In the clinical trial setting, even modest reductions in variability can translate to tangible savings in money, time, and patient enrollment.
On the problem of linearly dependent ML estimates of K mand V maxfrom noisy Michaelis-Menten observations
If an adequate probabilistic framework can be specified for a sample data set, maximum likelihood [ML] typically provides an efficient approach for parameter estimation. Equivalently, for data which are conditionally Gaussian-distributed, one may also use nonlinear least squares. However, due to the functional form of the Michaelis-Menten [MM] relationship, K m and V max are not uniquely estimable (from each other) from noisy MM observations. This problem is further exacerbated when the MM process is observed in a narrow glucose range (as is the case in our work). The problem can be understood by studying the information matrix for K m and V max, but we use a first-order expansion of the ML score function to heuristically verify that the ML parameter estimates of K m and V max are strongly co-linear.
We note that estimation of σ does not affect estimates of V max and K m, and its consideration is henceforth eschewed.
Taking expectations over the noise process, ε j yields that on the average, the score is minimized approximately when , where [m.glc] represents the mean glucose measurement. The result holds when the glucose measurements have low spread but can be shown to hold approximately even as the spread around [m.glc] increases. Thus, is linear in with a slope equal to V o/(K o+[m.glc]).
On the correlation between MRgluc and [glc]
As noted, several authors have described studies where the observed rates K i were negatively correlated with the glucose measurements [glc]. We show that when such a negative correlation exists, the correlation between MRgluc, defined as the product between K i and [glc], and [glc] cannot uniformly equal zero. Indeed, lack of correlation occurs but in a few special cases.
Thus, for cov(MRgluc, [glc]) = 0, we must have or This equality can clearly not hold uniformly across a broad range of realistic parameter choices of .
We illustrate the result by simulations. Our setting assumes that K i follows the MM form with constants K m and V max and that the observed rate is corrupted by noise. That is, K i = V max/(K m + [glc]) + ε, where ε is the random Gaussian with zero-mean and standard deviation σ. As is common, we further assume that the rate constant is observed (sampled) in a glucose range between 60 and 140. We note that when K i is observed in a limited range around some glucose midpoint [m.glc], K i ≈ (V max/(K m + [m.glc])+([m.glc]V max)/(K m + [m.glc])2)-V max/(K m + [m.glc])2[glc] + ε, i.e., K i is approximately linear in [glc].
The authors gratefully acknowledge the contributions of Annie Ogasawara, Alex Vanderbilt, Jeff Tinianow, Herman Gill, Leanne McFarland, and Karissa Peth who helped execute the imaging studies analyzed here.
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