Quantitative kinetic modelling and mapping of cerebral glucose transport and metabolism using glucoCESL MRI

Chemical-exchange spin-lock (CESL) MRI can map regional uptake and utilisation of glucose in the brain at high spatial resolution (i.e sub 0.2 mm3 voxels). We propose two quantitative kinetic models to describe glucose-induced changes in tissue R1ρ and apply them to glucoCESL MRI data acquired in tumour-bearing and healthy rats. When assuming glucose transport is saturable, the maximal transport capacity (Tmax) measured in normal tissue was 3.2 ± 0.6 µmol/min/mL, the half saturation constant (Kt) was 8.8 ± 2.2 mM, the metabolic rate of glucose consumption (MRglc) was 0.21 ± 0.13 µmol/min/mL, and the cerebral blood volume (vb) was 0.006 ± 0.005 mL/mL. Values in tumour were: Tmax = 7.1 ± 2.7 µmol/min/mL, Kt = 14 ± 1.7 mM, MRglc = 0.22 ± 0.09 µmol/min/mL, vb = 0.030 ± 0.035 mL/mL. Tmax and Kt were significantly higher in tumour tissue than normal tissue (p = 0.006 and p = 0.011, respectively). When assuming glucose uptake also occurs via free diffusion, the free diffusion rate (kd) was 0.061 ± 0.017 mL/min/mL in normal tissue and 0.12 ± 0.042 mL/min/mL in tumour. These parameter estimates agree well with literature values obtained using other approaches (e.g. NMR spectroscopy).

transporters have different K t values, and thus in-vivo measurement of K t may provide insight into their relative distribution on cellular membranes and alterations to this distribution caused by disease. The maximum transport capacity depends on the availability of transporters (i.e. their density), and similarly in-vivo measurements of T max could provide important information regarding the effects of ageing or pathology on glucose delivery to the brain.
In some solid tumours, glucose transporter expression is increased, 9 which has been shown to correlate with tumour hypoxia, a property associated with treatment resistance and tumour agressiveness. 10,11 Due to aberrant vessel formation during neo-angiogenesis, BBB integrity may also be compromised, leading to increased free diffusion of glucose across the bloodtumour interface. In neurodegenerative diseases such as Alzheimer's disease, glucose transporter expression is reduced 7,12 which may starve the brain of glucose, 13 and contribute towards neurodegeneration. 14 In many diseases, alterations to glucose transport and/or metabolism are regionally specific, and thus being able to detect glucose transport and utilisation at high spatial resolution across multiple brain regions, would greatly progress our ability to study deficiencies in glucose transport and metabolic dysfunction.
2-deoxy-2-[ 18 F]fluoro-D-glucose positron emission tomography (FDG-PET) is the gold-standard technique for imaging glucose transport and metabolism in-vivo, and has led to improved understanding of metabolic dysfunction across a range of brain diseases. 15,16 Typically, trace amounts of FDG are injected and late static tissue PET measurements made. For full kinetic modelling, and in order to derive rates of cerebral glucose transport and utilisation, FDG kinetics in arterial blood and tissue are needed from arterial blood sampling and dynamic PET scanner measurements. 17,18 FDG differs in its transport and phosphorylation properties relative to glucose and the estimated rate constants derived using FDG must be converted to glucose rate constants using an experimentally derived lumped constant. 19,20 Several attempts have been made to measure the lumped constant for FDG, with values ranging from 0.71-0.89 21,22 in rat and 0.68-0.89 20,[23][24][25] in human, presenting a significant potential source of bias. Quantification of glucose metabolic rates in brain tumours using FDG is further complicated by the unknown redistribution of glucose transporters.
Detection of glucose in-vivo in the brain is possible using magnetic resonance spectroscopy (MRS), as first reported by Frahm et al. in 1991. 3, 26 1 H MRS has been used to measure uptake and utilisation of glucose following hyperglycaemic challenge, measuring changes in the glucose signals at 3.44 and 5.23 ppm. [27][28][29] However, quantification is difficult due to spectral overlap with more concentrated metabolites such as myo-inositol, creatine/phosphocreatine and amino acids. Significant effort has been invested in developing methods to improve the specificity to glucose, and downstream metabolites using 13 C-labelling, 27,[30][31][32][33][34][35][36] and more recently, 2 H-labelled glucose. [37][38][39] In both cases, the label passes down the glycolytic pathway, and can be detected in glucose itself and downstream metabolites such as lactate and glutamate. A key advantage of these approaches is reduced spectral overlap, because unlabelled endogenous molecules do not contribute to the spectra. However, these methods are still limited by low sensitivity, meaning large voxel volumes are required to provide adequate signal to noise for precise quantification. Usually, specialised coils 36 are required for 13 C and 2 H approaches, limiting translation, though there are approaches to detect the presence of 13 C-labelled metabolites in 2 HMRS that do not require either a 13 C-channel or 13 C -tuned RF coil. 40 Nevertheless, this latter approach lacks the sensitivity to detect glucose itself, being limited to detection of the higher concentration downstream metabolite, glutamate.
More recently, chemical exchange saturation transfer (CEST) and chemical-exchange spin lock (CESL) MRI have been used to indirectly detect glucose via chemical exchange of protons in the hydroxyl groups ( À OH) of glucose with protons in water. These approaches are derived from techniques that image exchangeable protons in macromolecules 41,42 and more recently have been used to spatially map glucose uptake in living tissues, and appear to overcome the low detection sensitivity limitations of spectroscopic approaches. CEST was first applied to map uptake of intravenously administered glucose in tumours by Chan et al. 43 then Walker-Samuel et al.. 44 Since then, CEST and CESL have been applied to detect glucose uptake in rodent and human brain by a number of groups. [45][46][47][48][49][50][51][52][53] Jin et al. 46 and others 50,54-56 have performed key studies in normal brain, tumour, and stroke showing that glucoCESL and glucoCEST uptake curves are sensitive to both uptake and metabolism of glucose. 50,54 In these studies, the uptake curves of D-glucose and/or glucose-analog 3-Omethyl-D-glucose (3OMG) were monitored using CESL or CEST MRI. D-glucose and 3OMG are taken up by tissues at similar rates, but 3OMG, which is not metabolised, was found to remain at high levels, while the curves for D-glucose decreased to pre-injection levels. Despite these studies showing the clear sensitivity of CESL and CEST to the transport and metabolism of glucose, and the potential for mapping these processes at high spatial resolution, kinetic modelling to determine physiological rate constants has not been applied to this type of data.
In this study, we introduce the theoretical background for quantitative kinetic modelling of glucoCESL MRI data. We evaluate the sensitivity of R 1q to all kinetic model parameters, evaluate bias introduced due to inaccurate glucose input functions, and apply the models to data acquired in tumour-bearing (9 L glioma) and healthy rats. We evaluate two kinetic models describing purely saturable transport (model 1) and mixed transport with saturable and free diffusion components (model 2). The fit quality of each model is compared using the Akaike information criterion on a voxel wise level. Finally, we evaluate differences in the central tendency and distribution (standard deviation) of estimated transport and metabolic parameters between tumour and normal tissue.

Relaxation model
In the presence of an on-resonant spin-lock pulse, the spin-lock relaxation rate R 1q can be written in terms of chemical-exchange and non-chemical exchange contributions: 46 where R ex is the contribution due to chemical exchange between labile protons in biomolecules, and protons in water, and R 20 is the spin-spin relaxation rate of water without chemical exchange contributions. Following administration of glucose into the bloodstream, the amount of labile proton exchange between hydroxyl groups ( À OH) in glucose and water will theoretically increase in proportion with the concentration of glucose, leading to an increase in voxel R 1q via alteration of R ex . The effect of glucose concentration on R 20 is assumed negligible.
Changes in R 1q may occur due to increased/ decreased proton exchange with glucose or any other detectable (via exchange) metabolic product of glucose, i. We can therefore write the total change in R 1q as the sum of these contributions: where DC i is the change in concentration of molecule i (i includes any detectable molecule, including glucose or metabolic products of glucose) and r i is the R 1q relaxivity of each of these molecules.
Assuming the population of labile proton i is much smaller than the population of water protons, the relaxivities r i [(s mM) À1 ] are theoretically dependent on the spin-lock frequency x 1 (set during the experiment), the resonance frequency gap between molecule i and water, d i [s À1 ] (fixed), the number of labile protons per molecule n, and the exchange rate between these labile protons and water, k i [s À1 ] (fixed, but known to vary with tissue pH 46 ) The factor 1.11 Â 10 5 is the concentration of protons in pure water in units of mM (assuming a pure water concentration of 55 M) and ensures the relaxivity has units of [(s mM) À1 ]. For accurate modelling of the relaxivity, the concentration of water protons in the relevant brain structures should be used if known.
In healthy brain tissue, the majority of glucose undergoes glycolysis. In neurons and astrocytes, glucose is phosphorylated by hexokinase into glucose-6-phosphate (G6P), which at rest inhibits hexokinase activity to maintain a constant metabolic rate of glucose consumption, regardless of tissue glucose concentrations. 57 There may therefore be relatively large fluctuations in G6P following intravenous injection of glucose (in percentage terms), but the basal concentration will remain very low relative to glucose 58 (<2-3%), and therefore these changes will have only a minor effect on measured DR 1q , even if r i of 6GP is of a similar magnitude to glucose. At resting state, the concentrations of other downstream metabolic products (e.g. glycogen, glutamate) should remain fairly stable 59 and independent of changing tissue glucose levels, and thus should also not contribute to measured DR 1q . In tumours, metabolism is reprogrammed to support anabolic growth in nutrient deplete environments. A key component of this transformation is for tumour cells to convert pyruvate to lactate, which then accumulates extracellularly. Fortunately, for glucoCESL studies, the exchange rate of lactate is much lower than the resonance frequency separation between lactate and water 60 meaning increases in lactate concentration due to increased glucose supply should not contribute significantly to DR 1q (i.e., k i ( d i in equation (3)).
Taken together, we assume that changes in DR 1q following intravenous glucose injection are predominantly driven by changes in glucose concentration only (i.e. effects of changing metabolite concentrations are negligible). We can therefore rewrite equation (2) to include contributions from glucose only, and rearrange to show how measured DR 1q can be converted to changes in glucose concentration: where r g is the R 1q relaxivity of glucose, and DC is the change in tissue glucose concentration. Hence by tracking dynamic changes in R 1q we can infer changes in glucose concentration. Kinetic models can be fit to the concentration time-course data to extract quantitative parameters relating to glucose transport and metabolism.

Kinetic models
Two kinetic models are proposed to describe changes in tissue glucose concentration ( Figure 1). Both models assume the metabolic rate of glucose consumption, MR glc ¼ C 1 Á k 3 , is constant, as would be expected at rest. In the brain, glucose is phosphorylated into glucose-6-phosphate primarily by hexokinase I, and then stored intracellularly before being utilised in the glycolytic or pentose-phosphate pathways. 57 Hexokinase exhibits unique regulatory properties, in that its rate of activity is potently inhibited by its product. 57,61 Thus, when glucose-6-phosphate is required by the cell, local glucose-6-phosphate levels become depleted, and the rate of glucose phosphorylation increases to restore glucose-6-phosphate levels (increased k 3 ). When glucose-6-phosphate levels are high, hexokinase inhibition increases and glucose s.p phosphorylation is reduced (reduced k 3 ). Thus, when intracellular glucose concentrations are increased (as would occur during an i.v. bolus injection of glucose), the amount of glucose-6-phosphate being produced will momentarily increase, then immediately self-regulate via increased inhibition of hexokinase (reduced k 3 ). 61 This mechanism ensures that, at rest and at equilibrium, the metabolic rate of glucose consumption MR glc remains constant, even in the presence of increased intracellular glucose levels. Model 1 assumes the influx and efflux rates of glucose across the blood-brain barrier (BBB), K 1 and k 2 , are time-dependent and governed by Michaelis-Menten kinetics. 8 where C a (t) [mM], and C 1 (t) [mM] are the arterial and tissue glucose concentrations respectively, T max [mmol/ min/mL] is the maximal transport capacity, and K t [mM] is the half saturation constant of the glucose transporters. Model 2 assumes uptake of glucose is governed by a mixture of Michaelis-Menten kinetics and passive diffusion. 62 An extra term k d [mL/min/ mL] is added to the right-hand side of equations (5) and (6) (see Figure 1). The mass transport equation for both models is: Both models are defined diagrammatically in Figure 1. The total glucose concentration in a given brain voxel is modelled as a volume weighted average of blood (capillary) and tissue glucose concentrations: where v b [mL mL À1 ] is the fractional blood volume. The capillary glucose concentration is assumed to be equal to the glucose of the feeding artery (C a ¼ C c ), which is a good assumption in highly perfused tissues such as the brain. 63 To calculate the change in voxel glucose concentration following intravenous administration of glucose solution, we subtract the modelled baseline glucose concentration: where Cðt ¼ 0Þ can be derived for any compartment model at steady state by assuming dC 1 ðtÞ dt ¼ 0. Figure 1. Diagram of the kinetic models. Arterial and tissue glucose have concentrations C a (t) and C 1 (t) respectively. The rate of glucose uptake across an intact blood-brain barrier is modelled with Michaelis-Menten kinetics (model 1), where k 1 is parameterised by the maximal transport capacity T max , half saturation constant K t , and is dependent on the arterial glucose concentration C a (t). In addition to transporter mediated uptake, glucose may also freely diffuse across the BBB into tissue with rate constant k d (model 2). We assume that k 3 (t) decreases in response to an increase in C 1 (t) to maintain a constant metabolic rate of glucose consumption, MR glc (i.e., MR glc ¼ k 3 (t)C 1 (t)).
Derivations of Cðt ¼ 0Þ for model 1 and model 2 are given in Supplementary materials.

Sensitivity analysis
Model sensitivity analysis was performed to determine the relative contribution of each model parameter and the rate of glucose clearance from the blood on DR 1q . For both kinetic models, a 50% change in each parameter was simulated using equation (9), and the effect of these changes on DR 1q curves were plotted using equation (4). The initial values of each parameter from which perturbations were made were based on previously published literature data 31,58,64 and set to T max ¼ 3.3 mmol/min/mL,

Animals
All animal studies were approved by the Institutional Animal Care and Use Committee at the University of Pittsburgh in accordance with the National Institutes of Health guide for the care and use of laboratory animals. All procedures were conducted in accordance ARRIVE (Animals in Research: Reporting In Vivo Experiments) guidelines (www.nc3rs.org.uk/arrive-guidelines). Animal preparation and MRI experiments have been described previously. 46,50 Adult male Fischer-344 rats (n ¼ 4, Charles River, Wilmington, MA) weighing between 200-250 g (approx. 12 weeks of age) were anesthetized with 2% isoflurane. A burr hole was drilled into the right hemisphere at stereotaxic coordinates of 0 to 0.5 mm from bregma, and 3 to 3.5 mm from the midline. Inoculation of 100,000 to 300,000 9 L cells (ATCC CRL-2200, Manassas, VA) in a total volume of 5 mL was performed 4 mm below the surface of the dura. After the induction of tumour cells, rats were imaged 3 to 5 weeks after (approx. age 15-17 weeks), when the tumour reached appropriate volume. Healthy adult male Sprague-Dawley (n ¼ 3, Charles River, Wilmington, MA) rats weighing 335-388 g (approx. 12 weeks of age) were also imaged. An additional n ¼ 3 Fischer-344 rat (254-270 g) were purchased specifically for estimation of a Fischer-344 specific population glucose input function. These animals were not scanned using MRI. For MRI scans, rats were anesthetized with 2% isoflurane in a mixture of O 2 and air gases and the right femoral vein catheterised for delivery of glucose. After catheterisation, isoflurane levels were reduced to 1.5%. The dynamic blood pressure, end-tidal CO 2 , and rectal temperature were monitored throughout the duration of the scan. The end-tidal CO 2 level was kept within 3-4% and the rectal temperature was maintained at 37.5 AE 0.1 C using a feedback-controlled heating pad. MRI data was acquired on a 9.4 T small animal MRI system (Agilent Technologies, Santa Clara, CA, USA) . Radiofrequency transmission was provided by a volume coil (6.4 cm diameter) and received by a surface coil (2.2 cm diameter; coil combination from Nova Medical, MA, USA). Spin-echo EPI CESL images with and without R 1q -weighting (i.e., spin-lock pulse While not being varied, parameters were fixed to: duration TSL ¼ 0 and 50 ms) were acquired in an interleaved manner using a spin-lock frequency x 1 $ 500 Hz. Each set of TSL ¼ 0 ms and TSL ¼ 50 ms images constituted a block, and were used to calculate a single R 1q image as described below.
MRI data for tumour-bearing and healthy rats were acquired in two separate experiments, and some aspects have been previously reported. 46,65 The time difference between these studies meant there were some differences in the scanning and injection protocols. Table 1 shows the MRI acquisition parameters used for acquisition of tumour-bearing and healthy rat data. All experiments were performed in the non-fasted state. For tumour-bearing rats, glucose was dissolved in distilled water to 30% weight by volume concentration. For healthy rats, glucose was dissolved in distilled water to 20% weight by volume concentration. A glucose bolus (1 g/kg) was injected after 20 minutes (tumour bearing) or 30 minutes (healthy) of baseline data over approximately 30-60 seconds, after which R 1q was measured for 50 minutes. R 1q time-courses were estimated using the following equation:

Arterial glucose input functions
Image-based measurement of arterial glucose concentration, C a (t) (via measurement of DR 1q (t)), was attempted in each rat by choosing a voxel containing only arterial blood, but results were not reliable. Instead, population level arterial input functions were derived via benchtop arterial blood sampling. For the Sprague-Dawley strain, arterial input functions were obtained the day after MRI scanning under identical experimental conditions. This data has been previously reported in Jin et al. 46 For the Fischer-344 rats, glucose input functions were not measured at the time of imaging. Three additional Fischer-344 rats were purchased specifically to measure the arterial glucose concentration timecourse, C a (t).
For each animal, the right femoral artery was catheterized and blood samples acquired every 10 minutes (Sprague-Dawley) or 5 minutes (Fischer-344) following injection of D-glucose. As with MRI experiments, all blood sampling was performed under non-fasting conditions. The same volume, concentration, and injection rate as used during the imaging studies was used to ensure matched experimental conditions. Three blood glucose readings were acquired prior to the glucose injection and averaged to determine the baseline. Before kinetic model fitting, population input functions were smoothed by fitting a bi-exponential model with amplitude and rate parameters A 1 , A 2 , k 1 , k 2 .

Error analysis
Systematic error introduced into kinetic model parameters due to use of inaccurate arterial glucose input functions was determined by simulating tissue concentration curves with arterial glucose clearance rates of [A 1 , A 2 , 0.9k 1 0.9k 2 ] and [A 1 , A 2 , 1.1k 1 1.1k 2 ] as described above, then fitting back to the simulated curve assuming the original arterial glucose input function (i.e. using [A 1 , A 2 , k 1 k 2 ]). Zero mean gaussian noise with standard deviation, r, was added to the synthetic curves to give a contrast to noise ratio (DC max /r) of 30.

Kinetic model fitting
Voxelwise estimates of DR 1q (t) were computed by subtracting the mean pre-injection R 1q and converting to estimates of DC(t) using equation (4) and r g ¼ 0.066 (s mM) À1 derived from Figure 1D of Jin et al. 46 Kinetic models were then fit to DC(t) time-courses using the ordinary differential equation solver ode45 and lsqcurvefit function in Matlab (Mathworks, version 5 2017a). A delay parameter was included in the fitting routine to adjust for differences in bolus arrival In tumour-bearing rats, tumour regions of interest (ROIs) and normal tissue ROIs were outlined manually on the TSL ¼ 0 ms image (i.e. the T 2 -weighted image) in MRIcron (version 1.0.2). Normal tissue ROIs were drawn on the contralateral side, as to best match the shape and extent of the tumour ROI, but avoiding bright CSF signal. In general these ROIs covered cortical and subcortical gray matter. In healthy rats, normal tissue ROIs were drawn on both sides of the brain and summed together into a single region. These ROIs were drawn to best cover the same regions as the contralateral ROIs drawn for tumour-bearing animals. Prior to comparing tumour and normal tissue parameter values, tests for normality were performed using the Shapiro-Wilks test. The null hypothesis of normality was rejected in less than 5% of data groupings, and thus parametric tests were used throughout.

ROI-level analyses
The median value of each kinetic parameter was calculated for normal tissue and tumour ROIs. Normal tissue values in healthy rats were combined with contralateral normal tissue values from tumour-bearing rats, and the difference between tumour (n ¼ 4) and normal tissue (n ¼ 7) evaluated using t-tests for partially overlapping samples. To assess heterogeneity in glucose uptake and metabolism within each ROI, the standard deviation of each parameter within tumour and normal tissue ROIs was calculated 66 and the null hypothesis of no difference tested using t-tests for partially overlapping samples.
The Akaike information criterion (AIC) adjusted for small sample sizes and DAIC (DAIC ¼ AIC 1 -AIC 2 ) were computed on a voxelwise level to compare the fit quality between models, accounting for differences in the number of fit parameters. To test which model was preferred in tumour and normal tissue, t-tests were performed on DAIC to test the null hypothesis that DAIC ¼ 0. A t-test for partially overlapping samples was also performed to determine if there was a difference in DAIC between tumour and normal tissue. Statistical analysis was done in R (version 4.0.2). Data and analysis scripts used in this paper can be obtained by request to the corresponding author.

Measurement of the R1 q relaxivity of lactate
To assess the potential contribution of lactate to DR 1q we measured the R 1q relaxivity of lactate in phantom at 9.4T. Lactate phantoms were created with 5 mM, 10 mM, 20 mM and 40 mM of lactate added to 1Â phosphate-buffered saline and titrated to pH ¼ 7.0. Phantoms were bundled together and heated to 37 C degree for CESL measurement. The R 1q of each phantom was measured using the following spin-lock pulse parameters: x 1 ¼ 500 Hz, varied spin-locking durations (TSL) of 0 to 1 s in 100 ms step. Images were acquired with a spin-echo EPI sequence with the following parameters: one 5-cm slice with FOV ¼ 5 cm Â 5 cm, matrix size 64 Â 64, TR ¼ 15 s, and TE ¼ 20 ms. The relaxivity of lactate was estimated by plotting R 1q vs lactate concentration and performing linear regression to find the gradient. Figure 2(a) shows how the Sprague-Dawley arterial glucose concentration varies with k 1 and k 2 set to 0.90 (black dashed) and 1.1 (yellow dashed), simulating slower and faster glucose clearance (for example due to slower and faster insulin responses). Figure 2(b) and (c) show the results of the DR 1q sensitivity analysis for model 1 (saturable transport) and model 2 (mixed saturable transport and free diffusion) respectively. For both models, increases in T max , K t , v b produced positive DR 1q , whereas MR glc produced negative DR 1q . A slower glucose clearance from blood produced positive DR 1q , and faster glucose clearance from blood produced negative DR 1q . For both models T max produced the largest change in DR 1q and K t the smallest. For model 2, k d produced an effect similar in magnitude to T max but with a longer tail. For both models, v b produced a sharp effect during the first 10 minutes post injection, which rapidly decreased with the concentration of glucose in blood. Changes in the rate of glucose clearance from blood were greater than those due to changes in K t (both models) and MR glc (model 2 only). Figure 3 shows biases in kinetic model parameters due to use of an inaccurate arterial glucose input function. For model 1 (Figure 3(a)), biases were generally less than 40% for all parameters; biases were smallest for T max (median less than AE2.6%), and largest for K t (À2.8 to þ22%), MR glc (À13 to þ29%), and v b (À36% to þ6%). For model 2 (Figure 3(b)), T max (À17 to À9%), K t (À12 to 11%) and v b (À2.8 to þ2.9%) were least affected, and MR glc (À21 to þ45%) and k d (þ12 to þ27%) most affected. Errors in T max and k d were approximately equal but in opposite directions,  Example voxelwise data and kinetic model fits in normal tissue (c) and tumour (e). Example ROI averaged data and fits in the normal tissue (d) and tumour (f). Both models provide a good fit to normal tissue and tumour data. Model 2 appears to provide a better fit to tumour data at a ROI level. possibly reflecting degeneracy in these parameters for describing changes in DR 1q . Figure 4 shows an example of tumour and normal tissue region of interests (ROIs), the population-level input functions and corresponding bi-exponential fit for Sprague Dawley and Fischer-344 rats, and example voxelwise and ROI time-courses and model fits. The estimated parameters from the bi-exponential fits were A 1 ¼ 12.4 mM, A 2 ¼ 8.6 mM, k 1 ¼ 0.18 min À1 , k 2 ¼ 0.0086 min À1 for Sprague-Dawley, and A 1 ¼ 12.5 mM,

Results
for Fischer-344, where A 1 and A 2 are the exponential amplitudes, and k 1 and k 2 are the decay constants. As expected, ROI-level curves (Figures 4(d) and (f)) had higher signal to noise ratio than voxel-level curves (Figures 4(c) and (e)). In both normal and tumour tissue, model 1 and model 2 provided adequate fit at all timepoints in both voxel and ROI level data. In tumour tissue, model 2 was able to occasionally provide a more accurate description of the peak of the curve ( Figure 4E). Figure 5 shows spatial maps of each kinetic parameter for a tumour bearing rat, and mean values (AEsd) of each kinetic parameter averaged across all rats in normal (n ¼ 7) and tumour tissue (n ¼ 4). P-values show the results of t-tests comparing mean parameters in normal tissue and tumour. Mean values AE sd and p-values are also given for each parameter in Table 2. The difference in AIC between model 1 and model 2 in normal tissue and tumour is also shown as a parameter Figure 5. GlucoCESL parameter maps from a tumour bearing rat using model 1 (a-d) and model 2 (i-m), and mean parameter values in normal tissue (healthy -triangle; tumour bearing -circle; n ¼ 7) and tumour (n ¼ 4) for model 1 (e-h) and model 2 (n-r). Far right is the corresponding DAIC map (s) and mean DAIC in normal tissue and tumour (t). T-tests for partially overlapping samples were used to test the null hypothesis of no difference in parameter values between normal tissue and tumour. Error bars show group standard deviation. map ( Figure 5(s)), and summarised across all rats as a barplot ( Figure 5(t)). Visually, both models produce high fidelity maps of transport and metabolic parameters. For model 1, normal tissue/tumour contrast can be observed in images of T max and K t , whereas little or no contrast is observed for MR glc and v b . This is reflected in the group comparisons. There were significant differences between tumour and normal tissue for T max (54% increase; p ¼ 0.0059) and K t (40% increase; p ¼ 0.011). On the contrary, there was no significant difference in MR glc . When evaluating model 2 parameters, there was a borderline significant reduction in MR glc in tumour tissue (21% reduction; p ¼ 0.045), and a borderline significant increase in k d (49% increase; p ¼ 0.061). Model 2 estimates of T max , K t , and v b were unchanged between tumour and normal tissue. Averaged across all animals, the difference in AIC between models (AIC model 1-AIC model 2) was significantly less than zero for both normal tissue and tumour tissue (DAIC < 0; p ¼ 4.1 Â 10 À5 and p ¼ 0.0076 for normal and tumour tissue respectively).
Supplementary Figure 1 shows the standard deviation of glucose uptake and metabolic parameters in tumour and normal tissue ROIs. There was no differences in the standard deviation of parameters between normal and tumour tissue except for v b from model 1 (p ¼ 0.016; greater sd in tumour tissue), and k d from model 2 (p ¼ 0.014; greater sd in tumour tissue).
Supplementary Figure 2 shows how R 1q varies with lactate concentration in phantom. The R 1q relaxivity of lactate was found to be 0.0018 (s mM) À1 .

Discussion
We have presented the theory for quantitative kinetic modelling of glucoCESL MRI data, and applied two kinetic models to data from tumour-bearing and healthy rats. Parameter estimates from both models agree well with previously published values. 31 Mason et al. summarised NMR spectroscopy estimates of T max , K t , and k d in rat brain from 19 prior studies. In studies that used model 1 (n ¼ 14), estimates of T max and K t were 3.9 AE 0.5 lmol/min/mL (assuming 0.83 mL water/g tissue) and 8.1 AE 0.6 mM which agree well with our normal tissue estimates of 3.2 AE 0.62 lmol/min/mL and 8.8 AE 2.2 mM. In studies that used model 2 (n ¼ 5), estimates of T max , K t , and k d were 1.56 AE 0.61 lmol/ min/mL, 6.8 AE 0.6 mM, and k d ¼ 0.025 AE 0.0074 mL/ min/mL, which also agree well with our normal tissue estimates from model 2 of T max ¼ 1.7 AE 0.67 lmol/min/ mL, K t ¼ 4.7 AE 0.34 mM and k d ¼ 0.061 AE 0.017 mL/ min/mL. While there is greater discrepancy between model 2 parameters and available literature data than for model 1, the direction of change in T max and K t when using model 2 versus model 1 is consistent (i.e. both are reduced), indicating that our data is also consistent with prior studies.
Model 1 estimates of MR glc in normal tissue (0.21 AE 0.13 mmol/min/mL) were also within the range of previously published values. In laboratory rats, the effects of anesthesia on MR glc are profound, and must be considered when comparing MR glc between studies. 67 In fasted rats anesthetised with 2% isoflurane, Du et al. measured MR glc in a large region covering the cortex and subcortical brain and found values of 0.53 AE 0.20 mmol/min/mL. 28 Lu et al. used similar anaesthesia and deuterium spectroscopic imaging of deuterated glucose in unfasted rat, and found MR glc values in whole rat brain of 0.28 AE 0.13 mmol/min/g. 38 Shimoji et al. used FDG-PET scanning and halothane anesthesia (1-1.5%) in fasted rat to estimate MR glc in the cortex and found values of 0.24 AE 4.8 mmol/min/g. 68 Suzuki et al. investigated the effects of different anesthesia on MR glc using FDG-PET, including scanning of conscious rat. In conscious rat, much higher values of 0.93 AE 0.18 mmol/min/g were reported, which reduced to 0.54 AE 16 mmol/min/g with isoflurane anethesia. 67 69 Crane et al. also used pentabarbitol anesthesia but also studied conscious rat and found values of MR glc of 0.27 lmol/min/ g 21 and 1.02 lmol/min/g respectively. Cremer studied conscious rat only and found values of 1.24 lmol/min/ mL. 70 While data is not available on isofluraneanesthetised rats, all these values are not substantially different from literature values using model 1 indicating that model 2 estimates of MR glc should not be affected substantially by modelling of free diffusion, particularly in tissues with intact BBB. It is therefore important that further work is done to validate model 2 estimates of MR glc obtained using glucoCESL MRI. The maximal transport capacity (T max ) and the half saturation constant (K t ) from model 1 were significantly higher in tumour than normal tissue, indicating increased uptake of glucose. Higher T max may reflect an increased density of glucose transporters on the vasculature and tumour cell membranes. However, in the context of increased K t , the interpretation must be carefully considered. K t will trade-off saturable and nonsaturable kinetics, and as K t increases, K 1 becomes less dependent on C a , and begins to mimic the behaviour of the free diffusion constant k d . Thus, increased T max and K t together may reflect the presence of free diffusion and blood-brain barrier breakdown. This interpretation is supported by model 2 parameters, where differences in T max and K t between tumour and normal tissue do not exist or are much smaller, and variability in uptake of glucose is described mainly by k d (Figure 5 (a), (b), (i), (j) and (m)). We did not find any difference in metabolic rate of glucose consumption between tumours and normal tissue for model 1, which is consistent with the clinical observation that uptake of FDG in many brain tumours is often no higher than surrounding normal tissue. 16 On the contrary, a small but significant reduction in model 2 MR glc was observed in tumour tissue. However, given the uncertainty associated with model 2 estimates of MR glc as described previously, this result should be treated with caution until verified by further studies. Model 1 and model 2 predicted no difference in blood volume between normal tissue and tumour. Model 1 estimates of blood volume were much lower than known values (v b < 0.005 mL/mL and v b < 0.01 mL/mL for normal tissue and tumour respectively). Estimates for model 2 were more reasonable but still low $ 0.02-0.03 mL/ mL. The underestimation could be a result of the shorter T 2 of the blood compartment relative to tissue at 9.4 T 71 which might cause glucose in the blood compartment to be less visible than glucose in tissue.
Estimates of the Akaike information criterion (AIC) in normal and tumour tissue showed that model 1 had significantly lower AIC than model 2 (DAIC < 0; p ¼ 4.1 x10 À5 and p ¼ 0.0076 for normal and tumour tissue respectively), indicating that the additional free diffusion parameter of model 2 did not describe any additional variability above that described by model 1. This does not mean that model 2 is less accurate or did not fit the data as well as model 1, but that within the conditions of our experiment (i.e. temporal resolution, SNR), the additional parameter did not provide additional predictive power over parameters already present in model 1. It is clear from our results that the normal tissue values of transport and metabolic parameters from model 1 are more consistent with literature values. It was also shown in previous work that the same tumour-bearing rats as scanned in this study exhibit leakage of DCE-MRI contrast agents (see Figure 3C of Jin et al. 65 ) indicative of BBB breakdown. This suggests model 2 may more appropriately ascribe changes in glucose uptake in tumour tissue to free diffusion rather than changes to saturable transport mechanisms. Further validation work is needed to confirm these points.
We have assumed that contributions from metabolic products of glucose to DR 1q were negligible compared to the effects of glucose itself. It is shown in previous work that changes in concentrations of glucose-6-phosphate 58 and neurotransmitters such as glutamate 59 are very small, and should not affect DR 1q . Since tumours can produce high levels of lactate, we performed an additional phantom study to demonstrate that rising lactate levels will also not appreciably affect DR 1q . The R 1q relaxivity of lactate was found to be 0.0018 (s mM) À1 , 37 times smaller than glucose. Using the reported values of the chemical shift and exchange of lactate from recent literature, 60 predicted R 1q relaxivity is even smaller than our measured value ($0.0005 (s mM) À1 , 134 times smaller than glucose). It is not yet clear what effects lactate will have on CEST detection, which is more sensitive to slower exchange rates than CESL. Changes to DR 1q from osmolality effects and differences in tissue pH were also ignored, which may have a larger effect on DR 1q. 50 These factors, and the impact of a wider range of metabolites, should be investigated in future work to further refine the modelling approach.
The study has the following limitations. We have used a population-based arterial glucose input function instead of using individually sampled, or image-derived arterial input functions. We attempted to obtain imagederived AIFs from the carotid artery, but it was not possible to obtain reliable results. This is presumably due to inflow of fresh unsaturated blood into the imaging slice (i.e. blood that has not undergone spinlocking) during the time between spin-locking and the SE-EPI readout. It is possible the shape of the population level curves may have deviated from the true (individual) input function of the rats investigated in this study. Our simulations showed that deviations between assumed and true arterial glucose concentrations can have appreciable effects on DR 1q , particularly at later time points (the peak effect was observed $20 minutes post glucose injection), which then propagate through to biases in estimates of all kinetic parameters, but predominantly affected K t , MR glc , and v b for model 1 and MR glc and k d for model 2. These results highlight the importance of accurate input function characterisation when using such models, particularly for inter-subject or group-level comparisons. Accurate knowledge of input functions may not be so important for intrasubject comparisons (e.g. tumour vs normal tissue) as it could be expected the glucose input to different brain regions will be similar. Despite the potential for large biases, the excellent correspondence between our kinetic parameter estimates (particularly for model 1) and literature values suggest that the input functions used in this work are an accurate representation of the true arterial glucose time-courses. Future work should investigate the use of CESL and CEST approaches for measuring an image-derived input functions, and/or further validate the accuracy of the population based input function approach for group level comparisons. It is possible that adjustment (e.g. scaling) of population level input functions using pre-and post-scan venous blood glucose measurements (e.g. using a simple blood glucose monitor) may be adequate to account for inter-individual or group variability in blood glucose concentration timecourses. There were some small differences in the acquisition parameters and injection protocols between tumour bearing and healthy rats. For example, the weight by volume concentration, and hence the administered volumes of glucose solution were different. It is possible these effects led to small differences in osmolality of blood. However, given the estimated kinetic parameter values for normal tissue in healthy and tumour bearing rats were not appreciably different, we are confident these effects were small. For future glucoseenhanced MRI studies, including translational work in humans, a standardised set of optimal acquisition parameters are needed.
To conclude, we have for the first time demonstrated the feasibility of applying kinetic models to glucoCESL MRI data and show the potential of the approach to probe glucose transport and metabolism in-vivo at high spatial resolution.

Declaration of conflicting interests
The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: GJMP is a shareholder and director of Bioxydyn Limited, a company with an interest in quantitative imaging biomarkers. He is also a shareholder and Director of Queen Square Analytics, a company with an interest in MRI applied to neuroscience.