Determination of local brain glucose level with [14C]methylglucose: effects of glucose supply and demand

Gerald A. Dienel1, Nancy F. Cruz1, Keiji Adachi1, Louis Sokoloff1, and James E. Holden2

1 Laboratory of Cerebral Metabolism, National Institute of Mental Health, Bethesda, Maryland 20892; and 2 Department of Medical Physics, University of Wisconsin Medical School, Madison, Wisconsin 53706

    ABSTRACT
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Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References

Methylglucose can be used to assay brain glucose levels because the equilibrium brain-to-plasma distribution ratio for methylglucose (C*e/C*p) is quantitatively related to brain (Ce) and plasma (Cp) glucose contents. The relationship between Ce and C*e/C*p predicted by Michaelis-Menten kinetics has been experimentally confirmed when glucose utilization rate (CMRGlc) is maintained at normal, resting levels and Cp is varied in conscious rats. Theoretically, however, Ce and C*e/C*p should change when CMRGlc is altered and Cp is held constant; their relationship in such conditions was, therefore, examined experimentally. Drugs were applied topically to brains of conscious rats with fixed levels of Cp to produce focal alterations in CMRGlc, and Ce and C*e/C*p were measured. Plots of Ce as a function of C*e/C*p for each Cp produced straight lines; their slopes decreased as Cp increased. The results confirm that a single theoretical framework describes the relationship between Ce and C*e/C*p as either glucose supply or demand is altered over a wide range; they also validate the use of methylglucose to estimate local Ce under abnormal conditions.

cerebral glucose utilization; brain hexose transport

    INTRODUCTION
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Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References

MANY ABNORMAL and pathophysiological conditions alter local rates of glucose utilization (CMRGlc) in brain, which in turn cause the level of glucose in the tissue to vary. Metabolic mapping techniques are often used to detect and quantify abnormal brain function in affected structures, monitor the evolution of the condition, and evaluate the efficacy of treatment in experimental animals and human subjects. Determination of CMRGlc, particularly under abnormal conditions, requires knowledge of the local tissue glucose content so that the appropriate value of the lumped constant is used to calculate CMRGlc when [14C]deoxyglucose ([14C]DG) or 2-[14F]fluoro-2-deoxy-D-glucose is used as the tracer (3, 10, 11, 23, 24, 26, 28-30). The lumped constant of the [14C]DG method is relatively stable in normoglycemia and hyperglycemia, but it rises sharply when the brain glucose level drops below ~1 mmol/g (e.g., as in Refs. 3, 6, 23, 24, 26, and 30). Also, when labeled glucose is used as the tracer, local glucose levels must be determined so that the proper value for the brain-to-plasma distribution ratio for glucose is used to calculate CMRGlc in each structure (13).

Radiolabeled 3-O-methyl-D-glucose (methylglucose) can be used to assay local glucose levels in brain by direct measurement or by quantitative autoradiography, because the equilibrium brain (C*e)-to-plasma (C*p) distribution ratio for methylglucose (C*e/C*p) varies quantitatively with brain (Ce) and plasma (Cp) glucose contents (1, 10). The methylglucose distribution ratio is used to determine the local glucose content in brain, which in turn specifies the appropriate value for the lumped constant (10, 11). When the relationship between C*e/C*p and Ce was experimentally determined by changing Cp to alter Ce (i.e., control of Ce mainly by delivery), C*e/C*p progressively increased as Cp and Ce were reduced from hyperglycemic to hypoglycemic levels, in good agreement with results predicted by Michaelis-Menten kinetics for glucose and methylglucose transport across the blood-brain barrier (1, 6, 10, 11, 14). Thus these studies tested and validated the use of Eq. A2 (see APPENDIX for equations throughout paper) to estimate Ce from measured values of Cp and C*e/C*p when metabolic demand for glucose in brain was normal.

Theoretical considerations predict, however, that if metabolic demand were increased and Cp were kept constant, C*e/C*p would fall rather than increase with decreasing Ce (11, 14); conversely, C*e/C*p should increase with increasing Ce as metabolic demand decreases (see Eq. A1). Our preliminary experiments confirmed these predictions and demonstrated that, when Cp was constant and in the normoglycemic range, C*e/C*p decreased rather than increased when Ce fell because of stimulation of CMRGlc (8, 20). Thus measured values for C*e/C*p and Ce determined when glucose delivery is the predominant factor controlling brain glucose levels (6, 11) would not be expected to be appropriate for determination of local glucose levels under conditions in which CMRGlc is abnormal.

Because their values cannot be directly measured in all possible conditions, brain glucose levels must be computed from measured values for Cp and C*e/C*p, with presumed values for the half-saturation concentration for the transport of glucose (Kt) and the maximal distribution space for hexose in tissue relative to that in plasma (S), i.e., relative brain water content. The accuracy of calculated Ce depends, in part, on the assumption that Cp and C*p, measured in blood drawn from a peripheral artery, closely approximate the concentrations of these hexoses in brain capillaries, which cannot be directly measured. The objective of the present study was to determine whether measured and model-predicted glucose levels are equivalent when both glucose supply and glucose demand are varied over a wide range. CMRGlc was, therefore, altered in brains of conscious rats with constant plasma glucose levels (within the range of ~4-18 mM) by sustained drug-induced focal convulsive activity and focal depression of metabolism. The results of the present study demonstrate good agreement between measured and calculated values for Ce and C*e/C*p and confirm that relationships among Ce, Cp, and C*e/C*p predicted by Eq. A2 remain valid when tissue glucose content is altered over a wide range by changes in either glucose supply or demand. The results also identify conditions in which accurate estimates of tissue glucose level might not be obtained with the methylglucose method.

    MATERIALS AND METHODS
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Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References

Theory and experimental design. Equations describing the relationships among Ce, Cp, and C*e/C*p, with Kt and S as parameters, are derived in the APPENDIX.

Previous work (6, 10, 11, 14) tested the relationships among Ce, Cp, and C*e/C*p in the case of fixed demand and variable supply (Eqs. A10 and A11). The current experiments were designed to test these relationships in the alternative case of varying demand and fixed supply (Eqs. A1 and A2) by varying CMRGlc while Cp was maintained at different but constant levels within the range of 4-18 mM. Local CMRGlc rates in cerebral cortex of conscious rats were changed by topical application of drugs to the dura via four burr holes through the skull to produce simultaneously increases and decreases in CMRGlc in different regions of cerebral cortex and thus cause secondary changes in Ce in the drug-treated tissue regions. Then tracer amounts of [14C]methylglucose were infused intravenously for ~60 min to achieve a steady state, the brains were frozen in situ, and drug-treated tissue samples were dissected out and assayed for their hexose contents; untreated samples of cerebral cortex from the same brains were also dissected out to obtain tissue with normal or near-normal CMRGlc and little, if any, change in Ce. Thus many pairs of Ce and C*e/C*p were measured in extracts of dissected tissue samples exposed to the same Cp, all obtained from the same brain. The experimentally determined relationships between Ce, Cp, and C*e/C*p were analyzed by model-dependent fitting routines, and a contour map was constructed so that Ce could be obtained from measured values of Cp and C*e/C*p. The combined results of the present and previous studies were used to establish and validate a contour map to determine Ce from the relationship between measured values for C*e/C*p and Cp that can be applied to assays of local glucose levels when either or both supply and demand are altered.

Surgical procedures. Normal male Sprague-Dawley rats weighing 300-450 g were obtained from Taconic Farms (Germantown, NY) and were fed rat chow ad libitum until the day before the experiment. Fed rats were used to obtain higher steady-state levels of glucose in plasma, whereas other groups were fasted overnight to reduce their plasma glucose levels. On the day of the experiment the rats were anesthetized with halothane (1-1.5%, maintenance dose) in 70% N2O-30% O2, and catheters were inserted into a femoral artery and vein. Four burr holes ~2 mm in diameter and placed ~2.5 mm lateral to the sagittal suture and ~2 mm caudal or rostral to bregma were drilled through the skull with a trephine; the drill was periodically cooled in ice-cold saline to minimize heating that could damage underlying tissue, and care was taken not to cut or damage the dura. The burr holes were covered with Gelfoam (Upjohn, Kalamazoo, MI) soaked in 0.9% saline, the rats were restrained with a loose-fitting plaster cast around the lower torso, and >= 2.5 h were allowed for recovery from surgery before the experimental procedure. Rectal temperature was monitored with a thermistor (Yellow Springs Instrument, Yellow Springs, OH) and maintained at 37°C with a thermostatically controlled heating lamp. Arterial blood PO2, PCO2, and pH were determined with a model 170 pH-blood gas analyzer (Corning Medical Scientific, Medfield, MA). Arterial blood hematocrit was determined from blood samples after centrifugation. Mean arterial blood pressure was measured with an air-damped Hg manometer. Arterial plasma glucose levels were assayed with a Glucose Analyzer 2 (Beckman Instruments, Fullerton, CA).

All animal use procedures were in strict accordance with the National Research Council Guide for the Care and Use of Laboratory Animals and were approved by the local animal care committee.

Experimental procedures. Fed and fasted rats were used to obtain arterial plasma glucose levels in the mild hyperglycemic range (i.e., 13-18 mM) and normoglycemic range (i.e., 8-10 mM), respectively. Insulin (Regular Iletin I; Eli Lilly, Indianapolis, IN; 0.25-0.5 U/kg iv) was given to fasted rats to further depress their plasma glucose concentration to levels in the lower normoglycemic range (i.e., 6.5-7.4 mM) and the mild-to-moderate hypoglycemic range (i.e., 4.3-5.7 mM). Plasma glucose levels were monitored at 5- to 10-min intervals to establish the baseline value for each animal, to track changes after insulin injection, and to verify constancy throughout the experimental interval. About 30 min after the plasma glucose concentration had been maintained at a stable level, bicuculline methiodide (Sigma Chemical, St. Louis, MO; 2 or 10 mM freshly prepared stock solution in 0.9% NaCl containing 10 mM sodium phosphate, pH 7.2-7.4; total dose, 8 or 40 nmol to produce focal seizures) and muscimol (Sigma Chemical; 20 mM freshly prepared stock solution dissolved in 0.9% NaCl containing 10 mM sodium phosphate, pH 7.2-7.4; total dose, 20 or 80 nmol to produce focal depression of metabolism) were topically applied to the surface of the intact dura. The bicuculline and muscimol were applied either once (lower doses) or four times (1/4 of total dose per application) at intervals throughout the experimental period. The first dose of each drug was applied 20-30 min before the intravenous injection of [14C]DG or [14C]methylglucose; subsequent applications were given immediately before injection of the tracer and at ~20-min intervals thereafter, depending on the dose schedule for each drug at each burr hole. Plasma glucose levels usually remained constant if the animals were handled gently, particularly during application of drugs, and if a quiet laboratory environment was maintained; rats were not included in the study if their arterial plasma glucose or [14C]methylglucose levels deviated by more than ±10% during the experimental period.

The possibility of damage to the blood-brain barrier in the proximity of the burr hole because of high local levels of bicuculline and muscimol was examined in preliminary experiments. Rats were injected intravenously with Evans blue dye [1 ml of 2% (wt/vol) dissolved in 0.9% NaCl] 10 min before application of the drugs and were killed 15-60 min later; there was no visible penetration of Evans blue dye into the tissue under the burr holes (results not shown), indicating that these drug treatments did not cause gross damage to the blood-brain barrier.

Determination of CMRGlc and [14C]methylglucose distribution ratio. Radiochemical purities of 2-deoxy-D-[1-14C]glucose (51 mCi/mmol, Du Pont-NEN, Boston, MA) and 3-O-[methyl-14C]methyl-D-glucose (57 mCi/mmol, Du Pont-NEN) were assayed before use by thin-layer chromatography and/or by high-performance liquid chromatography, as previously described (6, 7), and found to be >98%. Initial studies established the magnitude of focal activation or depression of metabolism required to produce changes in the [14C]methylglucose distribution ratio. CMRGlc was determined with the routine [14C]DG procedure by use of a 45-min experimental period (29). Briefly, timed samples of arterial blood were drawn at frequent intervals after a pulse of 125 µCi/kg [14C]DG for determination of plasma [14C]DG and glucose contents; levels of 14C in plasma were assayed by liquid scintillation counting (Beckman model LS5801) with external standardization. About 45 min after the pulse of tracer, the rats were given a lethal dose of pentobarbital sodium and their brains were rapidly removed, frozen in isopentane chilled to -40 to -50°C with dry ice, and stored at -80°C. Each brain was then cut into 20-µm-thick sections in a cryostat at about -20°C, dried at 60°C on a hot plate, and exposed to SB-5 X-ray film (Kodak, Rochester, NY). Local tissue concentrations of total 14C were determined by autoradiography, and CMRGlc was calculated with the operational equation of the method, with a value of 0.48 for the lumped constant (29).

Steady-state C*e/C*p values were determined by quantitative autoradiography in separate groups of rats. [14C]Methylglucose (50-70 µCi/kg) was infused intravenously according to a program designed to maintain a constant concentration of 14C in arterial plasma throughout the experimental period (6); arterial blood was drawn at 5- to 10-min intervals to monitor plasma glucose and 14C levels. Sixty minutes after initiation of the infusion, a sample of arterial blood was drawn and the rats were given a lethal dose of pentobarbital; their brains were rapidly removed and processed for autoradiography, as described above. The steady-state C*e/C*p was calculated by dividing the brain 14C level by that in the last plasma sample.

To determine relationships among Cp, Ce, and C*e/C*p, brain glucose levels and C*e/C*p values for [14C]methylglucose were determined in parallel in the same tissue samples by direct chemical measurement. About 60 min after initiation of the programmed infusion of [14C]methylglucose, rats were anesthetized rapidly with intravenous thiopental (25 mg/kg), and their brains were immediately funnel-frozen in situ with liquid nitrogen (25). When frozen, the heads were removed and transferred to liquid nitrogen and stored at -80°C. Oxygen (100%) was provided during the funnel-freezing procedure via a nose cone to minimize tissue hypoxia. Immediately before and ~1 min after the the start of the freezing procedure, additional samples of arterial blood were taken for determination of the glucose and 14C contents of plasma; the mean of these two samples was used for the plasma glucose level and to calculate the C*e/C*p. Also, a second arterial blood sample was taken at 1 min after the start of funnel-freezing for determination of blood pH and gases.

Sampling, extraction, and analysis of brain tissue. Frozen heads from 21 rats were warmed to about -25°C in a cryostat; skin, muscle, and skull were removed; and superficial membranes and blood vessels were carefully scraped from the surface of the cerebral cortex. Samples of frozen cerebral cortex (5-12 from each brain) weighing ~5-15 mg were dissected out from cortex directly under the burr holes and from untreated cortical tissue located at various distances from the burr holes, weighed (at -25°C), and stored at -80°C until extracted with aqueous ethanol as previously described (7). Each of the extracts was stored at -80°C until it was analyzed for its [14C]methylglucose level by liquid scintillation counting and for glucose content by the standard fluorometric enzymatic assay with hexokinase and glucose-6-phosphate dehydrogenase (18). The measured concentrations of glucose and [14C]methylglucose in each of the samples derived from each brain were then corrected for corresponding hexose contents of residual blood [this value was assumed to be 2.6% (31) because of the presence of both large vessels and capillaries in the dissected samples] in brain, and C*e/C*p values were calculated from the corrected brain concentrations of glucose and [14C]methylglucose and the respective arterial plasma values (i.e., means of samples drawn immediately before freezing and 1 min after initiation of funnel-freezing). To summarize, a pair of measured values for glucose and [14C]methylglucose was obtained from each of the drug-treated (i.e., those with altered CMRGlc) or untreated (i.e., control) tissue samples dissected out of each brain; thus many pairs of tissue samples were assayed after exposure to the same measured plasma glucose level in each rat.

    RESULTS
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Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References

Physiological variables. Topical application of bicuculline and muscimol did not cause appreciable changes in physiological variables, and values of all variables were similar in each of the experimental groups (Table 1).

                              
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Table 1.   Physiological variables

Determination of drug dosages to alter CMRGlc. Experimental conditions for control of brain glucose concentration mainly by substrate demand were first established in preliminary experiments by topical application of various doses of different drugs (results not shown). We previously used penicillin and barbital to produce focal changes in rates of glucose utilization in cerebral cortex and tissue glucose content, but we found that penicillin produced only small reductions in glucose level and that barbital sometimes caused vascular damage (20). Bicuculline and muscimol, therefore, were tested for their ability to alter CMRGlc and found to be appropriate for use in the present study; two doses of each drug were used to produce a range of focal increases (up to ~2.5-fold) and decreases (20-50%), respectively, in CMRGlc in the cerebral cortex of each rat (Table 2). Values of C*e/C*p were highest when CMRGlc was normal (i.e., in untreated tissue) or lowered (i.e., in muscimol-treated tissue), and C*e/C*p and Ce fell as CMRGlc rose (i.e., in bicuculline-treated tissue) (Table 2). Because the normal value of the lumped constant of the [14C]DG method (i.e., 0.48) was used, calculated CMRGlc for some samples of bicuculline-treated tissue shown in Table 2 were probably somewhat overestimated. Any such errors would, however, be expected to be <30%, because the brain glucose content exceeded 0.8 µmol/g in almost all samples obtained from normoglycemic rats with Cp ~10 mM. (Compare values in Table 2 with those in Fig. 1 and Table 1 of Ref. 6). C*e/C*p did not change in the muscimol-treated samples shown in Table 2, indicating that Ce can be relatively stable even with a large (i.e., ~35%) decrease in CMRGlc for 60 min; C*e/C*p did, however, rise above normal in muscimol-treated samples from the other animals that were used to obtain the data shown in Figs. 1, 3, and 4. Thus topical application of different doses of drugs through the burr holes provides the means to produce focal variations in CMRGlc in the brain of each conscious rat over approximately a threefold range, broad enough to obtain 5-12 pairs of values from tissue with normal and altered CMRGlc for Ce and C*e/C*p for the same Cp in one rat.

                              
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Table 2.   Influence of bicuculline and muscimol on rate of glucose utilization and brain-to-plasma distribution ratio for methylglucose


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Fig. 1.   Influence of glucose demand on relationship between brain glucose level and steady-state brain-to-plasma distribution ratio for [14C]methylglucose. Solid and dotted contour lines are theoretical relationships predicted by Eqs. A2 and A8. Vm/Vt, ratio of Michaelis-Menten kinetic constants for maximal velocity of phosphorylation of glucose by hexokinase to that of transport of glucose across the blood-brain barrier. Solid lines, conditions of fixed supply of glucose as demand is varied; dotted lines, conditions of fixed demand for glucose (expressed in our theoretical model as Vm/Vt) as supply is varied. Data points, measured pairs of values for brain glucose content (Ce) and brain-to-plasma distribution ratio for [14C]methylglucose (C*e/C*p) assayed in individual tissue samples from the funnel-frozen brains of 4 representative rats, each with a different plasma glucose level. Local rates of glucose utilization in cerebral cortex were altered by topical application of bicuculline and muscimol while plasma glucose concentration (Cp) was maintained at level shown throughout experiment. Data points are thus aligned along solid lines. Under these conditions of fixed supply and variable demand, relationship between Ce and C*e/C*p has a positive slope, and the slope value increases as the level of Cp increases. The 4 animals were selected from the total of 21 studied, with fixed Cp values ranging from 4 to 18 mM.

Influence of CMRGlc and Cp on the relationship between Ce and C*e/C*p. Representative results from both the current and previous experiments are presented as data points on plots of Ce vs. C*e/C*p in Figs. 1 and 2. These figures also include solid and dotted contour lines calculated according to the theory developed in the APPENDIX. The solid lines show predicted relationships of Ce vs. C*e/C*p calculated by Eq. A2; there is a different line for each Cp value (see DISCUSSION). The dotted lines were calculated according to Eq. A8 and show lines for different values of the metabolic demand parameter R = Vm/Vt, where Vm and Vt are the Michaelis-Menten maximal velocity constants for phosphorylation and blood-brain barrier transport of glucose, respectively (see APPENDIX and DISCUSSION). The Vm/Vt rises above the normal value of ~0.32 (1, 14) (also see Fig. 2) when the maximal rate of glucose phosphorylation by hexokinase increases relative to that of glucose transport across the blood-brain barrier (e.g., during seizures when tissue glucose levels tend to fall), and it falls below that value when glucose metabolism is reduced relative to its maximal rate of transport (e.g., during coma or anesthesia when tissue glucose levels tend to rise).


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Fig. 2.   Influence of glucose supply on relationship between brain glucose level and steady-state brain-to-plasma distribution ratio for [14C]methylglucose. Solid lines, theoretical relationships in conditions of fixed supply of glucose; dotted lines, those in conditions of fixed demand (see Fig. 1). Two sets of data points correspond to conditions of fixed demand as Cp (supply) is varied. Thus relationships between the measured Ce and C*e/C*p values are oriented along dotted lines. Relationship has a negative slope; the magnitude of the slope decreases as demand increases. black-square, Values from 49 individual animals obtained in our previous study (6, 14) in normal, conscious rats. [14C]Methylglucose was infused for 60 min while Cp was clamped at values ranging from 4 to 28 mM. black-triangle, Means of groups of rats that were lightly sedated with pentobarbital sodium (15 mg/kg), injected iv with [14C]methylglucose, and brains sampled 10 min later by funnel-freezing (11). Points from normal, conscious rats (rate of glucose utilization is ~0.7 µmol · g-1 · min-1) fall on a contour corresponding to a Vm/Vt approximately equal to 0.32; for those from sedated rats, the apparent contour corresponds to a Vm/Vt approximately equal to 0.22.

When CMRGlc is varied while Cp is fixed at a constant level, Ce and C*e/C*p are predicted to change in the same direction, i.e., they increase and decrease in parallel in a Cp-dependent manner (Figs. 1 and 2, solid lines). Focal changes in CMRGlc were experimentally induced in brains of 21 rats with constant but different Cp levels ranging from ~4 to 18 mM, and glucose content and C*e/C*p were determined in each of the 5-12 samples of tissue dissected out of each brain. For clarity, pairs of measured glucose levels and [14C]methylglucose C*e/C*p, determined in each tissue sample dissected from brains of only four representative rats with different plasma glucose levels, are plotted in Fig. 1. The values for each of these animals fell along or parallel to the solid theoretical lines corresponding to the measured Cp for that animal (i.e., 18, 9.6, 6.8, and 5.5 mM). For example, the two bicuculline-treated tissue samples (i.e., those with focal seizures) dissected out of the moderately hyperglycemic rat with Cp = 18 mM had the lowest values for glucose (<2 µmol/g, Fig. 1; for clarity, drug-treated samples are not identified in the figure) and C*e/C*p (0.3-0.33) compared with all other samples obtained from that animal; these two glucose-methylglucose pairs fell on the dotted line corresponding to a higher-than-normal Vm/Vt, i.e., ~0.52 (Fig. 1, triangles). Untreated and muscimol-treated samples dissected out of the same brain had higher glucose levels and C*e/C*p that corresponded to normal or subnormal values for Vm/Vt; the highest brain glucose levels in this brain exceeded 4.5 µmol/g and corresponded to a below-normal Vm/Vt, i.e., ~0.23 (Fig. 1, triangles). Similar results were obtained in the three other normoglycemic and moderately hypoglycemic animals (Fig. 1), as well as in the other 17 animals assayed in the present study (data not shown). The slope of the relationship between glucose content and methylglucose distribution volume progressively decreased as the level of Cp fell (Fig. 1), as predicted by Eq. A2.

When Cp is varied while metabolic demand is fixed at a constant value, Ce and C*e/C*p values are predicted to change in opposite directions; the value for one variable increases as that for the other decreases, and vice versa (Figs. 1 and 2, dotted lines). Measured values obtained in our previous study (6, 14) from 49 normal conscious rats, which had their plasma glucose levels clamped at different levels to change Ce mainly by supply, are plotted in Fig. 2 (squares). When Cp concentrations were clamped at increasingly higher levels within the range of 4 to 26 mM, the measured level of glucose in brain of each animal progressively increased, whereas the C*e/C*p values measured in the same samples became smaller; all measured points fell along a line corresponding to a Vm/Vt of ~0.32 (Fig. 2, squares). On the other hand, measured values obtained by Gjedde and Diemer (11) from rats given barbiturate (and therefore expected to have a lower metabolic demand) fell along a line corresponding to a lower Vm/Vt, i.e., ~0.22 (Fig. 2, triangles). Thus the highest values for glucose-methylglucose distribution ratio pairs were associated with lowest values for Vm/Vt; conversely, the lowest levels for these pairs corresponded to higher Vm/Vt values (Figs. 1 and 2).

When measured and calculated values obtained in all samples from all experimental animals were compared, good agreement was found for Ce (Fig. 3; predicted results were calculated with Eq. A2) and C*e/C*p (Fig. 4; predicted results were calculated with Eq. A1). There was somewhat more scatter about the line of identity for the 167 samples from 21 rats in which focal changes in metabolic rates were induced compared with the 49 rats in which plasma glucose was clamped at different levels [Figs. 3 and 4; compare filled ("Vary CMRGlc...") to open ("Vary Cp...") symbols in each figure]. Also, estimates of C*e/C*p calculated from measured Ce and Cp values (Fig. 4) appear to be somewhat less variable than model-predicted glucose concentrations calculated from measured C*e/C*p (Fig. 3 and see DISCUSSION).


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Fig. 3.   Comparison of measured and model-predicted glucose levels in brain. Ce levels were calculated with Eq. A2 by use of C*e/C*p and Cp measured in present study and values for half-saturation constant for glucose transport (Kt, 6.56 mM) and for physical distribution space for hexose in tissue (S, 0.89 ml/g) determined from our previous work (6, 14; also see Fig. 5). CMRGlc, glucose utilization rate. Solid line, line of identity. Regression line (not shown) calculated for values obtained when control of brain glucose level is mainly by demand (black-lozenge ) is y = 0.71x + 0.76 (r = 0.77, P < 0.001, n = 167 from 21 rats); SE values of slope and intercept are 0.05 and 0.10, respectively. Regression line calculated for values obtained when control of brain glucose level is mainly by supply (open circle ) is y = 1.02x + 0.11 (r = 0.99, P < 0.001, n = 49 rats); SE values of slope and intercept are 0.02 and 0.06, respectively.


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Fig. 4.   Comparison of measured and model-predicted methylglucose distribution ratios. Methylglucose distribution ratios were calculated with Eq. A1 by use of Ce and Cp measured in the present study and values for Kt (6.56 mM) and for S (0.89 ml/g) determined from our previous work (6, 14; also see Fig. 5). Solid line, line of identity. Regression line (not shown) calculated with values obtained when control of Ce is mainly by demand (black-lozenge ) is y = 1.02x - 0.02 (r = 0.83, P < 0.001, n = 167); SE values of slope and intercept are 0.05 and 0.03, respectively. Regression line calculated with values obtained when control of Ce is mainly by supply (open circle ) is y = 0.95x + 0.02 (r = 0.98, P < 0.001, n = 49); SE values of slope and intercept are 0.03 and 0.02, respectively.

    DISCUSSION
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References

Influence of glucose supply and demand on relationship between glucose level and C*e/C*p: theory. The combined results of our present and previous (6, 14, 20) studies extend the elegant pioneering work of Buschiazzo et al. (1) and Gjedde (10) and Gjedde and Diemer (11) to establish quantitative relationships between the steady-state C*e/C*p and Cp and Ce levels under various experimental conditions. The experimental results shown in Figs. 1 and 2 confirm predictions that, when CMRGlc is relatively constant, Ce and C*e/C*p change in opposite directions as Ce is varied by clamping Cp at different levels. In contrast, when Ce is altered mainly by variations in CMRGlc, C*e/C*p changes in the same direction as Ce, with a different slope for each value of Cp. Thus a single theoretical framework derived from Michaelis-Menten competitive kinetics for hexose transport across the blood-brain barrier (see APPENDIX) provides reliable estimates of steady-state brain tissue glucose levels over a wide range of changes in glucose supply and/or demand. Either Eq. A2 or a contour map can be used to determine local glucose concentrations in brain under many, but not all, possible conditions.

The lines in the contour map shown in Fig. 5 were calculated according to the theory developed in the APPENDIX and illustrate the interrelationships between Ce and C*e/C*p at different values for Cp. The solid lines were calculated by Eq. A2; twelve different lines are shown for Cp values, ranging from a minimum of 4 mM (line at far right) in 2-mM steps to a maximum of 26 mM (line at far left). The dotted lines were calculated according to Eq. A8; eight lines are shown, with the metabolic demand parameter R = Vm/Vt ranging from a minimum value of 0.10 (line at far right) in steps of 0.06 to a maximum value of 0.52. To calculate these contour lines (Figs. 1, 2, and 5), the values used for Kt (6.56 mM) and for S (0.89 ml/g) were determined in the present study by substitution of 49 triplets of measured values (Cp, Ce, C*e/C*p) from our previous work (6, 14) into Eq. A1, followed by evaluation of Kt and S by least squares optimization. The value for S represents the physical distribution space for glucose and methylglucose in brain, i.e., the water content of brain relative to that in plasma (14); thus the value for S is slightly larger than the actual tissue water content used in previous studies (3, 10, 24). The value for the half-maximal saturation constant for phosphorylation (Km, 0.063 mM), also derived from our measured data, was from our previous report (14). As described in the APPENDIX, the straight-line segments of the dotted lines that appear to crosshatch with the solid curves are well approximated by Eq. A11.


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Fig. 5.   Contour map to determine Ce when either or both glucose supply and demand are varied. Solid contour lines, relationship between Ce and methylglucose distribution ratio when Cp is held constant at values shown (4-26 mM in 2-mM increments). Dotted lines, relationship when Vm/Vt is held constant at values shown (0.10 to 0.52 in increments of 0.06). Solid and dotted lines were calculated with Eqs. A2 and A8, respectively. Two of 3 model parameters (Kt: 6.56 mM; S: 0.89 ml/g) were determined by substitution of 49 triplets of measured values (Cp, Ce, C*e/C*p) from our previous work (6, 14) into Eq. A1, followed by evaluation of Kt and S by least squares optimization. The 3rd parameter [half-maximal saturation constant for phosphorylation (Km), 0.063 mM], also derived from our data, was taken from our previous report (14). As described in APPENDIX, straight-line segments of dotted curves are well approximated by Eq. A11.

When CMRGlc is varied while Cp is fixed at a constant level, Ce and C*e/C*p are predicted to change in the same direction, i.e., they increase and decrease in parallel (Fig. 5, solid lines). For example, when Cp = 8 mM and demand for glucose is increased relative to the rate of glucose supply (i.e., Vm/Vt rises from 0.1 to 0.52), Ce is predicted to fall from ~4.8 to 0.5 µmol/g, causing the theoretical C*e/C*p to decrease from ~0.73 to 0.43 ml/g (Fig. 5). Also, when Cp is clamped at progressively lower levels from 26 to 4 mM, the slopes of the solid lines are predicted to decrease, indicating that C*e/C*p is most sensitive to changes in Ce when Cp is lowest (Fig. 5, solid lines).

When Cp is varied while metabolic demand is fixed at a constant value, Ce and C*e/C*p are predicted to change in opposite directions; the value for one variable increases as that for the other decreases, and vice versa (Fig. 5, dotted lines). For example, when Vm/Vt is fixed at 0.34 and Cp is clamped at progressively decreasing levels from 26 to 4 mM, Ce should decrease from ~5 to 0.5 µmol/g, causing the theoretical C*e/C*p to rise from ~0.33 to 0.6 ml/g (Fig. 5). Also, when metabolic demand rises relative to supply (i.e., when Vm/Vt increases from 0.10 to 0.52), the slope of each dotted line is predicted to decrease progressively, indicating that C*e/C*p is most sensitive to changes in tissue glucose concentration when consumption of glucose is highest relative to its rate of influx into brain (Fig. 5, dotted lines).

In summary, changes in the equilibrium C*e/C*p as a function of Ce level are greatest when metabolic demand for glucose is highest and Cp levels are lowest. This sensitivity of C*e/C*p to tissue glucose level is apparent in Fig. 5 when it is viewed by rotation of the figure 90° counterclockwise so that glucose level is the abscissa and C*e/C*p is the ordinate. The method is most sensitive to changes in glucose level that would have the highest impact on calculated CMRGlc values.

Application of C*e/C*p to determine glucose level. Use of methylglucose to estimate glucose level in brain tissue in vivo (1, 6, 10, 11, 14, 20) or to evaluate hexose flux to and from tissue or distribution kinetics and volume in vitro (21) or in vivo (22, 32) does not depend on identity of the theoretical model and the biological system with many cellular compartments; the simplest model that provides accurate predicted values is most appropriate. In the present study, the transport-based model can be applied to a wide range of conditions; predicted glucose levels were, however, sometimes negative, i.e., when measured Ce values were very low (Fig. 3). Discrepant results could arise for various reasons: increased hexose extraction fraction, inaccuracies in estimates of the true values of S (relative brain water content) or Kt, changes in S or Kt under the experimental condition, and differences in the distribution of methylglucose and glucose into various brain compartments.

Use of the methylglucose method is appropriate when the concentrations of glucose and methylglucose in blood samples drawn from a large artery approximate those in brain capillaries. Because the experimental results obtained in the present study are in good agreement with most of the theoretically predicted values (Figs. 1-4), this assumption appears to hold under nearly all experimental conditions employed in this study for the Cp range of 5.5-18 mM. Accurate estimates for Ce should also be obtained when plasma glucose exceeds 18 mM, because the hexose extraction fractions would be expected to decrease as plasma glucose levels rise, and the arterial and capillary hexose levels would be essentially the same. On the other hand, when CMRGlc exceeds the rate of glucose supply to the brain in hypoglycemic animals, the extraction fraction for glucose would increase, causing overestimation of the true capillary level and errors in calculated Ce.

The brain compartments into which glucose and [14C]methylglucose might distribute (e.g., extracellular fluid spaces, different cell types, and various intracellular spaces) appear to be "kinetically invisible" over nearly the entire range of glucose levels investigated in the present study. The experimental data obtained in normal, conscious rats fall along a line corresponding to a theoretical Vm/Vt approximately equal to 0.32 (Fig. 2; also see Refs. 1 and 14), supporting the assumption that transport is not rate limiting for glucose utilization and that intracellular concentrations of glucose and methylglucose can be maintained at levels close to those in the extracellular fluid over a wide supply-demand range (14, 19, 27). Regardless of site(s) of the rate-limiting step(s) that ultimately control brain tissue and plasma hexose levels, the theoretical model accurately takes into account effects of well-established differences in Cp and Ce (1, 3, 6, 11, 19, 20, 23, 24, 27). However, at very low glucose levels, intracellular glucose would approach zero faster than the extracellular glucose level, and compartmentation might contribute to errors in model-predicted glucose concentrations.

Values for S [i.e., 0.7-0.8 (1, 11, 14)] and Kt [i.e., 6-14 mM (1, 4, 11, 14, 19, 23, 24)] have been determined in different laboratories by direct measurement and/or by model-dependent fitting of experimental results. Good agreement between measured and theoretical results in the present study indicates that values for S and Kt, derived from the data sets in our present and previous studies (6, 14) and used to calculate the predicted values, were sufficiently close to the true values and that they could be used to accurately estimate Ce levels. Because calculation of Ce with Eq. A2 includes the factor (Kt + Cp), the calculated value of Ce from measured values of Cp and C*e/C*p is expected to be least sensitive to inaccurate estimates of S or Kt when Cp is high. Thus, as Cp increases, the sum (Kt + Cp) would progressively diminish the fractional effect of errors in the estimate of the true value of Kt on calculated Ce; the magnitude of error would be expected to increase as Cp decreases.

Similar Cp-dependent sensitivity to errors in calculated Ce would be expected with any unknown changes in the values of S or Kt under abnormal or pathophysiological conditions. For example, pentobarbital treatment decreases Kt and Vt of the blood-brain barrier transporter for glucose (12, 15) and methylglucose (22), probably by direct interaction with the GLUT-1 transporter in a concentration-dependent manner (15). These effects, if any, of pentobarbital on determination of tissue glucose levels with methylglucose would, however, be expected to be "dampened" by 1) simultaneous interference with transport of glucose and methylglucose; 2) the slowing of metabolism by pentobarbital, which tends to stabilize tissue glucose concentration at higher levels; and 3) the factor in Eq. A2 that includes Kt as a sum with Cp. In fact, in our previous study barbital sometimes damaged the blood-brain barrier but did not appear to alter significantly the relationship between C*e/C*p and Ce (20), probably for the above reasons. Better "buffering" of inaccuracies in estimates of the true values of S and Kt by the ratio (KtS + Ce)/ (Kt + Cp) in Eq. A1 compared with that by the sum (Kt + Cp) in Eq. A2 would be expected to contribute to superior "goodness of fit" for predicted and measured values for C*e/C*p (Fig. 3) compared with glucose concentration (Fig. 4). Thus comparison of Figs. 3 and 4 illustrates another important issue in modeling: goodness of fit of theoretical to measured data can depend on the form of an equation derived from the model; poorer agreement between model-predicted and measured values can be apparent in one form of the equation (i.e., Fig. 3) but not when the equation is rearranged and tested (Fig. 4).

Estimation of intracellular water space. Methylglucose is also used to determine intracellular water space in cultured cells (e.g., Refs. 5, 9, 16, 17). Extracellular and intracellular methylglucose concentrations are assumed to be equal at equilibrium, and the intracellular water space is calculated from the known extracellular methylglucose concentration and the measured intracellular level of labeled methylglucose. This assumption may be valid only if there is no glucose transporter and/or no competition between glucose and any other substrate for the hexose transporter. Omission of glucose from the test medium would, however, cause energy failure, loss of ion homeostasis, and cell swelling. Figure 4 shows that C*e/C*p can vary over a twofold range when glucose levels are varied by supply or demand. Thus failure to account for competition of methylglucose with glucose for transport into and from the cell could lead to errors in estimates of intracellular water space.

Determination of the lumped constant of the DG method. The lumped constant is the factor that converts the rate of deoxyglucose phosphorylation to the CMRGlc; it accounts for kinetic differences in the rates of transport and phosphorylation of the two hexoses. When Ce exceeds 1 µmol/g, the value of the lumped constant is relatively stable, and any errors in determination of local tissue glucose level with methylglucose would therefore be expected to have a small effect on calculated CMRGlc. For example, the magnitude of change in the value of the lumped constant is only about one-tenth that of the corresponding change in tissue glucose content in normoglycemic and hyperglycemic animals; an increase in Ce level of 550% (e.g., from 1 to 5.5 µmol/g) causes the lumped constant to decrease by 40% (i.e., from 0.55 to 0.33) (6). Furthermore, compensatory mechanisms can maintain Ce at or above the "threshold level" at which changes in the lumped constant can occur when metabolic demand is increased in brains of normoglycemic and hyperglycemic rats. In the present study, a sustained two- to threefold stimulation of CMRGlc did not reduce local Ce below 0.8 µmol/g so long as Cp exceeded ~7 mM (Figs. 1 and 3 and Ref. 21).

The value of the lumped constant is most sensitive to incremental changes in Cp and Ce when the brain glucose level is <1 µmol/g. Below this level, the value of the lumped constant rises abruptly and steeply, almost in proportion to the fall in Ce. For example, when Ce drops by 50% (e.g., from 1 to 0.5, or from 0.5 to 0.25 µmol/g), the lumped constant increases by 30-40% [i.e., from 0.55 to 0.77, or from 0.77 to 1.0, respectively (6); for graphic plots of this relationship, see Refs. 3, 6, 23, 24, 26, 28, and 29]. Thus errors in determination of Ce would have the highest impact on calculated CMRGlc when Ce levels are in the hypoglycemic range.

The C*e/C*p value is also most sensitive to changes in Ce during hypermetabolic states, particularly in the presence of hypoglycemia (6, 14; Figs. 1, 2, and 5), and methylglucose would therefore be most useful for detection of changes in glucose level in the range that has the most significant impact on the value of the lumped constant. Unfortunately, the extreme conditions in which tissue glucose levels are the lowest are also those in which estimates of glucose level are least reliable (Fig. 3). No analytic method, however, is universally applicable to all possible circumstances, and demonstration that local tissue glucose levels are above or below the threshold level for uncertainty in the value of the lumped constant would help interpretation of results of metabolic studies under abnormal and pathophysiological states. The possibility of incurring errors in determination of CMRGlc in the hypoglycemic state suggests a simple strategy to avoid potential problems when CMRGlc must be accurately determined under conditions in which neural pathways are activated by physiological stimulation or mental tasks that would cause large increases in cerebral metabolic rate. Unless a study is specifically designed to examine conditions in which the brain glucose level becomes limiting, fed or mildly hyperglycemic subjects should be used so that compensatory mechanisms can maintain the Ce near normal, thereby preventing significant changes in Ce and value of the lumped constant of the DG method.

    APPENDIX
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References

Theory

The rates of transport of substances that compete for transport across the blood-brain barrier by the same transporter can be modeled by use of simple competitive Michaelis-Menten kinetics. Thus the expression for transport to and from the brain of tracer amounts of [14C]methylglucose in competition with the natural substrate, glucose, can be expressed as follows. At equilibrium, rates of transport of methylglucose to (v*in) and from (v*out) the brain tissue are equal
<IT>v</IT>*<SUB>in</SUB> = <FR><NU><IT>V</IT>*<SUB>t</SUB>C*<SUB>p</SUB></NU><DE><IT>K</IT>*<SUB>t</SUB>[1 + (C<SUB>p</SUB>/<IT>K</IT><SUB>t</SUB>)] + C*<SUB>p</SUB></DE></FR> = <IT>v</IT>*<SUB>out</SUB> = <FR><NU><IT>V</IT>*<SUB>t</SUB>C*<SUB>e</SUB>/<IT>S</IT></NU><DE><IT>K</IT>*<SUB>t</SUB>[1 + (C<SUB>e</SUB>/<IT>S</IT>)/<IT>K</IT><SUB>t</SUB>] + C*<SUB>e</SUB>/<IT>S</IT></DE></FR>
where Kt and K*t are Michaelis-Menten half-maximal saturation constants for transport of glucose and methylglucose, respectively; Vt and V*t are Michaelis-Menten maximal velocity constants for transport of glucose and methylglucose, respectively; Ce, C*e, Cp, and C*p are the hexose levels in brain (Ce, C*e) and arterial plasma (Cp and C*p); and S is the physical distribution space for glucose and methylglucose in brain (i.e., brain water content relative to that in plasma; see Refs. 1, 10, 11, 14). The expressions 1+(Cp/Kt) and 1+(Ce/S)/Kt take into account the competition of methylglucose with glucose by increasing the apparent value of K*t  for methylglucose.

At tracer concentrations, C*p and C*e are approximately equal to zero. Canceling and rearranging
Equilibrium methylglucose distribution ratio
 = <FR><NU>C*<SUB>e</SUB></NU><DE>C*<SUB>p</SUB></DE></FR> = <FR><NU><IT>K</IT><SUB>t</SUB><IT>S</IT> + C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB> + C<SUB>p</SUB></DE></FR> (A1)
Rearranging
C<SUB>e</SUB> = (C*<SUB>e</SUB>/C*<SUB>p</SUB>)(<IT>K</IT><SUB>t</SUB> + C<SUB>p</SUB>) − <IT>K</IT><SUB>t</SUB><IT>S</IT> (A2)
These equations express the relationships among Ce, Cp, and C*e/C*p, with Kt and S serving as constant parameters. The transport of hexoses back and forth across the blood-brain barrier is taken to be a process that obeys saturable kinetics according to the simplest Michaelis-Menten prediction. Our goal is to present a theoretical framework for understanding the relationships between C*e/C*p and Ce in two distinct circumstances: first, when Cp is clamped at a specific constant value (as in the current experiments), and C*e/C*p and Ce vary in response to changes in metabolic demand; second, when the level of metabolic demand is maintained at an essentially constant level [as in our previous work (6, 14) and that of others (1, 11)] and C*e/C*p and Ce vary in response to changes in glucose delivery, i.e., changes in Cp. This framework is the basis for comparison of calculated and experimentally determined relationships among Ce, Cp, and C*e/C*p in the present study. Methods for determination of parameter values from our data and used in these calculations are described in the DISCUSSION and legend to Fig. 5. The values used were as follows: Kt (6.56 mM), S (0.89 ml/g), and Km (0.063 mM).

The situation with constant supply is best described by Eqs. A1 and A2 themselves. If Cp is constant, Ce is shown by Eq. A2 to have a positive linear dependence on C*e/C*p with a slope (Kt+Cp) that increases as Cp increases. Because the blood-brain transport barrier for hexoses is passive and concentration driven, it follows that as the metabolic rate becomes very small, Ce will increase only to its limiting value of SCp, the hexose distribution space times the plasma concentration (please see text that follows Eq. A4 below for further explanation). This limiting value of Ce defines the equilibrium distribution space S, i.e., Ce/Cp = S. Substitution of this maximum into Eq. A1 shows that the corresponding maximum for C*e/C*p is equal to S. As metabolic rate increases with Cp held constant, Ce falls, ultimately to its lower limiting value of zero, when all transported glucose is consumed by tissue metabolism. Equation A1 predicts that C*e/C*p falls accordingly, until it reaches its smallest possible value of KtS/(Kt+Cp) in the limit of zero tissue glucose. Thus, in the situation of constant plasma glucose concentration, C*e/C*p and Ce increase and decrease together in response to changes in metabolic rate.

Equations A1 and A2 are not useful for illuminating the relationship between C*e/C*p and Ce in the second situation, in which we allow glucose delivery to vary while maintaining metabolic demand at some fixed level. This is because these two equations are explicitly dependent on Cp, which in those circumstances is not a fixed parameter but a variable. Rather, we substitute for Cp a model prediction for the Cp value that would be expected to attain steady-state equilibrium with the given value of Ce at some specific level of metabolic demand. The resulting relationship, no longer explicitly dependent on Cp, would then show the mutual dependencies of C*e/C*p and Cp as demand is held fixed. When glucose is in the steady state, the rate of transport from plasma to brain is equal to the sum of the rate of return from brain back to plasma and the rate of the (irreversible) phosphorylation reaction
<FR><NU><IT>V</IT><SUB>t</SUB>C<SUB>p</SUB></NU><DE><IT>K</IT><SUB>t</SUB> + C<SUB>p</SUB></DE></FR> = <FR><NU><IT>V</IT><SUB>t</SUB>C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB><IT>S</IT> + C<SUB>e</SUB></DE></FR> + <FR><NU><IT>V</IT><SUB>m</SUB>C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>m</SUB><IT>S</IT> + C<SUB>e</SUB></DE></FR> (A3)
where Vm and Km are the maximal velocity and half-maximal saturation constraints for phosphorylation of glucose by hexokinase, respectively. By dividing all terms in Eq. A3 by the maximum velocity for transport Vt, and defining the ratio R = Vm/Vt of the maximal velocities for phosphorylation and transport, we get
<FR><NU>C<SUB>p</SUB></NU><DE><IT>K</IT><SUB>t</SUB> + C<SUB>p</SUB></DE></FR> = <FR><NU>C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB><IT>S</IT> + C<SUB>e</SUB></DE></FR> + <FR><NU><IT>V</IT><SUB>m</SUB></NU><DE><IT>V</IT><SUB>t</SUB></DE></FR> ⋅ <FR><NU>C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>m</SUB><IT>S</IT> + C<SUB>e</SUB></DE></FR>
 = <FR><NU>C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB><IT>S</IT> + C<SUB>e</SUB></DE></FR> + <IT>R</IT> <FR><NU>C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>m</SUB><IT>S</IT> + C<SUB>e</SUB></DE></FR> (A4)
In this derivation, as in our previous report (14), changes in metabolic demand are modeled only as changes in the ratio R. Conversely, the condition of fixed metabolic demand is that R is a constant. Equation A4 provides a theoretical basis for the predictions made above, that Ce increases only to its limit SCp as R goes to zero, and that it declines monotonically from that maximum as R increases. As in the fixed-supply case, the assumption is again made that half-saturation constants and the distribution volume S do not vary. For a given pair of values of Ce and R, the equation predicts the steady-state value of Cp to be consistent with those values. If the right side of Eq. A4 is temporarily represented by the dummy variable F
<FR><NU>C<SUB>p</SUB></NU><DE><IT>K</IT><SUB>t</SUB> + C<SUB>p</SUB></DE></FR> = <IT>F</IT> (A5)
then
C<SUB>p</SUB> = <FR><NU><IT>K</IT><SUB>t</SUB><IT>F</IT></NU><DE>1 − <IT>F</IT></DE></FR> (A6)
Substitution of this expression into Eq. A1 above yields
<FR><NU>C*<SUB>e</SUB></NU><DE>C*<SUB>p</SUB></DE></FR> = <FR><NU><IT>K</IT><SUB>t</SUB><IT>S</IT> + C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB> + C<SUB>p</SUB></DE></FR> = <FR><NU><IT>K</IT><SUB>t</SUB><IT>S</IT> + C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB> + [(<IT>K</IT><SUB>t</SUB><IT>F</IT>)/(1 − <IT>F</IT>)]</DE></FR> 
= <FR><NU><IT>K</IT><SUB>t</SUB><IT>S</IT> + C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB></DE></FR> (1 − <IT>F</IT>) (A7)
Therefore
<FR><NU>C*<SUB>e</SUB></NU><DE>C*<SUB>p</SUB></DE></FR> = <FR><NU><IT>K</IT><SUB>t</SUB><IT>S</IT> + C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB></DE></FR>  ⋅ <FENCE>1 − <FR><NU>C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB><IT>S</IT> + C<SUB>e</SUB></DE></FR> − <FR><NU><IT>R</IT>C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>m</SUB><IT>S</IT> + C<SUB>e</SUB></DE></FR> </FENCE>
 = <IT>S</IT> − <FR><NU><IT>R</IT>C<SUB>e</SUB><IT>S</IT></NU><DE><IT>K</IT><SUB>m</SUB><IT>S</IT> + C<SUB>e</SUB></DE></FR> − <FR><NU><IT>R</IT>C<SUP>2</SUP><SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB>(<IT>K</IT><SUB>m</SUB><IT>S</IT> + C<SUB>e</SUB>)</DE></FR> (A8)
Although this expression appears to be very complex, its predictions are quite simple. The relationship between C*e/C*p and Ce for a given value of R is essentially composed of two distinct straight line segments. One segment describes their behavior in the case of very low Cp values, such that Ce is driven to such small values that hexokinase becomes unsaturated. If Ce actually becomes small relative to KmS, then the last term of the expression becomes negligible, and C*e/C*p is well approximated as
<FR><NU>C*<SUB>e</SUB></NU><DE>C*<SUB>p</SUB></DE></FR> = <IT>S</IT> − <FR><NU><IT>R</IT>C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>m</SUB></DE></FR>  C<SUB>e</SUB> << <IT>K</IT><SUB>m</SUB><IT>S</IT> (A9)
As supply is increased, the resulting level of tissue glucose Ce saturates hexokinase, the second term in the expression for C*e/C*p becomes independent of Ce, and the last term becomes proportional to Ce. The resulting prediction for C*e/C*p is again a straight line
<FR><NU>C*<SUB>e</SUB></NU><DE>C*<SUB>p</SUB></DE></FR> = <IT>S</IT>(1 − <IT>R</IT>) − <FR><NU><IT>R</IT>C<SUB>e</SUB></NU><DE><IT>K</IT><SUB>t</SUB></DE></FR>  C<SUB>e</SUB> >> <IT>K</IT><SUB>m</SUB><IT>S</IT> (A10)
As with Eqs. A1 and A2 we rearrange to get
C<SUB>e</SUB> = [<IT> 
S</IT>(1 − <IT>R</IT>) − (C*<SUB>e</SUB>/C*<SUB>p</SUB>)](<IT>K</IT><SUB>t</SUB>/<IT>R</IT>) (A11)
Because the desaturation of hexokinase occurs only in the most extreme cases of low supply or high demand, Eqs. A10 and A11 provide the practical working relationships between C*e/C*p and Ce when demand is fixed and supply varied. They are the counterparts of Eqs. A1 and A2 above, except with a parametric dependence on the demand parameter R rather than the supply parameter Cp. Note that in the circumstance of fixed demand the slopes of their relationships are negative, with one falling as the other increases and vice versa.

In the first situation (fixed supply and variable demand), the value of Ce had a maximum imposed by the fixed Cp value. As noted above, even in the event of zero glucose utilization, Ce could not rise above SCp because the transport barrier is passive. In the case of fixed demand, however, Ce (at least in principle) can be increased without limit, and therefore it is interesting to remark on the consequent behavior of Ce predicted by the current theory. For fixed demand and variable Cp, Ce rises in value as Cp is increased, and C*e/C*p falls accordingly. However, zero is the smallest physically allowable value for the methylglucose distribution space, and thus Ce does not increase without limit but is predicted by Eq. A11 to have the maximum value
Maximum C<SUB>e</SUB> = <IT>K</IT><SUB>t</SUB><IT>S</IT>(1 − <IT>R</IT>)/<IT>R</IT> (A12)
For low values of demand (small R), this limiting value can become quite large; however, for the level of demand determined in normal conscious rats in our previous report (R = 0.34), the maximum value of Ce is predicted to be the surprisingly small value of ~15 µmol/g, as Cp is increased without limit (14).

    FOOTNOTES

Address for reprint requests: G. Dienel, Dept. of Neurology, Slot 500, University of Arkansas for Medical Sciences, 4301 W. Markham St., Little Rock, AR 72205-7199.

Received 12 February 1997; accepted in final form 7 July 1997.

    REFERENCES
Top
Abstract
Introduction
Materials & Methods
Results
Discussion
Appendix
References

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AJP Endocrinol Metab 273(5):E839-E849