Abstract
To quantify kinetics of mammary glucose utilization in vivo, 24 paired glucose and extracellular indicator (paminohippuric acid) dilution curves across intact bovine mammary glands were obtained after bolus injections into the external iliac artery. Dilution curves were analyzed using a compartmental capillary, convolution integration model. Four candidate submodels of glucose transport and metabolism in capillary supply zones were fit to the glucose dilution curves and evaluated. Model I, with one extracellular compartment for glucose and firstorder unidirectional uptake, failed, indicating that efflux of glucose from the intracellular space could not be ignored. Model II, with firstorder exchanges between extracellular and intracellular compartments and sequestration from the latter, was overdefined because unidirectional clearance of glucose was at least five times the blood flow rate and 20 times the net clearance rate. Model III, combining extracellular and intracellular space into one compartment, was superior in its goodnessoffit to curves and identifiability of parameters. MichaelisMenten parameters of sequestration were not identifiable. Parameters of the optimal compartmental capillary, convolution integration model were applicable to both the dynamics of injected glucose dilution and the steadystate background arteriovenous difference of glucose. Glucose sequestration followed firstorder kinetics between 0 and 7 mM extracellular glucose with an average rate constant of 0.006 s^{−1} or a clearance of 44 ml/s. The ratio of intracellular to extracellular glucose distribution space was 0.34, which is considerably lower than the expected intracellular volume and suggests an intracellular occlusion compartment with which extracellular glucose rapidly exchanges.
 lactose synthesis
 glucose uptake
 arteriovenous differences
net uptake of glucose by the mammary glands of a lactating cow consumes 60 to 70% of the whole body glucose turnover and up to 90% of that uptake is used for lactose synthesis (1, 18). As a major osmolyte in milk, lactose concentration changes little and lactose synthesis rate is the main determinant of overall milk yield (13). Glucose is also metabolized in mammary tissue to facilitate synthesis of other milk components such as fatty acids (14). Thus the control of glucose transport and metabolism by mammary glands is important to milk synthesis and the glucose economy of the lactating animal.
Infusion of glucose for 10 h into the arterial supply of bovine mammary glands to increase blood plasma concentration from 3.6 to 6.4 mM caused only an 18% increase in milk lactose secretion rate (5). Similarly, lactose synthesis rate was unrelated to circulating glucose concentration between 4.0 and 5.0 mM but it varied 25% at a given glucose concentration (4). Throughout a 40wk lactation, the typical monoexponential decline in milk yield of 1 to 2% per week (19) results in a halving of lactose synthesis rate. These observations suggest that lactose synthesis in the cow responds not to circulating concentrations of glucose but to a centrally dictated mammary setpoint. Kinetically, the insensitivity to glucose concentration suggests that lactose synthesis operates close to its V_{max}, meaning the K_{m} is very low relative to physiological glucose concentration. However, the K_{m} of lactose synthesis by cultured bovine mammary epithelial cells was 3.5 and 4.7 mM (8, 24) and the K_{m} of 3Omethylglucose transport in vitro was 8 mM (23), whereas normal extracellular mammary glucose is 1.5 to 3.5 mM. The in vitro kinetics of mammary lactose synthesis appear to contradict the lack of sensitivity in vivo.
The balance of mammary glucose uptake not sequestered in lactose is oxidized. In the 10h arterial glucose infusion experiment, where lactose synthesis only increased 18%, net uptake of glucose by the mammary glands increased 39% (5), indicating responsiveness to substrate concentration in the glucose oxidation pathways. Again, the opposite is evident from in vitro experiments in which the K_{m} for glucose oxidation was 1.1 mM (7, 24).
Although kinetics of metabolic transformations are readily measured in vitro, they are not quantitatively applicable to in vivo rates of metabolism. In the current study, to resolve the apparent contradiction between in vitro and in vivo responses to glucose, a paired indicator/nutrient dilution technique was implemented to measure glucose transport and metabolism kinetics in intact lactating bovine mammary glands in vivo. Paraaminohippuric acid (PAH) was used as the extracellular indicator and dglucose as the nutrient of interest. Venous PAH and glucose dilution curves were analyzed using the novel compartmental capillary, convolution integration (CCCI) model described in a companion paper (17). In the CCCI model, all capillaries are identical and consist of wellmixed compartments representing extracellular and intracellular spaces and the various fluxes between and without. The heterogeneity of nonexchanging vessel transit times is used to transform concentrations of glucose in each single capillary outflow, c_{sc_v}(t) (mM), to concentrations in the whole organ outflow, c_{v}(t) (mM), in a linear accumulative manner. A normal distribution function, f(t), describes the density distribution of nonexchanging vessel transit times with mean t̄_{μ} (s) and standard deviation σ (s) obtained from the PAH curve (17). These parameters of the mammary extracellular space were presented by Qiao et al. (17). The concentration of glucose in the organ outflow is then a convolution of f(t) with c_{sc_v}(t), (1) where τ is the dummy variable of integration. This formulation of the outflow problem allows transport and sequestration of glucose within a single capillary to be described with relative ease by a compartmental submodel. In this paper, we tested several candidate submodels for their ability to describe dynamics of glucose exchange during a single pass through the mammary glands with the objectives of evaluating the CCCI model in a paired indicator/nutrient dilution experiment, selecting the best submodel of mammary glucose dynamics, and quantifying kinetics of glucose transport and metabolism in intact bovine mammary glands in vivo.
MATERIALS AND METHODS
A total of 33 paired PAH and glucose bolus injections into the external iliac artery of four lactating cows was conducted as described previously (17). The PAH dose was 0.06 to 0.1 g, and the glucose dose ranged from 1.1 to 22.2 mmol. Doses were administered over a period of 1 to 2 s. Glucose concentrations in each of 25 venous samples collected at 5s intervals from the start of injection and a background arterial sample were analyzed by the glucose oxidase method (Sigma Diagnostics Procedure No. 510, SigmaAldrich Canada, Oakville, Ontario, Canada).
Candidate submodels of glucose dynamics in the single capillary were based on two compartments to represent extracellular and intracellular distribution spaces (Fig. 1). Sequestration of glucose occurs from the intracellular compartment. After conservation of mass, the general differential equations to describe the extracellular and intracellular concentration functions c_{sc_v}(t) and c_{sc_c}(t), respectively, were (2) where q_{0} is the dose injected (mmol), F is the blood flow rate (l/s), t_{dose} is the noninstantaneous duration of the injection (s), p(t) is a rectangular pulse equal to 1 between time 0 and t_{dose} and 0 otherwise, c_{a}(0) is the background arterial concentration (mM), V_{e} is the extracellular distribution volume (liter), V_{c} is the intracellular volume (liter), and j_{1}(t), j_{2}(t), and j_{3}(t) are intercompartmental fluxes (mmol/s; Fig. 1).
Four candidate submodels (Table 1) were fit to the glucose dilution curves and evaluated. Parameters for each model were estimated using a fourthorder RungeKutta algorithm to integrate Eqs. 1 and 2 and an iterative LevenbergMarquardt algorithm to minimize the residual sum of squares. Because a portion of blood in the external iliac artery into which indicator and nutrient were injected does not flow to the mammary glands, F, V_{e}, and V_{c} could not be directly estimated so the area iA and monoexponential downslope b of the PAH curve were substituted for iq_{0}/F and F/V_{e}, respectively, in Eq. 2. The parameters t̄_{μ} and σ for simulation of glucose dilution (Eq. 1) were also obtained from the PAH curve (17). On the basis of leastsquares estimates from PAH curves of the time for rapid injection (17), t_{dose} was given a value of 1.5 s for all glucose dilution curve fits.
In model I, only influx according to firstorder kinetics is proposed, which was the basis for the paired indicator/nutrient dilution model of Yudilevich and Mann (27) and the irreversible compartmental model of Goresky (9). This model is simple, considers one wellmixed extracellular compartment and only one parameter, k_{1} (s^{−1}), and has to be estimated from glucose enrichment curves. It is the same model as that proposed by Hanigan et al. (11) for analysis of steadystate net arteriovenous differences. The k_{1} at steady state, before injection of the dose, was calculated by setting Eq. 2 to 0 to yield (3) The k_{1} estimated from glucose dilution curves should match the k_{1_bg}. Otherwise, model I is considered to have failed.
In model II, an intracellular compartment is considered in addition to the extracellular one. Both influx j_{1}(t) and efflux j_{2}(t) between these two compartments are assumed to follow identical firstorder kinetics where k_{1}V_{e} = k_{2}V_{c}. Rate of glucose metabolism also follows firstorder kinetics. The transport parameter k_{1} (s^{−1}), the ratio of intracellular to extracellular distribution volumes r_{v}, and the metabolism rate constant k_{3} (s^{−1}) were estimated by fitting glucose enrichment curves. The basal intracellular glucose concentration c_{c}(0) (mM) was calculated from the fitted k_{1} value based on the fact that, at steady state, the net uptake of glucose across the mammary glands can be defined as either the balance between arterial and venous flows c_{a}(0)F − c_{v}(0)F or the balance between extracellular and intracellular compartments k_{1}V_{e}c_{v}(0) − k_{2}V_{c}c_{c}(0). Where k_{1}V_{e} = k_{2}V_{c}, the equality of these two definitions of net uptake yields, upon rearranging, (4) where k_{1} > 0. The basal metabolism rate constant is then (5) Again, the relationship between k_{3_bg} and k_{3} is a test of the model’s ability to describe glucose uptake.
Model III is a special case of model II in which k_{1} and k_{2} are high enough to ignore the barrier between extracellular and intracellular compartments and combine these two compartments into one. It differs from model I in that an intracellular compartment is considered. The parameters k_{3} and r_{v} were estimated from glucose dilution curves. The steadystate glucose metabolism rate constant is (6) Model IV is the same as model III except that glucose metabolism follows MichaelisMenten kinetics rather than firstorder kinetics. To satisfy both the enrichment curve and the basal arteriovenous difference of glucose, the parameters K_{m} (mM) and r_{v} were estimated by fitting the glucose curves and V_{max} (mM s^{−1}) was calculated from the basal arteriovenous difference constraint as (7) In addition to the model validation criteria specified above, each model was evaluated for goodnessoffit to observations, randomness of residuals on the time course, and identifiability and plausibility of parameters.
RESULTS
The ratio of glucose to PAH in the injectates (r) and basal glucose concentrations in the artery and vein are listed in Table 2. The r and iA values were used to represent the exact glucose dose delivered to the mammary glands.
Thirtythree paired PAH/glucose curves were obtained from four lactating cows. A typical set of curves is shown in Fig. 2. Low doses (<1 g) of injected glucose did not result in sufficient elevation of venous glucose over background to be precisely detected by the enzymatic assay, so nine sets of curves (from cows 2410 and 382) were not analyzed. Leastsquares estimates of parameters of the four tentative glucose transport and metabolism models (Table 1) are summarized in Table 3 and final fits to selected glucose dilution curves are shown in Fig. 3.
The goodnessoffit of each model was evaluated from both r^{2} and the square root of the mean square prediction error (rMSPE). Using Tukey’s multiple range test, the r^{2} of model I (Table 3) was significantly lower (P < 0.001) than those of the other models, and its rMSPE was significantly higher (P < 0.001). There were no significant differences in r^{2} and rMSPE between models II, III, and IV or between cows. Because models I, II, and III are nested, fits to each curve were compared by partial Ftests, which yielded, on average, P < 0.001 for comparisons of models I and II, P < 0.001 between models I and III, and P = 0.441 between models III and II. Model II provided a significantly better fit than model III to 4 of the 24 glucose curves. According to the coefficient of variation of parameter estimation (approximate standard error/final estimate), only models I and III converged well and were, therefore, identifiable.
Model I did not describe the glucose dynamics as well as the other three. According to the assumption that efflux is negligible, the tail of the glucose curve must decline faster than that of the paired PAH curve because unidirectional uptake occurs in addition to simple wash out. However, the paired glucose and PAH curves actually became closer to each other with time (Fig. 2). Underestimation of tail concentrations was compensated by overestimation of peak concentrations in the leastsquares parameterization. Despite the poor fits, model I converged on a k_{1} solution, which agreed with k_{1_bg} calculated from basal arteriovenous differences. The slope of the relationship between these two parameters was not different from 1.0 (P = 0.69; Fig. 4A).
With compartments for both the extracellular and intracellular spaces, model II explained the glucose curves as well as models III and IV (Fig. 3). However, although the nonsteadystate k_{3} was correlated with k_{3_bg} (Fig. 4B), the slope was different from 1.0 (P = 0.03). In addition, the parameter k_{1} was not identifiable from most of the glucose curves (Table 3). Analysis of residual sums of squares indicated that there was no optimum k_{1} value (Fig. 5). Increasing k_{1} beyond a value of 0.5 s^{−1} had little impact on the goodnessoffit because at such high turnover, the intracellular and extracellular glucose concentrations become essentially equal. The average monoexponential washout slope b from the 24 PAH curves was 0.033 s^{−1} (17), whereas k_{1} was greater than 0.5 s^{−1} for nine of the curves and averaged 0.18 s^{−1} for the others. Unidirectional clearance was thus at least five times the rate of plasma flow through the mammary glands. Although glucose uptake is flow limited, transport was apparently not a limiting factor for glucose utilization in the bovine mammary glands. Model II was overdefined, and it was decided to combine extracellular and intracellular spaces into one wellmixed compartment for models III and IV.
Model III predicted the glucose curve well (Fig. 3) and possessed the most identifiable parameters (Tables 3 and 4). This model overcame the limitation of model I that the intracellular glucose distribution volume and efflux from it were ignored. The relationship between steadystate and nonsteadystate parameters, k_{3_bg} and k_{3}, of glucose metabolic rate (Fig. 4C) had a slope not different from 1.0 (P = 0.63) and a nonsignificant intercept (P = 0.6). The fact that model III with essentially instantaneous transport described both injected dose and basal arteriovenous differences best supports the conclusion from model II that glucose transport did not limit glucose utilization in the mammary glands.
Model IV assigned MichaelisMenten kinetics to the glucose sequestration function instead of firstorder kinetics, but K_{m} and V_{max} were not well identified by many glucose curves (Table 3). Using a nonlinear approach to describe a linear system often results in multiple parameter solutions. The k_{3_bg} estimated from model III was equivalent to the term V_{max}/K_{m} + c_{v}(0) from model IV (Fig. 4D), which indicates that glucose sequestration in the mammary gland did not follow MichaelisMenten kinetics, but rather simple linear kinetics.
Because model III was superior to the other models in identifiability of parameters and goodnessoffit to the observed data, further analysis focused only on model III. There was no correlation between residuals and corresponding sampling time (Fig. 6A). In addition, there was no correlation between parameter estimates and the glucose dose (Fig. 6, B and C). These results demonstrate that the indicator dilution technique yielded parameters representing instantaneous physiological status without influencing it.
DISCUSSION
Mammary glucose compartmentalization.
The failure of model I suggested that efflux of injected glucose from within cells had to be considered to fit the venous dilution curves. The primary glucose transporter in lactating mammary glands is GLUT1, the bidirectional facilitative and insulininsensitive transporter of erythrocytes (2, 3, 28). Although the exact mechanism of GLUT1 function remains to be elucidated, it is clear from the time course of uptake by erythrocytes that the transporter translocates extracellular sugar across the plasma membrane into an intracellular compartment from which the sugar may diffuse into the bulk cytosol or, with a greater likelihood, escape back to the extracellular space (12, 16). The same behavior was observed in bovine mammary epithelial cells in vitro (25) and is now apparent from in vivo mammary dilution curves.
Parameters for model II indicated that the unidirectional clearance of glucose by the mammary glands (k_{1}·V_{e}) was at least five times the plasma flow rate and 20 times the net clearance (k_{3}·V_{c}). Similarly, Goresky et al. (10) found from modeling of multiple indicator dilution across the liver in vivo that the capacity for influx of galactose into liver cells was 40 times that for galactose sequestration by phosphorylation. In human erythrocytes in vitro, the rate of sugar exchange between the extracellular and intermediate intracellular compartments was 23 times faster than the rate of exchange between the intermediate compartment and the bulk cytosol (12). Occlusion of glucose in an intermediate compartment did not appear to prevent phosphorylation in bovine mammary epithelial cells in vitro (24) but glucose utilization rate at 3.0 mM glucose was 62 nmol·h^{−1}·mg protein^{−1}, which, assuming 286 g mammary protein (22), represents a clearance of 1.6 ml/s. The clearance from extracellular to occlusion compartments was 6.5 ml/s, or four times the lactose synthesis rate. Thus, although the parameters of exchange with intracellular space in model II were difficult to estimate, the conclusion that translocation of glucose into mammary cells is many times faster than its rate of metabolism is supported by previous observations. Model III, which was superior in identifiability, assumed instantaneous transport.
The estimated ratio r_{v} of intracellular glucose distribution volume to extracellular volume for model III was 0.34 ± 0.09. There were no significant differences between cows in r_{v}. The value of r_{v} is low compared with an intracellulartoextracellular water ratio of 1.5 for the whole body (21), 2.8 for the heart (20), and 4.7 for the liver (26). The extracellular volume of a 22kg udder was estimated at 5.5 liters from the PAH dilution curves (17). Assuming 75% of udder mass is water leaves 11.0 liters of intracellular water, which is two times the extracellular volume. The estimated r_{v} for glucose of 0.34 is thus only 17% of the expected ratio and likely represents the size of the intermediate occlusion compartment. Occlusion volume was estimated at 30–40% of intracellular glucose space in human erythrocytes (6, 12) and 35% in cultured bovine mammary epithelial cells (25). Intracellular ATP will influence the size of the occlusion space, possibly through allosteric interaction with the GLUT1 homotetramer. Occlusion volume was 40% of intracellular in erythrocyte ghosts containing 4 mM ATP and 20% in its absence (12). The glucose dilution curves are consistent with the presence in vivo of an occlusion compartment, smaller than the total cytosolic volume, with which extracellular glucose rapidly exchanges and from which glucose is relatively slowly metabolized.
Kinetics of mammary glucose sequestration.
The modeling suggested linear kinetics of glucose sequestration between 0 and peak extracellular concentrations of 7 mM glucose. Although K_{m} and V_{max} are difficult to estimate from a time course curve even when saturation kinetics are in effect, one would not expect, with saturation, k_{3_bg} representing sequestration between 0 and 3 mM glucose to match k_{3} representing sequestration above 3 mM. For example, from the in vitro data of Xiao and Cant (24) where the K_{m} of glucose utilization by mammary epithelial cells was 4.2 mM, the glucose sequestration constant between 0 and 3 mM glucose, assuming 286 g mammary protein, was 1.6 ml/s and that between 3 and 5 mM was reduced to 1.0 ml/s. The lack of difference in vivo suggests a K_{m} for glucose utilization greater than the in vitro K_{m} of 4.2 mM. Multiplying the average k_{3} value of 0.006 s^{−1} by the compartment volume of (1 + r_{v}) 5.5 liters gives an in vivo clearance of 44 ml/s, which is 28 times the in vitro rate.
A primary objective of obtaining the glucose dilution curves was to reconcile kinetics of mammary glucose utilization in vitro with rates measured at steady state in vivo. Rates of glucose sequestration and secretion of lactose in milk are not correlated with circulating glucose concentrations (4, 7, 15), suggesting the K_{m} for glucose utilization is low. However, K_{m} values for transmembrane glucose transport and lactose synthesis from extracellular glucose estimated in isolated mammary epithelial cells in vitro were higher than normal extracellular glucose concentrations (8, 23, 24). In general, the in vivo kinetics of glucose fluxes in the mammary glands, estimated from the paired PAH/glucose dilution curves, support observations in vitro that there is an occlusion space for rapid, bidirectional translocation of glucose across the plasma membrane of cells and that the K_{m} for intracellular sequestration is higher than the normal concentration of extracellular glucose. The apparently low K_{m} implicated in correlation analysis of point estimates of plasma glucose concentration and mammary uptake could be due to control of sequestration efficiency. In a previous experiment, elevation of mammary venous glucose concentration, representing extracellular concentration, from 2.9 to 5.2 mM by infusion of 90 g/h glucose into the external iliac artery resulted in a 39% increase in net glucose uptake and an 18% increase in incorporation into lactose (5). At a constant clearance rate across glucose concentrations, one would expect net glucose to have increased to the same degree as venous glucose concentration, i.e., 80%. That it only increased 39% suggests a downregulation of glucose sequestration capacity during the 10 h of glucose infusion.
Sequestration of free glucose in mammary epithelial cells is accomplished in two ways: phosphorylation by hexokinase (EC 2.7.1.1) or hexosylation by lactose synthase (EC 2.4.1.22). Approximately 70% of the net glucose uptake from blood by lactating bovine mammary glands is used for lactose synthesis and the remainder is oxidized in pentose and TCA cycles (14). The lactose is synthesized in the Golgi apparatus from one free glucose and one uridine diphosphate galactose arising from glucose6phosphate. Thus about 65% of net glucose uptake is phosphorylated by hexokinase. Xiao and Cant (24), applying the methods of metabolic control analysis, estimated that hexokinase exerted 80% of the control over the flux of extracellular glucose to lactose in vitro and lactose synthase exerted <1%. Downregulation of glucose sequestration capacity of the mammary glands is therefore likely to be primarily due to hexokinase activity.
The linear kinetics observed here, where k_{3} for an exogenous dose was equal to k_{3_bg}, allow net clearance to be calculated from a single steadystate measurement of arteriovenous difference and blood flow according to Eq. 6, first used by Hanigan et al. (11), as (8) In the 10h arterial infusion experiment of Cant et al. (5), mammary glucose clearance during saline infusion, calculated according to Eq. 8, was 58 ml/s and, in the last 3 h of glucose administration, clearance was 44 ml/s. Blood flow rate was reduced on the glucose treatment, which may have contributed to the decreased clearance; V_{e} and r_{v} were not measured. From a number of point measures of mammary arteriovenous differences and blood flows, Hanigan et al. (11) calculated an average glucose clearance of 40 ml/s. The 95% confidence interval around the estimate was ±8.4 ml/s, indicating a large variation in kinetic parameters for a narrow range of extracellular glucose concentrations. Regulation of sequestration capacity is implicated. Periodic measures of glucose arteriovenous differences and blood flows throughout lactation in sheep (7) yield clearance estimates of 5.1 ml/s at peak lactation, declining to 2.3 ml/s when milk yield had fallen by 44%. The conclusion to be drawn from these observations is that, at any one instant, the kinetics of glucose sequestration by mammary glands are essentially linear between 0 and 7 mM glucose, but the longterm consequence of up and downregulation of sequestration capacity is a lack of relationship between glucose concentration and net glucose uptake.
GRANTS
This work was funded by a grantinaid from Agribrands International, a wholly owned subsidiary of Cargill, NSERC Canada, and the Ontario Ministry of Agriculture and Food.
Acknowledgments
Thanks to Donna Benschop, Scott Cieslar, Heather Copland, Torben Madsen, Norm Purdie, Lori Sheehan, Andrea Sotirakopoulos, Ying Wu, and Gao Yingxin for assistance in sample collection and analysis. Thanks also to Peter deVries and Jim van Dusen for animal care.
Present address of C. Xiao: Division of Clinical Investigation and Human Physiology, Toronto General Hospital, 200 Elizabeth St., Toronto, Ontario, M5G 2C4 Canada.
Footnotes

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 Copyright © 2005 the American Physiological Society