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【关键词】 lactate
Department of Chemical and Biological Engineering, Tufts University, Medford, Massachusetts
ABSTRACT
In this study, we modeled mathematically the transport of glucose across renal medullary vasa recta and its conversion to lactate by anaerobic glycolysis. Uncertain parameter values were determined by seeking good agreement between predictions and experimental measurements of lactate generation rates, as well as glucose and lactate concentration ratios between the papilla and the corticomedullary junction; plausible kinetic rate constant and permeability values are summarized in tabular form. Our simulations indicate that countercurrent exchange of glucose from descending (DVR) to ascending vasa recta (AVR) in the outer medulla (OM) and upper inner medulla (IM) severely limits delivery to the deep inner medulla, thereby limiting medullary lactate generation. If the permeability to glucose of OMDVR and IMDVR is taken to be the same and equal to 4 x 104 cm/s, the fraction of glucose that bypasses the IM is calculated as 54%; it is predicted as 37% if the presence of pericytes in OMDVR reduces the glucose permeability of these vessels by a factor of 2 relative to that of IMDVR. Our results also suggest that red blood cells (RBCs) act as a reservoir that reduces the bypass of glucose from DVR to AVR. The rate of lactate generation by anaerobic glycolysis of glucose supplied by blood from glomerular efferent arterioles is predicted to range from 2 to 8 nmol/s, in good agreement with lower estimates obtained from the literature (Bernanke D and Epstein FH. Am J Physiol 208: 541545, 1965; Bartlett S, Espinal J, Janssens P, and Ross BD. Biochem J 219: 7378, 1984).
kidney; glycolysis; mathematical model
THE URINARY CONCENTRATING mechanism is made possible by the steep corticomedullary osmolality concentration gradient, which is created and maintained by countercurrent exchange between the descending and ascending limbs of the loops of Henle, the collecting ducts, and the descending (DVR) and ascending vasa recta (AVR). However, the manner in which the hyperosmolality in the inner medullary interstitium is built up and preserved remains unclear. A hypothesis involving an external osmolyte proposed by Niesel and Roeskenbleck (30) and summarized by Kriz and Lever (23) has recently been examined by Jen and Stephenson (20) and Thomas and Wexler (43). These investigators showed that an unspecified solute, external to the tubular fluid, that can provide an added 20100 mosmol/l to the interstitial osmolality throughout the inner medulla (IM), may very significantly increase water reabsorption.
Because of hypoxia in the IM, anaerobic glycolysis appears to be the dominant energy supply for IM metabolism (37). Lactate is produced by anaerobic glycolysis, and it may later be used for gluconeogenesis in the outer medulla (OM) and cortex (37). Because anaerobic glycolysis generates two lactate molecules per molecule of glucose consumed, it increases the amount of interstitial solute; lactate could thus play a significant role as an external osmolyte. Although Kriz and Lever (23) dismissed this possibility more than three decades ago, recent studies have confirmed the potential for lactate to play a role in the urinary concentration mechanism. Thomas and colleagues (42) first developed a medullary microcirculation model to simulate lactate accumulation in IM, then incorporated the microcirculatory model into a urinary concentration model (19, 42). Their predictions suggested that lactate accumulation in the interstitium could be high enough to amplify axial IM NaCl concentration gradients (and therefore the corticomedullary osmolality gradient) under a number of conditions: if the DVR permeability to lactate is as high as 103 cm/s to accumulate lactate in significant levels in IM, if that to glucose is low (i.e., on the order of 105 cm/s) to prevent the significant radial diffusion of glucose from DVR to AVR, if lactate exerts its full osmotic effect across thin descending limbs (DTLs) and IM collcting ducts (IMCD), and if IM blood flow is reduced during antidiuresis.
Vasa recta (VR) permeability to glucose and lactate has not been measured in rats. As discussed below, the glucose permeability of microvascular walls of skeletal muscle has been reported as 105 cm/s (29), whereas extrapolation of mice permeability data (33, 36) suggests that DVR permeability to glucose is 7 x 104 cm/s in rats. The kinetics of anaerobic glycolysis of glucose in the renal medulla have not been determined experimentally either. Given the potential importance of lactate to the urinary concentration mechanism and the scarcity of experimental data on lactate accumulation, in this study we extended our mathematical model of transport in the renal medullary microcirculation to examine the conversion of glucose to lactate. Our objective was to determine parameter values that yield good agreement between predictions and experimental measurements of glucose and lactate concentrations, as well as lactate production rates.
Glucose is supplied to the renal medulla by the medullary microcirculation, which consists of DVR and AVR and smaller capillaries. In the OM, those DVR that are destined to the IM form vascular bundles with AVR and remain separate from the peripheral capillaries, which irrigate nephron tubules (35). In the IM, VR, IMCDs, DTLs and thin ascending limbs (ATLs) of the loops of Henle are interspersed, so that water and solutes (e.g., NaCl and urea) reabsorbed from nephron tubules and IMCDs diffuse into VR and are cleared away from the medulla (35).
While blood flows down from the cortex to the papilla in DVR, water is transported radially to AVR, thereby gradually decreasing single-vessel blood flow rates to the papillary tip and concentrating solutes in blood. At different levels between the IM-OM junction and the papilla, DVR divide into simple capillary networks that reconverge to form AVR, thereby progressively decreasing the overall blood flow to the deeper portions of the IM and papilla (35). In addition, solutes reabsorbed from nephron loops and CDs are partly trapped in the medulla through radial (i.e., transverse) exchange between DVR and AVR. Both phenomena help maintain the corticomedullary osmolality gradient.
Glossary
Ci0
Initial concentration of solute i in plasma, that is, in DVR at corticomedullary junction
Cij
Concentration of solute i in compartment j, j = R, P, and I
D
Diameter of vasa recta
f
Volume fraction of water in red blood cells
Gij
Overall medullary generation (or consumption) rate of component i in compartment j, j = R, I
h
Hematocrit
Jij
Flux of solute i through membrane j, j = R, W
Km
Michaelis-Menten constant for glucose consumption rate
L
Length of renal medulla, or position of the papillary tip when shown in parentheses
Lkj
Hydraulic conductivity of pathway k in compartment j, j = R, W
Pij
Permeability of membrane j to component i, j = R, W, D, A
Qj
Overall volume flow rate in compartment j, j = R, P
q
Volume flow rate in single vas rectum
rj
Ratio of lactate generation-to-glucose consumption rate in compartment j, j = R, I
Greek Symbols
Red blood cell-to-vessel surface area ratio
Oncotic pressure of proteins
i
Activity coefficient of solute i
i
Reflection coefficient to solute i
ij
Volumetric generation rate of component i in compartment j, j = R, I
max
Maximum volumetric glucose consumption rate in Michaelis-Menten equation
Subscripts and Superscripts
0
Initial value (in DVR at corticomedullary junction)
A
Ascending vasa recta
B
Whole blood
D
Descending vasa recta
I
Interstitium
P
Plasma
R
Red blood cell
V
Water volume
W
Vascular wall
MODEL AND PARAMETERS
In this study, we consider the entire medulla, that is, both the OM and IM; we do not take into account the peripheral capillaries in the outer medulla, however. Three compartments are considered in this model: red blood cells (RBCs), plasma, and interstitium. Figure 1 illustrates the model geometry, and the main assumptions and equations are given in the APPENDIX. The model consists of 1) a set of differential mass conservation equations for water and solutes (i.e., NaCl, urea, glucose, lactate, plasma proteins and hemoglobin) in RBCs and plasma, which take into account axial convection, radial transport (that is, across RBC membranes and vascular walls), and generation or consumption; 2) nonlinear mass conservation equations for water and solutes in the interstitium, which relate fluxes from (or toward) DVR and AVR, interstitial generation or consumption rates, and reabsorption rates from nephron loops and collecting ducts; and 3) flux equations across RBC membranes and VR walls, which couple the differential and nonlinear conservation equations. The consumption of glucose and generation of lactate are accounted for in conservation equations in RBCs and interstitium. Described below are the equations and parameters related to glucose and lactate.
Medullary hypoxia is a consequence of decreased blood flow in the papilla, the bypass of oxygen from DVR to AVR by radial diffusion, and the high energy requirement of active NaCl transport in the medullary thick descending limbs (mTALs) (6). Despite some controversy (9), the main source of ATP for IM cell metabolism is thought to be anaerobic glycolysis (37). As described below, we consider anaerobic glycolysis in erythrocytes and in the abluminal cells of the inner medullary interstitium. The conversion of glucose to lactate is simplified as
(1)
Thus two lactate molecules are produced for each glucose molecule that is consumed by anaerobic glycolysis.
Anaerobic Glycolysis in the Medulla
Glucose metabolism in the renal medulla involves glucose synthesis by gluconeogenesis and glucose consumption by oxidation, glycolysis, and the pentose phosphate shunt. The contribution of this last pathway appears to be small (37). Glucose oxidation has been shown to occur in the glomerulus, the OM, and the distal convoluted tubules (37). It is thought to play a critical role in providing energy for the urine concentrating mechanism (5). The main site of glycolysis in the kidney appears to be the IM and the papilla, as indicated by the mapping of glycolytic enzymes (37). Finally, glucose synthesis has been found to occur in the superficial and deep cortex and in the OM (17); Roxe et al. (38), however, observed that the rate of renal gluconeogenesis in the dog is extremely low. As reviewed by Laris (24), glucose that is added to plasma can be entirely recovered, indicating that there is no consumption of glucose in plasma. Taken together, these observations suggest that in the kidney, the production of lactate by anaerobic glycolysis occurs mainly in RBCs throughout the medulla, and in the IM abluminal cells; the latter include interstitial cells, the vascular wall cells of VR, and the epithelial cells of nephron tubules. Given the scarcity of experimental data related to reaction rates in these different types of cells (see below), and because tubular walls are impermeable to glucose and lactate (42), the term "interstitial lactate generation rate" in this study denotes the combined rate of lactate production by all three cell types.
Estimates of the rate of anaerobic glycolysis in the IM interstitium vary by about one order of magnitude. Bartlett et al. (3) measured the medullary lactate production rate in rat kidneys perfused by solutions with biological concentrations of glucose (5 mM) and lactate (2 mM) as 176 μmol?h1?g kidney dry wt1 for male Wistar rats with a body weight of 280350 g. If the weight of the kidney is taken as 0.36% that of the body (32), and the kidney wet-to-dry weight ratio as 4 (31), the overall lactate production rate in the kidney is calculated to be 1215 nmol/s. This value represents an upper limit for IM lactate generation because 1) glucose was perfused to the kidney in these experiments and 2) lactate may also be produced in parts of the kidney other than the IM. Another estimate can be obtained by assuming that all the glucose present in blood at the corticomedullary junction is entirely converted to lactate: if glucose concentration at the junction is taken as 6 mM in plasma and 3 mM in RBCs (see below), the initial hematocrit as 0.25 (35), and blood flow rate in a single vas rectum as 9 nl/min (35), the corresponding rate of lactate production is then estimated as 10 nmol/s. Alternatively, measurements by Bernanke and Epstein (5) indicated that in the IM of dog kidney, the oxygen consumption-to-lactate production ratio is 3 when the medullary slice is incubated with both 100 and 5% saturated oxygen. Our recent study suggested that IM oxygen consumption in rats is on the order of 1 nmol/s (46), yielding an overall IM lactate production rate of 3 nmol/s.
Other estimates of the IM interstitial lactate generation rate may be derived from the study of Bastin et al. (4), who incubated isolated rat kidneys, dissected tissue slices, and measured lactate concentration using an enzyme reagent. The investigators reported that lactate concentrations in papillary tissues increased from 22.5 to 33.3 and 57.7 mmol/kg papillary tissue dry wt over a period of 7.5 and 120 s, respectively. During the first period, the lactate generation rate can therefore be estimated as (33.3 22.5)/7.5 = 1.44 mmol?kg tissue dry wt1?s1; similar calculations yield a generation rate of 0.293 mmol?kg tissue dry wt1?s1 during the second period. In the absence of data on the density of dry papillary tissue, estimates of lactate generation rates based on unit tissue water volume can be obtained as follows. The mass fraction of water in rat papillary interstitium has been reported as 77.4% (7), i.e., the ratio of water volume-to-tissue dry weight is 3.45 liters water/kg dry wt, if water density is taken as 0.993 g/ml. The lactate generation rate is thus calculated as 414 and 84 nmol?ml tissue water1?s1 for the 7.5- and 120-s periods, respectively. If the lactate generation rate is taken to be the same throughout the IM as that measured in papillary tissue and the tissue volume is approximated as that of the tissue water, extrapolation of the data of Bastin et al. (4) using Eq. 3 below suggests that the overall lactate generation rate is on the order of 1.57.5 nmol/s. Because these measurements were obtained in vitro, in the absence of blood flow and reabsorption from nephrons, and given the simplifying assumptions made in converting these data, these estimates remain highly uncertain. Bastin et al. (4) also observed that over the 120-s period, glucose and glycogen concentrations decreased from 20.4 and 18.7 to 8.1 and 14.0 mmol/kg dry wt, that is, from 5.9 and 5.4 to 2.3 and 4.0 mmol/l of tissue water, respectively. The overall consumption rate of glucose and glycogen is thus calculated as 42 nmol?ml tissue water1?s1, exactly half of lactate generation rate, as would be expected under ischemic conditions (see below).
As described above, in this study the contribution of the tubular system to lactate generation and transport is implicitly taken into account in the interstitial glycolytic rate. Bagnasco et al. (2) measured the anaerobic glycolytic rate of dissected rat nephron segments incubated with glucose with and without antimycin A, an inhibitor of oxidative metabolism. They found that all distal tubules produce lactate; among them, the rate in IMCDs is three to seven times that in OM and cortical tubules. Using the rates reported by these investigators (in Table 1 of their study, also cited in Table 1 in this study), as well as the numbers of collecting ducts and mTALs we previously calculated (48) based on the measurements of Mejia and colleagues (26, 27), we estimate that the overall lactate generated by the IMCDs and mTALs is 1.4 nmol/s.
As shown in Eq. 1, two lactate molecules are produced for each glucose molecule that is consumed by anaerobic glycolysis. Because glucose molecules may also be oxidized, the ratio of lactate production-to-glucose consumption in the interstitium (rI) may differ from 2. Nevertheless, in the IM interstitium where oxygen is scarce, we assume a ratio of 2 as the baseline value of rI.
To account for the dependence of the volumetric glucose consumption rate (glu, in nmol?cm3?s1) on glucose concentration (Cglu), especially at low glucose concentration values, a Michaelis-Menten equation is used for glu, following the approach of Thomas (42)
(2)
where Km is the Michaelis-Menten constant and max is the maximum volumetric glycolysis rate. A value of 0.1 mM is chosen for Km (42). As shown by Edwards et al. (15), the overall medullary interstitial solute generation rate is obtained by integrating the product of the local generation rate and the interstitial cross-sectional area, adjusted for the number of vessels. We apply their final result (Eq. A12 in their study) to the production of lactate to relate the overall interstitial lactate production rate (G) to glu
(3)
where A is the medullary cross-sectional area at the IM-OM junction (estimated as 17.5 mm2 from Eq. A20), and Lim is the axial length of inner medulla of the rat kidney, taken as 5.9 mm (34). When glucose concentrations are high enough, glu max according to Eq. 2; with G taken as 3 nmol/s (see above), max is then calculated as 83 nmol?cm3?s1 using Eq. 3. The latter value corresponds to the baseline case in this study. As described below, toward the papillary tip, our results suggest that glucose concentrations become small enough that the dependence of glu on glucose concentration turns out to be significant; consequently, if max is maintained at 83 nmol?cm3?s1, the overall lactate generation rate calculated using our model, G, is lower than 3 nmol/s.
The glycolytic rate in red blood cells is estimated as follows. Messana et al. (28) reported that the rates of glucose consumption and lactate production in human erythrocytes vary with pH but remain independent of oxygenation. They measured the glucose consumption rate as 1560 nmol?ml RBCs1? min1 (i.e., 0.93.6 mmol?liter RBCs1?h1) and the ratio of lactate production to glucose consumption (rR) as close to 1. Ataullakhanov et al. (1) reported that glucose consumption rates in human erythrocytes range from 0.24 to 2.6 mmol?liter RBCs1?h1, whereas rR varies between 1.5 and 2.7 depending on arsenate and orthophosphate concentrations. In rats, the RBC glucose consumption rate has been reported as 5.6 mmol?liter RBCs1?h1 (22). The baseline values for max and rR in rat erythrocytes are therefore chosen as 5.6 mmol?liter RBCs1?h1 (that is, 1.56 nmol?cm3?s1) and 1.5, respectively. Simulation results (not shown) suggest that variations in these two RBC parameters have negligible effects, because the overall lactate production rate is 102-103 lower in RBCs than in the IM interstitium.
Initial Concentrations
Lactate concentration has been reported as 0.92 mM in plasma from the femoral artery of mongrel dogs (14), 1.7 mM in the OM of dogs (11), and 2.6 mM in the serum of rat medullary blood (41). In the absence of more specific data, we assume an initial (i.e., at the corticomedullary junction in DVR plasma) lactate concentration of 2 mM. As reviewed by Kaneko (22), lactate concentration in RBCs has been reported as 0.078 mM in humans. Lacking experimental data for rats, we assume an initial lactate concentration of 2 mM in RBCs as well as in plasma. Our simulations indicate that twofold variations in these two parameter values have a negligible effect on lactate generation rates, as well as glucose and lactate concentrations in the IM (results not shown).
Glucose concentration in human whole blood has been reported as 5.6 mM (45). As reviewed by Kaneko (22), RBC glucose concentration ranges between 0.67 and 3.7 mM in mammals. Glucose concentrations in rat arterial blood have been measured by Laris (24) as 114 mg/100 ml water in plasma and 78 mg/100 ml water in RBCs; the investigator also determined the average water fraction in plasma and RBCs as 92 and 65%, respectively. Based on these experimental values, we estimate glucose concentration in renal arterial blood as 5.8 mM in the plasma phase and 2.8 mM in the RBCs. Roxe et al. (38) also reported that glucose concentration in the renal arteriole of dog ranges from 3.2 to 7.7 mM and averages 5.8 mM. In the absence of more specific data in the glomerular efferent arterioles, the initial (i.e., at the corticomedullary junction in DVR) glucose concentration is taken as 6 mM in plasma and 3 mM in RBCs. The effects of varying these parameter values on concentration profiles are described below.
Permeability
Lactate and glucose transport across the erythrocyte membrane is mediated by carriers (8, 12, 21), making it difficult to measure the permeability of the RBC membrane to these two solutes. Experimental values are spread over a wide range, depending on the methodologies and solute concentrations used (8).
Deuticke et al. (13) estimated the human erythrocyte permeability to lactic acid at 30°C as 3.7 x 105 cm/s, and the activation enthalpy as 24 kcal/mol. Using these data and the Arrhenius equation, we calculate the RBC permeability to lactate at 37°C as 104 cm/s.
According to the review of Jung (21), the permeability of the human RBC membrane to glucose ranges from 2 x 106 to 4 x 104 cm/s. In the absence of rat data, we use 105 cm/s as our baseline value.
In their review, Michel and Curry (29) reported that the glucose permeability of the microvascular wall in skeletal muscle, which is dominated by that across paracellular pathways, is 105 cm/s. According to these investigators, the permeability across transcellular pathways is at least three orders of magnitude lower and can be neglected. Pallone et al. (33) measured mice OMDVR permeabilities to sodium and glucose as 3 x 103 and 2.8 x 103 cm/s, respectively, which suggests that DVR permeability to glucose is comparable to that to sodium. In rats, the DVR permeability to sodium was measured as 0.75 x 103 cm/s (36), that is, four times lower than that in mice. Assuming that the glucose-to-sodium permeability ratio is the same in these two species, we estimate the DVR permeability to glucose in rats as 7 x 104 cm/s, that is, 70 times higher than the value given by Michel and Curry (29) for skeletal muscle microvascular walls. This may not be unreasonable given the simple structure of DVR walls, which in OM are composed of a single layer of endothelial cells surrounded by a discontinuous layer of smooth muscle cell-like pericytes; the latter gradually disappear in the IM (35).
As an alternative approach, pore theory can be used to estimate the permeability and reflection coefficients of the VR paracellular pathway to glucose and lactate, assuming that this route consists of homogeneous slit pores in parallel. This approach is described in Zhang and Edwards (47) and can be summarized as follows. Using the measured DVR and AVR reflection coefficients to albumin, the half-width of the slit pore in DVR and AVR walls is estimated as 3.8 and 4.4 nm, respectively (47). The Stokes-Einstein diffusivities of glucose, lactate, and urea are calculated as 1.16, 1.44, and 1.89 x 105 cm2/s, and the molecular radii of those solutes as 0.35, 0.31, and 0.27 nm, respectively, using the online property calculator SPARC (http://ibmlc2.chem.uga.edu/sparc). The resulting permeabilities and reflection coefficients to glucose and lactate are shown in Table 1.
Because estimates of the permeability of VR walls to glucose range from 105 to 7 x 104 cm/s, and given that permeability is a determinant factor, a wide range of values is considered in this study (see RESULTS). We predict that the reflection coefficients of DVR and AVR paracellular pathways to both solutes are very small, on the order of 0.01. These estimates are much lower than the value of 0.5 used by Thomas (42), because our model distinguishes between transport across aquaporin 1 (AQP1) water channels, which are impermeable to solutes, and that across paracellular pathways, which are shared by water and solutes. Simulations in which we increased and decreased reflection coefficients by a factor of 10 suggest that these parameters have an insignificant effect on glucose and lactate concentration distributions, as well as on water fluxes (results not shown). These predictions agree with those of Thomas (42), who also found that lactate concentration profiles are essentially independent of the reflection coefficient of VR to lactate.
RESULTS
In this study, we examined the conversion of glucose to lactate in the medulla, assuming that blood from the glomerular efferent arterioles is the only source of glucose consumed by anaerobic glycolysis in the IM interstitium. Predicted glucose and lactate concentration ratios were compared with experimental measurements (see immediately below) to estimate uncertain parameter values and to assess the determinant factors in lactate generation.
To the best of our knowledge, there have been no direct measurements of glucose and lactate concentrations in the renal medulla of rats. As discussed by Thomas (42), Ruiz-Guinazu et al. (39) found that glucose and lactate concentrations at the papillary tip of the golden hamster kidney are one-third and twice those in aortic blood, respectively. It should be noted that concentrations may be slightly higher in DVR at the corticomedullary junction than in the arteriole due to glomerular filtration. Scaglione et al. (41) reported that lactate concentration in rats is about two to five times times greater at the papillary tip than that at the corticomedullary boundary, in good agreement with measurements in dogs performed by Dell and Winter (11). Glucose concentration has been reported to average 5.8 mM in the renal arteriole of the dog (38) and is expected to be lower in the renal medullary interstitium. As described in MODEL AND PARAMETERS, extrapolation of measurements by Bastin et al. (4) suggests that that glucose concentration in rat kidney decreases from 5.9 to 2.3 mmol/l of interstitial tissue water over a 2-min period following dissection and under full ischemia, whereas lactate concentration increases from 6.5 to 16.6 mmol/l of interstitial tissue water over the same period. The molar concentration of glucose and lactate (i.e., per unit volume of tissue) should be lower than these values, because water is only a fraction of interstitial tissue volume. Given the in vitro conditions of the study of Bastin et al. (4) as well as the numerous assumptions needed to extrapolate their results, we do not believe that accurate estimates of glucose and lactate medullary concentration can be obtained based on their data.
Baseline Results
We first performed simulations with baseline parameters. Shown in Fig. 2, A and B, are predicted glucose and lactate concentration profiles along the corticomedullary axis. As illustrated in Fig. 2A, glucose diffuses from DVR to the interstitium down its concentration gradient. In the IM, part of the glucose is converted to lactate by anaerobic glycolysis, whereas the rest diffuses back into AVR; as described by Eq. 2, the conversion rate depends on the local glucose concentration in interstitium (C). It is noticeable that in DVR, the predicted concentration of glucose in RBCs (C) rapidly increases in the outer medulla and remains significantly higher than that in plasma (C) throughout the IM; this latter fact suggests that RBCs may act as a reservoir for glucose, as discussed below. The initial, rapid rise in C results from our assumption, based on literature measurements, that glucose concentration at the corticomedullary junction is much lower in RBCs than in plasma; this would suggest the existence of a mechanism that maintains this concentration difference in the general circulation. Because our model did not account for such a mechanism in the renal medulla, the initial rise in C represents equilibration with the higher external glucose concentration, that is, C. With baseline values, the predicted ratio of glucose concentration at the corticomedullary junction to that at the papillary tip is >10, that is, much higher than expected from the data of Ruiz-Guinazu et al. (39), indicating a need to adjust some of the uncertain parameter values.
The lactate that is produced in the interstitium diffuses down its concentration gradient to DVR and is carried down to the papillary tip; as blood flows back up along AVR, lactate diffuses back to the interstitium (Fig. 2B). In the baseline case, the concentration of lactate in the interstitium (C) is predicted to be about four times greater at the papillary tip than at the corticomedullary boundary, in good agreement with experimental data (11, 41).
At steady state, the net amount of solute generated (or reabsorbed) is equal to the AVR-to-DVR solute mass flow difference at the corticomedullary boundary. For baseline parameter values, the net amount of lactate generated by anaerobic glycolysis in the IM is predicted to be 1.6 nmol/s. Because the fraction of lactate generated in RBCs is <1%, the overall and interstitial glycolytic rates are not distinguished in the remainder of this study, unless stated otherwise.
Shown in Fig. 2C are predicted interstitial sodium and urea concentrations along the corticomedullary axis. Throughout most of the medulla, plasma and interstitial concentrations of sodium and urea are calculated to be 10100 times larger than those of glucose and lactate. As opposed to the latter solutes, NaCl and urea significantly affect transversal water transport (see Eq. A10) and, therefore, blood flow to the deep medulla. Hence, NaCl and urea have an indirect effect on glucose supply to the IM and on the lactate generation rate. Lactate may, in turn, affect interstitial sodium and urea reabsorption rates, a phenomenon that is beyond the scope of this study; these rates were maintained constant in our simulations.
The flatness of the predicted sodium concentration profile in the OM interstitium within a vascular bundle (Fig. 2C) contradicts experimental results obtained by Gottschalk and Mylle (18). Indeed, the measurements of these investigators were performed in a portion of the OM interstitium interspersed with nephrons, whereas our model applies only to those DVR that traverse the vascular bundles, which we assumed are completely isolated from nephrons (see the APPENDIX). This is obviously an oversimplification, but we do not believe there are enough experimental data at present to allow us to obtain an accurate estimate of the amount of water and small solutes taken up by those DVR that form part of vascular bundles, versus those that irrigate the capillary plexus. Moreover, interstitial solute concentrations in these two separate regions are expected to be different. Nevertheless, it is likely that VR uptake of water and sodium in the OM affects the countercurrent exchange of glucose, as discussed further below.
Illustrated in Fig. 2D are the predicted overall and single-vessel plasma volume flow rates along the corticomedullary axis. Water reabsorbed from tubules is driven into AVR and carried away to the general circulation. As opposed to the overall flow rate, the single-vessel flow rate is lower in AVR than in DVR; this is because the number of AVR is more than twice that of DVR.
Glycolytic Rate
As described above, several parameters related to the anaerobic glycolysis rate remain difficult to evaluate precisely. In the absence of experimental data, the maximum volumetric rate (max) was taken as a constant in the IM interstitium; the actual rate depends on the local C as described by Eq. 2. We first explored the effect of variations in the three parameters related to the glycolytic rate, Km, max, and rI.
The Michaelis-Menten equation was used because glucose concentration may be too low to maintain a constant local glycolytic rate. Following the approach of Thomas (42), the baseline value of Km was taken as 0.1 mM, meaning that if glucose concentration is much higher than 0.1 mM, glu = max, whereas at low glucose concentration values (<0.1 mM), glu is proportional to glucose concentration. Shown in Fig. 3, A and B, are the predicted interstitial glucose concentration (C) distribution, overall lactate generation rate, and interstitial lactate concentration (C) at the papillary tip as a function of Km. A 10-fold increase in Km significantly raises predicted papillary C to 1 mM and lowers the corticomedullary junction-to-papillary tip glucose concentration ratio to 7, whereas a 10-fold decrease in Km slightly lowers C. Conversely, G and the concentration of lactate at the papillary tip, C(L), are found to be inversely proportional to Km. Indeed, increasing Km raises the glucose concentration threshold beyond which glu remains constant and equal to the maximum volumetric rate max.
Shown in Fig. 4 are the predicted overall lactate generation rate and C at the papillary tip as a function of max, with Km fixed at 0.1 mM. When the glucose conversion rate is small (i.e., max is less than half the baseline value, taken as 83 nmol?cm3?s1), C is much higher than Km, so that throughout most of the IM, the volumetric lactate production rate remains constant, with lac = rI glu = rI max. C is then proportional to max. Conversely, when the glucose conversion rate is high (i.e., max is greater than the baseline value), predicted glucose concentrations are very low near the papillary tip, so that glu and lac are proportional to C and close to 0 in the deep IM; the actual volumetric consumption of glucose is then calculated to be significantly lower than max, particularly near the papillary tip. Hence, further increases in max would only raise G slightly. According to our simulations, multiplying the baseline value of max by a factor of 10 would raise G by only 52%. Increasing the value of max significantly beyond its baseline value thus would have a limited effect on G and lactate accumulation in the medulla but would deplete glucose near the papillary tip.
Analysis of the combined effect of changes in both max and Km shows that even if max is elevated, the predicted glucose concentration can be maintained at high levels in the IM interstitium provided that Km is large enough. Our model suggests that in that case, further increases in max will have a more significant effect on the overall generation rate of lactate (see Table 3).
As expected, reducing the lactate production-to-glucose consumption ratio (rI) from 2 to 1 does not change the predicted glucose concentration distribution in the medulla, but decreases both G and C(L) by a factor of 2.
VR Permeability to Glucose
The permeability of VR to glucose has not been determined experimentally. As described above, baseline permeabilities to glucose were estimated as 4 x 104 and 6 x 104 cm/s for DVR and AVR, respectively, based on pore theory. From the measured DVR permeability to glucose and sodium in mice (33) and that to sodium in rats (36), the permeability of DVR to glucose (P) in rats was calculated as 7 x 104 cm/s. However, Michel and Curry (29) reported in their review that skeletal muscle microvessel permeability to glucose is on the order of 105 cm/s, the value used by Thomas in his model of lactate transport in the medulla (19, 42). Given these uncertainties, we varied permeability values over a wide range. Because glucose appears to permeate across microvascular walls mainly through paracellular pathways (29), it seemed reasonable to assume a fixed AVR-to-DVR permeability ratio.
Shown in Fig. 5A are predicted C profiles for different values of the DVR permeability to glucose: 7 x 104, 1 x 104, 5 x 105, and 1 x 105 cm/s. The AVR-to-DVR permeability ratio was maintained at 1.5, as in the baseline case. Illustrated in Fig. 5B is the predicted overall lactate generation rate as a function of P. We used a logarithmic scale for the abscissa to clearly show trends for low permeability values. A decrease in P diminishes the radial diffusion of glucose from DVR to AVR, thereby reducing glucose bypass from DVR to AVR in the OM and preserving delivery of glucose to IMDVR; hence the predicted increase in glucose concentration in the IM interstitium (Fig. 5A) and in the overall lactate generation rate (Fig. 5B).
However, if P is further reduced below 5 x 105 cm/s, our model suggests that the transport of glucose across the vessel walls becomes severely limited, and C(hence G) decreases with decreasing P. Under these conditions, glucose is transported to the interstitium not only from DVR but also from AVR to satisfy metabolic demands, as reflected by the fact that C is then lower than both C and C(not shown). The overall lactate generation rate appears to reach a maximum for P values on the order of 5 x 105 cm/s.
It is also interesting to note that C may increase toward the papillary tip when the VR permeability to glucose is low enough. This phenomenon stems from countercurrent exchange, the finite permeability of VR, and the resulting lag in concentration equilibration between DVR, AVR, and the interstitium. The papilla-to-corticomedullary junction glucose concentration ratio is therefore not necessarily a well-defined measure of axial glucose concentration gradients.
Because the pericytes that surround OMDVR gradually disappear in the IM (35), it is possible that the glucose permeability of OMDVR is lower than that of IMDVR. To examine this assumption, we lowered the parameter P in the OM to half its baseline value without changing IM permeability values. As a consequence of the OM permeability reduction, the fraction of glucose that bypasses the IM was predicted to decrease from 54 to 37%, the glucose conversion ratio to increase from 18 to 21%, and the glucose concentration at the papillary tip to increase from 0.07 to 0.10 mM.
Reservoir Effect of RBCs
As shown in Fig. 2A, with the baseline RBC permeability to glucose (i.e., 105 cm/s), our model predicts that in DVR, C rapidly increases in the OM to surpass C, whereas in AVR C rapidly decreases near the papillary tip to become lower than C. Thus in DVR glucose is transported from RBCs to plasma throughout the inner medulla, and in AVR, glucose is initially transported from RBCs to plasma near the papillary tip and then in the reverse direction. These effects result from countercurrent exchange and the high interstitial glucose consumption rate. The baseline model predicts that at the papillary tip, glucose concentrations in RBC, plasma, and interstitium are 0.8, 0.1, and 0.07 mM, respectively, and the hematocrit (that is, the ratio of RBC-to-whole blood flow rate) is 0.16. The amount of glucose is thus calculated to be 1.1 times higher in RBCs than in plasma at the papillary tip.
The low permeability to glucose of the RBC membrane (P), as well as the moderate cell-to-wall surface area ratio, which is close to 2 throughout the medulla, allows erythrocytes to preserve glucose in blood, as shown in Fig. 6. Predicted glucose concentration profiles are plotted for two values of P: 2 x 106 and 4 x 104 cm/s, the two limiting values reported by Jung (21). As expected, the RBC-to-plasma glucose concentration gradient increases significantly as P decreases (Fig. 6A) and decreases as P increases (Fig. 6B). However, predicted glucose concentrations in plasma and in interstitium remain unaffected. P, nevertheless, has a significant effect on lactate generation: when P is low, the high C values at the papillary tip mean that throughout most of the IM, glucose is transported from AVR RBCs to plasma, and henceforth to the interstitium. Our model thus suggests that RBCs in AVR act as storage of glucose rather than a sink, thereby raising the overall generation rate of lactate. As P decreases from 4 x 104 to 5 x 106 cm/s, G is calculated to rise from 1.3 to 1.7 nmol/s, a 31% increase. Further decreases in P below 5x106 cm/s will reduce G because a low RBC permeability to glucose helps to preserve it within the microcirculation, but it also limits glucose transport to the interstitium, where it is needed for metabolic purposes.
We also examined the effect of varying the initial (i.e., at the corticomedullary junction in DVR) RBC concentration of glucose between 1 and 4 mM, that is, within the range of reported values (22). The initial plasma glucose concentration, C, was kept fixed at 6 mM. Our model suggests that G increases linearly with the increase in initial C(results not shown). As the initial C increases from 1 to 4 mM, the overall amount of glucose supplied to the renal medulla increases by 16% (the initial hematocrit is taken as 0.25), yet G is predicted to rise from 1.3 to 1.7 nmol/s, a 31% increase. For comparison (see below), a fourfold (i.e., 400%) increase in the overall amount of glucose supplied to the renal medulla obtained by simultaneously increasing the initial C and C baseline values by a factor of 4 is predicted to raise G from 0.48 to 1.6 nmol/s, a 233% increase. RBCs therefore appear to play a significant role in storing glucose for metabolic purposes.
The C also remains uncertain, as described above. We therefore varied the initial C from 1 to 18 mM; the initial C/C ratio was kept fixed at 2, as in the baseline case. As expected, the higher the C, the higher the predicted values of C and G, as illustrated in Fig. 7, A and B, respectively. As C increases beyond 10 mM, the lactate generation rate is found to reach an asymptote corresponding to G= 3 nmol/s, the overall generation rate value that we used to estimate max. Indeed, glucose is then so plentiful in the IM that the volumetric glucose consumption rate is equal to its maximum value everywhere in the IM.
Lactate Permeability
To the best of our knowledge, there are no reported measurements of VR wall permeability to lactate (P and P, respectively) in rats. In the absence of evidence of significant transport through lactate transporters in the microvascular wall, we assumed that lactate permeates across DVR and AVR walls mainly through paracellular pathways, given its water solubility, and we estimated P and P based on pore theory. We then investigated the effect of varying both permeabilities while maintaining P/P= 1.6, as in the baseline case.
Our model suggests that changes in P and P have a negligible effect on glucose concentration and lactate generation rate but significantly affect lactate concentration and axial concentration gradients. As shown in Fig. 8, predicted lactate concentration at the papillary tip and the papillary tip-to-corticomedullary junction lactate concentration ratio both increase roughly exponentially with permeability. With other parameters equal to their baseline values, C(L)/C is predicted to be between 2 and 5, that is, within the range of reported experimental values (11, 39, 41), if P remains between 5 x 105 and 6 x 104 cm/s. As P increases from 5 x 105 to 6 x 104 cm/s, C(L) increases from 5 to 15 mM. The strong dependence of C on VR permeability to lactate, already predicted by previous investigators (42), further demonstrates that the accumulation of lactate in IM depends not only on the amount of lactate that is generated but also on the rate at which it is exchanged across medullary countercurrent systems.
Similarly, changes in the permeability of the RBC membrane to lactate (P) were found to have a negligible effect on glucose concentration and lactate generation rate. A 10-fold increase in P from the baseline value raised C at the papillary tip slightly, whereas a 10-fold decrease lowered it from 11.7 to 10.0 mM.
Blood Flow Rate
Measured blood flow rates in a single DVR mostly range from 5 to 14 nl?min1?vas rectum1 in the antidiurectic kidney (35). The baseline blood flow rate was taken as 9 nl?min1?VR1 (35). The effects of increasing the initial blood flow rate in a single DVR (q) from 5 to 14 nl/min on interstitial glucose and lactate concentrations, as well as the lactate generation rate, are shown in Fig. 9. Predicted glucose concentration at the papillary tip increases moderately with increasing q(Fig. 9A). Predicted lactate concentration at the papillary tip, however, reaches its maximum when q= 1011 nl/min, even though G increases proportionally with q(Fig. 9B). Indeed, an increase in q results in opposite effects: it raises the net amount of glucose supplied by the microcirculation, thereby increasing the generation rate of lactate and corticomedullary lactate concentration gradients; but it also dilutes blood (i.e., solute washout), thereby reducing axial solute concentration gradients. When q is <10 nl/min, the former effect dominates; when q is >11 nl/min, the reverse is true.
Effect of Reabsorption Rates
Several investigators (3, 14, 41) have reported that the relationship between lactate generation and water and solute reabsorption is complicated, in that solute diuresis, but not water diuresis, affects lactate generation in a significant way. The diuretic state affects the amount of water and solute reabsorbed from nephron loops and collecting ducts. In this study, reabsorption into the interstitium was taken into account by means of interstitial generation rates (see the APPENDIX), and in the simulations above, the overall reabsorption rate of water (G) was kept constant. Because the relationship between the reabsorption rate of water and the accumulation of lactate in the medulla cannot be explicitly determined in this model, the effects of isolated changes in G on glucose metabolism were investigated as illustrated in Fig. 10. The water reabsorption rate was decreased or increased by a factor 4 relative to its baseline value, 0.13 μl/s (15). As shown in Fig. 10B, our model suggests that increasing G dilutes interstitial solute concentrations, hence the reduction in C at the papillary tip, but has little effect on the lactate generation rate. Variations in C are also significant, except near the papillary tip (Fig. 10A); indeed, the low glucose concentration at the papillary tip is controlled by the high max value and/or the low Km value. If we either decreased max or increased Km, similar changes in G would result in more significant variations in C at x = L. If max were lowered by a factor 3 and Km were kept at its baseline value (0.1 mM), increasing G fourfold would lower C(L) from 1.5 to 0.4 mM; if Km were increased by a factor 10 and max were kept at its baseline value, increasing G fourfold would lower C(L) from 0.6 to 0.4 mM.
In the above simulations, the interstitial generation (i.e., reabsorption) rates of water, sodium, and urea were taken to be zero in the OM vascular bundles, hence the absence of an axial sodium gradient in OM, as discussed above and illustrated in Fig. 2C. Given that the amount of water and sodium reabsorbed within the vascular bundles may not be negligible and yet remains difficult to estimate accurately, we performed limiting-case simulations. First, we assumed constant volumetric reabsorption rates of water, sodium, and urea in the OM; the OM generation rates of water and urea were taken to be equal to their baseline values at the OM-IM junction, whereas the reabsorption rate of sodium was taken as a constant throughout the entire medulla (in the baseline case, it is 0 at the OM-IM junction and increases linearly toward the papillary tip, as expressed in Eq. A8), so that the total amount of sodium reabsorbed in the renal medulla was 1.1 times that in the baseline case. With those assumptions, the predicted sodium concentration profile remained flat in the OM VR, due to the simultaneous reabsorption of water and sodium in the vascular bundles; variations in the fractional amount of glucose that bypasses the IM were then found to be insignificant.
In the second case, we increased the rate of volumetric sodium reabsorption throughout the medulla by a factor of 2 and kept the baseline water and urea interstitial generation rates (that is, the latter were taken as 0 in the OM vascular bundles), to generate a significant axial sodium concentration gradient in the OM. The calculated amount of reabsorbed sodium was then 2.2 times greater than in the baseline case. The concentration of sodium was then predicted to increase from 164 mM at the corticomedullary junction to 251 mM at the OM-IM junction, and the fraction of glucose that is transported from DVR to AVR in the OM (i.e., that bypasses the IM) was predicted to increase by <1%. Indeed, increased OM interstitial sodium concentrations result in higher osmotic pressure differences across OMDVR walls, hence greater water efflux from DVR lumen to interstitium through both AQP-1 water channels and paracellular pathways, which raises the amount of glucose carried by convection across OMDVR walls. Because diffusion appears to be the predominant mode of transport across VR for small solutes (see Peclet number calculations below), the convection-induced increase in the fraction of glucose that bypasses the IM remains small.
DISCUSSION
Given the potential importance of lactate to the urinary concentrating mechanism, in this study we sought to better characterize the conversion of glucose to lactate in the medullary microcirculation by finding parameter values that yield good agreement between predictions and experimental measurements of glucose and lactate concentrations.
Our study is different from that of Thomas (42) in several respects, as summarized in Table 2. We considered the entire medulla, not just the IM, so that the effect of the partial bypass of glucose from DVR to AVR in the OM could be examined. Because we took into account all the DVR that are destined to the IM as well as the returning AVR, as opposed to the single-unit approach used by Thomas (42), we were able to predict an overall lactate generation rate, instead of specifying the glucose conversion ratio. In our model, AVR were modeled separately from the interstitium, a more accurate representation than that of Thomas (42). Our model also distinguished between RBCs and the plasma compartment, allowing us to investigate the specific role of erythrocytes in glucose transport along VR. In addition, we calculated radial water fluxes at every depth along the corticomedullary axis, by evaluating Starling forces and distinguishing between paracellular and transcellular (i.e., AQP-1 water channels and UTB urea transporters) pathways, whereas Thomas (42) assumed that the radial flux is a fixed fraction of the axial flow. Finally, parameters such as initial glucose concentrations and vascular wall permeabilities varied considerably between the two studies. Consequently, 1) we predicted generally lower glucose concentrations at the papillary tip; 2) we could examine the effect of glucose bypass in the OM and the role of RBCs; and 3) we were able to estimate the overall rate of lactate generation and compare it with experimental data.
Reported measurements of glucose and lactate concentrations in the renal medulla appear to be scarce. As described above, the experimental data of Ruiz-Guinazu et al. (39), Scaglione et al. (41), and Dell and Winter (11) suggest that the interstitial concentration of glucose is about one-third lower at the papilla than at the corticomedullary junction, and that of lactate is two to five times greater. To the best of our knowledge, there have been no direct measurements of lactate generation rate in the renal medulla of rats. As described in MODEL AND PARAMETERS, indirect estimates of the overall lactate generation rate vary between 1.5 and 15 nmol/s and are most likely on the lower end of that interval.
Our simulations suggest that the profile of C is strongly dependent on the values of max, Km, the vascular wall permeability to glucose (P and P), and C. G was found to be significantly affected by the values of P and P, C, and the initial blood flow rate (q). Finally, C was found to be mostly a function of vascular permeability to lactate and glucose, of C, and of the rate of water uptake from the nephrons and collecting ducts (G). With baseline parameter values, the papillary tip-to-corticomedullary junction glucose concentration ratio was predicted to be <0.1, far below experimental determinations. Our sensitivity analysis thus indicated that the baseline values of Km and C were underestimated, and/or those of P and P were overestimated. The papillary tip-to-corticomedullary junction lactate concentration ratio and G were calculated as 3.8 and 1.6 nmol/s, respectively, that is, within the experimental range.
As described above, the DVR permeability to glucose is highly uncertain; estimates based on the literature range from 105 to 7 x 104 cm/s (29, 33, 36). As reviewed by Michel and Curry (29), glucose transport across the microvascular wall is mainly through paracellular pathways; we therefore maintained the P/P ratio constant as we varied permeabilities. Our simulations indicate that a decrease in permeability reduces the bypass of glucose from OMDVR to OMAVR through radial diffusion, thereby increasing glucose supply in the IM and lactate generation. However, as the permeability of DVR to glucose is lowered below 5 x 105 cm/s (and that of AVR below 7.5 x 105 cm/s), glucose transport from the vascular lumen to the interstitium is predicted to become rate limiting, leading to a decrease in the glycolytic reaction rate. Given that 1) the low glucose permeability value (i.e., 105 cm/s) given by Michel and Curry (29) is for skeletal muscle microvessels; 2) the ultrastructure of DVR and AVR walls is relatively simple (35); 3) extrapolation of glucose permeability measurements from mice (33, 36) suggests that P is on the order of 7 x 104 cm/s in rats; and 4) lactate and glucose are of similar size so that vascular wall permeabilities to these two molecules are expected to be of similar magnitude in the absence of transporters, it is unlikely that the DVR permeability to glucose is as low as 105 cm/s and that to lactate as high as 103 cm/s, as assumed by Thomas and colleagues (19, 42).
To obtain the best agreement between our predictions and experimental data, we sought to find parameter values that generate C and C papillary tip-to-corticomedullary junction ratios of about one-third and comprised between 2 and 5, respectively. To obtain high enough values of C, we fixed Km at 1 mM (see RESULTS). The initial blood concentration of glucose, C, was set at either 6 or 10 mM, and P at either 104 or 5 x 105 cm/s, whereas the P/P ratio was kept equal to 1.5. For each set of (C, P) values, we determined the values of max and P that yield C(L)/ C 0.33 and C(L)/C(0) 4. The lactate permeability ratio was also kept constant, such that P/P= 1.6. Results are shown in Table 3. Among the parameters C, P(hence P), and max, the latter is the one that has the most significant effect on corticomedullary glucose concentration gradients. On the other hand, corticomedullary lactate concentration gradients depend mostly on P(hence P); the value of P that yields C(L)/C(0) 4 was consistently found to be 2 x 104 cm/s in all the cases examined.
The results shown in Table 3 also suggest that to increase lactate generation rate (G), it is necessary that both the glucose content in the blood that enters the medulla (i.e., C) be high, and that glucose bypass in the outer medulla be limited (that is, lower values of P and P are more favorable). To further illustrate