Renal actions of atrial natriuretic factor: a mathematical modeling study

Flow resistance along the rat renal tubule

The Reynolds number in the renal tubule is extremely low, consistent with laminar flow. Consequently, luminal flow can be described by the Hagen-Poiseuille laminar flow equation. This equation calculates the volumetric flow rate from the axial pressure gradient and flow resistance, which is dependent on the length and diameter of each renal tubule segment. Our goal was to calculate the pressure drop along each segment of the renal tubule and to determine the points of highest resistance. When the Hagen-Poiseuille equation was used for rat superficial nephrons based on known tubule flow rates, lengths, and diameters, it was found that the maximum pressure drop occurred in two segments: the thin descending limbs of Henle and the inner medullary collecting ducts. The high resistance in the thin descending limbs is due to their small diameters. The steep pressure drop observed in the inner medullary collecting ducts is due to the convergent structure of the tubules, which channels flow into fewer and fewer tubules toward the papillary tip. For short-looped nephrons, the calculated glomerular capsular pressure matched measured values, even with the high collecting duct flow rates seen in water diuresis, provided that tubule compliance was taken into account. In long-looped nephrons, the greater length of thin limb segments is likely compensated for by a larger luminal diameter. Simulation of the effect of proximal diuretics, namely acetazolamide or type 2 sodium-glucose transporter inhibitors, predicts a substantial back pressure in Bowman's capsule, which may contribute to observed decreases in glomerular filtration rate.

An optimization algorithm for a distributed-loop model of an avian urine concentrating mechanism

To better understand how the avian kidney's morphological and transepithelial transport properties affect the urine concentrating mechanism (UCM), an inverse problem was solved for a mathematical model of the quail UCM. In this model, a continuous, monotonically decreasing population distribution of tubes, as a function of medullary length, was used to represent the loops of Henle, which reach to varying levels along the avian medullary cones. A measure of concentrating mechanism efficiency - the ratio of the free-water absorption rate (FWA) to the total NaCl active transport rate (TAT) - was optimized by varying a set of parameters within bounds suggested by physiological experiments. Those parameters include transepithelial transport properties of renal tubules, length of the prebend enlargement of the descending limb (DL), DL and collecting duct (CD) inflows, plasma Na(+) concentration, length of the cortical thick ascending limbs, central core solute diffusivity, and population distribution of loops of Henle and of CDs along the medullary cone. By selecting parameter values that increase urine flow rate (while maintaining a sufficiently high urine-to-plasma osmolality ratio (U/P)) and that reduce TAT, the optimization algorithm identified a set of parameter values that increased efficiency by approximately 60% above base-case efficiency. Thus, higher efficiency can be achieved by increasing urine flow rather than increasing U/P. The algorithm also identified a set of parameters that reduced efficiency by approximately 70% via the production of a urine having near-plasma osmolality at near-base-case TAT. In separate studies, maximum efficiency was evaluated as selected parameters were varied over large ranges. Shorter cones were found to be more efficient than longer ones, and an optimal loop of Henle distribution was found that is consistent with experimental findings.

A model of glucose transport and conversion to lactate in the renal medullary microcirculation

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 10(-4) 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: 541-545, 1965; Bartlett S, Espinal J, Janssens P, and Ross BD. Biochem J 219: 73-78, 1984).

A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. I. Formulation and base-case results

We have developed a highly detailed mathematical model for the urine concentrating mechanism (UCM) of the rat kidney outer medulla (OM). The model simulates preferential interactions among tubules and vessels by representing four concentric regions that are centered on a vascular bundle; tubules and vessels, or fractions thereof, are assigned to anatomically appropriate regions. Model parameters, which are based on the experimental literature, include transepithelial transport properties of short descending limbs inferred from immunohistochemical localization studies. The model equations, which are based on conservation of solutes and water and on standard expressions for transmural transport, were solved to steady state. Model simulations predict significantly differing interstitial NaCl and urea concentrations in adjoining regions. Active NaCl transport from thick ascending limbs (TALs), at rates inferred from the physiological literature, resulted in model osmolality profiles along the OM that are consistent with tissue slice experiments. TAL luminal NaCl concentrations at the corticomedullary boundary are consistent with tubuloglomerular feedback function. The model exhibited solute exchange, cycling, and sequestration patterns (in tubules, vessels, and regions) that are generally consistent with predictions in the physiological literature, including significant urea addition from long ascending vasa recta to inner-stripe short descending limbs. In a companion study (Layton AT and Layton HE. Am J Physiol Renal Physiol 289: F1367-F1381, 2005), the impact of model assumptions, medullary anatomy, and tubular segmentation on the UCM was investigated by means of extensive parameter studies.

Determinants of basal nitric oxide concentration in the renal medullary microcirculation

In this study, we modeled the production, transport, and consumption of nitric oxide (NO) in the renal medullary microcirculation under basal conditions. To yield agreement with reported NO concentrations of ∼60–140 nM in medullary tissues (Zou AP and Cowley AW Jr. Hypertension 29: 194–198, 1997; Am J Physiol Regul Integr Comp Physiol 279: R769–R777, 2000) and 3 nM in plasma (Stamler JS, Jaraki O, Osborne J, Simon DI, Keaney J, Vita J, Singel D, Valeri CR, and Loscalzo J. Proc Natl Acad Sci USA 89: 7674–7677, 1992), the permeabilities of red blood cells (RBCs), vascular walls, and pericytes to NO are all predicted to lie between 0.01 and 0.1 cm/s, and the NO production rate by vasa recta endothelium is estimated to be on the order of 10−14μmol·μm−2·s−1. Our results suggest that the concentration of NO in RBCs, which is essentially controlled by the kinetics of NO scavenging by hemoglobin, is ∼0.01 nM, that is, 103times lower than that in plasma, pericytes, and interstitium. Because the basal concentration of NO in pericytes is on the order of 10 nM, it may be too low to active guanylate cyclase, i.e., to induce vasorelaxation. Our simulations also indicate that basal superoxide concentrations may be too low to affect medullary NO levels but that, under pathological conditions, superoxide may be a very significant scavenger of NO. We also found that although oxygen is a negligible NO scavenger, medullary hypoxia may significantly enhance NO concentration gradients along the corticomedullary axis.

Immunomorphometric study of rat renal inner medulla

We utilized immunofluorescent immunolabeling of renal tissue sections to identify and count tubules at specified depths of the rat renal inner medulla. We used primary antibodies to aquaporin-1 (AQP1; labeling thin descending limbs), aquaporin-2 (AQP2; labeling inner medullary collecting ducts), the rat kidney-specific chloride channel (ClC-K1; labeling thin ascending limbs), and von Willebrand factor (labeling descending vasa recta). Secondary antibodies conjugated to different fluorophores were used, giving up to a three-color display. Labeled structures were then identified and counted. At each level sampled in the inner medulla, many more thin limbs were labeled by ClC-K1 than AQP1. In addition, thin limbs were found to label with antibodies to ClC-K1 on both sides of their hairpin turns. We conclude that the descending thin limbs shift from expressing AQP1 to expressing ClC-K1 some distance before the point where they turn and begin to ascend. Mathematical models can use our quantitative data to explore implications for the urine-concentrating mechanism.