Urine concentrating mechanism in the inner medulla of the mammalian kidney: role of three-dimensional architecture.

The urine concentrating mechanism in the mammalian renal inner medulla (IM) is not understood, although it is generally considered to involve countercurrent flows in tubules and blood vessels. A possible role for the three-dimensional relationships of these tubules and vessels in the concentrating process is suggested by recent reconstructions from serial sections labelled with antibodies to tubular and vascular proteins and mathematical models based on these studies. The reconstructions revealed that the lower 60% of each descending thin limb (DTL) of Henle's loops lacks water channels (aquaporin-1) and osmotic water permeability and ascending thin limbs (ATLs) begin with a prebend segment of constant length. In the outer zone of the IM (i) clusters of coalescing collecting ducts (CDs) form organizing motif for loops of Henle and vasa recta; (ii) DTLs and descending vasa recta (DVR) are arrayed outside CD clusters, whereas ATLs and ascending vasa recta (AVR) are uniformly distributed inside and outside clusters; (iii) within CD clusters, interstitial nodal spaces are formed by a CD on one side, AVR on two sides, and an ATL on the fourth side. These spaces may function as mixing chambers for urea from CDs and NaCl from ATLs. In the inner zone of the IM, cluster organization disappears and half of Henle's loops have broad lateral bends wrapped around terminal CDs. Mathematical models based on these findings and involving solute mixing in the interstitial spaces can produce urine slightly more concentrated than that of a moderately antidiuretic rat but no higher.

Mathematical model of an avian urine concentrating mechanism.

A mathematical model was used to investigate how concentrated urine is produced within the medullary cones of the quail kidney. Model simulations were consistent with a concentrating mechanism based on single-solute countercurrent multiplication and on NaCl cycling from ascending to descending limbs of loops of Henle. The model predicted a urine-to-plasma (U/P) osmolality ratio of approximately 2.26, a value consistent with maximum avian U/P osmolality ratios. Active NaCl transport from descending limb prebend thick segments contributed 70% of concentrating capability. NaCl entry and water extraction provided 80 and 20%, respectively, of the concentrating effect in descending limb flow. Parameter studies indicated that urine osmolality is sensitive to the rate of fluid entry into descending limbs and collecting ducts at the cone base. Parameter studies also indicated that the energetic cost of concentrating urine is sensitive to loop of Henle population as a function of medullary depth: as the fraction of loops reaching the cone tip increased above anatomic values, urine osmolality increased only marginally, and, ultimately, urine osmolality decreased.

Limit-cycle oscillations and tubuloglomerular feedback regulation of distal sodium delivery.

A mathematical model was used to evaluate the potential effects of limit-cycle oscillations (LCO) on tubuloglomerular feedback (TGF) regulation of fluid and sodium delivery to the distal tubule. In accordance with linear systems theory, simulations of steady-state responses to infinitesimal perturbations in single-nephron glomerular filtration rate (SNGFR) show that TGF regulatory ability (assessed as TGF compensation) increases with TGF gain magnitude gamma when gamma is less than the critical value gamma(c), the value at which LCO emerge in tubular fluid flow and NaCl concentration at the macula densa. When gamma > gamma(c) and LCO are present, TGF compensation is reduced for both infinitesimal and finite perturbations in SNGFR, relative to the compensation that could be achieved in the absence of LCO. Maximal TGF compensation occurs when gamma approximately gamma(c). Even in the absence of perturbations, LCO increase time-averaged sodium delivery to the distal tubule, while fluid delivery is little changed. These effects of LCO are consequences of nonlinear elements in the TGF system. Because increased distal sodium delivery may increase the rate of sodium excretion, these simulations suggest that LCO enhance sodium excretion.

Nonlinear filter properties of the thick ascending limb.

A mathematical model was used to investigate the filter properties of the thick ascending limb (TAL), that is, the response of TAL luminal NaCl concentration to oscillations in tubular fluid flow. For the special case of no transtubular NaCl backleak and for spatially homogeneous transport parameters, the model predicts that NaCl concentration in intratubular fluid at each location along the TAL depends only on the fluid transit time up the TAL to that location. This exact mathematical result has four important consequences: 1) when a sinusoidal component is added to steady-state TAL flow, the NaCl concentration at the macula densa (MD) undergoes oscillations that are bounded by a range interval envelope with magnitude that decreases as a function of oscillatory frequency; 2) the frequency response within the range envelope exhibits nodes at those frequencies where the oscillatory flow has a transit time to the MD that equals the steady-state fluid transit time (this nodal structure arises from the establishment of standing waves in luminal concentration, relative to the steady-state concentration profile, along the length of the TAL); 3) for any dynamically changing but positive TAL flow rate, the luminal TAL NaCl concentration profile along the TAL decreases monotonically as a function of TAL length; and 4) sinusoidal oscillations in TAL flow, except at nodal frequencies, result in nonsinusoidal oscillations in NaCl concentration at the MD. Numerical calculations that include NaCl backleak exhibit solutions with these same four properties. For parameters in the physiological range, the first few nodes in the frequency response curve are separated by antinodes of significant amplitude, and the nodes arise at frequencies well below the frequency of respiration in rat. Therefore, the nodal structure and nonsinusoidal oscillations should be detectable in experiments, and they may influence the dynamic behavior of the tubuloglomerular feedback system.

Spectral properties of the tubuloglomerular feedback system.

A simple mathematical model was used to investigate the spectral properties of the tubuloglomerular feedback (TGF) system. A perturbation, consisting of small-amplitude broad-band forcing, was applied to simulated thick ascending limb (TAL) flow, and the resulting spectral response of the TGF pathway was assessed by computing a power spectrum from resulting TGF-regulated TAL flow. Power spectra were computed for both open- and closed-feedback-loop cases. Open-feedback-loop power spectra are consistent with a mathematical analysis that predicts a nodal pattern in TAL frequency response, with nodes corresponding to frequencies where oscillatory flow has a TAL transit time that equals the steady-state fluid transit time. Closed-feedback-loop spectra are dominated by the open-loop spectral response, provided that gamma, the magnitude of feedback gain, is less than the critical value gamma c required for emergence of a sustained TGF-mediated oscillation. For gamma exceeding gamma c, closed-loop spectra have peaks corresponding to the fundamental frequency of the TGF-mediated oscillation and its harmonics. The harmonics, expressed in a nonsinusoidal waveform for tubular flow, are introduced by nonlinear elements of the TGF pathway, notably TAL transit time and the TGF response curve. The effect of transit time on the flow waveform leads to crests that are broader than troughs and to an asymmetry in the magnitudes of increasing and decreasing slopes. For feedback gain magnitude that is sufficiently large, the TGF response curve tends to give a square waveshape to the waveform. Published waveforms and power spectra of in vivo TGF oscillations have features consistent with the predictions of this analysis.

Permeability criteria for effective function of passive countercurrent multiplier.

The urine concentrating effect of the mammalian renal inner medulla has been attributed to countercurrent multiplication of a transepithelial osmotic difference arising from passive absorption of NaCl from thin ascending limbs of long loops of Henle. This study assesses, both mathematically and experimentally, whether the permeability criteria for effective function of this passive hypothesis are consistent with transport properties measured in long loops of Henle of chinchilla. Mathematical simulations incorporating loop of Henle transepithelial permeabilities idealized for the passive hypothesis generated a steep inner medullary osmotic gradient, confirming the fundamental feasibility of the passive hypothesis. However, when permeabilities measured in chinchilla were used, no inner medullary gradient was generated. A key parameter in the apparent failure of the passive hypothesis is the long-loop descending limb (LDL) urea permeability, which must be small to prevent significant transepithelial urea flux into inner medullary LDL. Consequently, experiments in isolated perfused thin LDL were conducted to determine whether the urea permeability may be lower under conditions more nearly resembling those in the inner medulla. LDL segments were dissected from 30-70% of the distance along the inner medullary axis of the chinchilla kidney. The factors tested were NaCl concentration (125-400 mM in perfusate and bath), urea concentration (5-500 mM in perfusate and bath), calcium concentration (2-8 mM in perfusate and bath), and protamine concentration (300 micrograms/ml in perfusate). None of these factors significantly altered the measured urea permeability, which exceeded 20 x 10(-5) cm/s for all conditions. Simulation results show that this moderately high urea permeability in LDL is an order of magnitude too high for effective operation of the passive countercurrent multiplier.

Instantaneous and steady-state gains in the tubuloglomerular feedback system.

The load of water and solute entering each nephron of the mammalian kidney is regulated by the tubuloglomerular feedback (TGF) mechanism, a negative feedback loop. Experiments in rats have shown that key variables of this feedback system may exhibit TGF-mediated oscillations. Mathematical modeling studies have shown that the open-feedback-loop gain is a crucial parameter for determining whether oscillations will emerge. However, two different formulations of this gain have been used. The first is the steady-state gain, a readily measurable quantity corresponding to the steady-state reduction in single-nephron glomerular filtration rate (SNGFR) subsequent to a sustained increased in ascending limb flow rate. The second is an instantaneous gain, a variable arising from theoretical considerations corresponding to the maximum reduction in SNGFR resulting from an instantaneous shift of the ascending limb flow column, with the assumption that the SNGFR response is also instantaneous. Here we show by an analytic argument how the steady-state and instantaneous open-feedback-loop gains for the ascending limb are related. In the case of no solute backleak into the ascending limb, the two formulations of gain are equivalent; however, in the presence of solute backleak, the instantaneous gain is larger in magnitude than the steady-state gain. With typical physiological parameters for the rat, calculations with a model previously devised by us show that the gains differ by 5-10%. Hence, experimental measurements of the steady-state gain may provide useful lower-bound estimates of the instantaneous gain of the feedback system in the normal rat. However, the gains may diverge significantly in pathophysiological states where ascending limb transport is compromised by abnormally high NaCl permeability.

A dynamic numerical method for models of renal tubules.

We show that an explicit method for solving hyperbolic partial differential equations can be applied to a model of a renal tubule to obtain both dynamic and steady-state solutions. Appropriate implementation of this method eliminates numerical instability arising from reversal of intratubular flow direction. To obtain second-order convergence in space and time, we employ the recently developed ENO (Essentially Non-Oscillatory) methodology. We present examples of computed flows and concentration profiles in representative model contexts. Finally, we indicate briefly how model tubules may be coupled to construct large-scale simulations of the renal counterflow system.

Numerical simulation of propagating concentration profiles in renal tubules.

Method-dependent mechanisms that may affect dynamic numerical solutions of a hyperbolic partial differential equation that models concentration profiles in renal tubules are described. Some numerical methods that have been applied to the equation are summarized, and ways by which the methods may misrepresent true solutions are analysed. Comparison of these methods demonstrates the need for thoughtful application of computational mathematics when simulating complicated time-dependent phenomena.

Distributed solute and water reabsorption in a central core model of the renal medulla.

In this model study we investigate the dependence of urine concentrating capability on the spatial distribution of solute and water reabsorption from Henle's loops. Within the context of model assumptions, urine concentrating capability is increased by exponential decline in loop population as a function of medullary depth and by solute efflux localized near loop bends, in accordance with earlier, but less comprehensive, studies. Further, we find that water-impermeable prebend enlargements of the descending limb may release urine concentrating capacity that would otherwise be needed to concentrate the fluid flowing in the prebend enlargements. Calculations reported here suggest that without some distributed features, even vigorous net active transport of solute from the ascending limbs of the inner medulla would not be sufficient to explain the large concentration gradients generated by some mammals. We consider the significance of distributed reabsorption for the operation of the concentrating mechanisms of the mammalian inner medulla, the mammalian outer medulla, and the avian medullary cone.

Bifurcation analysis of TGF-mediated oscillations in SNGFR.

Recent micropuncture studies in rats have demonstrated the existence of oscillatory states in nephron filtration mediated by tubuloglomerular feedback (TGF). We develop a minimal mathematical model of the TGF system, consisting of a first-order hyperbolic partial differential equation describing thick ascending limb (TAL) NaCl reabsorption and an empirical feedback relation. An analytic bifurcation analysis of this model provides fundamental insight into how oscillatory states depend on the physiological parameters of the model. In the special case of no solute backleak in the TAL, the emergence of oscillations explicitly depends on two nondimensional parameters. The first corresponds to the delay time of the TGF response across the juxtaglomerular apparatus, and the second corresponds to the product of the slope of the TGF response curve at the steady-state operating point and the space derivative of the steady-state NaCl concentration profile in the TAL at the macula densa. Numerical calculations for the case without TAL backleak are consistent with this result. Numerical simulation of the more general case with TAL backleak shows that the bifurcation analysis still provides useful predictions concerning nephron dynamics. With typical parameter values, the analysis predicts that the TGF system will be in oscillatory state. However, the system is near enough to the boundary of the nonoscillatory region so that small changes in parameter values could result in nonoscillatory behavior.

Urea transport in a distributed loop model of the urine-concentrating mechanism.

Continuously distributed loops of Henle were used in a central core model of the rat kidney's urine-concentrating mechanism to investigate the importance of overlapping loops for three different modes of urea transport in the long loops of Henle: 1) urea-impermeable loops, 2) urea-permeable loops (as indicated by perfused tubule experiments), and 3) loops with urea-permeable descending limbs and active urea transport out of thin ascending limbs. Mode 1 produces high papillary tip osmolality in accordance with tissue slice experiments, but the relative contribution of urea to the osmolality of the central core and the long descending limbs is below experimental measurements. Mode 2 generates no significant osmolality increase in the inner medulla, in agreement with other model studies. Mode 3 produces high papillary tip osmolality with a substantial contribution of urea to the osmolality of the core and the descending limbs, which is more in accordance with experiments. The results suggest that 1) overlapping loops may produce a cascade effect that contributes to the inner medullary concentrating mechanism and that 2) new experiments are needed to more certainly ascertain the urea transport characteristics of the thin ascending limbs.

Distribution of Henle's loops may enhance urine concentrating capability.

A simple mathematical model that was developed by Charles S. Peskin (unpublished manuscript) for a single nephron is introduced and then extended to reflect the decreasing loop of Henle population as a function of increasing medullary depth. In the model, if all the loops turn at the same depth, the concentrating capability is limited by a factor of e over plasma osmolality. However, a decreasing loop population causes a multiplier effect that greatly enhances the concentrating capability. Using the loop distribution of the rat, the model produces a sigmoidal osmolality profile similar to the profiles found in tissue-slice studies of rat kidneys. These model calculations suggest that the decreasing nephron population found in vivo may be an important factor in the concentrating mechanism of the mammalian kidney.