Electrolyte transport in a central core model of the renal medulla

3D simulation of solutes concentration in urinary concentration mechanism in rat renal medulla

In this work, a mathematical model was developed to simulate the urinary concentration mechanism. A 3-D geometry was derived based on the detail physiological pictures of rat kidney. The approximate region of each tubule was obtained from the volume distribution of structures based on Walter Pfaller's monograph and Layton's region-based model. Mass and momentum balances were applied to solve for the change in solutes concentration and osmolality. The osmolality of short and long descending nephrons at the end of the outer medulla was obtained to be 530 mOsmol/kgH2O and 802 mOsmol/kgH2O, respectively, which were in acceptable agreement with experimental data. The fluid osmolality of the short and long ascending nephrons was also compatible with experimental data. The osmolality of CD fluid at the end of the inner medulla was determined to be 1198 mOsmol/kgH2O which was close the experimental data (1216 ± 118). Finally, the impact of the position of each tubule on the fluid osmolality and solutes concentration were obvious in the results; for example, short descending limb a1, which is the closest tubule to the collecting duct, had the highest urea concentration in all tubules. This reflects the important effect of the 3D modeling on the precise analysis of urinary concentration mechanism.

Advances in Understanding the Urine-Concentrating Mechanism

The renal medulla produces concentrated urine through the generation of an osmotic gradient that progressively increases from the cortico-medullary boundary to the inner medullary tip. In the outer medulla, the osmolality gradient arises principally from vigorous active transport of NaCl, without accompanying water, from the thick ascending limbs of short- and long-looped nephrons. In the inner medulla, the source of the osmotic gradient has not been identified. Recently, there have been important advances in our understanding of key components of the urine-concentrating mechanism, including (a) better understanding of the regulation of water, urea, and sodium transport proteins; (b) better resolution of the anatomical relationships in the medulla; and (c) improvements in mathematical modeling of the urine-concentrating mechanism. Continued experimental investigation of signaling pathways regulating transepithelial transport, both in normal animals and in knockout mice, and incorporation of the resulting information into mathematical simulations may help to more fully elucidate the mechanism for concentrating urine in the inner medulla.

The physiology of urinary concentration: an update

The renal medulla produces concentrated urine through the generation of an osmotic gradient extending from the cortico-medullary boundary to the inner medullary tip. This gradient is generated in the outer medulla by the countercurrent multiplication of a comparatively small transepithelial difference in osmotic pressure. This small difference, called a single effect, arises from active NaCl reabsorption from thick ascending limbs, which dilutes ascending limb flow relative to flow in vessels and other tubules. In the inner medulla, the gradient may also be generated by the countercurrent multiplication of a single effect, but the single effect has not been definitively identified. There have been important recent advances in our understanding of key components of the urine concentrating mechanism. In particular, the identification and localization of key transport proteins for water, urea, and sodium, the elucidation of the role and regulation of osmoprotective osmolytes, better resolution of the anatomical relationships in the medulla, and improvements in mathematic modeling of the urine concentrating mechanism. Continued experimental investigation of transepithelial transport and its regulation, both in normal animals and in knock-out mice, and incorporation of the resulting information into mathematic simulations, may help to more fully elucidate the inner medullary urine concentrating mechanism.

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.

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.

Externally driven countercurrent multiplication in a mathematical model of the urinary concentrating mechanism of the renal inner medulla

Substitution of measured permeabilities into mathematical models of the concentrating mechanism of the renal inner medulla yields less than the known urine osmolalities. To gain a better understanding of the mechanism we analyse a model in which a force of unspecified origin [expressed as fraction, epsilon, of entering descending thin limb (DTL) concentration] drives fluid from DTL to interstitial vascular space (CORE), thus concentrating the solution in DTL. When flow in the DTL reverses at the hairpin bend of the loop of Henle, the high solute permeability of ascending thin limb (ATL) permits solute to diffuse into the CORE thus permitting epsilon to be multiplied many-fold. Behavior of the model is described by two non-linear differential equations. In the limit for infinite salt permeability of ATL the two equations reduce to a single equation that is formally identical with that for the Hargitay and Kuhn multiplier, which assumes fluid transport directly from DTL to ATL (Z. Electrochem. Angew. Phys. Chem. 55, 539, 1951). Solutions of the equations describing the model with parameters taken from perfused thin limbs show that urine osmolalities of the order of 5000 mosm L-1 can be generated by forces of the order of 20 mosm L-1. It seems probable that mammals including desert rodents use some variant of this basic mechanism for inner medullary concentration.