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

A new, region-based mathematical model of the urine concentrating mechanism of the rat renal medulla was used to investigate the significance of transport and structural properties revealed in anatomic studies. The model simulates preferential interactions among tubules and vessels by representing concentric regions that are centered on a vascular bundle in the outer medulla (OM) and on a collecting duct cluster in the inner medulla (IM). Particularly noteworthy features of this model include highly urea-permeable and water-impermeable segments of the long descending limbs and highly urea-permeable ascending thin limbs. Indeed, this is the first detailed mathematical model of the rat urine concentrating mechanism that represents high long-loop urea permeabilities and that produces a substantial axial osmolality gradient in the IM. That axial osmolality gradient is attributable to the increasing urea concentration gradient. 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 that the interstitial NaCl and urea concentrations in adjoining regions differ substantially in the OM but not in the IM. In the OM, active NaCl transport from thick ascending limbs, at rates inferred from the physiological literature, resulted in a concentrating effect such that the intratubular fluid osmolality of the collecting duct increases ~2.5 times along the OM. As a result of the separation of urea from NaCl and the subsequent mixing of that urea and NaCl in the interstitium and vasculature of the IM, collecting duct fluid osmolality further increases by a factor of ~1.55 along the IM.     

A region-based model framework for the rat urine concentrating mechanism

The highly structured organization of tubules and blood vessels in the outer medulla of the mammalian kidney is believed to result in preferential interactions among tubules and vessels; such interactions may promote solute cycling and enhance urine concentrating capability. In this study, we formulate a new model framework for the urine concentrating mechanism in the outer medulla of the rat kidney. The model simulates preferential interactions among tubules and vessels by representing two concentric regions and by specifying the fractions of tubules and vessels assigned to each of the regions. The model equations are based on standard expressions for transmural transport and on solute and water conservation. Model equations, which are derived in dynamic form, are solved to obtain steady-state solutions by means of a stable and efficient numerical method, based on the semi-Lagrangian semi-implicit method and on Newton’s method. In this application, the computational cost scales as O(N ²), where N is the number of spatial subintervals along the medulla. We present representative solutions and show that the method generates approximations that are second-order accurate in space and that exhibit mass conservation.     

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, ɛ, 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 ɛ 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.     

An inverse algorithm for a mathematical model of an avian urine concentrating mechanism

A nonlinear optimization technique, in conjunction with a single-nephron, single-solute mathematical model of the quail urine concentrating mechanism, was used to estimate parameter sets that optimize a measure of concentrating mechanism efficiency, viz., the ratio of the free-water absorption rate to the total NaCl active transport rate. The optimization algorithm, which is independent of the numerical method used to solve the model equations, runs in a few minutes on a 1000 MHz desktop computer. The parameters varied were: tubular permeabilities to water and solute; maximum active solute transport rates of the ascending limb of Henle and the collecting duct (CD); length of the prebend enlargement (PBE) of the descending limb; fractional solute delivery to the CD; solute concentration of tubular fluid entering the CD at the cortico-medullary boundary; and rate of exponential CD population decrease along the medullary cone. Using a base-case parameter set and parameter bounds suggested by physiologic experiments, the optimization algorithm identified a maximum-efficiency set of parameter values that increased efficiency by 40% above base-case efficiency; a minimum-efficiency set reduced efficiency by about 41%. When maximum-efficiency parameter values were computed as medullary length varied over the physiologic range, the PBE was found to make up 88% of a short medullary cone but only 8% of a long medullary cone.     

The effect of solution non-ideality on membrane transport in three-dimensional models of the renal concentrating mechanism

Previous models of the renal concentrating mechanism employ ideal approximations of solution thermodynamics for membrane transport calculation. In three-dimensional models of the renal medulla, predicted urine concentrations reach levels where there idealized approximations begin to break down. In this paper we derive equations that govern membrane transport for non-dilute solutions and use these equations in a three-dimensional model of the concentrating mechanism. New numerical methods were employed that are more stable than those employed previously. Compared to ideal solution models, the urea non-ideality tends to increase predicted osmolarities, whereas NaCl non-ideality decreases predictions.     

Effect of varying salt and urea permeabilities along descending limbs of Henle in a model of the renal medullary urine concentrating mechanism

Modelling studies have played an important role in research on the mechanism of urine concentration and dilution by the medulla of the kidney ever since Hargitay and Kuhn (1951,Z. Elektrochem. 55, 539–558) first proposed that the parallel tubular structures in the kidney medulla must function as a “countercurrent multiplication” system. Present-day models, in keeping with our considerably improved understanding of most aspects of medullary structure-function relationships, have evolved into rather sophisticated systems of parallel tubes. In spite of this increasing complexity, it has remained the case that “model medullas” do not concentrate as well as the real kidney, especially in the inner medulla where only passive, diffusional transport occurs. Inasmuch as these models take into account the majority of contemporary ideas making up our global hypothesis about the functioning of this system, their failure to behave physiologically indicates that our understanding remains incomplete. The purpose of the present modelling study was to evaluate the implications of some recent measurements showing that permeabilities of NaCl (P _s ) and urea (P _u ) vary along the length of the descending thin limbs of Henle (Imaiet al., 1988,Am. J. Physiol. 254, F323–F328), rather than being constant throughout this segment as had been assumed earlier. It was hoped that these newly measured values might explain, by a passive, diffusional process, the net solute addition at the bend of Henle’s loop observed under some circumstances and heretofore attributed (though without any supporting experimental evidence) to active transport into the descending limb. The results of the present study show that whereas incorporation of the new values forP _s andP _u in the descending limbs of short nephrons does indeed improve the concentrating power of the model, these new values are nonetheless not sufficient to allow the model to build an osmolarity gradient that increases all the way through the inner medulla. This failing, which is common to virtually all modelling studies to date using measured values from rat kidneys, probably points to a key role for preferential exchange supposed by some to exist among certain tubule segments within vascular bundles in species whose kidneys have the highest concentrating power.     

A mathematical model of C4 photosynthesis: The mechanism of concentrating CO2 in NADP-malic enzyme type species

A computer model comprising light reactions in PS II and PS I, electron-proton transport reactions in mesophyll and bundle sheath chloroplasts, all enzymatic reactions and most of the known regulatory functions of NADP-ME type C₄ photosynthesis has been developed as a system of differential budget equations for intermediate compounds. Rate-equations were designed on principles of multisubstrate-multiproduct enzyme kinetics. Some of the 275 constants needed (ΔG₀′ and K _m values) were available from literature and others (V _m) were estimated from reported rates and pool sizes. The model provided good simulations for rates of photosynthesis and pool sizes of intermediates under varying light, CO₂ and O₂. A basic novelty of the model is coupling of NADPH production via NADP-ME with ATP production and regulation of the C₃ cycle in bundle sheath chloroplasts. The functional range of the ATP/NADPH ratio in bundle sheath chloroplasts extends from 1.5 to 2.1, being energetically most efficient around 2. In the presence of such stoichiometry, the CO₂ concentrating function can be explained on the basis of two processes: (a) extra ATP consumption for starch and protein synthesis in bundle sheath leads to a faster NADPH and CO₂ import compared with CO₂ fixation in bundle sheath, and (b) the residual photorespiratory activity consumes RuBP by oxygenation, NADPH and ATP and causes the imported CO₂ to accumulate in bundle sheath cells. As a wider application, the model may be used for predicting results of genetic engineering of plants. This revised version was published online in June 2006 with corrections to the Cover Date.     

A mathematical model of the mechanism of vertebrate somitic segmentation.

A mathematical model for the mechanism of periodic pattern formation in the process of somitogenesis is proposed. It is assumed that the metameric arrangement first appears before somite formation at the stage of transition of mesodermal cells into a polarized state. The model is based on the assumption that besides the mechanism of contact cell polarization there exists a mechanism of polarization suppression due to excretion of some chemical substance by polarized cells. Periodicity appears as a result of interaction of a kinematic wave of somitogenic cell determination with the cell cycles of mesodermal cells.     

On a simplified model of the renal medulla.

A set of equations that approximate the central core model of the human kidney are solved. It is noted that the solution indicates a limit to the increase in solute concentration in the descending limb of Henle for a given metabolic pump term. Comparison is made with the results of an alternative theory, and numerical results are presented for the increase in solute concentration in the descending limb of Henle.     

Endothelin B receptor antagonism in the rat renal medulla reduces urine flow rate and sodium excretion

It is well established that activation of endothelin B (ETB) receptor induces natriuresis and diuresis and thus reduces blood pressure. However, the site of action of ETB receptor is debatable. The present study was undertaken to address the role of renal medullary ETB receptor in renal excretory function. In volume-expanded Sprague-Dawley rats, infusion of the ETB antagonist A192621 at 0.5 mg/kg/hr to the renal medulla induced an immediate and significant reduction of urine flow rate that was 87.5% +/- 7.1%, 68% +/- 20%, and 58.3% +/- 15.5% of the control value at 10, 30, and 60 mins, respectively (n=5, P < 0.05 at each time point). Following intramedullary infusion of A192621, urinary sodium excretion remained unchanged during the first 20 mins but started to decline thereafter with a maximal effect at 60 mins. Changes in urinary excretion of potassium and chloride followed the same pattern of changes as for urinary sodium. In contrast, urinary osmolality gradually and significantly increased (control: 419 +/- 66; A192621 at 60 mins: 637 +/- 204 mOsm/kg H2O, P < 0.05). Over a 60-min period of intramedullary infusion of A192621, none of the hemodynamic parameters examined, including mean arterial pressure, renal blood flow, or medullary blood flow, were affected. These data suggest that: (i) intramedullary blockade of ETB receptor produces antidiuresis and antinatriuresis independently of hemodynamic changes, and (ii) the immediate response to intramedullary blockade of ETB receptor is the reduction of water excretion followed by the reduction of sodium excretion.     

Role of structural organization in the urine concentrating mechanism of an avian kidney

The organization of tubules and blood vessels in the quail medullary cone is highly structured. This structural organization may result in preferential interactions among tubules and vessels, interactions that may enhance urine concentrating capability. In this study, we formulate a model framework for the urine concentrating mechanism of the quail kidney. The model simulates preferential interactions among renal tubules by representing two concentric cores and by specifying the fractions of tubules assigned to each of the concentric cores. The model equations are based on standard expressions for transmural transport and on solute and water conservation. Model results suggest that the preferential interactions among tubules enhance the urine concentration capacity of short medullary cones by reducing the diluting effect of the descending limbs on the region of the interstitium where the collecting ducts are located; however, the effects on longer cones are unclear.     

Phospholipid metabolism in the rat renal inner medulla

In view of the importance of phospholipids as a source of precursor fatty acids for the high prostaglandin synthesis in the renal inner medulla, we studied pathways of phospholipid esterification and degradation in the rat inner medulla. De novo acylation of [14C]arachidonate occurred predominantly in position 2 of phosphatidylcholine in the microsomal fraction. This newly esterified [14C]arachidonate was accessible to deacylation by a microsomal phospholipase A2 (EC 3.1.1.4) with alkaline optimum which was Ca2+-dependent and resistant to 0.1% deoxycholate. No phospholipase A1 (EC 3.1.1.32) activity against endogenous labeled phosphatidylcholine could be demonstrated in the microsomal fraction. When exogenous phosphatidylcholine labeled at position 2 was deacylated by renomedullary homogenates, labeled free fatty acid but no labeled lysophosphatidylcholine was recovered in the reaction products. This could be attributed to further degradation of generated lysophosphatidylcholine by a cytosolic lysophospholipase (EC 3.1.1.5). Sodium deoxycholate at a concentration of 0.1% or higher inhibited the lysophospholipase and allowed the demonstration of both A2 and A1 alkaline phospholipase activities in the homogenate. The major in vitro pathway of lysophosphatidylcholine disposition is further degradation by a cytosolic lysophospholipase, while reutilization for phosphatidylcholine synthesis through the action of a predominantly microsomal acyltransferase appears to be a minor pathway. In the presence of several acyl-CoAs, reutilization of lysophosphatidylcholine is significantly increased by an acyl-CoA:lysophosphatidylcholine acyltransferase (EC 2.3.1.23) but there is no preferential transfer of arachidonyl-CoA compared to other acyl-CoAs.     

A mathematical model of hematopoiesis: II. Cyclical neutropenia

Cyclical neutropenia is a dynamical disease of the hematopoietic system marked by an oscillation in circulating leukocyte (e.g. neutrophil) numbers to near zero levels and then back to normal. This oscillation is also mirrored in the platelets and reticulocytes which oscillate with the same period. Cyclical neutropenia has an animal counterpart in the grey collie. Using the mathematical model of the hematopoietic system of Colijn and Mackey [A mathematical model of hematopoiesis: I. Periodic chronic myelogenous leukemia. Companion paper to the present paper.] we have determined what parameters are necessary to mimic laboratory and clinical data on untreated grey collies and humans, and also what changes in these parameters are necessary to fit data during treatment with granulocyte colony stimulating factor (G-CSF). Compared to the normal steady-state values, we found that the major parameter changes that mimic untreated cyclical neutropenia correspond to a decreased amplification (increased apoptosis) within the proliferating neutrophil precursor compartment, and a decrease in the maximal rate of re-entry into the proliferative phase of the stem cell compartment. For the data obtained during G-CSF treatment, good fits were obtained only when parameters were altered that would imply that G-CSF led to higher amplification (lower rate of apoptosis) in the proliferating neutrophil precursors, and a elevated rate of differentiation into the neutrophil line.     

An efficient numerical method for distributed-loop models of the urine concentrating mechanism

In this study we describe an efficient numerical method, based on the semi-Lagrangian (SL) semi-implicit (SI) method and Newton's method, for obtaining steady-state (SS) solutions of equations arising in distributed-loop models of the urine concentrating mechanism. Dynamic formulations of these models contain large systems of coupled hyperbolic partial differential equations (PDEs). The SL method advances the solutions of these PDEs in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The SI approach controls stiffness arising from transtubular transport terms by averaging these terms in time along flow trajectories. An approximate SS solution of a dynamic formulation obtained via the SLSI method can be used as an initial guess for a Newton-type solver, which rapidly converges to a highly accurate numerical approximation to the solution of the ordinary differential equations that arise in the corresponding SS model formulation. In general, it is difficult to specify a priori for a Newton-type solver an initial guess that falls within the radius of convergence; however, the initial guess generated by solving the dynamic formulation via the SLSI method can be made sufficiently close to the SS solution to avoid numerical instability. The combination of the SLSI method and the Newton-type solver generates stable and accurate solutions with substantially reduced computation times, when compared to previously applied dynamic methods.     

Some mathematical aspects of the diagnostic process. II: A mathematical model of electrocardiographic diagnosis

In a previous paper (Bartholomay, 1971), a general mathematical model of the medical diagnostic process was described. The present paper amounts to a relization of that process in terms of conventional 12-lead electrocardiographic diagnosis as enunciated by Dr. Harold D. Levine (1966) in the course of a collaborative study by Dr. Levine and the present author at the Peter Bent Brigham Hospital of the Harvard Medical School between 1963 and 1966. The main details of the cognitive component of that model are described in detail here. The model has been programmed onto a computer system consisting of an analog-digital converter and general purpose digital computer and amounts to a simulation of Dr. Levine’s electrocardiographic analysis procedure.     

Autoregulation of blood flow in renal medulla of the rat: no role for angiotensin II

Autoregulation of blood flow was assessed by a dual-slit technique in descending and ascending vasa recta of the exposed renal papillae of antidiuretic rats. There was complete autoregulation of blood flow in descending vasa recta. The lower limit of autoregulation was approximately 85 mmHg (1 mmHg = 133.3 Pa) and the upper limit was greater then 160 mmHg. Autoregulation in ascending vasa recta was also good. To test the role of angiotensin II in this autoregulation, the converting enzyme inhibitor captopril was infused. Captopril had no effect on autoregulation of blood flow in either descending or ascending vasa recta. We conclude that blood flow in vasa recta of renal medulla is efficiently autoregulated and that this autoregulation is independent of angiotensin II.