A Model of the Renal Cortex and Medulla*

Mathematical models of the renal cortex and medulla are developed. The glomerular filtration rate is treated as an input parameter. The differential equations for transfer of water and solutes are developed under the assumption that the primary driving forces are the osmotic effect of the plasma proteins and the active transport of one solute across the walls of the nephric tubules. The system of simultaneous ordinary differential equations obtained for the stationary state for constant inputs in the medulla present a multiple-point boundary-value problem of some complexity. In the course of developing the model a number of problems were unearthed. First, it became apparent that the commonly accepted countercurrent exchange model for the medullary capillaries is not supported by anatomical studies and that a distributed capillary bed model might give a truer picture. Second, it became obvious that the information available on the sodium pump is as yet insufficient to let us decide unambiguously whether the pump is reversible or irreversible. * This work was supported in part by grant GM 892 from the National Institutes of Health. Mathematical Biosciences 1, 221261 (1967) 228 JoHs .4. j~4CQUEZ, BRICE CRRNAHAN, AND PETER ABBRECHT Qur present knowledge of renal structure and function is well reviewed in the text by Bitts [l?‘J. I[n brief, the currently accepted picture of the way rhe kidney functions is as follows. Approximately 80% of the glomerular filtrate is reabsorbed in the proximal tubules in an ess‘entially isosmotic manner. Since practically all of the proximal tubules are in the corte:rt, this may be viewed as a cortical function. Concentration of the urine occurs in the medulla. Two hypotheses dominate our present view of m:dullary function. One is that the WenZe loop acts as a countercurrent multiplier by actively pumping sodium ions out of the ascending limb. The other and independent hypothesis is that the medullary capillaries are also arranged in a hairpin configuration, like the loops of PI :nie, and act as countercurrent exchangers. The evidence supporting the first of these hypotheses is strong; that supporting the second is weak, and in fact recent anatomical studies suggest that the medullary capillaries are got arranged in the hairpin configuration [la, 181. It would be most useful, in our thinking and speculation about renal function, to have available a mathematical model of the whole kidney that incorporates the major features of renal structure and the major hypotheses of renal function. Many models of particular features of renal structure and function have been proposed [14, 12, 13, 16, 23, 241. In all of these only ;a portion of the kidney, for example the proximal tubules or Wenle’s loop, is modeled. The model of the medulla of Pinter and ‘ihohet [16] appears to be unrealistic to us because it neglects water movtcment:s. A complete model with both cortex and medulla is needed. St:;!1 a mctiel should incorporate only the major features of renal structure ;itnd should be capable of showing the typical response of the mammalian kidlney in water deprivation, water diuresis, and osmotic diuresis. In this paper we attempt to formulate such a model and we discuss in some detail the reasons for including various features in the model. FORil~ULATION OF MODEL: ANI~TOMIICAL AND PHYSIOLOGICAL CONSIDERATIONS Although structurally there is a small transition zone between the cortex and medulia, for the most part the cortex and medulla are separa.te and distinctively different in structure; we neglect the transition zone and assume that the cortex and medulla are separated by a sharp boundA MODEL OF THr?: RENAL CORTEX AND MEDULLA 229 ary. Although Henle’s loop as commonly described in anatomical texts includes portions of both proximal and distal tubules, it is more convenient to treat it as a distinct unit. Hence for the purposes aIf this paper we consider the nephric tubules to consist of three portions, proximal tubule, Henle’s loop, and distal tubule. We include the proximal and distal tubules of the juxtamedullary nephrons in the model of the cortex, and assume that the strictly cortical nephrons consist primarily of proximal and distal tubules with a small Henle’s section. The ccwtex The basic assumptions used to derive a model of the cortex follow. Assuqbtion 1. Flow in the tubules and capillaries is a lbulk displacement flow with uniform mixing in a direction perpendicular to the axes of these tubes. In effect we assume a uniform fluid velocity at any cross section in one of the tubes and an average concentration for a solute at any fixed distance along a tubule or capillary. This is a reasonable approximation because the capillaries and nephrons have very small diameter-to-length ratios. This assumption has been used by Bergmann and Dikstein [l], Burgen [2], Dole [3], Kelman [12], Kuhn and Ramel [13], Wesson [24], and Pinter and Shohet [lG]. We further assume that volume changes are entirely attributable to the water movements. Assamfition 2. We assume uniform ;;,ld rapid mixing of the interstitial fluid of the cortex. In effect we assume that there are no significant concentration gmdients in the interstitial fluid and that we can use a mean concentration of any solute in the interstitial fluid in our equations. The fact that there is no structural orientation in the cortex favors this assumption. The close juxtaposition of different elements in the cortex also provides support for this assumption since a small portion of interstitial fluid may contact portions of proximal tubules, distal tubules, and the capillary network. Furthermore the capillary net from one glomerulus anastomoses freely with those from neighboring nephrons. Physiological evidence in support of this assumption is provided by observations mentioned by Pitts ([17], page 102). If inulin and labeled sulfate are injected into the renal artery, sulfate appears in the urine before inulin does. Inulin does not cross the wall of the nephron but sulfate does, hence this observation provides some evidence for downMathmntical Biosciemes I, 227 281 (1967) 230 JOHN A. JACQ’UEZ, BRICE CARNAHAN, AND PETEH ABBKECHT stream transfer of sulfate through the interstitial fluid. Wesson [24] used this assumption in his treatment of urea excretion. Assamy!&n 3. We neglect the thin segment of Henle’s loop for the cortical nephrons but allow for a short thick segment. These are reasonable assumptions because the cortical nephrons have rather small loops of Henle and many have no thin segments. The thin segments and distinctive hairpin arrangement are a unique structural feature of the medulla and come primarily from the juxtamedullary glomeruli. Assun.q!hw 4. The proximal and distal tubules of the juxtamedullary glomeruii are included in the cortex. This seems reasonable since the medulla consists almost entirely of Henle’s loops, collecting tubules, and blood vessels. We include the capillary networks of the proximal and distal tubules of the juxtamedullary nephrons with the cortical capillary beds and treat the medullary capillary networks as separate and restricted to the medulla. Assumption 5. The functioning of the cortex should be represented as a summation of effects contributed by each of the cortical nephrons and capillaries. We replace the approximately 1.05 106 cortical nephrons $9 ii FIG. ‘1. Schematic of model of cortex: 0. interstitial fluid; 1, cortical capillary bed; 2, cortical nephrons; 4, ‘7. proximal and distal tubules of juxtamedullary glomeruli. [l’?] by one lumped tubule of length equal to the average length of these tubules and with total surface arca and cross-sectional area per unit length equal. to that of the sum for these tubules. The cortical capillary beds are similarly lumped. This assumption has been used in many previous treatments [3, 23, 241. For the stationary state @de in/m), we assume Mathematical Biosciences 1, 227 261 (1967) A MODEL OF THE RENAL CORTEX AND MEDULLA 231 that the cross-sectional area and surface area per unit length are constant along the capillary bed and for the proximal and distal tubules separately. The model of the cortex that has just been outlined is summarized schematically in Fig. 1. Because of the unique structural arrangements in the medulla some of the assumptions made for the cortex do not hold. The collecting ducts and Henle’s loops run parallel to one another in the medulla. We refer to the direction parallel to these tubes as the longitudinal axis of the medulla. The basic assumptions are again presented. Asswwption I. Although the loops of Henle are not all of the same length, we assume that they are and use one lumped model for all of the descending limbs and another for all of the ascending limbs. We use the commonly accepted hypothesis that active transport is restricted to the ascending limb; however, the model can be modified to include active transport in the descending limb of Henle’s loop if necessary. Assumption 2. The collecting tubules are assumed to begin at the medullary boundary and to be of the same length as the limbs of the loop of Henle. Asswnption 3. As for the cortical model, we assume bulk displacement flow (plug flow) in each tubule and uniform concentration of any solute inside a tube throughout a cross section at any distance along the length of each tubule. AssumPtiotr 4. The interstitial fluid of the medulla is assumed uniformly mixed only in a direction perpendicu!ar to the longitudinal axis. The interstitial fluid may be viewed as contained in another impermeable tubule running in the direction of the longitudinal axis, a tubule that contains all the other tubules and capillaries and with which all the other tubules interchange solutes and water. So as not to prejudge the issue, we include the possibility of a longitudinal flow of interstitial fluid in the model. Assuvq%ion 5. There is some doubt about the structure of the medullary capillary beds. At least two models are possible and we will examine both. Mat2ematicaE Biosciemes 1, 227-261 (1965) 232 JOI-IEU’ A. JACQUEZ, BRICE CAHNAHAK, AND PETE3 ABBRECHT The picture commonly presented in the textbooks [17] is that the blood vessels form hairpin ,100~s similar in shape to Henle’s loop and act as countercurrent exchangers. M’e refer to this as the cozcntercwent exchange plodel of the medulla. However, there is now anatomical evidence for L qui+ different arrangement. Plakke and Pfeiffer [18] report that, 0 , . .the va~a recta arise in leashes from the efferent arterioles of the juxtamedullary glomeruli and descend, many as unbranched vessels, in parallel bundles to break up into capillary plexuses at different levels of the meldulla.” They further report that the capillary plexuses are most dense in the outer zone of the medulla, adjacent to the cortex, and that this zonation appears to be correlated with the ability to form a concentrated urine. In the model based on this picture we assume that the vasa recta are nonexchanging conduits and lump the capillary plexuses as a set of capillaries that run perpendicular to the longitudinal axis of the medulla and are distributed at different levels with mosr lying in the outer zone. Because we have assumed complete mixing in the interstitial fluid at any distance down the longitudinal axis, capillaries at one level are all bathed by interstitial fluid of constant composition. We call this model the distributed capillary bed model of the medulla. Schematic diagrams of these two models are shown in Figs. 2 and 3. In the model we consider only three solutes. One, M, is actively transported across the walls of the tubules of the nephrons; the other two, s and u’, are not actively transported. One of these, x, may be thought of as urea; the other, ZU, will be needed when we consider osmotic diuresis. Although zt may be thought of as sodium, we treat ZI as a nonionic solute. Including the ionic nature of the transported solute would make this already complicated model even more difficult to handle and we suspect that the essential feature required is the active transport of an osmotically active solute. In any case, we believe that the model can be expanded to include this feature later. Thus only two driving forces that can lead to sohite and solvent movement are included in the model, active transport of solute tt between the nephric tubules and interstitial fluid and the osmotic effect of the proteins in the capillary plasma. The interstitial fluid and gIomerular filtrate are treated as free of protein. Except for the active transport of u, the movement of soltite and solvent is assumed to follow Fick’s law. Two effects are neglected in this first model, the Donnan effect and the interaction between solure and solvent movement described by the cross coefficients in the phenomenological laws for diffusion ias postulated .Vf u!hemuticul l?iosciencrs I, 227 261 (1967) A MODEL OF THE RENAL CORTEX AND MEDULLA 233 in irreversible thermodynamics, Hence we do not include a reflection coefficient fcr each solute. The interaction terms might be important for the more permeable solutes. \Qe have not tried to model the process of glomerular filtration; the glomerular filtration rates are treated as input parameters of the model. Possible effects of pressure gradients on solute and solvent movement across the tubular walls have not been included in ,this model.