Proc. Nat. Acad. Sci. USA Vol. 73, No. 1, pp. 252-256, January 1976 Physiology Model of solute and water movement in the kidney (concentration of /renal blood flow/glomerular //hydrostatic pressure) JOHN L. STEPHENSON*, RAYMOND MEJIA*, AND R. P. TEWARSONt * Section on Theoretical Biophysics, National Heart and Lung Institute and Mathematical Research Branch, National Institute of Arthritis, Metabolism, and Digestive Diseases, National Institutes of Health, Bethesda, Maryland 20014; and t Applied Mathematics and Statistics Department, State University of New York, Stony Brook, N.Y. 11790 Communicated by Robert W. Berliner, October 2, 1975

ABSTRACT Finite difference equations describing salt Differential Equations. The fundamental mass balance and water movement in a model of the mammalian kidney equations for solute in a system of flow tubes are (2), have been solved numerically by an extension of the Newton- Raphson method used for the medullary counterflow system. -aFik/ax + - Jik = (AiCi) /8t, 1I] The method permits both steady-state and transient solu- Aisik tions. It has been possible to simulate behavior of the whole kidney as a function of hydrostatic pressures in renal artery, where Fik is the axial flow of the kth solute in the ith flow vein, and pelvis; protein and other solute concentrations in tube, Ai is the cross-sectional area of the tube, Sik is the aver- arterial blood; and phenomenological e uations describing age net rate at which material is being produced or de- transport of solute and water across neron and capillary stroyed by chemical or physical reaction, Jik is net outward walls. With the model it has been possible to compute con- transmural flux, Cik is concentration, x is normalized axial centrations, flows, and hydrostatic pressures in the various tube, and t is time. The corresponding nephron segments and in cortical and medullary capillaries distance along the and interstitium. In a general way, calculations on the model equations for volume flow are: have met intuitive expectations. In addition, they have re- emphasized the critical dependence of renal function on the -dFi, / ax -Ji, = aA /at; [21 hydraulic and solute permeabilities of glomerular, postglom- erular, and medullary capillaries. These studies provide addi- here F1V is the axial volume flow and J1v is the transmural tional support for our thesis that the functional unit of the volume flux. Pressure drop along the tubes is given by kidney is not the single nephron, but a nephrovascular unit consisting of a group of and their tightly coupled aPilax = -RIFFi, [31 vasculature. where Pi is hydrostatic pressure and RjF is flow resistance. In a previous paper (1) we formulated finite difference For all tubes except the proximal convoluted tubules, equations for solute and water movement in the medullary transmural fluxes are given by counterflow system of the mammalian kidney and described a modified Newton-Raphson method for solving these equa- Jik = Jil (1 - Ok)(CIk + Cpk)/2 + hik(Cik - Cpk) + 3ik, [41 tions. In this paper we describe the extension of this method to give both steady-state and transient solutions for differ- ence equations describing water and solute movement in a model of the whole kidney. THE MODEL The model is illustrated in Fig. 1. Input data for the model are hydrostatic pressures in renal artery, renal vein, and renal pelvis; protein and other solute concentrations in en- tering arterial blood. Parameters of the model are hydrody- namic flow resistance of glomerular capillaries, postglomer- ular capillaries, and renal tubules; and the parameters of the phenomenological equations describing transmural transport of water and solutes. The cortical interstitium is assumed to be a single well-mixed compartment. The medullary inter- stitium is divided into five to twenty compartments. These are assumed to be well mixed in a plane perpendicular to the cortical-papillary axis, but to have (in each compart- ment) a concentration gradient in a direction parallel to the cortical-medullary axis. In this model each flow tube is di- vided into a number of segments, which exchange with cor- tical or medullary interstitium, except for glomerular capil- laries, which exchange with an initial segment of the proxi- mal convoluted tubule corresponding to Bowman's space. FIG. 1. Solute and water movement in the kidney. Water movement is indicated by white arrows and solute movement by Abbreviations: PT, proximal tubule; PG, postglomerular; CD, col- black arrows. DT, distal tubule; PT, proximal tubule; CD, collect- lecting duct; AHL, ascending Henle's limb; DHL, descending ing duct; AHL and DHL, ascending and descending Henle's limb, Henle's limb; TF/P, tubular flow/plasma. respectively; PG, postglomerular. 252 Downloaded by guest on September 24, 2021 Physiology: Stephenson et al. Proc. Nat. Acad. Sci. USA 73 (1976) 253 and space, JBk is transmural flux from Bowman's space to corti- Jat = - Cik)Oik + Pi -Pp [5J cal interstitium, Fp, (0) is volume flow from Bowman's hi,[12kRT(cpk space into the first segment of the proximal tubule. For the medullary interstitium we have the equations: where a& is the Staverman reflection coefficient of the wall of the ith tube for the kth solute, hk is its passive permeabili- ty for the kth solute, hi, is its hydraulic permeability coeffi- -UFr /ax + Alslk + 2;iWiJik = (Ajck)/at, [14] cient, IJA is the metabolically driven transport out of the ith and tube, R is the gas constant, T is the absolute temperature, -dFI1/d9x + 2i wiJiaAM /at. [15] and subscript p refers to interstitium. The metabolically driven transport is assumed to obey ap- In all tubes and in the interstitium, axial solute and axial proximate Michaelis-Menten kinetics, i.e., volume flows are related by the equation

Fik = Ficik -Dik&ik ax Jlk = aik / [1 + bik/Ck], [6] dI [16] in the where as is the maximum rate of transport and b& is the Mi- where D& is the diffusion coefficient for the kth solute chaelis constant. All of the membrane parameters may be ith flow tube (or interstitium). in paper we have made functions of distance along the tube, but are assumed not to In the calculations described this the vol- depend on concentrations, flows, or pressures. certain simplifying assumptions: We have assumed In modeling proximal tubule transport, it is assumed that ume of the kidney and the cross-sectional area of the various are no transmural transport is isotonic, i.e., tubes to be constant; we have also assumed that there chemical sources, that diffusion is negligible in tubes and in- terstitium; and that axial bulk flow in the interstitium is also JIM = JiuciM, [7] negligible. Under these assumptions some of the above equa- tions simplify considerably; e.g. Eq. 1 becomes where CiM is total osmolality of tubular fluid. In our calcula- tions on the model we have assumed various empirical laws for Ji, in the proximal tubule; in particular -aFici)l ax J-ik = AiaCikl/n [17] To solve the above systems of equations they are replaced J= A + BFi, [8] with a system of finite difference equations as described or previously, the only exception being in the difference quo- Ji=a + AP, [9] tient corresponding to acac/ot, for which we use a quotient spatially averaged and backward in time. Thus, the differ- where A, B, a and are arbitrary constants. ence equation corresponding to [17] is: Eqs. 7, 8, and 9 are not intended as a substitute for a de-

tailed model of proximal tubule transport. In the develop- + Fin j + 1) Cnk(j + 1) - FiVn(j)cik () + [Jn (j 1) ment of the model they serve a "dummy" role for which a more sophisticated model of tubule solute and water trans- + Jik (J)]/(2N) + Ai[Cikn(j + 1) + Cik (j) can be substituted. port eventually + (j)]/(2NAt,) 0, [18] It will be noted that implicit in the above equations is the -Cikn' 1)-Cik assumption that at a given position along a capillary or tu- where N is the number of segments into which the ith flow and volume exchange are radially bule transmural solute tube is divided, and the superscript n refers to the nth time symmetric. step. Thus, if we designate the vector of concentrations and The equations for the cortical interstitium are pressures for the nth time step as yan, and the system of of equations by ), we seek a solution 'ny, of the system equa- XVICIk)dat 2iWi f'Jik(x)dx [10] tions

( n fyf = and ) 0, [19] / It = Y2iwi (x) dx, [11] vI fJi, where -yn- is known either as a set of initial values or from a previous time iteration. In the steady state ,n = 'y-1, So where VI is the volume of the cortical interstitium, clk is the we seek a solution of the system of equations interstitial concentration of the kth solute, w1 a weighting factor for the ith tube (w, = 0 for tubes that do not exchange #eY,, )=0, [20] with the interstitium). For Bowman's space we have the equations where by y' we indicate the steady state vector of concen- trations and pressures. of 19 or 20 we as YXVBCBk) /a = JG(X)dx JBk- CBkFpV (0), [12] To solve the system Eqs. proceed exactly for the steady-state equations of the medulla. Thus, to solve and the transient Eq. 19 we make an initial estimate of -yo of lyn. If the norm of the vector 0(-yoe, ay-l) is less than some [13] aVB/at = J(X)dxx JB- FPU (O), preset tolerance we are through. If not, we improve our esti- mate of ey by solving the system of linear equations where VB is the volume of Bowman's space, CBk is the con- centration of the kth solute in Bowman's space, JGk is transmural flux from glomerular capillary to Bowman's (_onn)- 0G Ayon = o0 [21] Downloaded by guest on September 24, 2021 254 Physiology: Stephenson et al. Proc. Nat. Acad. Sci. USA 73 (1976)

80

60 -4

z 0 , < ulz - UI.2 .; _3 x z s U-22 40 L _ COtn -8 - E

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20 -12 1

II

A 0 2 .4 .6 .8 PT REABSORPTION PT REABSORPTION FIG. 2. Effect of fractional proximal tubule reabsorption on FIG. 3. Effect of proximal tubule reabsorption on cortical in- glomerular plasma flow (0), glomerular filtration rate (A), urine terstitial pressure (A), medullary interstitial pressure at junction volume flow (a), and urine concentration (0). of cortex and medulla (0), and papillary interstitial pressure (0). where G is the Jacobian matrix 0)/a3y, evaluated at -fe. Keller for two point boundary value problems (3). In a general way it can be said that the iterative sQheme In practice, solutions are built up from known solutions ei- converges satisfactorily if a solution of the difference equa- ther by following the transient response after variation of a tions exists and if the initial estimate is close enough to the boundary condition or parameter, or by computing a new solution. How closely solutions of the difference equations steady state solution vector directly after a small change in a approximate solutions of the parent differential equations is boundary condition or parameter. Both methods were used a much more difficult question. Both problems have been in calculating the results that follow. treated by us in detail elsewhere for the medullar equation (Stephenson, Tewarson, and Mejia, unpublished) and by RESULTS Representative values of flow, concentration, and pressure Table 1. Representative values are shown in Table 1, for the normalized parameter values given in Table 2. The parameters were adjusted so that fil- Hydro- tration pressure equilibrium was attained in the efferent Volume Oncotic static outflow of the glomerular capillaries, the flow, Salt conc, pressure, pressure, was approximately one third, the tubular flow/plasma nl/min TF/P mm Hg mm Hg (TF/P) osmolality was about 4 in the final urine, and the hy- Aff 65.19* 1.000 22.61 77.35 drostatic pressure in cortical and medullary interstitium was Ga 65.19 1.000 22.61 42.84 everywhere less than that in tubules and capillaries. In these Ge 46.68 1.000 31.58 41.03 calculations fractional proximal tubule (PT) reabsorption Eff 46.68 1.000 31.58 23.73 was fixed at 0.5 and PT reabsorbate was assumed to be iso- PGCa 32.89 1.000 31.58 23.73 tonic. Water reabsorption from distal nephron makes corti- PGCe 40.84 0.992 25.43 5.95 cal interstitium, postglomerular (PG) capillaries, and ascend- DVRt 13.79 1.000 31.58 23.73 ing vasa recta (which are allowed to exchange with both AVR§ 24.17 0.990 18.01 5.95 medullary and cortical interstitium) slightly hypotonic. This CI 0.980 - -9.67 is an obviously correct result, which to-our knowledge has BC 18.52t 1.000 9.54 never been considered in cortical function. Since the hypoto- PT¶ 18.52 1.000 9.54 DHL 9.26 1.000 9.30 Table 2. Normalized parameters for representative values AHL 2.17 4.277 DN 2.17 0.346 8.65 R CD 0.76 0.991 8.57 hv (X 104) a hs Urine 0.18 4.274 8.54 MI(0) 1.018 6.61 G 1000 2 RA = 0.003 MI(L) 4.288 9.51 PGC 300 28.5 0 1 RE = 0.0021 DVR 100 28.5 0 100 Abbreviations: Aff and Eff, afferent and efferent arteriole of CAVR 200 4.9 0 20 glomerular capillaries, respectively; G. and Ge, afferent and AVR 100 24.5 0 100 efferent end of glomerulus; PGCG and PGCe, afferent and efferent BC 0 0 1 0 end of postglomerular capillaries; DVR and AVR, descending and PT ascending vasa recta; CI, cortical interstitium; BC, Bowman's 1 a 0.5 capsule; DN, distal cortical nephron; MI(0), interstitium value at DHL 20 5.7 1 0 corticomedullary junction; MI(L), interstitium value at papilla. 1 AHL 0 6 0 0 a=0.27,b= mm Hg = 133 Pa. 0.1 * Equals glomerular plasma flow. DN 0.2 4 1 0 t Equals glomerular filtration rate. CD 0.2 5 1 0 I At junction with afferent glomerular arteriole. § At junction with venous return. For abbreviations see Table 1 and equations. Additionally, G. ¶ Flow, concentration and pressure at proximal end of each nephron glomerulus; CAVR, cortical ascending vasa recta; RA and RE, af- segment. ferent and efferent flow resistance from glomerulus. Downloaded by guest on September 24, 2021 Physiology: Stephenson et al. Proc. Nat. Acad. Sci. USA 73 (1976) 255

z 0 z 0 0 0 -I c _L Urc. LW c A, wU.E z LL EE z - UA -JS z -J L.) 0 z 0) 0(,) 0 C.

0 .1 2 27 AHL SOURCE PA(mm Hg) FIG. 4. Effect of fractional solute transport out of ascending FIG. 6. Effect of arterial pressure on glomerular plasma flow limb of Henle on glomerular filtration rate (0), urine flow (A), and (A), glomerular filtration rate (0), urine flow (a), and urine con- urine concentration (0). centration (-). PT absorption was 0.5, and maximum rate in AHL source was 0.27. nicity corresponds to an osmotic force of 3 to 6 meq across PT wall, it is clearly of some functional significance. crease in hydrostatic pressure in Bowman's capsule. Concen- Fig. 2 shows the effect of varying fractional reabsorption tration and volume of urine flow vary as expected although in the proximal tubule while other parameters are held con- the change is less than if PT delivery to DHL is fixed as it is stant. As can be seen, increasing PT reabsorption increased in models of the medulla alone. glomerular plasma flow very slightly, glomerular filtration Fig. 5 shows the effects on interstitial pressures of varying rate somewhat more, and decreased urine flow and in- net solute transport out of AHL. As might be expected, as creased urine concentration. The effect on urine concentra- the AHL source increases, the medullary pressure increases tion and flow increases very markedly as PT reabsorption at both corticomedullary junction and papilla. The cortical rises above 0.5. pressure rises slightly, reflecting slightly greater net uptake In Fig. 3 is shown the effect on interstitial hydrostatic of fluid by PG capillaries. pressure of increasing fractional PT reabsorption. As one The effects of varying arterial pressure are illustrated in might expect, the cortical interstitial pressure increases. This Fig. 6. As must happen in the totally unregulated kidney, reflects the fact that an increased volume of fluid is being glomerular plasma flow and glomerular filtration rate both pumped from PT into cortical interstitium whose volume is increase sharply. So does urine flow. Concentration drops fixed. The hydrostatic pressure must increase until the net because the fixed AHL source is overwhelmed and fraction- driving force is sufficient to move an exactly balancing vol- al reabsorption falls. ume of water per unit time into the PG capillaries. In short, Fig. 7 demonstrates the effects of concurrently varying the interstitial hydrostatic pressure adjusts so as to maintain the hydraulic permeabilities of PG capillaries and of the . The same phenomenon is observed in the me- vasa recta. As would be anticipated, the driving force re- dulla. Here, because of the increased corticomedullary con- quired to move water increases as the hydraulic permeabili- centration gradient, the volume flux of water into the outer ty hk decreases. Hence, the interstitial pressure increases. medullary interstitium from descending Henle's limb For h0 less than some critical value, interstitial pressure rises (DHL) and collecting duct (CD) increases as PT reabsorp- above both capillary and tubular pressures and a compliant tion increases, but the volume flux into the inner medulla rather than a rigid system would shut down in the absence decreases. of some compensatory mechanism. In Fig. 4 we examine the effects of varying the ascending As was pointed out above, the methods used to obtain Henle's limb (AHL) solute source. As net solute transport out steady-state solutions extend readily to time-dependent of AHL increases, GFR increases slightly, reflecting a de-

5 7

D01 cn E Lu E X2 -7

-14 F

-21 200 400 600 Boo 1000 CAPILLARY hv AHL SOURCE FIG. 7. Effect on interstitial pressures of concurrently varying FIG. 5. Effect of ascending limb solute transport on cortical hydraulic permeabilities of postglomerular and medullary capillar- interstitial pressure (A), interstitial pressure at corticomedullary ies; (0) cortical interstitial pressure, (0) interstitial pressure at border (-), and papillary interstitial pressure (0). corticomedullary junction, (A) papillary interstial pressure. Downloaded by guest on September 24, 2021 256 Physiology: Stephenson et al. Proc. Nat. Acad. Sci. USA 73 (1976) i10 tering the capillaries. In our model this balance is attained primarily by adjustment of the interstitial hydrostatic pres- sure to give the required driving force across the capillary walls. The magnitude of this required force varies inversely with the hydraulic permeability-area product of the capil- laries. In the limit, as this product goes to infinity, the re- EU. quired force goes to zero and the sum of interstitial hydro- z Z-. static and oncotic pressures will approach that in the capil- 30 laries. Virtually, capillaries and interstitium have been z merged into a single functional space. We called this space 0 u the "central core" in the medullary models and will extend the terminology to the whole kidney. In the limit, transport of water and solutes from the neph- 10 rons to the central core depends only on the transport pa- TIME rameters of the nephrons, but the closeness of approach to FIG. 8. Change in urine flow (------) and concentration this limit is critically dependent on the permeabilities of the (-_-) on turning off and then on the ascending limb solute postglomerular capillaries and vasa recta. Unless these ex- source. One unit of time corresponds to approximately 10 seconds ceed some minimum value the kidney could not function at of real time. all. In kidney function the tubular and core space play a com- equations for the model and permit us to study its transient plementary role in which any decrement of volume flow in behavior. Fig. 8 shows the effect on urine concentration and the tubular space must be balanced by an increment of vol- volume flow of abruptly turning the AHL source off and on. ume flow in the core space. It is via the core that many of One unit of time on our scale corresponds to about 10 sec. It the intricate feedback and coupling mechanisms of the kid- is clear that in real time the response of the model is quite ney operate. Thus, it is by core coupling that solute transport rapid-particularly the volume flow. We are only starting to out of AHL drives water transport out of DHL and CD. It is look at the time-dependent behavior of the model in a sys- also via the core that urea transport out of CD can drive salt tematic way, but the computations we have done so far transport out of AHL (4-6). In the whole kidney model it is suggest that the time of transition from one physiological clear that transport out of distal nephron affects the osmolal- state to another depends on the rate of change of the param- ity of the cortical core and consequently transport out of PT. eter involved, rather than the intrinsic response time of the There are numerous directions in which this model of the model. whole kidney can be extended, but even in its present skele- DISCUSSION tal form it has given us considerable insight into the details of the coupled behavior of cortex and medulla and of the in- From the detailed calculations on this model and our earlier tricate fluid circuit of the whole kidney, and has given a analysis of medullary models, we are able to make some synthesis of the transport characteristics of nephrovascular generalizations on over-all kidney function. components into over-all function. Following glomerular filtration the bulk of the filtered fluid is returned to the systemic circulation by osmotic ex- 1. Stephenson, J. L., Tewarson, R. P. & Mejia, R. (1974) "Quanti- traction secondary to active NaCl transport. In its return to tative analysis of mass and energy balance in non-ideal models of the renal counterflow system," Proc. Nat. Acad. Sci. USA, the systemic circulation this fluid must traverse two mem- 71, 1618-1622. brane systems. The first is that separating tubular lumen 2. Stephenson, J. L. (1973) "Concentrating engines and the kid- from interstitial space; the second is that separating intersti- ney: I Central core model of the ," Biophysical J. tium from capillary lumen. 13,512-545. The forces moving fluid across the tubular membranes are 3. Keller, H. B. (1974) "Accurate difference methods for nonlin- a combination of osmotic and hydrostatic pressures, with the ear two-point boundary value problems," SIAM J. Numer. osmotic forces certainly playing the predominant role. The Anal. 11, 305-320. forces moving the fluid across capillary walls are a combina- 4. Stephenson, J. L. (1972) "Concentration of urine in a central tion of and oncotic. In our the reflection core model of the renal counterflow system," Kidney Int. 2, hydrostatic view, 85-94. coefficient for small solutes approaches zero for the capillar- 5. Stephenson, J. L. (1973) "Concentrating engines and the kid- ies, and solute transport across capillary walls is primarily by ney. II Multisolute central core systems," Biophysical J. 13, solvent drag. 546-567. In the steady state the primary consideration is that fluid 6. Kokko, J. P. & Rector, F. C., Jr. (1972) "Countercurrent multi- and solutes entering the interstitium from the nephrons ex- plication system without active transport in inner medulla," actly equal fluid and solutes leaving the interstitium and en- Kidney Int. 2,214-223. Downloaded by guest on September 24, 2021