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 urine/renal blood flow/glomerular filtration/nephron/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 nephrons 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.