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A comparison of multinephron and shunt models of the renal concentrating mechanism
08939659/93 $6.09 + 0.00 Pergamon Press Ltd
Appl. Moth. Leit. Vol. 6, No. 2, pp. 61-65, 1993 Printed in Great Britain
A COMPARISON OF MULTINEPHRON OF THE RENAL CONCENTRATING H. WANG AND
AND SHUNT MODELS MECHANISM*
R. P. TEWARSON
Institute of Mathematical Modeling, Department of Applied Mathematics and Statistics State University of New York, Stony Brook, NY 117943600, J. F.
JEN AND
U.S.A.
J. L. STEPHENSON
Department of Physiology, Cornell University Medical College 1300 York Avenue, New York, NY 10021, U.S.A. (Received
and accepted November
1992)
Abstract-A multinephron model of the inner medullary urinary concentrating mechanism in the mammalian kidney is described. A procedure to represent the distribution of nephrons with a finite number of different nephron types is introduced. The model is compared with a shunt model. These studies show that, for uniform transport permeabilities: (1) 40 properly weighted nephron types are sulhcient to represent the distribution of nephrons; (2) the two models give essentially the same results.
INTRODUCTION Mammals utilize the kidney to maintain the volume and composition of their body fluids within very narrow limits. Osmolality, in particular, is very closely regulated by production of urine that is either more or less concentrated than plasma or other body fluids. Mathematical models have been responsible for many of the basic ideas leading to our understanding of the urinary concentrating mechanism. The kidney consists of a large number of distinct functional units called nephrons, operating in parallel. The rat kidney contains 30,000 to 50,000 nephrons, while a human kidney contains roughly one million nephrons. Each nephron is a tube approximately and is made up of four main separate sections (each of onecm long and 100~ cm in diameter, which can be further subclassified): the renal corpuscle, the prozimal tubule, the loop of Henle and the distal tubule. The distal tubules of several nephrons join together to form the collecting ducts which take the urine to the bladder. A lengthwise cross-section of the kidney reveals two distinct parts: the cor2ez and the medulla. In view of differing physiological characteristics of the tubules, the medulla is subdivided into the upper and the lower parts. The cortex contains the renal corpuscles and the proximal and distal tubules. The loops of Henle (which turn at different levels) and the collecting ducts reside in the medulla. Blood gets filtered in the renal corpuscle; the filtrate then flows successively through the proximal tubule (which winds around the renal corpuscle), Henle’s loop (the descending and ascending parts are joined by a hairpin turn), the distal tubule (which winds around in the cortex and makes contact with the Juzta-glomerular apparatus related to the renal corpuscle), and the collecting duct to emerge as the final urine. Blood also flows down into the medulla and then back to the cortex through the vasa recta. This retrieves the water and solutes absorbed from the tubules. By varying the composition of the final urine, the composition of the interstitial fluid bathing the cells of the body is maintained within the narrow limits compatible with life. Computer models [l-17] that use experimentally available parameters have generated concentration gradients in the outer medulla consistent with experimental data by including metabollically driven (active) salt transport out of the thick ascending limb of Henle’s loop, but have been unable to generate any significant concentration gradient in the inner medulla where there *Research supported by NIH Grants DK17593, DK 31550, RR 06589.
61
H. b%‘ANQ
62
et rd.
is no active transport. How the inner medulla concentrates urine is still one of the major unsolved problems in renal physiology. Hypotheses continue to be proposed to answer this question. Mathematical models are essential in the testing of such alternate hypotheses. See [15] for more details and an extensive list of references. The loops of Henle (each loop consists of a descending and an ascending part--DHL and AHL) by merging turn at various medullary levels. In some models [12,17], this fact is incorporated all the loops into a single composite loop, and allowing part of the axial flow at each level to be shunted from DHL to AHL. Such models are called Shunt Models-SM. Naturally, in such models, the cross-sectional areas of the DHL and AHL in the composite loop progressively decrease in accordance with the original nephron distribution. The vasa recta have a shunting mechanism via the capillaries. This fact has been utilized to develop a composite vas rectum with shunts [1,4]. Some models have utilized multi-vasa recta [12]. Multinephron models, all of which use Stephenson’s central core assumption [13], have also been developed. Two representations have been used in such models. One is to use a small number (no more than six) of discrete loops of Henle [9-11,141. Layton has introduced the idea of continuous distribution of loops represented by weighting the integrals when computing convective and transmural fluxes [6-81. To derive his closed-form formula for the core osmolality, which extends Stephenson’s formula for a single nephron, Layton assumed that the descending limbs and collecting duct system are infinitely permeable to water so that descending limbs, collecting ducts, and central core all have the same osmolality at each medullary level. In this paper, we describe a procedure to represent the distribution of nephrons with a finite number of nephron types turning at various medullary levels. We feel that our approach has the following advantages: Since no closed-form formula for the core osmolality is sought, more detailed experimental data on renal tubule membrane transport, as it becomes available, can be easily incorporated into our model through numerical simulations. Furthermore, our multinephron model is extremely suitable for parallel computing because each individual loop can be solved separately against the central core. Two problems are addressed in this paper: First, how many different loop types are required to get a limiting solution? Second, how does the limiting solution for the multinephron modelMNM compare with the solution for the shunt model SM described earlier [16]? MULTINEPHRON
MODEL
We are concerned with only the inner medulla. Let A(z) be a weighted least square fitting [3] of the experimental data of the loop distributions in the inner medulla given in [2], where z is the normalized depth into the inner medulla, with z = 0 at the junction of the inner and outer medulla and z = 1 at the papillary tip. We want to approximate the continuous distributions of loops with a model of finite (p) types of loops classified according to their turning points. Given 0 = cc < 21 < . .a < 2) < -a. < zp = 1 with the kth type of loop turning at zk for k = ,p, the average total number of loops in interval [X&r, Xk] is I/(X, - XL-r) Jsy_l h(z) B + A(tk))/z) (the trapzoidal rule is used to compute the integral). To determine the (‘;Z. zk__1) . weighting factor & for the kth type of loop, or the number of loops of the kth type, we have Ao+A1 Xl + x2 + *. . + A, = 2 Al +A2 AZ+...+& = 2 ... A,-1 Ak+“‘+&
&-I
=
+ Ap = Ap=
A,-2
&-I
’ ’
+ hk
2
,
+&-I
2 ’ + A, 2 .
63
Multinephron and shunt models
Back to solving
the above triangular
xk =
system
Ak-1
-
of linear
Ak+l
we have
k = 1,2,...,p-
’
2
equaions,
1
and
Let us consider a multi-nephron model, consisting of p types of loops and a collecting duct system (CD)-interacting with a central core. The distribution of collecting ducts in the inner medulla is obtained by a weighted least square fitting [3] of the experimental data in [2], and the resulting continuous function n&(z) is used to approximate the successive fusions of collecting ducts in the inner medulla, as has been done by other modelers [8,17]. Let i denote the tube index, where DHL =S i = 2s - 1, AHL + i = 27, CD =+ i = 2p + 1, CORE =+ i = 2p + 2, x being the distance measured from to the bottom (z = 1). The variables are: Fi,(x) = total volume flows of tubes (except for r = 2p + 1,2p + 2; in these cases, they are the collecting where and central core flows, respectively), Cik(2) = so 1u t e concentrations, (urea). As boundary conditions, the entering flows for a single descending a single collecting duct f2r_l,v(0), and concentrations &-r+(O), C&,+l,k(O) Fz+IJO) = &f~+-l,~(O), J’z~+I,~(O) = ncd(O)f2,+1,,(0). at t = r/p, in the r th loop the DHL makes a hairpin F2r,v
(7/p)
= 4’2744~)
The differential
in,
equations
C2r,k(r/p)
= C2r--l,k(r/P).
de;;:,
+ AiJi,(+)
d(Fi,Cik) + AiJik(x) dx dFzp+l,v + ncd(x)Jzp+l,,(x) dx d(F2p+l,vCzp+l,t) dx
+ nCd(x)b+l,k(x) dFzp+z,L. + dx
dC2p+2,k dx
Also, Fz~+z.,~(~) = 0, C&+2,&) = 0; turn to become AHL and, therefore,
are
x
D2p+2,k
(T = 1,2,. . . ,p), the top (z = 0) the kth type of ducting system k = s (salt), u limb fzr-l+ (0), are given, thus,
+ F2p+2,k
-
J2p+2,k(x)
F2p+2,vC2p+2,k
= 0,
i=
1,2,...,2p
= 0,
i=
1,2,...
(1) ,2p;
k=s,u
(2)
(3)
= o, k=s,u
= 0,
(4
= 0,
= 0,
where Jiv(X) and Jjk(X) are, respectively, volume and solute transmural fluxes, diffusion coefficients and Fzp+2,k(x) are the axial solute flows, Ji”(t) and Jik(x)
Dzp+2,k are the are functions of
only cik(s) and C2p+2,k(d). Mass balance requires that
chJi.(z) +
nCd(x)J2p+l,v(x)
+ J2p+2,v(x)
=
0,
n4x)J2p+l,k(x)
-I- J2p+2,k(x)
=
0,
(7)
i= 1 2P xhJik(t)
+
k = s,u.
(8)
i=l
Equations (7) and (8) enable us to compute the total transmural fluxes into the central core which are needed in equations (5) and (6). W e solve the above system of differential equations using the numerical method described in [16].
64
H. WANG et al.
Table
1. Common boundary
values.
NaCl concetration (n&f/L)
Flow (10 --T mllsec)
Urea concetration (n&f/L)
DHL
0.609
410.
30
CD
0.160
63.
661
Table 2. Uniform Water
(10e5 cm/set. atm)
DHL
permeabilities.
NaCl
( low5 cm/set)
18.645
1.0
AHL
0
90.0
CD
2.00
0
Table 3. Concentration 1 Descending
Loops
ratios for
Henle’s Limb (DHL) NaCl
40
1
2.560
2.207
11.328
80
I
2.561
2.208
2.561
2.208
Table 4. Concentration Descending Osmolality
2.570
2.0 50.0
* ,d’ -2)
1 Collecting Duct (CD) NaCl
Urea
2.409
8.914
1.281
11.335
2.409
8.914
1.281
11.337
I 2.409
8.913
1.281
Urea
ratios for
cm/set)
2.0
using uniform penneabilities.
MNM
1 Osmolality
160
Urea (lo-’
using uniform permeabilities.
SM
Henle’s Limb (DI-ZL)
Collecting
Duct
(CD)
NaCl
Urea
Osmolality
NaCl
Urea
2.218
11.340
2.420
9.014
1.276
COMPUTATIONAL
RESULTS
In all of our computations, urea transport is assumed to be carrier mediated passive diffusion. NaCl and urea reflection coefficients are all equal to 1. NaCl and urea rational osmotic coefficients are, respectively, 1.82 and 1. DHL and AHL have the same radius 0.001 cm, and CD has the radius 0.0015cm. The length of the inner medulla is 0.4cm. The Michaelis constant for NaCl and urea are 52mM/L and 800mM/L, respectively. No active solute transport is assumed. The collecting duct urea permeability was obtained by using the formula (exponential interpolation) We used p = 0.176 and h(1) = 50. lo-‘cm/set. The h(z) = h(l)@‘-“1, where /3 = h(O)/h(l). diffusion coefficients are set to virtually zero. The boundary values and parameters are shown in Table 1 and Table 2. We computed the ratios of osmolalities, salt and urea concentrations of DHL and CD at x = 1 with the corresponding values at 2 = 0. The membrane transport properties of each tubule are uniform along its entire length (except CD urea permeability), see Table 2. The concentration ratios for MNM are given in Table 3 for p = 40, 80, and 160, and those for SM are given in Table 4. These tables suggest that: (1) as far as representing the distribution of nephrons with a finite number types is concerned, 40 properly weighted nephron types are sufficient; (2) SM is a very good approximation to the MiVM. Further studies are needed to verify if the SM is still a reasonable for non-uniform permeabilities.
approximation
of nephron
to the MNM
REFERENCES
1. 2. 3. 4. 5. 6.
P.S. Chandhoke
and G.M.
Saidel,
Mathematical
model of maas transport
through
the kidney:
Effects of
nephron heterogeneity and tubular-vascular organization, Ann. Biomed. Eng. 9, 263-301 (1981). J.S. Han, K.A. Thompson, C.L. Chou and M.A. Knepper, Experimental teats of three-dimensional model of urinary concentrating mechanism, .I. Am. Sot. Nephrol. 2, 1677-1688 (1992). J.F. Jen, H. Wang, HP. Tewarson and J.L. Stephenson, Effect of variation of tubular permeabilities and other parameters on concentrating ability of a model of the renal inner medulla, (Submitted for publication). J.A. Jacquez, D. Foster and E. Daniels, Solute concentration in the kidney-I: A model of the renal medulla and its limit cases, Math. Bioaci. 32, 307-335 (1976). J.P. Kokko and F.C. Rector, Jr., Countercurrent multiplication system without active transport in inner medulla, Kidney Znt. 2, 214-223 (1972). H.E. Layton, Distribution of Henle’s loops may enhance urine concentrating capability, Biophya. .I. 49, 1033-1040 (1986).
Multinephron and shunt models
65
7. H.E. Layton, Concentrating urine in the imer meduIIa of the kidney, Comments Thcorelicol Biolosq 1, 179-196 (1989). a. H.E. Layton, Urea transport in a distributed loop model of the urine concentrating mechanism, Am. J. Phyaiof. 258, FlllO-F1124 (1990). 9. P. Lory, Effectiveness of sdt transport cascade in the renal medulla: Computer simulations, Am. J. Phyriol. 252, Flo95-F1102 (1987). 10. R. Mejia and J.L. Stephenson, Numerical solution of multinephron kidney equations, J. Comput. Phqaica 32, 235-246 (1979). 11. Ft.Mejia and J.L. Stephenson, Solution of multinephron, multisolute model of the mammalian kidney by Newton and continuation methods, Math. Biorci. 68, 279-298 (1984). 12. L.C. Moore and D.J. Marsh, How descending limb of Herde’s loop permeability effects hypertonic urine formation, Am. J. Phyriol. 239, F57-F71 (1980). 13. J.L. Stephenson, Concentration of urine in a central core model of the renal counterflow system, Kidney Int 2, 85-94 (1972). 14. J.L. Stephenson, Y. Zhang and R.P. Tewarson, Electrolyte, urea, and water transport in a two nephron central core model of the medulla, Amer. J. Physiol. 257, F399-F413 (1989). 15. J.L. Stephenson, Urinary concentration and dilution: Models, In Handbook of Physiology, (Edited by Erich Windhag=), Chapter 30, pp. 1349-1408, Oxford University Press, New York, (1992). 16. R.P. Tewarson, H. Wang, J.L. Stephenson and F. Jen, Efficient solution of differential equations for kidney concentrating mechanism analyses, Appl. Math. Letters 4, 69-72 (1991). 17. A.S. WexIer, R.E. Kalaba and D.J. Marsh, Three-dimensional anatomy and renal concentrating mechanism-1: Modeling results, Am. J. Physiol. 260, F368-F383 (1991).
AM. 6:2-E
Appl. Moth. Leit. Vol. 6, No. 2, pp. 61-65, 1993 Printed in Great Britain
A COMPARISON OF MULTINEPHRON OF THE RENAL CONCENTRATING H. WANG AND
AND SHUNT MODELS MECHANISM*
R. P. TEWARSON
Institute of Mathematical Modeling, Department of Applied Mathematics and Statistics State University of New York, Stony Brook, NY 117943600, J. F.
JEN AND
U.S.A.
J. L. STEPHENSON
Department of Physiology, Cornell University Medical College 1300 York Avenue, New York, NY 10021, U.S.A. (Received
and accepted November
1992)
Abstract-A multinephron model of the inner medullary urinary concentrating mechanism in the mammalian kidney is described. A procedure to represent the distribution of nephrons with a finite number of different nephron types is introduced. The model is compared with a shunt model. These studies show that, for uniform transport permeabilities: (1) 40 properly weighted nephron types are sulhcient to represent the distribution of nephrons; (2) the two models give essentially the same results.
INTRODUCTION Mammals utilize the kidney to maintain the volume and composition of their body fluids within very narrow limits. Osmolality, in particular, is very closely regulated by production of urine that is either more or less concentrated than plasma or other body fluids. Mathematical models have been responsible for many of the basic ideas leading to our understanding of the urinary concentrating mechanism. The kidney consists of a large number of distinct functional units called nephrons, operating in parallel. The rat kidney contains 30,000 to 50,000 nephrons, while a human kidney contains roughly one million nephrons. Each nephron is a tube approximately and is made up of four main separate sections (each of onecm long and 100~ cm in diameter, which can be further subclassified): the renal corpuscle, the prozimal tubule, the loop of Henle and the distal tubule. The distal tubules of several nephrons join together to form the collecting ducts which take the urine to the bladder. A lengthwise cross-section of the kidney reveals two distinct parts: the cor2ez and the medulla. In view of differing physiological characteristics of the tubules, the medulla is subdivided into the upper and the lower parts. The cortex contains the renal corpuscles and the proximal and distal tubules. The loops of Henle (which turn at different levels) and the collecting ducts reside in the medulla. Blood gets filtered in the renal corpuscle; the filtrate then flows successively through the proximal tubule (which winds around the renal corpuscle), Henle’s loop (the descending and ascending parts are joined by a hairpin turn), the distal tubule (which winds around in the cortex and makes contact with the Juzta-glomerular apparatus related to the renal corpuscle), and the collecting duct to emerge as the final urine. Blood also flows down into the medulla and then back to the cortex through the vasa recta. This retrieves the water and solutes absorbed from the tubules. By varying the composition of the final urine, the composition of the interstitial fluid bathing the cells of the body is maintained within the narrow limits compatible with life. Computer models [l-17] that use experimentally available parameters have generated concentration gradients in the outer medulla consistent with experimental data by including metabollically driven (active) salt transport out of the thick ascending limb of Henle’s loop, but have been unable to generate any significant concentration gradient in the inner medulla where there *Research supported by NIH Grants DK17593, DK 31550, RR 06589.
61
H. b%‘ANQ
62
et rd.
is no active transport. How the inner medulla concentrates urine is still one of the major unsolved problems in renal physiology. Hypotheses continue to be proposed to answer this question. Mathematical models are essential in the testing of such alternate hypotheses. See [15] for more details and an extensive list of references. The loops of Henle (each loop consists of a descending and an ascending part--DHL and AHL) by merging turn at various medullary levels. In some models [12,17], this fact is incorporated all the loops into a single composite loop, and allowing part of the axial flow at each level to be shunted from DHL to AHL. Such models are called Shunt Models-SM. Naturally, in such models, the cross-sectional areas of the DHL and AHL in the composite loop progressively decrease in accordance with the original nephron distribution. The vasa recta have a shunting mechanism via the capillaries. This fact has been utilized to develop a composite vas rectum with shunts [1,4]. Some models have utilized multi-vasa recta [12]. Multinephron models, all of which use Stephenson’s central core assumption [13], have also been developed. Two representations have been used in such models. One is to use a small number (no more than six) of discrete loops of Henle [9-11,141. Layton has introduced the idea of continuous distribution of loops represented by weighting the integrals when computing convective and transmural fluxes [6-81. To derive his closed-form formula for the core osmolality, which extends Stephenson’s formula for a single nephron, Layton assumed that the descending limbs and collecting duct system are infinitely permeable to water so that descending limbs, collecting ducts, and central core all have the same osmolality at each medullary level. In this paper, we describe a procedure to represent the distribution of nephrons with a finite number of nephron types turning at various medullary levels. We feel that our approach has the following advantages: Since no closed-form formula for the core osmolality is sought, more detailed experimental data on renal tubule membrane transport, as it becomes available, can be easily incorporated into our model through numerical simulations. Furthermore, our multinephron model is extremely suitable for parallel computing because each individual loop can be solved separately against the central core. Two problems are addressed in this paper: First, how many different loop types are required to get a limiting solution? Second, how does the limiting solution for the multinephron modelMNM compare with the solution for the shunt model SM described earlier [16]? MULTINEPHRON
MODEL
We are concerned with only the inner medulla. Let A(z) be a weighted least square fitting [3] of the experimental data of the loop distributions in the inner medulla given in [2], where z is the normalized depth into the inner medulla, with z = 0 at the junction of the inner and outer medulla and z = 1 at the papillary tip. We want to approximate the continuous distributions of loops with a model of finite (p) types of loops classified according to their turning points. Given 0 = cc < 21 < . .a < 2) < -a. < zp = 1 with the kth type of loop turning at zk for k = ,p, the average total number of loops in interval [X&r, Xk] is I/(X, - XL-r) Jsy_l h(z) B + A(tk))/z) (the trapzoidal rule is used to compute the integral). To determine the (‘;Z. zk__1) . weighting factor & for the kth type of loop, or the number of loops of the kth type, we have Ao+A1 Xl + x2 + *. . + A, = 2 Al +A2 AZ+...+& = 2 ... A,-1 Ak+“‘+&
&-I
=
+ Ap = Ap=
A,-2
&-I
’ ’
+ hk
2
,
+&-I
2 ’ + A, 2 .
63
Multinephron and shunt models
Back to solving
the above triangular
xk =
system
Ak-1
-
of linear
Ak+l
we have
k = 1,2,...,p-
’
2
equaions,
1
and
Let us consider a multi-nephron model, consisting of p types of loops and a collecting duct system (CD)-interacting with a central core. The distribution of collecting ducts in the inner medulla is obtained by a weighted least square fitting [3] of the experimental data in [2], and the resulting continuous function n&(z) is used to approximate the successive fusions of collecting ducts in the inner medulla, as has been done by other modelers [8,17]. Let i denote the tube index, where DHL =S i = 2s - 1, AHL + i = 27, CD =+ i = 2p + 1, CORE =+ i = 2p + 2, x being the distance measured from to the bottom (z = 1). The variables are: Fi,(x) = total volume flows of tubes (except for r = 2p + 1,2p + 2; in these cases, they are the collecting where and central core flows, respectively), Cik(2) = so 1u t e concentrations, (urea). As boundary conditions, the entering flows for a single descending a single collecting duct f2r_l,v(0), and concentrations &-r+(O), C&,+l,k(O) Fz+IJO) = &f~+-l,~(O), J’z~+I,~(O) = ncd(O)f2,+1,,(0). at t = r/p, in the r th loop the DHL makes a hairpin F2r,v
(7/p)
= 4’2744~)
The differential
in,
equations
C2r,k(r/p)
= C2r--l,k(r/P).
de;;:,
+ AiJi,(+)
d(Fi,Cik) + AiJik(x) dx dFzp+l,v + ncd(x)Jzp+l,,(x) dx d(F2p+l,vCzp+l,t) dx
+ nCd(x)b+l,k(x) dFzp+z,L. + dx
dC2p+2,k dx
Also, Fz~+z.,~(~) = 0, C&+2,&) = 0; turn to become AHL and, therefore,
are
x
D2p+2,k
(T = 1,2,. . . ,p), the top (z = 0) the kth type of ducting system k = s (salt), u limb fzr-l+ (0), are given, thus,
+ F2p+2,k
-
J2p+2,k(x)
F2p+2,vC2p+2,k
= 0,
i=
1,2,...,2p
= 0,
i=
1,2,...
(1) ,2p;
k=s,u
(2)
(3)
= o, k=s,u
= 0,
(4
= 0,
= 0,
where Jiv(X) and Jjk(X) are, respectively, volume and solute transmural fluxes, diffusion coefficients and Fzp+2,k(x) are the axial solute flows, Ji”(t) and Jik(x)
Dzp+2,k are the are functions of
only cik(s) and C2p+2,k(d). Mass balance requires that
chJi.(z) +
nCd(x)J2p+l,v(x)
+ J2p+2,v(x)
=
0,
n4x)J2p+l,k(x)
-I- J2p+2,k(x)
=
0,
(7)
i= 1 2P xhJik(t)
+
k = s,u.
(8)
i=l
Equations (7) and (8) enable us to compute the total transmural fluxes into the central core which are needed in equations (5) and (6). W e solve the above system of differential equations using the numerical method described in [16].
64
H. WANG et al.
Table
1. Common boundary
values.
NaCl concetration (n&f/L)
Flow (10 --T mllsec)
Urea concetration (n&f/L)
DHL
0.609
410.
30
CD
0.160
63.
661
Table 2. Uniform Water
(10e5 cm/set. atm)
DHL
permeabilities.
NaCl
( low5 cm/set)
18.645
1.0
AHL
0
90.0
CD
2.00
0
Table 3. Concentration 1 Descending
Loops
ratios for
Henle’s Limb (DHL) NaCl
40
1
2.560
2.207
11.328
80
I
2.561
2.208
2.561
2.208
Table 4. Concentration Descending Osmolality
2.570
2.0 50.0
* ,d’ -2)
1 Collecting Duct (CD) NaCl
Urea
2.409
8.914
1.281
11.335
2.409
8.914
1.281
11.337
I 2.409
8.913
1.281
Urea
ratios for
cm/set)
2.0
using uniform penneabilities.
MNM
1 Osmolality
160
Urea (lo-’
using uniform permeabilities.
SM
Henle’s Limb (DI-ZL)
Collecting
Duct
(CD)
NaCl
Urea
Osmolality
NaCl
Urea
2.218
11.340
2.420
9.014
1.276
COMPUTATIONAL
RESULTS
In all of our computations, urea transport is assumed to be carrier mediated passive diffusion. NaCl and urea reflection coefficients are all equal to 1. NaCl and urea rational osmotic coefficients are, respectively, 1.82 and 1. DHL and AHL have the same radius 0.001 cm, and CD has the radius 0.0015cm. The length of the inner medulla is 0.4cm. The Michaelis constant for NaCl and urea are 52mM/L and 800mM/L, respectively. No active solute transport is assumed. The collecting duct urea permeability was obtained by using the formula (exponential interpolation) We used p = 0.176 and h(1) = 50. lo-‘cm/set. The h(z) = h(l)@‘-“1, where /3 = h(O)/h(l). diffusion coefficients are set to virtually zero. The boundary values and parameters are shown in Table 1 and Table 2. We computed the ratios of osmolalities, salt and urea concentrations of DHL and CD at x = 1 with the corresponding values at 2 = 0. The membrane transport properties of each tubule are uniform along its entire length (except CD urea permeability), see Table 2. The concentration ratios for MNM are given in Table 3 for p = 40, 80, and 160, and those for SM are given in Table 4. These tables suggest that: (1) as far as representing the distribution of nephrons with a finite number types is concerned, 40 properly weighted nephron types are sufficient; (2) SM is a very good approximation to the MiVM. Further studies are needed to verify if the SM is still a reasonable for non-uniform permeabilities.
approximation
of nephron
to the MNM
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