Tohoku

J. Exp.

Med., 1992, 168, 47-62

Simulation

of the

Urea Transport Multiplication Limb

and

Profile

of Water,

NaCI,

in the Countercurrent System between Thin

Inner

Medullary

Collecting

and

Ascending Duct

YASUKAZU RAMADA, MASASHI IMAI*,TAKAOAOKI,RYOJI SUZUKItand AKIRAKAMIYA Institute of Medical Erectronics, School of Medicine, University of Tokyo, Tokyo 113; *Department of Pharmacology,Jichi Medical school, Tochigi 329-04; and 'Department of Mathematical Engineering and Imformation Physics, Fculty of Engineering, University of Tokyo, Tokyo 113

RAMADA, Y., IMAI,M., AOKI,T., SUzUKI,R. and KAMIYA,A. Simulation of the Profile of Water, NaCI, and Urea Transport in the Countercurrent Multiplication System between Thin Ascending Limb and Inner Medullary Collecting Duct. Tohoku J. Exp. Med., 1992, 168 (1), 47-62 We simulated the profiles of water, NaCI, and urea transport in the countercurrent multiplication system between thin ascending limb (TAL) and inner medullary collecting duct (IMCD) by a mathematical model consisting of three compartments (TAL, IMCD, and CNW [capillary network] ), using phenomenological coefficients for hamsters. They are separated by two membranes with distinct permeability properties. The primary driving force which generates "single effect" has a lower reflection coefficient for urea than for NaCI in IMCD. The difference in urea and NaCI concentrations between CNW and IMCD provides an effective osmotic driving force which is favorable for water absorption from IMCD without physicochemical osmotic gradient. The entry of water in the CNW reduces the concentration in CNW and generates the concentration gradients which are favorable for these solutes to diffuse out of TAL. Thus, the fluid in IMCD is concentrated and that in TAL is diluted. The results of simulation showed that the concentration gradients were generated along the medullary axis, resulting in excretion of hypertonic urine. In addition, we examined effects of changes in phenomenological coefficients of IMCD on this concentrating system. Decreases in permeability and in reflection coefficient for urea and increase in hydraulic conductivity increased the osmotic gradients along each compartment. urine concentration mechanism ; countercurrent multiplication system ; inner medullary collecting duct ; thin ascending limb

The countercurrent system in the renal medulla is essential for the generation of an osmotic gradient along the renal medulla (Jamison and Ktiz 1982). Received

November

22, 1991;

revision

accepted 47

for publication

September

1, 1992.

48

Y. Hamada

et al.

Although the classical active NaCI transport model proposed by Kuhn and Ramel (1959) provided a theoretical base for the countercurrent multiplication system in the renal medulla and explain the accumulation of NaCI in the renal medulla, it could not explain the importance of urea in the urine concentration mechanism. In addition, characteristics of solute and water transport across the thin descending and ascending limb of Henle's loop do not fit well with the active NaCI transport model (Gottschalk et al. 1963 ; Jamison 1968 ; Morgan and Berliner 1968 ; Marsh 1970 ; Kokko 1970, 1972 ; Pennel et al. 1974 ; Imai and Kokko 1974, 1976 ; Elalouf et al. 1985, 1986a, b). Stephenson (1972) and Kokko and Rector (1972) independently proposed models which incorporated passive movement of NaCl and urea into a "single effect" operating in thin ascending limb of Henle's loop (TAL). These models explained the role of urea in the formation of concentrated urine. The essential feature which is common to these models is that dilution of the fluid in the TAL occurs by extrusion of NaCI in excess of urea entry. Although permeability properties of TAL are compatible with these models (Imai and Kokko 1974, ; Imai 1977), the entry of considerable amount of urea as observed by in vitro micropuncture studies (Jamison 1968 ; Pennel et al. 1974) is unfavorable for the operation of these models (Jamison and Kriz 1982 ; Imai et al. 1987 ; Imai and Yoshitomi 1990). Apart from these passive models, several investigators (Jaenike 1961; Rabinowitz 1970) have proposed a hypothesis that in the innermedullary collecting duct (IMCD) the lower reflection coefficient for urea than for NaCI could generate a driving force for water absorption from the collecting duct fluid if the concentration of NaCI in the interstitium is higher than that in the tubular lumen even though osmolality of the luminal fluid is counterbalanced by increased urea concentration. Imai et al. (1987) extended this model to the counterflow system between TAL and IMCD. According to the proposed model, water driven from the collecting duct by the above mentioned mechanism dilutes interstitium around the TAL, providing driving forces favoring for diffusion of both NaCI and urea out of the TAL. Imai et al. (1988) provided experimental evidence that osmotic work could be accomplished across the hamster IMCD in the absence of transmural osmotic gradient but in the presence of different solute composition between bathing and luminal fluid. However, it has not been tested whether the proposed single effect is actually multiplied to generate osmotic gradients along the axis of inner medulla. In order to address this issue, we made a computer simulation study under a limited and simplified condition where three compartments consisting of TAL, capillary network (CNW) and IMCD are separated from otherr renal medullary structure. The results show that the countercurrent multiplication system works under our model. By using this model, we analyzed contribution of urea permeability, reflection coefficient for urea and hydraulic conductivity of IMCD to the urine concentrating capacity.

Mathematical

Model

of Urine

Concentrating

Mechanism

49

The model to be analyzed in this paper is critically dependent on the feasibility of the lower reflection coefficient for urea of the IMCD. Recently, Knepper and his associates (Knepper et al. 1989 ; Chou et al. 1990) criticized this model by reporting that the reflection coefficient for urea in the rat IMCD was not less than unity. If urea and water pass through separate pathways, the reflection coefficient for urea must be unity in theory, However, as was analyzed in detail by Barry and Diamond (1984), under certain conditions reflection coefficients for solutes having no interaction with solvent is calculated to be less than unity, because the unstirred layer adjacent to the membrane generates a solute concentration gradient within the layer, decreasing effective osmotic gradient across the membrane. They called this phenomenon pseudo solvent drag. It is possible that cellular constraints to diffusion (Schafer and Andreoli 1972) may have additional contribution to this pseudo solvent drag. Therefore, for the practical purpose we used apparent reflection coefficient for urea and Na+ reported by Imai et al. (1988) rather than the reflection coefficient strictly defined by non-equilibrium thermodynamics. MODEL AND EQUATIONS The model predicts that the countercurrent multiplication is operated between TAL and IMCD with interposition of CNW (Fig. 1). The three compartments are separated by two membranes (M-1 and M-2), of which membrane properties represent those of TAL and IMCD, respectively. The compartments of TAL and IMCD are circular cylinders with 6 mm-length and 10 p m-radius enclosed in the compartment CNW. Sodium, chloride and urea are all the solutes considered in this model. The membrane M-1 (TAL) has a high permeability to sodium and to chloride (PNa=87X10-5 cm/sec, Pct-196X10-5 cm/sec), moderate permeability to urea (Purea =19 X 10-5 cm/sec) and very low hydraulic conductivity that is not different from zero (Imai and Kokko 1974 ; Imai 1977). The membrane M-2 (IMCD) has a moderate permeability to urea (Purea=20X 10-5 cm/sec) in the lower half portion and very low permeability (Purea = 2 X 10-5 cm/sec) in the upper half, a high hydraulic conductivity (Lp=10 X 10-5 cm/sec/atm) and negligible permeabilities to sodium and chloride (Imai et al. 1988). The reflection coefficient of the membrane M-2 for sodium and chloride (o =1.0, cyC1=1.0) are higher than the reflection coefficient of M-2 to urea (Ourea=0.70) (Imai 1977; Imai et al. 1988). We set up the initial concentration as follows. Both fluids in TAL and CNW consist of 0.3 mol/liter NaCI and 0.4 mol/liter urea, whereas the fluid in IMCD consists of only 1.0 mol/liter urea. Thus, the osmolalities in all starting fluids are identical. The initial flow rate of urine is 3.3 X 10-' cm3/sec (20 nl/min) from the papillary tip to the base of the inner medulla in TAL and CNW, and from the base to papillary tip in IMCD. The simulation study was conducted to demonstrate solute concentration profiles along the axis of each compartment as a function of time. The issues we addressed in this simulation study are three-fold. 1) Are osmotic gradients generated among these compartment? 2) Are axial osmotic gradients generated, and is final urine concentrated? 3) How changes in phenomenological coefficients of IMCD affect overall solute concentration profiles and hence urine concentrating capacity? The differential equations (Mejia and Stephenson 1979, 1984) that describe movement of solutes and water in the i-th compartment are

a axCa kFav-DZ k 2ax k +Jzk+

at a

AiCi,k -AiSi,k

(1)

50

Y. Hamada

Fig.

et al.

1. Schematic illustration of the model of countercurrent multiplication between thin ascending limb of Henle and inner medullary collecting duct. The left panel in the figure shows structural organization of renal medulla including long-loop nephron, vasa recta, capillary networks and inner medullary collecting duct. Open arrows mean the direction of urine flow and solid arrows that of blood flow. The right panel in the figure shows the three compartment model consisting of the thin ascending limb (TAL), the capillary network (CNW) and the inner medullary collecting duct (IMCD). In the right panel, open arrows mean urine flow and closed arrows transmural transports of solutes and water. M-1 shows the membrane between TAL and CNW. M-2 shows the membrane between IMCD and CNW.

ax

a~ ape +R ax 1F1,v=0

(3)

where x is the distance from the base of inner medulla along the length of each compartment tube ; ci,kis the concentration of the k-th solute in the i-th compartment ; Fi,,, is the axial volume flow ; Di,k is the diffusion coefficient of the k-th solute in the i-th compartment ; Ji,k is the transmembrane flux of the k-th solute ; Ai is the cross sectional area of the i-th compartment ; Si,k the average net rate at which the k-th solute is produced by physical or chemical reactions ; Ji ,,, is the transmembrane volume flux of water ; Fi is the hydrostatic pressure in the i-th compartment ; and Ri is the resistance to flow in the i-th compartment. We consider three compartments (TAL, CNW, and IMCD) and three solutes (sodium, chloride and urea) in this model. Therefore i (1, 2, 3) represent respective compartment and k (1, 2, 3) respective solutes. The equation (1) is the expression for solutes conservation, the equation (2) for volume conservation, and the equation (3) is the equation of pressure-flow relationship. To simplify the calculations, we set up several assumptions : Regarding the equations (1) and (2), we assumed the cross sectional area of all compartments to be constant throughout the time course. We also assumed that diffusion of solutes in each compartment was rapid and instantaneous. We further assumed that

k

Mathematical

Model

of Urine

Concentrating

Mechanism

there was no chemical or physical reaction in each compartment. the equation (1) becomes

51

Under these assumptions,

aFi,yCi,k +J ax i,k+Ai aCl'k at =~

(4)

and the equation (2) becomes al'i ,V+J ax ZU=0

(5)

The equation (3) can be neglected if we assume that the hydrostatic pressure is constant along the length of each compartment. The transmural fluxes of water and solutes can be expressed as (Taniguchi et al. 1987) Ji,,=-Lp1RTA*

(~o,kdCI,k)

Z!Ci,k` Ci,k Ci',k Ji,k`Pi,kA* (/JCi,k Cmi,kFVT/RT) Cmi,k(1-oi,k)Ji,v Cmi,k-ZJCi,k/ln(Ci,k/Ci',k)

(6) (7) (8) (9)

C 1 mi k` 2 (Ci,k+Ci', k)

(10)

V `_ RT I~k1PI,k,CI,ki+~kZPi,k2Ci',k2 [~ L ~1

(11T

,ik1Pi,k1Ci',kl+L~k2Pi,k2Ci,k2

where Ji, v is the transmembrane volume flux of water ; LPi is the hydraulic conductivity of i-th compartment ; R is gas constant ; T is the absolute temperature ( T = 310°) ; A is the lateral area of the i-th compartment ; ~j,k is the reflection coefficient of the i-th compartment for the k-th solute ; Pis the permeability of i-th compartment to the k-th solute ; Ji ,k is the transmembrane k-th solute flux of the i-th compartment. Cm is the average intramural concentration ; VT is the transmural electrical potential difference between the i-th compartment and the i'-th compartment which is next to the i-th compartment ; F is the Faraday's constant. The equation (6) gives the transmural volume flux of water driven by the effective osmotic pressure difference. The first term of the right hand of equation (8) expresses the transmural flux driven by the concentration gradient ; the second term expresses the passive diffusion driven by the transmural electrical potential difference ; the third term expresses the transmural flux caused by the solvent drag. The equation (9) gives the average of the intramural concentration. In our simulation study, we used the approximate equation (10) instead of the equation (9) by assuming that the membranes are very thin. The equation (11) is the solution of Goldmann's equation and gives the transmural electrical potential difference generated by passive diffusion of NaCI. In this equation, kl means a cation and k2 means an anion. Because we considered three compartments, three solutes and water, we solved twelve simultaneous nonlinear partial differential equations. The system of these differential equations was replaced with a system of finite differential equations. We selected a proper mesh spacing (J x = 0.2 mm) and a time increment (zit = 0.025 sec), adopting the marching method (Euler method). And we used 32bit personal computer NEC PC-9801RA with a co-processor for numerical simulation. In this calculation, we used the phenomenological coefficientsreported for hamsters (Imai 1977; Imai et al. 1988) and selected the appropriate initial and boundary conditions as already indicated in the description of the model.

n

Y. Hamada

52

et al.

RESULTS Formation of osmotic gradient along the axis of each compartment We simulated the profiles of water, NaCI and urea transport in three compartments and examined whether osmotic gradients are generated by the multiplication of the "single effect". It took two seconds to arrive at the equilibrium. Fig. 2-Fig. 5 show the transition of generation of osmolality and concentration gradients along the axis of each compartment. Fig. 2 shows the transition of the osmolality in TAL, CNW and IMCD. The osmolality in TAL decreased from 1.000Osm/kgH20 to 0.969Osm/kgH20 as the urine flows from the papillary tip to the base. The osmolality in CNW also decreased from 1.000Osm/kgH20 to 0.934Osm/kgH20. By contrast, the osmolality in IMCD increased from 1.000Osm/kgH20 to 1.076Osm/kgH20 as the urine flows to the papillary tip. There are diluting processesin TAL and CNW and a concentrating process in IMCD. As predicted from the hypothesis, the initial event was the rapid increase in water absorption out of IMCD which was dependent on the hydraulic conductivity of IMCD and the osmotic gradient. The outward flux of water from IMCD to CNW concentrated the fluid in IMCD and diluted fluid in CNW. This dilution in CNW fluid caused a concentration difference between TAL and CNW, and the solutes were passively moved from TAL to CNW. As a result, the osmolality of TAL became lower. Because the membrane M-1 has no hydraulic conductivity, water can not move across TAL. Thus, the hypotonicity in TAL caused by the passive diffusion of solutes was maintained without dissipating by water abstraction. The diluting process was more prompt in CNW than in TAL. At equilibrium state, axial osmotic gradients were generated along both TAL and CNW. In early period, the fluid in IMCD was transiently diluted. This is mainly

Fig. 2.

Osmolality

and axial length. on the horizontal

profiles

of fluid in TAL,

CNW

and IMCD

as functions

of time

The distance from the base of the inner medulla is shown axis. The left and right ends of this axis corresponds to the

papillary tip and the base, respectively. Time shows the passage start of simulation. The arrows show the direction of the volume

from flow.

the

Mathematical

Fig. 3.

Sodium

of sodium Fig. 2.

Model

concentration in IMCD

of Urine

profiles

is low enough

along

Concentrating

TAL

Mechanism

and CNW.

to be ignored.

The

The others

53

concentration are same

as in

accounted for by diffusion of urea out of IMCD. Solvent drag might not contribute to this phenomenon, because solute movement associated with solvent drag may be restricted as long as the reflection coefficient is not zero. Fig. 3 shows the concentration profile of sodium in TAL and CNW. Because we assumed that amount of sodium in IMCD is negligible in this model, the figure for IMCD is omitted. The concentration of sodium in TAL decreased from 0.300 mol/liter to 0.289 mol/liter and the concentration in CNW also decreased from 0. 300 mol/liter to 0.281 mol/liter at the equilibrium state. The concentration of sodium in CNW decreased more promptly than in TAL, reflecting the preferential water flux from IMCD. Thus the concentration gradients for sodium were also generated along the axis of TAL and CNW. Although similar profiles were obtained for chloride, we do not present the data because they are essentially the same as those observed for sodium. Fig. 4 shows the concentration profiles of the urea in TAL, CNW and IMCD. The urea concentration in TAL decreased from 0.400 mol/liter to 0.394 mol/liter and that in CNW decreased from 0.400 mol/liter to 0.370 mol/liter as the urine flows upward. By contrast, the urea concentration in IMCD increased from 1.000 mol/liter to 1.076 mol/liter as the urine flows downward.

Fig. 4. same

Urea concentrations as in Fig. 2.

profiles

along TAL,

CNW and IMCD.

The others

are

Y. Hamada

54

et al.

Fig. 5.

The volumetric flow rate in IMCD. The boundary value (3.3 X 10-' cm3/ sec) is given at the base of the inner medulla. As water is absorbed from IMCD, the flow rate decreases in the lower potion.

Fig. 5 shows the transition of the urine flow rate in IMCD. The boundary value (3.3 X 10-' cm3/sec) was given at the base. Water was absorbed along the length of IMCD. At the papillary tip, the flow rate was equal to 1.59 X 10-' cm3/ sec and it was about a half of the initial boundary value. In early period, much water was absorbed from IMCD, and therefore the flow rate in IMCD decreased very much. After about one second, a constant flow reduction was observed along the length of IMCD, indicating that almost constant fraction of water was reabsorbed along IMCD. Fig. 6 compares profiles of osmotic gradients among three compartments in steady state. Osmolality, sodium and urea concentrations were higher in TAL than in CNW at all levels, indicating that osmolality of the TAL decreased as a consequence of diffusion of both NaCI and urea from TAL to CNW.

Fig.

6. Comparison of osmotic gradients among TAL (o), CNW (.) (.) in steady state.

and IMCD

Mathematical

Model

of Urine

Concentrating

Mechanism

55

Effect of membrane properties of IMCD on osmotic gradient in renal medulla The above analyses were made by applying two different urea permeability values for IMCD, i.e. low permeability for the upper part and high permeability for the lower part. This was based on the actual experimental data (Imai et al.

Fig.

7. Urea concentration profile in CNW and osmolality profile in IMCD when homogeneously high urea permeability was applied to IMCD (solid symbols). The data are compared with thosd obtained in the standard condition where low and high urea permeabilities were applied to the upper and lower portions of IMCD, respectively (open symbls).

Fig. 8. The effects of changes in transport concentrating capacity. High hydraulic urea,

and low reflection

in three

compartments.

coefficient

parameters of IMCD on the urine conductivity, low permeability to

for urea generate

, a Urea ;

large

osmolality

, P Urea ; .....

Lp.

gradients

56

Y. Hamada

et al,

1988). To evaluate the physiological significance of this axial heterogeneity of urea permeability, we made simulation study by applying single uniform value of urea permeability to IMCD. The results are summarized in Fig. 7 comparing with the standard condition. Surprisingly, the profiles of urea concentration gradient in the CNW changed very much, and became flat in the upper portion of the medulla. This is associated with generation of a less steeper osmotic gradient along IMCD. In order to assess significance of transport parameters of IMCD in this model, we examined effects of changes in permeability of urea, reflection coefficient for urea and hydraulic conductivity on concentration profiles in three compartments. Calculation was made at several levels of parameters and the results were interpolated by the cubic spline function. Fig. 8 summarizes the crucial parts of these analyses. All values were depicted from steady state values. For TAL and CNW, osmolality values at the base are shown, whereas for IMCD the values at the papillary tip are shown. When reflection coefficient for urea was varied from 0.5 to 0.9 with fixed values for other parameters, an increase in reflection coefficient for urea was associated with a decrease in final urine osmolality. This is probably due to a decrease in water abstraction from the collecting duct, since osmolality of the CNW remained high. When permeability of urea was varied, an increase in permeability was associated with a small decrease in urine concentrating capacity. As was easily expected, an increase in hydraulic conductivity was associated with an increase in urine concentrating capacity in combination with marked decreases in osmolality CNW and TAL. DISCUSSION Countercurrent multiplication between TAL and IMCD The single effect for the countercurrent multiplication system dealing with this simulation study is based primarily on the model proposed by Imai et al. (1987). The initial driving force is water absorption from the IMCD by effective osmotic gradient generated by difference in apparent reflection coefficients between urea and NaCI and difference in solute composition in IMCD and CNW. This principle has been already proposed by Jaenike (1961) and Rabinowitz (1970). The principle of the proposed single effect is very similar to those proposed by Stephenson (1972) and Kokko and Rector (1972) in that we consider three compartments separated by two membranes with distinct permeability properties and two different kinds of solutes, i.e., NaCI and urea. However, the assigned compartments are different. Fig. 9 compares the principle of single effect proposed by Imai et al. (1987) with those proposed by Stephenson (1972)/ Kokko and Rector (1972). The latter models assume that the single effect operates between descending and thin ascending limbs of Henle loop with interposition of the capillary core. On the other hand, the present model postulates that the single effect operates between thin ascending limb and inner medullary

Mathematical

Model of Urine

Concentrating

Mechanism

57

Fig. 9. Schematic illustration of principle of the model of Stephenson/KokkoRector (left) and the present model (right). Abbreviations and symbols : DLH' descending limb of Henle ; CNW, capillary network ; TAL, thin ascending limb of Henle ; IMCD, inner medullary collecting duct ; P, permeability ; 0, very low permeability ; +, moderate permeability ; +, high permeability

; Arrows with half triangles

indicate direction of flow.

collecting duct with the interposition of capillary networks. In the Stephenson/ Kokko-Rector model, the primary driving force which initiates the single effect is diffusion of NaCI out of the TAL in excess of urea entry owing to a higher permeability of TAL to NaCI than to urea. Thus, the fluid in descending limb is concentrated by absorption of water. For the efficient operation of this system, it is essential that the concentration of urea at the tip of the loop is kept low. However, in vivo micropuncture studies (Jamison 1968; Pennel et al. 1974) as well as computer simulation stuies (Foster and Jacquez 1978; Chandhoke and Saidel 1981; Layton 1986, 1990; Lory 1987; Sands and Knepper 1987; Taniguchi et al. 1987 ; Wexler et al. 1987; Stephenson et al. 1989) have disclosed that a considerable amount of urea entry occurs along the descending limb. Therefore, both NaCI and urea must diffuse out of the TAL for the luminal fluid to be diluted along its axis. In addition, many trials of computer simulation of such passive model failed to demonstrate reasonable osmotic gradients in renal inner medulla (Foster and Jacquez 1978; Chandhoke and Saidel 1981; Layton 1986, 1990; Wexler et al. 1987; Lory 1987; Stephenson et al. 1989). The model of Imai et al. (1987) has been proposed to solve this problem. The initial driving force for this system to operate is the osmotic force generated by the difference in reflection coefficients of IMCD for NaCI and for urea. If one assume that the route of urea across the IMCD is distinct from that of water, the reflection coefficient for urea ought to be unity as strongly argued by Knepper et al. (1989) and Chou et al. (1990). However, as briefly mentioned in Introduction, apparent

58

Y. Hamada

et al.

reflection coefficient for highly permeable solute inevitably becomes less than unity due to the presence of unstirred layer in the vicinity of the membrane or cellular constraints to diffusion (Barry and Diamond 1984). In fact, our careful study in hamster IMCD (Imai et al. 1988) consistently yielded the reflection coefficient for urea to be about 0.7, a value sufficient for the generation of and effective osmotic force, though we have no good explanation for the discrepancy with the data reported by Knepper et al. (1989) and Chou et al. (1990). Although we have also reported experimental evidence that osmotic work in fact was accomplished across the in vitro perfused hamster IMCD when urea in the bathing fluid was replaced isotonically with NaCI, it was uncertain whether the proposed single effect is effectively multiplied along the medullary axis. The single effect proposed in this model may be less stable as compared to that of Kokko-Rector because diffusion of urea out of the IMCD would diminish or reverse the driving force for diffusion of urea out of the TAL. In that case, the model would not make any difference from that of Stephenson/Kokko-Rector. In the non-flowing system as shown in Fig. 9, such case may occur as the contact time is prolonged. Therefore, the present study was designed to demonstrate that the proposed single effect could actually be multiplied in limited environment among TAL, CNW and IMCD by considering time transient from the initial condition under the reasonable flow rate. The parameters utilized in this analysis are mostly derived from the data obtained in hamsters (Imai 1977 ; Imai et al. 1988). For the simplicity, we assumed that both TAL and IMCD are simple cylinder surrounded by the compartment of CNW (Fig. 1). Since one IMCD is comprised of several nephrons and thus is surrounded by several ascending limbs (Kriz 1981; Lemley and Kriz 1987), the surface area of the TAL must be greater than that of the corresponding IMCD. In this regard, we assumed that six nephrons comprise one IMCD. In other word, we set the surface area of IMCD to be one sixth of the TAL. We gave a value of 3.3 X 10-7 cm3/sec (20.0 nl/min) for the initial flow rate of the system. Although this value may be too high for the single nephron flow rate, it may not be unrealistic value for 6 nephrons. Under these limited conditions as mentioned above, we found that the solute concentration profile of each compartment reached a steady state within 2 sec. In the steady state, axial osmotic gradients were generated in all compartments. It is important to note that osmolality of the fluid in the IMCD was increased, indicating that osmotic work was accomplished by this system. When we compare profiles of Na+ and urea in the steady state between TAL and CNW, we find that concentrations of these solutes are always higher in TAL than in CNW, indicating that the driving forces are favorable for Na and urea to diffuse out of TAL. Although the magnitude of the generated osmotic gradients were very small, our analysis clearly shows that under these limited conditions the proposed three compartments system has capability to generate axial as well as transmural osmotic gradients, leading to formation of concentrated urine. The model of countercurrent multiplication system between

Mathematical

Model of Urine

Concentrating

Mechanism

59

TAL and IMCD also in accordance with the structural organization of the inner medulla (Kriz 1981; Lemley and Kriz 1987). According to the analysis of Lemley and Kriz (1987) in rat renal inner medulla, both TAL and ascending (venous) vasa recta are distributing close to IMCD. On the other hand, both descending (arterial) vasa recta and descending thin limb are separated from IMCD. Imai et al. (1987) also proposed that combination of vascular bundle and descending thin limb may constitute passive equilibrium system to maintain the generated osmotic gradient. It is possible that jointing the out flow of the descending limb from such a system to the inflow of TAL in the present system may modify the solute concentration process. Such a issue, however, remains to be elucidated in the present study. Recently, Wexler et al. (1991) conducted a large scale computer simulation study by considering these three dimensional organization, and succeeded to obtain osmotic gradients along the renal medulla in the absence of active solute transport. However, unfortunately, a part of parameters they used was unphysiological. Parameter sensitivity of the system Sands and Knepper (1987) reported that vasopressin increases urea permeability only in the terminal two third of the rat IMCD. Imai et al. (1988) confirmed that the same holds true for the hamster IMCD. For the standard simulation, we used such heterogeneous values for urea permeability of IMCD. In order to evaluate physiological significance of this heterogeneity, we computed solute concentration profile by applying high urea permeability to the entire length of IMCD. The finding that high urea permeability in IMCD was unfavorable for generation of osmotic gradient indicates that heterogeneity of urea permeability might be physiologically relevant. Although the solute concentration profiles were analyzed in the standard condition at fixed parameters obtained from hamsters, we tested whether changes in transport parameters of IMCD influence solute concentration profiles at the papillary tip in the three compartments (Fig. 8). It is natural that an increase in hydraulic conductivity is associated with an increase in osmolality of IMCD. The consequence of increase in urea permeability in IMCD is difficult to predict. On one hand, the increase in urea permeability might increase diffusion of urea from IMCD to CNW, and thereby dissipate urea concentration gradient favorable for urea diffusion out of TAL. On the other hand, the increase in urea concentration in the CNW might be favorable for water absorption from the IMCD even if the reflection coefficient for urea is dot unity. The result of our computer simulation showed that an increase in this parameter slightly decreased urine concentrating capacity. In marked contrast, an increase in reflection coefficient for urea greatly decreased urine concentrating capacity. These findings indicate that low reflection coefficient for urea is much more efficient than high urea permeability for the operation of this countercurrent multiplication system.

60

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et al.

However, it is unlikely in theory that the reflection coefficient for urea is not unity if the route of urea transport is distinct from that of water. This difficulty can be easily overcome by the view that apparent reflection coefficient actually measured in a biological system with unstirred layer should be functionally operative even in the absence of solute and solvent interaction (pseudo solvent drag). The detailed theoretical analysis on this phenomenon has been reported by Barry and Diamond (1984) in a model with simple one membrane system. If we consider cellular barrier having two membranes in series containing cytoplasma, the phenomenon of pseudo solvent drag would become more predominant. A simulation study in such a system is now in progress in our laboratory. Acknowledgments We would like to express our thanks to the staffs of Department of Pharmacology, Jichi Medical School (Imai laboratory), of Department of Mathematical Engineering and Information Physics, Faculty of Engineering, University of Tokyo (Suzuki laboratory) and of Institute of Medical Electronics, School of Medicine, University of Tokyo (Kamiya laboratory) for active discussion. We also thank Dr. Yoshizawa of University of Tokyo for his valuable advice. References

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