*Tel.: #44-171-9755369; fax: #44-181-9831007.E-mail address: wen.wang@qmw.ac.uk (W. Wang).

Journal of Biomechanics 33 (2000) 543}548

A critical parameter for transcapillary exchange of small solutesin countercurrent systems

Wen Wang*Medical Engineering Division, Department of Engineering, Queen Mary and Westxeld College, London E1 4NS, UK

Accepted 9 November 1999

Abstract

Small solute transport by a countercurrent capillary loop was studied using a theoretical model. In the model, the a!erent and thee!erent limbs of the loop share a common interstitial space, with which exchange of solute occurs. Sources of solute, epithelial cells,exist near capillaries and secret solute into the interstitial #uid. Parameters based on experimental measurements on youngSprague}Dawley rats were used in the model, and asymptotic solutions were derived. Comparison of the solute distribution in theinterstitium between a capillary loop and a single capillary reveals that the ratio of the product of permeability (P

1) and surface area

(A1) to #ow (F

1) of the a!erent limb, c

1"P

1A

1/F

1is a critical parameter for the countercurrent exchange system. It alone

determines whether the countercurrent arrangement of capillaries facilitates clearance of solute from the interstitial #uid, a greateraxial gradient of solute in the interstitium from the base to the tip of the capillary loop and a greater e!ect of #ow, F, upon thisgradient. The properties of the e!erent limb a!ect the results, but it is c

1that determines the characteristic di!erence between

a capillary loop and a single capillary. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Transcapillary exchange; Renal medulla; Theoretical model

1. Introduction

Countercurrent exchange systems consist of parallelvessels running close to each other with opposing #ows.Exchange of solutes, gas or heat between the vesselsoccurs either directly or through an intermediate me-dium. Such arrangements are widely used in engineeringto minimise heat loss or to maximise cooling. Counter#ow in nearby arterial and venous vessels facilitates gasand solute exchanges between them (e.g. Marsh andSegel, 1971; Piiper et al., 1984; Henderson and Daniel,1984; Rasio et al., 1994). In the mammalian kidney, forexample, extensive countercurrent exchange of ions,water, urea and proteins occurs in the medulla betweenthe descending and the ascending vasa recta and theloops of Henle. High concentrations of salt and ureaaccumulated during antidiuresis are believed to be sus-tained by the countercurrent exchange between the de-scending and the ascending vasa recta (e.g. Jamison and

Kriz, 1982; Pallone et al., 1990; Layton, 1990; Stephensonet al., 1995; Michel, 1995).

Although there have been several analyses of counter-current exchange systems, most have considered arrange-ments where the exchange vessels lie between the sourceand the sink of the solute or the energy (eg. heat) which isbeing exchanged. These analyses have been used to inves-tigate and describe the exchange of oxygen in the retemirabile of the swim bladders of teleost "sh (Rasio et al.,1994), exchange of heat between deep vessels of the limbsleading to and from the distal exchange surface (Mitchelland Myers, 1968), gas exchange between arterial andvenous vessels (Piiper et al., 1984). In super"cial vessels ofthe vascular bundles of the outer medulla, in the vasarecta of the inner medulla and in continuous portalsystems, the source of solute which is secreted into theinterstitial #uid lies parallel to the exchange vessels. Thissuggests a di!erent arrangement of the countercurrentexchange system, where the source or sink of the soluterun parallel to the vessels. In our previous studies, weinvestigated the extent to which convective transport anddi!usive transport interact under these conditions in thedevelopment of the solute concentration and concentration

0021-9290/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 1 - 9 2 9 0 ( 9 9 ) 0 0 2 2 0 - 1

Nomenclature

A1

surface area of the a!erent limb, cm2

A2

surface area of the e!erent limb, cm2

C0

solute concentration at the entrance of the a!erentlimb, mmol cm~3

C1

solute concentration in the a!erent limb, mmol cm~3

C2

solute concentration in the e!erent limb, mmol cm~3

C*

solute concentration in the interstitium, mmolcm~3

CMS

mean solute concentration in the ISF surroundinga single capillary

CML

mean solute concentration in the ISF surroundinga capillary loop

D#

di!usion coe$cient of solute in plasma, cm2 s~1

D*

di!usion coe$cient of solute in the interstitium,cm2 s~1

F #ow rate, cm3 s~1

GMS

e!ective gradient of solute concentration in the ISFsurrounding a single capillary

GML

e!ective gradient of solute concentration in the ISFsurrounding a capillary loop

¸ length of the exchange system, cmP1

permeability of the a!erent limb to solute, cm s~1

P2

permeability of the e!erent limb to solute, cms~1

r1

radius of the a!erent limb, cmr2

radius of the e!erent limb, cmS1

cross-sectional area of the a!erent limb cm2

S2

cross-sectional area of the e!erent limb, cm2

;1

average #ow velocity in the a!erent limb, cm s~1

;2

average #ow velocity in the e!erent limb, cm s~1

x length in the axial direction of the unit, cm

Greek lettersa solute input rate into a unit interstitial space,

mmol cm~3 s~1

c1

("A1P

1/F)

c2

("A2P

2/F)

Fig. 1. Schematic geometry of the countercurrent capillary loop* in-terstitium unit. xH is the axial distance along the direction of the #ow inthe a!erent limb (AL). The length of the unit is ¸, the cross-sectionalareas of the ISF, the AL and the EL are S

*, S

1and S

2. The average #ow

velocities in the AL and the EL are ;1

and ;2. Solute is secreted into

the ISF from an external source at a constant rate, aH.

gradient along the axis of the #ow (Wang et al., 1996;Wang and Michel, 1997).

This study focuses on a critical parameter, the ratio ofthe product of permeability (P

1) and surface area (A

1) to

#ow (F) of the a!erent limb, c"P1A

1/F, which deter-

mines characteristic di!erences between a single capillaryand a countercurrent capillary loop. As in previous stud-ies, small solute exchange between vasa recta and inter-stitial #uid in the renal medulla is used as an example.Data from experimental measurements on youngSprague}Dawley rats (15 d old) are used to estimate theorder of the magnitude of the parameters. In order tohave a clearer picture of the interactions between theparameters of physiological interest, several features ofthe renal medulla, e.g. the changing ratio of the cross-sectional areas between the ISF and capillaries at di!er-ent depths in the medulla and the exchange of #uid andlarge molecules between the ISF and vasa recta, werepurposely omitted from the model. The coupled trans-port of large molecules and #uid with small solutes in-creases the complexity of the system. In such a case,transport equations are no longer linear and we have toseek numerical solutions (Pallone et al., 1984; Wang andMichel, 1995).

2. Theoretical model

In the renal medulla, the loops of Henle of the nephronact as the source of solute that is secreted into themedullary ISF. The descending and ascending vasa rectaare, on average, only a few micrometers apart, and runparallel to the loops of Henle for several millimetres.They form a countercurrent exchange system for solutes

such as Na`, K`, Cl~ and urea in the renal medulla. Forsmall solutes, their transcapillary exchange is dominantlydi!usive. In order to simplify the medullary microcircula-tion using a basic unit, the following assumptions weremade about the arrangement of exchange vessels andtheir properties: the vasa recta are simple loops bearinga constant relation to each other and to the neighbouringnephron segments, secretion of solute from epithelia intothe ISF is uniform and constant, and #ow in the vasarecta is steady. These simpli"cations are supported byexperimental observation on the anatomical structure ofthe vasa recta and other vessels in the renal medulla andsteady #ows in the vasa recta (Jamison and Kriz, 1982;Michel, 1995). The basic unit includes a capillary loopwith an a!erent limb (AL), an e!erent limb (EL), a com-mon interstitium (ISF) and an external source (Fig. 1).The cross-sectional areas of the AL, the EL and the ISF

544 W. Wang / Journal of Biomechanics 33 (2000) 543}548

are S1, S

2and S

*respectively. The size of the cross-

section of the basic unit is indeed very small (approxim-ately 20}30 lm) compared to its length, ¸ (2}5 mm), andthe transient time for solute di!usion in this plane ismuch smaller than that of the solute transport in theaxial direction. Inside each compartment (i.e. ISF, AL orEL), variation of solute concentration in the transversesection is negligibly small compared to that over thelength of the unit. Solute concentrations in the ISF,the AL and the EL are therefore assumed to vary only inthe axial direction. Flow enters the AL at a given concen-tration of the solute, C

0and an average velocity of

#ow, ;1. AL and EL can have di!erent properties

such as cross-sectional area and solute permeability. Vol-ume #ow rate, F, is constant in the capillary loop, so that;

1S1";

2S2"F. In the analysis, we use subscripts

1 for the AL, 2 for the EL and i for the ISF.Conservation of solute in the ISF gives

S*aH!2pr

1P

1(CH

*!CH

1)!2pr

2P2(CH

*!CH

2)

#DH*SH*

d2CH*

dxH2"0, (1)

where xH is the length in the axial direction of the unit,aH is the constant solute input rate into the ISF per unitaxial length per unit cross-sectional area of the ISF. Ithas the dimension of mmol cm~3 s~1. CH is the concen-tration of the solute, DH is the di!usion coe$cient of thesolute, P is the permeability of the solute and r is theradius of the capillary. In the capillary loop, there areboth convective and di!usive transports of solute. Massconservation in the AL and EL gives

2pr1P

1(CH

*!CH

1)!F

dCH1

dxH#S

1DH

#

d2CH1

dxH2"0, (2)

2pr2P

2(CH

*!CH

2)#F

dCH2

dxH#S

2DH

#

d2CH2

dxH2"0, (3)

where DH#

is the di!usion coe$cient of the solute in blood.Nondimensionalisation is carried out using ;

1, C

0and

¸, and the nondimensional variables are x"xH/¸,C*"

CH*/C

0, C

1"CH

1/C

0and C

2"CH

2/C

0. The nondimen-

sional parameters which arise are

a"aHS*¸/(C

0F), c

1"2pr

1¸P

1/F,

c2"2pr

2¸P

2/F, D

*"DH

*S*/(¸F) and Pe";

1¸/DH

#.

Typical values of Pe and Dican be found using values

for solute transport in and across the vasa recta andin the ISF. The di!usion coe$cient for a small solute,such as Na`, is of the order of 10~6 cm2 s~1 in the ISFand of the order of 10~5 cm2 s~1 in blood. In the renalmedulla of rat, the axial length from the cortical}medul-lary junction to the tip of the papilla is several mil-limetres, the radii of the vasa recta are about 10 lm andthe ratio of the cross-sectional areas of the ISF and thecapillary is about 1 (Knepper et al., 1977). The average

#ow velocity in the vasa recta is about 10~2 cms~1 (Marshand Segel, 1971). Thus Pe (+102)<1 and D

*(+10~3)(1.

Di!usive transport of solute in the axial direction ofthe unit is several hundred times smaller compared toeither the convective transport of solute by circulationor the transcapillary transport of solute between theISF and the vasa recta. Di!usive terms in Eqs. (1)}(3) are,therefore, neglected and the simpli"ed nondimensionalequations are

a!c1(C

*!C

1)!c

2(C

*!C

2)"0, (4)

c1(C

*!C

1)!

dC1

dx"0 (5)

c2(C

*!C

2)#

dC2

dx"0 (6)

The boundary conditions are

at x"0: C1"1, (7)

C2"1#a. (8)

Solute concentrations in the AL, EL and ISF are

C1"1#a

c1

c1#c

2

x#ac1c2

c1#c

2Ax!

x2

2 B, (9)

C2"1#a!a

c2

c1#c

2

x#ac1c2

c1#c

2Ax!

x2

2 B (10)

C*"1#a

1#c2

c1#c

2

#ac1!c

2c1#c

2

x#ac1c2

c1#c

2Ax!

x2

2 B(11)

The term c1c2/(c

1#c

2) is the equivalent permeability of

the two membranes, the walls of the AL and EL, acting inseries. The concentration di!erence between the AL andthe EL, C

2!C

1"a(1!x), and the `di!usive shunta

between the AL and the EL or equivalently the solutetrapping by the countercurrent exchange is a(c

1c2/

(c1#c

2))(1!x). Thus, the last term in Eqs. (9)}(11)

represents the e!ect of the countercurrent exchange ofsolute between the AL and the EL. c

1/(c

1#c

2) and

c2/(c

1#c

2) are the fraction of the overall permeability of

the capillary loop represented by the AL and EL, respec-tively, and represent their contribution to solute ex-change between the vasa recta and the ISF in the absenceof the di!usive shunt.

For comparison, the solute exchange between a singlecapillary and the surrounding ISF is derived similarly.Solute concentrations in the single capillary, c and in theISF, c

*are

c"1#ax, (12)

c*"1#a/c

1#ax. (13)

W. Wang / Journal of Biomechanics 33 (2000) 543}548 545

Fig. 2. E!ects of #ow on the mean solute concentration in the ISF. Inthe "gure, S

*"S

1"S

2, ¸"2 mm, r

1"10 lm, P

1"5]10~4 cm s,

P2"10~3 cm s, C

0"0.15 mmol cm~3 and the solute secretion rate is

assumed to be 0.03 mmol cm~3 s~1. (***) the mean concentrationof solute in the ISF surrounding a capillary loop. (- - - - - - -) the meanconcentration of solute in the ISF surrounding a single capillary.

Fig. 3. E!ects of c1

and c2

on the ratio of the mean axial gradients ofsolute concentration in the ISF, GM

L/GM

S. c

1"4 is a critical value. When

c1'4, GM

L/GM

S*1 for all c

2, when c

1(4, GM

L/GM

S)1.

3. Results and discussion

Small solutes distribution in the renal medulla, parti-cularly, the axial gradients of the concentration of solutesfrom the cortical}medullary junction to the tip of thepapilla play a major role in the urine concentratingmechanism of the kidney. In this section, we compare thesolute clearance from the ISF by a countercurrent capil-lary loop to that by a single capillary, and investigate theaxial gradient of the solute in the ISF and the e!ect of#ow rate, F, upon this gradient in the two di!erentexchange systems.

In the ISF surrounding a single capillary, the meansolute concentration

CMS,P

1

0

c*(x) dx"1#

a2#

ac1

. (14)

In the ISF surrounding a capillary loop, the meansolute concentration

CML"P

1

0

C*(x) dx"1#

a2#

ac1#c

2

#

a3

c1c2

c1#c

2

. (15)

Considering the de"nition of nondimensional para-meters, a"aHS

*¸/(C

0F), c

1"P

1A

1/F and c

2"P

2A

2/

F, a, c1

and c2

are all inversely proportional to F. Thelast term in Eq. (15), however, is inversely proportional toF2. In Fig. 2, e!ects of #ow on the mean solute concentra-tion in the ISF are presented. Parameters used in thecalculation are based on data from experimental measure-ments on young Sprague-Dawley rats: ¸"2 mm,

r1"10 lm, r

2"10 lm, P

1"5]10~4 cms~1, P

2"

10~3 cm s~1, C0"0.15 mmol cm~3 and;

1"400 lm s~1

S*/S

1is assumed to be 1. There are no available data on

the solute secretion rate by the loops of Henle in the renalmedulla, so a value of 0.03 mmol cm~3 s~1 was used foraH which gives a"1 when the above values are used. InFig. 2, all parameters were kept the same apart from the#ow velocity, ;

1. It is seen that changes in #ow velocity

have a much greater e!ect on the mean concentration ofthe solute in the ISF surrounding a capillary loop. Whenthe velocity increases, the countercurrent arrangement ofa capillary loop causes a bigger decrease of the meanconcentration of the solute in the ISF than that ofa single capillary, and when #ow decreases, the capillaryloop is much more e!ective than a single capillary infacilatating solute accumulation in the ISF.

The simple formulae for the distribution of solute inthe ISF for a single capillary Eqs. (12) and (13) anda capillary loop Eqs. (9)}(11) make it convenient to com-pare the two exchange systems. The e!ective axial gradi-ent of the solute in the ISF surrounding a capillary loopis de"ned as

GML,

C*(1)!C

*(0)

1"a

c1#1

2c1c2!c

2c1#c

2

. (16)

In a single capillary, the axial gradient of solute in theISF,

GMS,

c*(1)!c

*(0)

1"a. (17)

Comparison between GML

and GMS

shows that c1

isa critical parameter, which alone determines whether thecountercurrent exchange system has a greater axialgradient of solute in the ISF. c

1"4 is a critical value. As

shown in Fig. 3, when c1'4, for all values of c

2, the

546 W. Wang / Journal of Biomechanics 33 (2000) 543}548

Fig. 4. E!ects of c1

and c2

on the ratio DLGML/LFD/DLGM

S/LFD. c

1"2 is

a critical value. When c1'2, the e!ect of #ow on the axial gradient of

the solute concentration in the ISF is greater for a capillary loop thana single capillary, when c

1(2, the e!ect of #ow is greater in a single

capillary.

countercurrent capillary loop has an e!ective concentra-tion gradient in the ISF greater than that of a singlecapillary, and when c

1(4, countercurrent solute ex-

change between the AL and EL results in a smaller solutegradient in the ISF than that of a single capillary. c

2af-

fects the magnitude of the ratio, but it is c1

alone thatdetermines the characteristic di!erence betweena countercurrent exchange system and a single capillary.In the renal medulla, c

1is at least 5}10, so that the

countercurrent arrangement of vasa recta there facilitateshigh gradient of solutes in the ISF.

We now compare the e!ects of #ow rate, F, upon theaxial gradients of the solute in the ISF surroundinga single capillary and a countercurrent capillary loop. Ina single capillary, the mean concentration gradient in theISF GM

S"a, and its magnitude of the rate of change of

#ow DLGMS/LFD"DLa/LFD. In a capillary loop, the mean

axial gradient of the solute in the ISF GML"a(c

1#

12c1c2!c

2)/(c

1#c

2), and its magnitude of the rate of

change of #ow DLGML/LFD"D(L/LF)(a(c

1#1

2c1c2!c

2)/

(c1#/c

2)D. Considering the nondimensional parameters,

a"aHS*¸/(C

0F), c

1"P

1A

1/F and c

2"P

2A

2/F,

KLaLF K"

aF

, (18)

KL

LFAa#a12c1!2

c1/c

2#1BK"

aF K

c1#c

1c2!c

2c1#c

2K. (19)

Thus

KLGM

LLF KNK

LGMS

LF K"Kc1!c

2#c

1c2

c1#c

2K.

When the capillary loop has a greater sensitivity tochanges in #ow, D(c

1!c

2#c

1c2)/(c

1#c

2)D'1.

Again, it is seen that c1

is the critical parameter, whichdetermines whether the countercurrent arrangement en-ables a greater e!ect of #ow upon the mean solute con-centration gradient in the ISF. In Fig. 4, sensitivity ofthe mean ISF concentration gradient to #ow in a capil-lary loop and in a single capillary is compared. It is seenthat c

1"2 is a critical value. When c

1'2, the mean

axial gradient of C*

around a capillary loop is moresensitive to changes in #ow than is the axial gradientaround a single capillary. When c

1(2, changes in #ow

have a larger e!ect upon the axial gradient of c*around

the single capillary. As with the concentration gradientin Fig. 3, c

2a!ects the magnitude of the ratio, but

c1

alone determines whether the ratio is greater or lessthan 1 when changes in #ow rate occur.

4. Conclusion

This paper investigated countercurrent exchange sys-tems by comparing solute distribution in the ISF sur-rounding a capillary loop to that surrounding a singlecapillary. It is found that although the ratio of the prod-uct of permeability and surface area to #ow in the e!erentlimb, c

2, a!ects the magnitude of the solute concentra-

tion in the ISF, the ratio of the product of permeabilityand surface area to #ow in the a!erent limb, c

1, is much

more important and alone determines whether:

(a) the axial gradient of solute concentration in the ISFis greater in a countercurrent exchange loop thanthat in a single exchange vessel (c

1'4); and

(b) the #ow, F, has a greater e!ect on this axial gradientin a countercurrent exchange system (c

1'2).

In the renal medulla, c1

has a value of at least 5}10, sothat compared to a single capillary, countercurrentexchange of solute between the descending and the as-cending vasa recta facilitates a higher gradient of soluteconcentrations in the interstitium. Furthermore, whenantidiurtic hormone reduces medullary #ow, the counter-current arrangement of the vasa recta favours a muchmore rapid accumulation of solutes in the ISF and en-sures a rapid development of an axial gradient of solute.Although transcapillary exchange of solute in the renalmedulla was used as an example to demonstrate thecritical parameter and its e!ects on solute distribution incountercurrent exchange systems, results from the modelhave a much more general application in exchange sys-tems of similar arrangement.

Acknowledgements

This work was carried out while the author worked atthe Imperial College Medical School at St. Marys, Lon-don with support from the Wellcome Trust.

W. Wang / Journal of Biomechanics 33 (2000) 543}548 547

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