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The osmotic permeability of a tubule wall
Bulletin of Mathematical Biology Vol.50, No. 5, pp. 547-558, 1988.
Printedin Great Britain,
0092-8240/88$3.00+0.00 PergamonPressplc Societyfor MathematicalBiology
THE OSMOTIC PERMEABILITY OF A TUBULE WALL GEOFFREY K. A L D I S
Department of Mathematics, University College, The University of New South Wales, Australian Defence Force Academy, Campbell, A.C.T. 2600, Australia The influence of unstirred layers in osmotic experiments designed to measure the osmotic permeability of a tubule wall is considered. Results are given in the form of a set of graphs whose axes are closely related to observed and k n o w n experimental parameters. These enable osmotic permeability values to be o b t a i n e d which are closer to the true values.
1. Introduction. Osmotic inflow experiments on tubules aimed at determining the osmotic permeability of the tubule wall are affected by unstirred layers (USLs) on both the internal and external surfaces of the tubule wall. Unstirred layers result in an underestimate for the osmotic permeability value unless allowance is made for their influence. Errors due to USLs can be reduced by stirring the solutions vigorously and allowing only weak osmotic flows but can never be completely eliminated. This paper applies the analysis of USLs given in Aldis (1988) to osmotic experiments on tubules. The Aldis (1988) analysis describes the interaction between osmotic inflow and the bulk flow in a tubule and models the experiments by O'Donnell et al. (1982). This arrangement aims to almost eliminate external USLs and allows us to consider the effect of the inner USL only. A consequence of the non-linear nature of osmosis problems is that the results of this paper are not valid for outflow problems. A length l of a tubule with radius a is assumed to be exposed to a perfectly-stirred external solution with solute concentration C o. A bulk flow in the tubule with upstream concentration C i is augmented by osmotic inflow through the wall at a local flux rate J given by
J=e(Cw-Col
11)
where P is the osmotic permeability of the tubule wall and C w is the solute concentration at the inside wall of the tubule. The internal and external solutions have the same single, uncharged solute, with diffusion coefficient D in water. Flow in the tubule is assumed to have a parabolic axial flow profile and the shear rate at the wall, ~, is linearly related to the volume flow rate, Q, and the mean axial velocity U by 547
548
GEOFFREY K. ALDIS
4Q
4u
7za 3
a
(2)
In Aldis (1988) the transformed distance variable ~ was related to the dimensional distance variable along the tubule, x, by (3) where AC is the upstream concentration difference C i - - C o and c~o is the upstream shear rate. In the O'Donnell et al. (1982) experiments the external bathing solution was a stream of fluid between two blunted syringe tips lying within a bath of paraffin oil (see Fig. 1).
ST
- "~
ST
Figure 1. A Malpighian tubule (MT) lies in the jet of fluid between two syringe tips (ST). Fluid is collected as droplets (D) from the cut end of the tubule.
The tubule was placed across the stream and surface tension confined the bathing solution to a short length of the tubule. The solution external to the tubule cannot be perfectly stirred but this experimental arrangetnent enables the effect of the external USL to be greatly reduced. Values of P obtained by the O'Donnell et al. procedure are up to five times higher than for previous methods using a stagnant external bathing solution (Maddrell, 1980). The influence of an internal USL acting in the absence of an external USL is treated in Section 2. The external USL is discussed briefly in Section 3 and in more detail in an Appendix. The O'Donnell et al. (1982) experiments are considered in detail in Section 4. 2. Internal Unstirred Layers. Parameters corresponding to experimental inputs are the perfusion length l, tubule radius a the initial tubule flow rate Qi, solute concentration in the tubule upstream of the inflow region Ci, external solution concentration C O and the solute diffusivity D. The principal experimental output is the emergent flow rate Qe (or equivalently the emergent
T H E O S M O T I C PERMEABILITY OF A T U B U L E WALL
549
concentration C¢ since Ce/Ci m Qi/Qe). An important nondimensional parameter which arises in the analysis of Aldis (1988) is k, where PACa k- - D
(4)
The relative concentration difference between the two solutions is another important quantity. In Aldis (1988) fl, the reciprocal of the relative concentration difference is defined by Co Co /~ = C i - C o - AC"
(5)
Results are given here for fi=0, 2 and 10, i.e. no solute in the external solution, and internal solution concentration raised by 50% and 10% over the external concentration, respectively. Figure 2 shows plots of Qe/Qi against lD/Q i for various values of k and fi = 0, 2 and 10. Both the abscissa and the ordinate are nondimensional and the ordinate could equally be CI/C ¢ or ~(~max)/% with the same scale where ~max is the transformed length of the perfused region. Since the flow inside the tubule is parabolic the abscissa is equivalent to 4~3max/grck3. An experiment corresponds to a point in (Q¢/Qi, ID/Qi) space and only one k curve, k -- ko say, passes through that point. The experimental inputs AC, a and D are then used to determine the osmotic permeability P from equation (4). This method of determining P is clearly less useful whefi the resolution between the different k curves is poor, i.e. when k is large or when e/ao is near 1 or near (1 + fl)/fl. For some physiological experiments k ~ 1 and direct application of a onedimensional model will give accurate approximations to k and P. Onedimensional models make the assumption that there are no gradients in solute concentration across the tubule, i.e. concentration varies only with distance along the tubule. Recasting equations (23) and (24) of Aldis (1988) gives ko = 2- ~l l / ~ ] { ~ - l + ~ - ~ l o g l i + f l - f l - ~ . ~ ]
}.
(6)
This is equivalent to equation (7) of Du Bois et al. (1976) except that inflow is being modelled here. When k is large Fig. 2 gives poor resolution and other procedures may be necessary. For sufficiently large k the small ~ power series solutions (Aldis, 1988) with k = oo can be used. These are strictly only valid for constant a, i.e. for an infinite radius tubule but they are close approximations provided a/a o ~ 1. Re-expressing equation (33) of Aldis (1988), a value of k is sought to satisfy
550
GEOFFREY
K. A L D I S
2.0 1.8
1.6 Qe
Qi
1.4
1.2 1.0 1.5 b
1.4
1.3 Qe
Qi
1.2
1.1 1.0
1.10
1.08
Q~-i
1.06 1.04
1.02
1.00 0,01
I 1
0.1 '
i
J i i
i i
10
~.D/Q i
Figure 2. Computed solutions for QJQi represented in terms of experimental parameters. The curves represent (equally log-spaced) lines of constant k. (a) fl = 0, (b)/~ = 2, (c)/~ = 10.
Qe 1 -F 8 [~j~(91Dn/4Qi)I/3 Qi
3-~
¢2~(¢) de
(7)
where 7(~) is a nondimensional form of the wall concentration and P is then obtained from k. The terms on the right hand side of equation (7) can be tabulated against k and upper and lower bounds can be placed on P. If values of
THE OSMOTIC PERMEABILITY OF A TUBULE WALL
551
lie outside the validity of the small ~ expansions then other methods are required. Some alternatives to a full numerical solution are given in Aldis (1983). The information from Fig. 2 is represented in another form suitable for experimental use in Fig. 3. The curves have been obtained from cubic spline interpolation of the information contained in Fig. 2. In a set of experiments l, D and Qi are often fixed, the experimental output is Qe/Qi and k is required. Therefore, once an experimental set-up has been established, one only needs to follow events along a single lD/Qi-curve. The simplest model commonly used to determine P for a tubule wall assumes uniform solute concentration across the tubule at any position along the perfusion length and assumes a linear fall in concentration along the inflow length. This model was used for example by Grantham and Orloff (1963). The average concentration difference across the wall is taken to be the average of the bulk solution differences at the two ends points. Ps is then inferred from ks where ks is given by Qe m
ks__!FQi~FQe l I - r e LUALQii
Qi
.
(8)
Ii+fl+ (1-]~)~i]
This model can give large errors since it ignores two major points. The first is that osmotic inflow is most rapid at the start of the perfused region and becomes progressively slower downstream. The assumption of a linear fall is clearly not suited to this type of problem. This difficulty is easily met by using the one-dimensional mode ! [equation (6)]. The other difficulty is that osmotic inflow depends on the concentration at the wall and that may not be adequately represented by any one-dimensional model. As k increases the bulk concentration becomes a progressively poorer estimate of the wall concentration since the osmotic disturbance is confined to a thin USL. The ratio of equations (8) and (6) indicates how different the estimates for P are when calculated using the simplest model (Ps) and the one-dimensional (Du Bois) model (PD). We have Qe
(9) PD
kl)
Ii+]~+(l_fl)~il{_~_l+~logIl+iff_
fl~l)"
As Du Bois pointed out, the one-dimensional model, equation (6), should always be used in preference to the simplest model, equation (8), and only the
552 GEOFFREY K.ALDIS 10
1
k 0.1
0.01
1.0
1.2
1,4
1.6
1.8
2.0
1.4
1.5
LoU//oJ
1
k 0.1
0.01
1.0
1.1 o
1
1.2
//
1.3
/ 0.01
~
k
j
0.1
0.1 0.01 1.00
1.02
1.04
1.06
1.08
1.10
Qe/Qi Figure 3. Computed solutions from Fig. 2 represented in an alternative manner. The curves represent (equally log-spaced) lines of constant lD/Q i. (a) fl = 0, (b) fl = 2, (c)/~ = 10. one-dimensional model and the numerical results from Aldis (1988) are compared further here. The positions where the one-dimensional and numerical estimates for k differ by 10 and 20% are shown in Fig. 2 by dashed and dotted curves, respectively. These curves and the curves for other percentage deviations are almost parallel to the solid curves over a large part of the figure. Assuming the curves are parallel it is natural to inquire h o w the percentage deviation varies with k. Figure 4 shows this dependence for the three cases/Y = 0, 2 and 10. The curves
THE OSMOTIC PERMEABILITY OF A TUBULE WALL
553
have been drawn with (Qe/Qi)=l.5, 1.25 and 1.05 for //=0, 2 and 10, respectively (i.e. assuming the percentage deviation along the horizontal line halfway down each of the graphs in Fig. 2 is representative). The ordinate kD/k is identical to PD/P. 1.0
ko -~
0.5
0
,
0.001
b
q l l t H
i
0.01
,
i,,,,,i
,
0.1
,
,,,,,,I
,
1
,
,
,,,,,
10
k
Figure 4. O n e - d i m e n s i o n a l model estimate of k over k estimated from the full
convection-diffusionsolution of Aldis (1988) against the latter estimate for/3= 0, 2 and 10 as shown. The choices of Qe/Qi=1.5 (/3=0), 1.25 (/3=2)and 1.05 (/3= 10) were made to obtain these representative curves. A 10% error in estimating P is therefore expected when using the onedimensional model wh6n (the true) k ~ 0 . 3 f o r / 3 = 0 , k ~ 0 . 0 9 f o r / ~ = 2 and k ~ 0 . 0 2 for /~=10. Above these values of k the predictions of the onedimensional and the full convection-diffusion model diverge further.
3. External Unstirred Layers. The comments here are confined to inflow situations. External USLs are likely to be more severe in outflow situations. External USLs cannot always be neglected but the experimental arrangement shown in Fig. 1 tends to reduce their influence. Firstly the osmotic flow outside the tubule is directed towards the tubule wall, so that it thins the USL and therefore enhances back-diffusion of solute. Secondly, the length of the USL on the exterior surface (half the outer circumference) will usually be much less than the inflow length, l, so that the USL will be less developed. Thirdly the bulk flow around the tubule can be increased until the external USL effect is almost eliminated without otherwise disturbing the experiment. In practice an upper limit on the external flow is forced by the need to keep the tubule in place without damaging it and to prevent a hydrostatic pressure driven flow of water across the tubule wall. The strength of the external USL is estimated for an experimental example in the Appendix and it is found to be weaker than the
•
554
°
G E O F F R E Y K. ALDIS
internal USL. Schwartz et al. (1981 ) found a significant external USL effect was present in a tubule diffusion study conducted in a well-bubbled medium. The stirring flow of O'Donnell et al. (1982) between two syringe tips concentrates the stirring at the tubule and would be expected to reduce the USL effect in those experiments. 4. Examples• We describe the procedure used to determine P from the O'Donnell et al. experiments on Rhodnius Malpighian tubules assuming that there is no external USL. The simplest estimate of osmotic permeability, Ps=4.3 x 10 -3 cm litre/s osmol, reported in O'Donnell et al. t1982) is an average value obtained using the linear model, equation (8), for the fall in osmotic gradient on each experiment. In order to calculate a more accurate value of Ps with the full convection-diffusion model one needs a single set of parameters representing the experiments• The set used was a i (inner radius): 32.5 #m; a o (outer radius): 50 #m; l (length of permeable segment): 0.09 cm; C i (initial bulk concentration): 340 mosmol/litre; Co (external bath concentration): 170 mosmol/litre; D (diffusivity of NaCI): 1.5 x 10 -5 cmZ/s; Qi (initial flow): 4 x 10 - 6 c m 3 / s ; C e (emergent concentration): 247 mosmol/litre; Qe (emergent flow): 5.5 x 10 - 6 c m 3 / s . Since the main solute in the bathing solutions was NaC1 we assume it is the only solute• In the experiments a i, a o, l, C i, C o, D and Q~ were usually kept constant. Ce and Q, have been chosen here to give the average Po value based on the external area of the tubule• The theory presented earlier is concerned with the interaction between the bulk flow and the inner USL and an equivalent problem for the tubule lumen is formulated here. Basmadjian and Baines (1980) showed in experiments with models of a brushborder-lined tubule that typically less than 1% of the bulk flow passes between the microvilli. The microvilli can therefore be ignored for our purposes and the tubule wall is replaced by a zero-thickness wall at the level of the microvfllous tips (at radius ai). In order to generate the correct flow through the resulting smaller surface area the permeability of the wall P~ is scaled up with P~= 50/32.5 Ps. The internal concentration C i was twice the external concentration and so /3 = 1. Graphs summarizing the/~ = 1 case are not given in this paper as it corresponds to an extreme osmotic gradient not likely to be commonly used. The estimation of P therefore requires numerical solution of the convectiondiffusion equation as described in Aldis (1988). The parameter k = P~ACa/D and transformed length of the perfused segment ~max=k(91D/o~oa3) 1/3 were varied until the numerical solution reproduced the correct osmotic inflow, i.e. o~(~max)/o~o=Qe/Qi. The theoretical estimate Pts=32.5/50 Pi for the outer surface of the tubule was found to be related to P~ by Pt~ = 1.35 P,, therefore predicting a 35% increase to obtain the true osmotic permeability. The one-
THE OSMOTIC PERMEABILITY OF A TUBULE WALL
555
dimensional model, equation (6), was also applied to this problem giving PD = 1.09 Ps. This is well short of the predicted value Pts= 1.35 Ps and therefore an USL must be causing a major part of the discrepancy. The value of k is 0.33 and the one-dimensional model cannot be expected to give accurate results. The case /3= 1 corresponds to a severe osmotic gradient for physiological tissues and, although the Rhodnius Malpighian tubule can withstand this gradient, the same experiments can be reliably performed using a smaller osmotic gradient. Since most tissues will require lower gradients, full results were not included in this paper for the case/~ = 1. Figure 5 shows plots of the dimensional osmotic gradient C w - C O for the linear model, one-dimensional model and the full convection-diffusion solution as a function of distance along the perfused length. An USL is clearly present since even at the end of the perfused region the wall concentration differs significantly from the bulk concentration. Figure 6 shows the nondimensional concentration profiles across the tubule predicted by the convection-diffusion model at positions one quarter, one half, three quarters and the full length of the tubule. I
200
I
I
I
[
~..~'-.,.
/
P
,s0
~
lO0 ~
50 0
J
I
I
0.2
0.4
0.6
I~"~ 0.8
1.0
X Figure 5. Three models of the O'Donnell et al. (1982) experiments with 8 = 1
showing dimensional concentration differential(mosmol) at the tubule wall against ; full numerical solution. ; a one-dimensional model with linear behaviour between the end points assumed. ; the one-dimensional model of Aldis (1988) with P adjusted to give the observed cumulative inflow at X=I.
X= x/l.
The validity of the convection-diffusion model is checked by noting that l>> a o, that the osmotic inflow is weak compared to the bulk flow, and that Pe=eoa2/2D=52>> 1. The Reynolds number for the tube flow is small at approximately 0.1. Although this example shows that a USL may be present in physiological tubules under experimental conditions it m a y not be typical.
556
GEOFFREY K. ALDIS 1.0
I
I
I
I
o.a [0.6
0.4
0.2
0 0
I
I
I
I
0.2
0.4
0.6
0.8
; .0
R
Figure 6. Radial concentration profilesfor the O'Donnell et al. (1982)experiments at positions (top to bottom in the Figure) I/4, l/2, 31/4, I. Abscissa, R, is nondimensionalized with the tube radius and the ordinate is nondimensional concentration 0 = ( C - C o )/AC. Malpighian tubules have a high osmotic permeability and here the osmotic gradient AC is large, k can be reduced by decreasing AC. Another set of parameters was supplied by Whittembury (private communication) in order to assess the likely influence of an USL in N e c t u r u s proximal tubule experiments. The set gave k--0.02,/3 = 10, Pe = 230 and R e = 0.2. The one-dimensional model would give a better estimate for P than in the previous example, but there would still be an USL. R h o d n i u s Malpighian tubules and rat proximal tubules have relatively high k values associated with them because of their high osmotic permeabilities. Values of k for other epithelial tubules including N e c t u r u s proximal tubule would typically be lower. Rhodnius
Experimenters need to know how large USL effects are and also how strongly they respond to changes in the experimental conditions. The figures included here enable USL effects to be estimated for osmotic experiments on tubules when the luminal USL is the major USL influence. Basmadjian and Baines (1978) and Friedlander and Walser (1965) concluded that various parts of the nephron can be adequately modelled by a onedimensional theory during normal operation. However, the results given here enable experiments to be performed beyond physiological conditions or on artificial tubules with knowledge of the likely internal USL effect. If the value of k in an experiment is large then an attempt should be made to reduce it since USL effects are likely to be important. However in practice there is limited scope for this since P and a are tubule constants and the solutes are usually chosen for physiological compatability. Therefore changing AC is the main way of changing k in practice. Once an initial experiment is performed the 5. D i s c u s s i o n .
THE OSMOTIC PERMEABILITY OF A TUBULE WALL
557
e x p e r i m e n t a l p a r a m e t e r s m a y be a d j u s t e d to give p r e d i c t a b l e results, e.g. to be m a d e to lie in a suitable r e g i o n of a g r a p h in Fig. 2 t h e r e b y a l l o w i n g a n easy e s t i m a t i o n of P. T h e a p p l i c a b i l i t y of the t h e o r y to a given s i t u a t i o n c a n also be tested b y the following p r o p e r t i e s . I f all p a r a m e t e r s except Qi are held fixed then e x p e r i m e n t a l results s h o u l d lie a l o n g a single k-curve. T h i s is also true w h e n l a l o n e is c h a n g e d . I f A C is v a r i e d while/~ a n d the o t h e r p a r a m e t e r s are k e p t c o n s t a n t t h e n the Qe/Qi values o b t a i n e d s h o u l d rise or fall to a c o r r e s p o n d i n g k-curve. It is w o r t h n o t i n g a g a i n t h a t the analysis is only valid w h e n l o n g i t u d i n a l diffusion c a n be neglected. This c o r r e s p o n d s to the n o n - d i m e n s i o n a l Peclet n u m b e r P e = e o a 2 / 2 D ~ > 1, i.e. Qi/aD~> 1 a n d Qi m u s t be sufficiently l a r g e for this c o n d i t i o n to be satisfied. T h i s w o r k w a s s u p p o r t e d b y a T r i n i t y College E x t e r n a l R e s e a r c h S t u d e n t s h i p . T h e a u t h o r also wishes to t h a n k D r s T. J. Pedley, S. H . P. M a d d r e l l a n d M . J. O ' D o n n e l l a n d the referees for helpful c o m m e n t s . T h i s w o r k was p a r t o f the r e q u i r e m e n t s for the d e g r e e of D o c t o r o f P h i l o s o p h y , at the U n i v e r s i t y of Cambridge.
APPENDIX
The External Unstirred Layer. A fully three-dimensional numerical solution is required to treat the external USL in detail. However to estimate its greatest effect in the following example we choose simplifications which tend to make the effect greater than it actually is. The external flow rate used in the O'Donnell et al. (1982) experiments was approximately 3.3 × 10 - 4 c m 3 / s . The area of a syringe tip orifice is 6.4 x 10- 3 cm z so the average flow speed U of the emerging jet is 5.2 x 10 -2 cm/s. To maximize the external USL effect we thicken the USL by reversing the direction of the osmotic flow, i.e. the external bath is initially at concentration C i and the solution inside the tubule is uniformly at CO. Since the tubule diameter is 0,1 mm and the syringe tip diameter is 0.9 mm we will ignore edge effects at x = 0 and x = l and assume the tubule is immersed in an infinite body of fluid. A typical slice across the tubule is considered in what follows. The Reynolds number for the flow is much less than one (Re = 0.05) so the flow is assumed to be a Stokes-Oseen flow past a cylinder, with flow speed at infinity equal to U. By symmetry we need only consider the upper half of the cylinder. There will be no separation of the flow and since we are considering a thin USL we will ignore the curvature of the tubule wall and flatten the wall out to a permeable section of length 5n x 10- 3 cm. A representative value for the shear rate on the notional plate is now obtained from the solution for flow around a cylinder. The stream function for Stokes-Oseen flow using polar coordinates (r, ~0) (e.g. Batchelor, 1967) is 0(r, ~o) = U sin q) - ~ C r
log -a + r 1 +
---
where C = 2/log(7.4/Re) and the shear rate c~is then given by
10uo 1~321~ C U s i n O [ 1 e(r, q~) - r ~r
r a r -f -
2
a2]
-~ + ~
"
558
G E O F F R E Y K. ALDIS
The mean value of a on the semi-circle is then 2CU ~ag and for this problem 8 = 530/s. The downstream end of the USL is therefore calculated to be at 4" =0.1 and the average value of 7 along the flat tubule wall, ~7,is given (Pedley,.1981) by
=~
u2~(u) du.
The k = oo,/~ = 1 small ~ expansion for 7 was computed and the integration resulted in y = 0.90. The external USL could therefore result in at most a 10% underestimate of P. Its effect is small compared with the internal USL in this case and is probably negligible since the model we used sought an upper bound only.
LITERATURE Aldis, G. K. 1983. Concentration Fields Near Transportin 9 Epithelia. PhD thesis. University of Cambridge. - - . 1988. "The Unstirred Layer during Osmotic Flow into a Tubule." Bull. math. Biol. 50, 531-545. Basmadjian, D. and A. D. Baines. 1978. "Examination of Transport Equations Pertaining to Permeable Elastic Tubules such as Henle's Loop." Biophysical J. 24, 629-643. - - , D. S. Dykes and A. D. Baines. 1980. "Flow Through Brushborders and Similar Protruberant Wall Structures." J. memb. Biol. 56, 183-190. Batchelor, G. K. 1967. An Introduction to Fluid Mechanics. Cambridge: Cambridge University Press. Du Bois, R., A. Verniory and M. Abramov. 1976. "Computation of the Osmotic Water Permeability of Perfused Tubule Segments." Kidney Int. 10, 478-479. Friedlander, S. K. and M. Walser. 1978. "Some Aspects of Flow and Diffusion in the Proximal Tubule of the Kidney." J. theor. Biol. 8, 87-96. Grantham, J. J. and J. Orloff. 1968. "Effect of Prostaglandin E 1 on the Permeability Response of the Isolated Collecting Tubule to Vasopressin, Adenosine 3'5'-Monophosphate and Theophylline." J. Clin. Invest. 47, 1154-1161. Maddrell, S. H. P. 1980. "Characteristics of Epithelial Transport in Insect Malpiphian Tubules." In: Current Topics in Membranes and Transport I4, 427-463. O'Donnell, M. J., G. K. Aldis and S. H. P. Maddrell. 1982. "Measurements of Osmotic Permeability in the Malpiphian Tubules e r an Insect, Rhodnius prolixus Stal." Pro¢. R. Soc. B216, 267-277. Pedley, T. J. 198t. "The Interaction Between Stirring and Osmosis. Part 2." J. Fluid Mech. 107, 281-296. Schwartz, G. J., A. M. Weinstein, R. E. Steele, J. L. Stephenson and M. B. Burg. 1981. "Carbon Dioxide Permeability of Rabbit Proximal Convoluted Tubules." Am. J. Physiol. 240, F231-F244.
R e c e i v e d 28 O c t o b e r 1987 R e v i s e d 19 A p r i l 1988
Printedin Great Britain,
0092-8240/88$3.00+0.00 PergamonPressplc Societyfor MathematicalBiology
THE OSMOTIC PERMEABILITY OF A TUBULE WALL GEOFFREY K. A L D I S
Department of Mathematics, University College, The University of New South Wales, Australian Defence Force Academy, Campbell, A.C.T. 2600, Australia The influence of unstirred layers in osmotic experiments designed to measure the osmotic permeability of a tubule wall is considered. Results are given in the form of a set of graphs whose axes are closely related to observed and k n o w n experimental parameters. These enable osmotic permeability values to be o b t a i n e d which are closer to the true values.
1. Introduction. Osmotic inflow experiments on tubules aimed at determining the osmotic permeability of the tubule wall are affected by unstirred layers (USLs) on both the internal and external surfaces of the tubule wall. Unstirred layers result in an underestimate for the osmotic permeability value unless allowance is made for their influence. Errors due to USLs can be reduced by stirring the solutions vigorously and allowing only weak osmotic flows but can never be completely eliminated. This paper applies the analysis of USLs given in Aldis (1988) to osmotic experiments on tubules. The Aldis (1988) analysis describes the interaction between osmotic inflow and the bulk flow in a tubule and models the experiments by O'Donnell et al. (1982). This arrangement aims to almost eliminate external USLs and allows us to consider the effect of the inner USL only. A consequence of the non-linear nature of osmosis problems is that the results of this paper are not valid for outflow problems. A length l of a tubule with radius a is assumed to be exposed to a perfectly-stirred external solution with solute concentration C o. A bulk flow in the tubule with upstream concentration C i is augmented by osmotic inflow through the wall at a local flux rate J given by
J=e(Cw-Col
11)
where P is the osmotic permeability of the tubule wall and C w is the solute concentration at the inside wall of the tubule. The internal and external solutions have the same single, uncharged solute, with diffusion coefficient D in water. Flow in the tubule is assumed to have a parabolic axial flow profile and the shear rate at the wall, ~, is linearly related to the volume flow rate, Q, and the mean axial velocity U by 547
548
GEOFFREY K. ALDIS
4Q
4u
7za 3
a
(2)
In Aldis (1988) the transformed distance variable ~ was related to the dimensional distance variable along the tubule, x, by (3) where AC is the upstream concentration difference C i - - C o and c~o is the upstream shear rate. In the O'Donnell et al. (1982) experiments the external bathing solution was a stream of fluid between two blunted syringe tips lying within a bath of paraffin oil (see Fig. 1).
ST
- "~
ST
Figure 1. A Malpighian tubule (MT) lies in the jet of fluid between two syringe tips (ST). Fluid is collected as droplets (D) from the cut end of the tubule.
The tubule was placed across the stream and surface tension confined the bathing solution to a short length of the tubule. The solution external to the tubule cannot be perfectly stirred but this experimental arrangetnent enables the effect of the external USL to be greatly reduced. Values of P obtained by the O'Donnell et al. procedure are up to five times higher than for previous methods using a stagnant external bathing solution (Maddrell, 1980). The influence of an internal USL acting in the absence of an external USL is treated in Section 2. The external USL is discussed briefly in Section 3 and in more detail in an Appendix. The O'Donnell et al. (1982) experiments are considered in detail in Section 4. 2. Internal Unstirred Layers. Parameters corresponding to experimental inputs are the perfusion length l, tubule radius a the initial tubule flow rate Qi, solute concentration in the tubule upstream of the inflow region Ci, external solution concentration C O and the solute diffusivity D. The principal experimental output is the emergent flow rate Qe (or equivalently the emergent
T H E O S M O T I C PERMEABILITY OF A T U B U L E WALL
549
concentration C¢ since Ce/Ci m Qi/Qe). An important nondimensional parameter which arises in the analysis of Aldis (1988) is k, where PACa k- - D
(4)
The relative concentration difference between the two solutions is another important quantity. In Aldis (1988) fl, the reciprocal of the relative concentration difference is defined by Co Co /~ = C i - C o - AC"
(5)
Results are given here for fi=0, 2 and 10, i.e. no solute in the external solution, and internal solution concentration raised by 50% and 10% over the external concentration, respectively. Figure 2 shows plots of Qe/Qi against lD/Q i for various values of k and fi = 0, 2 and 10. Both the abscissa and the ordinate are nondimensional and the ordinate could equally be CI/C ¢ or ~(~max)/% with the same scale where ~max is the transformed length of the perfused region. Since the flow inside the tubule is parabolic the abscissa is equivalent to 4~3max/grck3. An experiment corresponds to a point in (Q¢/Qi, ID/Qi) space and only one k curve, k -- ko say, passes through that point. The experimental inputs AC, a and D are then used to determine the osmotic permeability P from equation (4). This method of determining P is clearly less useful whefi the resolution between the different k curves is poor, i.e. when k is large or when e/ao is near 1 or near (1 + fl)/fl. For some physiological experiments k ~ 1 and direct application of a onedimensional model will give accurate approximations to k and P. Onedimensional models make the assumption that there are no gradients in solute concentration across the tubule, i.e. concentration varies only with distance along the tubule. Recasting equations (23) and (24) of Aldis (1988) gives ko = 2- ~l l / ~ ] { ~ - l + ~ - ~ l o g l i + f l - f l - ~ . ~ ]
}.
(6)
This is equivalent to equation (7) of Du Bois et al. (1976) except that inflow is being modelled here. When k is large Fig. 2 gives poor resolution and other procedures may be necessary. For sufficiently large k the small ~ power series solutions (Aldis, 1988) with k = oo can be used. These are strictly only valid for constant a, i.e. for an infinite radius tubule but they are close approximations provided a/a o ~ 1. Re-expressing equation (33) of Aldis (1988), a value of k is sought to satisfy
550
GEOFFREY
K. A L D I S
2.0 1.8
1.6 Qe
Qi
1.4
1.2 1.0 1.5 b
1.4
1.3 Qe
Qi
1.2
1.1 1.0
1.10
1.08
Q~-i
1.06 1.04
1.02
1.00 0,01
I 1
0.1 '
i
J i i
i i
10
~.D/Q i
Figure 2. Computed solutions for QJQi represented in terms of experimental parameters. The curves represent (equally log-spaced) lines of constant k. (a) fl = 0, (b)/~ = 2, (c)/~ = 10.
Qe 1 -F 8 [~j~(91Dn/4Qi)I/3 Qi
3-~
¢2~(¢) de
(7)
where 7(~) is a nondimensional form of the wall concentration and P is then obtained from k. The terms on the right hand side of equation (7) can be tabulated against k and upper and lower bounds can be placed on P. If values of
THE OSMOTIC PERMEABILITY OF A TUBULE WALL
551
lie outside the validity of the small ~ expansions then other methods are required. Some alternatives to a full numerical solution are given in Aldis (1983). The information from Fig. 2 is represented in another form suitable for experimental use in Fig. 3. The curves have been obtained from cubic spline interpolation of the information contained in Fig. 2. In a set of experiments l, D and Qi are often fixed, the experimental output is Qe/Qi and k is required. Therefore, once an experimental set-up has been established, one only needs to follow events along a single lD/Qi-curve. The simplest model commonly used to determine P for a tubule wall assumes uniform solute concentration across the tubule at any position along the perfusion length and assumes a linear fall in concentration along the inflow length. This model was used for example by Grantham and Orloff (1963). The average concentration difference across the wall is taken to be the average of the bulk solution differences at the two ends points. Ps is then inferred from ks where ks is given by Qe m
ks__!FQi~FQe l I - r e LUALQii
Qi
.
(8)
Ii+fl+ (1-]~)~i]
This model can give large errors since it ignores two major points. The first is that osmotic inflow is most rapid at the start of the perfused region and becomes progressively slower downstream. The assumption of a linear fall is clearly not suited to this type of problem. This difficulty is easily met by using the one-dimensional mode ! [equation (6)]. The other difficulty is that osmotic inflow depends on the concentration at the wall and that may not be adequately represented by any one-dimensional model. As k increases the bulk concentration becomes a progressively poorer estimate of the wall concentration since the osmotic disturbance is confined to a thin USL. The ratio of equations (8) and (6) indicates how different the estimates for P are when calculated using the simplest model (Ps) and the one-dimensional (Du Bois) model (PD). We have Qe
(9) PD
kl)
Ii+]~+(l_fl)~il{_~_l+~logIl+iff_
fl~l)"
As Du Bois pointed out, the one-dimensional model, equation (6), should always be used in preference to the simplest model, equation (8), and only the
552 GEOFFREY K.ALDIS 10
1
k 0.1
0.01
1.0
1.2
1,4
1.6
1.8
2.0
1.4
1.5
LoU//oJ
1
k 0.1
0.01
1.0
1.1 o
1
1.2
//
1.3
/ 0.01
~
k
j
0.1
0.1 0.01 1.00
1.02
1.04
1.06
1.08
1.10
Qe/Qi Figure 3. Computed solutions from Fig. 2 represented in an alternative manner. The curves represent (equally log-spaced) lines of constant lD/Q i. (a) fl = 0, (b) fl = 2, (c)/~ = 10. one-dimensional model and the numerical results from Aldis (1988) are compared further here. The positions where the one-dimensional and numerical estimates for k differ by 10 and 20% are shown in Fig. 2 by dashed and dotted curves, respectively. These curves and the curves for other percentage deviations are almost parallel to the solid curves over a large part of the figure. Assuming the curves are parallel it is natural to inquire h o w the percentage deviation varies with k. Figure 4 shows this dependence for the three cases/Y = 0, 2 and 10. The curves
THE OSMOTIC PERMEABILITY OF A TUBULE WALL
553
have been drawn with (Qe/Qi)=l.5, 1.25 and 1.05 for //=0, 2 and 10, respectively (i.e. assuming the percentage deviation along the horizontal line halfway down each of the graphs in Fig. 2 is representative). The ordinate kD/k is identical to PD/P. 1.0
ko -~
0.5
0
,
0.001
b
q l l t H
i
0.01
,
i,,,,,i
,
0.1
,
,,,,,,I
,
1
,
,
,,,,,
10
k
Figure 4. O n e - d i m e n s i o n a l model estimate of k over k estimated from the full
convection-diffusionsolution of Aldis (1988) against the latter estimate for/3= 0, 2 and 10 as shown. The choices of Qe/Qi=1.5 (/3=0), 1.25 (/3=2)and 1.05 (/3= 10) were made to obtain these representative curves. A 10% error in estimating P is therefore expected when using the onedimensional model wh6n (the true) k ~ 0 . 3 f o r / 3 = 0 , k ~ 0 . 0 9 f o r / ~ = 2 and k ~ 0 . 0 2 for /~=10. Above these values of k the predictions of the onedimensional and the full convection-diffusion model diverge further.
3. External Unstirred Layers. The comments here are confined to inflow situations. External USLs are likely to be more severe in outflow situations. External USLs cannot always be neglected but the experimental arrangement shown in Fig. 1 tends to reduce their influence. Firstly the osmotic flow outside the tubule is directed towards the tubule wall, so that it thins the USL and therefore enhances back-diffusion of solute. Secondly, the length of the USL on the exterior surface (half the outer circumference) will usually be much less than the inflow length, l, so that the USL will be less developed. Thirdly the bulk flow around the tubule can be increased until the external USL effect is almost eliminated without otherwise disturbing the experiment. In practice an upper limit on the external flow is forced by the need to keep the tubule in place without damaging it and to prevent a hydrostatic pressure driven flow of water across the tubule wall. The strength of the external USL is estimated for an experimental example in the Appendix and it is found to be weaker than the
•
554
°
G E O F F R E Y K. ALDIS
internal USL. Schwartz et al. (1981 ) found a significant external USL effect was present in a tubule diffusion study conducted in a well-bubbled medium. The stirring flow of O'Donnell et al. (1982) between two syringe tips concentrates the stirring at the tubule and would be expected to reduce the USL effect in those experiments. 4. Examples• We describe the procedure used to determine P from the O'Donnell et al. experiments on Rhodnius Malpighian tubules assuming that there is no external USL. The simplest estimate of osmotic permeability, Ps=4.3 x 10 -3 cm litre/s osmol, reported in O'Donnell et al. t1982) is an average value obtained using the linear model, equation (8), for the fall in osmotic gradient on each experiment. In order to calculate a more accurate value of Ps with the full convection-diffusion model one needs a single set of parameters representing the experiments• The set used was a i (inner radius): 32.5 #m; a o (outer radius): 50 #m; l (length of permeable segment): 0.09 cm; C i (initial bulk concentration): 340 mosmol/litre; Co (external bath concentration): 170 mosmol/litre; D (diffusivity of NaCI): 1.5 x 10 -5 cmZ/s; Qi (initial flow): 4 x 10 - 6 c m 3 / s ; C e (emergent concentration): 247 mosmol/litre; Qe (emergent flow): 5.5 x 10 - 6 c m 3 / s . Since the main solute in the bathing solutions was NaC1 we assume it is the only solute• In the experiments a i, a o, l, C i, C o, D and Q~ were usually kept constant. Ce and Q, have been chosen here to give the average Po value based on the external area of the tubule• The theory presented earlier is concerned with the interaction between the bulk flow and the inner USL and an equivalent problem for the tubule lumen is formulated here. Basmadjian and Baines (1980) showed in experiments with models of a brushborder-lined tubule that typically less than 1% of the bulk flow passes between the microvilli. The microvilli can therefore be ignored for our purposes and the tubule wall is replaced by a zero-thickness wall at the level of the microvfllous tips (at radius ai). In order to generate the correct flow through the resulting smaller surface area the permeability of the wall P~ is scaled up with P~= 50/32.5 Ps. The internal concentration C i was twice the external concentration and so /3 = 1. Graphs summarizing the/~ = 1 case are not given in this paper as it corresponds to an extreme osmotic gradient not likely to be commonly used. The estimation of P therefore requires numerical solution of the convectiondiffusion equation as described in Aldis (1988). The parameter k = P~ACa/D and transformed length of the perfused segment ~max=k(91D/o~oa3) 1/3 were varied until the numerical solution reproduced the correct osmotic inflow, i.e. o~(~max)/o~o=Qe/Qi. The theoretical estimate Pts=32.5/50 Pi for the outer surface of the tubule was found to be related to P~ by Pt~ = 1.35 P,, therefore predicting a 35% increase to obtain the true osmotic permeability. The one-
THE OSMOTIC PERMEABILITY OF A TUBULE WALL
555
dimensional model, equation (6), was also applied to this problem giving PD = 1.09 Ps. This is well short of the predicted value Pts= 1.35 Ps and therefore an USL must be causing a major part of the discrepancy. The value of k is 0.33 and the one-dimensional model cannot be expected to give accurate results. The case /3= 1 corresponds to a severe osmotic gradient for physiological tissues and, although the Rhodnius Malpighian tubule can withstand this gradient, the same experiments can be reliably performed using a smaller osmotic gradient. Since most tissues will require lower gradients, full results were not included in this paper for the case/~ = 1. Figure 5 shows plots of the dimensional osmotic gradient C w - C O for the linear model, one-dimensional model and the full convection-diffusion solution as a function of distance along the perfused length. An USL is clearly present since even at the end of the perfused region the wall concentration differs significantly from the bulk concentration. Figure 6 shows the nondimensional concentration profiles across the tubule predicted by the convection-diffusion model at positions one quarter, one half, three quarters and the full length of the tubule. I
200
I
I
I
[
~..~'-.,.
/
P
,s0
~
lO0 ~
50 0
J
I
I
0.2
0.4
0.6
I~"~ 0.8
1.0
X Figure 5. Three models of the O'Donnell et al. (1982) experiments with 8 = 1
showing dimensional concentration differential(mosmol) at the tubule wall against ; full numerical solution. ; a one-dimensional model with linear behaviour between the end points assumed. ; the one-dimensional model of Aldis (1988) with P adjusted to give the observed cumulative inflow at X=I.
X= x/l.
The validity of the convection-diffusion model is checked by noting that l>> a o, that the osmotic inflow is weak compared to the bulk flow, and that Pe=eoa2/2D=52>> 1. The Reynolds number for the tube flow is small at approximately 0.1. Although this example shows that a USL may be present in physiological tubules under experimental conditions it m a y not be typical.
556
GEOFFREY K. ALDIS 1.0
I
I
I
I
o.a [0.6
0.4
0.2
0 0
I
I
I
I
0.2
0.4
0.6
0.8
; .0
R
Figure 6. Radial concentration profilesfor the O'Donnell et al. (1982)experiments at positions (top to bottom in the Figure) I/4, l/2, 31/4, I. Abscissa, R, is nondimensionalized with the tube radius and the ordinate is nondimensional concentration 0 = ( C - C o )/AC. Malpighian tubules have a high osmotic permeability and here the osmotic gradient AC is large, k can be reduced by decreasing AC. Another set of parameters was supplied by Whittembury (private communication) in order to assess the likely influence of an USL in N e c t u r u s proximal tubule experiments. The set gave k--0.02,/3 = 10, Pe = 230 and R e = 0.2. The one-dimensional model would give a better estimate for P than in the previous example, but there would still be an USL. R h o d n i u s Malpighian tubules and rat proximal tubules have relatively high k values associated with them because of their high osmotic permeabilities. Values of k for other epithelial tubules including N e c t u r u s proximal tubule would typically be lower. Rhodnius
Experimenters need to know how large USL effects are and also how strongly they respond to changes in the experimental conditions. The figures included here enable USL effects to be estimated for osmotic experiments on tubules when the luminal USL is the major USL influence. Basmadjian and Baines (1978) and Friedlander and Walser (1965) concluded that various parts of the nephron can be adequately modelled by a onedimensional theory during normal operation. However, the results given here enable experiments to be performed beyond physiological conditions or on artificial tubules with knowledge of the likely internal USL effect. If the value of k in an experiment is large then an attempt should be made to reduce it since USL effects are likely to be important. However in practice there is limited scope for this since P and a are tubule constants and the solutes are usually chosen for physiological compatability. Therefore changing AC is the main way of changing k in practice. Once an initial experiment is performed the 5. D i s c u s s i o n .
THE OSMOTIC PERMEABILITY OF A TUBULE WALL
557
e x p e r i m e n t a l p a r a m e t e r s m a y be a d j u s t e d to give p r e d i c t a b l e results, e.g. to be m a d e to lie in a suitable r e g i o n of a g r a p h in Fig. 2 t h e r e b y a l l o w i n g a n easy e s t i m a t i o n of P. T h e a p p l i c a b i l i t y of the t h e o r y to a given s i t u a t i o n c a n also be tested b y the following p r o p e r t i e s . I f all p a r a m e t e r s except Qi are held fixed then e x p e r i m e n t a l results s h o u l d lie a l o n g a single k-curve. T h i s is also true w h e n l a l o n e is c h a n g e d . I f A C is v a r i e d while/~ a n d the o t h e r p a r a m e t e r s are k e p t c o n s t a n t t h e n the Qe/Qi values o b t a i n e d s h o u l d rise or fall to a c o r r e s p o n d i n g k-curve. It is w o r t h n o t i n g a g a i n t h a t the analysis is only valid w h e n l o n g i t u d i n a l diffusion c a n be neglected. This c o r r e s p o n d s to the n o n - d i m e n s i o n a l Peclet n u m b e r P e = e o a 2 / 2 D ~ > 1, i.e. Qi/aD~> 1 a n d Qi m u s t be sufficiently l a r g e for this c o n d i t i o n to be satisfied. T h i s w o r k w a s s u p p o r t e d b y a T r i n i t y College E x t e r n a l R e s e a r c h S t u d e n t s h i p . T h e a u t h o r also wishes to t h a n k D r s T. J. Pedley, S. H . P. M a d d r e l l a n d M . J. O ' D o n n e l l a n d the referees for helpful c o m m e n t s . T h i s w o r k was p a r t o f the r e q u i r e m e n t s for the d e g r e e of D o c t o r o f P h i l o s o p h y , at the U n i v e r s i t y of Cambridge.
APPENDIX
The External Unstirred Layer. A fully three-dimensional numerical solution is required to treat the external USL in detail. However to estimate its greatest effect in the following example we choose simplifications which tend to make the effect greater than it actually is. The external flow rate used in the O'Donnell et al. (1982) experiments was approximately 3.3 × 10 - 4 c m 3 / s . The area of a syringe tip orifice is 6.4 x 10- 3 cm z so the average flow speed U of the emerging jet is 5.2 x 10 -2 cm/s. To maximize the external USL effect we thicken the USL by reversing the direction of the osmotic flow, i.e. the external bath is initially at concentration C i and the solution inside the tubule is uniformly at CO. Since the tubule diameter is 0,1 mm and the syringe tip diameter is 0.9 mm we will ignore edge effects at x = 0 and x = l and assume the tubule is immersed in an infinite body of fluid. A typical slice across the tubule is considered in what follows. The Reynolds number for the flow is much less than one (Re = 0.05) so the flow is assumed to be a Stokes-Oseen flow past a cylinder, with flow speed at infinity equal to U. By symmetry we need only consider the upper half of the cylinder. There will be no separation of the flow and since we are considering a thin USL we will ignore the curvature of the tubule wall and flatten the wall out to a permeable section of length 5n x 10- 3 cm. A representative value for the shear rate on the notional plate is now obtained from the solution for flow around a cylinder. The stream function for Stokes-Oseen flow using polar coordinates (r, ~0) (e.g. Batchelor, 1967) is 0(r, ~o) = U sin q) - ~ C r
log -a + r 1 +
---
where C = 2/log(7.4/Re) and the shear rate c~is then given by
10uo 1~321~ C U s i n O [ 1 e(r, q~) - r ~r
r a r -f -
2
a2]
-~ + ~
"
558
G E O F F R E Y K. ALDIS
The mean value of a on the semi-circle is then 2CU ~ag and for this problem 8 = 530/s. The downstream end of the USL is therefore calculated to be at 4" =0.1 and the average value of 7 along the flat tubule wall, ~7,is given (Pedley,.1981) by
=~
u2~(u) du.
The k = oo,/~ = 1 small ~ expansion for 7 was computed and the integration resulted in y = 0.90. The external USL could therefore result in at most a 10% underestimate of P. Its effect is small compared with the internal USL in this case and is probably negligible since the model we used sought an upper bound only.
LITERATURE Aldis, G. K. 1983. Concentration Fields Near Transportin 9 Epithelia. PhD thesis. University of Cambridge. - - . 1988. "The Unstirred Layer during Osmotic Flow into a Tubule." Bull. math. Biol. 50, 531-545. Basmadjian, D. and A. D. Baines. 1978. "Examination of Transport Equations Pertaining to Permeable Elastic Tubules such as Henle's Loop." Biophysical J. 24, 629-643. - - , D. S. Dykes and A. D. Baines. 1980. "Flow Through Brushborders and Similar Protruberant Wall Structures." J. memb. Biol. 56, 183-190. Batchelor, G. K. 1967. An Introduction to Fluid Mechanics. Cambridge: Cambridge University Press. Du Bois, R., A. Verniory and M. Abramov. 1976. "Computation of the Osmotic Water Permeability of Perfused Tubule Segments." Kidney Int. 10, 478-479. Friedlander, S. K. and M. Walser. 1978. "Some Aspects of Flow and Diffusion in the Proximal Tubule of the Kidney." J. theor. Biol. 8, 87-96. Grantham, J. J. and J. Orloff. 1968. "Effect of Prostaglandin E 1 on the Permeability Response of the Isolated Collecting Tubule to Vasopressin, Adenosine 3'5'-Monophosphate and Theophylline." J. Clin. Invest. 47, 1154-1161. Maddrell, S. H. P. 1980. "Characteristics of Epithelial Transport in Insect Malpiphian Tubules." In: Current Topics in Membranes and Transport I4, 427-463. O'Donnell, M. J., G. K. Aldis and S. H. P. Maddrell. 1982. "Measurements of Osmotic Permeability in the Malpiphian Tubules e r an Insect, Rhodnius prolixus Stal." Pro¢. R. Soc. B216, 267-277. Pedley, T. J. 198t. "The Interaction Between Stirring and Osmosis. Part 2." J. Fluid Mech. 107, 281-296. Schwartz, G. J., A. M. Weinstein, R. E. Steele, J. L. Stephenson and M. B. Burg. 1981. "Carbon Dioxide Permeability of Rabbit Proximal Convoluted Tubules." Am. J. Physiol. 240, F231-F244.
R e c e i v e d 28 O c t o b e r 1987 R e v i s e d 19 A p r i l 1988