Clinical Science (1971) 40, 137-158.

HUMAN GLOMERULAR MEMBRANE POROSITY A N D FILTRATION PRESSURE:

DEXTRAN CLEARANCE DATA ANALYSED BY THEORETICAL MODELS

G. ARTURSON, T. GROTH AND G. GROTTE

Burn Centre and Department of Paediatric Surgery, University Hospital, and Institute of Physics, University of Uppsala

(Received 2 February 1970)

SUMMARY

1. The renal clearance of dextran of different molecular sizes has been measured in normal humans from 6 days to 61 years of age. Gel chromatography of dextran has been used for determination of molecular weight distribution.

2. Information about the functional ultrastructure of the glomerular membrane has been deduced from the experimental clearance data in the light of theoretical models. The glomerular membrane was here visualized as a gel filter, localized in the basement membrane. The physical analogue of this membrane was interpreted as a somewhat heteroporous structure of quite well defined pore sizes: one system of smaller pores in the range of 20-28 A radius and an additional system of larger pores of radii up to 80 A. These larger pores are quite few in number with an approximate ratio of one large pore per 10 OOO small pores.

3. The values obtained for the transglomerular pressure difference were low, about 1 cm of water or less. This supports the concept that tubular reabsorption may be the rate limiting factor in the process of urine formation and may also control the glomerular filtration rate.

4. The changes of the dextran clearance with ageing found in this investigation may be explained by an increase of the pore radii of the glomerular membrane and a concomitant decrease of the transglomerular pressure difference.

The sieving effect of the glomerular membrane has in electron microscopic studies been at- tributed to three different sites : fenestrae between endothelial cells, the basement membrane, and/or slits between epithelial cells in the urine space (Fig. 1). The functional ultrastructure of the glomerular membrane has also been evaluated by studying the transport of molecules of different sizes into urine.

In the present study dextran was used as a test substance in order to investigate glomerular permeability, i.e. glomerular membrane porosity and fltration pressure. The high precision of

Correspondence: Professor G. Arturson, Swedish Medical Research Council, University Hospital, 750-14- Uppsala, Sweden.

C 137

138 G. Arturson, T. Groth and G. Grotte concentration measurements of the various sizes of dextran molecules has made a more de- tailed mathematical analysis possible. Dextran clearance data were analysed by different theoretical models for molecular sieving through porous membranes and the adequacy of the models in relation to the data was investigated.

Capillary lumen

Fenestra 480 f 160 ; @ Endothelium

Basement membrane

@ Visceral epithelium j_

Bowman's space

FIG. 1. Schematic drawing of glomerular membrane (from electron microscopy).

M E T H O D S

Fourteen healthy humans of both sexes of age between 6,days and 61 years were given 10 ml of water per kg body weight by mouth in the morning. Thirty minutes later Rheomacrodex@ [molecular weight 10000-80000, 10% in NaCl), in a dose of 3 nll/kg body weight was given intravenously. This small dose minimizes the influence of any change in plasma volume. In children but not in adults the bladder was catheterized. After about 30 min the urinary bladder was emptied as completely as possible. About 20 min later the urine was collected quantitatively. Three minutes before the middle of this urine collection period a venous blood sample was drawn. Four to five similar clearance periods were repeated in each subject.

The total concentration of dextran in plasma and urine was determined by the anthrone method (Wallenius, 1954). Because of the uncertain stability of dextran in the urine (Petrie, MacLean & Robson, 1968) the dextran concentration in urine was always determined within 2 weeks after collection of the specimens.

The molecular weight distribution of dextran in each sample was estimated by gel chroma- tography (Granath & Kvist, 1967) adapted to automated routine and computer analysis (Arturson, Granath & Jenner, unpublished observations). The sensitivity, reproducibility and resolving power of the gel chromatography method has been thoroughly investigated. In this connection a distribution of dextran molecules calculated by summing the individual chromatograms of the constituent fractions, coincides well with the curve of the parent sample (Granath & Kvist, 1967; Granath & Laurent, 1967). In gel chromatography (Sephadex, Sepharose) no leakage of carbohydrate material originating from the gel bed occurs.

Human glomerular membrane porosity 139 Endogenous creatinine in plasma and urine was simultaneously determined using a Tech-

nicon AutoAnalyzer (method N I1 a). The clearance was calculated in the standard way. The accuracy of the clearance data for dextran of different classes of molecular weight was evaluated as follows. The total error in the creatinine clearance technique in infants and children has been estimated to be 5.8-6.7% SD (Winberg, 1959). The total concentration of dextran in plasma and urine has an error of 1.8 % SD (Wallenius, 1954). The accuracy of the determination of dextran molecular weight distribution is in the range of 3 % SD (Granath, personal communication). The error of all measurements combined was calculated according to the Gauss error propagation formula and found to be less than 10% SD. For molecular weights < 10 000 and > 40000 the error may be considerably greater because of low con- centrations of these molecular classes (cf. Figs. 2c and d). An estimate of maximum error was deduced from the variation of measurements of different clearance periods in the same patient (Fig. 2f) and used in the comparisons with the simulated curves (see later).

RESULTS

The results of measurements of dextran clearance are illustrated in Figs. 2-4. The disappearance from plasma and the cumulative excretion in urine is shown in Figs. 2(a) and 2(b) as well as changes of molecular weight distribution in plasma and urine with time (Figs. 2c and d). In Fig. 2(e) the rates of disappearance of different molecular weight classes are shown and in Fig. 2(f) the clearance data from four clearance periods in one patient (age 38 years, endogenous creatinine clearance 140 ml/min).

Urinary clearance of dextran of various molecular sizes, corrected to 1-73 m2 body surface area, is compared in patients of varying ages in Fig. 3. At a molecular weight of 15000 this clearance averages 95 ml/min for adults. For a molecular weight of 55000, corresponding to an Einstein-Stoke’s radius of about 52 A, the renal excretion approaches zero. For children below the age of about 3 years the dextran clearance for all molecular weight classes is lower than for adults. Comparisons are also made with data from a similar study by Mogensen (1968). The increase of clearance of the various molecular sizes with increasing age is clearly shown.

Fig. 4 illustrates dextran clearance expressed as percentage of endogenous creatinine clear- ance as a function of molecular weight and Einstein-Stoke’s radius (a). Data from four age groups (6 days, 3 months, 3 years and 61 years) are chosen and compared with those of Mogensen (1968) for 20-26-year-old subjects. A gradual shift of the dextran clearance curves to the right with increasing age up to about 3 years is found.

THEORY

In the present study the urine is considered to be a filtrate from a molecular sieving process through the glomerular membrane. The clearance data have been analysed by using:

(i) the molecular sieving theory for an isoporous membrane by Pappenheimer, Renkin & Borrero (1951), and

(ii) a theoretical model for a heteroporous membrane formulated by Winne (1965) from the general expressions given by Manegold & Solf (1932).

The steady state composition of the filtrate is determined by two processes, filtration (bulk

140 G. Arturson, T. Groth and G. Grotte

mg ,'lOOml

0 I 2 3 Time after dextron injection (h)

M ~ I . w t 10-3

dc dM -

C

% I0C

8 C

6C

4c

2c

I I I I I I I 2 3

Time after dextran injeciiop ( b l

C

Time after dextran injection ( h )

1 10 20 30 40 50 60 70 M ~ I . wt 10-3

i \

I I I I I @=. \, 1 15 20 25 30 40 50 60 10

ivl0i w t 10-3

FIG. 2. Clearance measurements of dextran in an adult (age 38 years). (a) Total dextran concentra- tion in plasma versus time after single injection. (b) Cumulative urinary excretion of dextran in per cent of injected dose. (c) and (d) Molecular weight frequency curves of dextran in plasma (c) and in urine (d). (e) Disappearance from plasma of dextran of different molecular weights. (f) Dextran clearance in per cent of endogenous creatinine clearance.

Human glomerular membrane porosity 141

flow) and restricted diffusion. The sieving coefficient, which is generally expressed as the ratio (Cz/Cl) of the concentration of solute in the filtrate (C,) to its concentration in the filtrand (Cl), is in this case equal to the dextran clearance as a percentage of the glomerular filtration

+ loor

Mol.wt x

FIG. 3. Dextran clearance in ml/min, corrected to 1.73 m2 body surface area, versus molecular weight for healthy subjects of different ages (0, 6 days; 0, 3 months; A, 3 years; V, 61 years). For comparison the data of Mogensen (1968) are added (+, 20-26 years).

rate (creatinine clearance). According to the molecular sieving theory of Pappenheimer et al. (1951) the sieving coefficient is given (Renkin, 1954) by

A, Q Ax where D, (cm2/s) is the free diffusion coefficient of the solute in water,

is the filtration rate, is the effective pore area per unit path length for water, and

is the effective pore area for the solute.

Q (ml/s) A , (cm) Ax As (cm2>

-

This equation is built on the assumption of a linear concentration gradient in the membrane. The pore size (r) is calculated by using Equation (1) and the relation

A , - B w (2) -- - As P s

/3 = [2(1 -a/r)2 - (1 -a/r)4][1-2*10(a/r) +2-09(0l/r)~ -0.95(0l/r)~] where

(3)

142 G. Arturson, T. Groth and G. Grotte and is the restriction factor for molecules of Einstein-Stoke’s radius (a) as expressed by Renkin (1954), taking into account steric hindrance at the entrance of the pore and friction against the walls of the pore.

The model of Winne (1965) is not primarily derived on the assumption of a linear concentra- tion gradient in the membrane but is based on general expressions given by Manegold & Solf

MOI. w t x 1 0 - 3

~~ ~

6 days 3 months 3 years 61 years 20-26 years 0 A V Mogensen (1968)

t

Radius of small pore : (A) 19.6 f0.2 25.0 1 0.1 26.5 f0.1 28.4rt0.1 28.3f0.1 Radius of large pore : ;; (a) 5 9 1 3 7 2 1 3 53f1 5 1 1 1 50f2

Transglomerular pressure diff.: An (cmHzO) 1 . 8 1 0 . 3 0.9zk0.2 (5.1zk0.7)10-2 (1.7f0.3)10-2 (1.2f0,5)10-2 Largelsmall pore ratio :vs/vA (04f0.2)10-4 (1~110~2)10-4 (0.8f0.1)10-4 (1~010.2)10-4 (1.690.5)10-4

FIG. 4. Dextran clearance in per cent of endogenous creatinine clearance as a function of molecular weight of dextran (and Einstein-Stokes radius a) in different age groups. The drawn lines are simulated curves according to an adequate two-pore model. Symbols are experimental values. The two filled symbols are extrapolated. In the Table are shown the values of the discrete radii of the two sets of pores, the density ratio large/small pores and the transglomerular pressure difference which were calculated in the model. Comparisons were made with the data of Mogensen (1968). The errors of the parameters are expressed as standard deviations.

(1932). The equations for the net transport of water soluble molecules through capillary membranes take into account transport by diffusion and filtration through a system of pores of different sizes. The sieving coefficient in the case of two discrete pore classes is given by

Human glomerular membrane porosity 143 (For r A = rB, one-pore system, this Equation (4) becomes equivalent to Equation (1) with Equation (2) and (3) included where

rA and rg (A) are the radii of two different discrete pore classes, vA and vB are the number of pores in these classes; (for rA = rB, one-pore system, this Equation (4) becomes equivalent to Equation (1) with Equation (2) and (3) included)

An (atm) is the mean transglomerular pressure dzflerence of the hydrostatic and colloid osmotic pressures ; Di (cm2/s) is the diffusion coefficient of dextran molecules in water at 37°C (Granath, 1958);

P' -- AT = + (<+Pv)-P"-T'+T"

P" 77''

W Parabolic flow - profile

Steric hindrance

Friction aaoinst the wall

Fro. 5. Model assumptions, P hydrostatic and z colloid osmotic pressure on plasma side and extravascular side of capillary membrane. A = arterial and V = venous end of capillary. For further symbols see text.

q (cpoise) is the viscosity of plasma at 37°C; p is the restriction factor; the indices A, B, i and w refer to the two different classes (A, B) of

pores, dextran molecules of different sizes (i) and water (w); the restriction was calculated according to Equation (3);

a (A) is the radius of a dextran molecule as calculated from Einstein-Stoke's formula using known values of the diffusion coefficient (Granath & Kvist, 1967). The dextran molecule is assumed to be a hydrated sphere (Ogston & Woods, 1953).

The model of Winne includes a number of assumptions (Fig. 5 and Winne, 1965): (a) the pores are of cylindrical shape and oriented perpendicularly to the membrane, (b) the flow through the pores is governed by Poiseuillks law giving a parabolic flow profile, (c) there is no back filtration from urine to plasma, (d) the test substance is water soluble; the transport is not influenced by electric charges and

144 G. Arturson, T. Groth and G. Grotte the passing molecules are not distorted to any significant degree (for discussion see Grotte, 1956),

(e) the hydrostatic pressure on the plasma side falls linearly along the capillary length, (f) the colloid osmotic pressure is constant within and outside the capillary, (g) the filtrate is considered to be the primary urine before any tubular reabsorption has

taken place (in accordance with the generally accepted clearance concept).

COMPUTATIONS The aim of the following computations has been to find an adequate physical model for the clearance data. The adequacy of a model infers two things:

(1) the model is able to match the clearance data within measuring errors, and (2) the parameters of the model are well determined by the data. The residuals, i.e. the differences between measured and calculated values, contain all

information about the adequacy of a model. If the model is adequate the residual function oscillates in a random manner with alternating sign. Large systematic errors, if any, are easily detected if the residuals are plotted in various manners. However, an acceptable residual function is no guarantee for adequacy, because a large systematic deviation can be shifted out of the range of observation by choosing suitable values of the parameters (Fig. 6).

The adequacy of a model in relation to the data is best revealed by the shape of the likelihood function or the sum of squares surface, if the errors are normally distributed. By plotting contours of constant likelihood (or constant sum of squares) the estimation situation can be vizualized in an illustrative way (Barnard, Jenkins & Winsten, 1962; Box, 1960). In such con- tour diagrams the inadequacy of a model is revealed by long oblique ridges, which means that independent estimates of some parameters cannot be obtained from the actual data. These diagrams are thus helpful in the design of experiments.

METHODS The parameters (0) of the two models A (0) were evaluated as least-square estimates, using available clearance data for all different molecular sizes ( A j , j = 1 , 2,. . . n). Because of the non- linearity of the model the parameters were evaluated by an iterative parameter search procedure (Moore & Zeigler, 1960) working according to Gauss’s method (Gauss, 1821). These estimates are maximum likelihood estimates under the assumption that the errors in the clearance data are independent random variables from a Gaussian distribution and that there are no systematic errors.

The confidence limits of the estimates of the parameters (6) were calculated from the covariance matrix V and the least square sum S (G),

R

S(B) = c ( n j ( e ) - n j ) 2

v = { V i j } = (x’.x)-l j = l

anj j = 1,2, ... n XIi i = 1,2,...p (7)

B , & ( V i i . S(B)/(n-p))1’2 (8)

x = (-)

Human glomerular membrane porosity 145

corresponding, in a linear case and under the assumptions above, to approximately 68.3 % confidence limits.

The estimation situation was also investigated by drawing likelihood contours. The region enclosed by the sum of squares contour S = S (0)

(9)

gives an approximate 100 (l-a)% confidence region, if the model is reasonably linear ip 8 within the region of interest (Beale, 1960). Fa(p,n-p) is the a significance level of the F dis- tribution with p and n - p degrees of freedom.

4 0 0

FIQ. 6. (a) The inadequacy of the isoporous model when applied to the w e aged 3 years. A = experimental points. Thick line = simulated curve. Thin line = the relative difference in per cent between calculated and experimental values. Shaded area = maximum error limits of the dextran clearance method. (b) Misleading compatibility between isoporous model and incomplete set of data.

146 G. Arturson, T. Groth and G. Grotce

RESULTS The isoporous model

The pore radius, r , and the effective pore area per unit path length, A,/Ax, were estimated from the clearance data for the case aged 3 years. It was not possible to get an acceptable agreement with the data as illustrated in Fig. 6(a).

By limiting the range of observation to include only the fifteen points in the molecular weight region 25000-50000, it is possible to get a good fit to the data as shown in Fig. 6(b) (r = 53 A, A,/Ax = lo6 cm). However, this circumstance does not infer that this isoporous model is adequate, but illustrates the importance of a complete range of observation. In order to be adequate this model must be modified to include heteroporosity.

The heteroporous model (a) Two pore-system with pores of discrete sizes. With rA, rB , vB/vA and An as free parameters,

the goodness of the fit could be increased considerably as shown in Fig. 4, where the drawn lines stand for simulated curves. The parameters turned out to be well determined by the

FIG. as in

Anhn + 100 1::

7. Adequate fit of a two-pore system (case 3 years of age). Symbols, lines and shaded Fig. 6.

area

clearance data, as seen from the error limits shown in the table in this figure. A strong negative correlation (-0.99) was observed between r B and vB/vA. This means that the estimate of one parameter highly depends on the value of the other. Furthermore it is possible to keep a good fit by increasing one parameter, if the other is decreased at the same time.

In order to investigate the adequacy of a two pore-system the residuals (Ai(@ - Ai)/Ai, i = 1 , 2,. . . n, were plotted against the molecular weight for the case aged 3 years as shown in Fig. 7.

As can be seen there is a large systematic deviation in the high molecular weight range. This

Human glomerular membrane porosity 147

figure also shows that the errors in the clearance data are largest in this range. The discrep- ancies are thus not critical. The errors of the clearance data were estimated as maximum errors from the data in Fig. 2(f), where four clearance periods were calculated in the same patient. The magnitude of these errors is in the range of general estimates of clearance measurements in humans.

The adequacy of the model in relation to the data is best revealed by likelihood contours in the four-dimensional parameter space (rA, rB, vB/vA, An). The correlation between rB and vs/vA is illustrated in Fig. 8, where the approximate 95% and 99% confidence regions are

13

12

II

-

-

- L n

o_ x 10- ha

9 -

8-

7-

6 -

\

FIG. 8. Likelihood contours in the (rA, rB, VB/VA, An)-space corresponding to 99% and 95% con- fidence limits. Lower part shows the correlation between the parameters rB and vB/vA. Upper part shows five projections on the rA-rB plane for five different values V&A.

plotted for the case aged 3 years in an rB-vB/vA-plane for r, = 26.5 A and An = 5.1.10-2 cm HzO. The five cuts for different values of vB/vA show the high precision of the estimate of r,. As rA and An both are determined mainly from the steep slope and bending of the &curve,

148 G. Arturson, T. Groth and G . Grotte

An is probably also estimated with good precision. In Fig. 9 likelihood contours are shown in r,-An-planes for seven different vBlvA values and rA = 26.5 A. These plots confirm this expectation.

(b) The influence of some model assumptions on the parameter estimates. The assumption of pore-classes of discrete sizes is probably not very realistic and, of course, one can get as good and even better agreement with the clearance data by assuming proper distributions of the

pore radii. Mathematically this is done by introducing two more parameters oA and 0, as measures of the width of the two distributions and replacing the terms r" /? in Equation (4) by integrals l:"/? h(r)dr

where h(r) is the assumed distribution, e.g. for a normal distribution

1 h(r) = -

or for a lognormal one

1 oJ2n

(log r - log T)' h(1ogr) = -

Human glomerular membrane porosity ' 149

where 7 is the mean radius and Q is the standard deviation or the logarithmic standard devia- tion. In the log-normal case the integral is written

The influence of different values of oA and ng on the residuals is illustrated in Figs. lO(a) and lO(b), and in Table 1 the corresponding values of the parameters are given.

FIG. 10. The influence of different normal distributions of pore sizes. (a) The influence of a distribution of the small pores (uA = 28, and 5 A uB = 0). (b) The influence of a distribution of the larger pores as = 10 and 13 8, (on = 1). Symbols, lines and shaded area as in Fig. 6.

From this it can be seen that the fit is improved by a narrow distribution of the radii (oA < 2 A) for the small pores, and at the same time the mean radius is shifted towards smaller values (= 25.5 A). In the high molecular weight range the best improvements are obtained

150 G. Arturson, T. Groth and G. Grotte

with rather broad distributions (rB N 10-13 A) of the radii for the larger pores. For this group of pores the mean radius is also shifted towards smaller values (= 41 A). No drastic changes in the parameters vs/vA and An are observed (Table 1).

5-

4 -

h c " - 3

TABLE 1. The influence of different values of oA and oB on the estimates of the parameters rA, rB, vB/vA and A z

-

-

3-

26.5 0 53 0 8.10-5 5.1.10-' 25.5 1 53 0 5-7.10-' 17.5 5 53 0 8.10-5 7.4.10-' 25.5 1 50 5 9.9.10-5 5.7.10-' 25.5 1 41 10 2.10-4 5.7.10-' 25.5 1 41 13 1.2.10-4 5.1.10-'

Distribution of small pore radii

- Frequency x 10

4 -

3-

2-

I -

22 23 24 25 26 27 28 29 30 Pore radius (8)

Pore radius (8)

FIG. 11. The distribution of small and large pores for the case shown in Fig. 10(b).

In view of the small difference between the mean radii of the two classes of pores and the large overlap between the distributions of radii (Fig. 11 and Table I), it seems plausible to assume that the data could fit a one pore-model with a skew distribution, e.g. log-normal. However, it was not possible to fit this model within estimated measuring errors (Fig. 12). Of course, this fact does not exclude a one pore-model with a skew distribution of another type.

Human glomerular membrane porosity 151

The influence of the viscosity of plasma was investigated by using the extreme values 0.7 cpoise (H20 at 37") and 1.6 cpoise (maximal limit of human plasma at 37"). This only affects the pressure parameter An (from 3.5.10-' cmH,O to 8.2. lo-' cmH,O).

(c) Conclusion. To sum up the influence of uncertainty in data and different model assump-

FIG. 12. The incompatibility of a one-pore system with a log-normal distribution. Symbols, lines and shaded area as in Fig. 6. Case 3 years of age. np

tions on the estimates of the parameters the following limits can be given for the case aged 3 years.

rA: 25'5 f (1-2) A r g : 41 +(lo-13) A vB/vA : (0-8-2). An: (2.3-9.7).10-2 cmH,O (Fig. 9) For other age groups the parameters and their error estimates in Fig. 4 and Table 1 will be

modified to a similar extent.

DISCUSSION

The present investigation indicates that the normal human glomerular membrane is an isopor- ous membrane with pore radii in the range of 20-28 A. The data however are not consistent with a completely isoporous system but additional larger pores of radii up to 80 A must be assumed. These larger pores however are few in number with an approximate ratio of one large pore per loo00 small pores (Fig. 11). Structural differences in the different age groups were also found, i.e. pore radii increase with advancing age. The transglomerular pressure difference was calculated to be less than 1 cmH20 and showed significant decrease with advancing age (Table in Fig. 4).

Our estimates of the glomerular membrane porosity and the transglomerular pressure difference are based upon the assumption of equivalent pores, i.e. the membrane is functionally

152 G. Arturson, T. Groth and G. Grotte

described by parallel cylindrical pores perpendicular to the surface. A self-consistent descrip- tion of the membrane pore is then obtained even if the membrane as such consists of a gel structure. The validity of this semi-empirical concept has recently been discussed in detail by Solomon (1969) and seems to justify this approach to describe the transport of lipid-insoluble molecules through biological membranes. All the different aspects of the application of Poiseuille’s law, estimation of restriction factors, etc., have been discussed by Solomon.

The physico-chemical nature of dextran is today quite well known and makes it well suited as a test substance for renal clearance measurements (Wallenius, 1954; Grotte, 1956; Ingelman, Gronwall, Gelin & Eliasson, 1969). Tubular absorption and/or excretion does not seem to influence such measurements (Engberg, 1969). Two main disadvantages of the use of dextran as a test substance have been proposed by Hardwicke, Hulme, Jones & Ricketts (1968): ‘Large infusions of dextran are necessary, which may significantly alter plasma volume’ and ‘the method of fractionation is tedious and relatively inaccurate’. In the present investigation a negligible increase of the plasma volume was obtained. The plasma concentration of the test substance was 200-400 mg/100 ml. Also the new method of determination of molecular size distribution of dextran, i.e. gel chromatography, has great accuracy (+ 3 %) and reproducibility (Granath & Kvist, 1967; Granath & Laurent, 1967; Arturson, Granath & Grotte, 1966).

Measurements of renal clearances by single injection of a polymer instead of a constant infusion technique introduces an error due to the rapidly decreasing plasma concentration for smaller molecules (Brun, Hilden & Raaschou, 1949) but becomes progressively less for larger molecules (dextran molecular weight > 10 000). However, it does not seem possible to maintain a constant plasma concentration for all the different molecular sizes of a polymer because of their great variability in plasma disappearance rate.

Considering the validity of the model concepts and the great accuracy of the measurements performed, the reliability of the parameters here calculated, pore radii and mean transglome- rular pressure difference, seems well founded. The present and earlier investigations are sum- marized in Table 2. Significant differences in estimated pore radii are found. The earlier investigations have used a one-pore model and an incomplete range of observation (Pappen- heimer, 1955; Mogensen, 1968) which may lead to erroneous conclusions (cf. Figs. 6a and b). The experimental data of Mogensen (adults) are similar to ours (Fig. 4) and are probably better explained by the two-pore model. It is interesting to compare our results with those of Hulme & Hardwicke (1968) using a different test substance (’ ‘I- or ‘251-polyvinyl-pyrrolidone) (Table 2). Effective pore radius was not calculated, but it was concluded that ‘macromolecules of molecular radius less than 24 A had a clearance identical to the glomerular filtration rate, while molecules greater than 60 A were virtually excluded from the urine’. Data for proteins are also available. Differences in sieving ratio, in spite of similar Einstein-Stoke’s radii, for dextran and proteins are most probably due to electric charges (Areekul, 1969).

How can our results be correlated to the present knowledge of the ultrastructure of the glomerular membrane ? Electron microscopic studies have shown mainly three layers of the glomerular membrane (Fig. 1) :

1. The endothelial cells. 2. The basement membrane. 3. The epithelial cells. Widely differing figures for the porosities of these layers have been published. It is generally

agreed that the endothelial fenestrae (1) are too big, 480+ 160 A diameter (Pitts, 1964; Gekle

Human glomerular membrane porosity 153 & Merker, 1966; Osterby-Hansen, 1965) for any molecular sieving effect. As regards the functional ultrastructure of the two other layers the reports are conflicting. Some reports (Hall, 1955, 1957; Graham & Karnowsky, 1966) favour the concept that the epithelial slits (3) represent filtration pores, restricting the passage of colloids. These slits according to Hall are remarkably uniform and of approximate dimension 100 A. On the other hand, h e r b y - Hansen (1965) gives this width to 278k60 A. Graham & Karnowsky (1966) found that myeloperoxidase (molecular weight 160 000) was restricted in its passage by these epithelial

TABLE 2. Human glomerular porosity as estimated by macromolecular clearance

Author Test substance Theoretical model Estimate of pore- radius

Pappenheimer (1955)

Pappenheimer (1955) Lambert & Grkgoire (1955)

Mogensen (1968)

Hulme & Hardwicke (1968)

Arturson, Groth & Grotte (1971) (Present investigation)

Dextran (Wallenius, 1954)

MyogIobin Egg albumin Haemoglobin Dextran

Polyvinyl- pyrrolidone ~ Z ~ I - P V P Dextran

Isoporous model (Pappenheimer et al., 1951), two-point fit Isoporous model (Pappenheimer et al., 1951), two-point fit Isoporous model (Pappenheimer et al., 1951), 3 x two-point fit Maximum estimate of passing molecules

Heteroporous model (Winne, 1965)

42 A

35 A

46 A 53 A 57 A u p to 60.4 100% GFR = 24A

Small pores 26 f 2 A Large pores 41 L- 13 A Number ratio large/small (Case 3 years of age, cf. p. 151)

slits. This does not necessarily mean a ‘restriction effect’, where the pore dimensions approach those of the passing molecules, but may simply be due to the ‘steric effect’ of a low number of epithelial slits. On a photomicrograph this will show an accumulation of colloidal molecules in front of these slits.

The present results favour the concept that the filtering barrier of the glomerular membrane is situated in the basement membrane. In support of this conclusion the work of Gekle & Merker (1966) may be mentioned. In studies of the sieving action of isolated fractions of basal membranes in the rat they found a mean pore equivalent radius of 29+ 10 A. Similar studies have not been performed in humans. Of the three layers of the glomerular membrane here discussed, the basement membrane then seems to be the most probable place for a sieving effect to macromolecules. This basement membrane may well consist of a gel structure, the functional geometry of which is here described by cylindrical pores.

In the present investigation surprisingly low figures (about 1 cmJ3,O) has been found for the

D

154

100-

80

60

40

20

0-

G. Arturson, T. Groth and G. Grotte

- - - - - - - - -

IOC C e

4-

0 U Y-

c b

e g 5c c m ._ c - 0) IL:

C

O Q 0 6 r, (a) 26.5 27.4 30.4 53

53 53

ArrrnrnHg 0.039 0.39 3.9 39 uB/un 8.10-5 8~.10-4 1.4.10-2 -

- - 40 - - -20 e

2 - 0 :

--20 5

L -

m - .- c

LL - - -40 -

( b) ... .... .....a... * * .a1 111 W88 1 1 S 1 1 1 1 1 8 8 8 t 8 .......... - . . . . . . . . ‘a*.. ... .. ........................................ .... :..- . . . . . .@ ” ‘m.. *; ** . . . .

Transport of dextran ,:

- Diffusion, small pores Diffusion, large pores

- Filtration, small pares -Filtrotion,large pores

-- -- - _ _ _ _ _ _ _ - --- -- ---

10 20 30 40 50

FIG. 13. (a) Simulated dextran clearance curves for different values of the transglomerular pressure difference (An). This figure illustrates the importance of the clearance data in the low molecular weight range (5000-15000) corresponding to E-S radii, a approx. 15-30 8, for the calculation of An. A indicates experimental values for case 3 years of age. Higher pressures necessitate in- creasingly larger pore dimensions ending up in a one-pore system (4). (b) The relative transport of dextran by diffusion and filtration through small and large pores for values of rA, rB, vB/vA and An given in Fig. 13(a).

transglomerular pressure difference in contrast to the values of about 30 mmHg generally accepted in the literature (Pitts, 1964; Balint, 1969). The precision of the calculated pressure parameter is highly dependent on the accuracy of the dextran clearance data in the low mole- cular weight range 5000-15000 as illustrated in Fig. 13(a). Dextran clearance curves were here simulated for An values 10,100 and 1000 times our estimate (case 3 years of age). Data in the

Human glomerular membrane porosity 155 higher molecular weight range only, giving sieving ratios below 0.5, do not allow an accurate estimate of AJAx and the pressure parameter Ax (Figs. 6b and 13a). The estimate of 25 mmHg for Ax calculated in this way by Lambert, Hulme, GassCe, Askenasi, Ficheroulle, Fafchamps & Verniory (1970) is thus invalidated. Our data in the low molecular weight range (5000-15000) are to some extent corroborated by those of Hulme & Hardwicke (1968) quoted above. They found no restriction to PVP-molecules up to radii of 24 A. However, this problem calls for more experiments and interesting results regarding Ax should be obtained by using a narrow fraction of dextran in this molecular weight range.

The transglomerular transport of dextran, as calculated from the model using the semi- empirical restriction factors, is illustrated in Fig. 13(b). For molecules of molecular weight <10000, filtration is the dominant mode of transport. As soon as molecular restriction occurs in the pores, a concentration gradient is obtained and diffusion becomes increasingly important.

In the process of urine formation the filtration of plasma across the glomerular membranes has always been considered as a primary process. A coupling exists between this glomerular filtration process and the secondary process of tubular reabsorption. This concept of glomerulo- tubular balance was introduced by Smith (1951). Evidence in support of this hypothesis has been presented by Giebisch & Windhager (1964), Dirks, Cirksena & Berliner (1965), Rector, Brunner & Seldin (1966) and Thurau & Schnermann (1965). The quite low filtration pressure here found supports the concept of Bojesen (1954a, b) and Leyssac & Bojesen (1967) that tubular reabsorption may be the rate limiting process in urine formation.

The rate of glomerular filtration (GFR) corrected for body size is lower in infants than in older children and adults (Barnett, 1940; Davies & Shock, 1950; Edelman, 1967). The reasons for the low GFR in infants are discussed in terms of:

(1) a small porosity of the glomerular membrane, and (2) a low filtration pressure. No difference in the permeability of the glomerular membranes with ageing has been found

earlier (Lowenstein, Faulstick, Yiengst & Shock, 1961 ; Faulstick, Yiengst, Oursler & Shock, 1962; Hulme & Hardwicke, 1968). However, these studies were made on humans above the age of 15 years only. In the present study the dextran clearance as a function of its molecular weight, expressed as a percentage of endogenous creatinine clearance (Fig. 4), showed a gradual increase with increasing age. Correction of GFR to body size was here eliminated by using the sieving ratio. Thus the present results reflect changes in the functional ultrastructure of the glomerular membrane independent of body size.

In terms of the two-pore model used there is a significant increase of the radii of the smaller pores concomitant with a significant decrease of the transglomerular pressure difference. The radii of the larger pores and the number ratio largelsmall pores do not show significant changes with age. From research on animals it seems that a direct relationship exists between glomerular pore size and the size of the animal (Berglund, 1965), the pore size being smaller in the rat than in the dog and smaller in the dog than in man. A relationship between glomerular pore size and body size may exist.

A parameter of interest is also the effective pore area per unit path length (AJAx), which can be estimated from GFR-data by using the law of Poisseuille and the values of Ax found here (see Table 3). These estimates are highly dependent on the pressure parameter Ax. By histological techniques the total capillary area has been estimated to be 1-56. lo4 cm2 for

156 G. Arturson, T. Groth and G. Grotte adult kidneys (Vimtrup, 1928), i.e. a considerably lower value. The error of this estimate was not reported, but is probably considerable. Accepting Az values for curves 2 and 3 in Fig. 13(a) the values for A,/Ax will be reduced by a factor 10-100.

TABLE 3

Case GFR (ml/s) AW/Ax (cm) Aw (cm2) A x = 2000A

6 days 0-6 5.108 25.104 3 months 1 *o 1 . l o9 50.104 3 years 1.3 2.1010 iooo.104

61 years 2.3 9.1010 4500.104

ACKNOWLEDGMENTS

This work was supported by the Swedish Medical Research Council (Project Nos B70-40X- 676-05B and B70-17X-28 10-Ol), Ollie and Elof Ericssons Stiftelse, Atvidaberg, Pharmacia AB, Uppsala, and Uppsala Data Center, Sweden.

Valuable technical assistance was given by Miss Katarina Hallin.

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