A network model of glomerular function

MICROVASCULAR

RESEARCH

23, %128 (1982)

A Network Model of Glomerular Function’ P. P. LAMBERT, B. AEIKENS, A. BOHLE, F. HANUS, S. PEGOFF AND M. VAN DAMME Queen Elisabeth Medical Foundation, 1020 Brussels, Belgium, Department of Urology, Medical School of Hannover, 3000 Hannover 61, West Germany and Department of Pathology, Eberhard-Karls Universitiit Tiibingen, 7400 Tiibingen I, West Germany Received April IS, 1981

A model has been developed to establish the determinants of glomerular function by means of a network analysis. The topological and dimensional parameters of the capillary network were obtained by reconstructing the lobular structures of two Wistar rat glomeruli. Calculation of the hydrostatic pressure drop from the afferent to the efferent extremity of the network (E = (P, - P-)/P,,) was based on the concept of the additional pressure drop per red cell in single-file flow, or, in multiple-layer Bow, on in vitro experimental data relating apparent viscosity of blood to dynamic hematocrit and capillary radius. Partition of red cells at bifurcations was calculated as a function of the velocity ratio within the branches. The micropuncture data obtained in Munich-Wistar rats at distal pressure disequilibrium (net ultrafiltration pressure greater than 0) were used to establish the validity of the model. E was found to be 3.1%. The efferent ultrafiltration pressure was close to the value predicted from the micropuncture data. At distal pressure disequilibrium, 100% of the glomerular surface area was utilized for filtration. In this case, the mean integrated ultrafiltration pressure and the filtration coefficient were close to the values reported by R. C. Blantz, F. C. Rector, Jr., and D. W. Seldin ((1974) Kidney Int. 6, 209-225) using the single-tube model of W. M. &en, C. R. Robertson, and B. M. Brenner ((1972) Amer. J. Physiol. 223, 1178-l 183). The hydraulic conductance was estimated between 0.046 and 0.09 ul set-’ mm Hg-’ cm-* S.A. but was dependent on the value chosen for the glomerular surface area.

In this study, a microrheological model will be developed to predict local hemodynamic parameters within a filtering capillary bed by means of a network analysis. The choice of the renal glomerulus for the development of the model is justified by the availability of the topological and physiological data in this experimental preparation. These data form the basis of this analysis and serve for an assessment of its validity. Several models have been proposed to compute the pressure drop along unbranched capillaries in the presence of red cells traveling in a single file. The erythrocytes are modeled as rigid spheres (38) coins (39), or undeformed or even deformed red-cell-shaped particles (23, 33). But to date, such studies have not been extended to networks. A major difficulty is to estimate red-cells partition at bifurcations. ’ Supported by the National Foundation for Medical Research (Belgium), Grant 3.4581.75 (P.P.L.) and by the German Foundation for Medical Research (West Germany), Grant A e 118(B.A.). 99 0021%2862/8210KlO99-30$02.00/O Copyright B 1982 by Academic Press. Inc. All rights of reproduction in any form reserved. Printed in U.S.A.

100

LAMBERT

ET AL.

Our model is based on the concept of additional pressure drop per particle (30, 33, 38). The intracapillary hydrostatic pressure is calculated step by step along the capillaries; the apparent viscosity of blood k is estimated at each step by taking into account local values of the intracapillary hematocrit HT (also called dynamic or tubular), the plasma viscosity t.~~,and h the ratio between cell radius and capillary radius; X accounts for cell deformation as a function of cell velocity and capillary radius. Although recent studies in vivo have formulated the partition of red cells at bifurcations (30), an uneven distribution of the cells (screening effect) is simulated according to the results of model experiments using disk-like flexible gelatin pellets (40). Glomerular filtration in the rat amounts to 30 to 40% of plasma flow rate. A model is introduced into the microrheological model to predict the filtration as a function of the effective ultrafiltration pressure PUFand L, the hydrodynamic conductance of the capillary wall. Water filtration (gfr) along a short capillary segment of length dl is calculated according to Starling’s hypothesis as gfr = 2 IT R dl*L;puF

= 2 T R dl.L, (p, - P,, - IT, + IT”,),

(1)

where R is the capillary radius; pC is the local value of the intracapillary hydrostatic pressure; P,,, the hydrostatic pressure in Bowmann space, which is assumed to be constant; IT, and IT,, are the oncotic pressures for plasma and the filtrate, and P,, may be neglected because the protein concentration in the filtrate is very low. Other models of glomerular function have been proposed previously (9, 12, 15, 26-28, 35, 37). In these models the vascular bed was idealized as a single tube or as a system of parallel tubes of unknown length and radius but of surface area (S.A.) equivalent to that of the network. The governing equations were nondimensionalized; this prevented the estimation of L,, but allowed the calculation of an ultrafiltration coefficient per glomerulus, K,, equal to L, * S.A. Papenfuss and Gross were the first to bring together in a dimensionless model recent accomplishments in the fields of microrheology and of glomerular dynamics (28). The present study extends their work to a well-defined finite network allowing the determination of L,. DEVELOPMENT Morphological

OF THE MODEL

Basis

Aeikens et al. (1) have reconstructed the vascular network of Wistar (W.) rats glomeruli (B. W. = 200 g), while Shea (32) described the network arrangement of a Sprague-Dawley (S.D.) rat glomerulus; the latter was found much larger than the former. Since the physiological data to be introduced in the calculations were obtained from Munich-Wistar rats (5), Shea’s model has not been utilized in the present study. Using serial OS-km-thick sections, Aeikens et al. measured the surface area and the contour length of each capillary lumen on each section and derived the total capillary volume V, and surface area S.A. per glomerulus. Total capillary length L was estimated from the ratio between V, and the volume of 200 well-oriented sections, perpendicular to the vessel axis. The mean values of V,, S.A., and L for four outer-cortex glomeruli were: 1.75 x lo-’ t 0.27

NETWORK MODEL OF GLOMERULAR FUNCTION

101

x lo-’ cm3, 0.11 x lo-* 2 0.01 x lo-* cm*, and 0.46 + 0.05 cm. The radii were asymmetrically distributed around a mean value of 3.8 km. The W. rat glomerulus is made of five lobular-like structures. In each lobule an atferent vessel divides, most generally in dichotomies, in order to converge again at an efferent vessel. A few interlobular connecting vessels are found passing from one lobule to the other. Ten lobules belonging to two glomeruli, denoted as Gl. I and Gl. II in this study, have been graphically reconstructed. Table 1 shows the number of “nodes”* and branches per lobule for Gl. I and Gl. II. The capillaries were classified according to their radius (R > 5.5 pm, 2.8 km < R < 5.5 pm, and R < 2.8 pm). From one lobule to the other, L varied between 259 and 557 pm in Gl. I and between 434 and 1340 pm in Gl. II. Total length was 0.23 cm for Gl. I and 0.39 cm for Gl. II. There is no doubt that Gl. I was smaller than the four glomeruli from which the average morphometric parameters have been derived. In order to obtain models accessible to numerical computations, three simplifications were introduced in each of these 10 networks (Table 2): First, the number of nodes was reduced to between 32 and 58 and the number of branches to between 46 and 85. For Gl. II, the length of the individual branches was kept unchanged, while it was increased by 30% in the models derived from the Gl. I lobules. This increase compensates partly the difference in L between Gl. I and the average outer-cortex glomerulus. Second, the interlobular connecting vessels were reanastomosed inside the network from which they had originated. Third, the vessels were not allowed to divide into more than two branches; if more vessels happened to be present, they were anastomosed at a short distance from the node. Figure 1 illustrates the lobular model derived from Gl. I lobe A. The length of each segment on the graph is proportional to its experimental value. Our models take into account four categories of vessels: large capillaries (5 pm < R < 6 pm), two classes of middle-sized capillaries (R between 4 and 5 pm and R between 3 and 4 Fm), and finally narrow capillaries (R < 3 pm>. The fourth class of capillaries (R between 4 and 5 pm) was introduced to obtain a progressive decrease of R from the large to the middle-sized capillaries having a radius between 3 and 4 km. The values of R in each category (Table 2) were computed to best fit the observed ratio between V, and S.A. Table 2 also shows the values of F, the ratio between the capillary volume of the average outercortex glomerulus (1.75 x lo-’ cm3) and the model volume. Introducing the lobular models one by one into the computer program, F represents the number of identical parallel model units to simulate an average outer-cortex glomerulus of the W. rat. Finally, the five lobular models derived from Gl. I were assembled to derive a “glomerular model” representative of the glomerulus as a whole. Its characteristics are reported in the first column of Table 3. There is evidence to suggest that the surface glomeruli of the Munich-Wistar (M.W.) rats from which the physiological parameters were obtained, could be smaller than the outer-cortex Wistar glomeruli from which we derived the average * A node indicates a place along the network where capillaries either bifurcate or converge.

93.5 522.0

105.0 557.0

38.8 259.5

167.0 501.5

115 7.0 421.5

72

74 114 8.0 444.0

D

C

2.5 218.2

76

52

B

I

12.0 322.5

137

Branches

(N)

92

Nodes (N)

Length (pm) Large cap. Middle-sized cap. Narrow cap. Total length

A

Lobule:

Glomerulus

THE MORPHOMETRIC

TABLE

77.5 449.0

11.5 360.0

106

68

E

PARAMETERS

481.8 2289.0

41.0 1766.2

548

358

Total

DESCRIBING

1

290.0 1340.2

9.5 1040.7

137

87

A

161.0 757.5

17.5 579.0

87

56

B

THE LOBULES

43.0 434.5

23.0 368.5

39

27

C

Glomerulus

75.5 612.0

31.5 505.0

71

47

D

II

183.0 722.0

39.0 500.0

119

74

E

752.5 3866.2

120.5 2993.2

453

291

Total

::.

i

F z E z m

F

0.72

13.90

1.08

9.28

10 24 249 54

0.126

23 26 366 88

Capillary length (wn)

76 5.5 4.3 3.4 2.6

52

B

0.188

76 5.5 4.3 3.4 2.6

Branches (N) Capillary radius (Irm)

Model volume (cm’ x 10’) Model surface (cm* x lo*)

52

A

Nodes (N)

Lobule:

6.94

1.38

0.252

30 . 90 423 81

79 5.5 4.3 3.4 2.6

54

C

Glomerulus I

THE MORPHOMETRIC

TABLE 2

9.11

1.10 8.53

1.12

0.205

48 18 369 68

3 66 377 66 0.192

85 5.5 4.3 3.4 2.6

58

E

67 5.5 4.3 3.4 2.6

46

D

PARAMETERS DESCRIBING

4.70

2.15

0.372

9 65 859 66

85 5.5 4.2 3.4 2.7

58

A

THE LOBULAR

6.58

1.54

9.55

1.05

6 184 264 39 0.183

3.0 2.4

3.4 2.6 18 58 501 157 0.266

46 5.6 4.1

32

C

Glomerulus II

82 5.5 4.3

56

B

MODELS

7.26

1.34

32 53 453 71 0.241

3.4 2.6

73 5.5 4.3

50

D

9.07

1.10

21 43 368 87 0.193

3.4 2.6

82 5.5 4.3

56

E

2

ii 2 g

E!

s si

E ; w

5

i

s 2

104

LAMBERT

ET AL.

FIG. 1. A simplified representation of a glomerulus lobe capillary network (GI. I lobe A) as it is introduced in the computer program. A: afferent extremity; E: efferent extremity. The largest capillaries, close to A and E, are represented as thick continuous lines; the thinnest capillaries as interrupted lines. The length of the lines is proportional to the length of the capillaries. Hachured lines, crosses, and half-ringed lines point to two short-circuiting pathways (1 and 2) and to a longer route (3). The distribution of blood and plasma flows between these routes and their contribution to SNGFR have been calculated (see text).

morphometric parameters. Indeed glomerular size is known to decrease from the inner to outer cortex (3). Therefore, simulation studies were undertaken to examine the effects of a decrease in glomerular size on the calculated parameters. L may be decreased in three ways: First, by decreasing the individual length of the branches, keeping F constant (Table 3, column 2). Second, by decreasing F, keeping individual length of the branches unchanged (Table 3, column 3). Third, by decreasing both parameters simultaneously. This was achieved by

NETWORK

MODEL

OF GLOMERULAR

105

FUNCTION

TABLE 3 THE MORPHOMETRICPARAMETERSDESCRIBING THE GLOMERULAR MODELAS APPLIEDTO THE AVERAGED OUTER-CORTEX GLOMERULUS (COLUMN 1) AND IN THE SIMULATION STUDY (COLUMNS 2 to 5)

1

2

3

Average Glomerulus

11 F-t

l+ F\

120 178

Nodes (N) Branches (N)

21 67 1643 45

Capillary length (wn)

1 and F\r

5 R/

As in column 1 As in column 1

5.5 4.3 3.4 2.6

Capillary radius(ym)

4 Simulations

6.3 4.9 3.9 3.0

As in column 1

16 52 1264 34

21 67 1643 45

16 52 1264 34

16 52 1264 34

Model volume (cm’ x 10’)

0.66

0.51

0.66

0.51

0.68

Model surface (cm2 X 104)

3.8

2.9

3.8

2.9

3.4

F

2.63

2.63

2.0

1.68

1.68

introducing into the computer program the Gl. I network model, as it was derived from the morphological study, with L = 0.23 cm instead of 0.46 cm (Table 3, column 4). It may also be argued that fixation and embedding of the tissue could result in underestimation of the capillary radii. Actually, the risk of tissue shrinkage was reduced by the perfusion of the fixative, glutaraldehyde, into the renal artery at physiological pressure. A simulation study was nevertheless performed to examine the effects of a 15% increase in R on the hemodynamic parameters, keeping L and F as in column 4 (Table 3, column 5). Rheological

Basis

In spite of the non-Newtonian flow properties of blood, Poiseuille’s law may be used to calculate the pressure drop AP, along the vessels of a network, provided the apparent blood viscosity TV,is correctly evaluated at each point along the vessels (28); p varies along the network due to filtration, hematocrit variations from one branch to the other, and the degree of cell deformation.

where Q is blood flow and dl is length of the capillary segment studied. This approach neglects blood pulsatility which, although small, is present in the glomerular capillaries (7). The calculation of p depends on the type of flow: If R is lower than 4 pm, cells flow in a single file. In large capillaries, they usually overlap and a cell-rich

106

LAMBERT ET AL. f-+-we

pR

Ic

---3

-----m+

FIG. 2. Schematic drawing of a capillary with single-file flow. Abbreviations and symbols are defined in the text and Appendix 1.

core flows along the axis surrounded by a cell-deficient wall layer. This is the so-called “two-layer flow” (28). We shall consider both patterns of flow separately. Single-file flow. The concept of additional pressure drop per particle has been useful to account for the effects of cells on apparent blood viscosity (30, 33, 38). In our model, red cells are assumed to be equally spaced coins moving axially in a cylindrical tube with a radius R. A typical unit of flow with length pR is shown in Fig. 2. There p is a dimensionless cell spacing. The volume of the coin is constant but coin radius a and coin thickness 1, are assumed to depend on R and on U, the red-cell velocity. The ratio between the coin radius and the capillary radius, A = a/R, is a dimensionless parameter. According to Wang and Skalak (38), we shall denote G,, a dimensionless coefficient measuring the additional viscosity due to the presence of the cell. Then, l.~, equals P = pot1 + GvJ.

(3)

According to Whitmore (39) if the coins are stacked (f3R = 1,)

If the coins are not stacked, the pressure drop AP,,, along the length PR can be expressed as the sum of the pressure drop for stacked coins on one coin length 1, and the pressure drop across the plasma space with length (PR - 1,) 8 PO q1, (PR- 1,)--cl+ @cam = - 8 ~04 7FR4

L)

1 - h4.

(5)

In this situation the hematocrit is

Combining Eqs. (5) and (6) (7)

NETWORK

MODEL

OF GLOMERULAR

FUNCTION

107

Gv, is then A2 GVo= HT 1 _ x4

(8)

and

p = pO(l +

HT1

x2)

_

h4*

Equation (9) differs by a factor 2 from that proposed by Papenfuss and Gross (28). They introduced this factor in the estimation of Gv, to take into account other factors which are neglected in the stacked-coins theory. As A will be made to vary with U, this factor will not be introduced here. Our approach is similar to that proposed by Schmid-Schonbein et al. (30). These authors derive the relative apparent viscosity which equals (1 + G,,,) from the numerical solutions obtained by Skalak ef al. for rigid particles shaped as undeformed red cells (33). The contribution of white cells to apparent viscosity has not been taken into account in our model (31). The three parameters of Eq. (9), b, HT, and A vary along the capillaries. We shall compute their value at each point of the network. The plasma viscosity. p,Oincreases as plasma protein concentration increases due to water filtration. Kawai et al. expressed p,. (centipoises) as a function of the plasma protein concentration C (g%) according to Eq. (10) (25) /J,~= 0.204 + 0.177 C.

(10)

The capillary hemutocrit. In computing HT two factors have to be considered: the so-called “screening effect” and the well-known “Fahraeus effect” (10, 17, 19, 21). Microrheological studies with red cells flowing along a glass capillary connecting a feed reservoir to a discharge reservoir have shown the following relationship among the tubular (HT), discharge (H,), and feed reservoir (HF) hematocrits: HT < HD < HF. HD is lower than HF because cells are excluded from the capillary as blood flows along the feed channel (screening effect). The ratio HJHF measures the exclusion of cells at the entry of the capillary. HT is lower than H,, because the red-cell velocity in a capillary is greater than that of plasma due to the more axial flow of the cell. This is the “Fahraeus effect”; it is measured as H,/H,. The “screening effect.” Two formulations have been developed: First, Yen and Fung examined the distribution of gelatin pellets between the branches of an inverted T tube as a function of flow velocities within the daughter branches. Pellet flexibility was comparable to that of red cells (40). Assuming that the flow in branch 1 is faster than that in branch 2 and calling HF, and HFz the fractional pellets volume flow at the entry of these branches, Yen and Fung proposed the following relationship between H,,IH,, and the mean velocity ratio V,/V,:

Hh -HF*

1= b(F-

1). 2

(11)

This linear relationship is only valid for VI/V2 lower than a critical value. Above this value, all the pellets flow to the branch with the greatest mean flow

108

LAMBERT

ET AL.

velocity. The factor b in Eq.(ll) depends on HFo, the ratio between pellets volume flow and the flow of the suspension at the distal end of the bifurcating tube. The factor b also depends on 6, the ratio between undeformed pellet radius and tube radius. From Yen and Fung’s data we calculated that, for 6 = 1, b was related to HP,,by b = 1.47 - 1.95 HFO.

WI

Second, Schmid-Schonbein et al. examined in vivo the partition of blood flow and of red cells at bifurcations in the rabbit ear chamber (30). The fractional fluxes of red cells into the right and left daughter branches, @)F and @Lc, were expressed as a function of JIR and +L, the fractional bulk flows. The authors determined for three bifurcations, the curvilinear relationship between cpkrLand3rRorL. One of these bifurcations had capillary sizes at the bifurcation comparable to the most proximal part of the glomerular network. At this branch they found @i” and @t’ equal to 0 and 1, respectively, for GR = 0.3 and +L = 0.7, i.e., when all the cells enter the left vessel. Schmid-Schonbein and his co-workers developed a mathematical model to represent the cell distribution function, using Fourier series to account for deviation from the linear relationship QRC= JI. The HF,/HF2ratio has been derived according to Schmid-Schiinbein’s model (Eq. (13), see Appendix 2) H -5

- 1 = 2 (e $ - 1),

HF*

(13)

2

where z and e are functions of V,,, V,, V,, R,, R,, R2. The subscript 0 refers to the feeding vessel; z implicitly depends on the parent vessel hematocrit, HFO. In the present study, the Yen and Fung equation will be used to account for red-cells partition at bifurcations although there is no doubt that SchmidSchonbein’s approach is closer to reality. Unfortunately only one of the three bifurcations studied by Schmid-Schonbein et al. has size parameters suitable for application to the glomerular network. It will be shown that under certain conditions, the Fourier coefficients derived for this bifurcation may be substituted for the Yen and Fung partition factors without modifying significantly the final results. The “Fahraeus effect.” HTIHD depends on the vessel radius R. If R is lower than 4 to 5 urn, HT/HD decreases when R increases; conversely, in larger vessels, HT/HD increases with capillary size (2, 11, 17). Papenfuss (29) has derived from Gaehtgens’s data Eq.(l4) relating HT/HD to 6, the ratio between undeformed cell radius (3.8 urn) and capillary radius. HT = H’F (1 - 0.125 (2.963 - 6) - 0.0017 (2.963 - 6)‘).

(14)

H’F, which has taken the place of H,, is a “fictitious feeding hematocrit” defined by Papenfuss and Gross (28) as “the feeding hematocrit, HF, which would yield HT if the upstream section of the capillary was replaced by a fictitious reservoir.” At any point x along a capillary, H’F is calculated as H’

F (x)

= HF, $, 4x

NETWORK

MODEL

OF GLOMERULAR

FUNCTION

109

where qa is blood flow at the entry of the branch and qx is blood flow at point x. &,, in this equation, is the ratio between the red-cell mass entering the branch per unit time and qa. According to Eq.(14), HT/H’F reaches a minimum for R = 3.8 km (6 = 1). Calculation of A as a function of the cell velocity. According to Papenfuss and Gross (28) the cell velocity U is equal to the product of bulk velocity, V, and the ratio HIFIHT.

+.v.T

(16)

Two approaches are available to relate A to U. First, Schmid-Schonbein et al. derived an empirical equation from a theoretical study on rigid red cells (30). u -=-

2 1 + X”’

V

(17)

where OLis a nondimensional coefficient equal to 2 for the stacked-coin model and 2.4 for rigid red cells. Second, Hochmuth et al. studied the deformation of red cells as they moved through glass capillaries having a radius between 2.2 and 4.8 pm (24). High-speed motion photography was used to estimate the plasma layer thickness (w) as a function of R and of U: A=---

R-w R

’

(18)

We derived the following relation from Hochmuth’s data: A = C + eAu + ‘,

(19)

where C, A, and B are functions of R, C = 1.051 - 1128.9R, A = 89944 R - 50.39, B = 956.2 R - 1.672,

R and U have the units of centimeters and centimeters per second, respectively. Figure 3 allows for comparison of the two approaches. The calculations were made according to Eq.(19) for three arbitrarily chosen values of U, 0.05, 0.12, and 0.25 cm set-‘, and five values of R between 4 and 2.4 pm. According to the values given to U, curves 1, 2, and 3 were obtained. Also represented are the curves derived according to Eq.(17) assuming OLeither equal to 2 or to 2.4. It appears that, at cell velocities between 0.2 and 0.08 cm set-‘, the relationships derived according to both approaches almost superimpose. At velocities lower than 0.08 cm set-‘, at any given value of U/V, higher A values are calculated according to Eq.( 19) as compared to those derived from the rigid-red-cells model (cx = 2.4 in Eq.(17)). Equation (19) was chosen rather than Eq.(17) because it yields A values varying according to U. Two-layer flow. Gaehtgens has calculated the relative flow resistance of redcells suspensions flowing in glass capillaries under controlled driving pressure (20, 22). Capillary radius varied between 5.5 and 1.65 pm. Equations (20) to

110

LAMBERT

ET AL.

1 2 3

‘\ \ ‘, ‘\ \ y

V1,6-

'\

u.cmsd

Oll5

u=cm se& 012 u=cmsd

025

'\ '\\

IaS-

\

'1

u-

tz-

‘\\ \\

U-

I

1

I

2

I

.3

I

A

I

.5

I

.6

I

-7

I

A

\\

‘\

‘&

I

.9

I

1

FIG. 3. The normalized cell velocity U/V as a function of the normalized deformed cell radius A. The interrupted lines represent this relationship at left, for stacked coins: a = 2 in Eq.(17) and at right, for disk-like particles u = 2.4 (30). Curves 1, 2, and 3 illustrate the relationship according to our model for three different values of CJ.U/V and A were calculated applying Eq.(16) and Eq.(19), respectively. For any given value of U/V, A increases as U decreases; A is inversely related to cell deformation.

(23) relate relative viscosity of blood, t.r,/l~,,,,to HT for various values of R. Introducing HT as a fraction, for R = 5.5 pm

pr = 0.906 c9 HT,

(20)

for R = 3.0 p,rn

/J,~= 1.076 e”.’ HT,

(21)

for R = 2.2 p.rn

pr = 0.878 e’.’ HT,

cm

for R = 1.65 pm

p, = 1.048 e3.’HT.

(23

To obtain a unique equation including R, we derived from Gaehtgens’s data a curvilinear relationship between (A In I.L,)/AH, and R (Fig. 4). This relationship was extended with results obtained by Cokelet for 7.5pm tubes (10). Finally, we derived the best fit of a second-order equation for R between 2.8 and 7.5 w p, = (1.28 - 680 R) ex HT, (24) where x = (2.791 x IO’ R* - 18924 R + 3.865) and R is in centimeters. Equation (24) is valid for the two flow patterns when capillary radius is between 2.8 and 6 pm. But we only used it when R was larger than 4 p,rn or if cell overlap

NETWORK MODEL OF GLOMERULAR

c

I 25

I 5

111

FUNCTION

I 75

R hicrons) 10

FIG. 4. Equations (20) to (23) relating pr to HT for different sizes of glass capillaries (20) were used to calculate the (A In p,)IA HT ratio which, thereafter, is related to R. From the data of Gaehtgens et al., the dots G were obtained (20). A similar relation was derived from Cokelet’s data for a tube 7.5 p,m in radius (Figs. 4 and 10 in Ref. (10) (dot C)). The best-fit second-degree equation was then calculated (Eq(24)). The crosses represent the (A In Q/A HT values derived according to Eq. (24) for R between 2.5 and 6 pm.

occurred in smaller branches. The condition for overlapping was pR-I,=y--&O,

(25)

where VRcis the volume of a red cell and qRC the local flow rate of cells. Figure 5 shows a satisfactory agreement between the apparent viscosity p calculated according to both approaches (Eq.(9) or Eq.(24) and values of R between 2.8 and 4 pm. In Fig. 5, pr is related to R at three values of HT : 0.4, 0.3, and 0.2. Let us first consider, from the right to the left, the curves derived from Eq.(24) (continuous lines) : l~,~decreases with R to a minimum at R = 4 Frn. For R < 4 pm, F, increases as R decreases. This relation illustrates the Fahraeus-Lindqvist effect (18). The curves derived according to Eq.(9) for R s 4 pm (interrupted lines) superimpose the left branch of the curves obtained from Eq.(24). Deriving p.r from Eq.(9), U was assumed equal to 0.04 cm set-’ in the calculation of A. Cell velocities are not reported in the Albrecht et al.

112

LAMBERT ET AL. 2

H, .0,3

HT=02 125,

1.

1 2

1 3

1 4

I 5

R (microns) 1 6

FIG. 5. Relative blood viscosity, p,,, as a function of R for different values of HT. Continuous lines: p, has been calculated applying Eq.(24) as derived from Gaehtgens et al.‘s (20) and Cokelet’s data (10). Interrupted lines: p, was calculated according to the model developed for single-file flow assuming U = 0.04 cm set-’ (Eqs. (9) and (19)).

study (2). It is said, however, that in most experiments, U exceeded 0.07 cm set-‘. Then, Eq.(9) would underestimate CL,by approximately 6%. The equations of glomerular$ltration. The relationships have previously been presented in detail (9, 12, 15) SNGFR, single-nephron glomerular filtration, is obtained by integration of Eq.(l). SNGFR

= Kf ; LL (PC - P,, - ‘rr,) dx.

(26)

In normal rats, 7~~(mm Hg) is related to the local protein concentration, W% by Eq.W’) (5). nTTp = 1.736 c + 0.281 c*.

c

(27)

As filtration proceeds, protein concentration increases. At point x along a branch c, = c,

gPfa gPf - Y’

(28)

where c, and gplf, are protein concentration and plasma flow at the entry of the branch and y the cumulative filtration rate from a to x. The hydrodynamic conductance L, which equals K,IS.A., is assumed to be constant along the network and independent of R. The computational procedure. All input parameters are listed in Table 4.

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TABLE 4 INPUT PARAMETERS

I. Morphometric V,, capillary volume per glomerulus R,, R,, R,, R,, radii as determined by the preliminary program The parameters describing the network Length of the segments Interrelations Radii Red-cell volume (0.92 x IO-” cm’) Undeformed cell radius (3.8 urn) F, number of identical network units put in parallel to simulate the glomerulus II. Physiological P, and P., SNGBF, SNGPF, SNGFR as derived from the micropuncture data C,,, protein concentration of arterial blood

The single-nephron blood and plasma flows (SNGBF, SNGPF) and SNGFR are first divided by the factor F, taking into account the volume reduction of the model as compared to the whole glomerulus. L,, PC, (pc at the entry of the network), and the fractions of blood flow distributed to daughter branches at the divergent nodes, are given arbitrary values. The condition for conservation of mass for cells and plasma is introduced at each node. The local values of p along the branches are calculated in steps of 0.01 to 0.5 pm. The step is chosen according to plasma flow rate in order to keep gfr lower than 5% of gpf in the segment under study. L,, Pea,the blood flow and cells distribution are iteratively modified until three conditions are satisfied: First, identity of the pc values at each recollecting node whatever the pathway followed to reach that node. Second, SNGFR and Is, (mean integrated pE value) equal the experimental values. Convergence is obtained applying the Newton-Raphson method (4). The calculated parameters are listed in Table 5. The input data. We introduced in the program, Blantz et al. micropuncture data obtained in hydropenic Munich-Wistar rats (M.W.) belonging to the socalled group 2 (5). PCaveraged 48.6 mm Hg; P,, : 16 mm Hg; SNGFR : 22.8 nl min-‘; SNGBF : 155 nl min-‘; and SNGPF : 78.4 nl min-‘. SNGFR, SNGBF, TABLE 5 CALCULATED

PARAMETERS

PC.- PC,

PC,,PC,,i,, and E = p

p

%

P”,,, P”F

K>nd L, % S.A., the percentage of the capillary surface area contributing to SNGFR c, HT, and U at the entry and at the distal end of each segment A, when the cells flow in single file f3R - I,, the length of the plasma column between the red cells CL 7

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LAMBERT ET AL.

and SNGPF were normalized by Blantz for kidney weight. As W. rats weighing 200 g were found to have a kidney of about 1 g, Blantz data were not modified prior to being introduced in the program. Plasma protein concentration was 5.1 g%. RESULTS The Determinants of Glomerular Function Solution for the average outer-cortex glomerulus of the W. rat. (V, = 1.75 x lo-’ cm3, S.A. = 0.1 x lo-’ cm*, L = 0.47 cm.) The parameters calculated according to the different lobular models and the glomerular model, are reported in Fig. 6. The values obtained applying the glomerular model are also listed in Table 6 column 1. Applying the lobular models, PC8is found between 48.8 and 49.8 mm Hg. P, equals 49.2 mm Hg according to the glomerular model. These slight variations are not surprising if one remembers that PC,the mean integrated intracapillary hydrostatic pressure, and not P,, is adjusted to the experimental value P,. P, is found between 47.2 and 48.3 mm Hg; the glomerular model value is 47.7 mm Hg. The pressure drop ratio, E, defined in Table 5, varies between 1.15 and 4.8%. A value close to the average of the individual values is obtained applying the glomerular model (3.1% as compared to 2.9%).

10

6-

.

03 I

02

Kf mIxlO sod mm H$

m-$X 00. . LP mlxd sod mmH$ .

60. O/oSA

FIG. 6. The determinants of glomerular function were derived applying the 10 Gl. I and Gl. II lobular models. The Blantz et al. data for hydropenic M.W. rats belonging to group 2 (5) were introduced in the program. E, PUFe,PUP, Kf, L,, % S.A.: see Appendix 1. Solid symbols: values obtained applying the 5 Gl. I lobular models; open symbols: applying the 5 Gl. II lobular models [ l , o lobes A; n , 0 lobes B; A, A lobes C; D, o lobes D; n, 6 lobes E 1. Crosses represent the values obtained applying the glomerular model derived from Gl. I.

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TABLE 6 THE DETERMINANTS OF GLOMERLJLARFUNCTION APPLYING THE GLOMERULAR MODEL. RESULTS OF THE SIMULATION STUDY

V, (cm’ X 10’): L (cm) F R (w)

1

2

3

4

5

1.75

1.34 0.36

1.34 0.36

0.86 0.23

1.13 0.23

0.47 2.63 2.6 < R < 5.5

2.63 2.02 1.68 1.68 2.6 s R s 5.5 2.6 s R c 5.5 2.6 =SR G 5.5 3 c R s 6

PC,(mm&I PC,(mmI-k?)

49.2

49.1

49.4

49.3

49.0

47.7

47.9

47.4

47.5

47.9

PC(mmHg) E (%o) P,,, (mmHg) f’,,, (mmW f’,, (mmHg)

48.6

48.6

48.6

48.6

48.6

3.1

2.4

4.0

3.7

2.3

17.0

16.9

17.2

17.2

16.9

4.8

5.0

4.5

4.6

5.0

8.4

8.4

8.3

8.4

8.5

Kf (cm’ x lo7

0.45

0.45

0.46

0.45

0.45

0.46

0.60

0.60

0.94

0.80

see-’ mm Hg-‘)

L, (cm’ x 10” set-’ mm Hg-‘/cm’ S.A.) % S.A.

100

100

100

100

100

The effective ultrafiltration pressures at the entry of the network are directly related to Pea.They vary little according to the lobular model applied. The value derived from the glomerular model is 17.0 mm Hg. PuF at the distal end of the network is found between 4.3 and 5.3 mm Hg; the “glomerular” value is 4.8 mm Hg. PUF, the mean integrated ultrafiltration pressure, strongly depends on the topology of the model; PUF varies between 4.9 and 10.1 mm Hg. The value derived according to the glomerular model is 8.4 mm Hg. The capillary surface area contributing to filtration (% S.A.) is close to 100% when it is calculated according to 5 of the 10 lobular models; it varies between 92 and 98% according to 4 and is only 79.5% when derived from the Gl. I lobe B model. Applying the glomerular model, percentage S.A. attains 100%. The filtration coefficient Kf varies in the opposite direction as compared to PUF. For the lobular models, the limits are 0.077 and 0.038 nl set-’ mm Hg-‘. According to the glomerular model, Kf equals 0.045 nl set-’ mm Hg-‘. The L, values vary between 0.081 and 0.038 l.~l set-’ mm Hg-‘/cm’ of S.A. L, equals 0.046 l.~l set-’ mm Hg-‘/cm’ of S.A. when it is derived from the glomerulus model. The simulation study. The effects of the imposed changes in the morphometric parameters have been examined applying the glomerular model. The results of the simulations are reported in Table 6. Decreasing by 23% the length of each segment in the network and keeping F unchanged, E decreases from 3.1 to 2.4% (column 2) because the distance

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between the afferent and the efferent extremities of the network is made shorter. pUF and Kf are not modified but L, increases from 0.046 to 0.060 pl set-’ mm Hg-‘/cm2 S.A. Decreasing F and keeping the length of the branches as in column 1, r’,, and Kr do not vary but E increases to 3.99% (column 3). L, has the same value as in column 2 as L has the same value in both models. The actual length and volume parameters which characterize Gl. I (V, = 0.86 x lo-’ cm3, S.A. = 0.049 x lo-’ cm’, L = 0.23 cm) were used to simulate a simultaneous reduction of L and F (column 4). p,, and Kr are unaltered but L, increases to 0.094 p,l set-’ mm Hg-‘/cm2 S.A. E equals 3.7%, a value between those derived in the two situations simulated previously. Finally, R was increased by 15% in large, middle-sized, and narrow vessels. L and F were kept constant as compared to the values given in column 4. Increasing R, E is lower (2.3% to be compared to 3.7%). As S.A. increases from 0.049 x lo-’ to 0.057 x lo-’ cm’, L, decreased from 0.094 to 0.080 ~1 set-’ mm Hg-‘/cm2 S.A. P,, and Kr show only negligible changes (column 5). It is noteworthy also that percentage S.A. equals 100% whatever the changes imposed to the morphometric parameters. Distribution of Blood and Plasma Flows within the Network, Contribution of the Segments to Filtration Our analysis is based on the results obtained applying the Gl. I lobe A model (Fig. 1) and solving for the average outer-cortex glomerulus. We first isolated three pathways within the network (1, 2, and 3 in Fig. 1) and calculated the percentages of blood and plasma flows distributed to these routes and their contribution to filtration. It will be noticed that the two first pathways are formed by the junction of two branches originating close to A. The shortest distances from A to E along these routes are 53, 66, and 79 pm, respectively. The percentages of blood flow distributed to these routes attain 35.6, 34.3, and 11%; plasma flow rates are 32.1, 34, and 13.3%. These routes contribute 20.1, 24, and 18.8% of the network filtration. To cancel out the effects of length and radius, the contribution to filtration has been calculated per unit of surface area (&r/s a). Pathways 1, 2, and 3 contribute 0.79, 0.68, and 0.56 ~1 set-’ cm-‘. Pathways 1 and 2 constitute short-circuiting routes within the network; together they admit 69.9 and 66.0% of network blood and plasma flows. Therefore, they may be considered as preferential channels. The branches may also be distributed into three groups with regard to their localization close to or far from A and E. Twenty-five segments were considered as “proximal” and 16 as “distal.” These two groups incorporate all the segments of the two short-circuiting pathways described above. The other segments (n = 30) form the third group. Groups 1, 2, and 3 contribute 66.7, 22.6, and 10.6% of the network filtration rate; per unit of S.A. the figures are 0.74, 0.39, and 0.11 l.~lset-’ cm-*. Filtration per unit of surface area is correlated to oncotic pressure and to plasma flow rate at the entry of the branches 7~~~ and qPa,respectively. Figure 7 shows a linear relationship between gfrls. a. and IT,,~;ITCH is the lowest in the proximal branches and the highest in the branches of group 3. A linear relationship also exists between gfrls. a. and log qPa(not represented). A multiplecorrelation-coefficient analysis has been performed according to Edwards (16),

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117

1.

075

05

425

W

FIG. 7. Relationship between filtration rate per unit of surface area in the individual branches (gfrls.a.) and the plasma oncotic pressure at the entry of these branches V~ @I. I lobe A model) (n = 71). 0, Proximal branches; x , distal branches; 0, branches distant from A and E in Fig. 1.

considering gfrls. a. as Y, 7~, and log qp. as X, and AC,.The multiple correlation coefficient (Rr.,J equals 0.948, The semipartial coefficients T~,,.~)and T,,(~.,,equal 0.802 and 0.315, respectively. Both are highly significant but the weight of IT,,~ in the correlation is much larger than that of log qp.. For T,,~,.~), F = 252, P < 0.001; for T~(~.~),F = 38.9, P c 0.001; IT,,,and log qPaare not correlated (Y = 0.22, P > 0.10). No correlation could be found between girls. a. and HT or H, at the entry of the vessels. The Rheological

Parameters

The apparent viscosity of blood. In 5&m capillaries, l.r,averages 1.8 CP (n = 3) and 2.7 CP (n = 3) at the distal end of the proximal and the distal branches of the network. For R = 4.3 pm, the corresponding values are 1.5 + 0.06 (SD) CP (n = 8) and 2.06 + 0.12 CP (n = 6). In 3.4~pm capillaries the figures are 1.41 + 0.08 CP (n = 11) and 1.9 + 0.09 CP (n = 11). The six narrow vessels (R = 2.6 pm) located in the center of the network have p values ranging between 1.6 and 2.5 cP. In proximal and distal branches TVvaries with R, this relationship accounts for the Fahraeus-Lindqvist effect (18). The form of the relation is given

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LAMBERT ET AL.

1.4

1.8

2.2

2.6

0.1

0.3

0.5

0.7

0.j cm,s

CP.

-kkF%-w 20

dynes&m-'

FIG. 8. Frequency distribution curves for p,, U, and T; p. and U were calculated at the proximal and the distal ends of the 152 branches 3.4 )*m in radius (glomeruiar model) (interrupted lines: p and II at the proximal end; continuous lines: p and U at the distal end); T was obtained from the pressure drop (p,, - p,,) along the branches.

by Eq. (14). Figure 8 illustrates the distribution curves of p, at the entry and at the distal end of the branches 3.4 pm in radius. Filtration produces a moderate shift of the distribution curve to the right. The non-Newtonian variation of l.~along the 3.4~Km branches has been estimated by calculating the ratio (& - t.~J(t$ - ~.LJ);t$ (blood viscosity at the distal end of the branch assuming the flow to be Newtonian) is calculated as (~-l~jp,~~ * pa. The ratio equaled 1.16 +- 0.06 (SD) (n = 125) for the branches showing no cell overlap and was inversely related to U. For U < 0.25 cm set-’ Pe - Pa N

Pe - )I=

= 1.006 u-O.“,

r = 0.88; n = 107; p < 0.001. Cell velocities. The distribution

curve of U at the distal end of 3.4~km capillaries is represented in Fig. 8. In most of the branches, U varies between 0.05 and 0.2 cm set-‘; the values obtained close to the entry of the network, are much higher than those in the distal branches (0.35 r 0.21 cm set-’ (n = 11) as compared to 0.23 +- 0.15 cm set-’ (n = 11)). The distribution curve is not different if the values for U are derived at the entry of the branches. Wall shear stress. For each branch of the network, T has been calculated according to Eq. (30)

where pC, and pECare the intracapillary pressures at both ends of the vessel and 1 is its length. Estimates of 7 are shown in Fig. 8; the peak of the distribution is located between 10 and 20 dyn cm-‘. Shear stress at the wall decreases with R, averaging 107 & 46, 60 t 33, 23 + 19, and 17 + 11 dyn cme2 in 5.5-, 4.3-, 3.4-, and 2.6+m capillaries, respectively. The screening effect. The partition of red cells into the daughter branches 1 and 2 of a bifurcation is shown in Fig. 9. The fractional fluxes of red cells @Tcand (I$” (ordinate) are related to fractional bulk flows & and +2 (abscissas).

NETWORK

.75

MODEL

OF GLOMERULAR

.50

119

FUNCTION

.25

0

a!

.75

.25

.5

.50

.2F

75

% I .25

.50

75

(0) @y” and @fc, fractional fluxes of red cells in the daughter branches of a bifurcating FIG 9 vesse; are related to JI, and &, fractional bulk flows, applying Yen and Fung’s approach (Eq.(ll)) to solve the glomerular model (40). Each point corresponds to a diverging node of the model (n = 59). No distinction has been made according to the radius of the feeding vessel or of the daughter branches. Squares: Comparison between the (I)?%2and IJJ,w 1 relationships according to the model used to calculate the distribution of red cells. Open symbols: Yen and Fung’s approach (Eq.(ll)). Solid symbols: Schmid-Schonbein e? al. approach (Eq.(13)). A simplified model (15 diverging nodes) derived from the Gl. II lobe E model has been used for this study. At a given divergent node (1, 2, or 3) a simultaneous shift of @ and IJIin opposite directions is observed.

Each point on the graph corresponds to a diverging node of the glomerular model. If the cells, for any given value of IJJ,were distributed in the same proportions as bulk flow, the points would fall on the identity line. Two peculiarities of the curve are noteworthy: First, QRCis lower than I$ for $ < 0.5. Second, the curve is symmetrical to the identity line; for $ = 0.5, @ycand @Fc equal 0.5; then HFI = HF2 = &,, when &,, is the feed hematocrit at the distal end of the parent vessel. Application of the Schmid-Schdnbein approach of the cell distribution function. The Fourier coefficients are functions of branch geometry and of cell eccentricities, factors which are not taken into account in Yen and Fung’s study. The Schmid-Schonbein approach also simulates the exclusion of cells from one of

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LAMBERT ET AL.

FIG. 10. 0: Distal end of a feeding vessel; I and 2: entry of the daughter branches.

the daughter branches if bulk flow in that branch is much smaller than in the other one ($ < 0.3). Substituting Eq.(13) for Eq.(ll) in the calculation of the &,I& ratio we introduced in our program the Fourier coefficients which characterize a symmetrical bifurcation with daughter branches 3.8 p,rn in radius. We used for this purpose, a simplified model derived from the Gl. II lobe E (number of divergent nodes reduced to 15). The results obtained according to Eq.(13) and to Eq.(ll) are illustrated in Fig. 9. In contrast with the symmetrical relationship between aRC and J, obtained according to the equation derived from Yen and Fung’s study (open squares), the Schmid-Schonbein approach yielded a relationship asymmetrical with respect to the identity line (solid squares). At any given node, simultaneous changes of both parameters are observed substituting Eq.(13) for Eq.(ll). If @&,, is lower than CD&~,,,+Eq.C13j is greater than il~~~,(~,), and conversely. The inverse relationship between the changes of aRC and JI, according to the model used, results from the necessity of equilibrating intracapillary hydrostatic pressure at the converging nodes. The glomerular function parameters derived according to both methods are very similar (Table 7). DISCUSSION The Rheological

Approach

The validity of our model with the steady flow relationships depends on their applicability to the in vivo situation. A comprehensive review of this problem has recently been published by Cokelet (11) who emphasized the role of factors which are neglected in the steady flow approach. We shall not discuss them here. Cokelet’s conclusion was that “although based on limited information, such analyses have been qualitatively successful.” Cell distribution at bifurcations is the main determinant of HT and t.~. The estimation of red-cells partition at the diverging nodes probably constitutes the most controversial part of our study. It is based on simulation models experiments

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FUNCTION

TABLE 7 THE DETERMINANTS OF GLOMERULAR FUNCTION: THE CELL DISTRIBUTION FUNCTION IS CALCULATED FROM YEN AND FUNG’S DATA (EQ. (11)) (40) OR ACCORDING TO SCHMID~CH~NBEIN ET AL. (EQ.

(13)) (30).

Eq.(ll)

Eq.W

Et%)

0.77

0.77

P,,, (mm Hg)

5.49

5.49

p,, (mm Hg)

7.15

7.23

Kf (cm3 x 10’ set-’ mm Hg-‘)

0.53

0.53

by Yen and Fung (40) who studied the distribution of flexible gelatin pellets between the symmetrical branches of an inversed T tube (branching angle 180”). A linear relationship could be established between &,I& and V,lV, at three values of 6 : 1, 0.67, and 0.5; the relationship was not studied for 8 > 1. The results for 6 = 1 were used to derive the constant terms in Eq.(12). Equations (11) and (12) have been applied at any bifurcation, even if the branching was asymmetrical and if one of the daughter branches was narrow enough to make 6 > 1. Besides, applying Eq. (1 I), we neglect the fact that above a critical velocity ratio all the cells are swept into the faster vessel. The Schmid-Schonbein approach (30) accounts for the exclusion of cells at JI < 0.3. However, the Fourier coefficients describing cell partition at nonsymmetrical bifurcations have not been calculated. Such bifurcations are frequently encountered in the glomerulus. It appears from the data presented in Table 7 that the glomerular function parameters would probably not be much different if Schmid-Schonbein’s approach would be applied to the glomerular model instead of Yen and Fung’s approach, even if the Fourier coefficients were available for a larger class of bifurcation geometries. However, to obtain a solution with Schmid-Schonbein’s approach, it has been necessary to decrease the value of HF at the entry of the network to approximately 0.27 instead of 0.51. Indeed gpf was found negative at the entry of one of the daughter branches for 8 of the 15 diverging nodes when HF equaled 0.51 at the entry of the model. This observation raises the question: Is it justified to consider &, in the afferent vessel, equal to the systemic hematocrit. A positive answer may be given because Brenner and Galla (6) and Stein et al. (34) have shown that the filtration fraction could be correctly estimated from the comparison between postglomerular and systemic hematocrits, and from the comparison between pre- and postglomerular protein concentrations. Another point deserves comment. According to Albrecht et al., we assumed that the ratio HT/HD is independent of flow in the tubes < 7.5 p,rn in radius (2). Cokelet on the contrary found HJHD flow dependent in this range of capillary sizes (10,ll). According to Cokelet, HTIHD is expected to increase as velocity decreases. No direct relation of HJHD with velocity appears in our model; however, since A depends on U, p varies according to U.

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ET AL.

of Flows and of Filtration Capacity within the Network The short-circuiting pathways are responsible for the low drop of the hydrostatic pressure E within the network. In its turn, a low E contributes to keep pUF greater than 0 along the whole length of the capillaries. It is beyond the scope of the present study to examine to what extent the distribution of blood and plasma flows and the contribution of the different segments to filtration are modified by increasing SNGBF. The model can be easily transformed to calculate GFR as a function of glomerular blood and plasma flows, assuming L, and PC, constant. It may be predicted that the contribution to filtration of the branches belonging to group 3 will increase as compared to that of the short-circuiting pathways; indeed qp at the entry of these branches will be lower than in the present study. The network analysis probably will help in understanding the regulation of SNGFR in relation with perfusion. Distribution

of Glomerular Function K,, and percentage S.A. have been shown to depend strongly on the ET &, topological characteristics of the lobular networks. Thus the lobular models are not representative of the glomerular network as a whole. For instance, introducing the Gl. I lobe B model, percentage S.A. was found to be only 79.5% while percentage S.A. raises to between 92 and 100% applying the nine other lobular models. The presence of nonfiltering segments in the network seriously modifies the values found for P,, and K,. Indeed, absence of filtration within a branch occurs at the pressure equilibrium (PC - P,,) = TV. P,, is integrated along the whole length of the network, and not only along the branches which contribute to GFR. Therefore P,, is found lower if an increasing percentage of S.A. does not take part in filtration. Returning to Eq.(26), it is clear that Kf has to be increased to obtain the experimental value of SNGFR. Pressure equilibrium or disequilibrium at the distal end of the network has to be distinguished from local achievement of equilibrium. In discussing the micropuncture data, pressure equilibrium at the distal end of the capillary bed is (14, 36). P, is a local value obtained by puncturing achieved if P, - P,, = 7~~~ a capillary at random, while IT, is the oncotic pressure in the postglomerular vessel. The assumption that P, = P, is justified only if E is very small. Our study emphasizes the fact that pressure disequilibrium in the distal vessels is not synonymous with complete utilization of the network for filtration. With these limitations in mind, P,, has been calculated as proposed by Deen et al. (12) idealizing the glomerular network as a single tube of unknown length and radius but of equivalent S.A. Applying Deen’s model to the data in hydropenic M.W. rats at disequilibrium, Blantz and his co-workers obtained the following values: P,, = 8.8 mm Hg and Kf = 0.043 nl set-’ mm Hg-’ 9 P, - P,, - IQ,,averaged 5 mm Hg. These values are in excellent agreement with ours applying the glomerular model (8,, = 8.4 mm Hg, Kf = 0.046 nl see-’ mm Hg-‘, and PuF. = 4.8 mm Hg). As far as PUF and Kf are concerned, the agreement results from the fact that percentage S.A. was indeed found to be equal to 100% in the glomerular model. Agreement between (P, - P,, - n,J and PuF, results from the low value calculated for E (3.1%). The results of our calculations thus confirm the conclusion drawn from the micropuncture measurements that the axial pressure drop was very small The Determinants

NETWORK

MODEL

OF GLOMERULAR

FUNCTION

123

(8). This situation is related to the high degree of branching in the glomerular capillary network to be contrasted with observations made on the cat mesentery (41). From the simulation study we may conclude that E and L, are dependent on the morphometric parameters, especially L and R. PUF, K,, and percentage S.A. are not greatly influenced. E decreases as the distance between the afferent and the efferent extremities of the network is reduced; E increases as capillary volume is decreased due to decreasing F. It is also noteworthy that a 15% increase of R has no other effect on the determinants of GFR than a decrease of E, in spite of the fourth power of R in Poiseuille’s law. We shall return to L, later in the discussion. The network model offers three advantages in comparison with the “singletube” model: First, it allows the calculation of E. Applying Deen’s model, E is either estimated equal to 0 or given a small arbitrarily chosen value. Second, the network model allows the estimation of percentage S.A. Although the network analysis yields a value of 100 for percentage S.A. with the Blantz group 2 rats data, one cannot expect to see the same result with physiological data obtained under other experimental conditions. Third, the network model allows the hydraulic conductance of the glomerular wall (L,) to be estimated. This estimation, however, has to be interpreted with caution. Introducing the parameters which characterize the average outer-cortex glomerulus of W. rats, L, equals 0.046 l.~l set-’ mm Hg-‘/cm2 S.A. Assuming the surface glomerulus of M.W. rats to be smaller and comparable by size to Gl. I, L, equals 0.094 l.~l set-’ mm Hg-‘/cm2 S.A. We cannot say presently which of the two sets of morphometric data better represents the surface glomeruli of M.W. rats. The question will remain unanswered until the morphology of these glomeruli has been studied. However, the microrheological model developed here allows a correct estimation of the ultrafiltration pressure at the distal end of the glomerular network. This observation confirms the validity of our approach. Indeed PUFeand (PC P,, - a,,) would not be in close agreement if the drop of the intracapillary hydrostatic pressure had not been correctly calculated or if the length of the network had obviously been overestimated.

APPENDIX 1: LIST OF SYMBOLS Subscripts: a and e indicate afferent and efferent extremities of the network or of a branch within the network. x refers to any point along a branch. 0 refers to the distal end of a feeding vessel at a divergent node. 1 and 2 refer to the daughter branches at a divergent node. Lower-case letters are used to indicate the local values of the parameter under consideration. A : B. W. : C : CP : E :

afferent extremity of the network body weight (g) plasma protein concentration (g%) centipoise efferent extremity of the network

124

LAMBERT

ET AL.

F : ratio between capillary volume of the glomerulus and capillary volume of the network model GFR : glomerular filtration rate (ml set-‘) GPF : glomerular plasma flow (ml set-‘) Gl : glomerulus GVO: additional viscosity coefficient to account for the presence of a particle in a liquid flowing in a cylindrical tube H D : hematocrit in the “discharge” reservoir H F : “feeding” hematocrit, i.e., hematocrit in the feeding reservoir; at the entry or at the distal end of a branch, HF equals cell volume flow rate/blood flow HfF : “ fictitious” feeding hematocrit according to Papenfuss and Gross (see text and Eq.(15)) H T : intracapillary hematocrit, also called “tubular” or “dynamic” Kr : ultrafiltration coefficient (ml set-’ mm Hg-‘) L : total length of the glomerular capillaries (cm) L p : hydraulic conductance of the glomerular wall, i.e., K4S.A. (ml set-’ mm Hg-‘/cm2) 1 : length of a capillary segment (cm) 1, : length of the coin in the single-file flow model (cm) M.W . : Munich-Wistar rats PC : intracapillary hydrostatic pressure (mm Hg) P, : mean integrated hydrostatic pressure along the network (mm I-M AP, : drop of intracapillary hydrostatic pressure along a short segment of a capillary (dyn cm-* in Poiseuille’s law) P UF : effective glomerular ultrafiltration pressure (mm Hg) &, : mean integrated ultrafiltration pressure (mm Hg) pus : hydrostatic pressure in Bowman space (mm Hg) Q : blood flow in Poiseuille’s law (ml see-‘) qp (or gpj) : local plasma flow rate (ml set-‘) 4RC: local flow rate of red cells (ml set-‘) R : capillary radius (cm) S.A . : surface area of the glomerular capillaries (cm2) % S.A. : percentage of S.A. contributing to filtration s. a. : surface area of a branch (cm2) S.D . : Sprague-Dawley rats SNGBF : single-nephron glomerular blood flow (nl min-‘) SNGFR : single-nephron glomerular filtration rate (nl min-‘) SNGPF : single-nephron glomerular plasma flow (nl min-‘) U : cell velocity (cm set-‘) V : mean blood velocity (cm see-‘) VC : capillary volume of the glomerulus (cm3) VRC : red-cell volume (cm3> W. : Wistar rats w : plasma layer thickness at the wall of a capillary (cm) Y : cumulative filtration rate along a branch of the network (ml see-‘)

NETWORK

a RC R or

L,

9 R or LI

MODEL

OF GLOMERULAR

125

FUNCTION

8 : a dimensionless factor such that f3R represents the length of the flow unit illustrated in Fig. 3 6 : ratio between undeformed cell radius (3.8 km) and capillary radius E : drop of intracapillary hydrostatic pressure from the afferent to the efferent end of the capillary network (P,, - PC,) as a percentage of P, h : ratio between deformed cell radius and capillary radius CL: apparent blood viscosity (poises) )I~ : plasma viscosity (poises) CL,: relative viscosity equal to $L/IL~ t.$ : blood viscosity at the distal end of a branch, assuming that the flow showed Newtonian character (poises) T, : plasma oncotic pressure (mm Hg) T,, : oncotic pressure of the glomerular filtrate (mm Hg) T : shear stress at walls (dyn cm-*) QRC, or2 : fraction of the red-cells volume entering the right or left branches per unit time; branches may also be referred to as 1 or 2 : fractional bulk flow in the daughter branches: R, L, 1, 2 as *,or* above

APPENDIX 2 Let us call q:‘, qyc, and qFc the mass of red cells flowing per unit time at 0, 1, and 2 of a bifurcating vessel, and qo, q,, and q2 the corresponding blood flows. By definition, and

q,RC 2

=

e

(14

RC’ 40

In the same way, and

J12= 42. 40

and

aRC -L--

CW

Dividing (la) by (2a): RC

@I

-=-

91

4PC* 90 4F.41

$2

RC 42

* 40

4F

* 42’

Assuming a uniform distribution of cells in vessel cross section,

H,, = 4pc 41 @RC

‘=H +,

and

HF2 = c,

(44

42 RC

FI ’ -$

and

@2

-= 312

Hp-**G. 90

Va>

126

LAMBERT

ET AL.

Dividing (5a) for branch 1 by (5a) for branch 2:

According to Schmid-Schonbein et al. (30) a RC lor2

I or 2 +

= dJ

c

(a,

24~~ or2 + h sin 2nNl

~0s

b

or2),

(74

where a,, and b, are the coefficients of the Fourier series as calculated by SchmidSchonbein et al. Replacing in Eq.(6a) a:“,, 2 by their values derived from Eq.(7a) one obtains I 1 + (l/9,) c (a, cos 27rn+, + b, sin 2an14,) H 0 -5 P (84 HF~ = 1 + (I/$,) c (a, cos 21~rn1),+ b, sin 214~~) 0

which shall be written H F, HF~ -

1

+

(lh,)fl+l)

1

+

wJ2mJ2)’

(94

Subtracting 1 from both terms of Eq.(9a) one obtains

where wJ,M*l) z

=

-

1

+

(**a)

W21.m2~

with *, =

V, TR: V, TR:

and

$2

=

V, TR: V, TR:’

(124

where V,, V,, and V, are the blood mean velocities. By substitution into Eq.(lOa) -HFI - 1 = z($ Hh and

- 1)

(134

2

R:.fW e=R:.-Al),)

Wa)

ACKNOWLEDGMENTS We are thankful to Professors K. Thurau and P. Gaehtgens, and to H. D. Papenfuss for advice during the elaboration of the models. We also acknowledge the help of the Computer Center of Brussels University. We sincerely thank Professor G. W. Schmid-Schonbein and Dr. P. Bergmann for their careful revision of the manuscript.

NETWORK

MODEL

OF GLOMERULAR

FUNCTION

127

REFERENCES 1. AEIKENS, B., EENBOOM, A., AND BOHLE, A. (1979). Untersuchungen zur Struktur des Glomerulum, Rekonstruktion eines Ratten glomerulum am 0,5 mu dicken Serien Schnitten. Virchows Arch. A: Pathol. Anat. 381, 283-293. 2. ALBRECHT, K. H., GAEHTGENS, P., PRIES, A., AND HEUSER, M. (1979). The Fahraeus effect in narrow capillaries. Microvasc. Res. 18, 33-47. 3. ARATAKI, M. (1926). On the postnatal growth of the kidney, with special reference to the number and size of the glomeruli. Amer. J. Anat. 36, 399-436. 4. BAKHVALOV, N. (1976). Resolution des systemes d’equations non lineaires et des probltmes d’optimisation. In “Methodes numtriques” (M.I.R.: Moscou, eds.), Chap. VII, pp. 396-401. 5. BLANTZ, R. C., RECTOR, F. C., JR., AND SELDIN, D. W. (1974). Effect of hyperoncotic albumin expansion upon glomerular ultrafiltration in the rat. Kidney Int. 6, 209-225. 6. BRENNER, B. M., AND GALLA, J. H. (1971). Influence of postglomerular hematocrit and protein concentration on rat nephron fluid transfer. Amer. J. Physiol. 220, 148-161. 7. BRENNER, B. M., TROY, J. L., AND DAUCHARTY, T. M. (1971). The dynamics of glomerular ultrafiltration in the rat. J. C/in. Invest. 50, 1776-1780. 8. BRENNER, B. M., TROY, J. L., DALJGHARTY, T. M., AND DEEN, W. M. (1972). Dynamics of glomerular ultrafiltration in the rat. II. Plasma flow dependence of GFR. Amer. J. Physiol. 223,

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