Journal of Membrane Science, 44 (1989) 35-46 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

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CONTROLLING FACTORS IN HEMOFILTRATION

WILLIAM J. DORSON* and JEANNE S. PIERSON

Chemical and Bioengineering Department, Engineering Center, Arizona State University, Tempe, AZ 85287 (U.S.A.)

(Received December 10,1987; accepted in revised form January 1,1989)

Summary

Ultrafiltration of bovine plasma and blood (hemofiltration) was performed in a dual-parallel- membrane test device over a wide range of hematocrits (O-85% ) , plasma protein concentrations (3-20 gm% ), channel heights (0.022-0.073 cm) and flow rates. The plasma ultrafiltration data was used to determine film theory equation constants by regression techniques similar to that which had been done by others in the past. The ratios of hemofiltration to plasma ultrafiltration rates were then correlated by an independent analytical function of hematocrit, total protein con- centration, and the superficial wall shear rate. The filtration flux ratio has a Fricke equation limit at low shear rates which would be below unity while a maximum filtration flux ratio greater than one was observed (and correlated) at high shear rates with hematocrits below 42%. The form of the hemofiltration correlation was also applied to previously published augmentation data in a pure shear field without plasma proteins.

The design ramifications of the correlation include the determination of when addition aug- mentation methods are needed. The combination of both plasma ultrafiltration and hemofiltra- tion correlations show how each variable affects and controls hemofiltration. The correlations are for small percentages of the incoming flow being filtered. For large flow fractions being filtered the correlations would have to be integrated along the axial length combined with mass balances. The hemofiltration equations can yield slightly lower fluxes at the lowest channel heights and slightly higher fluxes for the larger channel heights, presumably due to the Fahraeus-Lindqvist effect. The channel heights and shear rates utilized included conditions where this effect should be consid- erable as well as conditions where it should not be present. The regression techniques were applied to all the data.

Introduction

During hemofiltration blood cells and proteins are brought to the separating membrane by convection caused by the filtration. In the absence of any assis- tance only inefficient diffusion is available for the return of rejected species from the membrane surface back to the bulk stream. This limits the hemofil- tration process and results in plateaus of filtration flux versus transmembrane pressure. The value of the plateau flux depends upon many factors and the

*To whom correspondence should be addressed.

0376-7388/89/$03.50 0 1989 Elsevier Science Publishers B.V.

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velocity of the blood stream is an important variable. As the velocity is in- creased so is the limiting filtration flux. However, it is not the velocity alone that determines this limiting flux but a number of grouped dimensional and dimensionless variables. In addition, the membrane may either interact with or allow slight passage of protein molecules. These factors will be separately discussed and an analytical function correlating hemofiltration fluxes will be developed. The experimental data and predictive equations cover a wide range of hematocrit, total protein concentration, and shear rates. The analytical form should be useful in both design calculations and considering the effect of pre- or post-dilution hemofiltration. Additionally, some insight into the physical events accompanying hemofiltration may evolve.

Membrane effects

An extensive series of tests allowed the correlation of deposited-protein ef- fects in a stagnant filtration cell [ 1,2]. By using a wide variety of membranes a distinct categorization of protein interactions was possible as follows: (a) non-ingesting; (b) surface ingesting; (c ) protein transmitting. Protein transmitting membranes allowed a small but measurable leakage of albumin through the membrane. The protein ingesting membranes displayed penetra- tion of protein molecules into the surface matrix of membrane without detect- able transmission. In both cases the in. situ membrane-limited flux was appre- ciably lower than the pre-test isotonic buffered saline (IBS) value. In addition, these two types of membranes could not be easily restored to the initial IBS value following exposure to plasma or blood. In contrast, non-ingesting mem- branes maintained their IBS membrane filtration limits during testing with plasma or blood at very low transmembrane pressure. Also, after extensive testing with plasma and blood, the original IBS limits were restored by simple rinsing of the membrane. All of the filtration results herein were obtained with non-ingesting membranes. This allows the membrane effects to be separated from phenomena occurring within the flow channel. Membrane effects also account for some of the differences observed by different investigators.

Prior investigation

Ultrafiltration from flowing plasma has been extensively studied and cor- related by analytical functions [3,4] with excellent results. Predictive design equations for plasma in hollow fibers were developed. When blood was studied the filtration equations were less general [ 41 and restricted to the introduction of empirical factors [ 51. Most of the important controlling phenomena were identified but not correlated. Also, controlled experiments over a wide range of variation for each factor were lacking. Modeling of hemofiltration included the variation of viscosity with protein concentration and hematocrit in the

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flat-plate geometry [6]. The physical interpretation of events as the trans- membrane pressure, dP,, increased gradually was that the concentration of rejected proteins, C,, on the membrane surface increased until a gel limit, C,, was reached. No further increase in protein concentration was possible and the hemofiltration flux, Nf, became independent of dP, at that point. Further compression of the protein gel was in evidence as the passage of solutes was inhibited as LIP, was further increased [ 71. The early work of Blatt et al. (3)) Colton et al. [ 41 and others has been reviewed [ 11. In summary, the design equations covering the ultrafiltration of plasma are adequate and contain the major controlling variables. Such is not the case for ultrafiltration of blood. A correlation of controlling factors for the flat-plate geometry with dual filtering membrane walls will be developed herein.

Experimental

A flat-channel device was constructed as shown in Fig. 1. With two filtering membranes the total active area defined by the porous glass membrane sup- ports was 562 cm’. The channel height was varied with different spacer thick- nesses. The channel height itself was determined by both subtraction from micrometer measurements over the center of the assembled device and by a post-test injection of a casting plaster-mortar-water mixture. This was al- lowed to cure for 24 h before dismantling the assembly. Measurement of the cast thickness showed excellent uniformity of channel height and agreed with the overall device micrometer-based height determinations. The test system

Fig. 1. Assembly view of the parallel-plate hemofilter test device (porous supports were incom- pressible sintered glass beads).

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allowed for continuous recirculation through the device back to a 2200-ml res- ervoir, variation of transmembrane pressure, and return of filtrate to the res- ervoir. Pressures were measured at the reservoir and at the entrance and exit of the flow channel. Filtrate was collected at atmospheric pressure. Transmem- brane pressure, blood or plasma flow rates, total protein concentration, he- matocrit, and channel heights were varied. Bovine blood and plasma were used for all experiments.

Model development

Plasma ultrafiltration models were used as the basis for the development of hemofiltration correlations. Both a film theory [ 41 and variable diffusivity [ 1 ] model gave similar results. The advantage to the latter is the availability of a single analytical function which correlates the transition from membrane-lim- ited filtration at low P, to gel-limited conditions. The disadvantage is a com- plicated plasma ultrafiltration equation which includes a logarithmic average of the protein diffusivities evaluated at both the bulk and wall (gel) concen- trations. Since the gel-limited filtration is the maximum attainable, the devel- opment herein will be restricted to this condition. This allows the simpler film theory to be used for the plasma ultrafiltration limit.

For dual-membrane plasma ultrafiltration the equation format

Nfp=m, (gr331n ($)

was used with m, and m2 unknown coefficients to be determined by regression techniques. Nf, is the ultrafiltration flux ( ml/min-cm2 ), L the active channel length (cm), C,, the bulk protein concentration (gm% ) and j, the superficial wall shear rate (set -* ) calculated from

. 6Q Y=s (2)

where Q is the volumetric flow rate (ml/set), W is the active channel width (cm) and h is the channel height (cm), The format for hemofiltration was

Nfb -= NfP

[l+gW,j&,) 1 (3)

where Nm is the blood ultrafiltration flux, H the fractional bulk hematocrit (O- 1) and g (H,j,C,,) are hematocrit, shear rate, and total protein functions to be determined. The first bracket term on the right hand side of eqn. (3) is a simplified Fricke term for blood [ 81. Under conditions where the ultrafiltra-

39

tion of blood would not be augmented by red blood cell rotations and/or trans- verse motions, the plasma would have to flow around the cells towards the membrane wall. Thus, this limiting condition was adapted from the theory of disperse systems and was supported by the data.

Results

The plasma protein concentration was varied between 3 and 20 gm% and the channel height between 0.022 and 0.073 cm. A total of 127 data values for plasma ultrafiltration flux were used to determine the two floating constants in eqn. (1) to yield

N,=O.O0129 (-j!>,.,‘ln(z) (4)

with an average sum of the square difference of 6.61 x 10P7. Equation (4) along with the experimental data is shown in Fig. 2.

A separable function of augmentation with hematocrit and true shear rate was possible from the data of Wang [ 91. In the absence of plasma protein the analogous format to eqn. (3 ) was

l+A(%) -= 100

o.o?a cu- 5

f 0.020

g

8

s 0.010 6

K

z"

0

1-H

H 1+-

1.05

[l+gWW,,=o) 1 (5)

h+ EXPERI~~~JTAL(MVMIN-CM~)

Fig. 2. Comparison of eqn. (4) with experimental plasma ultrafiltration data.

SHEAR RATE (SEC-‘)

Fig. 3. Comparison of eqns. (5) and (6) with the data of Wang [9] which measured mass transfer augmentation in the absence of plasma proteins. (0 ) H= 0.10; (0 ) H= 0.40.

0.0200 -

HEMATOCRIT (%.)

Fig. 4. Comparison of eqns. (4) and (8) with hemofiltration data at normal total protein concen- trations. (0): j, 2: 230 set-‘, C,, z 7.0 gm%, h=0.072 cm; (0): ) z 860 set-‘, C,, z 6.9 gm%, h=O.O48cm; (A):j,z 1700sec-‘,Cp,=7.4gm%,h=0.027cm.

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I 1 1 , I , I

0203040 5060 70

HEMATOCRIT (%I

Eli

Fig. 5. Comparison of eqns. (4) and (8) with hemofiltration data at high and low total protein concentrations. (0): Jo z 760 see-‘, C,, z 19.5 gm%, h=0.069 cm; (0): i, = 2300 set-‘, C,, g 12.4 gm%, h=0.024 cm; (A): i, sz 1550 set-‘, C,, g 3.15 gm%, h=0.048 cm; (0): ) z 2160 set -I, C,, z 3.46 gm%, h=0.024 cm.

where A (% ) is their reporting of augmentation caused by increasing shear rate. A surprisingly straightforward function of g(H,j) evolved and, again, regression analysis yielded

g(H,j) =0.148H1.08jo.638 (6)

Of significant note is the proximity of the exponent on H to unity and the closeness of 0.638 to 2/3. The combination of eqns. (5) and (6) is shown in Fig. 3 compared to Wang’s data [ 91. The success of this approach led to a direct modification for hemofiltration in parallel-plate designs of the form

Nb -=

i I l-H [l+Iz,exp( -k,CP,)Hjk3] Nf, 1+H

1.05

(7)

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where kl, k, and k, are constants to be obtained by nonlinear regression tech- niques and exp ( - kpCpb) is an empirical form of the bulk protein concentration effect compatible with measurements of changes in physical property [ 11. Us- ing eqn. (4) for the Nfp, the derived model constants were

Nfl3 -=

i -1 l-H %J 1+ Jf

[ 1+ 0.383exp ( - 0.0496CPb)Hjo~420]

1.05

(8)

The function has a maximum at sufficiently high flow rates (Jo) and the cal- culated hemofiltration fluxes versus hematocrit are shown in Fig. 4 for normal plasma protein concentrations. A similar comparison at both high and low total protein is shown in Fig. 5.

Discussion

Considering the scatter in the hemofiltration data eqn. (8) describes the functional variation in the gel limited flux over a wide range of hematocrit, shear rate, and plasma protein concentration. A systematic underprediction of the filtration flux is observed at the lowest plasma protein concentration (3 gm% > coupled with the highest shear rate (2150 see-‘). This may, in part, be due to the exponential function of protein concentration contained in eqn. (8). The variation of diffusivity with protein concentration follows a tanh variation [ 1 ] which would be higher at low concentrations than the simple exponential function used in eqn. (8). However, the differences in channel heights in Fig. 5 are also considered important. At a channel height of 240 pm the Fahraeus- Lindqvist effect should be significant.

With an overall correlation of hemofiltration in a parallel-plate configura- tion the controlling factors evolve. The symbol 9 is referred to as shear rate and would be the wall shear rate for Newtonian fluid flow in the absence of filtering membrane boundaries. Blood is both multiphase and non-Newtonian. During hemofiltration a protein-rich boundary exists which is highly viscous and the velocity near the filtering membrane is distorted. Therefore, j, should be viewed as a similarity parameter best interpreted as 6 times the average

Q inlet velocity - expressed as the number of channel heights traversed per

Wh unit time. At all but the lowest channel height the percentages of the inlet flow rate being filtered were below 5.5% and at the lowest channei height they never exceeded 13.2%. Therefore, the bulk flow conditions changed little along the channel length and all calculations utilized average values between the inlet and outlet. The magnitude of j, affected both the plasma filtration flux and augmentation of protein diffusion caused by transverse motion of the red cells.

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In the absence of plasma proteins and in a constant shear field the exponent on j in the augmentation term was ca. 2/3 (eqn. 6). During hemofiltration with both shear rate and protein concentrations varying significantly in the channel cross-section the exponent on the average similarity parameter, +, was 0.42.

The gel-limited hemofiltration data that was used to derive the exponents and constants in eqn. (8) included 96 values with channel heights between 0.048 and 0.073 cm. The ) range was from 47 to 1560 set -’ for these data. There were 67 hemofiltration values with channel heights between 0.022 and 0.027 cm yielding a range of i, from 1660 to 7440 set-‘. Again, the smaller channel heights are in the range where the Fahraeus-Lindqvist non-contin- uum effects may be important [lo]. Jaffrin found great similarity between hemofiltration fluxes in channels from 0.0160 to 0.0450 cm in channel height [ 61. Lower channel heights led to fluxes substantially lower than either the theoretical predictions or data obtained at the larger channel heights. The data herein were obtained at much higher normalized flow rates. In terms of his similarity parameters WL/Qh our data range from 510 to 5110 set-cmm2 while his data range from 2 x lo* to 2.6 x lo5 set-cme2. Therefore, he may not have observed augmented hemofiltration. The data shown in Fig. 5 at C,, = 3 gm% are compatible with additional augmentation at the lowest channel heights. Thus, eqn. (8) may tend to underestimate fluxes below h 1: 0.04 cm and over- estimate fluxes at higher channel heights.

Equations (4) and (8) can be used to determine the relative performance of pre- versus post-dilution hemofiltration. The dominant plasma protein effect favors pre-dilution which is in agreement with the clinical preference for the pre dilution hemofiltration artificial kidney mode. For practical hemofiltration design calculations eqns. (4) and (8) would have to be combined with differ- ential mass balances in the channel and integrated over the channel length. The plasma filtration data fit resulted in a gel concentration estimate of 58.5 gm% in eqn. (4) and by comparison to theory a diffusion coefficient for plasma proteins of 1.4~ 10m7 cm2/sec is estimated. Using the log mean diffusivity plasma filtration model [l] on the same data yielded a gel concentration es- timate of 70 gm% along with a diffusivity function of

D 10.89 tanh (0.0918 C,,) -= D, c pb

with D, = 4.2 x 10W7 cm2/sec. Proteins interfere with hemofiltration augmen- tation caused by transverse movement and/or rotation of red cells in the chan- nel as witnessed by the exponential function needed to correlate the data in eqn. (8). The choice of an exponential function was compatible with measured exponential increases in viscosity as a function of protein concentrations above normal [ 1,6].

In the absence of shear, plasma proteins would have to diffuse through a red

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cell matrix. This concept was first applied to effective conductivity by using disperse systems theory [ 81. The Fricke equation for blood was also found to describe the decremental change in the diffusion of dissolved gases. Although many different forms of a hematocrit function were attempted none were as satisfactory as a simplified Fricke equation in the absence of augmentation. Interestingly, subtraction of a Fricke-based limit from N&/N, ratios was un- satisfactory compared to the final multiplied function. The augmentation term was then found to be linear with increasing H. The experimental data covered the range of O-O.85 fractional hematocrits with low hematocrit hemofiltration starting at 0.04. The interesting observation was that an optimum hematocrit was often observed at higher shear rates. The hemofiltration flux would be a maximum at this optimum hematocrit which was invariably less than 0.42. This lends additional support for the practice of pre-dilution hemofiltration. The hematocrit function in eqn. (8) was used to determine the conditions for maximum hemofiltration flux compared to the plasma values. The results are shown in Fig. 6 at three different plasma protein concentrations, 0, 7 and 15 gm%. At a given shear rate, hematocrit values higher or lower than the opti- mum will result in a lower hemofiltration flux than the maximum. Another interpretation is that optimum conditions move towards higher hematocrits as the shear rate increases. This would favor the practice of recirculation to independently increase the shear rate. Again, optimum hematocrits in chan-

i

E G f I+ g 0.50-

I

5

g 0.40 - WSHEAR RATE LIMIT---,

IO00

+ = 6Q/Wh* (SEC-‘)

Fig. 6. Variation of the optimum hematocrit for maximum hemofiltration with superficial wall

shear rate and total protein concentration.

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nels below 0.04 cm may be slightly higher and optimum hematocrits in larger channels may be lower for reasons already discussed. Also, the conditions for which there would be no maximum hemofiltration flux include low shear rate, high hematocrit and high total protein. For these conditions an increase in hematocrit can only decrease the hemofiltration flux compared to the plasma filtration limit.

The combination of eqns. (8) and (4) contain all the controlling factors in hemofiltration for the parallel membrane geometry. They represent the design limit for open unimpeded bulk flow of blood in a channel. Actual designs may contain transport-enhancing channel spacers (e.g. meshes or screens). Their performance can be compared to the open channel limits to determine the effectiveness of different screens on the blood side. This should also hold true for the spiral membrane geometry when blood flow is parallel to the spiral axis. The design equations coupled with hydraulic pressure drop versus flow can be used together to determine when enhancement techniques are necessary. For unobstructed channels the pressure drop and flow should match the calcula- tions using the Casson model [ 11. At low channel heights a deviation toward lower pressure drops for a given flow rate is caused by non-continuum effects, migration of red cells away from the walls [lo]. When this becomes substantial (h << 0.04 cm) augmented hemofiltration should occur and eqns. (4) and (8) could underestimate the fluxes. Conversely, for larger channels the pressure drop should be well accounted for with the integrated Casson equation relating hydraulic pressure drop with volumetric channel flow rate. Under these con- ditions eqns. (4) and (8) should yield good estimates of the hemofiltration fluxes and represent a limit for practical hemofilter designs. As an example, for an artificial kidney application a filtrate rate of 80 ml/min would be desir- able. Without employing augmentation means a surface area of at least 3.4 m2 with a channel height of 0.04 cm is calculated by using eqns. (4) and (8). By employing multiple augmentation methods of pre-dilution, recirculation, and an efficient eddy producing blood channel spacer (screen) these hemofiltra- tion rates can be achieved with membrane areas at and below 1 m2 [ 111.

Conclusions

The format for the correlation of hemofiltration with the ratio to the plasma ultrafiltration yielded reasonable equations which contain the flux as a func- tion of the controlling variables. The superficial wall shear rate not only con- trols the plasma ultrafiltration rate but also the augmentation of protein dif- fusion caused by transverse red cell motions and/or rotations. The Fricke equation appears to be a reasonable low shear rate limit as witnessed by:

(a) the excellent correlation of Wang’s data from a completely different mass transfer augmentation experiment;

(b) the resultant linear augmentation with hematocrit;

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(c) the prediction of optimum hematocrits and the conditions for the lack of an optimum;

(d) correlation of the dual-membrane gel-limited hemofiltration over the experimental ranges. The resultant eqns. (4) and (8) are correlations which apply to low filtration percentages of the flow rate. In order to use these equations for a design which filters a large percent of the blood they would have to be combined with protein and red cell differential balances and integrated over the channel length.

References

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5

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8

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10

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W.J. Dorson, Jr. and V.B. Pizziconi, Ultrafiltration of plasma and blood, in: D.O. Cooney (Ed.), Advances in Biomedical Engineering, Part II, Marcel Dekker, New York, NY, 1980, Chap. 11. W.J. Dorson, Jr., V.B. Pizziconi, M.H. Ferdman and C.N. Sizto, Quantitation of membrane- protein-solute interactions during ultrafiltration, Trans. Am. Sot. Artif. Intern. Organs, 24 (1978) 155. W.F. Blatt, A. Dravid, A.S. Michaels and L. Nelsen, Solute polarization and cake formation in membrane ultrafiltration: causes, consequences, and control techniques, in: J.E. Flinn (Ed.), Membrane Science and Technology, Plenum Press, New York, NY, 1970, p. 47. C.K. Colton, L.W. Henderson, C.A. Ford and M.J.Lysaght, Kinetics of hemodiafiltration. 1. In vitro transport characteristics of a hollow-filter blood ultrafilter, J. Lab. Clin. Med., 85 ( 1975) 355. M. Okazaki and F. Yoshida, Ultrafiltration of blood: effect of hematocrit on ultrafiltration rate, Ann. Biomed. Eng., 4 (1976) 138. M.Y. Jaffrin, Y. Butruille, A. Granger and G. Vantard, Factors governing hemofiltration (HF) in a parallel plate exchanger with highly permeable membranes, Trans. Am. Sot. Artif. Intern. Organs, 24 (1978) 448. W.J. Dorson, Jr., D.J. Cotter and V.B. Pizziconi, Ultrafiltration of molecules through depos- ited protein layers, Trans. Am. Sot. Artif. Intern. Organs, 21 (1975) 132. H. Fricke, A mathematical treatment of the electrical conductivity and capacity of disperse systems, Phys. Rev., 24 (1924) 575. N.L. Wang, Effects of fluid shear on the mass transportpropertiesof erythrocyte suspensions, Ph.D. Thesis, University of Minnesota, Minneapolis, 1978. R. Fahraeus and T. Lindqvist, The viscosity of the blood in narrow capillary tubes, Am. J. Physiol., 96 (1931) 562. W.J. Dorson and V.B. Pizziconi, High-efficiency hemofiltration, Trans. Am. Sot. Artif. In- tern. Organs, 33 (1987) 765.