Ind. Eng. C h e m . Fundam. 1986, 25, Butler, W. T.; Cottlove, E. J . Infect. Dis. 1971, 723, 341-350. Hellums, J. D.; Brown, C. H. I n Cardiovascular Flow Dynamics and Measurements; Hwang, N. H. C., Norman, N. A,, Eds.; University Park: Baltimore, MD, 1977;Chapter 20. Lijana, R. C.; Williams, M. C. Cell Biophys. 1986, 8 , 223-242. McLaughlin, S.;Eisenberg, M. Annu. Rev. Biophys. Bioeng. 1975, 4 ,
335-366. Mel, H. C.; Yee, J. P. Blood Cells 1975, 7 , 391-399. Monroe, J. M.; Lijana, R. C.; Williams, M. C. Biomafer., Med. Devices, Artif. Organs 1980, 8 , 103-144. Monroe. J. M.; True, D. E.; Williams, M. C. J . Biomed. Mater. Res. 1981, 75, 923-939. Nevaril, C. G.;Lynch, E. C.; Alfrey, C. P.; Hellums. J. D. J . Lab. Clin. Med. 1968, 77,784-790. Nichols, A. R.; Williams, M. C. Biomafer., Med. Devices, Artifif.Organs 1978, 4 , 21-48.
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633-635
Offeman, R. D.;Williams, M. 1978, 4 , 49-79. Offeman, R. D.; Williams, M. I979a, 7 , 359-391. Offeman, R. D.; Williams, M. 1979b, 7,393-420. Rubin, C. S.; Erlichman, J.;
C. Biomafer., Med. Devices, Artif. Organs
C. Biomafer., Med. Devices, Arfif. Organs C. Biomater ., Med. Devices, Arfif. Organs Rosen, 0. M. J . Biol. Chem. 1972, 2 4 7 ,
6135-6 139. ShaDiro, S. I.; Williams, M. C. AIChE J . 1970, 76.575-579. Tadano. K.;Hellums, J. D.; Lynch, E. C.; Peck, E. J.; Alfrey, C. P. Blood Cells 1977, 3 , 163-174. Yee, J. P.; Mel, H. C. Biorheology 1978, 75,321-339.
Received for review J u n e 4, 1986 Accepted July 21, 1986
Rates of Blood Filtration. A Brief Review Fumltake Yoshidat Kyoto University, Kyoto 606, Japan
Filtration of blood has two major categories: (a) ultrafiltration (“hemofiltration”) used in some types of artificial kidneys and (b) microfiltration (membrane “plasmapheresis”) used to separate blood into cells and plasma for therapeutic purposes or in pharmaceutical processing. The filtrate fluxes in both (a) and (b), which vary with the channel dimensions and the wall shear rates, can be correlated by assuming the concentration polarization of proteins in (a) and of blood cells in (b). I n (a) the effective diffusivity of proteins in plasma varies with the shear rate and the hematocrit (volume percentage of red cells). I n (b) the effective diffusivity of cells in plasma varies with the shear rate, the cell size, and the hematocrit. Only studies of chemical engineering interest are reviewed.
General Filtration of blood is increasingly important in medical technology. It has two major categories. One is ultrafiltration of blood, i.e., “hemofiltration” in medical terminology. In the artificial kidney using this principle, the blood of a patient is recirculated by a pump through an extracorporeal system which includes an ultrafilter. Blood cells and macromolecules in plasma, such as proteins, do not pass through the filter membrane. The filtrate containing micromolecular solutes, including urea and other uremic toxins, is continuously discarded. The loss of body fluid is made up by diluting the blood returning to the blood vessel of the patient with a physiological saline solution, either before or after filtration. Because of the higher costs of the equipment and the diluting fluid, such a system is not yet widely used, although it has merit over the conventional artificial kidney of the hemodialyzer type in that toxins larger than urea in molecular size can be removed from blood by convective transport through the membrane a t the same rates as urea. Of more recent development is the spontaneous continuous “AV (arteriovenous) hemofilter” for the removal of excessive body fluid of patients suffering from edema and other disorders. In this case, the driving forces for blood flow through the filter and for filtrate flow through the membrane are provided by the pumping action of the human heart. It might be mentioned that the glomerular basement membrane of the human kidney is also an ultrafilter. The other category of blood filtration is microfiltration, or, in medical terminology, membrane “plasmapheresis” Professor emeritus, C h e m i c a l Engineering. Address correspondence to: 2 Matsugasaki-Yobikaeshicho, K y o t o 606, J a p a n .
(from the Greek word aphairesis meaning removal). In this case, formed blood elements, i.e., erythrocytes (red blood cells (RBC), ca. 8 pm), leukocytes (while blood cells, ca. 10 pm), and platelets (ca. 3 pm), are filtered out by a microporous membrane. The filtrate is plasma, which contains all the macro- and micromolecular solutes outside the cells. Plasmapheresis has two major applications. In donor plasmapheresis, also performed by centrifugation, blood cells are returned to the blood vessel of the donor, and plasma is used for transfusion or large-scale fractionation of plasma components such as albumin. Intensive research is in progress for therapeutic applications of membrane plasmapheresis, in which plasma is continuously separated from the blood of patients with various serious diseases due to abnormality in blood components, e.g., myasthenia gravis, macroglobulinemia, rheumatoid arthritis, hyperleukocytosis, etc. Plasma containing pathogenic molecules is either replaced by plasma from healthy donors or is further treated to remove pathogens before it is returned to the patient. Various methods of plasma treatment, such as adsorption, affinity chromatography, and cascade filtration, are being developed. In cases in which disorder is with blood cells, removal of diseased cells could be performed after plasma separation. For blood ultrafiltration, various anisotropic membranes of polysulfones,poly(acrylonitri1e) (PAN),cellulose acetate, etc., are available. For membrane plasma separation, various mkroporous membranes, normally with mean pore sizes of 0.2-0.6 pm, made of cellulose acetate, PAN, poly(propylene), poly(methy1 methacrylate), poly(viny1 alcohol), etc., are used. Ozawa et al. (1986) report data on plasma separation by ceramic membranes, which could be used repeatedly after regeneration. As filter modules, the hollow-fiber (capillary) type and the flat-membrane type
0196-4313/86/1025-0633$01.50/00 1986 American Chemical Society
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are widely used, particularly the former. In both types blood flows along the membrane surface in cross-flow to the filtrate. The present review is intended to cover only a small number of studies of chemical engineering interest on the rates of ultrafiltration and microfiltration of blood, omitting numerous reports from the medical and chemical viewpoints.
Ultrafiltration of Blood Blood purification by ultrafiltration, i.e., “hemofiltration”, was first proposed by Henderson et al. (1967). In an article describing ultrafiltration as a new separation technique, Michaels (1968) presented a few data on ultrafiltration of human plasma and predicted largescale fractionation of human blood proteins by the technique. He also gave a model of concentration polarization, i.e., accumulation of solute on the membrane surface. Blatt et al. (1970) and Porter (1972) refined this model and also gave some data on ultrafiltration of blood and plasma. Up to a certain transmembrane pressure (TMP), usually 200-500 mmHg in the case of plasma or blood, the ultrafiltrate flux J increases almost linearly with TMP. In this region the membrane offers the controlling resistance to filtrate flow. At higher TMP’s, however, a dynamic gel layer of proteins is formed on top of the membrane, offering major resistance to filtrate flow. The gel layer grows in thickness until the convective transport of the solute toward the membrane becomes equal to the diffusive transport of the solute away from the gel layer surface due to the concentration gradient. In the plateau region of the J vs. T M P curve, increasing T M P will not help, since the gel layer only grows in thickness to offer more resistance in proportion to increased TMP. Thus, the dynamic balance of the solute transfer for the plateau region is given as
JC = D(dC/dx)
(11
where C is the solute concentration, D is the solute diffusivity, and x is the distance perpendicular to the gel layer surface. Integration of of eq 1 gives
J = ( D / A x ) In (C,/C) = k In (C,/C)
(2)
where Ax is the effective thickness of the boundary layer on the gel layer surface, C, is the solute concentration at the surface, and k is the mass-transfer coefficient for the solute. For correlating k in laminar flow in a capillary or a thin rectangular channel, which is commonly used in blood filtration, the classical Leveque (1928) type equation, originally obtained for heat transfer, is useful, i.e. Sh = 1.62[ReSc(dh/Z,)]’i”
(3)
where Sh, Re, and Sc are the dimensionless Sherwood, Reynolds, and Schmidt numbers, respectively; dh/L is the ratio of the hydraulic diameter to the channel length. From eq 3, Blatt et al. 11970) obtained for the local value of k
h = B(y,D2/L)’!3(y/L)-l/3
(4)
where B is a constant dependent on the wall boundary conditions, L is the membrane length, and y is the axial coordinate along the length. The wall shear rate y u is 8UId for a circular tube with diameter d and is 6Ulh for a rectangular channel with height h, where U is the average fluid velocity over the cross section. Kozinski and Lightfoot (1972) carried out an analysis of their data on ultrafiltration of albumin solutions, which included the effects of concentration-dependent physical
properties of protein solutions. From eq 2 and 4, Colton et al. (1975) derived the following equation for the length-averaged ultrafiltrate flux J from blood or plasma, assuming that C,/C was independent of y
J = 0.807(ywD,2/L)1’3In (C,/C)
(5)
where J , yw,De, and L should be in consistent units. It should be noted that in the case of whole blood, which is a suspension of blood cells in plasma, viscosity and the effective protein diffusivity De depend not only on the protein concentration but also on the hematocrit, i.e., the volume percentage of RBC, which normally ranges from 35% to 45%. Colton et al. (1975) observed a greater dependence of the filtrate flux on the wall shear rate with whole blood than with plasma. They suspected that the effective protein diffusivity was augmented by the rotational and translational movements of red cells. They also stated that the decrease in the flux with increasing hematocrit was possibly due to the damping out of such a augmentation mechanism by cell-cell interactions. Okazaki and Yoshida (1976) proposed the following semiempirical equation based on their experiments on ultrafiltration of blood a t 37 “C and for Re of 0.13-17, in which the hematocrit, Ht, was varied from 0 to 44%
J = 3.03 X 10-5f(Hty,,)(y,/l)1’3 In (C,/C)
(6)
where J , yw,and L are in cm/s, s-l, and cm, respectively. The variation of the empirical function f(Htyw)with H t and ywis shown by a graph. Filtrate fluxes, J , at wall shear rates, yw,over 3000 s-l increase with Ht, and those at yw below 2000 s-l increase and then decrease with increasing Ht. They attribute the increase and the decrease in J with increasing H t to the turbulence-promoting and the barrier effects of the red cells, respectively. They also showed that the data of Colton et al. (1975) could be correlated by eq 6. Dorson and Pizziconi (1980) published an extensive review on ultrafiltration of plasma and blood, covering reports that had appeared before 1978, by which time most of the fundamental research on this subject had been reported. In case of the spontaneous “AV hemofiltration”, the pressure drop available for blood flow is always smaller than the difference between the arterial and venous pressures, and the T M P is rather low, say 50-70 mmHg, since no pump is used for filtrate withdrawal. Thus, AV hemofilters are usually operated in the TMP-dependent region of the J vs. T M P curve, where no gel layer is formed and membrane permeability is important. As shown by Lysaght et al. (19851, the plasma oncotic pressure, Le., osmotic pressure due to the plasma colloids, cannot be neglected and must be deducted from the TMP. The intercept of the experimental J vs. T M P straight line on the TMP axis gives the oncotic pressure, which is normally ca. 25 mmHg.
Microfiltration of Blood Plasma separation from whole blood by microporous membranes was first reported by Solomon et al. (1978). Microporous membranes have much larger pore size, porosity, and water permeability compared with membranes for ultrafiltration. The plasma flux in blood microfiltration is several orders of magnitude smaller than water flux under the same conditions. Formation of a protein gel layer is inconceivable, since practically all plasma proteins go into the filtrate. The flux becomes independent of the transmembrane pressure (TMP) a t a lower T M P than in ultrafiltration, say 70-200 mmHg. In view of these facts,
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
it seems probable that, in the plateau region of the J vs. T M P curve, concentration or deposition of blood cells on the membrane surface offers the major resistance to the filtrate flow. The filtrate flux in blood microfiltration usually increases with the wall shear rate to the 0.7-1.0 power. In analogy to the concentration polarization of proteins in blood ultrafiltration, Zydney and Colton (1982) assumed the concentration polarization of blood cells, mainly red cells, in blood microfiltration. The problem was what sort of diffusivity should be used as the effective diffusivity, De, of RBC. They adopted eq 7, based on the data of Eckstein et al. (1977) for the enhanced diffusive motion of particles as a result of lateral migrations in a shear flow De = 0.025a2y,
(7)
where a is the particle radius and ywis the wall shear rate. From eq 2,4, and 7 Zydney and Colton derived eq 8 for the length-averaged flux in blood microfiltration, assuming that yw and C, were independent of the axial position
J = 0.070(a4/L)1/3y, In (Cw/C)
(8)
where J , a, yw, and the membrane length L should be in consistent units. Experimental data with hollow-fiber and flabmembrane devices showed agreement with predictions by eq 8. Martin et al. (1983) and Malbrancq et al. (1984) with flat membranes and Jaffrin et al. (1984) with hollow fibers also assumed concentration polarization. However, these investigators suspect that the filtrate flux is mainly governed by the concentration polarization of platelets rather than of RBC, in view of the variation of the filtrate flux with platelet concentrations (Martin e t al., 1983). For the enhanced effective solute diffusivity De in flowing blood a t 40% hematocrit,, Wang and Keller (1979) obtained the dimensionless equation (De - D , ) / D , = 0.061(a2y,/D,)0~9
(9)
where D, is the effective solute diffusivity in the stationary suspension and a is the sphere-equivalent red cell radius (2.75 pm for human blood). From eq 7 and 9, Malbrancq et al. (1984) suggested the use of eq 10 De = D,
+ 0.025a2yw
(10)
For relatively large particles, eq 10 would be practically the same as eq 7. If the filtrate flux is proportional to yw, as is indicated by eq 8, the number of hollow fibers in parallel or the width of a flat membrane, both for a given length, would not affect the total filtration rate, since an increase in the total membrane area would be offset by a decrease in the shear rate and, hence, the filtrate flux. This is supported by the experiments of Malbrancq et al. (1984). However, Jaffrin et al. (1984) observed that J was proportional to ywto the 0.74 power, which could be explained if De varies with yw to the 0.61 power. A t any rate, the effective particle diffusivity De, an empirical factor for convenience, increases with the shear rate 7,. High accuracy in correlating De with ywcannot be expected because of experimental difficulties.
635
It should be noted that an excessive blood velocity through a filter module would result in too high a pressure drop and, hence, too high an average TMP, which should be kept low, preferably under 100 mmHg to avoid too much hemolysis, i.a., rupture of RBC, as was shown by the data of Malbrancq et al. (1984). Other models have been proposed for microfiltration of blood. They differ from the concentration polarization model in explaining how blood cells migrate away from the membrane surface or from the cell layer which exists an the membrane. Dunleavy et al. (1984) believe that sliding or convection along the membrane of the RBC layer on the membrane surface counterbalances the convective flux of cells with the filtrate flow toward the membrane. They assume that the thickness of the cell layer varies with the wall shear rate, membrane length, and other factors. It is well-known that, when a suspension of particles flows through a thin tube in laminar flow, particles tend to migrate to an equilibrium position between the wall and the tube axis. Malchesky et al. (1984) considered that the cell migration away from the membrane surface was caused mainly by the lift force to such a “tubular pinch” effect. Final Remarks Other technical problems, such as biocompatible membrane materials, sieving or rejection characteristics of membranes, and design of filter modules, are important in the development of blood filters. Further treatment of plasma filtrate or separated blood cells for therapeutic purposes or pharmaceutical processing would impose many problems challenging to the chemical engineer. Literature Cited Blatt, W. F.; Dravid, A,; Michaels, A. S.; Neisen, L. I n Membrane Science and Technology; Flinn, J. E., Ed.; Plenum: New York, 1970; pp 47-97. Colton, C. K.; Henderson, L. W.; Ford, C. A,; Lysaght, M. J. J . Clin. Exp Med. 1975, 8 5 , 355-371. Dunleavy, M. J.; Leonard, E. F.; Vassilief, C. S. Trans-Am. SOC. Arfif. Intern. Organs 1984, 3 0 , 657-664. Dorson, W. J.; Plzziconi, V. B. I n Advances in Biomedical Engineering; Cooney, D. O., Ed.; Marcel Dekker: New York, 1980; Vol. 2, pp 561-683. Eckstein, E. C.; Bailey, D. G.; Shapiro, A. H. J . Fluid Mech. 1977, 79, 19 1-208. Henderson, L. W.; Besarab, A,; Michaels, A. S.; Bluemle, L. W. Trans.-Am. SOC.Arfif. Intern. Organs 1987, 13, 216-226. Jaffrin, M. Y.; Gupta, B. B.; Ding, L. H.; Garreau, M. Trans.-Am. SOC.Arfif. Intern. Organs 1984, 30, 401-405. Kozinski, A. A,; Lightfoot, E. N. AIChE J . 1972, 18, 1030-1038. Leveque, J. Ann. Mines. 1928, 73, 201, 305, 381. Lysaght, M. J.; Schmidt, B.; Gurland, H. J. I n Continuous Arteriovenous H e mofiflration; Kramer, P., Ed.; Springer-Verlag: Heidelberg, 1985; pp 3-13. Malbrancq, J.-M.; Jaffrin, M. Y.; Bouveret, E.; Angeraud, R.; Vantard, G. ASAIO J . 1984, 7 , 16-24. Malchesky, P. S.; Wojcicki, J.; Nos& Y. I n Progress in Arfificial Organs; Atsumi, K., et al., Eds.; ISAO: Cleveland, 1984; pp 649-654. Martin, T.; Jaffrin, M. Y.; Faure, A. Trans.-Am. SOC.Artif. Intern. Organs 1983, 29, 735-738. Michaels, A. S. Chem. Eng. Prog. 1968, 6 4 , 31-43. Okazaki, M.; Yoshida, F. Ann. Biomed. Eng. 1976, 4 , 138-150. Ozawa, K.; Sakurai, H.; Takesawa, S.; Sakai, K. Abstr.-Am. SOC. Artif. Intern. Organs Annu. Meet. 1986, 15, 31. Porter, M. C. Ind. Eng. Chem. Prod. Res. D e v . 1972, 1 1 , 234-248. Solomon, B. A.; Castino, F.; Lysaght, M. J.; Coiton, C. K.; Friedman, L. I. Trans.-Am. SOC.Artif. Intern. Organs 1978, 2 4 , 21-26. Wang, N.-H., L.; Keiler, K. H. Trans.-Am. SOC.Artif. Intern. Organs 1979, 2 5 , 14-18. Zydney, A. L.;Coiton, C. K. Trans.-Am. SOC. Artif. Intern. Organs 1982. 28. 408-412.
Received
reuieui June 10, 1986 Accepted June 30, 1986
for
