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A mechanistic model of plasma filtration
Medical Engineering & Physics 20 (1998) 383–392
Communication
A mechanistic model of plasma filtration Charles A. Fernandez a, Gerald M. Saidel a
a,*
, Paul S. Malchesky b, Maciej Zborowski
b
Department of Biomedical Engineering, Case Western Reserve University, Cleveland, OH, USA b Department of Biomedical Engineering, Cleveland Clinic Foundation, Cleveland, OH, USA Received 15 August 1997; accepted 21 January 1998
Abstract A model describing the sieving and transmembrane pressure behavior of plasma filtration is developed and numerically simulated. The model assumes a mechanistic criteria for particle passage through a membrane with cylindrical pores. The initial pore diameter distribution and porosity are assumed to be known. Model inputs include the particle diameter distribution, concentration and total flow rate of the permeate plasma solution. Outputs of the model include transmembrane pressure, the time-averaged sieving coefficients, and size distributions of the deposited particles and accumulated filtrate particles. Optimal filtration is characterized by high, stable sieving coefficients for desired particles, high retention of larger particles and relatively small increases in transmembrane pressure. These characteristics are realized for membranes with mean pore diameters equal to or slightly larger than mean permeate particle diameters. Simulations demonstrate that the incorporation of membrane properties into models of plasma filtration is both significant and readily possible. 1998 IPEM. Published by Elsevier Science Ltd. All rights reserved. Keywords: Plasma filtration; Plasma fractionation; Cryofiltration; Membrane; Sieving coefficient; Constant rate filtration; Transmembrane pressure; Modelling
1. Introduction Certain auto-immune disease states, such as rheumatoid arthritis and cryoglubulinemia, are characterized by high concentrations of immunoglobulins and immune complexes in the blood [1,2]. Generally accepted therapy for these conditions includes the removal of these pathogenic plasma components. For three decades, plasmapheresis therapy has involved separation and replacement of plasma from whole blood using centrifugation and membrane filtration techniques, i.e. plasma exchange [3,4]. Plasma exchange, however, necessitates the replacement of essential plasma volumes and constituents such as albumin, manifesting the possibility of infection and contributing to a high economic expense. To increase the efficiency of fractionation of plasma components by membrane filtration, the filtration process can be carried out at lower than physiologic temperature. In the treatment of cryoglubulinemia, plasma is cooled to a temperature of about 4°C and filtered to take advantage of the formation of macro-aggregates of immuno-
* Corresponding author.
globulins and immune complexes typical for this disease at low temperature [5]. The process of filtering plasma at reduced temperatures is called cryofiltration. Proteins exhibiting aggregation at temperatures below physiologic are called cryoprecipitable proteins (CPP). In cryofiltration, selectivity is increased because of the increased size differential between the aggregates of pathologic macromolecules and essential plasma constituents. Studies have shown that some of the most important considerations in cryofiltration are the temperature of the operation, the concentration and types of the plasma constituents, the type of anti-coagulant used, the membrane and module, and the flow dynamics [6,7]. The ideal membrane separation of CPP from albumin and immunoglobulins (IgG) would be completely selective, removing only the largest plasma component, CPP, with little pore plugging and a moderate transmembrane pressure increase. Transmembrane pressure increases limit the amount of plasma that can be processed with a given filter module. In a complex protein solution like plasma, containing many different types and sizes of molecules, separation between low molecular weight albumin and larger molecules is never complete and separation efficiency is determined by both the membrane
1350-4533/98/$19.00 1998 IPEM. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 0 - 4 5 3 3 ( 9 8 ) 0 0 0 2 1 - 6
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structure and the diameter distribution of molecules and molecular complexes [6,8,9]. The membranes used for plasma fractionation are characterized by a molecular weight cutoff point and a pore diameter distribution. However, the techniques used to measure membrane pore diameters, such as mercury intrusion and water permeability, may not give results directly applicable to cryofiltration due to uncertainty regarding the molecular diameter of CPP particles and the effects of membrane plugging. It is therefore considerably more enlightening to view membrane filtration of CPP as an interaction between a membrane having certain pore characteristics and CPP characterized by a particle diameter distribution. In this study, we model the dead-end cryofiltration process of CPP by incorporating permeability and sieving characteristics of the native membrane. The objective is to gain a qualitative understanding of favorable membrane properties that will provide high retention of CPP without a large increase in transmembrane pressure and, thereby, increase filtration capacity for a given solution. The mathematical model developed here incorporates the membrane pore diameter distribution and solute diameter distribution, which permits a quantitative assessment of specific measurable outputs. Outputs are generated through computer simulation of the model for various conditions including those mimicking experimental conditions. Comparison of experimental data with simulations gives insight into the contribution of CPP deposition on the membrane and the effectiveness of filtration.
phases: the membrane; the plasma solvent impinging on the membrane (permeate); and the filtered particles (solute) that deposit on the membrane (deposited layer). The filtered solution or filtrate passes through the membrane with less unwanted CPP. Plasma flows through the membrane at a constant flow rate Q. CPP particles containing plasma proteins (mostly immunoglobulins and immune complexes) are regarded as rigid spheres (including steric hindrance). The CPP aggregates, consisting of complex proteins, are approximated by assuming the particle diameters to be a Gaussian distribution. Particle rejection and pore blockage occurs when a particle is moved by convection to a pore whose opening is of equal or smaller diameter. This particle will adhere and block the pore entrance due to the transmembrane pressure. Smaller particles will pass through unhindered. This is equivalent to the assumption that the contact forces between membrane surfaces and the particle are significantly smaller than the fluid drag forces. In a limiting case of a monodispersed particle suspension filtered through a membrane with a monodispersed pore size distribution, this model predicts an increase in the sieving coefficient from 0 to 1 as the particle diameter decreases below the pore diameter. Consequently, the total pore area available for solution to flow through the membrane will decrease with time causing an increase in transmembrane pressure. The effects of cake formation are assumed to be negligible in this model because the particle concentration is low for the applications of interest. 2.1. Membrane pore diameter distribution
2. Model development The membrane is modeled as a thin disk having cylindrical pores and a Gaussian pore diameter distribution as illustrated in Fig. 1. The system consists of three
The membrane pore diameter (MPD) distribution is represented by the number, n(zj,t), of open pores of diameter zj (j = 1,2,...,m) per unit membrane surface area at time t. The distribution will change during the filtration depending on the particle diameter distribution of the input plasma. The model assumes that different particles in the plasma have the same transport rate determined solely by convection and that the effects of diffusion, adsorption and intermolecular electrostatic effects are negligible. If we define the fraction of open pores of diameter zj to be Nj(t) =
n(zj,t) n0(t)
(1)
where
冘 m
Fig. 1. Schematic representation of filtration model. Physical correspondence of the membrane pore diameter (MPD) distribution and the deposit, permeate and filtrate particle size distributions.
Nj(t) = 1
j=1
then the total number of pores per unit area at time t is
C.A. Fernandez et al. / Medical Engineering & Physics 20 (1998) 383–392
冘 m
n0(t) =
n(zj,t)
(2)
j=1
Since the membrane is modeled as having cylindrical pores, it follows that volume porosity and surface porosity are the same. For this model, the surface porosity, ⑀(t), is defined as the ratio of open pore area to the total membrane surface area:
冘
冘
z2n(z ,t) = n0(t) z2jNj(t) 4 j=1 j j 4 j=1 m
⑀(t) =
Qj dn(zj,t) =− dt A
冘
h(xi)U(xi−zj)
(6)
i=1
m
4⑀(t)
(3)
冘 k
Ch =
h(xi)
(7)
i=1
(4)
m
k
where h(xi) equals the number concentration of particles xi in the permeate. By incorporating a unit step function U(zj−xi) in Eq. (6), the pore-blocking particles are restricted to those equal to or larger than the pore diameter zj. The total number of particles present in the permeate per unit volume is
By rearrangement, we find n0(t) =
冘
385
We can define the fraction of particles of diameter xi in the permeate as:
2 j
z Nj(t) Hi =
j=1
2.2. Pore size dynamics
h(xi) Ch
(8)
To complete the model, we must relate Qi to the known flow rate Q via pressure and flow relations.
As the filtration proceeds, the membrane pore diameter distribution changes due to pore blockage. Pore blockage affects both the permeability characteristics of the membrane (viz. a higher hydraulic resistance at a given plasma flow rate) and its sieving characteristics. Both of these effects can be quantified by the permeate particle diameter distribution and the pore diameter distribution. The pore size dynamics depend on membrane–particle interactions. The particles are carried to the membrane by the convective flow of the solution, which passes through the open pores of the membrane. We shall consider the special case of particles flowing only into open pores via curved streamlines. In this case, there is minimal deposition in accordance with the assumption of negligible particle–membrane attractive forces compared to fluid drag [10]. Particle retention occurs only at membrane pores when the particle is of equal or greater (effective) diameter to that of the pore. For this minimal deposition model, we consider the number of open pores of diameter zj blocked in a short time dt to be − A dn(zj), where A is the membrane surface area. The decrease in the number of open pores of diameter zj is equal to the number of particles with diameter xi ⱖ zj that impinge upon the open pores in dt, multiplied by the fraction of total flow Qj entering open pores of diameter zj:
2.3. Flow and pressure relations During filtration, pores will be blocked by particles of equal or greater diameter causing a change in the MPD distribution. For steady flow through a cylindrical pore of diameter zj and length L, the flow rate through the pore qj is related to the transmembrane pressure ⌬P according to the Hagen–Poiseuille relation: qj = ␥⌬Pz4j
(9)
where ␥ = /128TL is a conductance coefficient in which is the absolute fluid viscosity and T is a tortuosity factor [11]. The flow in all pores of diameter zj is Qj(t) = qjn(zj ,t)A
(10)
where A is the membrane surface area. Substituting Eq. (9) for qj into Eq. (10) gives Qj(t) = ␥⌬Pz4jn(zj,t)A
(11)
The flow through all open pores is
冘 m
冘
Q=
k
− Adn(zj,t) = Qjdt
h(xi)U(xi−zj)
(5)
Qj
(12)
j=1
i=1
or
By substituting Eq. (11) into Eq. (12), we obtain the flow:
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冘 m
Q = ␥A⌬P
z4jn(zj,t)
(13)
j=1
Q/(␥A)
冘 m
(14)
4 j
z n(zj,t)
j=1
Substituting Eq. (14) for ⌬P into Eq. (11) gives the flow through a single pore: Qj(t) = Q
z4jn(zj,t)
冘 m
(18)
When combined with Eq. (16) for sieving, this becomes
or, by rearrangement, the transmembrane pressure: ⌬P =
dg(xi,t) = Q[h(xi) − f(xi,t)] dt
dg(xi) = Q{h(xi)[1 − (xi,t)]} dt
(19)
This can be solved with the condition that initially the membrane surface is free of particles: g(xi,0) = 0
(20)
(15)
z4jn(zj,t)
3. Simulation strategy
j=1
This expression relates the flow entering all pores of diameter zj at time t to the total flow rate Q and the MPD distribution. When combined with Eq. (6) and its initial condition, n(zj,0), the MPD distribution can be evaluated at any time. 2.4. Particle distribution dynamics
Solving the model equations above, we can analyze how the membrane pore distribution and the permeate particle distribution interact to produce the filtrate. In this simulation study, we assume a constant flow Q of permeate and compute the transmembrane pressure ⌬P(t). Another output function that can be determined experimentally is the time-averaged sieving coefficient: tf
The output distribution f(xi,t) of filtrate particles of diameter xi per unit volume at time t can to related to the input distribution of the permeate particles h(xi) according to f (xi,t) ⬅ (xi,t)h(xi)
(16)
where (xi,t) is the sieving coefficient. The sieving characteristics of a membrane determine the fraction of the concentration in the permeate present in the filtrate. In our membrane model, these characteristics are governed solely by the MPD distribution. Assuming a random distribution of particles to pores, each particle of a given diameter is subjected to the rejection characteristics of the same MPD. According to our model, the fraction of particles passed by the membrane equals the fraction of open pores in the membrane with diameters greater than the particle:
冘
S(xi,tf) =
Nj(t)U(zj−xi)
(21)
0
Characteristic of the filtrate is the number concentration distribution of the accumulated filtrate particles. This is obtained from the number of particles of diameter xi in the accumulated filtrate when the plasma volume V is processed: tf
F(xi,tf) =
冕
Q f(xi,t) dt V
(22)
0
Since Q is a constant in this model, V(t) = Qt. Also, we compute the total mass of permeate particles deposited on the membrane at any time (assuming spherical particles):
m
(xi,t) =
冕
1 (xi,t) dt tf
冘 k
(17)
j=1
where Nj(t) is the fraction of open pores of diameter zj and U(zj−xi) is the unit step function. To evaluate the distribution of deposited particles, g() of diameter x at time t, we make a particle number balance:
M(t) =
1 g(x ,t)x3 6 i=1 i i
(23)
where is the mass density of permeate particles. To simulate the constant flow filtration process, we solve the open pore size distribution simultaneously with the deposited particle size distribution, the flow relation and the sieving relation. The sequence of calculations is
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shown in the flow chart of Fig. 2. The differential equations are solved using LSODES, a well-tested integration program for sparse and stiff systems that includes error bounds [12]. The model inputs and known parameters are: the initial fraction Nj(0) of open pores of diameter zj; permeate particle diameter distribution, Hi, as a fraction of particles of diameter xi; initial membrane porosity, ⑀(0); initial transmembrane pressure, ⌬P(0); membrane area A; total permeate flow Q; and permeate particle concentration, Ch. Based on experimental distribution data for permeate particle diameter and membrane pore diameter [9], we assumed both Hi and Nj(0) to be Gaussian and characterized by distribution means and standard deviations. These parameters were chosen such that 99% of the distributions lie within a range of pore and particle diameters of 2–6 m. Forty discrete particle and pore diameters between 2 and 6 m (in 0.1 m steps) were defined for the simulation. The fractional permeate particle diameter distribution can be expressed as: Hi =
e−(xi−p)
冘
2/2s 2 p
41
e
, 2 ⱕ xi ⱕ 6
(24)
−(xi−p)2/2sp2
i=1
where p is the mean and sp is the standard deviation. The initial distribution of membrane pore diameters is Nj(0) =
e−(zj−n)
冘
2/2s 2 n
41
, 2 ⱕ zj ⱕ 6
(25)
−(zj−n)2/2sn2
e
j=1
where n is the mean and sn is the standard deviation. The model equations were solved repeatedly to show the theoretical effects of changes in the diameter distri-
387
Table 1 Parameter values for runs 1–9
⑀(0) A (m2) Q (ml/min) ⌬P(0) (mmHg) (gm/ml) Ch (#/m3) p(m) sp (m)
0.5 1 20 1 1.1 0.0035 4.0 0.4
bution of membrane pores. A special simulation was considered with parameter values typical of a particular set of experimental data [9]. The model parameter values used in these simulations are given in Tables 1 and 2. The simulations were stopped when the transmembrane pressure reached 300 mmHg.
4. Results Sieving coefficients for the specified range of particle sizes averaged over time, S(xi) are shown in Fig. 3. These S(xi) distributions differ significantly depending on the initial membrane pore diameters (MPD). When the mean (n) or standard deviation (sn) of the MPD is increased, the larger particles are able to pass through the membrane unhindered. This behavior is related to the g(xi) distribution of particles deposited on the membrane (Fig. 4). When the standard deviation sn is increased, g(xi) shifts toward the larger diameter particles and the number of deposited particles decreases. As n is increased, the total number of particles retained by the membrane decreases, reflecting the greater facility of the membrane to pass particles of the fixed size distribution of permeate particles. When the initial mean pore diameter is less than the mean permeate particle diameter (n⬍p), the particle deposition distribution tends to mirror the permeate distribution because the membrane is capable of retaining nearly all the particle sizes of the permeate. For n>p, a larger standard deviation (sn) of Table 2 Membrane distribution parameters for simulations with particles of mean diameter, p = 4.0, and standard deviation, sp = 0.4 Run
Fig. 2. Flowchart of simulation. Sequence of solution for model equations.
1 2 3 4 5 6 7 8 9
n
sn 3.5 3.5 3.5 4.0 4.0 4.0 4.5 4.5 4.5
0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6
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Fig. 3. Effect of initial membrane pore diameters on sieving. Each graph shows time-averaged sieving coefficients for different initial standard deviations (sn) of the MPD: (–) 0.2 m; (- - -) 0.4 m; (· · ·) 0.6 m. The three graphs correspond to initial mean n of membrane pore diameters (MPD): (a) 3.5 m, (b) 4.0 m and (c) 4.5 m.
Fig. 4. Effect of initial membrane pore diameters on deposition. Each graph shows the number distribution of particles retained by the membrane at the end of the filtration (⌬P = 300 mmHg) for different initial standard deviations (sn) of the MPD: (–) 0.2 m; (- - -) 0.4 m; (· · ·) 0.6 m. The three graphs correspond to initial means (n) of the MPD: (a) 3.5 m, (b) 4.0 m and (c) 4.5 m.
C.A. Fernandez et al. / Medical Engineering & Physics 20 (1998) 383–392
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pore diameters shifts g(xi) lower and tends to widen the deposition distribution, reflecting the existence of smaller pores in the membrane capable of blocking smaller particles. An increase of n at constant porosity implies fewer but larger diameter pores, which decrease membrane retention capacity. The anomalous appearance of the time-averaged sieving curve of Fig. 3(a) is a result of a middle range of pores becoming completely plugged over time. The smallest pores of the membrane do not become plugged because their share of the total permeate flow is small. This leads to two distinct pore size distributions that translate into a biphasic time averaged sieving curve. When sp of the permeate particles is greater than sn of the pores, more smaller particles are deposited on the membrane. The greater deposition with time caused by the faster plugging of pores occurs when n is smaller than p. The distributions of the particle deposited on the membrane are shown in Fig. 4. They show a shift depending on the initial membrane pore distribution. Fig. 5 shows filtrate particle concentration distributions F(xi) accumulated over time for different distributions of the membrane pore diameter. When the mean pore diameter is not greater than the mean particle diameter (n⬍p), the mean particle size in the accumulated filtrate (f) increases as the standard deviation of pore sizes (sn) increases. The accumulated filtrate distribution approaches that of the permeate as both membrane distribution parameters (n and sn) increase to allow easier passage of the permeate particles. The time course of the pressure drop (⌬P) across the membrane (Fig. 6) is sensitive to relative differences between the initial mean diameter of the membrane pores (n) and the mean diameter of the permeate particles (p) as well as to the standard deviation (sn) of the pore diameters. As expected for larger values of n and sn, the transmembrane pressure takes longer to reach the limit of 300 mmHg. The filtration capacity of the membrane can be characterized by the total mass of permeate particles M(t) removed over a time period t. Fig. 7 shows M(tf) for several values of n and sn, where tf is the time required for ⌬P to reach 300 mmHg. As n decreases or sn increases, the filtration capacity of the membrane increases. Optimal filtration is characterized by high, stable sieving coefficients for desired particles, high retention of larger particles and relatively small increases in transmembrane pressure. These characteristics are realized for membranes with mean pore diameters equal or slightly larger than mean permeate particle diameters. 5. Discussion Plasma filtration has been analyzed by mathematical models that either prescribe (a) the permeability charac-
Fig. 5. Effect of initial membrane pore diameters on filtrate size distribution. Each graph shows the number concentration distribution of particles in the filtrate accumulated over the entire filtration period for different initial sn of the MPD: (–) 0.2 m; (- - -) 0.4 m; (· · ·) 0.6 m. The three graphs correspond to different initial means (n) of the MPD: (a) 3.5 m, (b) 4.0 m and (c) 4.5 m.
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Fig. 7. Effect of initial membrane pore diameters on total mass removed. Total mass removed for different initial sn of the MPD: (䊊) 0.2 m; (쐌) 0.4 m; (ⵜ) 0.6 m; assuming a CPP mass density of 1.1 g/ml.
Fig. 6. Effect of initial membrane pore diameters on transmembrane pressure. Transmembrane pressure versus time for different initial sn of the MPD: (–) 0.2 m; (- - -) 0.4 m; (· · ·) 0.6 m. The three graphs correspond to different initial means (n) of the MPD: (a) 3.5 m, (b) 4.0 m and (c) 4.5 m.
teristics of the membrane or (b) the membrane’s selectivity characteristics. Cake Filtration, Polarization Boundary Layer and Gel Layer Polarization models have all been developed to explain and predict membrane permeability and how it changes during a filtration [13]. All of these models assume a series resistance to solvent flow. Usually the resistance due to the membrane is assumed to be constant, while the developing fouling layer determines the time dependence of the total resistance. Quantitative relations to describe membrane fouling during constant-rate filtration were developed by Hermans and Bredee [14]. Although the Hermans– Bredee filtration relations have been used to fit filtration data, there is no unique correspondence between different model parameter values and specific physical interpretations [15]. The problem with the series resistance approach is that it emphasizes solution properties over membrane properties. In a dynamic description of the filtration, the three models obscure the contribution of the membrane events that affect flow resistance at its surface. Therefore, the models predict that filtration systems identical except for the membrane will not affect the dynamic behavior of the filtration, e.g. the change of ⌬P with time. None of these models incorporate quantifiable membrane properties such as pore diameter distribution, or porosity implying that membrane contributions are either unimportant or unchanging. The models, therefore, give no insight into membrane choices for limiting ⌬P increases for a given permeate. In order to investigate membrane properties effects on filtration, we developed a mechanistic model of filtration where the formation of a cake layer is negligible. Discrepancies between experimental data and the predictions of previous models have stimulated more detailed, mechanistic studies of protein fouling of mem-
C.A. Fernandez et al. / Medical Engineering & Physics 20 (1998) 383–392
branes. The data of Turker and Hubble [16] indicate changes in the resistance of a fixed fouling layer in opposition to the GLP description. Dejmek and Nilsson [17] have shown that a series-resistance model does not appropriately describe data for filtration of whey protein mixtures with ultrafiltration membranes. Unfortunately, these models reveal nothing of the selective properties of the filtration. Changes in the separation characteristics of the membrane due to the phenomena proposed by the specific model (e.g. cake formation) are not addressed. Cherkasov [18] and Michaels [19] have noted that the rejection coefficient R is a sigmoidal function of particle diameter and have considered a log–normal probability function of rejection with respect to particle diameter. In our model the sigmoidal nature of the sieving coefficient (I–R) arose from the cumulative sum of the membrane pore diameter distribution. Zeman [20] addresses selectivity considerations, but does not incorporate pore diameter distributions. Sieving models such as Zeman’s predict selectivity, but not permeability. Conversely, permeability models fail to describe membrane selectivity. Cherkasov presents a model of pore size reduction by gel layer growth which predicts shifting with simultaneous sharpening of retention curves to the region of lower molecular weight. Our simulation showed the analogous behavior for sieving, namely, shifting with sharpening of sieving curves to regions of high molecular weight (greater diameter) as shown in Fig. 3. Additionally, Cherkasov generates rejection curves for heteroporous membranes by assuming a Gaussian distribution of pore sizes similar to our model. He explains the origin of the universal log–normal probability function for retention based on the plugging of smaller sized pores and the dynamic behavior of the pore size distribution. Cherkasov’s model, however, is developed only for constant pressure filtration. Additionally, the model does not have a clear time dependence and incorporates no permeate particle size distribution. A number of studies have qualitatively investigated the effect of membrane structure on filtration dynamics. Fane et al. [21] have shown data supporting a dependence of flux under gel-polarized conditions on pure water fluxes for unfouled membranes. These data support the notion of membrane properties playing an important role in the filtration process. To correct for membrane porosity differences, Fane et al. propose a modification of the gel-polarized model with an effective free area correction factor that accounts for increased lateral flow through deposited layers. Ethier and Kamm [22] describe membrane influences arising from solvent flow through the reduced cross-sectional area of membrane pores. Their model takes into account the higher flow velocities and thus increased flow resistance through deposited layers near membrane pores. Consequently, the effect of hydrodynamic resistance is greatest when cake or gel layer thickness is small compared to
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inter-pore spacing or when membrane porosities are relatively small due to increased lateral flow. In our model, however, hydrodynamic resistance decreases with greater interpore spacing (increasing n at constant porosity) because of the strong dependence of pore diameter on ⌬P predicted by the Hagen–Poiseuille relation as well as the absence of a dynamic cake layer that resists lateral flow. The sieving microfiltration model by Filippov et al. [10] applies basic concepts, which are similar to those of our model, to predict filtrate particle size distributions and flux dynamics. That model has been shown to correctly predict the results of filtrate flow changes with time at a constant transmembrane pressure. A significant difference is that our model predicts not only the filtrate size distribution dynamics, but also the dynamics of the size distribution of the particles deposited on the membrane and the dynamics of the transmembrane pressure for a constant permeate flow. In summary, compared to most models cited above, our model has the advantage of having clear transitions between physical phenomena and mathematical formulation. Expressions for rate of pore blocking follow from specified criteria for sieving and for flow through the cylindrical pore as specified by the Hagen–Poiseuille relation. Inputs of our model are total permeate flow rate, effective membrane area available for filtration, initial transmembrane pressure at the specified flow rate, particle number concentration in the permeate, and the fractional distributions of permeate particle size and of initial pore size. The model outputs include the number and size distribution of open pores and of deposited particles and the transmembrane dynamics. Both model inputs and key outputs are measurable [1,5,6,9]. References [1] Malchesky PS, Asanuma Y, Smith JW, Kayashima K, Zawicki I, Werynski A, Blumenstein M, Nose´ Y. Macromolecule removal from blood. Trans Am Soc Artif Int Organs 1981;27:439–43. [2] Smith JW. On-line plasma therapy: an assessment of current issues. In: Nose´ Y, Malchesky PS, Smith JW, editors. Therapeutic apheresis: a critical look. Cleveland: ISAO Press, 1984:155–60. [3] Agishi T, Mineshima M. Separation and fractionation of plasma by double filtration technique. J Meteor Sci 1989;44:47–54. [4] Horiuchi Y, Malchesky PS, Matsugane T, Abe Y, Smith JW, Sakai K, Nose´ Y. Effect of temperature and plasma constituents in plasma filtration. In: Atsulli K, Maekawa M, Ota K, editors. Progress in artificial organs. Cleveland: ISAO Press, 1984:691–7. [5] Malchesky PS, Horiuchi Y, Lewandowski JJ, Nose´ Y. Membrane plasma separation and the on-line treatment of plasma by membranes. J Membrane Sci 1989;44:55–88. [6] Malchesky PS, Wojcicki J, Horiuchi T, Lee JM, Nose´ Y. Membrane separation processes for macromolecule removal. In: Nose´ Y, Malchesky PS, Smith JW, editors. Plasmapheresis. Cleveland: ISAO Press, 1983:51–67. [7] Leypoldt JK, Jorstad S, Frigon RP, Henderson LW. Temperature dependence of macromolecule sieving across plasma fractionating membranes. Trans Am Soc Artif Int Organs 1988;34:420–4.
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[8] Horiuchi Y, Malchesky PS, Usami M, Emura M, Nose´ Y. Selective removal of high level cryoglobulins and immune complexes by a large pore membrane. Trans Am Soc Artif Int Organs 1987;33:374–9. [9] Zborowski M, Malchesky PS, Nose´ Y. Temperature dependent protein removal by large pore membrane filtration. Trans Am Soc Artif Int Organs 1989;35:572–5. [10] Fillipov AC, Starov VM, Lloyd DR, Chakravarti S, Glaser S. Sieve mechanism of microfiltration. J Membrane Sci 1994;89:199–213. [11] Johnston PR. Fluid filter media: measuring the average pore size, pore-size distribution, and correlation with results of filtration tests. J Texs Eval 1985;13:308–15. [12] Hindmarsh AC. ODEPACK, A systematized collection of ODE solvers. In: Stepleman RS, editor. Scientific computing. Amsterdam: North-Holland, 1983:509–16. [13] Reihanian H, Robertson CR, Michaels AS. Mechanisms of polarization and fouling of ultrafiltration membranes by proteins. J Membrane Sci 1983;16:237–58.
[14] Hermans PH, Bredee HL. Zur Kenntnis der Filtrationsgesetze. Recl Trav Chim 1935;54:680–700. [15] Grace HP. Structure and performance of filter media. AIChE J 1956;2:2307–36. [16] Turker M, Hubble J. Membrane fouling in a constant flux ultrafiltration cell. J Membrane Sci 1987;34:267–81. [17] Dejmek P, Nilsson JL. Flux-based measure of adsorption to ultrafiltration membranes. J Membrane Sci 1989;40:189–97. [18] Cherkasov AN. Selective ultrafiltration. J Membrane Sci 1990;50:109–30. [19] Michaels AS. Analysis and prediction of sieving curves for ultrafiltration membranes. Sep Sci Technol 1980;15:1305–22. [20] Zeman LJ. Adsorption effects in rejection of macromolecules by ultrafiltration membranes. J Membrane Sci 1983;15:213–30. [21] Fane AG, Fell CJD, Waters AG. Relationship between membrane surface pore characteristics and flux for ultrafiltration membranes. J Membrane Sci 1981;8:245–62. [22] Ethier CR, Kamm RD. Hydrodynamic resistance of filter cakes. J Membrane Sci 1989;43:19–30.
Communication
A mechanistic model of plasma filtration Charles A. Fernandez a, Gerald M. Saidel a
a,*
, Paul S. Malchesky b, Maciej Zborowski
b
Department of Biomedical Engineering, Case Western Reserve University, Cleveland, OH, USA b Department of Biomedical Engineering, Cleveland Clinic Foundation, Cleveland, OH, USA Received 15 August 1997; accepted 21 January 1998
Abstract A model describing the sieving and transmembrane pressure behavior of plasma filtration is developed and numerically simulated. The model assumes a mechanistic criteria for particle passage through a membrane with cylindrical pores. The initial pore diameter distribution and porosity are assumed to be known. Model inputs include the particle diameter distribution, concentration and total flow rate of the permeate plasma solution. Outputs of the model include transmembrane pressure, the time-averaged sieving coefficients, and size distributions of the deposited particles and accumulated filtrate particles. Optimal filtration is characterized by high, stable sieving coefficients for desired particles, high retention of larger particles and relatively small increases in transmembrane pressure. These characteristics are realized for membranes with mean pore diameters equal to or slightly larger than mean permeate particle diameters. Simulations demonstrate that the incorporation of membrane properties into models of plasma filtration is both significant and readily possible. 1998 IPEM. Published by Elsevier Science Ltd. All rights reserved. Keywords: Plasma filtration; Plasma fractionation; Cryofiltration; Membrane; Sieving coefficient; Constant rate filtration; Transmembrane pressure; Modelling
1. Introduction Certain auto-immune disease states, such as rheumatoid arthritis and cryoglubulinemia, are characterized by high concentrations of immunoglobulins and immune complexes in the blood [1,2]. Generally accepted therapy for these conditions includes the removal of these pathogenic plasma components. For three decades, plasmapheresis therapy has involved separation and replacement of plasma from whole blood using centrifugation and membrane filtration techniques, i.e. plasma exchange [3,4]. Plasma exchange, however, necessitates the replacement of essential plasma volumes and constituents such as albumin, manifesting the possibility of infection and contributing to a high economic expense. To increase the efficiency of fractionation of plasma components by membrane filtration, the filtration process can be carried out at lower than physiologic temperature. In the treatment of cryoglubulinemia, plasma is cooled to a temperature of about 4°C and filtered to take advantage of the formation of macro-aggregates of immuno-
* Corresponding author.
globulins and immune complexes typical for this disease at low temperature [5]. The process of filtering plasma at reduced temperatures is called cryofiltration. Proteins exhibiting aggregation at temperatures below physiologic are called cryoprecipitable proteins (CPP). In cryofiltration, selectivity is increased because of the increased size differential between the aggregates of pathologic macromolecules and essential plasma constituents. Studies have shown that some of the most important considerations in cryofiltration are the temperature of the operation, the concentration and types of the plasma constituents, the type of anti-coagulant used, the membrane and module, and the flow dynamics [6,7]. The ideal membrane separation of CPP from albumin and immunoglobulins (IgG) would be completely selective, removing only the largest plasma component, CPP, with little pore plugging and a moderate transmembrane pressure increase. Transmembrane pressure increases limit the amount of plasma that can be processed with a given filter module. In a complex protein solution like plasma, containing many different types and sizes of molecules, separation between low molecular weight albumin and larger molecules is never complete and separation efficiency is determined by both the membrane
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structure and the diameter distribution of molecules and molecular complexes [6,8,9]. The membranes used for plasma fractionation are characterized by a molecular weight cutoff point and a pore diameter distribution. However, the techniques used to measure membrane pore diameters, such as mercury intrusion and water permeability, may not give results directly applicable to cryofiltration due to uncertainty regarding the molecular diameter of CPP particles and the effects of membrane plugging. It is therefore considerably more enlightening to view membrane filtration of CPP as an interaction between a membrane having certain pore characteristics and CPP characterized by a particle diameter distribution. In this study, we model the dead-end cryofiltration process of CPP by incorporating permeability and sieving characteristics of the native membrane. The objective is to gain a qualitative understanding of favorable membrane properties that will provide high retention of CPP without a large increase in transmembrane pressure and, thereby, increase filtration capacity for a given solution. The mathematical model developed here incorporates the membrane pore diameter distribution and solute diameter distribution, which permits a quantitative assessment of specific measurable outputs. Outputs are generated through computer simulation of the model for various conditions including those mimicking experimental conditions. Comparison of experimental data with simulations gives insight into the contribution of CPP deposition on the membrane and the effectiveness of filtration.
phases: the membrane; the plasma solvent impinging on the membrane (permeate); and the filtered particles (solute) that deposit on the membrane (deposited layer). The filtered solution or filtrate passes through the membrane with less unwanted CPP. Plasma flows through the membrane at a constant flow rate Q. CPP particles containing plasma proteins (mostly immunoglobulins and immune complexes) are regarded as rigid spheres (including steric hindrance). The CPP aggregates, consisting of complex proteins, are approximated by assuming the particle diameters to be a Gaussian distribution. Particle rejection and pore blockage occurs when a particle is moved by convection to a pore whose opening is of equal or smaller diameter. This particle will adhere and block the pore entrance due to the transmembrane pressure. Smaller particles will pass through unhindered. This is equivalent to the assumption that the contact forces between membrane surfaces and the particle are significantly smaller than the fluid drag forces. In a limiting case of a monodispersed particle suspension filtered through a membrane with a monodispersed pore size distribution, this model predicts an increase in the sieving coefficient from 0 to 1 as the particle diameter decreases below the pore diameter. Consequently, the total pore area available for solution to flow through the membrane will decrease with time causing an increase in transmembrane pressure. The effects of cake formation are assumed to be negligible in this model because the particle concentration is low for the applications of interest. 2.1. Membrane pore diameter distribution
2. Model development The membrane is modeled as a thin disk having cylindrical pores and a Gaussian pore diameter distribution as illustrated in Fig. 1. The system consists of three
The membrane pore diameter (MPD) distribution is represented by the number, n(zj,t), of open pores of diameter zj (j = 1,2,...,m) per unit membrane surface area at time t. The distribution will change during the filtration depending on the particle diameter distribution of the input plasma. The model assumes that different particles in the plasma have the same transport rate determined solely by convection and that the effects of diffusion, adsorption and intermolecular electrostatic effects are negligible. If we define the fraction of open pores of diameter zj to be Nj(t) =
n(zj,t) n0(t)
(1)
where
冘 m
Fig. 1. Schematic representation of filtration model. Physical correspondence of the membrane pore diameter (MPD) distribution and the deposit, permeate and filtrate particle size distributions.
Nj(t) = 1
j=1
then the total number of pores per unit area at time t is
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冘 m
n0(t) =
n(zj,t)
(2)
j=1
Since the membrane is modeled as having cylindrical pores, it follows that volume porosity and surface porosity are the same. For this model, the surface porosity, ⑀(t), is defined as the ratio of open pore area to the total membrane surface area:
冘
冘
z2n(z ,t) = n0(t) z2jNj(t) 4 j=1 j j 4 j=1 m
⑀(t) =
Qj dn(zj,t) =− dt A
冘
h(xi)U(xi−zj)
(6)
i=1
m
4⑀(t)
(3)
冘 k
Ch =
h(xi)
(7)
i=1
(4)
m
k
where h(xi) equals the number concentration of particles xi in the permeate. By incorporating a unit step function U(zj−xi) in Eq. (6), the pore-blocking particles are restricted to those equal to or larger than the pore diameter zj. The total number of particles present in the permeate per unit volume is
By rearrangement, we find n0(t) =
冘
385
We can define the fraction of particles of diameter xi in the permeate as:
2 j
z Nj(t) Hi =
j=1
2.2. Pore size dynamics
h(xi) Ch
(8)
To complete the model, we must relate Qi to the known flow rate Q via pressure and flow relations.
As the filtration proceeds, the membrane pore diameter distribution changes due to pore blockage. Pore blockage affects both the permeability characteristics of the membrane (viz. a higher hydraulic resistance at a given plasma flow rate) and its sieving characteristics. Both of these effects can be quantified by the permeate particle diameter distribution and the pore diameter distribution. The pore size dynamics depend on membrane–particle interactions. The particles are carried to the membrane by the convective flow of the solution, which passes through the open pores of the membrane. We shall consider the special case of particles flowing only into open pores via curved streamlines. In this case, there is minimal deposition in accordance with the assumption of negligible particle–membrane attractive forces compared to fluid drag [10]. Particle retention occurs only at membrane pores when the particle is of equal or greater (effective) diameter to that of the pore. For this minimal deposition model, we consider the number of open pores of diameter zj blocked in a short time dt to be − A dn(zj), where A is the membrane surface area. The decrease in the number of open pores of diameter zj is equal to the number of particles with diameter xi ⱖ zj that impinge upon the open pores in dt, multiplied by the fraction of total flow Qj entering open pores of diameter zj:
2.3. Flow and pressure relations During filtration, pores will be blocked by particles of equal or greater diameter causing a change in the MPD distribution. For steady flow through a cylindrical pore of diameter zj and length L, the flow rate through the pore qj is related to the transmembrane pressure ⌬P according to the Hagen–Poiseuille relation: qj = ␥⌬Pz4j
(9)
where ␥ = /128TL is a conductance coefficient in which is the absolute fluid viscosity and T is a tortuosity factor [11]. The flow in all pores of diameter zj is Qj(t) = qjn(zj ,t)A
(10)
where A is the membrane surface area. Substituting Eq. (9) for qj into Eq. (10) gives Qj(t) = ␥⌬Pz4jn(zj,t)A
(11)
The flow through all open pores is
冘 m
冘
Q=
k
− Adn(zj,t) = Qjdt
h(xi)U(xi−zj)
(5)
Qj
(12)
j=1
i=1
or
By substituting Eq. (11) into Eq. (12), we obtain the flow:
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冘 m
Q = ␥A⌬P
z4jn(zj,t)
(13)
j=1
Q/(␥A)
冘 m
(14)
4 j
z n(zj,t)
j=1
Substituting Eq. (14) for ⌬P into Eq. (11) gives the flow through a single pore: Qj(t) = Q
z4jn(zj,t)
冘 m
(18)
When combined with Eq. (16) for sieving, this becomes
or, by rearrangement, the transmembrane pressure: ⌬P =
dg(xi,t) = Q[h(xi) − f(xi,t)] dt
dg(xi) = Q{h(xi)[1 − (xi,t)]} dt
(19)
This can be solved with the condition that initially the membrane surface is free of particles: g(xi,0) = 0
(20)
(15)
z4jn(zj,t)
3. Simulation strategy
j=1
This expression relates the flow entering all pores of diameter zj at time t to the total flow rate Q and the MPD distribution. When combined with Eq. (6) and its initial condition, n(zj,0), the MPD distribution can be evaluated at any time. 2.4. Particle distribution dynamics
Solving the model equations above, we can analyze how the membrane pore distribution and the permeate particle distribution interact to produce the filtrate. In this simulation study, we assume a constant flow Q of permeate and compute the transmembrane pressure ⌬P(t). Another output function that can be determined experimentally is the time-averaged sieving coefficient: tf
The output distribution f(xi,t) of filtrate particles of diameter xi per unit volume at time t can to related to the input distribution of the permeate particles h(xi) according to f (xi,t) ⬅ (xi,t)h(xi)
(16)
where (xi,t) is the sieving coefficient. The sieving characteristics of a membrane determine the fraction of the concentration in the permeate present in the filtrate. In our membrane model, these characteristics are governed solely by the MPD distribution. Assuming a random distribution of particles to pores, each particle of a given diameter is subjected to the rejection characteristics of the same MPD. According to our model, the fraction of particles passed by the membrane equals the fraction of open pores in the membrane with diameters greater than the particle:
冘
S(xi,tf) =
Nj(t)U(zj−xi)
(21)
0
Characteristic of the filtrate is the number concentration distribution of the accumulated filtrate particles. This is obtained from the number of particles of diameter xi in the accumulated filtrate when the plasma volume V is processed: tf
F(xi,tf) =
冕
Q f(xi,t) dt V
(22)
0
Since Q is a constant in this model, V(t) = Qt. Also, we compute the total mass of permeate particles deposited on the membrane at any time (assuming spherical particles):
m
(xi,t) =
冕
1 (xi,t) dt tf
冘 k
(17)
j=1
where Nj(t) is the fraction of open pores of diameter zj and U(zj−xi) is the unit step function. To evaluate the distribution of deposited particles, g() of diameter x at time t, we make a particle number balance:
M(t) =
1 g(x ,t)x3 6 i=1 i i
(23)
where is the mass density of permeate particles. To simulate the constant flow filtration process, we solve the open pore size distribution simultaneously with the deposited particle size distribution, the flow relation and the sieving relation. The sequence of calculations is
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shown in the flow chart of Fig. 2. The differential equations are solved using LSODES, a well-tested integration program for sparse and stiff systems that includes error bounds [12]. The model inputs and known parameters are: the initial fraction Nj(0) of open pores of diameter zj; permeate particle diameter distribution, Hi, as a fraction of particles of diameter xi; initial membrane porosity, ⑀(0); initial transmembrane pressure, ⌬P(0); membrane area A; total permeate flow Q; and permeate particle concentration, Ch. Based on experimental distribution data for permeate particle diameter and membrane pore diameter [9], we assumed both Hi and Nj(0) to be Gaussian and characterized by distribution means and standard deviations. These parameters were chosen such that 99% of the distributions lie within a range of pore and particle diameters of 2–6 m. Forty discrete particle and pore diameters between 2 and 6 m (in 0.1 m steps) were defined for the simulation. The fractional permeate particle diameter distribution can be expressed as: Hi =
e−(xi−p)
冘
2/2s 2 p
41
e
, 2 ⱕ xi ⱕ 6
(24)
−(xi−p)2/2sp2
i=1
where p is the mean and sp is the standard deviation. The initial distribution of membrane pore diameters is Nj(0) =
e−(zj−n)
冘
2/2s 2 n
41
, 2 ⱕ zj ⱕ 6
(25)
−(zj−n)2/2sn2
e
j=1
where n is the mean and sn is the standard deviation. The model equations were solved repeatedly to show the theoretical effects of changes in the diameter distri-
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Table 1 Parameter values for runs 1–9
⑀(0) A (m2) Q (ml/min) ⌬P(0) (mmHg) (gm/ml) Ch (#/m3) p(m) sp (m)
0.5 1 20 1 1.1 0.0035 4.0 0.4
bution of membrane pores. A special simulation was considered with parameter values typical of a particular set of experimental data [9]. The model parameter values used in these simulations are given in Tables 1 and 2. The simulations were stopped when the transmembrane pressure reached 300 mmHg.
4. Results Sieving coefficients for the specified range of particle sizes averaged over time, S(xi) are shown in Fig. 3. These S(xi) distributions differ significantly depending on the initial membrane pore diameters (MPD). When the mean (n) or standard deviation (sn) of the MPD is increased, the larger particles are able to pass through the membrane unhindered. This behavior is related to the g(xi) distribution of particles deposited on the membrane (Fig. 4). When the standard deviation sn is increased, g(xi) shifts toward the larger diameter particles and the number of deposited particles decreases. As n is increased, the total number of particles retained by the membrane decreases, reflecting the greater facility of the membrane to pass particles of the fixed size distribution of permeate particles. When the initial mean pore diameter is less than the mean permeate particle diameter (n⬍p), the particle deposition distribution tends to mirror the permeate distribution because the membrane is capable of retaining nearly all the particle sizes of the permeate. For n>p, a larger standard deviation (sn) of Table 2 Membrane distribution parameters for simulations with particles of mean diameter, p = 4.0, and standard deviation, sp = 0.4 Run
Fig. 2. Flowchart of simulation. Sequence of solution for model equations.
1 2 3 4 5 6 7 8 9
n
sn 3.5 3.5 3.5 4.0 4.0 4.0 4.5 4.5 4.5
0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6
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Fig. 3. Effect of initial membrane pore diameters on sieving. Each graph shows time-averaged sieving coefficients for different initial standard deviations (sn) of the MPD: (–) 0.2 m; (- - -) 0.4 m; (· · ·) 0.6 m. The three graphs correspond to initial mean n of membrane pore diameters (MPD): (a) 3.5 m, (b) 4.0 m and (c) 4.5 m.
Fig. 4. Effect of initial membrane pore diameters on deposition. Each graph shows the number distribution of particles retained by the membrane at the end of the filtration (⌬P = 300 mmHg) for different initial standard deviations (sn) of the MPD: (–) 0.2 m; (- - -) 0.4 m; (· · ·) 0.6 m. The three graphs correspond to initial means (n) of the MPD: (a) 3.5 m, (b) 4.0 m and (c) 4.5 m.
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pore diameters shifts g(xi) lower and tends to widen the deposition distribution, reflecting the existence of smaller pores in the membrane capable of blocking smaller particles. An increase of n at constant porosity implies fewer but larger diameter pores, which decrease membrane retention capacity. The anomalous appearance of the time-averaged sieving curve of Fig. 3(a) is a result of a middle range of pores becoming completely plugged over time. The smallest pores of the membrane do not become plugged because their share of the total permeate flow is small. This leads to two distinct pore size distributions that translate into a biphasic time averaged sieving curve. When sp of the permeate particles is greater than sn of the pores, more smaller particles are deposited on the membrane. The greater deposition with time caused by the faster plugging of pores occurs when n is smaller than p. The distributions of the particle deposited on the membrane are shown in Fig. 4. They show a shift depending on the initial membrane pore distribution. Fig. 5 shows filtrate particle concentration distributions F(xi) accumulated over time for different distributions of the membrane pore diameter. When the mean pore diameter is not greater than the mean particle diameter (n⬍p), the mean particle size in the accumulated filtrate (f) increases as the standard deviation of pore sizes (sn) increases. The accumulated filtrate distribution approaches that of the permeate as both membrane distribution parameters (n and sn) increase to allow easier passage of the permeate particles. The time course of the pressure drop (⌬P) across the membrane (Fig. 6) is sensitive to relative differences between the initial mean diameter of the membrane pores (n) and the mean diameter of the permeate particles (p) as well as to the standard deviation (sn) of the pore diameters. As expected for larger values of n and sn, the transmembrane pressure takes longer to reach the limit of 300 mmHg. The filtration capacity of the membrane can be characterized by the total mass of permeate particles M(t) removed over a time period t. Fig. 7 shows M(tf) for several values of n and sn, where tf is the time required for ⌬P to reach 300 mmHg. As n decreases or sn increases, the filtration capacity of the membrane increases. Optimal filtration is characterized by high, stable sieving coefficients for desired particles, high retention of larger particles and relatively small increases in transmembrane pressure. These characteristics are realized for membranes with mean pore diameters equal or slightly larger than mean permeate particle diameters. 5. Discussion Plasma filtration has been analyzed by mathematical models that either prescribe (a) the permeability charac-
Fig. 5. Effect of initial membrane pore diameters on filtrate size distribution. Each graph shows the number concentration distribution of particles in the filtrate accumulated over the entire filtration period for different initial sn of the MPD: (–) 0.2 m; (- - -) 0.4 m; (· · ·) 0.6 m. The three graphs correspond to different initial means (n) of the MPD: (a) 3.5 m, (b) 4.0 m and (c) 4.5 m.
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Fig. 7. Effect of initial membrane pore diameters on total mass removed. Total mass removed for different initial sn of the MPD: (䊊) 0.2 m; (쐌) 0.4 m; (ⵜ) 0.6 m; assuming a CPP mass density of 1.1 g/ml.
Fig. 6. Effect of initial membrane pore diameters on transmembrane pressure. Transmembrane pressure versus time for different initial sn of the MPD: (–) 0.2 m; (- - -) 0.4 m; (· · ·) 0.6 m. The three graphs correspond to different initial means (n) of the MPD: (a) 3.5 m, (b) 4.0 m and (c) 4.5 m.
teristics of the membrane or (b) the membrane’s selectivity characteristics. Cake Filtration, Polarization Boundary Layer and Gel Layer Polarization models have all been developed to explain and predict membrane permeability and how it changes during a filtration [13]. All of these models assume a series resistance to solvent flow. Usually the resistance due to the membrane is assumed to be constant, while the developing fouling layer determines the time dependence of the total resistance. Quantitative relations to describe membrane fouling during constant-rate filtration were developed by Hermans and Bredee [14]. Although the Hermans– Bredee filtration relations have been used to fit filtration data, there is no unique correspondence between different model parameter values and specific physical interpretations [15]. The problem with the series resistance approach is that it emphasizes solution properties over membrane properties. In a dynamic description of the filtration, the three models obscure the contribution of the membrane events that affect flow resistance at its surface. Therefore, the models predict that filtration systems identical except for the membrane will not affect the dynamic behavior of the filtration, e.g. the change of ⌬P with time. None of these models incorporate quantifiable membrane properties such as pore diameter distribution, or porosity implying that membrane contributions are either unimportant or unchanging. The models, therefore, give no insight into membrane choices for limiting ⌬P increases for a given permeate. In order to investigate membrane properties effects on filtration, we developed a mechanistic model of filtration where the formation of a cake layer is negligible. Discrepancies between experimental data and the predictions of previous models have stimulated more detailed, mechanistic studies of protein fouling of mem-
C.A. Fernandez et al. / Medical Engineering & Physics 20 (1998) 383–392
branes. The data of Turker and Hubble [16] indicate changes in the resistance of a fixed fouling layer in opposition to the GLP description. Dejmek and Nilsson [17] have shown that a series-resistance model does not appropriately describe data for filtration of whey protein mixtures with ultrafiltration membranes. Unfortunately, these models reveal nothing of the selective properties of the filtration. Changes in the separation characteristics of the membrane due to the phenomena proposed by the specific model (e.g. cake formation) are not addressed. Cherkasov [18] and Michaels [19] have noted that the rejection coefficient R is a sigmoidal function of particle diameter and have considered a log–normal probability function of rejection with respect to particle diameter. In our model the sigmoidal nature of the sieving coefficient (I–R) arose from the cumulative sum of the membrane pore diameter distribution. Zeman [20] addresses selectivity considerations, but does not incorporate pore diameter distributions. Sieving models such as Zeman’s predict selectivity, but not permeability. Conversely, permeability models fail to describe membrane selectivity. Cherkasov presents a model of pore size reduction by gel layer growth which predicts shifting with simultaneous sharpening of retention curves to the region of lower molecular weight. Our simulation showed the analogous behavior for sieving, namely, shifting with sharpening of sieving curves to regions of high molecular weight (greater diameter) as shown in Fig. 3. Additionally, Cherkasov generates rejection curves for heteroporous membranes by assuming a Gaussian distribution of pore sizes similar to our model. He explains the origin of the universal log–normal probability function for retention based on the plugging of smaller sized pores and the dynamic behavior of the pore size distribution. Cherkasov’s model, however, is developed only for constant pressure filtration. Additionally, the model does not have a clear time dependence and incorporates no permeate particle size distribution. A number of studies have qualitatively investigated the effect of membrane structure on filtration dynamics. Fane et al. [21] have shown data supporting a dependence of flux under gel-polarized conditions on pure water fluxes for unfouled membranes. These data support the notion of membrane properties playing an important role in the filtration process. To correct for membrane porosity differences, Fane et al. propose a modification of the gel-polarized model with an effective free area correction factor that accounts for increased lateral flow through deposited layers. Ethier and Kamm [22] describe membrane influences arising from solvent flow through the reduced cross-sectional area of membrane pores. Their model takes into account the higher flow velocities and thus increased flow resistance through deposited layers near membrane pores. Consequently, the effect of hydrodynamic resistance is greatest when cake or gel layer thickness is small compared to
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inter-pore spacing or when membrane porosities are relatively small due to increased lateral flow. In our model, however, hydrodynamic resistance decreases with greater interpore spacing (increasing n at constant porosity) because of the strong dependence of pore diameter on ⌬P predicted by the Hagen–Poiseuille relation as well as the absence of a dynamic cake layer that resists lateral flow. The sieving microfiltration model by Filippov et al. [10] applies basic concepts, which are similar to those of our model, to predict filtrate particle size distributions and flux dynamics. That model has been shown to correctly predict the results of filtrate flow changes with time at a constant transmembrane pressure. A significant difference is that our model predicts not only the filtrate size distribution dynamics, but also the dynamics of the size distribution of the particles deposited on the membrane and the dynamics of the transmembrane pressure for a constant permeate flow. In summary, compared to most models cited above, our model has the advantage of having clear transitions between physical phenomena and mathematical formulation. Expressions for rate of pore blocking follow from specified criteria for sieving and for flow through the cylindrical pore as specified by the Hagen–Poiseuille relation. Inputs of our model are total permeate flow rate, effective membrane area available for filtration, initial transmembrane pressure at the specified flow rate, particle number concentration in the permeate, and the fractional distributions of permeate particle size and of initial pore size. The model outputs include the number and size distribution of open pores and of deposited particles and the transmembrane dynamics. Both model inputs and key outputs are measurable [1,5,6,9]. References [1] Malchesky PS, Asanuma Y, Smith JW, Kayashima K, Zawicki I, Werynski A, Blumenstein M, Nose´ Y. Macromolecule removal from blood. Trans Am Soc Artif Int Organs 1981;27:439–43. [2] Smith JW. On-line plasma therapy: an assessment of current issues. In: Nose´ Y, Malchesky PS, Smith JW, editors. Therapeutic apheresis: a critical look. Cleveland: ISAO Press, 1984:155–60. [3] Agishi T, Mineshima M. Separation and fractionation of plasma by double filtration technique. J Meteor Sci 1989;44:47–54. [4] Horiuchi Y, Malchesky PS, Matsugane T, Abe Y, Smith JW, Sakai K, Nose´ Y. Effect of temperature and plasma constituents in plasma filtration. In: Atsulli K, Maekawa M, Ota K, editors. Progress in artificial organs. Cleveland: ISAO Press, 1984:691–7. [5] Malchesky PS, Horiuchi Y, Lewandowski JJ, Nose´ Y. Membrane plasma separation and the on-line treatment of plasma by membranes. J Membrane Sci 1989;44:55–88. [6] Malchesky PS, Wojcicki J, Horiuchi T, Lee JM, Nose´ Y. Membrane separation processes for macromolecule removal. In: Nose´ Y, Malchesky PS, Smith JW, editors. Plasmapheresis. Cleveland: ISAO Press, 1983:51–67. [7] Leypoldt JK, Jorstad S, Frigon RP, Henderson LW. Temperature dependence of macromolecule sieving across plasma fractionating membranes. Trans Am Soc Artif Int Organs 1988;34:420–4.
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