Affinity chromatography has been an industrial standard method used to economically purify high-value biomolecules present at very low concentrations in complex biological fluids because of its simplicity and high degree of specificity [1] . Conventionally, affinity purification is carried out in columns packed with porous beads to which the affinity ligand is immobilized.
However, the compressibility, slow mass transfer and adsorption kinetics of the traditional particle media for column chromatography have significantly limited their application in purification of bioproducts on a large scale [2] . In order to overcome these drawbacks, affinity-membrane chromatography, using porous structures with flat sheet and hollow fiber forms, introduces a different approach to exploit the biospecific interactions between a ligate and a ligand for biomedical applications. Affinity membranes are operated in convective mode, which can significantly reduce diffusion and pressure drop limitations commonly encountered in column chromatography [3] and [4] . In our previous work, polylysine-attached affinity membranes were prepared and were used to adsorb bilirubin from the bilirubin–phosphate solution and bilirubin–albumin solution [5] , [6] , [7] and [8] . The results showed that the polylysine-attached affinity membranes have a higher capacity for bilirubin.
Currently, the affinity-membrane chromatography is well accepted as a powerful technology for separation and purification of biomolecules.

The optimization and scale-up of affinity-membrane operations in the recovery, separation and purification of biochemical components is of major industrial importance [9] . The development of mathematical models to describe affinity-membrane processes and the use of these models in computer programs to predict membrane performance are an engineering approach that can help to attain these bioprocess engineering tasks successfully [1] and [10] . However, in literature, very few models have been developed so far to describe the observed breakthrough behavior. The
Thomas model [11] , which involves Langmuir reaction kinetics as the rate-limiting step, has been used frequently to describe the process of affinity chromatography in a packed column. In membrane column affinity chromatography, the Thomas solution can be used when working with axial
Peclet numbers greater than 40 and radial Peclet numbers smaller than 0.04, and good agreement is observed between experiments and theory, particularly in the early portion of the breakthrough curves. Tejeda-Mansir et al. [12] developed a method, which was based on the analytical solution of the Thomas model for frontal analysis in membrane column adsorption, for affinity-membrane column design. This method permits to find the operating conditions to reach 93.5% of the column capacity as operating capacity, using a sharpness restriction for the system breakthrough curve. Sridhar [13] proposed a model including convection, diffusion and rate kinetics, and used the model to analyze the design and operation of affinity-membrane

bioseparations. The results obtained from model simulation showed that the breakthrough of the solute is significantly influenced by Peclet number, feed protein concentration, ligand number, Damkőhler number, membrane thickness and flow rate. Suen and Etzel [14] extended the affinity-membrane chromatography model to study extra-column effects in membrane systems.
Their results showed that these effects are relevant for a proper description of performance in these types of separations. To evaluate nonlinear chromatographic performance, a multi-plate mathematical model for affinity-membrane was proposed through frontal analysis by Hao and Wang
[15] . The main advantage of this model is that the parameters can be easily calculated from experimental data. Due to good correlation between it and the equilibrium-dispersive or Thomas models, this model can be used to obtain information about band-broadening effects such as dispersion and sorption effects.
The equilibrium adsorption of a solute on an affinity matrix based on the above models is often described simply by the Langmuir equation, with the assumption that single site, homogeneous interaction occurs between the solute and the ligand, and that nonspecific interactions promoted by the support are absent [9] and [16] . However, homogeneity is seldom true in practical cases, which has been shown in some of the recent works on affinity adsorption, ion-exchange adsorption and adsorption to polymer surface [17] . Recently, we developed a new affinity-membrane model based

upon Freundlich equation [18] . The model was simulated by the experimental data of bilirubin adsorbed on affinity-membrane. The experimental and modeling results are in good agreement. This model can be used to describe the affinity-membrane processes in which the adsorption mechanism between ligand and ligate is Freundlich adsorption.
The key performance criteria for affinity-membrane processes are breakthrough curve sharpness and residence time at the adsorption stage
[1] . In fact, not only the feed solute concentration, ligand number, Peclet number, membrane thickness and flow rate but also the nonuniformites in membrane thickness [19] and [20] and pore-size [20] may have significant effects on the breakthrough curve sharpness and residence time. In this paper, variation analysis of the membrane thickness and pore size is proposed to predict the effects of nonuniformites on membrane performance by using our affinity-membrane model [18] .
The assumptions of the Langmuir model are that surface is homogeneous, and adsorption energy is constant over all sites. In fact, as the adsorbent surface is often heterogeneous and/or interaction among adsorbed molecules cannot be neglected, the heat of adsorption varies with the surface coverage.

When adsorption between ligand and ligate can be described by kinetics equation of Freundlich, we propose the following model to predict the breakthrough behavior of affinity-membrane [18] .
A mass balance over a section of the membrane gives the following continuity equation:
(1)
The kinetics equation of Freundlich is
(2) where c l
is the total adsorbed capacity, c and c s
the solute concentration in the fluid phase and concentration of solute–ligand complex in the solid phase, the void porosity, D the axial diffusion coefficient, v the flow velocity of the solute through the membrane, and k a0
and k d0
are association and dissociation rate constants of Freundlich adsorption equation, respectively.
Initial conditions: c =0 at z ≥0, t =0 (3) c s
=0 at z ≥0, t =0 (4)
Boundary conditions:
(5)
(6)

For convenience, Eqs. (1) , (2) , (3) , (4) , (5) and (6) can be converted to dimensionless groups as follows:
(7)
(8)
Initial conditions:
C =0 at ζ ≥0, τ =0 (9)
C s
=0 at ζ
≥0, τ
=0 (10)
Boundary conditions:
(11)
The definitions and physical meanings of the dimensionless parameters in
(12)
Eqs. (7) and (8) are summarized in Table 1 .
Table 1.
Definitions and physical meanings of dimensionless parameters

Dimensionless parameters and definitions their
Physical meanings
τ = vt / L Dimensionless time
ζ = z / L
C = c / c
0
Dimensionless spatial variable
Dimensionless ligate concentration in the fluid phase
C s
= c s
/ c l
Pe = vL / D m =((1− ε ) c l
)/ εc
0
Dimensionless concentration of ligand–ligate complex in the solid phase
Axial Peclet number
Ratio of total adsorbed capacity to feed solute concentration n =((1− ε ) k a0
L )/ εv Dimensionless number of transfer units r =1+( c
0
/ K d
) Dimensionless separation factor which determines the maximum ligate loaded onto the membrane matrix at equilibrium
We used the finite-difference method to solve the Eqs. (7) , (8) , (9) , (10) , (11) and (12) for the model of affinity-membrane performance.
The sample is fed continuously into the membrane column during the operation of affinity-membrane column. For a short time, the solute in the feed is taken up almost completely; however, after a while, solute breakthrough occurs and the effluent concentration of solute increases with time. Much of the information of column performance can be evaluated in plots of effluent concentration as a function of time or throughput volume, i.e., breakthrough curves. Fig. 1 shows a typical breakthrough curve for the

adsorption step. Breakthrough time is defined as the time required for the exit solute concentration to reach 10% of the inlet solute concentration
( C = 0.1). The breakthrough curve can be used to determine (1) the column capacity used, (2) solute lost in the effluent and (3) the processing time. This is precisely the performance information needed to optimize processing [21] .
For quantitative comparison purpose, the following parameters, which include solute recovery efficiency [22] , ligand utilization efficiency [22] , and thickness of unused membrane [13] , are defined.
(7K)
Fig. 1. Breakthrough curve for affinity adsorption.
The values S
1
, S
2
, and S
3
represent the areas of the shadowed portions in
Fig. 1 . These values of solute recovery efficiency, ligand utilization efficiency,

thickness of unused membrane and breakthrough time are important in the evaluation of the performance of affinity-membrane columns.
For convenience, our affinity-membrane model based upon Freundlich equation [18] is used to investigate the effects of the variations of thickness and pore-size on membrane performance under the condition of Pe ≥ 30.
Other parameter values used in variation analyses are taken from the literature [18] and are listed in Table 2 .
Table 2.
The values of model parameter used in variation analysis b
1 b
2 n r
Parameter Value m 38.57
23.36
5.52
0.8
1.5
The thickness of affinity membranes not only represents the length of membrane-pore but also is proportional to the capacity of the membranes.
Therefore, it is necessary to discuss the effect of thickness variations on the performance of affinity-membrane columns. We assume that the membrane

thickness increases linearly from a thin region to a thick region [19] . For a case of 10% variation in thickness, the membrane thickness linearly increases from 90 to 110% of the average thickness of L. Fluid flow through a microporous membrane in the laminar region can be expressed by the
Hagen–Poiseuille equation [23] :
(13) where Δ P is the pressure drop across the membrane, r a
the pore radius, Q v
the volumetric flow rate, μ the solution viscosity and L is the membrane thickness.
The flow velocity v through membrane is
When the pressure drop across the membrane and solution viscosity are
(14) kept constant, the flow velocity can be calculated using Eqs. (13) and (14) .
Eqs. (13) and (14) show that the flow velocity is inversely proportional to the membrane thickness. Furthermore, the capacity of the membranes is proportional to the membrane thickness. Thus, the time required for pores with thickness of L to reach saturation is proportional to L 2 .
The exit concentration in this model is the volume-averaged value. For a percentage thickness variation, δ, from the mean thickness L, the exit concentration can be calculated as follows:

The breakthrough curves for affinity membranes with known thickness variation can be obtained by using our affinity-membrane model [18] and
(15) carrying out an integration of Eq. (15) . Fig. 2 shows the effect of membrane-thickness variation on the shape of the breakthrough curve. As the percentage thickness variation increases, we can find that (1) the time of total saturation is delayed; (2) the loading capacity at the point of breakthrough is decreased; (3) solute recovery efficiency and ligand utilization efficiency are decreased; (4) the thickness of unused membrane is increased. Therefore, membrane-thickness variations of affinity membranes are an important factor affecting the overall performance of the membrane when adsorption between ligand and ligate can be described by kinetics equation of Freundlich. For thickness variations less than 3%, the effect of thickness variations is insignificant. This result is consistent with the membrane-thickness variation analysis based upon Thomas model [19] .
(7K)
Fig. 2. The effect of membrane-thickness variation on the shape of the breakthrough curve.

Pore-size variations over the membrane are unavoidable in the manufacturing process. Therefore, it is necessary to discuss the effect of pore-size distribution on the performance of affinity-membrane. The flow velocity of the solution through the membrane can also be calculated by Eqs.
(13) and (14) by assuming that the pressure drop across the membrane is constant and the membranes have uniform porosity and ligand density. From the Hagen–Poiseuille equation and Eq. (14) , we can show that the flow velocity is proportional to . Therefore, the residence time of fluid through a membrane with a pore radius r a
would be inversely proportional to . Due to the hypothesis of uniform ligand density of a membrane, the capacity of the membranes is proportional to the surface area of the membrane pore (2 πr a
L).
Thus, the time required for pores with radius r a
to reach saturation can be calculated from residence time and membrane capacity [20] .
We used normal distribution to describe the pore-size variation of microporous membrane and then discussed the effect of pore-size distribution on the affinity-membrane system. For membranes of mean pore radius r
0
and pore variation expressed in terms of the standard deviation, σ, from the mean value, the probability density function of membrane pores with pore radius r a
can be expressed as the following relation [24] :

(16)
The breakthrough curves for affinity membranes with known pore-size distribution can then be constructed by integrating the breakthrough for every pore [20] . Fig. 3 shows the effect of pore-size distribution on the shape of the breakthrough curve. The breakthrough curves broaden as the pore-size distribution increases. A 0.03 standard deviation of pore-size will cause a significant broad breakthrough cure. The pore-size standard deviation should be less than 0.01 for the effect to be insignificant. This is consistent with the pore-size distribution model proposed by Liu [20] based upon Thomas model. As the pore-size distribution increases, solute recovery efficiency and ligand utilization efficiency decreases and the thickness of unused membrane increases. So the membranes with a narrow pore-size distribution are more likely to be used as affinity-membrane. If pore-size of membranes, r a
, has a normal distribution with mean pore radius r a
and standard deviation σ, 68, 95 and 99% of the membrane pores are within σ,
2 σ and 3σ of membrane mean pore-size, respectively [24] . For example, with 1.00 μm mean pore radius and 0.03 standard deviation, 95% of the membrane pores are within 6% of the membrane mean pore-size. This is a very narrow pore-size distribution for phase inversion membranes. Therefore, the pore-size of membranes should be strictly controlled during membrane

manufacturing because even small variations will greatly degrade membrane performance.
(7K)
Fig. 3. The effect of membrane pore-size distribution on the shape of the breakthrough curve.
The nonuniformites in membrane thickness and pore-size may severely degrade the performance of affinity membranes. The main influences incarnate (1) the time of total saturation is delayed; (2) the loading capacity at the point of breakthrough is decreased; (3) solute recovery efficiency and ligand utilization efficiency are decreased; (4) the thickness of unused membrane is increased. In order to reduce the effect of the variations of thickness and pore-size, we should choose the uniform membrane, of which membrane-thickness variations and pore-size distribution are under 3% and
0.01. This variation analysis is carried out by using our affinity-membrane model based on Freundlich equation however, the results are basically consistent with the variation analysis by using Thomas model based on

Langmuir equation. Therefore, the membrane properties (thickness and pore-size) should be strictly controlled, no matter the adsorption mechanism of ligand and ligate is Freundlich or Langmuir adsorption. In the scale-up operations of affinity-membrane, the variation effects on membrane performance can be reduced by increasing the amount of membrane disks to average out the dispersion.
b
1 constant of Freundlich adsorption equation b
2 constant of Freundlich adsorption equation c solute concentration in the fluid phase (mol L −1 ) c
0 feed solute concentration in the fluid phase (mol L −1 ) c s solute concentration in the solid phase (mol L −1 )
C
C s dimensionless feed concentration

dimensionless concentration of solute–ligand complex in the solid phase
D axial diffusion coefficient (cm 2 s −1 ) k a0 association rate constant of Freundlich adsorption equation (s −1 )
K d dissociation equilibrium constant of Freundlich adsorption equation
(mol L −1 ) k d0 dissociation rate constant of Freundlich adsorption equation
(mol L −1 s −1 )
L membrane thickness (cm) m ratio of maximum concentration of solute adsorbed to feed solute concentration n
Δ P dimensionless number of transfer units pressure drop across the membrane (dyne cm −2 )
Pe

Q v axial Pelect number volumetric flow rate (cm 3 s −1 ) r dimensionless separation factor r a membrane pore radius (cm) r
0 membrane mean pore radius t time (s) v flow velocity (cm s −1 ) z axial distance along membrane (cm)
δ
μ percentage of thickness variation void porosity of membrane

viscosity of feed solution (g cm −1 s −1 )
σ standard deviation
τ dimensionless time
ζ dimensionless spatial variable
We are deeply indebted to the Natural Science Foundation of Fujian
Province (No. C0510005) and the National Nature Science Foundation of
China (No. 29776036) for supporting this research.
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