Efficiency Measurements for Chromatography Columns

Misinterpreting the effluent profiles obtained during tracer measurements performed for determining packing quality can often lead to excessively large percolation velocities and exaggeration of packing problems. Highly useful and reliable information can be obtained through characterization of tracer effluent curves using the method of moments, information that could be critical for successful scale-up of chromatographic steps. This is the sixth in the "Elements of Biopharmaceutical Production" series.

Anurag Rathore

PLATE THEORY

The plate theory of chromatography was introduced by Martin and Synge in 1941.¹ It offered the first description of the development of quasi-Gaussian bands in linear elution chromatography, which is defined as conditions where the solute is partitioned between the mobile phase and stationary phase in the presence of a linear isotherm. Plate theory describes a column as consisting of a number of theoretical plates such that at each plate, the solute is equilibrated between two phases. It is assumed that the band spreading due to diffusion of solute from one plate to another is negligibly small, and that at equilibrium the partition coefficient of a solute is independent of its concentration and of the presence of other solutes. It was observed that after a certain length of time, the zone profile became Gaussian in shape.

Martin and Synge defined a height equivalent to a theoretical plate (HETP) as "the thickness of a layer in the column such that the eluting mobile phase is in equilibrium with the solute concentration in the stationary phase." They argued that the HETP is a constant through a given column, except when the ratio of the mobile phase concentrations at the entrance and exit differ greatly from unity. The plate height depends on the solute diffusivity in the mobile phase, and is directly proportional to the flowrate (v₀) and the square of the particle diameter (d_p²). Though marred by several inconsistencies,^2,3 the originality and simplicity of the plate model made it very popular.

In 1955, Giddings and Eyring employed probability (stochastic) concepts to describe the differential migration process of chromatography.² They developed a rigorous probability theory that gave a precise description of the influence of simple adsorption-desorption processes on the zone profile. Their approach later led to the simplified and approximate theory of zone dispersion, often referred to as the random walk theory of chromatography,³ for the situation when the time spent by the solutes inside the column is large enough to allow each molecule to undergo a large number of individual sorption and desorption steps.

Let us delve into plate theory and the normal probability, or Gaussian, distribution upon which it is based.¹ Equation (1) describes the Gaussian distribution and Equation (2) the time elapsed. All symbols are defined in a sidebar.

We plotted two examples in Figure 1. The graphical interpretation of s is the half-width of the distribution at its inflection point. If the curves in this figure represent effluent concentrations from a tracer experiment on a chromatographic column, the number of plates is defined as:

Figure 1. A Gaussian Distribution

and the height of a plate, H is:

The vertical scale, corresponding to effluent concentration in our situation, does not affect calculation of H or N.

When time is expressed as in Figure 1, the value of s is independent of both the mass of pulse fed (if the effluent curve is Gaussian) and the column diameter. Moreover the mean residence time equals the time at which the maximum concentration, t_max, occurs. The number of plates is expressed in Equation (5).

The Short-cut Method and the Péclet Number

If effluent curves were always Gaussian, then calculating plate numbers and heights would be a simple process. There would, of course, be the nagging problem of determining standard deviations from experimental data, and it has been a common practice to use short-cut methods for this purpose. We discuss a set recommended by the American Society for Testing and Materials.⁴ Equation (6) from the ASTM is a variant of Equation (5).

If the peak is Gaussian, then the standard deviation is the half-width at 60.7% of peak height and it reduces to Equation (7).

Unfortunately no chromatographic peaks are truly Gaussian, and this is the problem we must deal with next. Figure 2 shows simulated solute concentrations along a column using L as the distance along the column from inlet.⁵ It makes evident that solute profiles within a column are functions of percolation velocity, adsorbent particle diameters and solute diffusivity as expressed via a dimensionless group known as the Péclet number.

Both peaks are left-skewed, but the magnitude of skewness increases with Pé, or inversely with diffusivity. The primary reason for this is intra-particle diffusional transients, and the degree of skewness depends primarily on a ratio of time constants as in Equation (9).

Equation (10) shows that k is always in the range 0 to 1.

The same value of K_D is used for both curves of Figure 2, so we can extract some useful information. First, we note that skewness becomes increasingly severe as T_D/T_v becomes large relative to the time required for solute to pass a particle. Next we see that the definitions of N and H are ambiguous, and no single measure is sufficient. Moreover the mean residence time is no longer the same as that for appearance of the concentration peak.

Figure 2. Tracer Distortion-simulation

A low-molecular weight tracer is commonly used to characterize a chromatographic column. Frequently, a low-molecular weight tracer will show less skewness that a larger solute. This is shown in Figure 3 for actual effluent curves. The acetone tracer has a sharper peak than for the desired product, a globular protein. Even the acetone tracer distribution is appreciably skewed, and this skewness shows a fundamental difference between the effluent curves of this figure and the intra-column profiles of Figure 2. The later bits of acetone to appear have been in the column longer and have therefore had more opportunity to disperse. In summary, no effluent curves are truly Gaussian, and the plate concept as defined earlier represents an ideal case that can only be approached as a limit for very large N.

Figure 3. Tracer Distortion - Effect of Experimental Conditions Column: 1.6 × 8.8 cm ToyoPearl SP 650M. Linear velocity: 200 cm/hr. Mobile Phase: 150mM Citrate, Solute Pulses: 500 mL of 10 mg/mL globular protein, 500 mL of 5% acetone.

Measures of Column Efficiency

There is no simple way to resolve this ambiguity, but it is clear that we now need at least two measures of tracer shape — one for the extent of dispersion about the mean residence time and another for skewness. In our experience, these have generally proved sufficient, and we recommend the use of statistical moments.These moments are described in Equations (11), (12), and (13).

The zeroeth moment is the area under the effluent curve, which is effectively the mass of the solute pulse fed. The first moment is equal to the mean residence time. M₂ is the second moment of concentration with respect to time about the mean residence time. It describes the breadth of the peak and can be calculated from the chromatogram and used to estimate the variance of Gaussian or near-Gaussian peaks. The first and the second moments of a peak are independent of the amount of the component as they are normalized to the peak area. It should be noted that Equations (11) thru (13) provide definitions for moments in the temporal time frame, whereas peak analysis can be performed in either the temporal or spatial timeframe. These moments can be related through the migration velocity, u.

While the peak traverses the column, its variance increases due to longitudinal diffusion and other dispersion effects.⁶ The variance of the peak is an additive quantity. That means the total variance can be expressed as the sum of variances coming from all the contributors to dispersion.^7,8

To characterize skewness we suggest using the third moment about the mean residence time according to Equation (14).

Skewness also can be characterized by differences between the right- and left-hand half widths, but there is a major advantage to using moments throughout. This is because M₁, M₂, and M₃ are additive.⁹ The observed values of these three moments are just the sums of those for the column itself and those for all of the auxiliary apparatus — tubing, headers, and even detectors. This is particularly important during scale-up as auxiliary apparatus can have major effects on effluent curves.

Figure 4. Comparison of Techniques to Characterize Column Effluent Curves Experimental Conditions: Column: 1.6 × 8.8 cm with ToyoPearl SP 650M. Mobile Phase: 150mM Citrate, Solute Pulse: 500 mL of 10 mg/mL globular protein.

Moments can be calculated by numerical integration, but care must be taken with the tails of the curves, especially for the second (and even more, the third). Here small errors are magnified at large time through multiplication by the square or cube of t. Extrapolations to infinite time are usually made from simple curve fits of the large-time data, and it often is sufficient to assume exponential decay with respect to t. One property of M₀ is that it is independent of system size and so is the same for the column proper and the whole experimental system (See Ref. 10, Ex. 23.6-3, for limitations on the use of moments. These are seldom important for our present purposes).

Table 1. ASTM Bases of N Determination

USING MOMENTS

To understand the benefits that the method of moments can offer, we base this part of our discussion on Figure 4, experimental data for a representative globular protein (molecular weight of about 150 kDa). The number of plates is calculated both by the method of moments (Equation 15), and by the widely used short-cut technique (at 50% of peak height from Table 1). The method of moments is seen to approach a straight-line asymptote on this figure, indicating that dispersion increases a bit faster than linearly with percolation velocity. This is in agreement with well-established theory as well as with common observation. To characterize the extent of dispersion, redefine plate number as per Equations (15) and (16).¹¹

The slope of a straight line from the origin to any point on this correlation (defined in Equation 17) can provide useful information for column scale-up. If we substitute Equation (17) and (18) into Equation (19), we are left with Equation (20).

The truest measure of column effectiveness is the number of plates, and Schneider found this to be proportional to solvent residence time, and inversely proportional to the slope g. It suggests, correctly, that solvent residence time is a safe scale-up criterion if velocity is no higher in the larger column, and the penalty for higher velocity can be predicted from the van Deemter plot of HETP as a function of u. Maintaining both solvent residence time and percolation velocity is almost always a reliable scale-up criterion — even for affinity columns with their notoriously slow kinetics. Moreover the effect of increasing velocity at constant solvent residence time is modest under most commonly observed conditions (high Pé).

On the other hand, the short-cut method falsely suggests that higher velocities yield a very large dividend on scale-up. They do not, and high velocities require expensive pumps and massive column walls. This is especially true for compressible particles.

CHARACTERIZING COLUMN FLOW DISTRIBUTION

We consider the problem of characterizing column packing. Poor flow distribution, resulting from either the column flow distributors or packing heterogeneity, can cause substantial deviations from the optimal performance of a chromatographic column. In a typical chromatographic process, flow in the distributors and the packed bed operates at very low Reynolds numbers, or in a laminar mode called creeping flow. Creeping flow through porous media is described by the pseudo-continuum Blake-Kozeny equation, presented here as Equation (21).¹⁰

Reversing the pressure drop, which corresponds to a reversal of the direction of flow, corresponds only to a change in the sign but not the magnitude of the velocity. For this reason, creeping flow is said to be reversible. It should be noted that this equation and the derived reversibility is valid only for a distance scale that is large in comparison to d_p. Characterizing heterogeneity can therefore be made with surprising ease via order-of-magnitude analysis because it usually occurs on a size scale, which is large compare to packing particle diameters.

The intrinsic broadening of a solute pulse, as predicted from linear chromatographic models, operates on a distance-scale comparable to a particle diameter, and it is irreversible. Thus, we may conclude that plate height is the sum of the dispersive contributions. All these dissipative processes occur independently of the flow direction.

These differences provide a very powerful tool for determining the quality of the flow distribution. Visualize a perfectly packed column with perfect headers, operated under normal conditions until a tracer pulse is halfway to the normal outlet, and then the flow is reversed — the effluent will now appear at the inlet. The pulse will have the same form as that leaving the full column under normal operation — small-scale contributions to band broadening will be the same as for normal operation. However, as per Equation (21), even if there is large-scale maldistribution of flow, the reversed-flow effluent will be that of a perfectly packed column with perfect headers.

A comparison of a pulse injected under normal conditions and under flow reversal at one-half of the retention time is shown in Figure 5. Flow reversal at one-half of the retention time mimics the effluent distribution of a perfectly packed column with perfect headers. The conventional, or forward-flow, effluent curve exhibits significant tailing for the system and column, which is not seen under reverse-flow conditions. This chromatographic column is only operating at 70% of its potential efficiency as determined by a comparison on the number of plates between the forward-flow and the reverse-flow case.

Figure 5. Comparison of Forward and Reverse-flow of an Acetone Pulse The conventional forward flow and reverse flow effluent curves have a plate count of 548 and 788. respectively.

A further extension of this reverse-flow technique has been developed that allows for the decoupling of the effects of non-uniform packing from poor distributor flow, with the latter solely dictated by the column design.¹² This extension provides a non-destructive test to characterize flow distribution.

FRONTAL ANALYSIS

In the majority of columns, performance is evaluated only once after the column is packed. Columns are often used multiple times, and it is essential to maintain packed- bed quality and efficiency throughout the column lifetime. Performing tracer analysis to measure the number of plates before every run can be time consuming and impractical for pilot- or commercial-scale columns. Frontal analysis, where a step change is applied to the packed bed rather than a pulse, is frequently used to estimate column performance without the need for extra lines or extra processing time.

It is easy to apply the established plate theory for linear chromatography to the response from a pulse. For linear systems, the response to a pulse input is equal to the derivative of the response to a step input.¹³ Thus numerical differentiation of the monitored output will provide the same information as a response to a non-interacting tracer pulse. This is shown in Figure 6, where the response to a pulse and the differentiated response to a step are overlaid. Both chromatograms show similar responses, particularly in the peak front. The differentiated step response shows a slight deviation at the tail, which would result in a slightly higher HETP.

Figure 6. Comparison of Column Performance as Determined from the Response to a Salt Pulse and a Salt Front. Column 10 × 29 cm PhenylSepharose FF(High Sub). Salt Pulse Conditions: Mobile Phase: 400 mM Citrate, Solute Pulse, 50 mL of 150mM Citrate. Salt Front Conditions: Mobile Phase: 1: 400 mM Citrate, Mobile Phase II, 150 mM Citrate. Chromatograms rescaled to overlay.

The two distributions contain the same information, but the curve for the differentiated step is more easily obtained. The situation for non-linear distributions is more complicated, but again the differential curve is generally more useful.

Nomenclature

In summary, the method of moments when applied to the reverse-flow technique allows for a non-intrusive and accurate method to determine the quality of flow distribution. This is particularly useful for preparatory or commercial columns, where column packing is a tedious and expensive task. The use of statistical moments to evaluate column performance can provide a great deal of information on column performance with little effort.

Anurag Rathore is a principal scientist at Amgen Inc., 30W-2-A, One Amgen Center Drive, Thousand Oaks, CA 91320, 805.447.4491, fax 805.499.5008, arathore@amgen.com.

John Moscariello is a scientist at Amgen Inc., Seattle, WA.

Edwin Lightfoot is a professor at Department of Chemical and Biological Engineering, University of Wisconsin, Madison.

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