What's Your Velocity?

By Dr. Scott Rudge

With the development of very high titer cell culture and fermentation processes, downstream processing has been identified as a new bottleneck in biotechnology. The productivity of chromatography in particular, has become a bottleneck. There are two schools of thought for scaling up chromatography: in one, linear velocity (flow rate divided by column cross sectional area) is held constant; in the other, the total (volumetric) flow rate divided by the volume of the column is held constant. In the former method, the length of the column has to be held constant. In the latter method, the geometry of the column is not important, as long as the column can be packed efficiently and flow is evenly distributed. This makes the latter method more flexible, and accommodating of commercially available off the shelf column hardware packages. But does it work?

In my experience, holding flow rate divided by column volume constant between scales works very well. There is plenty of theoretical basis for the methodology as well. Yamamoto has published extensively on the reasons that this technique works. This method is also the basis for scale up described in my textbook. Here, briefly, and using plate height theory, is the theoretical basis:

The basic goal in chromatography scale up is to maintain resolution. “Resolution” is a way to describe the power of a chromatography column to separate two components. It depends on the relative retention of the components, which is fixed by the thermodynamics of the column and remains constant as long as the chemistry (the resin type, the buffer composition) remains constant. It also depends on the peak dispersion in the column, which is a function of the transport phenomena, and is only related to the chemistry by the inherent diffusivity of the molecules involved. Otherwise, it is dependent on mass transfer, flow rate, temperature, flow distribution. Treating the thermodynamics as constant, we can say:

where Rs is Resolution, and N is the number of theoretical plates. N is the ratio of the column length L to the plate height, H, so

 

In liquids, H is approximately a linear function of linear velocity v, according to van Deemter, as discussed in a previous post. So we can say that H = Cv (where C= the van Deemter constant). Now, the linear velocity is the flow rate divided by the cross sectional area of the column, A, and the column volume, V, is the cross sectional area times the column length. The total flow rate (F) divided by the column volume is held constant between scales, we’ll call this constant “f”.

 

This bit of mathematical gymnastics says that Resolution only depends on two fundamental properties of the scale up, van Deemter’s term “C”, which considers the dispersion caused by convection relative to mass transfer to and from a resin particle, and “f”, the flow rate relative to the column volume that is chosen. There is no need to hold column length or linear velocity constant as long as the flow rate relative to column volume is held constant. You might also notice from the math that doubling the column length has the same effect on resolution as dropping the flowrate by half. However, doubling the length costs more in terms of resin, and consumes more solvent than decreasing the flow rate (that’s for you, Mike Klein!) and increases the pressure.

The plate count analysis is very phenomenological, but it does hold up under practice (otherwise it would be abandoned). And the more delicate mathematical models predict the same performance, so confidence in this scale up model is high.

One common mistake made by those using the constant linear velocity model is in adding extra column capacity. Since most people are unwilling to pay for a custom diameter column, but base their loadings on the total volume of resin, they add bed volume by adding length. But since they are unwilling to change the linear velocity, they end up decreasing the productivity of the column (because, for example, the resolution they achieved at a smaller scale in a 12 cm long column at 60 cm/hr is now being performed in a 15 cm long column, at 60 cm/hr, therefore taking 25% more time).

If the less well known constant F/V model is used for a process involving mammalian cells, it would be imperative to explain and demonstrate this model in the scale down validation that is a critical part of the viral clearance package.

But how can you get even more performance out of your chromatography? Treating the unit operation as an adsorption step, and scaling up using Mass Transfer Zone (MTZ) concepts will be treated in a future posting.

QbDer, Know Thy Model!

By Dr. Scott Rudge

Resolution in chromatography is critical, from analytical applications to large scale process chromatography. While baseline resolution is the gold standard in analytical chromatography, it is seldom achieved in process chromatography, where “samples” are concentrated and “sample volumes” represent a large fraction of the column bed volume, if not multiples of the bed volume. How do you know if your resolution is changing in process chromatography if you can’t detect changes from examining the chromatogram?


Many use the Height Equivalent to a Theoretical Plate technique to test the column’s resolution. In this technique, a small pulse of a non-offensive but easily detected material is injected onto the column, and the characteristics of the resulting peak are measured. The following equation is used:

Where H is the “height” in distance, t_r and t_w,1/2 are the retention time and time of the width of the peak at ½ height, L is the length of the column and N is the “number” of theoretical plates in the column. The lower the H, the smaller the dispersion, the greater the resolution.

It is well established that the flowrate and temperature affect the plate height. In fact, when the plate height is plotted against flow rate, we generate what is typically called the “van Deemter plot”, after Dutch scientists (van Deemter et al., Chem. Eng. Sci.,5, 271 (1956)) who established a common relationship in all chromatography (gas and liquid) according to the following equation:

Where A, B and C are constants and v is the linear velocity (flow rate divided by the column cross sectional area) of fluid in the column. It was later proposed by Knox (Knox, J., and H. Scott, J. Chromatog., 282, 297 (1983)) that van Deemter plots could be reduced to a common line for all chromatography if the plate height was normalized to resin particle size, and linear velocity was normalize to the resin particle size divided by the chemical’s diffusivity. While this did not turn out to be generally true, it is very close to true. Chemical engineers will recognize the ratio of linear velocity to diffusivity over particle size as the Peclet number, “Pe”, a standard dimensionless number used in many mass transfer analyses.

Since diffusivity is sensitive to temperature, it is logical that Pe is also sensitive to temperature, decreasing temperature decreases the diffusivity, and increases Pe. Thus, Pe is inversely proportional to temperature. We measured plate height as a function of linear flowrate and temperature in our lab on a Q-Sepharose FF column, using a sodium chloride pulse, and found the expected result, shown in the graph below.

We can easily use this graph as a measurement of van Deemter’s parameters A and C, and find the dependence of the diffusivity of sodium chloride on temperature. Based on these two points, we find the Peclet number proportional to 0.085/T where T is in degrees Kelvin. We also find the dependence of plate height on linear velocity, and we can predict that resolution will deteriorate in the column as flow rate increases.

We can also use Design of Experiments to find the same information. Analyzing the same data set with ANOVA yields significant factors for both flow rate and temperature, as shown in the following table:

Term	Coef	SE Coef	T	P
Constant	0.09756	0.05604	1.74	0.157
Temp	-0.007935	0.001858	-4.27	0.013
v	0.0497	0.02147	2.32	0.082

But since the statistics don’t know that the temperature dependence is inverse, or that the Peclet number is a function of both temperature and flowrate, the model yields no additional understanding of the process.

It is possible to use analysis of variance to fit a model other than linear, as the van Deemter model clearly is. But one must know that the phenomenon being measured behaves non-linearly in order to use the appropriate statistics. Using Design of Experiments blindly, without knowing the relationship of the factors and responses, leads to empirical correlations only relevant to the system studied, and should only be extended with caution.