Mathematical Modeling and Scale-up of Liquid Chromatography

High Performance Liquid Chromatography (HPLC) is undoubtedly one of the most important tools in chemical analysis. It has become increasingly popular at preparative- and large-scales, especially in purifying proteins. At such scales, larger particles are often used to pack the columns in order to reduce column pressure and facilitate column packing. Unlike small scale analytical HPLC columns that may give near plug flow performances, in large HPLC and lower pressure liquid chromatographic columns, dispersion and mass trans­fer effects are often important.

Many researchers have contributed to the modeling of liquid chromatography. There exist a dozen or more theories of different complexities. A comprehensive review on the dynamics and mathematical modeling of isothermal adsorption and chromatography has been given by Ruthven [5] who classified models into three general categories: equilibrium theory, plate models, and rate models.

For the modeling of multicomponent liquid chromatography, the column is divided into the bulk-fluid phase and the particle phase. The anatomy of a fixed-bed axial flow chromatography column is given in Fig. 3.1. To formulate a general rate model, the following basic assumptions are required.

For analytical and some preparative columns in liquid chromatography, mass transfer resistance is usually negligible and the equilibrium theory suffices [7]. But for preparative columns with smaller plate numbers and large-scale columns, mass transfer effects are often significant and cannot usually be neglected.

Analytical chromatography usually involves small and dilute samples. Thus, interference effects among different sample components are often negligible. With the rapid growth of biotechnology, preparative- and large-scale chromatography become more and more important. High feed concentrations and large sample volumes are often used to increase productivity. In such cases, interference effects may no longer be ignored.

In isocratic elution chromatography, a modifier is often added to the mobile phase in order to compete with sample solutes for binding sites [2]. This helps reduce the retention time and band spreading of the sample solutes.

In chromatographic separations of large biomolecules, such as proteins, using porous adsorbents, a size exclusion effect may be significant. Some large molecules cannot access either part of the small macropores in the adsorbent particles or all the macropores. This is especially possible in chromatographic separations of large proteins. For a multicomponent system involving components with very different molecular sizes, the extent of size exclusion is not the same for all the components. This causes uneven adsorption saturation capacities (based on moles) for the components. The least excluded component tends to have the highest saturation capacity and vice versa.

Affinity chromatography has seen rapid growth in recent years. It is a powerful tool for the purification of enzymes, antibodies, antigens, and many other proteins and macromolecules that are of important use in scientific research and development of novel biological drugs. Affinity chromatography not only purifies a product, but also concentrates the product to a considerable extent [32]. Over the years, this subject has been reviewed by many people, including Chase [32], and Liapis [93]. Affinity chromatography is also called biospecific adsorption, since it utilizes the biospecific binding between the solute molecules and immobilized ligands. The monovalent binding between a ligand and a solute macromolecule is generally considered as following second order kinetics expressed by Eqs. (3–32) and (3–33).

Gradient elution chromatography is a very important method in chromatographic separations, especially for proteins, because they have a wide range of retentivity. In gradient elution, a modulator is used in the mobile phase to adjust eluent strength. The modulator can be acetonitrile in reversed phase chromatography, or ammonium sulfate in hydrophobic interaction chromatography, or sodium chloride in ion-exchange chromatography. In ion-exchange chromatography, a pH gradient may also be used. The modulator concentration in the mobile phase is increased (or decreased as in hydrophobic interaction chromatography) continuously with time. This change in the strength of modulator allows gradient elution to separate components with widely different retentivities. In preparative- or large-scale operations, gradient elution can concentrate a sample while achieving a purification at the same time. In isocratic elution, a sample is always diluted to a certain degree. Because of this, gradient elution is often desired when handling large volumes of sample.

Radial flow chromatography (RFC) was introduced into the commercial market in the mid-1980s [100] as an alternative to the conventional axial flow chromatography (AFC) for preparative- and large-scale applications. Compared to AFC, the RFC geometry (Fig. 10.1) provides a relatively large flow area and a short flow path. It allows a higher volumetric flow rate with a lower bed pressure compared to longer AFC columns. If soft gels or affinity matrix materials are used as separation media, the low pressure drop of RFC helps prevent bed compression [21, 101]. RFC columns, both prepacked and unpacked, with a range of size from 50 milliliters to 200 liters in bed volume are commercially available. An experimental case study of the comparison of RFC and AFC was carried out by Saxena and Weil [102] for the separation of ascites using QAE cellulose packings. They reported that by using a higher flow rate, the separation time for RFC was a quarter of that needed for a longer AFC column with the same bed volume. It was claimed that by using RFC instead of AFC, separation productivity can be improved quite significantly [100].

One of the critical factors for a successful scale-up of liquid chromatography using the rates models is accurate parameter estimations. Three types of parameter are needed to carry out model calculations using the rate models. Isotherm parameters, the particle porosity and the bed void volume fraction are important to the accuracy of model calculations. Physical dimensions of the column are equally important, but they can be easily and precisely measured. Less important parameters are the mass transfer coefficients, which usually do not affect the general location of an elution peak. They affect the shape of the peak. However, such an influence is not very sensitive to the fluctuation of the mass transfer coefficients. Thus, parameter estimation of these coefficients does not have to be very stringent.