Algorithmic- and model-based methods

Throughout the bio-manufacturing process there is a tendency to progressively replace empirical correlations with mechanistic models in order to provide a higher level of process understanding (Hanke & Ottens, 2014). Mechanistic models are derived from fundamental (or first) principles (conservation of mass, heat and momentum) and thus have the potential to describe more accurately the phenomena occurring in any system.

The chemical industry has a long history and sufficient databases on thermodynamic properties exist to support the use of mechanistic models. In contrast, the biotechnology industry lacks the advanced models, software tools and simulation packages to improve and assist in process synthesis, design and optimisation mainly due to the complex nature of biomolecules (Nfor et al., 2009).

51 Significant research has been done since the late 1960s in the area of mathematical modelling for packed-bed adsorption operations (Bellot & Condoret, 1991; Bellot &

Condoret, 1993). A comprehensive overview has been given by Ruthven (1984) classifying different modelling approaches into three main categories; equilibrium, plate and rate models. Mathematical models based on the equilibrium theory make the assumption of rapid equilibrium achieved between the stationary and the mobile phase neglecting the effects of mass transfer resistance and axial dispersion. Models based on the plate theory adopt a different approach by dividing the column into a series of theoretical plates within which equilibrium has been achieved. The empirical nature of plate models cannot relate them to first principles (Gu et al., 1993). Nevertheless, their application has been demonstrated on the separation of multicomponent systems (Gu et al., 1993; Guiochon, 2002).

Rate models are expressed mathematically through a set of differential equations that describe the phenomena acquiring between the stationary and mobile phase for each component (Gu et al., 1990). Among rate models, the general rate model is considered one of the most comprehensive that attempts to capture the effect of all possible contributions such as axial dispersion, intra-particle diffusion, external firm mass transfer resistance and adsorption/desorption kinetics. The complexity and non-linearity of the general rate model have posed great challenges in solving the system of the differential equations required to describe the operation and efficient algorithms are necessary to provide a solution (Gu et al., 1990; Gu et al., 1993; Guiochon, 2002). Gu et al. (1990) proposed a numerical procedure to solve the general rate model using the finite element and the orthogonal collocation methods. Later the authors extended their work with the addition of second order kinetics and size exclusion effects in their model (Gu et al., 1993).

Other studies have used the general rate model or a variation of it to address various aspects of the chromatographic performance. Degerman et al. (2006) used the general rate model with Langmuir isotherm kinetics and a modulator to optimise the separation

52 of an IgG from bovine serum albumin. Their model was able to simulate real gradient elution profiles using a constraint on the slope of the gradient embedded in their optimisation procedure, thus avoiding making assumptions on ideal gradients. The authors demonstrated their approach using six decision variables with a fixed bed-height and confirmed graphically the success of their optimisation method. Gerontas et al.

(2010) integrated scale-down experimental data with general rate modelling to scale-up and predict the chromatographic separation at manufacturing scale. The general rate model with Langmuir kinetics and a mobile phase modulator was used and solved using the finite element method. Additionally, the authors demonstrated the use of genetic algorithms to calibrate model parameters and showed the validity and applicability of their approach using three different CEX resins. Boushaba et al. (2011) used the same modelling approach described by Gerontas et al. (2010) to evaluate the effect of fouling in the chromatographic performance. The authors demonstrated the use of windows of operation to visualise the trade-offs between the effort and the benefits of feed clarification prior to chromatography.

A systematic approach to model chromatography operations has been developed by Chan et al. (2008). Their approach consists of three main parts. First a methodology was proposed to determine the feed concentration and identify significant components in the feed mixture that are required for the development of an accurate model. Then, the second part involves the estimation of model parameters for different rate models. The authors identified two models that have been studied extensively; the equilibrium-dispersive and the general rate model and employed them to demonstrate their approach. The general rate model offers a more accurate description of the separation process however at a high computational cost. In contrast, the equilibrium-dispersive model makes further assumptions to simplify the general rate model thus mitigating the time required to solve it with potential consequences on the accuracy of the model. At the third and last part the authors utilised the fractionation diagram method to compare the accuracy of the mechanistic models they included. Three case-studies were

53 presented to illustrate the applicability and usefulness of their approach. The fractionation diagram method was employed due to its sensitivity to capture efficiently the trade-offs between purity and recovery.

Origin of the fractionation diagram approach

The final output of a chromatography cycle is a chromatogram, which presents how the concentration of the eluate changes against volume or time. However, it is difficult to extract straightforward relationships that can correlate performance with operating conditions and moreover, this task becomes more difficult when product and impurities concentrations are expressed in different units and when inadequate data are available (Ngiam et al., 2001). To address this issue, Ngiam et al. (2001) proposed a method to manipulate the information provided by chromatograms and construct graphical representations to visualise the trade-offs between purity and product recovery.

Originally, the graphical method was developed by Richardson et al. (1990) to correlate product recovery and purity in protein precipitation and furthermore, to optimise the precipitation conditions and identify the precipitant cut points. Later a similar graphical method was developed to optimise high-performance tangential flow filtration operations (Reis & Saksena, 1997).

Examples of implementation of the fractionation diagram approach

The fractionation diagram approach in chromatography development was first used by Ngiam et al. (2001) with a hypothetical three component mixture. The authors utilised a mathematical model of size exclusion chromatography (SEC) to describe and simulate the separation of the desired component at different flow rates of the mobile phase. The resulting chromatograms where then translated into fractionation diagrams and maximum purification factor (PF) versus recovery diagrams were generated to visualise the effect of flow rate on the chromatographic performance. In the same study, the team conducted experiments with an IEX column to evaluate the removal of endotoxin and plasmid DNA. The fractionation approach was used to identify the collection points on

54 the chromatogram that will provide the desired level of purity and yield. In a subsequent study, the same approach was employed to optimise a two-column purification sequence (HIC followed by SEC). The authors identified the need to perform an integrated optimisation approach and develop the purification sequence as a whole and not as individual steps, which may lead to a sub-optimal result. Initially, the mobile phase flow rate was chosen as the investigated variable and how it affects HIC performance in terms of purity and yield. Subsequently, the effect of each HIC flow rate on the following SEC step, was evaluated in order to make decisions regarding the operating flow rate of the HIC step. The authors concluded that SEC is strongly affected by HIC (Ngiam et al., 2003).

Salisbury et al. (2006) considered the need to optimise the chromatographic purification sequence as a whole, utilised the fractionation diagram approach and introduced 2D windows of operation, capturing the trade-offs between productivity and purity. To generate a window of operation, initially a range of product breakthrough levels and operating flow rates have to be chosen based on the desired level of resolution. Then the time to achieve each breakthrough level was determined and the elution profiles were generated for each flow rate/breakthrough level combination. Subsequently, PF versus recovery diagrams were created for each combination of flow rate and breakthrough level and purity was estimated at the desired level of product recovery. Finally, productivity was calculated for each flow rate and breakthrough level combination. A window of operation for the first chromatography step was created and a feasible region within the window of operation was identified for the operation of the next chromatography step.

The above studies demonstrate that the fractionation diagram approach is a powerful tool in purification process development and optimisation. The method offers an efficient solution to the issues arising from the insufficient approaches on how to extract rapidly useful information from a plethora of chromatograms. Moreover, it considers the importance of optimising each chromatography step as part of the whole purification sequence and not individually. Incorporation of the fractionation diagram approach with

55 other techniques has been proven to be an excellent tool in process development that can help make feasible decisions.