Multivariate Parameter Determination of Multi-Component Isotherms for Chromatography Digital Twins

Abstract

Many fundamental decisions in the process design of a separation task are conducted in an early stage where, unfortunately, process simulation does not have the highest priority. Subsequently, during the setup of the digital twin, dedicated experiments are carried out in the design space that was established earlier. These experiments are most often too complicated to conduct directly. This paper addresses the idea of a combined approach. The early-stage buffer screening and optimization experiments were planned with the Design of Experiments, carried out and then analyzed statistically to extract not only the best buffer composition but also the crucial model parameters, in this case the isotherm dependency on the buffer composition. This allowed the digital twin to predict the best buffer composition, and if the model-predicted control was applied to keep the process at the optimal productivity at a predetermined purity. The methodology was tested with an industrial peptide purification step.

1. Introduction

Chromatography is a widely used unit operation in chemical and pharmaceutical engineering with broad fields of application from low-cost bulk chemicals to high potential pharmaceuticals. Because of the variety of potential chromatography processes such as batch chromatography or various continuous process options, process modelling has been state of the art for decades [1,2,3,4,5,6]. With increasing computational power, model parameter determination has become the bottleneck for fast and efficient model implementation. There is an increasing number of different approaches that can, in general, be separated into two categories.

One approach is to measure each model parameter individually, preferably decoupled from the other effects. Fluid dynamics, e.g., the axial dispersion coefficients measured with tracer experiments [3,7,8,9]; thermodynamics, e.g., isotherms measured with dedicated experiments such as shaking flask experiments, frontal analysis or perturbation [1,10,11]; and mass transfer are evaluated separately. Although delivering precise and accurate results, this approach is time consuming.

The other approach is obtaining parameters from simple experiments, ideally directly from chromatograms. This is possible with fitting routines [12,13,14], neuronal networks [15,16,17] or alike. This is a fast approach; the resulting quality, however, might not be the best, especially if two parameters can describe the same effect. Peak shape, for example, is influenced by fluid dynamics, thermodynamics and mass transfer. Of course, a combined approach is also possible.

Great efforts have been devoted to the development of different isotherm models and the corresponding parameter determination methods, especially for gradient separations. In most cases, the thermodynamic behaviour of the analytes/compounds is only investigated for the modifier. In some cases, isotherms are determined for two different parameters, such as the pH and ionic strength. This was obviously needed for mixed-mode chromatography [18,19,20,21] but was also undertaken for other media [22,23]. The determination of isotherm dependency for more buffer components is rarely undertaken since the experimental effort increases exponentially. Nevertheless, most buffer components do have a significant influence on retention behaviour. In most cases, however, the buffer composition is examined in an early process stage and is kept untouched later. Often, only the gradient gets optimized and modelled thoroughly. Fluctuations in the buffer composition are, therefore, often not representable in chromatography models.

For this case study, a reversed-phase polishing step for an industrial peptide production was investigated. To evaluate the influences of buffer composition, a Design of Experiments approach was taken. Three components, counter ion, stabilizer and the pH value were investigated. The DoE was performed with downscaled preparative chromatography runs and first analysed for the significance of each parameter. The preparative DoE runs were further evaluated to obtain the model parameters to describe the isotherm dependency on each buffer compound.

2. Modelling Chromatography: General Rate Model

The chromatography model used throughout this work, namely the general rate model, as well as the general modelling approach, is described in detail in Zobel-Roos et al. [24]. The model was used and parametrized in Zobel-Roos et al. [25].

In order for the model to become a digital twin, it was necessary for the real process to provide information back to the model. In this case, this was accomplished through the use of Process Analytical Technologies (PAT) tools that converted the Diode Array Detector (DAD) signals to product and impurity concentrations in-line and in real time [26,27]. Furthermore, neural networks are sometimes used to determine or adjust model parameters during the operation, such as the fluid dynamic parameters [15]. The coupling with the process control system was described by the authors in [28,29].

The general rate model can be separated into three parts: the mass balance for the mobile phase, the mass balance for the light phase and the description of the equilibrium. The authors of [1,2,5,30,31,32] provide more detail regarding derivation, assumptions and further information.

Mass balance of mobile phase:

The mass balance of the mobile phase consists of four terms reading from left to right: storage, convective flow, axial dispersion and mass transport [1]:

$$\frac{\partial c_i}{\partial t}=-u_{int}·\frac{\partial c_i}{\partial x}+D_{ax}·\frac{{\partial }^2{c}_i}{\partial x^2}-\frac{6}{d_p}·\frac{\left(1-{\epsilon }_s\right)}{{\epsilon }_s}·k_{f,i}·\left(c_i-{\left.c_{p,i}\right|}_{r=R_p}\right)$$

(1)

with $u_{int}$ as interstitial velocity, $D_{ax}$ as axial dispersion coefficient, ${\epsilon }_s$ as voidage, $d_p$ as particle diameter and $k_{f,i}$ as film mass transport coefficient. The use of film mass transport coefficient demands the consideration of pore diffusion in the mass balance of the stationary phase. However, film mass transport and pore diffusion can be combined, resulting in the lumped pore diffusion model [32]. Here, the film mass transport coefficient $k_{f,i}$was replaced with an effective mass transport coefficient$k_{eff}$. This simplification is often applied in the early process development to reduce the model parameter determination efforts at the expense of the model accuracy and process understanding. An even further simplification is the lumped kinetic model that neglects the intraparticle pores completely [32].

Mass balance of stationary phase:

The mass balance of the stationary phase is mostly dominated by pore diffusion D_p,i and surface diffusion D_S,i [30,33]:

$${\epsilon }_{p,i}·\frac{\partial c_{p,i}}{\partial t}+\left(1-{\epsilon }_{p,i}\right)·\frac{\partial q_i}{\partial t}=\frac{1}{r^2}\frac{\partial }{\partial }\left[r^2\left({\epsilon }_{p,i}·D_{p,i}·\frac{\partial c_{p,i}}{\partial r}+\left(1-{\epsilon }_{p,i}\right)·D_{S,i}\frac{\partial q_i^{*}}{\partial r}\right)\right]$$

(2)

with $c_{p,i}$ as the concentration of component $i$ within the pores and $q_i$ as the surface loading of component i. For larger molecules, surface diffusion is often neglected or combined with pore diffusion into one effective diffusion coefficient D_eff [10,33].

$$D_{eff,i}={\epsilon }_{p,i}·D_{p,i}+\left(1-{\epsilon }_p,i\right)·D_{S,i}\frac{\partial q_i^{*}}{\partial c_{p,i}}$$

(3)

Combining Equations (2) and (3) results in:

$${\epsilon }_{p,i}·\frac{\partial c_{p,i}}{\partial t}+\left(1-{\epsilon }_{p,i}\right)·\frac{\partial q_i}{\partial t}=D_{eff,i}\left(\frac{{\partial }^2{c}_{p,i}}{\partial r^2}+\frac{2}{r}·\frac{\partial c_p}{\partial r}\right)$$

(4)

For the lumped pore diffusion model, the mass balance for the stationary phase reads [32]:

$${\epsilon }_{p,i}·\frac{\partial c_{p,i}}{\partial t}+\left(1-{\epsilon }_{p,i}\right)·\frac{\partial q_i}{\partial t}=\frac{6}{d_p}·\frac{\left(1-{\epsilon }_s\right)}{{\epsilon }_s}·k_{f,i}·\left(c_i-\left.c_{p,i}\right|\right)$$

(5)

Adsorption equilibrium:

There is a vast number of approaches to describe the adsorption equilibrium, mostly depending on the adsorption mechanism and mode of operation [11,34,35,36,37,38,39,40,41,42,43,44,45]. For this simulation study, competitive Langmuir isotherms were used [10,46]:

$$q_i=\frac{q_{max,i}·K_i·c_i}{1+{{\displaystyle \sum }}_{j=1}^n{K}_j·c_j}$$

(6)

Here, $K_i$ is the Langmuir coefficient and $q_{max,i}$ the maximum loading capacity of component i. There are different notations found in the literature, e.g., with the use of the Henry coefficient H_i. All notations can be transferred into the other with:

$${\mathrm{H}}_i=q_{max,i}·K_i$$

(7)

All necessary parameters for fluid dynamic ($D_{ax}, {\epsilon }_s \mathrm{and} {\epsilon }_{p,i})$, mass transfer $(k_{f,i})$ and adsorption equilibrium were measured in a previous work [25]. The Langmuir isotherm was measured for the modifier, a short-chain aliphatic alcohol.

3. Materials and Methods

3.1. Feed, Buffer Components and Columns

The feed solution was taken from an industrial peptide process. The short-chain aliphatic alcohol, buffer salts and stabilisers were obtained in pharmaceutical production quality from Sanofi-Aventis Deutschland GmbH. Short-chain aliphatic alcohol was taken from the production process.

Preparative chromatography was performed with silica-based reversed-phase media in self-packed glass columns (Götec-Labortechnik GmbH, Bickenbach, Germany). Analytical chromatography was performed with a RP-18 column.

3.2. Preparative Chromatography

All preparative runs were performed with the same method. Each run started with 1 column volume (CV) equilibration followed by 14 CV loading. The gradient was started directly afterwards. A run was terminated with a regeneration step after the elution of the main peak, detected via UV–Vis at 280 nm.

3.3. Software

For process simulation and isotherm parameter fitting, Aspen Custom Modeler (Aspentech, Bedford, MA, USA) was used. The Design of Experiments plan was set up and evaluated with JMP (SAS Institute, Cary, NC, USA). JMP was also used to statistically evaluate the isotherm parameters using multiple linear regression.

4. Design of Experiments

To identify the significance of each parameter in the given range, the following was used:

• Counter ion: 25–100 mM;
• Buffer: 50–150 mM;
• pH: 3.35–3.7.

A three-factor, level two, full factorial Design of Experiments plan was carried out. It included one centre point with three repetitions to evaluate the reproducibility. The boundaries were conceived from prior knowledge. The pattern introduced in

Table 1. Design of Experiments plan.

	Factors			Target Values
	CI	Buffer	pH	Normalized Yield	Normalized Productivity
Pattern	[mM]	[mM]	[-]	[-]	[-]
(000)	62.5	100	3.53	1.01	1.00
(000)	62.5	100	3.53	0.99	0.99
(000)	62.5	100	3.53	1.00	1.00
(---)	25	50	3.35	0.73	0.61
(--+)	25	50	3.7	1.06	1.08
(-+-)	25	150	3.35	1.09	1.08
(-++)	25	150	3.7	0.93	1.02
(+--)	100	50	3.35	0.93	0.91
(+-+)	100	50	3.7	1.20	1.20
(++-)	100	150	3.35	1.23	1.22
(+++)	100	150	3.7	0.97	0.99

is used throughout the article to identify the experiments. The centre point is 000 and + and—indicate that the factor is at minimum or maximum. The order of the symbols represents the order as given in Table 1: first, the counter ion; second, the buffer; and third, the pH. Although Table 1 shows the entries in order, the experiments were randomized. Each run was fractionated in 30 s intervals and analysed offline for the target and side component concentrations. The product was pooled out of these fractions to surpass 99% purity. The target values were productivity and yield, normalized to the mean value of the three centre points. The experimental data and results are given in Table 1:

The pareto charts of the standardized effects, given in Figure 1, indicated that each single parameter and each parameter combination had a significant effect on both the yield and productivity. Obviously, a higher yield automatically resulted in a higher productivity. In addition, the changes in the buffer composition had an influence on the retention time as well (see also Figure 4). Therefore, productivity was more affected by the buffer composition.

The results showed that each parameter individually had a positive influence on the target values. Increasing these would increase the outcome. The parameter combinations, however, had a negative impact, especially the combination of the pH and buffer, which had the highest impact values. Although the absolute values given in the pareto charts did not necessarily give the correct ranking or strength of impact, one might assume that the best buffer composition would be:

• A high counter ion (CI) value; and
• A high pH value at a low buffer concentration; or
• A low pH value at a high buffer concentration.

The achieved quality of the DoE results was excellent. Figure 2 shows the plots of the measured over the predicted values. All the data points were well on or very close to the red line, indicating that there was a good correlation between the measured and predicted values. This was underlined by very narrow confidence intervals. The other evaluation metrics were also very good. The p-values were very low. Generally speaking, the p-values describe the accordance of the dataset with a potential explanation, such as that the null-hypothesis is true. Here, the p-values were with p ≤ 0.001 far below the significance boundary of p = 0.05 chosen for this DoE. The R², here RSq, is near 1 and the Root Mean Square Errors (RMSE) were below 1.5%.

5. Parameter Extraction

The good results of the DoE, especially the very narrow confidence intervals, indicated that there might be a linear correlation between each individual buffer component and the adsorption isotherm. Figure 1 also suggests that there were interactions between the two buffer parameters. It is worth noting that this assumption can only be made within the observed DoE design space. An extrapolation outside the measured parameter range is in general not recommended by statistic fundamentals.

In previous work [25], the isotherm dependency on the modifier concentration was implemented with 4 factors for each component, a_1,i, a_2,1, b_1,i and b_2,i. These factors influence the Langmuir parameters Henry coefficient H_i and maximum loading capacity q_max,I depending on the modifier concentration:

$${\mathrm{H}}_i=a_{1,i }·c_p{}^{a_{2,1}}$$

(8)

$$q_{max,i}=b_{1,i }+b_{2,i }·c_p$$

(9)

Note that $c_p$ is the concentration of the modifier inside the pores. For clarity, the correct indices for $c_p$ were left out here. In case of linear dependencies between the single buffer compounds and the adsorption equilibrium as well as the two parameter interactions, the four factors mentioned above can be described with an equation such as:

$$y=i_0+i_1{x}_1+i_2{x}_2+i_3{x}_3+i_{12}{x}_1{x}_2+i_{13}{x}_1{x}_3+i_{23}{x}_2{x}_3$$

(10)

where y is one of the four factors, $x_i$ is the concentration of the buffer component and i_i is the component dependent linear factor. The latter should be determinable directly with JMP. To do so, $a_{1,i }$, $a_{2,i }$, $b_{1,i }$ and $b_{2,i }$ were added as new target values to the DoE. The factors themselves were obtained by fitting the simulation results to the experimental chromatograms. Again, the linear correlations looked very good. The p-values were between 0.003 and 0.0009, R² was above 0.98 and the Root Mean Square Errors were also very low. All the values can be found in

Table 2. Evaluation metrics for the DoE parameter determination.

	p	R²	RMSE
a1	0.003	0.98	2.8 × 10−7
a2	0.0009	0.99	0.0494
b1	0.0023	0.98	0.007
b2	0.0006	0.99	0.0115

. The observed vs. predicted plots are given in Figure 3, exemplified with b1 and b2. Again, the confidence intervals were narrow. An interesting observation can be undertaken for data point (---). Despite being on the line with the others, it is relatively far outside.

6. Discussion

So far, the method was straight forward. The Design of Experiments plan that was set up to determine the dependence of the preparative chromatography runs on the buffer composition was extended by a few isotherm parameters and evaluated statistically to identify the correlations between these parameters and the buffer composition. Statistically, the results were excellent. The important part, however, was the implementation into the digital twin and that the simulations carried out with different buffer compositions matched the corresponding DoE experiments. In terms of implementation, equation 10 was added to the model for each isotherm parameter ($a_{1,i }$, $a_{2,i }$, $b_{1,i }$ and $b_{2,i }$).

A comparison between the measured and simulated chromatograms can be found in Figure 4. It can be seen that the digital twin covered the changes in the buffer composition very well. The R² values were between 0.858 and 0.998 with an average of 0.952. More importantly, the yield and productivity could be described with good accuracy. On average, the deviation for the yield was 3.6% and the deviation for the productivity was 2.76%. All the values are listed in

Table 3. Comparison between yield and productivity values for simulations and preparative runs.

	Simulation		Experiment		Deviation
Pattern	Normalized Yield	Normalized Productivity	Normalized Yield	Normalized Productivity	Yield	Productivity
	[-]	[-]	[-]	[-]	[%]	[%]
(000)_1	1.00	1.00	1.11	1.08	9.88	7.32
(000)_2	1.00	1.00	1.09	1.07	8.65	6.70
(000)_3	1.00	1.00	1.10	1.08	9.12	7.59
(--+)	1.24	1.23	1.17	1.16	−5.51	−5.60
(-+-)	1.22	1.21	1.20	1.16	−1.74	−4.52
(-++)	1.01	1.04	1.03	1.10	1.66	5.03
(+--)	1.00	0.98	1.03	0.98	2.59	−0.19
(+-+)	1.29	1.27	1.32	1.29	1.91	1.81
(++-)	1.28	1.26	1.35	1.32	5.11	4.23
(+++)	1.02	1.03	1.06	1.07	3.89	3.26

.

Of course, the digital twin could be used to optimize the buffer composition. Contour plots are shown in Figure 5. The top two rows show the values for the buffer and counter ion (top) and the pH over counter ion (middle). In accordance with the pareto charts (Figure 1), the trend for all the components was the more the better. For the combination of the pH and the buffer (bottom), however, the ideal spots are on the two end points, being high pH with a low buffer concentration or a high buffer concentration with low pH. Again, this opposing trend was predicted by the pareto charts. Optimization studies showed that the best result was reached for the combination of the high buffer concentration at low pH. Thus, the best composition was 100 mM counter ion and 150 mM buffer at pH 3.35. This increased the yield by 29% and the productivity by 27% compared to the center point.

7. Conclusions

Design of experiment plans to estimate the best buffer composition are state of the art as part of the early process development phase, as they are easy to set up and fast to execute. The DoE itself shows the dependency of the performance values on the buffer composition. If coupled with a digital twin, additional information can be extracted easily. In this work, it was shown that fitting the isotherm parameters to the DoE runs and feeding these values back into the DoE as target values generated good correlations. These can again be integrated into the digital twin, which was able to optimize the buffer composition. This provided an increase in yield of 29% and productivity of 27%.

Additionally, if used for advanced process control, the digital twin is now able to detect fluctuations in the buffer composition and optimize the system, especially the cut points, to maintain the purity at maximum performance.