Parameter-by-parameter estimation method for adsorption isotherm in hydrophobic interaction chromatography

Model-based process development for hydrophobic interaction chromatography by considering prediction uncertainty analysis

Mechanistic models offer powerful tools for process development and optimization of hydrophobic interaction chromatography (HIC). Suitable parameter estimation approaches can efficiently calibrate the models, but some unavoidable biases between model prediction and actual experiment would reduce the credibility of the model's applications. In this study, a well-calibrated HIC model was found some significant discrepancies between the predicted yield (97.3 %) and experimental yield (86.0 %) during the process optimization. Therefore, Bayesian inference with Markov Chain Monte Carlo method was employed to calculate the uncertainty of model parameters, which was then transformed into the uncertainty of model predictions. The results indicated that the model-predicted yield uncertainty interval was as large as 76.9∼96.5 %, which was consistent with the experiment. Moreover, the model prediction uncertainty analysis was integrated into process optimization to obtain a more reliable and low-risk separation condition. The re-optimized process significantly narrowed the uncertainty of the predicted yield (94.2∼98.9 %), and high experimental yield (95.8 %) was obtained. The results demonstrated that process optimization based on the uncertainty quantification could reasonably reflect model prediction deviations, assist process development and contribute to product quality improvement. Finally, a framework was proposed for process optimization based on the uncertainty analysis to improve the accuracy of model predictions and reducing the risk of model-based process development.

Hybrid Modeling of the Reversed-Phase Chromatographic Purification of an Oligonucleotide: Few-Shot Learning From Differentiable Physics Solver-in-the-Loop

Hybrid models integrate mechanistic and data-driven components, effectively addressing the challenges of limited process understanding and data availability typical of biopharmaceutical processes. In this study, we applied a hybrid modeling framework named differentiable physics solver-in-the-loop (DP-SOL) to describe the reversed-phase chromatographic purification of an oligonucleotide, overcoming the mentioned limitations of purely mechanistic and data-driven models. The framework establishes a connection between neural networks (NNs) and mechanistic models through differentiable physical operators and their gradients. We first collected a data set comprising six linear gradient elution experiments at different resin loadings and gradient slopes, split in three experiments each for training and testing, for few-shot learning. The hyperparameters were determined through a grid search, resulting in a NN with two hidden layers and 14 nodes. Compared to a calibrated mechanistic model used for initialization of NN, the DP-SOL hybrid model showed significant performance improvement on both training and testing sets, withR 2 > ${R}^{2}\,\gt $ 0.97 for the former. The good predictivity of DP-SOL is attributed to the combination of mechanistic models and NNs at the solver level. As a novel and versatile hybrid modeling paradigm, DP-SOL has the potential to significantly impact modeling approaches in the downstream processing field and the broader biopharmaceutical sector.

Prediction of pH Gradient Elution of Ion Exchange Chromatography for Antibody Charge Variants Separation Based on Salt Gradient Elution Experiments

Mechanistic modeling of ion exchange chromatography (IEC) is a promising technique to improve process development. However, when considering the pH influence, model prediction becomes challenging due to the multiple pH-dependent parameters and complex interactions. In order to more effectively predict the pH gradient elution behavior, a two-step model calibration strategy was proposed for the pH-dependent steric mass action (SMA) model with the empirical correlations of characteristic charge ν and equilibrium coefficient keq. The strategy was verified through a case study of monoclonal antibody charge variants purification with IEC. All nine calibration experiments were conducted using linear salt gradient elution at three fixed pH values. The average root mean square error (RMSE) was 14.28% between the model calculation and experiments. Both ν and ln(keq) exhibited good linear correlations with pH (R2 > 0.99). Then, the well-calibrated pH-dependent SMA model showed a satisfactory capability for predicting the pH gradient elution behaviors with an RMSE of 16.18%. Moreover, the model was used for process optimization under different elution modes, including salt gradient, pH gradient, and salt-pH dual gradient, improving the yield from 70.07% to 74.91%. The optimized linear pH gradient elution was verified by experiment (RMSE = 8.30%). Finally, a methodological framework for utilizing the simplified pH-dependent SMA model developed in this work was summarized to explore its practical applications. The two-step calibration strategy proposed significantly alleviates the workload for the pH-dependent IEC modeling. The model-based process optimization effectively enables faster pH-dependent process development with minimal experiments.

Mechanistic modeling of anti-Langmuirian to Langmuirian behavior of Fc-fusion proteins in cation exchange chromatography

Development of a next-generation chromatographic model, capable of simultaneously meeting academic demands for thermodynamic consistency and industrial requirements in everyday project work, has become a focal point of research. In this study, anti-Langmuirian to Langmuirian (AL-L) elution behavior was observed in cation-exchange chromatographic separation of charge variants of industrial Fc-fusion proteins. To characterize this behavior, the multi-protein Mollerup activity model was integrated into the steric mass action (SMA) model, resulting in a new model named the generalized ion-exchange (nGIEX) isotherm for multi-protein systems. An R2 exceeding 0.95 calibrated by three elution experiments indicates an effective description of the AL-L behavior (dynamic adsorption). Using isotherm sampling, the nGIEX model exhibited sigmoidal AL-L isotherms (static adsorption). Finally, the model's extrapolation capability was externally validated through process optimization, resulting in an optimal two-step elution condition and a yield improvement of the main variant from 25.9 % to 89.1 % within purity specifications (>70 %).

Enhancing thermodynamic consistency: Clarification on the application of asymmetric activity model in multi-component chromatographic separation

The single-component Mollerup model, with over 40 direct applications and 442 citations, is the most widely used activity model for chromatographic mechanistic modeling. Many researchers have extended this formula to multi-component systems by directly adding subscripts, a modification deemed thermodynamically inconsistent (referred to as the reference model). In this work, we rederived the asymmetric activity model for multi-component systems, using the van der Waals equation of state, and termed it the multi-component Mollerup model. In contrast to the reference model, our proposed model accounts for the contributions of all components to the activity. Three numerical experiments were performed to investigate the impact of the three different activity models on the chromatographic modeling. The results indicate that our proposed model represents a thermodynamically consistent generalization of the single-component Mollerup model to multi-component systems. This communication advocates adopting of the multi-component Mollerup model for activity modeling in multi-component chromatographic separation to enhance thermodynamic consistency.

Modeling multi-component separation in hydrophobic interaction chromatography with improved parameter-by-parameter estimation method

Mechanistic models are powerful tools for chromatographic process development and optimization. However, hydrophobic interaction chromatography (HIC) mechanistic models lack an effective and logical parameter estimation method, especially for multi-component system. In this study, a parameter-by-parameter method for multi-component system (called as mPbP-HIC) was derived based on the retention mechanism to estimate the six parameters of the Mollerup isotherm for HIC. The linear parameters (ks,i and keq,i) and nonlinear parameters (ni and qmax,i) of the isotherm can be estimated by the linear regression (LR) and the linear approximation (LA) steps, respectively. The remaining two parameters (kp,i and kkin,i) are obtained by the inverse method (IM). The proposed method was verified with a two-component model system. The results showed that the model could accurately predict the protein elution at a loading of 10 g/L. However, the elution curve fitting was unsatisfactory for high loadings (12 g/L and 14 g/L), which is mainly attributed to the demanding experimental conditions of the LA step and the potential large estimation error of the parameter qmax. Therefore, the inverse method was introduced to further calibrate the parameter qmax, thereby reducing the estimation error and improving the curve fitting. Moreover, the simplified linear approximation (SLA) was proposed by reasonable assumption, which provides the initial guess of qmax without solving any complex matrix and avoids the problem of matrix unsolvable. In the improved mPbP-HIC method, qmax would be initialized by the SLA and finally determined by the inverse method, and this strategy was named as SLA+IM. The experimental validation showed that the improved mPbP-HIC method has a better curve fitting, and the use of SLA+IM reduces the error accumulation effect. In process optimization, the parameters estimated by the improved mPbP-HIC method provided the model with excellent predictive ability and reasonable extrapolation. In conclusion, the SLA+IM strategy makes the improved mPbP-HIC method more rational and can be easily applied to the practical separation of protein mixture, which would accelerate the process development for HIC in downstream of biopharmaceuticals.

A parameter estimation method for chromatographic separation process based on physics-informed neural network

Chromatographic separation processes are most often modeled in the form of partial differential equations (PDEs) to describe the complex adsorption equilibria and kinetics. However, identifying parameters in such a model requires substantial computational effort. In this work, a novel parameter estimation approach using a Physics-informed Neural Network (PINN) model is developed and tested for a binary component system. Numerical accuracy of our PINN model is confirmed by validating its simulations against those of the finite element method (FEM). Furthermore, model parameters in the kinetic model are estimated by the PINN model with sufficient accuracy from the observed data at the column outlet, where parameter fitting error can be reduced by up to 35.0 % from the conventional method. In a comparison with the conventional numerical method, our approach can reduce the computational time by up to 95 %. The robustness of the PINN model has also been demonstrated by estimating model parameters from noisy artificial experimental data.

Residence time distribution in continuous virus filtration

Regulatory authorities recommend using residence time distribution (RTD) to address material traceability in continuous manufacturing. Continuous virus filtration is an essential but poorly understood step in biologics manufacturing in respect to fluid dynamics and scale-up. Here we describe a model that considers nonideal mixing and film resistance for RTD prediction in continuous virus filtration, and its experimental validation using the inert tracer NaNO3. The model was successfully calibrated through pulse injection experiments, yielding good agreement between model prediction and experiment (R 2 > ${R}^{2}\gt $  0.90). The model enabled the prediction of RTD with variations-for example, in injection volumes, flow rates, tracer concentrations, and filter surface areas-and was validated using stepwise experiments and combined stepwise and pulse injection experiments. All validation experiments achievedR 2 > ${R}^{2}\gt $  0.97. Notably, if the process includes a porous material-such as a porous chromatography material, ultrafilter, or virus filter-it must be considered whether the molecule size affects the RTD, as tracers with different sizes may penetrate the pore space differently. Calibration of the model with NaNO3 enabled extrapolation to RTD of recombinant antibodies, which will promote significant savings in antibody consumption. This RTD model is ready for further application in end-to-end integrated continuous downstream processes, such as addressing material traceability during continuous virus filtration processes.