The biopharmaceutical industry has received considerable attention recently, as can be witnessed by the growing market demands and approvals of biological drugs by the EU and the US regulatory agencies [1], [2], [3], [4], [5]. During biologics manufacturing, the formation of protein aggregates during production and separation processes remains a primary concern [6,7]. Hydrophobic interaction chromatography (HIC) is one of the widely used techniques in downstream polishing steps for the separation of targeted monomeric forms of protein therapeutics from the dimeric and/or multimeric species [8]. HIC is an entropically-driven process that exploits the difference between the hydrophobicity of the monomer and aggregates to achieve separation [9]. Despite being commonly employed as an efficient purification strategy to remove aggregates, the mechanism for HIC adsorption is quite complex, relying on various process parameters, like pH, salt concentration, and adsorbent ligand hydrophobicity [10,11]. Recently an automatic workflow for HIC method development is presented by combining screening with in-silico modeling by Barrientos et al. [12]. However, the current process development of HIC still largely depends on rules of thumb or tedious experimental screening.

Inspired by the quality by design (QbD) initiative and the need to reduce the effort of performing time- and resource-intensive experiments, mechanistic modeling has become an important tool for process characterization, development and optimization [13], [14], [15]. Mechanistic models are built based on the understanding of chromatographic process physics, which mainly focuses on describing the solute transport in the mobile phase and kinetic adsorption in the stationary phase [16], [17], [18]. With different assumptions and simplifications, several variants of mechanistic models are used to describe the fluid flow inside the moving phase, including the equilibrium dispersive model, lumped kinetic model, and general rate model (GRM) [19], [20], [21]. In HIC modeling, the fluid dynamics are well-described by physics, but the underlying adsorption mechanism for the salt-dependent protein-ligand interaction is still unclear, which makes it challenging to postulate appropriate mathematical equations to describe the overall process [22]. To capture the role of salt, various adsorption laws like solvophobic and preferential interaction theories have been used [23], [24], [25]. Although the simple Langmuir isotherm is popular and frequently considered, this equation cannot account for the effect of salt concertation on adsorption behavior [16]. To overcome this limitation, the Langmuir isotherm is often modified to capture the dependence of adsorption on salt concentrations for the representation of the HIC adsorption process [26]. Moreover, Wang et al. [22] derived a new isotherm by introducing the salt-dependent hydration number of ions and demonstrated the model's capability in modeling the system of glucose oxidase, bovine serum albumin, and lysozyme.

The challenge associated with the development of a precise mechanistic model is to identify the exact underlying physicochemical phenomena. Due to a limited understanding of the HIC adsorption mechanism, there is an absence of a reliable mechanistic model to describe the interaction between proteins and ligands under varying salt ions [26,27]. Consequently, hybrid modeling strategy is a promising alternative to accurately describe HIC process and reduce the model development effort as this approach can exploit the available information about the process and represent the missing knowledge by a data-driven component [28,29]. By combining the mechanistic and data-driven models, the constructed hybrid model can retain the advantages of the mechanistic model (i.e., physical interpretability and generalizability) while being able to extract information from available data that cannot be captured by the first-principle knowledge [30,31]. Neural network (NN) is a commonly used data-driven model, which is based on the neural structure of the human brain [32,33]. NN is composed of many interconnected neurons in layers, with weights assigned to each interconnection. Narayanan et al. [34,35] developed different hybrid models by combining varying degrees of process knowledge with NN and evaluated their performance in terms of the interpolation and extrapolation capabilities in Protein A chromatography process. It was found that the developed hybrid model outperformed the mechanistic model in terms of prediction accuracy and robustness. It should be noted though that the hybrid modeling strategy has not been applied to describe the complex salt-dependent adsorption mechanism for HIC process.

In this work, a mechanistic model is first developed to describe the HIC chromatographic process using an equilibrium dispersive model for hydrodynamics and modified isotherm derived by Wang et al. [22], followed by model validation with experimental data. Due to the limited understanding of the underlying adsorption mechanism, a hybrid model is proposed by combining a simpler multi-component Langmuir isotherm (MCL) with a NN. Different methods to integrate the MCL with NN are investigated to find the appropriate hybrid model structure. During parameter estimation, a regularization strategy is incorporated to avoid overfitting and the effect of different NN structures and regularization rates is comprehensively investigated to acquire the hybrid model with the best performance. To ensure the generalizability of the developed hybrid model, an in-silico dataset is generated using the mechanistic model to test the extrapolation capability of the hybrid model. Finally, process optimization is conducted to find the optimal operating conditions under product quality constraints, and the optimal results obtained from the mechanistic and hybrid models are compared thoroughly.