MATHEMATICAL MODELLING FOR QBD IMPLEMENTATION

1.3 Mathematical modelling for QbD implementation

As extensively discussed in the previous sections, the most important target of the QbD paradigm is to promote the adoption of rigorous scientific tools to assist the different stages (pharmaceutical development, technology transfer, commercial manufacturing) of the lifecycle of a new pharmaceutical drug. One of the most important tools that can be used to achieve this purpose is mathematical modelling (García-Muñoz and Oksanen, 2010).

The importance of mathematical modelling for a correct implementation of the QbD paradigm is extensively emphasized in the regulatory documents (FDA, 2004c; ICH, 2009a; ICH, 2009b,ICH, 2012a). Since the final scope of every development/manufacturing activity is the achievement of the desired product quality, models are classified by regulators according to their impact (low/medium/high) in assuring this target (ICH, 2012a). Following the regulatory parlance (ICH, 2012a), low-impact models are defined as models that are typically used in product/process development (e.g., for formulation optimization). Medium-impact models are models that can be useful in assuring quality of the product but are not the sole indicators of product quality (e.g., most design space models). High-impact models are models whose predictions are significant indicators of product quality (e.g., a chemometric model for product assay or a surrogate model for dissolution).

From a PSE perspective, a much more useful classification of the models that can be exploited in pharmaceutical development/manufacturing is based on the following two criteria (Bonvin et al., 2017):

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© 2018 Gabriele Bano, University of Padova (Italy)

 model scope, i.e. based on the final purpose of implementation (e.g., models for design space determination).

From a general perspective, a model is characterized by three elements: equations, variables and parameters. The model type describes the amount of knowledge embedded within the model. In this regard, models can be classified as knowledge-driven (also known as first- principles or mechanistic or deterministic), hybrid (also known as semi-empirical or gray-box) and data-driven (also known as data-based or empirical or black-box).

Knowledge-driven models are based on fundamental knowledge of the underlying physical phenomena that govern the system under investigation. For these models, the equations describe in mathematical terms the physical laws (e.g., mass and energy balances, heat/mass transfer mechanisms) representing the system; the variables represent the system states; parameters inform on how the mathematical description given by the equations should be tuned to match the actual system behavior. Data-driven models do not embed any knowledge on the physical mechanisms involved in the system. For these models, the equations simply represent a convenient representation of the dataset(s) collected for the system under investigation; variables collect the inputs and outputs of the dataset; parameters inform on how equations should be tuned to match the available data. Semi-empirical models represent an intermediate situation between knowledge-driven and data-driven models, i.e. they combine fundamental knowledge on the system to describe certain phenomena with empirical reasoning to describe others phenomena.

The model scope describes the purpose of implementation (i.e., the application) of the model. Models can be exploited to assist the three stages of the drug lifecycle (pharmaceutical development, technology transfer, commercial manufacturing and continual process improvement). In pharmaceutical development, models can be used to assist all the sequential activities discussed in § 1.2.1 (QTTP and CQAs identification; product and process understanding; product and process design, including design space identification and definition of a control strategy) with the purpose of accelerating the launch of new products in the market. With respect to technology transfer, models can be used to assist both the scale-up and scale- out of the manufacturing process (including its design space and its control strategy), with the purpose of facilitating its transfer between different scales or between different sites. Finally, with respect to commercial manufacturing and continual process improvement, models can be used to assist product quality monitoring and control, as well as to enhance process productivity (i.e., via process optimization) and to assist process intensification.

The increasing interest of the pharmaceutical community towards mathematical modelling has been thoroughly reviewed by Troup and Georgakis (2013), Rogers and Ierapetritou (2015) and, very recently, by Reklaitis et al. (2017).

In 2013, Troup and Georgakis (2013) presented the results of a survey involving 21 professionals from worldwide top-pharma companies on the use of mathematical modelling for

Motivation and state of the art 31

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© 2018 Gabriele Bano, University of Padova (Italy)

process analytics, process monitoring, plant-wide information system, unit operation modeling, quality control and process optimization. The results showed that, with respect to process analytics and monitoring, the use of multivariate chemometric models and multivariate statistical process control represented common practices in the industrial context (according to 67% of the respondents), as well as the use of statistical multivariate tool to analyze historical process data. With respect to process modelling and optimization, more than one third of the respondents revealed that data-driven approaches were adopted to model between 80-100% of the unit operations of the manufacturing lines, while the other two thirds revealed that data- driven approaches were exploited to model at least 60% of the unit operations. However, the use of first principles models were acknowledged for at least 10% of the unit operations, mainly involving secondary manufacturing activities. Moreover, the advantages (possibility to perform extrapolation, wider applicability range, possibility to include product physical properties) and disadvantages (costs and complexity of model development, computational burden) of first- principles models with respect to data-driven models were stressed out by the respondents. In 2015, a review by Rogers and Ierapetritou (2015) revealed the increasing interest of the pharmaceutical community towards the use of hybrid as well as first-principles models. This interest was partially driven by the increasing attention towards continuous manufacturing in the pharmaceutical community (§ 1.1.3). Very recently, Reklaitis et al. (2017) revealed that data-driven, hybrid and first-principles models are nowadays jointly exploited by pharmaceutical companies to tackle different problems of the drug lifecycle. Specifically, while data-driven approaches still represent the most adopted tools for process analytics, process monitoring and process control, hybrid as well as first-principles models are starting being exploited much more frequently to assist the different phases of pharmaceutical development. Moreover, due to the recent advancements in computational power and technology, it is expected that online applications of these model will play a key role in the next years (Reklaitis et al., 2017).

Within the different scopes of mathematical modeling in the pharmaceutical context, the identification of the design space of a new pharmaceutical product probably represents the most important one (García-Muñoz and Oksanen, 2010). For example, almost 70% of the respondents of the survey of Troup and Georgakis (2013) reported the use of multivariate approaches for DS identification, while Reklaitis et al. (2017) suggested that this activity is nowadays common routine in pharmaceutical R&D. Since the launch of the QbD initiative, different modelling strategies have been proposed to tackle this issue. The aim of the next sections is to propose a critical review of the most important contributions on this topic, from both industry and academia.

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© 2018 Gabriele Bano, University of Padova (Italy)