(636d) Application of Kriging for Dynamic Data-Driven Modeling of Pharmaceutical Processes

In emergent technologies it is very common that a functional form for the input-output relationships is unavailable and the process behavior is symbolically described using black-box models. This is the case for many pharmaceutical process operations, for which first-principles models describing the behavior of the processed powders do not exist. Surrogate based methodologies- such as Kriging, Response Surface Methodology (RSM) and High Dimensional Model Representations- have been previously used to model the steady-state performance of pharmaceutical unit operations such as powder feeding and mixing given experimentally sampled outputs at specific operating conditions [1]. These processes, however, operate sequentially in an integrated production system. Working towards modeling the performance and interactions of an integrated system, it is necessary to be able to capture the dynamic behavior of such operations caused by changes in their operating conditions (input parameters). Extending the capabilities of data-driven modeling techniques to dynamic predictors, can find significant applications in real time optimization and control of these systems.

Kriging- classified as a Gaussian Process- is a probabilistic, non- parametric black-box modeling technique that has been used in the literature to capture the behavior of dynamic non-linear systems [2, 3]. Advantages of Kriging comprise of computational efficiency, calculation of a variance estimate at each prediction and lastly the ability to model multidimensional black- box systems having several inputs and outputs. Kriging differs from other response surface methods since it is not based on the approximation of parameters of prepostulated basis functions (i.e. RSM). The output of Kriging consists of a mean value and a variance, which depends on the quantity and location of the sampled data. The most important step to this procedure is the determination of the covariance function which can be achieved using variogram characteristics [4]. The most interesting feature of Kriging, however, is the possibility to include prior knowledge of the system into the model [5]. In this Bayesian modeling framework, a future time step prediction is achieved based on prior knowledge of the system's behavior and output values sampled at the current time step. Using experimental results- performed to capture the dynamic response of the studied processes at different operating conditions- the output at each time step is modeled as a function of the previous time step output and the new experimental data set measured at the current time step. Thus if Kriging is applied as a dynamic predictor, a complete evolution of the system behavior can be captured.

Another approach proposed to capture the dynamic behavior of pharmaceutical unit operations is based on ideas of interpolation of geographic data, where spatio- temporal modeling has been applied [6]. Spatio- temporal interpolation refers to the accurate estimation of unknown states at unsampled location- time pairs, where the output is considered as a sequence of spatial interpolations. First, spatial interpolation is performed separately for different time points followed by temporal interpolation between the produced spatial surfaces. Applying this approach to pharmaceutical processes requires that the process output has two components: the steady-state and transient effect. The steady-state component is modeled in the spatial dimension, assuming that the process has a different dynamic behavior at different operating conditions. The transient effect is first modeled separately at different spatial locations (operating conditions) using surrogate-based methods. Then, interpolation in the time domain is performed. The result of this procedure is an interpolated surface representing the temporal and spatial evolution of an output.

The proposed approaches are used to describe two pharmaceutical case studies: powder feeding and blending. This is achieved by the use of carefully designed experimental results which are performed to capture the dynamic response of the processes when step changes are performed at different operating conditions. Considering all the advantages of Kriging algorithm and its ability to model black- box pharmaceutical processes operated at steady-state, its extension to an adaptive modeling tool that can capture the dynamic evolution of such processes can be considered as a significant step towards modeling and controlling integrated pharmaceutical systems.

References:

1. Jia Z, Davis E, Muzzio F, and Ierapetritou M, Predictive Modeling for Pharmaceutical Processes Using Kriging and Response Surface. Journal of Pharmaceutical Innovation, 2009; 4(4): 174-186.

2. Azman K and Kocijan J, Application of Gaussian processes for black-box modelling of biosystems. ISA Transactions, 2007; 46(4): 443-457.

3. Hernandes Andres and Martha GG (2008) An Exploratory Study of Discrete Time State-Space Models using Kriging. 2008 American Control Conference

4. Davis E and Ierapetritou M, A kriging based method for the solution of mixed-integer nonlinear programs containing black-box functions. Journal of Global Optimization, 2009; 43(2): 191-205.

5. Carl Edward Rasmussen and Williams CKI, Gaussian Processes for Machine Learning, ed. M. Press. Boston, 2006.

6. Li L and Revesz P, Interpolation methods for spatio-temporal geographic data. Computers, Environment and Urban Systems, 2004; 28(3): 201-227.