Stijn Van Hoey

A GLUE uncertainty analysis of a drying model of pharmaceutical granules

A shift from batch processing towards continuous processing is of interest in the pharmaceutical industry. However, this transition requires detailed knowledge and process understanding of all consecutive unit operations in a continuous manufacturing line to design adequate control strategies. This can be facilitated by developing mechanistic models of the multi-phase systems in the process. Since modelling efforts only started recently in this field, uncertainties about the model predictions are generally neglected. However, model predictions have an inherent uncertainty (i.e. prediction uncertainty) originating from uncertainty in input data, model parameters, model structure, boundary conditions and software. In this paper, the model prediction uncertainty is evaluated for a model describing the continuous drying of single pharmaceutical wet granules in a six-segmented fluidized bed drying unit, which is part of the full continuous from-powder-to-tablet manufacturing line (Consigma™, GEA Pharma Systems). A validated model describing the drying behaviour of a single pharmaceutical granule in two consecutive phases is used. First of all, the effect of the assumptions at the particle level on the prediction uncertainty is assessed. Secondly, the paper focuses on the influence of the most sensitive parameters in the model. Finally, a combined analysis (particle level plus most sensitive parameters) is performed and discussed. To propagate the uncertainty originating from the parameter uncertainty to the model output, the Generalized Likelihood Uncertainty Estimation (GLUE) method is used. This method enables a modeller to incorporate the information obtained from the experimental data in the assessment of the uncertain model predictions and to find a balance between model performance and data precision. A detailed evaluation of the obtained uncertainty analysis results is made with respect to the model structure, interactions between parameters and uncertainty boundaries.

Identification of temporal consistency in rating curve data: Bidirectional Reach (BReach)

In this paper, a methodology is developed to identify consistency of rating curve data based on a quality analysis of model results. This methodology, called Bidirectional Reach (BReach), evaluates results of a rating curve model with randomly sampled parameter sets in each observation. The combination of a parameter set and an observation is classified as nonacceptable if the deviation between the accompanying model result and the measurement exceeds observational uncertainty. Based on this classification, conditions for satisfactory behavior of a model in a sequence of observations are defined. Subsequently, a parameter set is evaluated in a data point by assessing the span for which it behaves satisfactory in the direction of the previous (or following) chronologically sorted observations. This is repeated for all sampled parameter sets and results are aggregated by indicating the endpoint of the largest span, called the maximum left (right) reach. This temporal reach should not be confused with a spatial reach (indicating a part of a river). The same procedure is followed for each data point and for different definitions of satisfactory behavior. Results of this analysis enable the detection of changes in data consistency. The methodology is validated with observed data and various synthetic stage-discharge data sets and proves to be a robust technique to investigate temporal consistency of rating curve data. It provides satisfying results despite of low data availability, errors in the estimated observational uncertainty, and a rating curve model that is known to cover only a limited part of the observations.

Dynamic identifiability analysis-based model structure evaluation considering rating curve uncertainty

When applying hydrological models, different sources of uncertainty are present, and evaluations of model performances should take these into account to assess model outcomes correctly. Furthermore, uncertainty in the discharge observations complicates the model identification, both in terms of model structure and parameterization. In this paper, the authors compare two different lumped model structures (PDM and NAM) considering uncertainty coming from the rating curve. Limits of acceptability for the model simulations were determined based on derived uncertainty bounds of the discharge observations. The authors applied the DYNamic Identifiability Approach (DYNIA) to identify structural failure of both models and to evaluate the configuration of their structures. In general, similar model performances are observed. However, the model structures tend to behave differently in the course of time, as revealed by the DYNIA approach. Based on the analyses performed, the probability based soil storage representation of the PDM model outperforms the NAM structure. The incorporation of the observation error did not prevent the DYNIA analysis to identify potential model structural deficiencies that are limiting the representation of the seasonal variation, primarily indicated by shifting regions of parameter identifiability. As such, the proposed approach is able to indicate where deficiencies are found and model improvement is needed.

pyIDEAS: an Open Source Python Package for Model Analysis

Abstract Mathematical models are used in many scientific areas such as enzyme kinetics and process engineering. They can be used for process analysis and optimization. However, a model is always a simplified representation of the real process and predictions always come with uncertainty. Therefore, the model building process should be performed thoroughly addressing calibration and validation procedures. Specific modeling tools (e.g. sensitivity analysis, optimization algorithms, experimental design techniques,…) to derive additional information (e.g. importance of parameters, estimate parameter uncertainty,…) are at hand and available in existing software. First, implementing these algorithms is time-consuming and often suboptimal in efficiency. Second, existing software is in many cases closed-source and not flexible in use. In both cases this results in the unavailability of the programmed algorithms in the corresponding articles making use of them. Therefore it is hard to validate the published findings and in some cases even impossible to reproduce the presented results. To address this problem the scientific community needs a certain critical mass of ‘off-the-shelf’ algorithms to perform model analyses which are available to the modeling community. To improve overall quality and reliability, such kind of code library should be open source and well documented. We hereby present pyIDEAS, an open source Python package to thoroughly but swiftly analyze systems represented by a set of (possibly mixed) differential and algebraic equations. The pyIDEAS package allows performing a model analysis in a straightforward and fast way. pyIDEAS provides a well-structured and logic framework which allows non-programmers to perform some model analysis and more advanced users to extend or adapt current functionality to their own requirements.

A Numerical Procedure for Model Identifiability Analysis Applied to Enzyme Kinetics

Abstract The proper calibration of models describing enzyme kinetics can be quite challenging. In the literature, different procedures are available to calibrate these enzymatic models in an efficient way. However, in most cases the model structure is already decided on prior to the actual calibration exercise, thereby bypassing the challenging task of model structure determination and identification. Parameter identification problems can thus lead to ill-calibrated models with low predictive power and large model uncertainty. Every calibration exercise should therefore be preceded by a proper model structure evaluation by assessing the local identifiability characteristics of the parameters. Moreover, such a procedure should be generic to make sure it can be applied independent from the structure of the model. We hereby apply a numerical identifiability approach which is based on the work of Walter and Pronzato (1997) and which can be easily set up for any type of model. In this paper the proposed approach is applied to the forward reaction rate of the enzyme kinetics proposed by Shin and Kim (1998). Structural identifiability analysis showed that no local structural model problems were occurring. In contrast, the practical identifiability analysis revealed that high values of the forward rate parameter V f led to identifiability problems. These problems were even more pronounced at higher substrate concentrations, which illustrates the importance of a proper experimental design to avoid (practical) identifiability problems. By using the presented approach it is possible to detect potential identifiability problems and avoid pointless calibration (and experimental!) effort.