Mathematical modelling of chemical processes-obtaining the best model predictions and parameter estimates using identifiability and estimability procedures

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.

Supercritical Carbon Dioxide Extraction of Coumarins from the Aerial Parts of Pterocaulon polystachyum

Pterocaulon polystachyum is a species of pharmacological interest for providing volatile and non-volatile extracts with antifungal and amebicidal properties. The biological activities of non-volatile extracts may be related to the presence of coumarins, a promising group of secondary metabolites. In the present study, leaves and inflorescences previously used for the extraction of essential oils instead of being disposed of were subjected to extraction with supercritical CO2 after pretreatment with microwaves. An experimental design was followed to seek the best extraction condition with the objective function being the maximum total extract. Pressure and temperature were statistically significant factors, and the optimal extraction condition was 240 bar, 60 °C, and pretreatment at 30 °C. The applied mathematical models showed good adherence to the experimental data. The extracts obtained by supercritical CO2 were analyzed and the presence of coumarins was confirmed. The extract investigated for cytotoxicity against bladder tumor cells (T24) exhibited significant reduction in cell viability at concentrations between 6 and 12 μg/mL. The introduction of green technology, supercritical extraction, in the exploration of P. polystachyum as a source of coumarins represents a paradigm shift with regard to previous studies carried out with this species, which used organic solvents. Furthermore, the concept of circular bioeconomy was applied, i.e., the raw material used was the residue of a steam-distillation process. Therefore, the approach used here is in line with the sustainable exploitation of native plants to obtain extracts rich in coumarins with cytotoxic potential against cancer cells.

Contributions towards variable temperature shielding for compact NMR instruments

The application of compact NMR instruments to hot flowing samples or exothermically reacting mixtures is limited by the temperature sensitivity of permanent magnets. Typically, such temperature effects directly influence the achievable magnetic field homogeneity and hence measurement quality. The internal-temperature control loop of the magnet and instruments is not designed for such temperature compensation. Passive insulation is restricted by the small dimensions within the magnet borehole. Here, we present a design approach for active heat shielding with the aim of variable temperature control of NMR samples for benchtop NMR instruments using a compressed airstream which is variable in flow and temperature. Based on the system identification and surface temperature measurements through thermography, a model predictive control was set up to minimise any disturbance effect on the permanent magnet from the probe or sample temperature. This methodology will facilitate the application of variable-temperature shielding and, therefore, extend the application of compact NMR instruments to flowing sample temperatures that differ from the magnet temperature.

Reaction kinetics determination and uncertainty analysis for the synthesis of the cancer drug lomustine

Fast and reliable model development frameworks are required to support current trends in modernization of pharmaceutical processing, promoting the use of digital platforms to assist process design and operation. In this work, we use a parameter estimation framework built into the PharmaPy library to determine rate parameters and uncertainty regions of different mechanistic and semi-empirical kinetic expressions for the synthesis of the drug lomustine. The parameter estimation procedure was complemented by identifiability analysis, resulting in simplified reaction mechanisms. Comparison of parameters and their uncertainty in process design was demonstrated through design space analysis, showing important differences in model prediction and the extent of their corresponding design spaces. The results of this work can serve to analyze lomustine manufacturing processes that include separation and isolation steps, where parametric sensitivity is expected to propagate along the manufacturing line and impact process feasible operation, and attainment of critical quality attributes of the product.

Uncertainty Propagation Based MINLP Approach for Artificial Neural Network Structure Reduction

The performance of artificial neural networks (ANNs) is highly influenced by the selection of input variables and the architecture defined by hyper parameters such as the number of neurons in the hidden layer and connections between network variables. Although there are some black-box and trial and error based studies in the literature to deal with these issues, it is fair to state that a rigorous and systematic method providing global and unique solution is still missing. Accordingly, in this study, a mixed integer nonlinear programming (MINLP) formulation is proposed to detect the best features and connections among the neural network elements while propagating parameter and output uncertainties for regression problems. The objective of the formulation is to minimize the covariance of the estimated parameters while by (i) detecting the ideal number of neurons, (ii) synthesizing the connection configuration between those neurons, inputs and outputs, and (iii) selecting optimum input variables in a multi variable data set to design and ensure identifiable ANN architectures. As a result, suggested approach provides a robust and optimal ANN architecture with tighter prediction bounds obtained from propagation of parameter uncertainty, and higher prediction accuracy compared to the traditional fully connected approach and other benchmarks. Furthermore, such a performance is obtained after elimination of approximately 85% and 90% of the connections, for two case studies respectively, compared to traditional ANN in addition to significant reduction in the input subset.

An optimization-based model discrimination framework for selecting an appropriate reaction kinetic model structure during early phase pharmaceutical process development

A model discrimination workflow to develop fit for purpose kinetic models of new pharmaceutical compounds in early stages of drug development involving complex reaction networks with limited prior information and provision to run new experiments.

Effect of organic loading rate on anaerobic digestion of raw cheese whey: experimental evaluation and mathematical modeling

Whey is a high strength waste product of cheese manufacture. Anaerobic digestion of cheese allows pollution potential reduction and simultaneously energy production. Organic loading rate (OLR) is an important operating variable for anaerobic digestion (AD) process related to system stability, waste treatment capacity and biogas production. The actual OLR depends on the types of wastes (i.e., content of chemical oxygen demand [COD]) fed into a digester. In this paper, the effect of OLR on the AD process of the raw cheese whey in a semi-continuous up-flow system is studied experimentally and with numerical simulations using a simple dynamical model calibrated with experimental data. The digester operation was performed for 90 days, doubling the OLR every 30 days, from OLR of 2.5–10 gCOD L−1 d−1. Experimental results show that the increase in OLR favors the production of biogas. However, the proportion of methane may decrease. The highest methane yield and the most considerable substrate degradation were obtained at OLR of 5 gCOD L−1 d−1 and 10 gCOD L−1 d−1, respectively. The proposed mathematical model is used to describe the dynamic behavior of key variables as COD, volatile fatty acids (VFA) and methane production. A good fit between the variables estimated by the mathematical model and experimental data was obtained, reaching determination coefficients (R2) greater than 0.8. Therefore, this model might be beneficial in predicting the maximum methane production rate and the maximum OLR that could be used without risking the AD process stability.

Forecasting leachate generation from pilot woodchip stockpiles using a three-dimensional transient flow model

Leachate generation from open stockpiles of recycled woodchip materials is potentially harmful to aquatic ecosystems. There is growing interest in using numerical models to simulate leachate generation from outdoor piles, but this requires information about the hydraulic properties of the materials. The objectives of this study were to simulate leachate from woodchip piles with the numerical model HYDRUS-3D and to optimize subsets of parameters for single (SPM) and dual (DPM) pore flow models with the Bayesian Markov Chain Monte Carlo algorithm DREAM_ZS. Three experimental piles, each approximately 30 m³, were setup with mixtures of either once (coarse) or twice (fine) ground woodchips. Leachate continuously collected over a period of six months was similar across piles. As a result, subsets of optimized flow parameters for the coarse and fine woodchips were not different. Leachate predictions by the two pore flow models were similar and agreed reasonably with the field measurements, as indicated by Nash-Sutcliffe efficiency values greater than 0.6. This result suggests the simpler SPM is adequate for field predictions of leachate. However, leachate was consistently under-predicted by both pore models by 13-27% during rainfall events with more than 1 cm in 6 h. The optimized flow models can be used as a tool for studying pile management strategies.

An Unstructured Model for Anaerobic Treatment of Raw Cheese Whey for Volatile Fatty Acids Production

The whey is a byproduct of the dairy industry that, if not treated properly, can cause serious environmental pollution problems. Anaerobic treatment is an alternative for its recovery, since, in addition to reducing the organic load. it allows the generation of value-added products such as volatile fatty acids (VFA) and biogas. However, the process is very complex and requires specific operating conditions that guarantee its stability and favor the production of value-added compounds. In this work, an unstructured mathematical model is proposed to evaluate the dynamic behavior of the stages of the anaerobic degradation process of the whey (i.e., hydrolysis, acidogenesis, acetogenesis and methanogenesis). The proposed model considers the dynamic variation in pH during the experiment. To validate the model, an experimental set was carried out at pH and temperature conditions that favor the production of VFAs. Experimental results show that the anaerobic treatment of the raw cheese whey favors pH = 5.5; for T = 40 °C, the maximum VFA production is obtained (30.71 gCOD L−1), and for T = 35 °C, a 45.81% COD degradation is reached. The proposed model considers the effect of pH and temperature and it is validated in the region where the experimental tests were carried out. The model parameters were estimated using the Levenberg–Marquardt method, obtaining coefficients of determination R2 > 0.94. The proposed model can describe the dynamic behavior of the key variables in the anaerobic treatment of raw cheese whey at different pH and temperature conditions, finding that VFA production is favored at pH ≥ 7, while the highest COD removal results in acidic conditions

Global optimization of the Michaelis-Menten parameters using physiologically-based pharmacokinetic (PBPK) modeling and chloroform vapor uptake data in F344 rats

Objective: To quantify metabolism, a physiologically based pharmacokinetic (PBPK) model for a volatile compound can be calibrated with the closed chamber (i.e. vapor uptake) inhalation data. Here, we introduce global optimization as a novel component of the predictive process and use it to illustrate a procedure for metabolic parameter estimation.Materials and methods: Male F344 rats were exposed in vapor uptake chambers to initial concentrations of 100, 500, 1000, and 3000 ppm chloroform. Chamber time-course data from these experiments, in combination with optimization using a chemical-specific PBPK model, were used to estimate Michaelis-Menten metabolic constants. Matlab^® simulation software was used to integrate the mass balance equations and to perform the global optimizations using MEIGO (MEtaheuristics for systems biology and bIoinformatics Global Optimization - Version 64 bit, R2016A), a toolbox written for Matlab®. The cost function used the chamber time-course data and least squares to minimize the difference between data and simulation values.Results and discussion: The final values estimated for V_max (maximum metabolic rate) and K_m (affinity constant) were 1.2 mg/h and a range between 0.0005 and 0.6 mg/L, respectively. Also, cost function plots were used to analyze the dose-dependent capacity to estimate V_max and K_m within the experimental range used. Sensitivity analysis was used to assess identifiability for both parameters and show these kinetic data may not be sufficient to identify K_m.Conclusion: In summary, this work should help toxicologists interested in optimization techniques understand the overall process employed when calibrating metabolic parameters in a PBPK model with inhalation data.

Design of Feedforward Neural Networks in the Classification of Hyperspectral Imagery Using Superstructural Optimization

Artificial Neural Networks (ANNs) have been used in a wide range of applications for complex datasets with their flexible mathematical architecture. The flexibility is favored by the introduction of a higher number of connections and variables, in general. However, over-parameterization of the ANN equations and the existence of redundant input variables usually result in poor test performance. This paper proposes a superstructure-based mixed-integer nonlinear programming method for optimal structural design including neuron number selection, pruning, and input selection for multilayer perceptron (MLP) ANNs. In addition, this method uses statistical measures such as the parameter covariance matrix in order to increase the test performance while permitting reduced training performance. The suggested approach was implemented on two public hyperspectral datasets (with 10% and 50% sampling ratios), namely Indian Pines and Pavia University, for the classification problem. The test results revealed promising performances compared to the standard fully connected neural networks in terms of the estimated overall and individual class accuracies. With the application of the proposed superstructural optimization, fully connected networks were pruned by over 60% in terms of the total number of connections, resulting in an increase of 4% for the 10% sampling ratio and a 1% decrease for the 50% sampling ratio. Moreover, over 20% of the spectral bands in the Indian Pines data and 30% in the Pavia University data were found statistically insignificant, and they were thus removed from the MLP networks. As a result, the proposed method was found effective in optimizing the architectural design with high generalization capabilities, particularly for fewer numbers of samples. The analysis of the eliminated spectral bands revealed that the proposed algorithm mostly removed the bands adjacent to the pre-eliminated noisy bands and highly correlated bands carrying similar information.

Experimental design for parameter estimation in steady-state linear models of metabolic networks

Metabolic networks are typically large, containing many metabolites and reactions. Dynamical models that aim to simulate such networks will consist of a large number of ordinary differential equations, with many kinetic parameters that must be estimated from experimental data. We assume these data to be metabolomics measurements made under steady-state conditions for different input fluxes. Assuming linear kinetics, analytical criteria for parameter identifiability are provided. For normally distributed error terms, we also calculate the Fisher information matrix analytically to be used in the D-optimality criterion. A test network illustrates the developed tool chain for finding an optimal experimental design. The first stage is to verify global or pointwise parameter identifiability, the second stage to find optimal input fluxes, and finally remove redundant measurements.

Scalable nonlinear programming framework for parameter estimation in dynamic biological system models

We present a nonlinear programming (NLP) framework for the scalable solution of parameter estimation problems that arise in dynamic modeling of biological systems. Such problems are computationally challenging because they often involve highly nonlinear and stiff differential equations as well as many experimental data sets and parameters. The proposed framework uses cutting-edge modeling and solution tools which are computationally efficient, robust, and easy-to-use. Specifically, our framework uses a time discretization approach that: i) avoids repetitive simulations of the dynamic model, ii) enables fully algebraic model implementations and computation of derivatives, and iii) enables the use of computationally efficient nonlinear interior point solvers that exploit sparse and structured linear algebra techniques. We demonstrate these capabilities by solving estimation problems for synthetic human gut microbiome community models. We show that an instance with 156 parameters, 144 differential equations, and 1,704 experimental data points can be solved in less than 3 minutes using our proposed framework (while an off-the-shelf simulation-based solution framework requires over 7 hours). We also create large instances to show that the proposed framework is scalable and can solve problems with up to 2,352 parameters, 2,304 differential equations, and 20,352 data points in less than 15 minutes. The proposed framework is flexible and easy-to-use, can be broadly applied to dynamic models of biological systems, and enables the implementation of sophisticated estimation techniques to quantify parameter uncertainty, to diagnose observability/uniqueness issues, to perform model selection, and to handle outliers.

Constructing and validating dynamic models of biological systems spanning biomolecular networks to ecological systems is a challenging problem. Here we present a scalable computational framework to rapidly infer parameters in complex dynamic models of biological systems from large-scale experimental data. The framework was applied to infer parameters of a synthetic microbial community model from large-scale time series data. We also demonstrate that this framework can be used to analyze parameter uncertainty, to diagnose whether the experimental data are sufficient to uniquely determine the parameters, to determine the model that best describes the data, and to infer parameters in the face of data outliers.

Identifiability Analysis of Three Control-Oriented Models for Use in Artificial Pancreas Systems

OBJECTIVE

Our aim is to analyze the identifiability of three commonly used control-oriented models for glucose control in patients with type 1 diabetes (T1D).

METHODS

Structural and practical identifiability analysis were performed on three published control-oriented models for glucose control in patients with type 1 diabetes (T1D): the subcutaneous oral glucose minimal model (SOGMM), the intensive control insulin-nutrition-glucose (ICING) model, and the minimal model control-oriented (MMC). Structural identifiability was addressed with a combination of the generating series (GS) approach and identifiability tableaus whereas practical identifiability was studied by means of (1) global ranking of parameters via sensitivity analysis together with the Latin hypercube sampling method (LHS) and (2) collinearity analysis among parameters. For practical identifiability and model identification, continuous glucose monitor (CGM), insulin pump, and meal records were selected from a set of patients (n = 5) on continuous subcutaneous insulin infusion (CSII) that underwent a clinical trial in an outpatient setting. The performance of the identified models was analyzed by means of the root mean square (RMS) criterion.

RESULTS

A reliable set of identifiable parameters was found for every studied model after analyzing the possible identifiability issues of the original parameter sets. According to an importance factor ([Formula: see text]), it was shown that insulin sensitivity is not the most influential parameter from the dynamical point of view, that is, is not the parameter impacting the outputs the most of the three models, contrary to what is assumed in the literature. For the test data, the models demonstrated similar performance with most RMS values around 20 mg/dl (min: 15.64 mg/dl, max: 51.32 mg/dl). However, MMC failed to identify the model for patient 4. Also, considering the three models, the MMC model showed the higher parameter variability when reidentified every 6 hours.

CONCLUSION

This study shows that both structural and practical identifiability analysis need to be considered prior to the model identification/individualization in patients with T1D. It was shown that all the studied models are able to represent the CGM data, yet their usefulness in a hypothetical artificial pancreas could be a matter of debate. In spite that the three models do not capture all the dynamics and metabolic effects as a maximal model (ie, our FDA-accepted UVa/Padova simulator), SOGMM and ICING appear to be more appealing than MMC regarding both the performance and parameter variability after reidentification. Although the model predictions of ICING are comparable to the ones of the SOGMM model, the large parameter set makes the model prone to overfitting if all parameters are identified. Moreover, the existence of a high nonlinear function like [Formula: see text] prevents the use of tools from the linear systems theory.

Biochemical methane potential (BMP) tests: Reducing test time by early parameter estimation

Biochemical methane potential (BMP) test is a key analytical technique to assess the implementation and optimisation of anaerobic biotechnologies. However, this technique is characterised by long testing times (from 20 to >100days), which is not suitable for waste utilities, consulting companies or plants operators whose decision-making processes cannot be held for such a long time. This study develops a statistically robust mathematical strategy using sensitivity functions for early prediction of BMP first-order model parameters, i.e. methane yield (B₀) and kinetic constant rate (k). The minimum testing time for early parameter estimation showed a potential correlation with the k value, where (i) slowly biodegradable substrates (k≤0.1d^-1) have a minimum testing times of ≥15days, (ii) moderately biodegradable substrates (0.1<k<0.2d^-1) have a minimum testing times between 8 and 15 days, and (iii) rapidly biodegradable substrates (k≥0.2d^-1) have testing times lower than 7days.