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

A confidence building exercise in data and identifiability: Modeling cancer chemotherapy as a case study

Mathematical modeling has a long history in the field of cancer therapeutics, and there is increasing recognition that it can help uncover the mechanisms that underlie tumor response to treatment. However, making quantitative predictions with such models often requires parameter estimation from data, raising questions of parameter identifiability and estimability. Even in the case of structural (theoretical) identifiability, imperfect data and the resulting practical unidentifiability of model parameters can make it difficult to infer the desired information, and in some cases, to yield biologically correct inferences and predictions. Here, we examine parameter identifiability and estimability using a case study of two compartmental, ordinary differential equation models of cancer treatment with drugs that are cell cycle-specific (taxol) as well as non-specific (oxaliplatin). We proceed through model building, structural identifiability analysis, parameter estimation, practical identifiability analysis and its biological implications, as well as alternative data collection protocols and experimental designs that render the model identifiable. We use the differential algebra/input-output relationship approach for structural identifiability, and primarily the profile likelihood approach for practical identifiability. Despite the models being structurally identifiable, we show that without consideration of practical identifiability, incorrect cell cycle distributions can be inferred, that would result in suboptimal therapeutic choices. We illustrate the usefulness of estimating practically identifiable combinations (in addition to the more typically considered structurally identifiable combinations) in generating biologically meaningful insights. We also use simulated data to evaluate how the practical identifiability of the model would change under alternative experimental designs. These results highlight the importance of understanding the underlying mechanisms rather than purely using parsimony or information criteria/goodness-of-fit to decide model selection questions. The overall roadmap for identifiability testing laid out here can be used to help provide mechanistic insight into complex biological phenomena, reduce experimental costs, and optimize model-driven experimentation.

Self-optimisation and model-based design of experiments for developing a C-H activation flow process

A recently described C(sp3)-H activation reaction to synthesise aziridines was used as a model reaction to demonstrate the methodology of developing a process model using model-based design of experiments (MBDoE) and self-optimisation approaches in flow. The two approaches are compared in terms of experimental efficiency. The self-optimisation approach required the least number of experiments to reach the specified objectives of cost and product yield, whereas the MBDoE approach enabled a rapid generation of a process model.

Structured model and parameter estimation in plant cell cultures of Thevetia peruviana

In this work, a mechanistic model for predicting the dynamic behavior of extracellular and intracellular nutrients, biomass production, and the main metabolites involved in the central carbon metabolism in plant cell cultures of Thevetia peruviana is presented. The proposed model is the first mechanistic model implemented for plant cell cultures of this species, and includes 28 metabolites, 33 metabolic reactions, and 61 parameters. Given the over-parametrization of the model, its nonlinear nature and the strong correlation among the effects of the parameters, a parameter estimation routine based on identifiability analysis was implemented. This routine reduces the parameter's search space by selecting the most sensitive and linearly independent parameters. Results have shown that only 19 parameters are identifiable. Finally, the model was used for analyzing the fluxes distribution in plant cell cultures of T. peruviana. This analysis shows high uptake of phosphates and parallel uptake of glucose and fructose. Furthermore, it has pointed out the main central carbon metabolism routes for promoting biomass production in this cell culture.

Estimability Analysis and Optimal Design in Dynamic Multi-scale Models of Cardiac Electrophysiology

We present an applied approach to optimal experimental design and estimability analysis for mechanistic models of cardiac electrophysiology, by extending and improving on existing computational and graphical methods. These models are 'multi-scale' in the sense that the modeled phenomena occur over multiple spatio-temporal scales (e.g., single cell vs. whole heart). As a consequence, empirical observations of multi-scale phenomena often require multiple distinct experimental procedures. We discuss the use of conventional optimal design criteria (e.g., D-optimality) in combining experimental observations across multiple scales and multiple experimental modalities. In addition, we present an improved 'sensitivity plot' - a graphical assessment of parameter estimability - that overcomes a well-known limitation in this context. These techniques are demonstrated using a working Hodgkin-Huxley cell model and three simulated experimental procedures: single cell stimulation, action potential propagation, and voltage clamp. In light of these assessments, we discuss two model modifications that improve parameter estimability, and show that the choice of optimality criterion has a profound effect on the contribution of each experiment.

A simple method for identifying parameter correlations in partially observed linear dynamic models

Background

Parameter estimation represents one of the most significant challenges in systems biology. This is because biological models commonly contain a large number of parameters among which there may be functional interrelationships, thus leading to the problem of non-identifiability. Although identifiability analysis has been extensively studied by analytical as well as numerical approaches, systematic methods for remedying practically non-identifiable models have rarely been investigated.

Results

We propose a simple method for identifying pairwise correlations and higher order interrelationships of parameters in partially observed linear dynamic models. This is made by derivation of the output sensitivity matrix and analysis of the linear dependencies of its columns. Consequently, analytical relations between the identifiability of the model parameters and the initial conditions as well as the input functions can be achieved. In the case of structural non-identifiability, identifiable combinations can be obtained by solving the resulting homogenous linear equations. In the case of practical non-identifiability, experiment conditions (i.e. initial condition and constant control signals) can be provided which are necessary for remedying the non-identifiability and unique parameter estimation. It is noted that the approach does not consider noisy data. In this way, the practical non-identifiability issue, which is popular for linear biological models, can be remedied. Several linear compartment models including an insulin receptor dynamics model are taken to illustrate the application of the proposed approach.

Conclusions

Both structural and practical identifiability of partially observed linear dynamic models can be clarified by the proposed method. The result of this method provides important information for experimental design to remedy the practical non-identifiability if applicable. The derivation of the method is straightforward and thus the algorithm can be easily implemented into a software packet.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0234-3) contains supplementary material, which is available to authorized users.

Identification of parameter correlations for parameter estimation in dynamic biological models

Background

One of the challenging tasks in systems biology is parameter estimation in nonlinear dynamic models. A biological model usually contains a large number of correlated parameters leading to non-identifiability problems. Although many approaches have been developed to address both structural and practical non-identifiability problems, very few studies have been made to systematically investigate parameter correlations.

Results

In this study we present an approach that is able to identify both pairwise parameter correlations and higher order interrelationships among parameters in nonlinear dynamic models. Correlations are interpreted as surfaces in the subspaces of correlated parameters. Based on the correlation information obtained in this way both structural and practical non-identifiability can be clarified. Moreover, it can be concluded from the correlation analysis that a minimum number of data sets with different inputs for experimental design are needed to relieve the parameter correlations, which corresponds to the maximum number of correlated parameters among the correlation groups.

Conclusions

The information of pairwise and higher order interrelationships among parameters in biological models gives a deeper insight into the cause of non-identifiability problems. The result of our correlation analysis provides a necessary condition for experimental design in order to acquire suitable measurement data for unique parameter estimation.