Sensitivity analysis and practical identifiability of the mathematical model for partial differential equations

A sensitivity-based identifiability analysis of mathematical model for partial differential equations is carried out using an orthogonal method and an eigenvalue method. These methods are used to study the properties of the sensitivity matrix and the effects of changes in the model coefficients on the simulation results. Practical identifiability is investigated to determine whether the coefficients can be reconstructed with noisy experimental data. The analysis is performed using correlation matrix method with allowance for Gaussian noise in the measurements. The results of numerical calculations to obtain identifiable sets of parameters for the mathematical model arising in social networks are presented and discussed.

Practical identifiability of mathematical models of biomedical processes

The paper is devoted to a numerical study of the uniqueness and stability of problems of determining the parameters of dynamical systems arising in pharmacokinetics, immunology, epidemiology, sociology, etc. by incomplete measurements of certain states of the system at fixed time. Significance of parameters difficult to measure is very high in many areas, as their definition will allow physicians and doctors to make an effective treatment plan and to select the optimal set of medicines. Due to the fact that the problems under consideration are ill-posed, it is necessary to investigate the degree of ill-posedness before its numerical solution. One of the most effective ways is to study the practical identifiability of systems of nonlinear ordinary differential equations that will allow us to establish a set of identifiable parameters for further numerical solution of inverse problems. The paper presents methods for investigating practical identifiability: the Monte Carlo method, the matrix correlation method, the confidence intervals method and the sensitivity based method. There is presented two mathematical models of the pharmacokinetics of the C-peptide and mathematical model of the spread of the COVID — 19 epidemic. The identifiability investigation will allow us to construct a regularized unique solution of the inverse problem.

Practical identifiability in the frame of nonlinear mixed effects models: the example of the in vitro erythropoiesis

Background Nonlinear mixed effects models provide a way to mathematically describe experimental data involving a lot of inter-individual heterogeneity. In order to assess their practical identifiability and estimate confidence intervals for their parameters, most mixed effects modelling programs use the Fisher Information Matrix. However, in complex nonlinear models, this approach can mask practical unidentifiabilities. Results Herein we rather propose a multistart approach, and use it to simplify our model by reducing the number of its parameters, in order to make it identifiable. Our model describes several cell populations involved in the in vitro differentiation of chicken erythroid progenitors grown in the same environment. Inter-individual variability observed in cell population counts is explained by variations of the differentiation and proliferation rates between replicates of the experiment. Alternatively, we test a model with varying initial condition. Conclusions We conclude by relating experimental variability to precise and identifiable variations between the replicates of the experiment of some model parameters.

Practical Identifiability in the Frame of Nonlinear Mixed Effects Models: the Example of the in vitro Erythropoiesis

Nonlinear mixed effects models provide a way to mathematically describe experimental data involving a lot of inter-individual heterogeneity. In order to assess their practical identifiability and estimate confidence intervals for their parameters, most mixed effects modelling programs use the Fisher Information Matrix. However, in complex nonlinear models, this approach can mask practical unidentifiabilities. Herein we rather propose a multistart approach, and use it to simplify our model by reducing the number of its parameters, in order to make it identifiable. Our model describes several cell populations involved in the in vitro differentiation of chicken erythroid progenitors grown in the same environment. Inter-individual variability observed in cell population counts is explained by variations of the differentiation and proliferation rates between replicates of the experiment. Alternatively, we test a model with varying initial condition. We conclude by relating experimental variability to precise and identifiable variations between the replicates of the experiment of some model parameters.

Confidence intervals by constrained optimization—An algorithm and software package for practical identifiability analysis in systems biology

Practical identifiability of Systems Biology models has received a lot of attention in recent scientific research. It addresses the crucial question for models’ predictability: how accurately can the models’ parameters be recovered from available experimental data. The methods based on profile likelihood are among the most reliable methods of practical identification. However, these methods are often computationally demanding or lead to inaccurate estimations of parameters’ confidence intervals. Development of methods, which can accurately produce parameters’ confidence intervals in reasonable computational time, is of utmost importance for Systems Biology and QSP modeling. We propose an algorithm Confidence Intervals by Constraint Optimization (CICO) based on profile likelihood, designed to speed-up confidence intervals estimation and reduce computational cost. The numerical implementation of the algorithm includes settings to control the accuracy of confidence intervals estimates. The algorithm was tested on a number of Systems Biology models, including Taxol treatment model and STAT5 Dimerization model, discussed in the current article. The CICO algorithm is implemented in a software package freely available in Julia (https://github.com/insysbio/LikelihoodProfiler.jl) and Python (https://github.com/insysbio/LikelihoodProfiler.py).