Application of Structural Identifiability Analysis and Parametrisation of a Linear Mixed Effects Mirabegron Model
Mar Ribera Armengol (1), Brian J. Andrews (2), Michael J. Chappell (1)
(1) School of Engineering, University of Warwick, Coventry, CV4 7AL, United Kingdom (corresponding author e-mail: mar.ribera-i-armengol@warwick.ac.uk), (2) Nuffield Department of Surgical Sciences, University of Oxford, Oxford, OX3 9DU, UK.
Objectives: The application of structural identifiability analysis of a mathematical model is an essential step in its development to ensure that the parameter estimation process can be confidently and successfully performed whilst also supporting experiment design. A model is structurally globally identifiable if all of its unknown parameters can be uniquely determined for the given model observations/outputs. A locally identifiable model allows its parameters to take on countable numbers of values for the given system observations without affecting the goodness of fit. Finally, a model is structurally unidentifiable if it has an uncountable number of parameter values which can generate the same output data. Such an outcome would imply that model reparameterisation or additional system observations might need to be considered in order for an identifiable model to be generated. Hence, a structural identifiability analysis is of significant importance since it can confirm whether the unknown parameters in the studied model can be uniquely identified, or otherwise, from the given observations or not [1]. An investigation was performed applying structural identifiability analysis techniques including both fixed and random effects, to a 3-compartment population PK Mirabegron (MBG) model developed in this study. The drug examined, MBG, is employed to treat overactive bladder issues in spinal cord injury patients.
The first objective of this study was to develop a mathematical model that characterises the kinetics of orally administered MBG in the detrusor of the bladder. The second objective was to perform a structural identifiability analysis of the developed model to study whether the fixed and random effects incorporated in the model structures can be uniquely identified in order to proceed with the parameter estimation step with greater confidence.
Methods: The 3-compartment population PK model of MBG is formed of 5 unknown fixed effects and 4 unknown random effects. The Laplace Transform mixed-effects extension method [2] was chosen from other existing methods [2], [3] to use for the structural identifiability analysis for the model. This method was applied manually using the symbolic calculation software, MATHEMATICA. In addition, population parametrisation of the model was performed in Monolix and NONMEM in order to compare the results across both tools and to add robustness to the parameter estimation process.
Results: The structural identifiability analysis accounting for the mixed effects of a 3-compartment population PK model of MBG where the model observation was given by the concentration of MBG in plasma, revealed that the model is at least locally identifiable. This finding increased the confidence in the parameter estimation that followed. The population estimates of the absorption rate (ka1), MBG flow from plasma to detrusor rate (k23), MBG flow from detrusor back into plasma rate (k32), elimination rate (kel) and apparent volume of plasma compartment (Vi) obtained in Monolix were 0.38 h-1 (74% IIV), 0.1 h-1 (37% IIV), 0.046 h-1, 0.063 h-1 (18% IIV) and 1.5 L (43% IIV), respectively. The maximum RSE (%) value of the fixed effects was 20.9 for ka1, while the maximum RSE (%) value of the standard deviation of the random effects was 42.9 for the parameter kel. Therefore, the RSE revealed that all parameters were accurately estimated. Furthermore, the population parameter estimates obtained using Monolix were very similar to those estimated in NONMEM. The accuracy of the MBG population PK model in reproducing known plasma concentration profiles of MBG was confirmed by the support of a goodness-of-fit plot, VPC, and individual fit plots performed in Monolix.
Conclusions: Overall, the two main contributions of this study are, first, the development and parametrisation of a population PK MBG model with 3 compartments. The second contribution is the novel demonstration that both the fixed effects and random effects of the linear mixed effects model of MBG developed are at least structurally locally identifiable for the given model observation, offering greater confidence in the population parameters estimated. The results of this study have significant implications for the accurate and precise estimation of MBG pharmacokinetic population parameters, which could help improve the safety and efficacy of MBG treatment in spinal cord injury patients that suffer from overactive bladder.
References:
[1] M. P. Saccomani, “Structural vs Practical Identifiability in System Biology,” in International Work-Conference on Bioinformatics and Biomedical Engineering, {IWBBIO} 2013, Granada, Spain, March 18-20, 2013. Proceedings, 2013, pp. 305–313
[2] D. L. I. Janzén, M. Jirstrand, M. J. Chappell, and N. D. Evans, “Three novel approaches to structural identifiability analysis in mixed-effects models,” Comput Methods Programs Biomed, vol. 171, pp. 141–152, 2019.
[3] D. L. I. Janzén, M. Jirstrand, M. J. Chappell, and N. D. Evans, “Extending existing structural identifiability analysis methods to mixed-effects models.,” Math Biosci, vol. 295, pp. 1–10, Jan. 2018.