David Janzén
DMPK, Cardiovascular, Renal and Metabolic diseases, IMED Biotech Unit, AstraZeneca, Gothenburg, Sweden
Introduction: Mathematical modelling is widely applied as a tool for gaining insights into a biological system by means of formalizing the underlying biological mechanisms into a set of ordinary differential equations (ODE). These equations contains parameters of which some are known, whilst the remaining parameters are estimated using data from experiments. A subset of the estimated parameters are often subject to biological interpretations, e.g., potency or bioavailability of a drug.
Parameter identifiability, of which there are two types, is in this context important as it concerns how well-determined the parameters are. Practical identifiability is the study of how lack of information and noise levels in the measurements translates into the uncertainty in the parameters estimates. Structural identifiability however, is the study of whether there exist a unique, or otherwise, solution to the inverse problem under the condition of continuous noise-free data given a particular set of input and output functions to a model [1]. Structural identifiability is therefore a prerequisite for successful parameter estimation and subsequent biological interpretations. This is because structurally unidentifiable (SU) parameters can have any numerical value whilst the observed model output remains the same. Furthermore, while a model with a unique solution to the inverse problem is called a structurally globally identifiable model (SGI), a model with more than one solution, but a finite number of solutions, is called structurally locally identifiable (SLI). The one-compartment absorption model is an example of a SLI model, but is in a drug discovery context more commonly known as a flip-flop model.
Structural identifiability for models defined by ODE’s has extensively been studied and several methods have been developed, see for instance [4,5]. However, up until recently no theoretical framework or analytical methods for studying structural identifiability in population models has existed. In [2,3] methods for analyzing both linear and nonlinear models in a population setting are described. Results from applying these methods so far indicates that structural identifiability results in a deterministic setting does not necessarily translate to the population setting. In this work we have studied this by focusing particularly on the one-compartment absorption model in a population setting.
Objectives:
- Show analytically that identifiability of bioavailability is dependent on the postulated distribution model in the one-compartment absorption model
- Evaluate the error of the estimated bioavailability when postulating the wrong distribution model
- Study numerically the required numbers of samples and subjects necessary to estimate the bioavailability with a logit-normal distribution, while clearance, volume and absorption rate are log-normally distributed, when IV data is lacking
Methods: Recently published analytical methods [2,3] applicably to population models were here applied to the one-compartment absorption model. In summary, by generating the exhaustive summary [6] for the deterministic version of the population model and studying its distribution the identifiability of the population model can be determined analytically. For the numerical approach, data was generated with the one-compartment absorption model using known values of the population parameters and the distribution parameters. The parameters where then re-estimated using different numbers of samples and subjects with the correct and incorrect postulated parameter distributions.
Results: The analytical result show that if a lognormal distribution of bioavailability is used the model will be unidentifiable. However, if a logit-normal distribution is postulated for the bioavailability then the model is (at least) SLI. The numerical part of this study confirms that while the true bioavailability can indeed be estimated with a logit-normal distribution it is highly affected by the quality of the data. In addition, if a lognormal distribution is used but the true distribution is logit-normal, the error in the estimate of the bioavailability will depend on how similar the logit-normal distribution is to a lognormal distribution.
Conclusion: The presented results concludes that the population bioavailability, but not individual estimates, can be estimated despite lacking IV data if the distribution is postulated to have a logit-normal distribution.
References:
[1] Bellman R, Åström KJ., On structural identifiability. Math Biosci. 1970;7(3):329–339
[2] Janzén D et al., Extending existing structural identifiability analysis methods to mixed-effects models. Math Biosci. 2018:295:1-10
[3] Janzén D et al., Three novel approaches to structural identifiability analysis in mixed-effects models Comput. Methods Progr. Biomed 2016 In press
[4]O. Chis, J.R. Banga, E. Balsa-Canto., Structural identifiability of systems biology models: a critical comparison of methods PLoS ONE, 6 (2011), p. e27755
[5] D. Bearup et al., The input-output relationship approach to structural identifiability analysis. Comput. Methods Programs Biomed., 109 (2013), pp. 171-181
[6] Walters, E Identifiability of State Space Models. Springer-Verlag 1982
Reference: PAGE 27 (2018) Abstr 8589 [www.page-meeting.org/?abstract=8589]
Poster: Methodology - Model Evaluation