Structural and practical identifiability analysis in bioengineering: a beginner's guide

Abstract

Advancements in digital technology have brought modelling to the forefront in many disciplines from healthcare to architecture. Mathematical models, often represented using parametrised sets of ordinary differential equations, can be used to characterise different processes. To infer possible estimates for the unknown parameters, these models are usually calibrated using associated experimental data. Structural and practical identifiability analyses are a key component that should be assessed prior to parameter estimation. This is because identifiability analyses can provide insights as to whether or not a parameter can take on single, multiple, or even infinitely or countably many values which will ultimately have an impact on the reliability of the parameter estimates. Also, identifiability analyses can help to determine whether the data collected are sufficient or of good enough quality to truly estimate the parameters or if more data or even reparameterization of the model is necessary to proceed with the parameter estimation process. Thus, such analyses also provide an important role in terms of model design (structural identifiability analysis) and the collection of experimental data (practical identifiability analysis). Despite the popularity of using data to estimate the values of unknown parameters, structural and practical identifiability analyses of these models are often overlooked. Possible reasons for non-consideration of application of such analyses may be lack of awareness, accessibility, and usability issues, especially for more complicated models and methods of analysis. The aim of this study is to introduce and perform both structural and practical identifiability analyses in an accessible and informative manner via application to well established and commonly accepted bioengineering models. This will help to improve awareness of the importance of this stage of the modelling process and provide bioengineering researchers with an understanding of how to utilise the insights gained from such analyses in future model development.

Fig. 1

Brief overview of the model development process for predictive models

Fig. 2

Simplistic overview of practical identifiability analysis using the profile likelihood process. NLL: negative log likelihood. A Example model with parameter estimate. B Example model fitting to observed data. C Chi- squared (${\chi }^2$) distribution (orange) for the associated model. D Model simulation based on different values for $a$. E Different negative loglikelihood values based on the different parameter estimates. F: profile likelihood of $a$. Note that some tools will use ${\chi }^2$ and negative loglikelihood interchangeably for the profile likelihood plots. Each of the steps are explained below

Fig. 3

Change in mRNA (A and C) and Protein (B and D) activity due to change in $k_1$ and $k_3$ values. Initial values for the simulation are as follows: $k_1$ and $k_3$ = 0.25, $k_2$ and $k_4$ = 0.5, $mRNA(0)$ = 2.5 and $Protein(0)$ = 6.5

Fig. 4

Predictions vs observations for glucose (A), lactate (B) and cell concentration (C) using the estimates summarised in Table 1. Dashed lines represent the model simulation whereas the dots represent the extracted experimental data

Fig. 5

Profile likelihood for model 2 case 1. The Red line depicts the 95% CI threshold which is at 77.259. The Blue line depicts the estimated values for each of the parameters of interest. The 95% CI for each parameter can be found in Table 1

Fig. 6

Predictions vs observations for cell concentration using the estimates summarised in Table 3. Dashed lines represent the model simulation whereas the dots represent the extracted experimental data

Fig. 7

Profile likelihood for model 3 case 1. The red line depicts the 95% CI threshold which is at 801.553 and the blue line depicts the estimated values for each of the parameters of interest. The confidence intervals for each parameter can be found in Table 3