Calculating all multiple parameter solutions of ODE models to avoid biological misinterpretations

Biological system's dynamics are increasingly studied with nonlinear ordinary differential equations, whose parameters are estimated from input/output experimental data. Structural identifiability analysis addresses the theoretical question whether the inverse problem of recovering the unknown parameters from noise-free data is uniquely solvable (global), or if there is a finite (local), or an infinite number (non identifiable) of parameter values that generate identical input/output trajectories. In contrast, practical identifiability analysis aims to assess whether the experimental data provide information on the parameter estimates in terms of precision and accuracy. A main difference between the two identifiability approaches is that the former is mostly carried out analytically and provides exact results at a cost of increased computational complexity, while the latter is usually numerically tested by calculating statistical confidence regions and relies on decision thresholds. Here we focus on local identifiability, a critical issue in biological modeling. This is the case when a model has multiple parameter solutions which equivalently describe the input/output data, but predict different behaviours of the unmeasured variables, often those of major interest. We present theoretical background and applications to locally identifiable ODE models described by rational functions. We show how structural identifiability analysis completes the practical identifiability results. In particular we propose an algorithmic approach, implemented with our software DAISY, to calculate all numerical parameter solutions and to predict the corresponding behaviour of the unmeasured variables, which otherwise would remain hidden. A case study of a locally identifiable HIV model shows that one should be aware of the presence of multiple parameter solutions to comprehensively describe the biological system and avoid biological misinterpretation of the results.