## Abstract

**Background**: Models of dynamical systems described by ordinary differential equations often contains a number of parameters to be determined by time-series measurements and non-linear regression. A necessary condition for successful non-linear regression is that parameters are identifiable. One usually distinguish between practical identifiability and structural identifiability. The former concerns to what extent the system is sufficiently perturbed and if uncertainty (noise) in measurements makes estimating parameters a feasible task while the latter is a property of the parametrized equations of the dynamical systems model. A model is structurally identifiable if its parameters can be uniquely determined under the assumption of perfect knowledge of the set of measured variables. Standard models in pharmacokinetics and pharmacodynamics are usually structurally identifiable since years of use and development have decontaminated the area of nonidentifiable models. However, with the advent of quantitative systems pharmacology and efforts to increase the level of mechanistic detail in PK/PD models, novel and unexplored parametrized systems of ordinary differential equations appears. Hence, there is need for easy-to-use and efficient methods to decide structural identifiability of a proposed parametrized models prior to the application of numerical fitting and non-linear regression procedures.

**Aim**: To extend an existing highly efficient probabilistic method for structural identifiability analysis to include parametrized initial conditions and to provide an easy-to-use implementation in Mathematica to facilitate the use of identifiability analysis in systems pharmacology.

**Methods**: We describe the implementation of a recent probabilistic semi-numerical method for testing local structural identifiability based on computing the rank of a numerically instantiated Jacobian matrix (observability/identifiability matrix). To obtain this, parameters and initial conditions are specialized to random integer numbers, inputs are specialized to truncated random integer coefficient power series, and the corresponding output of a state space system is computed in terms of a truncated power series, which then is utilized to indirectly calculate the elements of the Jacobian matrix. To reduce the memory requirements and increase the speed of the computations
all operations are done modulo a large prime number.

**Results**: A target mediated drug disposition model, a dose-response-time model, and four signaling
pathway models (Ras, JAK-STAT, MAP Kinase Cascade, and NF-kB) have been analyzed with respect to structural identifiability.

**Conclusion**: Structural identifiability analysis can be carried out on a standard desktop computer on dynamical system models in the order of hundred parameters and equally many state variables.

## Authors and Affiliations

- Johan Karlsson, Fraunhofer-Chalmers Centre
- Milena Anguelova, SAAB AB
- Mats Jirstrand, Fraunhofer-Chalmers Centre