[Paper Review] Determining Structurally Identifiable Parameter Combinations Using Subset Profiling

Marisa C. Eisenberg, Michael A. L. Hayashi|arXiv (Cornell University)|Jul 8, 2013

Control Systems and Identification30 references3 citations

TL;DR

This paper proposes a numerical method that combines Fisher Information Matrix (FIM)-based subset selection with profile likelihood to identify structurally identifiable parameter combinations in nonlinear ODE models. By preconditioning the profile likelihood using rank-selected parameter subsets, the method efficiently reveals functional relationships between unidentifiable parameters, enabling reparameterization and improved identifiability in models from pharmacokinetics, cellular biology, and physiology.

ABSTRACT

Identifiability is a necessary condition for successful parameter estimation of dynamic system models. A major component of identifiability analysis is determining the identifiable parameter combinations, the functional forms for the dependencies between unidentifiable parameters. Identifiable combinations can help in model reparameterization and also in determining which parameters may be experimentally measured to recover model identifiability. Several numerical approaches to determining identifiability of differential equation models have been developed, however the question of determining identifiable combinations remains incompletely addressed. In this paper, we present a new approach which uses parameter subset selection methods based on the Fisher Information Matrix, together with the profile likelihood, to effectively estimate identifiable combinations. We demonstrate this approach on several example models in pharmacokinetics, cellular biology, and physiology.

Motivation & Objective

• Address the gap in numerical methods for determining identifiable parameter combinations in nonlinear ODE models, especially where analytical methods are computationally infeasible.
• Overcome limitations of existing profile likelihood and FIM approaches by integrating them to improve detection of functional dependencies between unidentifiable parameters.
• Provide a computationally tractable, general-purpose method applicable to a wide range of dynamic system models without requiring prior knowledge of parameter combinations.
• Enable model reparameterization and experimental design guidance by revealing identifiable functional forms of parameter dependencies.
• Extend the applicability of numerical identifiability analysis to complex models beyond simple or linear structures.

Proposed method

• Use the Fisher Information Matrix (FIM) to perform a rank search on parameter subsets, identifying those with high information content for subsequent profile likelihood analysis.
• Select parameter subsets that are nearly full rank to reduce excess degrees of freedom and improve the stability and interpretability of profile likelihood results.
• Apply the profile likelihood method to each selected subset to explore the functional relationship between unidentifiable parameters.
• Fit rational functions to pairwise profile plots to estimate explicit functional forms of identifiable combinations, such as ratios or linear combinations.
• Use the resulting functional forms to reparameterize models or compute FIM-based variances for the combinations, enhancing estimation accuracy.
• Leverage the parallelizability of subset selection and profile computation to improve scalability for larger models.

Experimental results

Research questions

• RQ1 Can a numerical method be developed to systematically identify structurally identifiable parameter combinations in nonlinear ODE models where analytical methods are intractable?
• RQ2 How can the Fisher Information Matrix be used to precondition profile likelihood analysis and improve the detection of functional parameter dependencies?
• RQ3 To what extent can this hybrid approach detect identifiable combinations involving more than two parameters, and how can the relationships be algorithmically reconstructed?
• RQ4 How does the method perform across diverse biological and physiological models with varying complexity and parameter count?
• RQ5 Can the identified combinations be used to reparameterize models to achieve identifiability or guide experimental measurements to recover unidentifiable parameters?

Key findings

• The method successfully identified three identifiable combinations in a three-gene repressilator model: $ K_1/\beta_1 $, $ K_2/\beta_2 $, and $ K_3/\beta_3 $, with functional forms $ K_i = \frac{c_i \beta_i}{d_i} $, where $ c_i $ and $ d_i $ are fitted constants.
• In the pharmacokinetic model, the approach revealed that the product $ k_{el} \cdot V $ was structurally identifiable, even though $ k_{el} $ and $ V $ individually were not.
• The FIM-based subset selection effectively reduced the number of profile likelihood calculations needed and improved convergence by avoiding ill-conditioned parameter pairs.
• The method was effective on models with up to 19 parameters and six ODEs, demonstrating feasibility on moderately complex systems with common hardware.
• The functional forms of identifiable combinations were consistently recoverable through rational function fitting of profile plots, enabling direct reparameterization.
• The approach is generalizable to ODEs, delay differential equations, and discrete models, provided the FIM and likelihood profiles are computable.