[Paper Review] Systems Biology: Identifiability analysis and parameter identification via systems-biology informed neural networks

Mitchell Daneker, Zhen Zhang+2 more|arXiv (Cornell University)|Feb 3, 2022

Gene Regulatory Network Analysis4 citations

TL;DR

This paper introduces systems-biology informed neural networks (SBINN) for parameter estimation in ordinary differential equation (ODE)-based biological models, integrating structural and practical identifiability analysis to ensure reliable inference. It demonstrates the framework on the ultradian glucose-insulin model, showing that SBINN enables accurate parameter recovery and FIM-based practical identifiability analysis reveals that parameter $ C_2 $ is practically unidentifiable due to near-zero sensitivity in the Fisher information matrix.

ABSTRACT

The dynamics of systems biological processes are usually modeled by a system of ordinary differential equations (ODEs) with many unknown parameters that need to be inferred from noisy and sparse measurements. Here, we introduce systems-biology informed neural networks for parameter estimation by incorporating the system of ODEs into the neural networks. To complete the workflow of system identification, we also describe structural and practical identifiability analysis to analyze the identifiability of parameters. We use the ultridian endocrine model for glucose-insulin interaction as the example to demonstrate all these methods and their implementation.

Motivation & Objective

• To develop a unified workflow for systems biology model identification that integrates structural and practical identifiability analysis with parameter estimation.
• To address the challenge of estimating unknown parameters in ODE models from noisy and sparse experimental data in systems biology.
• To demonstrate the effectiveness of systems-biology informed neural networks (SBINN) in estimating parameters for complex biological systems with high accuracy.
• To identify and diagnose unidentifiable parameters using Fisher information matrix (FIM)-based practical identifiability analysis, particularly in the context of real-world biological data limitations.
• To extend the SBINN framework to handle time-dependent parameters by incorporating learnable functions within the neural network architecture.

Proposed method

• Integrate the system of ODEs directly into a physics-informed neural network (PINN) framework, enforcing the ODE dynamics as a loss term during training to estimate unknown parameters.
• Apply structural identifiability analysis prior to parameter estimation to determine whether parameters are uniquely identifiable in the absence of noise.
• Use a neural network to approximate the state variables and their time derivatives, with the ODE residuals penalized via a loss function to enforce model consistency.
• Perform practical identifiability analysis a posteriori using the Fisher information matrix (FIM), computed via sensitivity analysis of the ODE system with respect to parameters.
• Compute the correlation matrix of parameters from the inverse FIM to detect high correlations indicating non-identifiability, and analyze null eigenvectors of the FIM to identify parameters with negligible influence on observables.
• Extend SBINN to time-dependent parameters by introducing additional output neurons that represent time-varying parameter functions, enabling dynamic parameter inference.

Experimental results

Research questions

• RQ1 Can systems-biology informed neural networks (SBINN) effectively estimate unknown parameters in ODE-based biological models from noisy and sparse data?
• RQ2 Which parameters in the ultradian glucose-insulin model are structurally identifiable, and what constraints are needed to ensure uniqueness of the solution?
• RQ3 How can practical identifiability be assessed in the presence of measurement noise, and which parameters are sensitive to data uncertainty?
• RQ4 What role does the Fisher information matrix (FIM) play in diagnosing parameter unidentifiability, and how do null eigenvectors reveal non-informative parameters?
• RQ5 Can SBINN be extended to infer time-varying parameters by modeling them as learnable functions within the neural network architecture?

Key findings

• SBINN successfully estimated all 21 parameters of the ultradian glucose-insulin model with high accuracy, as validated by comparison to nominal values in Table 1.
• Structural identifiability analysis confirmed that all parameters were structurally identifiable, enabling reliable parameter estimation under ideal conditions.
• Practical identifiability analysis using the FIM revealed that parameter $ C_2 $ was practically unidentifiable, as indicated by a near-zero eigenvalue in the FIM and a dominant component in the associated null eigenvector.
• The correlation matrix showed high correlations between certain parameters, suggesting potential identifiability issues, though this was less conclusive than the null eigenvector analysis.
• The null eigenvector analysis identified $ C_2 $ as having negligible influence on the observable state variable $ G $, confirming it as practically unidentifiable from the dataset.
• The framework successfully demonstrated the integration of FIM-based practical identifiability into the SBINN pipeline, enabling early detection of unreliable parameter estimates before model deployment.