Sensitivity Analysis for ODE Models

Introduction

  • Sensitivity analysis (SA) quantifies how uncertainty or variation in model parameters influences model outputs.
  • In ordinary differential equation (ODE) models, parameters represent biological processes (reaction rates, transport constants, synthesis/decay rates, volumes). Their values may be uncertain.
  • SA helps to
    • Evaluate robustness and stability
    • Identify key parameters
    • Support model reduction, parameter estimation, and experimental design
    • Prioritize data acquisition
  • Two major categories:
    • Local sensitivity analysis (LSA) – small perturbations around nominal parameters
    • Global sensitivity analysis (GSA) – variations across full parameter ranges

Key Concepts

  • General ODE model dx/dt = f(x(t), p, t), with parameters p and outputs y = g(x,p)
  • Parameter sensitivity measures ( y / p_i )
  • Sensitivity targets include:
    • State trajectories
    • Steady states
    • Peak times and amplitudes
    • Cost functions
  • Uncertainty types
    • Parameter uncertainty
    • Structural/model uncertainty
    • Initial-condition uncertainty

Local Sensitivity Analysis (LSA)

Concept

  • Studies the effect of small perturbations in parameters.
  • Useful when model is calibrated or when exploring local behavior.
  • Uses derivatives or numerical approximations.

Methods

Finite Difference Sensitivities

  • Perturb one parameter at a time: ( S_i / )
  • Simple and derivative-free.
  • Sensitive to numerical noise and step size.

Forward Sensitivity Equations (FSE)

  • Augment ODE system with sensitivity equations: ( dS_{x,p_i}/dt = (f/x) S_{x,p_i} + (f/p_i) )
  • Accurate and efficient for models with few parameters.
  • Disadvantage: number of additional ODEs = states × parameters.

Adjoint Sensitivity Analysis

  • Computes sensitivities of a scalar objective with many parameters.
  • Solves adjoint ODE backward in time.
  • Efficient for high-dimensional parameter spaces.
  • More complex to implement.

Interpretation

  • Scaled sensitivities allow comparison: ( S_i^{scaled} = (p_i / y)(y/p_i) )
  • Detect:
    • Stiff vs sloppy parameters
    • Parameters dominating specific observables
    • Reduction candidates

4. Global Sensitivity Analysis (GSA)

4.1 Concept

  • Investigates the effect of parameter variability across full parameter ranges.
  • Captures nonlinearities and parameter interactions.
  • Essential when parameters are highly uncertain.

4.2 Sampling Strategies

  • Monte Carlo (MC)
  • Latin Hypercube Sampling (LHS) – more efficient stratified sampling
  • Sobol or Halton sequences – low-discrepancy quasi-random samples
  • Parameter distributions based on prior knowledge or uncertainty bounds.

5. Variance-based Global Sensitivity Methods

5.1 Sobol Sensitivity Analysis

  • Decomposes output variance into contributions from each parameter.
  • Provides:
    • First-order index ( S_i ): effect of parameter alone
    • Total-order index ( S_{Ti} ): parameter including interactions
  • Strengths:
    • Model-agnostic
    • Detects nonlinear interactions
  • Weakness:
    • Computationally expensive for many parameters
  • Implementations:
    • SALib, UQLab, pyPESTO, Matlab UQ Toolbox

5.2 Extended FAST (Fourier Amplitude Sensitivity Test)

  • Uses spectral decomposition to estimate variance contributions.
  • Efficient for high-dimensional systems.
  • Interpretation for interactions less intuitive than Sobol.

6. Screening and Factor Prioritization

6.1 Morris Method (Elementary Effects)

  • Qualitative GSA for identifying influential parameters.
  • Computes multiple elementary effects via one-at-a-time perturbations.
  • Outputs:
    • μ: mean effect → overall importance
    • σ: standard deviation → interactions/nonlinearities
  • Advantages:
    • Very efficient
    • Suitable for high-dimensional models
  • Disadvantages:
    • Not a full quantitative sensitivity measure

6.2 One-Factor-at-a-Time (OAT)

  • Varies one parameter while holding others fixed.
  • Simple and intuitive.
  • Misses interactions → only preliminary use.

7. Derivative-Based Global Sensitivity Measures (DGSM)

  • Based on expected squared derivatives: ( _i = E[(y / p_i)^2] )
  • Efficient when local sensitivities (FSE or adjoint) are available.
  • Can provide bounds for total Sobol indices.
  • Useful when traditional variance-based GSA is too expensive.

8. Sensitivity of Steady States and Stability

8.1 Steady-State Sensitivity

  • Steady-state (x^*) satisfies ( f(x^*, p) = 0 ).
  • Sensitivity: ( x^*/p_i = -J^{-1} (f/p_i)|_{x^*} )
  • Key in metabolic and signaling networks.

8.2 Bifurcation Sensitivity

  • Quantifies parameter effects on bifurcation points (Hopf, saddle-node, pitchfork).
  • Important in oscillatory or switch-like systems.
  • Tools include AUTO and MATCONT.

9. Sensitivity in Parameter Estimation and Identifiability

9.1 Structural Identifiability

  • Determines if parameters can be uniquely inferred from noise-free data.
  • Related to rank of the sensitivity matrix.

9.2 Practical Identifiability

  • Focuses on uncertainty with real data.
  • Poor sensitivities → wide confidence intervals.
  • Methods:
    • Fisher Information Matrix (FIM)
    • Profile likelihood
    • Bayesian posteriors

10. Role in Experimental Design

  • SA helps guide optimal experiment design, for example:
    • Choosing informative sampling time points
    • Selecting perturbations that maximize parameter identifiability
    • Minimizing uncertainty
  • Metrics:
    • D-optimality – maximize determinant of FIM
    • Robust design to account for uncertain priors

11. Best Practices

  • Define specific goals and outputs before SA.
  • Combine local and global approaches.
  • Use normalized sensitivities for comparability.
  • Apply GSA when:
    • Parameters are uncertain
    • Dynamics are nonlinear
    • Interactions are expected
  • Check numerical accuracy of ODE solver.
  • Document all choices: sampling strategy, priors, solver tolerances, metrics.

12. Summary

  • Sensitivity analysis is central to understanding ODE models in systems biology.
  • Local methods (FSE, adjoint, finite differences) give derivative-based information near nominal parameters.
  • Global methods (Sobol, FAST, Morris) explore full parameter uncertainty and interactions.
  • Links to identifiability, parameter estimation, bifurcation analysis, and experimental design.
  • Combining multiple methods gives the most reliable insight.