## Sensitivity Analysis in SimBiology

### Sensitivity Analysis

Sensitivity analysis lets you explore the effects of variations in model quantities (species, compartments, and parameters) on a model response. You can use the analysis to validate preexisting knowledge or assumption about influential model quantities on a model response or to find such quantities. You can use the information from sensitivity analysis for decision making, designing experiments, and parameter estimation. SimBiology® supports two types of sensitivity analyses: local sensitivity analysis and global sensitivity analysis.

Global sensitivity analysis uses Monte Carlo simulations, where a representative (global) set of parameter sample values are used to explore the effects of variations in model parameters of interest on the model response. GSA provides insights into relative contributions of individual parameters that contribute most to the overall model behavior.

On the other hand, local sensitivity analysis is derivative based. This technique analyzes the effect of one model parameter at a time, keeping the other parameters fixed. Local sensitivities are dependent on a specific choice of parameter values at a time point where the analysis is performed and do not capture how parameters interact with each other during simulation when they are varied jointly.

### Global Sensitivity Analysis (GSA)

In GSA, model quantities are varied together to simultaneously evaluate the relative contributions of each quantity with respect to a model response. SimBiology provides two ways to perform GSA.

#### Sobol Indices

In this approach, SimBiology performs a decomposition of the model output (response) variance by calculating the first- and total-order Sobol indices [1]. The first-order Sobol indices give the fractions of the overall response variance that can be attributed to variations in an input parameter alone. The total-order Sobol index gives the fraction of the overall response variance that can be attributed to joint parameter variations. For details, see Saltelli Method to Compute Sobol Indices.

Use `sbiosobol` to compute the Sobol indices. The function requires Statistics and Machine Learning Toolbox™.

#### Multiparametric GSA (MPGSA)

MPGSA lets you study the relative importance of parameters with respect to a classifier defined by model responses. SimBiology implements the MPSA method proposed by Tiemann et. al. [2]. For details, see Multiparametric Global Sensitivity Analysis (MPGSA).

Use `sbiompgsa` to perform MPGSA. The function requires Statistics and Machine Learning Toolbox.

### Local Sensitivity Analysis (LSA)

In this analysis, SimBiology calculates the time-dependent sensitivities of all the species states with respect to species initial conditions and parameter values in the model.

Thus, if a model has a species `x`, and two parameters `y` and `z`, the time-dependent sensitivities of `x` with respect to each parameter value are the time-dependent derivatives

`$\frac{\partial x}{\partial y},\frac{\partial x}{\partial z}$`

where, the numerator is the sensitivity output and the denominators are the sensitivity inputs to sensitivity analysis. For more information on the calculations performed, see [3][4][5].

#### Model Requirements for LSA

LSA is supported only by the ordinary differential equation (ODE) solvers. SimBiology calculates local sensitivities by combining the original ODE system for a model with the auxiliary differential equations for the sensitivities. The additional equations are derivatives of the original equations with respect to parameters. This method is sometimes called forward sensitivity analysis or direct sensitivity analysis. This larger system of ODEs is solved simultaneously by the solver.

SimBiology sensitivity analysis calculates derivatives by using a technique called complex-step approximation. This technique yields accurate results for the vast majority of typical reaction kinetics, which involve only simple mathematical operations and functions. However, this technique can produce inaccurate results when analyzing models that contain mathematical expressions that involve nonanalytic functions, such as `abs`. In this case, SimBiology either disables the sensitivity analysis or warns you that the computed sensitivities may be inaccurate. If sensitivity analysis gives questionable results for a model with reaction rates that contain unusual functions, you may be running into limitations of the complex-step technique. Contact MathWorks Technical Support for additional information.

### Note

Models containing the following active components do not support sensitivity analysis:

• Nonconstant compartments

• Algebraic rules

• Events

### Note

You can perform sensitivity analysis on a model containing repeated assignment rules, but only if the repeated assignment rules do not determine species or parameters used as inputs or outputs in sensitivity analysis.

#### SUNDIALS as Default Solver

SimBiology always uses the SUNDIALS solver to perform sensitivity analysis on a model, regardless of what you have selected as the `SolverType` in the configuration set.

In addition, if you are estimating model parameters using `sbiofit` or the Fit Data program with one of these gradient-based estimation functions: `fmincon`, `fminunc`, `lsqnonlin`, or `lsqcurvefit`, SimBiology uses the SUNDIALS solver by default to calculate sensitivities and use them to improve fitting. If you are using `sbiofit`, you can turn off this sensitivity calculation feature by setting the 'SensitivityAnalysis' name-value pair argument to `false`. However, if you are using the Fit Data program, you cannot turn off this feature. It is recommended that you keep the sensitivity analysis feature on whenever possible for more accurate gradient approximations and better parameter fits.

#### Calculate Local Sensitivities Using sbiosimulate

Set the following properties of the `SolverOptions` property of your `configset` object, before running the `sbiosimulate` function:

• `SensitivityAnalysis` — Set to `true` to calculate the time-dependent sensitivities of all the species states defined by the `Outputs` property with respect to the initial conditions of the species and the values of the parameters specified in `Inputs`.

• `SensitivityAnalysisOptions` — An object that holds the sensitivity analysis options in the configuration set object. Properties of `SensitivityAnalysisOptions` are:

After setting `SolverOptions` properties, calculate the sensitivities of a model by providing the `model object` as an input argument to the `sbiosimulate` function.

The `sbiosimulate` function returns a `SimData object` containing the following simulation data:

• Time points, state data, state names, and sensitivity data

• Metadata such as the types and names for the logged states, the configuration set used during simulation, and the date of the simulation

A `SimData object` is a convenient way of keeping time data, state data, sensitivity data, and associated metadata together. A `SimData object` has properties and methods associated with it, which you can use to access and manipulate the data.

For illustrated examples, see:

#### Calculate Local Sensitivities Using SimFunctionSensitivity object

Create a `SimFunctionSensitivity object` using the `createSimFunction` specifying the `'SensitivityOutputs'` and `'SensitivityInputs'` name-value pair arguments. Then execute the object. For an illustrated example, see Calculate Sensitivities Using SimFunctionSensitivity Object.

#### Calculate Local Sensitivities Using SimBiology Model Analyzer App

For a workflow example using the app, see Find Important Parameters with Sensitivity Analysis Using SimBiology Model Analyzer App.

## References

[1] Saltelli, Andrea, Paola Annoni, Ivano Azzini, Francesca Campolongo, Marco Ratto, and Stefano Tarantola. “Variance Based Sensitivity Analysis of Model Output. Design and Estimator for the Total Sensitivity Index.” Computer Physics Communications 181, no. 2 (February 2010): 259–70. https://doi.org/10.1016/j.cpc.2009.09.018.

[2] Tiemann, Christian A., Joep Vanlier, Maaike H. Oosterveer, Albert K. Groen, Peter A. J. Hilbers, and Natal A. W. van Riel. “Parameter Trajectory Analysis to Identify Treatment Effects of Pharmacological Interventions.” Edited by Scott Markel. PLoS Computational Biology 9, no. 8 (August 1, 2013): e1003166. https://doi.org/10.1371/journal.pcbi.1003166.

[3] Martins, Joaquim, Ilan Kroo, and Juan Alonso. “An Automated Method for Sensitivity Analysis Using Complex Variables.” In 38th Aerospace Sciences Meeting and Exhibit. Reno,NV,U.S.A.: American Institute of Aeronautics and Astronautics, 2000. https://doi.org/10.2514/6.2000-689.

[4] Martins, J., Peter Sturdza, and Juan Alonso. “The Connection between the Complex-Step Derivative Approximation and Algorithmic Differentiation.” In 39th Aerospace Sciences Meeting and Exhibit. Reno,NV,U.S.A.: American Institute of Aeronautics and Astronautics, 2001. https://doi.org/10.2514/6.2001-921.

[5] Ingalls, Brian P., and Herbert M. Sauro. “Sensitivity Analysis of Stoichiometric Networks: An Extension of Metabolic Control Analysis to Non-Steady State Trajectories.” Journal of Theoretical Biology 222, no. 1 (May 2003): 23–36. https://doi.org/10.1016/S0022-5193(03)00011-0.

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