Calculating sensitivities lets you determine which species or parameter in a model is most sensitive to a specific condition (for example, a drug), defined by a species or parameter. Calculating sensitivities 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 References.

Sensitivity analysis is supported only by the ordinary differential equation (ODE) solvers. The software 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.

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

Nonconstant compartments

Algebraic rules

Events

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.

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.

You can calculate sensitivities using `sbiosimulate`

or the ```
SimFunctionSensitivity
object
```

.

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:`Outputs`

— Specify the species and parameters for which you want to compute the sensitivities. This is the numerator as described in About Calculating Sensitivities.`Inputs`

— Specify the species and parameters with respect to which you want to compute the sensitivities. Sensitivities are calculated with respect to the`InitialAmount`

property of the specified species. This is the denominator, described in About Calculating Sensitivities.`Normalization`

— Specify the normalization for the calculated sensitivities:`'None'`

— No normalization`'Half'`

— Normalization relative to the numerator (species output) only`'Full'`

— Full dedimensionalization

For more information about normalization, see

`Normalization`

.

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:

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.

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

Martins, J.R.R.A., Kroo, I.M., and Alanso, J.J. (Jan. 2000). An automated method
for sensitivity analysis using complex variables. AIAA Paper
*2000–0689*.

Martins, J.R.R.A., Sturdza, P., and Alanso, J.J. (Jan. 2001). The connection
between the complex-step derivative approximation and algorithmic differentiation.
AIAA Paper *2001–0921*.

Ingalls, B.P, and Sauro, H.M. (2003). Sensitivity analysis of stoichiometric
networks: an extension of metabolic control analysis to non-steady state
trajectories. J Theor Biol. *222(1)*, 23–36.