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
z, the time-dependent sensitivities
x with respect to each parameter value are the time-dependent derivatives
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:
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 task with one of these gradient-based estimation functions:
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 task, 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.
Set the following properties of the
SolverOptions property of your
configset object, before running the
— 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
SensitivityAnalysisOptions — An object that holds
the sensitivity analysis options in the configuration set object.
Specify the species and parameters with respect to which you
want to compute the sensitivities. Sensitivities are calculated
with respect to the
property of the specified species. This is the denominator,
described in About Calculating Sensitivities.
— Specify the normalization for the calculated
'None' — No
'Half' — Normalization
relative to the numerator (species output) only
'Full' — Full
For more information about normalization, see
SolverOptions properties, calculate the
sensitivities of a model by providing the
model object as an
input argument to 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
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:
'SensitivityInputs' name-value pair arguments. Then
execute the object. For an illustrated example, see Calculate Sensitivities Using SimFunctionSensitivity Object.
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.