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 the following features to perform GSA.
Sobol Indices
In this approach, SimBiology performs a decomposition of the model output (response) variance by calculating the first and totalorder Sobol indices [1]. The firstorder Sobol indices give the fractions of the overall response variance that can be attributed to variations in an input parameter alone. The totalorder 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.
Elementary Effects
sbioelementaryeffects
lets you assess the global sensitivity of
a model response with respect to variations in model parameters by computing the
means and standard deviations of the elementary effects of input parameters. An
elementary effect (EE) of an input parameter
P with respect to a model response R
is defined as: $$E{E}_{P}\left(x\right)=\frac{R\left(x\right)R\left(x+delta\right)}{delta}$$.
Here, EE_{P}(x) is the elementary
effect of P. R(x) and
R(x+delta) are model responses at specific time or the
value of an observable, evaluated for parameter values
x
and
x+delta
. For
details, see Elementary Effects for Global Sensitivity Analysis.
Comparison of GSA Functions
GSA Function  Sensitivity Measure  Considerations 

sbiosobol  It computes the fractions of total variance of a model response (sensitivity output) that can be attributed to individual model parameters (sensitivity inputs). 

sbiompgsa  It answers the question of whether variations in a model parameter (sensitivity input) have an influence on answering a modeling question. For example, the question might be: does a model parameter have an effect on the model response exceeding or falling below a target threshold? You can define such a question using
a mathematical expression (classifier). For example, the
following classifier defines an exposure (area under the
curve) threshold for the target occupancy


sbioelementaryeffects 
It computes the means and standard deviations of elementary effects of sensitivity inputs with respect to a model response. It assesses the average sensitivity by linear approximations of model responses, similar to averaged local sensitivities. It also assesses if the sensitivity of a model response is the same across the input parameter domain or if there is a spread of sensitivity values across the parameter domain. 

Local Sensitivity Analysis (LSA)
In this analysis, SimBiology calculates the timedependent 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 timedependent sensitivities
of x
with respect to each parameter value are the timedependent derivatives
$$\frac{\partial x}{\partial {y}_{0}},\frac{\partial x}{\partial {z}_{0}}$$
where, the numerator is the sensitivity output and the denominators are the sensitivity inputs to sensitivity analysis. y_{0} and z_{0} represent the initial conditions of y and z, respectively. 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.
LSA calculates derivatives by using a technique called complexstep
approximation. This technique yields accurate results for the vast majority of
typical reaction kinetics, which involve only simple mathematical operations and
functions. However, the complexstep approximation requires the functions to be
complex analytic, that is, to be infinitely differentiable in the complex plane.
This technique can produce inaccurate results when analyzing models that contain
mathematical expressions that involve nonanalytic functions, except
min
, max
, and
abs
. If the model contains nonanalytic functions,
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 complexstep technique.
Contact MathWorks Technical Support for additional information.
Although the min
, max
, and
abs
are nonanalytic functions and are not compatible
with the complexstep approximation, SimBiology provides replacements of these
functions so that they become compatible and can be used for LSA. Note that:
The replacement function
simbio.complexstep.abs(x)
is not differentiable when the real part of x is 0.The replacement functions
simbio.complexstep.min(x,y)
andsimbio.complexstep.max(x,y)
are not differentiable when x is equal to y.
For details, see [3].
If your model uses the abs
, min
,
and max
functions, SimBiology automatically replaces them
when:
You calculate local sensitivities using
sbiosimulate
, aSimFunctionSensitivity
object, or the Calculate Sensitivities program.sbiofit
orfitproblem
uses local sensitivity analysis to determine the gradients of the objective function during parameter estimation.
Note
Models containing the following active components do not support local 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 gradientbased 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 namevalue 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 with sbiosimulate
For illustrated examples, see:
More information on sensitivity analysis options and sensitivity outputs are described next.
The following properties of the SolverOptions
property of your
configset
object define the sensitivity analysis options.
You need to set these options before you run
sbiosimulate
.
SensitivityAnalysis — Set to
true
to calculate the timedependent sensitivities of all the species states defined by theOutputs
property with respect to the initial conditions of the species and the values of the parameters specified inInputs
.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 Local Sensitivity Analysis (LSA).
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 Local Sensitivity Analysis (LSA).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 using sbiosimulate
.
You can use one of the following syntaxes to get the sensitivity results.
Refer to sbiosimulate
for a complete
input argument list.
simData = sbiosimulate(_)
[time,x,names] = sbiosimulate(_)
.
The first syntax returns a scalar 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.
The second syntax returns three output arguments as follows:
time
is annby1
vector, wheren
is the number of steps taken by the ode solver andt
defines the time steps of the solver.x
is annbym
matrix, wheren
is the number of steps taken by the ode solver andm
is computed as: number of species and parameters specified inStatesToLog
+ number of sensitivity outputs * number of sensitivity inputs. A SimBiology state includes species and nonconstant parameters.names
is the list of states logged and the list of sensitivities of the species specified inStatesToLog
with respect to the input factors.
Calculate Local Sensitivities Using SimFunctionSensitivity
object
Create a SimFunctionSensitivity object
using the createSimFunction
specifying
the 'SensitivityOutputs'
and
'SensitivityInputs'
namevalue pair arguments. Then
execute the object. For an illustrated example, see Calculate Local Sensitivities Using SimFunctionSensitivity Object.
Calculate Local Sensitivities Using SimBiology Model Analyzer App
For a workflow example using the app, see Find Important Tumor Growth Parameters with Local Sensitivity Analysis Using SimBiology Model Analyzer.
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/10.2514/6.2000689.
[4] Martins, J., Peter Sturdza, and Juan Alonso. “The Connection between the ComplexStep 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/10.2514/6.2001921.
[5] Ingalls, Brian P., and Herbert M. Sauro. “Sensitivity Analysis of Stoichiometric Networks: An Extension of Metabolic Control Analysis to NonSteady State Trajectories.” Journal of Theoretical Biology 222, no. 1 (May 2003): 23–36. https://doi.org/10.1016/S00225193(03)000110.