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Plot quantile summary of model simulations from global sensitivity analysis (requires Statistics and Machine Learning Toolbox)

plots quantiles and model responses of simulated samples and returns the figure handle
`h`

= plotData(`resultsObj`

)`h`

.

uses additional options specified by one or more name-value pair arguments.`h`

= plotData(`resultsObj`

,`Name,Value`

)

Load the Tumor Growth Model.

`sbioloadproject tumor_growth_vpop_sa.sbproj`

Get a variant with the estimated parameters and the dose to apply to the model.

```
v = getvariant(m1);
d = getdose(m1,'interval_dose');
```

Get the active configset and set the tumor weight as the response.

```
cs = getconfigset(m1);
cs.RuntimeOptions.StatesToLog = 'tumor_weight';
```

Simulate the model and plot the tumor growth profile.

sbioplot(sbiosimulate(m1,cs,v,d));

Perform global sensitivity analysis (GSA) on the model to find the model parameters that the tumor growth is sensitive to.

First, retrieve model parameters of interest that are involved in the pharmacodynamics of the tumor growth. Define the model response as the tumor weight.

modelParamNames = {'L0','L1','w0','k1','k2'}; outputName = 'tumor_weight';

Then perform GSA by computing the first- and total-order Sobol indices using `sbiosobol`

. Set `'ShowWaitBar'`

to `true`

to show the simulation progress. By default, the function uses 1000 parameter samples to compute the Sobol indices [1].

rng('default'); sobolResults = sbiosobol(m1,modelParamNames,outputName,'Variants',v,'Doses',d,'ShowWaitBar',true)

sobolResults = Sobol with properties: Time: [444x1 double] SobolIndices: [5x1 struct] Variance: [444x1 table] ParameterSamples: [1000x5 table] Observables: {'[Tumor Growth Model].tumor_weight'} SimulationInfo: [1x1 struct]

You can change the number of samples by specifying the `'NumberSamples'`

name-value pair argument. The function requires a total of `(number of input parameters + 2) * NumberSamples`

model simulations.

Show the mean model response, the simulation results, and a shaded region covering 90% of the simulation results.

plotData(sobolResults);

You can adjust the quantile region to a different percentage by specifying `'Alphas' `

for the lower and upper quantiles of all model responses. For instance, an alpha value of 0.1 plots a shaded region between the `100 * alpha`

and `100 * (1 - alpha)`

quantiles of all simulated model responses.

`plotData(sobolResults,'Alphas',0.1);`

Plot the time course of the first- and total-order Sobol indices.

```
h = plot(sobolResults);
% Resize the figure.
h.Position(:) = [100 100 1280 800];
```

The first-order Sobol index of an input parameter gives the fraction of the overall response variance that can be attributed to variations in the input parameter alone. The total-order index gives the fraction of the overall response variance that can be attributed to any joint parameter variations that include variations of the input parameter.

From the Sobol indices plots, parameters `L1`

and `w0`

seem to be the most sensitive parameters to the tumor weight before the dose was applied at t = 7. But after the dose is applied, `k1`

and `k2`

become more sensitive parameters and contribute most to the after-dosing stage of the tumor weight. The total variance plot also shows a larger variance for the after-dose stage at t > 35 than for the before-dose stage of the tumor growth, indicating that `k1`

and `k2`

might be more important parameters to investigate further. The fraction of unexplained variance shows some variance at around t = 33, but the total variance plot shows little variance at t = 33, meaning the unexplained variance could be insignificant. The fraction of unexplained variance is calculated as 1 - (sum of all the first-order Sobol indices), and the total variance is calculated using `var(response)`

, where `response`

is the model response at every time point.

You can also display the magnitudes of the sensitivities in a bar plot. Darker colors mean that those values occur more often over the whole time course.

bar(sobolResults);

You can specify more samples to increase the accuracy of the Sobol indices, but the simulation can take longer to finish. Use `addsamples`

to add more samples. For example, if you specify 1500 samples, the function performs `1500 * (2 + number of input parameters)`

simulations.

gsaMoreSamples = addsamples(gsaResults,1500)

The SimulationInfo property of the result object contains various information for computing the Sobol indices. For instance, the model simulation data (SimData) for each simulation using a set of parameter samples is stored in the `SimData`

field of the property. This field is an array of `SimData`

objects.

sobolResults.SimulationInfo.SimData

SimBiology SimData Array : 1000-by-7 Index: Name: ModelName: DataCount: 1 - Tumor Growth Model 1 2 - Tumor Growth Model 1 3 - Tumor Growth Model 1 ... 7000 - Tumor Growth Model 1

You can find out if any model simulation failed during the computation by checking the `ValidSample`

field of `SimulationInfo`

. In this example, the field shows no failed simulation runs.

all(sobolResults.SimulationInfo.ValidSample)

`ans = `*1x7 logical array*
1 1 1 1 1 1 1

`SimulationInfo.ValidSample`

is a table of logical values. It has the same size as `SimulationInfo.SimData`

. If `ValidSample`

indicates that any simulations failed, you can get more information about those simulation runs and the samples used for those runs by extracting information from the corresponding column of `SimulationInfo.SimDat`

a. Suppose that the fourth column contains one or more failed simulation runs. Get the simulation data and sample values used for that simulation using `getSimulationResults`

.

[samplesUsed,sd,validruns] = getSimulationResults(sobolResults,4);

You can add custom expressions as observables and compute Sobol indices for the added observables. For example, you can compute the Sobol indices for the maximum tumor weight by defining a custom expression as follows.

% Suppress an information warning that is issued during simulation. warnSettings = warning('off', 'SimBiology:sbservices:SB_DIMANALYSISNOTDONE_MATLABFCN_UCON'); % Add the observable expression. sobolObs = addobservable(sobolResults,'Maximum tumor_weight','max(tumor_weight)','Units','gram');

Plot the computed simulation results showing the 90% quantile region.

h2 = plotData(sobolObs); h2.Position(:) = [100 100 1280 800];

You can also remove the observable by specifying its name.

`gsaNoObs = removeobservable(sobolObs,'Maximum tumor_weight');`

Restore the warning settings.

warning(warnSettings);

Load the Target-Mediated Drug Disposition (TMDD) Model.

`sbioloadproject tmdd_with_TO.sbproj`

Get the active configset and set the target occupancy (`TO`

) as the response.

```
cs = getconfigset(m1);
cs.RuntimeOptions.StatesToLog = 'TO';
```

Simulate the model and plot the `TO`

profile.

sbioplot(sbiosimulate(m1,cs));

Define an exposure (area under the curve of the TO profile) threshold for the target occupancy.

`classifier = 'trapz(time,TO) <= 0.1';`

Perform MPGSA to find sensitive parameters with respect to the TO. Vary the parameter values between predefined bounds to generate 10,000 parameter samples.

% Suppress an information warning that is issued during simulation. warnSettings = warning('off', 'SimBiology:sbservices:SB_DIMANALYSISNOTDONE_MATLABFCN_UCON'); rng(0,'twister'); % For reproducibility params = {'kel','ksyn','kdeg','km'}; bounds = [0.1, 1; 0.1, 1; 0.1, 1; 0.1, 1]; mpgsaResults = sbiompgsa(m1,params,classifier,'Bounds',bounds,'NumberSamples',10000)

mpgsaResults = MPGSA with properties: Classifiers: {'trapz(time,TO) <= 0.1'} KolmogorovSmirnovStatistics: [4x1 table] ECDFData: {4x4 cell} SignificanceLevel: 0.0500 PValues: [4x1 table] SupportHypothesis: [10000x1 table] ParameterSamples: [10000x4 table] Observables: {'TO'} SimulationInfo: [1x1 struct]

Plot the quantiles of the simulated model response.

plotData(mpgsaResults);

Plot the empirical cumulative distribution functions (eCDFs) of the accepted and rejected samples. Except for `km`

, none of the parameters shows a significant difference in the eCDFs for the accepted and rejected samples. The `km`

plot shows a large Kolmogorov-Smirnov (K-S) distance between the eCDFs of the accepted and rejected samples. The K-S distance is the maximum absolute distance between two eCDFs curves.

```
h = plot(mpgsaResults);
% Resize the figure.
pos = h.Position(:);
h.Position(:) = [pos(1) pos(2) pos(3)*2 pos(4)*2];
```

To compute the K-S distance between the two eCDFs, SimBiology uses a two-sided test based on the null hypothesis that the two distributions of accepted and rejected samples are equal. See `kstest2`

(Statistics and Machine Learning Toolbox) for details. If the K-S distance is large, then the two distributions are different, meaning that the classification of the samples is sensitive to variations in the input parameter. On the other hand, if the K-S distance is small, then variations in the input parameter do not affect the classification of samples. The results suggest that the classification is insensitive to the input parameter. To assess the significance of the K-S statistic rejecting the null-hypothesis, you can examine the p-values.

bar(mpgsaResults)

The bar plot shows two bars for each parameter: one for the K-S distance (K-S statistic) and another for the corresponding p-value. You reject the null hypothesis if the p-value is less than the significance level. A cross (`x`

) is shown for any p-value that is almost 0. You can see the exact p-value corresponding to each parameter.

[mpgsaResults.ParameterSamples.Properties.VariableNames',mpgsaResults.PValues]

`ans=`*4×2 table*
Var1 trapz(time,TO) <= 0.1
________ _____________________
{'kel' } 0.0021877
{'ksyn'} 1
{'kdeg'} 0.99983
{'km' } 0

The p-values of `km`

and `kel`

are less than the significance level (0.05), supporting the alternative hypothesis that the accepted and rejected samples come from different distributions. In other words, the classification of the samples is sensitive to `km`

and `kel`

but not to other parameters (`kdeg`

and `ksyn`

).

You can also plot the histograms of accepted and rejected samples. The historgrams let you see trends in the accepted and rejected samples. In this example, the histogram of `km`

shows that there are more accepted samples for larger `km`

values, while the `kel`

histogram shows that there are fewer rejected samples as `kel`

increases.

```
h2 = histogram(mpgsaResults);
% Resize the figure.
pos = h2.Position(:);
h2.Position(:) = [pos(1) pos(2) pos(3)*2 pos(4)*2];
```

Restore the warning settings.

warning(warnSettings);

Load the Tumor Growth Model.

`sbioloadproject tumor_growth_vpop_sa.sbproj`

Get a variant with estimated parameters and the dose to apply to the model.

```
v = getvariant(m1);
d = getdose(m1,'interval_dose');
```

Get the active configset and set the tumor weight as the response.

```
cs = getconfigset(m1);
cs.RuntimeOptions.StatesToLog = 'tumor_weight';
```

Simulate the model and plot the tumor growth profile.

sbioplot(sbiosimulate(m1,cs,v,d));

Perform global sensitivity analysis (GSA) on the model to find the model parameters that the tumor growth is sensitive to.

First, define model parameters of interest, which are involved in the pharmacodynamics of the tumor growth. Define the model response as the tumor weight.

modelParamNames = {'L0','L1','w0','k1'}; outputName = 'tumor_weight';

Then perform GSA by computing the elementary effects using `sbioelementaryeffects`

. Use `100`

as the number of samples and set `ShowWaitBar`

to `true`

to show the simulation progress.

```
rng('default');
eeResults = sbioelementaryeffects(m1,modelParamNames,outputName,Variants=v,Doses=d,NumberSamples=100,ShowWaitbar=true);
```

Show the median model response, the simulation results, and a shaded region covering 90% of the simulation results.

plotData(eeResults,ShowMedian=true,ShowMean=false);

You can adjust the quantile region to a different percentage by specifying `Alphas `

for the lower and upper quantiles of all model responses. For instance, an `alpha`

value of 0.1 plots a shaded region between the `100*alpha`

and `100*(1-alpha)`

quantiles of all simulated model responses.

plotData(eeResults,Alphas=0.1,ShowMedian=true,ShowMean=false);

Plot the time course of the means and standard deviations of the elementary effects.

```
h = plot(eeResults);
% Resize the figure.
h.Position(:) = [100 100 1280 800];
```

The mean of effects explains whether variations in input parameter values have any effect on the tumor weight response. The standard deviation of effects explains whether the sensitivity change is dependent on the location in the parameter domain.

From the mean of effects plots, parameters `L1`

and `w0`

seem to be the most sensitive parameters to the tumor weight before the dose is applied at t = 7. But, after the dose is applied, `k1`

and `L0`

become more sensitive parameters and contribute most to the after-dosing stage of the tumor weight. The plots of standard deviation of effects show more deviations for the larger parameter values in the later stage (t > 35) than for the before-dose stage of the tumor growth.

You can also display the magnitudes of the sensitivities in a bar plot. Each color shading represents a histogram representing values at different times. Darker colors mean that those values occur more often over the whole time course.

bar(eeResults);

You can also plot the parameter grids and samples used to compute the elementary effects.

plotGrid(eeResults)

You can specify more samples to increase the accuracy of the elementary effects, but the simulation can take longer to finish. Use `addsamples`

to add more samples.

eeResults2 = addsamples(eeResults,200);

The `SimulationInfo`

property of the result object contains various information for computing the elementary effects. For instance, the model simulation data (SimData) for each simulation using a set of parameter samples is stored in the `SimData`

field of the property. This field is an array of `SimData`

objects.

eeResults2.SimulationInfo.SimData

SimBiology SimData Array : 1500-by-1 Index: Name: ModelName: DataCount: 1 - Tumor Growth Model 1 2 - Tumor Growth Model 1 3 - Tumor Growth Model 1 ... 1500 - Tumor Growth Model 1

You can find out if any model simulation failed during the computation by checking the `ValidSample`

field of `SimulationInfo`

. In this example, the field shows no failed simulation runs.

all(eeResults2.SimulationInfo.ValidSample)

`ans = `*logical*
1

You can add custom expressions as observables and compute the elementary effects of the added observables. For example, you can compute the effects for the maximum tumor weight by defining a custom expression as follows.

% Suppress an information warning that is issued. warnSettings = warning('off', 'SimBiology:sbservices:SB_DIMANALYSISNOTDONE_MATLABFCN_UCON'); % Add the observable expression. eeObs = addobservable(eeResults2,'Maximum tumor_weight','max(tumor_weight)','Units','gram');

Plot the computed simulation results showing the 90% quantile region.

h2 = plotData(eeObs); h2.Position(:) = [100 100 1500 800];

You can also remove the observable by specifying its name.

`eeNoObs = removeobservable(eeObs,'Maximum tumor_weight');`

Restore the warning settings.

warning(warnSettings);

`resultsObj`

— Global sensitivity analysis results`SimBiology.gsa.Sobol`

object | `SimBiology.gsa.MPGSA`

object | `SimBiology.gsa.ElementaryEffects`

Global sensitivity analysis results, specified as a `SimBiology.gsa.Sobol`

, `SimBiology.gsa.MPGSA`

, or `SimBiology.gsa.ElementaryEffects`

object.

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`h = plotData(results,'ResponseLeap',10)`

specifies to plot
every 10th model response. `Observables`

— Model responses or observables to plotcharacter vector | string | string vector | cell array of character vectors | vector of positive integers

Model responses or observables to plot, specified as a character vector, string, string
vector, cell array of character vectors, or vector of positive integers indexing into
`resultsObject.Observables`

. By default, the function plots GSA
results for all model responses or observables.

**Data Types: **`double`

| `char`

| `string`

| `cell`

`Alphas`

— Size of shaded region0.05 (default) | positive scalar between 0 and 1 | numeric vector

Size of the quantile region in the plot, specified as a positive scalar between 0
and 1 or numeric vector. The percentage of a region is calculated as ```
100 * (1
- 2 * Alpha)
```

. Hence, the default alpha value of 0.05 corresponds to the
90% quantile region.

You can specify multiple values as a vector to plot multiple regions. For
instance, `'Alphas',[0.05 0.1]`

shows both 90% and 80%
regions.

**Data Types: **`double`

`FaceColor`

— Color of shaded regionsthree-element row vector

Color of the shaded regions, specified as a three-element row vector. By default,
the function uses the first MATLAB^{®} default color. To view the default color order, enter
`get(groot,'defaultAxesColorOrder')`

or see the ColorOrder property.

**Data Types: **`double`

`MedianColor`

— Color of median model responsethree-element row vector

Color of the median model response, specified as a three-element row vector. By
default, the function uses the second MATLAB default color. To view the default color order, enter
`get(groot,'defaultAxesColorOrder')`

or see the ColorOrder property.

**Data Types: **`double`

`ShowMedian`

— Flag to plot median model response`false`

(default) | `true`

Flag to plot the median model response, specified as `true`

or
`false`

.

**Data Types: **`logical`

`MeanColor`

— Color of mean model responsethree-element row vector

Color of the mean model response, specified as a three-element row vector. By
default, the function uses the second MATLAB default color. To view the default color order, enter
`get(groot,'defaultAxesColorOrder')`

or see the ColorOrder property.

**Data Types: **`double`

`ShowMean`

— Flag to plot mean model response`true`

(default) | `false`

Flag to plot the mean model response, specified as `true`

or
`false`

.

**Data Types: **`logical`

`ResponseColor`

— Color of model responses`[0.3 0.3 0.3]`

(default) | three-element row vectorColor of model responses or simulations, specified as a comma-separated pair
consisting of `'ResponseColor'`

and a three-element row vector. By
default, the function uses the color gray `[0.3 0.3 0.3]`

.

**Data Types: **`double`

`ResponseLeap`

— Every positive integer

Every *n*th response to plot, specified as the comma-separated
pair consisting of `'ResponseLeap'`

and a positive integer. By
default, the function plots 10% of all model responses.

**Data Types: **`double`

`h`

— Handlefigure handle

Handle to the figure, specified as a figure handle.

`plotData`

will show median response instead of mean response*Behavior change in future release*

`plotData`

currently shows the mean model response by default. In
a future release, it will show the median model response instead. Use the name-value
arguments `ShowMedian`

or `ShowMean`

to plot either
the mean or median response.

[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] Morris, Max D. “Factorial Sampling Plans for Preliminary
Computational Experiments.” *Technometrics* 33, no. 2 (May
1991): 161–74.

[4] Sohier, Henri, Jean-Loup Farges, and Helene Piet-Lahanier.
“Improvement of the Representativity of the Morris Method for Air-Launch-to-Orbit Separation.”
*IFAC Proceedings Volumes* 47, no. 3 (2014):
7954–59.

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