sbiofit

fitResults = sbiofit(sm,grpData,responseMap,estiminfo) estimates parameters of a SimBiology model sm using nonlinear least-squares regression.

grpData is a groupedData object specifying the data to fit. responseMap defines the mapping between the model components and response data in grpData. estimatedInfo is an EstimatedInfo object that defines the estimated parameters in the model sm. fitResults is a OptimResults object or NLINResults object or a vector of these objects.

sbiofit uses the first available estimation function among the following: lsqnonlin (Optimization Toolbox required), nlinfit (Statistics and Machine Learning Toolbox required), or fminsearch.

By default, each group in grpData is fit separately, resulting in group-specific parameter estimates. If the model contains active doses and variants, they are applied before the simulation.

fitResults = sbiofit(sm,grpData,responseMap,estiminfo,dosing) uses the dosing information specified by a matrix of SimBiology dose objects dosing instead of using the active doses of the model sm if there is any.

fitResults = sbiofit(sm,grpData,responseMap,estiminfo,dosing,functionName) uses the estimation function specified by functionName. If the specified function is unavailable, a warning is issued and the first available default function is used.

fitResults = sbiofit(sm,grpData,responseMap,estiminfo,dosing,functionName,options) uses the additional options specified by options for the function functionName.

fitResults = sbiofit(sm,grpData,responseMap,estiminfo,dosing,functionName,options,variants) applies variant objects specified as variants instead of using any active variants of the model.

fitResults = sbiofit(_,Name,Value) uses additional options specified by one or more Name,Value pair arguments.

[fitResults,simdata] = sbiofit(_) also returns a vector of SimData objects simdata using any of the input arguments in the previous syntaxes.

Note

sbiofit unifies sbionlinfit and sbioparamestim estimation functions. Use sbiofit to perform nonlinear least-squares regression.
sbiofit simulates the model using a SimFunction object, which automatically accelerates simulations by default. Hence it is not necessary to run sbioaccelerate before you call sbiofit.

Examples

Fit a One-Compartment Model to an Individual's PK Profile

Background

This example shows how to fit an individual's PK profile data to one-compartment model and estimate pharmacokinetic parameters.

Suppose you have drug plasma concentration data from an individual and want to estimate the volume of the central compartment and the clearance. Assume the drug concentration versus the time profile follows the monoexponential decline $C_{t} = C_{0} e^{- k_{e} t}$ , where $C_{t}$ is the drug concentration at time t, $C_{0}$ is the initial concentration, and $k_{e}$ is the elimination rate constant that depends on the clearance and volume of the central compartment $k_{e} = C l / V$ .

The synthetic data in this example was generated using the following model and parameters:

One-compartment model with bolus dosing and first-order elimination
Volume of the central compartment (Central) = 1.70 liter
Clearance parameter (Cl_Central) = 0.55 liter/hour
Constant error model

Load Data and Visualize

The data is stored as a table with variables Time and Conc that represent the time course of the plasma concentration of an individual after an intravenous bolus administration measured at 13 different time points. The variable units for Time and Conc are hour and milligram/liter, respectively.

clear all
load(fullfile(matlabroot,'examples','simbio','data15.mat'))

plot(data.Time,data.Conc,'b+')
xlabel('Time (hour)');
ylabel('Drug Concentration (milligram/liter)');

Convert to groupedData Format

Convert the data set to a groupedData object, which is the required data format for the fitting function sbiofit for later use. A groupedData object also lets you set independent variable and group variable names (if they exist). Set the units of the Time and Conc variables. The units are optional and only required for the UnitConversion feature, which automatically converts matching physical quantities to one consistent unit system.

gData = groupedData(data);
gData.Properties.VariableUnits = {'hour','milligram/liter'};
gData.Properties

ans = struct with fields:
                Description: ''
                   UserData: []
             DimensionNames: {'Row'  'Variables'}
              VariableNames: {'Time'  'Conc'}
       VariableDescriptions: {}
              VariableUnits: {'hour'  'milligram/liter'}
         VariableContinuity: []
                   RowNames: {}
           CustomProperties: [1x1 matlab.tabular.CustomProperties]
          GroupVariableName: ''
    IndependentVariableName: 'Time'

groupedData automatically set the name of the IndependentVariableName property to the Time variable of the data.

Construct a One-Compartment Model

Use the built-in PK library to construct a one-compartment model with bolus dosing and first-order elimination where the elimination rate depends on the clearance and volume of the central compartment. Use the configset object to turn on unit conversion.

pkmd                    = PKModelDesign;
pkc1                    = addCompartment(pkmd,'Central');
pkc1.DosingType         = 'Bolus';
pkc1.EliminationType    = 'linear-clearance';
pkc1.HasResponseVariable = true;
model                   = construct(pkmd);
configset               = getconfigset(model);
configset.CompileOptions.UnitConversion = true;

For details on creating compartmental PK models using the SimBiology® built-in library, see Create Pharmacokinetic Models.

Define Dosing

Define a single bolus dose of 10 milligram given at time = 0. For details on setting up different dosing schedules, see Doses in SimBiology Models.

dose                = sbiodose('dose');
dose.TargetName     = 'Drug_Central';
dose.StartTime      = 0;
dose.Amount         = 10;
dose.AmountUnits    = 'milligram';
dose.TimeUnits      = 'hour';

Map Response Data to the Corresponding Model Component

The data contains drug concentration data stored in the Conc variable. This data corresponds to the Drug_Central species in the model. Therefore, map the data to Drug_Central as follows.

responseMap = {'Drug_Central = Conc'};

Specify Parameters to Estimate

The parameters to fit in this model are the volume of the central compartment (Central) and the clearance rate (Cl_Central). In this case, specify log-transformation for these biological parameters since they are constrained to be positive. The estimatedInfo object lets you specify parameter transforms, initial values, and parameter bounds if needed.

paramsToEstimate    = {'log(Central)','log(Cl_Central)'};
estimatedParams     = estimatedInfo(paramsToEstimate,'InitialValue',[1 1],'Bounds',[1 5;0.5 2]);

Estimate Parameters

Now that you have defined one-compartment model, data to fit, mapped response data, parameters to estimate, and dosing, use sbiofit to estimate parameters. The default estimation function that sbiofit uses will change depending on which toolboxes are available. To see which function was used during fitting, check the EstimationFunction property of the corresponding results object.

fitConst = sbiofit(model,gData,responseMap,estimatedParams,dose);

Display Estimated Parameters and Plot Results

Notice the parameter estimates were not far off from the true values (1.70 and 0.55) that were used to generate the data. You may also try different error models to see if they could further improve the parameter estimates.

fitConst.ParameterEstimates

ans=2×4 table
         Name         Estimate    StandardError      Bounds  
    ______________    ________    _____________    __________

    {'Central'   }     1.6993       0.034821         1      5
    {'Cl_Central'}    0.53358        0.01968       0.5      2

s.Labels.XLabel     = 'Time (hour)';
s.Labels.YLabel     = 'Concentration (milligram/liter)';
plot(fitConst,'AxesStyle',s);

Use Different Error Models

Try three other supported error models (proportional, combination of constant and proportional error models, and exponential).

fitProp = sbiofit(model,gData,responseMap,estimatedParams,dose,...
                      'ErrorModel','proportional');
fitExp  = sbiofit(model,gData,responseMap,estimatedParams,dose,...
                      'ErrorModel','exponential');
fitComb = sbiofit(model,gData,responseMap,estimatedParams,dose,...
                      'ErrorModel','combined');

Use Weights Instead of an Error Model

You can specify weights as a numeric matrix, where the number of columns corresponds to the number of responses. Setting all weights to 1 is equivalent to the constant error model.

weightsNumeric = ones(size(gData.Conc));
fitWeightsNumeric = sbiofit(model,gData,responseMap,estimatedParams,dose,'Weights',weightsNumeric);

Alternatively, you can use a function handle that accepts a vector of predicted response values and returns a vector of weights. In this example, use a function handle that is equivalent to the proportional error model.

weightsFunction = @(y) 1./y.^2;
fitWeightsFunction = sbiofit(model,gData,responseMap,estimatedParams,dose,'Weights',weightsFunction);

Compare Information Criteria for Model Selection

Compare the loglikelihood, AIC, and BIC values of each model to see which error model best fits the data. A larger likelihood value indicates the corresponding model fits the model better. For AIC and BIC, the smaller values are better.

allResults = [fitConst,fitWeightsNumeric,fitWeightsFunction,fitProp,fitExp,fitComb];
errorModelNames = {'constant error model','equal weights','proportional weights', ...
                   'proportional error model','exponential error model',...
                   'combined error model'};
LogLikelihood = [allResults.LogLikelihood]';
AIC = [allResults.AIC]';
BIC = [allResults.BIC]';
t = table(LogLikelihood,AIC,BIC);
t.Properties.RowNames = errorModelNames;
t

t=6×3 table
                                LogLikelihood      AIC        BIC  
                                _____________    _______    _______

    constant error model            3.9866       -3.9732    -2.8433
    equal weights                   3.9866       -3.9732    -2.8433
    proportional weights           -3.8472        11.694     12.824
    proportional error model       -3.8257        11.651     12.781
    exponential error model         1.1984        1.6032     2.7331
    combined error model            3.9163       -3.8326    -2.7027

Based on the information criteria, the constant error model (or equal weights) fits the data best since it has the largest loglikelihood value and the smallest AIC and BIC.

Display Estimated Parameter Values

Show the estimated parameter values of each model.

Estimated_Central       = zeros(6,1);
Estimated_Cl_Central    = zeros(6,1);
t2 = table(Estimated_Central,Estimated_Cl_Central);
t2.Properties.RowNames = errorModelNames;
for i = 1:height(t2)
    t2{i,1} = allResults(i).ParameterEstimates.Estimate(1);
    t2{i,2} = allResults(i).ParameterEstimates.Estimate(2);
end
t2

t2=6×2 table
                                Estimated_Central    Estimated_Cl_Central
                                _________________    ____________________

    constant error model             1.6993                0.53358       
    equal weights                    1.6993                0.53358       
    proportional weights             1.9045                0.51734       
    proportional error model         1.8777                0.51147       
    exponential error model          1.7872                0.51701       
    combined error model             1.7008                0.53271

Conclusion

This example showed how to estimate PK parameters, namely the volume of the central compartment and clearance parameter of an individual, by fitting the PK profile data to one-compartment model. You compared the information criteria of each model and estimated parameter values of different error models to see which model best explained the data. Final fitted results suggested both the constant and combined error models provided the closest estimates to the parameter values used to generate the data. However, the constant error model is a better model as indicated by the loglikelihood, AIC, and BIC information criteria.

Fit a Two-Compartment Model to PK Profiles of Multiple Individuals

Suppose you have drug plasma concentration data from three individuals that you want to use to estimate corresponding pharmacokinetic parameters, namely the volume of central and peripheral compartment (Central, Peripheral), the clearance rate (Cl_Central), and intercompartmental clearance (Q12). Assume the drug concentration versus the time profile follows the biexponential decline $C_{t} = A e^{- a t} + B e^{- b t}$ , where C_t is the drug concentration at time t, and a and b are slopes for corresponding exponential declines.

The synthetic data set contains drug plasma concentration data measured in both central and peripheral compartments. The data was generated using a two-compartment model with an infusion dose and first-order elimination. These parameters were used for each individual.

	`Central`	`Peripheral`	`Q12`	`Cl_Central`
Individual 1	1.90	0.68	0.24	0.57
Individual 2	2.10	6.05	0.36	0.95
Individual 3	1.70	4.21	0.46	0.95

The data is stored as a table with variables ID, Time, CentralConc, and PeripheralConc. It represents the time course of plasma concentrations measured at eight different time points for both central and peripheral compartments after an infusion dose.

clear all
load(fullfile(matlabroot,'examples','simbio','data10_32R.mat'))

Convert the data set to a groupedData object which is the required data format for the fitting function sbiofit for later use. A groupedData object also lets you set independent variable and group variable names (if they exist). Set the units of the ID, Time, CentralConc, and PeripheralConc variables. The units are optional and only required for the UnitConversion feature, which automatically converts matching physical quantities to one consistent unit system.

gData = groupedData(data);
gData.Properties.VariableUnits = {'','hour','milligram/liter','milligram/liter'};
gData.Properties

ans = 

  struct with fields:

                Description: ''
                   UserData: []
             DimensionNames: {'Row'  'Variables'}
              VariableNames: {'ID'  'Time'  'CentralConc'  'PeripheralConc'}
       VariableDescriptions: {}
              VariableUnits: {1x4 cell}
         VariableContinuity: []
                   RowNames: {}
           CustomProperties: [1x1 matlab.tabular.CustomProperties]
          GroupVariableName: 'ID'
    IndependentVariableName: 'Time'

Create a trellis plot that shows the PK profiles of three individuals.

sbiotrellis(gData,'ID','Time',{'CentralConc','PeripheralConc'},...
            'Marker','+','LineStyle','none');

Use the built-in PK library to construct a two-compartment model with infusion dosing and first-order elimination where the elimination rate depends on the clearance and volume of the central compartment. Use the configset object to turn on unit conversion.

pkmd                 = PKModelDesign;
pkc1                 = addCompartment(pkmd,'Central');
pkc1.DosingType      = 'Infusion';
pkc1.EliminationType = 'linear-clearance';
pkc1.HasResponseVariable = true;
pkc2                 = addCompartment(pkmd,'Peripheral');
model                = construct(pkmd);
configset            = getconfigset(model);
configset.CompileOptions.UnitConversion = true;

Assume every individual receives an infusion dose at time = 0, with a total infusion amount of 100 mg at a rate of 50 mg/hour. For details on setting up different dosing strategies, see Doses in SimBiology Models.

dose             = sbiodose('dose','TargetName','Drug_Central');
dose.StartTime   = 0;
dose.Amount      = 100;
dose.Rate        = 50;
dose.AmountUnits = 'milligram';
dose.TimeUnits   = 'hour';
dose.RateUnits   = 'milligram/hour';

The data contains measured plasma concentrations in the central and peripheral compartments. Map these variables to the appropriate model species, which are Drug_Central and Drug_Peripheral.

responseMap = {'Drug_Central = CentralConc','Drug_Peripheral = PeripheralConc'};

The parameters to estimate in this model are the volumes of central and peripheral compartments (Central and Peripheral), intercompartmental clearance Q12, and clearance rate Cl_Central. In this case, specify log-transform for Central and Peripheral since they are constrained to be positive. The estimatedInfo object lets you specify parameter transforms, initial values, and parameter bounds (optional).

paramsToEstimate   = {'log(Central)','log(Peripheral)','Q12','Cl_Central'};
estimatedParam     = estimatedInfo(paramsToEstimate,'InitialValue',[1 1 1 1]);

Fit the model to all of the data pooled together, that is, estimate one set of parameters for all individuals. The default estimation method that sbiofit uses will change depending on which toolboxes are available. To see which estimation function sbiofit used for the fitting, check the EstimationFunction property of the corresponding results object.

pooledFit = sbiofit(model,gData,responseMap,estimatedParam,dose,'Pooled',true)

pooledFit = 

  OptimResults with properties:

                   ExitFlag: 3
                     Output: [1x1 struct]
                  GroupName: []
                       Beta: [4x3 table]
         ParameterEstimates: [4x3 table]
                          J: [24x4x2 double]
                       COVB: [4x4 double]
           CovarianceMatrix: [4x4 double]
                          R: [24x2 double]
                        MSE: 6.6220
                        SSE: 291.3688
                    Weights: []
              LogLikelihood: -111.3904
                        AIC: 230.7808
                        BIC: 238.2656
                        DFE: 44
             DependentFiles: {1x3 cell}
    EstimatedParameterNames: {'Central'  'Peripheral'  'Q12'  'Cl_Central'}
             ErrorModelInfo: [1x3 table]
         EstimationFunction: 'lsqnonlin'

Plot the fitted results versus the original data. Although three separate plots were generated, the data was fitted using the same set of parameters (that is, all three individuals had the same fitted line).

plot(pooledFit);

Estimate one set of parameters for each individual and see if there is any improvement in the parameter estimates. In this example, since there are three individuals, three sets of parameters are estimated.

unpooledFit =  sbiofit(model,gData,responseMap,estimatedParam,dose,'Pooled',false);

Plot the fitted results versus the original data. Each individual was fitted differently (that is, each fitted line is unique to each individual) and each line appeared to fit well to individual data.

plot(unpooledFit);

Display the fitted results of the first individual. The MSE was lower than that of the pooled fit. This is also true for the other two individuals.

unpooledFit(1)

ans = 

  OptimResults with properties:

                   ExitFlag: 3
                     Output: [1x1 struct]
                  GroupName: 1
                       Beta: [4x3 table]
         ParameterEstimates: [4x3 table]
                          J: [8x4x2 double]
                       COVB: [4x4 double]
           CovarianceMatrix: [4x4 double]
                          R: [8x2 double]
                        MSE: 2.1380
                        SSE: 25.6559
                    Weights: []
              LogLikelihood: -26.4805
                        AIC: 60.9610
                        BIC: 64.0514
                        DFE: 12
             DependentFiles: {1x3 cell}
    EstimatedParameterNames: {'Central'  'Peripheral'  'Q12'  'Cl_Central'}
             ErrorModelInfo: [1x3 table]
         EstimationFunction: 'lsqnonlin'

Generate a plot of the residuals over time to compare the pooled and unpooled fit results. The figure indicates unpooled fit residuals are smaller than those of pooled fit as expected. In addition to comparing residuals, other rigorous criteria can be used to compare the fitted results.

t = [gData.Time;gData.Time];
res_pooled = vertcat(pooledFit.R);
res_pooled = res_pooled(:);
res_unpooled = vertcat(unpooledFit.R);
res_unpooled = res_unpooled(:);
plot(t,res_pooled,'o','MarkerFaceColor','w','markerEdgeColor','b')
hold on
plot(t,res_unpooled,'o','MarkerFaceColor','b','markerEdgeColor','b')
refl = refline(0,0); % A reference line representing a zero residual
title('Residuals versus Time');
xlabel('Time');
ylabel('Residuals');
legend({'Pooled','Unpooled'});

This example showed how to perform pooled and unpooled estimations using sbiofit. As illustrated, the unpooled fit accounts for variations due to the specific subjects in the study, and, in this case, the model fits better to the data. However, the pooled fit returns population-wide parameters. If you want to estimate population-wide parameters while considering individual variations, use sbiofitmixed.

Estimate Category-Specific PK Parameters for Multiple Individuals

This example uses:

This example shows how to estimate category-specific (such as young versus old, male versus female), individual-specific, and population-wide parameters using PK profile data from multiple individuals.

Background

Suppose you have drug plasma concentration data from 30 individuals and want to estimate pharmacokinetic parameters, namely the volumes of central and peripheral compartment, the clearance, and intercompartmental clearance. Assume the drug concentration versus the time profile follows the biexponential decline $C_{t} = A e^{- a t} + B e^{- b t}$ , where $C_{t}$ is the drug concentration at time t, and $a$ and $b$ are slopes for corresponding exponential declines.

Load Data

This synthetic data contains the time course of plasma concentrations of 30 individuals after a bolus dose (100 mg) measured at different times for both central and peripheral compartments. It also contains categorical variables, namely Sex and Age.

clear
load(fullfile(matlabroot,'examples','simbio','sd5_302RAgeSex.mat'))

Convert to groupedData Format

Convert the data set to a groupedData object, which is the required data format for the fitting function sbiofit. A groupedData object also allows you set independent variable and group variable names (if they exist). Set the units of the ID, Time, CentralConc, PeripheralConc, Age, and Sex variables. The units are optional and only required for the UnitConversion feature, which automatically converts matching physical quantities to one consistent unit system.

gData = groupedData(data);
gData.Properties.VariableUnits = {'','hour','milligram/liter','milligram/liter','',''};
gData.Properties

ans = struct with fields:
                Description: ''
                   UserData: []
             DimensionNames: {'Row'  'Variables'}
              VariableNames: {1x6 cell}
       VariableDescriptions: {}
              VariableUnits: {1x6 cell}
         VariableContinuity: []
                   RowNames: {}
           CustomProperties: [1x1 matlab.tabular.CustomProperties]
          GroupVariableName: 'ID'
    IndependentVariableName: 'Time'

The IndependentVariableName and GroupVariableName properties have been automatically set to the Time and ID variables of the data.

Visualize Data

Display the response data for each individual.

t = sbiotrellis(gData,'ID','Time',{'CentralConc','PeripheralConc'},...
                'Marker','+','LineStyle','none');
% Resize the figure.
t.hFig.Position(:) = [100 100 1280 800];

Set Up a Two-Compartment Model

pkmd                                    = PKModelDesign;
pkc1                                    = addCompartment(pkmd,'Central');
pkc1.DosingType                         = 'Bolus';
pkc1.EliminationType                    = 'linear-clearance';
pkc1.HasResponseVariable                = true;
pkc2                                    = addCompartment(pkmd,'Peripheral');
model                                   = construct(pkmd);
configset                               = getconfigset(model);
configset.CompileOptions.UnitConversion = true;

For details on creating compartmental PK models using the SimBiology® built-in library, see Create Pharmacokinetic Models.

Define Dosing

Assume every individual receives a bolus dose of 100 mg at time = 0. For details on setting up different dosing strategies, see Doses in SimBiology Models.

dose             = sbiodose('dose','TargetName','Drug_Central');
dose.StartTime   = 0;
dose.Amount      = 100;
dose.AmountUnits = 'milligram';
dose.TimeUnits   = 'hour';

Map the Response Data to Corresponding Model Components

The data contains measured plasma concentration in the central and peripheral compartments. Map these variables to the appropriate model components, which are Drug_Central and Drug_Peripheral.

responseMap = {'Drug_Central = CentralConc','Drug_Peripheral = PeripheralConc'};

Specify Parameters to Estimate

Specify the volumes of central and peripheral compartments Central and Peripheral, intercompartmental clearance Q12, and clearance Cl_Central as parameters to estimate. The estimatedInfo object lets you optionally specify parameter transforms, initial values, and parameter bounds. Since both Central and Peripheral are constrained to be positive, specify a log-transform for each parameter.

paramsToEstimate    = {'log(Central)', 'log(Peripheral)', 'Q12', 'Cl_Central'};
estimatedParam      = estimatedInfo(paramsToEstimate,'InitialValue',[1 1 1 1]);

Estimate Individual-Specific Parameters

Estimate one set of parameters for each individual by setting the 'Pooled' name-value pair argument to false.

unpooledFit =  sbiofit(model,gData,responseMap,estimatedParam,dose,'Pooled',false);

Display Results

Plot the fitted results versus the original data for each individual (group).

t = plot(unpooledFit);
% Resize the figure.
t.hFig.Position(:) = [100 100 1280 800];

For an unpooled fit, sbiofit always returns one results object for each individual.

Examine Parameter Estimates for Category Dependencies

Explore the unpooled estimates to see if there is any category-specific parameters, that is, if some parameters are related to one or more categories. If there are any category dependencies, it might be possible to reduce the number of degrees of freedom by estimating just category-specific values for those parameters.

First extract the ID and category values for each ID

catParamValues = unique(gData(:,{'ID','Sex','Age'}));

Add variables to the table containing each parameter's estimate.

allParamValues            = vertcat(unpooledFit.ParameterEstimates);
catParamValues.Central    = allParamValues.Estimate(strcmp(allParamValues.Name, 'Central'));
catParamValues.Peripheral = allParamValues.Estimate(strcmp(allParamValues.Name, 'Peripheral'));
catParamValues.Q12        = allParamValues.Estimate(strcmp(allParamValues.Name, 'Q12'));
catParamValues.Cl_Central = allParamValues.Estimate(strcmp(allParamValues.Name, 'Cl_Central'));

Plot estimates of each parameter for each category. gscatter requires Statistics and Machine Learning Toolbox™. If you do not have it, use other alternative plotting functions such as plot.

h           = figure;
ylabels     = {'Central','Peripheral','Cl\_Central','Q12'};
plotNumber  = 1;
for i = 1:4
    thisParam = estimatedParam(i).Name;
    
    % Plot for Sex category
    subplot(4,2,plotNumber);
    plotNumber  = plotNumber + 1;
    gscatter(double(catParamValues.Sex), catParamValues.(thisParam), catParamValues.Sex);
    ax          = gca;
    ax.XTick    = [];
    ylabel(ylabels(i));
    legend('Location','bestoutside')
    % Plot for Age category
    subplot(4,2,plotNumber);
    plotNumber  = plotNumber + 1;
    gscatter(double(catParamValues.Age), catParamValues.(thisParam), catParamValues.Age);
    ax          = gca;
    ax.XTick    = [];
    ylabel(ylabels(i));
    legend('Location','bestoutside')
end
% Resize the figure.
h.Position(:) = [100 100 1280 800];

Based on the plot, it seems that young individuals tend to have higher volumes of central and peripheral compartments (Central, Peripheral) than old individuals (that is, the volumes seem to be age-specific). In addition, males tend to have higher clearance rates (Cl_Central) than females but the opposite for the Q12 parameter (that is, the clearance and Q12 seem to be sex-specific).

Estimate Category-Specific Parameters

Use the 'CategoryVariableName' property of the estimatedInfo object to specify which category to use during fitting. Use 'Sex' as the group to fit for the clearance Cl_Central and Q12 parameters. Use 'Age' as the group to fit for the Central and Peripheral parameters.

estimatedParam(1).CategoryVariableName = 'Age';
estimatedParam(2).CategoryVariableName = 'Age';
estimatedParam(3).CategoryVariableName = 'Sex';
estimatedParam(4).CategoryVariableName = 'Sex';
categoryFit = sbiofit(model,gData,responseMap,estimatedParam,dose)

categoryFit = 
  OptimResults with properties:

                   ExitFlag: 3
                     Output: [1x1 struct]
                  GroupName: []
                       Beta: [8x5 table]
         ParameterEstimates: [120x6 table]
                          J: [240x8x2 double]
                       COVB: [8x8 double]
           CovarianceMatrix: [8x8 double]
                          R: [240x2 double]
                        MSE: 0.4362
                        SSE: 205.8690
                    Weights: []
              LogLikelihood: -477.9195
                        AIC: 971.8390
                        BIC: 1.0052e+03
                        DFE: 472
             DependentFiles: {1x3 cell}
    EstimatedParameterNames: {'Central'  'Peripheral'  'Q12'  'Cl_Central'}
             ErrorModelInfo: [1x3 table]
         EstimationFunction: 'lsqnonlin'

When fitting by category (or group), sbiofit always returns one results object, not one for each category level. This is because both male and female individuals are considered to be part of the same optimization using the same error model and error parameters, similarly for the young and old individuals.

Plot Results

Plot the category-specific estimated results.

t = plot(categoryFit);
% Resize the figure.
t.hFig.Position(:) = [100 100 1280 800];

For the Cl_Central and Q12 parameters, all males had the same estimates, and similarly for the females. For the Central and Peripheral parameters, all young individuals had the same estimates, and similarly for the old individuals.

Estimate Population-Wide Parameters

To better compare the results, fit the model to all of the data pooled together, that is, estimate one set of parameters for all individuals by setting the 'Pooled' name-value pair argument to true. The warning message tells you that this option will ignore any category-specific information (if they exist).

pooledFit = sbiofit(model,gData,responseMap,estimatedParam,dose,'Pooled',true);

Warning: CategoryVariableName property of the estimatedInfo object is ignored when using the 'Pooled' option.

Plot Results

Plot the fitted results versus the original data. Although a separate plot was generated for each individual, the data was fitted using the same set of parameters (that is, all individuals had the same fitted line).

t = plot(pooledFit);
% Resize the figure.
t.hFig.Position(:) = [100 100 1280 800];

Compare Residuals

Compare residuals of CentralConc and PeripheralConc responses for each fit.

t = gData.Time;
allResid(:,:,1) = pooledFit.R;
allResid(:,:,2) = categoryFit.R;
allResid(:,:,3) = vertcat(unpooledFit.R);

h = figure;
responseList = {'CentralConc', 'PeripheralConc'};
for i = 1:2
    subplot(2,1,i);
    oneResid = squeeze(allResid(:,i,:));
    plot(t,oneResid,'o');
    refline(0,0); % A reference line representing a zero residual
    title(sprintf('Residuals (%s)', responseList{i}));
    xlabel('Time');
    ylabel('Residuals');
    legend({'Pooled','Category-Specific','Unpooled'});
end
% Resize the figure.
h.Position(:) = [100 100 1280 800];

As shown in the plot, the unpooled fit produced the best fit to the data as it fit the data to each individual. This was expected since it used the most number of degrees of freedom. The category-fit reduced the number of degrees of freedom by fitting the data to two categories (sex and age). As a result, the residuals were larger than the unpooled fit, but still smaller than the population-fit, which estimated just one set of parameters for all individuals. The category-fit might be a good compromise between the unpooled and pooled fitting provided that any hierarchical model exists within your data.

Estimate a Parameter from the Yeast G Protein Model

This example uses the yeast heterotrimeric G protein model and experimental data reported by [1]. For details about the model, see the Background section in Parameter Scanning, Parameter Estimation, and Sensitivity Analysis in the Yeast Heterotrimeric G Protein Cycle.

Load the G protein model.

sbioloadproject gprotein

Store the experimental data containing the time course for the fraction of active G protein.

time = [0 10 30 60 110 210 300 450 600]';
GaFracExpt = [0 0.35 0.4 0.36 0.39 0.33 0.24 0.17 0.2]';

Create a groupedData object based on the experimental data.

tbl = table(time,GaFracExpt);
grpData = groupedData(tbl);

Map the appropriate model component to the experimental data. In other words, indicate which species in the model corresponds to which response variable in the data. In this example, map the model parameter GaFrac to the experimental data variable GaFracExpt from grpData.

responseMap = 'GaFrac = GaFracExpt';

Use an estimatedInfo object to define the model parameter kGd as a parameter to be estimated.

estimatedParam = estimatedInfo('kGd');

Perform the parameter estimation.

fitResult = sbiofit(m1,grpData,responseMap,estimatedParam);

View the estimated parameter value of kGd.

fitResult.ParameterEstimates

ans=1×3 table
     Name      Estimate    StandardError
    _______    ________    _____________

    {'kGd'}    0.11307      3.4439e-05

Suppose you want to plot the model simulation results using the estimated parameter value. You can either rerun the sbiofit function and specify to return the optional second output argument, which contains simulation results, or use the fitted method to retrieve the results without rerunning sbiofit.

[yfit,paramEstim] = fitted(fitResult);

Plot the simulation results.

sbioplot(yfit);

Estimate Time Lag and Duration of a Dose

This example shows how to estimate the time lag before a bolus dose was administered and the duration of the dose using a one-compartment model.

Load a sample data set.

load lagDurationData.mat

Plot the data.

plot(data.Time,data.Conc,'x')
xlabel('Time (hour)')
ylabel('Conc (milligram/liter)')

Convert to groupedData.

gData = groupedData(data);
gData.Properties.VariableUnits = {'hour','milligram/liter'};

Create a one-compartment model.

pkmd                    = PKModelDesign;
pkc1                    = addCompartment(pkmd,'Central');
pkc1.DosingType         = 'Bolus';
pkc1.EliminationType    = 'linear-clearance';
pkc1.HasResponseVariable = true;
model                   = construct(pkmd);
configset               = getconfigset(model);
configset.CompileOptions.UnitConversion = true;

Add two parameters that represent the time lag and duration of a dose. The lag parameter specifies the time lag before the dose is administered. The duration parameter specifies the length of time it takes to administer a dose.

lagP = addparameter(model,'lagP');
lagP.ValueUnits = 'hour';
durP = addparameter(model,'durP');
durP.ValueUnits = 'hour';

Create a dose object. Set the LagParameterName and DurationParameterName properties of the dose to the names of the lag and duration parameters, respectively.

dose                = sbiodose('dose');
dose.TargetName     = 'Drug_Central';
dose.StartTime      = 0;
dose.Amount         = 10;
dose.AmountUnits    = 'milligram';
dose.TimeUnits      = 'hour';
dose.LagParameterName = 'lagP';
dose.DurationParameterName = 'durP';

Map the model species to the corresponding data.

responseMap = {'Drug_Central = Conc'};

Specify the lag and duration parameters as parameters to estimate. Log-transform the parameters. Initialize them to 2 and set the upper bound and lower bound.

paramsToEstimate    = {'log(lagP)','log(durP)'};
estimatedParams     = estimatedInfo(paramsToEstimate,'InitialValue',2,'Bounds',[1 5]);

Perform parameter estimation.

fitResults = sbiofit(model,gData,responseMap,estimatedParams,dose,'fminsearch')

fitResults = 
  OptimResults with properties:

                   ExitFlag: 1
                     Output: [1x1 struct]
                  GroupName: One group
                       Beta: [2x4 table]
         ParameterEstimates: [2x4 table]
                          J: [11x2 double]
                       COVB: [2x2 double]
           CovarianceMatrix: [2x2 double]
                          R: [11x1 double]
                        MSE: 0.0024
                        SSE: 0.0213
                    Weights: []
              LogLikelihood: 18.7511
                        AIC: -33.5023
                        BIC: -32.7065
                        DFE: 9
             DependentFiles: {1x2 cell}
    EstimatedParameterNames: {'lagP'  'durP'}
             ErrorModelInfo: [1x3 table]
         EstimationFunction: 'fminsearch'

Display the result.

fitResults.ParameterEstimates

ans=2×4 table
      Name      Estimate    StandardError    Bounds
    ________    ________    _____________    ______

    {'lagP'}     1.986        0.0051568      1    5
    {'durP'}     1.527         0.012956      1    5

plot(fitResults)

Input Arguments

`sm` — SimBiology model
SimBiology model object

SimBiology model, specified as a SimBiology model object. The active configset object of the model contains solver settings for simulation. Any active doses and variants are applied to the model during simulation unless specified otherwise using the dosing and variants input arguments, respectively.

`grpData` — Data to fit
groupedData object

Data to fit, specified as a groupedData object.

The name of the time variable must be defined in the IndependentVariableName property of grpData. For instance, if the time variable name is 'TIME', then specify it as follows.

grpData.Properties.IndependentVariableName = 'TIME';

If the data contains more than one group of measurements, the grouping variable name must be defined in the GroupVariableName property of grpData. For example, if the grouping variable name is 'GROUP', then specify it as follows.

grpData.Properties.GroupVariableName = 'GROUP';

A group usually refers to a set of measurements that represent a single time course, often corresponding to a particular individual, or experimental condition.

Note

sbiofit uses the categorical function to identify groups. If any group values are converted to the same value by categorical, then those observations are treated as belonging to the same group. For instance, if some observations have no group information (that is, an empty character vector ''), then categorical converts empty character vectors to <undefined>, and these observations are treated as one group.

`responseMap` — Mapping information of model components to `grpData`
character vector | string | string vector | cell array of character vectors

Mapping information of model components to grpData, specified as a character vector, string, string vector, or cell array of character vectors.

Each character vector or string is an equation-like expression, similar to assignment rules in SimBiology. It contains the name (or qualified name) of a quantity (species, compartment, or parameter) in the model sm, followed by the character '=' and the name of a variable in grpData. For clarity, white spaces are allowed between names and '='.

For example, if you have the concentration data 'CONC' in grpData for a species 'Drug_Central', you can specify it as follows.

responseMap = 'Drug_Central = CONC';

To name a species unambiguously, use the qualified name, which includes the name of the compartment. To name a reaction-scoped parameter, use the reaction name to qualify the parameter.

If the model component name or grpData variable name is not a valid MATLAB^® variable name, surround it by square brackets, such as:

responseMap = '[Central 1].Drug = [Central 1 Conc]';

If a variable name itself contains square brackets, you cannot use it in the expression to define the mapping information.

An error is issued if any (qualified) name matches two components of the same type. However, you can use a (qualified) name that matches two components of different types, and the function first finds the species with the given name, followed by compartments and then parameters.

`estiminfo` — Estimated parameters
`estimatedInfo` object | vector of `estimatedInfo` objects

Estimated parameters, specified as an EstimatedInfo object or vector of estimatedInfo objects that defines the estimated parameters in the model sm, and other optional information such as their initial estimates, transformations, bound constraints, and categories. Supported transforms are log, logit, and probit.

You can specify bounds for all estimation methods. The lower bound must be less than the upper bound. For details, see boundValues.

When using scattersearch, you must specify finite transformed bounds for each estimated parameter.

When using fminsearch, nlinfit, or fminunc with bounds, the objective function returns Inf if bounds are exceeded. When you turn on options such as FunValCheck, the optimization might error if bounds are exceeded during estimation. If using nlinfit, it might report warnings about the Jacobian being ill-conditioned or not being able to estimate if the final result is too close to the bounds.

If you do not specify Pooled name-value pair argument, sbiofit uses CategoryVariableName property of estiminfo to decide if parameters must be estimated for each individual, group, category, or all individuals as a whole. Use the Pooled option to override any CategoryVariableName values. For details about CategoryVariableName property, see EstimatedInfo object.

Note

sbiofit uses the categorical function to identify groups or categories. If any group values are converted to the same value by categorical, then those observations are treated as belonging to the same group. For instance, if some observations have no group information (that is, an empty character vector '' as a group value), then categorical converts empty character vectors to <undefined>, and these observations are treated as one group.

`dosing` — Dosing information
`[]` | 2-D matrix of dose objects

Dosing information, specified as an empty array, or 2-D matrix of dose objects (ScheduleDose object or RepeatDose object).

If empty, no doses are applied during simulation, even if the model has active doses.
If not empty, the matrix must have a single row or one row per group in the input data. If it has a single row, same doses are applied to all groups during simulation. If it has multiple rows, each row is applied to a separate group, in the same order as the groups appear in the input data.
Multiple columns are allowed so that you can apply multiple dose objects to each group. All doses in a column must reference the same components in the model. For example, doses in the same column must have the same dose target (TargetName). If you parameterize any property of a dose, then all doses within the column must have this property parameterized to the same parameter. If some groups require more doses than others, then fill in the matrix with default (dummy) doses.
A default dose has default values for all properties, except for the Name property. Create a default dose as follows.
```
d1 = sbiodose('d1');
```
In addition to manually constructing dose objects using sbiodose, if grpData has dosing information, you can use the createDoses method to construct doses.

`functionName` — Estimation function name
character vector | string

Estimation function name, specified as a character vector or string. Choices are as follows.

'fminsearch'
'nlinfit' (Statistics and Machine Learning Toolbox is required.)
'fminunc' (Optimization Toolbox is required.)
'fmincon' (Optimization Toolbox is required.)
'lsqcurvefit' (Optimization Toolbox is required.)
'lsqnonlin' (Optimization Toolbox is required.)
'patternsearch' (Global Optimization Toolbox is required.)
'ga' (Global Optimization Toolbox is required.)
'particleswarm' (Global Optimization Toolbox is required.)
'scattersearch'

By default, sbiofit uses the first available estimation function among the following: lsqnonlin, nlinfit, or fminsearch. The same priority applies to the default local solver choice for scattersearch.

For the summary of supported methods and fitting options, see Supported Methods for Parameter Estimation.

`options` — Options specific to estimation function
struct | optimoptions object

Options specific to the estimation function, specified as a struct or optimoptions object.

statset struct for nlinfit
optimset struct for fminsearch
optimoptions object for lsqcurvefit, lsqnonlin, fmincon, fminunc, particleswarm, ga, and patternsearch
struct for scattersearch

See Default Options for Estimation Algorithms for more details and default options associated with each estimation function.

`variants` — Variants
`[]` | vector of variant objects

Variants, specified as an empty array or vector of variant objects. If empty, no variants are used even if the model has active variants.

Name-Value Pair Arguments

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.

Example: 'ErrorModel','constant','UseParallel',true specifies a constant error model and to run simulations in parallel during parameter estimation.

`'ErrorModel'` — Error model
`'constant'` (default) | character vector | string | string vector | cell array of character vector | categorical vector | table

Error models used for estimation, specified as a character vector, string, string vector, cell array of character vectors, categorical vector, or table.

If it is a table, it must contain a single variable that is a column vector of error model names. The names can be a cell array of character vectors, string vector, or a vector of categorical variables. If the table has more than one row, then the RowNames property must match the response variable names specified in the right hand side of responseMap. If the table does not use the RowNames property, the nth error is associated with the nth response.

If you specify only one error model, then sbiofit estimates one set of error parameters for all responses.

If you specify multiple error models using a categorical vector, string vector, or cell array of character vectors, the length of the vector or cell array must match the number of responses in the responseMap argument.

You can specify multiple error models only if you are using these methods: lsqnonlin, lsqcurvefit, fmincon, fminunc, fminsearch, patternsearch, ga, and particleswarm.

There are four built-in error models. Each model defines the error using a standard mean-zero and unit-variance (Gaussian) variable e, simulation results f, and one or two parameters a and b.

'constant': $y = f + a e$
'proportional': $y = f + b | f | e$
'combined': $y = f + (a + b | f |) e$
'exponential': $y = f * \exp (a e)$

Note

If you specify an error model, you cannot specify weights except for the constant error model.

`'Weights'` — Weights
matrix | function handle

Weights used for estimation, specified as a matrix of real positive weights, where the number of columns corresponds to the number of responses, or a function handle that accepts a vector of predicted response values and returns a vector of real positive weights.

If you specify an error model, you cannot use weights except for the constant error model. If neither the 'ErrorModel' or 'Weights' is specified, by default the function uses the constant error model with equal weights.

`'Pooled'` — Fit option flag
`false` (default) | `true`

Fit option flag, specifying whether to fit each individual (false) or pool all individual data (true).

When true, the function performs fitting for all individuals or groups simultaneously using the same parameter estimates, and fitResults is a scalar results object.

When false, the function fits each group or individual separately using group- or individual-specific parameters, and fitResults is a vector of results objects with one result for each group.

Note

Use this option to override any CategoryVariableName values of estiminfo.

`'UseParallel'` — Flag for parallel simulations
`false` (default) | `true`

Flag to enable parallelization, specified as true or false. If true and Parallel Computing Toolbox™ is available, sbiofit supports several levels of parallelization, but only one level is used at a time.

For an unpooled fit ('Pooled' = false) for multiple groups, each fit is run in parallel.

For a pooled fit ('Pooled' = true), parallelization happens at the solver level. In other words, solver computations, such as objective function evaluations, are run in parallel. For details, see Multiple Parameter Estimations in Parallel.

`'SensitivityAnalysis'` — Flag to use parameter sensitivities to determine gradients of the objective function
`true` | `false`

Flag to use parameter sensitivities to determine gradients of the objective function, specified as true or false. By default, it is true for fmincon, fminunc, lsqnonlin, and lsqcurvefit methods. Otherwise, it is false. If it is true, SimBiology always uses the sundials solver, regardless of what you have selected as the SolverType property in the Configset object.

SimBiology uses the complex-step approximation method to calculate parameter sensitivities. Such calculated sensitivities can be used to determine gradients of the objective function during parameter estimation to improve fitting. The default behavior of sbiofit is to use such sensitivities to determine gradients whenever the algorithm is gradient-based and if the SimBiology^® model supports sensitivity analysis. For details about the model requirements and sensitivity analysis, see Sensitivity Calculation.

`'ProgressPlot'` — Flag to show the progress of parameter estimation
`false` (default) | `true`

Flag to show the progress of parameter estimation, specified as true or false. If true, a new figure opens containing plots that show the log-likelihood, termination criteria, and estimated parameters for each iteration. This option is not supported for the nlinfit method.

If you are using the Optimization Toolbox functions (fminunc, fmincon, lsqcurvefit, lsqnonlin), the figure also shows the First Order Optimality (Optimization Toolbox) plot.

For an unpooled fit, each line on the plots represents an individual. For a pooled fit, a single line represents all individuals. The line becomes faded when the fit is complete. The plots also keep track of the progress when you run sbiofit (for both pooled and unpooled fits) in parallel using remote clusters. For details, see Progress Plot.

Output Arguments

`fitResults` — Estimation results
`OptimResults object` | `NLINResults object` | Vector of results objects

Estimation results, returned as a OptimResults object or NLINResults object or a vector of these objects. Both results objects are subclasses of the LeastSquaresResults object.

If the function uses nlinfit, then fitResults is a NLINResults object. Otherwise, fitResults is an OptimResults object.

When 'Pooled' is set to false, the function fits each group separately using group-specific parameters, and fitResults is a vector of results objects with one results object for each group.

When 'Pooled' is set to true, the function performs fitting for all individuals or groups simultaneously using the same parameter estimates, and fitResults is a scalar results object.

When 'Pooled' is not used, and CategoryVariableName values of estiminfo are all <none>, fitResults is a single result object. This is the same behavior as setting 'Pooled' to true.

When 'Pooled' is not used, and CategoryVariableName values of estiminfo are all <GroupVariableName>, fitResults is a vector of results objects. This is the same behavior as setting 'Pooled' to false.

In all other cases, fitResults is a scalar object containing estimated parameter values for different groups or categories specified by CategoryVariableName.

`simdata` — Simulation results
Vector of `SimData` objects

Simulation results, returned as a vector of SimData objects representing simulation results for each group or individual.

If the 'Pooled' option is false, then each simulation uses group-specific parameter values. If true, then all simulations use the same (population-wide) parameter values.

The states reported in simdata are the states that were included in the responseMap input argument and any other states listed in the StatesToLog property of the runtime options (RuntimeOptions) of the SimBiology model sm.

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