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estimatedInfo

Object containing information about estimated model quantities

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Description

The estimatedInfo object contains information about estimated model quantities (species, parameters, or compartments). Use this object to specify which quantities in a SimBiology® model are estimated, what parameter transforms are used, and optionally, the initial estimates for these quantities when using sbiofit or sbiofitmixed.

Creation

Syntax

estimInfo = estimatedInfo
estimInfoArray = estimatedInfo(transformedNames)
estimInfoArray = estimatedInfo(___,Name,Value)

Description

estimInfo = estimatedInfo creates an empty estimatedInfo object.

estimInfoArray = estimatedInfo(transformedNames) creates a vector of estimatedInfo objects for quantities specified in transformedNames. The initial values for these quantities are obtained from the SimBiology model when you run sbiofit or sbiofitmixed.

example

estimInfoArray = estimatedInfo(___,Name,Value) specifies additional options using one or more name-value arguments. For example, you can define the initial values or the initial transformed values of model quantities, the lower and upper bounds or the transformed lower and upper bounds for parameter estimation, and the groups to have separate estimated parameters.

example

Input Arguments

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Names of estimated model quantities, specified as a character vector, string, string vector, or cell array of character vectors. To name a species unambiguously, use the qualified name, which includes the name of the compartment that the species is in. To name a reaction-scoped parameter, use the reaction name to qualify the parameter. Each character vector (or string) must be in one of these formats:

  • Name or qualified name of a model quantity, such as 'Cl', 'Reaction1.k','[c 1].[r 1]'

  • Name of a supported parameter transform (log, logit, or probit) followed by a quantity name in parentheses, such as 'log(Cl)', 'logit(Reaction1.k)', 'probit([c 1].[r 1])'

For details, see Parameter Transformations.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: estimated = estimatedInfo('log(Central)','InitialValue', 1,'Bounds',[0 10])

Initial transformed values of model quantities, specified as a vector or cell array. It must have the same length as transformedNames. If it is a cell array, each element of the cell must be a scalar or the empty matrix [].

You cannot specify this name-value argument along with the 'InitialValue' name-value argument.

Initial values of model quantities, specified as a vector or cell array. It must have the same length as transformedNames. If it is a cell array, each element of the cell must be a scalar or the empty matrix [].

You cannot specify this name-value argument along with the 'InitialTransformedValue' name-value argument.

Lower and upper bounds for estimated parameters (boundValues), specified as a matrix or cell array. If boundValues is a matrix, it is an N-by-2 matrix of numbers, where N is either 1 or the length of transformedNames. If it is a cell array, each element must be a vector of size 1-by-2 or empty [].

Each row of boundValues corresponds to the lower (the first number) and upper (the second number) bounds of each element (such as a parameter) of estimInfo. The lower bound must be less than the upper bound. If you specify a single row, these bounds are applied to all elements of estimInfoArray.

All methods support parameter bounds in sbiofit (that is, fminsearch, nlinfit (Statistics and Machine Learning Toolbox), fminunc (Optimization Toolbox), fmincon (Optimization Toolbox), lsqcurvefit (Optimization Toolbox), lsqnonlin (Optimization Toolbox), patternsearch (Global Optimization Toolbox), ga (Global Optimization Toolbox), particleswarm (Global Optimization Toolbox), and scattersearch). 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 may error if bounds are exceeded during estimation. If using nlinfit, it may report warnings about the Jacobian being ill-conditioned or not being able to estimate if the final result is too close to the bounds. sbiofitmixed does not support parameter bounds.

You cannot specify this name-value argument along with the 'TransformedBounds' name-value argument.

Transformed lower and upper bounds for estimated parameters (tBoundValues), specified as a matrix or cell array. tBoundValues is a N-by-2 matrix of numbers, where N is either 1 or the length of transformedNames. If it is a cell array, each element must be a vector of size 1-by-2 or empty [].

Each row of tBoundValues corresponds to the lower (the first number) and upper (the second number) bounds of each element (such as a parameter) of estimInfo. The lower bound must be less than the upper bound. If you specify a single row, the bounds are applied to all elements of estimInfoArray.

You cannot specify this name-value argument along with the 'Bounds' name-value argument.

Group names for estimated parameters, specified as a character vector, string, string vector, or cell array of character vectors. Each character vector (or string) must be one of the following.

  • Name of a variable in the data used for fitting

  • '<GroupVariableName>' (default)

  • '<None>'

'<GroupVariableName>' indicates that each group in the data uses a separate parameter estimate. '<None>' indicates that all groups in the data use the same parameter estimate.

If the data you plan to use for fitting contains variables that group data into different categories, you can specify the names of those variables. For instance, if you have a variable called Sex which indicates male and female individuals, you can specify 'Sex' as the 'CategoryVariableName'. This means that all male individuals have one set of parameter estimates and all females have a separate set.

Output Arguments

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Estimated model quantity, returned as an estimatedInfo object.

Estimated model quantities, returned as an estimatedInfo object or vector of estimatedInfo objects. If transformedNames is a single character vector, estimInfoArray is a scalar estimatedInfo object. Otherwise, estimInfoArray is a vector of estimatedInfo objects with the same length as the input argument transformedNames.

Properties

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Name of an estimated model quantity, specified as a character vector. Changing this property also updates the TransformedName property.

Applied transformation for the quantity value during estimation, specified as '', 'log', 'logit', or 'probit'. An empty character vector '' indicates that no transform is applied.

A log transform ensures that the component value is always positive during estimation. The logit and probit transforms constrain component values to lie between 0 and 1.

The probit function is the inverse cumulative distribution function associated with the standard normal distribution. For the probit transform, SimBiology uses the norminv (Statistics and Machine Learning Toolbox) function. Hence Statistics and Machine Learning Toolbox™ is required for the transform.

The logit function, which is the inverse of sigmoid function, is defined as logit(x) = log(x) – log(1 – x).

This property is read-only.

Combined transform name (such as 'log') and quantity name (such as 'Central'), specified as a character vector (such as 'log(Central)').

Initial values of model quantities used for estimation, specified as an empty matrix [] or a real, finite, scalar value. The empty matrix indicates that the initial values for estimation are obtained from the relevant quantity property (Value for parameters, InitialAmount for species, and Capacity for compartments).

Changing this property automatically updates the InitialTransformedValue property of corresponding model quantities.

Initial transformed values of model quantities used for estimation, specified as an empty matrix [] or a scalar value. The empty matrix indicates that the initial transformed values for estimation are obtained by transforming the relevant quantity property (Value for parameters, InitialAmount for species, and Capacity for compartments).

Changing this property automatically updates the InitialValue property of corresponding model quantities.

Lower and upper bounds for an estimated parameter, specified as an empty matrix [] or a 1-by-2 vector of real, finite values. The empty matrix [] indicates that the only bound constraints are those introduced by the value of Transform. For example, setting Transform to 'log' constrains the parameter to the range [0,inf]. Changing this property also updates TransformedBounds.

The lower bound must be less than the upper bound.

Transformed lower and upper bound for an estimated parameter, specified as an empty matrix [] or a 1-by-2 vector of real, finite values. The empty matrix [] indicates that the value of the parameter in transformed space is not constrained. Changing this property also updates Bounds.

The lower bound must be less than the upper bound.

Data groups to have separate estimated parameters, specified as a character vector or a cell array of character vectors. The character vector can be the name of a variable in the data used for fitting or one of the keywords: '<GroupVariableName>' or '<None>'.

'<GroupVariableName>' indicates that each group in the data uses a separate parameter estimate. '<None>' indicates that all groups in the data use the same parameter estimate.

If you specify 'Pooled' name-value argument (to either true or false) when you run sbiofit, then the function ignores this variable. sbiofitmixed always ignores this property.

Examples

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Open Live Script

Create a one-compartment PK model with bolus dosing and linear clearance.

pkmd                    = PKModelDesign;
pkc1                    = addCompartment(pkmd,'Central');
pkc1.DosingType         = 'Bolus';
pkc1.EliminationType    = 'linear-clearance';
pkc1.HasResponseVariable = true;

Suppose you want to estimate the volume of the central compartment (Central). You can specify such estimated model quantity as well as appropriate parameter transform (log transform in this example), initial value, and parameter bound using the estimatedInfo object.

estimated = estimatedInfo('log(Central)','InitialValue', 1,'Bounds',[0 10])
estimated = 
  estimatedInfo with properties:

                       Name: 'Central'
                  Transform: 'log'
            TransformedName: 'log(Central)'
               InitialValue: 1
    InitialTransformedValue: 0
                     Bounds: [0 10]
          TransformedBounds: [-Inf 2.3026]
       CategoryVariableName: '<GroupVariableName>'
Open Live Script

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 Ct=C0e-ket, where Ct is the drug concentration at time t, C0 is the initial concentration, and ke is the elimination rate constant that depends on the clearance and volume of the central compartment ke=Cl/V.

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

  • 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

  • Bolus dose of 10 mg

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.

load('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: [1×1 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.

This example uses:

Open Live Script

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 Ct=Ae-at+Be-bt, where Ct 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('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: {'ID'  'Time'  'CentralConc'  'PeripheralConc'  'Sex'  'Age'}
              VariableTypes: ["double"    "double"    "double"    "double"    "categorical"    "categorical"]
       VariableDescriptions: {}
              VariableUnits: {''  'hour'  'milligram/liter'  'milligram/liter'  ''  ''}
         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];

Figure contains 30 axes objects. Axes object 1 with title ID 1 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 2 with title ID 2 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 3 with title ID 3 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 4 with title ID 4 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 5 with title ID 5 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 6 with title ID 6 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 7 with title ID 7 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 8 with title ID 8 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 9 with title ID 9 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 10 with title ID 10 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 11 with title ID 11 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 12 with title ID 12 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 13 with title ID 13 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 14 with title ID 14 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 15 with title ID 15 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 16 with title ID 16 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 17 with title ID 17 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 18 with title ID 18 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 19 with title ID 19 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 20 with title ID 20 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 21 with title ID 21 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 22 with title ID 22 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 23 with title ID 23 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 24 with title ID 24 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 25 with title ID 25 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 26 with title ID 26 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 27 with title ID 27 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 28 with title ID 28 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 29 with title ID 29 contains 2 objects of type line. One or more of the lines displays its values using only markers Axes object 30 with title ID 30 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent CentralConc, PeripheralConc.

Set Up a Two-Compartment Model

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                         = '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).

plot(unpooledFit);

Figure contains 32 axes objects. Axes object 1 with title 30 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 2 with title 29 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 3 with title 28 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 4 with title 27 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 5 with title 26 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 6 with title 25 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 7 with title 24 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 8 with title 23 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 9 with title 22 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 10 with title 21 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 11 with title 20 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 12 with title 19 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 13 with title 18 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 14 with title 17 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 15 with title 16 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 16 with title 15 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 17 with title 14 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 18 with title 13 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 19 with title 12 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 20 with title 11 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 21 with title 10 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 22 with title 9 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 23 with title 8 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 24 with title 7 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 25 with title 6 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 26 with title 5 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 27 with title 4 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 28 with title 3 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 29 with title 2 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 30 with title 1 contains 4 objects of type line. One or more of the lines displays its values using only markers Hidden axes object 31 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Predicted (Predicted.Central.Drug_Central), Observed (Observed.CentralConc). Hidden axes object 32 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Predicted (Predicted.Peripheral.Drug_Peripheral), Observed (Observed.PeripheralConc).

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","Q12","Cl\_Central"];
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];

Figure contains 8 axes objects. Axes object 1 with ylabel Central contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Female, Male. Axes object 2 with ylabel Central contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Old, Young. Axes object 3 with ylabel Peripheral contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Female, Male. Axes object 4 with ylabel Peripheral contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Old, Young. Axes object 5 with ylabel Q12 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Female, Male. Axes object 6 with ylabel Q12 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Old, Young. Axes object 7 with ylabel Cl\_Central contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Female, Male. Axes object 8 with ylabel Cl\_Central contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Old, Young.

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 lower 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}
                       Data: [240x6 groupedData]
    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.

plot(categoryFit);

Figure contains 32 axes objects. Axes object 1 with title 30 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 2 with title 29 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 3 with title 28 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 4 with title 27 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 5 with title 26 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 6 with title 25 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 7 with title 24 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 8 with title 23 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 9 with title 22 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 10 with title 21 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 11 with title 20 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 12 with title 19 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 13 with title 18 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 14 with title 17 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 15 with title 16 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 16 with title 15 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 17 with title 14 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 18 with title 13 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 19 with title 12 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 20 with title 11 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 21 with title 10 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 22 with title 9 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 23 with title 8 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 24 with title 7 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 25 with title 6 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 26 with title 5 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 27 with title 4 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 28 with title 3 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 29 with title 2 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 30 with title 1 contains 4 objects of type line. One or more of the lines displays its values using only markers Hidden axes object 31 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Predicted (Predicted.Central.Drug_Central), Observed (Observed.CentralConc). Hidden axes object 32 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Predicted (Predicted.Peripheral.Drug_Peripheral), Observed (Observed.PeripheralConc).

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).

plot(pooledFit);

Figure contains 32 axes objects. Axes object 1 with title 30 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 2 with title 29 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 3 with title 28 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 4 with title 27 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 5 with title 26 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 6 with title 25 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 7 with title 24 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 8 with title 23 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 9 with title 22 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 10 with title 21 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 11 with title 20 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 12 with title 19 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 13 with title 18 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 14 with title 17 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 15 with title 16 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 16 with title 15 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 17 with title 14 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 18 with title 13 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 19 with title 12 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 20 with title 11 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 21 with title 10 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 22 with title 9 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 23 with title 8 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 24 with title 7 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 25 with title 6 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 26 with title 5 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 27 with title 4 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 28 with title 3 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 29 with title 2 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 30 with title 1 contains 4 objects of type line. One or more of the lines displays its values using only markers Hidden axes object 31 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Predicted (Predicted.Central.Drug_Central), Observed (Observed.CentralConc). Hidden axes object 32 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Predicted (Predicted.Peripheral.Drug_Peripheral), Observed (Observed.PeripheralConc).

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];

Figure contains 2 axes objects. Axes object 1 with title Residuals (CentralConc), xlabel Time, ylabel Residuals contains 4 objects of type line. One or more of the lines displays its values using only markers These objects represent Pooled, Category-Specific, Unpooled. Axes object 2 with title Residuals (PeripheralConc), xlabel Time, ylabel Residuals contains 4 objects of type line. One or more of the lines displays its values using only markers These objects represent Pooled, Category-Specific, Unpooled.

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.

Version History

Introduced in R2014a

See Also

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Topics