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 $$k_e=Cl/V$$.

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

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('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.

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 $$C_t=A{e}^{-at}+B{e}^{-bt}$$, 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('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

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).

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.