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Fit PK Parameters Using SimBiology Problem-Based Workflow

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This example shows how to estimate PK parameters of a SimBiology® model using a problem-based approach.

Load a synthetic data set. It contains drug plasma concentration data measured in both central and peripheral compartments. 

load('data10_32R.mat')

Convert the data set to a groupedData object.

gData = groupedData(data);
gData.Properties.VariableUnits = ["","hour","milligram/liter","milligram/liter"];

Display the data.

sbiotrellis(gData,"ID","Time",["CentralConc","PeripheralConc"],...
            Marker="+",LineStyle="none");

Figure contains 4 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 These objects represent CentralConc, PeripheralConc. Axes object 4 is empty.

Use the built-in PK library to construct a two-compartment model with infusion dosing and first-order elimination. 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");
model2cpt            = construct(pkmd);
configset            = getconfigset(model2cpt);
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";

Create a problem object.

problem = fitproblem
problem = 
  fitproblem with properties:

   Required:
                   Data: [0x0 groupedData]
              Estimated: [1x0 estimatedInfo]
            FitFunction: "sbiofit"
                  Model: [0x0 SimBiology.Model]
            ResponseMap: [1x0 string]

   Optional:
                  Doses: [0x0 SimBiology.Dose]
           FunctionName: "auto"
                Options: []
           ProgressPlot: 0
            UseParallel: 0
               Variants: [0x0 SimBiology.Variant]

   sbiofit options:
             ErrorModel: "constant"
                 Pooled: "auto"
    SensitivityAnalysis: "auto"
                Weights: []

Define the required properties of the object.

problem.Data        = gData;
problem.Estimated   = estimatedInfo(["log(Central)","log(Peripheral)","Q12","Cl_Central"],InitialValue=[1 1 1 1]);
problem.Model       = model2cpt;
problem.ResponseMap = ["Drug_Central = CentralConc","Drug_Peripheral = PeripheralConc"];

Define the dose to be applied during fitting.

problem.Doses        = dose;

Show the progress of the estimation.

problem.ProgressPlot = true;

Fit the model to all of the data pooled together: that is, estimate one set of parameters for all individuals by setting the Pooled property to true.

problem.Pooled      = true;

Perform the estimation using the fit function of the object.

pooledFit = fit(problem);

Figure Progress Plot for lsqnonlin contains 7 axes objects and other objects of type uicontrol. Axes object 1 with title Log Likelihood, xlabel Iteration contains 2 objects of type line, text. Axes object 2 with title First Order Optimality contains an object of type line. Axes object 3 with title Central, ylabel Current Parameter Value contains an object of type line. Axes object 4 with title Peripheral contains an object of type line. Axes object 5 with title Q12 contains an object of type line. Axes object 6 with title Cl_Central contains an object of type line. Hidden axes object 7 contains an object of type text.

Display the estimated parameter values.

pooledFit.ParameterEstimates
ans=4×3 table
         Name         Estimate    StandardError
    ______________    ________    _____________

    {'Central'   }     1.6627        0.16569   
    {'Peripheral'}     2.6864         1.0644   
    {'Q12'       }    0.44945        0.19943   
    {'Cl_Central'}    0.78497       0.095621

Plot the fitted results.

plot(pooledFit);

Figure contains 6 axes objects. Axes object 1 is empty. Axes object 2 with title 3 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 3 with title 2 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 4 with title 1 contains 4 objects of type line. One or more of the lines displays its values using only markers Hidden axes object 5 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 6 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).

Estimate one set of parameters for each individual and see if the parameter estimates improve.

problem.Pooled  = false;
unpooledFit     = fit(problem);

Figure Progress Plot for lsqnonlin contains 12 axes objects and other objects of type uicontrol, uipanel. Axes object 1 with title Log Likelihood, xlabel Iteration contains 3 objects of type line. Axes object 2 with title First Order Optimality contains 3 objects of type line. Axes object 3 with title Termination Conditions contains an object of type text. These objects represent Failed, Converged. Axes object 4 with title Central, ylabel Current Parameter Value contains 3 objects of type line. Axes object 5 with title Final Estimated Parameter Values contains an object of type histogram. Axes object 6 with title Peripheral contains 3 objects of type line. Axes object 7 with title Final Estimated Parameter Values contains an object of type histogram. Axes object 8 with title Q12 contains 3 objects of type line. Axes object 9 with title Final Estimated Parameter Values contains an object of type histogram. Axes object 10 with title Cl_Central contains 3 objects of type line. Axes object 11 with title Final Estimated Parameter Values, ylabel Frequency contains an object of type histogram. Hidden axes object 12 contains an object of type text.

Display the estimated parameter values.

unpooledFit.ParameterEstimates
ans=4×3 table
         Name         Estimate    StandardError
    ______________    ________    _____________

    {'Central'   }      1.422        0.12334   
    {'Peripheral'}     1.5619        0.36355   
    {'Q12'       }    0.47163        0.15196   
    {'Cl_Central'}     0.5291       0.036978   


ans=4×3 table
         Name         Estimate    StandardError
    ______________    ________    _____________

    {'Central'   }     1.8322       0.019672   
    {'Peripheral'}     5.3364        0.65327   
    {'Q12'       }     0.2764       0.030799   
    {'Cl_Central'}    0.86035       0.026257   


ans=4×3 table
         Name         Estimate    StandardError
    ______________    ________    _____________

    {'Central'   }     1.6657       0.038529   
    {'Peripheral'}     5.5632        0.37063   
    {'Q12'       }    0.78361       0.058657   
    {'Cl_Central'}     1.0233       0.027311   


plot(unpooledFit);

Figure contains 6 axes objects. Axes object 1 is empty. Axes object 2 with title 3 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 3 with title 2 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 4 with title 1 contains 4 objects of type line. One or more of the lines displays its values using only markers Hidden axes object 5 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 6 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).

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 the 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(:);
figure;
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"]);

Figure contains an axes object. The axes object with title Residuals versus Time, xlabel Time, ylabel Residuals contains 3 objects of type line. One or more of the lines displays its values using only markers These objects represent Pooled, Unpooled.

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. As an alternative, if you want to estimate population-wide parameters while considering individual variations, you can perform nonlinear mixed-effects (NLME) estimation by setting problem.FitFunction to sbiofitmixed.

problem.FitFunction = "sbiofitmixed";
NLMEResults = fit(problem);

Figure contains 9 axes objects. Axes object 1 with title theta1 contains an object of type line. Axes object 2 with title theta2 contains an object of type line. Axes object 3 with title theta3 contains an object of type line. Axes object 4 with title theta4 contains an object of type line. Axes object 5 with title Psi indexOf 1 _ 1 baseline contains an object of type line. Axes object 6 with title Psi indexOf 2 _ 2 baseline contains an object of type line. Axes object 7 with title Psi indexOf 3 _ 3 baseline contains an object of type line. Axes object 8 with title Psi indexOf 4 _ 4 baseline contains an object of type line. Axes object 9 with title loglikelihood contains an object of type line.

Display the estimated parameter values.

NLMEResults.IndividualParameterEstimates
ans=12×3 table
    Group         Name         Estimate
    _____    ______________    ________

      1      {'Central'   }     1.4623 
      1      {'Peripheral'}     1.5306 
      1      {'Q12'       }     0.4587 
      1      {'Cl_Central'}    0.53208 
      2      {'Central'   }      1.783 
      2      {'Peripheral'}     6.6623 
      2      {'Q12'       }     0.3589 
      2      {'Cl_Central'}     0.8039 
      3      {'Central'   }     1.7135 
      3      {'Peripheral'}     4.2844 
      3      {'Q12'       }    0.54895 
      3      {'Cl_Central'}     1.0708

Plot the fitted results.

plot(NLMEResults);

Figure contains 6 axes objects. Axes object 1 is empty. Axes object 2 with title 3 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 3 with title 2 contains 4 objects of type line. One or more of the lines displays its values using only markers Axes object 4 with title 1 contains 4 objects of type line. One or more of the lines displays its values using only markers Hidden axes object 5 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Predicted - individual (Predicted - individual.Central.Drug_Central), Observed (Observed.CentralConc). Hidden axes object 6 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent Predicted - individual (Predicted - individual.Peripheral.Drug_Peripheral), Observed (Observed.PeripheralConc).

Plot the conditional weighted residuals (CWRES) and individual weighted residuals (IWRES) of model predicted values.

plotResiduals(NLMEResults,'predictions')

Figure contains 3 axes objects. Axes object 1 with ylabel PeripheralConc contains 3 objects of type line. One or more of the lines displays its values using only markers Axes object 2 with ylabel CentralConc contains 3 objects of type line. One or more of the lines displays its values using only markers Hidden axes object 3 contains 2 objects of type line. One or more of the lines displays its values using only markers These objects represent IWRES, CWRES.

See Also