Considerable research has been done in the area of optimal designs for linear models. Most methods involve computation of the variance covariance matrix.
The “optimal” design is usually one in which replicate samples are taken at a lim-ited number of combinations of experimental conditions. Unfortunately, these methods are of little or no value when designing experiments involving nonlinear models for a number of reasons, including:
» It can be difficult or, in the case of a pharmacokinetic study, impossible to obtain replicate observations.
» The primary interest often is not in the model parameters but in some func-tions of the model parameters such as AUC, t1/2, etc.
When Phoenix performs a simulation the output includes information on precisely how parameters in the model can be estimated for specified values of the inde-pendent variables such as time.
Comparison of two designs
Assume that a study is being planned and that the data produced by this study should be consistent with Phoenix PK model 3. Assume also that the parameter values should be approximately:
V_F 10
K01 3
K10 0.05
or
0, 0.5, 1, 2, 4, 8, 12, 24, and 36 hours.
Simulation can be used to determine which set of sampling times would produce the more precise estimates of the model parameters. This example will use Phoenix to simulate the model with each set of sampling times, and compare the variance inflation factors for the two simulations.
Create a new project:
1. Select File > New Project to create a new project. A new project is created in the Object Browser.
2. Name the new project Study Design.
The data set
First create a data set with the following column headers and data:
Group Times
Create the data set:
1. Right-click the Data folder in the Object Browser and select New > Work-sheet.
2. Name the new worksheet Example Data.
The new worksheet is automatically displayed in the Grid tab, which is located in the right viewing panel.
The Columns tab is located underneath the Grid tab. The Columns tab is used to add columns to a worksheet.
3. Click the Add button underneath the Columns box. The New Column Proper-ties dialog is displayed.
The New Column Properties dialog is used to define the data type and the name of a new column.
4. The Numeric option button is selected by default. Do not change this setting.
5. In the Column Name field type Group and click OK.
A new column is displayed in the Columns box and in the Grid tab.
6. In the first cell under Group, type 1 and press ENTER. Repeat for cells 2 through 9.
7. In cells 10 - 18 type 2 in the Group column.
8. Click the Add button underneath the Columns box.
9. In the Column Name field type Times. Leave the data type set to Numeric and click OK.
10.Type the values from the Times column in the table on page 160 in the cells in the Times column.
The finished worksheet looks like the table on page 160.
• Users can also import the data set Example Data.csv from the Phoe-nix examples directory, which by default is located at C:\Program
Files\Pharsight\Phoenix\application\Examples.
Insert and map the PK model
1. Select the workflow in the Object Browser and then select Insert > tWNL5 Classic Modeling > PK Model.
The PK Model object is added to the workflow in the Object Browser.
2. Map the data set Example Data as the input source for the PK Model object:
• Use the pointer to drag the Example Data worksheet from the Data folder to the PK Model object’s Main Mappings panel.
OR
• In the PK Model Main Mappings panel click the Select source button to open the Select Object dialog.
• Select the Example Data worksheet and click Select.
The Example Data data set is mapped to the PK Model object.
3. Use the option buttons in the Main Mappings panel to map the data types to the following contexts:
• Map Group to the Sort context.
• Map Times to the Time context.
4. Use the Model Selection tab to specify which PK model Phoenix uses in the analysis. The Model Selection tab is located underneath the Setup tab.
5. Select the Number 3 model check box in the Options tab.
6. Select the Simulation check box in the Options tab.
7. In the Y Units field, type ng/mL.
Enter the dosing data
1. Select the PK Model's Dosing panel.
2. Select the Use internal Worksheet check box.
The Select sorts dialog is displayed. The Select sorts dialog prompts a user to select the sort variables to use to create the internal dosing worksheet.
Select sorts dialog
3. Click OK to accept the default sort variable.
4. In the Time column type 0 for both groups.
5. In the Dose column type 100 for both groups.
Note: The number of rows in the Group column corresponds to the number of doses received. For example, if group 1 had 10 doses, there would be 10 rows of dosing information for group 1. In Phoenix this grouping of data is referred to as stacking data.
6. Select the Weighting/Dosing Options tab to specify settings for the PK Model dosing options.
Dosing options are located in the Dosing area in the Weighting/Dosing Options tab.
7. In the Unit field type mg.
Model parameters and simulation
Parameter values must be specified for simulations.
1. Select the Parameter Options tab, which is located underneath the Setup tab.
• The User Supplied Initial Parameter Values option button is selected by default. This setting cannot be changed.
• The Do Not Use Bounds option button is selected by default. This setting cannot be changed.
Selecting the Simulation check box makes the parameter calculation and bound-ary selection options unavailable. If the Simulation check box is selected then users must supply initial parameter values, and parameter boundaries are not used.
Enter the initial estimates:
1. Select the PK Model's Initial Estimates panel.
2. Select the Use internal Worksheet check box.
The Select sorts dialog is displayed. The Select sorts dialog prompts a user to select the sort variables to use to create the internal dosing worksheet.
3. Click OK to accept the default sort variable.
• V_F = 10
• K01 = 3
• K10 = 0.05
All the settings are complete and the model can be executed.
5. Click the Execute button. The results are displayed on the Results tab.
Results
The variance inflation factors (VIF) for each dosing scheme (groups 1 and 2) are located in the Final Parameters worksheet in the PK worksheet results, and are summarized in the following table.
In practice it is useful to vary the values of V_F, K01, and K10 and repeat the sim-ulations to determine if the first set of sampling times consistently yields less pre-cise estimates than the second set.
Designing the sampling plan
Note that for the parameters V_F and K10, the estimated variances would be approximately 15% lower using the second set of times, while the difference is much more dramatic for the parameter K01. These sets of variance inflation fac-tors indicate that the second set of sampling times would provide tighter esti-mates of the model parameters.
The partial derivatives plots for this model explain this result. The locations at which the partial derivative plots reach a maximum or a minimum indicate times the model is most sensitive to changes in the model parameters, so one approach to designing experiments is to sample where the model is most sensi-tive to changes in the model parameters.
Parameter Estimate Group 1 VIF Group 2 VIF
V_F 10 0.779 0.657
K01 3 68.48 1.176
K10 0.05 0 0
Partial Derivatives plot Group 1
Partial Derivatives plot Group 2
Note that in the first plot of the partial derivatives the model is most sensitive to changes in K10 at about 20 hours. Both sampling schemes included times near 20 hours, so therefore the two sets of sampling times were nearly equivalent in
For both V_F and K01 the model is most sensitive to changes very early, at about 0.35 hours for K01 and about 1.4 hours for V_F. The first set of sampling times does not include any post-zero points until hour 3, long past these areas of sensi-tivity. Even the second set of times could be improved if samples could be taken earlier than 0.5 hours.
• This same technique could be used for other models in Phoenix or for user-defined models.
Note: It is not necessary to keep a project open after completing each chapter. This project is not required when working in the next chapter. To close a project right-click the project and select Close Project.