A. Normal Fitting - Two Compartment I.V. Bolus Model
After a 250 mg iv bolus administration the data below were collected.
Plotting the data on semi-log graph paper gives the `X' below.
Semi-log linear regression of the last five data points gives; B = 5.8 mg/L and [[beta]] = 0.089 hr. The residual line results in; A = 9.3 mg/L and [[alpha]] = 2.3 hr. Manipulation of these results gives the values; k21 = 0.938 hr, k10 = 0.218 hr, k12 = 1.233 hr, and V1 = 16.6L. The next step is to define the model on paper. This is shown at the right.
Number each component (circles) of the model in sequence. The central compartment is represented by component 1. The tissue compartment is component 2. There is one data set or data line. This is Data Set Number 1. The model is further defined by three first order rate constants (arrows) and one apparent volume of distribution (triangle). The apparent volume of distribution relates the amount of drug in the central compartment to the plasma concentrations in the data set. Including the dose there are five parameters needed to fully define the model. These are summarized in the table, below.
Start Boomer and select Keyboard entry. Select normal fitting (0), screen output (0) and proceed to parameter definition and entry. Enter each of the parameters from the table below. Enter -1 for parameter type to complete this section. Choose Runge-Kutta-Fehlberg (2) as the numerical integration technique and press return twice to accept the default values for relative and absolute error. Choose Damping Gauss-Newton (1) as the fitting algorithm and again press return twice to accept the default values of DT and PC. Enter a title for this run `Two compartment Fit to Data' and select keyboard data entry. Enter the x-value and y-value data. Enter -1 for x-value to finish the data entry. Press return (for 0), unless you need to correct some of the data points. Press return if you don't want to save the data. If you want to save the data enter a file name and choose the required format for the x and y-values.
The available formats are G14.1 to G14.6 and F14.0 to F14.6. All of these formats produce data within 14 columns. The G format changes from fixed format to scientific format as necessary and is better if a wider range of values are expected. The number after the decimal point indicates the number of digits saved. The F format is fixed format with the number after the decimal point indicating the number of digits after the decimal point in the number. Examples include
G14.2 12. 1.2 .12E+10 .12E-04 G14.4 1234. 12.34 .1234E+10 .1234E-05 G14.6 123456. 12.3456 .123456E+10 .123456E-06 F14.0 12. 1. 0. F14.2 12.34 1.234 .12 F14.4 12.3456 1.2345 .1234 F14.6 12.345678 1.234567 .123456 *******.******
The last entry indicates any number greater than 9999999.999999 in F14.6 format.
A variety of data weighting schemes is possible. Enter 0 to select equal weights. The program will now undertake the data analysis and finally produce the output information. This includes the model specification, final parameter values and calculated data. Choose 0 for AUC line and enter 2 to produce a linear and semi-log plot of the observed and calculated data, and weighted residual plots. This completes the analysis.
A list of the keyboard entries required for this analysis are included in the file Tutorial_1a.BAT. You may want to either open the file in the editor or print it out for reference. It is also possible to run this analysis by starting Boomer, selecting Batch File run and enter Tutorial_1a as the filename.
An alternative approach to the analysis of these data is to fit the data with a sum of exponential equation such as:
This is carried out in the same way as the first example except for the model definition and the lack of numerical integration routine (and error terms) specification.
B. Bayesian Analysis - I.V. Infusion with kel function of Creatinine Clearance
This example set involves the analysis of plasma concentrations measured after an i.v. infusion of 200 mg/hr for 30 minutes to a patient with a creatinine clearance of 30 ml/min. The measured concentrations were:
Plasma concentration 1 hour after the start of the infusion = 4.0 mg/L Plasma concentration 12 hours after the start of the infusion = 1.7 mg/L
From previous experiments with this drug in similar patients it has been found that:
kel = a * CL(Cr) + b
where a = 0.0028 +/- 0.00028 b = 0.02 +/- 0.002 and V(1) = 21 +/- 2.1 L
We can set up the model first on paper.
Component 1 represents the one compartment model with a first order elimination rate constant k10. Linking the amount of drug in component 1 with the concentration in data set 1 is the apparent volume of distribution, V1. By putting the value of creatinine clearance in component 2 without any rate constants (it will stay fixed) we can now calculate k10 as a function of creatinine clearance (using parameter type 23).
Start Boomer and select Keyboard entry. Select Bayesian fitting (1), screen output (0) and proceed to parameter definition and entry. Enter each of the parameters from the table below.
After entering all the parameters enter -1. Choose integration method 2 and choose the default for the relative and absolute error terms by pressing return. Use the Marquardt fitting algorithm and the default DT and PC values. After entering the title for this analysis choose data from keyboard (1). Enter time and Cp values of 1 hr, 4 mg/L and 12 hr, 1.7 mg/L, respectively. Accept the data `as is' unless you need to make a correction. Continue without saving the data. With a Bayesian analysis a little more attention needs to be taken regarding the weighting of the data. When the parameters were entered a standard deviation value for each parameter is entered and then used as a weighting factor during the fitting process. We must also use a `suitable' weighting scheme for the plasma data. One scheme might be to assume that the standard deviation for each data point is 10% of the data value. Thus
Standard Deviation = 0.1 x Value
Variance = 0.01 x Value
And if the weight is to be 1/Variance a weighting scheme type 3 may be most useful where
Weight = 1/(a x Value) with a = 0.01 and b = 2.
A batch file, Tutorial_2.BAT, includes all the commands for this analysis.
C. Working with Batch (.BAT) files - Simulation
In the two previous tutorials all the program entries were made from the keyboard. There are two other possibilities when working with Boomer (and MultiForte). A batch file can be created while you are entering the instructions from the keyboard or a batch file can be used as the source of all the instructions. Once a batch file is created it is possible to edit the batch file and perform a sequence of similar runs.
For example, fitting data from several subjects to the same model can be accomplished by creating a batch file during the analysis of the first subject. This batch file can be duplicated for as many subjects as available and edited appropriately.
Another example might to perform a sequence of simulations with different values for selected parameters of the model. Thus you could simulate drug effect after an i.v. bolus administration using a two-compartment model. By editing the resultant batch file the effect of varying some of the parameters can be determined. For example the effect of ke0 on the Effect versus time curve can be explored.
Start Boomer and select Keyboard --> .BAT entry. Select Simulation (2), file output (1) and enter Tutorial_3 as the output file name. Proceed to parameter definition and entry. Enter each of the parameters from the table below.
After entering all the parameters enter -1. Choose numerical integration method 2 and select the default values for relative and absolute errors. After entering a suitable title enter 1 for data from keyboard and enter time values of 0, 0.5, 1, 2, 3, 4, 6, 9, and 12 hours for each data set. Enter zero for the y-values. Enter 2 to plot the simulated data for concentration in the central compartment and the effect. The batch file Tutorial_3.BAT includes all the entries needed for this simulation.
After the first simulation, enter -2 to quit the run without exiting the program. Using the editor it is possible to alter the value of ke0, save the new batch file and then choose batch file run to the new simulation.
D. Simultaneous Fit to IV and Oral Data - IRWLS
Boomer (and MultiForte) will allow the simulation and analysis of up to 20 data set lines. Boomer also allows setting a parameter equal to another parameter or pair of parameters. Thus fitting iv and oral data simultaneously is not difficult. Again, the first step is to develop the model on paper. For example a model with separate iv and oral administration is shown in the figure below.
There will be two types of parameter dependencies used to define this model. First, we can set the kel_po and V_po equal to kel_iv and V_iv, respectively. Thus the values of kel_iv and V_iv will be adjustable and the oral parameter values will follow along during the analysis. The second type of dependence is a double dependence. The oral dose and F value can be set-up as `dummy' parameter and a type 1 (dose/initial condition) can be set equal to F x Dose_po. This way the dose_po can be fixed at the dose given and the bioavailability, F, can be adjustable.
For this tutorial we will use the iteratively reweighted least squares method. This is very similar to the normal fitting method except that the data weight is recalculated during the fitting process as the calculated concentration changes. Additionally the iv and oral data will be read from two disk files, Tutorial_4iv.DAT and Tutorial_4po.DAT, respectively.
Start Boomer and select Keyboard entry. Select IRWLS (3), file output (1) and enter Tutorial_4 as the output file name. Proceed to parameter definition and entry.
Enter each of the parameters from the table above. After entering all the parameters enter -1. Choose numerical integration method 2 and select the default values for relative and absolute errors. Choose the simplex fitting algorithm (3) and select the default value for PC. After entering a suitable title enter 0 for data from disk file. Enter disk filenames for the iv data and the oral data, respectively (or select these files from the `Open' dialog). Choose weight type 2, (1/val), for both data set lines. Once the fitting is complete choose 1 then 2 to calculate the AUC, AUMC, and MRT for the iv and oral data sets. These sets are included in the file Tutorial_4.BAT.
E. Simulation with Error and Monte-Carlo Simulations
On occasion it may be useful to simulate models and be able to add error to the data. On other occasions it may be useful to simulate data where the parameters have different, random values. This is possible with the Simulation with Error option. Also, by editing a previously created batch file it is possible to perform many repeated simulations or analyses. In this tutorial you will simulate a two-compartment model after iv bolus administration and add a random error to the three rate constants. Once the initial batch file is created it will be edited to allow repeated simulation and production of a series of data files.
Start Boomer and select Keyboard entry creating a .BAT file (2). Name this .BAT file, Tutorial_5. Select Simulation with random error (3), screen output (0). Proceed to parameter definition and entry. Enter each of the parameters from the table below.
After entering all the parameters enter -1. Choose numerical integration method 2 and select the default values for relative and absolute errors. After entering a suitable title for the run enter the times 0, 0.5, 1, 2, 3, 4, 6, 9, and 12 with zero for the y-value from the keyboard. Accept the data (0) and don't save the data at this point. Enter error type 0 so no error is added to the data calculated. After the simulation enter 0 for the AUC line and 1 to save the calculated data. Enter Tutorial_5 as the data file name (use tut5 with DOS version) and -2 to finish the simulation run. This gives us an initial simulation run with one data set. The next step is to edit the file, Tutorial_5.BAT by adding a new second line with ` 10'. The first four lines are now:
Boomer Batch File 10 4 wls,bayes,sim,irwls,sim+error,grid 0 Screen, diskfile, printer
This will cause Boomer to run the simulation 10 times with different values for k10, k12, and k21, each time producing a new Tutorial_5 x.DAT file where the x is the sequence number.
F. Grid Search Method
The final tutorial will use the grid search method of fitting a set of data. This method breaks the range from lower to upper limit into a number of steps. With more than one parameter a number of grids are produced and the program will methodically calculate the WSS at each point on each grid. This can be quite slow unless a coarse grid is used. Again using the iv bolus two compartment model you can explore the grid search method by allowing the parameters k12 and k21 to be `grid searched'.
Start Boomer and select Keyboard entry. Select Grid Search Method (5) and screen output (0). Proceed to parameter definition and entry. Enter each of the parameters from the table below.
After entering all the parameters enter -1. Choose numerical integration method 2 and select the default values for relative and absolute errors. After entering a suitable title enter the data from the keyboard and -1 at the end of data entry. Accept the data and continue without saving. Enter weight type 1 then 1 to save a WSS file to disk. Next enter 20 as the maximum WSS for the grid. Enter a filename for the WSS file, however, only the first three letters are significant. After the run is complete the file Tut01_02.WSS contains the WSS data in a comma delimited format. Tut01_02.SFF, a SYLK format file, is also produced. The batch file, Tutorial_6.BAT, includes the steps required to perform this analysis.
Next Chapter - Chapter SEVEN
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