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## Mwpharm document

MediWare BV, Groningen, The Netherlands
MW\Pharm: Version 3.30
Date:

KinPop: Iterative two-stage Bayesian population procedure
A completely new module 'KinPop' has been added to MW\Pharm. KinPop performs a
population pharmacokinetic analysis by an 'Iterative two-stage Bayesian population procedure'
(ITSB).
The ITSB procedure in KinPop is similar to that described by Prévost (ref. 1), which has been
used also by others (ref. 2-6). It has been demonstrated that the ITSB procedure performs
well in comparison to other methods including standard two-stage (STS), global two-stage
(GTS), and nonlinear mixed-effect modelling (NONMEM) (ref. 2-5).
The ITSB procedure is close to the method in the program IT2B, the 'front end' program of
NPEM2 in USC*PACK (ref. 7). Details of the procedures in IT2B have not been published.
An example of the practical use is given in ref. 6.
An overview of the various methods for population analysis can be found in ref. 8.
For a detailed description of several procedures, and a comparison by Monte Carlo simulation,
see ref. 2-5.
Principle
The principle of ITSB may be understood from the following scheme:
Assume a reasonable set of population data (e.g. from STS):
means and SDs (or complete variance-covariance matrix).
Step 1 Perform Bayesian analysis on each subjects separately
Step 2 Calculate new set of population data
means and SDs (or complete variance-covariance matrix) Repeat steps 1 and 2 until convergence is reached. (in KinPop, each repetition of steps 1 an 2 is called a 'cycle'). If convergence is reached, the assumed population data, as used in step 1, are equal to the population data obtained in step 2. It can be shown by Monte Carlo simulation (see elsewhere) that this procedure provides
accurate and precise estimates of the means and SDs (or complete variance-covariance matrix)
of the population.
The final outcome is not dependent on the initial set of population means and SDs (or
complete variance-covariance matrix). However, small insignificant differences will occur due
to rounding off and the stop criteria. In exceptional cases the procedure may convergence to a
different solution (local minimum). The 'optimal' solution is the parameter set with the highest
value for the summed log-likelihood.
To diminish the risk of convergence in a local minimum, it is advised to start the analysis with
a reasonable set of initial estimates. Also, one may start the analysis with a different set of
initial estimates.
User manual
KinPop can be started via the KinMenu (Main menu option 8, 3).
1 Make patient selection file
First, a selection of patient records must be made. The patient database window is shown
(similar to that in the Main menu, option 1), and patients can be selected by pressing the Space
bar. Marked patients are indicated in bold. After <F1> or <F10> the patient selection is saved
automatically to a file. A previously saved patient selection can be loaded and modified.
A previously saved patient selection can be loaded. This option is also useful for resetting all
values to their original values before restarting a population analysis.
The data of the selected drug, which is read from the first patient of the patient selection, are
show and can be changed if necessary.
The following drug data are used during the population analysis:
- population mean values and SDs of model parameters are used as initial estimates of the
population means and SDs. Missing SDs are replaced automatically by the corresponding
population mean (CV 100%).
(KinPop does not check whether or not the data of each patient refers to the same drug; this
must be done by the user!)
- SD assay error
- distribution of assay error (normal or log-normal)
- distribution of parameters (see option 4)
4 Parameters distribution
Normal distribution or log-normal distribution.
The choice made here is similar to that in the medication screen (after <PgDn>, last line).
In general, a log-normal distribution is more suited than a normal distribution for
pharmacokinetic parameters. The actual distribution can be checked in the output after the
fitting procedure.
Even more important than the choice between normal and log-normal distribution is that
Bayesian forecasting should be performed with the same type of distribution as was used for
the generation of the population data. The type of distribution is saved with the population
data (see below).
5 New population parameters based on
Options:
- mean  sd
- median and df(mean) (see below option 9, and manual of NPEM2)
The option 'means  sd' is the default and more natural choice. In the manual of NPEM2,
Prof. R. Jelliffe advises the use of 'median and df(mean)' as the best population values to be
used in Bayesian forecasting. However, using this option, the calculation may converge less
efficiently (see below).
6 Output to file
Intermediate and final results of calculations can be saved in a text file if a filename is chosen
(extension 'PRT').
This may be useful for checking irregularities, but is not necessary for normal procedures
(default: results not saved in a text file).
7 Mode
KinPop will stop to allow user interaction or to show intermediate results:
1 interactive mode
2 pause after each patient and after each cycle
3 pause after each cycle
4 automatic mode (does not pause)
The automatic mode is the default setting.
8 Stop criterion
KinPop stops the iterative procedure if the relative change of each of the calculated population
values is less than the stop criterion (default is 0.0001).
9 Start KinPop (Iterative two-stage procedure)
First the screen of the fitting procedure of the medication history (similar to that of option 5 of
Main menu) is shown. The entries and selections are now used for all patients (except for the
interactive mode, see option 7).
If 'Bayes' is set OFF, KinPop performs a standard two-stage procedure (STS).
If a parameter is set 'fixed', its value will not be fitted, and the population value will be kept at
its initial value.
Initial individual parameter estimates are ignored. For each patient and for each cycle, the
initial parameter values are set equal to the population mean.
After <F10> the correlation matrix between the population parameters is shown. A detailed
description is given below. Default: all correlations are '0' and 'fixed'.
After again <F10> the iterative analysis starts. The screen shows the progress in the current
cycle number, current patient, and parameter set of current patient.

If the stop criterion is reached, or if the calculation is interrupted by the user by pressing
<F1>, the results are shown (see below).
The new population values (means and SDs) are copied automatically to the medication
screen. To save these values, e.g. for future Bayesian forecasting, the drug data have to be
save to the database by <Ctrl><s> in the medication screen (option 3 in KinPop, or option 2 in
Main menu). To keep old values as well, choose a new name for the drug.
Overview of results shown on successive screens:
Table 1
- old population values (initial values of Bayesian procedure)
- new population values (shown in yellow):
mean
(for log-normal distribution: geometric mean) (for log-normal distribution: log-sd * mean) median median of individual parameters. df(mean) (DF50) standard deviation of a (log-)normal distribution which has the same values for the 25- and 75-percentiles as the actual distribution (see also manual of NPEM2). (DF95) idem, for the 2.5- and 97.5-percentiles. skewness of the distribution (a true (log-)normal distribution gives a value 0; a negative value indicates a skewness to the left, a positive value to the right). kurtosis of the distribution (a measure of the degree of similarity with the shape of a (log-)normal distribution; a true (log-)normal distribution has a value of 3).
Also the summed 'log-likelihood' of all patients is given.
Table 2
- Correlation matrix of the individual patient parameters (Covarj,k is calculated as given below,
without the factor Covarj,k(i) in the equation).
- Correlation matrix of t he population parameters 'calculated' (Covarj,k is calculated as given
below).
- Correlation matrix of the 'actual' population parameters, including fixed values replacing the
'calculated' values Covarj,k; This is the correlation matrix of the population parameters which is
used during the Bayesian fit procedure, and is copied to the medication screen (and can be
saved to the drug database).
Graphs
- For each parameter a histogram of the individual values of each patient. This allows an 'eye
'bimodal').
The graph is scaled automatically, but the scaling can be changed, including the number of
bars. The X-axis is scaled logarithmically if a log-normal distribution is chosen (see option 4).
For comparison, also the frequency distribution the (log-)normal distribution corresponding to
the calculated population parameters is shown.
- For each combination of two parameters a scatterplot: for each patient a circle is plotted .
This allows an 'eye judgement' of the joint distribution of both parameters, e.g. correlation and
clustering.
For comparison, also the boundaries of the 50% and 95% probabilities corresponding to the
calculated distribution and correlation of both parameters are shown, i.e. the 50% and 95% of
the patients should lie within these boundaries.
- The change of the calculated population parameters during the iterative procedure.
Parameter values are scaled relative to the initial value.
- Idem, for the calculated standard deviations of the population parameters. Parameter values
are scaled relative to the initial value.
- Idem, for the correlation coefficients between the calculated parameters (actual values).
- Idem, for the summed 'log-likelihood' (actual value).
Table 3
For all selected patients the individual parameter values.
Showing the results can be interrupted by pressing <F1>.
All results are shown again after pressing <a> in the KinPop menu.
Description of the procedures
Method meanSD
The population values of the pharmacokinetic parameters are calculated as the mean of the
individual parameters of N patients (j).
The standard deviations of the population values are calculated as the square root of the
variance (Covarj,j).
The covariance matrix of the population parameters is calculated from:
[ ( Pj(i) - P j ( ) Pk(i) - Pk Co covariance of population parameters j and k mean value of parameter j (in N patients) covariance of parameters j and k in patient i In case of a log-normal distribution of the parameter, the values Pj refer to the logarithm (base e) of the parameter value. The standard error of parameter j in patient i, as shown after fitting of each patient is equal to the square root of Covarj,j(i)). Note: The individual parameter estimates obtained by the Bayesian procedure are different from the estimates of that parameter obtained with 'Bayes OFF'. The parameter values are moved into the direction of the population mean, and represent the most likely values given the measurement for that patient and given the distribution of the parameter in the population; this is the Bayesian principle. The less information is obtained for the individual patient, the more the parameter for that patient is moved to the population mean. The more information is obtained for the individual patient, the more the parameter for that patient moves to the value which would be obtained with 'Bayes OFF'. As a result, the standard deviation of the individual parameter values is not an unbiased estimate of the population standard deviation: it is too small. By taking into account the term Covarj,k(i) in the equation given above, an unbiased estimate of the interindividual standard deviation is obtained. The correlation matrix of the population parameters is calculated from: correlation coefficient between parameters j and k
Method medianDFmean
The population values of the pharmacokinetic parameters are calculated as the median of N
patients.
The standard deviations of the population parameters are calculated from:
DFmean average value of DF50 and DF95 DF50 standard deviation of a normal distribution which has the same values for the 25- and 75-percentiles as the actual distribution (similar values are used in the NPEM2 program of USC*PACK) The correlation matrix of the population parameters is calculated as indicated for the method 'meansd'. Correlation between population parameters ITSB can be performed in various ways with respect to the correlation between population parameters: (A) Assuming NO CORRELATION between the population parameters. In this case the 'classical' Bayesian fitting is applied. The correlations between the population parameters are calculated from the individual patient parameters and shown on the screen, but these values are not used or copied to the medication screen. (B) Assuming a KNOWN (fixed) CORRELATION between the population parameters. The Bayesian fitting procedure takes this correlation into account. Before the iterative procedure starts, the fixed values for the correlations can be entered and set 'fixed'. As in case A), the correlations between the population parameters are also calculated from the individual patient parameters and shown on the screen, but these values are not used or copied to the medication screen. (C) Assuming an UNKNOWN CORRELATION between the population parameters. ITSB estimates the correlation during the iterative procedure. The Bayesian fitting procedure takes this correlation into account. Before the iterative procedure starts, initial values can be entered (a value '0' is the most objective choice), and set 'variable'. If appropriate, some correlations can be set 'variable' and others 'fixed' (see under B). The correlations between the population parameters are calculated from the above equation (Covarj,k). The values are shown with the results, and are copied to the medication screen. The boundaries for correlations are -0.999 to +0.999 (see below). Method C) performs the analysis with less assumptions than methods B) and C). As a result of the increased flexibility, irregularities may occur occasionally. In exceptional cases, the correlation between two parameters increases, and may approach -1 or +1 (for technical reasons the actual boundaries in the program are -0.999 and +0.999). Such extremely high correlations are unlikely, and should be regarded carefully. References:
1. Prévost G. Estimation of a normal probability density function from samples measured with
non-negligible and non-constant dispersion. Internal report, Adersa-Gerbios, 2 Avenue du 1er
Mai, F-91120, Palaiseau, June 1977.
2. Steimer JL, Mallet A, Golmard JL, Boisvieux JF. Alternative approaches to estimation of
population pharmacokinetic parameters: comparison with the nonlinear mixed-effect model.
Drug Metab Rev 1984;15:265-292.
3. Racine-Poon A, Smith AFM. Population models. In: Statistical Methodology in the
Pharmaceutical Sciences, Chapter 5. Ed. Berry DA. Marcel Dekker Inc, New York, 1990.
4. Mentré F, Gomeni R. A two-step iterative algorithm for estimation in nonlinear
mixed-effect models with an evaluation in population pharmacokinetics. J Biopharm Stat
1995;5:141-158.
5. Bennett JE, Wakefield JC. A comparison of a Bayesian population method with two
methods as implemented in commercially available software. J Pharmacokinet Biopharm
1996;24:403-432.
6. Forrest A, Ballow CH, Nix DE, Brimingham MC, Schentag JJ. Development of a
population pharmacokinetic model and optimal sampling strategies for intravenous
ciprofloxacin. Antimicrob Agents Chemother 1993;37:1065-1072.
7. Jelliffe RW, Schumitzky A, Van Guilder M. User manual for version 10.7 of the
USC*PACK collection of PC programs. Laboratory for Applied Pharmacokinetics, USC
School of Medicine, Los Angeles, CA, December 1, 1995.
8. Rosenbaum SE, Carter AA, Dudley MN. Population pharmacokinetics: fundamentals,
methods and application. Drug Dev Ind Pharm 1995;21:1115-1141.