The basic premise for the clinical utility of pharmacokinetics is that there is a clearly defined relationship between drug concentration in readily available samples and drug response. Application of pharmacokinetic methods allows us to account for the variability in an individual's ability to absorb, distribute, metabolize and excrete drugs. The objective of these methods is to control the drug concentration in blood or plasma and potentially other fluids and tissues. For this approach to be effective there needs to be a relationship between these concentrations and the response to the drug. In some cases the response is direct and reversible and the Hill equation or some variation of it may be be applied.
In general the receptor might be 'mathematically' within the central compartment, a peripheral compartment or a separate effect compartment. In each case the relationship between concentration at the receptor and the response might be described with a Sigmoid Emax model (Hill equation)
In Equations 1 and 2, Emax is the maximum response, CR, 50% is the concentration which produces a 50% of maximum response (often called EC50%) and γ is a slope factor. For responses which increases from zero, Equation 1 is more appropriate. In other cases where the response is an increase or decrease from a baseline value Equation 2 may be a better choice. Another version of this equation, Equation 1, is the Emax model but this is just the same except that γ is set to 1.
A two compartment PK/PD model with an effect compartment
| Parameter | Value | Units |
|---|---|---|
| Dose | 200 | mg |
| k10 | 0.2 | hr-1 |
| k12 | 1.3 | hr-1 |
| k21 | 1.1 | hr-1 |
| V1 | 12.0 | L |
| k1e | 0.01 | hr-1 |
| -k1e | -0.01 | hr-1 |
| keo | 0.8 | hr-1 |
| Emax | 100 | percent |
| CR, 50% | 0.25 | mg/L |
| γ | 1.5 |
A two compartment PK/PD model with a response compartment
>cd /Directory/For/This/Analysis
>boomer
Boomer v3.4.5
Ref: Comput.Meth.Prog.Biomed.,29 (1989) 191-195
David W.A. Bourne
Copyright 1986-2018 D.W.A. Bourne
See http://www.boomer.org or email david@boomer.org for more info
Based on the original MULTI by K. Yamaoka, et.al.
J. Pharmacobio-Dyn., 4,879 (1981); ibid, 6,595 (1983); ibid, 8,246 (1985)
DATA ENTRY
0) From KEYBOARD
1) From .BAT file -1) From .BAT file (with restart)
2) From KEYBOARD creating .BAT file
3) From .BAT file (quiet mode) -3) From .BAT file (quiet mode-with restart)
4) to enter data only 5) to calculate AUC from a .DAT file
-9) to quit -8) Registration Information
Enter choice (0-5, -1, -3, -8 or -9) 1
Enter .BAT filename
sim
The input .BAT filename is sim.BAT
METHOD OF ANALYSIS
0) Normal fitting
1) Bayesian
2) Simulation only
3) Iterative Reweighted Least Squares
4) Simulation with random error or sensitivity analysis
5) Grid Search
-5) To perform Monte Carlo run (Only once at the start of BAT file)
-4) To perform multi-run (End of BAT file only)
-3) To run random number test subroutine
-2) To close (or open) .BAT file
-1) To finish
Enter choice (-3 to 5) 2
Where do you want the output?
0) Terminal screen
1) Disk file
Enter choice (0-2) 1
Enter .OUT filename
The output .OUT filename is sim.OUT
Enter the parameters for the run.
Parameters currently available in Boomer for model building
Enter type# for parameter 1 (-5 to 51) 1
Enter parameter name Dose
Enter Dose value 200.0
Enter component to receive dose 1
Enter component for F-dependence ( 1 to - 1 or 0 for no dependence) 0
Input summary for Dose (type 1)
Fixed value is 200.0
Dose/initial amount added to 1
Enter 0 if happy with input, 1 if not, 2 to start over 0
Enter pharmacokinetic parameters (k10, k12, k21 and V1) for the two compartment model. Note the line name, Cp, for compartment one.
Enter type# for parameter 2 (-5 to 51) 2
Enter parameter name k10
Enter k10 value 0.2000
Enter component to receive flux 0
Enter component to lose flux 1
Input summary for k10 (type 2)
Fixed value is 0.2000
Transfer from 1 to 0
Enter 0 if happy with input, 1 if not, 2 to start over 0
Enter type# for parameter 3 (-5 to 51) 2
Enter parameter name k12
Enter k12 value 1.300
Enter component to receive flux 2
Enter component to lose flux 1
Input summary for k12 (type 2)
Fixed value is 1.300
Transfer from 1 to 2
Enter 0 if happy with input, 1 if not, 2 to start over 0
Enter type# for parameter 4 (-5 to 51) 2
Enter parameter name k21
Enter k21 value 1.100
Enter component to receive flux 1
Enter component to lose flux 2
Input summary for k21 (type 2)
Fixed value is 1.100
Transfer from 2 to 1
Enter 0 if happy with input, 1 if not, 2 to start over 0
Enter type# for parameter 5 (-5 to 51) 18
Enter parameter name V1
Enter V1 value 12.00
Enter data set (line) number 1
Enter line description Cp
Enter component number (0 for obs x) 1
Input summary for V1 (type 18)
Fixed value is 12.00
Component 1 added to line 1
Enter 0 if happy with input, 1 if not, 2 to start over 0
Next we enter the parameters for the effect compartment. Note that k1e is described as k1e/V1 so that the drug in the effect compartment is in concentration units relative to the concentration in compartment 1, the plasma compartment. Also inclusion of the -k1e term means that the effect compartment amount/concentration doesn't influence the pharmacokinetic results.
Enter type# for parameter 6 (-5 to 51) 2
Enter parameter name k1e
Enter k1e/V1 value 0.0100
Enter component to receive flux 3
Enter component to lose flux 1
Input summary for k1e (type 2)
Fixed value is 0.0100
Transfer from 1 to 3
Enter 0 if happy with input, 1 if not, 2 to start over 0
Enter type# for parameter 7 (-5 to 51) 2
Enter parameter name -k1e
Enter -k1e/V1 value -0.0100
Enter component to receive flux 0
Enter component to lose flux 1
Input summary for -k1e (type 2)
Fixed value is -0.0100
Transfer from 1 to 0
Enter 0 if happy with input, 1 if not, 2 to start over 0
Enter type# for parameter 8 (-5 to 51) 2
Enter parameter name ke0
Enter ke0 value 0.8000
Enter component to receive flux 0
Enter component to lose flux 3
Input summary for ke0 (type 2)
Fixed value is 0.8000
Transfer from 3 to 0
Enter 0 if happy with input, 1 if not, 2 to start over 0
Finally we can enter the Hill equation parameters, eMax, EC50% and gamma (here S or i for slope term). The last two parameters, Receptor and Tissue are included here so we can 'sample' the concentration or amount in the receptor and tissue components of the model.
Enter type# for parameter 9 (-5 to 51) 11
Enter parameter name Effect
Enter Emax or a Effect value 100.0
Enter data set (line) number 2
Enter line description Effect
Enter component number (0 for obs x) 3
Input summary for Emax or a Effect (type 11)
Fixed value is 100.0
Component 3 added to line 2
Enter 0 if happy with input, 1 if not, 2 to start over 0
Enter Ec(50%) or b Effect value 0.2500
Input summary for Ec(50%) or b Effect (type 12)
Fixed value is 0.2500
Enter 0 if happy with input, 1 if not, 2 to start over 0
Enter S or i Effect value 1.500
Input summary for S or i Effect (type 13)
Fixed value is 1.500
Enter 0 if happy with input, 1 if not, 2 to start over 0
Enter type# for parameter 12 (-5 to 51) 18
Enter parameter name Receptor
Enter Receptor value 1.000
Enter data set (line) number 3
Enter line description Receptor
Enter component number (0 for obs x) 3
Input summary for Receptor (type 18)
Fixed value is 1.000
Component 3 added to line 3
Enter 0 if happy with input, 1 if not, 2 to start over 0
Enter type# for parameter 13 (-5 to 51) 18
Enter parameter name Tissue
Enter Tissue value 1.000
Enter data set (line) number 4
Enter line description Tissue
Enter component number (0 for obs x) 2
Input summary for Tissue (type 18)
Fixed value is 1.000
Component 2 added to line 4
Enter 0 if happy with input, 1 if not, 2 to start over 0
Next the numerical integration method is chosen and a description entered before entering the data. Compartmental pharmacokinetic models are typically not stiff systems so the Fehlberg RKF45 method is usually a good choice.
Method of Numerical Integration
0) Classical 4th order Runge-Kutta
1) Runge-Kutta-Gill
2) Fehlberg RKF45
3) Adams Predictor-Corrector with DIFSUB
4) Gears method for stiff equations with PEDERV
5) Gears method without PEDERV
Enter choice (0-5) 2
Enter Relative error term for
Numerical integration (0.0001) 0.000
Enter Absolute error term for
Numerical integration (0.0001) 0.000
Enter description for this analysis: Simulation of Effect (Two compartment Model)
Enter data from
0) Disk file
1) Keyboard
Enter Choice (0-1) 1
Enter data for Cp
Enter x-value (time) = -1 to finish data entry
X-value (time) 0.000
Y-value (concentration) 0.000
After entering values for each data set exit by entering -1 for the last X-value. For these simulations Y-values can be entered as 0. Check for errors, save or not and move to the next data set. After all the data sets are entered you can choose to calculate a AUC values or other options such as graphs.
X-value (time) 24.00
Y-value (concentration) 0.000
X-value (time) -1.000
Data for Cp
DATA # Time Concentration
1 0.000 0.000
2 0.1000E-01 0.000
3 0.2000E-01 0.000
4 0.5000E-01 0.000
5 0.1000 0.000
6 0.1500 0.000
7 0.2000 0.000
8 0.2500 0.000
9 0.5000 0.000
10 0.7500 0.000
11 1.000 0.000
12 2.000 0.000
13 3.000 0.000
14 4.000 0.000
15 6.000 0.000
16 9.000 0.000
17 12.00 0.000
18 15.00 0.000
19 18.00 0.000
20 21.00 0.000
21 24.00 0.000
Do you want to
0) Accept data
1) Correct data point
2) Delete data point
3) Insert new data point
4) Add offset to x-value
Enter choice (0-3) 0
Save Observed Data to Disk Module
0) Continue without saving
1) Save data for on disk
Enter choice 0
Calculation of AUC and AUMC section
0) Exit this section
1) Cp
2) Effect
3) Receptor
4) Tissue
Enter line # for required AUC (0- 4) 0
Additional Output
Enter -> 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Save Data x x x x x x x x
Graphs x x x x x x x x
Extra files x x x x x x x x
Sensitivity x x x x x x x x
Save calculated data files, Produce linear, semi-log and
weighted residual plots, save extra data files or perform
sensitivity or optimal sampling analysis
Enter choice (0-15) 2
Among other output information Boomer provides a summary of the model as entered. Here we can check for errors. The BAT file can be edited and re-run as needed.
Model and Parameter Definition
# Name Value Type From To Dep Start Stop
1) Dose = 200.0 1 0 1 0 0 0
2) k10 = 0.2000 2 1 0 0 0 0
3) k12 = 1.300 2 1 2 0 0 0
4) k21 = 1.100 2 2 1 0 0 0
5) V1 = 12.00 18 1 1 0 0 0
6) k1e = 0.1000E-01 2 1 3 0 0 0
7) -k1e = -0.1000E-01 2 1 0 0 0 0
8) ke0 = 0.8000 2 3 0 0 0 0
9) Emax or a Effect = 100.0 11 3 2 0 0 0
10) Ec(50%) or b Effect = 0.2500 12 0 0 0 0 0
11) S or i Effect = 1.500 13 0 0 0 0 0
12) Receptor = 1.000 18 3 3 0 0 0
13) Tissue = 1.000 18 2 4 0 0 0
Data for Cp :-
DATA # Time Observed Calculated (Weight) Weighted residual
1 0.000 0.00000 16.6667 0.00000 -0.00000
2 0.1000E-01 0.00000 16.4197 0.00000 -0.00000
3 0.2000E-01 0.00000 16.1787 0.00000 -0.00000
4 0.5000E-01 0.00000 15.4902 0.00000 -0.00000
5 0.1000 0.00000 14.4492 0.00000 -0.00000
6 0.1500 0.00000 13.5276 0.00000 -0.00000
7 0.2000 0.00000 12.7114 0.00000 -0.00000
8 0.2500 0.00000 11.9882 0.00000 -0.00000
9 0.5000 0.00000 9.42473 0.00000 -0.00000
10 0.7500 0.00000 7.99131 0.00000 -0.00000
11 1.000 0.00000 7.16223 0.00000 -0.00000
12 2.000 0.00000 5.90452 0.00000 -0.00000
13 3.000 0.00000 5.35621 0.00000 -0.00000
14 4.000 0.00000 4.90285 0.00000 -0.00000
15 6.000 0.00000 4.11486 0.00000 -0.00000
16 9.000 0.00000 3.16424 0.00000 -0.00000
17 12.00 0.00000 2.43323 0.00000 -0.00000
18 15.00 0.00000 1.87109 0.00000 -0.00000
19 18.00 0.00000 1.43882 0.00000 -0.00000
20 21.00 0.00000 1.10641 0.00000 -0.00000
21 24.00 0.00000 0.850795 0.00000 -0.00000
22 30.00 0.00000 0.503112 0.00000 -0.00000
23 36.00 0.00000 0.297486 0.00000 -0.00000
Linear and semi-log printer-style plots can be output for a quick review of the results. Note that you can select (above) to save the calculated data for plotting with more advanced graphing software. Below some of these data were plotted using a spreadsheet program.
Plots of observed (*) and calculated values (+)
versus time for Cp. Superimposed points (X)
16.67 Linear 16.67 Semi-log
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|***** * * * * * * * X X | +
|_____________________________________ |*****_*__*__*__*__*__*__*_____*_____*
0.000 0.2975
0 <--> 36. 0 <--> 36.
Plots of observed (*) and calculated values (+)
versus time for Effect. Superimposed points (X)
86.85 Linear 86.85 Semi-log
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|X**** * * * * * * * * * |+
|_____________________________________ |X****_*__*__*__*__*__*__*_____*_____*
0.000 2.176
0 <--> 36. 0 <--> 36.
It might interesting to view some of the plots that could be prepared from this output. Plasma concentration falling rapidly at first during the rapid phase.
The BAT and OUT files can be dowloaded.
The BAT file can be saved as sim.BAT and run with Boomer. With the current version of Boomer for Windows the lineEnter component for F-dependence ( 1 to - 1 or 0 for no dependence) 0will need to be removed.