Homework #2

PHAR 7633 Fall 1999

Answer

Question 1

Given the data in Table 1 and 2, calculate ka, F, kel, and V using graph paper and calculator. Assume a one compartment linear model.

Table 1
Plasma Concentration after an IV Bolus Dose of 200 mg

Time (hr)
Cp (mg/L)
0

1
2.9
3
2.1
6
1.3
°

First plot the IV data to get:

From the slope we can calculate kel = 0.160 hr-1. The intercept is 3.40 and the dose is 200 mg, thus V = Dose/Cp(0) = 200/3.40 = 58.8 L.

Table 2
Plasma Concentration after an Oral Dose of 450 mg

Time (hr)

Cp (mg/L)

0

0.25

0.6

0.5

1.1

0.75

1.5

1

1.9

1.5

2.5

2

2.8

3

3.2

4

3.2

6

2.8

9

1.9

12

1.3

°

Since we have a kel value from the IV data we can use the Wagner-Nelson method. This involves calculating A/V = Cp + kel • AUC for each time point. The maximum value is Amax/V. Subtracting each the A/V values from Amax/V give the amount remaining to be absorbed = (Amax-A)/V.

The Wagner-Nelson Table

Time (hr)
Cp (hr)
delta AUC
AUC
kel • AUC
Col 2 + Col 5
(Amax-A)/V
0
0
0.000
0
0
0
5.702
0.25
0.6
0.075
0.075
0.012
0.612
5.090
0.5
1.1
0.213
0.288
0.046
1.146
4.556
0.75
1.5
0.325
0.613
0.098
1.598
4.104
1
1.9
0.425
1.038
0.166
2.066
3.636
1.5
2.5
1.100
2.138
0.342
2.842
2.860
2
2.8
1.325
3.463
0.554
3.354
2.348
3
3.2
3.000
6.463
1.034
4.234
1.468
4
3.2
3.200
9.663
1.546
4.746
0.956
6
2.8
6.000
15.663
2.506
5.306
0.396
9
1.9
7.050
22.713
3.634
5.534
0.168
12
1.3
4.800
27.513
4.402
5.702
0.000
Infinity

8.125
35.638
5.702
5.702
0.000

Amax/V

Plotting (Amax-A)/V versus time give:

Thus ka = 0.405 hr-1. Note the Oral AUC is 35.6 mg.hr/L (from the table above using the trapezoidal rule). The IV AUC can be calculated using the trapezoidal rule with Cp(0) = 3.40 to give 21.4 mg.hr/L.

F(oral) = [AUC(oral) • Dose (IV)] / [Dose(oral) • AUC(IV)] = (35.6 x 200) / (450 x 21.4) = 0.739

Question 2.

Use a non-linear regresion program to fit both the IV and oral data from question 1 simultaneously.

Batch File

Boomer Batch File
    0                             wls,bayes,sim,irwls,sim+error,grid
    1                             Screen, diskfile, printer
HW9912fit
    1                             Parameter type
IV_Dose
   200.0                          Parameter value
    0                             Fixed,adjust,depend1,depend2
    1                             To
    0                             happy or not
    2                             Parameter type
kel_IV
  0.1600                          Parameter value
    1                             Fixed,adjust,depend1,depend2
  0.0000                          Lower limit
   5.000                          Upper limit
    0                             To
    1                             From
    0                             happy or not
   18                             Parameter type
V_IV
   58.80                          Parameter value
    1                             Fixed,adjust,depend1,depend2
   1.000                          Lower limit
   100.0                          Upper limit
    1                             To
[Cp] - IV
    1                             From
    0                             happy or not
   19                             Parameter type
F
  0.7390                          Parameter value
    1                             Fixed,adjust,depend1,depend2
  0.1000                          Lower limit
   1.200                          Upper limit
    0                             happy or not
   19                             Parameter type
Dose_PO
   450.0                          Parameter value
    0                             Fixed,adjust,depend1,depend2
    0                             happy or not
    1                             Parameter type
F*Dose_PO
   332.6                          Parameter value
    3                             Fixed,adjust,depend1,depend2
    5                             Double dependence type
    4                             Dependence-para1
    5                             Dependence-para2
    3                             To
    0                             happy or not
    2                             Parameter type
ka
   0.405                          Parameter value
    1                             Fixed,adjust,depend1,depend2
  0.0000                          Lower limit
   15.00                          Upper limit
    4                             To
    3                             From
    0                             happy or not
    2                             Parameter type
kel_PO
  0.1600                          Parameter value
    2                             Fixed,adjust,depend1,depend2
    2                             Dependence-para
    0                             To
    4                             From
    0                             happy or not
   18                             Parameter type
V_PO
   58.80                          Parameter value
    2                             Fixed,adjust,depend1,depend2
    3                             Dependence-para
    2                             To
[Cp] - PO
    4                             From
    0                             happy or not
   -1                             Parameter type
    2                             Integration method
  0.0000                          Relative error
  0.0000                          Absolute error
    4                             Fitting algorithm
  0.0000                          PC value
Fit to IV and Oral Data
    0                             Data from disk or keyboard
dataiv
    0                             Accept or edit data
datapo
    0                             Accept or edit data
    2                             Weight type
    2                             Weight type
    1                             AUC line number
    2                             AUC line number
    0                             AUC line number
    2                             Continue,save,plot,both
   -2                             wls,bayes,sim,irwls,sim+error,grid

Output File

 ** FINAL OUTPUT FROM Boomer (v2.7.8) **      29-Sep-1999 --- 10:43:13 pm

 Title:  Fit to IV and Oral Data                                     
 Input: From HW9912fit.BAT                                                   
 Output:  To HW9912fit.OUT                                                   
 Data for [Cp] - IV came from dataiv.DAT                                                      
 Data for [Cp] - PO came from datapo.DAT                                                      
 Fitting algorithm: Simplex Method             
 Weighting for [Cp] - IV       by 1/Cp(Obs )^2                                      
 Weighting for [Cp] - PO       by 1/Cp(Obs )^2                                      
 Numerical integration method: 2) Fehlberg RKF45                          
          with  4 de(s)
 With relative error   0.1000E-03
 With absolute error   0.1000E-03
 PC =   0.1000E-04

                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) kel_IV                0.15939                           0.00       5.0    
  2) V_IV                   58.988                            1.0      0.10E+03
  3) F                     0.75249                           0.10       1.2    
  4) ka                    0.44561                           0.00       15.    

 AIC =   -71.3801     Final WSS =   0.344784E-02
               R-squared =   0.9996     Correlation Coeff =   0.9991    

 ** FINAL OUTPUT FROM Boomer (v2.7.8) **      29-Sep-1999 --- 10:43:14 pm

 Title:  Fit to IV and Oral Data                                     
 Input: From HW9912fit.BAT                                                   
 Output:  To HW9912fit.OUT                                                   
 Data for [Cp] - IV came from dataiv.DAT                                                      
 Data for [Cp] - PO came from datapo.DAT                                                      
 Fitting algorithm: DAMPING-GAUSS/SIMPLEX      
 Weighting for [Cp] - IV       by 1/Cp(Obs )^2                                      
 Weighting for [Cp] - PO       by 1/Cp(Obs )^2                                      
 Numerical integration method: 2) Fehlberg RKF45                          
          with  4 de(s)
 With relative error   0.1000E-03
 With absolute error   0.1000E-03
 DT =   0.1000E-02     PC =   0.1000E-04 Loops =     2
 Damping =     1

                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) kel_IV                0.15946      0.389E-02   2.4      0.00       5.0    
  2) V_IV                   58.976      0.960       1.6       1.0      0.10E+03
  3) F                     0.75260      0.104E-01   1.4      0.10       1.2    
  4) ka                    0.44545      0.114E-01   2.6      0.00       15.    

 AIC =   -71.3806     Final WSS =   0.344771E-02
               R-squared =   0.9996     Correlation Coeff =   0.9991    

 Model and Parameter Definition

  #  Name                    Value       Type From To     Dep  Start Stop

  1) IV_Dose             =   200.0        1    0    1       0    0    0
  2) kel_IV              =  0.1595        2    1    0       0    0    0
  3) V_IV                =   58.98       18    1    1       0    0    0
  4) F                   =  0.7526       19    0    0       0    0    0
  5) Dose_PO             =   450.0       19    0    0       0    0    0
  6) F*Dose_PO           =   338.7        1    0    3 5004005    0    0
  7) ka                  =  0.4454        2    3    4       0    0    0
  8) kel_PO              =  0.1595        2    4    0 1002000    0    0
  9) V_PO                =   58.98       18    4    2 1003000    0    0

 Data for [Cp] - IV       :-

 DATA #   Time       Calculated     Observed      (Weight)   Weighted residual

     1   0.0000       3.39121      0.000000      0.000000     -0.000000    
     2    1.000       2.89137       2.90000      0.344828      0.297546E-02
     3    3.000       2.10185       2.10000      0.476191     -0.879765E-03
     4    6.000       1.30271       1.30000      0.769231     -0.208295E-02

     WSS for data set  1 =   0.1397E-04
               R-squared =    1.000     Correlation Coeff =    1.000    

 Data for [Cp] - PO       :-

 DATA #   Time       Calculated     Observed      (Weight)   Weighted residual

     1   0.0000      0.000000      0.000000      0.000000      0.000000    
     2   0.2500      0.593056      0.600000       1.66667      0.115734E-01
     3   0.5000       1.10044       1.10000      0.909091     -0.396208E-03
     4   0.7500       1.53207       1.50000      0.666667     -0.213830E-01
     5    1.000       1.89682       1.90000      0.526316      0.167157E-02
     6    1.500       2.45633       2.50000      0.400000      0.174661E-01
     7    2.000       2.83223       2.80000      0.357143     -0.115111E-01
     8    3.000       3.19304       3.20000      0.312500      0.217348E-02
     9    4.000       3.22091       3.20000      0.312500     -0.653565E-02
    10    6.000       2.81817       2.90000      0.344828      0.282164E-01
    11    9.000       1.96721       1.90000      0.526316     -0.353718E-01
    12    12.00       1.27721       1.30000      0.769231      0.175294E-01

     WSS for data set  2 =   0.3434E-02
               R-squared =   0.9996     Correlation Coeff =   0.9990    

 Maximum value for [Cp] - IV       is      2.9000     at       1.000    

 Calculation of AUC and AUMC based on trapezoidal rule

 AUC and AUMC for [Cp] - IV       using Observed data

       Time         Concentration      AUC             AUMC

     0.000000        0.000000    
      1.00000         2.90000         3.14561         1.45000    
      3.00000         2.10000         8.14561         10.6500    
      6.00000         1.30000         13.2456         31.8000    
                                      21.3983         131.843    

 MRT =      6.1614    

 Calculation of AUC and AUMC using Method 9 of R.D. Purves

 AUC and AUMC for [Cp] - IV       using Observed data

       Time         Concentration      AUC             AUMC

     0.000000        0.000000    
      1.00000         2.90000    
      3.00000         2.10000         8.10000         13.7250    
      6.00000         1.30000         13.1045         35.6473    
                                      21.2571         135.691    

 MRT =      6.3833    


 Maximum value for [Cp] - PO       is      3.2000     at       3.000    

 Calculation of AUC and AUMC based on trapezoidal rule

 AUC and AUMC for [Cp] - PO       using Observed data

       Time         Concentration      AUC             AUMC

     0.000000        0.000000    
     0.250000        0.600000        0.750000E-01    0.187500E-01
     0.500000         1.10000        0.287500        0.106250    
     0.750000         1.50000        0.612500        0.315625    
      1.00000         1.90000         1.03750        0.693750    
      1.50000         2.50000         2.13750         2.10625    
      2.00000         2.80000         3.46250         4.44375    
      3.00000         3.20000         6.46250         12.0438    
      4.00000         3.20000         9.66250         23.2438    
      6.00000         2.90000         15.7625         53.4438    
      9.00000         1.90000         22.9625         105.194    
      12.0000         1.30000         27.7625         154.244    
                                      36.7916         325.305    

 MRT =      8.8418    

 Calculation of AUC and AUMC using Method 9 of R.D. Purves

 AUC and AUMC for [Cp] - PO       using Observed data

       Time         Concentration      AUC             AUMC

     0.000000        0.000000    
     0.250000        0.600000    
     0.500000         1.10000        0.291667        0.958333E-01
     0.750000         1.50000        0.618750        0.302344    
      1.00000         1.90000         1.04479        0.677214    
      1.50000         2.50000         2.15451         2.07687    
      2.00000         2.80000         3.49062         4.42131    
      3.00000         3.20000         6.54618         12.0935    
      4.00000         3.20000         9.79062         23.4491    
      6.00000         2.90000         16.1017         54.9046    
      9.00000         1.90000         23.7101         111.217    
      12.0000         1.30000         28.6642         162.786    
                                      37.6934         333.847    

 MRT =      8.8569    

Plots of observed (*) and calculated values (+)
           versus time for [Cp] - IV      . Superimposed points (X)

    3.391      Linear                      3.391      Semi-log
 |+                                      |+                                    
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |      X                                |      X                              
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                  X                    |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                  X                  
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                    X  |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |*                                      |                                    X
 |_____________________________________  |*____________________________________
   0.0000                                  1.300    
 0              <-->             6.0     0              <-->             6.0    
 Plot of Std Wtd Residuals (X)         Plot of Std Wtd  Residuals (X)
   versus time for [Cp] - IV             versus calcd Cp(i) for [Cp] - IV      

   0.1755                                 0.1755    
 |      X                                |                           X         
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 0X====================================  0====================================X
 |                                       |                                     
 |                                       |                                     
 |                  X                    |             X                       
 |                                       |                                     
 |                                       |                                     
 |                                    X  |X                                    
 |                                       |                                     
  -0.1229                                -0.1229    
     0.00       <-->             6.0          1.3       <-->             3.4    
Plots of observed (*) and calculated values (+)
           versus time for [Cp] - PO      . Superimposed points (X)

    3.221      Linear                      3.221      Semi-log
 |            +                          |            +                        
 |         X  *                          |         X  *                        
 |                                       |                  *                  
 |                  *                    |      X           +                  
 |      X           +                    |                                     
 |                                       |    X                                
 |                                       |                                     
 |    *                                  |                                     
 |    +                                  |                                     
 |                                       |                           +         
 |                                       |   X                       *         
 |                                       |                                     
 |                           +           |                                     
 |   X                       *           |                                     
 |                                       |  X                                  
 |                                       |                                     
 |  +                                    |                                     
 |  *                                    |                                    X
 |                                    *  |                                     
 |                                    +  |                                     
 | X                                     | X                                   
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |X                                      |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |                                       |                                     
 |X                                      |X                                    
 |_____________________________________  |X____________________________________
   0.0000                                 0.5931    
 0              <-->             12.     0              <-->             12.    
 Plot of Std Wtd Residuals (X)         Plot of Std Wtd  Residuals (X)
   versus time for [Cp] - PO             versus calcd Cp(i) for [Cp] - PO      

    1.665                                  1.665    
 |                  X                    |                               X     
 |                                       |                                     
 |                                       |                                     
 |    X                               X  |              X            X         
 |                                       |                                     
 |X                                      |      X                              
 |                                       |                                     
 |   X     X                             |                     X             X 
 0XX===================================  0X===========X========================
 |            X                          |                                    X
 |      X                                |                               X     
 |                                       |                                     
 |                                       |                                     
 |  X                                    |                 X                   
 |                                       |                                     
 |                                       |                                     
 |                           X           |                     X               
 |                                       |                                     
   -2.087                                 -2.087    
     0.00       <-->             12.         0.00       <-->             3.2    
 Plot of Std Wtd Residuals (X)        Plot of Std Wtd  Residuals (X)
   versus time for all data              versus calcd Cp(i) for all data       

    1.665                                  1.665    
 |                  X                    |                             X       
 |                                       |                                     
 |                                       |                                     
 |    X                               X  |             X            X          
 |                                       |                                     
 |X                                      |      X                              
 |                                       |                                     
 |   X     X                             |                    X         X  X   
 0XX=======X========X==================  0X==========X=X========X=============X
 |            X                          |                                  X  
 |      X                                |                              X      
 |                                       |                                     
 |                                       |                                     
 |  X                                    |                X                    
 |                                       |                                     
 |                                       |                                     
 |                           X           |                    X                
 |                                       |                                     
   -2.087                                 -2.087    
     0.00       <-->             12.         0.00       <-->             3.4    

 

Question 3.

Given the model in Figure 1 and the data in Table 3 determine which of the parameters of this model are identifiable. Use either the numerical / empical approach and/or the analytical / Laplace transform approach. [Some additional information: The parameter values that I used to generate the data below are ke = 0.1 hr-1, km1 = 0.05 hr-1, km2 = 0.1 hr-1, and V1 = 20 L.]

Figure 1

Table 3
Plasma Concentration after an IV Bolus Dose of 200 mg 

Time
(hr)
Plasma Concentration (mg/L)
Time
(hr)
Amount of Unchanged Drug in Urine (mg)
0.5
8.8
1.0
17.7
1.0
7.8
2.0
31.5
1.5
6.9
4.0
50.6
2.0
6.1
6.0
62.2
4.0
3.7
9.0
71.6
6.0
2.2
12.0
76.0
9.0
1.1

12.0
0.5

Using the Numerical Simulation Approach

The Batch file:

Boomer Batch File
    0                             wls,bayes,sim,irwls,sim+error,grid 
    1                             Screen, diskfile, printer          
HW9912ident                                                 
    1                             Parameter type                     
Dose                                                        
   200.0                          Parameter value                    
    0                             Fixed,adjust,depend1,depend2       
    1                             To                                 
    0                             happy or not                       
    2                             Parameter type                     
ke                                                          
  0.1000                          Parameter value                    
    1                             Fixed,adjust,depend1,depend2       
  0.0000                          Lower limit                        
   5.000                          Upper limit                        
    2                             To                                 
    1                             From                               
    0                             happy or not                       
    2                             Parameter type                     
km1                                                         
  0.5000E-01                      Parameter value                    
    1                             Fixed,adjust,depend1,depend2       
  0.0000                          Lower limit                        
   5.000                          Upper limit                        
    0                             To                                 
    1                             From                               
    0                             happy or not                       
    2                             Parameter type                     
km2                                                         
  0.1000                          Parameter value                    
    1                             Fixed,adjust,depend1,depend2       
  0.0000                          Lower limit                        
   5.000                          Upper limit                        
    0                             To                                 
    1                             From                               
    0                             happy or not                       
   18                             Parameter type                     
V                                                           
   20.00                          Parameter value                    
    1                             Fixed,adjust,depend1,depend2       
   1.000                          Lower limit                        
   100.0                          Upper limit                        
    1                             To                                 
Cp                                                          
    1                             From                               
    0                             happy or not                       
   18                             Parameter type                     
One                                                         
   1.000                          Parameter value                    
    0                             Fixed,adjust,depend1,depend2       
    2                             To                                 
U                                                           
    2                             From                               
    0                             happy or not                       
   -1                             Parameter type                     
    2                             Integration method                 
  0.0000                          Relative error                     
  0.0000                          Absolute error                     
    4                             Fitting algorithm                  
  0.0000                          PC value                           
HW9912 - Identifiabilty Problem                             
    1                             Data from disk or keyboard         
  0.0000                          X value                            
  0.0000                          Y value                            
  0.5000                          X value                            
   8.800                          Y value                            
   1.000                          X value                            
   7.800                          Y value                            
   1.500                          X value                            
   6.900                          Y value                            
   2.000                          X value                            
   6.100                          Y value                            
   4.000                          X value                            
   3.700                          Y value                            
   6.000                          X value                            
   2.200                          Y value                            
   9.000                          X value                            
   1.100                          Y value                            
   12.00                          X value                            
  0.5000                          Y value                            
  -1.000                          X value                            
    0                             Accept, correct, delete, insert, of
    0                             Continue or save data              
  0.0000                          X value                            
  0.0000                          Y value                            
   1.000                          X value                            
   17.70                          Y value                            
   2.000                          X value                            
   31.50                          Y value                            
   4.000                          X value                            
   50.60                          Y value                            
   6.000                          X value                            
   62.20                          Y value                            
   9.000                          X value                            
   71.60                          Y value                            
   12.00                          X value                            
   76.00                          Y value                            
  -1.000                          X value                            
    0                             Accept, correct, delete, insert, of
    0                             Continue or save data              
    3                             Weight type                        
  0.2500E-02                      Weight value - A                   
   2.000                          Weight value - B                   
    3                             Weight type                        
   16.00                          Weight value - A                   
  0.0000                          Weight value - B                   
    0                             AUC line number                    
    2                             Continue,save,plot,both            
   -4                             wls,bayes,sim,irwls,sim+error,grid 
HW9912ident

This .BAT file will run until the program is quit by the user so you can do 10 or more fits/runs in one session. [Note the -4 on the second last line and the name of the .BAt on the last line.]

The Final Parameter Values from 10 runs:

                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.99660E-01  0.735E-03  0.74      0.00       5.0    
  2) km1                   0.33915E-01   1.87      0.55E+04  0.00       5.0    
  3) km2                   0.11505       1.88      0.16E+04  0.00       5.0    
  4) V                      20.013      0.126      0.63       1.0      0.10E+03

 AIC =    1.61444     Final WSS =   0.633743    
               R-squared =   0.9999     Correlation Coeff =    1.000    
--
                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.99667E-01                       0.00       5.0    
  2) km1                   0.54763E-01                       0.00       5.0    
  3) km2                   0.94187E-01                       0.00       5.0    
  4) V                      20.013                            1.0      0.10E+03

 AIC =    1.61402     Final WSS =   0.633724    
               R-squared =   0.9999     Correlation Coeff =    1.000    
---
                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.99666E-01                       0.00       5.0    
  2) km1                   0.66782E-01                       0.00       5.0    
  3) km2                   0.82166E-01  0.230E+09  0.28E+12  0.00       5.0    
  4) V                      20.013      0.230       1.1       1.0      0.10E+03

 AIC =    1.61409     Final WSS =   0.633728    
               R-squared =   0.9999     Correlation Coeff =    1.000    
---
                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.99678E-01                       0.00       5.0    
  2) km1                   0.48799E-01                       0.00       5.0    
  3) km2                   0.10014                           0.00       5.0    
  4) V                      20.013                            1.0      0.10E+03

 AIC =    1.61402     Final WSS =   0.633724    
               R-squared =   0.9999     Correlation Coeff =    1.000    
---
                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.99678E-01                       0.00       5.0    
  2) km1                   0.48799E-01                       0.00       5.0    
  3) km2                   0.10014      0.230E+09  0.23E+12  0.00       5.0    
  4) V                      20.013      0.230       1.1       1.0      0.10E+03

 AIC =    1.61402     Final WSS =   0.633724    
               R-squared =   0.9999     Correlation Coeff =    1.000    
---
                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.99672E-01                       0.00       5.0    
  2) km1                   0.49467E-01                       0.00       5.0    
  3) km2                   0.99489E-01                       0.00       5.0    
  4) V                      20.012                            1.0      0.10E+03

 AIC =    1.61402     Final WSS =   0.633724    
               R-squared =   0.9999     Correlation Coeff =    1.000    
---
                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.99673E-01  0.708E-03  0.71      0.00       5.0    
  2) km1                   0.77701E-01                       0.00       5.0    
  3) km2                   0.71251E-01                       0.00       5.0    
  4) V                      20.011      0.123      0.61       1.0      0.10E+03

 AIC =    1.61403     Final WSS =   0.633725    
               R-squared =   0.9999     Correlation Coeff =    1.000    
---
                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.99666E-01                       0.00       5.0    
  2) km1                   0.48992E-01                       0.00       5.0    
  3) km2                   0.99991E-01                       0.00       5.0    
  4) V                      20.008                            1.0      0.10E+03

 AIC =    1.61570     Final WSS =   0.633800    
               R-squared =   0.9999     Correlation Coeff =    1.000    
---
                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.99669E-01  0.126E-02   1.3      0.00       5.0    
  2) km1                   0.14682      0.159      0.11E+03  0.00       5.0    
  3) km2                   0.21498E-02  0.160      0.74E+04  0.00       5.0    
  4) V                      20.010      0.174      0.87       1.0      0.10E+03

 AIC =    1.61441     Final WSS =   0.633742    
               R-squared =   0.9999     Correlation Coeff =    1.000    
---
                    ** FINAL PARAMETER VALUES ***

  #  Name                  Value       S.D.       C.V. %  Lower <-Limit-> Upper

  1) ke                    0.99671E-01                       0.00       5.0    
  2) km1                   0.50268E-01                       0.00       5.0    
  3) km2                   0.98686E-01                       0.00       5.0    
  4) V                      20.012                            1.0      0.10E+03

 AIC =    1.61405     Final WSS =   0.633725    
               R-squared =   0.9999     Correlation Coeff =    1.000    

Note that the values of ke and V are consistent across all 10 runs. Thus they are (probably) identifiable. Are the kel values (= ke + km1 + km2) values consistent? kel was consistently 0.2486.

Using the Laplace Transfrom Approach

Write the differential equation for the data supplied and take the Laplace of these equations before evaluating whether parameters are identifiable.

Starting with Xp and Cp

From the 's' term we can see that ke + km1 + km2 is identifiable in total but not the separate components. That is kel (= ke + km1 + km2) is identifiable. From the 'intercept' term Dose/V we know that V is identifiable since we know Dose.

Continuing with U

From the 's' term we can see that ke + km1 + km2 is identifiable in total but not the separate components. That is kel (= ke + km1 + km2) is identifiable (again). From the 'intercept' term ke • Dose we know that ke is identifiable since we know Dose.


Copyright 1999 David W.A. Bourne