Jonsson EN, Wade JR and Karlsson MO Nonlinearity Detection: Advantages of Nonlinear Mixed-Effects Modeling AAPS PharmSci 2000;
2
(3)
article 32
(https://www.pharmsci.org/scientificjournals/pharmsci/journal/32.html).
Figures and Tables
Table 1.
Parameter values used during data simulation
|
Parameter
|
Value
| |
Pharmacokinetic model I
|
| |
Km
|
10
| |
Vmax
|
10
| |
V
|
10
| |
Clint
|
1
| |
Pharmacokinetic model II
|
| |
Km
|
10
| |
Vmax
|
10
| |
k
|
0.05
| |
V
|
10
| |
Clint
|
1.5
| |
Pharmacodynamic model
|
| |
CL
|
1
| |
V
|
10
| |
Base
|
100
| |
Emax
|
0.5
| |
EC50
|
0.7
| |
γ
|
2
|
*Clint is the ratio of Vmax and
Km and the ratio of Vmax and Km plus k over V for pharmacokinetic models I and II,
respectively.
Figure 1.Dose-specific profiles for the 3 simulation models.
Each line corresponds to 1 dose level (1, 10, 35, 50, 200, 600, and 1,000 and 3,
5, 7, 10, 15, 20, 30, and 60 for the 2 pharmacokinetic models and the
pharmacodynamic models, respectively). In the panel for the pharmacodynamic
model, the curves are separated by a small shift in concentration to avoid
superposition.
Table 2.
Median parameter estimates (and range) when fitting
competing linear and nonlinear models to the data simulated using
pharmacokinetic model I
|
Method
|
Dose
|
Km
|
Clint
|
No. data sets best described by
| |
|
|
(True value 10)
|
(True value 1)
|
Linear model (simple model)
|
Nonlinear model (true model)
| |
STS
|
1
|
-*
|
-*
|
30
|
0
| |
|
10
|
-*
|
-*
|
30
|
0
| |
|
35
|
-*
|
-*
|
29
|
1
| |
|
50
|
-*
|
-*
|
28
|
2
| |
|
200
|
10.6 (9.3-14.8)
|
0.98 (0.85-1.53)
|
5
|
25
| |
|
600
|
9.9 (9.3-13.6)
|
1.05 (0.97-1.46)
|
7
|
23
| |
|
1000
|
10.3 (8.8-13.1)
|
0.95 (0.85-1.53)
|
5
|
25
| |
FOCE
|
1
|
-*
|
-*
|
30
|
0
| |
|
10
|
-*
|
-*
|
28
|
2
| |
|
35
|
-*
|
-*
|
26
|
4
| |
|
50
|
10.5 (6.43-16.0)
|
0.93 (0.69-1.47)
|
0
|
30
| |
|
200
|
10.2 (7.67-12.5)
|
0.94 (0.71-1.50)
|
0
|
30
| |
|
600
|
10.2 (7.66-12.4)
|
0.93 (0.72-1.48)
|
0
|
30
| |
|
1000
|
10.1 (8.14-12.3)
|
0.93 (0.73-1.44)
|
0
|
30
|
The number of the 30 data sets best described by each of the
linear and nonlinear models is also shown. The FO method failed to detect the
true nonlinear model at all dose levels; thus, no results are presented for this
method.
*No results for Km and Clint are
presented when the nonlinear model did not best describe the 30 data sets
overall (as described in Materials and Methods).
Table 3.
Median parameter estimates (and range) when fitting
competing linear and nonlinear models to the data simulated using
pharmacokinetic model II
|
Method
|
Dose
|
Km
|
Clint
|
No. data sets best described by
| |
|
|
(True value 10)
|
(True value 1.5)
|
Linear model (simple model)
|
Nonlinear model (true model)
| |
STS
|
1
|
-*
|
-*
|
29
|
1
| |
|
10
|
-*
|
-*
|
29
|
1
| |
|
35
|
-*
|
-*
|
29
|
1
| |
|
50
|
-*
|
-*
|
29
|
1
| |
|
200
|
-*
|
-*
|
18
|
12
| |
|
600
|
11.7 (6.0-33.8)
|
1.0 (0.8-1.5)
|
2
|
28
| |
|
1000
|
10.4 (4.3-24.9)
|
1.1 (0.7-1.3)
|
0
|
30
| |
FOCE
|
1
|
-*
|
-*
|
29
|
1
| |
|
10
|
-*
|
-*
|
29
|
1
| |
|
35
|
-*
|
-*
|
16
|
14
| |
|
50
|
8.5 (2.8-19.0)
|
1.4 (1.2-1.7)
|
8
|
22
| |
|
200
|
10.1 (4.5-27.8)
|
1.4 (1.2-2.0)
|
0
|
30
| |
|
600
|
10.6 (7.0-40.4)
|
1.4 (1.2-2.0)
|
0
|
30
| |
|
1000
|
11.2 (7.1-26.0)
|
1.4 (1.1-2.0)
|
0
|
30
|
The number of the 30 data sets best described by each of the
linear and nonlinear model is also shown. The FO method failed to detect the
true nonlinear model at all dose levels; thus, no results are presented for this
method.
*No results for Km and Clint are
presented when the nonlinear model did not best describe the 30 data sets
overall (as described in Materials and Methods).
Table 4.
Median parameter estimates (and range) when fitting
competing linear and nonlinear models to the data simulated using
pharmacodynamic model
|
Method
|
Dose
|
EC50
|
Emax
|
γ
|
No. data sets best described by:
| |
|
|
(True value 0.7)
|
(True value 0.5)
|
(True value 2)
|
Log linear model (simple model)
|
Nonlinear model (true model)
| |
STS
|
3
|
-*
|
-*
|
-*
|
29
|
1
| |
|
5
|
-*
|
-*
|
-*
|
29
|
1
| |
|
7
|
-*
|
-*
|
-*
|
29
|
1
| |
|
10
|
-*
|
-*
|
-*
|
29
|
1
| |
|
15
|
-*
|
-*
|
-*
|
29
|
1
| |
|
20
|
-*
|
-*
|
-*
|
29
|
1
| |
|
30
|
-*
|
-*
|
-*
|
29
|
1
| |
|
60
|
-*
|
-*
|
-*
|
25
|
5
| |
FO
|
3
|
-*
|
-*
|
-*
|
29
|
1
| |
|
5
|
-*
|
-*
|
-*
|
28
|
2
| |
|
7
|
-*
|
-*
|
-*
|
29
|
1
| |
|
10
|
0.87 (0.71-0.97)
|
0.62 (0.56-0.73)
|
2.17 (1.97-2.44)
|
14
|
16
| |
|
15
|
0.75 (0.49-1.29)
|
0.51 (0.42-0.72)
|
1.89 (1.39-2.56)
|
5
|
25
| |
|
20
|
0.78 (0.34-1.07)
|
0.50 (0.39-0.65)
|
1.81 (1.33-2.51)
|
0
|
30
| |
|
30
|
0.82 (0.63-1.23)
|
0.52 (0.47-0.63)
|
1.85 (1.58-2.24)
|
5
|
25
| |
FOCE
|
3
|
-*
|
-*
|
-*
|
30
|
0
| |
|
5
|
-*
|
-*
|
-*
|
28
|
2
| |
|
7
|
-*
|
-*
|
-*
|
23
|
7
| |
|
10
|
0.95 (0.84-1.25)
|
0.67 (0.55-0.89)
|
2.23 (1.99-2.43)
|
15
|
15
| |
|
15
|
0.74 (0.52-1.32)
|
0.51 (0.38-0.76)
|
1.96 (1.40-2.54)
|
3
|
27
| |
|
20
|
0.75 (0.50-1.45)
|
0.50 (0.38-0.65)
|
2.02 (1.48-2.48)
|
0
|
30
| |
|
30
|
0.77 (0.51-1.08)
|
0.50 (0.41-0.60)
|
2.03 (1.47-2.44)
|
0
|
26 †
|
The number of the 30 data sets best described by each of the
linear and nonlinear model is also shown.
*No results for EC50, Emax and γ are presented when the nonlinear model did not best describe the thirty data
sets overall (as described in Materials and Methods).
†At the highest dose level and FOCE, there were 4 instances when
neither of the power or nonlinear model terminated successfully, which is why
the totals in the 2 rightmost columns do not add up to 30.
| 
