PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 1
PhUSE 2012
Paper SP08
Estimating terminal half life by
non-compartmental methods with
some data below the limit of
quantification
Jochen Mueller-Cohrs
Director Biostatistical Methods
CSL Behring, Marburg, Germany
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 2
! Estimation of terminal half-life by
non-compartmental methods
! Generalisation in case some data are
below the quantification limit (QL)
! Realisation with SAS/STAT
! Results of some simulation studies
Outline
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 3
Concentration
0
20
40
60
80
100
120
Time0 12 24 36 48 60 72 84 96
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 4
Concentration
0
20
40
60
80
100
120
Time0 12 24 36 48 60 72 84 96
C = A exp ( – T ) + B exp ( – T )
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 5
Concentration
0
20
40
60
80
100
120
Time0 12 24 36 48 60 72 84 96
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 6
Concentration
0
20
40
60
80
100
120
Time0 12 24 36 48 60 72 84 96
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 7
Con
cent
ratio
n
0.25
0.5
1
2
4
8
16
32
64
128
Time0 12 24 36 48 60 72 84 96
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 8
Con
cent
ratio
n
0.25
0.5
1
2
4
8
16
32
64
128
Time0 12 24 36 48 60 72 84 96
adjusted R-square:terminal half-life:
0.94013.5
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 9
Con
cent
ratio
n
0.25
0.5
1
2
4
8
16
32
64
128
Time0 12 24 36 48 60 72 84 96
adjusted R-square:terminal half-life:
0.97714.2
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 10
Con
cent
ratio
n
0.25
0.5
1
2
4
8
16
32
64
128
Time0 12 24 36 48 60 72 84 96
adjusted R-square:terminal half-life:
0.97112.7
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 11
Con
cent
ratio
n
0.25
0.5
1
2
4
8
16
32
64
128
Time0 12 24 36 48 60 72 84 96
adjusted R-square:terminal half-life:
0.97812.2
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 12
Con
cent
ratio
n
0.25
0.5
1
2
4
8
16
32
64
128
Time0 12 24 36 48 60 72 84 96
adjusted R-square:terminal half-life:
0.97411.6
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 13
Con
cent
ratio
n
0.25
0.5
1
2
4
8
16
32
64
128
Time0 12 24 36 48 60 72 84 96
adjusted R-square:terminal half-life:
0.96010.8
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 14
Con
cent
ratio
n
0.25
0.5
1
2
4
8
16
32
64
128
Time0 12 24 36 48 60 72 84 96
adjusted R-square:terminal half-life:
0.96710.7
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 15
Con
cent
ratio
n
0.25
0.5
1
2
4
8
16
32
64
128
Time0 12 24 36 48 60 72 84 96
adjusted R-square:terminal half-life:
0.96710.4
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 16
Start of Number Estimated terminal of Adjusted terminal phase points R-squared half-life (h) (h)
36 3 0.940 13.5 24 4 0.977 14.2 16 5 0.971 12.7 12 6 0.978 * 12.2 8 7 0.974 11.6 4 8 0.960 10.8 2 9 0.967 10.7 0.1 10 0.967 10.4
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 17
Concentration
1
2
4
8
16
32
64
128
Time0 6 12 18 24 30 36 42 48 54 60
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 18
Concentration
1
2
4
8
16
32
64
128
Time0 6 12 18 24 30 36 42 48 54 60
QL
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 19
Concentration
1
2
4
8
16
32
64
128
Time0 6 12 18 24 30 36 42 48 54 60
QL
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 20
Concentration Censoring -log(C) TIME C QL Z Y
8 91.3 16 0 -4.5142
12 49.6 16 0 -3.9040
16 54.1 16 0 -3.9908
24 40.9 16 0 -3.7111
36 16.1 16 0 -2.7788
48 16.0 16 1 -2.7726
60 16.0 16 1 -2.7726
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 21
Without censored observations
PROC REG;
MODEL Y = TIME ;
PROC LIFEREG;
MODEL Y * Z(1) = TIME / D = normal ;
With censored observations
The regression coefficient of TIME is the terminal elimination rate λ
Terminal half-life is t½ = log(2) / λ
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 22
In case of no censoring:
! maximum likelihood (PROC LIFEREG) and
ordinary least squares (PROC REG) yield
identical estimates
! R2 = 1 – exp (– 2 Λ / n )
where Λ is the likelihood ratio statistic
for testing the regression coefficient;
n is the number of observations
! R2-adj = 1 – (1 – R2 ) (n – 1) / (n – 2)
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 23
In case of censoring a natural choice is:
! R2 = 1 – exp (– 2 Λ / m )
m is the number of observations above QL
! R2-adj = 1 – (1 – R2 ) (m – 1) / (m – 2)
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 24
Calculation of the likelihood-ratio statistic Λ
PROC LIFEREG;
MODEL Y * Z(1) = TIME / D = normal;
Full model
PROC LIFEREG;
MODEL Y * Z(1) = / D = normal;
Null model
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 25
Full Model Model Information Data Set WORK.PK Dependent Variable Y Censoring Variable Z Censoring Value(s) 1 Number of Observations 7 Noncensored Values 5 Right Censored Values 2 Left Censored Values 0 Interval Censored Values 0 Number of Parameters 3 Name of Distribution Normal Log Likelihood 1.9695339985 Analysis of Maximum Likelihood Parameter Estimates Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 -4.8295 0.1586 -5.1404 -4.5187 927.23 <.0001 TIME 1 0.0547 0.0073 0.0403 0.0690 55.67 <.0001 Scale 1 0.1631 0.0515 0.0879 0.3028
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 26
Null Model Model Information Data Set WORK.PK Dependent Variable Y Censoring Variable Z Censoring Value(s) 1 Number of Observations 7 Noncensored Values 5 Right Censored Values 2 Left Censored Values 0 Interval Censored Values 0 Number of Parameters 2 Name of Distribution Normal Log Likelihood -8.314262737 Analysis of Maximum Likelihood Parameter Estimates Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 -3.3418 0.3492 -4.0262 -2.6575 91.60 <.0001 Scale 1 0.8731 0.2969 0.4484 1.7001
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 27
Λ = Log-likelihood ( full model)
– Log-likelihood ( null model)
= 1.9695339985 – ( -8.314262737 )
= 10.28
Calculation of the likelihood-ratio statistic Λ
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 28
! m is the number of observations above QL
! The terminal elimination rate λ is determined by maximizing adjusted R2
! R2-adj = 1 – exp (– 2 Λ / m ) (m – 1) / (m – 2)
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 29
Con
cent
ratio
n
1
2
4
8
16
32
64
128
256
Time (h)0 6 12 18 24 30 36 42 48 54 60
Mono-exponential: half-life 12 h
QLQL
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 30
Con
cent
ratio
n
1
2
4
8
16
32
64
128
256
Time (h)0 6 12 18 24 30 36 42 48 54 60
Bi-exponential: half-lives 1 h / 12 h
QLQL
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 31
Con
cent
ratio
n
1
2
4
8
16
32
64
128
256
Time (h)0 6 12 18 24 30 36 42 48 54 60
Bi-exponential: half-lives 4 h / 12 h
QLQL
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 32
Censored Deleted
1 1.10.90 1.25.80 1.50.67Estimated half-life / true half-life
0.10
0.25
0.50
0.75
0.90
Mono-exponential: half-life 12 h QL at 36 h
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 33
1 1.10.90 1.25.80 1.50.67Estimated half-life / true half-life
0.10
0.25
0.50
0.75
0.90
Mono-exponential: half-life 12 h QL at 42 h
Censored Deleted
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 34
1 1.10.90 1.25.80 1.50.67Estimated half-life / true half-life
0.10
0.25
0.50
0.75
0.90
Bi-exponential: half-lives 1 h / 12 h QL at 36 h
Censored Deleted
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 35
1 1.10.90 1.25.80 1.50.67Estimated half-life / true half-life
0.10
0.25
0.50
0.75
0.90
Bi-exponential: half-lives 1 h / 12 h QL at 42 h
Censored Deleted
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 36
1 1.10.90 1.25.80 1.50.67Estimated half-life / true half-life
0.10
0.25
0.50
0.75
0.90
Bi-exponential: half-lives 4 h / 12 h QL at 36 h
Censored Deleted
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 37
1 1.10.90 1.25.80 1.50.67Estimated half-life / true half-life
0.10
0.25
0.50
0.75
0.90
Bi-exponential: half-lives 4 h / 12 h QL at 42 h
Censored Deleted
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 38
Con
cent
ratio
n
1
2
4
8
16
32
64
128
256
Time (h)0 6 12 18 24 30 36 42 48 54 60
Bi-exponential: half-lives 4 h / 12 h
QLQL
PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data below QL, Jochen Mueller-Cohrs 17-Oct-12: 39
! In non-compartmental PK analysis data below the
quantification limit can be dealt with by maximum
likelihood methods for censored data
Summary
! In the scenarios considered here, there was no major
difference between ignoring data below the
quantification limit and using censored data methods
! It is equally important for either method to
cover the terminal phase with data above
the quantification limit