Estimating terminal half life by non-compartmental methods with … · 2014. 10. 27. · PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data

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


!   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


Concentration

0

20

40

60

80

100

120

Time0 12 24 36 48 60 72 84 96


Concentration

0

20

40

60

80

100

120

Time0 12 24 36 48 60 72 84 96

C = A exp ( – T ) + B exp ( – T )


Concentration

0

20

40

60

80

100

120

Time0 12 24 36 48 60 72 84 96


Concentration

0

20

40

60

80

100

120

Time0 12 24 36 48 60 72 84 96


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


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


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


0.97714.2


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


0.97112.7


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


0.97812.2


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


0.97411.6


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


0.96010.8


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


0.96710.7


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


0.96710.4


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


Concentration

1

2

4

8

16

32

64

128

Time0 6 12 18 24 30 36 42 48 54 60


Concentration

1

2

4

8

16

32

64

128

Time0 6 12 18 24 30 36 42 48 54 60

QL


Concentration

1

2

4

8

16

32

64

128

Time0 6 12 18 24 30 36 42 48 54 60

QL


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


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


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)


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)


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


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


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


Λ = Log-likelihood ( full model)

– Log-likelihood ( null model)

= 1.9695339985 – ( -8.314262737 )

= 10.28

Calculation of the likelihood-ratio statistic Λ


!   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)


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


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


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


QLQL


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



0.10

0.25

0.50

0.75

0.90

Mono-exponential: half-life 12 h QL at 42 h

Censored Deleted



0.10

0.25

0.50

0.75

0.90

Bi-exponential: half-lives 1 h / 12 h QL at 36 h

Censored Deleted



0.10

0.25

0.50

0.75

0.90


Censored Deleted



0.10

0.25

0.50

0.75

0.90


Censored Deleted



0.10

0.25

0.50

0.75

0.90


Censored Deleted


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


QLQL


!   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

Documents

Estimating terminal half life by non-compartmental methods with … · 2014. 10. 27. · PhUSE 2012, SP08 Estimating terminal half life by non-compartmental methods with some data