From: Martin Bergstrand <*martin.bergstrand*>

Date: Fri, 20 Nov 2009 11:44:08 +0100

Dear Leonid,

The extra error term for the BQL observations was estimated. This extra

error term was only added for Model A. In the other examples there was

already an additive part in the residual error that sufficiently relaxed =

the

assumption of the imputation.

I am not a big fan of imputations of BQL samples and think that they =

should

be avoided in favor of the M3 or M4 method as often as possible. However =

if

practical issues (probably disappearing as estimation methods and

computational power are improving) hinder you from using the likelihood

based methods I suggest to do a small sensitivity analysis regarding =

what

imputation to chose. There are no possibility to make general

recommendations on what imputations (LOQ/2 etc.) that will result in

unbiased estimates.

The first thing you should do is to evaluate the simulation properties =

of

the obtained model parameters. In the article that you refer to we have

described how we think that VPC are best done for datasets with censored

observations such as BQL. If a certain imputation is chosen and the =

final

model demonstrate good simulation properties both for the contentious

observations (above LOQ) and for the fraction of BQL samples this is a =

good

indication that the obtained parameter estimates are likely to be fairly

unbiased.

I have also thought of more elaborate approaches with simulation and

re-estimation with different imputations of BQL observations to evaluate =

the

possible biased introduced by substitution the BQL samples with an =

assumed

value (similar principle as the Back step method described by Kjellsson =

MC

et.al. (1)). However this is nothing that I have tested or think is an

attractive solution for many cases.

(1) Kjellsson MC, Jönsson S, Karlsson MO. The back-step method--method =

for

obtaining unbiased population parameter estimates for ordered =

categorical

data. AAPS J. 2004 Aug 11;6(3):e19.

By the way I don't agree with your conclusion that LOQ/2 substitution

provides reasonable results in the indirect response model example (C). =

It

induces a non negligible amount of bias in both the concentration effect

parameter (SLOP) and Kout (5-15%). In my opinion substitution with LOQ/2 =

are

just as bad as omitting the BQL data (just bad in another way) for that =

very

example.

Kind regards,

Martin Bergstrand, MSc, PhD student

-----------------------------------------------

Pharmacometrics Research Group,

Department of Pharmaceutical Biosciences,

Uppsala University

-----------------------------------------------

P.O. Box 591

SE-751 24 Uppsala

Sweden

-----------------------------------------------

martin.bergstrand

-----------------------------------------------

Work: +46 18 471 4639

Mobile: +46 709 994 396

Fax: +46 18 471 4003

-----Original Message-----

From: Leonid Gibiansky [mailto:LGibiansky

Sent: den 20 november 2009 04:01

To: Mats Karlsson

Cc: 'Doshi, Sameer'; 'nmusers'; Martin Bergstrand

Subject: Re: [NMusers] Modeling biomarker data below the LOQ

Mats, Martin,

In the paper, you mentioned that "an extra additive error

model term was added for samples substituted with LOQ/2". Was it fixed

or estimated? If fixed, how? Have you tried to vary this level?

In many of your examples, LOQ/2 imputations and exclusion of BQL samples =

seen to lead to bias in opposite directions; if so, it could be an

optimal value (relative to LOQ) of the fixed extra error term that

provides the least biased parameters.

Laplacian method is often not feasible for receptor (target) models

since they are strongly nonlinear (thus requiring differential

equations) and stiff. Based on the the indirect-response model

simulations considered in your paper, LOQ/2 substitution seems to

provide reasonable results if Laplacian (and thus M2-M3-M4) cannot be =

used.

Thanks

Leonid

--------------------------------------

Leonid Gibiansky, Ph.D.

President, QuantPharm LLC

web: www.quantpharm.com

e-mail: LGibiansky at quantpharm.com

tel: (301) 767 5566

Mats Karlsson wrote:

*> Dear Sameer,
*

*>
*

*>
*

*>
*

*> We’ve had this problem with biomarker data and published experiences =
*

in

*> terms of a methodological paper (below). Maybe it can give you some =
*

ideas.

*>
*

*>
*

*>
*

*> Handling data below the limit of quantification in mixed effect =
*

models.

*>
*

*> Bergstrand M, Karlsson MO.
*

*>
*

*> AAPS J. 2009 Jun;11(2):371-80. Epub 2009 May 19.
*

*>
*

*>
*

*>
*

*> Best regards,
*

*>
*

*> Mats
*

*>
*

*>
*

*>
*

*> Mats Karlsson, PhD
*

*>
*

*> Professor of Pharmacometrics
*

*>
*

*> Dept of Pharmaceutical Biosciences
*

*>
*

*> Uppsala University
*

*>
*

*> Box 591
*

*>
*

*> 751 24 Uppsala Sweden
*

*>
*

*> phone: +46 18 4714105
*

*>
*

*> fax: +46 18 471 4003
*

*>
*

*>
*

*>
*

*> *From:* owner-nmusers *

*> [mailto:owner-nmusers *

*> *Sent:* Wednesday, November 18, 2009 6:53 PM
*

*> *To:* nmusers
*

*> *Subject:* [NMusers] Modeling biomarker data below the LOQ
*

*>
*

*>
*

*>
*

*> Hello,
*

*>
*

*> We are attempting to model suppression of a biomarker, where a number =
*

of

*> samples (40-60%) are below the quantification limit of the assay and
*

*> where 2 different assays (with different quantification limits) were
*

*> used. We are trying to model these BQL data using the M3 and M4 =
*

methods

*> proposed by Ahn et al (2008).
*

*>
*

*>
*

*>
*

*> I would like to know if anyone has any comments or experience
*

*> implementing the M3 or M4 methods for biomarker data, where levels are =
*

*> observed at baseline, are supressed below the LOQ for a given =
*

duration,

*> and then return to baseline.
*

*>
*

*>
*

*>
*

*> Also please advise if there are other methods to try and incorporate
*

*> these BQL data into the model.
*

*>
*

*>
*

*>
*

*> I have included the relevant pieces of the control file (for both M3 =
*

and

*> M4) and data from a single subject.
*

*>
*

*>
*

*>
*

*> Thanks for your review/suggestions.
*

*>
*

*>
*

*>
*

*> Sameer
*

*>
*

*>
*

*>
*

*> DATA:
*

*>
*

*> #ID TIME AMT DV CMT EVID TYPE ASSY
*

*>
*

*> 1 0 0 65.71 0 0 0 1
*

*>
*

*> 1 0 120 0 3 1 0 1
*

*>
*

*> 1 168 0 10 0 0 1 1
*

*>
*

*> 1 336 0 10 0 0 1 1
*

*>
*

*> 1 336 120 0 3 1 0 1
*

*>
*

*> 1 504 0 12.21 0 0 0 1
*

*>
*

*> 1 672 120 0 3 1 0 1
*

*>
*

*> 1 1008 0 10 0 0 1 1
*

*>
*

*> 1 1008 120 0 3 1 0 1
*

*>
*

*> 1 1344 0 10 0 0 1 1
*

*>
*

*> 1 1344 120 0 3 1 0 1
*

*>
*

*> 1 1680 0 10 0 0 1 1
*

*>
*

*> 1 1680 120 0 3 1 0 1
*

*>
*

*> 1 2016 0 10 0 0 0 1
*

*>
*

*> 1 2352 0 25.64 0 0 0 1
*

*>
*

*> 1 2688 0 59.48 0 0 0 1
*

*>
*

*>
*

*>
*

*> MODEL M3:
*

*>
*

*> $DATA data.csv IGNORE=#
*

*>
*

*> $SUB ADVAN8 TRANS1 TOL=6
*

*>
*

*> $MODEL
*

*>
*

*> COMP(central)
*

*>
*

*> COMP(peri)
*

*>
*

*> COMP(depot,DEFDOSE)
*

*>
*

*> COMP(effect)
*

*>
*

*>
*

*>
*

*> $DES
*

*>
*

*> DADT(1) = KA*A(3) - K10*A(1) - K12*A(1) + K21*A(2)
*

*>
*

*> DADT(2) = K12*A(1) - K21*A(2)
*

*>
*

*> DADT(3) = -KA*A(3)
*

*>
*

*> CONC = A(1)/V1
*

*>
*

*> DADT(4) = KEO*(CONC-A(4))
*

*>
*

*>
*

*>
*

*> $ERROR
*

*>
*

*> CALLFL = 0
*

*>
*

*>
*

*>
*

*> LOQ1=10
*

*>
*

*> LOQ2=20
*

*>
*

*>
*

*>
*

*> EFF = BL* (1 - IMAX*A(4)**HILL/ (IC50**HILL+A(4)**HILL))
*

*>
*

*> IPRED=EFF
*

*>
*

*> SIGA=THETA(7)
*

*>
*

*> STD=SIGA
*

*>
*

*> IF(TYPE.EQ.0) THEN ; GREATER THAN LOQ
*

*>
*

*> F_FLAG=0
*

*>
*

*> Y=IPRED+SIGA*EPS(1)
*

*>
*

*> IRES =DV-IPRED
*

*>
*

*> IWRES=IRES/STD
*

*>
*

*> ENDIF
*

*>
*

*> IF(TYPE.EQ.1.AND.ASSY.EQ.1) THEN ; BELOW LOQ1
*

*>
*

*> DUM1=(LOQ1-IPRED)/STD
*

*>
*

*> CUM1=PHI(DUM1)
*

*>
*

*> F_FLAG=1
*

*>
*

*> Y=CUM1
*

*>
*

*> IRES = 0
*

*>
*

*> IWRES=0
*

*>
*

*> ENDIF
*

*>
*

*> IF(TYPE.EQ.1.AND.ASSY.EQ.2) THEN ; BELOW LOQ2
*

*>
*

*> DUM2=(LOQ2-IPRED)/STD
*

*>
*

*> CUM2=PHI(DUM2)
*

*>
*

*> F_FLAG=1
*

*>
*

*> Y=CUM2
*

*>
*

*> IRES = 0
*

*>
*

*> IWRES=0
*

*>
*

*> ENDIF
*

*>
*

*>
*

*>
*

*> $SIGMA 1 FIX
*

*>
*

*>
*

*>
*

*> $ESTIMATION MAXEVAL=9990 NOABORT SIGDIG=3 METHOD=1 INTER =
*

LAPLACIAN

*>
*

*> POSTHOC PRINT=2 SLOW NUMERICAL
*

*>
*

*> $COVARIANCE PRINT=E SLOW
*

*>
*

*>
*

*>
*

*> MODEL M4:
*

*>
*

*> $DATA data.csv IGNORE=#
*

*>
*

*> $SUB ADVAN8 TRANS1 TOL=6
*

*>
*

*> $MODEL
*

*>
*

*> COMP(central)
*

*>
*

*> COMP(peri)
*

*>
*

*> COMP(depot,DEFDOSE)
*

*>
*

*> COMP(effect)
*

*>
*

*>
*

*>
*

*> $DES
*

*>
*

*> DADT(1) = KA*A(3) - K10*A(1) - K12*A(1) + K21*A(2)
*

*>
*

*> DADT(2) = K12*A(1) - K21*A(2)
*

*>
*

*> DADT(3) = -KA*A(3)
*

*>
*

*> CONC = A(1)/V1DADT(4) = KEO*(CONC-A(4))
*

*>
*

*>
*

*>
*

*> $ERROR
*

*>
*

*> CALLFL = 0
*

*>
*

*>
*

*>
*

*> LOQ1=10
*

*>
*

*> LOQ2=20
*

*>
*

*>
*

*>
*

*> EFF = BL* (1 - IMX*A(4)**HILL/ (IC50**HILL+A(4)**HILL))
*

*>
*

*> IPRED=EFF
*

*>
*

*> SIGA=THETA(7)
*

*>
*

*> STD=SIGA
*

*>
*

*> IF(TYPE.EQ.0) THEN ; GREATER THAN LOQ
*

*>
*

*> F_FLAG=0
*

*>
*

*> YLO=0
*

*>
*

*> Y=IPRED+SIGA*EPS(1)
*

*>
*

*> IRES =DV-IPRED
*

*>
*

*> IWRES=IRES/STD
*

*>
*

*> ENDIF
*

*>
*

*> IF(TYPE.EQ.1.AND.ASSY.EQ.1) THEN
*

*>
*

*> DUM1=(LOQ1-IPRED)/STD
*

*>
*

*> CUM1=PHI(DUM1)
*

*>
*

*> DUM0=-IPRED/STD
*

*>
*

*> CUMD0=PHI(DUM0)
*

*>
*

*> CCUMD1=(CUM1-CUMD0)/(1-CUMD0)
*

*>
*

*> F_FLAG=1
*

*>
*

*> Y=CCUMD1
*

*>
*

*> IRES = 0
*

*>
*

*> IWRES=0
*

*>
*

*> ENDIF
*

*>
*

*> IF(TYPE.EQ.1.AND.ASSY.EQ.2) THEN
*

*>
*

*> DUM2=(LOQ2-IPRED)/STD
*

*>
*

*> CUM2=PHI(DUM2)
*

*>
*

*> DUM0=-IPRED/STD
*

*>
*

*> CUMD0=PHI(DUM0)
*

*>
*

*> CCUMD2=(CUM2-CUMD0)/(1-CUMD0)
*

*>
*

*> F_FLAG=1
*

*>
*

*> Y=CCUMD2
*

*>
*

*> IRES = 0
*

*>
*

*> IWRES=0
*

*>
*

*> ENDIF
*

*>
*

*>
*

*>
*

*> $SIGMA 1 FIX
*

*>
*

*>
*

*>
*

*> $ESTIMATION MAXEVAL=9990 NOABORT SIGDIG=3 METHOD=1 INTER =
*

LAPLACIAN

*>
*

*> POSTHOC PRINT=2 SLOW NUMERICAL
*

*>
*

*> $COVARIANCE PRINT=E SLOW
*

*>
*

*>
*

*>
*

*>
*

*>
*

*>
*

*>
*

*>
*

*>
*

*> Sameer Doshi
*

*>
*

*> Pharmacokinetics and Drug Metabolism, Amgen Inc.
*

*>
*

*> (805) 447-6941
*

*>
*

*>
*

*>
*

*>
*

*>
*

*>
*

*>
*

*>
*

*>
*

Received on Fri Nov 20 2009 - 05:44:08 EST

Date: Fri, 20 Nov 2009 11:44:08 +0100

Dear Leonid,

The extra error term for the BQL observations was estimated. This extra

error term was only added for Model A. In the other examples there was

already an additive part in the residual error that sufficiently relaxed =

the

assumption of the imputation.

I am not a big fan of imputations of BQL samples and think that they =

should

be avoided in favor of the M3 or M4 method as often as possible. However =

if

practical issues (probably disappearing as estimation methods and

computational power are improving) hinder you from using the likelihood

based methods I suggest to do a small sensitivity analysis regarding =

what

imputation to chose. There are no possibility to make general

recommendations on what imputations (LOQ/2 etc.) that will result in

unbiased estimates.

The first thing you should do is to evaluate the simulation properties =

of

the obtained model parameters. In the article that you refer to we have

described how we think that VPC are best done for datasets with censored

observations such as BQL. If a certain imputation is chosen and the =

final

model demonstrate good simulation properties both for the contentious

observations (above LOQ) and for the fraction of BQL samples this is a =

good

indication that the obtained parameter estimates are likely to be fairly

unbiased.

I have also thought of more elaborate approaches with simulation and

re-estimation with different imputations of BQL observations to evaluate =

the

possible biased introduced by substitution the BQL samples with an =

assumed

value (similar principle as the Back step method described by Kjellsson =

MC

et.al. (1)). However this is nothing that I have tested or think is an

attractive solution for many cases.

(1) Kjellsson MC, Jönsson S, Karlsson MO. The back-step method--method =

for

obtaining unbiased population parameter estimates for ordered =

categorical

data. AAPS J. 2004 Aug 11;6(3):e19.

By the way I don't agree with your conclusion that LOQ/2 substitution

provides reasonable results in the indirect response model example (C). =

It

induces a non negligible amount of bias in both the concentration effect

parameter (SLOP) and Kout (5-15%). In my opinion substitution with LOQ/2 =

are

just as bad as omitting the BQL data (just bad in another way) for that =

very

example.

Kind regards,

Martin Bergstrand, MSc, PhD student

-----------------------------------------------

Pharmacometrics Research Group,

Department of Pharmaceutical Biosciences,

Uppsala University

-----------------------------------------------

P.O. Box 591

SE-751 24 Uppsala

Sweden

-----------------------------------------------

martin.bergstrand

-----------------------------------------------

Work: +46 18 471 4639

Mobile: +46 709 994 396

Fax: +46 18 471 4003

-----Original Message-----

From: Leonid Gibiansky [mailto:LGibiansky

Sent: den 20 november 2009 04:01

To: Mats Karlsson

Cc: 'Doshi, Sameer'; 'nmusers'; Martin Bergstrand

Subject: Re: [NMusers] Modeling biomarker data below the LOQ

Mats, Martin,

In the paper, you mentioned that "an extra additive error

model term was added for samples substituted with LOQ/2". Was it fixed

or estimated? If fixed, how? Have you tried to vary this level?

In many of your examples, LOQ/2 imputations and exclusion of BQL samples =

seen to lead to bias in opposite directions; if so, it could be an

optimal value (relative to LOQ) of the fixed extra error term that

provides the least biased parameters.

Laplacian method is often not feasible for receptor (target) models

since they are strongly nonlinear (thus requiring differential

equations) and stiff. Based on the the indirect-response model

simulations considered in your paper, LOQ/2 substitution seems to

provide reasonable results if Laplacian (and thus M2-M3-M4) cannot be =

used.

Thanks

Leonid

--------------------------------------

Leonid Gibiansky, Ph.D.

President, QuantPharm LLC

web: www.quantpharm.com

e-mail: LGibiansky at quantpharm.com

tel: (301) 767 5566

Mats Karlsson wrote:

in

ideas.

models.

of

methods

duration,

and

LAPLACIAN

LAPLACIAN

Received on Fri Nov 20 2009 - 05:44:08 EST