From: Paul.Tanswell@bc.boehringer-ingelheim.com
Subject: residual error model, endogeneous drug
Date: Thu, 3 May 2001 09:46:45 +0200

NONMEM users:

I am modeling single dose, 2-compartment i.v. pharmacokinetics of a drug X which exhibits endogenous plasma levels, in >100 individuals. In each subject, the baseline concentration of X was measured in one pre-dose sample, and multiple samples were subsequently measured for X post dose. The baseline levels are low compared to Cmax after exogenous dosing, but not negligible compared to the plasma levels measured at later time points.

I followed the procedure suggested for such data in the NONMEM help guide (p312) and used the following code for the residual error (power model; was better than the additive + proportional model)

$ERROR
FTOT=F+CBAS
W=FTOT**THETA(6)
Y=FTOT+W*(EPS(1))
IPRED=FTOT
IRES=CP-IPRED
IWRES=IRES/W

THETA(6) is the exponent in the power model. FTOT is the total plasma concentration estimate, comprising the sum of the exogenous (F) and endogenous (CBAS) drug concentrations. CBAS was allowed to vary between individuals in the $PK block (CBAS=THETA(5)*EXP(ETA(4)). The PK model is ADVAN 3 TRANS 4 with ETAs for CL, Q and V1, and a block structure for OMEGA.

THE PROBLEM:
1. Although the model ran well and the plots of PRED vs DV and IPRED vs DV were satisfactory, the individual values of CBAS, which are a small cluster of values at the lower end of the DV range, were systematically underestimated by about 30%. A plot of IPRED (y) vs DV (x) just for the CBAS yielded an approximately linear relationship but with slope 0.68.

2. The code does not seem quite right to me. Considering CBAS, which is a single pre-dose endogenous measurement per subject, it seems logical to give it an ETA, but it does not seem reasonable to allow an EPS additionally. However, in the code as written, the estimate for the sum of the endogenous and exogenous drug (FTOT) allows both levels of random variability. When FTOT decreases to near CBAS or equals CBAS, then CBAS will effectively have both an ETA and an EPS assigned to it.

My questions to the group:
- is there a method of coding such that F but not CBAS is allowed to have an EPS assigned to it
- would this remove the bias in the estimation of CBAS,
- can one still get correct values for W and IWRES?

Thanks for any help and best regards
Paul Tanswell

 

*****

 

From: LSheiner <lewis@c255.ucsf.edu>
Subject: Re: residual error model, endogeneous drug
Date: Thu, 03 May 2001 14:04:40 -0700

See below.

Paul.Tanswell@bc.boehringer-ingelheim.com wrote:


> THE PROBLEM:
> 1. Although the model ran well and the plots of PRED vs DV and IPRED vs DV
> were satisfactory, the individual values of CBAS, which are a small cluster
> of values at the lower end of the DV range, were systematically
> underestimated by about 30%. A plot of IPRED (y) vs DV (x) just for the CBAS
> yielded an approximately linear relationship but with slope 0.68.

This is puzzling, as the IPRED should certainly be centered about the actual observations. It may have to do with the fact that the model you use for the obserfations involves "interaction" between eta(4) and eps(1), for example, which is famous for giving problems if it is not accounted for (although I do not recall this particular bias). To see if this is what is going on, try an additive error model, or try using 'FOCE with interaction' with the model you have, or see below.

> 2. The code does not seem quite right to me. Considering CBAS, which is a
> single pre-dose endogenous measurement per subject, it seems logical to give
> it an ETA, but it does not seem reasonable to allow an EPS additionally.
> However, in the code as written, the estimate for the sum of the endogenous
> and exogenous drug (FTOT) allows both levels of random variability. When
> FTOT decreases to near CBAS or equals CBAS, then CBAS will effectively have
> both an ETA and an EPS assigned to it.
>
> My questions to the group:
> - is there a method of coding such that F but not CBAS is allowed to have
> an EPS assigned to it

It could be done, ... but it would not be right (in the famous words of Richard Nixon, who claimed the last clause was said, but not recorded on tape, when he was asked by Haldeman/Erlichman whether they should pay off the Watergate burglars to be quiet ..).

The baseline measurement is a measurement like any other: Under your model the observed baseline value has a mean (theta(5)), interindividual variability (eta(4)), and measurement error (eps(1)). Seems just right to me.

However, if you did not care about this nicety, you could certainly let all the variability of the baseline be "absorbed" by the eta. This would also remove the interaction, and would not increase the number of parameters. You might do this with code such as

$ERROR
> W=F**THETA(6)
> FTOT=F+CBAS
> Y=FTOT+W*(EPS(1))
> IPRED=FTOT
> IRES=CP-IPRED
> IWRES=IRES/W

Two final points:
1. This model does not allow for feedback inhibition of endogenous substance production.
2. There is another way to deal with this problem that I actually prefer. It involves conditioning on the baseline observation, but recognizing that it has error of the same magnitude as any other. Unfortunately, the code is tricky enough (one must constrain the variance of an eta to be the same as that of eps, and must recognize (NEWIND) and fix initial conditions), especially for a 2-compartment model (because both compartments must be initialized) that I don't trust myself to write it off the top of my head...

--
_/ _/ _/_/ _/_/_/ _/_/_/ Lewis B Sheiner, MD (lewis@c255.ucsf.edu)
_/ _/ _/ _/_ _/_/ Professor: Lab. Med., Bioph. Sci., Med.
_/ _/ _/ _/ _/ Box 0626, UCSF, SF, CA, 94143-0626
_/_/ _/_/ _/_/_/ _/ 415-476-1965 (v), 415-476-2796 (fax)

*****

 

From: "R. E. Port" <R.Port@DKFZ-Heidelberg.de>
Subject: residual error model, endogeneous drug
Date: Tue, 22 May 2001 12:25:42 +0200 (CEST)

Dear Paul,

a few weeks ago you posed a question to the group about how to model an endogenous baseline concentration (CBAS) of a compound which is also administered exogenously; specifically, the question was how to model the interindividual variation of this baseline concentration:

> My questions to the group:
> - is there a method of coding such that F but not CBAS is allowed to have
> an EPS assigned to it
> - would this remove the bias in the estimation of CBAS,
> - can one still get correct values for W and IWRES?

Lewis, in his reply, mentioned a way of modeling which

...
> involves conditioning on the baseline observation, but recognizing that it
> has error of the same magnitude as any other.

My own experience with modeling an endogenous baseline value ("BASE") has been like yours, namely that describing BASE using a theta and some kind of random variation didn't satisfying results, especially in the presence of large interindividual variation in BASE. On the other hand, I get nice results (with $PRED, I haven't tried $PK) with a procedure where no mean baseline (theta) is estimated at all and where "all the variability of the baseline (is) "absorbed" by the eta ... recognizing that it has error of the same magnitude as any other (observation)" (Lewis). The code is as follows (assuming that your baseline measurements were taken at time zero and that there are no data records at times < 0):

IF (TIME.EQ.0) BASEfee = DV ; fixed-effects estimate of baseline
BASEmee = BASEfee + THETA(1)*ETA(1) ; mixed-effects estimate of baseline

...

; mixed-effects prediction for what was measured at later times:
mep = BASEmee + increment ; "increment": effect of treatment
Y = mep + THETA(1)*EPS(1)

$OMEGA 1 FIXED
$SIGMA 1 FIXED

This code sets PRED = DV for time zero. Setting MDV=1 for this data record prevents it from having an EPS assigned to it, such as you wanted. This lets NMTRAN issue a warning message which, I hope, doesn't matter in this case. ETA(1) and EPS(1) have the same standard deviation (THETA(1)) recognizing that the baseline measurement has the same residual error as all the others. The procedure has been mentioned in a publication (Port et alii, Br J Clin Pharmacol 46, 461 - 466, 1998) but may not have been thoroughly examined by a statistician, so if someone finds it flawed for some statistical reason I'ld be grateful to learn about.

With best regards, Ruedi

--------------------------------------------------------------------------
Dr. R.E. Port, German Cancer Research Center, D-0200
P.O. Box 10 19 49, D-69009 Heidelberg, Germany
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