From: Paul Laub <P_Laub@fccc.edu>
Subject: model building question

Date: 19 Aug 1997 12:27:44 -0400

Dear NONMEM users,

This multiple part question concerns the development on an initial PK model of population data believed to have much interindividual variability. Issues raised here may be of general interest to anyone developing structural models of PK data.

My approach is motivated by chapter 11, "Model Building", in part V of the "NONMEM Users Guide" (Nov. 1994), particularly pages 122-3 and 139. I am using NONMEM IV level 2.1 on a PC with the MS Powerstation FORTRAN 1.0 compiler.

In assessing alternative structural models, I want to examine ETAs from each individual. To do this I use the following:

\$ESTIMATION POSTHOC
\$TABLE ID TIME ETA(1) ETA(2) FILE=somefile.txt

in conjunction with a candidate structural model such as the following:

\$PK
CL = THETA(1)*(1+ETA(1))
V1 = THETA(2)*(1+ETA(2))
Q = THETA(3)
V2 = THETA(4)
S1 = V1
\$ERROR
Y = F*(1+ERR(1))

I have reliable initial estimates for THETA vector. I also supply two estimates for the diagonal matrix OMEGA.

The estimation step proceeds to convergence, then at posthoc NONMEM fails. The error message is -

> 0PRED EXIT CODE = 1
> 0INDIVIDUAL NO. 1 ID= .17000000E+02 (WITHIN-INDIVIDUAL) DATA REC NO. 1
> THETA=
> 3.14E+01 4.11E+00 1.95E+02 9.08E+01
> OCCURS DURING SEARCH FOR ETA AT A NONZERO VALUE OF ETA
> ERROR IN TRANS4 ROUTINE: V1 IS NEGATIVE
> 0PROGRAM TERMINATED BY FNLETA
> MESSAGE ISSUED FROM TABLE STEP

Patient number 17 is the first patient; thus posthoc thus appears to fail on the first record of the data set. I suspect that the variability might be so great that ETA(2) < -1 for patient 17 so that V1 < 0 thereby causing the error. The same PRED error occurred for the additive ETA representation: V1 = THETA(2) + ETA(2).

QUESTION 1: Is my interpretation correct? If not, then what is happening?

QUESTION 2: Is there still some way of getting individual ETAs at this early stage of model development?

One answer to question 2 may be the following. Somebody somewhere (at the NONMEM course?) once mentioned the following:

CL = THETA(1)*DEXP(ETA(1))
V1 = THETA(2)*DEXP(ETA(2))

where DEXP() is the FORTRAN double precision exponential function.

Now, CL and V1 will be positive-valued for any value of the ETAs. Consequently, both the estimation and posthoc steps worked, and as desired the ETAs were listed in the table file. The final objective function value was identical to that obtained using the above proportional model for ETAs.

Surprisingly(!?), the two diagonal OMEGA elements returned for ETAs coded as

; exponential representation
CL = THETA(1)*DEXP(ETA(1))
V1 = THETA(2)*DEXP(ETA(2))

were exactly identical to the OMEGA elements with ETAs coded as

; proportional representation
CL = THETA(1)*(1+ETA(1))
V1 = THETA(2)*(1+ETA(2))

in the run where estimation worked but posthoc didn't.

QUESTION 3: How do I explain the equality of ETAs returned from the exponential and proportional representation?

My first thought is that NONMEM is linearizing the exponential function, ie., from the Taylor series expansion for any x,

exp(x) = 1 + x + [higher order terms]

But I expect that such an approximation should be poor when it is suspected from the outset that there is much interindividual variability in the data.

Finally I would like to gain so physical intuition about the meaning of individual ETAs and elements of the OMEGA matrix. I can readily do this for the additive and proportional representations but have no idea how to do this for the exponential representation.

QUESTION 4: Can anyone help me here?

Thank you for reading all of the way through this long and complicated problem.

Sincerely,

Paul (Sisyphus) B. Laub
mathematical modeling of biomedical data
Dept. of Pharmacology
328 West Bldg.
Fox Chase Cancer Center
7701 Burholme Ave.
Phila. PA 19111 USA
p_laub@fccc.edu
(215) 728-4743 (voice)
(215) 728-2741 (fax)

****

From: alison@c255.ucsf.EDU (ABoeckmann)
Subject: model building question
Date: 19 Aug 1997 18:42:44 -0400

Attached is a question that was sent today by Paul Laub. Full text of the question is attached below. My comments are marked with >.

QUESTION 1: Is my interpretation correct?

>Yes.

QUESTION 2: Is there still some way of getting individual ETAs at this early stage of model development?

>Yes, use Exponential moel for eta, as you suggested.
>This is suggested in Guide VII, Conditional Estimation Methods,
>Chapter III, p. 8.

QUESTION 3: How do I explain the equality of ETAs returned from the exponential and proportional representation?

>This is not so surprising. Guide VII, above, tells you to expect just
>what you have seen. Please look through Guide VII - it is only 12
>pages long and contains much valuable information about posthoc and
>other conditional methods.

Finally I would like to gain so physical intuition about the meaning of individual ETAs and elements of the OMEGA matrix. I can readily do this for the additive and proportional representations but have no idea how to do this for the exponential representation.

QUESTION 4: Can anyone help me here?

>With first order estimation method, there is no difference between
>proportional and exponental models for eta. The interpretation of
>OMEGA is the same. In Guide IV, Chapt. 3, it is stated:
>
>These estimates are empirical Bayesian esti- mates, conditional not
>only on the data, but, importantly, also on values for the
>population parameters. If the first-order estimation method is
>used, they may be obtained after the population parameter
>estimates have themselves been obtained.
>
>I don't know what "physical intuition" you are looking for. Many
>physiological characteristics (e.g., body weight) have skewed rather
>than symmetric distributions in the population, so it is not such
>an odd way of modelling them.
>One further note:
>
>Guide V was for the most part written prior to the developement
>of the conditional estimation methods, and before people realized
>how useful posthoc etas etc. can be early in model development.
>The short course content has changed since it was written.
>That is why the additive/proportional models for eta appear in
>Guide V rather than the (under some circumstances) equivalent
>exponential model. Sorry if it you found it misleading in that
>respect.

Alison Boeckmann

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