Hello
Nele,
You can try Ke, V2, K23 and K32 instead of Cl, Q, V2 and V3. It
may mostly remove the correlation between V2 and Cl because
Cl=V2*Ke. If F1 is not a parameter in your model and you give
SC doses, it is the same. You have (Cl/F1) = (V2/F1)*Ke, i.e.
your parameters are "apparent". So, using different
parameterizations may help in both cases assuming that Ke and
V2 are not correlated, of course. Unless there are physiological
reasons for them to be correlated, which should be rare, the
correlation between them may be a result of noise in a small
datasets.
Take care,
Pavel
On Tue, Nov 26, 2013 at 10:18 AM, Bob Leary wrote:
Peiming –
Thanks – you are right. The only constraint that the diagonal 3-eta
parameterization parameterization places is that the covariance term of
the block 2-eta
Omega is non-negative.
From: [email protected] [mailto:[email protected]]
On Behalf Of Peiming Ma
Sent: Monday, November 25, 2013 8:55 PM
To: 'nmusers'
Subject: RE: [NMusers] Getting rid of correlation issues between CL and
volume parameters
Dear Bob,
The 3-eta parameterization really is mathematically equivalent to a
2-eta parameterization that has a non-negative covariance term. Here the
3-eta form is just two linear combinations of normal random variables,
which form a bivariate normal with non-negative covariance. No other
restrictions are there.
Regards,
Peiming
From: [email protected] <mailto:[email protected]>
[mailto:[email protected]
<mailto:[email protected]> ] On Behalf Of Bob Leary
Sent: Tuesday, November 26, 2013 5:09 AM
To: Nick Holford; 'nmusers'
Subject: RE: [NMusers] Getting rid of correlation issues between CL and
volume parameters
Nick –
I defer to you and the undoubtedly many other readers who know far more
about pharmacokinetic theory than I do as to which particular
formulation is more appropriate from a PK theoretic point of view. I
was merely trying to note (and as I point out below, incorrectly) that
something like the 2-eta formulation
CL=THETA(1)*EXP(ETA(1))V=THETA(2)*EXP(ETA(2)) Where ETA(1) and ETA(2)
have a full 2 by 2 block correlation matrix so that correlation between
ETA(1) and ETA(2) is Handled by an OMEGA(1,2) parameter
Is ‘mathematically equivalent’ to a 3-eta formulation with a 3 by 3
diagonal Omega (ETA(1), ETA(2), ETA(3) independent)
FF1=EXP(ETA(3))CL=THETA(1)*EXP(ETA(1))/FF1V=THETA(2)*EXP(ETA(2))/FF1
(The fact that FF1 formally looks like a bioavailability is irrelevant
here, since I was not really intending to make any specific comments or
recommendations with respect to how best to
deal with bioavailabilities)
Now that I look at it a bit more closely, the formulations actually are
not at all mathematically equivalent (the 2 by 2 block formulation is
much more
General than the 3 by 3 diagonal formulation, even though they have the
same number of parameters). While all 3 by 3 diagonal Omegas have
Equivalent 2 by 2 block Omegas, the reverse is clearly not true. This
is most easily seen in
in the second 3 by 3 diagonal formulation where
CL=THETA(1)*EXP(ETA(1)-ETA(3)),
and V=THETA(2)*EXP(ETA(2)-ETA(3)),
so cov(log CL, log V) = var(ETA(3)) >0. Thus in the second diagonal
3-eta formulation, the log CL-log V correlation must be positive (or at
least non-negative), while there is no such restriction
on the full block 2-eta formulation. So in fact the 2-eta block
formulation is more general. I think it is even worse than this –
there appear to be some regions of
the block 2 eta parameter space that do not have equivalents in the
diagonal 3-eta space even when the correlations are positive. (For
example, if log CL and log V are highly correlated,
then the variance of ETA(3) must be very large relative to the variance
of ETA(2) and ETA(1) in the 3-eta formulation. But this means the
variance of ETA(1) and ETA(2) in an equivalent two eta formulation must
be relatively similar and roughly equal to the variance of ETA(3) in the
3-eta formulation.
So without working out the details, I think there are regions of the
block 2-eta space corresponding to highly correlated log CL and log V
but with very different log CL and log V variances that
are unattainable in the 3-eta formulations. So in fact the second 3
eta diagonal formulation is fundamentally different and less general
than the first 2eta block formulation. But this just means that if CL
and V are correlated only thru
the F11 bioavailability like mechanism posited in the 3-eta formulation,
there are restrictions as to what the corresponding 2 by 2 full block
omega matrix can looks like. This leaves open the interesting point –
run it both ways, and
then see if the 2 by 2 and 3 by 3 methods produce compatible Omegas. If
not, then this might provide some evidence that the coupling is more
complicated than just that posited in the 3 by 3 diagonal model
But in any event, the EM methods are not well suited to the second
case, and will be inefficient relative to the first case if indeed they
work at all (which may depend on the particular implementation)
One problem is that the EM update of THETA(1) in the second case
depends on the means for the various subjects of the posterior
distributions of both ETA(1) and ETA(3) – most EM implementations
usually have one or possibly several fixed effects coupled to a single
random effect, and the update of that fixed effect, at least in the
simple mu-modeled case, comes from a simple linear regression of the
associated fixed effects on the posterior means of the single random
effect. The fact that now there are multiple random effects paired
with a single fixed effect is unusual
and may not in fact be handled (I am not sure what NONMEM IMPEM will do
with this; I am pretty sure that the analogous Phoenix NLME QRPEM will
reject it).
Bob
From: [email protected] <mailto:[email protected]>
[mailto:[email protected]
<mailto:[email protected]> ] On Behalf Of Nick Holford
Sent: Monday, November 25, 2013 1:43 PM
To: 'nmusers'
Subject: Re: [NMusers] Getting rid of correlation issues between CL and
volume parameters
Bob,
You use an estimation method justification for choosing between
estimating the covariance of CL and V and estimating the variance of F.
An alternative view is to apply a fixed effect assumption based on
pharmacokinetic theory. The fixed effect assumption is that some of the
variation in CL and V is due to differences in bioavailability and other
factors such as linear plasma protein binding and differences in the
actual amount of drug in the oral formulation. This fixed effect
assumption is described in the model by the variance of F.
It is quite plausible to imagine that there is still some covariance
between CL and V that is not related to the differences in F. For
example, if you did not know the subject's weights and therefore could
not account for the correlated effects of weight on CL and V. The
estimation of the variance of F would only partly account for this
because of the non-linear correlation of weight with CL and V. Another
non-linear correlation would occur if plasma protein binding was
non-linear in the range of measured total concentrations.
In such case one might propose trying to estimate the covariance of CL
and V as well as including F as a fixed effect and estimating the
variance of F. Do you think that SAEM or IMP would be able to come up
with a reasonable estimate of the covariance of CL and V?
Best wishes,
Nick
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