RE: VD as a fraction of another VD
Dear all,
>From this thread, I wanted tangentially to broach some issues/thoughts with
covariate analysis, and to a lesser extent, initial OMEGA matrix formulation
when dealing with parent-metabolite models. For simplicity, assume only one
metabolite, IV injection of parent with clearance and clearance to
metabolite of CLo and CLm, respectively, (total parent clearance is
CLt=Clo+Clm) and one compartment disposition models. Then let k = CLt/Vp
and Vp (Vm) is the volume of the parent (metabolite).
The model is well known:
Cm = Dose*k12/Vm*(exp(-k*t)-exp(k20)*t)/(k20-k) [k20 is metabolite
elimination rate constant], but to focus the discussion use
Cm = Dose*k12/Vm*A(t) to simplify, since we will not need A(t) .
Since metabolite is not dosed, as discussed, we have an identifiability
issue. The model can be rewritten as discussed,
Cm = Dose*Fm/Vm*k*A(t) where Fm = k12/k = Clm/(Clo+Clm) because k12 =
Clm/Vp. An estimable form of this model is
Cm = Dose/Vm0*k*A(t) where Vm0 = Vm/Fm, ie, the fraction metabolized is
absorbed into the volume
It would seem to me that covariates assumed to have the relationship
(Clm+Clo)*f(x) [f(x) is the covariate function for x] are not necessarily
required to be tested on Vm0 (the parameter we can estimate) because it
would cancel. For example, if f(x) = (WT/70)^x, then it is not a component
of Vm0 because of Clm/(Clo+Clm) and its cancelation. However, if a
covariate affects only one of Clm or Clo, or effects these in a different
way (or to a different extent), then one should evaluate this covariate on
either Vm0, or using a relative version of Fm, eg, Fm = 1 * f(x), (ideally
based on the interpretation the analyst wants to imply), because the
relationship will be implicit and not cancel. Carrying this to random
effects, if one is to fit a model to the parent with CLt*exp(etaCLt) then
this would not induce an implicit correlation with Vm0 (as above). If there
is variability in the fraction metabolized between subjects, then even if
CLt*exp(etaCLt) is used for modeling the parent, the eta in Vm0*exp(etaVm0)
should be evaluated for correlation with etaCLt, because of the underlying
variablity in Clo and Clm. Additionally, it would seem that becuase Vp
factors out of Fm, covariates influencing Vp would not necessarily need to
be tested (from an implicit viewpoint) on Vm0, and a priori correlation
between Vm0 and Vp need not be applied as there is not underlying implicit
relationship induced by the inclusion of Fm into Vm0, but that correlation
could still be evaluated.
Best regards,
Matt
Quoted reply history
From: [email protected] [mailto:[email protected]] On
Behalf Of Martin Bergstrand
Sent: Wednesday, May 23, 2012 1:56 PM
To: 'Nick Holford'; 'e.krekels'; 'Carlos Orlando Jacobo Cabral'; 'nonmem
users'
Subject: RE: [NMusers] VD as a fraction of another VD
Dear Elke, Orlando and Nick,
I have to give Nick my full hearted support in this question. Parent
drug/metabolite models are common practice in population PK and there should
be a kind of best practice for how to parameterize these instead of
inventing one new way after another. I do not doubt that the Knibbe model
gives an excellent fit to that data and is predictive with respect to
external data. That is not the point, the point is that an identical fit to
the data could have been obtained by another parameterization that makes for
a much more straight forward interpretation.
The volume of distribution for the metabolite (e.g. M3G) is unidentifiable
in the exact same way that the volume of distribution is unidentifiable for
any drug where only data following oral administration is available. The
estimate of both Volume and CL for the metabolites will be estimates over
Fmet (i.e. the fraction of the parent compound that forms the metabolite).
To estimate V2 as a fraction of V1 is a pointless parameterization that
serves no purpose. It is reasonable to believe that there will be a high
correlation between the volumes of distribution (e.g. V1 and V2) and this
can be assessed by applying an OMEGA BLOCK to estimate the covariance (e.g.
OMEGA1 and OMEGA2, see below).
V1 = THETA(1)*EXP(ETA(1)) ; central volume for morphine
V2 = THETA(2)*EXP(ETA(2)) ; central volume for M3G/Fm3g
$OMEGA BLOCK(2) 0.1 ; VAR_V1
0.08 ; COVAR_V1_V2
0.1 ; VAR_V2
The outcome of this could be that the estimated covariance corresponds to
approximately 100% correlation. In this case it is still not clearly
justified to reduce the model to assume the same OMEGA variance for both
parameters since the magnitude of variability could still differ between the
two parameters. To assume 100% correlation but different variances can be
done with this parameterization:
V1 = THETA(1)*EXP(ETA(1)) ; central volume for morphine
V2 = THETA(2)*EXP(ETA(1)*THETA(3)) ; central volume for M3G/Fm3g
Where THETA(4) relates the standard deviation of V2 to the standard
deviation of V1 random effect. This model is hierarchically related to a
parameterization that is mathematically equivalent to the parameterization
in the Kibbe model:
V1 = THETA(1)*EXP(ETA(1)) ; central volume for morphine
V2 = THETA(2)*EXP(ETA(1)) ; central volume for M3G/Fm3g
This parameterization could very well turn out to be a sufficient
characterization of the system but it is not true that it cannot be tested
if a more complex model is better (see above steps).
When it comes to the fraction of morphine that is metabolized into M3G and
M6G it can as pointed out not be estimated without access to data following
iv. administration of the metabolites or making very strong assumption such
as fixing distribution volumes etc. Instead it is better to in the model
have all morphine that is eliminated forms both M3G and M6G. This way the
estimated clearance parameters for the metabolites will be (CLm3g/Fm3g and
CLm6g/Fm6g). By the same logic that it isn't identifiable to quantify the
relative formation of M3G and M6G it is also impossible to characterize any
additional rout of elimination.
Reducing the model by setting similar volumes of distribution to the one and
same parameter is nothing that I would practice and I think that it is more
transparent to show the certainty estimates for each parameter in the model.
Let me again stress that I do not question the predictive performance of the
Knibbe model or that it has been useful for it's purposes. I have no insight
to this . However I don't think that it has applied a type of
parameterization that should be put forward as a good example since it has
no advantages compared to the standard parameterization that I suggest that
does facilitate a straight forward interpretation and easy comparison to
results from other studies (with or without data following iv administration
of M3G/M6G).
Regards,
Martin Bergstrand, PhD
Pharmacometrics Research Group
Dept of Pharmaceutical Biosciences
Uppsala University
Sweden
[email protected]
Visiting scientist:
Mahidol-Oxford Tropical Medicine Research Unit,
Bangkok, Thailand
Phone: +66 8 9796 7611