Dear all,
I have come across an interesting proposal to account for correlation between
CL and volume parameters by dividing by bioavailability within the NONMEM
control stream:
http://www.wright-dose.com/tip2.php
I liked the approach, however I have been wondering how exactly to interpret
the resulting parameter values for CL and V.
As an illustration, a potential problem might be that we have doses of 10, 25
and 50 mg with a fixed bioavailability of 100% for the 10 mg dose, and
bioavailabilities of 80% and 50% for the doses of 25 and 50 mg, respectively.
In addition, a between-subject variability on F1 of ~30% would be present.
If I now code my CL and V as follows:
CL=THETA(1)/F1
V=THETA(2)/F1,
to account for the correlation between CL and V, what exactly would be the
meaning/interpretation of THETA(1) and THETA(2)?
As the THETAs would be the same for all doses, the CL of 50 mg would be twice
as high as the one for the 10 mg dose – does that make sense, as we already
estimated the reduced relative bioavailability using parameter F1?
Any comments would be very much appreciated.
Thanks and best
Nele
Dr. Nele Müller-Plock, CAPM
Principal Scientist Modeling and Simulation
Pharmacometrics
Experimental Medicine
Takeda Pharmaceuticals International GmbH
8152 Glattpark-Opfikon (Zürich)
Switzerland
Visitor address:
Alpenstrasse 3
8152 Glattpark-Opfikon (Zürich)
Switzerland
Phone: (+41) 44 / 55 51 404
Mobile: (+41) 79 / 654 33 99
mailto: [email protected]
http://www.takeda.com
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copy, disclose or forward all or any of its contents. Please immediately notify
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Getting rid of correlation issues between CL and volume parameters
Nele,
I am not sure why would you like to divide by F1.
Can we just do it explicitly?
F1=EXP(ETA(1))
(or F1=function(dose)*EXP(ETA(1))
CL=..
V=..
F1 can be > 1 as it is not absolute but relative (to the other subjects); I assume that this is oral dose, not IV, correct?
In your code, be careful not to call it F1 as the nonmem will interpret it as bioavailability parameter, and you should not account for it twice.
So it should be either
F1=EXP(ETA(1))
CL=THETA()*EXP(ETA())
V=THETA()*EXP(ETA())
or
F1=1 (can me implicit and omitted)
FF1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/FF1
V=THETA()*EXP(ETA())/FF1
but not
F1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/F1
V=THETA()*EXP(ETA())/F1
Leonid
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
Quoted reply history
On 11/22/2013 12:14 PM, Mueller-Plock, Nele wrote:
> Dear all,
>
> I have come across an interesting proposal to account for correlation between
> CL and volume parameters by dividing by bioavailability within the NONMEM
> control stream:
>
> http://www.wright-dose.com/tip2.php
>
> I liked the approach, however I have been wondering how exactly to interpret
> the resulting parameter values for CL and V.
>
> As an illustration, a potential problem might be that we have doses of 10, 25
> and 50 mg with a fixed bioavailability of 100% for the 10 mg dose, and
> bioavailabilities of 80% and 50% for the doses of 25 and 50 mg, respectively.
> In addition, a between-subject variability on F1 of ~30% would be present.
>
> If I now code my CL and V as follows:
> CL=THETA(1)/F1
> V=THETA(2)/F1,
> to account for the correlation between CL and V, what exactly would be the
> meaning/interpretation of THETA(1) and THETA(2)?
> As the THETAs would be the same for all doses, the CL of 50 mg would be twice
> as high as the one for the 10 mg dose – does that make sense, as we already
> estimated the reduced relative bioavailability using parameter F1?
>
> Any comments would be very much appreciated.
> Thanks and best
> Nele
>
> Dr. Nele Müller-Plock, CAPM
> Principal Scientist Modeling and Simulation
> Pharmacometrics
> Experimental Medicine
>
> Takeda Pharmaceuticals International GmbH
> 8152 Glattpark-Opfikon (Zürich)
> Switzerland
>
> Visitor address:
> Alpenstrasse 3
> 8152 Glattpark-Opfikon (Zürich)
> Switzerland
>
> Phone: (+41) 44 / 55 51 404
> Mobile: (+41) 79 / 654 33 99
> mailto: [email protected]
> http://www.takeda.com
> --------------------------------------------------------------------
>
> The content of this email and of any files transmitted may contain
> confidential, proprietary or legally privileged information and is intended
> solely for the use of the person/s or entity/ies to whom it is addressed. If
> you have received this email in error you have no permission whatsoever to use,
> copy, disclose or forward all or any of its contents. Please immediately notify
> the sender and thereafter delete this email and any attachments.
>
>
Dear Leonid,
Thanks for your answer. Maybe I was not completely clear about the reasons why
I tried to account for F1. The reason is that after oral dosing, a correlation
between CL and should be present, as these parameters in reality represent CL/F
and V/F. One way to account for this would be to estimate the correlation via
the $OMEGA BLOCK syntax. As this is sometimes hard to estimate, I looked if any
alternative is available, and then found the discussion of this topic in the
provided link ( http://www.wright-dose.com/tip2.php).
From your answer, I would conclude that the proposed code should only account
for random between-subject variability, i.e. it should only consider the ETA on
F1, but not the THETA (which in my example had values of 1, 0.8 and 0.5). Is
this correct?
So whereas an increase in ETA on F1 without accounting for the correlation
would automatically result in positive ETA values for CL and V, even without
any inherent variability in true CL and V, with the code
F1=1
FF1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/FF1
V=THETA()*EXP(ETA())/FF1
this would already be taken care of, and the $OMEGA BLOCK could be omitted.
Right?
Thanks and best
Nele
______________________________________________________________
Dr. Nele Mueller-Plock, CAPM
Principal Scientist Modeling and Simulation
Global Pharmacometrics
Therapeutic Area Group
Takeda Pharmaceuticals International GmbH
Thurgauerstrasse 130
8152 Glattpark-Opfikon (Zürich)
Switzerland
Visitor address:
Alpenstrasse 3
8152 Glattpark-Opfikon (Zürich)
Switzerland
Phone: (+41) 44 / 55 51 404
Mobile: (+41) 79 / 654 33 99
mailto: [email protected]
http://www.takeda.com
Quoted reply history
-----Original Message-----
From: Leonid Gibiansky [mailto:[email protected]]
Sent: Freitag, 22. November 2013 19:44
To: Mueller-Plock, Nele; 'nmusers'
Subject: Re: [NMusers] Getting rid of correlation issues between CL and volume
parameters
Nele,
I am not sure why would you like to divide by F1.
Can we just do it explicitly?
F1=EXP(ETA(1))
(or F1=function(dose)*EXP(ETA(1))
CL=..
V=..
F1 can be > 1 as it is not absolute but relative (to the other subjects); I
assume that this is oral dose, not IV, correct?
In your code, be careful not to call it F1 as the nonmem will interpret it as
bioavailability parameter, and you should not account for it twice.
So it should be either
F1=EXP(ETA(1))
CL=THETA()*EXP(ETA())
V=THETA()*EXP(ETA())
or
F1=1 (can me implicit and omitted)
FF1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/FF1
V=THETA()*EXP(ETA())/FF1
but not
F1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/F1
V=THETA()*EXP(ETA())/F1
Leonid
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
On 11/22/2013 12:14 PM, Mueller-Plock, Nele wrote:
> Dear all,
>
> I have come across an interesting proposal to account for correlation between
> CL and volume parameters by dividing by bioavailability within the NONMEM
> control stream:
>
> http://www.wright-dose.com/tip2.php
>
> I liked the approach, however I have been wondering how exactly to interpret
> the resulting parameter values for CL and V.
>
> As an illustration, a potential problem might be that we have doses of 10, 25
> and 50 mg with a fixed bioavailability of 100% for the 10 mg dose, and
> bioavailabilities of 80% and 50% for the doses of 25 and 50 mg, respectively.
> In addition, a between-subject variability on F1 of ~30% would be present.
>
> If I now code my CL and V as follows:
> CL=THETA(1)/F1
> V=THETA(2)/F1,
> to account for the correlation between CL and V, what exactly would be the
> meaning/interpretation of THETA(1) and THETA(2)?
> As the THETAs would be the same for all doses, the CL of 50 mg would be twice
> as high as the one for the 10 mg dose – does that make sense, as we already
> estimated the reduced relative bioavailability using parameter F1?
>
> Any comments would be very much appreciated.
> Thanks and best
> Nele
>
>
>
> Dr. Nele Müller-Plock, CAPM
> Principal Scientist Modeling and Simulation Pharmacometrics
> Experimental Medicine
>
> Takeda Pharmaceuticals International GmbH
> 8152 Glattpark-Opfikon (Zürich)
> Switzerland
>
> Visitor address:
> Alpenstrasse 3
> 8152 Glattpark-Opfikon (Zürich)
> Switzerland
>
> Phone: (+41) 44 / 55 51 404
> Mobile: (+41) 79 / 654 33 99
> mailto: [email protected]
> http://www.takeda.com
> --------------------------------------------------------------------
>
> The content of this email and of any files transmitted may contain
> confidential, proprietary or legally privileged information and is intended
> solely for the use of the person/s or entity/ies to whom it is addressed. If
> you have received this email in error you have no permission whatsoever to
> use, copy, disclose or forward all or any of its contents. Please immediately
> notify the sender and thereafter delete this email and any attachments.
>
> --------------------------------------------------------------------
>
--------------------------------------------------------------------
The content of this email and of any files transmitted may contain
confidential, proprietary or legally privileged information and is intended
solely for the use of the person/s or entity/ies to whom it is addressed. If
you have received this email in error you have no permission whatsoever to use,
copy, disclose or forward all or any of its contents. Please immediately notify
the sender and thereafter delete this email and any attachments.
Nele,
Basically what you have done is traded an off diagonal parameter in a two
dimensional Omega matrix for an extra on-diagonal parameter in a three
dimensional diagonal Omega matrix.
Y0u still have 3 Omega parameters either way.
For methods like SAEM and IMP, the two-dimensional formulation is much
preferable since you end up in a lower 2-d dimensional eta space which a) is
easier to sample,
b) is easily mu-modeled (not the case for the 3-d formulation) , and c) SAEM
and IMP methods handle full block Omegas very naturally, in fact more
naturally than
diagonal Omegas. With FOCEI it is not so clear if there would be any
difference at all.
Quoted reply history
-----Original Message-----
From: [email protected] [mailto:[email protected]] On
Behalf Of Mueller-Plock, Nele
Sent: Monday, November 25, 2013 2:05 AM
To: Leonid Gibiansky; 'nmusers'
Subject: RE: [NMusers] Getting rid of correlation issues between CL and volume
parameters
Dear Leonid,
Thanks for your answer. Maybe I was not completely clear about the reasons why
I tried to account for F1. The reason is that after oral dosing, a correlation
between CL and should be present, as these parameters in reality represent CL/F
and V/F. One way to account for this would be to estimate the correlation via
the $OMEGA BLOCK syntax. As this is sometimes hard to estimate, I looked if any
alternative is available, and then found the discussion of this topic in the
provided link ( http://www.wright-dose.com/tip2.php).
From your answer, I would conclude that the proposed code should only account
for random between-subject variability, i.e. it should only consider the ETA on
F1, but not the THETA (which in my example had values of 1, 0.8 and 0.5). Is
this correct?
So whereas an increase in ETA on F1 without accounting for the correlation
would automatically result in positive ETA values for CL and V, even without
any inherent variability in true CL and V, with the code
F1=1
FF1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/FF1
V=THETA()*EXP(ETA())/FF1
this would already be taken care of, and the $OMEGA BLOCK could be omitted.
Right?
Thanks and best
Nele
______________________________________________________________
Dr. Nele Mueller-Plock, CAPM
Principal Scientist Modeling and Simulation Global Pharmacometrics Therapeutic
Area Group
Takeda Pharmaceuticals International GmbH Thurgauerstrasse 130
8152 Glattpark-Opfikon (Zürich)
Switzerland
Visitor address:
Alpenstrasse 3
8152 Glattpark-Opfikon (Zürich)
Switzerland
Phone: (+41) 44 / 55 51 404
Mobile: (+41) 79 / 654 33 99
mailto: [email protected]
http://www.takeda.com
-----Original Message-----
From: Leonid Gibiansky [mailto:[email protected]]
Sent: Freitag, 22. November 2013 19:44
To: Mueller-Plock, Nele; 'nmusers'
Subject: Re: [NMusers] Getting rid of correlation issues between CL and volume
parameters
Nele,
I am not sure why would you like to divide by F1.
Can we just do it explicitly?
F1=EXP(ETA(1))
(or F1=function(dose)*EXP(ETA(1))
CL=..
V=..
F1 can be > 1 as it is not absolute but relative (to the other subjects); I
assume that this is oral dose, not IV, correct?
In your code, be careful not to call it F1 as the nonmem will interpret it as
bioavailability parameter, and you should not account for it twice.
So it should be either
F1=EXP(ETA(1))
CL=THETA()*EXP(ETA())
V=THETA()*EXP(ETA())
or
F1=1 (can me implicit and omitted)
FF1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/FF1
V=THETA()*EXP(ETA())/FF1
but not
F1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/F1
V=THETA()*EXP(ETA())/F1
Leonid
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
On 11/22/2013 12:14 PM, Mueller-Plock, Nele wrote:
> Dear all,
>
> I have come across an interesting proposal to account for correlation between
> CL and volume parameters by dividing by bioavailability within the NONMEM
> control stream:
>
> http://www.wright-dose.com/tip2.php
>
> I liked the approach, however I have been wondering how exactly to interpret
> the resulting parameter values for CL and V.
>
> As an illustration, a potential problem might be that we have doses of 10, 25
> and 50 mg with a fixed bioavailability of 100% for the 10 mg dose, and
> bioavailabilities of 80% and 50% for the doses of 25 and 50 mg, respectively.
> In addition, a between-subject variability on F1 of ~30% would be present.
>
> If I now code my CL and V as follows:
> CL=THETA(1)/F1
> V=THETA(2)/F1,
> to account for the correlation between CL and V, what exactly would be the
> meaning/interpretation of THETA(1) and THETA(2)?
> As the THETAs would be the same for all doses, the CL of 50 mg would be twice
> as high as the one for the 10 mg dose – does that make sense, as we already
> estimated the reduced relative bioavailability using parameter F1?
>
> Any comments would be very much appreciated.
> Thanks and best
> Nele
>
>
>
> Dr. Nele Müller-Plock, CAPM
> Principal Scientist Modeling and Simulation Pharmacometrics
> Experimental Medicine
>
> Takeda Pharmaceuticals International GmbH
> 8152 Glattpark-Opfikon (Zürich)
> Switzerland
>
> Visitor address:
> Alpenstrasse 3
> 8152 Glattpark-Opfikon (Zürich)
> Switzerland
>
> Phone: (+41) 44 / 55 51 404
> Mobile: (+41) 79 / 654 33 99
> mailto: [email protected]
> http://www.takeda.com
> --------------------------------------------------------------------
>
> The content of this email and of any files transmitted may contain
> confidential, proprietary or legally privileged information and is intended
> solely for the use of the person/s or entity/ies to whom it is addressed. If
> you have received this email in error you have no permission whatsoever to
> use, copy, disclose or forward all or any of its contents. Please immediately
> notify the sender and thereafter delete this email and any attachments.
>
> --------------------------------------------------------------------
>
--------------------------------------------------------------------
The content of this email and of any files transmitted may contain
confidential, proprietary or legally privileged information and is intended
solely for the use of the person/s or entity/ies to whom it is addressed. If
you have received this email in error you have no permission whatsoever to use,
copy, disclose or forward all or any of its contents. Please immediately notify
the sender and thereafter delete this email and any attachments.
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
Quoted reply history
On 26/11/2013 4:04 a.m., Bob Leary wrote:
> Nele,
> Basically what you have done is traded an off diagonal parameter in a two
> dimensional Omega matrix for an extra on-diagonal parameter in a three
> dimensional diagonal Omega matrix.
> Y0u still have 3 Omega parameters either way.
> For methods like SAEM and IMP, the two-dimensional formulation is much
> preferable since you end up in a lower 2-d dimensional eta space which a) is
> easier to sample,
> b) is easily mu-modeled (not the case for the 3-d formulation) , and c) SAEM
> and IMP methods handle full block Omegas very naturally, in fact more
> naturally than
> diagonal Omegas. With FOCEI it is not so clear if there would be any
> difference at all.
>
> -----Original Message-----
> From: [email protected] [mailto:[email protected]] On
> Behalf Of Mueller-Plock, Nele
> Sent: Monday, November 25, 2013 2:05 AM
> To: Leonid Gibiansky; 'nmusers'
> Subject: RE: [NMusers] Getting rid of correlation issues between CL and volume
> parameters
>
> Dear Leonid,
>
> Thanks for your answer. Maybe I was not completely clear about the reasons why
> I tried to account for F1. The reason is that after oral dosing, a correlation
> between CL and should be present, as these parameters in reality represent CL/F
> and V/F. One way to account for this would be to estimate the correlation via
> the $OMEGA BLOCK syntax. As this is sometimes hard to estimate, I looked if any
> alternative is available, and then found the discussion of this topic in the
> provided link ( http://www.wright-dose.com/tip2.php).
> From your answer, I would conclude that the proposed code should only account
> for random between-subject variability, i.e. it should only consider the ETA on
> F1, but not the THETA (which in my example had values of 1, 0.8 and 0.5). Is
> this correct?
>
> So whereas an increase in ETA on F1 without accounting for the correlation
> would automatically result in positive ETA values for CL and V, even without
> any inherent variability in true CL and V, with the code
>
> F1=1
> FF1=EXP(ETA(1))
> CL=THETA()*EXP(ETA())/FF1
> V=THETA()*EXP(ETA())/FF1
>
> this would already be taken care of, and the $OMEGA BLOCK could be omitted.
> Right?
>
> Thanks and best
> Nele
> ______________________________________________________________
>
> Dr. Nele Mueller-Plock, CAPM
>
> Principal Scientist Modeling and Simulation Global Pharmacometrics Therapeutic
> Area Group
>
> Takeda Pharmaceuticals International GmbH Thurgauerstrasse 130
>
> 8152 Glattpark-Opfikon (Zürich)
> Switzerland
>
> Visitor address:
> Alpenstrasse 3
> 8152 Glattpark-Opfikon (Zürich)
> Switzerland
>
> Phone: (+41) 44 / 55 51 404
> Mobile: (+41) 79 / 654 33 99
>
> mailto: [email protected]
>
> http://www.takeda.com
>
> -----Original Message-----
> From: Leonid Gibiansky [mailto:[email protected]]
> Sent: Freitag, 22. November 2013 19:44
> To: Mueller-Plock, Nele; 'nmusers'
> Subject: Re: [NMusers] Getting rid of correlation issues between CL and volume
> parameters
>
> Nele,
> I am not sure why would you like to divide by F1.
> Can we just do it explicitly?
>
> F1=EXP(ETA(1))
> (or F1=function(dose)*EXP(ETA(1))
> CL=..
> V=..
>
> F1 can be > 1 as it is not absolute but relative (to the other subjects); I
> assume that this is oral dose, not IV, correct?
>
> In your code, be careful not to call it F1 as the nonmem will interpret it as
> bioavailability parameter, and you should not account for it twice.
>
> So it should be either
> F1=EXP(ETA(1))
> CL=THETA()*EXP(ETA())
> V=THETA()*EXP(ETA())
>
> or
>
> F1=1 (can me implicit and omitted)
> FF1=EXP(ETA(1))
> CL=THETA()*EXP(ETA())/FF1
> V=THETA()*EXP(ETA())/FF1
>
> but not
>
> F1=EXP(ETA(1))
> CL=THETA()*EXP(ETA())/F1
> V=THETA()*EXP(ETA())/F1
>
> Leonid
>
> --------------------------------------
> Leonid Gibiansky, Ph.D.
> President, QuantPharm LLC
> web: www.quantpharm.com
> e-mail: LGibiansky at quantpharm.com
> tel: (301) 767 5566
>
> On 11/22/2013 12:14 PM, Mueller-Plock, Nele wrote:
>
> > Dear all,
> >
> > I have come across an interesting proposal to account for correlation between
> > CL and volume parameters by dividing by bioavailability within the NONMEM
> > control stream:
> >
> > http://www.wright-dose.com/tip2.php
> >
> > I liked the approach, however I have been wondering how exactly to interpret
> > the resulting parameter values for CL and V.
> >
> > As an illustration, a potential problem might be that we have doses of 10, 25
> > and 50 mg with a fixed bioavailability of 100% for the 10 mg dose, and
> > bioavailabilities of 80% and 50% for the doses of 25 and 50 mg, respectively.
> > In addition, a between-subject variability on F1 of ~30% would be present.
> >
> > If I now code my CL and V as follows:
> > CL=THETA(1)/F1
> > V=THETA(2)/F1,
> > to account for the correlation between CL and V, what exactly would be the
> > meaning/interpretation of THETA(1) and THETA(2)?
> > As the THETAs would be the same for all doses, the CL of 50 mg would be twice
> > as high as the one for the 10 mg dose – does that make sense, as we already
> > estimated the reduced relative bioavailability using parameter F1?
> >
> > Any comments would be very much appreciated.
> > Thanks and best
> > Nele
> >
> > Dr. Nele Müller-Plock, CAPM
> > Principal Scientist Modeling and Simulation Pharmacometrics
> > Experimental Medicine
> >
> > Takeda Pharmaceuticals International GmbH
> > 8152 Glattpark-Opfikon (Zürich)
> > Switzerland
> >
> > Visitor address:
> > Alpenstrasse 3
> > 8152 Glattpark-Opfikon (Zürich)
> > Switzerland
> >
> > Phone: (+41) 44 / 55 51 404
> > Mobile: (+41) 79 / 654 33 99
> > mailto: [email protected]
> > http://www.takeda.com
> > --------------------------------------------------------------------
> >
> > The content of this email and of any files transmitted may contain
> > confidential, proprietary or legally privileged information and is intended
> > solely for the use of the person/s or entity/ies to whom it is addressed. If
> > you have received this email in error you have no permission whatsoever to use,
> > copy, disclose or forward all or any of its contents. Please immediately notify
> > the sender and thereafter delete this email and any attachments.
> >
> > --------------------------------------------------------------------
>
> --------------------------------------------------------------------
>
> The content of this email and of any files transmitted may contain
> confidential, proprietary or legally privileged information and is intended
> solely for the use of the person/s or entity/ies to whom it is addressed. If
> you have received this email in error you have no permission whatsoever to use,
> copy, disclose or forward all or any of its contents. Please immediately notify
> the sender and thereafter delete this email and any attachments.
>
> --------------------------------------------------------------------
>
>
Hello all,
I am a little concerned that this is more complicated than the simple trick we
are talking about implementing. I think the trick can be used, but I am
concerned about the empirical Bayes predictions using three random effects
here, when thinking about the structural model, one could not fit such profiles
individually using least squares due to lack of identifiability. The
information is likely blurred between these random effects such that any plots
of the empirical Bayes estimates would be likely misleading (shrinkage could be
distributed in an odd way throughout). I am also not sure what the
consequences would be for estimation for the FOCE case (maybe a multimodal
issue?). If we ignore the extra correlation discussed by Nick below for now
(this would require more thought) and just deal with correlation induced by F
alone (and we ignore the constrain the F<=1 such that log-normal eta for F
could be entertained), then this in my mind is a constrained OMEGA matrix
optimization problem. That is, the variability in F places a specific
structure on OMEGA. Only one more parameter (variance component) needs to be
estimated than the number of compartments. For example, take a 2 compartment
model (CL,Vc,Q,Vp). One can write the model as:
CL=theta1*EXP(eta1-etaF)
Vc=theta2*EXP(eta2-etaF)
Q =theta3*EXP(eta3-etaF)
Vp=theta4*EXP(eta4-etaF)
The OMEGA matrix in terms of V11=Var(eta1) … V44=Var(eta4), VFF=Var(etaF), with
Covar(etai,etaF)=0 is
V11+VFF VFF VFF VFF
VFF V22+VFF VFF VFF
VFF VFF V33+VFF VFF
VFF VFF VFF V44+VFF
And one can see 5 identifiable parameters. To avoid 5 etas, maybe we can
reparameterize using the (Log-)Cholesky decomposition. Let OMEGA=S’S where S
is an upper triangular matrix, then S11^2=V11+VFF, S11*S12=VFF, S11*S13=VFF,
S11*S14=VFF, etc (unless someone is really interested I will leave this out),
such that elements of S, Sij, can be calculated in terms of V11, .. VFF. Then
we can reparameterize the model
ET1=S11*eta1
ET2=S12*eta1+S22*eta2
ET3=S13*eta1+S23*eta2+S33*eta3
ET4=S14*eta1+S24*eta2+S34*eta3+S44*eta4
Where these etas, are N(0,1) and CL=theta1*EXP(ET1), etc. Sorry to make this
more complicated. I have not really examined this other than mere
contemplation. Would be interested to know what people think.
Kind regards,
Matt
Quoted reply history
From: [email protected] [mailto:[email protected]] On
Behalf Of Nick Holford
Sent: Monday, November 25, 2013 13:43
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
On 26/11/2013 4:04 a.m., Bob Leary wrote:
Nele,
Basically what you have done is traded an off diagonal parameter in a two
dimensional Omega matrix for an extra on-diagonal parameter in a three
dimensional diagonal Omega matrix.
Y0u still have 3 Omega parameters either way.
For methods like SAEM and IMP, the two-dimensional formulation is much
preferable since you end up in a lower 2-d dimensional eta space which a) is
easier to sample,
b) is easily mu-modeled (not the case for the 3-d formulation) , and c) SAEM
and IMP methods handle full block Omegas very naturally, in fact more
naturally than
diagonal Omegas. With FOCEI it is not so clear if there would be any
difference at all.
-----Original Message-----
From: [email protected] [mailto:[email protected]] On
Behalf Of Mueller-Plock, Nele
Sent: Monday, November 25, 2013 2:05 AM
To: Leonid Gibiansky; 'nmusers'
Subject: RE: [NMusers] Getting rid of correlation issues between CL and volume
parameters
Dear Leonid,
Thanks for your answer. Maybe I was not completely clear about the reasons why
I tried to account for F1. The reason is that after oral dosing, a correlation
between CL and should be present, as these parameters in reality represent CL/F
and V/F. One way to account for this would be to estimate the correlation via
the $OMEGA BLOCK syntax. As this is sometimes hard to estimate, I looked if any
alternative is available, and then found the discussion of this topic in the
provided link ( http://www.wright-dose.com/tip2.php).
>From your answer, I would conclude that the proposed code should only account
>for random between-subject variability, i.e. it should only consider the ETA
>on F1, but not the THETA (which in my example had values of 1, 0.8 and 0.5).
>Is this correct?
So whereas an increase in ETA on F1 without accounting for the correlation
would automatically result in positive ETA values for CL and V, even without
any inherent variability in true CL and V, with the code
F1=1
FF1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/FF1
V=THETA()*EXP(ETA())/FF1
this would already be taken care of, and the $OMEGA BLOCK could be omitted.
Right?
Thanks and best
Nele
______________________________________________________________
Dr. Nele Mueller-Plock, CAPM
Principal Scientist Modeling and Simulation Global Pharmacometrics Therapeutic
Area Group
Takeda Pharmaceuticals International GmbH Thurgauerstrasse 130
8152 Glattpark-Opfikon (Zürich)
Switzerland
Visitor address:
Alpenstrasse 3
8152 Glattpark-Opfikon (Zürich)
Switzerland
Phone: (+41) 44 / 55 51 404
Mobile: (+41) 79 / 654 33 99
mailto: [email protected]
http://www.takeda.com
-----Original Message-----
From: Leonid Gibiansky [mailto:[email protected]]
Sent: Freitag, 22. November 2013 19:44
To: Mueller-Plock, Nele; 'nmusers'
Subject: Re: [NMusers] Getting rid of correlation issues between CL and volume
parameters
Nele,
I am not sure why would you like to divide by F1.
Can we just do it explicitly?
F1=EXP(ETA(1))
(or F1=function(dose)*EXP(ETA(1))
CL=..
V=..
F1 can be > 1 as it is not absolute but relative (to the other subjects); I
assume that this is oral dose, not IV, correct?
In your code, be careful not to call it F1 as the nonmem will interpret it as
bioavailability parameter, and you should not account for it twice.
So it should be either
F1=EXP(ETA(1))
CL=THETA()*EXP(ETA())
V=THETA()*EXP(ETA())
or
F1=1 (can me implicit and omitted)
FF1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/FF1
V=THETA()*EXP(ETA())/FF1
but not
F1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/F1
V=THETA()*EXP(ETA())/F1
Leonid
Another argument in favor of using F1 ~ EXP(ETA(1)) instead of block OMEGA matrix is the covariate modeling. In cases where variability in apparent CL and V is due to the F1 variability, this formulation allows for more mechanistic interpretation of the covariate effects and ETA dependencies on covariates. For example, one can easily explain why ETA_F1 may depend on food while it is less straightforward to interpret ETA_V dependence on food. So while these models (with F1=1 and OMEGA block versus F1=EXP(ETA(1)) and diagnonal OMEGA), may be numerically similar if not equivalent, it could be better to use more mechanistically relevant model and put the variability where it would be expected from the mechanistic point of view.
Regards,
Leonid
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
Quoted reply history
On 11/25/2013 1:43 PM, Nick Holford wrote:
> 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
>
> On 26/11/2013 4:04 a.m., Bob Leary wrote:
>
> > Nele,
> > Basically what you have done is traded an off diagonal parameter in a two
> > dimensional Omega matrix for an extra on-diagonal parameter in a three
> > dimensional diagonal Omega matrix.
> > Y0u still have 3 Omega parameters either way.
> > For methods like SAEM and IMP, the two-dimensional formulation is much
> > preferable since you end up in a lower 2-d dimensional eta space which a) is
> > easier to sample,
> > b) is easily mu-modeled (not the case for the 3-d formulation) , and c) SAEM
> > and IMP methods handle full block Omegas very naturally, in fact more
> > naturally than
> > diagonal Omegas. With FOCEI it is not so clear if there would be any
> > difference at all.
> >
> > -----Original Message-----
> > From:[email protected] [mailto:[email protected]] On
> > Behalf Of Mueller-Plock, Nele
> > Sent: Monday, November 25, 2013 2:05 AM
> > To: Leonid Gibiansky; 'nmusers'
> > Subject: RE: [NMusers] Getting rid of correlation issues between CL and volume
> > parameters
> >
> > Dear Leonid,
> >
> > Thanks for your answer. Maybe I was not completely clear about the reasons why
> > I tried to account for F1. The reason is that after oral dosing, a correlation
> > between CL and should be present, as these parameters in reality represent CL/F
> > and V/F. One way to account for this would be to estimate the correlation via
> > the $OMEGA BLOCK syntax. As this is sometimes hard to estimate, I looked if any
> > alternative is available, and then found the discussion of this topic in the
> > provided link ( http://www.wright-dose.com/tip2.php).
> > >From your answer, I would conclude that the proposed code should only account
> > for random between-subject variability, i.e. it should only consider the ETA on
> > F1, but not the THETA (which in my example had values of 1, 0.8 and 0.5). Is this
> > correct?
> >
> > So whereas an increase in ETA on F1 without accounting for the correlation
> > would automatically result in positive ETA values for CL and V, even without
> > any inherent variability in true CL and V, with the code
> >
> > F1=1
> > FF1=EXP(ETA(1))
> > CL=THETA()*EXP(ETA())/FF1
> > V=THETA()*EXP(ETA())/FF1
> >
> > this would already be taken care of, and the $OMEGA BLOCK could be omitted.
> > Right?
> >
> > Thanks and best
> > Nele
> > ______________________________________________________________
> >
> > Dr. Nele Mueller-Plock, CAPM
> > Principal Scientist Modeling and Simulation Global Pharmacometrics Therapeutic
> > Area Group
> >
> > Takeda Pharmaceuticals International GmbH Thurgauerstrasse 130
> > 8152 Glattpark-Opfikon (Zürich)
> > Switzerland
> >
> > Visitor address:
> > Alpenstrasse 3
> > 8152 Glattpark-Opfikon (Zürich)
> > Switzerland
> >
> > Phone: (+41) 44 / 55 51 404
> > Mobile: (+41) 79 / 654 33 99
> >
> > mailto:[email protected]
> > http://www.takeda.com
> >
> > -----Original Message-----
> > From: Leonid Gibiansky [mailto:[email protected]]
> > Sent: Freitag, 22. November 2013 19:44
> > To: Mueller-Plock, Nele; 'nmusers'
> > Subject: Re: [NMusers] Getting rid of correlation issues between CL and volume
> > parameters
> >
> > Nele,
> > I am not sure why would you like to divide by F1.
> > Can we just do it explicitly?
> >
> > F1=EXP(ETA(1))
> > (or F1=function(dose)*EXP(ETA(1))
> > CL=..
> > V=..
> >
> > F1 can be > 1 as it is not absolute but relative (to the other subjects); I
> > assume that this is oral dose, not IV, correct?
> >
> > In your code, be careful not to call it F1 as the nonmem will interpret it as
> > bioavailability parameter, and you should not account for it twice.
> >
> > So it should be either
> > F1=EXP(ETA(1))
> > CL=THETA()*EXP(ETA())
> > V=THETA()*EXP(ETA())
> >
> > or
> >
> > F1=1 (can me implicit and omitted)
> > FF1=EXP(ETA(1))
> > CL=THETA()*EXP(ETA())/FF1
> > V=THETA()*EXP(ETA())/FF1
> >
> > but not
> >
> > F1=EXP(ETA(1))
> > CL=THETA()*EXP(ETA())/F1
> > V=THETA()*EXP(ETA())/F1
> >
> > Leonid
> >
> > --------------------------------------
> > Leonid Gibiansky, Ph.D.
> > President, QuantPharm LLC
> > web:www.quantpharm.com
> > e-mail: LGibiansky at quantpharm.com
> > tel: (301) 767 5566
> >
> > On 11/22/2013 12:14 PM, Mueller-Plock, Nele wrote:
> >
> > > Dear all,
> > >
> > > I have come across an interesting proposal to account for correlation between
> > > CL and volume parameters by dividing by bioavailability within the NONMEM
> > > control stream:
> > >
> > > http://www.wright-dose.com/tip2.php
> > >
> > > I liked the approach, however I have been wondering how exactly to interpret
> > > the resulting parameter values for CL and V.
> > >
> > > As an illustration, a potential problem might be that we have doses of 10, 25
> > > and 50 mg with a fixed bioavailability of 100% for the 10 mg dose, and
> > > bioavailabilities of 80% and 50% for the doses of 25 and 50 mg, respectively.
> > > In addition, a between-subject variability on F1 of ~30% would be present.
> > >
> > > If I now code my CL and V as follows:
> > > CL=THETA(1)/F1
> > > V=THETA(2)/F1,
> > > to account for the correlation between CL and V, what exactly would be the
> > > meaning/interpretation of THETA(1) and THETA(2)?
> > > As the THETAs would be the same for all doses, the CL of 50 mg would be twice
> > > as high as the one for the 10 mg dose – does that make sense, as we already
> > > estimated the reduced relative bioavailability using parameter F1?
> > >
> > > Any comments would be very much appreciated.
> > > Thanks and best
> > > Nele
> > >
> > > Dr. Nele Müller-Plock, CAPM
> > > Principal Scientist Modeling and Simulation Pharmacometrics
> > > Experimental Medicine
> > >
> > > Takeda Pharmaceuticals International GmbH
> > > 8152 Glattpark-Opfikon (Zürich)
> > > Switzerland
> > >
> > > Visitor address:
> > > Alpenstrasse 3
> > > 8152 Glattpark-Opfikon (Zürich)
> > > Switzerland
> > >
> > > Phone: (+41) 44 / 55 51 404
> > > Mobile: (+41) 79 / 654 33 99
> > > mailto:[email protected]
> > > http://www.takeda.com
> > > --------------------------------------------------------------------
> > >
> > > The content of this email and of any files transmitted may contain
> > > confidential, proprietary or legally privileged information and is intended
> > > solely for the use of the person/s or entity/ies to whom it is addressed. If
> > > you have received this email in error you have no permission whatsoever to use,
> > > copy, disclose or forward all or any of its contents. Please immediately notify
> > > the sender and thereafter delete this email and any attachments.
> > >
> > > --------------------------------------------------------------------
> >
> > --------------------------------------------------------------------
> >
> > The content of this email and of any files transmitted may contain
> > confidential, proprietary or legally privileged information and is intended
> > solely for the use of the person/s or entity/ies to whom it is addressed. If
> > you have received this email in error you have no permission whatsoever to use,
> > copy, disclose or forward all or any of its contents. Please immediately notify
> > the sender and thereafter delete this email and any attachments.
> >
> > --------------------------------------------------------------------
> >
> >
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
Quoted reply history
From: [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))/FF1
V=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]] 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
Dear all,
Thanks for picking up this discussion, and bringing in so many points of view.
When I started the discussion I had in mind the physiological viewpoint, from
which we know that if there is between-subject variability in F1, this must
result in a correlation between volume and CL parameters. From the discussions
I would conclude that the group would favor to account for this correlation via
inclusion of ETA on F1 and then a coding of
FF1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/FF1
V=THETA()*EXP(ETA())/FF1
whereas this does not mean that there is no additional correlation between the
parameters which needs to be accounted for in the off-diagonal OMEGA BLOCK
structure? Also, I am afraid I was not able to completely follow Matt's
argumentation, but would also be interested to hear if implementing the code
above might lead to misleading plots.
Thanks and best
Nele
______________________________________________________________
Dr. Nele Mueller-Plock, CAPM
Principal Scientist Modeling and Simulation
Global Pharmacometrics
Therapeutic Area Group
Takeda Pharmaceuticals International GmbH
Thurgauerstrasse 130
8152 Glattpark-Opfikon (Zürich)
Switzerland
Visitor address:
Alpenstrasse 3
8152 Glattpark-Opfikon (Zürich)
Switzerland
Phone: (+41) 44 / 55 51 404
Mobile: (+41) 79 / 654 33 99
mailto: [email protected]
http://www.takeda.com
Quoted reply history
-----Original Message-----
From: [email protected] [mailto:[email protected]] On
Behalf Of Leonid Gibiansky
Sent: Dienstag, 26. November 2013 00:51
To: 'nmusers'
Subject: Re: [NMusers] Getting rid of correlation issues between CL and volume
parameters
Another argument in favor of using F1 ~ EXP(ETA(1)) instead of block OMEGA
matrix is the covariate modeling. In cases where variability in apparent CL and
V is due to the F1 variability, this formulation allows for more mechanistic
interpretation of the covariate effects and ETA dependencies on covariates.
For example, one can easily explain why
ETA_F1 may depend on food while it is less straightforward to interpret ETA_V
dependence on food. So while these models (with F1=1 and OMEGA block versus
F1=EXP(ETA(1)) and diagnonal OMEGA), may be numerically similar if not
equivalent, it could be better to use more mechanistically relevant model and
put the variability where it would be expected from the mechanistic point of
view.
Regards,
Leonid
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
On 11/25/2013 1:43 PM, Nick Holford wrote:
> 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
>
>
> On 26/11/2013 4:04 a.m., Bob Leary wrote:
>> Nele,
>> Basically what you have done is traded an off diagonal parameter in a two
>> dimensional Omega matrix for an extra on-diagonal parameter in a three
>> dimensional diagonal Omega matrix.
>> Y0u still have 3 Omega parameters either way.
>> For methods like SAEM and IMP, the two-dimensional formulation is
>> much preferable since you end up in a lower 2-d dimensional eta space
>> which a) is easier to sample,
>> b) is easily mu-modeled (not the case for the 3-d formulation) , and c)
>> SAEM and IMP methods handle full block Omegas very naturally, in fact more
>> naturally than
>> diagonal Omegas. With FOCEI it is not so clear if there would be any
>> difference at all.
>>
>>
>>
>> -----Original Message-----
>> From:[email protected]
>> [mailto:[email protected]] On Behalf Of Mueller-Plock,
>> Nele
>> Sent: Monday, November 25, 2013 2:05 AM
>> To: Leonid Gibiansky; 'nmusers'
>> Subject: RE: [NMusers] Getting rid of correlation issues between CL
>> and volume parameters
>>
>> Dear Leonid,
>>
>> Thanks for your answer. Maybe I was not completely clear about the reasons
>> why I tried to account for F1. The reason is that after oral dosing, a
>> correlation between CL and should be present, as these parameters in reality
>> represent CL/F and V/F. One way to account for this would be to estimate the
>> correlation via the $OMEGA BLOCK syntax. As this is sometimes hard to
>> estimate, I looked if any alternative is available, and then found the
>> discussion of this topic in the provided link
>> ( http://www.wright-dose.com/tip2.php).
>> >From your answer, I would conclude that the proposed code should only
>> >account for random between-subject variability, i.e. it should only
>> >consider the ETA on F1, but not the THETA (which in my example had values
>> >of 1, 0.8 and 0.5). Is this correct?
>>
>> So whereas an increase in ETA on F1 without accounting for the
>> correlation would automatically result in positive ETA values for CL
>> and V, even without any inherent variability in true CL and V, with
>> the code
>>
>> F1=1
>> FF1=EXP(ETA(1))
>> CL=THETA()*EXP(ETA())/FF1
>> V=THETA()*EXP(ETA())/FF1
>>
>> this would already be taken care of, and the $OMEGA BLOCK could be omitted.
>> Right?
>>
>> Thanks and best
>> Nele
>> ______________________________________________________________
>>
>> Dr. Nele Mueller-Plock, CAPM
>> Principal Scientist Modeling and Simulation Global Pharmacometrics
>> Therapeutic Area Group
>>
>> Takeda Pharmaceuticals International GmbH Thurgauerstrasse 130
>> 8152 Glattpark-Opfikon (Zürich)
>> Switzerland
>>
>> Visitor address:
>> Alpenstrasse 3
>> 8152 Glattpark-Opfikon (Zürich)
>> Switzerland
>>
>> Phone: (+41) 44 / 55 51 404
>> Mobile: (+41) 79 / 654 33 99
>>
>> mailto:[email protected]
>> http://www.takeda.com
>>
>> -----Original Message-----
>> From: Leonid Gibiansky [mailto:[email protected]]
>> Sent: Freitag, 22. November 2013 19:44
>> To: Mueller-Plock, Nele; 'nmusers'
>> Subject: Re: [NMusers] Getting rid of correlation issues between CL
>> and volume parameters
>>
>> Nele,
>> I am not sure why would you like to divide by F1.
>> Can we just do it explicitly?
>>
>> F1=EXP(ETA(1))
>> (or F1=function(dose)*EXP(ETA(1))
>> CL=..
>> V=..
>>
>> F1 can be > 1 as it is not absolute but relative (to the other subjects); I
>> assume that this is oral dose, not IV, correct?
>>
>> In your code, be careful not to call it F1 as the nonmem will interpret it
>> as bioavailability parameter, and you should not account for it twice.
>>
>> So it should be either
>> F1=EXP(ETA(1))
>> CL=THETA()*EXP(ETA())
>> V=THETA()*EXP(ETA())
>>
>> or
>>
>> F1=1 (can me implicit and omitted)
>> FF1=EXP(ETA(1))
>> CL=THETA()*EXP(ETA())/FF1
>> V=THETA()*EXP(ETA())/FF1
>>
>> but not
>>
>> F1=EXP(ETA(1))
>> CL=THETA()*EXP(ETA())/F1
>> V=THETA()*EXP(ETA())/F1
>>
>> Leonid
>>
>>
>>
>>
>> --------------------------------------
>> Leonid Gibiansky, Ph.D.
>> President, QuantPharm LLC
>> web:www.quantpharm.com
>> e-mail: LGibiansky at quantpharm.com
>> tel: (301) 767 5566
>>
>>
>>
>> On 11/22/2013 12:14 PM, Mueller-Plock, Nele wrote:
>>> Dear all,
>>>
>>> I have come across an interesting proposal to account for correlation
>>> between CL and volume parameters by dividing by bioavailability within the
>>> NONMEM control stream:
>>>
>>> http://www.wright-dose.com/tip2.php
>>>
>>> I liked the approach, however I have been wondering how exactly to
>>> interpret the resulting parameter values for CL and V.
>>>
>>> As an illustration, a potential problem might be that we have doses of 10,
>>> 25 and 50 mg with a fixed bioavailability of 100% for the 10 mg dose, and
>>> bioavailabilities of 80% and 50% for the doses of 25 and 50 mg,
>>> respectively. In addition, a between-subject variability on F1 of ~30%
>>> would be present.
>>>
>>> If I now code my CL and V as follows:
>>> CL=THETA(1)/F1
>>> V=THETA(2)/F1,
>>> to account for the correlation between CL and V, what exactly would be the
>>> meaning/interpretation of THETA(1) and THETA(2)?
>>> As the THETAs would be the same for all doses, the CL of 50 mg would be
>>> twice as high as the one for the 10 mg dose – does that make sense, as we
>>> already estimated the reduced relative bioavailability using parameter F1?
>>>
>>> Any comments would be very much appreciated.
>>> Thanks and best
>>> Nele
>>>
>>>
>>>
>>> Dr. Nele Müller-Plock, CAPM
>>> Principal Scientist Modeling and Simulation Pharmacometrics
>>> Experimental Medicine
>>>
>>> Takeda Pharmaceuticals International GmbH
>>> 8152 Glattpark-Opfikon (Zürich)
>>> Switzerland
>>>
>>> Visitor address:
>>> Alpenstrasse 3
>>> 8152 Glattpark-Opfikon (Zürich)
>>> Switzerland
>>>
>>> Phone: (+41) 44 / 55 51 404
>>> Mobile: (+41) 79 / 654 33 99
>>> mailto:[email protected]
>>> http://www.takeda.com
>>> --------------------------------------------------------------------
>>>
>>> The content of this email and of any files transmitted may contain
>>> confidential, proprietary or legally privileged information and is intended
>>> solely for the use of the person/s or entity/ies to whom it is addressed.
>>> If you have received this email in error you have no permission whatsoever
>>> to use, copy, disclose or forward all or any of its contents. Please
>>> immediately notify the sender and thereafter delete this email and any
>>> attachments.
>>>
>>> --------------------------------------------------------------------
>>>
>> --------------------------------------------------------------------
>>
>> The content of this email and of any files transmitted may contain
>> confidential, proprietary or legally privileged information and is intended
>> solely for the use of the person/s or entity/ies to whom it is addressed. If
>> you have received this email in error you have no permission whatsoever to
>> use, copy, disclose or forward all or any of its contents. Please
>> immediately notify the sender and thereafter delete this email and any
>> attachments.
>>
>> --------------------------------------------------------------------
>>
>>
>>
Hi Nele,
I believe Matt's point was more to the situation where any remaining
correlation between CL and V random components can not be accounted for by
covariates, so that both eta on F and block2 on CL and V is used?
If eta on F and covariates takes care of the correlation between CL and V: I
would say that you may get even more informative diagnostics with this
implementation.
For example, if you have not yet taken dose/formulation into account and this
affects only F, it would come out as a clearer trend on the eta1 (relative F).
This would help in interpretation (but I would highlight Nick's earlier point
that eta on F may capture other nonlinearities that are shared between CL and
V; like degree of protein binding for a low-extraction drug).
Best
Jakob
Quoted reply history
-----Original Message-----
From: [email protected] [mailto:[email protected]] On
Behalf Of Mueller-Plock, Nele
Sent: 26 November 2013 08:21
To: Leonid Gibiansky; 'nmusers'
Subject: RE: [NMusers] Getting rid of correlation issues between CL and volume
parameters
Dear all,
Thanks for picking up this discussion, and bringing in so many points of view.
When I started the discussion I had in mind the physiological viewpoint, from
which we know that if there is between-subject variability in F1, this must
result in a correlation between volume and CL parameters. From the discussions
I would conclude that the group would favor to account for this correlation via
inclusion of ETA on F1 and then a coding of
FF1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/FF1
V=THETA()*EXP(ETA())/FF1
whereas this does not mean that there is no additional correlation between the
parameters which needs to be accounted for in the off-diagonal OMEGA BLOCK
structure? Also, I am afraid I was not able to completely follow Matt's
argumentation, but would also be interested to hear if implementing the code
above might lead to misleading plots.
Thanks and best
Nele
______________________________________________________________
Dr. Nele Mueller-Plock, CAPM
Principal Scientist Modeling and Simulation
Global Pharmacometrics
Therapeutic Area Group
Takeda Pharmaceuticals International GmbH
Thurgauerstrasse 130
8152 Glattpark-Opfikon (Zürich)
Switzerland
Visitor address:
Alpenstrasse 3
8152 Glattpark-Opfikon (Zürich)
Switzerland
Phone: (+41) 44 / 55 51 404
Mobile: (+41) 79 / 654 33 99
mailto: [email protected]
http://www.takeda.com
-----Original Message-----
From: [email protected] [mailto:[email protected]] On
Behalf Of Leonid Gibiansky
Sent: Dienstag, 26. November 2013 00:51
To: 'nmusers'
Subject: Re: [NMusers] Getting rid of correlation issues between CL and volume
parameters
Another argument in favor of using F1 ~ EXP(ETA(1)) instead of block OMEGA
matrix is the covariate modeling. In cases where variability in apparent CL and
V is due to the F1 variability, this formulation allows for more mechanistic
interpretation of the covariate effects and ETA dependencies on covariates.
For example, one can easily explain why
ETA_F1 may depend on food while it is less straightforward to interpret ETA_V
dependence on food. So while these models (with F1=1 and OMEGA block versus
F1=EXP(ETA(1)) and diagnonal OMEGA), may be numerically similar if not
equivalent, it could be better to use more mechanistically relevant model and
put the variability where it would be expected from the mechanistic point of
view.
Regards,
Leonid
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
On 11/25/2013 1:43 PM, Nick Holford wrote:
> 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
>
>
> On 26/11/2013 4:04 a.m., Bob Leary wrote:
>> Nele,
>> Basically what you have done is traded an off diagonal parameter in a two
>> dimensional Omega matrix for an extra on-diagonal parameter in a three
>> dimensional diagonal Omega matrix.
>> Y0u still have 3 Omega parameters either way.
>> For methods like SAEM and IMP, the two-dimensional formulation is
>> much preferable since you end up in a lower 2-d dimensional eta space
>> which a) is easier to sample,
>> b) is easily mu-modeled (not the case for the 3-d formulation) , and c)
>> SAEM and IMP methods handle full block Omegas very naturally, in fact more
>> naturally than
>> diagonal Omegas. With FOCEI it is not so clear if there would be any
>> difference at all.
>>
>>
>>
>> -----Original Message-----
>> From:[email protected]
>> [mailto:[email protected]] On Behalf Of Mueller-Plock,
>> Nele
>> Sent: Monday, November 25, 2013 2:05 AM
>> To: Leonid Gibiansky; 'nmusers'
>> Subject: RE: [NMusers] Getting rid of correlation issues between CL
>> and volume parameters
>>
>> Dear Leonid,
>>
>> Thanks for your answer. Maybe I was not completely clear about the reasons
>> why I tried to account for F1. The reason is that after oral dosing, a
>> correlation between CL and should be present, as these parameters in reality
>> represent CL/F and V/F. One way to account for this would be to estimate the
>> correlation via the $OMEGA BLOCK syntax. As this is sometimes hard to
>> estimate, I looked if any alternative is available, and then found the
>> discussion of this topic in the provided link
>> ( http://www.wright-dose.com/tip2.php).
>> >From your answer, I would conclude that the proposed code should only
>> >account for random between-subject variability, i.e. it should only
>> >consider the ETA on F1, but not the THETA (which in my example had values
>> >of 1, 0.8 and 0.5). Is this correct?
>>
>> So whereas an increase in ETA on F1 without accounting for the
>> correlation would automatically result in positive ETA values for CL
>> and V, even without any inherent variability in true CL and V, with
>> the code
>>
>> F1=1
>> FF1=EXP(ETA(1))
>> CL=THETA()*EXP(ETA())/FF1
>> V=THETA()*EXP(ETA())/FF1
>>
>> this would already be taken care of, and the $OMEGA BLOCK could be omitted.
>> Right?
>>
>> Thanks and best
>> Nele
>> ______________________________________________________________
>>
>> Dr. Nele Mueller-Plock, CAPM
>> Principal Scientist Modeling and Simulation Global Pharmacometrics
>> Therapeutic Area Group
>>
>> Takeda Pharmaceuticals International GmbH Thurgauerstrasse 130
>> 8152 Glattpark-Opfikon (Zürich)
>> Switzerland
>>
>> Visitor address:
>> Alpenstrasse 3
>> 8152 Glattpark-Opfikon (Zürich)
>> Switzerland
>>
>> Phone: (+41) 44 / 55 51 404
>> Mobile: (+41) 79 / 654 33 99
>>
>> mailto:[email protected]
>> http://www.takeda.com
>>
>> -----Original Message-----
>> From: Leonid Gibiansky [mailto:[email protected]]
>> Sent: Freitag, 22. November 2013 19:44
>> To: Mueller-Plock, Nele; 'nmusers'
>> Subject: Re: [NMusers] Getting rid of correlation issues between CL
>> and volume parameters
>>
>> Nele,
>> I am not sure why would you like to divide by F1.
>> Can we just do it explicitly?
>>
>> F1=EXP(ETA(1))
>> (or F1=function(dose)*EXP(ETA(1))
>> CL=..
>> V=..
>>
>> F1 can be > 1 as it is not absolute but relative (to the other subjects); I
>> assume that this is oral dose, not IV, correct?
>>
>> In your code, be careful not to call it F1 as the nonmem will interpret it
>> as bioavailability parameter, and you should not account for it twice.
>>
>> So it should be either
>> F1=EXP(ETA(1))
>> CL=THETA()*EXP(ETA())
>> V=THETA()*EXP(ETA())
>>
>> or
>>
>> F1=1 (can me implicit and omitted)
>> FF1=EXP(ETA(1))
>> CL=THETA()*EXP(ETA())/FF1
>> V=THETA()*EXP(ETA())/FF1
>>
>> but not
>>
>> F1=EXP(ETA(1))
>> CL=THETA()*EXP(ETA())/F1
>> V=THETA()*EXP(ETA())/F1
>>
>> Leonid
>>
>>
>>
>>
>> --------------------------------------
>> Leonid Gibiansky, Ph.D.
>> President, QuantPharm LLC
>> web:www.quantpharm.com
>> e-mail: LGibiansky at quantpharm.com
>> tel: (301) 767 5566
>>
>>
>>
>> On 11/22/2013 12:14 PM, Mueller-Plock, Nele wrote:
>>> Dear all,
>>>
>>> I have come across an interesting proposal to account for correlation
>>> between CL and volume parameters by dividing by bioavailability within the
>>> NONMEM control stream:
>>>
>>> http://www.wright-dose.com/tip2.php
>>>
>>> I liked the approach, however I have been wondering how exactly to
>>> interpret the resulting parameter values for CL and V.
>>>
>>> As an illustration, a potential problem might be that we have doses of 10,
>>> 25 and 50 mg with a fixed bioavailability of 100% for the 10 mg dose, and
>>> bioavailabilities of 80% and 50% for the doses of 25 and 50 mg,
>>> respectively. In addition, a between-subject variability on F1 of ~30%
>>> would be present.
>>>
>>> If I now code my CL and V as follows:
>>> CL=THETA(1)/F1
>>> V=THETA(2)/F1,
>>> to account for the correlation between CL and V, what exactly would be the
>>> meaning/interpretation of THETA(1) and THETA(2)?
>>> As the THETAs would be the same for all doses, the CL of 50 mg would be
>>> twice as high as the one for the 10 mg dose – does that make sense, as we
>>> already estimated the reduced relative bioavailability using parameter F1?
>>>
>>> Any comments would be very much appreciated.
>>> Thanks and best
>>> Nele
>>>
>>>
>>>
>>> Dr. Nele Müller-Plock, CAPM
>>> Principal Scientist Modeling and Simulation Pharmacometrics
>>> Experimental Medicine
>>>
>>> Takeda Pharmaceuticals International GmbH
>>> 8152 Glattpark-Opfikon (Zürich)
>>> Switzerland
>>>
>>> Visitor address:
>>> Alpenstrasse 3
>>> 8152 Glattpark-Opfikon (Zürich)
>>> Switzerland
>>>
>>> Phone: (+41) 44 / 55 51 404
>>> Mobile: (+41) 79 / 654 33 99
>>> mailto:[email protected]
>>> http://www.takeda.com
>>> --------------------------------------------------------------------
>>>
>>> The content of this email and of any files transmitted may contain
>>> confidential, proprietary or legally privileged information and is intended
>>> solely for the use of the person/s or entity/ies to whom it is addressed.
>>> If you have received this email in error you have no permission whatsoever
>>> to use, copy, disclose or forward all or any of its contents. Please
>>> immediately notify the sender and thereafter delete this email and any
>>> attachments.
>>>
>>> --------------------------------------------------------------------
>>>
>> --------------------------------------------------------------------
>>
>> The content of this email and of any files transmitted may contain
>> confidential, proprietary or legally privileged information and is intended
>> solely for the use of the person/s or entity/ies to whom it is addressed. If
>> you have received this email in error you have no permission whatsoever to
>> use, copy, disclose or forward all or any of its contents. Please
>> immediately notify the sender and thereafter delete this email and any
>> attachments.
>>
>> --------------------------------------------------------------------
>>
>>
>>
Hi Jacob, and everyone,
Sorry to be unclear and if I have added any confusion. My derivation was for
the oral/SC administration (into a depot compartment) case with no IV data and
with no extra CL/V correlation. If there were extra correlation, the OMEGA
matrix would look like
V11+VFF VFF+COV(eta1,eta2) VFF+COV(eta1,eta3)
VFF+COV(eta1,eta4)
VFF+COV(eta2,eta1) V22+VFF VFF+COV(eta2,eta3)
VFF+COV(eta2,eta4)
VFF+COV(eta3,eta1) VFF+COV(eta3,eta2) V33+VFF VFF
VFF+COV(eta3,eta4)
VFF+COV(eta4,eta1) VFF+COV(eta4,eta2) VFF+COV(eta4,eta3) V44+VFF
which would not be identifiable without the IV data. In my opinion, if there
is no IV data, the F is really just conceptual. It is a a way of thinking
about certain covariates that affect both CL and V etc in an identical way.
Parameterization using covariates (which I do often) and an eta on F is just a
trick (in the no-IV data case) to get the OMEGA matrix as previously defined
and to avoid having to specify eg, CL=THETA(1)/(1+THETA(2)*FOOD)
V=THETA(3)/(1+THETA(2)*FOOD), in the model (which is equivalent). In this
case, I am concerned about adding the extra eta on F to constrain the OMEGA
matrix because of the whole identifiability issue. Plots would certainly be
affected (there really aren't 3 etas in the non-IV data case). In there is no
extra-correlation, and F is inducing a high degree of correlation, one might
consider putting the eta's on V, K, K12 and K21. The variability of F would be
lumped into V, and this would cancel from the K's allowing a diagonal matrix
(note that one would need to be careful how one parameterized this and it does
not preclude evaluating and estimating fixed effects on CL, V, etc.)
Best,
Matt
(I have trimmed some of the earliest emails from this note to ensure delivery).
Quoted reply history
-----Original Message-----
From: [email protected] [mailto:[email protected]] On
Behalf Of Ribbing, Jakob
Sent: Tuesday, November 26, 2013 05:46
To: Mueller-Plock, Nele; Leonid Gibiansky; 'nmusers'
Cc: Ribbing, Jakob
Subject: RE: [NMusers] Getting rid of correlation issues between CL and volume
parameters
Hi Nele,
I believe Matt's point was more to the situation where any remaining
correlation between CL and V random components can not be accounted for by
covariates, so that both eta on F and block2 on CL and V is used?
If eta on F and covariates takes care of the correlation between CL and V: I
would say that you may get even more informative diagnostics with this
implementation.
For example, if you have not yet taken dose/formulation into account and this
affects only F, it would come out as a clearer trend on the eta1 (relative F).
This would help in interpretation (but I would highlight Nick's earlier point
that eta on F may capture other nonlinearities that are shared between CL and
V; like degree of protein binding for a low-extraction drug).
Best
Jakob
-----Original Message-----
From: [email protected] [mailto:[email protected]] On
Behalf Of Mueller-Plock, Nele
Sent: 26 November 2013 08:21
To: Leonid Gibiansky; 'nmusers'
Subject: RE: [NMusers] Getting rid of correlation issues between CL and volume
parameters
Dear all,
Thanks for picking up this discussion, and bringing in so many points of view.
When I started the discussion I had in mind the physiological viewpoint, from
which we know that if there is between-subject variability in F1, this must
result in a correlation between volume and CL parameters. From the discussions
I would conclude that the group would favor to account for this correlation via
inclusion of ETA on F1 and then a coding of
FF1=EXP(ETA(1))
CL=THETA()*EXP(ETA())/FF1
V=THETA()*EXP(ETA())/FF1
whereas this does not mean that there is no additional correlation between the
parameters which needs to be accounted for in the off-diagonal OMEGA BLOCK
structure? Also, I am afraid I was not able to completely follow Matt's
argumentation, but would also be interested to hear if implementing the code
above might lead to misleading plots.
Thanks and best
Nele
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.
Quoted reply history
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]] 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))/FF1
V=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]] 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
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
Quoted reply history
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