Re: Time varying volume of distribution implementation
Dear Nick, Thorsten and others,
Nick's example is very good.
He models WT so that it increases linearly with TIME:
DWT_T=WT_ZERO + WT_ALPHA*T
where WT_ZERO and WT_ALPHA are thetas.
It occurs to me that some users may not have a model for WT vs. T,
but have only observed values of WT at fixed points.
In this case, WT can be interpolated within the $DES block.
Here is a small example of code that can be used to interpolate WT
between values that are recorded on the data records.
All the code in $PK to compute OLDTIME and OLDWT and SLOPE, and
the code for D_WT in $DES, could be copied to user's control stream.
Other code (for integrating D_WT in $DES and the analytic solution in
$ERROR)
is for testing and would not be part of the user's control stream.
Here is the control file:
$PROB INTERPOLATE WT IN $DES
; this example shows how to interpolate WT in $DES.
; it is assumed that WT is recorded on every data record.
; As a test, the value of D_WT in $DES is integrated to obtain AUC of WT
VS. T
; This is also calculated analytically in $ERROR.
$INPUT ID TIME WT DV
$DATA desinterp.dat
$SUBROUTINES ADVAN6 TOL=5
$MODEL
COMP=(AUC_WT DEFOBS)
$PK
; initialize OLDTIME and OLDWT
IF (NEWIND.LE.1) THEN
OLDTIME=TIME
OLDWT=WT
ENDIF
; calculate the slope for $DES
DELTA_TIME=TIME-OLDTIME
DELTA_WT=WT-OLDWT
IF (DELTA_TIME>0) THEN
SLOPE=DELTA_WT/DELTA_TIME
ELSE
SLOPE=0.
ENDIF
; save wt and time for next $PK record
OLDTIME=TIME
OLDWT=WT
$DES
D_WT=OLDWT+SLOPE*(T-OLDTIME) ; D_WT is the value of WT at time T
DADT(1)=D_WT ; compute AUC of D_WT as a test
$ERROR
Y=F+ETA(1)+EPS(1)
; Compute analytic solution as a test.
; Does not use compartment amounts.
; Uses only the values of WT and TIME on event records.
; Suppose WT vs T looks like this:
;
;
; WT
; |
; | w3 w4
; | w2 w5
; | w1
; |
; --------------------------------> TIME
; t1 t2 t3 t4 t5
;
; at t2, the contribution to the sum is
; the rectangle w1 x (t2-t1)
; plus the triangular piece
; (w2-w1)/(t2-t1) / 2
;
; w2
; /|
; / |
; /__|
; w1 |
; | |
; | |
; ------------
; t1 t2
IF (NEWIND.LE.1) THEN
PREV_WT=WT ; Initialize WT from previous data record
SUM=0
ELSE
SUM=SUM+PREV_WT*DELTA_TIME+DELTA_WT*DELTA_TIME/2
ENDIF
PREV_WT=WT ; save WT from previous data record
$OMEGA 1
$SIGMA 1
$TABLE ID TIME WT PRED=AUC_WT SUM FILE=desinterp.tbl NOPRINT NOAPPEND
; The following two values should always be equal:
; PRED (which is the AUC of WT obtained by integrating WT)
; SUM (which is the analytic solution) computed in $ERROR
Here is the data file for the first subject. Note that WT
sometimes is constant and sometimes decreases:
1 0. 10 0
1 1. 20 0
1 2. 35 0
1 2. 35 0
1 4. 45 0
1 5. 40 0
Here is the table file:
TABLE NO. 1
ID TIME WT AUC_WT SUM
1.0000E+00 0.0000E+00 1.0000E+01 0.0000E+00 0.0000E+00
1.0000E+00 1.0000E+00 2.0000E+01 1.5000E+01 1.5000E+01
1.0000E+00 2.0000E+00 3.5000E+01 4.2500E+01 4.2500E+01
1.0000E+00 2.0000E+00 3.5000E+01 4.2500E+01 4.2500E+01
1.0000E+00 4.0000E+00 4.5000E+01 1.2250E+02 1.2250E+02
1.0000E+00 5.0000E+00 4.0000E+01 1.6500E+02 1.6500E+02
Quoted reply history
On Sat, Apr 23, 2016, at 01:43 AM, Nick Holford wrote:
> Thorsten,
>
> Time varying V is no different from time varying CL (or any other
> parameter). You should use the variable T in $DES, not TIME, in order to
> have the time at the instant the DEQ solver evaluates $DES. T may occur
> anywhere in the interval between the previous record TIME and the
> current record TIME. TIME in $PK, $DES and $ERROR is the time at the end
> of the $DES solution interval.
>
> The other thing that you may wish to do is to assign the random effect
> expression for V and CL in $PK so that you can estimate the random
> variability after accounting for the fixed effect variability in WT. An
> expression involving ETA() cannot be used in $DES so it has to be
> assigned in $PK.
>
> e.g.
>
> $PK
> POP_V=THETA(1)
> POP_CL=THETA(2)
> WT_ZERO=THETA(3)
> WT_ALPHA=THETA(4)
> PPV_V=EXP(ETA(1)) ; random effect for V (PPV_V=population parameter
> variability for V)
> PPV_CL=EXP(ETA(2)) ; random effect for CLT (PPV_CL=population parameter
> variability for CL)
>
> $DES
> ;Variable names e.g. DWT_T are used in $DES because the same variable
> names cannot be assigned in both $DES and in $ERROR
>
> DWT_T=WT_ZERO + WT_ALPHA*T ; fixed effect prediction of WT at T
>
> ; Biology requires V and CL must both be functions of WT
> DGRP_V=POP_V*DWT_T/70
> DV=DGRP_V*PPV_V ; "individual" V at T using random effect for V
>
> DGRP_CL=POP_CL*(DWT_T/70)**(3/4)
> DCL=DGRP_CL*PPV_CL ; "individual" CL at T using random effect for CL
>
> DADT(1)= -DCL*A(1)/DV
>
> $ERROR
>
> WT_T=WT_ZERO + WT_ALPHA*TIME ; fixed effect prediction of WT at the TIME
> of the current record
>
> GRP_V=POP_V*WT_T/70
> GRP_CL=POP_CL*(WT_T/70)**(3/4)
>
> V=VT*PPV_V ; "individual" V at the TIME of the current record
> CL=GRP_CL*PPV_CL ; "individual" CL at the TIME of the current record
>
> C=A(1)/V
>
> You may, of course, add random effects to WT_ZERO and/or WT_ALPHA as
> well as having random effects on V and CL.
>
> BTW You should consider using the term postmenstrual age rather than
> gestational age. Gestational age is a single value defined at the time
> of delivery according to the American Academy of Pediatrics (Engle et al
> 2004). Postmenstrual age is a continuous variable which may be used
> during pregnancy and after birth to represent the biological age of the
> fetus/child.
>
> Best wishes,
>
> Nick
>
> Engle WA. Age terminology during the perinatal period. Pediatrics.
> 2004;114(5):1362-4.
>
>
> On 22-Apr-16 23:34, Thorsten Lehr wrote:
> >
> > Dear NMusers,
> >
> > I'm modeling a compound where body weight has a known impact on the
> > volume of distribution. This compound is investigated in pregnant
> > women over a long period (from gestational age of 8 weeks until they
> > give birth). Consequently, the body weight changes over time and I
> > have a decent formula to describe the individual body weight change.
> > The PK model has to be coded by ODEs. Does anyone has experience how
> > to integrate a time varying volume of distribution if differential
> > equations are used?
> >
> > Best regards
> >
> > Thorsten
> >
> > --
> >
> > Thorsten Lehr, PhD
> > Junior Professor of Clinical Pharmacy
> > Saarland University
> > Campus C2 2
> > 66123 Saarbrücken
> > Germany
> >
> > Office: +49/681/302-70255
> > Mobile: +49/151/22739489
> > [email protected]
> > www.clinicalpharmacy.me
>
> --
> Nick Holford, Professor Clinical Pharmacology
> Dept Pharmacology & Clinical Pharmacology, Bldg 503 Room 302A
> University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
> office:+64(9)923-6730 mobile:NZ+64(21)46 23 53 FR+33(6)62 32 46 72
> email: [email protected]
> http://holford.fmhs.auckland.ac.nz/
>
> "Declarative languages are a form of dementia -- they have no memory of
> events"
>
> Holford SD, Allegaert K, Anderson BJ, Kukanich B, Sousa AB, Steinman A,
> Pypendop, B., Mehvar, R., Giorgi, M., Holford,N.H.G. Parent-metabolite
> pharmacokinetic models - tests of assumptions and predictions. Journal of
> Pharmacology & Clinical Toxicology. 2014;2(2):1023-34.
> Holford N. Clinical pharmacology = disease progression + drug action. Br
> J Clin Pharmacol. 2015;79(1):18-27.
>
--
Alison Boeckmann
[email protected]