From rs@chdr.leidenuniv.nl Mon Dec 11 16:40:20 1995
Subject: modelling flow-dependent clearance

To anyone who can give me a clue...

I'm trying to model the kinetics of a test-drug with high hepatic clearance given as an intravenous infusion (single compartment kinetics to start with). Clearance of the drug is assumed dependent on hepatic blood flow which is measured using echo-doppler techniques, and which can vary continously over time. It seems to me that I need to model the kinetics using a differential equation (e.g. using ADVAN6) in which clearance is some function (presumably linear) of measured flow. So far so good. The trouble is that I'm pretty certain that I need flow-estimates in between the actual measurements in order to adequately define the process. Something as simple as linear interpolation or else some smooth function would suffice but I have no clue as to the implementation of such a function in NONMEM. Naturally, if someone could convince me that this is not necessary, I'd be more than pleased...

Grateful for any response,

Rik Schoemaker
CHDR, Leiden, The Netherlands

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From lewis Mon Dec 11 17:59:29 1995
Subject: Re: modelling flow-dependent clearance

Sorry, but you need exactly what you think you need - a way to interpolate flow so that the diff eqn solver can have a value at any time. That's the bad news.

The good news is that it is very simple to do. The idea is to transform your actual flow measurements into a slope and intercept such that for any time between the "current" one (the one on the current data record) and the previous data record's time, flow = int + slope*TIME , where TIME IS ACTUAL TIME, NOT TIME ELAPSED SINCE LAST RECORD. Then the diff eqn uses this computed flow.

Here is a template control stream to give you the idea:

\$PROBLEM An Example of interpolated "driver" of Diff Eqn
\$INPUT ID TIME DV INT SLOP
\$DATA data
\$MODEL
COMP=CENTRAL ; Just an example
; Define other compartments as needed
\$PK
B=INT
M=SLOP
CL = THETA(1)*EXP(ETA(1)) ; Just an example
; Define other parameters as needed
\$DES
X=B+(M*T) ; X = time varying factor that is interpolated
DADT(1)= -X*CL ; Just an example
\$ERROR
Y= F*(1+ERR(1)) ; Just an example
;Add \$THETA, \$OMEGA, etc as needed

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From R.Port@dkfz-heidelberg.de Tue Dec 12 00:55:16 1995
Subject: interpolation with diff. eqn.

Re.: Interpolated "driver" of differential equation, Rik Schoemaker's message of Tue, Dec 12; Lewis' message of Mon, Dec 11

Hi Lewis,

do you think one could use (T - TIME) for time elapsed since last record, e.g.

> \$PROBLEM An Example of interpolated "driver" of Diff Eqn
> \$INPUT ID TIME DV INT SLOP ; SLOP = increment per
time elapsed since last record
> \$DATA data
>
> \$MODEL
> COMP=CENTRAL

>
> \$PK
> B=INT
> M=SLOP
> CL = THETA(1)*EXP(ETA(1))

\$DES
TT = T - TIME
X = B + M*TT

??

Ruedi
-------------------------------------------------------------------------------
R.E. Port, Dept. 0420, German Cancer Research Center
P.O. Box 10 19 49, D-69009 Heidelberg
phone: x49-6221 42-3385
-3347
fax: -3346
e-mail: r.port@dkfz-heidelberg.de

****

From lewis Tue Dec 12 04:19:11 1995
Subject: Interpolation with Diff Eqn

Hesterday, in answering a question about interpolating hepatic blood flow in a differential equation moel, I discussed the inberpolation correctly, but presented a kinetic model that was quite wrong, even as an example. Herewith follows a better example (although I still don't guarantee it is correct), and an example of the way to do the interpolation. I hope this is more helpful than yesterday's hasty reply.
================================================================
\$PROBLEM An Example of interpolation for hep bl flow
\$INPUT ID TIME INT SLOP DV AMT
\$DATA data IGNORE=#
\$MODEL
COMP=CENTRAL
; Define other compartments as needed
\$PK
B=INT
M=SLOP
CLI = THETA(1)*EXP(ETA(1)) ; Intrinsic Clearance
V = THETA(2)*EXP(ETA(2))
S1 = V
; Define other parameters as needed
\$DES
Q = B+(M*T) ; Q = time varying flow that is interpolated
CL = CLI*Q/(CLI+Q)
; Add / modify diff Eq's as needed
\$ERROR
Y= F*(1+ERR(1)) ; For example
; Add \$THETA, \$OMEGA, etc as needed
;
; As an example of how to compute (off-line) INT and
; SLOP, Imagine the following data:
; (1) Flow is meawsured at times 1, 5, and 10, with values
; 3, 4, and 1, respectively.
; (2) DV is observed at times 1, 2, 4, 6, 8
;
; The formulas for SLOP, INT to be recorded at DV times >t1 and
; <= t2, where t1 and t2 are onbservation times of flow (Q) are:
; SLOP = (Q(t2)-Q(t1))/(t2-t1)
; INT = Q(t2)-SLOP*t2
;
; Therefore partial data recrods are:
;
; TIME INT SLOP ...
; # Assume flow = value at first measurement time before that time
; # First record willbe dose record perhaps
; 0 . .
; 1 3 0
; 2 3.75 .25
; 4 3.75 .25
; 6 7 -.6
; 8 7 -.6

` `