RE: Steady state model
Hello Sherwin,
SS=1 infers a perfect steady state situation, i.e. a particular dosing event has been occurring with a particular time interval for infinitely long - and nothing else is happening. For the "standard PK model library" in NONMEM, i.e. for those linear ODE systems with their analytical (closed form) solutions implemented in specific NONMEM Advans, the single dose (SD) solution is a sum of exponential terms e.g. for a 1st order absorption 1st order elimination 1-cmpt model the solution for conc in the central cmpt is
C(t)= (exp(-K*t)-exp(-Ka*t)) *D*F*Ka/V/(Ka-K). Because all processes in these models are linear, when you add more doses their individual contributions to C(t) are additive. So for n doses interspaced by a constant interval tau, with the example model we get
C(t)= (exp(-K*t)+exp(-K*(t+tau))+exp(-K*(t+2tau))...+exp(-K*(t+n*tau)) -exp(-Ka*t)-exp(-Ka*(t+tau))-exp(-Ka*(t+2tau))...-exp(-Ka*(t+n*tau))) *D*F*Ka/V/(Ka-K).
This is the sum of two geometric series multplied by a constant, which reduces to
C(t)= (exp(-K*t)*(1-exp(-K*n*tau))/(1-exp(-K*tau)) -exp(-Ka*t)*(1-exp(-Ka*n*tau))/(1-exp(-Ka*tau)) *D*F*Ka/V/(Ka-K).
As n tends to infinity, 1-exp(-K*n*tau) and 1-exp(-Ka*n*tau) both tend to 1 so for true steady state i.e. infinite n you get the simpler expressions referred by Samer.
I believe the Gabrielsson & Weiner PKPD book has some of the SS solutions in it.
Best,
Andreas
Quoted reply history
-----Original Message-----
From: owner-nmusers
Behalf Of Mouksassi Mohamad-Samer
Sent: 6. august 2009 08:23
To: nmusers
Cc: n.holford
Subject: RE: [NMusers] Steady state model
Hello Sherwin,
All SS routines source code are located at the C:\nmvi\pr folder.
Each Advan closed form model has specific routines (and equations for SS) that can be used with it.
For linear models the magic factor for steady state computation will be:
exp(-rate.constant.time)/(1-exp(-rate.constant.Tau).
Monolix guide: Monolix31_PKPD_library.pdf has a lot of SS equations for commonly used models.
Of course general Advans have their general SS routines too and as Nick mentioned there is some root finding going on:
a comment from the SS6.FOR routine reads
C SS IS SOLUTION A OF: 0DT(A)+R
C APPROXIMATION: DADT(A)DT(0)+DA*A
C 0DT(0)+DA*A+R
C A=-DAINV*(DADT(0)+R)
...
Happy Reading !
Bests,
Samer
-----Original Message-----
From: owner-nmusers
Sent: Wed 8/5/2009 15:44
To: nmusers
Subject: Re: [NMusers] Steady state model
Sherwin,
I dont understand exactly how NONMEM computes the steady state value but with ODEs it seems to be done using a numerical root finding procedure i.e. solves for the amt in each of the compartments when all the DEs have a value of zero.
The amt in each compartment is set to the steady state value. There is no initial 'parameter' for the compartment. Compartment amounts are variables. Parameters are constants. Parameters (e.g. THETA values) are used in the ODEs to define the DE values.
Perhaps Alison Boechmann (who wrote the code) could give a more thorough answer?
Nick
Sherwin K Sy wrote:
> Dear NONMEM users,
>
> I'm wondering what equation or ODE is used in NONMEM when the steady
> state is set (i.e. SS = 1). Is it the case that the initial parameter
> for the compartment is set to a different value? If so, how does
> NONMEM set this value?
>
> I would appreciate if anyone can provide me with a reference or point
> me to where I can find this information, including the type of
> equation used for extravascular, iv bolus and iv infusion models.
>
> Thanks,
>
> Sherwin
>
>
--
Nick Holford, Professor Clinical Pharmacology Dept Pharmacology & Clinical Pharmacology University of Auckland, 85 Park Rd, Private Bag 92019, Auckland, New Zealand n.holford
90
mobile: +64 21 46 23 53
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford