From: musor000@optonline.net
Subject: [NMusers] target-mediated model - steady state - unstable system?
Date: Thu, 27 Oct 2005 19:01:57 -0400

Hello everyone,

I try to run a simple target-mediated model.  It converges well. Plots look OK.  
There is an association-dissociation process.  In the model, 

A2 - drug concentration,
A3 - hormone concentration,
A4 - concentration of complex (hormone+drug)
Ket - elimination constant for A2
Kev - elimination constant for A3
Ketv - elimination constant for complex (TV)
Tin - A2 infusion rate
Vin - A3 production rate
Kon - association rate constant
Koff - dissociation rate constant

DADT(2) =  - KET *A(2) - KON*A(2)*A(3) + KOFF*A(4) + Tin 
DADT(3) =  - KEV *A(3) - KON*A(2)*A(3) + KOFF*A(4) + Vin
DADT(4) =  - KETV*A(4) + KON*A(2)*A(3) - KOFF*A(4)  

There is one thing I do not fully understand.  If I assume that patients get infinite
infusion (Tin is the infusuin rate), then concentrations can achieve steady state.  The
system of differential equatuions transformes into system of 3 algebraic equations
because in steady-state all derivavtives are equal to zero.  I tried to get A3 (hormone
which we try to eliminate but cannot measure) as a function of Tin.  I had to solve a
simple quadratic equation to get function A3 = A3(Tin, Ket,Kev,Ketv,Kon,Koff).
Surprisingly, when Tin is large enough, disctiminant is negative.  Possibly, this means
that the steady-state does not exist, i.e. concentration of A4 (A4 is complex:  A2
bound to A3) grows infinitely.  When I studied control systems, we called it "unstable
system."  Does it sound familiar to you?  Is there anyone who is familiar with properties
of this model?  Can this system get unstable?  

Thanks!
Pavel
_______________________________________________________

From: Leonid Gibiansky leonidg@metrumrg.com
Subject: Re: [NMusers] target-mediated model - steady state - unstable system?
Date: Thu, 27 Oct 2005 23:40:25 -0400

Pavel,
I think you made a mistake somewhere. Determinant can be presented as
x(Tin-Vin)2 + y, where x and y are some positive combination of the parameters
(assuming that all Ks and Ts  are positive). Also, it is unlikely that A4 grows
unbounded. Indeed, let KETV=0 (no A4 elimination). Then on steady state
KON*A(2)*A(3) = KOFF*A(4),  KET *A(2)  = Tin;  KEV *A(3) = Vin. System with
extra elimination (KETV > 0) should have lower A4 level.
Leonid
_______________________________________________________