target-mediated model - steady state - unstable system?

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 _______________________________________________________