Note on comp1.react:
1. With G = zero, Flow =0.01 ml/sec and V1 = .05 ml, the mean
transit time through the mixing chamber is 5 seconds.
With CinA = 0, and A10 = 1 mM, the concn should fall to 1/e times
the initial concentraion in 5 sec.
The solution shows that A(t=5sec) = 0.36787944; 1/e = 0.367879
so the solutioon seems good to six figures at least.
2. With an input CinA = 0 mM, G = 0.01 ml/sec, Rev = 0 ml/sec,
and V1=0.05 ml, A falls as it is converted to B. With Flow set to 0,
the time to A10*(1/e) is 5.0 sec. as before, and the fraction of A
converted to B is 1-1/e = 0.632121.
3.With an input CinA = 1 mM, G = 0.01 ml/sec, Rev = 0.01 ml/sec,
and V1=0.05 ml, and with Flow set to 0.01 ml/sec, A falls as it
is converted to B. It levels off at A = 0.6666666 mM and B levels off
at 0.3333333 mM. Calculate to see if this is correct.
(Hint: In steady state, the derivatives = zero, so set A:t and B:t
to zero and solve the two simultaneous equations for A and B.
The answer is 2/3 and 1/3.) Did your computation depend on the value
of V1? (No it didn't.) Does the form of the solution depend on V1?
JSim v1.1
import nsrunit;
unit conversion on;
// MODEL NUMBER: 0040
// MODEL NAME: Anderson_JC_2007_fig3
// SHORT DESCRIPTION: Volume of Distribution of equilibrium binding and unsteady state.
// Figure 3 of "Tracers in Physiological Systems Modeling".
math Vdist.bind {
realDomain t sec; t.min=0; t.max=30; t.delta=0.1; // independent variable
real C1(t) mM, // concn of C in compartment 1
B(t) mM, CB(t) mM, // concn of binder B and CB complex in V1
Vdist(t) ml, // steady state volume of distibution (free + bound C)
Vdist2(t) ml, // transient state volume of distibution (free + bound C)
C10 = 1 mM, // initial concn C in compartment 1
kon = 0.01 1/(mM*sec), // rate of C + B -> CB
kd = 1 mM, // equilibrium dissociation const
koff = kd*kon, // rate of dissociation of CB into B and C
Btot =.1 mM, // total binding substance
V1 = 1 ml, // volume of compartment 1
Rate = 0.1 pmol/s^2; // rate of addition of C to V1
// extern real CinA(t) mM;
// Initial conditions
when(t=t.min) {C1=C10; CB = 0;}
real C1tr(t) pM, CBtr(t) pM, fluxtr1(t);//tracers for C
real C1tr0 = 1 pM;
// real C1T(t) pM, C1T0 = 1 pM; //
when(t=t.min) {C1tr = C1tr0; CBtr = 0;
/* C1T = C1T0; */ } //
// ODE's from mass balance
// Non-tracer xolute equations:
C1:t = - kon*C1*B + koff*CB + Rate*t/V1;
B = Btot - CB - CBtr;
CB:t = kon*C1*B - koff*CB;
Vdist = V1*(1 + Btot/(kd + C1 + C1tr));
Vdist2 = V1*(1+ CB/C1); // Volume of distribution at equilibrium.
//Conservation expession: C1:t + CB:t = 0.
C1tr:t = - kon*C1tr*B + koff*CBtr;
//B = Btot - CB -CBtr;
CBtr:t = kon*C1tr*B - koff*CBtr; //bound tracer
fluxtr1 = (C1tr/C1)*C1:t;
// Equations assuming instantaneous binding instead of slow binding for tracer:
// C1T:t = -PS/(V1*(1+ Btot/kd))*(C1T-C2T);
// C2T:t = (PS/V2)*(C1T-C2T);
}
/*
DETAILED DESCRIPTION:
One compartment model with constant volumes, V1, but
concentration-dependent volume of distribution.
B is an uncomplexed binding site which reacts, first order, with C1 to form CB
The reverse dissociation reaction converts CB to C1 and B.
The dissociation const Kd = koff/kon and at equilibrium = C1*B/CB
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: tracer, tracee, metabolic physiologic modeling, lumped compartmental versus
spatially distributed systems, capillary-tissue exchange, membrane transporters, enzyme
reactions, steady state versus transient states.
REFERENCES:
Anderson JC and Bassingthwaighte JB: "Tracers in physiological systems modeling".
In: Mathematical Modeling in Nutrition and Agriculture. Proc 9th International Conf on Mathematical
Modeling in Nutrition, Roanoke, VA, August 14-17, 2006, edited by Mark D. Hanigan JN and Casey L Marsteller.
Virginia Polytechnic Institute and State University, Blacksburg, VA 2007, pp 125-159.
REVISION HISTORY:
Original Author : JBB Date: 08/12/08
Revised by : BEJ Date: 05/26/2009
Revision: 1) Update comments and format
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2009 University of Washington. From the National Simulation Resource,
Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Academic use is unrestricted. Software may be copied so long as this copyright notice is included.
This software was developed with support from NIH grant HL073598.
Please cite this grant in any publication for which this software is used and send one reprint
to the address given above.
*/