Questions:
(1) Does the integrated output area (Area_out) for C+D+E
and the transit time depend on the conversion rates?
(2) How does the output area for each curve depend on
the model parameters?
(3) What are the steady state concentrations when Cin
is a constant?
Figure 1A: Concentrations: Default parameter set.
Cout (purple), Dout (red), Eout (orange), and their sum
(blue dashed) are plotted as functions of time.
Figure 1B: Areas: Default parameter set
The integrated inflow and outflow concentrations with
respect to time are plotted as functions of time. Note
That the Area of the sum of the outputs eventually equals
the area of the inflow concentration.
Figure 1C: Transit times: Default parameter set
The transit time calculation reaches a final value
after the transient concentrations have washed out.
Tbar_sys is also given as
Tbar_sys=V/F.
Figure 2: Ratio of Output Areas, D/C and E/D, Default
parameter set
Determine the functional relationship
for the ratio of the area of Dout divided by the area of
Cout, and the ratio of the area Eout divided by the area of
Dout.
Area_out(D) Gc2d
----------- = ------------,
Area_out(C) F + Gd2e
Area_out(E) Gd2e
----------- = ------------.
Area_out(D) F
Does this relationship hold when Cin is a constant?
Figure 3: Steady State, Cin Constant: Fig_3 parameter set
The steady state concentrations for Cin = 3 are calculated
numerically and compared with solving the ordinary
differential equations when the rate of change is 0, i.e.
solving
F (Cin - C) Gc2d C
----------- - ------ = 0,
V V
F D Gc2d C Gd2e D
- --- + ------ - ------ = 0,
V V V
F E Gd2e D
- --- + ------ = 0,
V V
for C, D, and E, yielding
F Cin
C = --------,
F + Gc2d
F Cin Gc2d
D = ----------------------, and
(F + Gc2d) (F + Gd2e)
Cin Gc2d Gd2e
E = ----------------------.
(F + Gc2d) (F + Gd2e)
Prove: C+D+E=Cin.
/* MODEL NUMBER: 0236
MODEL NAME: Comp1FlowReactions2
SHORT DESCRIPTION: Single Compartment with flow and
irreversible conversion of C to D and D to E.
*/
import nsrunit; unit conversion on;
math Comp1FlowReactions2 {
// INDEPENDENT VARIABLE
realDomain t sec; t.min=0; t.max=60; t.delta=0.1;
// PARAMETERS
real F = 0.01 ml/sec, // Flow
V = 0.05 ml, // Volume
Gc2d = 0.03 ml/sec, // Clearance rate C->D
Gd2e = 0.015 ml/sec, // Clearance rate D->E
C0 = 0 mM, // Initial concentration of C
D0 = 0.0 mM, // Initial concentration of D
E0 = 0.0 mM; // Initial concentration of E
extern real Cin(t) mM; // Inflow Concentration
// VARIABLES
real C(t) mM, // Concentration of C
D(t) mM, // Concentration of D
E(t) mM, // Concentration of E
Cout(t) mM, // Outflow concentration of C, Cout=C
Dout(t) mM, // Outflow concentration of D, Dout=D
Eout(t) mM; // Outflow concentration of E, Eout=E
// INITIAL CONDITIONS
when(t=t.min) { C = C0; D = D0; E = E0; }
// ORDINARY DIFFERENTIAL EQUATIONS
C:t = F/V*(Cin-C) -Gc2d/V*C;
D:t = F/V*( -D) +Gc2d/V*C-Gd2e/V*D;
E:t = F/V*( -E) +Gd2e/V*D;
/* Outflow concentration equals compartment concentration
because compartments are instantaneously well mixed. */
Cout=C;
Dout=D;
Eout=E;
//ADDITIONAL CALCULATIONS
real Qint(t) mmol,
Q(t) mmol;
when(t=t.min) {Qint=V*(C+D+E);}
Qint:t=F*(Cin-Cout-Dout-Eout);
Q=V*(C+D+E);
// AREA AND TRANSIT TIME OF INFLOW AND OUTFLOW CONCENTRATIONS
real Tout(t) mM;
Tout = Cout+Dout+Eout;
private real S2_in(t) mM*sec^2, S2_out(t) mM*sec^2; // First moments
real Area_in(t) mM*sec, Area_out(t) mM*sec, // Areas
Tbar_in(t) sec, Tbar_out(t) sec, Tbar_sys(t) sec; // Transit times
when(t=t.min) {Area_in = 0; S2_in = 0;
Area_out = 0; S2_out = 0; }
Area_in:t = Cin;
Area_out:t = Tout;
S2_in:t = Cin*t;
S2_out:t = Tout*t;
Tbar_in = if(Area_in>0) S2_in /Area_in else 0;
Tbar_out = if(Area_out>0) S2_out/Area_out else 0;
Tbar_sys = Tbar_out-Tbar_in;
real Area_outC(t) mM*sec,
Area_outD(t) mM*sec,
Area_outE(t) mM*sec;
when(t=t.min) { Area_outC=0; Area_outD=0; Area_outE=0;}
Area_outC:t=Cout;
Area_outD:t=Dout;
Area_outE:t=Eout;
}
/*
DIAGRAM:
F(flow) +-------------------------------+
Cin(t) ----> | F(flow)
| Gc2d Gd2e -->Cout(t)=C1(t)
| C(t)----->D(t)----->E(t) -->Dout(t)=D1(t)
| -->Eout(t)=E1(t)
| V (volume) |
| Gc2d, Gd2E (conversion rates) |
| instantaneously well mixed |
+-------------------------------+
DETAILED DESCRIPTION:
This is a one compartment model. F is flow, Cin is inflow
concentration, Cout, Dout, and Eout are outflow concentrations, V
is volume, Gc2d is the rate at which substance C is converted
to substance D, andGd2e is the rate at which substance D is
converted to substance E. The reactions are irreversible.
C0, D0, and E0 are the initial concentrations of
C, D, and E respectively. The amount of material in the compartment
is calculated by multiplying the volume by the sum of the
concentrations and also by integrating the flow multiplying the
difference of what flows in minus what flows out.
SHORTCOMINGS/GENERAL COMMENTS:
KEY WORDS: Compartmental, one compartment, single compartment,
flow, reactions, conversion, irreversible, Tutorial
REFERENCES:
Jacquez JA. Compartmental Analysis in Biology
and Medicine. Ann Arbor: University of Michigan Press, 1996.
REVISION HISTORY:
JSim SOFTWARE COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
JSim software was developed with support from NIH grants HL088516,
and HL073598. Please cite these grants in any publication for which
this software is used and send one reprint of published abstracts or
articles to the address given below. Academic use is unrestricted.
Software may be copied so long as this copyright notice is included.
Copyright (C) 1999-2009 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle, WA
98195-5061
*/