DEFAULT PLOT PAGE:
Figure 1: Inflow and Outflow (LOOPS): Default parameter set
Run Loops with PSg = 0 and 1.
Loop 1 (PSg=0) plots the inflow (solid black) and outflow (solid red)
concentrations for the A solute in the flowing region as a
function of time. The inflowing concentration is given by the
function generator as a lagged normal density input function.
The outflow represents an intravascular tracer such as
C-14 labeled albumin. The outflow has been dispersed somewhat by
including axial diffusion in the calculation with a diffusion
coefficient equal to 1e-5 cm^2/sec.
Loop 2 (PSg=1) represents an extracellular or permeant tracer, such as
labeled sucrose or adenosine. The A solute outflow(dashed red line) is
reduced because some of the A solute has been transported to the second
compartment and converted to B. Both A and B are transported from the
second compartment back to the first compartment. Notice that the tail
of Aout, the A outflow is greater than the tail of a intravascular because
of the backflux of A.
EQUIL PLOT PAGE:
Figure 2: Inflow and Outflow: Equil parameter set
With a constant infusion, all concentrations reach steady state after ~60 seconds.
Figure 3: Ratio of Aisf/Ap at Equilibrium: Equil parameter set
Visfp = lambda*Visf, where lambda is the isf/plasma partition coefficient, that is
the ratio of Aisf/Ap at equilibrium, and Visf is the anatomic volume of the
interstitial space. Visfp > Visf if the "solubility" for a solute in isf is higher
than in plasma. Normally for small solutes Visfp = Visf, but for large solutes, or
those with the same charge as ISF matrix (normally negatively charged) then
Visfp < Visf. For albumin, Visfp ~(1/2)*Visf/2.
(L.E. Feinendegen, W.W. Shreeve, W.C. Eckelman, Yong Whee Bahk, H.N. Jr. Wagner;
Molecular Nuclear Medicine: The Challenge of Genomics and Proteomics to
Clinical Practice, page 177: 2003).
In what direction do you have to move PSg in order to make the ratio of Aisf/Ap
approach 1?
ENVEL PLOT PAGE
Figure 4: Inflow and Outflow: Envel parameter set
The inflow and outflow of A and B, and also A and B at the end of the capillary in
the isf region are plotted as a function of time.
Figure 5: Spatial profiles: Envel parameter set
The plasma concentration of A, Ap, is plotted as a function of position along the
capillary at times 3.5, 4.0, 4.5, 5.0, 5.5, and 6.0 seconds.
The envelope for Ap as a function of distance is given as
envelope1 =(peakHtCin(t.max)/(1 mM))*exp(-(PSg+PScmax)*x/(Fp*L))
when PScmax is relatively constant (KmA>Ap).
Change KmA = 0.05 and run again. Now the envelope is better approximated as
envelope2 =(peakHtCin(t.max)/(1 mM))*exp(-PSg*x/(Fp*L))*exp(-PSc(5.5,x)*x/(Fp*L));
FLUX PLOT PAGE:
Figure 6: Flux(KmA) (LOOPS): Flux parameter set
Ratio(KmA/Ap): Flux parameter set
Set PSg=0, PScmax=1, KmA=1, Ga2b=0. LOOP on KMa = 1, 0.1, 0.01, and 0.001.
The flux in the middle of the capillary between the compartments is plotted
as a function of time. As KmA is set to smaller values, the flux decreases
proportionately. Ap(t,L/2) is ~1 mM, so the ratio of KmA/Ap(t,L/2) is KmA/(1 mM).
/* MODEL NUMBER: 0025
MODEL NAME: TranspMM.2sol2sided.Distrib2F
SHORT DESCRIPTION: Two region capillary-tissue exchange model with both passive and Michaelis-Menton (MM)
transport of two solutes with MM reaction of A to B in interstitial fluid region.
*/
import nsrunit; unit conversion on;
math TranspMM2sol2sidedDistrib2F {
// INDEPENDENT VARIABLES
realDomain t sec ; t.min=0; t.max=30; t.delta=0.05;
realDomain x cm; real L=0.1 cm, Ngrid=61; x.min=0; x.max=L; x.ct=Ngrid;
private x.min, x.max, x.ct; //Ngrid must be odd number
// PARAMETERS
real Fp = 1 ml/(g*min), // Plasma flow: Subscript p for plasma
Vp = 0.05 ml/g, // Plasma volume
Visf = 0.15 ml/g, // ISF volume of distribution (virtual)
PSg = 1.0 ml/(g*min), // Permeability-surface area product cleft between p and ISF
GpA = 0 ml/(g*min), // Plasma consumption rate for metabolite A
GisfA = 0 ml/(g*min), // ISF consumption rate for metabolite A
GpB = 0 ml/(g*min), // Plasma consumption rate for metabolite B
GisfB = 0 ml/(g*min), // ISF consumption rate for metabolite B
DpA = 1e-06 cm^2/sec,// Plasma axial diffusion coefficient A
DisfA = 1e-05 cm^2/sec,// ISF axial diffusion coefficient A
DpB = 1e-06 cm^2/sec,// Plasma axial diffusion coefficient B
DisfB = 1e-05 cm^2/sec,// ISF axial diffusion coefficient B
PScmax= 2 ml/(g*min), // Maximum PSc for Michaelis-Menten transporter
KmA = 1 mM, // Michaelis constant for membrane transporter
KmB = 1 mM, // Michaelis constant for membrane transporter
Ga2b = 1.0 ml/(g*min), // Maximum reaction rate for A2 to B2 in V2
Kma2b = 0.1 mM; // Michaelis constant for reaction
extern real Ain(t) mM, // A inflow concentration
Bin(t) mM; // B inflow concentration
// DEPENDENT VARIABLES
real Ap(t,x) mM, // A plasma concentration
Aisf(t,x) mM, // A isf concentratioin
Aout(t) mM, // A outflow concentration
Bp(t,x) mM, // B plasma concentration
Bisf(t,x) mM, // B isf concentration
Bout(t) mM, // B outflow concentration
PSc(t,x) ml/(g*min); // Concentration dependent membrane exchange coefficient
// Boundary Conditions for PDEs:
when (x=x.min) { Aisf:x = 0; Bisf:x = 0;}
when (x=x.max) { Aisf:x = 0; Bisf:x = 0;}
when (x=x.max) { DpA*Ap:x = 0; Aout = Ap; } // Right Hand
when (x=x.min) { (-Fp*L/Vp)*(Ap-Ain)+DpA*Ap:x =0;} // Left Hand Total flux BC.
when (x=x.max) { DpB*Bp:x = 0; Bout = Bp; } // Right Hand
when (x=x.min) { (-Fp*L/Vp)*(Bp-Bin)+DpB*Bp:x =0;} // Left Hand Total flux BC.
// Initial Conditions for PDEs:.
when (t=t.min) { Ap = if (x=x.min) Ain else 0; }
when (t=t.min) { Aisf = 0; }
when (t=t.min) { Bp = if (x=x.min) Bin else 0; }
when (t=t.min) { Bisf = 0; }
// Partial differential equation, PDE
PSc = PScmax/(1 + (Ap + Aisf)/KmA + (Bp + Bisf)/KmB );
Ap:t = -Fp*L/Vp*Ap:x - (PSg+PSc)*(Ap-Aisf)/Vp - GpA*Ap/Vp + DpA*Ap:x:x;
Aisf:t = (PSg+PSc)*(Ap-Aisf)/Visf- GisfA*Aisf/Visf -Ga2b/Visf*Aisf/(1+Aisf/Kma2b)+ DisfA*Aisf:x:x;
Bp:t = -Fp*L/Vp*Bp:x - (PSg+PSc)*(Bp-Bisf)/Vp - GpB*Bp/Vp + DpB*Bp:x:x;
Bisf:t = (PSg+PSc)*(Bp-Bisf)/Visf- GisfB*Bisf/Visf +Ga2b/Visf*Aisf/(1+Aisf/Kma2b)+ DisfB*Bisf:x:x;
choice STATISTICS("A only","B only","A and B")=1;
real stat = STATISTICS;
real Cin(t) mM, //Model input
Cout(t) mM; //Model output
Cin = (if(stat=1 or stat=3) 1 else 0)*Ain
+(if(stat=2 or stat=3) 1 else 0)*Bin;
Cout = (if(stat=1 or stat=3) 1 else 0)*Aout
+(if(stat=2 or stat=3) 1 else 0)*Bout;
private real S1_in(t) mM*sec, S2_in(t) mM*sec^2, S3_in(t) mM*sec^3;
real area_in(t) mM*sec, tbar_in(t) sec, RD_in(t) ;
when(t=t.min) {S1_in=0;S2_in=0;S3_in=0;} // Initial conditions for integrators
S1_in:t = Cin; // Integrating to get Area
S2_in:t = Cin*t; // Integrating to get mean time
S3_in:t = Cin*t*t; // Integrating to get variance
area_in = S1_in;
tbar_in = S2_in/S1_in; // tbar_in is mean time of input function
RD_in=sqrt((abs(S3_in/S2_in)/tbar_in)-1); // RD_in = relative dispersion of input
private real S1_out(t) mM*sec, S2_out(t) mM*sec^2, S3_out(t) mM*sec^3;
real area_out(t) mM*sec, tbar_out(t) sec, RD_out(t);
when(t=t.min) {S1_out=0;S2_out=0;S3_out=0;}
S1_out:t=Cout;
S2_out:t=Cout*t;
S3_out:t=Cout*t*t;
area_out=S1_out;
tbar_out=S2_out/S1_out;
RD_out=sqrt((abs(S3_out/S2_out)/tbar_out)-1);
//Calculate system moments from input & output moments
real SD_op(t), tbar_op(t), RD_op(t);
SD_op= sqrt(abs( (RD_out*tbar_out)^2 - (RD_in*tbar_in)^2 ));
tbar_op=tbar_out-tbar_in; // Should = Vp/Fp when PSg = 0, else (Vp+Visfp)/Fp
RD_op=SD_op/tbar_op; // but these are only correct if S1_out =S1_in
// ADDITIONAL CALCULATIONS
realState peakHtCin(t) mM;
when(t=t.min) peakHtCin=0;
event( t>t.min and peakHtCin<Cin) peakHtCin=Cin;
real envelope1(x) dimensionless, envelope2(x) dimensionless;
envelope1 =(peakHtCin(t.max)/(1 mM))*exp(-(PSg+PScmax)*x/(Fp*L));
envelope2 =(peakHtCin(t.max)/(1 mM))*exp(-PSg*x/(Fp*L))*exp(-PSc(5.5,x)*x/(Fp*L));
}
/*
Fp ________________________________________
Cin(t) ---> |Vp Bp(t) Ap(t)|---> Cout(t)
|Gp ^ ^ |
|Dp | | PLASMA|
___________PSg___PSc___________________|
|Visfp | | Bisf(t Aisf(t)|
|Gisf v v INTERSTITIAL|
|Disf Aisf->Bisf FLUID REGION|
________________________________________
|<----------------L------------------->|
|--> x
Fp : Plasma Flow Rate, (ml/g)/min
Vp : Plasma Volume, ml/g
Visfp: Volumes of Distribution, ml/g
PSg: Permeability-surface area product exchange
coefficients, (ml/g)/min
Gp, Gisf: Consumption rates for metabolite, (ml/g)/min
Dp, Disf: Axial Diffusion Rate, cm^2/sec
Cin: Plasma metabolite inflow, mmol/ml
Cout: Plasma metabolite outflow, mmol/ml
Cp, Cisf: metabolite concentration, mmol/ml
DETAILED DESCRIPTION:
This is a two solute (A and B), one-dimensional, two region convection-permeation-
diffusion-reaction model. The plasma region has flow, Fp, volume Vp, first order
consumption, GpA and GpB, axial diffusion coefficients, DpA and DpB, and two
exchange mechanisms between the plasma and interstitial fluid(isf) regions: a
passive exchange process governed by PSg through the interendothelial clefts,
and a Michaelis-Menten membrane 2-sided competitive transport process for both
solutes in both regions. The parameter units are physiological, that is, "per gram
of tissue" to facilitate using this model to represent a homogenously perfused organ.
The (isf) region has volume Visf, first order consumption, GisfA and GisfB, axial
diffusion coefficients, DisfA and DisfB, the two exchange mechanisms already mentioned,
and a Michaelis-Menten conversion of Aisf to Bisf, governed by Ga2b and Kma2b.
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: transporter, Michaelis-Menten, capillary-tissue exchange,
axial gradients, solute-solute competition, permeability surface area,
BTEX spatially distributed, convection, diffusion reaction, tutorial
REFERENCES:
Sangren WC and Sheppard CW. A mathematical derivation of the
exchange of a labeled substance between a liquid flowing in a
vessel and an external compartment. Bull Math Biophys 15: 387-394, 1953
(This gives an analytic solution for the two-region model.)
Goresky CA, Ziegler WH, and Bach GG. Capillary exchange modeling:
Barrier-limited and flow-limited distribution. Circ Res 27: 739-764, 1970.
(This gives another derivation of the analytical form, and uses the model in
both single and multicapillary models.
Bassingthwaighte JB. A concurrent flow model for extraction
during transcapillary passage. Circ Res 35: 483-503, 1974.
(This gives numerical solutions, which are faster than the analytic solutions,
and imbeds the model in an organ with tissue volums conserved, and with arteries
and veins. The original Lagrangian sliding fluid element model with diffusion.)
Guller B, Yipintsoi T, Orvis AL, and Bassingthwaighte JB. Myocardial
sodium extraction at varied coronary flows in the dog: Estimation of
capillary permeability by residue and outflow detection.
Circ Res 37: 359-378, 1975.
(Application to sodium exchange in the heart.)
Goresky CA. Hepatic membrane carrier transport processes: Their involvement
in bilirubin uptake. In: Chemistry and Physiology of Bile Pigments.
Washington, D.C.: Publishing House U.S. Government, 1977, p. 265-281.
Silverman M and Goresky CA. A unified kinetic hypothesis of carrier-mediated
transport: Its applications. Biophys J 5: 487-509, 1965.
REVISION HISTORY:
Original Author : JBB Date: Dec/08
Revised by : BEJ Date: 16/Dec/09
REvised by : GMR Date: 16/Jul/10 : Major revision
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 an email
with the citation and, if possible, a PDF file of the paper to: staff@bioeng.washington.edu.
*/