Figure 1: Inflow and Outflow: Orig parameter set
The input function Cin(t) can been chosen to be a Lagged
Normal Density Curve, selected by clicking on the circled
tilde (~) near the bottom of the Input list and setting
fgen_1 as LagNormal instead of the default Pulse1. The
parameters of this function are: the area, the mean transit
time (tMean), the relative dispersion (RD) which is the
standard deviation divided by tmean; the skewness (skewn), and
the height of the curve as a fraction of its peak height at
which the function is terminated (frpeak). The area under the
curve has units of mmol/ml as defined in the MML code by
"extern real Cin(t) mmol/ml;".
1. Run loops changing both PSg and PScmax from 1 to 0. The second
solution with no exchange between the plasma and isf regions,
mimics a reference intravascular solute. In multiple indicator
dilution experiments there are normally two tracers injected. The
input function, Cin (black curve) would be the same for both. Cout
is the concentration-time curve of response observed at the
outflow. The intravascular tracer (e.g. albumin) is represented by
the dashed red line on the plot labeled Cout, while the continuous
red curve is the permeant tracer.
2. Visfp = lambda*Visf, where lambda is the isf/plasma partition
coefficient, that is the ratio of Cisf/Cp at equilibrium, and Visf
is the anatomic volume of the interstitial space. Visfp is greater
than Visf if the "solubility" for a solute in the isf is higher than
in the 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 is less than Visf.
Figure 2: Spatial Profiles: Orig
(Change Cin (fgen_1) to Pulse1 and do Single Run.) (Parameter set Pars2)
3. How Figure 2 was displayed:
View: Reset Number of Rows: 2
Toggle "plot 1" (beneath File menu) and select plot2
For Curve [1] Enter Cp(3.5,x).
Toggle the [1] in Curve[1] to be Curve[2].
For Curve [2] enter Cp(4,x).
And so forth. You can also use the down arrow button,
[V] Curve [1] to bring up the menu of variables, select
Cp(t,x) and then change the t to the appropriate numeric value.
Figure 2 is set up to show a sequence of plasma curves as a function of
position, x, at different times.Change PSg or PScmax to see effects.
Set PScmax=0.
When Gisf is large or Visf is large, so that there is little return flux
from the ISF to the capillary plasma, then the envelope of the
spatial curves at the different times is:
C(x) = (Peak Height of the input curve)*exp(-(PSg)*x/(Fp*L)).
To simplify our understanding, we shall keep the Pulse1 amplitude equal to 1.
Similarly when PSg = 0 and back diffusion is zero, and Kmc is >> Cp(t,x), then
C(x) = (Peak Height of the input curve)*exp(-PScmax*x/(Fp*L)).
When both PSg and PScmax are > 0 and Kmc >> Cp, and there is no back
diffusion, then the envelope of C(t,x) is given by
C(x) = (Peak Height of the input curve)*exp(-(PSg+PScmax)*x/(Fp*L)).
Set PSg to zero, amd reduce Kmc toward the levels of Cp and examine the
effects on the spatial profiles (parameter set Pars3) . Why do the peaks go
above the analytical equation. When the Kmc = Cp we expect 50% inhibition
of the flux, when Kmc/Cp is 10 we expect a 10% reduction in PS from
PScmax and when Kmc = 100*Cp we expect a 1% inhibition. Test these ideas.
Add another curve to this spatial plot or create another plot and put
Cisf(3.5,x) and othe curves of Cisf(t,x) in order to compare Cisf(t,x)
and Cp(t,x). Do this for different permeabilities, by putting PSg into
the "Loops" and raising or lowering it a little. Adjust PSg to make the
ISF profiles closer to the Plasma profiles. In what direction does each
move as PSg is raised ... lowered?
/* MODEL NUMBER: 0016
MODEL NAME: TranspMM.1sided.Distrib2F
SHORT DESCRIPTION: An axially distributed two region model with a two-sided passive transporter (PSg)
through clefts and a one-sided Michaelis-Menten transporter (PSc) for membrane transport.
*/
import nsrunit; unit conversion on;
math TranspMM1sidedDistrib2F {
// 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 = 0.25 ml/g, // ISF volume of distribution (virtual)
PSg = 1.0 ml/(g*min), // Permeability-surface area product cleft between p and ISF
Gp = 0 ml/(g*min), // Plasma consumption rate for metabolite
Gisf = 100 ml/(g*min), // ISF consumption rate for metabolite
Dp = 1e-06 cm^2/sec, // Plasma axial diffusion coefficient
Disf = 1e-05 cm^2/sec, // ISF axial diffusion coefficient
PScmax= 1 ml/(g*min), // Max PSc across endo cells
Kmc = 1 mM; // Km, Michaelis const for transporter
// Using hVolumes protects against setting V = 0.
private real hVp = if(Vp>0) Vp else (1e-6 ml/g);
private real hVisf = if(Visf>0) Visf else (1e-6 ml/g);
// DEPENDENT VARIABLES
real PSc(t,x) ml/(g*min); // PS via MM transporter across endothelial cell
extern real Cin(t) mM;
real Cp(t,x) mM, Cisf(t,x) mM, Cout(t) mM; //Concentration plasma,
// isf and outflow
// BOUNDARY CONDITIONS FOR PDEs
when (x=x.min) { (-Fp*L/hVp)*(Cp-Cin)+Dp*Cp:x =0;} // Left Hand Total flux BC.
when (x=x.max) { Cp:x = 0; Cout = Cp; } // Right Hand (no flux)
when (x=x.min) { Cisf:x = 0;} // Left Hand (no flux)
when (x=x.max) { Cisf:x = 0;} // Right Hand (no flux)
// INITIAL CONDITIONS FOR PDEs:.
when (t=t.min) { Cp = if (x=x.min) Cin else 0; }
when (t=t.min) { Cisf = 0; }
// PARTIAL DIFFERENTIAL EQUATIONS
PSc = PScmax/(1 + Cp/Kmc );
Cp:t = -Fp*L/hVp*Cp:x + (PSg+PSc)*(Cisf-Cp)/hVp - Gp*Cp/hVp + Dp*Cp:x:x;
Cisf:t = (PSg+PSc)*(Cp-Cisf)/hVisf- Gisf*Cisf/hVisf + Disf*Cisf:x:x;
// 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(4.5,x)*x/(Fp*L));
// STATISTICS from calculated moments:
// Cin is the model input, Cout is the model output
real S1_in(t) mM*sec, S2_in(t) mM*sec^2, S3_in(t) mM*sec^3;
real 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
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
//
real S1_out(t) mM*sec, S2_out(t) mM*sec^2, S3_out(t) mM*sec^3;
real 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;
tbar_out=S2_out/S1_out;
RD_out=sqrt((abs(S3_out/S2_out)/tbar_out)-1);
//Calculate moments of the operator from the input & output moments
real SD_op(t), tbar_op(t), RD_op(t);
SD_op= sqrt( (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
} // END OF MML MODEL
/*
DIAGRAM:
Fp ________________________________________
Cin(t) ---> |Vp Cp(t)|---> Cout(t)
|Gp ^ ^ |
|Dp | | PLASMA|
___________PSg_____PSc_________________|
|Visfp | | Cisf(t)|
|Gisf V V INTERSTITIAL|
|Disf 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 an axially distributed 2-region capillary-tissue exchange model with
permeation across the capillary wall via clefts (PSg) and cell transporters (PSc).
The capillary plasma region has volume Vp, flow Fp, first order consumption Gp,
and axial diffusion Dp. Units are physiological (i.e. per gram of tissue) so that
this can represent a homogeneously perfused organ. Radial diffusion is assumed
instantaneous (short radial distances).
This interstitial fluid region, isf, has volume Visf, first order consumption Gisf,
and axial diffusion Disf. Capillary-tissue exchange is modeled by two parallel routes:
1. PSg: Passive exchange between plasma and surrounding non-flowing interstitial
fluid is through interendothelial clefts. PSg is Permeability-Surface area product.
2. PSc: Facilitated transport occurs via a transporter on the capillary membrane
with PScmax as maximal conductance at low concentrations.
Transporter is modified from TranspMM.1sided.Distrib2F--facilitated transport can go either way.
SHORTCOMINGS/GENERAL COMMENTS:
KEY WORDS: Axially Distributed, two region, capillary-tissue exchange, facilitated transport,
plasma, interstitial fluid region, radial diffusion, tutorial. Michaelis-Menten
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 embeds 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: Nov/08
Revised by : BEJ Date: 16/Dec/08
Revision: 1) Update format of comments
Revised by : GMR Date 11/May/10
Revision: 2) Standard 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 an email
with the citation and, if possible, a PDF file of the paper to: staff@bioeng.washington.edu.
*/