// MODEL NUMBER: 0333
// MODEL NAME: BTEX10_Terminology
// SHORT DESCRIPTION: Uses a single capillary convection diffusion model to
// generate functions of linear stationary systems: h(t), H(t), R(t), eta(t).
//*****************************************************************************
import nsrunit; unit conversion on;
math BTEX10_Terminology {
// INDEPENDENT VARIABLES
realDomain t sec ; t.min=0; t.max=30; t.delta=0.1;
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; // removes thewe from parameter menu
// PARAMETERS:
real F = 1 ml/(g*min), // Flow
V = 0.05 ml/g, // Volume of tube p
G = 0 ml/(g*min), // Gulosity, consumption, first order clearance
D = 1.0e-5 cm^2/sec; // Diffusion coefficient
private real hV =if(V>0) V else (1e-6 ml/g);
// Define Input fucntion: INFLOWING CONCENTRATION
extern real Cin(t) mM;
// CONCENTRATION VARIABLES
real C(t,x) mM, // Concentration
Cout(t) mM; // Outflow Concentration from tube
// BOUNDARY CONDITIONS (Note total flux BC for inflowing region.)
when (x=x.min) { (-F*L/hV)*(C-Cin)+D*C:x = 0; }
when (x=x.max) { C:x = 0; Cout = C;}
// INITIAL CONDITIONS
when (t=t.min) { C = 0; }
// PARTIAL DIFFERENTIAL EQUATIONS
C:t = -F*L/hV*C:x -G/hV*C+ D*C:x:x ;
// Mass Conservation
real Qinject(t) umol/g; // Total Mass injected
real Q(t) umol/g; // Mass remaining in capillary
when(t=t.min) {Q=0; Qinject=0;}
Q:t = F*(Cin-Cout);
Qinject:t = F*Cin;
// TRANSPORT TERMINOLOGY
real h(t) sec^(-1); // Transport function
real H(t); // Residence Time Distribution Function
real R(t); // Residue Function
real eta(t) ; // Emergence Function
when(t=t.min) { H=0;}
real Qtot = Qinject(t.max); // Total mass entering by time t.max
h = F * Cout / Qtot;
H:t = h ;
R = 1 - H ; //== Q/Qinject +0.0005; //
eta = if(R <> 0.0) h/(R+0.002) else 0;
//Gaussian g(x)
real g(x) 1/cm, sig =0.02 cm, xbar = 0.05 cm /* scalar dimensionless*/;
g = (1/29.6)/((2*PI)^(1/2)*sig) * exp(- 0.5*((x-xbar)/sig)^2);
}
/*
FIGURE:
Fp ________________________________________
| V |
Cin(t) --->| G C(t) |---> Cout(t)
| D |
|______________________________________|
|<--------------- x ----------------->|
0 --> x L
DETAILED DESCRIPTION:
This partial differential equation models a "tissue cylinder"
consisting of a single capillary, and is mathematically like BTEX10.
The model has been parameterized to produce the operator functions
from the Terminology paper of Bassingthwaighte et al.(1986).
wIth a Dirac delta function input (unity area, input at t = 0) the
response functions are:
h(t) is the transport function (or transfer function) the probability
density function of arrival times at the outflow, fraction of dose per second.
H(t) is the cumulative residence time distribution function, and equals
the fraction of the dose accumulated in a bucket at the outflow.
R(t) is the residue function, the fraction of the injected mass retained
in the system at time t.
eta(t) is the emergence function or the fractional escape rate, the
fraction of the retainied mass emerging from the outflow per second.
It is equivalent to the risk function of the life insur=ance industry,
the risk of dying at age t.
Three situations are illustrated:
1. A general situation described above. Plot TERM, Parameter set Term.
This uses a very short pulse injection. A modest diffusion coefficient
illustrates the standard. If D is small the output is a delayed pulse,
but numerical inaccuracies distort a very narrow pulse; for D > 0 the
outflow, h(t), is nearly a Gaussian pdf, as exemplified by the closeness
of fit of the spatial distribution by a Gaussian density funnction, on
the spatial plot, SPACE.
2. A stirred tank. Use plot TANK and parameter set Tank. The ridiculously
high diffusion coefficient disperses the matrial almost uniformly throughout
the lngth of the tube, approximiating a stirred tank within the axially
distributed region. The downslope of h(t) and of R(t) are almost
monoexponenetial with time constabt = V/F.
3. a dispersed input function. USe plot DISP and parameter set Disp.
Becasue the input is not a DIrac delta function the responses are not formally
the "transport function", h(t), H(t), R(t), and eta(t), but are the delayed
equivaalents. Using a lagged normal density function as input avoids the sharp
discontinuities so that the numerical methods work better. (The equations
are not so "stiff".)
KEY WORDS:
BTEX10,PDE,convection,diffusion,reaction,distributed,capillary,
plasma, terminology, Tutorial
REFERENCES:
Key reference: J.B. Bassingthwaighte, F.P. Chinard, C. Crone, C.A. Goresky,
N.A. Lassen, R.S. Reneman, and K.L. Zierler. Terminology for
mass transport and exchange. Am. J. Physiol. 250 (Heart. Circ.
Physiol. 19): H539-H545, 1986.
Selected early references on convection -diffusion situations:
Taylor G. Dispersion of soluble matter in solvent flowing slowly
through a tube. Proc R Soc Lond A 219: 186-203, 1953.
Taylor G. The dispersion of matter in turbulent flow through a pipe.
Proc R Soc Lond A 223: 446-468, 1954.
Danckwerts PV. Continuous flow systems: Distribution of residence times.
Chem Eng Sci 2: 1-13, 1953.
Andres R, Zierler KL, Anderson HM, Stainsby WN, Cader G, Ghrayyib AS, and
Lilienthal JL Jr. Measurement of blood flow and volume in the forearm of man;
with notes on the theory of indicator-dilution and on production of turbulence,
hemolysis, and vasodilatation by intra-vascular injection.
J Clin Invest 33: 482-504, 1954.
Meier P and Zierler KL. On the theory of the indicator-dilution method for
measurement of blood flow and volume. J Appl Physiol 6: 731-744, 1954.
References for Operational Analysis:
Zierler KL. Theoretical basis of indicator-dilution methods for measuring
flow and volume. Circ Res 10: 393-407, 1962.
Cox DR. Renewal Theory. New York: Wiley, 1962, 142 pp.
Other references re Convection Diffusion
W.C. Sangren and C.W. Sheppard. A mathematical derivation of the
exchange of a labelled substance between a liquid flowing in a
vessel and an external compartment. Bull Math BioPhys, 15, 387-394,
1953.
C.A. Goresky, W.H. Ziegler, and G.G. Bach. Capillary exchange modeling:
Barrier-limited and flow-limited distribution. Circ Res 27: 739-764, 1970.
J.B. Bassingthwaighte. A concurrent flow model for extraction
during transcapillary passage. Circ Res 35:483-503, 1974.
B. Guller, T. Yipintsoi, A.L. Orvis, and J.B. Bassingthwaighte. 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.
C.P. Rose, C.A. Goresky, and G.G. Bach. The capillary and
sarcolemmal barriers in the heart--an exploration of labelled water
permeability. Circ Res 41: 515, 1977.
J.B. Bassingthwaighte, C.Y. Wang, and I.S. Chan. Blood-tissue
exchange via transport and transformation by endothelial cells.
Circ. Res. 65:997-1020, 1989.
Poulain CA, Finlayson BA, Bassingthwaighte JB.,Efficient numerical methods
for nonlinear-facilitated transport and exchange in a blood-tissue exchange
unit, Ann Biomed Eng. 1997 May-Jun;25(3):547-64.
REVISION HISTORY:
Written and formatted by JBB 10jan2015
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2015 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.
When citing JSim please use this reference: Butterworth E, Jardine BE, Raymond GM, Neal ML, Bassingthwaighte JB.
JSim, an open-source modeling system for data analysis [v3; ref status: indexed, http://f1000r.es/3n0]
F1000Research 2014, 2:288 (doi: 10.12688/f1000research.2-288.v3)
This software was developed with support from NIH grants HL088516 and HL073598, NIBIB grant BE08417
and the Virtual Physiological Rat program GM094503 (PI: D.A.Beard). 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@physiome.org.
*/