// MODEL NUMBER: 0277
// MODEL NAME: CTEX20_5path
// SHORT DESCRIPTION: Multipath Capillary-tissue exchange unit accounting
// for intra-organ flow heterogeneity.
import nsrunit; unit conversion on;
math CTEX20_5path {
// INDEPENDENT VARIABLES
realDomain t sec ; t.min=0; t.max=38; t.delta=0.1;
realDomain N; N.min=1; N.delta=1; // private N.min, N.delta, N.ct;
real Ntank = 1;
private int iNtank = floor(Ntank); // non-integer values of Ntank -> lower integer
N.max = if(iNtank=1) 2 else iNtank;// N.max needed for the realDomain N
// PARAMETERS
real Fp = 1 ml/(g*min), // Flow per gram tissue
PSg = 0 ml/(g*min), // Permeability-Surface Area product for Clefts
Gp = 0 ml/(g*min), // plasma consumption rate coeff (Gulosity) for N isf comps
Gisf = 0 ml/(g*min), // isf consumption rate coeff (Gulosity) for N isf comps
Vp = 0.05 ml/g, // Total Volume for N capillary plasma tanks
Visfp = 0.15 ml/g; // Total Volume for N capillary isf tanks
real hVp ml/g, // Volume of the tanks comprising the capillary
hVisfp ml/g; // Volume of the tanks comprising the extravasc region,
hVp = if(Vp>0) Vp/iNtank else (1e-6 ml/g); //protects vs Vp = 0
hVisfp = if(Visfp>0) Visfp/iNtank else (1e-6 ml/g); //protects vs Visf = 0
real hPSg ml/(g*min), // PSg for each tank
hGp ml/(g*min), // Gp for each tank
hGisf ml/(g*min); // Gisf for each tank
hPSg = PSg/iNtank;
hGp = Gp/iNtank;
hGisf = Gisf/iNtank;
// INPUT FUNCTION
real Cinn(t,N) mM; // Inflow for all serial tanks
extern real Cin(t) mM; // Inflow for first tank
// VARIABLES
real Cp(t,N) mM, // Plasma concentration for each tank
Cisf(t,N) mM, // Extravascular concentration for each tank
Cout(t) mM; // Outflow from last tank
// INITIAL CONDITIONS
when(t=t.min) {Cp = 0; Cisf =0;}
Cinn = if(N=1) Cin else Cp(t,N-1) ; // Inflow for tanks is outflow from previous tank
// ORDINARY DIFFERENTIAL EQUATIONs (Applies to all tanks.)
hVp*Cp:t = Fp * (Cinn - Cp) + hPSg*(Cisf - Cp) - hGp*Cp; // Capillary plasma
hVisfp*Cisf:t = hPSg*(Cp - Cisf) - hGisf*Cisf; // Extravascular region
Cout = Cp(t,Ntank) ; // Outflow
/* To approx a 5-path organ, add taps along the capillary to represent paths with shorter
transit times. The algorithm uses linear interpolation to find Ctap at points between
the computed just up and downstream from the tap.
*/
real L = Ntank;
real C1(t) mM, C2(t) mM, C3(t) mM, C4(t) mM, C5(t) mM; // points along cap
real Cwout(t) mM; //weighted sum of tapped points
real w1 =0.15, w2 = 0.2, w3 = 0.5, w4 = 0.2, w5=0.1 ;// weighting
real rw1, rw2,rw3,rw4,rw5, wsum;
wsum=w1+w2+w3+w4+w5;
rw1=w1/wsum;
rw2=w2/wsum;
rw3=w3/wsum;
rw4=w4/wsum;
rw5=w5/wsum;
real L1 =0.6, L2 = 0.7, L3= 0.8, L4 = 0.9, L5= 1.00; //points of tapping the CTEX20
C1 = Cp(t,L1*L);
C2 = Cp(t,L2*L);
C3 = Cp(t,L3*L);
C4 = Cp(t,L4*L);
C5 = Cp(t,L5*L);
Cwout = rw1*C1+rw2*C2+rw3*C3+rw4*C4+rw5*C5; //weighted sum of tapped points
real Fbar ml/(g*min);
Fbar = Fp*(rw1/L1 + rw2/L2 + rw3/L3 + rw4/L4 + rw5/L5);
} //program END
/*
FIGURE:
Tank 1 Tank 2 Tank N
+---------+ +---------+ +---------+
Fp*Cin -> ->Fp*CP(1)-> -> .....->Fp*Cp(N-1)-> ->Fp*Cp(N)
| Cp(1) | | Cp(2) | | Cp(N) | =
| V/N | | V/N | | V/N | Fp*Cout
| Gp/N | | Gp/N | | Gp/N |
+---------+ +---------+ +---------+
^ ^ ^
| PSg/N | PSg/N | PSg/N
v v v
+---------+ +---------+ +---------+
| Cisf(1) | | Cisf(2) | | Cisf(N) |
| Visfp/N | | Visfp/N | | Visfp/N |
| Gisf/N | | Gisf/N | | Gisf/N |
+---------+ +---------+ +---------+
DETAILED DESCRIPTION:
This single path model for blood-tissue exchange serves as a multipath
model representing an organ with heterogeneous regional flows. By setting
the flow to a low value, as a low as the lowest expected in any region, the
model outflow concentration-time curve C(L,t) represents only that at the end,
L, of this slowest pathway. However by tapping a sequence of positions along
the vessel to get the concentration-time curves, C(ai*L,t), where 0 < ai < L,
one obtains a family of curves representing regions with higher than the
lowest flow. The weighted array of the SUM of the Wi*C(ai*L,t) can be fitted
to the experimentally observed outflow curve, Cdata(t), for the whole organ by
optimization of the set of wi's. The result provides an estimate of the mean
flow for the organ, Fbar, and the Wi/ai define the probability density
function of regional flows, whose variance describes the heterogeneity of
intraorgan flows per gram of tissue.
This strategy assumes linear, concentration independent transport across
the capillary wall. It is exactly correct for tracer studies, but for
chemical level studies where transporter or enzymes may be saturated, it
is still a good approximation.
SHORTCOMINGS/GENERAL COMMENTS: See Notes page for additional info.
KEY WORDS: CTEX20, BTEX20, Compartmental, Stirred Tanks,mulitpath model
heterogeneous flow, vascular dispersion, optimization for determining PDFs of
regional flows, tutorial
REFERENCES:
Knopp TJ, Dobbs WA, Greenleaf JF, and Bassingthwaighte JB.
Transcoronary intravascular transport functions obtained via a stable
deconvolution technique. Ann Biomed Eng 4: 44-59, 1976.
REVISION HISTORY:
Original Author : JBB Date: 3/Dec/09
Revised by : BEJ Date: 31/Dec/09 Update comment format
Revised by : JBB Date: 6jan10 Revise description and BestFit optim
Revised by: BEJ: 08jul15: Add consumption terms and individ tank values for PSg
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.
*/