// MODEL NUMBER: 0085
// MODEL NAME: Vessel_Resistance_Only
// SHORT DESCRIPTION: Model rigid vessel flow with vessels of varying diameter and length
// in series and in parallel. Model 'Two_Resistors' uses two vessels in series and
// model 'Four_resistors' has two vessels in parallel and two more in series.
import nsrunit;
unit conversion on;
math Rigid_Vessel{
// DOMAIN VARIABLE
realDomain t sec; t.min = 0; t.max = 10; t.delta = 0.1;
// -----------------------------------------------------------------------------
// PARAMETERS for model Two_Resistors
// -----------------------------------------------------------------------------
// MODEL PARAMETERS
real mu = 3 cP, // Fluid viscosity
P1(t) mmHg, // Press after resistance 1
Pout = 0 mmHg, // Pressure at circuit exit
L1 = 1500 um, // Vessel 1 length
D1 = 500 um, // Vessel 1 diameter
L2 = 1500 um; // Vessel 2 length
extern real D2(t) um; // Vessel 2 diameter - variable
real // CALCULATED PARAMETERS
R1 dyne*s/(mL*cm^2), // Vessel 1 resistance to flow
R2(t) dyne*s/(mL*cm^2); // Vessel 2 resistance to flow (time dependent)
// -----------------------------------------------------------------------------
// VARIABLES for model Two_Resistors
// -----------------------------------------------------------------------------
real // MODEL VARIABLES
Delta_P(t) mmHg, // Pressure drop along whole circuit
Delta_P1(t) mmHg, // Pressure drop after first vessel resistance
F(t) mL/s; // Flow through the vessel
// INPUT FUNCTION
extern real Pin(t) mmHg; // Driving pressure at vessel entrance
// -----------------------------------------------------------------------------
// SYSTEM OF EQNS for model Two_Resistors
// -----------------------------------------------------------------------------
R1 = (128*mu*L1) / (PI*D1^4);
R2 = (128*mu*L2) / (PI*D2^4); // To use only R1 make L2 ~ 0 and D2 >>0
Delta_P1 = F*R1;
P1 = Pin - Delta_P1;
Delta_P = Pin - Pout;
F = Delta_P/(R1+R2); // resistors in series
}
/*
FIGURE: F
Pin R1 P1 ---> R2 Pout
o-------/\/\/\/\----o-------/\/\/\/\-----o
LEGEND:
/\/\/\/\ = resistance
DETAILED DESCRIPTION:
The model simulates fluid flow, F, through rigid vessels, in series, of resistance,
R1 and R2, given a pressure drop across the length of the vessel Delta_P. The
flow is related to the resistance by the fluid equivalent of Ohm's Law.
F = Delta_P / (R1+R2)
where Delta_P is the difference in pressure between the beginning and
end of the vessel and R is determined from Poiseuille's Law as:
R = 128*mu*L / pi*D^4
where mu is the fluid viscosity, L is the vessel length, and D is the
vessel diameter.
Model 'Four_resistors' (a part of this JSim project file) simulates flow through
four resistors (four vessels of varying diameter and length), two of which are in parallel.
The geometry is of one vessel emptying into two separate vessels which empty into one.
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: blood flow, series, parallel resistors, circulatory system, rigid vessel, vessel network,
two resistors, four resistors, tutorial, tube impedence
REFERENCES:
Ohm GS. Die galvanische Kette mathematisch bearbeitet, 1827
Burattini R, Di Salvia PO, Development of systemic arterial mechanical properties
from infancy to adulthood interpreted by four-element windkessel models,
J Appl Physiol 103:66-79, 2007
REVISION HISTORY:
Original Author : BEJ Date: 01/jun/11
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2011 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@physiome.org.
*/