// MODEL NUMBER: 0218
// MODEL NAME: Single_Vessel
// SHORT DESCRIPTION:
// Fluid flow from an open, compliant vessel, driven only by the
// energy stored inthe compliant vesel wall.
import nsrunit; unit conversion on;
math Single_Vessel { realDomain t sec; t.min=0; t.max=5.0; t.delta=0.1;
// PARAMETERS:
real
resistance = 0 mmHg*sec/ml, // Resistance of the vessel
compliance = 0 ml/mmHg, // Compliance of the vessel wall
restvolume = 0 ml; // vessel volume at zero pressure
// VARIABLES:
real
volume(t) ml, // Vessel Volume
flowout(t) ml/sec; // Flow from Vessel
// INITIAL CONDITIONS:
when (t=t.min) volume = 0;
// ALGEBRAIC AND ODE EQUATIONS:
flowout = (volumerestvolume) / (resistance*compliance);
volume:t = (1)*flowout;
} // END OF MML CODE
/*
FIGURE:
compliance, volume
/
 
.o
  

 resistance
> /
>
> flowout
 /
v



.
LEGEND:
C ^ 
C or CCCCC = Inertance < > = direction of flow  or  = Compliance
C v  
>  
0 = Junction ^^^^^ or > = Resistance . or  = Ground
>  .
_ 
> or < or ^ = diode (1way valve) VE = Varying elastance

: = indicates a labeling arrow, not a path of flow
DETAILED DESCRIPTION:
This model simulates current flow generated from the discharging
of a charged capacitor through a resistance element. It is
analogous in the fluid flow context to applying a pressure across
an open, compliant vessel that contains a volume of fluid and
then letting the fluid drain out driven only by the energy
stored in the compliant vessel wall. For example, a balloon is
filled with water with its outlet held closed which generates an
internal pressure, P. At a time, t=0, the outlet is allowed to open
and the time course for the balloon outflow can be recorded as a
function of time.
The simplest description of an elastic vessel under the influence
of timevarying pressure must have a resistance and compliance
element such as in this model. Here the simulation begins with
a given volume of fluid in the vessel and at time t=0 the outlet
of the vessel is opened and the compliant vessel drains. The flow,
F is a function of the difference in the current volume and the
volume at rest of the vessel as well as the compliance, C, of the
vessel and the resistance, R, to flow out of the vessel. The
change in vessel volume as a function of time is equal to the
flow out of the vessel, F. External
pressure is assumed to be zero.
The model uses a constant compliance to create a linear
relationship between pressure and volume. In reality the
pressurevolume curves of flexible tubes are nonlinear and
transmural pressure trends towards negative infinity as volume
goes to zero. However, there are linear portions of the PV
curve which can be approximated using a constant compliance or
elastance value. A constant resistance that is independent of
vessel geometry is also used in this model. For laminar flow
through a cylindrical tube, resistance is dependent on fluid
viscosity, tube length and tube radius (Poiseiulle's Law);
however, resistance in this model remains independent of these
properties.

REFERENCE EQUATIONS:
Eq. A) Flow (mL/unit time) = change in volume / change in time
Basis: Definition of flow
Eq. B) Compliance = Change in volume / Change in transmural pressure
Basis: Fluid analog of capacitance
Eq. C) Pressure drop = Resistance * Flow
Basis: Fluid analog of Ohm's Law
Eq. D) (Sum of flows entering junction = sum of flows leaving junction)
Basis: Kirchhoff Junction rule
Eq. E) Pressure drop = (change in Flow/change in time)*Inertance
Basis: Fluid analog of inductance

SHORTCOMINGS/GENERAL COMMENTS:
KEY WORDS: Cardiovascular system, single, vessel, compliant wall, flow,
Poiseiulle's Law, Hemodynamics
REFERENCES:
Ohm GS. Die galvanische Kette, mathematisch bearbeitet, 1827
REVISION HISTORY:
Original Author : CarlsonB Date: 11/Jul/07
Revised by: Micah Nicholson Date: 4/2/2009
Revision: 1) CHange format for new model web page
Revised by :BEJ Date: 21/Apr/10
Revision: 1) Update format
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 19992010 University of Washington. From the National Simulation Resource,
Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 981955061.
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.
*/