# Nonlinear_Compliant_Vessel

Model number
0061

This model describes the flow characteristics of a single vessel with resistance to flow, R, nonlinear compliance, C, and input flow of Fin.

## Description

```  The model simulates pressure and division of fluid flow through a compliant
vessel with resistance to flow, R, and vessel compliance, C, given an input
flow of Fin.  The compliant vessel follows a nonlinear pressure volume
curve which takes into account the resistance to vessel collapse that
occurs at low or negative pressures and the concave form of the curve at
high pressures. The pressure-volume function used is that proposed by
Drzewiecki et al. (Am J Physiol, Heart Circ Physiol 273:H2030-H2043, 1997).

The flow out of the vessel is related to the resistance by the fluid
equivalent of Ohm's Law.

Fout = (Pin - Pout) / R

where Pin is the pressure at the vessel entrance, Pout is the pressure at
the 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.

Pin is given from the formulation by Drzewiecki et al. and is given by
the following expression:

_     _            _      _
|     |      V/L-Ab  |      |
Pin = a * <  exp<  b * --------  > - 1  >
|_    |_       Ab   _|     _|
_             _
|         n     |
- EIhat * <  (Ab*L/V)  - 1  >  + Pb
|_             _|

where a and b are constants which determine the nonlinear shape of the
vessel elastance, V is the vessel volume, Ab is the vessel cross-
sectional area at buckling, EIhat is the normalized flexural rigidity
of the vessel wall, n determines the curvature of the pressure volume
curve in compression and Pb is the intraluminal pressure at buckling
which is given by:
_     _           _      _
|     |      Ab-A0  |      |
Pb = a * <  exp<  b * -------  > - 1  >
|_    |_      A0   _|     _|

where A0 is the vessel cross-sectional area at an intraluminal
parameter values given for different vessels see Drzewiecki et al.

The flow into the vessel and the flow out of the vessel are different
because of the change in volume which adds or subtracts flow from that
leaving the vessel depending on whether the pressure is increasing or
decreasing in the vessel.  So we have for vessel compliance, Fcomp:

Fcomp = Fin - Fout

The change in vessel volume as a function of time is determined from
the flow attributed to the vessel diameter change and is given by:

dV/dt = Fcomp

It should be noted in the code that the compliance, C, is calculated by
taking the inverse of the derivative of the Pin with respect to V. ```

## Equations

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

References
```Drzewiecki G, Field S, Moubarak I and Li JKJ.
Vessel growth and collapsible pressure-area relationship
American Journal of Physiology, Heart and Circulatory Physiology
273:H2030-H2043, 1997.
```
Key terms
Compliance
Flow
Pressure
Ohm's Law
Acknowledgements

Please cite https://www.imagwiki.nibib.nih.gov/physiome in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.