This model describes a flow through vessels with resistance to flow, and compliance. Model 'Vessel_2RC' has two RC vessels in series while 'Vessel4RC has two RC vessels in parallel with two RC vessels in series.

## Description

The model simulates fluid flow, F, through a compliant vessel with resistance to flow, R, and vessel compliance, C, given a pressure drop across the length of the vessel equivalent to Pin - Pout. 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 - Pout 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. 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: Fin = Fout + Fcomp where the flow attributed to the vessel compliance, Fcomp, is given by: Fcomp = dV/dt and where V is the vessel volume and is related to the compliance, C and intraluminal pressure, Pin, by: V = Pin * C It should be noted in the code that the initial vessel volume is prescribed by Pin and the value of C calculated from the initial diameter and vessel length.

**Figure:**

2 RC vessels: Fin ----> Pin R1 P1 R2 Pout ---------o------/\/\/\/\---o-----/\/\/\/\----o | | ----> | | | | Fout | | F1comp | | F2comp | v | v C1,V1 ===== C2,V2 ===== | | | | | | o P1ext o P2ext ----- ----- --- --- - - 4RC vessels: - --- ----- P2ext o | ===== C2,V2 ^ | F2 F2comp | | R2 ----> | |----/\/\/\/\/\-----------| | | Fin | | ----> Pin R1 |P1 P2 R4 Pout ---------o------/\/\/\/\---o -----o-----/\/\/\/\----o | | | | ----> | | | R3 | | | Fout | | F1comp |-----/\/\/\/\/\------ | | F4comp | v | ----> | v C1,V1 ===== | F3 ===== C4,V4 | ===== C3,V3 | | | | | o P3ext | o P1ext ----- o P4ext ----- --- ----- --- - --- - -

## 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.

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Ohm GS. Die galvanische Kette mathematisch bearbeitet, 1827

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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.