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.