Vascular operator consists of a pure delay operator plus two second order operators, all in series to simulate the mean transit time and dispersion in arteries and veins.

Description

Figure shows output concentration when the relative dispersion, RD, is varied from 0.0 to 0.48. This figure matches figure 5A in the reference.

 This program is for a vascular operator using a pure delay operator and a
 4th order operator consisting of two second order operators, all in series.
 It is used to simulate the delay and dispersion of a substance passing through
 arteries, arterioles, venules, and veins.

 Figure shows operator pathways based on RD (relative dispersion input
 parameter) value. RD4th is the relative dispersion of a 4th order operator.
 RD4th=0.47994 for zeta1=0.95, zeta2=0.8, and wratio = w2/w1 = 1.82.

                       +--------+
       RD=<0   Cin---->| Cdelay |------------------------------------------>Cout
                       +--------+     A value of zero will be used for RD.

                                      +---------------------------------+
                                      |     4th order operator (op4)    |
                       +--------+     |   +----------+   +----------+   |
    0<RD<RD4th Cin---->| Cdelay |---->|-->| C1 (op2) |-->| C2 (op2) |-->|-->Cout
                       +--------+     |   +----------|   +----------+   |
                                      +---------------------------------+

                                      +---------------------------------+
                                      |     4th order operator (op4)    |
                                      |   +----------+   +----------+   |
                    RD=>RD4th Cin---->|-->| C1 (op2) |-->| C2 (op2) |-->|-->Cout
                                      |   +----------|   +----------+   |
                                      +---------------------------------+
                                      A value of RD4th will be used for RD.
        
 The user sets the total delay (totalDelay) and the relative dispersion (RD).
 The program calculates a relative dispersion, RD4th, based on the damping
 coefficients, zeta1 and zeta2, and the ratio of w2 to w1 (wratio).

                           2               2      2           2 1/2
                   (4 zeta2  - 2 + 4 wratio  zeta1  - 2 wratio )
           RD4th = ------------------------------------------------.
                               2 zeta2 + 2 zeta1 wratio
 
 The 4th order operator delay time is
           op4Delay = (1-RD/RD4th)*totalDelay.
 The pure delay time is
           pureDelay = totalDelay-op4Delay.
 The relative dispersion of the 4th order operator is RD4th.
 The relative dispersion of the pure delay operator is zero.
 The relative dispersion of the combined operator is RD.

                          Cdelay = Cin(t-pureDelay);
 
 The two second order operators solve the following two 2nd order ordinary
 differential equations:

               / 2    \
               |d     |              /d    \     2
               |--- C1| + 2 zeta1 w1 |-- C1| + w1  (C1 - Cdelay) = 0
               |  2   |              \dt   /
               \dt    /

 with initial conditions C1(0)=0 and dC1(0)/dt=0, and

               / 2    \
               |d     |              /d    \     2
               |--- C2| + 2 zeta2 w2 |-- C2| + w2  (C2 - C1) = 0
               |  2   |              \dt   /
               \dt    /

 with initial conditions C2(0)=0 and dC2(0)/dt=0.

 This version allows the user to vary the damping coefficients, zeta2
 and zeta2, and wratio, the ratio of w2/w1. zeta1 is protected from
 being out of range and RD4th is calculated.

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.