Problem 2

This problem considers a cardiovascular model due to Ottesen [16] involving the arterial pressure, Pa(t) = y1(t), the venous pressure, Pv(t) = y2(t), and the heart rate, H(t) = y3(t). Ottesen studies conditions under which the delay causes qualitative differences in the solution and in particular, oscillations in Pa(t). Delays t = 1.0, 1.4, 3.9, 5.0, 7.5,10 are considered in [16]. There are a number of parameters in the model, so you might wish simply to declare them as global values rather than pass them through dde23 to the function for evaluating the equations, or even to hard code them. Plot y1(t) = Pa(t) for several values of t. You should find that the solutions obtained for different values of t differ dramatically. Solve on [0,350] the equations
y1(t)
=
- 1
ca R
 y1(t) + 1
ca R
 y2(t) + 1
ca
 Vstr  y3(t)
y2(t)
=
1
cv R
 y1(t) -

1
cv R
+ 1
cv r


 y2(t)
y3(t)
=
f(Ts,Tp)
where
Ts
=
1
1 + ( y1(t-t)/as )bs
Tp
=
1
1 + ( ap/y1(t) )bp
f(Ts,Tp)
=
aH Ts
1 + gH Tp
 -   bH Tp .
For t 0, the solution has the constant value
y1(t)
=
P0
y2(t)
=


1
1 + R/r


 P0
y3(t)
=


1
R  Vstr


 

1
1 + r/R


  P0
As in [16], use ca = 1.55, cv = 519, R = 1.05,r = 0.068, Vstr = 67.9, a0 = as = ap = 93,aH = 0.84, b0 = bs = bp = 7, bH = 1.17,gH = 0, P0 = 93 . The following figures for t = 1 and t = 7.5 show qualitatively different solutions.

images/p2a.gif
images/p2b.gif
One of the figures of [16] shows the solution components when the peripheral pressure R is reduced exponentially from its value of 1.05 to 0.84 beginning at t = 600. For this computation the delay was 4 and the interval [0,1000]. You can easily modify the previous program to solve this problem. All you have to do is inform the solver of the low-order discontinuity at a known time by setting the value of the 'Jumps' option to 600, modify the function for evaluating the DDEs to include

 if t <= 600
    R = 1.05;
 else
    R = 0.21 * exp(600-t) + 0.84;
 end
and use the specified delay and interval. All the solution components are of interest. The figure shows the sharp change in the heart rate due to the change in R at t = 600.

images/p2c.gif

References

[16]
J.T. Ottesen, Modelling of the Baroflex-Feedback Mechanism With Time-Delay, J. Math. Biol., 36 (1997), 41-63.