which defines parametrically the envelope of intersection points as a function of t, the time of origin of one of the characteristics.
The important point to note is that the first point of shock formation is where xi is smallest. In other words, for data with s'(t) large, the shock develops quickly, and close to x = 0. Thus, generally speaking, the steeper the pulse generated by the heart, the sooner and closer a shock will form. This may explain why the pistol-shot occurs in patients with aortic insufficiency but not in other patients. Using numerical simulations of the model equations, Anliker et al. (1971a,b) have shown that under conditions of aortic insufficiency, a steep pressure gradient can develop within 40 cm of the heart, well within the physiological range.
It is also noteworthy that the slope s depends on diastolic pressure P0 through s = 3u+K(P0). Thus, a decrease in K(P0), caused either by a decrease of P0 or a decrease of the function A(P), leads to a decrease of the first location of shock formation x,.
Notice also that if s'(t) < 0, so that u(0, t) is decreasing, no shock can form for positive x. This can also be seen from Fig. 15.16, since, if u(0, t) is a decreasing function of t, the characteristics fan out and do not intersect for positive x.
1. Equation (15.7) was derived assuming that the radius of the vessel and pressure drop along the vessel were constant, but then it was used in (15.14) as if the pressure were variable. Under what conditions is this a valid approximation?
2. Show that so that (15.21) follows
3. Simplify the six-compartment model of the circulation by assuming that there are no pressure drops over the arterial and venous systems (either systemic or pulmonary), and thus, for example, Psa = Ps1. Solve the resultant equations and compare with the behavior of the three-compartment model presented in the text. How do the parameter values change? Are the sensitivities altered? (Calculate the sensitivities using a symbolic manipulation program.)
4. Showthat(15.67)-(15.75)can be derived from (15.54)-(15.66) by letting Rs and Rp approach zero and by letting Cpa = Cpv and Rpa = Rpv.
5. In the three-compartment circulatory model it was assumed that the base volume of the pulmonary circulation V0 was zero. Using the fact that the systolic pressure in the pulmonary artery is about 22 mm Hg and the diastolic pressure is about 7 mm Hg, determine the typical value for the volume of the pulmonary circulation. How does this change to the model affect the results?
6. Find the pressures as a function of cardiac output assuming Vi = 0, Vr = 0, where Vi is the basal volume of the left heart, and similarly for Vr (cf. equation (15.35)). Show that
7. Explore the behavior of the autoregulation model with R = R0 (1 + a[O2]v) /(1 + b[O2]v).
8. What symptoms in the circulation would you predict from anemia?
Hint: Anemia refers simply to an insufficient quantity of red blood cells, which results in decreased resistance and oxygen-carrying capacity of the blood.
9. Devise a simple single loop model for the fetal circulation that treats the systemic flow and the placental flow as parallel flows. How should the flow be split between system and placenta to support the highest metabolic rate?
10. In the model for autoregulation Pa and Pv are given. More realistically, they would be determined, in part at least, by Rs. Construct a more detailed model for autoregulation, including the effects of Rs on the pressures, and show how the arterial and venous pressures, and the cardiac output, depend on M and A.
11. Modify the model for capillary filtration by allowing the plasma osmotic pressure to vary along the capillary distance, by setting nc = RTccQ-, where cc is the local concentration of osmolites. If incoming pressure is unchanged from the first model, what is the effect of osmotic pressure on total filtration? What changes must be made to the incoming pressure to maintain the same total filtration?
Hint: Study the phase portrait for this system of equations.
12. Derive a simplified windkessel model by starting with a single vessel with volume V(t) = V0 + CP(t). Assume that the flow leaves through a resistance R and that there is an inflow (from the heart) of Q(t). Derive the differential equation for P and compare it to (15.141).
13. Frank (1899) described a method whereby the flux of blood out of the heart could be estimated from a knowledge of the pressure pulse, even when the arterial resistance is unknown. Starting with (15.141), assume that during the second part of the arterial pulse, Q(t) = 0. Write down equations for the first and second parts of the pulse, eliminate R, and find an expression for Q. Give a graphical interpretation of the expression for Q.
14. In the model for the arterial pulse, set u(0, t) = at for some constant a, and determine the curve in the (t,x) plane along which characteristics form an envelope. Determine the first value of x at which a shock forms.
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