Windkessel Model

Although our knowledge of pulse propagation in the arterial system is extensive, it has been useful to approximate the relationship between pressure and flow in a single artery or vascular bed. Such approximations have been referred to as reduced arterial system models of which the three element Windkessel is the most widely employed [Westerhof et al., 1971 Noordergraaf, 1978]. This Windkessel model is modified here to study the effects of peripheral resistance. 

The three element Windkessel model approximates the pulse propagation quality of the arterial system as a combination of an infinitely long tube, which is represented by its characteristic impedance, Zo,... 

In this chapter, the dynamics of autoregulation are incorporated into the modified Windkessel model. The frequency response of this autoregulating Windkessel is then predicted and compared with that of the standard three element model. The time response is also determined and discussed relative to the experimental observations in the literature. The stability of the model is also examined with respect to very low frequency oscillation in peripheral resistance. 

The physical relationships that govern the Windkessel model are now reviewed. Time dependency is provided by the blood volume storage property of arterial compliance, Cj. The time derivative of peripheral arterial pressure, Ps, is proportional to the difference in blood flow into and out of the compliance, according to,... 

The two differential equations (Equation 14.1 and Equation 14.7) govern the time dependent characteristics of the autoregulating Windkessel model. They were integrated numerically by a computer. All parameters were evaluated for the canine systemic arterial system (Table 14.1). 

As in much of hemodynamic research, the Windkessel model, as well as others, have been portrayed in electrical circuit form. The autoregulating Windkessel described here cannot be simplified completely in circuit form. But, a useful approximation was designed. It was assumed that the time constants of autoregulation... 

FIGURE 17.3 The coupled model of the left ventricle (LV) and the arterial system (AS). The LV is represented here by a time-varying compliance and a resistance. The AS is represented by the modified windkessel model with characteristic impedance, Zo, peripheral resistance, Rs, and compliance of the arterial system, C. C(P) denotes the case when compliance is allowed to change with blood pressure levels. 

To compare gross features of the arterial trees of different mammals, modeling approach can be particularly useful. The modified windkessel model of the systemic arterial system that is coupled to the heart (Figure 17.3) has been shown to represent well the features of the input impedance of the systemic arterial tree. For this representation, the input impedance is... 

FIGURE 3.16 (a) The Windkessel model of the aorta and peripheral circulation, (fe) Electrical model of the Windkessel model. 

To answer the question of optimal matching between the ventricle and arterial load, we developed a framework of analysis which uses simplified models of ventricular contraction and arterial input impedance. The ventricular model consists only of a single volume (or chamber) elastance which increases to an endsystolic value with each heart beat. With this elastance, stroke volume SV is represented as a linearly decreasing function of ventricular endsystolic pressure. Arterial input impedance is represented by a 3-element Windkessel model which is in turn approximated to describe arterial end systolic pressure as a linearly increasing function of stroke volume injected per heart beat. The slope of this relationship is E. Superposition of the ventricular and arterial endsystolic pressure-stroke volume relationships yields stroke volume and stroke work expected when the ventricle and the arterial load are coupled. From theoretical consideration, a maximum energy transfer should occur from the contracting ventricle to the arterial load under the condition E = Experimental data on the external work that a ventricle performed on extensively varied arterial impedance loads supported the validity of this matched condition. The matched condition also dictated that the ventricular ejection fraction should be nearly 50%, a well-known fact under normal condition. We conclude that the ventricular contractile property, as represented by is matched to the arterial impedance property, represented by a three-element windkessel model, under normal conditions. 

The conversion of the arterial impedance property into an effectixr elastance can be done by first representing the arterial impedance by a three-element Windkessel model (consisting of R, C, and R) and then by manipulating the model equations to arrive at an approximation of arterial end-systolic pressure PJ as... 

If the actual blood flow differs significantly from Qo, the regulatory error can be large. Since a linear model is being employed here, the value of R predicted by Equation 14.4 to Equation 14.7 may fall outside of the physiological range. Thus, the time dependent value of Rg was limited to maximum and minimum values of 1.6Jiso and 0.4J so> respectively. These values correspond with maximal vasoconstriction and vasodilation. When the computations reveal that the limit has been reached, the Hmited value of resistance was then inserted into Equation 14.1 of the Windkessel model. Otherwise, the time dependent value obtained from Equation 14.7 was employed. 

The three-element Windkessel model approximates the pulse propagation quahty of the arterial system as a combination of an infinitely long tube, which is represented by its characteristic impedance, Z , in series with a parallel arrangement of a peripheral flow resistance, R, and a total arterial compliance, C,. Within a single cardiac cycle, these three parameters are generally assumed to be constants (Toorop et al, 1987). Recent studies have scrutinized this assumption. For example, compliance has been shown to depend on the level of arterial pressure (Bergel, 1961). Other studies have called into question the accuracy of the various methods employed to experimentally determine their values (Stergiopulos et al.. 

Windkessel model (Figure 23.1). To reduce computation time, the time constants of the model were shortened. This does not affect the result, but simply shortens the time to reach steady state. 

---