5.1.1 The three-element Windkessel model

The three-element Windkessel model was introduced by Westerhof et al. as an improvement on the classical two-element Windkessel for modelling the systemic circulation [Westerhof N, Bosman F, DeVries CJ and Noordergraaf A. (1969) Analogue studies of the human systemic arterial tree. J. Biomech. 2, 121-143.]. It has been very successful and gained wide acceptance in the analogue modelling of the cardiovascular system.

The electrical analogue circuit for the three-element Windkessel is the RCR circuit shown to the right. Z₁ and C₁ together make up the classical Windkessel model introduced by Frank in 1899. C₁ represents the compliance (capacitance) of the elastic arteries and Z₁ represents the hydraulic resistance (impedance) of the vessels draining the arteries. In Westerhof's model, the extra resistance Z₀ represents the 'characteristic' impedance of the aorta; i.e., it is the impedance that the heart would see if there were no reflections in the aorta.

The identification of the first resistance with the characteristic impedance is consistent with our observation that the wave pressure in the ascending aorta is proportional to the measured flow where the constant of proportionality (the slope of the PU-loop) being the characteristic impedance. By the water hammer equations, the slope is ρc. When expressed in terms of flow rate, as is usually done in analogue modelling, this implies Z₀ = ρc/A which is the characteristic impedance.