Windkessel effect

Windkessel physiology remains a relevant yet dated description of important clinical interest. The historic mathematical definition of systole and diastole in the model are obviously not novel but are here elementally staged to four degrees. Reaching five would be original work.

2-Element Windkessel Circuit Analogy Illustrated

It is assumed that the ratio of pressure to volume is constant and that outflow from the Windkessel is proportional to the fluid pressure. Volumetric inflow must equal the sum of the volume stored in the capacitive element and volumetric outflow through the resistive element. This relationship is described by a differential equation:

${\displaystyle I(t)={P(t) \over R}+C{dP(t) \over dt}}$

I(t) is volumetric inflow due to the pump (heart) and is measured in volume per unit time, while P(t) is the pressure with respect to time measured in force per unit area, C is the ratio of volume to pressure for the Windkessel, and R is the resistance relating outflow to fluid pressure. This model is identical to the relationship between current, I(t), and electrical potential, P(t), in an electrical circuit equivalent of the two-element Windkessel model.

In the blood circulation, the passive elements in the circuit are assumed to represent elements in the cardiovascular system. The resistor, R, represents the total peripheral resistance and the capacitor, C, represents total arterial compliance.[7]

During diastole there is no blood inflow since the aortic (or pulmonary valve) is closed, so the Windkessel can be solved for P(t) since I(t) = 0:

${\displaystyle P(t)=P(t_{d})e^{-(t-t_{d}) \over (RC)}}$

where t_d is the time of the start of diastole and P(t_d) is the blood pressure at the start of diastole. This model is only a rough approximation of the arterial circulation; more realistic models incorporate more elements, provide more realistic estimates of the blood pressure waveform and are discussed below.

The three-element Windkessel improves on the two-element model by incorporating another resistive element to simulate resistance to blood flow due to the characteristic resistance of the aorta (or pulmonary artery). The differential equation for the 3-element model is:

${\displaystyle (1+{R_{1} \over R_{2}})I(t)+CR_{1}{dI(t) \over dt}={P(t) \over R_{2}}+C{dP(t) \over dt}}$

3-Element

where R₁ is the characteristic resistance (this is assumed to be equivalent to the characteristic impedance),[7] while R₂ represents the peripheral resistance. This model is widely used as an acceptable model of the circulation.[5] For example it has been employed to evaluate blood pressure and flow in the aorta of a chick embryo [8] and the pulmonary artery in a pig[8] as well as providing the basis for construction of physical models of the circulation providing realistic loads for experimental studies of isolated hearts.[9]

4-Element compared to the 2- and 3-Element Windkessel models

The three-element model overestimates the compliance and underestimates the characteristic impedance of the circulation.[7] The four-element model includes an inductor, L, which has units of mass per length, (${\displaystyle {M \over l^{4}}}$), into the proximal component of the circuit to account for the inertia of blood flow. This is neglected in the two- and three- element models. The relevant equation is:

${\displaystyle (1+{R_{1} \over R_{2}})I(t)+(R_{1}C+{L \over R_{2}}){dI(t) \over dt}+LC{d^{2}I(t) \over dt^{2}}={P(t) \over R_{2}}+C{dP(t) \over dt}}$

The 'Windkessel effect' becomes diminished with age as the elastic arteries become less compliant, termed hardening of the arteries or arteriosclerosis, probably secondary to fragmentation and loss of elastin.[10] The reduction in the Windkessel effect results in increased pulse pressure for a given stroke volume. The increased pulse pressure results in elevated systolic pressure (hypertension) which increases the risk of myocardial infarction, stroke, heart failure and a variety of other cardiovascular diseases.[11]