Windkessel effect (German: Windkesseleffekt) is a term used inmedicine to account for the shape of thearterial blood pressure waveform in terms of the interaction between thestroke volume and thecompliance of the aorta and largeelastic arteries (Windkessel vessels) and theresistance of the smaller arteries andarterioles. Windkessel when loosely translated fromGerman to English means 'air chamber',^[1][2] but is generally taken to imply anelastic reservoir.^[3] The walls of large elastic arteries (e.g.aorta,common carotid,subclavian, andpulmonary arteries and their larger branches) contain elastic fibers, formed ofelastin. These arteries distend when theblood pressure rises duringsystole and recoil when the blood pressure falls duringdiastole. Since the rate of blood entering these elastic arteries exceeds that leaving them via theperipheral resistance, there is a net storage of blood in the aorta and large arteries during systole, which discharges during diastole. The compliance (ordistensibility) of the aorta and large elastic arteries is therefore analogous to acapacitor (employing thehydraulic analogy); to put it another way, these arteries collectively act as ahydraulic accumulator.

The Windkessel effect helps indamping the fluctuation inblood pressure (pulse pressure) over thecardiac cycle and assists in the maintenance of organperfusion during diastole when cardiac ejection ceases. The idea of the Windkessel was alluded to byGiovanni Borelli, althoughStephen Hales articulated the concept more clearly and drew the analogy with an air chamber used infire engines in the 18th century.^[4]Otto Frank, an influential German physiologist, developed the concept and provided a firm mathematical foundation.^[2] Frank's model is sometimes called a two-element Windkessel to distinguish it from more recent and more elaborate Windkessel models (e.g. three- or four-element and non-linear Windkessel models).^[5][6]

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.^{[citation needed]}

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 adifferential equation:^{[citation needed]}

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

I(t) is volumetric inflow due to the pump (heart) and is measured in volume per unit time, whileP(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, andR is the resistance relating outflow to fluid pressure. This model is identical to the relationship between current,I(t), andelectrical potential,P(t), in an electrical circuit equivalent of the two-element Windkessel model.^{[citation needed]}

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

Duringdiastole there is no blood inflow since the aortic (or pulmonary valve) is closed, so the Windkessel can be solved forP(t) sinceI(t) = 0:

${\displaystyle P(t)=P(t_d)e^{\frac{-(t-t_d)}{(RC)}}}$

wheret_d is the time of the start ofdiastole andP(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). Thedifferential equation for the 3-element model is:^{[citation needed]}

${\displaystyle (1+\frac{R_1}{R_2})I(t)+C{R}_1\frac{dI(t)}{dt}=\frac{P(t)}{R_2}+C\frac{dP(t)}{dt}}$

whereR₁ is the characteristic resistance (this is assumed to be equivalent to the characteristic impedance),^[7] whileR₂ 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]

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

${\displaystyle (1+\frac{R_1}{R_2})I(t)+(R_{1}C+\frac{L}{R_2})\frac{dI(t)}{dt}+LC\frac{d^{2}I(t)}{d{t}^2}=\frac{P(t)}{R_2}+C\frac{dP(t)}{dt}}$

These models relate blood flow to blood pressure through parameters ofR, C (and, in the case of the four-element model,L). These equations can be easily solved (e.g. by employing MATLAB and its supplement SIMULINK) to either find the values of pressure given flow andR, C, L parameters, or find values ofR, C, L given flow and pressure. An example for the two-element model is shown below, whereI(t) is depicted as an input signal during systole and diastole. Systole is represented by thesin function, while flow during diastole is zero.s represents the duration of the cardiac cycle, whileTs represents the duration of systole, andTd represents the duration of diastole (e.g. in seconds).^{[citation needed]}

${\displaystyle I(t)=I_o\sin [\frac{(\pi \ast \frac{t}{s})}{Ts}]\text{ for }\frac{t}{s}\le Ts}$

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

Although the Windkessel is a simple and convenient concept, it has been largely superseded by more modern approaches that interpret arterial pressure and flow waveforms in terms of wave propagation and reflection.^[12] Recent attempts to integrate wave propagation and Windkessel approaches through a reservoir concept,^[13] have been criticized^[14][15] and a recent consensus document highlighted the wave-like nature of the reservoir.^[16]

• Hydraulic accumulator – Reservoir to store and stabilise fluid pressure

• Cardiovascular physiology
• Cardiology

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