Misplaced Pages

#558441

47-465: The Windkessel effect helps in damping the fluctuation in blood pressure ( pulse pressure ) over the cardiac cycle and assists in the maintenance of organ perfusion during diastole when cardiac ejection ceases. The idea of the Windkessel was alluded to by Giovanni Borelli , although Stephen Hales articulated the concept more clearly and drew the analogy with an air chamber used in fire engines in

94-512: A cardiac stress test is a good way to test for heart failure with preserved ejection fraction . Classification of blood pressure in adults: Brain natriuretic peptide (BNP) is a cardiac neurohormone secreted from ventricular myocytes (ventricular muscle cells) at the end of diastole—this in response to the normal, or sub-normal (as the case may be), stretching of cardiomyocytes (heart muscle cells) during systole. Elevated levels of BNP indicate excessive natriuresis (excretion of sodium to

141-476: A harmonic oscillator is losing energy faster than it is being supplied. A true sine wave starting at time = 0 begins at the origin (amplitude = 0). A cosine wave begins at its maximum value due to its phase difference from the sine wave. A given sinusoidal waveform may be of intermediate phase, having both sine and cosine components. The term "damped sine wave" describes all such damped waveforms, whatever their initial phase. The most common form of damping, which

188-399: A step input , the percentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the overshoot is just the maximum value of the step response minus one. The percentage overshoot (PO) is related to damping ratio ( ζ ) by: Conversely, the damping ratio ( ζ ) that yields a given percentage overshoot is given by: When an object

235-487: A numerator over a denominator, rather it is a medical notation showing the two clinically significant pressures involved. It is often shown followed by a third value, the number of beats per minute of the heart rate . Mean blood pressure is also an important determinant in people who have had certain medical interventions like Left Ventricular Assist Devices (LVAD) and hemodialysis that replace pulsatile flow with continuous blood flow. Examining diastolic function during

282-429: A second-order system has ζ < 1 {\displaystyle \zeta <1} (that is, when the system is underdamped), it has two complex conjugate poles that each have a real part of − α {\displaystyle -\alpha } ; that is, the decay rate parameter α {\displaystyle \alpha } represents the rate of exponential decay of

329-404: A value of less than one. Critically damped systems have a damping ratio of exactly 1, or at least very close to it. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. For a damped harmonic oscillator with mass m , damping coefficient c , and spring constant k , it can be defined as the ratio of the damping coefficient in

376-425: A variety of other cardiovascular diseases. 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. Recent attempts to integrate wave propagation and Windkessel approaches through a reservoir concept, have been criticized and a recent consensus document highlighted

423-424: Is 75 beats per minute (bpm), which means that the cardiac cycle that produces one heartbeat, lasts for less than one second. The cycle requires 0.3 sec in ventricular systole (contraction)—pumping blood to all body systems from the two ventricles; and 0.5 sec in diastole (dilation), re-filling the four chambers of the heart, for a total of 0.8 sec to complete the cycle. During early ventricular diastole, pressure in

470-402: Is a parameter, usually denoted by ζ (Greek letter zeta), that characterizes the frequency response of a second-order ordinary differential equation . It is particularly important in the study of control theory . It is also important in the harmonic oscillator . In general, systems with higher damping ratios (one or greater) will demonstrate more of a damping effect. Underdamped systems have

517-415: Is a suction mechanism between the atrial and ventricular chambers. Then, in late ventricular diastole, the two atrial chambers contract (atrial systole), causing blood pressure in both atria to increase and forcing additional blood flow into the ventricles. This beginning of the atrial systole is known as the atrial kick —see Wiggers diagram. The atrial kick does not supply the larger amount of flow (during

SECTION 10

#1732858879559

564-425: Is described by a differential equation : I ( t ) = P ( t ) R + C d P ( t ) d t {\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

611-434: Is dissipated as heat by electric eddy currents which are induced by passing through a magnet's poles, either by a coil or aluminum plate. Eddy currents are a key component of electromagnetic induction where they set up a magnetic flux directly opposing the oscillating movement, creating a resistive force. In other words, the resistance caused by magnetic forces slows a system down. An example of this concept being applied

658-539: Is falling through the air, the only force opposing its freefall is air resistance. An object falling through water or oil would slow down at a greater rate, until eventually reaching a steady-state velocity as the drag force comes into equilibrium with the force from gravity. This is the concept of viscous drag , which for example is applied in automatic doors or anti-slam doors. Electrical systems that operate with alternating current (AC) use resistors to damp LC resonant circuits. Kinetic energy that causes oscillations

705-482: Is not to be confused with friction , which is a type of dissipative force acting on a system. Friction can cause or be a factor of damping. The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium . A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce,

752-473: Is often of interest in a diverse range of disciplines that include control engineering , chemical engineering , mechanical engineering , structural engineering , and electrical engineering . The physical quantity that is oscillating varies greatly, and could be the swaying of a tall building in the wind, or the speed of an electric motor , but a normalised, or non-dimensionalised approach can be convenient in describing common aspects of behavior. Depending on

799-415: Is the brakes on roller coasters. Magnetorheological Dampers (MR Dampers) use Magnetorheological fluid , which changes viscosity when subjected to a magnetic field. In this case, Magnetorheological damping may be considered an interdisciplinary form of damping with both viscous and magnetic damping mechanisms. Diastole Diastole ( / d aɪ ˈ æ s t ə l i / dy- AST -ə-lee )

846-427: Is the characteristic resistance (this is assumed to be equivalent to the characteristic impedance), while R 2 represents the peripheral resistance. This model is widely used as an acceptable model of the circulation. For example it has been employed to evaluate blood pressure and flow in the aorta of a chick embryo and the pulmonary artery in a pig as well as providing the basis for construction of physical models of

893-454: 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

940-507: Is the relaxed phase of the cardiac cycle when the chambers of the heart are refilling with blood. The contrasting phase is systole when the heart chambers are contracting. Atrial diastole is the relaxing of the atria, and ventricular diastole the relaxing of the ventricles. The term originates from the Greek word διαστολή ( diastolē ), meaning "dilation", from διά ( diá , "apart") + στέλλειν ( stéllein , "to send"). A typical heart rate

987-462: 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

SECTION 20

#1732858879559

1034-672: Is usually assumed, is the form found in linear systems. This form is exponential damping, in which the outer envelope of the successive peaks is an exponential decay curve. That is, when you connect the maximum point of each successive curve, the result resembles an exponential decay function. The general equation for an exponentially damped sinusoid may be represented as: y ( t ) = A e − λ t cos ⁡ ( ω t − φ ) {\displaystyle y(t)=Ae^{-\lambda t}\cos(\omega t-\varphi )} where: Other important parameters include: The damping ratio

1081-790: The sin function, while flow during diastole is zero. s represents the duration of the cardiac cycle, while Ts represents the duration of systole, and Td represents the duration of diastole (e.g. in seconds). I ( t ) = I o sin ⁡ [ ( π ∗ t s ) T s ]  for  t s ≤ T s {\displaystyle I(t)=I_{o}\sin[{(\pi *{t \over s}) \over Ts}]{\text{ for }}{t \over s}\leq Ts} I ( t ) = 0  for  T s < ( T d + T s ) {\displaystyle I(t)=0{\text{ for }}Ts<(Td+Ts)} The 'Windkessel effect' becomes diminished with age as

1128-490: The 18th century. Otto Frank , an influential German physiologist, developed the concept and provided a firm mathematical foundation. 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). Windkessel physiology remains a relevant yet dated description of important clinical interest. The historic mathematical definition of systole and diastole in

1175-417: The amount of damping present, a system exhibits different oscillatory behaviors and speeds. A damped sine wave or damped sinusoid is a sinusoidal function whose amplitude approaches zero as time increases. It corresponds to the underdamped case of damped second-order systems, or underdamped second-order differential equations. Damped sine waves are commonly seen in science and engineering , wherever

1222-399: The cardiac cycle) as about 80 percent of the collected blood volume flows into the ventricles during the active suction period. At the beginning of the cardiac cycle the atria, and the ventricles are synchronously approaching and retreating from relaxation and dilation, or diastole. The atria are filling with separate blood volumes returning to the right atrium (from the vena cavae ), and to

1269-413: 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 and R, C, L parameters, or find values of R, C, L given flow and pressure. An example for the two-element model is shown below, where I(t) is depicted as an input signal during systole and diastole. Systole is represented by

1316-544: The characteristic resistance of the aorta (or pulmonary artery). The differential equation for the 3-element model is: ( 1 + R 1 R 2 ) I ( t ) + C R 1 d I ( t ) d t = P ( t ) R 2 + C d P ( t ) d t {\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}} where R 1

1363-820: The circuit to account for the inertia of blood flow. This is neglected in the two- and three- element models. The relevant equation is: ( 1 + R 1 R 2 ) I ( t ) + ( R 1 C + L R 2 ) d I ( t ) d t + L C d 2 I ( t ) d t 2 = P ( t ) R 2 + C d P ( t ) d t {\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}} These models relate blood flow to blood pressure through parameters of R, C ( and, in

1410-418: The circulation providing realistic loads for experimental studies of isolated hearts. The three-element model overestimates the compliance and underestimates the characteristic impedance of the circulation. The four-element model includes an inductor , L , which has units of mass per length, ( M l 4 {\displaystyle {M \over l^{4}}} ), into the proximal component of

1457-403: The cycle begins again. In summary, when the ventricles are in systole and contracting, the atria are relaxed and collecting returning blood. When, in late diastole, the ventricles become fully dilated (understood in imaging as LVEDV and RVEDV), the atria begin to contract, pumping blood to the ventricles. The atria feed a steady supply of blood to the ventricles, thereby serving as a reservoir to

Windkessel effect - Misplaced Pages Continue

1504-584: The definition of the damping ratio above, we can rewrite this as: This equation is more general than just the mass–spring system, and also applies to electrical circuits and to other domains. It can be solved with the approach where C and s are both complex constants, with s satisfying Two such solutions, for the two values of s satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes: The Q factor , damping ratio ζ , and exponential decay rate α are related such that When

1551-432: The elastic arteries become less compliant, termed hardening of the arteries or arteriosclerosis , probably secondary to fragmentation and loss of elastin. 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

1598-445: The heart is known as systole . Ejection causes pressure within the ventricles to fall, and, simultaneously, the atria begin to refill (atrial diastole). Finally, pressures within the ventricles fall below the back pressures in the aorta and the pulmonary arteries, and the semilunar valves close. Closure of these valves give the second heart sound (S2). The ventricles then start to relax, the mitral and tricuspid valves begin to open, and

1645-399: The left atrium (from the lungs). After chamber and back pressures equalize, the mitral and tricuspid valves open, and the returning blood flows through the atria into the ventricles. When the ventricles have completed most of their filling, the atria begin to contract (atrial systole), forcing blood under pressure into the ventricles. Now the ventricles start to contract, and as pressures within

1692-417: The model are obviously not novel but are here elementally staged to four degrees. Reaching five would be original work. 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

1739-650: The oscillations. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times. For example, a high quality tuning fork , which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer. For underdamped vibrations, the damping ratio is also related to the logarithmic decrement δ {\displaystyle \delta } . The damping ratio can be found for any two peaks, even if they are not adjacent. For adjacent peaks: where x 0 and x 1 are amplitudes of any two successive peaks. As shown in

1786-482: The right figure: where x 1 {\displaystyle x_{1}} , x 3 {\displaystyle x_{3}} are amplitudes of two successive positive peaks and x 2 {\displaystyle x_{2}} , x 4 {\displaystyle x_{4}} are amplitudes of two successive negative peaks. In control theory , overshoot refers to an output exceeding its final, steady-state value. For

1833-492: The right ventricle and right atrium through the tricuspid valve . The ventricular filling flow (or flow from the atria into the ventricles) has an early (E) diastolic component caused by ventricular suction, and then a late one created by atrial systole (A). The E/A ratio is used as a diagnostic measure as its diminishment indicates probable diastolic dysfunction , though this should be used in conjunction with other clinical characteristics and not by itself. Early diastole

1880-576: The system tends to return to its equilibrium position, but overshoots it. Sometimes losses (e.g. frictional) damp the system and can cause the oscillations to gradually decay in amplitude towards zero or attenuate . The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next. The damping ratio is a system parameter, denoted by ζ (" zeta "), that can vary from undamped ( ζ = 0 ), underdamped ( ζ < 1 ) through critically damped ( ζ = 1 ) to overdamped ( ζ > 1 ). The behaviour of oscillating systems

1927-470: The system's differential equation to the critical damping coefficient: where the system's equation of motion is and the corresponding critical damping coefficient is or where The damping ratio is dimensionless, being the ratio of two coefficients of identical units. Using the natural frequency of a harmonic oscillator ω n = k / m {\textstyle \omega _{n}={\sqrt {{k}/{m}}}} and

Windkessel effect - Misplaced Pages Continue

1974-505: The total peripheral resistance and the capacitor, C , represents total arterial compliance. 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: P ( t ) = P ( t d ) e − ( t − t d ) ( R C ) {\displaystyle P(t)=P(t_{d})e^{-(t-t_{d}) \over (RC)}} where t d

2021-437: The two ventricles begins to drop from the peak reached during systole. When the pressure in the left ventricle falls below that in the left atrium, the mitral valve opens due to a negative pressure differential (suction) between the two chambers. The open mitral valve allows blood in the atrium (accumulated during atrial diastole) to flow into the ventricle (see graphic at top). Likewise, the same phenomenon runs simultaneously in

2068-401: The urine) and decline of ventricular function, especially during diastole. Increased BNP concentrations have been found in patients who experience diastolic heart failure . Impaired diastolic function can result from the decreased compliance of ventricular myocytes , and thus the ventricles, which means the heart muscle does not stretch as much as needed during filling. This will result in

2115-411: The ventricles and ensuring that these pumps never run dry. This coordination ensures that blood is pumped and circulated efficiently throughout the body. Blood pressure is usually written with the systolic pressure expressed over the diastolic pressure or separated by a slash , for example, 120/80  mmHg . This clinical notation is not a mathematical figure for a fraction or ratio, nor a display of

2162-446: The ventricles rise, the mitral and tricuspid valves close producing the first heart sound (S1) as heard with a stethoscope. As pressures within the ventricles continue to rise, they exceed the "back pressures" in the aorta , and the pulmonary trunk . The aortic and pulmonary valves known as the semilunar valves open, and a defined fraction of blood within the heart is ejected into the aorta and pulmonary trunk. Ejection of blood from

2209-696: The wave-like nature of the reservoir. Damping ratio In physical systems , damping is the loss of energy of an oscillating system by dissipation . Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. Examples of damping include viscous damping in a fluid (see viscous drag ), surface friction , radiation , resistance in electronic oscillators , and absorption and scattering of light in optical oscillators . Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes (ex. Suspension (mechanics) ). Damping

#558441