Biomedical Engineering - Bioengineering of Physiological Control Systems

The windkessel model

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– – – – – THE WINDKESSEL MODEL A model for the heart and the systemic arterial system. In particular, it relates the Arterial Pressure and the incoming blood flow into the aorta, representing the load undertaken by the heart during the cardiac cycle . Its importance comes from the modeling of the mechanism that allows a continuous flow even at very low pressure during diastole , since this is the only moment when the hart can be nourished. It can be relevant in the analysis of the effects of vasodilator or vasoconstrictor drugs or in the development of mechanical hearts and heart-lung machines , for instance. It relates: blood pressure blood flow in the aorta and characterizes: arterial compliance peripheral resistance of the valves the inertia of the blood flow The Windkessel Model is analogous to the Poiseuille’s Law for a hydraulic system. It describes the cardiac system as a closed hydraulic circuit that contains a water pump connected to a chamber, filled with water except for a pocket of air. The flow of blood through the ar teries is represented by the flow of fluid through pipes. As it’s pumped, the water compresses the air, which in turn pushes the water out of the chamber. This analogy resembles the mechanics of the heart. ● ● ● ● ● ● The Windkessel model takes into consideration the following parameters while modeling the cardiac cycle: Arterial Compliance : refers to the elasticity and extensibility of the major artery during the cardiac cycle Peripheral Resistance : refers to the flow resistance encountered by the blood as it flows through the systemic arterial system. Inertia : simulates the inertia of the blood as it is cycled through the heart. We assume that: Cardiac cycle starts at systole. The period of the systole is 2/5th of the period of cardiac cycle (HR = 1 Hz Arterial Compliance, Peripheral Resistance, and Inertia are modeled as a capacitor, a resistor, and an inductor respectively ● ( ELECTRICAL CIRCUIT EQUIVALENT ) N number of elements - increasing complexity There are various kinds of Windkessel models, depending on the number of elements considered. N = # capacitors & inductors = model dimension As the number of elements in the model increases —> the number of physiological factor accounted for increases —> the accuracy of the curve that is determined increases when related to the original trend of arterial pressure. However, also the computational complexity increases and sometimes it is not necessary to have the highest accuracy . The 2-Element Windkessel Model The basic Windkessel model calculates the exponential pressure curve determined by the systolic and diastolic phases of the cardiac cycle. It well explain the buffer action provided by the arterial compliances C , even though the description provided is basic and highly simplified and it provides reasonable approximation only during diastole, not during systole . —> (INPUT) Flow of blood from the heart = current flowing in the circuit (I(t) in cm3/s) ** —> Blood pressure in the aorta = time-varying electric potential (P(t) in mmHg) ** Not precisely the cardiac output ● ● It takes into account the effect of: C - The arterial compliance represented as a capacitor with electric charge storage properties it represents the distributed compliances of the whole arterial tree, with a major contribution given by the aorta and the large arteries . It is defined as the ratio between the change in volume and the change in pressure [cm^3/mmHg] : C = ΔV / ΔP R - The Total Peripheral Resistance (TPR) of the systemic arterial system represented as an energy dissipating resistor [mmHg * s/ cm^3] R = MAP / CO MAP = Mean Arterial Pressure (or P_ao,mean) MVP = Mean Ventricular Pressure (or P_ven,mean) CO = Cardiac Output Since, the TPR should be the difference between the average pressure in the aorta and in the ventricle, divided by the cardiac output [ R = MAP – MVP / CO ] but we consider the latter as the reference point. TPR autonomic regulation The value of R (TPR) continuously changes by vasoconstriction or, conversely, vasodilation, in order to regulate the arterial pressure, given the cardiac output. (See next, the baroreflex chapter). !"# = $ ∙ %& However, R changes do also affect the timing of arterial pressure decay during the diastolic runoff, since ' = $% . In AP monitoring (e.g. in the intensive care unit, ICU) the evaluation of time constant ' is exploited to indirectly evaluate other variables harder to be measure, such as compliance C or, conversely, the stroke volume. During systole the blood pumped from the heart is stored in C. Then the diastolic phase (or “diastolic runoff” ) follows, while which the stored blood is gradually delivered to the peripheral resistances R, thus keeping a smooth flow dynamics ➥ the 2-element WM describes the exponential decay of pressure with time constant t = RC —> The RC is a characterization of the arterial system, the Heart Period, T, is a characterization of the heart. ● ● An idea of Normal values: IMPORTANT CONCLUSIONS: The WINDKESSEL EFFECT ensures that the pressure drop is limited to a 33% (where a 100% would be a total drop during diastole down to 0 mmHg) therefore the diastolic pressure would be at least 67% of the systolic pressure. A 100% drop would means a STOP in blood flow, therefore organs would run out of nourishment and the coronary arteries would stop refill the heart during diastole. the CUT-OFF FREQUENCY is an order of magnitude smaller than HR, this means that flow oscillations in the arterial tree are attenuated by an order of magnitude by the windkassel effect of large arteries, if compared to the sharp flow peak delivered by the heart in the systolic phase; further attenuation to an almost ● 1 . 2 . continuous flow is caused by peripheral resistances and compliances (see next, peripheral circulation WK). 2-elements PROBLEMS: For high frequencies of alternating current a capacitor becomes a short-cut , therefore for the high frequencies carried in systolic phase the 2-element WK model would predict a null impedance, while the real impedance settles at a low but non null value , which is the characteristic impedance of aorta Conversely, the experimental data showed amplitude tending to the constant ( c, and phase returning to the 0° line. The 2-element WK accounts only for the elastic energy stored by the arterial compliance C and not for the energy that is given to the blood mass from the heart to flow in the arterial tree (**) (**) Indeed, the heart works as a dynamic pump. The myocardium contraction, provoked by the depolarization, actively launches the SV mass towards the aorta, giving this mass the kinetic energy that makes it rapidly propagate in the aorta and the arterial tree. 3-elements Windkessel Model It takes into account also the flow/pressure wave propagating in the arterial tree —> Aortic characteristic impedance ( Zc ) It represents the transportation of energy from the central heart system to the periphery , we introduce this loss of energy in a concentrated variables model by simply dissipate this energy in heat energy in the electric model . The characteristic impedance is a typical element of fluid-dynamic models and wave transmission problems and it usually represents the general impedance in the “tubes system”. Also in this case the arterial tree is modeled as an INFINITE TUBE in which this impedance allows to model a travelling AP wave in phase with a travelling flow wave. By introducing some approximation with regard to the distributed variables and the wave transmission, that we won’t address in this course, it is possible to reach the well-known result: In general, ( c( ) ) is a complex function describing the pressure/flow amplitude and phase relationships at each frequency ) : ( c( ) ) = Pressure / flow ~ voltage / current But, according to this simplification the aortic characteristic impedance is a REAL and COSTANT number at each frequency. The practical ● ● consequence is that we are allowed to substitute a complex distributed load by a lumped resistor $ . Besides, this is in accordance with the experimental observation that the flow and the pressure wave travel synchronously (i.e., in-phase) at least within the aorta down the major branching point of the femoral arteries. IMPORTANT CONCLUSIONS: at very low frequencies the 3-elements WK model provides a WORSE MODEL than the 2-element WK model, since the total resistance becomes higher than just the TPR, even though the error is minimal since Zc is far smaller than Zr at high frequencies the 3-elements WK provides a FAR BETTER MODEL than the 2-elements WK model, since the overall resistance tends to a constant value that it is small, but NOT NULL. Moreover, at high frequencies, the phase shift between pressure and flow (V and i) tends to 0°, as emerged experimentally (flow and pressure resulted to be in phase waves) To c o n c l u d e , t h i s m o d e l a l t o g e t h e r m a n a g e s t o b e t t e r d e s c r i b e t h e Arterial Pressure dynamics during a cardiac cycle, since it provides a reasonable approximation also during systole. IMPORTANT NOTE – The characteristic impedance ( c, even though it is located at the exit of the aortic valve, must not be confounded with the resistance of the aortic valve, which is normally negligible or should be added in series when considering a stenotic valve. Higher order Windkessel Models Higher order models were introduced in order to have a better fitting of impedance at the high frequencies, without the complexity of distributed models. 1 . 2 . For instance, the 4-elements WK model, the inertance is modelled as an electrical inductance, that behaves as a short cut at DC, thus eliminating the effect of Zc and, conversely, it is an open circuit at high frequencies, thus letting the model impedance converge to ZC. However, in Westerhof’s review (2008, course material) it is claimed that the improvement given by the 4-element WK is negligible: the inertance value is hardly estimated in real cases; the 3-element WK perfectly works in mock loops; i.e., in physical hydraulic models were it is necessary to mimic the heart afterload In conclusion, the 3-element WK already provides a reasonable estimate of the AP wave given the flow wave, therefore it is a simple and effective way to model the arterial tree from the stand point of the pumping heart. LIMITATION OF THE WK MODELS It only deals with the Arterial Pressure wave shape in the aorta as seen by the heart. It doesn’t accounts for the pressure wave propagation towards the peripheral vessels. In WK model the energy is described as if it was “dissipated” by the lumped resistor, while in the cardiac system it is transported through the arterial tree by the blood-pressure (and flow) wave. This does not make any difference when studying the heart work load, e.g. in testing pumps substituting the heart (artificial heart) or supporting it, or when studying the behavior of artificial aortic valves in a mock-loop. Conversely, it makes a lot of difference when studying the pressure wave propagation in the arterial tree. (Recall, each model is valid in its specific application range!). Aortic Pulse Pressure and Arterial Compliance The aortic pulse pressure or PP is the maximal change in aortic pressure during systole, from the time the aortic valve opens until the peak aortic pressure is attained. PP = Systolic Pressure − Diastolic Pressure The rise in aortic pressure from its diastolic to systolic value is determined by both ARTERIAL COMPLIANCE (C) and VENTRICULAR STROKE VOLUME (SV) SV = Heart stroke volume = end-diastolic volume - end-systolic volume (~ blood volume pumped in arteries), it likely goes like CO As the left ventricle ejects blood into the aorta, the aortic pressure increases.  The greater the stroke volume, the greater the change in aortic pressure during ejection.  – – Although, the arterial compliance dampens (smorza) the pulse pressure of the cardiac output. If the Compliance decreases (increase of arterial stiffness), but the peripheral resistance remains constant, the systolic aortic pressure increases and the diastolic pressure decreases, therefore the PP increases. If the aorta were a rigid tube, the pulse pressure would be very high.  Because the aorta is compliant, as blood is ejected into the aorta, the walls of the aorta expand to accommodate the increase in blood volume. The more compliant the aorta, the smaller the pressure change during ventricular ejection (i.e., smaller pulse pressure). Viceversa, when we get older the aortic compliance decreases and pressure related problems arise (higher pressure peak and slower decrease, higher time constant) The 5-elements WK model In this case the further element is given by the distinction between the aortic section and the rest of the arterial vasculature. The AORTIC part is described by Aor tic compliance Aor tic resistance (negligible wr t the peripheral resistance) – – – Aor tic inductance The rest of the arterial tree is described as before by Peripheral compliance Peripheral resistance