1. 1 Role of the arterial system: The arteries (Fig.1) carry blood away from the heart to the tissues. The aorta branches toward periphery until blood reaches the arterioles and finally the capillaries. It is very interesting to notice that mankind was able to understand the heliocentrism before the working of the circulatory system (Fig.2). The role of the arterial system is to convert the high velocity (around 1 m/s) pulsatile flow at the level of the ascending aorta to a low velocity (around 0.01 cm/s) steady flow necessary to cellular exchanges. This is performed using the so called: Windkessel effect. 1473 Copernic 1543 1571 Kepler 1630 1578 Harvey 1657 Publication (1628) of “on the motion of the heart and blood in animals” Distance earth-sun: 150 10 6 Km Distance skin-heart: 3 cm

2. 2 Windkessel Effect Windkessel: a german word that can be translated as air (wind) chamber (kessel). A description of an early Windkessel effect was given by the German physiologist Otto Frank in 1899. It likens the heart and systemic arterial system to a closed hydraulic circuit comprised of a water pump connected to a chamber. The circuit is filled with water except for a pocket of air in the chamber, this was the same model used by firemen in the early XX century. As water is pumped into the chamber, the water both compresses the air in the pocket and pushes water out of the chamber. The compressibility of the air in the pocket simulates the elasticity and extensibility of the major artery, as blood is pumped into it by the heart ventricle. This effect is commonly referred to as arterial compliance. The resistance water encounters while leaving the Windkessel, simulates the resistance to flow encountered by the blood as it flows through the arterial tree from the major arteries, to minor arteries, to arterioles, and to capillaries, due to decreasing vessel diameter. This resistance to flow is commonly referred to as peripheral resistance. Compliance Resistance

3. 3 Theoretical development of the Windkessel effect We will consider here the simplest form of the Windkessel effect. This model called Windkessel 2-element considers only the arterial compliance (C) and the peripheral resistance (R). Hypotheses 1- Unsteady flow. 2- The pressure difference across the resistance is a linear function of the flow rate. 3- The working fluid is incompressible. 4- The flow is constant throughout the ejection phase. Symbols P: pressure generated by the heart (N m -2 ) [mmHg] Q: blood flow in the aorta (m 3 s -1 ) [l mn -1 ] R: peripheral resistance (N s m -5 ) [dyne s cm -5 ] C: arterial or systemic compliance (m 5 N -1 ) [ml mmHg -1 ] t: time [(s) T: period (s) Ts: ejection time (s)

4. 4 Theoretical development of the Windkessel effect PV QQ1Q1 R air Systolic phase Q t Ts T I- Systolic phase (valve in open position) Conservation of mass Q cc is the flow to the compliance chamber. Thus Hyp 2: P-P cv = R  Q 1 (P cv is the central veinus pressure: P cv << P [P cv  5 mmHg vs. P  100 mmHg ]) P cv Hyp 4: Q=C te throughout the systolic phase. Therefore Compliance (C) Then

5. 5 Theoretical development of the Windkessel effect PV QQ1Q1 R air Systolic phase Q t Ts T P cv Finally the equation to be solved for the systolic phase is Eq.I Initial condition: P(t=0)=P 0 - Solving eq. I a) Particular solution (Q=C te =0)

6. 6 Theoretical development of the Windkessel effect PV QQ1Q1 R air Systolic phase Q t Ts T P cv b) Method of variation of the paramter (  1 =  1 (t)) replace in eq.I Hence Then To be replaced in the particular solution

7. 7 Theoretical development of the Windkessel effect PV QQ1Q1 R air Systolic phase Q t Ts T P cv c) The general solution for the systolic phase is therefore: To determine the constant  2 we use the initial condition: Finally, the pressure waveform for the systolic phase can be written as:

8. 8 Theoretical development of the Windkessel effect P V QQ1Q1 R air Diastolic phase Q t Ts T II- Diastolic phase (valve in closed position) It is same thing as for the systolic phase but with Q=C te =0 Therefore; Eq.I  Initial condition: P(t=Ts)=P s (Ts) The solution for this equation is under the following form:  3 is determined using the initial condition: Then

9. 9 Theoretical development of the Windkessel effect P V QQ1Q1 R air Diastolic phase Q t Ts T Finally, the pressure waveform for the diastolic phase can be written under the form:

10. 10 Theoretical development of the Windkessel effect PV QQ1Q1 R air P V QQ1Q1 R Systolic phase Diastolic phase Q t Ts T To compute the solution, we need to know: P 0 ; Q; R; C; T; Ts. However, it is convenient to use a condition of recurrence to compute P 0 : P(0)=P(T)

11. 11 PV QQ1Q1 R air P V QQ1Q1 R Systolic phase Diastolic phase Q t Ts T with General solution for the Windkessel 2-element Model

12. 12 Analysis of the solution for the Windkessel 2-element Model We can notice from the analytical solution of the Windkessel 2-element model the importance of the term (R  C) because it determines the “speed” of the exponential rise or decay. This product is called the characteristic time and is usually noted (  ). Case  =0 t P Case  =  t P R  Q P0P0

13. 13 Analysis of the solution for the Windkessel 2-element Model Case  0 and   Mean Pressure C=Cte R  C  R  C  R=Cte Normal Hypertension Ps > 140 mmHg And/or Pd > 90 mmHg Hypertension Systolic pressure (Ps) Diastolic pressure (Ps)

14. 14 Windkessel Model: Practical application. Build a Windkessel model on a cardiac simulator Fluid Water Air Elastic membrane Fluid water air