Role of the arterial system:

The arteries (Fig.1) carry blood away from the heart to the tissues. The branches toward periphery until blood reaches the and finally the capillaries. It is very interesting to notice that mankind was able to understand the heliocentrism before the working of the (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. Publication (1628) of “on the motion of the heart and blood in animals”

6 Distance earth-sun: 150 10 Km 1578 Harvey 1657 Distance skin-heart: 3 cm 1571 Kepler 1630 1 1473 Copernic 1543 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

2 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 (m3 s-1) [l mn-1] R: peripheral resistance (N s m-5) [dyne s cm-5] C: arterial or systemic compliance (m5 N-1) [ml mmHg-1] t: time [(s) T: period (s) Ts: ejection time (s) 3 Theoretical development of the Windkessel effect

I- Systolic phase (valve in open position) 0 ≤ ≤ Tst

Conservation of mass air

Ts in out += QQQ cc Q PV Qcc is the flow to the compliance chamber. dV t Thus QQ += T R 1 dt QQP Hyp 2: P-Pcv= R×Q1 (Pcv is the central veinus pressure: Pcv << P 1 cv [Pcv≅5 mmHg vs. P≅100 mmHg ]) Systolic phase Hyp 4: Q=Cte throughout the systolic phase. Compliance (C) P dV P dV dP Therefore Q +=+= R dt R dP dt P dP Q += C Then R dt 4 Theoretical development of the Windkessel effect

Finally the equation to be solved for the systolic phase is

dP P Q air =+ Eq.I dt RC C Q Ts PV Initial condition: P(t=0)=P0

t - Solving eq. I T R

a) Particular solution (Q=Cte=0) QQ1 Pcv

t dP P − )(0 =⇔=+ α etP RC Systolic phase dt RC 1

5 Theoretical development of the Windkessel effect

b) Method of variation of the paramter (α1= α1(t)) replace in eq.I air t t d ⎛ − ⎞ 1 ⎛ − ⎞ Q ⎜α ()et RC ⎟ + ⎜α ()et RC ⎟ = Ts ⎜ 1 ⎟ ⎜ 1 ⎟ Q dt ⎝ ⎠ RC ⎝ ⎠ C PV t t t − − − 1 α1(td ) 1 ⎛ ⎞ Q − α () RC + eet RC + ⎜α ()et RC ⎟ = t 1 ⎜ 1 ⎟ T RC RCdt ⎝ ⎠ C R

t α ()td Q QQ1 Pcv Hence 1 = e RC dt C Systolic phase t RC Then α1() eRQt += α 2

To be replaced in the particular solution

6 Theoretical development of the Windkessel effect

c) The general solution for the systolic phase is therefore:

t t ⎛ − ⎞ − air )( = ⎜ eRQtP RC +α ⎟ e RC s ⎜ 2 ⎟ ⎝ ⎠ Q Ts PV To determine the constant α2 we use the initial condition: t T R ()0 PtP 0 α 02 −=⇒== RQP QQP Finally, the pressure waveform for the systolic phase can be 1 cv written as: Systolic phase

t − RC s 0 −+= )()( eRQPRQtP

7 Theoretical development of the Windkessel effect

II- Diastolic phase (valve in closed position) ≤ ≤ TtTs

te It is same thing as for the systolic phase but with Q=C =0 air

Therefore; Ts Q P V dP P Q Eq.I ⇔ ==+ 0 dt RC C t T R

Initial condition: P(t=Ts)=P (Ts) s QQ1 The solution for this equation is under the following form: Diastolic phase t − RC )( = α3 etP Ts Ts − − α is determined using the initial condition: RC RC 3 α3 ()0 −+= eRQPRQe Ts − Then α eP RC −+= )1( RQ 8 03 Theoretical development of the Windkessel effect

air

Finally, the pressure waveform for the diastolic phase Ts can be written under the form: Q P V

Ts t ⎛ − ⎞ − t R RC RC T d ⎜ 0 ePtP −+= )1()( ⎟ eRQ ⎜ ⎟ QQ ⎝ ⎠ 1 Diastolic phase

9 Theoretical development of the Windkessel effect air

Q Ts PV

t t ⎧ − T R RC ⎪ 0 −+ )( eRQPRQ 0 ≤≤ Tst tP )( = QQ1 ⎨ Ts t − ⎪ RC RC ⎩ 0 −+ ))1(( eRQeP ≤< TtTs Systolic phase air To compute the solution, we need to know: P0; Q; R; C; T; Ts. However, it is convenient to use a condition of recurrence to P V compute P0: P(0)=P(T) R

QQ1

10 Diastolic phase General solution for the Windkessel 2-element Model air

Q Ts PV

t t ⎧ − T R RC ⎪ 0 −+ )( eRQPRQ 0 ≤≤ Tst tP )( = QQ1 ⎨ Ts t − ⎪ RC RC ⎩ 0 −+ ))1(( eRQeP <≤ TtTs Systolic phase ⎛ Ts ⎞ air with ⎜ RC ⎟ ⎜ e − 1⎟ = RQP ⎝ ⎠ 0 ⎛ T ⎞ P V ⎜ RC ⎟ ⎜ e − 1⎟ ⎝ ⎠ R

QQ1

11 Diastolic phase 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 (τ). P P R ×Q

P0

t t

Case τ=0 Case τ=∞

12 Analysis of the solution for the Windkessel 2-element Model

Case τ≠0 and τ≠ ∞ 150 150

140 Normal C=Cte 140 130 R ⇑ Systolic pressure (Ps) 120 130 Mean Pressure 110 120 100 110 90

80 Pressure [mmHg] Pressure [mmHg] 100 Hypertension Diastolic pressure (Ps) 70 90 60 Ps > 140 mmHg 50 80 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Time [s] Time [s] And/or 150 180 Pd > 90 mmHg 140 C ⇓ 170 C ⇓ 130 R=Cte 160 R ⇑ 120 Hypertension 150 Hypertension 110 140

100 130

90 120

80 110 Pressure [mmHg] Pressure [mmHg]

70 100

60 90

50 80 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 13 Time [s] Time [s] Windkessel Model: Practical application.

Build a Windkessel model on a cardiac simulator

Air Fluid

Water

Elastic membrane

air

Fluid

water 14