3. Impedance. Lumped Parameters and Windkessel models

what affects the morphology of the measured pressure and flow waveforms

pressure and flow waves propagate along the arterial tree and get reflected with varying magnitudes

What are forward waves in the arterial system

Waves travelling away from the heart

What are backward waves in the arterial system

Waves travelling towards the heart due to reflection

What generates the forward propagating arterial waves

ventricular contraction during blood ejection

Where can arterial wave reflections occur

At bifurcations, junctions, changes in mechanical properties and arterioles

Why are arterial pressure and flow waveforms complex

Because many forward and reflected waves interact with different timings and magnitudes

Why is Fourier analysis used in arterial haemodynamics

To decompose measured pressure/flow waves into a mean term plus sinusoidal harmonics

what is the fundamental frequency in Fourier analysis of arterial waves

cardiac frequency

What is the fundamental frequency of the nth harmonic

n x fundamental frequency

What two conditions are needed for Fourier decomposition to apply to physiological signals

Steady-state measurement at constant heart rate and a sufficiently linear arterial system

How are pressure and flow harmonics written

As complex numbers with a modulus and phase angle

What is the fundamental angular frequency

w=2*pi/T where T is the heart cycle duration

For a heart rate pf 75 bpm, what is the fundamental frequency

f= 1.25Hz

For a heart rate of 75 bpm, what is the fundamental angular frequency

w= 7.85 rad/s

How many harmonics are usually sufficient to describe haemodynamic variables

About 10 to 15 harmonics

What is arterial impedance

The ratio of pressure to flow:

Z=P/Q

What are the usual units of impedance in haemodynamics

mmHgs/ml

What is the impedance of the nth harmonic

Zn=Pn/Qn

What is the modulus of impedance

|Zn|=|Pn|/|Qn|

What is the phase of impedance

ϕZ​=ϕP​−ϕQ​

What does the DC component of impedance represent

Total peripheral resistance

What is arterial input impedance

The impedance measured at the entrance of the systemic or pulmonary circulation

What does input impedance describe

The full relation between pressure and flow at the entrance of the arterial system

Why is input impedance important

Because if Zin is known, pressure can be calculated from flow and vice versa

Does input impedance include wave reflection effects

Yes it fully captures the effects of wave reflection

What happens to the impedance modulus at higher harmonics

It oscillates around a plateau called the characteristic impedance Z0

What is characteristic impedance Z0

The pressure-to-flow ratio in a system with forward waves only and no reflections

In what ideal system is Z0 defined

An infinitely long uniform tube with no reflections

What is the high-frequency approximation for characteristic impedance

Z0=pc/A

What do the symbols in Z0=pc/A mean

p=blood density, c=wave speed, A=cross-sectional area

How can Z0 be estimated in the ascending aorta

Using either a time-domain or frequency-domain approach

What is the time domain idea for estimating Z0

In early systole the pressure-flow relation is nearly linear because reflected waves have not yet returned

What part of systole is often assumed reflection-free in the ascending aorta

The first 50 ms of systole

In the time-domain method, what gives Z0

The slope of the regression line of pressure versus flow in early systole

In clinical practice, how is Z0 estimated in the frequency domain

By averaging the modulus of the higher harmonics, typically the 3rd to 10th

What is wave separation

The decomposition of measured pressure and flow into forward and backward components

How are total pressure and flow expressed in wave separation

P=P+ + P-

Q=Q+ + Q-

What are the wave separation formulas for pressure

P+=P+Z0Q/2

P-=P-Z0Q/2

What is the reflection coefficient T

The ratio of backward to forward pressure amplitude for a harmonic

What simpler quantity can be used as a surrogate of reflection magnitude

The ratio of backward pressure amplitude to forward pressure amplitude

What is the tube model of the arterial system

A model that treats arteries as a uniform or tapered elastic tube with an effective length and a single reflection site

What causes reflections in the tube model

An impedance mismatch between the tube and terminal resistance

What formula links the first minimum frequency and tube length in the tube model

fmin=c/4L

What does the quarter-wavelength formula estimate in vivo

The effective length to an apparent reflection site

What is the approximate effective arterial length found in humans in the lecture

About 44cm

Is the tube model more physical or conceptual

It is more conceptual than physical

What is a lumped parameter model

A model in which distributed arterial properties are grouped into a few discrete parameters with physical meaning

What is the Windkessel model

The basic lumped parameter model of the arterial system

What electrical elements are used in lumped arterial models

Resistance, inductance/inertance, and capacitance/compliance

What does resistance represent in arterial models

Friction in small arterioles and capillaries

What does inertance represent

The acceleration and deceleration of blood in arteries

What does compliance represent

The buffering capacity of large elastic arteries

What does the heart correspond to in the electrical analogy

A source that produces current Q(t)

What electrical component represents compliant arteries

A capacitor

What electrical component represents peripheral arterioles

A resistor

What quantity represents blood pressure in the electrical analogy

A potential difference

What quantity represents veins in the electrical analogy

Earth/zero pressure

What equation represents resistive pressure loss

ΔP=RQ

What equation represents inertial pressure loss

ΔP=L dQ/dt

How is compliance defined

C=dV/dP

or for finite changes

C=ΔV/ΔP

What is one advantage of lumped parameter models

They give qualitative understanding and account for vessel distensibility

What is one disadvantage of lumped parameter models

They are not very accurate and do not account for wave-like behaviour well

What are the elements of the 2-element Windkessel model

A resistance and a compliance/air chamber

What happens in systole in the 2-element Windkessel model

Part of stroke volume is buffered in the chamber, increasing pressure, while flow continues across the resistance

What happens in diastole in the 2-element Windkessel model

Aortic pressure decays exponentially

What is the diastolic pressure decay equation in the 2-element Windkessel model

P(t)=P0 exp(-t/r)

What is the Windkessel time constant

t=RC

What does P0 represent in the 2-element model

The aortic pressure at the onset of diastole, usually at the dicrotic notch

What does the 2-element Windkessel model capture well

The lowest 2-3 harmonics of input impedance reasonably well

Where does the 2-element Windkessel model fail

In the mid to high frequency range

Why is the 2 element Windkessel model still useful

Because the lowest harmonics carry most of the power and shape the main waveform features

Why was the 3-element Windkessel model introduced

To fix the poor high-frequency behaviour of the 2-element model

What is third element in the 3-element Windkessel model

The characteristic impedance Z0

What does Z0 represent in the 3-element model

The local inertia and compliance properties of the proximal ascending aorta

What is a strength of the 3-element Windkessel model

It gives realistic aortic pressure and flow waves and mimics input impedance reasonably well

What is a key drawback of the 3-element Windkessel model

It overestimates compliance C and underestimates "Z0

How does the 3-element model affect the mean pressure-flow relation

It predicts

Pao​​/Qao​​=R+Z0

when ideally it should be

Pao​​/Qao​​=R

What extra element is added in the 4-element Windkessel model

An inertial element L

Where is the inertial element placed in the 4 element model

In parallel with Z0

What does L represent in the 4 element model

The sum of all local inertances throughout the arterial tree

What improvement does the 4-element Windkessel model provide

It improves the low to mid frequency behaviour and better covers the whole frequency range

As the 4-element windkessel model now covers the whole frequency range in a more appropriate way, What happens at 0Hz in the 4 element model?

L shorts Z0 and Pao/Qao=R

What dominates in the mid frequency range in the 4 element model

Compliance C

What does the impedance approach at high frequencies in the 4 element model

Z0

What happens if L is placed in series with Z0 instead

Impedance becomes too high at all frequencies which is not appropriate

What is the trade-off of adding more Windkessel elements

Potentially better accuracy, but less clear physical meaning