Resonance and damping in arterial lines

The arterial line is a driven harmonic oscillator. The driving function is the arterial blood pressure. The response is the strain signal recorded by the Wheatstone Bridge resistor.

$$m\ddot{x} = \overbrace{F(t)}^{\text{driving function}}-\overbrace{\gamma\dot{x}}^{\text{damping}}−\overbrace{(2 \pi f_n)^2x}^{\text{oscillation}}$$

Definitions

The damping coefficient \(\gamma\) is the rapidity with which oscillations cease.

The natural frequency \(f_n\) is the frequency of oscillations after an impulse if there was no damping. $$f_n \propto \text{Tube radius} \cdot \sqrt{\frac{1}{\text{Compliance} \cdot \text{Tube length} \cdot \text{Fluid density}}}$$

The resonant frequency is the frequency of oscillations after an impulse. It is decreased by damping. If the driving function (blood pressure signal) or a large term in its fourier series is near the resonant frequency, resonance will occur (i.e. increased oscillation).

Critical damping occurs at the lowest \(\gamma\) such that the system does not overshoot, called \(\gamma_{crit}\)

The damping ratio \(\zeta\) is \(\frac{\gamma}{\gamma_{crit}}\).

Causes and effects of resonance

Resonant frequency increased by short, fat tube with noncompliant walls and minimal damping

If any large amplitude harmonics in the fourier series of the blood pressure waveform near resonant frequency, they will be amplified \(\to\) inaccurate waveform, widened pulse pressure

First 10 harmonics contain most amplitude so resonant frequency must be 10 times the heart rate \(\to\) 25hz.

Typical natural frequency 200hz.

Causes and effects of damping

Caused by bubbles/clots in line, vasospasm, 3 way taps, narrow/long/compliant tubing

Fast flush test \(\to\) \(\zeta\) is the amplitude ratio of 2 consecutive waves, resonant frequency is the frequency of these waves

Optimal damping occurs when \(\zeta \approx 0.6-0.7\).

Below this, pulse pressure is widened

Above this, pulse pressure narrowed

The importance of damping depends on \(f_n\).

\(\uparrow f_n \to\) signal less affected by resonance \(\to\) broader range of acceptable \(\zeta\).