Resonance, damping and frequency response

This chapter is relevant to Section G7(ii)  of the 2017 CICM Primary Syllabus, which asks the exam candidate to "describe the principles of measurement, limitations, and potential sources of
error for pressure transducers, and their calibration". The concepts of resonance and damping fall neatly into the category of limitations and potential sources of error. For a variety of sensible reasons, the college examiners have clearly prioritised this topic, and it appears in multiple past paper questions:

• Question 12 from the first paper of 2020
• Question 17 from the second paper of 2017
• Question 12 from the first paper of 2015
• Question 24 from the second paper of 2012

In spite of how common these questions have been, they are still done very poorly, with a pass rate ranging between 25% and 33%. The topic had also come up once in the Part II exam, in Question 11.2 from the first paper of 2010 where candidates were expected to comment on the "fidelity" of the pressure transducer system which was shown undergoing a fast flush test.  The interpretation of fast flush tests and other practical matters related to the "fidelity" of the arterial transducer system are discussed in the section on arterial line dynamic response testing. In many ways, this is a huge self-indulgent redux version of that chapter. From the standpoint of exam preparation, it would be possible to skip this entire chapter and only review the brief square wave test section in the Fellowship Exam required reading chapter on the information derived from arterial line waveforms.

In summary:

Moxham (2003) is probably the most useful single reference one can offer for this topic, and will satisfy the majority of readers. Stoker (2004) and Gilbert (2012) were also used to put this chapter together, but these are difficult to find in free full-text form. In general, one can find the best explanations of these matters in textbooks of anaesthesia, and if the reader is not satisfied with Moxham or Gilbert they can be referred to Miller's Anaesthesia (Chapter 40 in the 7th edition). This list of references will cover the material to an exhausting depth, but if the reader's need for detail is still somehow unsatisfied one can point them towards Jonathan B. Mark's Atlas of Cardiovascular Monitoring. This legendary book is somehow (accidentally?) available for free from the University of Montreal. It is definitive; when you see a pressure waveform in almost any other textbook, most frequently you will find that the images have been borrowed from Mark. Or from Leslie Geddes' Handbook of Blood Pressure Measurement (1991), which is the Iliad of haemodynamic monitoring literature, worth reading for the quality of the writing alone. The author's vocabulary and command of language betray a scientific upbringing from an earlier time, when education and culture could be expected to coexist. 

The pressure transducer system as a harmonic oscillator

The fluid between the artery and the transducer can be described as a simple harmonic oscillator, analogous to a pendulum or a mass hanging on a spring. When the pendulum is displaced, it undergoes simple harmonic motion: i.e. it oscillates around the equilibrium point. The mass hanging on a spring will oscillate up and down when disturbed, like a bungee jumper. Probably a better analogy is a rubber ball dropped from a height. 

In a totally frictionless system, after one heartbeat the system would just continue to oscillate forever at its natural frequency. In the real world, some friction occurs - fluid rubbing against the walls of the tubing, for example. As a result, after one displacement the system will oscillate briefly and come to rest.

Natural frequency and resonance of the transducer system

The natural frequency of a system is determined by the "stiffness" of the spring and the mass of the system, such that 

From this, one can see the sort of things that can change the natural frequency of a fluid-filled pressure transducer system. The density of the fluid is pretty well fixed unless you use some sort of weird fluid to prime your tubing, and so the remaining variables are the length and radius of the tubing and elasticity of the system.

• Lengthening the tubing increases the elasticity of the system and decreases the natural frequency
• Increasing the diameter of the tubing  increases the natural frequency
• Stretchy tubing increases the elasticity of the system and decreases the natural frequency, which is why transducer kits typically use relatively rigid non-compliant tubing.
• Air bubbles in the fluid increase the elasticity of the system and decrease the natural frequency.

Elasticity, the tendency of a volume to return to its initial shape after being distorted, is used in the formula given above (from Gilbert's paper), which is adapted to the specific case scenario of the fluid-filled transducer. However, Gilbert uses E to describe the deformability of the transducer diaphragm (the change in volume which occurs per unit change of pressure). On closer inspection, that's actually elastance.

The natural frequency of a system plays a role in how the system responds to sustained stimuli. Consider the system as it sits there after the last impulse, oscillating and slowly coming to rest. If another stimulus occurs (a new wave arrives), it interacts with the existing waveform. If the new waves arrive at the same frequency as the natural frequency of the system, the peaks will coincide and the sum of the amplitudes will be greater (the peaks will be higher). Similarly, the troughs will be lower. This tendency of system to oscillate with greater amplitude at the natural frequency than at other frequencies is resonance.

The frequency of the pulse, for example, is 1 Hz at a heart rate of 60.  If the natural frequency of your pressure transducer is also 1Hz, each peak of the pulse pressure wave can coincide with a peak of the system's own oscillation, increasing the amplitude of the measured peaks (systolic pressure) and decreasing the amplitude of the troughs (diastolic pressure).

Fourier analysis of physiological pressure waves

Most of the physiologically relevant waveforms are not sine waves but rather complex waves. Fourier demonstrated that any complex waveform can be constructed from a number of simple sinusoidal waveforms, the frequency of which is some multiple of the frequency of the complex wave (which is called the fundamental). The other component of the calculation is a constant, which is a mean value of the waveform over the duration of its cycle (in this case, this is the mean arterial pressure).

Thus, for arterial pressure measurement, the complex arterial pressure waveform is composed of one fundamental waveform (that's the pulse rate) and numerous other higher frequency waveforms (harmonics). At a heart rate of 60, the fundamental frequency is 1Hz, and the harmonic frequencies are 2Hz, 3Hz, 4 Hz, et cetera. The lower frequency harmonics tend to have the higher amplitude.

By adding all these waveforms together, one may approximate the true shape of the pulse waveform, which is recognizable as an arterial pulse waveform. Well, no. That idealised graphic is not what it looks like when you go full nerd and generate sine waves in a spreadsheet. It looks more like the Google-coloured graph on the left, which is less recognizable as an arterial waveform, but is more scientifically accurate. 

In actual fact, whenever you see these superimposed harmonics in a textbook, it is only the first and second, and the source of the image is usually Geddes (1991). It is usually in black and white, and usually has a little rectangular box to illustrate the resemblance between the summed waveform and the shape of the arterial pulse. In this graph, the fundamental waveform is added to 63% of the second harmonic. (i.e. the amplitude of the fundamental is taken as 100%). As far as can be reasonably reconstructed from historical records, Geddes and all subsequent authors who keep re-using this image do so not because it represents a realistic model of the human arterial pressure wave, but because the addition of these sine functions conveniently produces a waveform with a recognisable dicrotic notch.

So, what do the harmonics of the arterial pulse really look like? For this, one needs to look to an earlier, more honest time. The best representation of this comes from a 1949 book by Anders Tybjaerg Hansen, titled "Pressure Measurement in the Human Organism". It is reproduced here, with no permission whatsoever.

Six harmonics with their respective amplitudes are represented here, their summed waveform indeed looking very familiar and arterial-like. The book itself is so old that there is no digital copy anywhere within sight. It was published as a supplement of Acta Physiological Scandinavica. Without this, it is impossible to determine how Hansen came up with these phase shifts and amplitudes, except to guess that in the 1940s he must have laboriously and manually performed a Fourier analysis of recorded waveforms to plot the six harmonics.

How many harmonics does one need to analyse, and why

The answer depends on how much "resolution" is required. Realistically, you could make do with just five harmonics- that would be enough to have an accurate representation of the pulse pressure - but at least eight harmonics need to be analysed and added together in order to reproduce the pulse waveform with enough fidelity to discern such structures as the dicrotic notch. The higher you go in frequency of harmonics, the lower their amplitude, and therefore the smaller their contribution to the summed waveform - so to analyse anything beyond the tenth harmonic is pointless, as it will not add very much to the shape of your pulse waveform. 

The natural frequency of the transducer system needs to be much higher than the fundamental frequency of the pulse wave. With a low natural frequency, the fundamental frequency or some of the first few harmonics would end up being amplified by resonance, and because these are already high-amplitude waves the effect on the summed waveform would be quite significant. If the natural resonance of the transducer system is closer to the eighth harmonic, resonance will still amplify that waveform, but because the amplitude of this harmonic is very low, the effect on the summed waveform will be minimal. Ergo, transducer systems need to have a minimum natural frequency at least eight times the expected maximum frequency of the expected fundamental frequency of the measured system. Most commercially available systems analyse eight harmonics; thus to maintain accuracy for pulse rates up to 180 bpm (3 Hz), the natural frequency of the system needs to be at least (3 × 8) = 24 Hz.  Generally speaking commercially available systems have a natural frequency well above this value (usually 200Hz) but we interfere with this by adding tubing, stopcocks, cannulae, three-way taps and air bubbles. How much interference this causes can actually be measured - you can assess the natural frequency of the completed arterial line transducer setup by doing a fash flush test (the wavelength of the oscillations which occur after the square wave is the natural frequency).

Damping

Damping is the absorption of the energy of oscillations, by whatever means. Generally, this results in the decreased amplitude of the waves. The processes which result in damping also reduce the natural frequency of a system. It is a fairly complicated process which is mainly the consequence of fluid viscosity and its interaction with the walls of the tubing system into which it has been primed. One can make the general statement that the diameter of the tubing has the greatest effect on damping; damping increases by the third power of any decrease in the diameter of the tubing. In other words, narrower tubing increases damping. Which is to say, narrow tubing results in lower amplitude waveforms. One can model this. Say, the tubing decreases in diameter by 33%. The damping will increase by 135%. 

The natural frequency of the system decreases with damping. In fact, some damping is good, for this very reason. And from that statement it follows that, if some damping is good, then there must be some level of damping which is "best", for any given system. With this optimal damping, it is possible to maximise the frequency response along the "flat range" of frequencies (the range of frequencies over which the natural frequency does not amplify the signal by very much). This optimal level of damping corresponds to a "damping coefficient" of around 0.7, damping coefficient being an index of the tendency of the system to resist oscillations. The reason these terms appear in bold here is because they were specifically asked about in Question 17 from the second paper of 2017. The concise definitions which were asked for are as follows:

• Damping coefficient:
  • An index of the tendency of the system to resist oscillations
  • Given by the equation,
    γ = c / 2m
    where
    - γ is the damping coefficient,
    - c is the friction coefficient, and
    - m is the mass of the oscillating thing. 
  • A damping coefficient around 0.7 is optimal, >1.0 is overdamped, and <0.7 is underdamped,
• Optimal damping: A damping coefficient of  around 0.64-0.7
  • Maximises frequency response
  • Minimises overshoot of oscillations
  • Minimises phase and amplitude distortion
  • Corresponds to 2-3 oscillations following an arterial line flush test
• Critical damping: a damping coefficient of 1.0
  • The oscillator returns to the equilibrium position as quickly as possible, without oscillating, and passes it only once.
  • Occurs when the damping coefficient is equal to the resonant frequency of the oscillator

The optimal damping coefficient of a system therefore depends on the natural frequency. With a very low natural frequency, the flat range is very narrow, and no amount of damping can prevent the distortion of the measured waveform by resonant amplification. On the other hand, with a very high natural frequency, the system will never distort any waveforms within a clinically relevant range, and it doesn't matter what your damping coefficient is.

Observe above. Two systems, equally underdamped. One has a natural frequency of around 7Hz, well inside the clinically relevant range. Damping coefficient would have to be very high, and even then it would be unlikely to make the system useable (the waveform and measured pressure would always be a little inaccurate). In contrast, the system with the natural frequency of 70 Hz could have any damping coefficient whatsoever- within the relevant range of frequencies it is always going to give results unaffected by resonance. This dependence of optimal damping coefficient on natural frequency can be represented in a graph, which was first presented by Gardiner et al. A heavily modified version is presented here:

Underdamping leads to overshoot. A system with minimal damping will reverberate, the natural frequency of the system will amplify the measured waves and therefore the pressure registered by the tranducer will be higher than the actual pressure. Sharp jagged waveforms will be produced. Conversely, an over-damped system will produce underestimated values and slurred broad waveforms. With overdamping, some of the higher frequency harmonic waveforms will be lost (remember that these are the waveforms which already have a lower amplitude; with more damping their amplitude disappears to near zero).

The changes in waveform shape can be illustrated in a situation where the damping coefficient of a system is gradually increased while oscillations are occurring - in this case, a dog's pulse. Because damping increases dramatically with decreasing tube diameter, one may conveniently model an increasing damping coefficient by tightening a clamp over the arterial line tubing, just as Geddes et al did in 1984:

Note how first, the dicrotic notch is lost (because it is produced by high frequency, low amplitude elements). Then, the waveform begins to flatten as the amplitude of even the low-frequency waves is affected. At last, with the tubing clamped, the waveform flattens at the mean arterial pressure.

Frequency response

The frequency response of a system is the relationship between the frequency of the measured waves and the amount of amplitude amplification which might occur as the result of resonance. It can be represented on an amplitude/frequency graph:

From this, the beneficial effects of damping become clear. The damped system has a larger range over which there is little amplitude increase with increasing frequency, and even at its natural frequency the amplitude change is smaller. As a result, the measured pressure waves will not be overestimated as much. In an ideal system, all frequencies of clinical interest will lie within this flat range. For an arterial line, for example, that would be all eight harmonics - i.e. there should be little amplitude change up to a frequency of around 24 Hz, if the heart rate is 180. 

So, how do you know what the frequency response of a system is? Well. If you know what the natural frequency of the system is, you can predict that this is where the peak of the amplitude/frequency graph is going to fall, and from that, it is possible to predict where the flat range will be.

Testing for frequency response and damping

For arterial line pressure transducers, the "fast flush test" is the clinical bedside test which is used to assess the natural frequency of the system. To return to the model of the transducer system as a simple harmonic oscillator, this "fast flush" is dropping the ball from a height, or giving the pendulum a gentle nudge. Then you watch it and wait for it to swing through a few oscillations, and measure the frequency. In effect that is what you're doing when you open the fast flush valve on the arterial line transducer set. 

The "bounce" of oscillations after a fast flush can be recorded on graph paper, vellum parchment or wet clay if you have a dislike of computers.  More likely, as an intensivist you're an intensely technophilic organism, and prefer to measure the time interval between oscillations with the convenient digital calipers integrated into most monitoring software packages. You'd get a number, typically in milliseconds. That can be converted to a frequency in Hz (the number of oscillations per second). This is the natural frequency of your transducer system. If the natural frequency is over 30-40 Hz, you'd be able to confidently say that the clinically relevant range of frequencies (0-24 Hz) is well within the flat range of this system, and the pressure values you are recording are accurate. Generally speaking, at the bedside you will find most arterial line systems have a natural frequency somewhere between 10 and 25 Hz (Schwid et al, 1988).