## Dynamic Pressure Measurements

### UNDER CONSTRUCTION

This article is about the frequency response of a fluid filled catheter.  If you'd like to muck around with the $$\alpha$$ version of the Active Figure for this topic,  DOWNLOAD NOW. I don't have any instructions or text for the Figure yet so you're on your own. Here are some hints: On the "DISPLAY" tab of the "CONTROL" form you will select 1 of 2 images to display, either "Physiograph" ("real time" display of the pressure signals) or "Transfer Function" (frequency domain display of the the catheter's frequency response).  Don't forget to press the "RUN" button to see the real time pressure tracings; there's nothing displayed until you do. Change the catheter's characteristics on the "CATHETER" tab.  Change the input pressure on the "INPUT" tab.  That's all for now folks.

Above: Time domain representation of the measured pressure response of a fluid filled catheter to a square wave input pressure.  The measured pressure doesn't track the input pressure with good fidelity but overshoots the input every time there's a rapid change in pressure -- a transient.  I've heard this referred to as "catheter whip" at many a conference.  Just so we're on the same page, this is going to happen even if the catheter remains stock still.  It's a physics thing.

Above: Frequency domain response of the above same catheter in the form of a transfer function,  here showing both the modulus and phase of the response as a spectrum (a function of frequency).  The transfer function is literally of ratio of the output pressure signal divided by the input, sinusoid by sinusoid. Each sinusoidal input frequency is affected differently by the catheter, but each can be considered separately and the final result is the sum of the individual sinusoids (in a linear system). The square wave from the earlier figure can be represented by the sum of numerous sinusoidal waves (actually an infinite number) each with specific magnitude so that the summation results in the observed input.  Due to the physical properties of this particular catheter, some of the frequencies are amplified in the measured signal (modulus value > 1.0) and some are diminished (modulus < 1.0).  (We'll see other catheter examples where ALL the sinusoids are diminished relative to the input.) Sinusoids present in the square wave also are shifted in phase depending on frequency (lower part of figure).  High frequency sinusoids are shifted to a greater extent.  The phase plot shows negative values for all frequencies (but changing with frequency) meaning that the measured sinusoid always lags the corresponding input sinusoid;  the phase lag increases with increasing frequency approaching $$-\pi$$ asymptotically at sufficiently high frequency (it always does this for this type of 2nd order catheter system).  For sufficiently high frequency, a measured sinusoid will approach $$180^{\circ}$$ out of phase with the input sinusoid.

The fact that this particular catheter has amplitudes > 1.0 for some of the frequencies in the transfer function means it is underdamped and we'll see overshooting of the measured pressure if the input pressure contains frequencies subjected to this amplification.  A fluid-filled catheter acts as a filter that amplifies some of the input frequencies and diminishes others (except for the 0 frequency if the system is correctly calibrated).  Once you get used to the concept, it's probably easier to say simply that the filter multiplies each frequency by a (complex) number that depends on frequency and leave the word "amplification" out of the description; the amplitude factor can be greater OR less than 1.0.  The fact that the square wave contains a wide range of frequencies results in considerable distortion of the input waveform.

We can illustrate the same concepts from a different angle.  For the same catheter we'll switch to different type of input, a sine wave where the frequency can be specified.  The tracing below starts with a low frequency sine wave input pressure of 1.0 Hz (100 mmHg amplitude).  The frequency increases in sequence; 2.0 Hz, 4.0 Hz, 8.0 Hz, 16.0 Hz, and 32.0 Hz at the far right.  The amplitude of the input wave (black) remains constant but it's clear how the amplitude of the recorded pressure (blue) first increases, then decreases as the frequency increases.

The tracing below is from the 32.0 Hz input, essentially just a continuation of the previous tracing but the sweep speed is increased shortly into the trace so that you can appreciate the phase relationship between the input and measured sine waves (10X the sweep speed as the above).  The measured pressure is out of phase at sufficiently high frequency by an amount approaching $$180^{\circ}$$; the input sinusoidal peaks coincide in time with the troughs of the output.  It's clear in the above trace that the sine waves are in phase at low frequency.

This was just a time-domain recap of what the transfer function is telling us.  Pick a frequency on the transfer function and check the modulus and phase values.  It shows the amplification factor (what to multiply the input sine wave amplitude by to give the measured amplitude); the value of the phase tells how much the measured (output) pressure wave lags the input.

In the next example, physical properties of the catheter have been altered; specifically the catheter length has been shortened considerably (and nothing else). You might think offhand that this would improve the situation since the transducer is now "closer" to the pressure input but without changing anything else about the catheter or fluid properties.  The recording shows, however, that we now have a situation with marked ringing of the system -- marked overshoot with prolonged oscillation in the recorded pressure.  Inspection of the transfer function shows that there's a marked amplification of frequencies in the 60-80 Hz range.  Since the square wave includes frequencies in those ranges, they get amplified and show up in the recorded pressure as shown.