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The Importance of an Ideal Damping Coefficient
It has been established by researchers1 that in order to accurately recreate a blood pressure waveform, the sensing system must be able to measure up to the 20th harmonic of the observed waveform. In his textbook, Dr. William Grossman states that “If components of a particular frequency range are either suppressed or exaggerated by the transducer system, the resulting signals will be greatly distorted.”2
A rapid transient test is important for determining the suitability of a sensor system to measure a dynamic pressure signal. An accepted method for creating a transient pulse is to expose the sensor under test to the pressure created by an inflated balloon, and observe the transient response when the balloon is popped. The resulting waveform provides us insight into the sensor’s dynamic properties.
Figure 1.0: representative pressure wave during a “pop test” where a sensing system is underdamped (left) and optimally damped (right)
Figure 1.0 is from Dr. William Grossman’s3 studies of blood pressure measurements and suitability of sensing systems. The two traces represent an underdamped system response on the left tracing and an optimally damped system on the right. The observed values of X1, X2 and t are parameters used to extract Damping coefficient, D, Damped Natural Frequency Response, ND, Undamped Natural Frequency, N, and calculate a flat system frequency response to +/- 5% of the input signal. Based on these parameters a Scisense Pressure System and other competing pressure system were evaluated.
Figures 2.0 and 3.0 below represent the output of two pressure sensor systems currently available on the market. Both sensors were exposed to the same rapid pressure transient. Because of the rapid change in pressure, both sensors overshoot the target pressure of zero mmHg. The signals then resonate around the true (zero) pressure before settling to a steady state. From the manner in which the signals reach steady state, we can first measure and then calculate parameters of sensor performance. These parameters will in turn help us make decisions on suitability of sensor to measure a blood pressure signal.
Figure 2.0: Pressure wave generated by Competing Pressure System during rapid transient pop test
Figure 3.0: Pressure wave generated by Scisense Pressure System during rapid transient pop test
The calculations used below are from Dr. William Grossman’s Test4.
Damping Coefficient: D
The damping coefficient represents the amount of friction available to dissipate energy stored in the system as it transits from one level to another. It can be calculated by using the ratio of the transient signal above and below the zero axis X1 and X2. Figure 2.0 represents a competing micro-sensor system with an underdamped coefficient of 0.185, while Figure 3.0 shows a Scisense micro-sensor system with a near optimum coefficient of 0.623.
Natural frequency of the sensor: Damped and Un-damped.
As each sensor stabilizes, it repeatedly crosses the zero point. If we measure the time between zero crossing, and call it “t”, we can define the Damped Natural Frequency (ND) of the sensing system as 1/t. Therefore, the Competing micro-sensor system corresponds to a ND of 556 Hz, while the Scisense micro-sensor system corresponds to a ND of 294 Hz.
The Un-damped Natural Frequency (N) is calculated from ND and D as follows:
Therefore, the final calculated Un-damped Natural Frequency for the Competing micro-sensor system and the Scisense micro-sensor system are 565 Hz and 375 Hz, respectively.
What These Numbers Tell Us
In the case of a murine heartbeat, it is possible to have a fundamental frequency as high as 700 beats per minute (BPM), or 12 Hz. It has been determined that a pressure measurement with a frequency response flat to within +/- 5% of the first 20 harmonics is required for accurate reproduction of the amplitude of maximal dP/dt. This means that we require a pressure sensing system capable of maintaining a flat frequency response up to 233 Hz. So how do the two signals compare?
If we use the Damping Coefficient curves charted in Dr. Grossman’s test4, and accept +/- 5% distortion as our limit, we find that the Competing micro-sensor system, with a damping coefficient of 0.185, can accurately represent a signal up to approximately 35% of its natural frequency. Thirty-five percent of 565 Hz represents a signal that has a flat frequency response up to 197 Hz.
If we look at the Scisense sensor system with a damping coefficient of 0.623, we find that the sensor has a frequency response that is flat up to 85% of N or 0.85 x 376 = 319 Hz. It is interesting to note that while the Scisense system shows a lower natural frequency, by using optimum damping coefficients, it is able to use a higher percentage of its natural frequency than a system that is under damped. The frequency response limits of the Scisense system are sufficiently high to ensure that the sensor is never going to be operating in the non-linear frequency range when applied to a murine study. By definition, being linear to the 20th harmonic ensures that the sensor is capable of reporting accurate signals for both amplitude and dP/dt. The competing sensor is already working in the +/- 5% error band at 200 Hz; this neither satisfies the minimum requirements for 20 harmonics nor does it leave any overhead for higher frequencies to be encountered.
Figure 5.0: Comparison of the Scisense micro-sensor system and the Competing micro-sensor system where Scisense electronic damping has been removed.
The graph above shows the results of lowering the Scisense system’s damping coefficient to match that of the competing pressure system. While the Scisense system will now have a response that is slightly faster than the competing pressure system, the frequency response will be lowered to the same level as that of the competing system. By lowering the system’s damping coefficient to match the competitor’s, we would no longer be able to meet the frequency criteria demanded to reproduce the murine heartbeat.
Bibliography
- Bernard J. Gersh, Clive E.W. Hahn and Cedric Prys-Roberts: Physical Criteria for Measurement of Left Ventricular Pressure and its First Derivative. Cardiovascular Research, 1971, 5, 32-40.
- Dr. William Grossman, Grossman’s Cardiac Catheterization, Angiography, & Intervention, 7th Edition: 2006 Lippicott Williams & Wilkin. Page 134.
- Dr. William Grossman, Grossman’s Cardiac Catheterization, Angiography, & Intervention, 7th Edition: 2006 Lippicott Williams & Wilkin. Page 140.
- Dr. William Grossman, Grossman’s Cardiac Catheterization, Angiography, & Intervention, 7th Edition: 2006 Lippicott Williams & Wilkin. Page 137.
Solid State Pressure: