The principles of the acoustic method has been explained by a number of authors (Fredberg et al., 1980; Brooks et al., 1984; Hoffstein et al., 1991; Marshall et al., 1991; Marshall et al., 1993, Louis et al., 1994, Kamal, 2004c). The basic principle of the AR method has been described as follows by Hoffstein (Hoffstein et al., 1991):
- As a sound pulse travels along a tube and comes across a change in area from A1 to A2, part of the pulse is reflected and travels back along the tube, and part is transmitted. - With known wavespeed (c) and travel time (t), the length of the tube (d) can be calculated
to be d = ct.
- With one-dimensional wave propagation, the measurement of wave travel time is equivalent to the measurement of distance.
- The amplitude of the reflected pulse (Pr) is determined by the amplitude of the incident pulse (Po), and the physical property of the tube. Considering a tube with a single discrete area change from A1 to A2, and assuming constant and uniform gas composition, the amplitude of the reflected pulse is given by: Pr = Po[(A1 - A2) / (A1 + A2)].
The CSA of A2 can be calculated by measuring the amplitude of the incident and reflected pulses, since A1 is presumed known. Therefore, the determination of the length and area of the straight tube is reduced to measuring the travel time of the pressure pulses from the area change of the tube, and the amplitudes of the incident and reflected waves. In case of a duct consisting of many
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segments - each with different area – the incident sound wave (pressure wave) will be reflected in part every time for each new segment. Determination of the lengths and areas of the individual segments is based on measurement of arrival times and amplitudes of the reflections. This gives the area of the duct versus distance from inlet that is the area-distance function or the airway echogram (Hoffstein et al., 1991).
Kamal expanded on the principles of the AR method as used in the Eccovision equipment (Kamal, 2004c) as follows:
- An acoustic impulse traveling through a wavetube into an upper airway will undergo partial reflection and partial transmission at each change in the CSA creating a reflection sequence which will return through the wavetube without further reflection. The passage of the impulse is recorded through a microphone in the wavetube close to the connection between wavetube and the upper airway, the input impulse response. An area-distance relationship of the upper airway geometry can be created by comparing the incident and the reflected acoustic impulse.
- The input impulse response is a series of reflections created by changes in the impedance within the upper airway. The reflection can be either single due to a single change of a tube CSA or multiple as in a human upper airway. The input impulse response and the input impedance are closely related.
- A straight tube with a single change in CSA can be used as an example to highlight how acoustic reflection is used to obtain an area distance function (Figure 2.29).
- The pulse is recorded as it passes the microphone and when the pulse reaches the area of discontinuity some of it is reflected back from right to left (r0) and some continues through the discontinuity (1-r0). The amplitude of the reflected part is calculated as follows: r0 = (A0 – A1) / (A0 + A1) which can be rearranged as: A1 = A0 × (1 – r0) / (1 + r0). Assuming the pulse travels at a constant speed (C, meters per second) in the wavetube, the distance from the microphone to area change can be computed (Kamal, 2004c).
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Figure 2.29: The amplitudes of reflected and transmitted impulses (waves) for unit pulse arriving
from left (top) and right (bottom) of a single area change (Kamal, 2004c). Kamal expanded on the principles for a tube with variable CSAs as follows:
- A tube with variable CSAs can represent the upper airway as highlighted in a schematic space-time diagram (Figure 2.30). In this diagram the first reflection has amplitude r0, and with r0 and A0 we can get A1. Thus the amplitude of the pulse transmitted through the first area change is 1 –r0. In the second area change the reflected portion becomes (1 – r0) × r1. The pulse travels back and reaches the first area change and the amplitude which reaches the microphone is (1 – r0) × r1 × (1 + r0). As r0 is known r1 can be computed given the amplitude of the pulse reaching the microphone at time 2 × 2L / C where L is the length, and with r1 and A1 known, A2 can be computed. This is more complex with increasing number of segments as there are two components of the pulse arriving at the microphone at time 2 × 3L / C. The first is the part of the original impulse which is transmitted through the first two area changes, is reflected from the third area change and is then transmitted again through the first two area changes and reaches the microphone. This component has amplitude r2 × (1 – r12) × (1 – r02). The second component is due to the part of the impulse which was transmitted through area change 1 (A1), then reflected back and forth from A2 to A1 again and then through A1 to the microphone. As this component is determined by the known r0 and r1 , this can be subtracted from the impulse and solve for r2.
- In summary, the impulse response of an upper airway with multiple area changes consists of a series of impulses arriving at times 2 × n × L / C. The impulse arriving at tn = n × 2L
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/ C consists of two components of which one is due to the original impulse transmitted through area changes A1 through n-1, reflected back at area change n and then transmitted back through area change n-1 to A1 to the microphone. The amplitude of this component is rn-1 × (1 – rn -22) × (1 – rn – 2 2) ×…× (1 – r02). The other component is caused by reverberations between area changes A1 through n-1 and this component is determined by r0 through r n-2. The major assumption is that once a reflected impulse passes the microphone it does not return which can be assured by having a wavetube which is at least as long as the farthest area change measured (Kamal, 2004c). Thus the wavetube should be at least as long as the upper airway measured.
Figure 2.30: A schematic drawing of components of reflected waves as a function of multiple
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