The use of stringed instruments as an analogy to explain oscillation mechanics of the developing normal and diseased respiratory system

To the Editor:

The field of oscillation mechanics has both diagnostic and therapeutic applications, e.g., respiratory oscillometry techniques for measuring respiratory impedance, and high frequency ventilation, respectively). Yet, some of the physiologic concepts can be daunting because they involve the mathematics of complex numbers with two components, “real” (resistive impedance, or simply resistance, in which pressure and flow are in phase) and “imaginary” (reactive impedance, or simply reactance, in which pressure and flow are out of phase)(1). We have found that musical analogies are useful in explaining the concepts of oscillation mechanics. The clinical study of oscillation mechanics of the respiratory system usually refers to how the system behaves during oscillatory frequencies higher than the normal physiological range, e.g., between 1 and about 40 Hz. During oscillometry, high frequency low pressure signals are introduced down the airways through a mouthpiece, superimposed on the subject’s normal tidal breathing. Respiratory system impedance is calculated from the ratio of the peak- to- peak pressure signals to the ensuing peak- to- peak flow signals (2). The oscillation mechanics of the respiratory system are affected both by normal growth and by disease.

The concept of resistance, or the pressure “cost” of flow, ΔP/ΔF, is usually more readily understood than reactance, since resistance is often measured in the pulmonary function laboratory and in the intensive care unit during mechanical ventilation. This letter will focus on explaining the concept of reactance. Reactance incorporates both the pressure cost of a volume change (elastance, or stiffness), in which flow change precedes pressure change by 90 degrees (think of blowing up a balloon), and the pressure cost of acceleration (inertance), in which pressure change precedes flow change by 90 degrees (think of pushing a mass along a table). This means that in a simple single pathway model, inertive and elastic pressures are 180 degrees out of phase with each other, and tend to cancel each other out. By convention, on a graph of reactance magnitude vs oscillatory frequency (f), elastic reactance is assigned a negative sign and inertive reactance a positive sign to reflect their opposite phases relative to each other (Figure 1). The magnitude of the reactance at a given frequency is then given by the absolute value of the graphed curve. As the figure also shows, elastic reactance is proportional to 1/f, i.e., is greatest at low frequencies because volume change is greater when there is more time for flow to enter and exit respiratory pathways. Inertive reactance is proportional to f, i.e., is greatest at high frequencies because there are rapid alternating accelerations and decelerations of flow. The total reactance, “X,” is the arithmetic sum of the “Xelastic” and the “Xinertive” lines. When the magnitude of inertive and elastic reactance are equal, they do in fact cancel each other, and then only the resistive impedance is left. This occurs at the resonant, or “natural,” frequency, Fres (Figure 1). Analogies can be drawn between the mechanical properties of the respiratory system and properties of musical instruments. Both stringed instruments and the human respiratory system have a resonant frequency. In the instrument, Fres of the strings and the instrument itself determines the pitch.

Figure 1.

Elastic reactance and inertive reactance change with oscillatory frequency. Reactances at each frequency are represented by the absolute magnitude of the curves on this graph. Elastic reactance is highest at low frequencies and inertive reactance is highest at high frequencies. By convention, inertive reactance has a positive sign and elastic reactance has a negative sign; thus tending to cancel each other, a reflection that their pressure oscillations during periodic flow are 180° out of phase (see text). When their magnitudes are equal, the sum of these two reactances is zero; this occurs at the resonant frequency, Fres.

The resonance frequency of both the respiratory system and musical instruments is determined by the balance of the inertial and elastic reactance. It is given by the equation:

$$\mathrm{Fres}\hspace{0.17em}=\hspace{0.17em}1/(2\pi \sqrt{\left[\text{CI}\right]})$$ (Equation 1)

where C is compliance (change in volume/change in pressure,ΔV/ΔP, the reciprocal of elastance), and I is inertance. Any change in the stiffness of a violin string during tuning either increases the compliance/decreases the stiffness and therefore lowers the Fres) or decreases the compliance/increases stiffness and Fres (Audio file 1, violin A string tuning, and Figure 2). Conversely, increasing the inertance by switching between strings of low mass to high mass decreases the Fres of the instrument (Audio file 2 and Figure 2). Equation (1) can be better understood by further considering Figure 1. An increase in stiffness shifts the “Xelastic” line down and to the right, with a resulting shift to a higher Fres. An increase in inertance increases the slope of the “Xinertive” line, with a resulting shift to a lower Fres. Musically this is represented by the descending slope on a musical staff, which can be thought of as a graph of frequency vs time. Similarly, as instrument size increases, the larger volume of air means that the gas compliance in the instrument increases as well, which also lowers Fres (Equation 1). This is responsible for the decrease in resonance frequency as stringed instruments increase in size.

Audio File 1.

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E-audio file 1-Violin oscillometry analogy letter tuning A.mp3. Tuning of a violin A string to a reference tone. The violinist changes the tension on the A string with pegs until the pitch matches the reference tone of about 440 Hz, in this case given by a piano key “A.”

Figure 2.

Tuning a violin string by increasing or decreasing its tension by the pegs shown increases or decreases respectively Fres (pitch) (Audio file 1). Increasing the inertance by switching between strings of low mass to high mass, e.g. a violinist descending across the strings from the E string to the G string, decreases the Fres of the instrument (Audio file 2). In this passage from a Bach sonata for solo violin, this is represented by a descending slope on a musical staff, which can be thought of as a graph of frequency vs time.

Audio File 2.

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E-audio file 2-violin oscillometry analogy letter Bach exerpt.mp3. Descending passage from Bach Sonata # 1. The violinist crosses the strings with her bow from the string of least mass, “E,” to the string of most mass, “G.” (Figure 2)

In the respiratory system, Fres also decreases with growth (Figure 3a) (3), due to the relationship between lung volume and compliance. As lung volume increases, so does compliance. Thus while tidal pleural pressure swings increase little with growth, tidal volume (about 8 cc/kg) increases linearly with body weight. Similarly, a cello has a lower resonance frequency than a violin (figure 3b)(4). Considerations of inertial forces (related to mass: F = ma) also contribute to the decrease in resonance frequency with increased size seen in both the respiratory system and musical instruments. Just as the mass of a cello is greater than that of a violin, leading to greater inertance and a lower Fres, a larger chest wall and mass of gas volume also decrease Fres of the respiratory system. To summarize, the decrease in Fres with size of both musical instruments and the human respiratory system arises from increases in both volume (increased compliance) and mass (increased inertance) (Equation 1).

Figure 3:

3A. Resonant frequency decreases with growth. For explanation see text. From: Reference (3)

3B. The decrease in resonant frequency with growth is analogous to the lower resonance frequency of a cello compared with a violin. For explanation, see text. From: Reference (4)

Fres is also altered by disease. In diseases such as asthma during bronchospasm, respiratory muscle stiffness increases and the Fres, as measured by respiratory oscillometry is high. When a bronchodilator is administered , smooth muscle stiffness decreases and the resonant frequency does as well (Figure 4a)(5). This can also be appreciated by auscultation; wheeze pitch decreases when a patient with asthma is administered a bronchodilator (6).

Figure 4:

A. Resonant frequency is altered by disease. In diseases such as asthma during bronchospasm, respiratory muscle stiffness increases and Fres is high. When a bronchodilator is administered, smooth muscle stiffness decreases and the resonant frequency does as well. The upper two curves are resistance before (solid circles) and after (open circles) bronchodilator administration, and the lower two curves are reactance before (solid circles) and after (open circles) bronchodilator administration. From: Reference (5)

B. Disorders that stiffen the airways, parenchyma or chest wall increase the resonance frequency above normal, as demonstrated by a reactance curve that is shifted down and to the right. This example is from a subject with late effects of chemo- and radiation therapy. The curves marked “1” are the predicted values. LLN: lower limit of normal. ULN: upper limit of normal.

Oscillometry measures the pressure across the entire respiratory system, from the mouth through the airway, across the pleural space and across the chest wall. Elastances in series are additive; therefore the oscillometric elastance includes airway elastance, lung parenchyma elastance and chest wall elastance. Any disorders that stiffen the airways, parenchyma or chest wall will therefore result in an increased resonance frequency as the reactance curve is shifted down and to the right. Examples of this include asthma, kyphoscoliosis and abnormalities resulting from pulmonary or chest wall fibrosis, e.g., long term effects of chemo- and radiation therapy in children undergoing hematologic stem cell transplantation (Figure 4b).

By drawing analogies between the physics of sound production of stringed instruments and the growing and/or diseased respiratory system, we have found the concepts of respiratory oscillation mechanics can be easily explained by relatively simple equations that need not invoke the mathematics of complex numbers.