## Oscillation Mechanics of the Respiratory System

### Abstract

1 Modeling the Respiratory System as a Linear System
1.1 Models
1.2 Elemental Equations
1.3 Sinusoidal Forcing and Complex Impedance
1.4 Continuity and Compatibility Conditions
1.5 General Formulation of System Models
1.6 System Functions
1.7 Equivalent Circuits
1.8 Distributed‐Parameter Models
2 History
3 Experimental Methods
3.1 Equipment
3.2 Inputs and Data Processing
3.3 Upper Airway Variability and Shunt
4 Frequency Responses Below 100 HZ
4.1 Total Respiratory System
4.2 Input Impedance Forcing at the Airway Opening
4.3 Transfer Impedance Forcing at the Chest
4.4 Transfer Impedance Forcing at the Mouth
4.5 Pressure and Flow Transfer Functions
4.6 Lung Impedance
4.7 Chest Wall Impedance
4.8 Airway Impedance
5 Frequency Responses Above 100 HZ
5.1 Wave Propagation in the Airways
5.2 Input Impedance Forcing at the Airway Opening
5.3 Pressure Transfer Functions
5.4 Airway Area by Acoustic Reflections
6 Clinical Applications
6.1 Pulmonary Function in Children
6.2 Obstructive Lung Diseases
6.3 Restrictive Lung Diseases
6.4 Miscellaneous Lung Diseases
 Figure 1. Rohrer relation (Eq. ) taken as an example of nonlinear pressure‐flow relations linearized for small departures about any given bias flow rate (). Oscillatory resistance for slow oscillations is the local tangent slope of the pressure‐flow curve. For long straight tubes, slope at is the Poiseuille flow resistance. Figure 2. A: sinusoidal strip‐chart recording of flow (bottom) imagined to be the projection of a rotating ray of length (top). B: as in the case of flow, pressure may be thought of as the projection of a rotating ray of length Pao(ω). Phase angle between pressure and flow is Φrs. C: impedance may be decomposed into real and imaginary parts or into magnitude and phase. D: Pao(t) vs. (t) yields a Lissajous ellipse, from which impedance components may be deduced. Figure 3. A: simple series model of the respiratory system. B: impedance vector at several frequencies. Components of impedance vector: real (resistive) component (C); imaginary (reactive) component (D); magnitude (E); phase angle (F). Figure 4. Model of T network accounting for airway impedance (Zaw), tissue impedance (Zt), and gas compression (Zg). Forcing may be at mouth, body surface, or both. Either generator may be short‐circuited (no pressure difference) or open‐circuited (no flow) to deal with the case of a single input. Note the sign convention for flows: continuity applied at node PA (alveolar pressure) requires . Raw, airway resistance; Iaw, airway inertance; Rt, tissue resistance; It, tissue inertance; Ct, tissue compliance; Cg, gas compressibility; PB, barometric pressure; , alveolar gas flow. Figure 5. Equivalent circuits for one‐port systems (A) and two‐port systems (B). Figure 6. An airway of length L may be divided into infinitesimal segments of length dx. The equivalent circuit for such a segment incorporates airway inertance (Iaw) and viscous resistance (Raw) as series elements. Shunt elements can be subdivided into the gas compressibility pathway, consisting of gas compressibility (Cg) and gas thermal conductance (Gt), and the wall distension pathway, consisting of airway wall elastance (Eaww), inertance (Iaww), and resistance (Raww). Figure 7. Pressure and flow generators: reciprocating pump to apply pressure variations at the chest (A); loudspeaker to apply pressure variations at the mouth (B); loudspeakers arranged in series and in parallel to obtain larger pressures and volume changes (C). Figure 8. Model of T network including the airways (complex impedance Zaw), tissues (lung plus chest wall, complex impedance Zt), and compressible alveolar gas (complex impedance Zg). Functional arrangement of the compartments depends on where the pressure variations are applied. Equations on right express relationships between measured impedance and that of the compartments when the output variable is airway opening (A), body surface (B), and gas compression flow (C).Adapted from Peslin et al. Figure 9. A: modulus (|Zrs|), phase angle (θrs), and real (real [Zrs] or equivalent resistance) and imaginary (imag[Zrs] or equivalent reactance) parts of the relationship between transrespiratory pressure and flow when pressure is varied at the mouth, . Data from 9 healthy humans. [Upper part from Michaelson et al. ; lower part from E. D. Michaelson, unpublished observations.] B: modulus (|Zrs|), phase angle (θrs), and real (real [Zrs]) and imaginary (imag[Zrs]) parts of the total respiratory input impedance observed at 5 lung volumes in a healthy human.Adapted from Michaelson et al. Figure 10. Model proposed by Michaelson et al. to interpret input impedance measurements. The lung, distal to central airways [upper airway resistance (Ruaw) and inertance (Iuaw)], is represented by 2 resistance‐inertance‐compliance pathways arranged in parallel; also featured are the properties of extrathoracic airway walls [mouth resistance (Rm), inertance (Im), and compliance (Cm)] and chest wall compliance (Cw).From Michaelson et al. , by copyright permission of The American Society for Clinical Investigation Figure 11. Transfer impedance forcing at the chest, [Pbs/(–)]Pao=0. Modulus (|Z|), phase angle (θ), and real (real[Z]) and imaginary (imag[Z]) parts of the relationship between transrespiratory pressure and flow when pressure is varied around the chest. Data from 5 healthy humans. Note that – is the flow out of the mouth according to sign convention of Figs. and .Upper part adapted from Peslin et al. Figure 12. Real (real[Z]) and imaginary (imag[Z]) parts of total respiratory input impedance (continuous lines), lung impedance (dashed line), and an approximation to chest wall impedance (dotted lines) at 3 levels of vital capacity: 70% (open circles), 40% (closed circles), and 25% (triangles). Average values in 15 healthy subjects ±1 SD.From Nagels et al. Figure 13. Measured and theoretically determined phase velocity for an excised canine trachea with static distending pressure of zero.From Guelke and Bunn Figure 14. Measured input impedance components of human at tracheostomy opening.From Ishizaka et al. Figure 15. Measured and theoretically determined transfer functions of alveolar pressure to airway opening pressure in excised canine lung.From Fredberg Figure 16. Pressure distribution along the tracheobronchial tree varying with oscillatory frequency. Glottis corresponds to x = 0; open circles correspond to locations of successive bifurcations. As a result of standing waves, the pressure does not fall monotonically from glottis to alveolus except at low frequenciesFrom Fredberg and Moore Figure 17. Airway area by acoustic reflection (solid lines) yields good agreement with radiographic determinations (filled circles) of tracheal area (top). G, glottis; S, sternal notch; C, carina. Result is accurate with light gases (HeO2, top and middle) but erroneous when air is employed (bottom). Dashed lines indicate ±1 SD.From Fredberg et al.