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Oscillation Mechanics of the Respiratory System

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Abstract

The mechanical impedance of the respiratory system defines the pressure profile required to drive a unit of oscillatory flow into the lungs. Impedance is a function of oscillation frequency, and is measured using the forced oscillation technique. Digital signal processing methods, most notably the Fourier transform, are used to calculate impedance from measured oscillatory pressures and flows. Impedance is a complex function of frequency, having both real and imaginary parts that vary with frequency in ways that can be used empirically to distinguish normal lung function from a variety of different pathologies. The most useful diagnostic information is gained when anatomically based mathematical models are fit to measurements of impedance. The simplest such model consists of a single flow‐resistive conduit connecting to a single elastic compartment. Models of greater complexity may have two or more compartments, and provide more accurate fits to impedance measurements over a variety of different frequency ranges. The model that currently enjoys the widest application in studies of animal models of lung disease consists of a single airway serving an alveolar compartment comprising tissue with a constant‐phase impedance. This model has been shown to fit very accurately to a wide range of impedance data, yet contains only four free parameters, and as such is highly parsimonious. The measurement of impedance in human patients is also now rapidly gaining acceptance, and promises to provide a more comprehensible assessment of lung function than parameters derived from conventional spirometry. © 2011 American Physiological Society. Compr Physiol 1:1233‐1272, 2011.

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Figure 1. Figure 1.

Frequency dependence of total respiratory impedance (Zrs) of the healthy human respiratory system from low to high frequencies, indicating resonance (•) and antiresonance (°).

Figure 2. Figure 2.

Block diagram representation of the lung as a linear dynamic system (top) with three examples of possible input‐output pairs shown underneath. Input is the sum of inputs and . Likewise, output P(t) is the sum of outputs P1(t) and P2(t).

Figure 3. Figure 3.

The Argand diagram. A vector having length A and making an angle θ with the horizontal (real) axis projects a length x on the real axis and a length y on the imaginary axis. The real part of the vector is thus x and the imaginary part is iy, where i is the positive square root of −1.

Figure 4. Figure 4.

The discrete Fourier transform. Eight equally spaced data points (circles) have a discrete Fourier transform consisting of five sine waves including the zero frequency component (thin lines) whose sum (thick line) passes exactly through all eight points.

Figure 5. Figure 5.

Schematic arrangement for the measurement of input impedance Zin = Pao/ (top) and transfer impedance Ztr = Pbs/ (bottom). The diagrams in the left column show the respiratory system modeled with the DuBois’ T‐network comprising airway impedance (Zaw), tissue impedance (Zti) and the shunt impedance of alveolar gas compressibility (Zg). The schematics in the right column illustrate corresponding measurement configurations in which flow ( ) is measured with a pneumotachograph, while airway opening pressure (Pao) and body surface pressure (Pbs) are measured at the entrance to the mouth and inside a head‐out plethysmograph, respectively. Adapted from . ©Copyright 2009 J. Bates. Reprinted with the permission of Cambridge University Press.

Figure 6. Figure 6.

Schematic of the alveolar capsule technique for the explanation of the input impedance of the lungs Zin = Pao/ ; transfer airway impedances Zaw,i = (PaoPalv,i)/ , and transfer tissue impedances Zti,i = Palv,i/ .

Figure 7. Figure 7.

Schematic of the alveolar capsule oscillator for the measurement of the retrograde local input impedance of the lungs Zin,alv = (PalvPao)/ .

Figure 8. Figure 8.

Schematic of the pulmonary structures involved in the local input impedance of the regional lung (Zreg).

Figure 9. Figure 9.

Measurement of input impedance of the respiratory system as the load impedance on a wave tube. Reproduced by permission from Cauberghs and Van de Woestijne .

Figure 10. Figure 10.

Multiple‐frequency oscillatory driving pressure (P) waveforms. (A) segment of random noise (RN); (B) train of bipolar impulses (IMP); (C) pseudorandom noise (PRN); (D) optimum ventilatory waveform (OVW). The total signal energy was set equal for RN, IMP, and PRN. Note the different scales for OVW.

Figure 11. Figure 11.

Representative measurements of total Rrs and Xrs as functions of frequency in a patient before and after administration of a bronchodilator at end‐expiratory pressures of 7 and 10 cmH2O. Symbols are mean values from four measurements; bars show SD. Reproduced with permission from .

Figure 12. Figure 12.

Resistance (R) and reactance (X) of the single‐compartment linear model of the lung as a function of angular frequency (ω). Reproduced from . © Copyright 2009 J. Bates. Reprinted with the permission of Cambridge University Press.

Figure 13. Figure 13.

Resistance (R) and reactance (X) of the single‐compartment linear model with inertance in the airway. ωres is the resonant frequency. Reproduced from . © Copyright 2009 J. Bates. Reprinted with the permission of Cambridge University Press.

Figure 14. Figure 14.

Resistance (top) and elastance (bottom) versus frequency for the parallel two‐compartment model of the lung. R(0) and R(∞) are the values of resistance at frequencies of zero and infinity, respectively. E(0) and E(∞) are corresponding values of elastance. Reproduced from . © Copyright 2009 J. Bates. Adapted with the permission of Cambridge University Press.

Figure 15. Figure 15.

(A) Model of sub‐pleural lung region connected to an alveolar capsule oscillator via a short conduit with resistance Rhole and inertance Ihole. EA is the elastance of a region similar in size to a single acinus, while RA is the resistance of the terminal airway serving it. (B) Example of alveolar input impedance up to 200 Hz in a dog and the fit provided by the model. Reproduced with permission from .

Figure 16. Figure 16.

(A) Electrical circuit representation of the six‐element model. (B) The T‐network representation. Note that the capacitors are labeled as elastances, which are the inverses of the capacitances conventionally used in electric circuits. Reproduced from . © Copyright 2009 J. Bates. Reprinted with the permission of Cambridge University Press.

Figure 17. Figure 17.

High‐frequency input impedance in humans when breathing air (+) and a helium‐oxygen mixture (×). Reproduced with permission from .

Figure 18. Figure 18.

(A) Real (top) and imaginary (bottom) parts of Zrs from rats under control conditions and at increasing concentrations of intravenous methacholine infusion. The open circles are data with poor coherence due to interference by the heart beat. The lines are the fits provided by the constant phase model [Eq. ]. Reproduced with permission from . (B) Zrs obtained from a normal, anesthetized, paralyzed and tracheostomized mouse (symbols) together with the fit provided by the constant phase model in Eq. . Adapted with permission from .

Figure 19. Figure 19.

Estimates of Raw made directly by alveolar capsule versus RN estimates obtained by fitting Eq. to impedance measurements in mice. Adapted with permission from .

Figure 20. Figure 20.

Newtonian resistance (RN), as a function of body weight (BW) in different mammalian species. *Original impedance data fitted by Kaczka et al. .

Figure 21. Figure 21.

Tissue damping (G), as a function of body weight (BW) in different mammalian species. For symbol definitions and data source see Figure .

Figure 22. Figure 22.

Tissue elastance (H), as a function of body weight (BW) in different mammalian species. For symbol definitions and data source see Figure .

Figure 23. Figure 23.

Tissue hysteresivity (η), as a function of body weight (BW) in different mammalian species. For symbol definitions and data source see Figure .

Figure 24. Figure 24.

The parameters of respiratory input impedance (RN, G, and H) expressed as fractional changes above baseline (ΔRN, ΔG, and ΔH, respectively) in control and allergically inflamed mice following a 40 s challenge with an aerosol of methacholine. MCh indicates time of completion of delivery of methacholine. DI indicates time of delivery of two deep lung inflations to 25 cmH2O. Reproduced with permission from .

Figure 25. Figure 25.

Reference values of Rrs and Xrs at 6 Hz for 55 years old males and females at three representative weights. ULN= upper limit of normal. LLN = lower limit of normal. Reproduced from .

Figure 26. Figure 26.

Representative measurements of total resistance (Re(z)) and reactance (Im(z)) along the breathing cycle (V′ is flow) in a patient subjected to mechanical ventilation. Reproduced with permission from .

Figure 27. Figure 27.

Total respiratory resistance (Rrs) and reactance (Xrs) versus oscillatory frequency in patients with severe airway obstruction, measured with the head generator (open circles) and the conventional technique, with (closed circles) and without support of cheeks (closed triangles). Reproduced with permission from .

Figure 28. Figure 28.

Top: respiratory resistance (Rrs) and reactance (Xrs) in patients with obstructive disease in baseline (open circles) and after a bronchodilation test (closed circles). From Ref. . Reprinted with permission of the American Thoracic Society. © Copyright American Thoracic Society. Official Journal of the American Thoracic Society. Diane Gern, Publisher. Bottom: changes in different Zrs values after bronchodilation in COPD patients with or without expiratory flow limitation at baseline (black and white bars, respectively). All indices are mean values computed during the inspiratory (insp) or expiratory (exp) phase, or during the total breathing cycle (tot). 5 to 19 indicates mean difference at 5 and 19 Hz. Reproduced with permission from .

Figure 29. Figure 29.

Respiratory resistance (Rrs) and reactance (Xrs) when a patient with SAHS was subjected to different levels of constant negative pressure in the upper airways. Symbols correspond to 0, −5, −10, and −15 cmH2O. Reproduced with permission from .

Figure 30. Figure 30.

Breathing flow (V′), esophageal pressure (Pes), and respiratory resistance (Rrs) measured in a representative patient with SAHS at different levels of CPAP. From Ref. . Reprinted with permission of the American Thoracic Society. © Copyright American Thoracic Society. Official Journal of the American Thoracic Society. Diane Gern, Publisher.



Figure 1.

Frequency dependence of total respiratory impedance (Zrs) of the healthy human respiratory system from low to high frequencies, indicating resonance (•) and antiresonance (°).



Figure 2.

Block diagram representation of the lung as a linear dynamic system (top) with three examples of possible input‐output pairs shown underneath. Input is the sum of inputs and . Likewise, output P(t) is the sum of outputs P1(t) and P2(t).



Figure 3.

The Argand diagram. A vector having length A and making an angle θ with the horizontal (real) axis projects a length x on the real axis and a length y on the imaginary axis. The real part of the vector is thus x and the imaginary part is iy, where i is the positive square root of −1.



Figure 4.

The discrete Fourier transform. Eight equally spaced data points (circles) have a discrete Fourier transform consisting of five sine waves including the zero frequency component (thin lines) whose sum (thick line) passes exactly through all eight points.



Figure 5.

Schematic arrangement for the measurement of input impedance Zin = Pao/ (top) and transfer impedance Ztr = Pbs/ (bottom). The diagrams in the left column show the respiratory system modeled with the DuBois’ T‐network comprising airway impedance (Zaw), tissue impedance (Zti) and the shunt impedance of alveolar gas compressibility (Zg). The schematics in the right column illustrate corresponding measurement configurations in which flow ( ) is measured with a pneumotachograph, while airway opening pressure (Pao) and body surface pressure (Pbs) are measured at the entrance to the mouth and inside a head‐out plethysmograph, respectively. Adapted from . ©Copyright 2009 J. Bates. Reprinted with the permission of Cambridge University Press.



Figure 6.

Schematic of the alveolar capsule technique for the explanation of the input impedance of the lungs Zin = Pao/ ; transfer airway impedances Zaw,i = (PaoPalv,i)/ , and transfer tissue impedances Zti,i = Palv,i/ .



Figure 7.

Schematic of the alveolar capsule oscillator for the measurement of the retrograde local input impedance of the lungs Zin,alv = (PalvPao)/ .



Figure 8.

Schematic of the pulmonary structures involved in the local input impedance of the regional lung (Zreg).



Figure 9.

Measurement of input impedance of the respiratory system as the load impedance on a wave tube. Reproduced by permission from Cauberghs and Van de Woestijne .



Figure 10.

Multiple‐frequency oscillatory driving pressure (P) waveforms. (A) segment of random noise (RN); (B) train of bipolar impulses (IMP); (C) pseudorandom noise (PRN); (D) optimum ventilatory waveform (OVW). The total signal energy was set equal for RN, IMP, and PRN. Note the different scales for OVW.



Figure 11.

Representative measurements of total Rrs and Xrs as functions of frequency in a patient before and after administration of a bronchodilator at end‐expiratory pressures of 7 and 10 cmH2O. Symbols are mean values from four measurements; bars show SD. Reproduced with permission from .



Figure 12.

Resistance (R) and reactance (X) of the single‐compartment linear model of the lung as a function of angular frequency (ω). Reproduced from . © Copyright 2009 J. Bates. Reprinted with the permission of Cambridge University Press.



Figure 13.

Resistance (R) and reactance (X) of the single‐compartment linear model with inertance in the airway. ωres is the resonant frequency. Reproduced from . © Copyright 2009 J. Bates. Reprinted with the permission of Cambridge University Press.



Figure 14.

Resistance (top) and elastance (bottom) versus frequency for the parallel two‐compartment model of the lung. R(0) and R(∞) are the values of resistance at frequencies of zero and infinity, respectively. E(0) and E(∞) are corresponding values of elastance. Reproduced from . © Copyright 2009 J. Bates. Adapted with the permission of Cambridge University Press.



Figure 15.

(A) Model of sub‐pleural lung region connected to an alveolar capsule oscillator via a short conduit with resistance Rhole and inertance Ihole. EA is the elastance of a region similar in size to a single acinus, while RA is the resistance of the terminal airway serving it. (B) Example of alveolar input impedance up to 200 Hz in a dog and the fit provided by the model. Reproduced with permission from .



Figure 16.

(A) Electrical circuit representation of the six‐element model. (B) The T‐network representation. Note that the capacitors are labeled as elastances, which are the inverses of the capacitances conventionally used in electric circuits. Reproduced from . © Copyright 2009 J. Bates. Reprinted with the permission of Cambridge University Press.



Figure 17.

High‐frequency input impedance in humans when breathing air (+) and a helium‐oxygen mixture (×). Reproduced with permission from .



Figure 18.

(A) Real (top) and imaginary (bottom) parts of Zrs from rats under control conditions and at increasing concentrations of intravenous methacholine infusion. The open circles are data with poor coherence due to interference by the heart beat. The lines are the fits provided by the constant phase model [Eq. ]. Reproduced with permission from . (B) Zrs obtained from a normal, anesthetized, paralyzed and tracheostomized mouse (symbols) together with the fit provided by the constant phase model in Eq. . Adapted with permission from .



Figure 19.

Estimates of Raw made directly by alveolar capsule versus RN estimates obtained by fitting Eq. to impedance measurements in mice. Adapted with permission from .



Figure 20.

Newtonian resistance (RN), as a function of body weight (BW) in different mammalian species. *Original impedance data fitted by Kaczka et al. .



Figure 21.

Tissue damping (G), as a function of body weight (BW) in different mammalian species. For symbol definitions and data source see Figure .



Figure 22.

Tissue elastance (H), as a function of body weight (BW) in different mammalian species. For symbol definitions and data source see Figure .



Figure 23.

Tissue hysteresivity (η), as a function of body weight (BW) in different mammalian species. For symbol definitions and data source see Figure .



Figure 24.

The parameters of respiratory input impedance (RN, G, and H) expressed as fractional changes above baseline (ΔRN, ΔG, and ΔH, respectively) in control and allergically inflamed mice following a 40 s challenge with an aerosol of methacholine. MCh indicates time of completion of delivery of methacholine. DI indicates time of delivery of two deep lung inflations to 25 cmH2O. Reproduced with permission from .



Figure 25.

Reference values of Rrs and Xrs at 6 Hz for 55 years old males and females at three representative weights. ULN= upper limit of normal. LLN = lower limit of normal. Reproduced from .



Figure 26.

Representative measurements of total resistance (Re(z)) and reactance (Im(z)) along the breathing cycle (V′ is flow) in a patient subjected to mechanical ventilation. Reproduced with permission from .



Figure 27.

Total respiratory resistance (Rrs) and reactance (Xrs) versus oscillatory frequency in patients with severe airway obstruction, measured with the head generator (open circles) and the conventional technique, with (closed circles) and without support of cheeks (closed triangles). Reproduced with permission from .



Figure 28.

Top: respiratory resistance (Rrs) and reactance (Xrs) in patients with obstructive disease in baseline (open circles) and after a bronchodilation test (closed circles). From Ref. . Reprinted with permission of the American Thoracic Society. © Copyright American Thoracic Society. Official Journal of the American Thoracic Society. Diane Gern, Publisher. Bottom: changes in different Zrs values after bronchodilation in COPD patients with or without expiratory flow limitation at baseline (black and white bars, respectively). All indices are mean values computed during the inspiratory (insp) or expiratory (exp) phase, or during the total breathing cycle (tot). 5 to 19 indicates mean difference at 5 and 19 Hz. Reproduced with permission from .



Figure 29.

Respiratory resistance (Rrs) and reactance (Xrs) when a patient with SAHS was subjected to different levels of constant negative pressure in the upper airways. Symbols correspond to 0, −5, −10, and −15 cmH2O. Reproduced with permission from .



Figure 30.

Breathing flow (V′), esophageal pressure (Pes), and respiratory resistance (Rrs) measured in a representative patient with SAHS at different levels of CPAP. From Ref. . Reprinted with permission of the American Thoracic Society. © Copyright American Thoracic Society. Official Journal of the American Thoracic Society. Diane Gern, Publisher.

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Jason H.T. Bates, Charles G. Irvin, Ramon Farré, Zoltán Hantos. Oscillation Mechanics of the Respiratory System. Compr Physiol 2011, 1: 1233-1272. doi: 10.1002/cphy.c100058