## Dynamics of Respiration

### Abstract

1 Equation of Motion
1.1 Application of Equation of Motion to a Model
1.2 Determination of Resistance and Compliance
1.3 Inertance and Reactance
1.4 Impedance
1.5 Time Constant
1.6 Nonlinear Resistance and Hysteresis
1.7 Application of Equation of Motion to the Respiratory System
2 Applied Pressures
2.1 Relationship to Lung Volume
2.2 Tidal Breathing
3 Opposing Pressures
3.1 Flow Resistance
4 Maximal Expiratory Flow
4.1 Gas Compression and Measurement of Flow‐Volume Curves
5 Dynamics of Breathing
5.1 Breathing at Rest and During Exercise
5.2 Factors Affecting Respiratory Dynamics
 Figure 1. Equation of motion with mechanical analogues. A: mechanical system that has motion in 1 direction. Force (F) is applied to mass (M), causing it to be displaced along the coordinate x, sliding along a surface with a coefficient of friction (R) and distending the spring with a spring constant (K). B: three‐dimensional mechanical system. Pressure (P) causes gas to flow through a tube with resistance (R) into elastic bellows with compliance (C). Volume (V) is analogue of displacement. Figure 2. Equation of motion was used to compute pressure from volume (V), flow (), and acceleration () during sinusoidal oscillation for a sinusoidal volume change of 1 liter at a frequency of 10 cycles/min. Elastic pressure (Pel), flow‐resistive pressure (Pfr), and total pressure (P) have been computed assuming resistance = 2 cmH2O · liter−1 · s, compliance = 0.2 liter · cmH2O−1, and inertance = 0.01 cmH2O · liter · s2. Points a and b, midpoints of volume; points c and d, extremes of volume where flow = 0. A: all variables shown as function of time. B: top, flow vs. volume. Middle, pressure vs. volume. Because elastic pressure is directly determined by volume, relationship is represented by straight line (slope = 1/C). Flow‐resistive pressure‐volume relationship (Pfr) represented by loop similar to flow‐volume loop in top, because resistive pressure is the product RV. Bottom, pressure vs. flow. Because resistive pressure is directly proportional to flow, the relationship is represented by straight line (slope = R). Elastic pressure is directly proportional to volume and is represented by an ellipse similar to flow‐volume curve in top, except that flow is plotted on a horizontal rather than vertical axis and amplitude of pressure is 1/C times amplitude of volume. Figure 3. Impedance and phase. Length of vector Z represents the magnitude of impedance and θ represents the phase angle between pressure and flow. Horizontal component of impedance vector is magnitude of resistance (R). Vertical component is magnitude of reactance (X), which is vector sum of a component due to compliance (C) and a component due to inertance (I), ω, Angular velocity. Figure 4. Pressure (P)‐volume relationships. A: airway opening pressure during relaxation (Pel,rs) and during maximal static expiratory and inspiratory efforts shown as a function of lung volume (VC, vital capacity). Pressures generated by respiratory muscles (Pmus) during maximal efforts are computed from total pressure and passive pressure‐volume characteristics. Pel,rs represents static pressure‐volume relationship of respiratory system. B: pleural pressure (Ppl) plotted as function of lung volume during maximal static and dynamic inspiratory and expiratory efforts. Relationship between lung volume and Ppl when lung volume is increased voluntarily (–Pel,L) or by passive inflation (Pel,w) also shown. Functional residual capacity (FRC) is that lung volume at which inward recoil of lung (–Pel,L) is equal in magnitude and opposite in sign to outward recoil of the chest wall (Pel,w) and corresponds to volume at which passive recoil of respiratory system in Fig. 2A is zero. Horizontal separation between curves –Pel,L and Pel,w is the passive pressure‐volume relationship of respiratory system (Pel,rs), shown in Fig. 2A. Figure 5. Dynamic pressure‐volume relationship of respiratory system. Static pressure‐volume relationships of lung (–Pel,L) and of chest wall (Pel,w) plotted as in Fig. 4B. A: quiet inspiration. Heavy solid line to left of –Pel,L depicts dynamic pleural pressure‐volume relationship during spontaneous inspiration. Diagonally hatched area to left of –Pel,L, which has units of pressure × volume or work (∫PdV), equals flow‐resistive work done on lung during inspiration. Thin solid line to right of Pel,w depicts pleural pressure‐volume relationship that would occur if an identical inspiration had been produced by positive pressure applied at the airway opening. Diagonally hatched areas to right of Pel,w equals work done on chest wall to overcome its flow resistance. Horizontally hatched area between the 2 static curves is pressure required to overcome elastic recoil of lung and chest wall. B: quiet expiration. Heavy solid line to right of –Pel,L depicts dynamic pleural pressure‐volume relationship during a quiet expiration; thin solid line to left of Pel,w depicts pleural pressure‐volume relationship that would occur if an identical expiration were produced passively by reducing positive pressure at airway opening. Diagonally hatched areas between these dynamic and static curves represent the pressures required to overcome flow resistance of lung and chest wall, respectively. Horizontally hatched area between these 2 dynamic curves represents the pressure generated by inspiratory muscles required to retard expiratory flow rate. C: passive expiration. Curves for dynamic pressure‐volume relationship of lung and chest wall now coincide, indicating that there is no pressure produced by respiratory muscles. Diagonally hatched area between static and dynamic curves represents the pressure required to overcome flow resistances of lung and chest wall, and the sum of these resistances is equal to elastic recoil of respiratory system. D: active expiration. Pleural pressure‐volume relationships occurring during a modest expiratory effort. Cross‐hatched area between the 2 dynamic curves represents the pressure generated by expiratory muscle that is required to produce increased expiratory flow over that which would occur with a truly passive expiration. Figure 6. Flow‐volume‐pressure relationships during graded vital capacity (VC) maneuvers. A: flow ()‐volume relationships during a series of inspiratory and expiratory VC maneuvers with progressively increasing efforts. Expiratory flow plotted above and inspiratory flow below volume axis. B: pleural pressure (Ppl)‐volume relationships during the same maneuvers shown in Fig. 6A. Dotted lines, static pressure‐volume relationship of the lung. Over lower half of VC, large increases in expiratory effort, indicated by increasing pleural pressure, are not associated with increases in expiratory flow. Figure 7. Isovolume pressure‐flow curves. Flow (V) shown as a function of pleural pressure (Ppl). Expiratory flow plotted above horizontal axis and inspiratory flow below. Data obtained at 75%, 50%, and 25% vital capacity (VC) from graded VC efforts shown in Fig. 6 were used to construct these curves. Below 75% VC, there is a point (asterisk) beyond which further increases in Ppl are not associated with increases in . Figure 8. Flow‐volume curves of a maximal forced expiratory vital capacity (VC) maneuver. Flow () measured at mouth plotted as function of simultaneously determined expired gas volume (Vexp) and plethysmographically determined lung volume (Vpleth). At same flow, the 2 volume signals differ by amount equal to reduction of absolute lung volume due to gas compression; absolute lung volume can be detected only by plethysmograph. Figure 9. Flow‐volume‐pressure relationships during quiet and augmented breathing. A: flow ()‐volume relationships during quiet breathing (R), during progressive levels of exercise (E1 and E2), during maximal voluntary ventilation (MVV), and during maximal forced inspiratory and expiratory vital capacity (VC) efforts. B: pleural pressure (Ppl)‐volume relationships corresponding to flow‐volume curves in Fig. 9A. Curve –Pel,L is static pressure‐volume relationship of lung. Minimal pressure required to achieve maximal flow at a given lung volume (P*) is exceeded only during the MVV and maximal forced expiratory maneuvers but not during progressive levels of exercise.