The Equation of Motion, Compliance, Resistance, & Time Constants
The Mathematical Law Governing Every Mechanical Breath
Why This Matters in the ICU
Every breath delivered by any ventilator obeys the Equation of Motion. Total pressure is the sum of elastic pressure (volume / compliance) plus resistive pressure (flow × resistance) plus baseline PEEP.
Learning Objectives
- State and apply the Equation of Motion for the Respiratory System.
- Differentiate Static Respiratory Compliance (Cstat) from Dynamic Compliance (Cdyn).
- Define Airway Resistance (Raw) and factors altering resistance in intubated patients.
- Calculate the Respiratory Time Constant (τ = R × C) and apply it to exhalation mechanics.
The Equation of Motion
The total pressure required to deliver a breath into the respiratory system (Pvent + Pmus) must overcome elastic recoil of the lungs/chest wall, frictional resistance of airways and ETT, and baseline PEEP:
Ptotal = Pelastic + Presistive + Pbaseline = (VT / Crs) + (Flow × Raw) + PEEP
- Pelastic = VT / Crs: Governed by tidal volume and respiratory system compliance. Increases linearly as volume enters the lung.
- Presistive = Flow × Raw: Governed by gas flow rate and airway/endotracheal tube resistance. Present ONLY while gas is moving.
- Pbaseline = Total PEEP: Sum of set extrinsic PEEP and intrinsic Auto-PEEP.
Compliance & Resistance Defined
Compliance is the change in volume per unit change in pressure (distensibility). Resistance is the pressure drop per unit flow (friction).
| Metric | Formula | Normal Intubated Adult Value | Common Clinical Derangements |
|---|---|---|---|
| Static Compliance (Cstat) | VT / (Pplat - PEEP) | 50–80 mL/cmH2O | Decreased in ARDS (<30), pulmonary edema, atelectasis, pneumothorax, abdominal distension. |
| Airway Resistance (Raw) | (Ppeak - Pplat) / Flow (L/s) | 5–10 cmH2O/L/s | Elevated in bronchospasm (>15–25), small/kinked ETT, secretions, mucus plugs. |
The Respiratory Time Constant (τ = R × C)
One time constant (τ, in seconds) is the product of airway resistance and compliance. It represents the time required for the lung to fill or empty by 63.2% of a tidal volume change.
Time Constant (τ in seconds) = Raw (cmH2O/L/s) × Cstat (L/cmH2O)
- 1 τ = 63% lung volume change (inflow or outflow).
- 2 τ = 86% volume change.
- 3 τ = 95% volume change.
- 4–5 τ = >98–99% complete lung emptying.
- Clinical Rule: If expiratory time (Te) is shorter than 3–4 time constants, passive exhalation is interrupted, trapping gas (Auto-PEEP / Dynamic Hyperinflation).
Physiology Integration: Respiratory System Compliance
Compliance reflects total respiratory system distensibility (lungs + chest wall in series: 1/Crs = 1/CL + 1/Cw). Stiffening either the lung or chest wall drops Cstat and increases elastic airway pressures.
Interactive Waveform Analysis
Volume-Controlled Ventilation (VC-AC)
Constant square inspiratory flow with a progressive linear rise in airway pressure up to Ppeak (25 cmH₂O), relaxing to Pplat (18 cmH₂O) during an end-inspiratory pause.
The ventilator delivers a preset tidal volume at a constant flow. Pressure rises as work is done against airway resistance and lung/chest wall elastance.
Standard baseline. Perform an inspiratory pause to check Pplat (≤ 30 cmH₂O) and calculate driving pressure (Pplat - PEEP).
💡 Bedside Clinical Pearls
- In a patient with severe COPD or asthma, resistance is high (e.g., 20 cmH2O/L/s) and compliance is normal or high (e.g., 0.08 L/cmH2O), producing a long time constant of 1.6 seconds. Allowing 4 time constants for complete exhalation requires at least 6.4 seconds of expiratory time!
- In stiff ARDS lungs, compliance is very low (e.g., 0.02 L/cmH2O) and resistance is near normal, producing a short time constant (~0.2s). The lung fills and empties rapidly.