CCM fellow at Cleveland Clinic. Interests: Cardiopulmonary physiology, shock, mechanical ventilation, and ARDS. Music genres: Blues, Rock, and Heavy metal.

**The Pre-brief**

The equation of motion is a mathematical description of the forces that drive ventilation, and the way these forces are spent in doing so. For the uninitiated reader, this may sound a bit esoteric with little clinical relevance, but nothing could be further from the truth. This is a rather simple, yet powerful fundamental concept that is critical for understanding mechanical ventilation. It is the very foundation of respiratory mechanics. Once this concept is better appreciated, its applicability at the bedside can be found on a routine basis.

The equation of motion for inspiration can be written as follows:

*Pmus + Pvent = RxV̇ + ExV + Po + inertance [1]*

**Pmus** = Pressure generated by the inspiratory muscles

**Pvent** = Paw (airway pressure) = Pao (pressure at the airway opening)

**R** = Resistance of the respiratory system

**E** = Elastance of the respiratory system. This is the *inverse* of compliance

**V** (more accurately **ΔV**) = Change in volume from baseline (e.g. tidal volume)

**V̇** = Flow rate

**Po** = Pressure before initiation of inspiration = PEEP (let’s consider this to be zero for simplicity)

**Inertance** = Inertial components to move the gas – can be safely ignored except in HFOV[2]

Put in words, the equation states that the force utilized to drive ventilation (by the respiratory muscles and/or the ventilator) is equal to the force applied by the respiratory system in the opposite direction. In its purest sense, this equation is an application of Newton’s third law of motion. Let’s break it down in a systematic manner.

**Left side of the equation:**

This side describes the total pressure required to generate a certain level of ventilation. Note that this is a sum of the total pressure delivered by the ventilator (Pvent) and the pressure generated by the patient’s respiratory muscles (Pmus). The key point here is that the relative contribution of Pvent and Pmus varies. E.g. in a paralyzed patient, Pmus is zero. If the ventilator is disconnected, Pvent becomes zero and the required pressure has to be generated by the patient’s respiratory muscles. Most commonly, it is some combination of Pmus and Pvent.

**Right side of the equation:**

This delineates how the pressure described on the left side is utilized. As can be seen from the equation, this is also a sum of two components of pressure (or load):

*(a) Resistive (dynamic) load:*

- Mathematically defined as the product of resistance and flow.

- Before the inspired air reaches the gas-exchanging lung units, it has to pass through several generations of airways. Resistive pressure is the pressure generated due to the friction caused by any source of resistance from the airway opening up to the alveoli.

- Resistive pressure would increase with an increase in resistance (e.g. asthma) and/or flow (e.g. high flow set on the ventilator).

*(b) Elastic (static) load:*

- Mathematically defined as the product of respiratory system elastance and volume.
- Elastic load would be high in cases of increased lung and/or chest wall elastance (e.g. ARDS, IPF). Note that elastic load is also a function of volume. This is why patients with IPF (increased lung elastance) adapt their breathing pattern to rapid-shallow breathing as the small tidal volume reduces the elastic load per breath.[3]

These basic concepts of respiratory mechanics are best visualized using the single-compartment model of the respiratory system.

During inspiration, the equation of motion can be applied instantaneously as well as in steady-state conditions. The application of the equation in steady-state requires constant inspiratory flow. For example, when respiratory mechanics (elastance and resistance) are known, total pressure required to inhale a certain tidal volume can be deduced if the flow is constant (e.g. square waveform in volume control). In the absence of Pmus, this pressure would be equal to the peak inspiratory pressure.

The proportional assist ventilation (PAV) mode relies on instantaneous application of equation of motion.[4,5] Determination of resistance and elastance is prerequisite for PAV. During inspiration, instantaneous volume and flow are measured every 5 milliseconds. As elastance and resistance are known, the pressure required to generate these (Ptotal) can be calculated from the equation of motion. The proportion of Pvent/Ptotal is set by the clinician and Pvent is adjusted quasi-continuously. E.g. if ‘percent support’ is set to 50%, half the pressure for the instantaneous flow/volume will be applied by the ventilator (Pvent) and the remaining by the patient (Pmus).

**The Debrief**

While assessing a respiratory pathology causing high ventilatory load, it is important to analyze if that excess load is resistive or elastic, and if the bulk of the ventilatory work is performed by the patient or the ventilator. Practical details of objectively measuring resistance and elastance in mechanically ventilated patients will be discussed in the next post in this series. This information has important diagnostic and therapeutic implications.

**References:**

- Hess DR. Respiratory mechanics in mechanically ventilated patients. Respir Care. 2014;59(11):1773-1794. doi:10.4187/respcare.03410
- Akoumianaki E, Maggiore SM, Valenza F, et al. The application of esophageal pressure measurement in patients with respiratory failure. Am J Respir Crit Care Med. 2014;189(5):520-531. doi:10.1164/rccm.201312-2193CI
- Javaheri S, Sicilian L. Lung function, breathing pattern, and gas exchange in interstitial lung disease. Thorax. 1992;47(2):93-97. doi:10.1136/thx.47.2.93
- Akoumianaki E, Kondili E, Georgopoulos D. Proportional-assist ventilation. Eur Respir Soc Monogr. 2012;55:97–115 (New developments in Mechanical Ventilation). DOI: 10.1183/1025448x.10001911
- Xirouchaki N, Kondili E, Vaporidi K, et al. Proportional assist ventilation with load-adjustable gain factors in critically ill patients: comparison with pressure support. Intensive Care Med. 2008;34(11):2026-2034. doi:10.1007/s00134-008-1209-2

The equation is not correct in that inertance does not have units of pressure. Inertance times the rate of change of the volumetric flow rate is what has units of pressure. In physics (as best I understand it!) an equation of motion of a particle can involve position, speed and acceleration and it’s the same here, with the first term being like speed, the second like position, and the third relates to acceleration. Just as once you know the compliance of the system you have to divide volume by compliance to get the pressure needed, once you know the inertance of the system you have to multiply it by the acceleration of the system to get the pressure needed.

Thank you!