Respiratory Mechanics and the Equation of Motion

Educational reference — a way to reason about the physics underneath every assisted breath. It is not a bedside directive or a substitute for a device’s operator’s manual or for clinical judgment.

Why mechanics underlie everything

A ventilator has exactly one job: to move gas in and out of a patient’s lungs against the mechanical properties of that patient’s respiratory system. Every other thing it does — every mode name, every waveform on the screen, every alarm — is a consequence of that one job carried out against those mechanics. This is why the whole edifice of mode classification and waveform reading in this corpus is built on a single physical statement, the equation of motion for the respiratory system. Chatburn’s mode-classification system names it as its “underlying theoretical framework” [Chatburn 2007]; the 2014 taxonomy uses it to specify how a ventilator can assist a breath at all [Chatburn 2014]; the 2022 and 2026 waveform papers make it the thing you are literally looking at when you read a tracing [Mireles-Cabodevila 2022; Chatburn 2026 · Waveforms]; and a 2025 practical guide frames bedside ventilator adjustment around it directly [Mireles-Cabodevila 2025].

Hold one distinction throughout. The physics the equation describes is settled science [established]. Building mode classification and waveform interpretation on top of that physics is a well-argued proposal that the authors advocate [proposed] — the same “settled physics, proposed vocabulary” split that runs through The Taxonomy of Ventilator Modes. This chapter is about the settled part: learn it once here, and the taxonomy and the tracings both become readable.

One compartment, three variables

flowchart LR
  V["Ventilator<br/>Pvent"] --> S(("Respiratory<br/>system"))
  M["Patient muscles<br/>Pmus"] --> S
  S --> EL["Elastic load<br/>E · V"]
  S --> RES["Resistive load<br/>R · V̇"]

Figure 1 — the equation of motion as a balance: the ventilator (Pvent) and the patient’s muscles (Pmus) together overcome the elastic and resistive loads. Adapted from [Chatburn 2026 · Waveforms].

The starting move is a deliberate simplification. The equation of motion reduces the entire respiratory system — dozens of airway generations, hundreds of millions of alveoli, the chest wall, the diaphragm — to a single-compartment model: one resistance in series with one elastance, described by just three variables — pressure, volume, and flow [Chatburn 2007; Chatburn 2026 · Waveforms] [established]. Flow is the rate of change of volume (its first derivative in time), and volume is the running integral of flow, so the two are tied together and are never independent [Chatburn 2007].

Two lumped parameters characterize the patient:

• Resistance (R) — how hard it is to drive flow through the airways (natural and artificial). Its reciprocal is conductance.
• Elastance (E) — how hard it is to push volume into the lungs and chest wall, i.e. how stiff the system is. Its reciprocal, which clinicians more often quote, is compliance (C = 1/E) [Chatburn 2007; Chatburn 2026 · Waveforms].

The single-compartment picture is an idealization — real lungs are many compartments with different regional mechanics — but it is the right level of abstraction for reasoning at the bedside, and the paper is explicit that only a conceptual grasp of it is needed to use it well [Chatburn 2007].

The equation, in its several forms

The equation of motion is one statement written several ways. Each form makes a different idea obvious, so it is worth meeting all of them.

Conceptual (Newton’s-third-law) form. At every instant, the pressure applied to the system equals the pressure the system pushes back with — action equals reaction. Written at the airway opening:

Paw(t) = PE(t) + PR(t) + PEEPset + PEEPauto

where Paw is the pressure at the airway opening, PE is the pressure stored in elastic recoil, PR is the pressure spent driving flow through resistance, and the two PEEP terms are the set and intrinsic (auto) positive end-expiratory pressures. The identical relationship can be derived either from Kirchhoff’s Voltage Law applied to an equivalent electrical circuit (pressure ≈ voltage, flow ≈ current, so the airway “circuit” is a resistor in series with a capacitor) or from Newton’s third law directly [Chatburn 2026 · Waveforms] [established]. The electrical analogy is not decoration: it is how Mead brought the equation into ventilation in the first place (below), and it is why series and parallel mechanics behave as they do — resistances in series add, compliances in parallel add [Chatburn 2026 · Waveforms].

Elastance–resistance form. Substituting the physical definitions of the elastic and resistive pressures gives the form most people mean by “the equation of motion”:

Paw(t) = E · V(t) + R · V̇(t) + PEEPtot

with V volume, V̇ (= dV/dt) flow, and PEEPtot = PEEPset + PEEPauto. Mathematically this is a first-order, linear differential equation with constant coefficients — of the generic shape f(x) = a·y + b·(dy/dx), where the two coefficients are elastance and resistance [Chatburn 2026 · Waveforms] [established]. A fuller version in the 2007 glossary carries an inertance term, ΔPTR + Pmus = E·V + R·V̇ + I·V̈, but inertance (I·V̈, the pressure to accelerate the gas and tissue) is small and is routinely dropped [Chatburn 2007].

Compliance form. Because clinicians report compliance rather than elastance, the same equation is often written with C = 1/E:

Paw(t) = V(t)/C + R · V̇(t) + PEEPtot

[Chatburn 2026 · Waveforms] [established].

The working waveform-reading form. For reading tracings, two adjustments are made. First, the ventilator’s own contribution is separated out and called Pvent. Second, a muscle-pressure term Pmus is added as a second forcing function — because an awake patient’s inspiratory muscles push in parallel with the machine. Dropping the explicit “(t)”:

Pvent + Pmus = E·V + R·V̇ + PEEPauto

This is the form to memorize [Mireles-Cabodevila 2022; Chatburn 2026 · Waveforms] [established]. It says something simple and powerful: the pressure the ventilator supplies, plus whatever the patient’s muscles supply, must together overcome the elastic load and the resistive load (and any trapped-gas pressure). The 2022 companion states the same relation (Pmus + Pvent = E·V + R·V̇) and draws the consequence that makes waveform reading possible: every ventilator that plots pressure, volume, and flow is graphing the equation of motion. At any instant the height of the pressure curve equals the scaled height of the volume curve plus the scaled height of the flow curve — Pvent = PE + PR [Mireles-Cabodevila 2022] [established].

There is also a crucial degree-of-freedom consequence, and it is the hinge between this chapter and the next. During inspiration, if the ventilator predetermines any one of pressure, volume, or flow, the other two are no longer free — they fall out of the mechanics [Chatburn 2014]. Because volume and flow are locked together, the practical choice collapses to two: control pressure or control volume. That single fact is the root of the control variable, the first axis of the mode taxonomy.

Elastic load versus resistive load

The working equation splits the patient’s opposition to a breath into two physically distinct loads, and separating them is the entire aim of reading a tracing:

• Elastic load = E·V — the pressure needed to expand the lungs and chest wall to a given volume. It is zero at the start of inspiration and greatest at end-inspiration, when volume is largest [Mireles-Cabodevila 2022; Chatburn 2026 · Waveforms] [established].
• Resistive load = R·V̇ — the pressure needed to drive gas through the natural and artificial airways. It tracks flow, so it is largest where flow is largest [Mireles-Cabodevila 2022; Chatburn 2026 · Waveforms] [established].

A third contributor, auto-PEEP (PEEPauto), behaves as an additional elastic load, equal to E × the trapped volume [Chatburn 2026 · Waveforms].

The equation also delivers the single most useful reading rule in the corpus: look at the waveform opposite the control variable. In volume control the ventilator fixes volume and flow, so load and effort have nowhere to appear but the pressure curve; in pressure control the ventilator fixes pressure, so load and effort appear in the flow and volume curves [Mireles-Cabodevila 2022] [established]. That rule, and the eight reference points it is applied to, are the subject of Reading Ventilator Waveforms.

Recovering R and C from the machine

Because the equation of motion has only two unknown parameters, a ventilator can solve for them from data it already collects. Two routes [Chatburn 2026 · Waveforms] [established]:

• Static measurement. Under controlled, passive conditions you read the parameters off single pressure/volume/flow changes: resistance from a pressure drop divided by flow (R = ΔPTR / flow), compliance from a pressure change divided by the volume that produced it (C = ΔPTR / volume). At the bedside this is the familiar inspiratory-hold maneuver, where the peak-to-plateau pressure gap reflects R·V̇ and the plateau-above-PEEP reflects V/C.
• Dynamic linear regression. Because the equation is linear in E and R, a ventilator can fit a stream of pressure/volume/flow samples to P = E·V + R·V̇ and recover E (hence C) and R simultaneously, breath by breath, with no hold maneuver at all.

Both are just the equation of motion turned inside out — solving for the parameters instead of predicting the waveforms.

Figure 2 — the equation of motion drawn onto a single volume-control breath. With flow held constant, airway pressure is a resistive step (R·V̇, the jump above PEEP at onset) plus an elastic ramp (V/C) up to the peak (PIP). Add an end-inspiratory hold and flow → 0, so the resistive term vanishes and pressure falls to the plateau (Pplat = PEEP + V/C): the peak-to-plateau gap reads resistance, the plateau-above-PEEP reads elastance. Original diagram, computed from the equation of motion (C 50 mL/cmH₂O, R 10 cmH₂O/L/s); the hold maneuver follows [Chatburn 2026 · Waveforms].

PEEP, auto-PEEP, and the expiratory equation

Expiration is passive in most modes: the ventilator stops pushing and the elastic energy stored during inspiration drives gas back out. To describe it, assume Paw drops immediately to PEEPset and subtract PEEPset from both sides of the compliance form. The E·V and R·V̇ terms rearrange to:

V/C + PEEPauto = −R · V̇

The negative sign is the physics of exhalation: the pressure driving expiratory flow is the elastic pressure stored with the tidal volume, plus any trapped-gas pressure, and it drives flow in the negative (outward) direction [Chatburn 2026 · Waveforms] [established].

Auto-PEEP is what remains when a breath is not fully exhaled before the next one begins. Alveolar pressure decays exponentially during expiration, PA = (VT/C)·e^(−t/RC); the total PEEP is whatever that curve has reached when expiratory time runs out, and PEEPauto = PEEPtot − PEEPset [Chatburn 2026 · Waveforms] [established]. Auto-PEEP has a direct clinical corollary that comes straight out of the working equation: for inspiratory flow to begin at all, (Pvent + Pmus) − PEEPauto must be greater than zero [Chatburn 2026 · Waveforms]. A patient whose muscular effort cannot clear the trapped-gas threshold simply fails to trigger the machine — the mechanical root of a failed trigger, taken up in Reading Ventilator Waveforms.

The time constant

One derived quantity ties resistance and compliance together and governs how fast anything happens in the lung: the time constant,

τ = R × C

measured in units of time (seconds) [Mireles-Cabodevila 2022; Chatburn 2026 · Waveforms] [established]. It is the natural clock of the single-compartment system. Over one time constant, an exponential process completes about 63% of its change; setting inspiratory time equal to RC makes the decay exponent e^(−1) ≈ 0.37, which is the 63% figure seen from the other side (in expiration, 63% gone leaves 37% remaining) [Chatburn 2026 · Waveforms].

The practical rules of thumb follow directly:

• Inspiration or expiration is ~95% complete after 3 time constants, and effectively finished (under ~1% remaining) by about 5τ [Chatburn 2026 · Waveforms; Mireles-Cabodevila 2022].
• Setting inspiratory or expiratory time to at least 3τ is usually adequate; “adequate” ventilation broadly takes 3–5 time constants [Chatburn 2026 · Waveforms; Mireles-Cabodevila 2022].

And the time constant takes on characteristic bedside values [Mireles-Cabodevila 2022] [established]:

• A normal intubated, passive respiratory system on a heated humidifier has τ ≈ 0.5 s, so flow reaches zero in roughly 2.5 s (about 5τ). If flow returns to zero much sooner than ~1.5 s, compliance is very likely reduced.
• ARDS shortens τ (low compliance), so flow returns to baseline quickly.
• COPD lengthens τ (high resistance), so flow decays slowly and gas trapping / auto-PEEP is the standing danger.

The same τ explains a shape you see on every screen: because a pressure-controlled step produces exponential flow and volume, a long τ means a slow, drawn-out flow decay, and a short τ a fast one [Mireles-Cabodevila 2022]. When effort is present, approximating Pmus as a sinusoid makes the volume and flow solutions sinusoids too, each phase-shifted from Pmus by an angle set by τ and breathing frequency — with the general ordering that flow leads pressure and pressure leads volume (which is why waveforms are conventionally stacked pressure → flow → volume, and why volume, arriving last, carries the least new information) [Chatburn 2026 · Waveforms].

A warning: “pressure” is not one thing

Everything above hangs on the word pressure, and the 2026 review pauses to insist that the word is ambiguous — a point worth carrying into any measurement [Chatburn 2026 · Waveforms] [established]. “Pressure” may be:

• a point pressure — absolute (vs vacuum), gauge (vs atmosphere), or relative to a clinical reference such as PEEP;
• a difference between two points in space — trans-respiratory pressure PTR = Paw − PBS (airway opening minus body surface), or trans-pulmonary pressure PTP = Paw − Pes (airway opening minus esophageal, isolating the lung from the chest wall);
• a difference between two points in time — driving pressure, P0.1, or an occlusion pressure (Pocc);
• a difference in both space and time — for example plateau pressure, Pplat.

The organizing idea is that a structure of the respiratory system is defined as whatever lies between the two points of a pressure difference [Chatburn 2026 · Waveforms]. This is why transpulmonary pressure “sees” the lung while transrespiratory pressure “sees” lung-plus-chest-wall — and why choosing the wrong reference silently changes what a number means.

From pressure to energy: a bridge

Integrate the pressure terms of the equation of motion over the volume they act on and mechanics becomes energetics. The work of a breath is pressure integrated over volume (∫P dV); because the equation carries both a Pvent term and a Pmus term, that work partitions cleanly between ventilator and patient — the basis of the “work of breathing” and “work shifting” ideas the 2022 paper uses to describe patient effort [Chatburn 2014; Mireles-Cabodevila 2022] [established].

The natural next step — multiply energy per breath by respiratory frequency to get power, and ask how much mechanical power the lung can absorb before it is injured — leads into the mechanical-power / ventilator-induced-lung-injury literature. That framework reaches beyond this single-school corpus and enters only through Chatburn’s collaborations, so its energetics are developed (with that provenance flagged) in Bedside Monitoring of Respiratory Mechanics rather than here. This chapter’s job is only to show the door: work and power are the equation of motion, integrated.

Where it came from

The equation of motion is old, and knowing its lineage keeps its authority straight [Chatburn 2026 · Waveforms]:

• Otis, Fenn, and Rahn (1950) — the first English-language use, applied to unassisted normal breathing.
• Jere Mead (1961) — popularized it for mechanical ventilation using electrical analogs and circuit-analysis equations (the Kirchhoff route above).
• Marini and Crooke — expanded it into a general equation of motion applicable to any mode, first for the passive respiratory system and then for active ventilation, where patient effort interacts with ventilator output.

The taxonomy and the waveform method are recent proposals; the equation they rest on is three-quarters of a century old and thoroughly established.

How this chapter connects

• The equation’s one-degree-of-freedom consequence — control pressure or control volume — is the first axis of The Taxonomy of Ventilator Modes.
• “Read the waveform opposite the control variable,” the eight reference points, and the failed-trigger / auto-PEEP logic are worked out in Reading Ventilator Waveforms.
• Turning the pressure terms into energy — work of breathing, and the out-of-school mechanical-power extension — is carried into Bedside Monitoring of Respiratory Mechanics.

Open items and honest caveats

• The explicit solutions are [open]. The step-by-step pressure/volume/flow solutions for volume control and pressure control, and the sinusoidal unassisted-breathing solutions, live in the 2026 review’s Supplementary Figures S1–S6, which are not in the frozen main-text artifact — their exact algebra is held [open] (supplementary data needed) [Chatburn 2026 · Waveforms].
• The 2025 adjustment rules are [open]. The 2025 guide that frames bedside ventilator adjustment around the equation of motion is held metadata-only; its specific adjustment rules are [open] until full text is obtained, and only its title-level thesis is cited here [Mireles-Cabodevila 2025].

Sources

Every claim above is drawn from the frozen, human-reviewed primary literature in the machine layer (raw/literature/, one artifact per paper). The short keys used inline resolve to:

• [Chatburn 2007] — Chatburn RL. Classification of ventilator modes: update and proposal for implementation. Respir Care 2007;52(3):301–323. PMID 17328828.
• [Chatburn 2014] — Chatburn RL, El-Khatib M, Mireles-Cabodevila E. A taxonomy for mechanical ventilation: 10 fundamental maxims. Respir Care 2014;59(11):1747–1763. PMID 25118309.
• [Mireles-Cabodevila 2022] — Mireles-Cabodevila E, Siuba MT, Chatburn RL. A Taxonomy for Patient-Ventilator Interactions and a Method to Read Ventilator Waveforms. Respir Care 2022;67(1):129–148. PMID 34470804.
• [Mireles-Cabodevila 2025] — Mireles-Cabodevila E, Chatburn RL. The equation of motion: a brief guide to ventilator adjustment. Eur Heart J Acute Cardiovasc Care 2025;14(8):494–496. PMID 40631448. (Metadata-only artifact — cited only for its title-level thesis; specific adjustment rules held [open].)
• [Chatburn 2026 · Waveforms] — Chatburn RL. How to interpret ventilator waveforms using the taxonomy for modes of mechanical ventilation. Respir Care 2026;71(6):566–587. PMID 41631602.

This chapter is the human-layer synthesis of the machine-layer record it derives from — Equation of Motion for the Respiratory System — and the Lit — … source summaries that record cites (see the derived_from field).