Physics of respiration

The physics of respiration encompasses the physical principles and laws that govern gas exchange and breathing mechanics in living organisms. Respiration is fundamentally a biophysical process governed by classical gas laws, fluid dynamics, thermodynamics, and mechanics.^[1][2]

Physics of Respiration
Major structures of the respiratory system involved in physical gas exchange processes
Alveolar structure showing gas exchange surfaces governed by diffusion laws
Details
Synonyms	Respiratory physics, Biophysics of breathing
System	Respiratory system
Artery	Pulmonary artery
Vein	Pulmonary vein
Nerve	Phrenic nerve, Vagus nerve
Lymph	Bronchopulmonary lymph nodes
Anatomical terminology [edit on Wikidata]

Overview

The respiratory system functions to exchange oxygen (O₂) and carbon dioxide (CO₂) between the atmosphere and blood. This exchange depends entirely on pressure gradients, molecular diffusion, and mechanical forces.^[1] The physics of respiration can be analyzed through several fundamental domains: gas laws governing partial pressures, mechanics of breathing involving pressure-volume relationships, diffusion kinetics across membranes, fluid dynamics in airways, and surface tension effects in alveoli.

Gas Laws in Respiration

Dalton's Law of Partial Pressures

Main article: Dalton's law

Dalton's law states that in a mixture of non-reacting gases, the total pressure equals the sum of the partial pressures of individual gases:^[1]

${\displaystyle P_{total}=P_{1}+P_{2}+P_{3}+...+P_{n}}$

In respiratory physiology, this law explains how each gas in air (nitrogen, oxygen, carbon dioxide, water vapor) contributes to total atmospheric pressure. At sea level, atmospheric pressure is approximately 760 mmHg. The partial pressure of oxygen (P_O₂) is:^[1]

${\displaystyle P_{O_{2}}=F_{O_{2}}\times (P_{atm}-P_{H_{2}O})}$

where F_O₂ is the fractional concentration of oxygen (0.21), P_atm is atmospheric pressure, and P_H₂O is water vapor pressure. This yields an inspired P_O₂ of approximately 150 mmHg under standard conditions.

The alveolar gas equation applies Dalton's law to calculate alveolar oxygen tension:^[1]

${\displaystyle P_{A}O_{2}=P_{IO_{2}}-{\frac {P_{A}CO_{2}}{R}}}$

where P_AO₂ is alveolar oxygen tension, P_IO₂ is inspired oxygen tension, P_ACO₂ is alveolar carbon dioxide tension, and R is the respiratory quotient (typically 0.8).

Henry's Law

Main article: Henry's law

Henry's law describes gas solubility in liquids, stating that the amount of dissolved gas is proportional to its partial pressure:^[1][2]

${\displaystyle C=k_{H}\times P}$

where C is the concentration of dissolved gas, k_H is Henry's constant (solubility coefficient), and P is the partial pressure. Different gases have different solubility coefficients in blood. Carbon dioxide is approximately 20 times more soluble in blood than oxygen, which has important physiological implications for gas transport.^[1]

Boyle's Law

Main article: Boyle's law

Boyle's law states that for a fixed amount of gas at constant temperature, pressure and volume are inversely related:^[1]

${\displaystyle P_{1}V_{1}=P_{2}V_{2}}$

This fundamental relationship explains the mechanics of breathing. During inhalation, the thoracic cavity expands, increasing lung volume and decreasing intrapulmonary pressure below atmospheric pressure, causing air to flow inward. During exhalation, the thoracic cavity decreases in volume, increasing pressure and forcing air outward.^[1][2]

Mechanics of Breathing

Pressure-Volume Relationships

The mechanics of breathing involve coordinated changes in pressure and volume within the respiratory system. The key pressures involved are:^[1]

• Atmospheric pressure (P_atm): pressure of ambient air (760 mmHg at sea level)
• Alveolar pressure (P_alv): pressure within the alveoli
• Intrapleural pressure (P_pl): pressure in the pleural space
• Transpulmonary pressure (P_tp): difference between alveolar and intrapleural pressure

The transpulmonary pressure represents the distending pressure across the lung:^[1]

${\displaystyle P_{tp}=P_{alv}-P_{pl}}$

This pressure determines lung volume and is critical for understanding lung mechanics. The relationship between transpulmonary pressure and lung volume defines pulmonary compliance.

Compliance and Elastance

Compliance (C) is defined as the change in volume per unit change in pressure:^[1][2]

${\displaystyle C={\frac {\Delta V}{\Delta P}}}$

Normal lung compliance is approximately 200 mL/cmH₂O. Diseases such as pulmonary fibrosis decrease compliance (stiff lungs), while emphysema increases compliance (loss of elastic recoil).

Elastance (E) is the reciprocal of compliance:^[1]

${\displaystyle E={\frac {1}{C}}={\frac {\Delta P}{\Delta V}}}$

The lung and chest wall each have their own compliance values, and the total respiratory system compliance is determined by:^[1]

${\displaystyle {\frac {1}{C_{total}}}={\frac {1}{C_{lung}}}+{\frac {1}{C_{chestwall}}}}$

Work of Breathing

The work of breathing represents the energy required to overcome elastic and resistive forces during ventilation. It can be calculated as:^[1]

${\displaystyle W=\int P\,dV}$

This work has three components:^[1][2]

1. Elastic work: energy to overcome elastic recoil of lungs and chest wall
2. Resistive work: energy to overcome airway and tissue resistance
3. Inertial work: energy to accelerate gases and tissues (usually negligible)

Gas Diffusion

Fick's Laws of Diffusion

Main article: Fick's laws of diffusion

Gas exchange across the alveolar-capillary membrane is governed by Fick's first law of diffusion:^[1][2]

${\displaystyle V_{gas}={\frac {A\times D\times (P_{1}-P_{2})}{T}}}$

where:

• V_gas is the rate of gas transfer
• A is the surface area available for diffusion (~70 m² in human lungs)
• D is the diffusion coefficient (depends on gas properties and tissue)
• P₁ - P₂ is the partial pressure difference across the membrane
• T is the thickness of the membrane (~0.5 μm)

This equation can be simplified to:^[1]

${\displaystyle V_{gas}=D_{L}\times (P_{1}-P_{2})}$

where D_L is the diffusing capacity of the lung, which incorporates surface area, membrane thickness, and diffusion properties.

Diffusion Coefficients

The diffusion coefficient depends on:^[1]

1. Molecular weight: inversely proportional to the square root of molecular weight
2. Solubility: directly proportional to gas solubility in tissue
3. Temperature: increases with temperature

Carbon dioxide diffuses approximately 20 times faster than oxygen across the alveolar-capillary membrane due to its much higher solubility, despite its slightly larger molecular weight.^[1]

Diffusion Limitation vs. Perfusion Limitation

Gas transfer can be limited by either diffusion or perfusion:^[1][2]

• Diffusion-limited: Transfer rate limited by membrane properties (occurs with CO, O₂ during exercise or disease)
• Perfusion-limited: Transfer rate limited by blood flow (occurs with N₂O, O₂ at rest)

Airway Resistance and Fluid Dynamics

Poiseuille's Law

Main article: Hagen–Poiseuille equation

For laminar flow through cylindrical tubes, the Hagen-Poiseuille equation describes the relationship between flow rate and pressure:^[1][2]

${\displaystyle Q={\frac {\pi r^{4}\Delta P}{8\mu L}}}$

where:

• Q is flow rate
• r is radius of the airway
• ΔP is pressure difference
• μ is viscosity of the gas
• L is length of the airway

This equation demonstrates that flow is proportional to the fourth power of the radius, making airway radius the most critical determinant of resistance.

Airway resistance (R) is defined as:^[1]

${\displaystyle R={\frac {\Delta P}{Q}}}$

From Poiseuille's law:^[1]

${\displaystyle R={\frac {8\mu L}{\pi r^{4}}}}$

Normal total airway resistance is approximately 1-2 cmH₂O/L/s. Disease states such as asthma and chronic obstructive pulmonary disease significantly increase airway resistance.

Laminar and Turbulent Flow

Flow patterns in airways depend on the Reynolds number (Re):^[1][2]

${\displaystyle Re={\frac {\rho vd}{\mu }}}$

where:

• ρ is gas density
• v is velocity
• d is diameter
• μ is viscosity

• Laminar flow (Re < 2000): smooth, organized flow following Poiseuille's law
• Turbulent flow (Re > 4000): chaotic, disorganized flow with higher resistance
• Transitional flow (2000 < Re < 4000): mixture of laminar and turbulent

In turbulent flow, resistance increases proportionally to the square of flow rate, rather than being independent of flow as in laminar conditions.^[1]

Surface Tension and the Laplace Law

Young-Laplace Equation

Main article: Young–Laplace equation

Surface tension at the air-liquid interface in alveoli creates a pressure according to the Young-Laplace equation:^[1][2]

${\displaystyle P={\frac {2T}{r}}}$

where:

• P is pressure inside the alveolus
• T is surface tension
• r is radius of the alveolus

This equation predicts that smaller alveoli would have higher pressures and would tend to collapse into larger alveoli (a phenomenon that would lead to complete lung collapse). This is prevented by pulmonary surfactant.^[1]

Pulmonary Surfactant

Main article: Pulmonary surfactant

Pulmonary surfactant is a complex mixture of lipids and proteins secreted by type II pneumocytes. It reduces surface tension from approximately 70 dyn/cm (pure water) to as low as 5-10 dyn/cm.^[1][2]

Surfactant has two critical physical properties:^[1]

1. Reduces surface tension: Decreases the work of breathing
2. Provides stability: Surface tension varies with area, decreasing more in smaller alveoli, which prevents collapse

Without surfactant, the pressure required to inflate the lungs would be prohibitively high, and alveolar instability would result in atelectasis. Neonatal respiratory distress syndrome occurs when premature infants lack adequate surfactant production.^[1]

Ventilation-Perfusion Relationships

V/Q Ratio

The ventilation/perfusion ratio (V̇/Q̇) describes the relationship between alveolar ventilation and pulmonary blood flow. Ideal gas exchange requires matching of ventilation to perfusion.^[1][2]

For the whole lung:^[1]

• Normal alveolar ventilation (V̇_A) ≈ 4-5 L/min
• Normal cardiac output (Q̇) ≈ 5 L/min
• Normal V̇/Q̇ ratio ≈ 0.8-1.0

Different V̇/Q̇ ratios produce different gas exchange patterns:^[1]

• V̇/Q̇ = 0 (shunt): perfusion without ventilation → blood remains unoxygenated
• V̇/Q̇ = ∞ (dead space): ventilation without perfusion → no gas exchange
• V̇/Q̇ = 1 (ideal): optimal matching of ventilation and perfusion

Regional Differences

Due to gravity, both ventilation and perfusion are greater at the lung bases than the apices when upright. However, perfusion increases more than ventilation from apex to base, creating a gradient of V̇/Q̇ ratios:^[1]

• Apex: V̇/Q̇ ≈ 3.0 (high V̇/Q̇)
• Base: V̇/Q̇ ≈ 0.6 (low V̇/Q̇)

This regional variation is explained by the hydrostatic pressure gradient in pulmonary blood flow and the mechanical properties of the lung.^[1]

Gas Transport in Blood

Oxygen Transport

Oxygen is transported in blood in two forms:^[1][2]

1. Dissolved in plasma (~2%): Governed by Henry's law
2. Bound to hemoglobin (~98%): Governed by the oxygen-hemoglobin dissociation curve

The oxygen content of blood is:^[1]

${\displaystyle C_{O_{2}}=(1.34\times [Hb]\times S_{O_{2}})+(0.003\times P_{O_{2}})}$

where:

• 1.34 mL O₂/g Hb is the oxygen-carrying capacity
• [Hb] is hemoglobin concentration (g/dL)
• S_O₂ is hemoglobin saturation (%)
• 0.003 mL O₂/dL/mmHg is the solubility coefficient
• P_O₂ is partial pressure of oxygen (mmHg)

Carbon Dioxide Transport

Carbon dioxide is transported in three forms:^[1][2]

1. Dissolved in plasma (~7%): Governed by Henry's law
2. Bicarbonate (~70%): Formed by carbonic anhydrase reaction
3. Carbamino compounds (~23%): Bound to hemoglobin and plasma proteins

The Haldane effect describes how deoxygenated hemoglobin can carry more CO₂ than oxygenated hemoglobin, facilitating CO₂ transport from tissues to lungs.^[1]

Control of Breathing

Chemical Control

Respiratory control involves chemoreceptors that sense changes in blood gases and pH:^[1][2]

• Central chemoreceptors: Located in the medulla, respond primarily to pH changes in cerebrospinal fluid (reflecting CO₂ levels)
• Peripheral chemoreceptors: Located in carotid and aortic bodies, respond to O₂, CO₂, and pH

The ventilatory response to CO₂ is nearly linear:^[1]

${\displaystyle V_{E}=S\times (P_{CO_{2}}-B)}$

where V_E is minute ventilation, S is the slope (sensitivity), P_CO₂ is arterial CO₂, and B is the intercept.

Mechanical Control

Mechanoreceptors in the lungs and airways provide feedback:^[1]

• Stretch receptors: Detect lung inflation (Hering-Breuer reflex)
• Irritant receptors: Respond to noxious stimuli
• J-receptors: Respond to pulmonary congestion

Clinical Applications

Respiratory Mechanics in Disease

Understanding the physics of respiration is essential for managing respiratory diseases:^[1][2]

• Obstructive lung disease (asthma, COPD): Increased airway resistance due to narrowed airways (Poiseuille's law)
• Restrictive lung disease (pulmonary fibrosis): Decreased lung compliance due to stiff lungs
• Acute respiratory distress syndrome (ARDS): Decreased compliance and impaired gas exchange due to alveolar damage and surfactant dysfunction

Mechanical Ventilation

Mechanical ventilation applies physical principles to support gas exchange:^[1]

The equation of motion for the respiratory system describes the relationship between applied pressure and resulting volume:^[1]

${\displaystyle P=E\times V+R\times {\dot {V}}+P_{0}}$

where:

• P is applied pressure
• E is elastance
• V is volume
• R is resistance
• V̇ is flow rate
• P₀ is baseline pressure

This equation guides ventilator settings and monitoring.

Altitude Physiology

At high altitude, atmospheric pressure decreases, reducing inspired P_O₂:^[1]

${\displaystyle P_{IO_{2}}=F_{IO_{2}}\times (P_{barometric}-P_{H_{2}O})}$

At the summit of Mount Everest (8848 m), barometric pressure is approximately 253 mmHg, resulting in an inspired P_O₂ of only ~43 mmHg, barely sufficient to sustain life.^[1]

Historical Development

The understanding of respiratory physics developed through several key discoveries:^[1]

• 1660s: Robert Boyle formulated Boyle's law
• 1801: John Dalton proposed Dalton's law of partial pressures
• 1803: William Henry described Henry's law of gas solubility
• 1840s: Jean Poiseuille studied fluid flow in tubes
• 1906: August Krogh studied gas diffusion in lungs
• 1920s: Discovery of pulmonary surfactant
• 1940s-1960s: Development of modern respiratory physiology concepts

See Also

• Respiratory physiology
• Pulmonary function testing
• Gas exchange
• Spirometry
• Alveolar gas equation
• Oxygen-hemoglobin dissociation curve
• Ventilator-associated lung injury
• Diving physics
• Aviation medicine