Common respiratory equations

Boyle's Law:

• For a fixed mass of gas at constant temperature, the pressure (P) and volume (V) are inversely proportional, such that P ×V = k, where k is a constant.

Charles' Law:

• The volume occupied by a fixed mass of gas at constant pressure is directly proportional to its absolute temperature (V/T = k).

Third gas law (Gay-Lussac's Law):

• The pressure of a fixed mass of gas at constant volume is directly proportional to its absolute temperature (P/T = k).

Avogadro's Law:

• Equal volumes of gases at the same temperature and pressure contain the same number of molecules (6.023 × 10²³, Avogadro’s number).

Universal (Ideal) Gas Law:

• The state of a fixed mass of gas is determined by its pressure, volume and temperature(PV = nRT)

Henry's Law

• The amount of a given gas dissolved in a given liquid is directly proportional to the partial pressure of the gas in contact with the liquid:
  
  P = H_v × M
  
  Where
  • P is pressure
  • M is the molar concentration of gas
  • H_v is Henry's Proportionality Constant

Flow =  volume / time

Volume = flow × time

Pressure =  flow × resistance

Resistance = change in pressure / flow

Compliance = volume / change in pressure

Work of breathing =  pressure × volume

The Bohr equation for measuring dead space:

V_D/V_T = (F_ACO₂ - F_ECO₂) / F_ACO₂

Where:

• V_D = dead space volume
• V_T = tidal volume
• F_ECO₂ = fraction of expired CO₂
• F_ACO₂ = fraction of alveolar CO₂

Diffusing capacity = Net rate of gas transfer / Partial pressure gradient

Alveolar gas equation

PAO₂ = (FiO₂ × (P_atm - P_H2O) - (PaCO₂ / RQ)

Where 

• PAO₂ = Partial pressure of alveolar oxygen
• FiO₂ = fraction of inspired oxygen
• P_atm = Atmospheric pressure (usually 760 mmHg)
• P_H2O = partial pressure of water vapour at the alveolus (usually 47 mmHg)
• PaCO₂  = Partial pressure of arterial carbon dioxide
• RQ = respiratory quotient, usually 0.8

Oxygen content of whole blood

 

=  (sO₂ × ceHb × BO₂ ) + (PaO₂ × 0.03), where:

• ceHb = the effective haemoglobin concentration
  • i.e. concentration of haemoglobin species capable of carrying and releasing oxygen appropriately
• PaO₂ = the partial pressure of oxygen in arterial gas
• 0.03 = the content, in ml/L/mmHg, of dissolved oxygen in blood
  • Henry's law states that the amount of dissolved gas in a liquid is proportional to its partial pressure above the liquid;
  • Ergo the amount of oxygen dissolved in is proportional to its partial pressure, e.g for a PaO₂ of 100 mmHg the oxygen content is 0.03 × 100 = 3ml/L
• BO₂ =  the maximum amount of Hb-bound O₂ per unit volume of blood
  • normally 1.39 of dry Hb, or closer to 1.30 in "real" conditions
• sO₂ = oxygen saturation:
  • determined by the sigmoid oxygen-haemoglobin dissociation curve
  • Sigmoid shape of the curve comes from the positive cooperativityof oxygen binding
  • Once an oxygen molecule is bound to it, the oxygenated subunit increases the oxygen affinity of the three remaining subunits
  • This is because of a conformational change produced by each subunit binding oxygen, which mediates the transition from the T (tense, deoxygenated) state to the R (relaxed, oxygenated) state

The shunt equation

 

Qs/Qt = (Cc_O2 - Ca_O2) / (Cc_O2 - Cv_O2)

where

•     Qs/Qt = shunt fraction (shunt flow divided by total cardiac output)
•     Cc_O2 = pulmonary end-capillary O₂ content, same as alveolar O₂ content
•     Ca_O2 = arterial O₂ content
•     Cv_O2 = mixed venous O₂ content