# Common respiratory equations

This chapter is most relevant to Section F9(iii) from the 2023 CICM Primary Syllabus, which expects the exam candidates to  "understand respiratory equations that describe ventilation, perfusion, blood flow and respiratory mechanics".  Of course, no specific question has ever tested a syllabus item as nebulous as this, because how would it. "Describe and explain the common respiratory equations" would be cruel even for CICM. However, in the post apocalyptic world of the Second Part exam, it is implied that candidates already know these equations, as ABG questions occasionally call for their use. Thus, somewhere in this revision resource, some sort of short list of "respiratory equations" is expected.

What would go on that list? If one digs deep enough - the answer could be "everything". Bertrand Maury famously has a whole book where the respiratory system is reduced to maths (The respiratory system in equations, 2013). Clearly, the revision of a CICM First Part candidate should stop well short of this 275-page monstrosity. However, one is somewhat adrift if one looks for guidance, even among the usual suspects (Part One, cicmwrecks, icuprimaryprep). Ultimately, the author had no choice but to trawl the past papers, looking for SAQs which required even the slightest bit of calculation, and extrapolate speculatively from the syllabus document.  There is no guarantee that any of this is relevant; the emptor is left to caveat. The only essential element here, from the exam perspective, is the alveolar gas equation, which gains importance in the Fellowship exam.

## Gas laws

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).

• 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 = Hv × M

Where
• P is pressure
• M is the molar concentration of gas
• Hv is Henry's Proportionality Constant

## Relationships between common respiratory parameters

Flow =  volume / time

Volume = flow × time

Pressure =  flow × resistance

Resistance = change in pressure / flow

Compliance = volume / change in pressure

Work of breathing =  pressure × volume

## Lung volumes and capacities

### The Bohr equation for measuring dead space:

VD/VT = (FACO- FECO2) / FACO2

Where:

• VD = dead space volume
• VT = tidal volume
• FECO2 = fraction of expired CO2
• FACO2 = fraction of alveolar CO2

## Diffusion of gases

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

### Alveolar gas equation

PAO2 = (FiO2 × (Patm - PH2O) - (PaCO2 / RQ)

Where

• PAO2 = Partial pressure of alveolar oxygen
• FiO2 = fraction of inspired oxygen
• Patm Atmospheric pressure (usually 760 mmHg)
• PH2O = 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 carriage

### Oxygen content of whole blood

(sO2 × ceHb × BO) + (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
• BO 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 = (CcO2 - CaO2) / (CcO2 - CvO2)

where
•     Qs/Qt = shunt fraction (shunt flow divided by total cardiac output)
•     CcO2 = pulmonary end-capillary O2 content, same as alveolar O2 content
•     CaO2 = arterial O2 content
•     CvO2 = mixed venous O2 content

## References

Maury, Bertrand. The respiratory system in equations. Springer Science & Business Media, 2013.