Shunt, Dead space, and gasses - oh my!

vivianimbriotis | Aug. 13, 2025, 7:04 p.m.

The Riley model is a lie for children, in which the lung is conceptualized as consisting of three populations of alveoli:

1. Functional alveoli with perfect V/Q matching where perfect gas exchange takes place, such that the pulmonary venous drainage from them has equal oxygen and CO2 tension with the alveolar gas
2. Alveolar dead space, alveoli that are ventilated but NOT perfused, where no gas exchange takes place
3. Intrapulmonary shunt, alveoli that are perfused but NOT ventilated, such that the venous drainage has equal oxygen and CO2 tension with mixed venous blood

In this model, we can calculate two related terms:

1. The dead space fraction, the proportion of each tidal volume which is exhaled having never participated in gas exchange, which is alveolar dead space, anatomical dead space, and also mechanical dead space (say, from a ventilator circuit).
2. The total shunt fraction, the proportion of cardiac output that is shunted from the right ventricle to the left atrium without being oxygenated, which is from bronchial veins, Thesbian veins, intracardiac shunt (in Eisenmenger syndrome), and intrapulmonary shunt (atelectasis, pulmonary oedema, et al).

NB in the real world, the alveolar "dead space" actually includes some poorly or imperfectly perfused alveoli (i.e. with a V/Q > 1) and the "shunt fraction" includes poorly or imperfectly ventilated alveoli (V/Q < 1).

Nonetheless, sticking with the Riley model, how would we go about finding these quantities?

Dead space estimation

Here are the steps

1. You take a breath
2. The gas in your ideal alveoli equilibrates with the blood in the pulmonary capillaries, becoming loaded with CO2; the dead space gas still contains no CO2
3. You exhale, mixing together these two gases

The key insight is that ALL of the CO2 in the mixed gas came from the alveolar gas, and none came from the dead space. That is, the CO2 content in mixed expiratory gas equals the CO2 content in the alveolar gas, i.e. the tidal volume less the dead space. That is:

$${CO_2}_{breath} = P_ACO_2 \cdot \overbrace{(V_T - V_D)}^{\text{alveolar gas}} = P_{\bar{E}}CO_2 \cdot V_T$$

$$V_T - V_D = \frac{P_{\bar{E}}CO_2 \cdot V_T}{P_ACO_2}$$

$$1 - \frac{V_D}{V_T} = \frac{P_{\bar{E}}CO_2}{P_ACO_2}$$

$$\frac{V_D}{V_T} = 1 - \frac{P_{\bar{E}}CO_2}{P_ACO_2}$$

Or, if you prefer to think in terms of alveolar ventilation and minute volume, we can do this:

$$\frac{V_D \cdot RR}{V_T \cdot RR} = 1 - \frac{P_{\bar{E}}CO_2}{P_ACO_2}$$

$$\frac{1 - V_A}{MV} = 1 - \frac{P_{\bar{E}}CO_2}{P_ACO_2}$$

$$\frac{V_A}{MV} = \frac{P_{\bar{E}}CO_2}{P_ACO_2}$$

Now we can't usually measure \(P_ACO_2\), but we can measure the arterial CO2 tension \(P_aCO_2\), and unless there's a large shunt, those are close enough together for government work.

$$\frac{V_A}{MV} = \frac{P_{\bar{E}}CO_2}{P_aCO_2}$$

Shunt fraction estimation

Here are the steps:

1. Mixed venous blood surges into the right atrium, then divides into shunt blood and pulmonary blood.
2. The pulmonary blood equilibrates in the pulmonary capillaries.
3. The shunt blood, meanwhile, does not change its gas tension.
4. The pulmonary blood and the shunt blood then mix in the left atrium. Their volumes and oxygen content sum.
5. You, a sleep-deprived resident who has eaten two biscuits in the last 24 hours, take a blood gas from the radial artery, consisting of these mixed pulmonary and shunt blood samples.

Let's consider the oxygen content in these samples. Let \(C_aO_2\) be the oxygen content of the arterial sample, \(C_vO_2\) be the oxygen content of mixed venous (or shunted) blood, and \(C_pO_2\) be the oxygen content of blood in the pulmonary capillaries. Let Qp, Qs, and CO be the pulmonary capillary flow, shunt flow, and cardiac output respectively.

We wish to find

$$Fs = \frac{Q_s}{CO}$$

We have two relationships, refering to total flow and total oxygen flux:

$$CO = Q_p + Q_s $$

$$C_aO_2 \cdot CO = C_pO_2 \cdot Q_p + C_vO_2 \cdot Q_s$$

So diving by CO we have

$$C_aO_2 = \frac{C_pO_2 \cdot Q_p}{CO} + C_vO_2 \cdot F_s$$

And then using the total flow equation

$$C_aO_2 = C_pO_2 \frac{CO - Q_s}{CO} + C_vO_2 \cdot F_s$$

$$C_aO_2 = C_pO_2 \cdot (1 - F_s) + C_vO_2 \cdot F_s$$

$$C_aO_2 = C_pO_2 - C_pO_2 \cdot F_s + C_vO_2 \cdot F_s$$

$$(C_aO_2 - C_pO_2) = (C_vO_2- C_pO_2) \cdot F_s$$

$$F_s = \frac{C_aO_2 - C_pO_2}{C_vO_2- C_pO_2}$$

or equivalently

$$F_s = \frac{C_pO_2 - C_aO_2}{C_pO_2 - C_vO_2}$$

We can find some intuitive results from this relationship. If the shunt fraction is zero, then the numerator is zero, so the arterial blood is the same as ideal pulmonary capillary blood. If the shunt fraction is one, then:

$$F_s = 1 = \frac{C_pO_2 - C_aO_2}{C_pO_2 - C_vO_2}$$

$$C_pO_2 - C_aO_2 = C_pO_2 - C_vO_2$$

$$C_aO_2 = C_vO_2$$

...the arterial blood looks just like mixed venous blood, and no gas exchange has happened.

Okay, but how on earth do we figure out these oxygen contents? These refer to the total amount of oxygen in the blood. Oxygen can either be carried as a dissolved gas (in which case we can measure its partial pressure) or bound to haemoglobin. The maximum amount of oxygen carried by a gram of haemoglobin is around 3.9 ml/g. So:

$$C_{O_2} = (3.9 \cdot ceHb \cdot sO_2) + 0.03 \cdot P_aO_2$$

Where \(ceHb\) is the effective haemoglobin concerntration in g/L (effective meaning less dyshaemoglobinaemias like methaemoglobin or carboxyhaemoglobin). We assume that the sats in the pulmonary capillary blood are 100%.

Usually the term for dissolved gas is neglibile, except in weird situations (100% FiO2 with anaemia; or hyperbaric therapy), so we can limit ourselves to

$$C_{O_2} = 3.9 \cdot ceHb \cdot sO_2$$

This good approximation reduces our shunt equation to

$$F_s = \frac{1 - S_aO_2}{1 - S_{\bar{v}}O_2}$$

So for example, with mixed venous sats of 70%, and arterial saturations of 88%, we find:

$$F_s = \frac{1 - 0.88}{1 - 0.7}$$

$$F_s = 40\%$$