Medical Breathing

This page summarizes the mathematical models used by ACE Medical Breathing. For the numerical methods used to apply the differential equations, see General Considerations.

Respiratory System

The model uses two gas reservoirs: alveoli and blood, connected as follows:

Surroundings \overset{Vent.}{⇄} Alveoli \overset{Perf.}{⇄} Blood

Ventilation exchanges gas with the surroundings and perfusion drives exchange of gas between alveoli and blood. $O_{2}$ in blood gets continuously converted to $C O_{2}$ . Gas exchanges are simplified as continuous flows. No simulation of actual breathing cycles is done.

Lung Volumes

Tidal and capacity volumes of the lungs are scaled by pneumothorax (PTX):

V_{T} = V_{T, 0} \cdot (1 - ξ_{PTX}) V_{C} = V_{C, 0} \cdot (1 - ξ_{PTX})

$V_{T} [ml]$ : Tidal volume.
$V_{C} [ml]$ : Lung capacity.
$V_{T, 0} = 500 \cdot 2 / 3 ml$ : Default tidal volume.
$V_{C, 0} = 6000 \cdot 2 / 3 ml$ : Default lung capacity.
$ξ_{PTX} \in [0, 0.75]$ : PTX progress, expressed as the fraction of lung volume lost.

The factors of $2 / 3$ account for dead space.

Respiratory Rate

Respiratory rate $R R$ increases to compensate for the loss of lung volume:

R R = \min (R R_{\max}, \frac{R R_{0}}{1 - ξ_{PTX}})

$R R [m i n^{- 1}]$ : Respiratory rate in breaths per minute.
$R R_{0} [m i n^{- 1}]$ : Default respiratory rate (m_fDefaultRespiratoryRateBPM).
$R R_{\max} [m i n^{- 1}]$ : Maximum respiratory rate (m_fMaxRespiratoryRateBPM).

Ventilation

The volume-fraction-based ventilation rate is given below:

k_{V} = \frac{R R}{60 s / m i n} \cdot \frac{V_{T}}{V_{C}}

$k_{V} [s^{- 1}]$ : Fractional ventilation rate.

Ventilation equilibrates partial gas pressures between alveoli and surroundings:

Φ_{V, O_{2}} = k_{V} \cdot (p_{O_{2}}^{\max} - p_{O_{2}}) Φ_{V, C O_{2}} = k_{V} \cdot p_{C O_{2}}

$p_{O_{2}}$ , $p_{C O_{2}} [kPa]$ : Alveolar $O_{2}$ and $C O_{2}$ partial pressures.
$p_{O_{2}}^{\max} [kPa]$ : Supplied $O_{2}$ pressure, normally 19.925 kPa.
$Φ_{V, O_{2}} [k P a / s]$ : Ventilation rate of increase in alveolar $O_{2}$ partial pressure.
$Φ_{V, C O_{2}} [k P a / s]$ : Ventilation rate of decrease in alveolar $C O_{2}$ partial pressure.

The $C O_{2}$ partial pressure of the surroundings is currently approximated as 0 kPa. An equipped $O_{2}$ mask raises $p_{O_{2}}^{\max}$ to 95 kPa and enforces $k_{V} \geq 0.015 s^{- 1}$ .

Cellular Respiration

$O_{2}$ is consumed by cellular respiration and $C O_{2}$ is produced in return. The ratio between production and consumption depends on the chemical reactions that take place (i.e. aerobic vs anaerobic conditions) and the ratio is given by the respiratory exchange rate $R E R$ . The values are adjusted through linear interpolation between hypoxic and normal $S p O_{2}$ values:

φ_{C R, O_{2}} = lerp (φ_{C R, O_{2}}^{hypo}, φ_{C R, O_{2}}^{0}, λ) R E R = lerp (R E R_{hypo}, R E R_{0}, λ)

λ = clamp (\frac{S p O_{2} - S p O_{2, hypo}}{S p O_{2,0} - S p O_{2, hypo}}, 0, 1)

φ_{C R, C O_{2}} = φ_{C R, O_{2}} \cdot R E R

$φ_{C R, O_{2}} [m m o l / s]$ : $O_{2}$ consumption rate.
$φ_{C R, O_{2}}^{0}$ , $φ_{C R, O_{2}}^{hypo} [m m o l / s]$ : Normal and hypoxic consumption rates (m_fDefaultOxygenConsumptionMMOLPS and m_fHypoxicOxygenConsumptionMMOLPS).
$R E R$ , $R E R_{0}$ , $R E R_{hypo} [\cdot]$ : Current, normal, and hypoxic respiratory exchange ratios (m_fDefaultRER and m_fHypoxicRER).
$S p O_{2} [%]$ : Peripheral blood $O_{2}$ saturation.
$S p O_{2,0}$ , $S p O_{2, hypo} [%]$ : Normal and hypoxic saturation levels (m_fDefaultSpO2 and m_fHypoxicMetabolismSpO2).
$λ [\cdot]$ : Interpolation fraction.
$φ_{C R, C O_{2}} [m m o l / s]$ : $C O_{2}$ production rate.

Note that both consumption and production rates are defined as positive values. The sign convention will be important later for the correct signs in the ODEs.

Reduced saturation lowers consumption, which simulates the body prioritizing vital organs.

Perfusion: $O_{2}$ Exchange

Perfusion is driven by cardiac output and $O_{2}$ binding at the alveolar pressure follows a Hill function:

Q = \frac{C O}{60 s / m i n} \cdot {(1 - ξ_{PTX})}^{2 / 3} S_{alv} = \frac{{(p_{O_{2}} / K_{D})}^{γ}}{1 + {(p_{O_{2}} / K_{D})}^{γ}}

$Q [m l / s]$ : Lung perfusion.
$C O [m l / m i n]$ : Cardiac output.
$S_{alv} \in [0,1]$ : Equilibrium $O_{2}$ saturation at the alveolar $O_{2}$ pressure.
$K_{D} = 3.6 kPa$ : Hemoglobin $O_{2}$ dissociation constant.
$γ = 2.7$ : Hill coefficient for hemoglobin.

The $ξ_{PTX}$ -based term accounts for loss of surface area through pneumothorax. The exponent of $2 / 3$ comes from converting a volume-based quantity to an area-based quantity.

$O_{2}$ exchange tries to maintain a steady state by compensating for consumption and equilibrating alveolar and peripheral blood saturation:

φ_{Q, O_{2}} = clamp (r_{Q} \cdot 1 m m o l / s \cdot (S_{alv} - \frac{S p O 2}{100 %}) + \min (r_{Q}, 1) \cdot φ_{C R, O_{2}}, - φ_{Q, O_{2}}^{\max}, φ_{Q, O_{2}}^{\max})

φ_{Q, O_{2}}^{\max} = Q \cdot c_{heme} \cdot S_{alv}

r_{Q} = \frac{Q}{Q_{0}} Q_{0} = \frac{H R_{0} \cdot S V_{0}}{60 s / m i n}

$φ_{Q, O_{2}}$ , $φ_{Q, O_{2}}^{\max} [m m o l / s]$ : $O_{2}$ transfer from alveoli to blood and its maximum value.
$c_{heme} = 0.0098 m m o l / m l$ : Heme concentration in blood.
$r_{Q} [\cdot]$ : Relative perfusion to default.
$Q_{0} [m l / s]$ : Default perfusion.
$H R_{0} [m i n^{- 1}]$ : Default heart rate (m_fDefaultHeartRateBPM).
$S V_{0} [ml]$ : Default stroke volume (m_fDefaultStrokeVolumeML).

Perfusion: $C O_{2}$ Exchange

$C O_{2}$ exchange is driven by a deviation from Henry’s law equilibrium between venous concentration and alveolar partial pressure:

φ_{Q, C O_{2}} = k_{C} \cdot (c_{v} - K_{H} \cdot p_{C O_{2}})

$φ_{Q, C O_{2}} [m m o l / s]$ : $C O_{2}$ transfer rate from blood to alveoli.
$c_{v} [m m o l / m l]$ : Venous blood $C O_{2}$ concentration.
$K_{H} = 0.000225 m m o l / (m l \cdot k P a)$ : Henry constant for $C O_{2}$ in blood.
$k_{C} [m l / s]$ : Equilibration rate to restore Henry’s law between blood and alveolar $C O_{2}$ .

Positive $φ_{Q, C O_{2}}$ decreases venous $C O_{2}$ concentration and increases alveolar $C O_{2}$ partial pressure. Cellular production adds $C O_{2}$ to blood, while ventilation removes it from the alveoli.

Net Evolution of Gas Pressures and Concentrations

Ventilation, perfusion, and cellular respiration are combined in ODEs. For evolution of alveolar partial pressures:

\frac{d p_{O_{2}}}{d t} = Φ_{V, O_{2}} - \frac{R T}{V_{C}} \cdot φ_{Q, O_{2}}

\frac{d p_{C O_{2}}}{d t} = \frac{R T}{V_{C}} \cdot φ_{Q, C O_{2}} - Φ_{V, C O_{2}}

Peripheral blood $O_{2}$ saturation:

\frac{d S p O_{2}}{d t} = \frac{100 %}{V_{a} \cdot c_{heme}} \cdot (φ_{Q, O_{2}} - φ_{C R, O_{2}})

$C O_{2}$ concentration in venous blood:

\frac{d c_{v}}{d t} = \frac{1}{V_{v}} \cdot (φ_{C R, C O_{2}} - φ_{Q, C O_{2}})

$t [s]$ : Simulation time.
$T [K]$ : Body temperature.
$R [k P a \cdot m l / (m m o l \cdot K])$ : Universal gas constant.
$V_{a}$ , $V_{v} [ml]$ : Arterial and venous blood volumes, taken as 30% and 70% of total blood volume.

Conversion between partial pressure $p$ and amount of substance $n$ is done through the ideal gas law:

p = \frac{n R T}{V}

Obstructed Airway, Cardiac Arrest and CPR

Tongue-based obstruction and vomit-based occlusion completely shut down any ventilation. Cardiac arrest additionally shuts down perfusion. Cellular respiration still keeps consuming $O_{2}$ from stale blood, leading to $S p O_{2}$ depletion and acidosis from $C O_{2}$ accumulation. CPR can partially restore exchange with chest compression-induced blood flow and passive ventilation (m_fCPRPassiveVentilationRateBPM). An equipped $O_{2}$ mask additionally increases ventilation as described in Ventilation.

Changes to Cardiovascular System

An elevated respiratory rate raises the base heart rate target:

Δ H R_{base} = 2 \cdot \max (R R - R R_{0}, 0)

$Δ H R_{base} [m i n^{- 1}]$ : Shift added to base heart rate target.