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[[https://anesthesia.ucsf.edu/sites/anesthesia.ucsf.edu/files/wysiwyg/Ventilator_Pocket_Guide_2020.pdf | Ventilator Pocket Guide]]

==== Foundational Equations ====

^ Ohm's Law                         | $\Delta P = FR = P_{aw} - P_{alv} = P_{pl} - PEEP_{total}$                                               |
^ Equation of Motion                | $P_{aw} =  FR + \frac{V_{t}}{C} + PEEP_{total}$                                                          |
^ Compliance                        | $C = \frac{\Delta V}{\Delta P}$                                                                          |
^ Natural Decay Equation            | $V_i(t)= \frac{V_o}{e^{\frac{t}{RC}}} = \frac{V_o}{e^{\frac{t}{\tau}}}$                                  |
^ Calculating $\Tau$, General Case  | $ \tau = \frac{V_t}{F} \Bigg(\frac{PIP - P_{plt}}{P_{plt} - PEEP_{total}}\Bigg) $                        |
^ Alveolar Gas Equation             | $P_AO_2 = F_iO_2(P_{atm}-P_{H_2O}) - \frac{P_aCO_2}{RQ} $, where $RQ = 0.80$ |

  * [[https://xlung.net/en/mv-manual/basic-modes-of-mechanical-ventilation | Vent Waveforms]]
==== Alveolar Gas Equation====
$P_AO_2 = F_iO_2(P_{atm}-P_{H_2O}) - \frac{P_aCO_2}{RQ}$

substituting back in to $RQ$ equation:
$RQ = \frac{P_ACO_2}{\frac{V_AP_ACO_2}{kVO_2}}= \frac{VO_2}{V_a}k$

$V_T = V_A + V_D$, where $V_A = 350$ and $V_D = 150$


==== Dead Space Fraction ====
$\frac{V_D}{V_T} = \frac{P_ACO_2 - P_ECO_2}{P_ACO_2}$

Formal measurement of $P_ECO_2$ requires volumetric capnography, which requires a capable ventilator or a dedicated measurement device.

Thankfull, $P_ECO_2 \approx ETCO_2$, so an approimation would $\frac{V_D}{V_T} = \frac{P_ACO_2 - ETCO_2}{P_ACO_2}$



==== Alveolar ventilation ====
$P_{A}O_2 = F_iO_2(P_{atm}-P_{H_2O}) - \frac{P_AO2}{RQ}$
$\dot{V}_A=k\frac{\dot{V}CO_2}{P_ACO_2}$
$\implies \dot{V}CO2 = \frac{\dot{V}_AP_ACO_2}{k}$

To convert $F_ACO_2$ into $P_ACO_2$, we have $F_ACO_2(P_{atm} - PH_2O = P_ACO_2$
Similarly, using $F_ECO_2$, we can show $P_ECO_2 = F_ECO_2(P_{atm} - P_{H_2O})$

$Volume_{expiredCO2} = Volume_{producedAlvCO2}$

$V_TF_ECO_2 = V_AF_ACO_2$

$V_TF_ECO_2 = (V_T - V_D)F_ACO_2$, and we can convert $F_ACO_2$ into $P_ACO_2$
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