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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 $RQ = 0.80$                  |
^ Mech Power, VC                    | ${MP}_{VC} = 0.098 \cdot RR \cdot V_t[PIP-\frac{1}{2}(P_{plat}-PEEP)]$                        |
^ Mech Power, PC                    | ${MP}_{VC} = 0.098 \cdot RR \cdot V_t[PEEP + \Delta P_{insp}(1-e^{\frac{-T_{insp}}{RC}})]$  |

===== Respiratory Equations =====
==== Mechanical Power ====
=== Volume Control ===
${MP}_{VC} = 0.098 \cdot RR \cdot V_t[PIP-\frac{1}{2}(P_{plat}-PEEP)] \approx \frac{MV(P_{peak}+PEEP+\frac{Q_{insp}}{6})}{20}$

===Pressure Control ===
${MP}_{VC} = 0.098 \cdot RR \cdot V_t[PEEP + \Delta P_{insp}(1-\exp(\frac{-T_{insp}}{RC}))]$ 

${MP}_{VC} = 0.098 \cdot RR \cdot V_t[PEEP + \Delta P_{insp}(1-e^{\frac{-T_{insp}}{RC}})] \approx 0.098 \cdot RR \cdot V_t(PEEP + \Delta P_{insp})$
  * [[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 $V_D = 150$


==== Shunt Equation (Berggren Equation)====
$$\frac{Q_s}{Q_t} = \frac{C_{C_{O_2}} - C_{a_{O_2}}}{C_{C_{O_2}} - C_{v_{O_2}}}$$

where:
  * $Q_s=$ pulmonary physiology shunt $(\frac{mL}{min})$
  * $Q_t=$ cardiac output $(\frac{mL}{min})$
  * $C_{C_{O_2}} = $ end-pulmonary-capillary oxygen content
  * $C_{a_{O_2}} = $ arterial oxygen content
  * $C_{v_{O_2}} =$ mixed venous oxygen content

So, you will need an ABG and a true mixed VBG (art line + SGC).

=== Derivation ===
==== 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 $P_ACO_2$

















===== PULM =====
==== Equation of Motion ====
$P_{delivered} = P_{resistive} + P_{elastic}$

$P_{aw} = \dot VR + \frac{V_t}{C} + PEEP_{total} + P_{musc}$ 

==== CPET Testing====
===Heart rate reserve
$HRR = HR_{achieved}^{max} - HR_{predicted}^{peak}$,

where $HR_{predicted}^{peak} = 220 - age$

===Slope of work efficiency===
$m(work_e) = \frac{\Delta VO_2}{\Delta WR}$

===Slope of heart rate rise===
$\frac{\Delta HR}{\Delta VO_2}$
===== CARDS ===== 
$TPG = mPAP - PCWP$

$SVR =\frac{MAP-CVP}{CO}\cdot80$

$CO = LVOT_{area}\cdot LVOT_{VTI}\cdot HR$

====Swan-Ganz Equations====
$CO = \frac{VO_{2}}{C_a - C_v}$, where $C_v = ScvO_2$ (mixed venous oxygen content)
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