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Respiratory Equations – Behind the Numbers Southern African Journal of Anaesthesia and Analgesia. 2020;26(6 Suppl 3):S90-93 https://doi.org/10.36303/SAJAA.2020.26.6.S3.2546 South Afr J Anaesth Analg Open Access article distributed under the terms of the ISSN 2220-1181 EISSN 2220-1173 Creative Commons License [CC BY-NC 3.0] © 2020 The Author(s) http://creativecommons.org/licenses/by-nc/3.0 FCA 1 REFRESHER COURSE Respiratory equations – behind the numbers T Leonard Department of Anaesthesia, School of Clinical Medicine, Faculty of Health Sciences, Chris Hani Baragwanath Academic Hospital, University of the Witwatersrand, South Africa Corresponding author, email: [email protected] Summary Candidates for the FCA 1 exam will come across dozens of equations that eventually all merge into something complicated and daunting. The purpose of this review is to highlight some of the respiratory equations that are important and that candidates find confusing and explain the mathematical and physiological principles behind them. Keywords: equations, respiratory physiology, ventilation, perfusion, dead space Introduction VD / VT = (PACO2 – PECO2) / PACO2 P CO partial pressure of CO in alveolar gas There are many equations that candidates will come across in A 2 2 their study of respiratory physiology. These equations describe PECO2 partial pressure of CO2 in the total mixed principles of ventilation, perfusion and diffusion within the res- exhaled breath piratory system. This review attempts to explain the origins and A further modification was made by Henrik Enghoff due to make sense of the numbers in some of these equations. 2 difficulties measuring theA P CO2. This gives the physiological Equations to be covered: dead space equation:1 • Dead space equations VD / VT = (PaCO2 – PeCO2) / PaCO2 • The alveolar gas equation VD physiological dead space volume • Diffusion equations VT tidal volume • Ventilation-perfusion equations PaCO2 partial pressure of carbon dioxide in arterial blood Dead space equations PeCO2 partial pressure of carbon dioxide in expired Physiological dead space represents the portion of ventilation gas 1 that does not eliminate carbon dioxide (CO2). This consists of the anatomical dead space (the fraction of ventilation Important in this equation is understanding the derivation. 1,2 delivered to the conducting airways – roughly 150 ml) and The derivation is based on the principal that only the gases the alveolar dead space (the fraction of ventilation delivered to involved in alveolar ventilation (VA) are involved in gas alveoli with no pulmonary artery perfusion).2 In 1891, Christian exchange and produce CO2. The total tidal volume (VT) is made Bohr introduced his equation to represent the volume of gas 1,2,4 up of VA + VD; we can substitute VA for VT – VD. that constitutes the dead space. Now we must look at the fraction of expired (F ) and inspired The original version of this equation is:2 E (F) CO . This can be done for nitrogen and oxygen, but CO is V / V = (F CO – F CO ) / F CO i 2 2 D T A 2 E 2 A 2 most commonly used. VD respiratory dead space volume In one exhalation the expired CO2 = FE.VT and this must be VT tidal volume made of alveolar gas and dead space gas. Therefore VT.FE = FACO2 mean estimate of alveolar CO2 concentration VA.FA + VD.Fi and it is assumed that the FICO2 is 0. FECO2 CO2 concentration in the total mixed exhaled 1,2,4 breath Therefore: VT.FE = VA.FA This equation has undergone some changes due to difficulties Substitute VA for VT – VD in measuring FACO2 and FECO2. Using Dalton’s law (the concentration of a gas is proportional to its partial pressure)3 VT.FE = (VT – VD).FA 4 we can substitute FA and FE for partial pressures. Multiply out the brackets South Afr J Anaesth Analg 2020; 26(6)Supplement S90 http://www.sajaa.co.za Respiratory equations – behind the numbers VT.FE = (VT.FA) – (VD.FA) Input = VA x FIO2 Rearrange to get VD on the left of the equation Output = VO2 + (VA x FAO2) VD.FA = VT(FA – FE) VA x FIO2 = VO2 + (VA x FAO2) Divide VT and FA Solve for VO2 VD / VT = (FA – FE) / FA VO2 = VA x (FIO2 – FAO2) Substitute with partial pressure Respiratory quotient: VD / VT = (PACO2 – PECO2) / PACO2 Now use the Enghoff modification This is defined as “the volume of carbon dioxide released over the volume of oxygen absorbed during respiration. It VD / VT = (PaCO2 – PeCO2) / PaCO2 is a dimensionless number used in the calculation for basal metabolic rate.”6 The alveolar gas equation (AGE) R = VCO2 / VO2 The AGE describes the alveolar concentration (or partial pressure) of oxygen (O2) in terms of the inspired oxygen The final derivation: concentration, the alveolar concentration of CO2 and the R = VCO / VO = (F CO x V ) / [V x (F O – F O )] respiratory quotient (R).5 2 2 A 2 A A I 2 A 2 Cancel out VA in the numerator and denominator FAO2 = FIO2 – (FACO2 / R) R = FACO2 / (FIO2 – FAO2) Rewriting the equation applying Dalton’s law again gives this Solve for FAO2 equation: (FIO2 – FAO2) = FACO2 / R PAO2 = PIO2 – (PACO2 / R) FAO2 = FIO2 – FACO2 / R PAO2 alveolar partial pressure of O2 Convert to partial pressures and assume PACO2 = PaCO2 PIO2 inspired partial pressure of O2 PAO2 = PIO2 – (PaCO2 / R) PACO2 alveolar partial pressure of CO2 This is AGE that is most often used. However, there is a problem R respiratory quotient with this equation in that it may be too simplistic when the 1,5 Derivation of the AGE: value of R does not equal 1. In the derivation of the AGE we substituted R for VCO2 / VO2 and indeed when R = 0.8 as it most To derive and understand the AGE (as well as its limitations) we often does this would mean a VCO2 of 0.20 l/min and VO2 of need to look at each component of the equation. 0.25 l/min which is a discrepancy of 50 ml per minute. Carbon dioxide: Thus, the modified AGE is:1,5,7 At steady state all CO2 produced by the body (VCO2) must be PAO2 = PIO2 – (PaCO2 / R) + [FIO2 x PaCO2 x ( (1 – R) / R)] removed by the alveolar ventilation each minute and because The additional part of the equation has relatively minor effect CO2 is highly diffusible we can assume that AP CO2 very closely 5 in usual clinical practice. approximates PaCO2. VCO2 = FACO2 x VA Diffusion equations Dalton’s law of partial pressures needs to be applied again and In 1855 Adolf Fick described how a gas moves across a mem- states that in a mixture of gases the individual gas CO2 will brane. It is not an equation that can be solved with measurable be present in a concentration that is the same proportion as numbers but rather shows the factors that affect the move- 5 8 PACO2 is of the total pressure PI. ment of a gas (oxygen) across the alveolar membrane. VCO2 = (PACO2 / PI) x VA Flow of gas ∞ [A x D (P1 – P2)] / T The equation can be rearranged A area of the membrane PACO2 = (VCO2 x PI) / VA = PaCO2 D diffusion constant of the gas Oxygen: (P1 – P2) partial pressure (or concentration) gradient across the membrane All oxygen entering the alveoli must equal the oxygen leav- T thickness of the membrane ing the alveoli. Input into the alveoli is from the inspired air whereas the output of oxygen is a combination of oxygen D ∝ solubility of gas / √molecular weight of gas 5 consumption (VO2) and expired oxygen. This states that the rate of transfer of a gas is directly pro- Input = output portional to the area of a membrane, the diffusion constant South Afr J Anaesth Analg 2020; 26(6)Supplement S91 http://www.sajaa.co.za Respiratory equations – behind the numbers for that gas and the concentration gradient across the VA / Q = 8.63 x (CaO2 – CvO2)/(PIO2 – PAO2) membrane and it is inversely proportional to the thickness of The constant allows for standardisation when the units used for the membrane.1,8 VA and Q are l/min, CaO2 and CvO2 are ml/dl and for PIO2 and PAO2 are mmHg.10 Ventilation and perfusion relationship equations The same principle can be applied for CO except that the CO The concept of dead space has been discussed previously but 2 2 content of mixed venous and arterial blood are reversed because there are two other equations that can be used to describe CO is being eliminated.10 ventilation and perfusion relationships in the lung. 2 VA / Q = 8.63 x (CVCO2 – CaCO2) / (PACO2 – PICO2) These two equations explain why in an area of lung with reduced VA / Q ratio the PAO2 and CaO2 will fall greater than the PACO2 will rise while in areas with high VA / Q ratio the PAO2 rises while PACO2 falls. Therefore, low VA / Q areas affect O2 more and high VA / Q 10 areas affect CO2 more. The shunt equation: This equation gives a ratio of the shunt blood flow to total blood flow. Shunt blood flow is blood that is not exposed to any gas Figure 1: O2-CO2 diagram showing a ventilation-perfusion (V/Q) ratio line1 exchange. This may be areas of the lung with V/Q ratio of 0 or 11 venous blood that enters the arterial system directly. The figure illustrates the relationship between alveolar 2O and Important in understanding and deriving this equation is to be CO2 partial pressure at varying V/Q ratios. Numerically this can be described using the Bohr equation, V/Q ratio equation and able to draw a theoretical alveolus with blood being oxygenated the shunt equation.
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