[Music] [Applause] hello this is the first of several supplemental lectures to this course on understanding abgs these supplemental lectures are going to focus on topics that are not essential for common AB interpretations per se but which may simply be of interest people who have already shown interest in abnormalities of acidbase balance or gas exchange as you can see the specific topic of this lecture is the derivations of the equations of gas exchange the learning objectives of this lecture are to be able to derive the alveolar ventilation and Alvar gas equations from basic principles and to be aware of some of the Alternative forms of the alveolar gas equation regarding the basic principles from which these equations are derived they both depend upon Dalton's law and the conservation of a particular gas the Alvar ventilation equation is specifically based on the conservation of carbon dioxide while the alviola gas equation is based on the conservation of oxygen the first thing to review is Dalton's law John Dalton was an English chemist and physicist who laid much of the earliest groundwork for modern atomic theory and in this groundwork was the law of partial pressures more commonly referred to today as Dalton's law although Dalton discovered this equation empirically uh it was later shown that it could be mathematically derived from the kinetic theory of gases it states that the total pressure of a mixture of gases is equal to the summation of the partial pressures of each individual gas if the summation sign notation is unfamiliar or intimidating uh this can also be represented as such for the purposes of respiratory physiology there are normally only four gases present in significant quantities in the lungs thus barometric pressure commonly notated as p subb is equal to the sum of the partial pressures of nitrogen oxygen carbon dioxide and water vapor in the absence of Applied positive pressure such as would be seen by a patient on a mechanical ventilator or BiPAP barometric pressure is equal to inspite pressure denoted as P sub I a corlar to Dalton's law can be quickly derived from simple algebra that the partial pressure of a gas X is equal to the fractional concentration of that gas multiplied by the total pressure for example the partial pressure of carbon dioxide in alveolar gas is equal to the fractional concentration of CO2 in the alveoli times the total intraalveolar pressure and the average intraalveolar pressure must equal barometric pressure so now let's talk about the Alvar ventilation equation which allows one to calculate the partial pressure of CO2 and arterial blood as a function of barometric pressure in Alvar ventilation as I already mentioned this equation is based on the concervation of CO2 in other words the rate of CO2 production by the body must equal the rate of CO2 elimination by the lungs the rate of CO2 production is denoted as v. CO2 in this case the dot notation sometimes referred to as Newton's notation refers to the derivative as a function of time of whatever variable sits underneath the dot in this case for example the dot sits over V for volume the time derivative of volume is flow or in this case the rate of gas production so v.2 equals v. big a which is the rate of total gas flow from the alveoli which is another way of stating uh the alular ventilation multiplied by the F big a CO2 or the fractional concentration of CO2 in alveol or gas from Dalton's law we know that P big a CO2 equals P big a time fbig a CO2 therefore using substitution we can say that v. CO2 equals Alvar ventilation time P big a CO2 / intraalveolar pressure as has been empirically demonstrated in humans alv CO2 is an equilibrium with arterial CO2 that is there is no aa gradient with respect to carbon dioxide even in Most pathologic states therefore P big a CO2 is approximately equal to P little a CO2 so we'll make a simple substitution like so simple algebraic rearrangement leads us to this P little a CO2 equals the rate of CO2 production by the body times the intra alv pressure which is normally equal to the barometric pressure divided by Alvar ventilation turning to the Alvar gas equation which was originally known as the alol air equation it is a means to calculate the partial pressure of Alvar oxygen from which we can calculate the aa gradient its derivation utilizes the fact that the net rate of oxygen entering the lungs must equal the rate of oxygen consumption by the body the net rate of oxygen entering the lungs is equal to the oxygen inhaled which is Alvar ventilation times the fraction of o2 in inspired air minus the alv ventilation times the fraction of o2 in Alvar air this is equal to v.2 which is the rate of oxygen consumption by definition it's critical to realize that the fio2 in this equation is not exactly equal to the F2 to which we commonly refer this F2 is not the fraction of oxygen in the air that is breathed in through the nose and mouth but rather the fraction of oxygen in the air once it reaches the alveoli we will denote this with a red star the reason for this difference is that both ambient air and medical oxygen is relatively dry however before it reaches the alvioli it becomes saturated by water vapor that is picked up within the respiratory tract by Dalton's law if some of the total pressure is occupied by water vapor there must be an equal decrease in the other gases present since this decrease in each of the other gases is proportional to the relative concentration of water vapor F2 star is equal to F2 times the total pressure of inspired air or P subi minus the partial pressure of water vapor all divided by P subi we'll need to come back to this definition of F2 star at the end of our derivation from this first step we'll factor out the Alvar ventilation then we'll use the definition of the respiratory quotient if you remember from lecture 16 the respiratory quotient is defined as the ratio of the rate of carbon dioxide produced by metabolism to the rate of oxygen consumed respiratory quotient can be measured for an individual patient but this is an extremely cumbersome task and instead is a usually assumed to be 0.8 another supplementary lecture discusses the respiratory quotient in more detail with some substitution we'll replace v.2 with v. CO2 divided by the respiratory quotient remembering from our derivation of the Alvar ventilation equation uh v. CO2 equals alv ventilation times the fraction of CO2 in Alvar gas so we'll substitute that in and we get this and of course the alv ventilation of both sides cancels out we are now left with fi2 star minus F big A2 equals fbig a CO2 ided by the respiratory quotient and after another simple rearrangement the equation form is starting to look a little more familiar we're going to bring in Dalton's law once more and substitute for the fbig ao2 and the fbig a CO2 multiply everything by the total alveolar pressure now we're almost there just substitute in the expression for f io2 star to get this since the total inspired pressure is equal to the average total alveola pressure these two variables cancel out this leaves us with a familiar form of the alv gas equation I'd like to review two other forms of the alol gas equation first with some algebraic manipulation of the common form we can restate this equation as the partial pressure of oxygen in the alvioli equals the partial pressure of oxygen in inspired air minus the difference between inspired pressure and water vapor times uh the ratio of oxygen consumption to overall Alvar ventilation although more Awkward to use in practice and form this actually makes more intuitive sense as one can restate it as the amount of oxygen alveolar gas is what remains after the oxygen used from metabolism has been removed from the oxygen in inspired gas the second alternative form acknowledges that because the rate of o2's consumption exceeds CO2 production additional air passively enters the lungs during respiration that is in addition to active Alvar ventilation I won't go through the somewhat tedious derivation that accounts for this effect but here is the final result it's the same as above with the exception of a lengthy addition term thrown onto the end luckily although it technically makes the alv gas equation more accurate this additional term is too small to be clinically relevant for example for a patient with normal respiratory physiology breathing room air this final term equals 0.21 * 40 * 1 - 0.8 / 0.8 this equals a minuscule 2.1 mm of mercury compare the magnitude of this to that of the first term which is normally 150 and the second term which is normally 50 therefore for all practical purposes this last term can be ignored reducing the complete form back to the common form above finally there are several assumptions to briefly mention that are required by these derivations first gases must obey Dalton's law modern physics has demonstrated that Dalton's law does not exactly describe the properties of real gases at high pressure though the differences seen at typical barometric pressure are inconsequential second Alvar gas is saturated with water vapor this assumption is completely valid third carbon dioxide in the alvioli is in equilibrium with carbon dioxide in the pulmonary capillaries that is p big a CO2 must equal p little A2 this holds uh to be valid in all but the most extreme of pathologic processes and finally inspired gas contains no carbon dioxide with our atmospheric concentration of CO2 at less than 0.1% this is close enough to be true in all medical situations but may not hold true in experimental situations or other very unusual circumstances so that's it for the derivations of the equations of gas exchange I hope you found this course supplement interesting uh the remaining mental lectures will cover topics such as what factors can influence the respiratory quotient and how to interpret abgs at high altitude [Music]