THE FOUR MOST IMPORTANT EQUATIONS IN CLINICAL PRACTICE


Lawrence Martin, M.D., FACP, FCCP
Clinical Associate Professor of Medicine
Case Western Reserve University School of Medicine
Cleveland, Ohio



3. Alveolar Gas Equation

The alveolar gas equation for calculating PAO2 is essential to understanding any PaO2 value and in assessing if the lungs are properly transferring oxygen into the blood. Is a PaO2 of 28 mm Hg abnormal? How about 55 mm Hg? 95 mm Hg? To clinically interpret PaO2 one has to also know the patient's PaCO2, FIO2 (fraction of inspired oxygen) and the PB (barometric pressure), all components of the equation for PAO2: 

				1-FIO2
PAO2 = FIO2(PB-PH20) - PACO2[FIO2 + ------------- ]
					R

Despite this undisputed physiologic fact physicians sometimes make clinical decisions
based on PaO 2 alone, without reference to the calculated PAO2.

The abbreviated equation below is useful for clinical purposes; in this version alveolar PO2 equals inspired PO2 (PIO2) minus arterial PCO2 x 1.2, assuming the R value is 0.8 (and assuming identical values for arterial and alveolar PCO2. Water vapor pressure in the airways is dependent only on body temperature and is 47 mm Hg at normal body temperature (37 degrees C).

PAO2 = FIO2(PB-47) - 1.2(PaCO2)

Ambient FIO2 is the same at all altitudes, 0.21. It is usually not necessary to measure PB if you know its approximate average value where the blood was drawn (e.g. sea level 760 mm Hg; Cleveland 747 mm Hg; Denver 640 mm Hg). In the abbreviated equation PaCO2 is multiplied by 1.2, a factor based on assumed respiratory quotient (CO2 excretion over O2 uptake in the lungs) of 0.8; this factor becomes 1.0 when the FIO2 is 1.0.22 The following comments are meant to show how the alveolar gas equation can be clinically helpful without the need for anything more than mental calculation.

a) If PIO2 is held constant and PaCO2 increases, PAO2 and PaO2 will always decrease. Since PAO2 is a calculation based on known (or assumed) factors, its change is predictable. PaO2, by contrast, is a measurement whose theoretical maximum value is defined by PAO2 but whose lower limit is determined by ventilation-perfusion (V-Q) imbalance, pulmonary diffusing capacity and oxygen content of blood entering the pulmonary artery (mixed venous blood). In particular, the greater the imbalance of ventilation-perfusion ratios the more PaO2 tends to differ from the calculated PAO2. (The difference between PAO2 and PaO2 is commonly referred to as the 'A-a gradient.' However, 'gradient' is a misnomer since the difference is not due to any diffusion gradient, but instead to V-Q imbalance and/or right to left shunting of blood past ventilating alveoli. Hence 'A- a O2 difference' is the more appropriate term.)
b) The alveolar-arterial PO2 difference, notated P(A-a)O2, varies normally with age and FIO2. Up to middle age, breathing ambient air, normal P(A-a)O2 ranges between 5 and 20 mm Hg. Breathing an FIO2 of 1.0 the normal P(A-a)O2 ranges up to about 110 mm Hg23(Figure 2). If P(A-a)O2 is increased above normal there is a defect of gas transfer within the lungs; this defect is almost always due to V-Q imbalance.
CASE 3. A 27-year-old young woman came to the emergency room complaining of pleuritic chest pain of several hours duration. She was not a smoker but gave a history of using birth control pills. Her chest x-ray and physical exam were normal except for splinting with deep inspirations. Arterial blood gas showed pH 7.45, PaCO2 31 mm Hg, HCO3- 21 mEq/L, PaO2 83 mm Hg (breathing ambient air; PB 747 mm Hg). She was presumptively diagnosed as having pleurodynia and discharged with pain medication.
This young woman's PaO2 was initially judged 'normal' and so an abnormality in oxygen transfer was missed. The calculated PIO2 and PAO2 were 147 mm Hg and 110 mm Hg, respectively. Her P(A-a)O2 was elevated at 27 mm Hg (110 minus 83), indicating a state of V-Q imbalance, and therefore some parenchymal lung disease or abnormality. Indeed, she returned the next day with similar complaints, at which time a lung scan showed defects interpreted as high probability for pulmonary embolism.
c) Because of several assumptions in clinical use of the alveolar gas equation, precision in calculating PAO2 is not achievable.22 Fortunately an estimate of P(A-a) O2 is usually sufficient for clinical purposes. In Case 3, for example, the fact that the patient was hyperventilating and PaO2 was only 83 mm Hg indicates an elevated P(A-a)O2 and therefore a defect in gas exchange. The alveolar gas equation shows that with hyperventilation PaO2 should go up; PaO2 should be much higher than 83 mm Hg in a hyperventilating 27-year-old patient. Similarly, a patient breathing 40% oxygen whose PaO2 and PaCO2 are normal for room air (e.g., PaO2 90 mm Hg, PaCO2 40 mm Hg) has an elevated P(A-a)O2 and therefore a defect in gas exchange; with this FIO2, PAO2 should be over 200 mm Hg and PaO2 well over 100 mm Hg. These observations require nothing more than knowledge of the alveolar gas equation and simple mental calculation.
d) Since oxygen enters the pulmonary capillary blood by passive diffusion, it follows that in a steady state the alveolar PO2 m ust always be higher than the arterial PO2. This fact is useful to spot 'garbage' blood gas data, a not infrequent problem. For example, a PaO2 of 150 mm Hg in a patient breathing 'room air' at sea level (FIO2 = .21) must represent some kind of error, since at all conceivable PaCO2 values the P(A-a)O2 would have a negative value; even with extreme hyperventilation (PaCO2 10 mm Hg) the alveolar PO2 would be no higher than 140 mm Hg. A moment's reflection will reveal several possible explanations for the apparently negative alveolar-arterial PO2 difference: the patient was in fact breathing supplemental oxygen during or just prior to the sample drawing; an air bubble in the arterial sample syringe; a quality control or reporting error from the lab; a transcription error - someone wrote down the wrong number; etc.
What about the oxygen values mentioned at the beginning of this section? A PaO2 of 28 mm Hg would be normal on the summit of Mt. Everest for a climber breathing ambient air. At the summit barometric pressure is 253 mm Hg, which provides a PIO2 of only 43 mm Hg24 (Table V).


TABLE V. Gas Pressures at Various Altitudes*
LOCATION ALT. PB FIO2 PIO2 PaCo2 PAO2 PaO2
Sea Level 0 760 .21 150 40 102 95
Cleveland 500 747 .21 147 40 99 92
Denver 5280 640 .21 125 34 84 77
*Pikes's Peak 14114 450 .21 85 30 62 55
*Mt. Everest 29028 253 .21 43 7.5 35 28
*All pressures in mm Hg; Pike's Peak and Mt. Everest data from summits

ALT. = altitude in feet
PB = barometric pressure
FIO2 = fraction of inspired oxygen
PIO2 = pressure of inspired oxygen in the trachea
PaCO2 = arterial PCO2, assumed to = alveolar PCO2
PAO2 = alveolar PO2, PAO2 is calculated using an assumed R value of 0.8 except for the summit of Mt. Everest, where 0.85 is used 24
PaO2 = arterial PO2, assuming a P(A-a)O2 of 7 mm Hg at each altitude; each PaO2 value is normal for its respectove altitude

If the climber maintained PaCO2 at 40 mm Hg his PAO2 would be minus 5 mm Hg, a value wholly incompatible with life! Ability to oxygenate blood at this altitude without supplemental oxygen is made possible (in large part) by extreme hyperventilation. On one expedition to the summit, 10 minutes after supplemental oxygen was removed a climber's end-tidal PCO2 (equivalent to PACO2) was measured at 7.5 mm Hg; assuming an R value of 0.85, the PAO2 was only 35 mm Hg.24 Based on a theoretical alveolar-arterial PO2 difference of 7 mm Hg, the climber's PaO2 at the summit was estimated at 28 mm Hg - very low but 'normal' under the circumstances.24

A PaO2 of 55 mm Hg would likewise be normal at Pike's Peak, Colorado, assuming a PaCO2 of 30 mm Hg from modest hyperventilation and a P(A-a)O2 of 7 mm Hg (Table V). On the other hand, a PaO2 of 95 mm Hg would represent a serious abnormality in anyone breathing 100% oxygen near sea level, as under these conditions PaO should be over 500 mm Hg. In summary, to properly interpret PaO2 one needs to have some appreciation of the alveolar PO, which requires knowing (at least approximately) the barometric pressure, FIO2 and PaCO2.

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Copyright © 1996-2009 Lawrence Martin, M.D.