Four equations are taught briefly in medical school but are grossly under-emphasized in importance and are therefore invariably forgotten in later years, when they are most needed. The reasons why these highly important equations are 'under taught' in medical school are several:

• a crowded curriculum that must make room for immunology and cell biology
• the teachers may have little or no clinical experience with respiratory patients, and therefore can't possibly know how important these equations are in the everyday practice of medicine
• misguided leadership of curriculum committees that may feel every subject deserves equal balance, and thus leave it up to the student to 'learn it all' without anyone guiding them as to what is really important in the care of patients. (For example, one hour on surfactant may be equally weighted with one hour on gas exchange, which may be OK for training Ph.D.'s but is misguided for training physicians).

These four equations express relationships that are extremely important in clinical practice. They are the:

1. PCO₂ equation
2. Henderson-Hasselbalch equation
3. Alveolar Gas equation, and
4. Oxygen Content equation.

Emphasis should be placed on understanding the simple qualitative relationships expressed by these equations. Each equation can be clinically applied in the assessment of abnormal oxygenation, ventilation, or acid-base balance. For example, variables in the PCO₂ equation, and not any bedside observations, define the common terms hyperventilation and hypoventilation and explain why a dyspneic, tachypneic patient may be retaining CO₂. Ignorance of this and other relationships expressed in the four equations is reflected in some common diagnostic and therapeutic mistakes.

INTRODUCTION

There is disparity between the physiology we teach and expect medical students to learn and the physiology that medical residents and practicing physicians seem to know and understand. This disparity is perhaps best exemplified by four simple equations important in understanding cardiopulmonary and renal disorders (Table I). These equations are seldom emphasized beyond medical school, yet not appreciating the physiology behind them can (and often does) lead to clinical errors.

Intensive care units have contributed to the weakening knowledge of physiology among primary care physicians. Today, the more profound physiologic derangements are usually managed in ICUs by organ-specific specialists; these derangements (e.g., shock, pulmonary edema, acute ventilatory failure, acute renal failure) are literally outside the care of most physicians and surgeons. Not all serious physiologic problems are handled in ICUs however, and the need for understanding basic physiology - in the office, on the general medical wards - remains paramount.

The four equations in this paper (Table I, below) are important clinically not so much for the numbers they generate as for their qualitative relationships. All four equations can be abbreviated to simpler terms that are adequate for most clinical purposes.

TABLE I: THE FOUR MOST IMPORTANT EQUATIONS
IN CLINICAL PRACTICE

Equation Title	Complete Equation	Abbreviation Sufficient for Most Clinical Applications
1. PCO 2  equation	PACO2=VCO2 x 0.863 / VA where VA=VE-VD	PaCO2 ~ VCO2 / VA
2. Henderson- Hasselbalch equation	pH=pK + log HCO3- / 0.03(PaCO2)	pH ~ HCO3- / PaCO2
3. Alveolar gas equation	PAO2=FIO2(PB-PH2o)-PACO2[FIO2 + (1-FIO2) / R]	PAO2=FIO2(PB-47)-1.2(PaCO2)
4. Oxygen content equation	CaO2=(SaO2 x Hb x 1.34) + .003(PaO2) where: 1.34=ml O2/gram Hb .003=ml   O2/mm Hg  PaO2/dl Hb=content in grams/dl	CaO2=SaO2 x 1.34 x Hb

Obviously other equations besides those in Table 1 can be important in assessing disordered physiology. The point is not to belabor equations but to emphasize a few key relationships often overlooked or misapplied in the daily practice of medicine. By understanding them we can take better care of our sickest patients wherever they are encountered. Non-intensivists should be familiar with these particular equations and their clinical relevance. They were probably all learned by most physicians at one time. For physicians in training and practitioners alike, now is the time to review.

The PCO₂ equation puts into physiologic perspective one of the most common of all clinical observations: a patient's respiratory rate and breathing effort. The equation states that alveolar PCO₂ (PACO₂) is directly proportional to the amount of CO₂ produced by metabolism and delivered to the lungs (VCO₂) and inversely proportional to the alveolar ventilation (VA). While the derivation of the equation is for alveolar PCO₂, its great clinical utility stems from the fact that alveolar and arterial PCO₂ can be assumed to be equal. Thus:

From the PCO₂ equation it follows that the only physiologic reason for elevated PaCO₂ is a level of alveolar ventilation inadequate for the amount of CO₂ produced and delivered to the lungs. ¹ Thus arterial hypercapnia can always be explained by:

1. not enough total ventilation (as may occur from central nervous system depression or respiratory muscle weakness); or
2. too much of the total ventilation ending up as dead space ventilation (as may occur in severe chronic obstructive pulmonary disease, or from rapid, shallow breathing); or
3. some combination of 1) and 2).

Excess CO₂ production is omitted as a specific cause of hypercapnia because it is never a problem for the normal respiratory system unimpeded by a resistive load. During submaximal exercise, for example, where CO₂ production is increased, PaCO₂ stays in the normal range because VA rises proportional to the rise in VCO₂. With extremes of exercise (beyond anaerobic threshold) PaCO₂ falls as compensation for the developing lactic acidosis.² In health PaCO₂ may be reduced but is never elevated.

An important clinical corollary of the PaCO₂ equation is that we cannot reliably assess the adequacy of alveolar ventilation - and hence PaCO₂ - at the bedside. Although VE can be easily measured with a handheld spirometer (as tidal volume times respiratory rate), there is no way to know the amount of VE going to dead space or the patient's rate of CO₂ production. A common mistake is to assume that because a patient is breathing fast, hard and/or deep he or she must be "hyperventilating." Not so, of course.

Although nothing in the PCO₂ equation directly relates respiratory rate or depth of breathing to PaCO₂, physicians commonly (and mistakenly) use these observations to assess a patient's PaCO₂. The error in this case was to assume the patient was hyperventilating (because she was breathing fast) and could tolerate the sedative; in fact she was hypoventilating - her PaCO₂ was elevated (as will be explained further under Equation 2).

Hypercapnia represents a failure of the respiratory system in some aspect and therefore a state of severe organ system impairment. In addition to this clinical fact there are three physiologic reasons why elevated PaCO₂ is potentially dangerous. First, as PaCO₂ increases, unless HCO₃^- also increases by the same degree pH will fall (see Equation 2). Second, as PaCO₂ increases PAO₂ (and hence PaO₂) will fall unless inspired oxygen is supplemented (see Equation 3). Third, the higher the PaCO₂, the less defended is the patient against any further decline in alveolar ventilation.

This last point is graphically illustrated by plotting PaCO₂ against alveolar ventilation Figure 1. The higher the PaCO₂ is to begin with, the more it will rise for any given decrement in alveolar ventilation. For example a decrease in alveolar ventilation of one L/minute (as may occur from anesthesia, sedation, congestive heart failure, etc.) will increase a baseline PaCO₂ of 30 mm Hg to 36.3 mm Hg when VCO₂ is 200 ml/min; the same one L/min decline in VA will raise a baseline PCO₂ of 60 mm Hg to 92 mm Hg Figure 1). Whereas the hyperventilating or normally- ventilating patient can almost always tolerate sedating drugs (without clinically important hypoventilation), even a small amount of sedative may be dangerous in the hypercapnic patient.

Note also from Figure 1 that an increase in CO₂ production (e.g., from 200 to 300 ml/min) without concomitant increase in VA (as should occur normally) will cause PaCO₂ to increase. This situation is sometimes seen in patients with severe chronic obstructive lung disease when they exercise, and in artificially-ventilated patients who are carbohydrate loaded (which increases CO₂ production). The basic mechanism for hypercapnia in these and all other cases, however, is inadequate VA for the amount of CO₂ delivered to the lungs.

Of the four equations in this paper, the Henderson-Hasselbalch is the one with which physicians are most familiar. The H-H equation is repeatedly emphasized in basic science courses and in renal and pulmonary pathophysiology lectures; students hear about it on many occasions.
The bicarbonate buffer system, quantitatively the largest in the extracellular fluid, instantaneously reflects any blood acid-base disturbance in one or both of its buffer components (HCO₃^- and PACO₂). The ratio of HCO₃^- to PACO₂ determines pH and therefore the acidity of the blood:

Unfortunately, the logarithmic nature of pH and the fact that acid-base disorders involve simultaneous changes in three biochemical variables and in the function of two organ systems (renal and respiratory), have all combined to made acid-base a difficult subject for many clinicians. In the 1970s nomograms incorporating the H-H variables and compensation bands for the four primary acid-base disorders were introduced as aids to determining a patient's acid-base status.^3-8 While nomograms can be helpful if readily available and properly used, there is much to be gained by simply knowing the relationship among the three H-H variables and the type of changes expected with each disorder. In this regard the following items of clinical importance bear emphasis.

a) If any of the three H-H variables is truly abnormal the patient has an acid-base disturbance without exception. Thus any patient with an abnormal HCO₃^- or PaCO₂, not just abnormal pH, has an acid-base disorder. Most hospitalized patients have at least one bicarbonate measurement as part of routine serum electrolytes; this is usually called the 'CO₂' or 'total CO₂' when measured in venous blood. (Total CO₂ includes bicarbonate and the CO₂ contributed by dissolved carbon dioxide, the latter 1.2 mEq/L when PaCO₂ is 40 mm Hg. For this reason, and because bicarbonate concentration is slightly higher in venous than in arterial blood, total CO₂ runs a few mEq/L higher than the bicarbonate value calculated using the H-H equation.) If total CO₂ is truly abnormal the patient has an acid-base disorder. In Case 1 there were two sets of electrolyte measurements on the patient's chart when the sedative was ordered; both showed total CO₂ elevated at 34 mEq/L. The patient had been taking a diuretic so it was probably assumed that her elevated total CO₂ reflected a mild metabolic alkalosis. More likely, however, it represented chronic respiratory acidosis with renal compensation. When she arrived to the ICU her arterial blood gas showed pH 7.07, PaCO₂ 83 mm Hg, PaO₂ 55 mm Hg (breathing supplemental oxygen), HCO₃^- 23 mEq/L, values that reflected a worsening of previously- unrecognized respiratory acidosis plus a new metabolic acidosis (lactic acidosis from decreased organ perfusion). The patient's long smoking history and the physical findings suggested chronic obstructive lung disease (later confirmed by pulmonary function tests). Her anxiety prior to MICU transfer was related to worsening acidosis and dyspnea.

b) The simplified version of the H-H equation eliminates the log and the pK, and expresses the relationships among the three key values.

TABLE III. The four primary acid-base disorders and their compensatory changes. The primary event leads to a large change in pH (larger arrows). Compensation (changes in HCO₃^- and PaCO₂ represented by smaller arrows) attempts to normalize the ratio of HCO₃^-/PaCO₂ and bring the pH back toward normal (smaller arrows next to pH). Each primary disorder may be caused by a variety of specific clinical conditions (see text).

g) Acute, uncompensated respiratory alkalosis (acute hyperventilation) and acidosis (acute hypoventilation) cause predictable changes in pH and bicarbonate^11,12 (Table IV). Bicarbonate increases slightly from the biochemical reaction of acutely retained CO₂ and decreases when CO₂ is acutely excreted;^11,12 these changes are instantaneous and independent of any renal compensation. Extreme acute hyperventilation can lower the bicarbonate to about 15 mEq/L and extreme acute hypoventilation can raise it to about 29 mEq/L (Table IV); a bicarbonate value outside this range must indicate either a renal compensatory mechanism or a primary metabolic acid-base disorder. The biochemical changes in bicarbonate from acute shifts in PaCO₂ point to another particularly useful clue to the presence of a mixed disorder: a higher- or lower-than- expected bicarbonate value with any change in PaCO₂. Thus a slightly low HCO₃^- concentration in the presence of hypercapnia suggests a concomitant metabolic acidosis (e.g., PCO₂ 50 mm Hg, pH 7.27, HCO₃^- 22 mEq/L); a slightly elevated HCO₃^- in the presence of hypocapnia suggests a concomitant metabolic alkalosis (e.g. PCO₂ 30 mm Hg, pH 7.56, HCO₃^- 26 mEq/L).
TABLE IV. Changes in arterial pH and bicarbonate with acute changes in PaCO₂. The ranges represent the 95% confidence limits for pH and bicarbonate when PaCO₂ changes acutely (before any renal compensation takes place). Note that bicarbonate decreases with acute hyperventilation and increases with acute hypoventilation. (Data from references 11-12).

h) The bicarbonate (or total CO₂) should also be examined in relation to the other measured electrolytes, specifically to calculate the anion gap (AG). AG is the Na⁺ concentration minus (total CO₂ + Cl^-). The normal AG, 12 +/- 4 mEq/L, is an artifact of measurement since these three electrolytes are only the ones most commonly measured. (Since the value of K⁺ is small and relatively constant it is not usually used to calculate the AG; if K⁺ is used then the normal AG is about 16 +/- 4 mEq/L). If all the serum anions and cations were measured anions would equal cations and there would be no anion gap. The importance of the anion gap is that it can help both to diagnose the presence of a metabolic acidosis and characterize its cause. Thus, regardless of pH an elevated AG suggests a metabolic acidosis from unmeasured organic anions, e.g., lactic acidosis or ketoacidosis;^19-21 the higher the AG the more likely it reflects an organic acidosis.¹⁹ On the other hand a normal AG in a patient with metabolic acidosis indicates a hyperchloremic acidosis, most commonly from renal or gastrointestinal bicarbonate loss, e.g., renal tubular acidosis or diarrhea.