# Simple Arithmetic

Goethe

## Non-Visual Learners:

This section is aimed at people who don’t like maps and diagrams. The rest of the site makes extensive use of graphic images. This section focuses on using words and arithmetic as an alternative approach to understanding acid-base balance.

### The Simple Equation.

Non-visual learners may prefer this simple arithmetic relationship between PCO2, pH, and metabolic acid level (SBE). It allows the approximate level of metabolic acidosis to be calculated from the PCO2 and the pH. It is slightly easier to use when the PCO2 is measured in mmHg.

## Explanation.

The equation means that a change of 0.1 in the pH can be caused by either:

1. Respiratory: PCO2 change of 12mmHg (1.6 kPa), or
2. Metabolic: SBE change of 6 mEq/L.
3. Mixture of the two.

This relationship allows the components to be “added” and “subtracted”. For example, a pH of 7.2 (0.2 more “Acid”) can be caused by:

1. a PCO2 of 64 mmHg (8.5 kPa) with a BE =   0 mEq/L
2. a PCO2 of 52 mmHg (6.9 kPa) with a BE =  -6 mEq/L
3. a PCO2 of 40 mmHg (5.3 kPa) with a BE = -12 mEq/L
4. a PCO2 of 28 mmHg (3.7 kPa) with a BE = -18 mEq/L

Although this approximation provides acceptable clinical results in most circumstances, its real value is providing insight for the “non-visual” learner.

A Few Examples employing this equation may make these concepts easier to understand:

## Chronic (Compensated) Respiratory:

#### 1) PCO2 = 64 mmHg (8.5 kPa), pH = 7.3.

The PCO2 is raised by 24 mmHg or 3.2 kPa. This would be expected to cause an acid shift in the pH of 0.2 to pH 7.2. The actual pH (7.3) is about halfway back to the normal of 7.4. This is characteristic of chronic hypoventilation and might be seen in chronic obstructive pulmonary disease. The 0.1 difference between expected and actual pH corresponds to 6 mEq/L of compensatory metabolic alkalosis.

#### 2) PCO2 = 28 mmHg (3.7 kPa), pH = 7.45.

The PCO2 is low by 12 mmHg or 1.6 kPa. This would be expected to cause an alkaline shift in the pH of 0.1 to pH 7.5. The actual pH (7.45) is about halfway back to normal of 7.4. This is characteristic of chronic hyperventilation and occurs, for example, at high altitude in response to hypoxia. The 0.05 difference between expected and actual pH corresponds to 3 mEq/L of compensatory metabolic acidosis.

## Acute Pure (Uncompensated) Respiratory:

#### 1) PCO2 = 52 mmHg (6.9 kPa), pH = 7.3.

The PCO2 is raised by 12 mmHg or 1.6 kPa. This would be expected to cause an acid shift in the pH of 0.1 to pH 7.3. The actual pH (7.3) is as expected, i.e., there is no evidence of any metabolic compensation. Pure (acute) respiratory acidosis occurs with acute respiratory depression or by setting a ventilator to deliver a minute-ventilation smaller than the patient’s normal. It takes a day or two for a patient’s kidney to respond and produce the typical partial correction.

#### 2) PCO2 = 28 mmHg (3.7 kPa), pH = 7.5.

The PCO2 is low by 12 mmHg or 1.6 kPa. This would be expected to cause an alkaline shift in the pH of 0.1 to pH 7.5. The actual pH (7.5) is as expected, i.e., there is no evidence of any metabolic compensation. Pure (acute) respiratory alkalosis occurs in acute hyperventilation or by setting a ventilator to deliver a minute-ventilation greater than the patient’s normal. It takes a day or two for a patient’s kidney to respond and produce the typical partial correction.

## Compensated Metabolic:

#### 1) PCO2 = 28 mmHg (3.7 kPa), pH = 7.3.

The PCO2 is low by 12 mmHg or 1.6 kPa. This would be expected to cause an alkaline shift in the pH of 0.1 to pH 7.5. The actual pH is more acid than expected by 0.2 – equivalent to 12 mEq/L of metabolic acidosis. Because the metabolic change matches the acid pH change, it is the underlying problem. Alone it would be expected to cause a pH of 7.2. In fact the pH is half way back towards normal, a typical compensation which might be found in lactic acidosis following tissue ischemia.

#### 2) PCO2 = 52 mmHg (6.9 kPa), pH = 7.5.

The PCO2 is raised by 12 mmHg or 1.6 kPa. This would be expected to cause an acid shift in the pH of 0.1 to pH 7.3. The actual pH is, therefore, more alkaline than expected by 0.2 – equivalent to 12 mEq/L of metabolic alkalosis. Because the metabolic matches the alkaline pH change, it is the underlying problem. Alone it would be expected to cause a pH of 7.6. In fact the pH is half way back towards normal, a typical compensation which might be found in vomiting or gastric aspiration.

## Mixed Disturbances:

#### 1) PCO2 = 64 mmHg (8.5 kPa), pH = 7.1.

The PCO2 is raised by 24 mmHg or 3.2 kPa. This would be expected to cause an acid shift in the pH of 0.2 to pH 7.2. The actual pH is, therefore, more acid than expected by 0.1 – equivalent to 6 mEq/L of metabolic acidosis. This represents a combination of metabolic and respiratory acidosis. It might occur following trauma with tissue ischemia and respiratory depression.

#### 2) PCO2 = 28 mmHg (3.7 kPa), pH = 7.6.

The PCO2 is low by 12 mmHg or 1.6 kPa. This would be expected to cause an alkaline shift in the pH of 0.1 to pH 7.5. The actual pH is, therefore, more alkaline than expected by 0.1 – equivalent to 6 mEq/L of metabolic alkalosis. This represents a combination of metabolic and respiratory alkalosis. It might occur with gastric aspiration and mechanical hyperventilation.