## Abstract

Medical students, residents, and allied health professionals often have difficulty quantitating ventilation-perfusion mismatch in ill patients. This manuscript quantitates ventilation-perfusion mismatch using the underlying physiological concepts and equations that describe mismatch. In addition, clinical problems with diagrams and worked-out solutions are supplied to help students master these equations as well as their practical limitations.

- ventilation

When patients have hypoxemia due to ventilation-perfusion mismatch, clinicians need quantitative methods to measure the severity of illness as a function of time. Medical students, residents, and allied health professionals usually have some familiarity with the virtual shunt equation, shunt nomogram, alveolar-arterial gradient, and arterial/alveolar ratio. These techniques are all commonly used to assess respiratory failure. Nonetheless, students often have difficulty quantitating these measures and do not understand when these methods fail. Standard textbooks may provide the theoretical background for the material above, but they rarely provide clinical examples to solidify these concepts (1, 3, 4). This manuscript presents the derivation of equations describing ventilation-perfusion mismatch. In addition, clinical examples with worked solutions are provided to help students learn to manipulate the formulas necessary to assess respiratory failure. The practical limitations of these formulas are also discussed.

## MODEL

The formal method of modeling pulmonary dysfunction in a hypoxemic patient is the patient’s virtual shunt through his lungs. Figures 1 and 2 and *Equations 1*-*3* explain how this is done.

Figure 1*A* shows a healthy alveolus and pulmonary capillary. In this part of the lung, ventilation and perfusion are matched, and there is no venous admixture. Figure 1*B* shows a collapsed alveolus and a healthy capillary. Here, all of the mixed venous blood is shunted past the functional part of the lung. This is called venous admixture. Figure 1*C* shows a healthy alveolus and collapsed pulmonary capillary. This is the definition of dead space. It is included here to avoid confusion, although it is not part of the model.

Figure 2 shows how venous admixture to the systemic arterial blood can occur.

Conservation of blood flow dictates that the pulmonary capillary blood flow and shunted blood flow must equal the total cardiac output through the lungs (*Eq. 1*) 1 where Q̇_{T}, Q̇_{C}, and Q̇_{S} are cardiac output, pulmonary capillary blood flow, and shunted pulmonary capillary blood flow, respectively.

Conservation of mass dictates that the transport of oxygen through the lung is also conserved (*Eq. 2*) 2 where Ca_{O2} is the O_{2} content in the systemic arterial blood (pulmonary vein), C_{c}O_{2} is the O_{2} content in the pulmonary capillary, and C_{v̄}O_{2} is the O_{2} content in the shunted or mixed venous blood. Substituting *Eq. 1* into *Eq. 2* and rearranging yields the familiar virtual shunt equation. 3 To calculate shunt (*Eq. 3*), one must measure the partial pressure of O_{2} in the mixed venous and systemic arterial blood. In addition, one must estimate the partial pressure of O_{2} in the pulmonary capillary by calculating the partial pressure of oxygen delivered to the alveolus (Pa_{O2}; *Eq. 3A*) 3A where PB is atmospheric pressure, and PH_{2}O is the pressure of water vapor in the lung. Pa_{CO2} is the partial pressure of CO_{2} in the systemic arterial blood, and R is the respiratory quotient, which is assumed to be 1 in this model. The use of *Eq. 3* is impractical in routine clinical care because the mixed venous blood is rarely sampled.

To get around this, *Eq.* 3 was recast in nomogram form. Nunn assumed that the C_{v̄}O_{2} had a constant value and that the hemoglobin and Pa_{CO2} were within common ranges (2). Using these assumptions, he created the isoshunt lines shown in Fig. 3.

This nomogram allows one to estimate the patient’s virtual shunt and follow his pulmonary dysfunction as his Fi_{O2} and ventillatory therapy are adjusted by measuring his Pa_{O2}. Despite their simplicity, the isoshunt lines did not achieve popularity because pulmonary critical care involves treating more than a single parameter and because it was impractical to carry the nomogram around.

A number of years later, interest was renewed in *Eq. 3* with the advent of pulse oximeters and oxymetric pulmonary artery catheters. Manipulation of *Eq. 3* shows why. The oxygen content of blood is given by *Eq. 4* 4 where Po_{2} is the partial pressure of oxygen in the blood. The term (0.003 Po_{2}) represents dissolved O_{2} in the plasma and will be ignored here. If we assume that the amount of dissolved oxygen is negligible, then we can substitute *Eq. 4* into *Eq. 3* to yield. 5 With oximetry, both the arterial saturation and mixed venous saturation could be measured continuously to give a good estimate of shunt. Unfortunately, *Eq. 5* still involved a cumbersome calculation and the cost and morbidity of an oxymetric pulmonary artery catheter. For these reasons, *Eq. 5* never achieved common usage either. *Eq. 5*, however, can be simplified to make it useful for clinicians.

In patients with stable cardiac outputs who are administered a high concentration of O_{2} (Fi_{O2}), the difference between end-capillary saturation and mixed venous saturation is ∼0.25. Thus *Eq. 5* becomes 6 *Equation 6* is made usable by relating O_{2} saturation to Pa_{O2}. Reference to the oxyhemoglobin dissociation curve shows how this is done (Fig. 4).

At a Pa_{O2} >100 Torr, the graph is nearly a flat line. Thus O_{2} saturation and Pa_{O2} are related by the formula for a straight line (*Eq. 7*) 7 where *m* is the slope of the line, and* b *is the intercept on the saturation axis. Substituting *Eq. 7* into *Eq. 6* yields 8 where we have assumed that the Pa_{O2} is the same as Po_{2} in the pulmonary capillary. Using the slope (*m* = 1.4 × 10^{−4} saturation/Torr) on the flattest part of Fig. 4 allows *Eq. 8* to be recast as 9 If we desire to use shunt fraction rather than shunt, then *Eq. 9* becomes 10 *Equation 10* shows that the commonly used alveolar-arterial gradient divided by a constant yields an easy estimate of shunt when the patient’s cardiac output and hemoglobin are stable and when the Pa_{O2} and Pa_{O2} lie along the flat part of the oxyhemoglobin dissociation curve. Using a similar approach, the shunt fraction can be estimated when the Pa_{O2} < 100 Torr and the Pa_{O2} lies between 50 and 100 Torr. In this case, *Eq. 10* becomes 10A *Equations 10* and *10A* are only estimates of shunt fraction and are insensitive to larger gradients in the denominator of *Eq. 5*. Larger gradients in the denominator of *Eq. 5* may occur in some patients with decreased cardiac output or significant shunt. Nonetheless, as long as cardiac output is stable, *Eqs.* *10* and *10A* will track changes in shunt fraction as the patient’s pulmonary function improves or declines. If one wishes to account for larger gradients in *Eq. 5*, then the slope in *Eq. 7* must be adjusted to fit *Eqs. 10* and *10A* to clinical data.

The following clinical cases below are illustrative:

An 18-yr-old male with a history of asthma presents to the emergency department in respiratory distress. A nasal cannula supplies oxygen at 28%. His blood gas shows the following values: 7.32/31/74/23, O_{2} saturation 95%. In this case, his alveolar O_{2} (Pa_{O2}) is given by *Eq. 3A*. Thus Pa_{O2} = (760 − 47) · 0.28 − 31 = 168 Torr. The patients’ alveolar-arterial gradient is given by Pa_{O2} − [Pa_{O2}] = 168 − 74 = 94 Torr. Reference to Fig. 3 shows the patient has a virtual shunt of 15–20%. His estimated shunt using the Pa_{O2}-Pa_{O2} gradient (*Eq. 10*) is ∼6%.

Now consider a very different patient. A 74-yr-old male with a 50-pack/yr history of cigarettes undergoes a Whipple procedure. At the end of the surgery, the patient is intubated and ventilated with an Fi_{O2} of 50%. His blood gas shows the following values: 7.32/34/115/27, O_{2} saturation 98%. Using the same analysis as above, we find that this patient has Pa_{O2}-Pa_{O2} gradient of 207 Torr, a virtual shunt of 10–15%, and estimated shunt using the Pa_{O2}-Pa_{O2} gradient (*Eq. 10*) of 12%. This is summarized in Table 1.

Notice that *patient 1* (asthma) is sicker (virtual shunt = 15–20%) even though his Pa_{O2}-Pa_{O2} gradient is smaller than *patient 2*. This shows the value of the virtual shunt method, i.e., virtual shunt remains constant regardless of the Fi_{O2} administered. The Pa_{O2}-Pa_{O2} gradient, however, widens as the Fi_{O2} is increased, even in the absence of increased pulmonary dysfunction. The estimated shunt of *patient 1* (*Eq. 10*) is 6%, which compares poorly with his virtual shunt (*Eq. 3*). This is because his Pa_{O2} lies along the steep part of the oxyhemoglobin dissociation curve. In this situation, *Eq. 10* is not a good measure of venous admixture. Conversely, the estimated shunt of *patient 2* (12%) is similar to his virtual shunt. This is because both his Pa_{O2} and Pa_{O2} lie along the flat part of the oxyhemoglobin dissociation curve (Fig. 4).

Figure 5 shows the relationship between virtual shunt and estimated shunt. Notice that when the Pa_{O2} is greater than 100 Torr, the virtual shunt lines and estimated shunt lines have slopes and absolute values that are similar. Below 100 Torr, the estimated shunt (Pa_{O2} − Pa_{O2}) reproduces the virtual shunt poorly.

Now let’s look at the problem in a different way. Suppose we consider the right heart and lungs to be an engine whose job is to convert mixed venous blood into oxygenated blood. If the engine is ideal (Fig. 6*A*), then the blood will be fully saturated with oxygen when it leaves the engine.

If the engine is less than ideal (Fig. 6*B*), then the blood will be only partially oxygenated when it leaves the heart-lung engine. We can calculate an efficiency for this real engine by comparing it with the ideal engine (*Eq. 11*). 11 Substituting *Eq. 4* into *Eq. 11* yields 12 Now, recall that on the flat part of the oxyhemoglobin dissociation curve (Fig. 4), the O_{2} saturation = *m* × Po_{2} + *b* (*Eq. 7*). Substituting *Eq. 7* into *Eq. 11* yields 13 This is commonly called the a/A ratio. Our derivation shows that the a/A ratio is only a good measure of pulmonary dysfunction (venous admixture) when the Pa_{O2} and Pa_{O2} both lie along the flat part of the oxyhemoglobin dissociation curve.

Figure 7 shows the graphical relationship of Pa_{O2}/Pa_{O2} to the isoshunt lines. When the Pa_{O2}-Pa_{O2} ratio is near 1, isoshunt and Pa_{O2}/Pa_{O2} correlate well. As the ratio falls, the correlation becomes poorer because the Pa_{O2} no longer lies along the flat part of the oxyhemoglobin dissociation curve.

As an example, consider *patients 1* and *2* again. The efficiency of *patient 1* (Pa_{O2}/Pa_{O2}) is 0.44, whereas that of *patient 2* is 0.36. By comparing a/A ratios, *patient 1* appears healthier than *patient 2*, although his virtual shunt is greater. This is because his Pa_{O2} lies along the steep part of the oxyhemoglobin dissociation curve, and the a/A ratio is less accurate in this case. This is summarized in Table 2.

## SUMMARY

Virtual shunt remains the best way of quantitating venous admixture in patients with pulmonary pathology. It is a cumbersome technique that requires pulmonary artery catheterization and data from the oxyhemoglobin dissociation curve. The isoshunt lines approximate virtual shunt when the mixed venous saturation, hemoglobin, and Pa_{CO2} are within common ranges and are stable. Most clinicians do not carry this nomogram with them. The A-a gradient and a/A ratio both give a good estimate of venous admixture when the conditions for the isoshunt lines are met and when both Pa_{O2} and Pa_{O2} lie along the flat part of the oxyhemoglobin dissociation curve. Pa_{O2} is easily measured, and Pa_{O2} is easily calculated. For these reasons, they have become the standards in clinical care.

## PRACTICE PROBLEM

An 18-mo-old child presents to the emergency department with respiratory distress and a chest X-ray consistent with pneumonia. A facemask supplies oxygen at 35%. His arterial blood gas shows the following values: 7.31/29/67/23, O_{2} saturation 93%.

*1*) Use *Eq. 3A* to calculate the patient’s alveolar O_{2} (PAo_{2}). Answer, 220 Torr.

*2*) Calculate the patient’s alveolar arterial gradient, i.e., Pao_{2} − Pa_{O2}. Answer, 153 Torr.

*3*) Use Fig. 3 to estimate the patient’s virtual shunt. Answer, 25%.

*4*) Now use *Eq. 10* to estimate the patient’s shunt. Answer, 9%. Can you explain why *Eq. 10* gives such a poor estimate of virtual shunt? Hint, see the discussion of *patient 1*.

*5*) Now use *Eq. 19* to calculate the patient’s arterial-alveolar ratio. Answer, 0.3. Does *Eq. 13* overestimate or underestimate the patient’s illness and why? Hint, see the discussion of *patient 1*.

## Acknowledgments

Address for reprint requests and other correspondence: P. E. Bigeleisen, Dept. of Anesthesiology, Box 604, Strong Memorial Hospital, 601 Elmwood Ave., Rochester, NY 14642 (E-mail: Paul_Bigeleisenurmc.rochester.edu).

Received 12 November 2000; accepted in final form 4 May 2001

- © 2001 American Physiological Society