## Abstract

Functional values of LogSD of the ventilation distribution (σ_{V̇}) have been reported previously, but functional values of LogSD of the perfusion distribution (σ_{q̇}) and the coefficient of correlation between ventilation and perfusion (ρ) have not been measured in humans. Here, we report values for σ_{V̇}, σ_{q̇}, and ρ obtained from wash-in data for three gases, helium and two soluble gases, acetylene and dimethyl ether. Normal subjects inspired gas containing the test gases, and the concentrations of the gases at end-expiration during the first 10 breaths were measured with the subjects at rest and at increasing levels of exercise. The regional distribution of ventilation and perfusion was described by a bivariate log-normal distribution with parameters σ_{V̇}, σ_{q̇}, and ρ, and these parameters were evaluated by matching the values of expired gas concentrations calculated for this distribution to the measured values. Values of cardiac output and LogSD ventilation/perfusion (V̇a/Q̇) were obtained. At rest, σ_{q̇} is high (1.08 ± 0.12). With the onset of ventilation, σ_{q̇} decreases to 0.85 ± 0.09 but remains higher than σ_{V̇} (0.43 ± 0.09) at all exercise levels. Rho increases to 0.87 ± 0.07, and the value of LogSD V̇a/Q̇ for light and moderate exercise is primarily the result of the difference between the magnitudes of σ_{q̇} and σ_{V̇}. With known values for the parameters, the bivariate distribution describes the comprehensive distribution of ventilation and perfusion that underlies the distribution of the V̇a/Q̇ ratio.

- gas mixing
- gas exchange
- cardiac output

the concentration of oxygen in the pulmonary vein serving a region of the lung depends on the ratio of alveolar ventilation to perfusion in the region (V̇a/Q̇). If this ratio is non-uniform among regions of the lung, the partial pressure of oxygen in pulmonary mixed venous blood is lower than the concentration in mixed expired gas, and the efficiency of gas transport is less than ideal. In normal subjects, the difference in partial pressures is small, but in many pulmonary diseases, the nonuniformity of V̇a/Q̇ is increased, and lower efficiency of gas transport is a significant component of the disease. For the past four decades, the distribution of V̇a/Q̇ has been measured using the multiple inert gas elimination technique (MIGET), and the method has provided valuable information about this component of gas transport.

Much less is known about the distributions of ventilation and perfusion that underlie the V̇a/Q̇ distribution. Anatomic measurements of the ventilation and perfusion distributions have been made in animals and humans using a number of techniques. The resolution of all these anatomic techniques is limited; most of these techniques are restricted to use in animals, and in many cases the entire lung is not sampled. Because much of the non-uniformity of ventilation and perfusion occurs at small scale (6, 17), the anatomic measurements may well miss much of the variation. The functional distribution of ventilation has been measured in normal subjects using the helium wash-in technique (2), but we know of no reports that describe the functional variation of perfusion. Here, we report values for the widths of the ventilation and perfusion distributions and for the coefficient of correlation between ventilation and perfusion. These results were obtained by interpreting data on the wash-in of helium (He) and two soluble gases, acetylene (Ac) and dimethyl ether (DME).

## METHODS

#### Experiment.

Subjects were recruited from hospital staff of St. Mary's Hospital and Mayo Clinic (Rochester, MN). All subjects signed a written, informed consent that had been approved by the Mayo Institutional Review Board.

Subjects performed a graded cycle exercise test while expired gas was collected using a non-rebreathing valve that separated expired gas from inspired gas. A Hans-Rudolph screen-type pneumotachograph measured airflow at the mouth, and a respiratory mass spectrometer measured expired gas concentrations continuously. Data were sampled every 8 ms using a custom-data acquisition system running in Microsoft Windows. At selected times at rest and each work load, inspired gas was switched to a mixture of 9% He, 0.6% Ac, 0.5% DME, 21% O_{2} and nitrogen to perform an open-circuit gas uptake measurement. After 10 breaths, the valve was switched back to room-air inspirate. Analysis methods were similar to those described by Johnson et al. (10). Work loads were 0 (rest), 70, 140, and 210 W. Each exercise level was 4–5 min in duration. The first wash-in measurement was obtained 2 min after a change in intensity, and subsequent trials were obtained when end-tidal Ac concentration decreased to <10% of the inspired bag value. The protocol was not designed to determine maximal exercise capacity, although the 210-W level was likely close to maximal for most subjects.

Subject anatomic dead space was estimated for each trial by analyzing the He curve during expiration. For each breath, a value for the subject's serial dead space volume (V_{DS}^{S}) was obtained from the ratio V_{DS}^{S}/Vt = (C^{ET} − C^{E})/(C^{ET} − C^{0}), where Vt is tidal volume, and C^{ET}, C^{E}, and C^{0} denote end-tidal, mixed expired, and inspired concentrations of He, repectively. Values of V_{DS}^{S} were averaged over all the breaths for each trial. Equipment dead space (150 ml) was added to V_{DS}^{S} to obtain total dead space (V_{DS}).

#### Mathematical modeling and analysis.

The distributions of ventilation and perfusion are described as follows. The lungs are imagined to be divided into a large number of units with equal gas volumes at end expiration. The time-averaged ventilation of each unit, as a fraction of its end-expiratory volume, is denoted v̇. That is v̇ = *f* v_{T} where *f* is the breathing frequency and v_{T} is tidal volume of the unit divided by end-expiratory volume. Perfusion per unit end-expiratory volume is denoted q̇. The fraction of lung volume (*F*) occupied by units with given values of v̇ and q̇ is described by a bivariate log-normal distribution (16).
*Eq. 1*, μ_{v̇} and μ_{q̇} are the mean values of lnv̇ and lnq̇, respectively; σ_{v̇} and σ_{q̇} are the LogSD of v̇ and q̇; and ρ is the coefficient of correlation between the variations of lnv̇ and lnq̇ around their means. The values of the five parameters in *F*, namely μ_{v̇}, μ_{q̇}, σ_{v̇}, σ_{q̇}, and ρ, were determined by calculating the concentrations of the test gases at end-expiration for this distribution and minimizing the difference between the calculated and measured values.

The bivariate distribution was represented numerically by entries in a 33 × 33 array of cells in a spread sheet with lnv̇ and lnq̇ as the two variables that describe each cell. The range of the two variables extended from 4σ above the mean to 4σ below the mean. This array constituted a 1,089-compartment model of the lung, each with discrete values of v̇ and q̇.

The concentration of the test gases in each compartment were calculated by the following procedure. The rates of inspiration and expiration were assumed to be constant, and the duration of each phase was assumed to be half the period for a breath (*p* = 1/*f*). Thus, during inspiration, flow into a region is 2v̇, and during expiration flow is −2v̇. With this assumption, an analytical solution for the concentration as a function of time in a region with ventilation and perfusion, v̇ and q̇, can be constructed. For each breath, the time period of the breath is divided into three intervals: the first is the interval when gas is re-inspired from the dead space, the second is the inspiration of fresh test gas, and the third is expiration. The concentration in each interval was calculated as follows.

During inspiration, the volume of a compartment is 1 + 2v̇τ, where τ denotes time from beginning of inspiration. The mass balance for the region is expressed by *Eq. 2*, where α denotes the solubility of the test gas, v_{ti} denotes tissue volume per unit end-expiratory volume, *c*(*i*,τ) denotes the concentration of the test gas during the *i*th breath, and *C*^{in} denotes the concentration of the test gas in the gas flowing into the alveolar space.
*K*_{1} is a constant.

It was assumed that the volume of dead space gas that reenters each compartment is proportional to v̇ for the compartment and that the concentration of the test gas in the dead space equals that of mixed expired gas at the end of the previous expiration, denoted *C*^{exp}(*i* − 1, *p*). Therefore, in the first interval, 0 < τ < (V_{DS}/Vt)·*p*/2, *C*^{in} = *C*^{exp}(*i* − 1, *p*). At τ = 0, *c*(*i*,0) = *c*(*i* − 1, *p*). From this, *K*_{1} can be evaluated, and then *Eq. 3* can be used to obtain *c*[*i*,(V_{DS}/Vt)·*p*/2]. For the second interval, (V_{DS}/Vt)·*p*/2 < τ < *p*/2, *C*^{in} = *C*^{0}, where *C*^{0} is the concentration of the test gas in the inspired gas mixture. The value of *K*_{1} in this interval is obtained from *Eq. 3* and the value of *c* at the end of the previous interval. Then, *Eq. 3* is used to obtain the value of *c* at the end of the interval.

During expiration, *p*/2 < τ < *p*, regional volume is 1 + 2v̇·(*p* − τ), and the mass balance equation is the following.
*Eq. 4* has the following integral.
*K*_{2} is obtained from the value of *c*(*i*,τ) at end inspiration, and *Eq. 5* then gives *c*(*i*,τ) during expiration.

The concentration of the test gas in the mixed expired gas *C*^{exp}(*i*,τ) was calculated from the following equation, where v̇̄ denotes the mean value of v̇.

The Bunsen solubilities of Ac and DME were taken as 0.74 (3) and 9 ml/ml·atm (11, 14), respectively, the value of v_{ti} was assumed to be 0.2, and the concentration in each compartment at each breath was evaluated numerically, using the analytical equations for *c*(*i*, τ). Then, *Eq. 6* was evaluated numerically to obtain values for *C*^{exp}(*i*, τ), which were required for the calculation of *c*(*i* + *i*, τ), and for *C*^{exp}[*i*, (1 − V_{DS}^{s}/2Vt)*p*], where V_{DS}^{s} is the subject dead space. The second is the value of *C*^{exp}(*i*) that appears at the mouth at end-expiration, and these values were compared with the experimental values of *C*^{ee}(*i*).

The average values of ventilation and perfusion, v̇ and q̇, are related to the parameters of the distribution by the following equations (16).

Because total ventilation (V̇) was obtained independently from *f*·Vt and v̇̄ = V̇/V_{ee}, values of V_{ee} can be obtained from *Eq. 7*. Also, q̇̄ = Q̇/V_{ee} and values of Q̇ can be obtained from *Eq. 8*.

#### Data analysis.

To avoid the effect of recirculation of blood on expired gas concentration, data for the soluble gases for breaths that occurred later than the recirculation time were discarded. Recirculation time was estimated by assuming that blood volume was 5.5 liters and using the expression 5.5*f*/Q̇ to calculate the last acceptable breath number.

For each run, the values of μ_{v̇} and σ_{v̇} were obtained by fitting the calculated values of *C*_{He}^{ee}(*i*) to the measured values. Then the values of μ_{q̇}, σ_{q̇}, and ρ were chosen to obtain the best fits of the calculated values of *C*_{Ac}^{ee}(*i*) and *C*_{DME}^{ee}(*i*) to the measured values.

Nine subjects were studied. The data for two subjects were discarded because their breathing frequencies during exercise were low, and the acceptable breath numbers were <5, too few to allow values of μ_{q̇}, σ_{q̇}, and ρ to be determined accurately. Data are reported for the remaining seven subjects, four men and three women. All were reasonably fit young adults with no history of smoking or pulmonary disease. For these subjects, all 10 breaths were accepted for the resting state, and the average acceptable breath number decreased to ∼8.0 ± 0.4 for exercise.

Two of the seven subjects could not reach the highest exercise level, and three were not able sustain the highest level for more than one run. Also, the breathing patterns for some runs were irregular, and the data for four runs were discarded, two because the range of Vt was >0.5 times the mean Vt, and two because the range of breath durations exceeded 0.5 times the mean. For cases with two acceptable runs (19/26), the parameter values for the two runs were first averaged. Then, the values for each state for the seven subjects (five for heavy exercise) were averaged.

Sample data and fits are shown in Fig. 1. The average rms difference between the calculated and measured concentrations was 0.009 ± 0.001. The 95% confidence limits for the crucial parameters of the distribution are given in Table 1. Also, the SD of the values for the pairs of runs for given subject/exercise level cases are shown in Table 1. These are approximately half the size of the 95% confidence limits, as would be expected because the 95% confidence limits extend to 2 SDs of the expected error distribution.

All results are reported as means ± SE. Statistical significance was assessed by paired *t*-test.

## RESULTS

The whole lung properties, minute ventilation (V̇) and cardiac output (Q̇), are given in the first two lines of Table 2, and the plot of Q̇ vs V̇ is shown in Fig. 2. Other whole lung properties are the following. Vt was 1.1 ± 0.1 liter at rest and increased to 2.4 ± 0.1 liter at the highest exercise level. Breathing frequency was set at 20 breaths/min at rest and increased to a maximum of 35 ± 3 breaths/min. Subject dead space volume was 0.23 ± 0.01 liter at rest and increased with increasing exercise to 0.36 ± 0.01 liter. End-expiratory volume was 3.3 ± 0.5 liters at rest and decreased to 2.5 ± 0.2 liters with exercise.

The primary results of this study, the values of the parameters of the bivariate distribution of ventilation and perfusion, are given in the next five lines of Table 2. Because ln *v̇*/*q̇*= ln *v̇*-ln *q̇*. LogSD v̇/q̇, denoted σ_{v̇/q̇}, is related to σ_{v̇}, σ_{q̇}, and ρ by the following equation

The values of σ_{v̇/q̇} are given in the last line of Table 2, and the values of the σ are also shown in Fig. 3.

## DISCUSSION

In previous studies (e.g., Ref. 2), data on He wash-in have been used to obtain information about the width of the ventilation distribution. For an insoluble gas, the bimodal distribution (*Eq. 1*) reduces to a log-normal distribution for the variable v̇, and we have used the same method as that described in Ref. 2 to obtain values for the parameters μ_{v̇} and v̇. Previously (e.g., Refs. 5, 10), data on He and Ac wash-in have been used to obtain estimates of Q̇. Here, we have added a second soluble gas, DME, to the wash-in. The solubility of DME is ∼10 times that of Ac. With data for these two gases with quite different solubilities, values of the remaining parameters of the bimodal distribution, μ_{q̇}, σ_{q̇}, and ρ, could be obtained with reasonable accuracy. Values of Q̇ and σ_{v̇/q̇} follow from these.

Assumptions were made in the modeling, and the validity of the results depends on the accuracy of these assumptions. First, we assumed that the distribution of v̇ and q̇ is a bivariate log-normal distribution. We have no independent support for this assumption. However, for normal subjects, the distributions of alveolar ventilation (v̇_{A}) and q̇ as functions of v̇_{A}/q̇ obtained from MIGET are nearly log-normal distributions (14), and the bivariate log-normal distribution of v̇ and q̇ is consistent with these (see appendix). In disease, this assumption would be questionable. We used an average recirculation time to determine the number of acceptable breaths for the fit of the model. Circulation times for different organs are different, and partial recirculation would occur before the average. To assess the possible effect of recirculation, the 14 cases in which the last acceptable breath was the seventh or eighth breath were studied. The difference between the measured concentration for the second breath after the last acceptable breath was compared with the concentration predicted by the model for that breath. In all 14 cases, the measured concentration was higher than the predicted for both Ac and DME. This difference, apparently evidence of recirculation, was 0.026 ± 0.012 (SD) for Ac and 0.014 ± 0.011 for DME. The effect of recirculation would be expected to be greater for DME than for Ac, but it was smaller, perhaps because the solubility for DME in water is 20 times that for Ac, and more DME than Ac was removed from the blood during its transit through the body. The differences between measured and predicted concentrations for the last acceptable breath were both positive and negative and averaged 0.008 ± 0.008 for Ac and −0.001 ± 0.008 for DME. Thus partial recirculation may have affected the concentration of Ac for the last accepted breath, but the effect is smaller than the rms difference between measured and predicted concentrations. The same values of gas solubility were used for all subjects. Barker et al. (1) measured the solubility of Ac in the blood of individual subjects and found a variability among subjects of approximately ±10% of the mean. Individual variability of solubility would affect the values of the perfusion parameters for individual subjects but would have less effect on the average values. We assumed that the volume of the dead space was distributed in proportion to volume expansion, that the compartments were well mixed with no serial inhomogeneity, and that the compartmental inspirations and expirations occurred in phase. Diffusional transport was neglected. Thus, whatever the mechanisms that generate the slope of phase III in the wash-in of insoluble gases, those mechanisms are not included in our model. The magnitude of the effect of these mechanisms on our results is unknown. Finally, we assumed that inspiratory and expiratory flows were constant and that the breathing pattern was regular. In fact, irregularities occurred, and we think that the major source of the rms difference between the measured and model values of expired concentration is the breath-to-breath variability of Vt during the wash-in.

Some of the values listed in Table 2 can be compared with values in the literature. In the experiments, the subjects at rest were asked to breathe in time with a metronome set at 20 beats/min. As a result, the value of V̇ for rest is about twice the value that would be expected for natural breathing, but the value for Q̇ agrees with the expected value. During exercise, the subjects breathed freely, and the relation between Q̇ and V̇ shown in Fig. 2 lies within the narrow band of values obtained by MIGET (4, 7, 8, 12). In several studies (e.g., Refs. 5, 10), estimates of Q̇ have been obtained from data on wash-in of He and Ac. In this work, distributions of v̇ and q̇ are ignored, and the data are interpreted by a one-compartment model. It has been thought that neglecting the variations in v̇ and q̇ would lead to an underestimate of Q̇, but the magnitude of this error has been uncertain. We applied the methods of Gan et al. (5) and Johnson et al. (10) to our data for He and Ac wash-in. For exercise states, the values of Q̇ obtained by the method of Gan et al. were ∼2 liters/min lower than ours, but the values obtained by the method of Johnson et al. were nearly the same as ours.

The values of σ_{v̇} for rest and light and moderate exercise, ∼0.47 ± 0.10, are somewhat smaller than the values reported by Beck and Wilson (2), also obtained from He wash-in data. These lower values were the result of values of nearly zero for one subject. The average for the other six subjects is near the value of ∼0.6 reported by Beck and Wilson.

The decrease in the value of σ_{q̇}, from 1.08 ± 0.12 at rest to 0.85 ± 0.09 at the onset of exercise, is statistically significant (*P* < 0.01). For the exercise states, σ_{q̇} of ∼0.8 ± 0.1 is significantly larger than σ_{v̇}. For light and moderate exercise, ρ is quite large, ∼0.86 ± .06. There are no previous reports of the functional values of σ_{q̇} and ρ with which the values shown in Table 2 can be compared.

The inhomogeneities of ventilation and perfusion are large. For a variable with LogSD = 0.55, the value of the variable at one LogSD above the mean is three times the value at one LogSD below the mean. Also, although the LogSDs of v̇ and q̇ change little with increasing levels of exercise, the arithmetic SDs of v̇ and q̇ increase markedly because v̇̄ and q̇̄ increase. The width of the perfusion distribution is significantly larger than the width of the ventilation distribution, and ventilation and perfusion are highly correlated, particularly during light and moderate exercise. The bivariate distribution for parameter values for moderate exercise is shown in Fig. 4.

The relations between our comprehensive distribution of ventilation and perfusion and the distributions of ventilation and perfusion obtained from MIGET are analyzed in the appendix. This analysis shows that our parameter, σ_{v̇/q̇}, is equivalent to LogSD V̇_{A} obtained from MIGET and, therefore, that our value for this parameter can be compared with values obtained in studies of gas exchange during exercise using MIGET. Our values are higher than those reported in Refs. 7, 12, and 13, and about the same as those reported in Refs. 4, 8, and 9.

Although LogSD V̇_{A} can be obtained from multiple gas wash-in data, this method cannot match the capabilities of MIGET. It cannot discern changes in the shape of the V̇/Q̇ distribution that may occur in disease, and it does not detect shunt.

The parameter σ_{v̇/q̇}^{2} is particularly important because inefficiency of gas transport is proportional to σ_{v̇/q̇}^{2}. The primary significance of the results reported here is the information that they provide about the values of σ_{v̇}, σ_{q̇}, and ρ that underlie the value of σ_{v̇/q̇}^{2}. We find that σ_{q̇} is higher than σ_{v̇} and that, for light and moderate exercise, ρ is quite high. If ventilation and perfusion were uncorrelated, σ_{v̇/q̇}^{2} would be 0.9 for moderate exercise. Because of the correlation between the variations in ventilation and perfusion, the value of σ_{v̇/q̇}^{2} during moderate exercise is 0.3. With ventilation and perfusion highly correlated, the magnitude of σ_{v̇/q̇} is largely the result of the difference in the magnitudes of σ_{v̇} and σ_{q̇}.

## GRANTS

This work was supported by National Institutes of Health Grants HL-71478 and 1KL2 RR-024151.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: K.C.B., B.D.J., T.P.O., and T.A.W. conception and design of research; K.C.B., B.D.J., T.P.O., and T.A.W. performed experiments; K.C.B., B.D.J., T.P.O., and T.A.W. analyzed data; K.C.B., B.D.J., T.P.O., and T.A.W. interpreted results of experiments; K.C.B., B.D.J., T.P.O., and T.A.W. prepared figures; K.C.B., B.D.J., T.P.O., and T.A.W. drafted manuscript; K.C.B., B.D.J., T.P.O., and T.A.W. edited and revised manuscript; K.C.B., B.D.J., T.P.O., and T.A.W. approved final version of manuscript.

## ACKNOWLEDGMENTS

The authors thank the participants who took part in this research, and we thank Dr. James P. Butler for guidance in the formulation of the analysis in the appendix.

## APPENDIX

The relations between our description of the distribution of ventilation and perfusion [*F*(ln v̇, ln q̇)], and the distributions of ventilation and perfusion determined by MIGET can be obtained as follows. First, the variables of the MIGET distributions are regional alveolar ventilation (v̇_{A}) rather than regional ventilation (v̇) and perfusion (q̇). With our assumption that the dead space volume (V_{DS}) is distributed in proportion to v̇, v̇_{A} is simply related to v̇ by the following equation.
*F*(ln v̇, ln q̇) is the bivariate log-normal distribution given by *Eq. 1*, the distribution of v̇_{A} and q̇, denoted *F*(ln v̇_{A}, ln q̇) has the same form as *Eq. 1*, namely *Eq A2*.
*Eq A2*, μ_{v̇A} = μ_{v̇} − ln β, but the other parameters of the distribution are the same as those of *Eq. 1*.

The ventilation and perfusion distributions of MIGET, denoted V̇[ln(v̇_{A}/q̇)] and Q̇[ln(v̇_{A}/q̇)] here, are defined as the amounts of ventilation and perfusion feeding regions with values of v̇_{A}/q̇ within *d* v̇_{A}/q̇ of a given value of ln v̇_{A}/q̇. V̇[ln(v̇_{A}/q̇)] is obtained from *F*(ln v̇_{A}, ln q̇) by summing or integrating the ventilation along the lines of constant ln (v̇_{A}/q̇) in the ln (v̇_{A}, ln q̇) plane. Similarly, Q̇[ln(v̇_{A}/q̇)] is obtained by integrating V_{ee}·q̇·*d* ln(v̇_{A}/q̇) along a line of constant ln(v̇_{A}/q̇). These integrals can be expressed by multiplying the integrands by δ(ln v̇_{A} − ln q̇ − ξ) where δ(*x*) is the Dirac delta function, defined by the properties δ(*x*) = 0 for *x* ≠ 0 and ∫δ(*x*)·*dx* = 1. V̇(ξ) and Q̇(ξ) are then given by the following equations.
^{ln v̇A} = e^{ξ + ln q̇} along the line of integration, and ξ is treated as a parameter in evaluating these integrals, it can be seen that V̇(ξ) = e^{ξ} Q̇(ξ). Thus V̇(ξ) and Q̇(ξ) are not independent.

If *F*(ln v̇, ln q̇) is the bivariate ln-normal distribution (*Eq. A2*), the integrals in *Eqs. A3* and *A4* can be evaluated analytically, although the algebra is tedious. The results are the following.
_{A} and Q̇ is a bivariate log-normal distribution, both Q̇(ξ) and V̇(ξ) are log-normal distributions. Also, LogSD V̇_{A} = LogSD Q̇, and both equal σ_{v̇A/q̇} of the underlying distribution. The peaks of the two distributions are separated by σ_{v̇A/q̇}^{2}. The ventilation and perfusions distributions, as defined in MIGET, for our values of v̇̄_{A}/q̇̄ and σ_{v̇/q̇} for moderate exercise are shown in Fig. 5.

Nearly the same analysis was given in the appendix of Ref 16. It is repeated here because it is central to this paper and because of typographical errors in Ref 16.

- Copyright © 2012 the American Physiological Society