Annals of Biomedical Engineering, Vol. 13, pp. 475-490, 1985 Printed in the USA. All rights reserved.

0090-6964/85 $3.00 + .00 Copyright 9 1985 Pergamon Press Ltd.

CHARACTERISTICS OF ~ / Q DISTRIBUTIONS RECOVERED FROM INERT GAS ELIMINATION DATA Chi-Sang Poon Heung Kuk Kim Department of Electrical and Electronics Engineering North Dakota State University Fargo, North Dakota

The resolving powers of the enforced smoothing and log-normal parametric estimation techniques in recovering ventilation/perfusion ratio (I?A/(2) distributions were evaluated using noisy inert gas elimination data simulated from hypothetical distribution functions representing various degrees of heterogeneity. The resolving powers were assessed in terms of the statistical recoverabilities of the shunt, dead space, modality, and modal moments characterizing the perfusion distribution. For all distributions tested, both modal mean and shunt were estimated by either technique with sufficient accuracies. Modal dispersions (a) were consistently overestimated by up to 0.15 decade for narrow distributions, but the mean errors became negligible for o greater than 0.2 decade. As compared with the shunt, the dead space estimates were more variable and biased, probably due to their indirect estimation from the perfusion distribution, which was imperfectly recovered. Both broad unimodal and widely separated bimodal or trimodal distributions (o > O.6 decade) were recovered as bimodal distributions of similar forms, so that detection of modality was difficult. The recoveries by both techniques were comparable in most cases studied, except that parametric estimation generally tended to be more sensitive to measurement errors and was computationally less efficient. These results provide a useful basis for the interpretation of (/A/(2 distributions obtained from empirical inert gas data. Keywords - Enforced smoothing, Nonlinear parametric estimation, Pulmonary shunt, Pulmonary dead space, Pulmonary gas exchange.

INTRODUCTION The pattern of inert gas elimination from the lung (21,23) is a useful test for uneven distribution of pulmonary ventilation/perfusion ratios (I?A/Q). However, despite the great recent interest in this subject, the validity of the test in recovering the continuous I?A/Q distribution has remained uncertain. Acknowledgement-This work was supported by American Heart Association (Dakota Affiliate) Grant DA-G-19 and National Institutes of Health Grants HL-30794, RR-02142, and RR-07206. Dr. Poon was the recipient of the 1983 Harold Lamport Young Investigator Award, presented by the Biomedical Engineering Society for a preliminary report of this work. Address correspondence to C.-S. Poon, Department of Electrical and Electronics Engineering, North Dakota State University, Fargo, ND 58105. 475

476

C.-S. Poon and H.K. Kim

One major difficulty is that the unknown variables required for complete characterization of I?A/(2 distribution are generally in great excess of the available data; consequently, the measurement system is highly underdetermined with nonunique solutions (11,16). Evans and Wagner (4) and Olszowka and Wagner (12) suggested that a representative distribution compatible with the measured data could be found by using the numerical technique of enforced smoothing (ES). More recently, Stewart and Mastenbrook (15) proposed a nonlinear parametric estimation (PE) technique based on a log-normal model of the I?A/(2 distribution (17). These numerical procedures have been extensively tested with real inert gas data (4,5,14,15,22) but have not been fully validated against known distribution functions of the general form. Furthermore, whereas the limits of ES have been variously suggested in qualitative terms under specific circumstances, a quantitative and systematic evaluation of both numerical techniques under carefully controlled conditions (i.e., with known distribution data) is presently unavailable. Accordingly, the present study was undertaken to examine the resolving powers of the ES and PE techniques in recovering ~ / ( 2 distributions. Since complete recovery is generally impossible, a more useful approach is to recover some partial features of the distribution that might be of theoretical and clinical significance. We define the resolving power in terms of the recoverabilities of some configurational parameters characterizing the I?A/(2 distribution. Since the recoverabilities may vary with the particular form of distribution (4,14), the resolving powers of both techniques were examined using widely varying distribution characteristics of clinical interest. The theoretical results provide a useful basis for the interpretation of I?A/(2 distributions obtained from empirical inert gas data. METHODS

Computational Aigorithms In the mathematical formulation of Wagner et al. (4,12,21), the lung is divided into 50 compartments of ventilation/perfusion ratios (I?A/(2)j, j = 1, 2 . . . . . 50. Compartment numbers 1 and 50 correspond to the pulmonary shunt and dead space (12A/(2 = 0 and co), respectively. The other I/A/(2 compartments are assumed to be equally spaced on a logarithmic scale of I)A/(2 values ranging from 0.005 to 100. The ES algorithm used in this study was similar to that described by Wagner and associates (4,12). Briefly, the compartmental perfusion fractions (qj, j = 1, 2 . . . . , 50) were chosen to minimize an objective function: s , = [ I W ( R - A q ) l l = + ZllUqll ~

(1)

where q is the vector of 50 compartmental perfusions; R is the vector of six measured retentions of inert gases having blood/gas partition coefficients ki,

477

Characteristics of ~ /Q Distributions

i = 1, 2 . . . . . 6; A is a 6 x 50 matrix with elements A u = ki/[Xi + ( ~ / O ) j ] ; W and U are diagonal weighting matrices (with appropriate dimensions) for uniform error sensitivity and equal compartmental smoothing, respectively (4); and Z is a smoothing parameter equal to 40 for the above weighting factors. The area and nonnegativity constraints on compartmental perfusions were enforced by a Lagrangian and a systematic elimination method, respectively, as described by Evans and Wagner (4). As an alternative approach to recover the ~ / Q distribution, the compartmental perfusions were assumed to be described by a bimodal log-normal function of the form: l qj

=

A, exp

[ ,

2

(xj -

m,) z]

d~

J + A2 exp

[ ,

2

(xj

m )21

d~

J

j = 2, 3.... ,49

(2)

where xj = ln(PA/Q)j; the parameters Ak, mk and dk (k = 1, 2) are, respectively, the peak, mean, and standard deviation of the kth log-normal mode. The six model parameters, together with the shunt and dead space fractions, constitute a set of eight unknowns that must satisfy the six retention data and the area constraints on the compartmental ventilation and perfusion fractions (vj and qj, respectively). This was accomplished by the minimization of the following objective function: S2=IIW(R-Aq)II:+~

{( l - ~ q,9 i)2[+ 1--OD

qj( A/Q)j

]2}

(3)

j=l

where uD is the dead space fraction; I,>Tand QT are the total ventilation and perfusion, respectively; and/~ is a penalty factor for the area constraints on vj and qj and was arbitrarily set by a sequential search procedure (1). The minimization was performed using the Levenberg-Marquardt algorithm (8), which is known to be highly efficient for nonlinear least-squares problems of the given form. A solution was considered to be acceptable when all terms in Eq. 4 were satisfied to within 1~ accuracy. Test Protocols

The resolving powers of both numerical algorithms were evaluated using a wide range of test distributions with varying modalities, shunt fractions, modal locations, and dispersions. For simplicity and without loss of generality, the test distributions were taken to be composed of simple rectangular functions of various combinations. A rectangular distribution corresponds to complete heterogeneity of all compartments over a given I?A/Q range; this represents the greatest degeneration of gas exchange in these lung units. While 1With suitable overlapping of the two modes this model m a y also be used to describe any unimodal distribution o f log-normal form.

478

C.-S. Poon and H.K. Kim

such conditions may not be typical of ~ / Q maldistribution in general, these distributions provide the most severe tests of recoverability, so that the resolution limits with each numerical algorithm may be defined. 2 For the purpose of analysis, the resolution of each algorithm was evaluated as a function of the modality, shunt, and the modal mean and dispersion of different distribution forms. These variables were of particular interest because of their direct correspondence with visual inspection and possible relevance to clinical diagnosis. All distributions were assumed to have a total blood perfusion of 6 1/min and a shunt and dead space of 2 and 30~ respectively, unless otherwise specified. For each test distribution, the retentions and excretions of six inert gases with blood/gas partition coefficients of 0.0599, 0.08556, 0.5348, 2.339, 12.48, and 285.2 were computed using the well-known inert gas elimination equations (12,21). These k values correspond to the gases sulfur hexafluoride, ethane, cyclopropane, enflurane, ether, and acetone, respectively, which are the most frequently used gases experimentally. For the purpose of statistical analysis, 10 sets of noisy retentions and excretions were simulated by repeatedly adding normally distributed random numbers to the corresponding error-free data. The errors were assumed to be at a coefficient of variation of 3% for all gas concentrations (5% for sulfur hexafluoride) and 5~ for all partition coefficients (10% for sulfur hexafluoride), cardiac output, and minute ventilation. These error levels are consistent with the experimental limits of current practice (20). For minimum error sensitivity, corresponding errorcontaining retention and excretion were pooled together to give the corrected retention:

R~=(1-t)Ri+t 1 - XiQTE

i=1,2,...,

6

(4)

where t is a weighting parameter (0 ___t ___1) determined by the measured data and their error variances (4). Thus, only the numerical algorithms based on retention data were used to recover the perfusion distributions. As a basis of analysis, the resolutions of both numerical techniques were assessed in terms of the statistical accuracies of the shunt, dead space, modal mean, and modal dispersion of the resulting distributions. For each set of noisy data (n = 10) the sample mean and standard deviation of the recovered distributions and of the errors in each parameter estimate were computed using unbiased statistics. All computations were performed on an IBM 4341 mainframe computer in FORTRAN IV programming. PE usually required more extensive computation than did ES (typically 30 sec or more vs. 3 sec), depending on the initial parameter estimates. 2Among the class of distributions within a given ~ / Q range, the rectangular function always assumes the m i n i m u m norm (i.e., ~q} is minimum). Since ES and PE always result in relatively smooth functions with n o n m i n i m u m norms, the errors in recovering a rectangular distribution must also be the greatest. This represents a "worst case" test for the recovery procedures so that the resolution limits under all possible conditions may be defined.

Characteristics of ~ /Q Distributions

479

RESULTS

Modal Dispersion Test The variabilities of the resolutions for various modal dispersions were investigated using rectangular test functions with increasing widths (mean I?A/Q = 1). Figure 1 shows that for modal dispersions of up to 0.7 decade both the modal mean and shunt were accurately recovered by either technique. The errors in the dead space estimates were more significant and variable. The dispersion was overestimated by up to 0.15 decade for narrow distributions, but the mean errors became negligible for distribution dispersions of greater than 0.2 decade. For even broader distributions (a > 0.6 decade) both techniques resulted in distinctly bimodal functions (Fig. 2) so that inferences about the modality could be erroneous without prior knowledge of the true distribution.

Modal Location Test The dispersion test shows that for wide enough distributions ( a > 0.2 decade) the dispersion and other modal characteristics can be reliably recovered by both ES and PE techniques. To determine if this might depend on the location of the distribution relative to the six partition coefficients, we examined the resolving power of both techniques for a test distribution with

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Modal Dispersion (decade)

Modal Dispersion (decade)

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F I G U R E 1. Effects of modal dispersion (o) on the resolutions of characteristic parameters obtained by enforced smoothing (open circles) a n d p a r a m e t r i c estimation (closed circles). Data are mean errors • SD. The test distribution was a rectangular function with varying widths (mean VA/O = 1).

480

C.-S. Poon and H.K. Kim ENFORCED SMOOTHING

ENFORCED SMOOTHING

0,5' o.4' 0,3'

os1

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PARAMETRIC ESTIMATION

"c

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1;o

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,,

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....

o.o1 o.1 1.o I'o Ventilation-Perfueion Ratio

1;o

FIGURE 2. Recovery of a rectangular distribution (mean VA/(~ = 1, solid lines) by enforced smoothing (upper panel) and parametric estimation (lower panel). Left: a = 0 . 1 2 decade. Right: a = 0 . 7 2 decade. Data are means _+ SD. The distributions obtained by both techniques were unimodal when narrow and became distinctly bimodal for a > 0 . 6 decade.

critical dispersion (a = 0.2 decade) and varying mean ~ / Q . Figure 3 shows that for a wide range of mean ~ / Q values (from 0.025 to 5) the recoveries of all distribution parameters were quite satisfactory and similar to those obtained at a mean I?A/Q of 1. Since a wider distribution may be even more closely recovered (Fig. 1), the dispersion threshold of 0.2 may apply to all distributions with a mean ~ / Q in the above range. This shows that the chosen partition coefficients are satisfactory in providing uniform resolutions for a wide range of ~ / Q distributions.

Modal Separation Test The abilities of the two numerical techniques in resolving a bimodal distribution are demonstrated in Fig. 4. The original distribution consisted of two identical narrow modes 3 separated at various I?A/Q intervals centered at 1. Despite the difference in modality, the recovery patterns in all distribution parameters were remarkably similar to those for a unimodal distribution (Fig. 1). For modal separations of less than 1.2 decade (tr = 0.6 3The narrowest distribution that can be realized in a 50-compartment model has a span of approximately 0.1 decade on the given I?A/~ range.

Characteristics of fzA/() Distributions

481

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FIGURE 3, Effects of modal mean on the resolutions of characteristic parameters obtained by enforced smoothing (open circles) and parametric estimation (closed circles). Data are mean errors _+ SD. The test distribution was a rectangular function with critical dispersion (a = 0.2 decade) and varying mean MA/Q values.

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FIGURE 4, Effects of modal separation interval (S) on the resolutions of characteristic parameters obtained by enforced smoothing (open circles) and parametric estimation (closed circles). Data are mean errors _+ SD. The test distribution consisted of t w o identical n a r r o w modes (centered at 1 ) so that a = S/2. The recovery patterns are remarkedly similar to the unimodal case (Fig. 1).

482

C.-S. Poon and H.K. Kim

decade), each algorithm separately recovered a unimodal distribution which closely resembled that obtained from a unimodal distribution with similar dispersion (Fig. 5). Wider modal separations resulted in similar bimodal distributions that, however, were indistinguishable from those recovered from a broad unimodal distribution over the same ~ / Q range. Thus, discrimination of the modality by either technique was difficult at all separation intervals.

Modal Overlapping Test The recovery patterns with overlapping modes are displayed in Fig. 6. The original distributions were in the form of a narrow mode on top of a broad base, with varying widths but equal area and mean ( ~ / Q = 1). This situation may arise, for example, in unilateral pulmonary disease, where ~ / Q impairment is localized in only one lung lobe. As in the unimodal case, the errors in all parameter estimates were negligible for distribution dispersions of greater than 0.2 decade. Both techniques, however, failed to reliably resolve the overlapping narrow mode; the recovery by PE being somewhat closer but more variable (Fig. 7).

Multimodality Test Figure 8 shows the recoveries of a trimodal distribution consisting of three identical narrow modes separated at equal intervals of approximately 0.75

01 0.4

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0,2

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ENFORCED SMOOTHING

ENFORCED SMOOTHING

0.4

o.,

~ o.,

~= o.o "c

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o.'ol

o'.1

, 1.o

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t;o

a.

== o.o ~, ;j "c o

o.~1

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1.o

lo

o.

PARAMETRIC ESTIMATION

~

~. o.4

.'- 0.4

0.3

l

0.3

0.2

0.2

0.1 0.0

PARAMETRIC ESTIMATION

9. ~ 0

;" 0.;1

011

t 1.0

0.1 0.0 10

Ventilation-Perfusion Ratio

1;0

'11:~,| Ir ()

:~' 0,~)1

i 0.1

/ 1.0

~, /~ 10

100

Ventilation-Perfusion Ratio

F I G U R E 5. Recovery of a bimodal narrow distribution (mean VA/(2 = 1, solid lines) by enforced smoothing (upper panel) and parametric estimation (lower panel). L e f t : ~ = 0 . 3 5 decade. Right: a = 0 . 6 2 decade. Data are means _+ SD. The recovered distributions closely resemble those obtained from unimodal functions w i t h similar d i s p e r s i o n s (Fig. 2).

of ~ /0_ Distributions

Character&tics

483

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Modal Dispersion (decade)

Model Dispersion (decade)

4 0.15

0I

0.4

iii. 1311!~ O ......................... i ~ 9 O 0.1 0.2

Modal Dispersion (decade)

~ 0.3

i 0.4

Modal Dispersion (decade)

FIGURE 6. Effects of dispersion of overlapping modes on the resolutions of characteristic parameters obtained by enforced smoothing (open circles) and parametric estimation (closed circles). Data are mean errors _+ SD. The test distribution consisted of a narrow mode on top of a rectangular base with varying widths but equal area and mean (t/A/(2 = 1 ). The recovery patterns are remarkedly similar to the unimodal case (Fig. 1).

decade (o = 0.6 decade). Not surprisingly, the recovered distributions were all bimodal in form and were remarkably similar to those obtained from a broad unimodal or bimodal distribution with similar dispersions. Thus, the resulting distribution form was dictated by the distribution dispersion rather than the actual modality. Shunt Test

For a typical distribution (a = 0.12 decade) the shunt fraction was accurately recovered by either technique over a range of up to 40% of total blood flow (Fig. 9). The accuracies in the other parameter estimates were similar to those given in Fig. 1 and were not significantly affected by changes in the shunt. The complete recoveries with both techniques are illustrated in Fig. 10. DISCUSSION Application of the multiple inert gas elimination test in describing I?A/(~ distribution requires an understanding of the allowable resolution as well as its variability with different data and measurement errors. Although various numerical techniques have been developed and extensively employed in the analysis of clinical inert gas data, the validity of the resulting distributions has not been fully established. Thus, earlier studies on the resolution of the

484

C.-S. Poon and H.K. Kim

ENFORCED SMOOTHING 0.5

'

0.4' 0.3" 0.2" 0.1~

0.0 .

~

,

:1

0

o.s

,

9

0.01

0.1

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PARAMETRIC

'0

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ESTIMATION

LI. 0 . 4

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=, ,, 0

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0.01

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0.1

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1.0

10

100

Ventilation-Perfusion Ratio FIGURE 7. Recovery of a distribution with overlapping modes (solid lines) by enforced smoothing

(upper panel) and parametric estimation (lower panel). Data are means +_ SD.

multiple inert gas elimination test with a least-squares approach (7,18,21) were confounded by the nonuniqueness problem. Subsequently, Pimmel and coworkers (13,19) presented a theoretical analysis of the ES algorithm using simulated data but did not include the nonnegativity and area constraints, which were found to be critical for meaningful solutions. Furthermore, all previous studies have only considered the recoverabilities of relatively few test distributions of special forms so that a general characterization was not obtained. On the other hand, several investigators have addressed the resolutions of the ES (5,14) and PE (14,15) techniques by testing with real inert gas elimination data. However, since the true distributions representing the data are unavailable for verification, interpretation of the results is difficult. Until the present, systematic evaluations of the complete ES algorithm have not been reported other than in abstract form (3), and similar studies of the PE technique are lacking. The present study demonstrates the practical resolution limits of the ES and PE techniques in recovering ~ / Q distribution. The

Characteristics of ~ /Q Distributions

485

ENFORCED SMOOTHING 0.5-

0.4

"

0.3" 0.20.1-

:

"~ 0 . 0 qm

0

L.

0.01

O.

1.0

10

0 a.

t~ t,0 ;~ rJ 0 . 5 "

100

PARAMETRIC ESTIMATION

1.1. 0 . 4 0.3" Ii 0.2. I' I'

0.1. 0.0

t 0

/;

i 0.01

!

I

0."

1.0

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100

Ventilation-Perfusion Ratio FIGURE 8. Recovery of a trimodal narrow distribution (a = 0.6 decade, solid lines) by enforced smoothing (upper panel) and parametric estimation (lower panel), Data are means + SD. The distinctly bimodal distributions recovered by both techniques closely resemble those obtained from a unimodal or bimodal distribution with similar dispersions (Figs. 2 and 5).

results offer a useful basis for the interpretation of empirical ~ / Q distribution data. A major limitation of the inert gas elimination test is the problem of mathematical ill-conditioning inherent in the recovery of a continuous distribution function from finite measurements. Olszowka (11) showed that, for any given data set, a variety of compatible distributions could be found, which, in some cases, may assume grossly different shapes. Furthermore, for certain data sets the recovered distributions may be highly sensitive to experimental errors (4,7). It is therefore quite doubtful that the recovery by any numerical means may uniquely identify the true distribution. Whereas complete recovery may not be feasible in general, this study demonstrates that certain configurational features of ~ / Q distribution can be reliably resolved by the current techniques of inert gas analysis. Specifically, we have shown that both pure shunt and the leading moments (log mean and

486

C.-S. Poon and H.K. Kim 0.30 c

.2

o.15

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,,,r

~

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Shunt Fraction FIGURE 9. Recoverability of the shunt fraction with enforced smoothing (open circles) and parametric estimation (closed circles). Data are mean errors _+ SD, The test distribution was a rectangular function (mean ~/A/Q = 1, o = 0,12 decade) with varying shunt fractions.

ENFORCED SMOOTHING 0.5

~

0.4' 0.3" 0.2" C 0 r

0.1" 0.0

,

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Ventilation-Perfusion Ratio FIGURE 1 O, Recovery of the same test distribution as in Fig. 10 (40% shunt) by enforced smoothing (upper panel) and parametric estimation (lower panel). Data are means +_ SD.

Characteristics of ~ /Q Distributions

487

dispersion) of the perfusion distribution can be accurately identified over a wide range of characteristic values. The resolution limit of approximately 0.2 decade for dispersion (Fig. 1) is sufficient for the detection of I?A/Q heterogeneity in disease states that typically exhibit much wider dispersions. In fact, since the I?A/Q values are not likely to be completely uniform even in the normal lung, the resolution should be considered adequate in most cases. Compared with the shunt, dead space estimates were more variable and, in some cases, had greater biases. This is probably due to the fact that in the present numerical schemes, the dead space estimates are indirectly derived from the recovered perfusion distribution via the transformation between qj and vi implicit in Eq. 3. Thus, any deviations in the resulting perfusion distribution due to imperfect recovery would result in similar errors in the dead space estimate. This suggests that numerical schemes that are specifically intended for recovery of the perfusion distribution, while satisfactory for estimating the shunt, may not be suitable for determining the dead space. On the other hand, due to the duality between shunt and dead space it is possible that improved accuracies for the dead space estimates comparable to those for the shunt may be obtained by direct recovery of the ventilation distribution using similar numerical schemes. Because of the limited data available, it is not surprising that different numerical procedures may be more specific to the estimation of different distribution parameters. This set of identifiable features, while necessarily still limited in nature, provides useful quantifications of I?A/Q distribution that are not readily obtainable from classical approaches. It should be noted that the use of rectangular functions as test distributions provides the most severe tests for the recovery procedures. These functions have also been used by other investigators for I)A/Q analysis (2,9). Since both the log-normal and ES models are relatively smooth functions, it is expected that general distributions having contours smoother than a rectan= gular function may be even better recovered. Thus, our conclusion on the identifiability of the characteristic parameters is quite general and is not restricted to the specific test functions being chosen. Conversely, the use of a smoother function as test distribution may not allow definitive conclusions about the resolutions because of the predisposition to the estimation models. For example, an original distribution in the form of a log-normal function must necessarily be perfectly recovered by PE, which uses the same model, under all circumstances. Furthermore, although certain smooth functions such as those resulting from ES or log-normal modeling have generally been taken to represent VA/Q distribution, the true shape of any real distribution has, in fact, never been completely determined except in trivial cases. In the absence of a firm physiological basis for any test distribution, the extent of the errors with both algorithms, due to departure from the true distribution form, must be evaluated. For the purpose of the present analysis, therefore, it may be more meaningful to subject the numerical algorithms to the most severe tests by using rectangular functions rather than smoother distributions.

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While the present study indicates that the leading moments of ~ / Q distribution can be reliably recovered by these methods, it must be emphasized that the same may not apply to higher order moments. In particular, as the class of compatible distributions may consist of rectangular functions as well as relatively humped functions, such as those defined by the log-normal model or ES, estimation of the kurtosis (4th central moment), or peakedness, of a distribution may be difficult. Thus, additional data may be required to recover more subtle characteristics of the I?A/Q distribution. Previous studies have emphasized the importance of distribution modality in the characterization of ~ / Q abnormalities (14,22). Unfortunately, our results indicate that resolution for this feature is extremely difficult, if not impossible, using the present analytical techniques. Indeed, different distributions with the same dispersion factor may result in nearly identical recoveries, irrespective of the original modalities. In particular, the recovered distributions always assume a bimodal configuration when the distribution dispersion exceeds a critical value of 0.6 decade, and is unimodal otherwise (Figs. 2, 5, and 9). This finding suggests that the bimodal distribution forms widely reported in disease states (14,22) must be interpreted with caution. In fact, it is reasonable to assume that the observed modality might simply reflect the biasing nature of the particular estimation procedure under specific experimental conditions rather than a characteristic of the original distribution. For example, a parametric model with only six parameters is obviously inadequate for recovering distributions with more than two modes; similarly, an ES algorithm with a fixed parameter Z would always result in a distribution with the chosen degree of smoothness, thus limiting the modality. Furthermore, the errors in the predicted modality may not be eliminated by repeated trials such as Monte Carlo simulation (4) or multiple experiments because the modality and shape of the recovered distributions were shown to be statistically biased (Figs. 2, 5, and 8). For all practical purposes, therefore, it appears that the distribution dispersion may be a more meaningful index of I?A/Q heterogeneity than the specific modality of the recovered distribution. In a recent article, Stewart and Mastenbrook (15) proposed a systematic procedure for the determination of the number of log-normal modes and the confidence limits of the estimated modal parameters based on statistical residual analyses. Such information should be very useful, provided that the true distribution conforms to the log-normal model. Since this condition may not be satisfied in general, however, the validity of this approach is uncertain. Furthermore, it has been shown that both the retention and excretion curves for different log-normal distributions with one or two modes are nearly identical when the dispersions are equal (10). The statistical sensitivity of the residual tests should therefore be quite limited. Nevertheless, it is possible that improved sensitivity for modal detection may be achieved by other analytical techniques such as linear programming (4) or by suitable choice of the partition coefficients of the six inert gases. Further studies are needed to inves-

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tigate the identifiability of this distribution characteristic under general experimental conditions. Other investigators have proposed alternative analytical schemes for the determination of similar distribution characteristics. For example, Neufeld et al. (10) have shown that, for unimodal log-normal distributions with zero shunt and dead space, both the log variance and mean could be estimated directly from a plot of the arterial-to-alveolar (a-A) partial pressure difference versus solubility. Departure from unimodality, however, results in significant error in the mean estimate. Since the modality of the true distribution is usually unknown a priori, the estimate for the mean is unreliable with this method. Furthermore, it is not certain whether the same relationships between the 12A/Q distribution and the a-A difference plot would prevail in the presence of shunt and dead space, which are known to exist even in normal lungs and become a significant part of ~ / Q heterogeneity in disease states. The applicability of this method in practical situations is therefore questionable. In a similar fashion, Hlastala and Robertson (6) showed experimentally that a semiquantitative description of 12A/Q distribution could be obtained from direct analysis of the retention, excretion, and a-A gradient plots for six inert gases. None of these studies, however, considered the effects of measurement errors on the resulting resolutions. The present study demonstrates that quantitative information about I?A/Q distribution can be derived from compatible distributions defined by ES and PE modeling. The method is more reliable in the presence of measurement errors and multiple distribution modes than are the previous approaches. It has been suggested that ES stabilizes the resulting distribution in the presence of random error (4,12). While this is also true for our distribution data (Figs. 2, 5, 7, and 8), the resolution of a narrow distribution was more difficult, compared with PE (Fig. 7). Thus, different techniques could be used for the selection of representative distributions compatible with the measured data. A practical consideration is the computational efficiency, where ES has a definite advantage. REFERENCES 1. Bard, Y. Nonlinear Parameter Estimation. New York: Academic Press, 1974, pp. 159-160. 2. Dawson, S.V., J.P. Butler, and J. Reed. Indirect estimation of physiological distribution functions. Fed. Proc. 37:2803-2810, 1978. 3. Dawson, S.V., H. Ozkaynak, J.A. Reeds, and J.P. Butler. Evaluation of estimates of the distribution of ventilation-perfusion ratios from inert gas data. Fed. Proc. 35:453, 1976. 4. Evans, J.W. and P.D. Wagner. Limits on I2A/Qdistributions from analysis of experimental inert gas elimination. J. Appl. PhysioL 42(6):889-898, 1977. 5. Hendriks, F.F.A., B. Van Zomeren, K. Kroll, M.E. Wise, and Ph.H. Quanier. Distributions of ~ / ( ~ in dog lungs obtained with the 50 compartment and the log normal approach. Respir. Physiol. 38:267-282, 1979. 6. Hlastala, M.P. and H.T. Robertson. Inert gas elimination characteristics of the normal and abnormal lung. J. AppL Physiol. 44:258-266, 1978.

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7. Jaliwala, S.A., R.E. Mates, and F.J. Klocke. An efficient optimization technique for recovering ventilation-perfusion distributions from inert gas data. J. Clin. Invest. 55:188-192, 1975. 8. Marquardt, D.W. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Indust. Appl. Math. 11:431-441, 1963. 9. Mertens, P. A simple model of ~ / Q distribution for analysis of inert gas elimination data. J. Appl. Physiol. 55:562-568, 1983. 10. Neufeld, G.R., J.J. Williams, P.L. Klineberg, and B.E. Marshall. Inert gas a-A differences: A direct reflection of ~ / Q distribution. J. Appl. Physiol. 44:277-283, 1978. 11. Olszowka, A.J. Can ~ / Q distributions in the lung be recovered from inert gas retention data? Respir, Physiol. 25:191-198, 1975. 12. Olszowka, A.J. and P.D. Wagner. Numerical analysis of gas exchange. In Pulmonary Gas Exchange, Vol. 1, edited by J.B. West. New York: Academic Press, 1980, pp. 263-306. 13. Pimmel, R.L., M.J. Tsai, and P.A. Bromberg. Estimating ~ / Q distributions from inert gas data with an enforced smoothing algorithm. J. Appl. Physiol. 43:1106-1110, 1977. 14. Ratner, E.R., and P.D. Wagner. Resolution of the multiple inert gas method estimating ~ / Q map distribution. Respir. Physiol. 49:293-313, 1982. 15. Stewart, W.E., and S.M. Mastenbrook, Jr. Parametric estimation of ventilation-perfusion ratio distribution. J. Appl. Physiol. 55:37-51, 1983. 16, Teplick, R. and M. Snider. Letter to the editor. J. Appl. Physiol. 38:951-953, 1975. 17. Tham, M.K. Letter to the editor. J. Appl. Physiol. 38:950-951, 1975. 18. Tsai, M.J., R.L. Pimmel, and P.A. Bromberg. An evaluation of recovery of ventilation-perfusion ratios from inert gas data. Comp. Blomed. Res. 10:101-112, 1975. 19. Tsai, M.J., R.L. Pimmel, and P.A. Bromberg. Enforced smoothing techniques for recovering ~ / Q distributions from inert gas data. IEEE Trans. Biomed. Eng. BME-26:140-147, 1979. 20. Wagner, P.D., P.F. Naumann, and R.B. Laravuso. Simultaneous measurement of eight foreign gases in blood by gas chromatography. J. Appl. Physiol. 36:600-605, 1974. 21. Wagner, P.D., H. Saltzman, and J.B. West. Measurement of continuous distributions of ventilationperfusion ratios: Theory. J. Appl. Physiol. 36:588-599, 1974. 22. Wagner, P.D. and J.B. West. Ventilation-perfusion relationships. In: Pulmonary Gas Exchange, Vol. 1, edited by J.B. West. New York: Academic Press, 1980, pp. 219-262. 23. Yokoyama, T. and L.E. Farhi. Study of ventilation-perfusion ratio distribution in the anesthetized dog by multiple inert gas washout. Respir. Physiol. 3:166-176, 1967.