COMPUTERS

AND

BIOMEDICAL

An Evaluation

RESEARCH

10,

101-112

(1977)

of Recovery of Ventilation-Pe~usion Ratios from Inert Gas Data*

MING JER TSAI, RUSSELL L. PIMMEL, AND PHILIP A. BROMBERG Department of Medicine, University of North Carolina, Chapel Hill, North Carolina 27514 AND

ROBERTB. MCGHEE Department of Electrical Engineering, The Ohio State University, Columbus, Ohio 43210 Received August 22,1975 The recoverability of the distribution of ventilation-perfusion ratios from calculated retention (or excretion) for six inert gases was studied. Least square error minimization was investigated using noise-free artificial data and data with simulated experimental error. The accuracy of the recovered distribution was q~ntified by a distance function. Both unimodal and bimodal distributions were recovered from noise-free data and from data with a simulated It1 % error. Using data with a +3 % error, the original distributions were not recovered, in fact, the distance function increased while the error function decreased with repeated iterations. Since +3 % represents current experimental error in retention (or excretion) ratio measurements, great care must be taken in applying this technique to real data. INTRODUCTION

There is considerable current interest in quantitative techniques for estimating the distribution of the ventilation-perfusion ratios (p,JQ) in the lungs of normal subjects and in subjects with various lung diseases. Recently, Wagner and his associates (I, 2) have described a technique for determining virtually continuous distributions of VA/o. In their method, trace amounts of six inert gases dissolved in normal saline or 5 % dextrose are continuously infused intravenously. After equilibrium is established, the retention (the ratio of concentration in arterial blood to that in mixed venous blood) and excretion (the ratio of concentration in expired air to that in mixed venous blood) are determined for each gas. From the six values for retention and the known blood: gas partition coefficients for each gas, an iterative least squares regression technique estimates the fractional blood flow in each of 50 compartments. An identical approach is used to estimate the corresponding 50 fractional ventilation values from the 6 measured excretions. The above formulation amounts to an underspecified mathematical problem because 6 measured quantities are used to compute the 50 fractionai blood flows or * This work was supported by U.S. Public Health Service Grants HL-17585, HL-91118, and HL-00207. 101 Copyright 0 1977 by Academic Press, Inc. All rights of reproduction Printed in Great Britain

in any form

reserved.

ISSN

0010-4809

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TSAI ET AL.

ventilations. In such a situation, there are in fact an infinite number of solutions, constitutingasubspaceof44dimensions. Wagneretal.(2)wereawareofthislimitation and used a gradient descent method to obtain one element from this subspaceof solutions. They reported the results of an empirical study in which they tested the recoverability of various artificial distributions, including unimodal, bimodal, and trimodal distributions with both wide and narrow dispersionand with shunt and dead space. They also evaluated the effects of simulated measurement noise on the recovered distributions. In all cases,they were able to recover the shapeof original distributions. They speculated that although there are an infinite number of recoverable distributions, all of them are very similar. Jaliwala et al. (3) also investigated the recoverability of artificial distributions using the samemathematical model. They were able to duplicate Wagner’s results with noise-free data, but faiied to recover the distribution if artificial random noise was added to the measurements.They also found that with the noisy measurements. the discrepanciesbetween the original and the recovered distributions, while at first reduced, later increasedasthe number of iterations wasincreased,despite a decrease in the error function. The present study was intended to further investigate the effects of noise on the recoverability of artificial distributions and to quantify the discrepanciesbetween the initial and the recovered distribution with a distance function. Recent reports by Wagner (4, 5) have suggestedthat enforced smoothing eliminates the difficulty in recovering artificial distributions in the presenceof simulated experimental error. However, in preliminary studies in our laboratory, we have found that the quantitative and qualitative discrepanciesobserved with Wagner’s algorithm for enforced smoothing are greater than those seen in the gradient descent method. Thus. the method for quantifying the discrepancies which is introduced in this paper should provide a useful concept for further studiesin this area. METHODS

Since the present paper is concerned with the mathematics of the minimization schemesand since the samemethod is used in computing the distribution of blood flow and ventilation, we choose to consider only the blood flow problem. A mathematical model derived from the work of Farhi (7) describing the relationship between the retention of six inert gasesand the ventilation perfusion ratio r’,/g in 50 compartments hasbeenformulated (2). The model is summarizedin Eqs. [1], [2], and [3], inwhichR,istheretention of inert gas,Ai is theblood:gaspartitioncoefficientforinert gasi, fj is the fractional blood flow in compartmentj, and (PA/o)>, is the ventilationperfusion ratio in compartmentj:

[II

VENTILATION-PERFUSION

,g=

RATIOS

1;

103

PI

0 Gfj 6 1,

j= 1,2 ,..., 50. [31 We assumed various distributions of blood flow, and with six blood: gas partition coefficients (0.0076,0.092,0.415, 2.30, 11.7, and 333) we first computed a set of six retentions which served as “experimental” data for the noise-free condition. To simulate noisy measurements, small computer-generated errors were added to each of the calculated retentions. In two studies we utilized different levels of random error which were selected from normal distributions with a zero mean and a standard deviation equal to 1 or 3 % of the calculated retentions. In another series, we explored the sensitivity of the recoverability to experimental error on each retention measurement by individually adding &-3‘? error to each retention value, while the other five were kept at their ideal values. These “experimental” values for the six retentions were then used in our iterative numerical technique to obtain an optimum set of values for the 50 fractional blood flows. In our procedure, the optimum is determined by a constrained regression analysis approach in which the squared error function shown in Eq. [4] is minimized using a gradient descent method (6): 141

Each recovered distribution is then compared with the original distribution used to calculate the “experimental” retentions. With both levels of noise we investigated several sets of random errors as well as several starting conditions, including a uniform distribution. A root-mean-square distance function was used to quantify the discrepancy between the initial and the recovered distribution. This function is described by Eq. [5] in which S is the distance andhj and fRj are the initial and recovered fractional blood flow in compartmentj: S=

[A

,zI

(Xj

-fRj)zl”2*

[51

In addition, we varied the number of iterations used in the minimization scheme in order to investigate the effect of this parameter on the shape of the recovered distribution, on the magnitude of the error function, Eq. [4], and of the distance function, Eq. E51. RESULTS

Figure 1 shows an example of the recovered distribution of the fractional blood flows for a unimodal and a bimodal case. These were obtained with noise-free “experimental” retentions by minimizing the least square error function. In both

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TSAI ET AL.

cases,there is qualitative recovery of the general shapesof the original assumed distributions after 10,000iterations. However, in the bimodal case, significant discrepancies in numerical values occur. The accuracy of the recovered distribution is

FIG. 1. Two examples of recovered distributions of fractional blood flow after 10,000 iterations (*‘s) showing original artificial distributions for comparison (solid curve). Noise-free data were used in both cases.

VENTILATION-PERFUSION

105

RATIOS

quantified by the magnitude of the root-mean-square distance function S, which after 10,000 iterations, equals 0.00269 for the unimodal case and 0.00909 for the bimodal case. This means that for the given unimodal distribution, the expected error

Is . -

s-.

a I

0

31

1

100

VT?/6

r(.

,

E.

Y

i

33

53



I-2

5.

0

FIG. 2. Recovery of known distributions of fractional blood flow after 10,000 iterations in the presence of two levels of random error, 1% SD (l’s) and 3 % SD (3’s) in the “experimental” retentions. The artificial original distributions (solid curve) are the same as those shown in Fig. 1.

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TSAIET

AL.

in the value forfin a given r;h/Q compartment is 3.9% of the true peak value for.11 For the given bimodal distribution, it is 18 p/b’.Although a straight line starting condition (fj = 0.02 for allj’s) was used in obtaining these results, nearly identical results were obtained with several other continuous starting conditions. Figure 2 illustrates the effect of two levels of random error (1 and 3 1’6 standard deviation) in the “experimental” retentions on the recovered distributions. The original unimodal and bimodal distributions were the same as those shown in Fig. I. A comparison of the curves of Figs. 1 and 2 indicates that the lower level of random noise (1 ,% standard deviation) has comparatively little effect on the recoverability of the assumed distribution after 10,000 iterations. This was consistently the case for all the various sets of random error and starting conditions that we tried. When TABLE VARIATION

IN RECOVERED

1

UNIMODAL DBTRIBUT~ON AFTER 3000 ITERATIONS

Peak fractional flow” Trial

Magnitude

I 2 3 4 5 6 I

0.0892 0.1314 0.0674 0.06413 0.0940 0.0782 0.0835

Compartment 34 34 30 30 28 ‘8 ‘9

WITH 3% RANDOM

ERROR

Final error (x 10-5)

rms distance (x 10-A)

1.24 42.1 0.460 0.0507 75.9 II.0 20.3

9.52 24.06 9.09 7.23 23.71 10.24 15.09

’ The original magnitude and location are 0.0689 and 33, respectively.

the higher level of noise(3 yi standard deviation) wasused,the shapeof the recovered distribution after 10,000 iterations was qualitatively different from that obtained with no noise or with a lower level of noise. Perhaps just as importantly the shape of the recovered distribution varied among trials using different sets of random error, all selectedfrom the samedistribution. With the unimodal distribution, six of sevendistributions recovered with different setsof random error had multiple peaks, and Table I indicates the magnitude and location of the major peak as well as the final error and the rms distance. With the bimodal distribution the shape of the recovered distributions varied considerably among trials and it was impossible to characterize the results in a tabular form similar to Table 1. Figure 3 showshow the error function C#I for the two distributions, shown in Figs. 1 and 2, varies as the number of iterations is increased. The rapid reduction during the first few hundred iterations indicates that the algorithm has a high initial rate of

VENTILATION-PERFUSION

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convergence. After 10,000 iterations in the noise-free case, 4 is on the order of 1O-7 for the unimodal distribution and 10e6 for the bimodal case. One percent noise has little effect on the variation in the error function, but 3 % noise increases the final

I 1

2

1

1

1

b

6

6

NO OF

0

2

b

NO BF

ITER

6

ITER

0

I

*moo

l 1ooo

10

10

FIG. 3. Magnitude of the error function vs the number of iterations in recovering the two distributions, unimodal (a) and bimodal (b), shown in Figs. 1 and 2. For each distribution, the figure includes three cases, error-free data (A), 1% noisy data (*), and 3 % noisy data (C).

108

TSAI

ET AL.

error approximately two orders of magnitude. Similar results were seen in all trials with various sets of random errors at both the 1 and 3 y,’ level. The six noise-free “experimental” retentions used to obtain the distributions in Fig. 1 are shown in Table II, along with the recovered retentions which correspond to the distribution obtained by minimizing the least square error after 10,000 iterations. The three most significant digits of the corresponding retentions are nearly identical, and thus further iterations will not provide any appreciable improvement in these retentions, and consequently will not reduce the error function. Figure 4 shows S as a function of the number of iterations for the noise-free case and for the two levels of noise for the two cases shown in Figs. 1 and 2. With the higher level of noise, the magnitude of the distance function is first reduced and then increased as the iterative process proceeds, in this case, beyond about 200 iterations TABLE COMPARISON

OF “EXPERIMENTAL” ITERATIONS

II

AND RECOVERED WITH NOISE-FREE

RETENTIONS DATA

Unimodal “Experimental” 0.004907 0.050739 0. I65626 0.4373 13 0.734644 0.981752

AFTER

10,000

Bimodal Recovered

“Experimental”

0.005169 0.050692 0.165614 0.437337 0.734625 0.981758

0.29958 0.53241 0.67561 0.84708 0.95002 0.99769

Recovered 0.29960 0.53222 0.67590 0.8468 I 0.95026 0.99757

Although the particular iteration at which the distance function began to increase varied with different sets of random error, it always reached a minimum when 3 “(; random noise was used. It is important to recognize that this increase in discrepancy is occurring while the least square error is being monotonically reduced, as shown in Fig. 3. Thus, reducing the error function by further iterations does not necessarily imply that the accuracy of the recovered distribution is improving where there is a random error in the “experimental” retentions. Results obtained when only one retention value was perturbed with a 3 4,’ error were, in general, similar to those obtained with the noise-free data. The variability in the recovered distributions is summarized in Tables III and IV. For the unimodal distribution the location of the peak fractional flow was always within four compartments ofthe original location and its magnitude was within 20 1; of the original value. Most values of S were close to those found with the noise-free retention values; however, some were about three times larger and these occurred with errors in R.,, R,,

VENTILATION-PERFUSION

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!

RATIOS

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b

FIG. 4. rms distance between original and recovered distributions vs the number of iterations. a is for the unimodal distribution and b is for the bimodal one. For each distribution, the threecases, error free (A), and 1 ‘A error (*), and 3 ‘A error (G), are included.

110

TSAI ET AL. TABLE VARIATION

IN RECOVERED WITH

A

3%

UNIMODAL ERROR

III DISTRIBUTION

ON A SINGLE

Peak fractional flow” Perturbation

Magnitude

Compartment

0.0639 0.0639 0.0648 0.0636 0.0633 0.067 1 0.0710 0.0753 0.0843 0.0610 0.0616 0.0792

3000 VALUE

AFTER

RETENTION

33 33 33 33 34 31 ‘9 34 34 29 37 34

ITERATIONS

Final error (x 10-I)

rms

0.0278 0.0374 0.0482 0.0456 0.664 0.285 1.04 1.73 2.46 2.15 41.9 0.133

2.83 2.80 2.64 2.98 3.36 2.55 8.50 6.39 7.91 9.56 7.55 9.45

u The original magnitude and location are 0.0689 and 33, respectively. TABLE VARIATION

IV

IN RECOVERED BIMODAL DISTRIBUTIONS AFTER 3000 ITERATIONS 3 ‘4 ERROR ON A SINGLE RETENTION VALUE’

WITH

A

Peak fraction flow -Perturbation

Magnitude 1

Magnitude 2

Location 1

Location 2

0.0287 0.0341 0.0400

0.0422 0.0419 0.0456

26 27 27

0.0392 0.0368 0.0340 0.0404 0.0288 0.0329 0.0314

0.0585 0.0668 0.0469 0.0811 0.0439 0.0411 0.0501

20 20 19 Unimodal Unimodal 18 18 21 19 20 19 19

26 26 30 28 25 26 26

Final error (x 10-51

distance (x 10-a)

1.57 2.82 8.28 0.0276 2.76 16.1 19.1 17.9 38.6 16.8 49.9 30.3

11.2 10.9 10.3 13.3 15.5 9.50 9.73 13.6 13.8 12.0 10.7 10.8

a The original magnitudes and locations are 0.0508,0.0506, 15, and 33, respectively.

VENTILATION-PERFUSION

RATIOS

111

and R,, where the largest discrepancies in the location and magnitude of the peak fractional flows were seen. For the bimodal distribution the results were more varied. In two cases the recovered distribution was unimodal. If these two cases are disregarded then both peaks occurred within six compartments of the original location and the magnitudes were always within 60 “/d and generally within 30% of the true values. Values of S were comparable to or slightly larger than those obtained with noise-free retention values. DISCUSSION

The mathematical formulation used to compute the distribution of fractional blood flow from six retention measurements is an underspecified problem, and as such would have an infinite number of solutions. Wagner et al. (2) have speculated that all the possible distributions are very similar, and they attempted to verify this using an iterative numerical technique in an empirical study in which they assumed some distribution, calculated the associated retentions, and then used these “experimental” retentions to recover the distribution. They found that for noise-free “experimental” retentions they were able to recover distributions that qualitatively were similar to a wide variety of original assumed distribution. This result was confirmed by Jaliwala et al. (3) and is also supported by our study. Although in our study and in the previous ones (2,3), Gaussian distributions were utilized to calculate the “experimental” retention values, we also recovered an originally flat distribution with an accuracy comparable to that obtained with the Gaussian distributions. While the general shape of the distributions recovered from noise-free data agrees with the original ones, there are some significant discrepancies, especially in the bimodal distribution that appear to be larger than those shown in the previous studies (2, 3). These exist even though the error function was reduced to a value of 10v6 to 10-14, which corresponds to the values reported by Jaliwala et al. (3). In our study, these were quantified by the distance function which, in the example included in this report, was 18 “/i of the true peak value of the fractional blood flow for the bimodal distribution. Since this measure was not reported in the previous studies, we have nothing with which to compare it. There is no apparent explanation for these discrepancies except the details of the algorithms, which could play an important role in this problem because it is mathematically underspecified. Our algorithm differs from that of Jaliwala et al. (3), who used the highly efficient variable metric technique to solve the problem, and although it is very similar to that used by Wagner et al. (2), there most likely are differences in the details. In their original study, Wagner et al. (2) showed that nonrandom errors on the “experimental” retentions did not qualitatively affect the shape of the recovered distribution. Our results, obtained while changing only one retention value (Tables III and IV), generally support this conclusion although in certain cases of bimodal distribution, the recovered distributions were qualitatively different, suggesting that

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the more complicated distributions may be more difficult to compute. Jaliwala et al. (3) investigated the effects of truly random errors in the “experimental” retentions by adding errors that were randomly selected from the normal distributions with a zero mean and standard deviation of 6% of the retention for the lowest solubility gas, and 2.5 % for the others. This led to a significantly different conclusion, which was somewhat paradoxical in that, as the number of iterations increased, the error was monotonically reduced; but at the same time, the shape of the recovered distribution was less and less like that of the original assumed distribution. Our study, in which we used several sets of random error selected from a normal distribution with 3 % standard deviations for all gases, confirmed this result and quantified the discrepancy between the original and the recovered distribution. Figure 3 shows that the error is reduced as the number of iterations increases, and Fig. 4 shows that, at the same time, the distance between the original and the recovered distribution may actually increase. The errors used in this study and in that of Jaliwala et al. (3) reflect current experimental accuracies (8). We therefore agree with Jaliwala and associates that it is difficult to determine the significance of distributions that are computed using actual experimental data. They found that if the magnitude of the error was reduced by a factor of five, then the problems associated with noisy retention measurements were significantly reduced. When we used smaller errors (1% standard deviation), the magnitudes of the distance function were comparable to those obtained with noise-free retentions. Thus it seems that an improvement in experimental accuracy might remove some of the constraints on this approach. However, further study is needed to substantiate this possibility. REFERENCES 1. WAGNER, P. D., LARAVUSO, R. B., UHL, R. R., AND WEST, J. B. Continuous distributions of ventilation-perfusion ratios in normal subjects breathing air and lOOoh 02. J. Cfin. Invest. 54, 54-68

(1974).

WAGNER, P. D., SALTZMAN, H. A., AND WEST, J. B. Measurement of continuous distributions of ventilation-perfusion ratios: theory. J. Appl. Physiol. 36,588-599 (1974). 3. JALIWALA, S. A., MATES, R. E., AND KLOCKE, F. J. An efficient optimization technique for recovering ventilation-perfusion distributions from inert gas data. J. Clin. Znuest. 55, 188-192 2.

(1975).

4. WAGNER, P. D. Letters to the editor. J. Appl. Physiol. 38,950-953 (1975). 5. WAGNER, P. D., EVANS, J. W., AND WF..V, J. B. Analytically derived distributions of ventilationperfusion ratios in chronic lung disease. Fed. Proc. 34,451 (1975). 6. MCGHEE, R. B. Some parameter-optimization techniques. In “Digital Computer User’s Handbook,” Chapter 4.8 McGraw-Hill, New York, 1967. 7. FARHI, L. E. Elimination of inert gas by the lung. Respir. Physiol. 3,1-l 1 (1967).

An evaluation of recovery of ventilation-perfusion ratios from inert gas data.

COMPUTERS AND BIOMEDICAL An Evaluation RESEARCH 10, 101-112 (1977) of Recovery of Ventilation-Pe~usion Ratios from Inert Gas Data* MING JER T...
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