Estimating
VA@
with an enforced R. L. PIMMEL, Department
distributions from inert gas data smoothing algorithm
M. J, TSAI,
of Medicine,
AND
University
P. A. BROMBERG
of North
PIMMEL,
R. L., M. J. TSAI, AND P. A. BROMBERG. Estimatdistributions from inert gas data with an enforced smwthing algorithm. J. Appl. Physiol.: Fkspirat. Environ. Exercise Physiol. 43(6): 1106-1110, 1977. -The accuracy of a noniterative smoothing algorithm wjthout a nonnegativity constraint for estimating continuous VA/Q distributions from inert gas retentions was evaluated. Simulated retention measurements computed from three assumed distributions were processed by the algorithm. Original and recovered distributions were compared with the root-mean-square (rms) difference. With noise-free retentions, the rms differences were minimum with z = 0 (no smoothing). With simulated 3% noise, the minima occurred when 0.01 < z < U.10. Seven trials with independent simulated errors showed a large variation in the recovered distribution and the rms difference. All recovered distributions were qualitatively different from the originals especially at low and high VA/Q values. Negative fractional flows existed in all distributions that were not grossly different from the originals. We believe that this method, which did nut include a nonnegativity constraint, is inappropriate for processing inert gas retentions.
ing
VA@
inert gas retention; mathematical programming; ences; simulated experimental errors
rms differ-
RECENTLY, WAGNER ET AL. (10,lZ) described an iterative numerical method for obtaining estimates of virtually continuous distributions of ventilation-perfusion ratio (VA/Q) from retention measurements of six inert gases. Subsequent simulation studies showed that defining the proper number of iterations is problematic (3, 6) and that a set of retentions can be associated with several dissimilar distributions (4). In response to the criticisms, Wagner and associates (7, 9) proposed a finite search algorithm based on mathematical smoothing with a nonnegativity constraint which reportedly overcomes these prob1ems.l In the algorithm a “smoothness parameter,” x, is selected to provide the best recovered distribution. Tsai et al. (5), in a simulation study using a different but mathematically equivalent formulation of Wagner’s algorithm without the nonnegativity constraint, found that small values of z produce the best recovery, that with these values of z there were some compartments with negative flow, and that the recovered distributions were not better than I For algorithm to obtain distributions of VA and Q from inert gas data, order NAPS Document 03185 from Wcrofiche Publications, P.O. Box 3513, Grand Central Station, New York, N. Y. 10017.
Carolina,
Chapel Hill,
North
Carolina
27514
those achieved with the iterative technique (5). In the present simulation study, we show that the latter results are also true when Wagner’s formulation is used without the nonnegativity constraint. METHODS
The theory and simulation methods have been described previously (3, 6, 12). Briefly a VA/Q distribution with 50 compartments was assumed and used to compute six retentions with the mathematical model described by Farhi (2). These values were used as noisefree measurements or were perturbed by adding small random numbers to simulate experimental error. These “experimental retentions” were used with the algorithm to compute a recovered VA/Q distribution with 50 compartments. The recovered and original distributions were compared to assessthe accuracy of the algorithm. The discrepancy was quantified by the root-meansquare (rms) value of the difference between recovered and original distributions (6). In the present study we utilized narrow and broad unimodal distributions, and a broad bimodal distribution, identical to those originally utilized by Wagner et al. (12). The small numbers used to simulate realistic experimental error (3, 6, 11) were selected randomly by computer from a normal distribution with a zero mean and a standard deviation equal to 3% of the calculated ideal retentions. Equation 1, representing Wagner’s (7) formulation, was used to compute the recovered distribution.
QI = A+1 + AAf)-IR
(1)
In this equation 4 is a 50 x 1 vector with each component representing the fractional flow in compartmentj; A is a 6 x 50 matrix with qj = hi/[ 1 + (VA/ Qj] where Xi is the blood:gas partition coeffkient for gas i and (VA/Q& is the ventilation-perfusion ratio in compatiment j; At is the transpose of A; z is a constant which determines the amount of smoothing; I is a 6 x 6 identity matrix; and R is a 6 x 1 vector where Ri represents the retention of gas i. To quantify the discrepancy between the original and recovered distributions, the root-mean-square (rms) differences were computed using Eq. 2 + rms =
112 1
1,50 2 (Oj-, - Qj-(y (2) j=1 [ In this equation, qimRand QjmOrepresent the fractional
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 16, 2019.
ESTIMATING
DTSTRTBUTIONS
VA/Q
WTTH SMOOTHING
flows in compartment j in the recovered and original distributions, respectively. For each of the three distributions, we used the unperturbed (noise-free) retentions to obtain recovered distributions and rms differences with z ranging from 0 to 20. This was repeated for all three distributions with a single set of perturbed (noisy) retentions. The latter results were used to define an optimum range of z which was found to be 0.01~ z 5 0.10. Usingz = 0.02, seven independent sets of 3% random errors were added to a set of-ideal retentions for each of the three distributions. Resulting perturbed retentions were used to compute recovered distributions and rms differences to evaluate the effect of experimental error on the agreement between the original and recovered distributions. RESULTS
Figure 1 shows how the rms differences varied with z for the three distributions studied. With noise-free retentions, smaller values of z produced the lowest rms differences. With noisy retentions, the relationship between the rms difference showed a true minimum in the range 0.0X-0.10 for all three distributions. As z increased the rms differences for the noise-free and noisy cases for all three distributions approached each other and became coincident. For all three distributions, lower values of z produced recovered distributions with negative fractional flows in some compartments. All distributions obtained with values of Z that produced no negative flows were grossly distorted. Figure 2 shows the three original distributions and those recovered with noise-free retentions with no smoothing (z = 0), withz = 0.02, withz = 0.10, and with the values that produced the lowest rms difference without negative fractional flows (2 = 4 for narrow unimodal, z = 7 for broad unimodal and z = 2 for bimodal). Visually, x = 0 produced the best agreement between the original
0.
IO+
’
I
I
lO-4
I
I
I
iO-3
I
1107
ALGORITHM
I
and recovered distributions, and the discrepancy increased with z as the effect of smoothing flattened out the sharp peaks. This agrees with the variation in the rms differences shown in Fig. 1. Table I shows the ranked rms differences found with seven independent sets of noisy retention values for each of the three distributions with z = 0.02. For each distribution, the range of rms values indicates that the agreement between the original and recovered distributions is greatly influenced by the actual error in the retentions. Figure 3 shows the recovered distributions with the lowest, highest, and median rms differences for the three original distributions. This further illustrates that significantly different distributions can be obtained from the same noise-free retention values that have been perturbed by small random error. Also it indicates that there are some large qualitative differences between the originals and all recovered distributions. Similar variations in rms difference and recovered distributions were found with x = OJO. DISCUSSION
In this simulation study we evaluated an enforced smoothing algorithm for calculating VA/Q distributions from inert gas data by comparing assumed original distributions to those calculated using the algorithm. Differences between original and recovered distributions characterize the error introduced by applying this algorithm. We quantified these differences by their rms value to represent the “effective” difference between the two distributions. Although it is difficult to identify a threshold for acceptability of the recovered distribution based on the rms difference, these values provide a measure for comparing the goodness of various recovered distributions in a theoretical study of this type. The results with noise-free retention values indicate
1
m2
I
1
I
I
I
0’
I
1
IO0
i
I
IO’
z
FIG.
1. Graph showing effects of smoothing parameter z on rms difference between original noise-free retentions and those containing a given set of simulated 3% experimentalerrors.
and
recovered
distributions
obtained with
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 16, 2019.
1108
PIMMEL, --
ORIGINAL z=o
-----------
ORIGINAL z=o 2 = 0.02 z=O.lO 2 = 7.0
TSAI,
AND
BROMBERG
0.08 0.06
0.06 61
0.04 -
2. Original and recovered distributions obtained noise-free retentions and z = 0 (no smoothing), z = 0.02, 2 = 0.10 and with smoothing that produced a nonnegative distribution (z = 4 for narrow unimodal, z = 7 for broad unimodal, and z = 2 for bimodal). FIG.
with
0.02,
-o-02I
I
-
ORIGINAL
-.-"-----
r=O.lO
2=0.02 2 = 2.0 I
I
0.01
0.0001
I
I
I.0
1 I00
VA/G
TABLE 1. Ranked rms differences between original and recuvered distributions obtained with 7 independent sets of retentions containing simulated 3 % experimental error Narrow
Unimodal
0.0163 0.0166 0.0182 0.0220 0.0221 0.0229 0.0237
Broad
Unimodal
0.0058 0.0060 OeO061 0.0066 0.0071 0.0085 0.0094
Bimodal
0.0092 0.0109 0.0151 0.0160 0.0186 0.0219 0.0253
that smoothing did not improve the goodness of the calculated distribution for the three types we considered. However, when a particular set of noisy data were used for each distribution, smoothing with z in the range 0.01 < x < 0,lO reduced the rms difference and improved the agreement between the original and recovered distributions. This is considerably less than the value selected by Wagner and associates (8), who used z = 40 in their algorithm. In our study the
optimum range of z coincided for the three types of distributions we evaluated. Although this in no way proves that the range represents a universal optimum,, it leaves that possibility open. With our algorithm, we found small negative fractional flows in all recovered distributions that closely approximate the original. Wagner (7) indicated that, with this formulation, the nonnegativity constraint could be imposed by a rapid finite search procedure through the nonnegative subsets. The use of this nonnegativity constraint would eliminate compartments with negative fractional flows. In examining the effect of simulated experimental error on the shape of the recovered distribution, we found significant variation in the shape as actual values of simulated errors were altered randomly. This implies that repeated retention measurements obtained under identical experimental conditions could produce grossly different distributions. This is not surprising since the problem is ill conditioned and, as such, the solution will be sensitive to small random errors in the input data. Perhaps other algorithms with a nonnegativity constraint, such as the one used by Wagner and associ-
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 16, 2019.
ESTIMATING
0.06
-
0.04
-
0.06
irA/Q
DISTRIBUTIONS
- I-‘I----
ORtGINAL RMS =0.00580 RMS =0.0066 RMS=0.00941
-
ORiGINAL
WITH
SMOOTHING
1109
ALGORITHM
FIG. 3. Original distributions and worst, median, and best distributions recovered from retentions perturbed with independent sets of simulated 3% experimental errors with z = 0.02. Recovered distributions were selected from a set of 7 ranked by the rms difference between original and recovered distributions.
I
ates (7, 9), or by Dawson et al. (l), would be less sensitive to random errors in the input data. In support of our smoothing algorithm, we found that most of the recovered distributions qualitatively resemble their original. Although the rms differences were similar to those obtained with the original iterative approach (6) and with our earlier smoothing algorithm (5), this enforced smoothing algorithm provides a direct noniterative solution which can be obtained on any small computer capable of inverting a 6 x 6 matrix. Although our earlier smoothing algorithm (5) provided identical results with no noise and similar results with simulated error, it involved the inversion of a 50 x 50 matrix, which caused computational problems. For both unimodal cases, the recovered distributions were generally unimodal with slightly smaller peaks close to the correct location. Although distributions obtained with bimodal data varied more among trials and showed less agreement with the original, all were multimodal patterns. In general, the peaks of the recovered bimodal distributions were displaced considerably from those in the originals. For all three types of distributions, con-
siderable distortion at both extremes of VA/Q suggested that estimates of the fraction of flow in shunt and dead space may be especially inaccurate. Again, a different algorithm with a nonnegativity constraint could possibly eliminate some of these differences. In summary, our algorithm suffered three shortcomings: I ) recovered distributions had compartments with negative fractional flows, 2) they were extremely sensitive to small random variations in the input data, and 3) even in the noise-free situation, they were significantly different from the original distribution, especially for low and high VA/Q. Because of these, it appears that our algorithm is inappropriate for this application. Clearly, the nonnegativity constraint is important, and smoothing algorithms with such a constraint would eliminate the first of these problems. It is possible that the effects of the other two also could be reduced by the imposition of this constraint. This work was supported by the National Grant HL-19118. R. L. Pimmel is supported tutes of Health Research Career Development Received
for publication
27 February
Institutes of Health by the National InstiAward HL-00207.
1976.
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 16, 2019.
1110
PIMMEL,
TSAI,
AND
BROMBERG
REFERENCES 1. DAWSON, S. V., If. OZKAYNAK, J. A. REEDS, AND J, P. BUTLER. Evaluation of estimates of the distribution of ventilation-perfusion ratios from inert gas data (Abstract). IQderatiort Proc. 35: 453,1976. 2. FARHX, L. E. Elimination of inert gas by the lung. Respiration Physiol. 3: l-11, 1967. 3. JALIWALA, S. A., R. E. MATES, AND F. J. KLOCBE. An efficient optimization technique for recovering ventilation-perfusion distributions from inert gas data. J. CZin. Invesk 55: 188-192, 1975. 4. OLSZOWKA, A. J. Can VA/Q distributions in the lung be recovered from inert gas data? Respiration Physiol. 25: 191-198, 1975. 5. TSAI, M. J., R. L. PIMMEL, AND P. A. BROMBERG. A study of the recovery of the distribution of ventilation-perfusion ratios using enforced smoothing techniques. Digest 11th Intern. Conf. Med. Viol, Eng. 386-387, 1976. 6. TSAI M. J.,, R. L. PIMMEL, R. B. MCGHEE, AND P, A. BROMBERG. An evaluation of recovery of ventilation-perfusion ratios from inert gas data. Comp. Biomed. Res, 10: 101-112,1977.
7, WAGNER, P. D. Letters to the Editor. J. Appl. PhysioZ. 38: 950953, 1975. 8. WAGNER, P. D., D. R. DANTZKER, R. DUECK, J. L. CLAUSEN, AND J. B. WEST. Ventilation-perfusion inequality in chronic obstructive pulmonary disease. J. Chin. Invest. 59: 203-216, 1977. 9. WAGNER, P. D., J, W, EVANS, AND J. B. WEST. Analytically derived distributions of ventilation-perfusion ratios in chronic lung disease (Abstract). hderatisn Proc. 34: 351, 1975. 10. WAGNER, P. D., R. B. LARAVUSO, R. R. UHL, AND J. B. WEST. Continuous distributions of ventilation-perfusion ratios in normal subjects breathing air and 100% 0,. J. CLin. Inuest. 54: 5468, 1974. 11. 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. 12. WAGNER, P. D., H. A. SALTZMAN, AND J. B. WEST. Measurement of continuous distributions of ventilation-perfusion ratios: theory, J. Apple. Physiol. 36:588-599, 1974.
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 16, 2019.
VA@
with an enforced R. L. PIMMEL, Department
distributions from inert gas data smoothing algorithm
M. J, TSAI,
of Medicine,
AND
University
P. A. BROMBERG
of North
PIMMEL,
R. L., M. J. TSAI, AND P. A. BROMBERG. Estimatdistributions from inert gas data with an enforced smwthing algorithm. J. Appl. Physiol.: Fkspirat. Environ. Exercise Physiol. 43(6): 1106-1110, 1977. -The accuracy of a noniterative smoothing algorithm wjthout a nonnegativity constraint for estimating continuous VA/Q distributions from inert gas retentions was evaluated. Simulated retention measurements computed from three assumed distributions were processed by the algorithm. Original and recovered distributions were compared with the root-mean-square (rms) difference. With noise-free retentions, the rms differences were minimum with z = 0 (no smoothing). With simulated 3% noise, the minima occurred when 0.01 < z < U.10. Seven trials with independent simulated errors showed a large variation in the recovered distribution and the rms difference. All recovered distributions were qualitatively different from the originals especially at low and high VA/Q values. Negative fractional flows existed in all distributions that were not grossly different from the originals. We believe that this method, which did nut include a nonnegativity constraint, is inappropriate for processing inert gas retentions.
ing
VA@
inert gas retention; mathematical programming; ences; simulated experimental errors
rms differ-
RECENTLY, WAGNER ET AL. (10,lZ) described an iterative numerical method for obtaining estimates of virtually continuous distributions of ventilation-perfusion ratio (VA/Q) from retention measurements of six inert gases. Subsequent simulation studies showed that defining the proper number of iterations is problematic (3, 6) and that a set of retentions can be associated with several dissimilar distributions (4). In response to the criticisms, Wagner and associates (7, 9) proposed a finite search algorithm based on mathematical smoothing with a nonnegativity constraint which reportedly overcomes these prob1ems.l In the algorithm a “smoothness parameter,” x, is selected to provide the best recovered distribution. Tsai et al. (5), in a simulation study using a different but mathematically equivalent formulation of Wagner’s algorithm without the nonnegativity constraint, found that small values of z produce the best recovery, that with these values of z there were some compartments with negative flow, and that the recovered distributions were not better than I For algorithm to obtain distributions of VA and Q from inert gas data, order NAPS Document 03185 from Wcrofiche Publications, P.O. Box 3513, Grand Central Station, New York, N. Y. 10017.
Carolina,
Chapel Hill,
North
Carolina
27514
those achieved with the iterative technique (5). In the present simulation study, we show that the latter results are also true when Wagner’s formulation is used without the nonnegativity constraint. METHODS
The theory and simulation methods have been described previously (3, 6, 12). Briefly a VA/Q distribution with 50 compartments was assumed and used to compute six retentions with the mathematical model described by Farhi (2). These values were used as noisefree measurements or were perturbed by adding small random numbers to simulate experimental error. These “experimental retentions” were used with the algorithm to compute a recovered VA/Q distribution with 50 compartments. The recovered and original distributions were compared to assessthe accuracy of the algorithm. The discrepancy was quantified by the root-meansquare (rms) value of the difference between recovered and original distributions (6). In the present study we utilized narrow and broad unimodal distributions, and a broad bimodal distribution, identical to those originally utilized by Wagner et al. (12). The small numbers used to simulate realistic experimental error (3, 6, 11) were selected randomly by computer from a normal distribution with a zero mean and a standard deviation equal to 3% of the calculated ideal retentions. Equation 1, representing Wagner’s (7) formulation, was used to compute the recovered distribution.
QI = A+1 + AAf)-IR
(1)
In this equation 4 is a 50 x 1 vector with each component representing the fractional flow in compartmentj; A is a 6 x 50 matrix with qj = hi/[ 1 + (VA/ Qj] where Xi is the blood:gas partition coeffkient for gas i and (VA/Q& is the ventilation-perfusion ratio in compatiment j; At is the transpose of A; z is a constant which determines the amount of smoothing; I is a 6 x 6 identity matrix; and R is a 6 x 1 vector where Ri represents the retention of gas i. To quantify the discrepancy between the original and recovered distributions, the root-mean-square (rms) differences were computed using Eq. 2 + rms =
112 1
1,50 2 (Oj-, - Qj-(y (2) j=1 [ In this equation, qimRand QjmOrepresent the fractional
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 16, 2019.
ESTIMATING
DTSTRTBUTIONS
VA/Q
WTTH SMOOTHING
flows in compartment j in the recovered and original distributions, respectively. For each of the three distributions, we used the unperturbed (noise-free) retentions to obtain recovered distributions and rms differences with z ranging from 0 to 20. This was repeated for all three distributions with a single set of perturbed (noisy) retentions. The latter results were used to define an optimum range of z which was found to be 0.01~ z 5 0.10. Usingz = 0.02, seven independent sets of 3% random errors were added to a set of-ideal retentions for each of the three distributions. Resulting perturbed retentions were used to compute recovered distributions and rms differences to evaluate the effect of experimental error on the agreement between the original and recovered distributions. RESULTS
Figure 1 shows how the rms differences varied with z for the three distributions studied. With noise-free retentions, smaller values of z produced the lowest rms differences. With noisy retentions, the relationship between the rms difference showed a true minimum in the range 0.0X-0.10 for all three distributions. As z increased the rms differences for the noise-free and noisy cases for all three distributions approached each other and became coincident. For all three distributions, lower values of z produced recovered distributions with negative fractional flows in some compartments. All distributions obtained with values of Z that produced no negative flows were grossly distorted. Figure 2 shows the three original distributions and those recovered with noise-free retentions with no smoothing (z = 0), withz = 0.02, withz = 0.10, and with the values that produced the lowest rms difference without negative fractional flows (2 = 4 for narrow unimodal, z = 7 for broad unimodal and z = 2 for bimodal). Visually, x = 0 produced the best agreement between the original
0.
IO+
’
I
I
lO-4
I
I
I
iO-3
I
1107
ALGORITHM
I
and recovered distributions, and the discrepancy increased with z as the effect of smoothing flattened out the sharp peaks. This agrees with the variation in the rms differences shown in Fig. 1. Table I shows the ranked rms differences found with seven independent sets of noisy retention values for each of the three distributions with z = 0.02. For each distribution, the range of rms values indicates that the agreement between the original and recovered distributions is greatly influenced by the actual error in the retentions. Figure 3 shows the recovered distributions with the lowest, highest, and median rms differences for the three original distributions. This further illustrates that significantly different distributions can be obtained from the same noise-free retention values that have been perturbed by small random error. Also it indicates that there are some large qualitative differences between the originals and all recovered distributions. Similar variations in rms difference and recovered distributions were found with x = OJO. DISCUSSION
In this simulation study we evaluated an enforced smoothing algorithm for calculating VA/Q distributions from inert gas data by comparing assumed original distributions to those calculated using the algorithm. Differences between original and recovered distributions characterize the error introduced by applying this algorithm. We quantified these differences by their rms value to represent the “effective” difference between the two distributions. Although it is difficult to identify a threshold for acceptability of the recovered distribution based on the rms difference, these values provide a measure for comparing the goodness of various recovered distributions in a theoretical study of this type. The results with noise-free retention values indicate
1
m2
I
1
I
I
I
0’
I
1
IO0
i
I
IO’
z
FIG.
1. Graph showing effects of smoothing parameter z on rms difference between original noise-free retentions and those containing a given set of simulated 3% experimentalerrors.
and
recovered
distributions
obtained with
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 16, 2019.
1108
PIMMEL, --
ORIGINAL z=o
-----------
ORIGINAL z=o 2 = 0.02 z=O.lO 2 = 7.0
TSAI,
AND
BROMBERG
0.08 0.06
0.06 61
0.04 -
2. Original and recovered distributions obtained noise-free retentions and z = 0 (no smoothing), z = 0.02, 2 = 0.10 and with smoothing that produced a nonnegative distribution (z = 4 for narrow unimodal, z = 7 for broad unimodal, and z = 2 for bimodal). FIG.
with
0.02,
-o-02I
I
-
ORIGINAL
-.-"-----
r=O.lO
2=0.02 2 = 2.0 I
I
0.01
0.0001
I
I
I.0
1 I00
VA/G
TABLE 1. Ranked rms differences between original and recuvered distributions obtained with 7 independent sets of retentions containing simulated 3 % experimental error Narrow
Unimodal
0.0163 0.0166 0.0182 0.0220 0.0221 0.0229 0.0237
Broad
Unimodal
0.0058 0.0060 OeO061 0.0066 0.0071 0.0085 0.0094
Bimodal
0.0092 0.0109 0.0151 0.0160 0.0186 0.0219 0.0253
that smoothing did not improve the goodness of the calculated distribution for the three types we considered. However, when a particular set of noisy data were used for each distribution, smoothing with z in the range 0.01 < x < 0,lO reduced the rms difference and improved the agreement between the original and recovered distributions. This is considerably less than the value selected by Wagner and associates (8), who used z = 40 in their algorithm. In our study the
optimum range of z coincided for the three types of distributions we evaluated. Although this in no way proves that the range represents a universal optimum,, it leaves that possibility open. With our algorithm, we found small negative fractional flows in all recovered distributions that closely approximate the original. Wagner (7) indicated that, with this formulation, the nonnegativity constraint could be imposed by a rapid finite search procedure through the nonnegative subsets. The use of this nonnegativity constraint would eliminate compartments with negative fractional flows. In examining the effect of simulated experimental error on the shape of the recovered distribution, we found significant variation in the shape as actual values of simulated errors were altered randomly. This implies that repeated retention measurements obtained under identical experimental conditions could produce grossly different distributions. This is not surprising since the problem is ill conditioned and, as such, the solution will be sensitive to small random errors in the input data. Perhaps other algorithms with a nonnegativity constraint, such as the one used by Wagner and associ-
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 16, 2019.
ESTIMATING
0.06
-
0.04
-
0.06
irA/Q
DISTRIBUTIONS
- I-‘I----
ORtGINAL RMS =0.00580 RMS =0.0066 RMS=0.00941
-
ORiGINAL
WITH
SMOOTHING
1109
ALGORITHM
FIG. 3. Original distributions and worst, median, and best distributions recovered from retentions perturbed with independent sets of simulated 3% experimental errors with z = 0.02. Recovered distributions were selected from a set of 7 ranked by the rms difference between original and recovered distributions.
I
ates (7, 9), or by Dawson et al. (l), would be less sensitive to random errors in the input data. In support of our smoothing algorithm, we found that most of the recovered distributions qualitatively resemble their original. Although the rms differences were similar to those obtained with the original iterative approach (6) and with our earlier smoothing algorithm (5), this enforced smoothing algorithm provides a direct noniterative solution which can be obtained on any small computer capable of inverting a 6 x 6 matrix. Although our earlier smoothing algorithm (5) provided identical results with no noise and similar results with simulated error, it involved the inversion of a 50 x 50 matrix, which caused computational problems. For both unimodal cases, the recovered distributions were generally unimodal with slightly smaller peaks close to the correct location. Although distributions obtained with bimodal data varied more among trials and showed less agreement with the original, all were multimodal patterns. In general, the peaks of the recovered bimodal distributions were displaced considerably from those in the originals. For all three types of distributions, con-
siderable distortion at both extremes of VA/Q suggested that estimates of the fraction of flow in shunt and dead space may be especially inaccurate. Again, a different algorithm with a nonnegativity constraint could possibly eliminate some of these differences. In summary, our algorithm suffered three shortcomings: I ) recovered distributions had compartments with negative fractional flows, 2) they were extremely sensitive to small random variations in the input data, and 3) even in the noise-free situation, they were significantly different from the original distribution, especially for low and high VA/Q. Because of these, it appears that our algorithm is inappropriate for this application. Clearly, the nonnegativity constraint is important, and smoothing algorithms with such a constraint would eliminate the first of these problems. It is possible that the effects of the other two also could be reduced by the imposition of this constraint. This work was supported by the National Grant HL-19118. R. L. Pimmel is supported tutes of Health Research Career Development Received
for publication
27 February
Institutes of Health by the National InstiAward HL-00207.
1976.
Downloaded from www.physiology.org/journal/jappl by ${individualUser.givenNames} ${individualUser.surname} (129.081.226.078) on January 16, 2019.
1110
PIMMEL,
TSAI,
AND
BROMBERG
REFERENCES 1. DAWSON, S. V., If. OZKAYNAK, J. A. REEDS, AND J, P. BUTLER. Evaluation of estimates of the distribution of ventilation-perfusion ratios from inert gas data (Abstract). IQderatiort Proc. 35: 453,1976. 2. FARHX, L. E. Elimination of inert gas by the lung. Respiration Physiol. 3: l-11, 1967. 3. JALIWALA, S. A., R. E. MATES, AND F. J. KLOCBE. An efficient optimization technique for recovering ventilation-perfusion distributions from inert gas data. J. CZin. Invesk 55: 188-192, 1975. 4. OLSZOWKA, A. J. Can VA/Q distributions in the lung be recovered from inert gas data? Respiration Physiol. 25: 191-198, 1975. 5. TSAI, M. J., R. L. PIMMEL, AND P. A. BROMBERG. A study of the recovery of the distribution of ventilation-perfusion ratios using enforced smoothing techniques. Digest 11th Intern. Conf. Med. Viol, Eng. 386-387, 1976. 6. TSAI M. J.,, R. L. PIMMEL, R. B. MCGHEE, AND P, A. BROMBERG. An evaluation of recovery of ventilation-perfusion ratios from inert gas data. Comp. Biomed. Res, 10: 101-112,1977.
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