Continuous distributions of specific recovered from inert gas washout
ventilation
STEVEN MI, LEWIS, JOHN W. EVANS, AND ALFRED0 A, JALOWAYSKI Departments of Medicine and Mathematics, University of California, San Diego, La Jolla, California 92037
LEWIS, STEVEN M., JOHN W. EVANS, JALOWAYSKI. Continuous distributions of
AND
ALFREDO
A.
specific ventilation recovered from inert gas washout. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 44(3): 416-423, 1978. -We describe a new technique for recovering continuous distributions of ventilation (Q as a function of tidal ventilation/volume ratio (ir/V,) from th e nitrogen washout. The analysis yields a continuous distribution of v as a function of %7/V, represented as fractional ventilations of 50 compartments plus dead space. The procedure was verified by recovering known distributions from data to which noise had been added. Using an apparatus to control the subject’s tidal volume and FRC, mixed expired N, data gave the following results: a) the distributions of young, normal subjects were narrow and unimodal with a mean In standard deviation of 0.56 5 0.13; b) those of subjects over age 40 were broader (In SD 0.86 * 0.19) with more poorly ventilated units; c) patients with pulmonary disease of all descriptions showed enlarged dead space; d) patients with cystic fibrosis showed multimodal distributions with the bulk of the ventilation going to overventilated units; and e) patients with obstructive lung disease fell into several classes, three of which are illustrated. These results suggest that our approach is well suited for clinical investigation. nitrogen
washout;;
fibrosis; pulmonary
dead space; obstructive function testing
lung disease; cystic
INERT GAS WASHOUT SAMPLED by a rapid gas analyzer has proved to be a potent tool in the analysis of pulmonary function. Multibreath washout curves have conventionally been described in terms of a model with two or three compartments (1, 2, 6, 11, 15, 21, 23). Several groups have described analyses that recovered a continuous distribution of specific tidal volume, the ratio of the end-expiratory volume to the tidal volume. Two mathematical techniques were employed to generate these distributions. Nakamura et al. (16) and Okubo and Lenfant (18) used the direct inversion of the Laplace transform. Gomez and his colleagues (s-10) fit their curves with a continuous distrib&on which was described by a small number of parametera. These techniques have been criticized on four grounds (12, 17, 20). First,, the ability to recover physiologically relevant distributions has been questioned, especially narrow distributions as might be seen in a healthy young normal. Second, variations in tidal volume during the test make some of the underlying assumptions suspect. Third, since the direct inversion techniques THE
416
involve taking higher derivatives of the data, they are exquisitely sensitive to noise in the data. Fourth, the ability of a given parametric distribution to approximate well the actual distribution present in the lung has been questioned. We have undertaken to recover continuous distributions of specific tidal ventilation using an experimental and analytical approach which eliminates many of the objections previously raised and several others which we shall discuss below. METHODS
Nitrogen washouts were performed using the equipment shown in Fig. 1. This apparatus, a modification of that used by Martin and his colleagues (15), permits control of both the tidal volume and lung volume of untrained subjects, allowing greater accuracy and repeatability. The subject breathed air for several breaths until the operator was satisfied with his ability to breathe using the controller, at which point the inspired gas switched to oxygen. From 18 to 40 breaths of washout data were obtained. The mean expired nitrogen concentration before the washout and on each breath of the test was computed after correction for the analyzer delay and nitrogen delivery from the blood. Data from a single test are shown in Fig. 2. The data were analyzed by a ridge regression similar to that used by Evans and Wagner (5). The mean expired N, concentration is described as arising from a set of parallel compartments each having a ventilation of VT v(i), and a specific tidal ventilation (VT v(i)/ VO(i)) of S(i), where VT is the tidal volume, o(i) is the fraction of the ventilation going to ith compartment, and V, is the volume of the compartment at end expiration. The mean expired concentration on the jth breath (Cj) is thus described by Eq. I l
l
Cj C0
-=
l
c
l .
V(l)
= Mdj, ?)
i compartments
(1)
If the compartments are specified, fixing S(i), then the term on the right of Eq. 1 is a function of the breath number and the set of v(i) and may be written as M(j, 3> where v indicates the collection of V(i). The data were analyzed by minimizing over V(i) the expression
OOZl-8987/78/0000-OOOO$Ol.
25 Copyright
0 1978 the American
Physiological
Society
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RECOVERY
OF
CONTINUOUS
VENTILATORY
417
DISTRIBUTlONS
SPIROMETER
)W VOLUME
I
VOLUME
CONCENTRATION
LOGGER
FIG. 1. Experimental setup. Subjects begin test breathing through a port in mouthpiece (not shown). At end inspiration the port is closed. Volume controller is set at that point so that only the solenoid from air bag is open. Subject inhales from bag-in-box system whose volume is sensed by an electronic spirometer. After a predetermined volume has been inhaled, the controller switches so that both inspiration solenoids are closed, but expiratory solenoid is open, allowing the subject to expire into the bag-in-box. On return to end-expiratory volume, selected inspiratory solenoid is opened. Subjects are instructed to control flow rates by observing a meter. Once the operator is satisfied wit,h subject’s performance breathing air, the controller may be altered without subject’s knowledge so that subsequent inspirations are from the oxygen bag. Volume and nitrogen concentration are recorded continuously for subsequent analysis.
80X---’ ’
1;..
--
-----
JB. 2SVo Q NORMAL i-
t-
27.
>
A.--i-
_
T
--
-,--------i-_--.
FIG. 2. Sample data. Note uniformity flow (as slope of volume). Subject took before washout began.
c [W(j)-(D(j)
- Wj,
of both tidal volume and four practice breaths of air
v’,)lz
j bwaths
(2) +
2’
c i compartrrwnts
[WC(i)
’ V(i)]
where D(j) is the measured C&, W(j) and WC(i) are weights described in the text, and V(i) is subject to a nonnegativity constraint. The term on the right of expression 2 is a smoothing term which causes the procedure to choose a smooth distribution closely fitting the data. We used a value of z of 0.001 which was
chosen empirically by fitting known distributions in the presence of noise. The weighting of the data, W(j), was set at unity and the weighting of the compartments, WC(i), was set at I./C,* (1 - C,), where C, and C, are the concentrations in a given compartment on the first and last breaths of the test, respectivelv. This method of weighting chooses a model which is ,T;mooth and has a small lung volume and a large dead space from all possible models which are consistent with a close fit to the data. Effects of error and nonuniqueness of the recovered distribution are discussed in detail below. Except for the constraint that no compartment. be allowed to have a negative ventilation, expression 2 is minimized for a single set of V(i) which may be determined analytically. A search procedure was employed to seek the largest collection of compartments over which the nonnegativity constraint was satisfied and the resultant distribution was the desired one. The specified compartments were equally spaced on a log scale from a specific ventilation of 0.005 t.o 10 as shown in Fig. 3. Figure 3 shows the ability of the technique to recover several known distributions in the presence of added error. The data presented to the algorithm to generate Fig. 3 was D(j) 0(1 + (R/2) (1 + I./ JO(j)>), where D(j) is the error-free data for the jth breath and R is a normal random variable with the stated standard deviation. This function tends to approximate the observed distribution of residuals in real data. Errors of 0.5-l% approximate the size of residuals seen in real data. The calculated dead space is of particular interest. When mixed expired nitrogen concentration is used, dead space can be defined as the ventilation to a compartment having an infinite specific ventilation. In practice this may be computed as the ventilation not present in the other compartments. Thus the distributions shown have areas less than 1, the remaining area representing the ventilation of the dead space. This definition of dead space has been shown (4) to include any series dead space and will be equal to the ventilation of the series dead space in the absence of unusual regions such as respiratory units which collapse to zero vo1um.e on each breath. The dead space was first defined in this way by Young et al. (25) and has since been discussed by Tsunoda et al. (23) and Martin et al. (15). It differs from the Fowler dead space in that no assumption about the portion of the breath over which the dead space empties is necessary. The dead space as defined above is sensitive to noise on the early breaths which may cause some of the ventilation of the dead space to appear in compartments with very high specific ventilations. If the measured nitrogen on the first breath is erroneously high, this can be interpreted as indicating a very well ventilated compartment, which would have significant nitrogen only on the first one to two breaths. The ventilation of this spurious compartment would otherwise be assigned to the dead space. To estimate the amount of ventilation wasted in ventilating the dead space and the very well ventilated units another quantity DS’ was defined as
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LEWIS, BROf3D
UI G
BIMOOAL
DISTRIBUTION
0.5
J.B.
PCT NOISE
0 0 0
RECOVERED
-
GIVEN
x,
TRIMODAL
DISTRIBUTION
DISTRIBUTION
SCALE1
c
i compartments
WiP(
JALOWAYSKI
1 PCT NQFX
a.5
a 0 Q
RECOVERED
-
GWEN
PCT Narx
FIG. 3. Recovery of known distributions. Recovered distribution (points) are output of the algorithm when presented with data generated by a model of the given distribution (solid curves) to which noise was added as described in text. Axes used for all distributions are tidal ventilation/volume on a log scale from 0.005 to 10. Dead space is not shown on the distribution but was 25% of tidal volume in all cases and was recovered within 2% of tidal volume in all cases. Y-axis represents fraction of ventilation in each of 50 compartments, Note expanded scale on delta function and unimodal distribution. Also note the ability to recover a narrow distribution even in the presence of large amounts of error.
RESULTS
FIG, 4. Recovered distribution from data shown in Fig. 2. Distribution is a narrow unimodal pattern. Distribution is broader than that recovered with a uniform lung model (see upper left in Fig. 3), even though that model contained a large amount of noise. Small amount of ventilation in the very high ventilation/volume regions may represent noise bringing some of the dead space ventilation into these regions, or a very well ventilated compartment as seen in subj LK in Fig. 5.
DS ‘= DS+
AND
first one or two breaths will have a comparatively minor effect on this measurement. DS’ was very close to the dead space in most subjects. Normal subjects were chosen from workers in the hospital and research laboratories. Diseased subjects included patients from a group of chest clinic patients, primarily with obstructive lung disease, selected for long-term study and patients coming to a pediatric pulmonary function laboratory. Data were considered acceptable if they could be fit with an rms error of less than 0.75% nitrogen. In children under age 12 the data were, for technical reasons, somewhat noisier and an rms error of 1.0% was considered acceptable if replicate studies confirmed the broad features of the recovered distributions.
YOUNG NURMAL
VENTILATION/VULUME[LOG
EVANS,
l-
j&,)2
(3)
DS’ corresponds to the maximal dead space which might exist if the very well ventilated units were considered to be mixtures of dead space and alveolar gas. Because much of the ventilation of very well ventilated units will be included in DS’, noise in the
Nineteen normal subjects were under age 40 and had no respiratory symptoms. This group included several smokers who were indistinguishable from the rest of the group on the basis of the washout and other respiratory tests. Eighteen of the nineteen showed a narrow unimodal distribution with only insignificant ventilation outside the narrow peak. Examples of results from normal subjects are shown in Figs. 4-6. Some of the characteristics of the subjects and their distributions are listed in Table 1. One subject repeatedly showed ventilation to units with high specific ventilations as well as the narrow mode seen with the others. Two of the tests are shown in Fig. 5. The uniformity of these results is consistent with previous descriptions of the washout curves of young normals as essentially monoexponential. However, in
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RECOVERY
OF
CONTINUOUS
VENTILATORY
419
DISTRIBUTIONS
no case was the distribution recovered from the data of any subject as narrow as that shown in Fig. 3 for a truly monoexponential model. The standard deviations L.K.
‘I,,,(
1
I
YOUNG
I,lll,,
r
IO“
NORHRL
B.H.
>
0
lo-' VENTILRTION/VOLUHEILOG
W.R.
TEST
SC&l
flND ONE YEHR FOLLOWUP
of the recovered distributions shown in Table 1 give an indication of the spread of specific ventilation in young normal subjects. CYSTIC
FIBROSIS
3’
10' VENTILRTION/VOLUtlEILOG
0.0.
SCALE1
TEST F1ND ONE YEFtR FOLLOWUP -L
ID
DISTRIBUTIONS
5 PFITIENTS
FRON EIGHT YOUNG NORHF1LS
13.O.P.S.
PRTTERN 1
5 PFlTIENTS
D.O.P.S.
O.O.P.
FIG. 5. Four examples of repeatability of test. Tests on subjects BM and LK were repeated on same day. Subject LK is the only young normal in this series whose distribution demonstrates a substantial departure from a narrow unimodal distribution. Subjects WR and DD, both part of a group of subjects with obstructive lung disease being followed as part of a longitudinal study, were tested after an interval of 1 yr. Subjects examined on 1 yr follow-up generally kept the same pattern unless there were major changes in the shape of washout curve, possibly indicating a major change in subject’s lung function.
S PFlTTERN 2
TYPE 3
FIG. 6. Some classes of patterns observed. Upper left: 8 young normal subjects (note expanded scale). All show a narrow unimodal distribution with absence of any substantial ventilation to very poorly ventilated units. Ventilation to well ventilated units is occasionally observed. Lower Eefi: 5 different subjects with obstructive lung disease (DOPS) exhibiting what we call pattern 1, an asymmetric unimodal distribution lacking any ventilation to very well ventilated units. Upper right: 5 different subjects showing what we call pattern 2, a bimodal pattern with substantial ventilation going to very well ventilated units and lesser amounts of ventilation going to very poorly ventilated units. Lower right: pattern 3, possibly only a variation of pattern 2. Three subjects exhibited this pattern with two well-separated modes. Two of the subjects who had this pattern also showed the same pattern on 1 yr follow-up. Both distributions were used in figure.
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420
LEWIS,
Subjects without respiratory complaints over the age of 40 had distributions which were generally broader than those seen in young normals (see Table 1). In addition, ventilation of units with specific ventilations of 0.01 and below were sometimes observed. One of the aims of this work was to determine whether the results of these tests could be used to group subjects into meaningful categories. Figure 6 shows several categories of results which were found. Young normal subjects looked generally similar. Three types of pattern were observed in patients with obstructive disease. In pattern 1, subjects showed an absence of very well ventilated units and a broad asymmetric distribution of poorly ventilated units. Pattern 2 was bimodal with very well ventilated and moderately well 1. Parameters
TABLE
Group
of distributions
Age,yr
EVANS,
AND
JALOWAYSKI
ventilated units and comparatively few poorly ventilated units. Pattern 3 was bimodal with the modes widely separated, one of very well ventilated units and one of poorly ventilated units. Other subjects did not fall into the patterns shown. There is insufficient data to report differences in other physiological measurements between subjects showing different patterns. That will be an object of future investigations. The subjects with cystic fibrosis (Fig. 7) all showed distributions with distinct peaks of very well and very poorly ventilated units. One subject exhibited a repeatable trimodal pattern as shown in Fig. 5. The “poorly ventilated compartment” in these subjects is evident and examination of the continuous distribution justifies its characterization as a separate compartment.
studied DS
DS’
Mean
SD
Error
Young normal
27 4 4.5 (19-38)
2,770 k 510 (2,010-3,550)
123 -+ 33 (86-164)
130 in 32 (102-175)
0.19 2 0.03 (0.12-Oe25)
0.56 -t- 0.13 (0.30-O. 79)
0.41 + 0.22 (0.27-0.68)
Older normal
51 * 8 (40-61)
3,400 5 900 (2,400~5,700)
172 + 50 (76-276)
186 + 44 (93-276)
0.19 -t- 0.06 (0.08-0.27)
0.86 + 0.19 (0.57-l. 14)
0.39 -+ 0.14 (0.11-0.68)
Clinic patients
57 k 11 (26-76)
4,041 1 1,470 (2,090-7,170)
180 2 57 (104-402)
207 f 65 (131-448)
0.17 2 0.07 (0.07-0.35)
1.07 -t- 0.35 (0,60-2.04)
0.36 2 0.18 (0.X1-0.60)
Cystic fibrosis
11 f 7 (7-23)
1,281 -t- 888 (539-2,567)
108 -+ 105 (44-263)
144 f 115 (64-315)
0,44 k 0.77 (0.20-0.83)
1.23 2 0.24 (0.89-1.46)
0.54 k 0.39 (0.20-0.83)
Values estimated from the deviations
are means k SD, with ranges given in parentheses. FRC is estimated from the distribution; DS is the anatomic dead space from the multiple-breath washout; IX’ is an alternative estimation of the dead space which includes a fraction of the ventilation very well ventilated compartments; this estimate is more repeatable due to a lower sensitivity to noise. Means and standard are computed on a log scale of v/V, and include only the alveolar ventilation. Error is the rms residual error, K.H.
H.C. c
FRC
CYSTIC
CYSTIC
fIBROSIS
PIBROSIS
S.K.
CYSTIC
FIBROSIS
FIG. 7. Distributions recovered from 4 subjects with cystic fibrosis. All 4 subjects show ventilation to very poorly ventilated units but with the bulk of ventilation going to very well ventilated units. All subjects also showed a high dead space which is not seen in these figures. Three of four subjects exhibited a trimodal distribution with a third peak having approximately same ventilation/volume ratio as the single mode seen in young normal subjects.
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RECOVERY
OF
CONTINUOUS
VENTILATORY
DISTRIBUTIONS
The high dead space found in the diseased population (see Table 1) has previously been discussed (15, 23, 25). The dead space estimated by multiple-breath techniques may be appreciably higher than that estimated from single-breath measurements. This estimate is relatively insensitive to the analytical approach, compartmental or continuous.
The lung is known as a collection of both series and parallel compartments (i.e., trachea and large airways; upper and lower lobes). It may be shown that for any combination of series and parallel units an equivalent model consisting of a collection of parallel units, which behaves in a manner indistinguishable with respect to the washout test (3), can be described. The only assumptions required for this description are a) that all breaths of the washout may be considered identical and b) that the tracer takes the same path entering and leaving the lung, i.e., that there are no unidirectional gas flows in the lung. Diffusion and convection will meet these assumptions and cardiogenic mixing will meet them in a statistical sense. It is important, however, to remember that when compartments which are truly series are modeled as parallel units, the units in the model may not strictly correspond to any anatomic site. A given series unit may be modeled as a mixture of two parallel units such as the dead space and an alveolar compartment, In special cases such as the dead sp’ace and the most poorly ventilated units, some correspondence between series units and units in the model will be present, but this need not be the case. This lack of correlation to specific sites or mechanisms does not affect the model’s ability to describe or to make useful statements about gas exchange in the lung. It is important in considering the validity of an equivalent model to consider what information is to be obtained from it. Numbers such as the ventilation of the dead space and lung volume are relatively independent of the procedure chosen to fit the data and most models appear to give similar values for these. Further information is available in the shape and width of the continuous distribution or with a compartmental approach, in the location and ventilation of the compartments. We have demonstrated the ability of our algorithm to recover a number of known distributions from model data to which realistic amounts of noise have been added. The algorithm used in this work is able to recover a much narrower distribution when presented with a delta function than earlier techniques used to examine continuous distributions. We have chosen to test the technique using smooth and continuous distributions since we believe these distributions best represent what one might expect in a normal or a diseased lung. The technique has the power to determine the general shape of distribution and give an estimate of the number of modes present. For these techniques to be useful, it is necessary to demonstrate that the distributions recovered are reproducible within an individual. Clearly the repeatability
421 will not be exact; however, the broad features of the distribution should be preserved. Since all young normals show narrow distributions there is little question of repeatability in this group. Figure 5 shows that the more complex patterns seen in disease are also repeatable. Not all l-yr follow-up tests demonstrated the degree of repeatability shown in Fig. 5. In every case where there was a major change in the pattern, there was also a major change in the data, suggesting that a shift in the lung function rather than an instability in the technique was responsible for the observed change. Comparison with earlier work. Continuous distributions have been recovered before by Gomez and his colleagues (7-N), Nakamura and his colleagues (16), and Okubo and Lenfant (18). Gomez fit a continuous function of four parameters to smoothed, end-tidal data. This approach suffers from the fact that the distribution sought may not be adequately described by the parametric function. In particular, examination of the distributions published by the Gomez group shows almost no ability to resolve features below specific ventilations of 0.1. In addition the parametric function does not allow for the bi- and trimodal patterns so often observed in this work and by others (16, 18). Nakamura et al. (16) and Okubo and Lenfant (18) both used a direct inversion of the Laplace transform (22) performed on curves fit by hand to mixed data collected over fixed intervals. The direct inversion involves taking first or higher order derivatives of the function fit to the data. This makes the inversion exquisitely sensitive to small variations either in the data or in the curve which is fitted by hand to the data before this inversion is applied. The less sensitive lower order approximations which both groups chose to use suffer from an inability to resolve any but the broadest features. Thus the patterns that these groups report for young normal subjects are several times wider than those observed in this work, as is their recovery of data from a truly uniform lung model. In addition their use of mixed data collected at intervals rather than computed on a breath-by-breath basis restricts their ability to resolve the faster compartments and, as a result, the dead space could not be well described in subjects with obstructive disease. The techniques used in this work to recover the distributions are those of ridge regression. They are well known in the mathematical literature (14). Recovery of distributions of ventilation/perfusion ratios using similar techniques has generated considerable discussion of the uniqueness of the recovered distributions (5, 13, 19). Many of the points raised in those discussions are relevant to this work. The output of the present algorithm is unique? in the sense that a given set of data will be fitted with a given distribution. The recovered distributions themselves are not unique in that it is possible to demonstrate alternative distributions which fit the data within a reasonable tolerance. A trivial example is a distribution where the flow is zero in alternate compartments and roughly twice as large in the remaining compartments. For the ventilation/ perfusion problem Olszowka (19) presents several examples of alternative distributions which will fit a set
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422 of error-free data exactly. Close study has shown (5) that the alternatives consist of a local rearrangement of the original distribution and not a shift of major features. The distributions recovered by the algorithm may be considered to be representative of those distributions fitting the data. We have demonstrated that on repeated testing of a given individual similar distributions are recovered and therefore that a significant change in the recovered distribution can be attributed to a change in the subject’s lungs. The key is recognition of what constitutes a significant change. Significant changes involve alterations in the shape of the entire distribution rather than in the floti to a given compartment. Evans and Wagner (5) have used linear programming to examine this question for the ventilation/perfusion problem. They are able to assign probabilities for a given set of data coming from a distribution with a particular form. They find that if a well-separated bimodal distribution is seen, the underlying distribution has a very high probability of being multimodal, i.e., one cannot find a unimodal distribution which fits the data well. Conversely, if a narrow unimodal distribution is recovered, it is improbable that the data could have come from a bimodal distribution with well-separated modes. They find that a given peak may be separated into several subpeaks without affecting the fit- and that it is difficult to separate units with very low ventilation/perfusion ratios from shunt, Olszowka’s paper provides several examples of these effects. The nitrogen washout problem is not exactly analogous to the ventilation/perfusion problem. We use the same algorithm that Wagner’s group uses (5) with somewhat different weighting factors. In the washout there are at least 18 data points rather than only 6. Nitrogen concentrations may be measured more accurately than the retentions used in ventilation/perfusion studies. We believe for these reasons that the distributions recovered in this work should be better defined than those recovered in ventilation/perfusion studies. However, the same statements about uniqueness apply; where a mode is observed, there may be more than one mode in that region and where there is flow in the dead space and compartments with very high ventilation/ volume ratio, separation of the two is difficult. Alternative distributions which fit the data must be broadly similar to those we recover. It has been argued (17) that there is enough information in the washout curve to describe two or three
LEWIS,
EVANS,
AND
JALOWAYSKI
compartments and thus descriptions should be left at that. Three compartments plus dead space are described by six parameters representing the location and ventilation of the compartments. Solution of Eqs. 1 and 2 reveals that the ventilation of the compartments is a function of N parameters, where N is the number of breaths entered into the analysis. The number of breaths entered could be varied in each case, choosing N breaths spaced approximately equally in log time from the start to the end of the test. Adding more breaths may give new information about the distribution, or, in the case of data with noise, merely permit the effects of noise to be canceled. The degree of cancellations of noise goes as the square root of N. Patterns recovered by our algorithm were not grossly affected by decreasing the number of points entered, when errorfree data were used, until only five or six points were analyzed, implying that the number of parameters necessary to describe the distribution is five or six. Thus, the same information is probably present in a compartmental fit. However, much of the information in a compartmental fit is in the exact position of the compartments and is of limited use in describing the broad underlying distributions. While the compartmental methods may fit the data as well, or nearly as well, the information in the fit is not readily adapted to describing the broad, continuous distributions we believe are present in the lung. Nitrogen washout is a simple and sensitive test of lung function. The control of tidal volume to reduce variation in the data combined with a new analytic approach permits the recovery of continuous distributions of ventilation as a function of specific ventilation in untrained subjects. These distributions may prove useful in the diagnosis and classification of lung disease and in aiding our understanding of the underlying physiology of gas exchange. We thank Dr. C. J. Martin for allowing us to use data collected by him and his colleagues in this work. We also thank Drs. J. B. West and P. D. Wagner for their suggestions and support during the course of this work. The work presented here was supported by National Aeronautics‘ and Space Administration Grant NGL-05-009-109 and Public Health Service Grant HL-17731. Much of the data presented here were collected under National Heart and Lung Institute SCOR Grant
HL-1415-z. S. M. Lewis was Grants
Received
IT-32-HE-07212-01 for publication
supported
by
Public
Health
Service
and HL-05831-04. 24 January
1977.
REFERENCES M. R. New index of intrapulmonary mixture of air. Thorax 7: 111-116, 1952. BOUHUYS, A., and G. LUNDIN. Distribution of inspiratory gas in lungs. Physiol. Rev. 39: 731-750, 1959. EVANS, J. W. The gas washout determination under a symmetry assumption. BUZZ. Math. Biophys. 32: 59-63, 1970. EVANS, J. W., 0. G. CANTOR, AND J. R, NORMAN. Dead space in a compartmental lung model. Math. Biophys. 29: 711-718, 1967. EVANS, J. W., AND P. D. WAGNER. Limits on VA/~ distribution from analysis of experimental inert gas elimination. J. Appl. Physiol.: Respirat . Environ, Exercise PhysioZ. 42: 889-898, 1977. FOWLER, W. S. Lung function study. III, Uneven pulmonary
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ventilation in normal subjects and in patients with pulmonary disease. J. Appl. Physiol. 2: 283-299, 1949. 7, GOMEZ, D. M. A mathematical treatment of the distribution of tidal volume throughout the lung. Proc. N&Z. Acad. Sci., US 49: 312-319, 1963. 8. GOMEZ, D. M., W. A. BRISCOE, AND G. CUMMING. Continuous distribution of specific tidal volume throughout the lung. J. Appl. PhysioZ. 19: 683-692, 1964. 9. G~MEZ, D. M., AND J. FILLER. An analysis of the distribution of specific tidal volumes in children with bronchial asthma. J. Exptl. Med. 193: 514-541, 1965. 10. GOMEZ, D. M., AND J. FILLER. On the concept of continuous
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RECOVERY
11.
12.
13.
14, 15.
16.
17.
OF
CONTINUOUS
VENTILATORY
DISTRIBUTIONS
distribution of specific tidal volume throughout the lungs. Am. Rev+ Respirat. Diseases 93: 1-8, 1966, HASHIMOTO, T., A. C. YOUNG, AND C. J. MARTIN. Compartmental analysis of the distribution of gas in the lungs. J. Appl. Physiol. 23: 203-209, 1967. HEISE, M., H. KOPPE, AND K. SCHMIDT. Mathematical treatment of inert gas clearance curve as a method for studying regional inhomogeneity of alveolar ventilation in the lung. Respiration 31: 310-317,1974. JALIWALA, S. A., R. E, MATE, AND F. J. KLOKE. An efficient optimization technique for recovering ventilation-perfusion distributions from inert gas data: effects of random experimental error. J. CZin, Invest. 53: 188-192, 1975. LAWSON, C. L., AND R. J. HANSON. Solving Least Squares Problems, Englewood Cliffs, N. J.: Prentice-Hall, 1974. MARTIN, C. J., S. TSUNODA, AND A. C. YOUNG. Lung emptying patterns in diffuse obstructive pulmonary syndromes. Respiration. PhysioZ. 21: 156-168, 1974. NAKAMURA, T., T. TAKISHIMA, T. OKUBO, T. SASAKI, AND H. TAKAHASHI. Distribution function of the clearance time constant in lungs. J. AppZ. Physiol. 21: 227-232, 1966. NYE, R. E. Theoretical limits of the measurement of uneven ventilation. J. Appl. Physiol. 16: 1115-1123, 1961.
423 18. OKUBO, T., AND C. LENFANT. Distribution function of lung volume and ventilation determined by lung N, washout. J. Appl. Physiol. 24: 658-667, 1968. 19. OLSZOWKA, A. J. Can VA/Q distributions in the lung be recovered from inert gas retention data? Respiration Physiol. 25: 191-198, 1975. 20. PESLIN, R., S. DAWSON, AND J. MEAD. Analysis of multicomponent exponential curves by the Post-Widder’s equation. J. Appl. Physiol. 30: 462-472, 1971. 21. ROSSING, R. G. Evaluation of a computer solution exponential decay or washout curves. J. Appl. Physiol. 21: 1907-1910, 1966. 22. SCHWARZL, F., AND A. J, STAVERMAN. Higher approximation of relaxation spectra. Physica 18: 791-798, 1952. 23. TSUNODA, S., A. C. YOUNG, AND C. J. MARTIN. Emptying pattern of lung compartments in normal man. J. AppL Physiol. 32: 644-649, 1972. 24. WAGNER, P. D,, H. A. SALTZMAN, AND J, B. WEST. Measurement of continuous distributions of ventilation-perfusion ratios: theory. J. Appl. Physiol. 36: 588-599, 1974. 25. YOUNG, A. C., C. J. MARTIN, AND S. TSUNODA. Lung volumes in diffuse obstructive pulmonary syndromes. J, Clin. Invest. 53: 1178-1184,1974.
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ventilation
STEVEN MI, LEWIS, JOHN W. EVANS, AND ALFRED0 A, JALOWAYSKI Departments of Medicine and Mathematics, University of California, San Diego, La Jolla, California 92037
LEWIS, STEVEN M., JOHN W. EVANS, JALOWAYSKI. Continuous distributions of
AND
ALFREDO
A.
specific ventilation recovered from inert gas washout. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 44(3): 416-423, 1978. -We describe a new technique for recovering continuous distributions of ventilation (Q as a function of tidal ventilation/volume ratio (ir/V,) from th e nitrogen washout. The analysis yields a continuous distribution of v as a function of %7/V, represented as fractional ventilations of 50 compartments plus dead space. The procedure was verified by recovering known distributions from data to which noise had been added. Using an apparatus to control the subject’s tidal volume and FRC, mixed expired N, data gave the following results: a) the distributions of young, normal subjects were narrow and unimodal with a mean In standard deviation of 0.56 5 0.13; b) those of subjects over age 40 were broader (In SD 0.86 * 0.19) with more poorly ventilated units; c) patients with pulmonary disease of all descriptions showed enlarged dead space; d) patients with cystic fibrosis showed multimodal distributions with the bulk of the ventilation going to overventilated units; and e) patients with obstructive lung disease fell into several classes, three of which are illustrated. These results suggest that our approach is well suited for clinical investigation. nitrogen
washout;;
fibrosis; pulmonary
dead space; obstructive function testing
lung disease; cystic
INERT GAS WASHOUT SAMPLED by a rapid gas analyzer has proved to be a potent tool in the analysis of pulmonary function. Multibreath washout curves have conventionally been described in terms of a model with two or three compartments (1, 2, 6, 11, 15, 21, 23). Several groups have described analyses that recovered a continuous distribution of specific tidal volume, the ratio of the end-expiratory volume to the tidal volume. Two mathematical techniques were employed to generate these distributions. Nakamura et al. (16) and Okubo and Lenfant (18) used the direct inversion of the Laplace transform. Gomez and his colleagues (s-10) fit their curves with a continuous distrib&on which was described by a small number of parametera. These techniques have been criticized on four grounds (12, 17, 20). First,, the ability to recover physiologically relevant distributions has been questioned, especially narrow distributions as might be seen in a healthy young normal. Second, variations in tidal volume during the test make some of the underlying assumptions suspect. Third, since the direct inversion techniques THE
416
involve taking higher derivatives of the data, they are exquisitely sensitive to noise in the data. Fourth, the ability of a given parametric distribution to approximate well the actual distribution present in the lung has been questioned. We have undertaken to recover continuous distributions of specific tidal ventilation using an experimental and analytical approach which eliminates many of the objections previously raised and several others which we shall discuss below. METHODS
Nitrogen washouts were performed using the equipment shown in Fig. 1. This apparatus, a modification of that used by Martin and his colleagues (15), permits control of both the tidal volume and lung volume of untrained subjects, allowing greater accuracy and repeatability. The subject breathed air for several breaths until the operator was satisfied with his ability to breathe using the controller, at which point the inspired gas switched to oxygen. From 18 to 40 breaths of washout data were obtained. The mean expired nitrogen concentration before the washout and on each breath of the test was computed after correction for the analyzer delay and nitrogen delivery from the blood. Data from a single test are shown in Fig. 2. The data were analyzed by a ridge regression similar to that used by Evans and Wagner (5). The mean expired N, concentration is described as arising from a set of parallel compartments each having a ventilation of VT v(i), and a specific tidal ventilation (VT v(i)/ VO(i)) of S(i), where VT is the tidal volume, o(i) is the fraction of the ventilation going to ith compartment, and V, is the volume of the compartment at end expiration. The mean expired concentration on the jth breath (Cj) is thus described by Eq. I l
l
Cj C0
-=
l
c
l .
V(l)
= Mdj, ?)
i compartments
(1)
If the compartments are specified, fixing S(i), then the term on the right of Eq. 1 is a function of the breath number and the set of v(i) and may be written as M(j, 3> where v indicates the collection of V(i). The data were analyzed by minimizing over V(i) the expression
OOZl-8987/78/0000-OOOO$Ol.
25 Copyright
0 1978 the American
Physiological
Society
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RECOVERY
OF
CONTINUOUS
VENTILATORY
417
DISTRIBUTlONS
SPIROMETER
)W VOLUME
I
VOLUME
CONCENTRATION
LOGGER
FIG. 1. Experimental setup. Subjects begin test breathing through a port in mouthpiece (not shown). At end inspiration the port is closed. Volume controller is set at that point so that only the solenoid from air bag is open. Subject inhales from bag-in-box system whose volume is sensed by an electronic spirometer. After a predetermined volume has been inhaled, the controller switches so that both inspiration solenoids are closed, but expiratory solenoid is open, allowing the subject to expire into the bag-in-box. On return to end-expiratory volume, selected inspiratory solenoid is opened. Subjects are instructed to control flow rates by observing a meter. Once the operator is satisfied wit,h subject’s performance breathing air, the controller may be altered without subject’s knowledge so that subsequent inspirations are from the oxygen bag. Volume and nitrogen concentration are recorded continuously for subsequent analysis.
80X---’ ’
1;..
--
-----
JB. 2SVo Q NORMAL i-
t-
27.
>
A.--i-
_
T
--
-,--------i-_--.
FIG. 2. Sample data. Note uniformity flow (as slope of volume). Subject took before washout began.
c [W(j)-(D(j)
- Wj,
of both tidal volume and four practice breaths of air
v’,)lz
j bwaths
(2) +
2’
c i compartrrwnts
[WC(i)
’ V(i)]
where D(j) is the measured C&, W(j) and WC(i) are weights described in the text, and V(i) is subject to a nonnegativity constraint. The term on the right of expression 2 is a smoothing term which causes the procedure to choose a smooth distribution closely fitting the data. We used a value of z of 0.001 which was
chosen empirically by fitting known distributions in the presence of noise. The weighting of the data, W(j), was set at unity and the weighting of the compartments, WC(i), was set at I./C,* (1 - C,), where C, and C, are the concentrations in a given compartment on the first and last breaths of the test, respectivelv. This method of weighting chooses a model which is ,T;mooth and has a small lung volume and a large dead space from all possible models which are consistent with a close fit to the data. Effects of error and nonuniqueness of the recovered distribution are discussed in detail below. Except for the constraint that no compartment. be allowed to have a negative ventilation, expression 2 is minimized for a single set of V(i) which may be determined analytically. A search procedure was employed to seek the largest collection of compartments over which the nonnegativity constraint was satisfied and the resultant distribution was the desired one. The specified compartments were equally spaced on a log scale from a specific ventilation of 0.005 t.o 10 as shown in Fig. 3. Figure 3 shows the ability of the technique to recover several known distributions in the presence of added error. The data presented to the algorithm to generate Fig. 3 was D(j) 0(1 + (R/2) (1 + I./ JO(j)>), where D(j) is the error-free data for the jth breath and R is a normal random variable with the stated standard deviation. This function tends to approximate the observed distribution of residuals in real data. Errors of 0.5-l% approximate the size of residuals seen in real data. The calculated dead space is of particular interest. When mixed expired nitrogen concentration is used, dead space can be defined as the ventilation to a compartment having an infinite specific ventilation. In practice this may be computed as the ventilation not present in the other compartments. Thus the distributions shown have areas less than 1, the remaining area representing the ventilation of the dead space. This definition of dead space has been shown (4) to include any series dead space and will be equal to the ventilation of the series dead space in the absence of unusual regions such as respiratory units which collapse to zero vo1um.e on each breath. The dead space was first defined in this way by Young et al. (25) and has since been discussed by Tsunoda et al. (23) and Martin et al. (15). It differs from the Fowler dead space in that no assumption about the portion of the breath over which the dead space empties is necessary. The dead space as defined above is sensitive to noise on the early breaths which may cause some of the ventilation of the dead space to appear in compartments with very high specific ventilations. If the measured nitrogen on the first breath is erroneously high, this can be interpreted as indicating a very well ventilated compartment, which would have significant nitrogen only on the first one to two breaths. The ventilation of this spurious compartment would otherwise be assigned to the dead space. To estimate the amount of ventilation wasted in ventilating the dead space and the very well ventilated units another quantity DS’ was defined as
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LEWIS, BROf3D
UI G
BIMOOAL
DISTRIBUTION
0.5
J.B.
PCT NOISE
0 0 0
RECOVERED
-
GIVEN
x,
TRIMODAL
DISTRIBUTION
DISTRIBUTION
SCALE1
c
i compartments
WiP(
JALOWAYSKI
1 PCT NQFX
a.5
a 0 Q
RECOVERED
-
GWEN
PCT Narx
FIG. 3. Recovery of known distributions. Recovered distribution (points) are output of the algorithm when presented with data generated by a model of the given distribution (solid curves) to which noise was added as described in text. Axes used for all distributions are tidal ventilation/volume on a log scale from 0.005 to 10. Dead space is not shown on the distribution but was 25% of tidal volume in all cases and was recovered within 2% of tidal volume in all cases. Y-axis represents fraction of ventilation in each of 50 compartments, Note expanded scale on delta function and unimodal distribution. Also note the ability to recover a narrow distribution even in the presence of large amounts of error.
RESULTS
FIG, 4. Recovered distribution from data shown in Fig. 2. Distribution is a narrow unimodal pattern. Distribution is broader than that recovered with a uniform lung model (see upper left in Fig. 3), even though that model contained a large amount of noise. Small amount of ventilation in the very high ventilation/volume regions may represent noise bringing some of the dead space ventilation into these regions, or a very well ventilated compartment as seen in subj LK in Fig. 5.
DS ‘= DS+
AND
first one or two breaths will have a comparatively minor effect on this measurement. DS’ was very close to the dead space in most subjects. Normal subjects were chosen from workers in the hospital and research laboratories. Diseased subjects included patients from a group of chest clinic patients, primarily with obstructive lung disease, selected for long-term study and patients coming to a pediatric pulmonary function laboratory. Data were considered acceptable if they could be fit with an rms error of less than 0.75% nitrogen. In children under age 12 the data were, for technical reasons, somewhat noisier and an rms error of 1.0% was considered acceptable if replicate studies confirmed the broad features of the recovered distributions.
YOUNG NURMAL
VENTILATION/VULUME[LOG
EVANS,
l-
j&,)2
(3)
DS’ corresponds to the maximal dead space which might exist if the very well ventilated units were considered to be mixtures of dead space and alveolar gas. Because much of the ventilation of very well ventilated units will be included in DS’, noise in the
Nineteen normal subjects were under age 40 and had no respiratory symptoms. This group included several smokers who were indistinguishable from the rest of the group on the basis of the washout and other respiratory tests. Eighteen of the nineteen showed a narrow unimodal distribution with only insignificant ventilation outside the narrow peak. Examples of results from normal subjects are shown in Figs. 4-6. Some of the characteristics of the subjects and their distributions are listed in Table 1. One subject repeatedly showed ventilation to units with high specific ventilations as well as the narrow mode seen with the others. Two of the tests are shown in Fig. 5. The uniformity of these results is consistent with previous descriptions of the washout curves of young normals as essentially monoexponential. However, in
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RECOVERY
OF
CONTINUOUS
VENTILATORY
419
DISTRIBUTIONS
no case was the distribution recovered from the data of any subject as narrow as that shown in Fig. 3 for a truly monoexponential model. The standard deviations L.K.
‘I,,,(
1
I
YOUNG
I,lll,,
r
IO“
NORHRL
B.H.
>
0
lo-' VENTILRTION/VOLUHEILOG
W.R.
TEST
SC&l
flND ONE YEHR FOLLOWUP
of the recovered distributions shown in Table 1 give an indication of the spread of specific ventilation in young normal subjects. CYSTIC
FIBROSIS
3’
10' VENTILRTION/VOLUtlEILOG
0.0.
SCALE1
TEST F1ND ONE YEFtR FOLLOWUP -L
ID
DISTRIBUTIONS
5 PFITIENTS
FRON EIGHT YOUNG NORHF1LS
13.O.P.S.
PRTTERN 1
5 PFlTIENTS
D.O.P.S.
O.O.P.
FIG. 5. Four examples of repeatability of test. Tests on subjects BM and LK were repeated on same day. Subject LK is the only young normal in this series whose distribution demonstrates a substantial departure from a narrow unimodal distribution. Subjects WR and DD, both part of a group of subjects with obstructive lung disease being followed as part of a longitudinal study, were tested after an interval of 1 yr. Subjects examined on 1 yr follow-up generally kept the same pattern unless there were major changes in the shape of washout curve, possibly indicating a major change in subject’s lung function.
S PFlTTERN 2
TYPE 3
FIG. 6. Some classes of patterns observed. Upper left: 8 young normal subjects (note expanded scale). All show a narrow unimodal distribution with absence of any substantial ventilation to very poorly ventilated units. Ventilation to well ventilated units is occasionally observed. Lower Eefi: 5 different subjects with obstructive lung disease (DOPS) exhibiting what we call pattern 1, an asymmetric unimodal distribution lacking any ventilation to very well ventilated units. Upper right: 5 different subjects showing what we call pattern 2, a bimodal pattern with substantial ventilation going to very well ventilated units and lesser amounts of ventilation going to very poorly ventilated units. Lower right: pattern 3, possibly only a variation of pattern 2. Three subjects exhibited this pattern with two well-separated modes. Two of the subjects who had this pattern also showed the same pattern on 1 yr follow-up. Both distributions were used in figure.
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420
LEWIS,
Subjects without respiratory complaints over the age of 40 had distributions which were generally broader than those seen in young normals (see Table 1). In addition, ventilation of units with specific ventilations of 0.01 and below were sometimes observed. One of the aims of this work was to determine whether the results of these tests could be used to group subjects into meaningful categories. Figure 6 shows several categories of results which were found. Young normal subjects looked generally similar. Three types of pattern were observed in patients with obstructive disease. In pattern 1, subjects showed an absence of very well ventilated units and a broad asymmetric distribution of poorly ventilated units. Pattern 2 was bimodal with very well ventilated and moderately well 1. Parameters
TABLE
Group
of distributions
Age,yr
EVANS,
AND
JALOWAYSKI
ventilated units and comparatively few poorly ventilated units. Pattern 3 was bimodal with the modes widely separated, one of very well ventilated units and one of poorly ventilated units. Other subjects did not fall into the patterns shown. There is insufficient data to report differences in other physiological measurements between subjects showing different patterns. That will be an object of future investigations. The subjects with cystic fibrosis (Fig. 7) all showed distributions with distinct peaks of very well and very poorly ventilated units. One subject exhibited a repeatable trimodal pattern as shown in Fig. 5. The “poorly ventilated compartment” in these subjects is evident and examination of the continuous distribution justifies its characterization as a separate compartment.
studied DS
DS’
Mean
SD
Error
Young normal
27 4 4.5 (19-38)
2,770 k 510 (2,010-3,550)
123 -+ 33 (86-164)
130 in 32 (102-175)
0.19 2 0.03 (0.12-Oe25)
0.56 -t- 0.13 (0.30-O. 79)
0.41 + 0.22 (0.27-0.68)
Older normal
51 * 8 (40-61)
3,400 5 900 (2,400~5,700)
172 + 50 (76-276)
186 + 44 (93-276)
0.19 -t- 0.06 (0.08-0.27)
0.86 + 0.19 (0.57-l. 14)
0.39 -+ 0.14 (0.11-0.68)
Clinic patients
57 k 11 (26-76)
4,041 1 1,470 (2,090-7,170)
180 2 57 (104-402)
207 f 65 (131-448)
0.17 2 0.07 (0.07-0.35)
1.07 -t- 0.35 (0,60-2.04)
0.36 2 0.18 (0.X1-0.60)
Cystic fibrosis
11 f 7 (7-23)
1,281 -t- 888 (539-2,567)
108 -+ 105 (44-263)
144 f 115 (64-315)
0,44 k 0.77 (0.20-0.83)
1.23 2 0.24 (0.89-1.46)
0.54 k 0.39 (0.20-0.83)
Values estimated from the deviations
are means k SD, with ranges given in parentheses. FRC is estimated from the distribution; DS is the anatomic dead space from the multiple-breath washout; IX’ is an alternative estimation of the dead space which includes a fraction of the ventilation very well ventilated compartments; this estimate is more repeatable due to a lower sensitivity to noise. Means and standard are computed on a log scale of v/V, and include only the alveolar ventilation. Error is the rms residual error, K.H.
H.C. c
FRC
CYSTIC
CYSTIC
fIBROSIS
PIBROSIS
S.K.
CYSTIC
FIBROSIS
FIG. 7. Distributions recovered from 4 subjects with cystic fibrosis. All 4 subjects show ventilation to very poorly ventilated units but with the bulk of ventilation going to very well ventilated units. All subjects also showed a high dead space which is not seen in these figures. Three of four subjects exhibited a trimodal distribution with a third peak having approximately same ventilation/volume ratio as the single mode seen in young normal subjects.
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RECOVERY
OF
CONTINUOUS
VENTILATORY
DISTRIBUTIONS
The high dead space found in the diseased population (see Table 1) has previously been discussed (15, 23, 25). The dead space estimated by multiple-breath techniques may be appreciably higher than that estimated from single-breath measurements. This estimate is relatively insensitive to the analytical approach, compartmental or continuous.
The lung is known as a collection of both series and parallel compartments (i.e., trachea and large airways; upper and lower lobes). It may be shown that for any combination of series and parallel units an equivalent model consisting of a collection of parallel units, which behaves in a manner indistinguishable with respect to the washout test (3), can be described. The only assumptions required for this description are a) that all breaths of the washout may be considered identical and b) that the tracer takes the same path entering and leaving the lung, i.e., that there are no unidirectional gas flows in the lung. Diffusion and convection will meet these assumptions and cardiogenic mixing will meet them in a statistical sense. It is important, however, to remember that when compartments which are truly series are modeled as parallel units, the units in the model may not strictly correspond to any anatomic site. A given series unit may be modeled as a mixture of two parallel units such as the dead space and an alveolar compartment, In special cases such as the dead sp’ace and the most poorly ventilated units, some correspondence between series units and units in the model will be present, but this need not be the case. This lack of correlation to specific sites or mechanisms does not affect the model’s ability to describe or to make useful statements about gas exchange in the lung. It is important in considering the validity of an equivalent model to consider what information is to be obtained from it. Numbers such as the ventilation of the dead space and lung volume are relatively independent of the procedure chosen to fit the data and most models appear to give similar values for these. Further information is available in the shape and width of the continuous distribution or with a compartmental approach, in the location and ventilation of the compartments. We have demonstrated the ability of our algorithm to recover a number of known distributions from model data to which realistic amounts of noise have been added. The algorithm used in this work is able to recover a much narrower distribution when presented with a delta function than earlier techniques used to examine continuous distributions. We have chosen to test the technique using smooth and continuous distributions since we believe these distributions best represent what one might expect in a normal or a diseased lung. The technique has the power to determine the general shape of distribution and give an estimate of the number of modes present. For these techniques to be useful, it is necessary to demonstrate that the distributions recovered are reproducible within an individual. Clearly the repeatability
421 will not be exact; however, the broad features of the distribution should be preserved. Since all young normals show narrow distributions there is little question of repeatability in this group. Figure 5 shows that the more complex patterns seen in disease are also repeatable. Not all l-yr follow-up tests demonstrated the degree of repeatability shown in Fig. 5. In every case where there was a major change in the pattern, there was also a major change in the data, suggesting that a shift in the lung function rather than an instability in the technique was responsible for the observed change. Comparison with earlier work. Continuous distributions have been recovered before by Gomez and his colleagues (7-N), Nakamura and his colleagues (16), and Okubo and Lenfant (18). Gomez fit a continuous function of four parameters to smoothed, end-tidal data. This approach suffers from the fact that the distribution sought may not be adequately described by the parametric function. In particular, examination of the distributions published by the Gomez group shows almost no ability to resolve features below specific ventilations of 0.1. In addition the parametric function does not allow for the bi- and trimodal patterns so often observed in this work and by others (16, 18). Nakamura et al. (16) and Okubo and Lenfant (18) both used a direct inversion of the Laplace transform (22) performed on curves fit by hand to mixed data collected over fixed intervals. The direct inversion involves taking first or higher order derivatives of the function fit to the data. This makes the inversion exquisitely sensitive to small variations either in the data or in the curve which is fitted by hand to the data before this inversion is applied. The less sensitive lower order approximations which both groups chose to use suffer from an inability to resolve any but the broadest features. Thus the patterns that these groups report for young normal subjects are several times wider than those observed in this work, as is their recovery of data from a truly uniform lung model. In addition their use of mixed data collected at intervals rather than computed on a breath-by-breath basis restricts their ability to resolve the faster compartments and, as a result, the dead space could not be well described in subjects with obstructive disease. The techniques used in this work to recover the distributions are those of ridge regression. They are well known in the mathematical literature (14). Recovery of distributions of ventilation/perfusion ratios using similar techniques has generated considerable discussion of the uniqueness of the recovered distributions (5, 13, 19). Many of the points raised in those discussions are relevant to this work. The output of the present algorithm is unique? in the sense that a given set of data will be fitted with a given distribution. The recovered distributions themselves are not unique in that it is possible to demonstrate alternative distributions which fit the data within a reasonable tolerance. A trivial example is a distribution where the flow is zero in alternate compartments and roughly twice as large in the remaining compartments. For the ventilation/ perfusion problem Olszowka (19) presents several examples of alternative distributions which will fit a set
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422 of error-free data exactly. Close study has shown (5) that the alternatives consist of a local rearrangement of the original distribution and not a shift of major features. The distributions recovered by the algorithm may be considered to be representative of those distributions fitting the data. We have demonstrated that on repeated testing of a given individual similar distributions are recovered and therefore that a significant change in the recovered distribution can be attributed to a change in the subject’s lungs. The key is recognition of what constitutes a significant change. Significant changes involve alterations in the shape of the entire distribution rather than in the floti to a given compartment. Evans and Wagner (5) have used linear programming to examine this question for the ventilation/perfusion problem. They are able to assign probabilities for a given set of data coming from a distribution with a particular form. They find that if a well-separated bimodal distribution is seen, the underlying distribution has a very high probability of being multimodal, i.e., one cannot find a unimodal distribution which fits the data well. Conversely, if a narrow unimodal distribution is recovered, it is improbable that the data could have come from a bimodal distribution with well-separated modes. They find that a given peak may be separated into several subpeaks without affecting the fit- and that it is difficult to separate units with very low ventilation/perfusion ratios from shunt, Olszowka’s paper provides several examples of these effects. The nitrogen washout problem is not exactly analogous to the ventilation/perfusion problem. We use the same algorithm that Wagner’s group uses (5) with somewhat different weighting factors. In the washout there are at least 18 data points rather than only 6. Nitrogen concentrations may be measured more accurately than the retentions used in ventilation/perfusion studies. We believe for these reasons that the distributions recovered in this work should be better defined than those recovered in ventilation/perfusion studies. However, the same statements about uniqueness apply; where a mode is observed, there may be more than one mode in that region and where there is flow in the dead space and compartments with very high ventilation/ volume ratio, separation of the two is difficult. Alternative distributions which fit the data must be broadly similar to those we recover. It has been argued (17) that there is enough information in the washout curve to describe two or three
LEWIS,
EVANS,
AND
JALOWAYSKI
compartments and thus descriptions should be left at that. Three compartments plus dead space are described by six parameters representing the location and ventilation of the compartments. Solution of Eqs. 1 and 2 reveals that the ventilation of the compartments is a function of N parameters, where N is the number of breaths entered into the analysis. The number of breaths entered could be varied in each case, choosing N breaths spaced approximately equally in log time from the start to the end of the test. Adding more breaths may give new information about the distribution, or, in the case of data with noise, merely permit the effects of noise to be canceled. The degree of cancellations of noise goes as the square root of N. Patterns recovered by our algorithm were not grossly affected by decreasing the number of points entered, when errorfree data were used, until only five or six points were analyzed, implying that the number of parameters necessary to describe the distribution is five or six. Thus, the same information is probably present in a compartmental fit. However, much of the information in a compartmental fit is in the exact position of the compartments and is of limited use in describing the broad underlying distributions. While the compartmental methods may fit the data as well, or nearly as well, the information in the fit is not readily adapted to describing the broad, continuous distributions we believe are present in the lung. Nitrogen washout is a simple and sensitive test of lung function. The control of tidal volume to reduce variation in the data combined with a new analytic approach permits the recovery of continuous distributions of ventilation as a function of specific ventilation in untrained subjects. These distributions may prove useful in the diagnosis and classification of lung disease and in aiding our understanding of the underlying physiology of gas exchange. We thank Dr. C. J. Martin for allowing us to use data collected by him and his colleagues in this work. We also thank Drs. J. B. West and P. D. Wagner for their suggestions and support during the course of this work. The work presented here was supported by National Aeronautics‘ and Space Administration Grant NGL-05-009-109 and Public Health Service Grant HL-17731. Much of the data presented here were collected under National Heart and Lung Institute SCOR Grant
HL-1415-z. S. M. Lewis was Grants
Received
IT-32-HE-07212-01 for publication
supported
by
Public
Health
Service
and HL-05831-04. 24 January
1977.
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