V. FORCED
FLOW
RATES
George R. DeMuth, M.D., William F. Howatt, M.D., and Bruce Hill, Ph.D.
Departments of Pediatrics and Mathematics, University of Michigan, Ann Arbor
Subjects
The subjects were 147 normal school
chil-This study was supported by Crant A-3575 from the National Institute of Arthritis and Metabolic
Diseases.
PEDIATRICS, January, Part II, 1965
200
I
N THE srury of normal children wein-cluded several measurements of forced flow rates as indirect indicators of airway
resistance (or its reciprocal, airway conduct-ance, Cd). The measurements of airway resistance by use of body plethysmogra-phy or by intraesophageal pressure and air flow determinations are complicated, un-pleasant to the subject, and time consum-ing. Therefore, tests which reflect airway
conductance are commonly utilized. Of such indirect tests the maximal breathing
capacity has in the past been most promi-nent but is being displaced by single-breath tests. In general, the single-breath
tests are easier, less exhausting for the sub-ject, and more suitable to office and clinic work.
We have measured the peak flow rates during forced inspiration and during forced
expiration, as well as the forced expiratory flow rate at the mid-volume point and at the point where 75% of the vital capacity has been expired. Data for the last two
flow rates have not been published. These latter tests are especially important since they appear to be much less effort-depend-ent than the peak flow rates and may re-flect the more involved portions of the
lung. We have examined the relationships of all the tests to age, body size, and lung volumes. Further, we have postulated
a mathematical relationship between the forced flow rate and lung volume. By use of this relationship one can obtain an
“emp-tying factor” in normal individuals which is theoretically independent of the lung vol-ume at which it is measured.
METHOD AND MATERIAL
dren previously described.’ Longitudinal data for the peak flow rates are available from the studies repeated approximately 15 months after the first series. Not all
sub-jects participated in the second series and complete data were not obtained in either series. The number of subjects for each of the tests is given in the results.
Equipment
The previously described1 Krogh
spirom-eter was connected by a 12-inch straight tube with a minimal internal diameter of 4 cm to a Fleisch No. 1 heated
pneumotach-ograph.#{176} Pressure changes from the pneu-motachograph were sensed by a Statham
P97 strain gauge transducer connected to one channel of a recorder. The rectified output of the spirometer was simul-taneously recorded on a second channel. Approximately the first third of the first series were recorded on a Grass 4-channel oscillograph. The remainder of the first
#{176}The characteristics of this spirometer were as
follows: Its free vibration frequency (at about 4
liters of volume) was 8.9 cycles/second with a damping ratio of #{216}#{216}42 Opposing pressures
dur-ing ventilation occurred from three sources:
in-ertia, spirometer resistance to motion, and a
posi-tional pressure related to the tendency of the
spirometer to return to an equilibrium point. The
first two factors were assessed by measuring the
pressures developed when the spirometer was filled
by air flowing at 2 1/sec. The onset of flow was very
abrupt and the acceleration thus probably much
greater than occurred even with a forced
expira-tion. The pressure developed at the onset,
prin-cipally to overcome inertia, was 0.174 cmFIO. The
resistance of the spirometer to steady flow at 2 l/sec.
was about 0.015 cm H,O/h/sec. The pressure
re-quired to hold the spirometer away from the
equil-ibrium point was virtually linear in the range tested
and was 0.013 cm H,O/l. These pressures were all
small and should not have interfered appreciably
series and all the second series were record
on a Offner Dynograph, 4-channel
oscillo-graph. The paper speed was 25mm/sec.
The measured 90% response time of the Grass recorder was at about 0.01 see, of the Dynograph, about 0.003 sec. Since both times are short compared to the time to reach peak flow, it was felt that this
differ-ence would not affect the comparability of the data.
The pneumotachograph was calibrated by forcing air through it at different con-stant rates. Tile gas was collected for a timed interval in a Tissot spirometer and measured. The resistance to flow by the
apparatus was very low. The recorder de-flection per unit flow was not appreciably alinear until quite high flow rates were ob-tained (over 250 1/mm) and was but 4% deviant at 500 1/mm.
DEFINITIONS AND CALCULATIONS
The measurements were made on two forced inspirations and two forced expira-tions, starting from maximal expiratory and
maximal inspiratory points respectively. The subjects were rehearsed prior to the
test. In forced expiration the peak or high-est flow value occurred early and the flow rate quickly slowed. In contrast, in forced
inspiration the flow rate plateaued near its maximal value during the first third of the
inhalation. The expiratory tracing was not used unless tile curve was sufficiently smooth following the peak. For both peak flow
tests the reported values were the higher of two duplicates. Both of these tests were done in both series.
The flow rates were obtained from the tracings at the points when half and three-quarters of the vital capacity were expired. These have been designated as the E50 and E75.3 The values reported for each were the means of duplicates. These flows were measured only during the second series.
ANALYSIS
Unless otherwise stated all analyses were carried out after logarithmic transformation
of the variables. The relationships of the
peak expiratory flow rate and peak inspira-tory flow rates to height were analyzed by
covariance analysis. Many other relation-ships of these two dependent variables to age, body size, and lung volumes were analyzed by the calculation of ordinary regression lines. Both methods have been previously discussed.1 The latter method
was convenient but does not allow for the fact that some of the points were obtained from the same individuals. As has been pre-viously discussedl the regression coefficient from covariance analysis can be very
sen-sitive to small systematic changes; so its principal use has been to indicate the tend-ency of the individuals to maintain the:r
respective positions in the group (by its reduction in residual variance from the
ordinary regression). The E50, E75, and the ratios of these flows to the volumes at which
they were measured were analyzed by cal-culating ordinary regressions on independ-ent variables, since the results were
ob-tained from only one series. Unless stated otherwise the reported results of analyses are based upon ordinary regression after logarithmic transformations of both the dependent and independent variables utiliz-ing all the available data which were pooled. The standard deviations are ex-pressed as fractions of the predicted values.
RESULTS
In analyzing the data of the peak
expira-tory and of the inspiratory flow rates we have used the higher of duplicates; we felt
that the error in these tests obtained from exerting improper effort (which lowers the value) may greatly exceed that due to other
measurement errors. In general, the values were lower for girls than boys (at a given
age or size). The standard deviations about the regression lines (even after log trans-formation) also tended to be smaller for
girls.
202 FORCED FLO\V RATES
SLogarithmic transformation used throughout.
as they were for the lung volumes or the
diffusing capacity. Tile addition of
inde-pendent variables barely increases tile
mul-tiple correlation coefficients. The regression coefficients tend to be lower than compar-able ones obtained for lung volumes or for the diffusing capacity while the standard errors of the coefficients tend to be high. The lowest residual variances obtained by
employing two or less independent van-ables were found with age and surface area for boys and with age and vital capacity for girls. The standard deviations around the regression lines using these independent
variables were 0.23 and 0.21 expressed as a fraction of the predicted values for boys
and girls respectively. The residual variance from the ordinary regression of the peak expiratory flow with height was 0.01055 for boys and 0.01118 for girls. Comparable
values obtained from the analyses of
covari-ance were 0.00599 for boys and 0.00444 for
girls, being approximately half those
oh-tamed from the ordinary regression. This finding indicates that the subjects do tend to maintain their respective positions within
their size group during growth. Covaniance reduced the standard deviations to 0.18 (boys) and 0.15 (girls) expressed as fractions of the predicted values.
Our results at any given height are lower
than those obtained for school-age children using the Wright peak flow meter. This can be seen in Table II, in which, for compani-son, are given values from three recent papers on the peak flow rates. Our values differ from those of Naim et al. and from
those of Murray and Cook.5 This may be
due to peculiarities of the testing
appara-tuses. The sharp rise and fall in the
expira-TABLE I
EXPIRATORY Fww
PEAK
indep.
Boys (n=1O1) GirL, (n=18O)’
Variable Correlation ResidualRegression
Coefficient Variance Coefficient
St. Error
CoefficienE 1ercept
Correlation Corffieient
Ruidual Varianre
Regression
Coefficitni
SI. Error
Coefficieni Intercept
Height .8536 .01055 5.645 .161 -5.31908 .8399 .01118 1.864 .164 -3.84931
Age .8410 .01131 1.109 .071 1.S9l4 .8677 .00938 0.981 .050 1.31638
SitLht. .733 .01801 5.113 .197 -1.3797 .8314 .01173 .656 .137 -.66O95
Veig)it .8486 .01088 0.950 .060 0.87744 .796 .O141 0.916 .O6 0.87409
Surf.area .8561 .01038 1.404 .085 .49I0 .8159 .0I69 1.437 .01)0 .1945
Biacromial .8098 .01338 5.381 .173 -1.13833
Vit.aicap. .8498 .01079 0.918 .058 -0.77314 .8373 .01005 0.986 .03 -0.97438
Age& .8609 .01016 0.441 .103 -1.65714 .878 .0091 0.716 .131 -0.6769
height 1.665 .477 0.861 .394
Age& .8413 .01139 1.044 .136 1.01166 .871 .00916 0.75! .i4 0.11437
mitt. lit. 0.167 .199 0.707 .350
Age& .8670 .00975 0.464 .170 1.81884 .8703 .00918 0.815 .118 1.46053
surf.area 0.871 .111 0.184 .is:t
Height & .8399 .01013 0.337 .370 -1.39417 .8400 .01117 1.810 .487 -3.76494
weight 0.417 .106 0.019 .165
Age& .8615 .01003 0.510 .177 0.03861 .8861 .00811 0.569 .105 0.17708
vitalcap. 0.537 .147 0.467 .107
Age& .8687 .00973 0.491 .100 0.47155 .8718 .00919 0.716 .131 -0.17786
height & 0.310 .747 0.804 .573
L/mln.
600 500
400
300
200
GIRLS
00
/
/
HEIGHT IN CM.
j
160 ITO $0 PEAK EXPIRATORY
FLOW
Ioo (0 IO io i4o 50
HEIGHT IN CM.
Fia. 1. The relationship of peak expiratory flow rate to height (logarithmic scales).
tory flow pattern makes the measurement of this value very sensitive to the force of
acceleration, to any tendency to overshoot, and to any lag effect due to inertia, either
electrical or mechanical. Because of these factors any set of abnormal data must be compared to normals done on the same type of apparatus. The values obtained from the table of Rivera,6 who used a Lilly pneumotachograph, are closer to ours.
In Table III are given the results of the analyses of the peak inspiratory flow data. The correlation coefficients are a little high-er and the residual variance somewhat low-er than those of the peak expiratory flow
rates. It is likely that this occurred because
the forced inspiratory flow pattern tends to have a plateau at the high point. The values are thus less dependent upon the
rate of acceleration, the tendency to over-shoot or to lag, and so are more reproduc-ible. The regression coefficients from these analyses are also more nearly the same as those for the lung volumes than are those for the peak expiratory flow. The best single anthropomorphic measurement as an independent variable for boys was height; it was second to sitting height for
girls. The relationships of peak inspiratory flow to height for boys and girls are shown
in Figure 2. The residual variances from
the analyses of covariance of the flow
against height were 0.00527 (boys) and
0.00521 (girls). These represent a drop of about 50% for boys but only about 38% for girls from those obtained from ordinary re-gression. The standard deviations from the covariance analyses were 0.17 expressed as a fraction of the predicted value.
No combination of independent variables reduced the residual variance much. The
combination of two that produced the low-est residual variance in ordinary regression
equations was age and vital capacity for both boys and girls. The standard devia-tions around the regression line, calculated from these two variables were 0.22 and 0.19
TABLE II
COMPARISON OF PREDIcTED VALUES
PEAK EXPIRATORY Fww RATES
.
Height Sex Present
si
.
Rivera
Nairn
a a!.
Murray
& Cook
110cm
110 cm
140 cm
140 cm
170 cm
170 cm Boys
Girls
Boys
Girls
Boys
Girls
119
99
4
198
375
346
1
1
139
145
307
305
474
465
167
173
303
98
490
204 FORCED FLOW RATES
TABLE III
Indep.
Variable
Boys (n=116)
Correlation Residual Regression St. Error
Coefficient Variance Coefficient Coefficient intercept
Girls (n=184)
Correlation Residual Coefficient lariance
Regression .‘t. Error
Coefficient Coefficient lnfrrcrpt
2.717 .139 3.66313
2.656 .113 -2.76966
0.892 .048 1.29944
0.888 .053 0.80876
1.384 .076 2.08935
.877 .801 863 .855 .871 .841 887 .883 .873 .883 .895 .878 .879 .894 .885 Height Silt, lit. Age Weight Surf. nrea Biacrom.
Vitalcap. (I)
Age & height Age & silt. ht. Age & surf, area Age & vital cap. height & weight height & biacrom. Height & vital cap (1)
Age & height & weight .01065 .01650 .01180 .01137 .01114 .01351 .00986 .01025 .01109 01012 00925 01002 01060 .00935 .01020 2.914 2.557 1.229 1.009 1.515 2.665 0.998 0.450 1.921 0.927 0.798 0.560 0.880 0.440 0.078 2.341 0.211 2.411 0.502 1.192 0.615 0.459 1.284 0.118 .150 .179 .068 .057 .080 161 049 .193 .449 .113 .276 .167 .204 .151 .119 .527 187 .416 .406 .441 .150 .192 .680 .183 -4.03462 -2. 54276 1.00750 0.06008 2.11110 -1.68481 1.85724 -2.35018 -0.18420 1.60101 .53644 -3.13565 -3.71324 -0.56354 -1. 34473 .864 883 .852 .827 .847 .882 .875 .888 .869 .894 .865 .889 .876 .00831 .00722 .00894 .01033 .00927 .00726 .00773 .00698 .00800 .00661 .00831 .00688 .00775 0.921 0.386 1.649 0.259 1.968 0.500 0.673 0.337 0.612 1.348 0. 137 0.996 0.614 0.380 1.347 0.114 .043 .118 .355 .111 .319 109 171 090 .090 .408 139 .349 .115 .118 .501 .134 1.89901 -1.73470 -1.74370 1.64091 1.65561 -3.06238 -0.14293 -1.16043
for boys and girls respectively.
The duplicate values of the E50 (and the
L5) were averaged after logarithmic trans-formation. Tile estimated duplicability vari-ances of the E50 were 0.00112 for boys and
0.00270 for girls. The standard deviations
of the duplicates were 0.077 and 0.120 expressed as fractions of their mean values. The correlations between duplicates were .95 for boys and .90 for girls. The results of the regression analyses of the E50 data
are given in Table. IV The data in relation to height are shown in Figure 3. Although
no pair of slopes from boys and girls of the regressions on any single variable reveals a statistically significant difference between the sexes, the fact that each of the slopes
on an individual variable is larger for boys than girls suggests a real difference as was found for the FRC. However, we are hesi-tant to judge on the basis of these data
alone. In contrast there was a significant
difference between boys and girls when age was used as the independent variable. This may be attributed to the sex difference in body growth with increasing age. For both boys and girls the best correlations of the E50 with a single independent variable was with age, especially so for girls. The
stand-ard deviations around these regression lines
are 0.18 and 0.22 for boys and girls. Table V has been prepared to compare the E50 values predicated by our equations for different-sized boys and girls with values
BOYS
400
300
200
ISO
GIRLS
25 PK INSPIRATORY
SUPPLEMENT 205
FLOW L/MIN. 600
100
50
00 110 120 30 I’0 50 60 ITO 80
CM. CM.
Fic. 2. The relationship of peak inspiratory flow to height (logarithmic scales).
TABLE IV
ANALYSES OF THE E-50
indep.
Variable
Boys’
-Girist
Correlation Coefficient
Residual Variance
Regression
Coefficient
St. Error Coefficient Intercept.
Correlation
Coefficient
Residual
Variance
Regression
Coefficient
St. Error Coefficient mnt
height .8283 .00703 1.281 .173 -2.80281 .6741 .01217 1.870 .310 -1.91131
Age .8450 .00040 1.007 .119 1.03010 .7700 .00916 0.719 .091 1.30385
Weight .7924 .00833 0.785 .107 0.88419 .6150 .01397 0.561 .111 1.11415
Surf.area .8157 .00749 1.191 .149 1.01165 .0570 .01178 0.935 .106 1.01906
Vitalcap. .8332 .00085 0.925 .109 -1.02416 .6689 .01143 0.640 .110 -0.01884
Age& .8573 .00612 0.069 .280 -0.02313 .7735 .00910 0.870 .219 1.30050
height 0.950 .610 -0.509 .679
hleight& .8287 .00714 1.083 .864 -2.49349 .0754 .01253 2.165 .897 -2.39313
weight 0.075 .311 -0.104 .294
Age& .8580 .00010 0.714 .248 1,34911 .7728 .00928 0.841 .205 1.25591
surf.area 0.462 .187 -0.209 .312
Age& .8669 .00574 0.025 .234 -0.02780 .7714 .00933 0.808 .209 1.61337
vitalcap. 0.444 .205 -0.102 .214
Age& .8581 .00630 0.075 .284 -0.13538 .7730 .00949 0.878 .233 2.38305
height& 0.643 1.007 -0.561 1.004
weight 0.114 .290 0.010 .258
S 34 SUI)jacts.
t44 subjects.
, / , / / 0/ 0 , / / / / / / / FLOW L/MIN. / 300
0 GIRLS ,‘ .
250 #{149}BOYS 0,’ #{149}
.
, O 0.
2.00 / ‘
/ 0
// Oo o,,p
/ S
, . 0’OO
4 a. #{149} #{149}
/ 0 /
/ ,#{149} SO
/ . A
, S ‘ S /
, , ,
100 . :‘‘ 50o5
>
, 000 4’
/ / ‘3
/ S /
/ /
50
100 110 120 10. 140 50 160 ITO ISO 190
HEIGHT (CM)
Fic. 3. The relationship of the forced expiratory
flow after half of the vital capacity has been
ex-pired (E) to height (logarithmic scales).
for the maximal mid-expiratory flow (MMF) reported by Cherniak.7 It can be seen that
our values are about 13% lower than those from his equations. This proportionality holds approximately over the entire range.
In order to examine the relationship of
the E30 to airway conductance (Cd) a graph (log-log) of the relationship of Cd and E50
to height was constructed (Fig. 4). The combined data for boys and girls were used. The logarithmic spread of the two ordinate scales are identical so that the slopes of the lines may be compared directly.
C. L/S(C/CU NO 4.5 40 .35 .30 FLOW L /5CC
TABLE V .25
COMPARISON OF PREDICTED VALUES
(E50 AND MAXIMAL MID-EXPIRATORY FLOW)
/ i#{176}
P5.5% FLOW /
/ F40 / / 35 / / -30 25 / / / / / / / / / / .
Height Age Sex
E50 Present . ertes MMF . Chernzaek
110 cm 5 Boys 72 87
110 cm 5 Girls 79 85
130 cm 8 Boys 105 125
130 cm 8 Girls 108 121
150 cm 12 Boys 146 165
150 cm 12 Girls 141 160
/
/
V.
.20
ill
.IOL , ,,,,,
P00 PlO 20 30 I40 50 60 70 80
HCIGHT PP CId
Ftc. 4. A comparison of the regression lines of
airway conductance, C5,8 peak expiratory flow rate,
and the E50 on height (logarithmic scales).
206 FORCED FLOW RATES
,‘ The slope of E0 line is a little over 2. The
‘ data for the airway conductance was ob-tained from the paper of Helliesen et al. They chose to analyze their data by a semi-logarithmic type of equation which accounts
for the slight curve in the Cd line.
How-ever, it is apparent from the graph that the slopes of the curve in this range are very similar to that of the E0. This indicates
that both grow in a similar way in relation to the growth of height (at least in this
size range). The E0 in I/sec is equal to about 11 times the airway conductance in
1/sec/cm H2O. For contrast the pooled
slope (boys and girls) of the log of peak
expiratory flow rate in relation to the log height has been included. Its slope is about __________________________________ 2.8. It can be seen that this slope deviates
appreciably from those of the E50 and C,. Although we have felt that the
logarith-mic transformation of both the independent
and dependent variables used throughout this set of papers was the best approach, we
analyzed the E30 data in the semi-logarith-mic manner (logarithm of dependent vari-able only) used by Helliesen et al. for
air-way resistance in order to compare further
the growth of the E0 and Cd. The
relation-ship of the E50 to height then becomes
Boys (n=4?)
(‘orrelation Coefficient
95% (‘onfidence Limit?
Girls (n=49)
Age
height
Weight
Body surface area
Biacromial width
Vital capacity
Correlation
Coefficient
.035
.071
.028
.078
039 122
95% Confidence
Linux
- .28 to .34
- .24 to .37
- .28 to .33
- .24 to .38 - .27 to .34 - .20 to .42
.228
.411
.354
.373
.335
.451
- .06 to .48
.14 to .62
.07 to .58
.10 to .60
.05 to .57
.19 to .65
Flow = a X ht’ (Eq. 1)
TABLE VI
CoRRElATIoN OF “K” TO BODY SIZE AND AGF
The number n then shows the dependency of the logarithm of the E10 on height. The value obtained for this constant was
0.0061 ± 0.0006. This is not significantly different than the value 0.0068 obtained by Helliesen Ct a!. for airway resistance (and hence, airway conductance).
The peak expiratory flow rate, the E50 and the E75 were obtained from the same
tracing. Because of this the deflection for the E75 was quite small. As a result there was much less precision in its measurement.
This is probably the cause of the larger
estimated duplicability variances obtained
(0.00570 for boys and 0.00918 for girls). The standard deviations of duplication were 0.17 (boys) and 0.22 (girls). The correlations between duplicates were 0.83 and 0.88 for boys and girls respectively. This greater imprecision may also partly account for the lower correlation coefficients found be-tween the E75 and the independent
vari-ables. However, the general pattern was similar to that of the E50.
From theoretical reasons to be discussed,
the E75 should be about one-half of the E0. The average of the E75/E50 ratios over
all the children was actually 0.53 showing a good approximation to the theoretical (especially so in view of the previously
mentioned imprecision in the E75
measure-ment). The ratios of the E50 to one-half the
vital capacity and of the E75 to one-fourth
the vital capacity should be the same. In
most subjects we had four estimates of this
ratio (2 from E50 duplicates and 2 from the
E75 duplicates). In others we had fewer data. In each of 42 boys and of 49 girls we had at least two estimates of the ratio. For each of these subjects the mean ratio was obtained. The mean of the ratios for boys
was 104.6 min1 with a standard deviation
of 23.0 min, for the girls the mean was 128.3 mm1 with a standard deviation of 35.1 min. The standard error of the
differ-ence is 6.13. The difference between the mean ratios for boys and girls is significant
(p<O.OOl). The relationships of this ratio to age, height, weight, surface area, biacro-mial width, and vital capacity were
ana-lyzed. The correlations are given in Table VI. It can be seen that the ratios in boys
show no significant correlations with their ages, body sizes, or vital capacities. This is not true for girls where 95% confidence
limits of the correlation coefficients do not include 0 except for that with age. The ratio falls as the girls get larger. In small girls values are higher than for boys and fall to that for boys as the girls get larger.
The best correlation is with vital capacity. Even with a regression analysis of the ratio on this last variable the standard deviation
about the line is only about 10% less than the standard deviation of the group.
COMMENT
MAX.
INSPIRATION
VOLUME
208 FORCED FLOW RATES
F LOW
MAX.
EXPIRATION
Fic. 5. A diagram of the relationships of forced
flow to expired volume. The solid line represents
that found in normal subjects; the dashed line
that of patients with airway obstruction (induced
asthma).
the sizes of the exponents (b) are simi-lar to those of the vital capacity and of the diffusing capacity. The exponents of the peak flows for boys and girls are not
sig-nificantly different, but the constant multi-pliers (a’s) are, indicating that the girls’ growth of peak flow rates in relation to size is of the same form as that for boys, but at the same body size they have lower values. In contrast to the peak flow rate the E50 grows differently than the lung volumes and
diffusing capacity. The exponents are lower: the E50 growing as about the square of height rather than cube as do the peak
flows, lung volumes, and diffusing capacity. The growth of the E50 is similar in manner to that of the airway conductance.
The work of Hyatt and others has been very informative about the meaning of the rate of flow during forced expiration. They have noted that each normal individual has
a maximal rate of flow at a given lung volume which occurs at an optimal
transpul-monary pressure. Points representing these flows when plotted against volume form a
line which after the initial acceleration and before the last bit of the exhalation is nearly
a straight line. This is illustrated in Figure 5. They have shown that the optimal pres-sure early in expiration is large (much effort); whereas later it is relatively small
(less effort). At greater than optimal pres-sure the flow is lowered a small amount but the reduction in flow at any point by too
much effort is small compared to that by too little effort. Thus, flow rates measured late in the forced expiratory maneuver are less likely to deviate greatly from the opti-mal because of an improper amount of effort.
After the first part of the forced expira-tion in normal subjects the flow is nearly linearly related to the remaining vital capacity. That is, flow, the derivative in
respect to time of volume, is nearly linearly related to volume. If linearity over that range is assumed, then the relationship can
be described by a negative exponential equation such as the following:
and
Vol.(t) = ft = VoI.to X et (Eq. 2)
Flow(t) = f’(t) = - K(vol.1, X e) (Eq. 3)
where
Volume = portion of vital capacity remain-ing in the body
to = start of expiration (assuming
in-stantaneous acceleration)
t=time later
K=an emptying factor in units of the reciprocal of time (1/sec or 1/mm).
The assumption of instantaneous
accelera-tion was made. This, of course, is not true and could be allowed for by a somewhat more complex equation which would still be of the
same basic nature.
The foregoing mathematical model is
em-piric, only an approximation, and different from the model proposed by Hyatt, Schilder, and Fry.’#{176}In their empirical approach they assumed a linear relation in the last part of expiration between volume and the log of flow. This approximation appeared closer to the data in the abnormal than in the normal. In the model we propose, an extension of our basic equation is necessary to fit the abnor-mal. It is important to emphasize that our
non-uniform reduction in the airway con-ductance. After the peak the lung functions as if composed of multiple units in parallel,
(Eq. 4) each with its own K, or airway emptying fac-tor. This can be expressed as:
(Eq. 5)
(Eq. 6) SUPPLEMENT
By dividing Equation 3 by Equation , one obtains the following:
Flow at a given time
-K =
Volume at a given time
The negative sign means flow is outward. The K, which we will call the emptying fac-tor, under these assumptions is constant. The K can be found by looking at the flow at any point after the “linear” relationship is
ob-tamed. If taken where 50% or 75% of the
vital capacity has been expired, the following
relations are found:
. E50 (1/mm)
Jk=
VC/2 (I)
K = E75 (1/mm)
VC/4 (1)
Thus, in the healthy subject the ratio E50 to
one-half vital capacity (or Em to one-fourth
vital capacity) is an estimate of the K, or
emptying factor, for that individual. This
factor isrelated to, but not the same as, the
airway conductance. The K, or emptying
factor, reflects not only the airway
conduct-ance, but also reflects the way the airway
conductance changes with change in volume
during forced expiration. In view of the
rela-tions of the E50 and the vital capacity to
height (E50C1ht2, VCC2ht3) one might
have anticipated that the K values would
fall with increasing body size. In the
anal-yses of K, a negative regression was found
for boys but was not significant. A
signifi-cant negative regression was found for the girls. The use of the regression equations does
not lower the standard deviations greatly.
For practical purposes in this age group the
K’s can be considered to vary around a
con-stant.
The dotted line in Figure 5 represents the
type of flow-volume curve obtained in sub-jects with airway obstructive disease. We
have repeatedly seen this type of curve de-velop from a nearly straight line during
ex-perimentally induced asthma.1’ The volume-flow mathematical relationship above can be extended to cover this situation. With
air-way obstructive disease we assume there is a
= f(t) = Vi X e’t + V2 X 6K2t
+ V3 X e_K31, etc. (Eq. 7)
F8 = f’(l) = - K,(V, x e_K)
- K2(V2 x e2I)
- K3(V3 x eK8t), etc. (Eq. 8)
A graph of flow and volume based on these
formulas will give a curved line such as iii
Figure 5. The greater the portion of vital
capacity expired the more the points on the curve will be determined by the parts of the lung with the lower emptying factors. Thus,
flow rates measured at points in the latter
portions of the forced expirogram reflect the more involved portions of the lung. The
slight curvature sometimes found in normal
subjects can be explained in the same way.
Even in these subjects the greatest part of the line is practically straight, however, and
suggests the dominance by one compartment
with its emptying factor. The maximal
mid-expiratory flow (MMF)’2 is the average flow over the middle half of the forced expiration.
It can be obtained from a forced spirogram by dividing half the total volume by the time
between the points where p25% and 75% of the volume are expired. If the flow were linearly related to the volume (solid line, Fig. 5) the MMF and E50 would have
identi-cal values. When this relationship becomes curvilinear the E50 will be somewhat, but not
a.great deal, lower than the MMF.
There are several reasons to use tile E50, the E75, or the maximal mid-expiratory flow (MMF) for the evaluation of airway
resist-ance in clinical situations. Single-breath tests are much less complicated and foreboding to the patient than the direct measurements of
airway resistance. The maximal breathing
capacity (MBC) is more fatiguing than single-breath tests. In the ill patient the
MBC may easily be reduced because of
210 FORCED FLOW RATES
changes in airway resistance. As has been pointed out, tests, such as the E50, obtained at points after the initial acceleration are less
likely than the MBC or peak flow rates to be
lowered greatly because of too little effort.
In so far as the postulated mathematical
hypothesis holds, the flow rates determined
at the middle or later in the expiratory trac-ing reflect the more involved parts of the lung. Thus they should be more sensitive to disease than the peak flow rates. Lastly, the growth of the E50 (as opposed to the peak flow rates) in relation to height is similar to that of airway conductance.
CONCLUSIONS
1. Data from normal subjects on the peak expiratory flow rate, the peak inspiratory flow rate, and the flow rate at the point where half the vital capacity has been expired (E50), have been obtained as part of a
longi-tudinal study of the growth of lung function in school children.
2. The peak flow rates grow in a manner
similar to that of the lung volumes.
3. The E50, in contradistinction, grows in a manner similar to airway conductance (the reciprocal of airway resistance).
4. Longitudinal studies of the peak flow
rates indicate that individuals tend to main-tain their positions in relation to the group.
5. A theoretical postulation of the forced flow-volume relationship has been suggested. According to this postulation the lungs of the normal subject empty as a unit or nearly so. The flow (after the initial acceleration) at any point in time is dependent on his vital
capacity and an emptying factor. This emp-tying factor decreases somewhat with in-creasing body size. This is more evident in the girls.
6. In the patient with airway obstructive disease (as asthma) the lung can be thought of as composed of multiple units in parallel with varying amounts of reduction in their emptying factor.
7. The theoretical advantages of clinical tests based upon flow in the mid or later
por-tions of the forced expiration have been
given.
REFERENCES
1. DeMuth, G. R., Howatt, W. F., and Hill, B.:
The growth of lung function. Part I. Lung
volumes. PEDIAmIcs, 35: 162, 1965.
2. Wells, H. S., Steod, W. W., Rossing, T. D.,
and Oganovich, J.:Accuracy of an improved
spirometer for recording of fast breathing.
J. Appi. Physiol., 14:451, 1959.
3. Franklin, W., and Lowell, F. C. : The
expira-tory rate during the third quarter of a
maxi-ma! forced expiration (E50-7). Amer. J. Al-lergy, 32: 162, 1961.
4. Nairn, J. R., Bennett, A. J., Andrew, J. D., and
Mac Arthur, P. : A study of respiratory
func-tion in normal school children. The peak
expiration flow rate. Arch. Dis. Child.,
36:253, 1961.
5. Murray, A. B. , and Cook, C. D. : Measurement
of peak expiratory flow rates in 220 normal
children from 4.5 to 18.5 years of age. J.
Pediat., 62:186, 196.3.
6. Rivera, L. M., and Snider, C. L.: Ventilatory
studies in pre-school children. I. Peak
cx-piratory flow rate in normal and abnormal
pre-school children. PEDIATRICs, 30:117,
1962.
7. Cherniak, R. M.: Ventilatory function in
nor-mal children. Canad. Med. Ass. Jl., 87:80,
1962.
8. Helliesen, P. J., Cook, C. D., Friedlander, L.,
and Agathon, S.: Studies of respiratory
physiology in children. I. Mechanics of
respiration and lung volumes in 85 normal
children 5 to 17 years of age. PEDIATRICS, 22:80, 1958.
9. Fry, D. L., and Hyatt, R. E.: Pulmonary
me-chanics, a unified analysis of the relationship
between pressure, volume and gasfiow in the
lungs of normal and diseased subjects. Amer.
J. Med., 29:672, 1960.
10. Hyatt, R. E., Schilder, D. P., and Fry, D. L.:
Relationship bebveen maximum expiratorv
flow and degree of lung inflation. J. Appl.
Physiol., 13:331, 1958.
11. DeMuth, G. R., and Howatt, \V. F.:
Unpub-lished Work. The lung volume-flow rate
pat-tern in forced expiration and its significance
in applied tests in patients with asthma.
Ab-stract, J. Pediat., 61:282, 1962.
12. Leuahlen, E. C., and Fovler, W. S.: Maximal
midexpiratory flow. Amer. Rev. Tuberc.,
