JouRN.4L
OF
APPLIED
Vol. 36, No. 5, May
PHYSIOLOGY
1974.
Printed
in
U.S..jl.
Measurement
of continuous
ventilation-perfusion
ratios:
distributions
of
theory
PETER
D. WAGNER,
HERBERT
A. SALTZMAN,
AND JOHn’
B. WEST
Department of Medicine, University of California, San Diego, La Jolla, California
YZ&V
blood
flow;
gas exchange;
hypoxemia;
inert
gas
IT IS GENERALLY
ACCEPTED
that the major cause of hypoxemia in most types of lung disease is the existence of an
uneven distribution
of ventilation-perfusion
(VA/&)
ratios.
However,
in spite of a large variety of experimental
approaches,
the shapes of the distributions
of VA/Q
ratios
remain virtually
unknown,
both in health and disease.
Most investigators
have characterized
the lung as if it
consisted of two or three compartments.
Thus, Riley
and
his colleagues
(27, 28) used a combination
of Po2 and
PCO~ in arterial blood and mixed expired gas to divide the
lung into three functional
compartments:
one ideal, one
unventilated,
and the third unperfused.
Briscoe and his
coworkers
(3-5) divided the lung into two ventilated
compartments
on the basis of gas washout rate, and then calculated
the blood flow to each compartment.
A related
method was described
by Finley
(lo).
Lenfant
(17)
measured the alveolar-arterial
difference
for 02, COZ, and N2
with increasing
FIEF and proposed
the existence of VA&,
distributions
with two modes, the majority
of alveoli having
a VA/Q
ratio slightly above the mean, and the rest having
a very low VA/Q ratio.
Lenfant
and Okubo
(19, 25) have derived
continuous
distributions
of \;'A/Q
ratios, using the change in arterial
with
increasing
FIEF during
a nitrogen
02 saturation
washout.
Their method has been criticized
on theoretical
grounds
(26), and in addition
the distributions
were
assumed not to change with Fro2, which as Lenfant
(18)
showed may be unjustified
and lead to errors. It is also
possible that when room air is breathed,
some of the hypoxemia
results from diffusion
impairment
rather
than
VA/Q inequality.
If inert gas techniques
are used in place of the oxygen
methods described,
both of the above-mentioned
objections
are circumvented.
Kety (13), Noehren
(24),
and Farhi
(9) have given theoretical
equations
relating
inert gas
exchange
in the lungs to the ventilation-perfusion
ratio
and the solubility
of the gas. Measurements
with several
foreign inert gases such as krypton
and xenon (29) and
methane,
ethane, and nitrous oxide (34) have been used
to gain information
about distributions
of VA/~
ratios.
However,
these analyses, as with the oxygen methods,
have been limited to a small number
of compartments.
Measurements
with radioactive
tracers (2, 6, 14, 22,
32) have yielded useful topographical
information
about
ventilation
and blood flow in normal lungs. However,
even
in normal,
but especially
in diseased lungs, these techniques lack resolution
because of the large tissue volumes
that must be averaged. It is unlikely that the differences of
ventilation
and blood flow detected by external counters in
patients with lung disease throw much light on the VA/~
distributions
responsible
for their impaired
gas exchange.
This paper describes a method for determining
virtually
continuous
distributions
of VA/Q
ratios. The resolution
of
the technique
is sufficient to describe smooth distributions
containing
blood flow to unventilated
regions
(shunt),
ventilation
to unperfused
regions (dead space), and up to
three additional
modes over the range of finite VA/Q
ratios.
In particular,
areas whose VA/Q
ratios are low can be
separated
from unventilated
regions
and those whose
VA/Q
ratios are high can similarly
be distinguished
from
unperfused
areas. The technique
has been developed using
inert gases as the forcing function,
both because of the
simple relationship
governing
inert gas exchange,
and
because of the objections
to the oxygen method outlined
above.
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
PETER
D., HERBERT
A. SALTZMAN,
AND
JOHN
B.
Mgasuremcnt
of continuous
distributions
of ventilation-perfusion
1974.-Most
ratios.
theory.
J. Appl.
Physiol.
36(5):
588-599.
previous
descriptions
of the distribution
of ventilation-perfusion
ratios
(VA/Q)
divide
the lungs into only two or three uniform
However,
an analysis
which
would
result
in
compartments.
definition
of the position,
shape, and dispersion
of the distribution
would
be more
realistic.
We describe
here such a technique,
applicable
both in health
and disease, in which the characteristics
of distributions
containing
up to three modes can be determined.
In particular,
areas with low but finite VA&
ratios are separated
from
areas whose VA/Q
ratio
is zero (shunt),
and regions
with
high VA/Q
ratios
are differentiated
from
regions
that are unperfused
(dead
space).
To perform
the measurement,
dextrose
solution
or saline is equilibrated
with a mixture
of several
gases
of different
solubilities
and then infused
into a vein.
After
a
steady
state has been established,
the concentrations
of each gas
are measured
in the mixed
arterial
blood and mixed
expired
gas.
The curve
relating
arterial
concentration
and solubility
is transformed
into a virtually
continuous
distribution
of blood
flow
against
VA/Q,
using techniques
of numerical
analysis.
The relation between
expired
concentration
and solubility
is similarly
converted
into the distribution
of ventilation.
The
numerical
analysis
technique
has been tested
against
many
artificial
distributions
of VA/Q
ratios
and these have
all been accurately
recovered.
WAGNER,
WEST.
MEASUREMENT
OF
VENTILATION-PERFUSION
589
INEQUALITY
METHODS
I
PC
Fi
Or
PA
-TV
BLOOD
: GAS
PARTITION
COEFFICIENT
1. Relationship
between
inert
gas retention
Pc/PT
(or excretion
PA/Pv)
and blood-gas
partition
coefficient,
using
a logarithmic
scale for the abscissa.
Four
curves
are drawn,
each for homogeneous
lung
units
with
different
VA/i2
ratios.
Note
that
the curves
are all
smooth
and
monotonic.
Blood-gas
partition
coefficients
for human
blood
at 37°C
are also shown.
FIG.
in each VA/Q
unit the quantitative
exchange of any inert
gas depends only on the VA/Q
ratio of the unit and the
blood:gas
partition
coefficient
of the gas (X), as shown in
Eq. 1.
(0
where PA is the alveolar
partial
pressure,
PC the endcapillary
partial pressure, assumed equal to PA, and PV is
the mixed venous partial
pressure. It is more convenient
to divide by PV. If E = PA/PC and R = Pc/PV, Eq. I
may be rewritten
as Eq. 2:
(2)
Examples of the relationship
between R and X (the retention-solubility
curve) and between E and X (the excretionsolubility
curve) for lung units with low, normal,
and high
VA/Q
ratios are shown in Fig. 1.
One may now consider a lung containing
a distribution
of VA/Q
ratios as defined above. If the lung contains N
different
VA/Q
units with blood flow Qj and ventilation
Vj in the jth unit, then for any gas with blood : gas partition
coefficient X i, overall retention Ri and overall excretion Ei
are given by Eq. 3A and 3B, respectively:
1
(34
cj=l K&l
Vj
X' i
+
C
j=l
vj/Qj
1
[vj]
These equations state that the mixed arterial concentration
is a blood flow-weighted
mean of compartmental
values
while the mixed expired
level is similarly
a ventilationweighted
mean of compartmental
values.
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
Experimental
outline. The experimental
procedure
which
provides the data for the analysis is suitable both for human
subjects and experimental
animals in health and disease. A
mixture of several inert gases (six, for example)
whose solubilities in blood range from very low (e.g., sulfur hexafluoride)
to very high (e.g., acetone) is equilibrated
with a
suitable solution such as normal
saline or 5 % dextrose in
distilled
water. The relative
proportions
of the gases are
selected so as to result in approximately
equal arterial
concentrations
for each gas. The solution
is then infused
into a peripheral
vein, and, after a steady state within
the
lungs has been established,
mixed arterial blood and mixed
expired gas are simultaneously
sampled.
For each gas, its concentration
in both samples and its
blood: gas partition
coefficient are measured by gas chromatography (30). It is also necessary to know total pulmonary
blood flow and minute
ventilation.
Then,
by use of the
Fick principle,
the mixed venous concentration
of each
gas is calculated
from the arterial
and expired
values,
thereby avoiding
the necessity of direct sampling
via a
pulmonary
artery catheter
(see step 3 in APPENDIX
II).
The ratios of mixed arterial to mixed venous concentration (defined as retention,
R) and mixed expired to mixed
venous concentration
(defined
as excretion,
E) for each
gas are plotted against the solubility
of the gas in question.
The retention-solubility
plot is then processed by digital
computer
and yields the distribution
of blood flow with
respect
to VA/~,
and, independently,
the excretionsolubility
plot is similarly
converted
into the distribution
of ventilation
with respect to VA/Q.
It is then possible to
predict the arterial
and alveolar
Po2 and Pco2, and these
may be compared
with the measured values.
L4ssumptions and dejnitions. The lung is assumed to consist of a number of compartments
arranged
in parallel with
respect to both ventilation
and blood flow, but any number of compartments
is allowed.
Each compartment
is
taken to be homogeneous,
with single values for ventilation,
blood flow, and alveolar,
venous, and end-capillary
concentrations
of each gas. It is also assumed that diffusion
equilibration
between alveolar gas and end-capillary
blood
is complete,
and that the inspired
concentration
of each
of the inert gases is zero.
The distribution
of blood flow may then be defined from
this description
of the lung as the plot of blood flow against
the ratio of ventilation
to blood flow, compartment
by
compartment.
Similarly,
the ventilation
distribution
is
defined as the plot of ventilation
against the ratio of ventilation to blood flow.
A further
assumption
in this analysis is that the distribution
of ventilation
and blood flow in any particular
case have smooth contours and contain no sudden irregularities. The numerical
method is not capable of recovering
distributions
containing
such sudden irregularities
but on
intuitive
grounds we believe it is reasonable
to assume that
these do not occur either in the normal or diseased lung.
Equations. If in all VA/Q
units the amount
of any gas
exchanged
between pulmonary
capillary
blood and alveoli
equals the amount exchanged
between the alveoli and the
atmosphere,
the lung may be said to be in a steady state
of gas exchange.
Under these conditions,
it is known that
590
WAGNER,
Since total blood
flow,
Qt, is given
AND
WEST
by
j=N
.8 -
C
IQJ
j=l
and total ventilation,
SALTZMAN,
(30
Homogeneous
with
20%
lung
shunt
Vt, is given by
Homogeneous
lung,
i=N
C
[lvjl
j=l
W)
Eqs 3A and 3B may be rewritten:
Homogeneous
with
A
.Ol
.I
BLOOD : GAS PARTITION
0
%
:
w
0
0
BLOOD: GAS
FIG.
partition
perfusion
tonicity
shown
cretion
tributions
PARTITION
COEFFICIENT
VA/6
2. A: relationships
between
inert
gas retention
and blood-gas
coefficient
in three
examples
of lungs
with
ventilationinequality.
Note
the preservation
of smoothness
and monoin each case.
Corresponding
distributions
of blood
flow
are
with
the same symbols.
B: relationships
between
inert
gas exand blood-gas
partition
coefficient
in the corresponding
disof ventilation.
space
100
1000
10000
COEFFICIENT
FIG.
3. A: relationship
between
retention
and blood-gas
partition
coefficient
in a homogeneous
lung
with
and without
a 2Oa/o right
to
left shunt.
Note
that the greatest
difference
between
the curves
occurs
for insoluble
gases, for which
the retention
asymptotically
approaches
the shunt
fraction.
B: relationship
between
excretion
and solubility
in a homogeneous
lung
with
and without
20%
dead
space.
Here
the
difference
between
the curves
increases
as solubility
increases,
approaching
the dead
space
fraction.
It can be seen that
the presence
of both shunt
and dead space
does not alter
the smooth
nature
of the
curves.
veoli. If the anatomic
dead space is measured,
the mixed
expired gas concentrations
can be corrected
for the dilution produced,
before the analysis is performed,
so that
any dead space present now will refer only to alveoli which
are unperfused.
Using theoretical
models, the numerical
analytical
procedure
for recovering
the VA/Q
distribution
has been found to operate satisfactorily
under both circumstances. The correction
factor for the effect of anatomic
dead space is:
CA
FJ
F
I 0
dead
= CI&E/(\;TE
- f*VD)
where CA and CE are the corrected
and original
mixed expired concentrations,
respectively,
VE is the minute
ventilation,
f the respiratory
frequency,
and VD the anatomic
dead space.
Numerical analysis. Eq. 4,4 and 4B are solved by numerical
analysis in the following
way. The methods of solution of
Eq. 4A (blood flow) and 4B (ventilation)
are identical,
and for clarity only the solution for blood flow is described
here.
First a set of VA/Q
values is chosen to span the range of
interest. In practice,
50 values are used, 2 of these being
zero and infinity.
Then the range of ventilation-perfusion
ratios between 0.005 and 100.0 is divided equally on the
customary
logarithmic
scale into 48 additional
compartments giving 50 in all. Next a starting guess is made for
the distribution
of blood flow with respect to ventilationperfusion ratio. In practice, we choose a smooth curve which
covers the entire above-mentioned
VA/Q
range and which
assigns rather more blood flow to the middle of the VA/Q
range than to either end. However,
as shown in the DISCUSSION,
the shape of the starting guess is not critical and a
variety of initial distributions
will give final solutions which
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
Three examples of the overall retention-solubility
relationship and excretion-solubility
relationship
in lungs with
VA/Q
inequality
are given in Fig. 2, A and B, respectively.
Eq. 4A and 4B are valid both for lungs containing
regions that are totally
unventilated
(shunt)
and totally
unperfused
(dead space). Shunt can be regarded
as blood
flow through
a lung unit with a VA/()
ratio of zero, in
which the retention
for all gases, from Eq. 2, is 1.0, while
dead space corresponds
to a unit with a VA/Q
ratio of
infinity,
in which the retention
is zero, also by Eq. 2. In
Fig. 3A, the retention-solubility
curves of a homogeneous
lung with and without
20 % shunt are compared
while in
Fig. 3B the excretion-solubility
curves for a homogeneous
lung with and without
20% dead space are compared.
The definition
of dead space given above includes both
the anatomic
dead space and that fraction
of the alveolar
ventilation
distributed
to totally unperfused
alveoli. If the
anatomic
dead space is not measured,
the analysis to be
described
gives the total dead space ventilation,
which
is
made UD of the anatomic
dead space and unperfused
al-
lung
20%
MEASUREMENT
OF
VENTILATION-PERFUSION
591
INEQUALITY
c
.
l o
C
.
@
.
C
.
c
C
.
Q
.
$
0
.
c
l
Q
.
0
.
0..
2
E
c
.
Q
0
?
l c
ccc
E
0
ul
.025
@
0
.
00
0
0
Original
0
l
Dlstrlbutlon
L
.OOOl
Recovered
.
.OOl
F
Dlstributlon
61
8
I
.Ol
L
0.1
I .o
3 D
‘I
B
.
Dlstrlbutlon
0
Orcgml
Recovered
ca
Dtstrrbuhon
.
0
0’
.
0.
otJ* Q
l
Dlstributlon
r
0
Or~gtnol
Recovered
Distrlbutlon
9
0
.
.
@
C’
c
.
FIG. 4. Four
examples
of recovered
distributions
of blood
flow
(closed
circles)
showing
original
theoretical
distribution
for comparison
(open
circles).
In
each
case,
the curve
of fractional
pulmonary
blood
flow plotted
against
\iA/Q
is smooth
with
peaks
and
troughs
corresponding
to those
of the original
distribution.
Recover-v
is good
for both
narrow
and
broad
functions.
Note
that
in A, B, and C, there
is
no blood
flow
to unventilated
alveoli,
but,
in D,
there
is 30%.
.050
are indistinguishable
from each other. Using this starting
guess, the computer
calculates
the retentions
of each of
the six gases. This is possible because in each compartment
the retention
for each gas of known solubility
is calculated
from Eq. 2, and thus the overall retention
from Eq. 3A. The
computer
then compares these calculated
retentions
with
the measured
retentions
of the six experimental
gases.
This is done by calculating
the sum of squares of the differences for each gas. Naturally
this sum of squares will
be large, so the computer
then alters the blood flow in
each compartment
so as to reduce the differences between
measured
and calculated
retentions.
This is done by a
gradient
method
with the constraint
that no value of
blood flow is permitted
to become negative.
By use of this
new set of blood flows, retentions
are again calculated,
compared
with the measured
values, and once more the
blood flows are altered to reduce even further
the sum of
squares. In other words, the differences between the measured and calculated
retentions
are successively reduced, the
program
being allowed
to run until the sum of squares
cannot be materially
reduced any further. The number of
iterations
required
varies from 400 to 4,000 depending
on
the nature of the VA/Q
distribution
under analysis. Some
details of the entire process are given in APPENDIX
I.
The limits of the range of ventilation-perfusion
ratios
between 0.005 and 100 are dictated
by the solubilities
of
the least and most soluble gases. Thus, because of the finite
solubility
of sulfur hexafluoride
(0.0009
ml/l00
ml per
mmHg), regions with a VA/~
ratio of less than 0.005 cannot be distinguished
from regions whose VA/Q
ratio is
0. Similarly,
since the solubility
of acetone is finite (40
ml/l00
ml per mmHg),
regions whose VA/~
ratios exceed
100 cannot
be separated
from regions
with an infinite
\iAla
ratio.
Although
the analysis is performed
using 50 compartments, there are always fewer than 50 compartments
in
the final solution in which the blood flow is greater than
zero. In any particular
example,
the number
of compartments which do have positive blood flow is unknown
initially and is selected by the numerical
procedure.
By using
50 compartments,
a balance is achieved between the ability
to locate and describe the modes of the distribution,
and
can, however,
be made
computing
costs. The analysis
using any number of compartments.
Theoretically,
once the set of blood flows has been determined from Eq. 4A the set of ventilations
could be calculated directly
from the product
of blood flow and VA/Q
ratio in each compartment.
In practice,
however,
when
the VA/Q
ratio is high, a very small error in blood flow
will produce a large error in calculated
ventilation,
so that
more accurate results are obtained
when Eq. 4L4is solved
for blood flow, and Eq. 4B is solved independently
for
ventilation.
Empirically
this method
is successful as demonstrated
in the RESULTS
section of this paper and particularly
in
Figs. 4-7. It will be shown there that it is possible to recover
a range
of artificial
distributions
accurately
including
unimodal
and bimodal
distributions
with both wide and
In addition
distributions
containing
narrow
dispersions.
dead space and shunt can also be accurately recovered.
The question arises as to how it is possible to recover 50
compartments
with far fewer independent
equations.
The
answer to this is not clear and we do not propose to consider
it in any detail in this paper. However,
it is probably
not
correct to regard each individual
compartment
as being
completely
independent,
though
the formal
relationship
between the compartments
is not clear. It also seems likely
that the retention-solubility
curve has special properties
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
.025
592
WAGNER,
AND
WEST
VA/Q
VA/d
FIG. 6. Effect
of number
of compartments
on recovery
of a bimodal
distribution.
In each
case the original
distribution
is shown
by the
unbroken
line,
while
solid points
are those determined
by the analysisEven
with
10 compartments,
original
curve
is being
followed
closely,
so that with
50, the function
is adequately
described.
.045
a.
r
.0 0.
P
.oCl
0
.
0
.
0
.
0
0
.
.
.
.
0
.
0
0
.
0
0
.
0.
0
.
l
oa
.
Recovered
Dlstrlbutlon
.
A
.,
@I’1
3
7
0
.
.
.’
3
a
0
3
Dtstrlbutlon
Orlginol
0
cloa
.
1
1
.
. .
l
.
Ortgmal
Dlstrlbutaon
Dlslrlbutlon
aLI
Recovered
.
FIG. 7. Effect
of choice
of VA/Q
ratios.
Open
and
show
separate
analyses
of the same
bimodal
distribution
with
interdigitating
sets of VA/Q
ratios
chosen
equally
logarithmic
scale.
Same
curve
is recovered
in each case.
closed
circles
recovered
apart
on a
06
r o Orlolnol
,Ibutton
Dacrlbutlon
Recovered
Orqnol
Dlstrfbutlon
.,
.
.
0 1Y&6
01
10 01
100
VA/6
J
1000 3 I@
00
1..
oOOl*=
.
01
Dlstrlbutlon
Recovered
-.
.
IO
too
1000
C‘ L
\iA/6
FIG. 5. Four
examples
of recovered
distributions
of ventilation
(closed
circles)
showing
the original
theoretical
distributions
for comparison
(open
circles).
In the distributions
shown
in A and B, there
are no totally
unperfused
alveoli,
but in C and D, in the original
distribution,
30%
of ventilation
is to unperfused
alveoli
(dead
space).
In all cases,
adequate
recovery
of the distribution
has been achieved.
test distributions
were chosen, including
narrow and broad
types with single and double
modes, and distributions
including
shunt and dead space.
For both unimodal
and bimodal
distributions,
the effect
of carrying
out the analysis using different
numbers
of
compartments
was examined,
and for one bimodal
distribution,
the effect of using a different set of VA/Q
ratios
(still equally spaced logarithmically)
was investigated.
Finally,
effects of ordinary
experimental
errors in the
data were assessed using known
theoretical
distributions
and actual errors determined
from the measurement
of
inert gas concentrations
in experiments
with dogs.
RESULTS
Recovery of theoretical
recovery of distributions
distributions. Four examples
of the
of blood flow are given in Fig. 4,
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
which help to make possible the type of solution described
here. APPENDIX
II shows that all retention-solubilitv
curves
or excretion-solubility
curves have no maximums,
minimums, or points of inflection
throughout
the finite range of
solubility.
This means that the curve is tightly constrained
between
the actual data points. This behavior
might be
contrasted
with that of a high-order
polynomial
where an
infinite
number
of very different
expressions
could satisfy
the actual six data points while exhibiting
very different
behavior in the regions between these points. Thus, it may
be that while
the 50 compartment
solution
found by
numerical
analysis is not mathematically
unique,
all the
possible solutions lie very close to each other. At anv event
we shall show that it is possible to recover model distributions with a high degree of accuracy.
Calculations. The program
is written
in Fortran
and has
been executed on both Control
Data 3600 and Burroughs
6700 digital
computers.
For a 50-compartment
analysis
with six gases, execution
time on the CDC 3600 is about
50 s. This includes the solution
for both blood flow and
ventilation,
and the prediction
of arterial
Po2 and Pco2
from the recovered distributions.
Before the program
was used to recover distributions
from experimental
data, its ability
to recover known distributions
was tested extensively
using many theoretical
distributions
of ventilation
and blood flow. To accomplish
this, mixed arterial
and alveolar concentrations
were computed for each gas according
to the given distribution,
and
these values were used as data for the program.
For the
solution to be satisfactory,
the recovered distributions
and
original
distributions
had to correspond
closely. Several
SALTZRIAN,
MEASUREMENT
OF
VENTILATION-PERFUSION
593
INEQUALITY
arterial
samples are handled similarly,
many errors will be
avoided
that would
arise if absolute concentrations
were
required.
However,
there will still be ordinary experimental
error in determining
the retentions
and excretions.
Single
sourious points are ea sily recognized
beta use of the requirement that the curve be smoothly monotonic,
and will have
of measurement
of concentration
was determined
for eight
gases by dividing
a blood sample previously
equilibrated
with these gases into six aliquots. Their concentrations
were
measured in e ach sample and from each set of measurements
the retention
and excret ion that would exist according
to
two known theoretical
distributions
were computed.
Retention-solubility
and excretion
solubility
curves were then
constructed
using a) mean retentions
and excretions,
b)
alternating
mean & two standard
deviations
of a single
observation,
and c) alternating
mean =f two standard deviations. For all six situations,
the distributions
of blood flow
and ventilation
were subsequently
recovered and compared
with the original
distributions.
The results are shown in
Fig. 8.
It can be seen that the degree of error present under
actual conditions
of measurement
produces little change in
the shape of the curves (Fig. 8, ‘4 and C) and little change
in the distributions
recovered
(Fig. 8, B and D). In particular,
measurement
error is small at both ends of the
spectrum of solubility
and therefore estimates of both shunt
and dead space will be still less affected.
We conclude
that the shape of the retention-solubility
and excretion-solubility
curves is the major factor in determining the characteristics
of the distributions
of blood flow
and ventilation,
respectively
(see also Fig. 3), while measurement
errors on the other hand will give rise to little
inaccuracy,
both when the distribution
of VA/Q
ratios is
normal and abnormal.
DISCUSSION
Selection of gases. To afford maximum
resolution
between
regions with low VA/Q
ratios and shunt on the one hand
and between areas with high \;TA/Q
ratios and dead space
on the other, a poorly soluble and highly soluble gas, respectively,
are required.
With sulfur hexafluoride,
VA/Q
ratios as low as 0.005 can be distinguished
from shunt,
while with acetone, VA/Q
ratios as high as 100 can be separated from dead space. These limits were defined using
theoretical
distributions
containing
units with both very
low and very high \iA/(jratios. Gases of intermediate
solubility
are required
to define the retentions
and excretions between
the extremes. Their
solubilities
should be
chosen so as to evenly divide the solubility
range, thus permitting all sections of the retention-solubility
and excretionsolubility
curves to be defined. It is difficult
to state the
minimum
number
of gases required
for a satisfactory
analysis,
but in theoretical
studies six well-placed
gases
adequately
define the curves, so that given and recovered
distributions
correspond
closely. Gases that could be used
are shown in Table
1 together
with their blood: gas partition coefficients at 37°C. The six gases used routinely
are:
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
and four examples
of the recovery
of distributions
of
ventilation
in Fig. 5. It can be seen that correspondence
between the original
and recovered distributions
was very
satisfactory in all cases. In particular,
the shape and dispersion of the curves were accurately
determined,
so that
the mean and standard
deviation
of the recovered
and
original curves agreed to within 1 %.
Broad distributions
with a single mode can be recovered
with a smaller number of iterations of the numerical
process
than either narrow distributions
or bimodal
types. Therefore, in analyzing
a given set of data, the number
of iterations required
for the solution should be judged by failure
to further improve the result appreciably
with more iterations as judged by the residual sum of squares.
When data corresponding
to a truly homogeneous
lung
are analyzed, the numerical
technique
returns a distribution
that is almost but not quite homogeneous.
The narrowest
distribution
that could be accurately
recovered was therefore determined,
and found to be one which would give rise
to an alveolar-arterial
O2 difference
of 5 mmHg.
This is
within
the range of normality.
Such a distribution,
if
logarithmically
normal,
would have a log standard
deviation of 0.3.
Comklvtment number. Although
the numerical
technique
can be executed
using any number
of VA/Q
ratios, an
adequate
pictorial
representation
requires some minimum
number of compartments.
The greater the number of modes,
the more compartments
will be needed, but the precise
number
for a particular
distribution
is difhc.ult to define.
As more compartments
are used, the distance
between
adjacent
compartments
is reduced so that the location
of
the distribution
can be achieved more accurately,
but the
computing
costs rise at the same time.
In Fig. 6 is shown the result obtained
using diflerent
numbers of compartments
for a given bimodal distribution.
It can be seen that as long as there are sufficient compartments to follow the contours of the curve, adding compartments simply results in a scaling change on the ordinate,
with little improvement
in description.
In particular,
the
shape and position of the recovered
curve is not affected.
This was also the case for unimodal
distributions.
Based on these considerations,
there will be some intermediate number
of compartments
that is most practical.
In general, this has been found to be 50, and this number is
now used routinely.
C/lojce of VA/Q
ratios. A given bimodal
distribution
was
recovered using two sets of 50 compartments
whose interdigitating
VA/~
ratios were equally spaced logarithmically
along the VA&,
axis. It can be seen from Fig. 7 that the
shape did not depend on the choice of VA/Q
ratios. In fact,
it has been found that if a set of VA/Q
ratios which are not
equally logarithmically
spaced is used, the recovered mean
and standard deviation
are within
1 % of the values of the
given distribution,
although
the pictorial
representation
of
the distribution
depends on the manner in which the spacing
of compartments
is allocated.
In such a situation,
the
cumulative
frequency distribution
curves of the given and
recovered distributions
do correspond
closely.
Effects of experimental errors. Because only ratios of gas
concentrations
are needed,
rather
than absolute
values,
experimental
errors are reduced.
Since the expired
and
594
WAGNER,
.
SALTZMAN,
AND
WEST
0
X
ox
l
.
0
ox
0
MEAN
l
MEAN
x
MEAN
t2SD
-2SD
X
l
.
0
X
0
l
X
.
ox
.
ox.
oxOX
0
X
0’
OX
0:x
I .o
0.1
SOLUBILITY,
ml /IO0
/mm
100.0
h/b
Hg
.08
D
I
0
MEAN
l
MEAN
MEAN--SD
x
+
2SD
x
2
x0
9
;
zx
t
t3
.x
0
.x
3
.
:
.
J
I
.8Ol
8
10.0
0
.
4
.Ol
SOLUBILITY,
0. I
ml / IOOml
IO
IO.0
i/A/O
/ m m Hg
sulfur
hexafluoride,
ethane,
cyclopropane,
halothane,
diethyl
ether, and acetone. The others in Table
1 are
alternatives
whose choice would depend on the conditions
of measurement.
Limitations of method. The use of a large number of compartments
and small number of gases imposes certain limitations. First, the maximum
number of discrete modes that
can be described has been found to be three, which is consistent with the use of just six gases. When only one mode
was present in the given distribution,
its mean, standard
deviation,
and skewness could be accurately
determined
as
long as the log standard
deviation
exceeded 0.3, as discussed above. When two modes were present, the position,
dispersion,
and relative height of each mode could be established
adequately.
When three modes were given, their
existence could be identified
although
the regions between
the modes were not accurately
described. In 30 experiments
in both normal
and abnormal
dogs (some of which were
suffering
from lobar pneumonia,
some from artificial
or
natural
pulmonary
embolism,
and some from pulmonary
edema), the maximum
number
of modes seen was two for
either ventilation
or blood flow. In addition,
in 12 normal
human volunteers the distributions
were unimodal
in every
case, while in one patient with a lobar infiltrate
the distribution of blood flow was bimodal.
Second, only smoothly
contoured
curves can be described, but since we assume that real distributions
are
smooth, this is not seen as a disadvantage.
Finally, isolated
values of blood flow or ventilation
in any one compartment
TABLE
1. Gases suitable for use in method
Formula
Gas
Sulfur
hexafluoride
Methane
Ethane
Freon
12
(Dichlorodifluoromethane)
Cyclopropane
Acetylene
Fluroxene
Halothane
Diethyl
ether
Acetone
* Measured
t Measured
SF6
CH4
C2Hs
CClpF2
C3H6
C2H2
C4F3H
50
CzFsBrCIH
(C2H
(CH3)
5)20
2CO
in this
laboratory
in this
laboratory
MO1 ivt.
Blood: Gas Partition
Coefficient
(37 “C)
146
0.0076
16
30
121
0.038*
0 092*
0:26t
42
26
126
197.5
74
58
(human
(dog
blood).
0.415
0.842
1.37
2.30
11.7t
333
blood,
mean
(20;)
(16)
(12)
(23)
(15)
(33
of 8 subj).
should not be regarded
as having meaning
on their own.
Rather, the use of many points is a means of indicating
the
shape and dispersion of the distribution.
The advantages of
such an approach
are that neither the mathematical
form
of the distribution
nor the number
of modes need to be
known prior to the analysis, so that distributions
of many
different types can be described.
Choice of initial distribution.
Since the numerical
procedure
is an iterative
one which commences
from some starting
guess at the solution, the effect on the results of choosing
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
FIG. 8. Effects
of experimental
errors
on
recovery
of two
theoretical
distributions.
Using
errors
derived
from
actual
measurements,
corresponding
errors
in retention
for each
gas were
computed
(A and
C) and
distributions
corresponding
to these
various
curves
were
recovered
(B and
0).
There
is
relatively
little
change
in shape
or
position
of
either
distribution
even
when
curves
two
SD away
from
the
mean
are used.
Effects
on distributions
of ventilation
caused
by the same errors
in excretion-solubility
plots
are
very
similar.
.
Ox
MEASUREMENT
OF
VENTILATION-PERFUSION
595
INEQUALITY
z
MIXED
F
z
VENOUS
F
8
- 0.10
ii
-J
a
ik
RETENTION
F
2
MIXED
TIME
ARTERIAL
OF
INFUSION.
min
RETENTION
5I-
0’
0
FIG. 9. Theoretical
venous
partial
pressure
(B) during
continuous
severe
VA/Q
inequality
its equilibrium
value
librium
very
rapidly.
computations.
I
25
1
50
1
75
I
too
time
courses
of retention,
and arterial
and mixed
of sulfur
hexafluoride
(A) and diethyl
ether
infusion
of the gases.
In this analysis,
there
is
in the lung.
SF6 retention
has almost
reached
by 20 min
while
that
for ether
reaches
equiPeripheral
tissues
uptake
is considered
in the
this being the ratio of cardiac output (6 l/min)
to functional
residual capacity, (3 liters) although
it is possible in practice
that the ratio may be lower in units with low VA/Q.
Calculations were made from the tissue dimensions
given by Eger
(7). Figure 9B shows similar calculations
made for diethyl
ether. In both cases, the time course of retention
was also
computed
neglecting
uptake
by peripheral
tissues, and
although
not shown,
these were indistinguishable
from
those in Fig. 9, L4 and B. It may be seen that in the presence
of severe VA/Q
inequality,
even SF6 reaches a retention
of
0.050 in 20 min (the equilibrium
value being 0.056). This
small absolute
difference
is within
thei limits of experimental error. It is of interest that, although
the venous and
arterial
concentrations
for ether change slowly, calculated
retention
rapidly reaches a constant value.
We conclude
that unless there is a large amount of lung
volume
and blood flow associated
with regions
whose
VA/Q
ratios are very low, virtual equilibrium
for all gases
in the lung will be reached within
20 min of commencing
the infusion.
In the experimental
situation,
timed samples
can be drawn to determine
that equilibrium
does in fact
exist. At the usual infusion rate of 5 ml/min,
the total fluid
load to the patient from a 20-min infusion is 100 ml.
Henry’s law. Eq. I and all subsequent
equations
depend
on the abeyance of Henry’s law over the range of partial
pressures encountered.
This means that concentration
and
partial pressure are linearly related. Maharajh
and Walkley
(2 1) recently reported the failure of oxygen and nitrogen to
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
different
starting
distributions
was examined.
Four such
starting points were used, three of which were smooth functions. The fourth was a grossly irregular
oscillating
function
such that if in any one compartment
the blood flow or
ventilation
were small, that of the compartments
immediately preceding
and following
were large, and vice versa.
The three smooth curves were 1) horizontal,
so that blood
flow and ventilation
was the same in every compartment,
2) log parabolic,
so that blood flow and ventilation
at the
extremes of VA/Q
was very large, while that in the middle
of the range was very small, and 3) inverted log parabolic,
so that most of the ventilation
and blood flow was in the
physiological
range of VA/Q.
For each of the three smooth starting distributions,
the
resulting solutions were indistinguishable
from one another,
and the correspondence
with the given distributions
was
as close as depicted in Figs. 4 and 5. This was true for both
narrow
and broad distributions.
Even when the irregular
starting point was used, the solution was adequate
in that
the mean and standard
deviation
of the original
and recovered curves were within
1 % of each other. When the
recovered
distribution
was plotted,
the contours
were
irregular,
the points oscillating
about the given smooth
curve.
Thus an adequate
solution for smooth distributions
will
be obtained
from a smooth starting distribution
even when
that starting point has a shape grossly different from that of
the real curve. It should also be noted that the same starting
point
(inverted
log parabola)
was used in every case
depicted in Figs. 4 and 5. Since it is assumed that the distribution
to be measured
is smoothly
contoured,
then by
choosing a smooth starting guess, the result will be appropriate and not dependent
on the shape of the initial guess.
Assumptions of steady-state conditions. Eq. 1 and hence all
subsequent
expressions explicitly
depend on the existence
of steady-state
conditions
as defined earlier.
This means
that the ratio of arterial to venous and alveolar to venous
partial pressures for all gases must be constant, though it is
not necessary for the absolute partial
pressures to be constant. It is well known for example, that arterial and venous
concentrations
of halothane
and ether may not reach constant values for many hours following
the onset of continuous administration
(8).
The factors that affect the rate at which equilibrium
in
lung units is reached are the blood: gas partition
coefficient
of the gas concerned,
the VA/Q
ratio of the lung unit, and,
separately,
the blood flow per unit gas volume in that lung
unit. It can be shown that a poorly soluble gas in a lung
unit whose VA/Q
ratio and blood flow per unit volume are
both low will take the longest to equilibrate.
Factors such
as the fluid volume and blood flow of the tissues and the
various tissue: blood partition
coefficients
have almost no
effect on the rate of attainment
of equilibrium
in the lung,
although
they are critical in determining
the rate at which
absolute gas concentrations
change in the blood. In Fig.
9A are shown the rates of change of retention,
and of arterial
and mixed venous partial
pressures of the poorly soluble
gas SF6 in a lung with severe VA/Q
inequality
assuming
equilibration
between
alveolar
gas and lung tissue. The
VA/Q
distribution
is that shown in Fig. 4B. Blood flow
per unit volume was taken to be 2.0 in all VA/Q
units,
596
WAGNER,
Mlxed
Venous
Point
\
s
Alveoloi
P;lnt
TIME
ALONG
CAPILLARY,
set
FIG. 10. Time
course
along
the pulmonary
capillary
for a gas of
extremely
high
molecular
weight
(halothane)
expressed
as percentage
change
from
mixed
venous
to alveolar
pressure.
Five
such
courses
are shown,
for normal
and several
values
of reduced
membrane
difFor
convenience,
diffusing
capacity
is labeled
in
fusing
capacity.
terms
of that
for oxygen
in ml/min
per mmHg.
Even
when
Dmo,
is only
2 ml/min
per mmHg
(5yo of its normal
value
of 40 ml/min
per mmHg)
equilibration
is 98.9%
complete
in 0.75 s.
AND
WEST
O-
6Severe
combtned
serves
6-
Severe
.OOl
01
BLOOD : GAS
FIG. 11. Effects
equality
on inert
comparison.
Both
the
retention-solubility
has greater
effects
solubili
ty curve.
I
series
IO
PARTITION
lnequollty
alone
100
100
COEFFICIENT
of severe
series
and
parallel
gas retention.
*4 homogeneous
forms
increase
the retention
curve
remains
smooth.
than
series
on the displacement
forms
of VA/Q
inlung
is shown
for
of any given
gas, but
Parallel
inequality
of the retention-
the gas phase exists, and this would impair the elimination
of gases with high molecular
weight. The measured retentions would
be artificially
high and the resulting
VA/Q
distribution
would be in error. Adaro and Farhi (1) have
recently reported
in abstract that the elimination
of acetylene (mol wt 26) is 8 %, higher than that of monochlorodifluoromethane
(mol wt 86.5) in dogs in spite of their
similar solubilities.
The eight gases in Table 1 have molecular weights ranging from 16 (methane)
to 197.5 (halothane),
and if there are measurable
effects of gaseous diffusion
limitation,
halothane,
fluroxene,
and SF6 retentions
would
be artificially
high. We have looked carefully for evidence
of this in approximately
20 experiments
with these gases.
No irregularity
of retention
has been observed, all points
lying on a smooth curve.
Continuous ventilation und perfusion. Retention
calculated
from Eq. I is based on constant alveolar tensions. This implies continuous
rather than tidal ventilation
and steady
perfusion of all lun g units. Since in reality neither ventilation nor perfusion are constant, it is important
to determine
any effects of this on the analysis. Models of gas exchange
incorporating
cyclic ventilation
and perfusion
have been
examined
at a variety of breathing
frequencies,
and the
elimination
of several gases compared
with those predicted
bv Eq. I (W. E. Colburn,
personal communication).
At a frequency of lO/min, for all gases, the elimination
in the tidal
model differed from that in the continuous
model by a
maximum
of only 0.5 %. As the frequency
was increased,
agreement
between the models became closer. These results
justify the use of the continuous
model, and this greatly
simplifies the analysis.
of the model is
Inspired gus composition. An assumption
that the inspired
gas of all lung units contains none of the
gases being measured.
All lung units will inspire gas from
the anatomic
dead space, but it is not possible to take this
into account in the calculations.
If lung units with high
VA/Q
ratios inspire some dead space gas contributed
to by
units with lower VA/Q
ratios, the alveolar concentration
of
all gases in these units will be artificially
elevated
resulting
in an effectively lower VA/Q
ratio from the point of view of
gas exchange.
To the extent to which this occurs, this is a
source of error in the present method, and will lead to an
underestimate
of the ventilation
and blood flow to high
VA/Q ratio units.
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
obey Henry’s law over a I-atm range of partial pressure. In
the present situation,
the maximum
arteriovenous
partial
pressure difference of the infused gases is less than 1 mmHg.
Over this small range, any change in solubility
must be
very small. Measurements
in dogs and man with each of the
gases shown in Table 1 reveal no differences
between partition coefficients in arterial
and venous blood. However,
solubility
will depend to varying
degrees on hemoglobin,
hematocrit,
temperature,
protein, and lipid levels. It might
therefore be necessary to measure solubility
in the individual
situation.
D$kion
across alveoh-cu@lary
membrane. Also implicit
in
Eq. I is the assumption
that for each gas, the partial pressures are the same in both end-capillary
blood and alveolar
gas. Forster (11) calculated
that the time for 99 % equilibration
for nitrogen
along the capillary
was about 0.01
s under normal conditions,
and since the transit time along
the capillary
is of the order of 1 s, he concluded
that the
assumption
of equilibration
is reasonable.
We have examined the factors affecting diffusion and calculated
the time
courses of the gases in Table 1 under conditions
of normal
and reduced diflusing
capacity.
Because of Henry’s
law,
the time courses are exponential.
When expressed so that
the venous to alveolar difference is always 100 %, the time
courses are independent
of absolute values of either the
venous or alveolar tensions. As diffusing capacity is reduced
and molecular
weight increased, the rate of equilibration
is
less, but even when the membrane
diffusing
capacity for
oxygen (Dmo2) is only 2.0 ml/min
per mmHg and halothane (mol wt 197.5) is considered,
there is only a 1 .l %
reduction
in gas transfer compared
with that when diffusing capacity is infinitely
great. Figure 10 shows time courses
for halothane
when the Dmo2 is both normal and reduced.
Since halothane
represents
the worst case, it appears
justifiable to neglect alveolar-capillary
diffusion impairment
for inert gases.
Transport of gas from alveoli to the atmosphere. Eq. 1 assumes
that the only factors affecting the elimination
of a gas are
the solubility
of the gas and the VA/Q
ratio of the lung
unit concerned.
It is possible that diffusion
limitation
in
SALTZMAN,
MEASUREMENT
OF
VENTILATION-PERFUSION
597
INEQUALITY
APPENDIX
,tfethod
If
F =
then
Eq.
1,000
I can
.
273
+
body
temp
273
+
room
temp
be rewritten
with
Pv can
be isolated
R=-=
5. These
8. From
sum
of squares
PH20bodytemp
in ml/min,
= lO.Qt&[Pa
from
this
equation,
-
and
BTPS:
Pii]
retention
Pii
1
PT
FJ?E-PE
C 1o.Qt.S.
equations
Eq. 4A :
are
(PB
those
-
PH~O,,,,)*Pii
described
by Eq.
found:
1
4A.
of differences
iA1
=
Thus,
ment
.
(J =
sides
1
~~~~~~~~~~~~~
PH~%orn)
l+
Step
Step
-
PB -
both
F-\~E-PE
P B -
PB
S
=
’
A
2 [&
$t;Q)
-
Qt’Ri]
the partial
derivative
of s with
respect
to Qj in the jth
is given
in Eq. 5, noting
that in the present
formulation,
1, N) is a set of constants.
compartVj/‘Qj,
I
of Solution
for
Distribution
of Blood
Flow
The
solution
for the distribution
of ventilation
exactly
parallels
for blood
flow
and will
be omitted
here
for clarity.
The steps involved
in the solution
for blood
flow are as follows:
I. En&number
of gases (h(I) and number
of \jAIQ
compartments
N
2. Enter
solubilities,
and arterial
and
expired
partial
pressures
for
each
gas, and additional
data
including
VE, Qt,
anatomic
dead
space
3. Calculate
retention
(arterial-to-venous
ratio)
for each gas
4. Set up N-2 values
of VA/Q
between
0.005
and
100.0
(equally
spread
along
a log scale),
and add one compartment
with
\jA/
Q = 0.0 and one with
VA/Q
= CZI
5. Formulate
M equations
in N unknown
blood
flows,
corresponding
to N compartments
6. Set up initial
guess at the set of N blood
flows
(a convenient
initial
guess is a low broad
inverted
parabola)
7. Calculate
sum
of squares
of differences
between
observed
and
predicted
retentions
8. Calculate
for each
compartment
the partial
derivative
of this
sum of squares
with
respect
to blood
flow
9. Alter
the blood
flow
in each
compartment
so that
according
to
the sign and size of the partial
derivative
found
in step 8, the sum
of squares
is reduced
10. Test for lack of significant
further
reduction
in sum of squares
and
change
in blood
flows;
if reduction
continues,
go to step 7; otherwise to II
II. Print
and plot
blood
flow
(Y axis)
and \iA&
ratio
(X axis)
for
each compartment
12. Determine
end capillary
02 and CO2 contents
for each compartment
using
Kelman’s
subroutines
and
thereby
compute
the
mixed
arterial
PO:! and Pco:!
13. stop
that
Thus,
for any given
value
(i.e., initial
guess)
of Qj the partial
derivative
can be evaluated.
Step 9. From
any preceding
value
of Qi, the new value
is found
from Eq. 5 as follows.
a) The
direction
in which
Qj should
be altered
depends
on the
sign of &/aQj.
If the sign is positive,
Qj must be decreased
and conversely
Qj must be increased
if the sign is negative.
b) The magnitude
of the change
is also found
from Eq. 5, by summing
all the small
changes
As that would
result
from
a small
change
AQj in each compartment
and scaling
this sum to the current
value
of the sum of squares.
This is sufficient
constraint
to allow
computation
of the new Qj in each compartment.
Because
a change
in blood
flow
in any
one compartment
affects
the partial
derivative
in all
other
compartments,
this approach
does not yield
the correct
solution
in just one pass. Rather,
due to this interdependence,
many
iterations
are required.
Note
that
theoretically
it is possible
to solve the system
directly
withou!
an iterative
procedure
by writing
N simultaneous
equations
as/aQj
= 0, (j = 1, N) in the N unknown
blood
flows.
This,
being
a square
system,
should
be directly
solvable,
but
in
practice,
it is so poorly
conditioned
that the solution
cannot
be found.
APPENDIX
II
Properties
Solubility
of Relationships
and Between
Between
Retention
Excretion
and Solubility
It has been
established
gas with
partition
coefficient
R
in
the
X
text
X +
E
where
S is solubility
measured
in ml/min,
VE.FE
=
in
ml/100
LHS
BTPS;
lO*QtS[Pi?
ml blood
in l/min,
-
expired
partial
partial
pressure
ratio.
The
ATPS.
mmHg,
as in the text.
of R with
respect
to X in Eq. 9:
4A
. .
Vj/Qj
and
PB),
that
1
1
to X is given
for
any
(6)
(7)
in
Eq.
8, and
(8)
Pa]
per
using
the same notation
The first derivative
that for E with
respect
(Eq.
Qx
Notes
Step 3. Calculation
of retention
from
arterial
and
pressures
requires
computation
of the mixed
venous
since
retention
is the
arterial-to-venous
concentration
Fick principle
is used.
and
BTPS.
RHS
is
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
We have formulated
a model of series inequality
for inert
gases based on that developed
for O2 and CO2 (3 1). This
examines the situation
in which the inspired
gas of some
lung units comes from other units already participating
in
gas exchange.
Although
series inequality
resulted in impaired gas exchange,
the effects were much less marked
than with comparable
degrees of parallel
inequality
of
ventilation
and blood flow. Figure 11 shows the retentionsolubility
curve for a homogeneous
lung, a lung with severe
parallel
inequality
alone, one with severe series inequality
combined
series and parallel
inalone, and one with
equality.
The parallel
distribution
is logarithmically
normal with a log standard
deviation
of 2.0 while the series
distribution
is one in which 99 %I of both ventilation
and
blood flow are partitioned
to the parasitic
compartment.
Both distributions
produce
severe hypoxemia
(31). However, since series and parallel
inequality
both increase retention of gases, they can be handled
by the analysis when
present together although
the resulting
distribution
cannot
at the present time be separated
into its “series”
and
“ parallel”
components.
598
WAGNER,
The second
spectively,
derivatives
of R
in Eq. 10 and II:
and
E with
respect
to X are
given,
re-
d2R
-=-c-h2
d2E
-dX2
For
all
-
gases,
and
x >
For
all
possible
and
in at least
Vj>
0.0;
Qj
one
>
2
0.0;
0;
Qk
order
to have the existence
Therefore,
from Eq. 3,
from
Eq.
Vj/Qj
compartment,
dR
dX
and
from
Eq.
2
0.0;
j
=
0;
and
vk/Qk
>
0
of gas exchange.
> 0.0,
o<x<@J
4,
> 0.0,
distributions,
both
R and E are monoof X in the finite
range.
Thus,
as X inalways
increase,
and never
decrease
or
d2R
dA2
< 0.0,
o<x<oo
d2E
dX2
< 0.0,
o<x<
00
This proves
that not only
are R and E both
monotonic,
but that these
functions
can have no points
of inflection
throughout
the finite
range
of X. It is this
behavior
of the retention-solubility
and
excretionsolubility
curves
which
greatly
restricts
the amount
of variation
that
can occur
in the regions
between
actual
data
points.
The authors
express
appreciation
for the helpful
advice
Evans
at many
points
in the development
of this method.
This
study
was supported
by Public
Health
Service
13687-02
and HL-0593
l-02 and by National
Aeronautics
Administration
Grant
NGL-05-009109.
of Dr.
Grants
and
John
HLSpace
o<x<w
Received
for
publication
2 August
1973.
REFERENCES
1. ADARO, F., AND L. E. FARHI. Effects
of intralobular
gas diffusion
on alveolar
gas exchange
(Abstract).
Federation
Proc. 30 : 437, 197 1.
2. BALL,
W. C., P. B. STEWART,
L. G. S. NEWSHAM,
AND D. V.
BATES.
Regional
pulmonary
function
studied
with
xenonl33.
J. CZin. Invest. 41 : 519-53
1, 1962.
3. BRISCOE,
W. ,4. A method
for dealing
with
data
concerning
uneven ventilation
of the lung
and its effects
on blood
gas transfer.
J. AppZ. Physiol.
14 : 29 l-298,
1959.
4. BRISCOE,
W. A. Comparison
between
alveolo
arterial
gradient
predicted
from
mixing
studies
and
the
observed
gradient.
J.
Appl.
Physiol.
14 : 299-303,
1959.
5. BRISCOE,
W. A., E. CREE, J. FILLER,
H. E. J. HOUSSAY,
AND A.
COURNAND.
Lung
volume,
alveolar
ventilation,
and
perfusion
interrelationships
in chronic
pulmonary
emphysema.
J. A#.
Physiol.
15 : 785-795,
1960.
6. DYSON,
N. A., P. HUGH-JONES,
G. R. NEWBERY,
J. D. SINCLAIR,
AND J. B. WEST.
Studies
of regional
lung
function
using
radioactive
oxygen.
Brit.
Med.
J. 1: 231-238,
1960.
7. EGER,
E. I. II. In : Uptake
and Distribution
of Anesthetic
Agents,
edited
by E. M. Papper
and
R. J. Kitz.
New
York:
McGraw,
1963, p. 76.
8. EGER,
E. I. II. In : Uptake
and Distribution
of Anesthetic
Agents,
edited
by E. M. Papper
and
R. J. Kitz.
New
York:
RicGraw,
1963, p. 258.
9. FARHI,
L. E. Elimination
of inert
gas by the lung.
Respiration
Physiol.
3: l-l 1, 1967.
10. FINLEY,
T. N. The
determination
of uneven
pulmonary
blood
flow
from
the arterial
oxygen
tension
during
nitrogen
washout.
J. CZin. Invest. 40 : 1727-l
734, 196 1.
11. FORSTER,
R. E. Diffusion
of gases.
In:
Handbook
of Physiology.
Respiration.
Washington,
D.C.
: Am.
Physiol.
Sot.,
1964,
sect.
3, vol. I, chapt.
33, p. 839-872.
12. GROLLMAN,
A. The solubility
of gases in blood
and bloodfluids.
J. BioZ. C’hem. 82 : 317-325,
1929.
13. KETY,
S. S. The theory
and applications
of the exchange
of inert
gas at the lungs
and tissues.
Pharmacol.
Rev. 3 : l-41,
195 1.
14. KNIPPING,
H. W.,
W. BOLT,
H. VENRATH,
H. VALENTIN,
H.
LUDES,
AND P. ENDLER.
Eine
neue
Methode
zur
Priifung
der
Herzund
Lungenfunktion.
Die
regionale
Funktionsanalyse
in
der
Lungen-und
Herzklinik
mit
Hilfe
des radioaktiven
Edelgases Xenon
l 33 (1 isotopen
Thorakographie).
Deut. Med.
Wochschr.
80: 1146-l
147, 1955.
15. LARSON,
C. P., E. I. EGER,
AND J. W. SEVERINGHAUS.
Solubility
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
of halothane
in blood
and
tissue
homogenates.
Anesthesiology
23:
349-355,
1962.
LARSON,
C. P., E. I. EGER,
AND J. W. SEVERINGHAUS.
Ostwald
solubility
coefficient
for anesthetic
gases
in various
fluids
and
tissues.
Anesthesiology
23 : 686-689,
1962.
LENFANT,
C. Measurement
of ventilation/perfusion
distribution
with
alveolar-arterial
differences.
J. AppZ. Physiol.
18 : 10901094,
1963.
LENFANT,
C. Effect
of high
Fro, on measurement
of ventilation/
perfusion
distribution
in man
at sea level.
Ann. N. Y. Acad.
Sci.
121: 797-808,
1965.
LENFANT,
C., AND T. OKUBO.
Distribution
function
of pulmonary
blood
flow and ventilation-perfusion
ratio
in man. J. AppZ. Physiol.
24: 668-677,
1968.
LONGO,
L. D.,
M.
DELIVORIA-PAPADOPOULOS,
G. G. POWER,
E. P. HILL,
AND R. E. FORSTER
II. Diffusion
equilibration
of
inert
gases
between
maternal
and
fetal
placental
capillaries.
Am. J. Physiol.
2 19 : 561-569,
1970.
MAHARAJH,
D. M., AND J. WALKLEY.
Lowering
of the saturation
solubility
of oxygen
by the presence
of another
gas. Nature
236:
165, 1972.
MILIC-EMILI,
J., J. A. hl. HENDERSON,
hi. B. DOLOVICH,
D. TROP,
AND K. KANEKO.
Regional
distribution
of inspired
gas in the
lung.
J. APPZ. Physiol.
2 1 : 749-759,
1966.
MUNSON,
E. S., L. J. SAIDMAN,
AND E. I. EGER II. Solubility
of
fluoroxene
in blood
and
tissue
homogenates.
Anesthesiology
25:
638-640,
1964.
NOEHREN,
T. H. Pulmonary
clearance
of inert
gases with
particular
reference
to ethyl
ether.
J. A@Z. Physiol.
17 : 795-798,
1962.
OKUBO,
T., AND
C. LENFANT.
Distribution
function
of lung
volume
and
ventilation
determined
by lung
N2 washout.
J.
&II.
Physiol.
24: 658-667,
1968.
PESLIN,
R., S. DAWSON,
AND J. MEAD.
Analysis
of multicomponent
exponential
curves
by the Post-Widder’s
equation.
J.
AppZ. Physiol.
30: 462-472,
1971.
RILEY,
R. L., AND A. COURNAND.
Analysis
of factors
affecting
partial
pressures
of oxygen
and carbon
dioxide
in gas and
blood
of lungs : theory.
J. ApPZ. Physiol.
4: 77-102,
195 1.
RILEY,
R. L., A. COURNAND,
AND K. W. DONALD.
Analysis
of
factors
affecting
partial
pressures
of oxygen
and
carbon
dioxide
in gas and blood
of lungs : methods.
J. A@Z. Physiol.
4: 102-120,
1951.
ROCHESTER,
D. F., R. A. BROWN,
JR., W. A. WICHERN,
JR., AND
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017
dE
dX
WEST
1, N
k,
>
AND
6:
0.0
distributions,
vk
in
This proves
that for all VA/Q
tonically
increasing
functions
creases,
both
R and E must
remain
unchanged.
From
Eq. 5:
SALTZMAN,
MEASUREMENT
H. W.
trations
OF
VENTILATION-PERFUSION
FRITTS, JR. Comparison
of alveolar
and arterial
concen133Xe
infused
intravenously
in man.
J. A$$.
Physiol. 22 : 423-430,
1967.
30. WAGNER,
P. D., P. F. NAUMANN,
AND R. B. LARAVUSO.
Simultaneous
measurement
of eight
foreign
gases in blood
by gas chromatography.
J. AppZ. Physiol. 36: 600-605,
1974.
3 1. WEST, J. B. Gas exchange
when
one lung
region
inspires
from
another.
J. AppZ. Physiol. 30: 479-487,
1971.
of 85Kr
and
599
INEQUALITY
32. WEST,
J. B., AND C. T. DOLLERY.
Distribution
of blood
flowventilation-perfusion
ratio
in the lung,
measured
with
radioactive
COZ. J. A@Z. Physiol. 15 : 405-410,
1960.
33. WIDMARK,
E. hf.
Studies
in concentration
of indifferent
narcotics
in blood
and tissues.
Acta Med. Stand. 52 : 87-l 64, 19 19.
34. YOKOYAMA,
T., AND L. E. FARHI.
Study
of ventilation-perfusion
ratio
distribution
in the anesthetized
dog by multiple
inert
gas
washout.
Respiration Physiol. 3 : 166-l
76, 1967.
Downloaded from http://jap.physiology.org/ by 10.220.32.247 on July 28, 2017