Vol. 87, Issue 1, 161-169, July 1999
A tidal breathing model for the multiple inert gas elimination
technique
J. P.
Whiteley1,2,
D. J.
Gavaghan1,2, and
C.
E. W.
Hahn1
1 Nuffield Department of Anaesthetics,
University of Oxford, Radcliffe Infirmary, Oxford OX2 6HE; and
2 Oxford University Computing Laboratory, Oxford
OX1 3QD, United Kingdom
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ABSTRACT |
The tidal
breathing lung model described for the sine-wave technique (D. J. Gavaghan and C. E. W. Hahn. Respir. Physiol. 106: 209-221,
1996) is generalized to continuous ventilation-perfusion and
ventilation-volume distributions. This tidal breathing model is then
applied to the multiple inert gas elimination technique (P. D. Wagner,
H. A. Saltzman, and J. B. West. J. Appl. Physiol. 36:
588-599, 1974). The conservation of mass equations are solved, and
it is shown that 1) retentions vary considerably over the course of a breath, 2) the retentions are dependent on alveolar volume, and 3) the retentions depend only weakly on the width of the ventilation-volume distribution. Simulated experimental data
with a unimodal ventilation-perfusion distribution are inserted into
the parameter recovery model for a lung with 1 or 2 alveolar compartments and for a lung with 50 compartments. The parameters recovered using both models are dependent on the time interval over
which the blood sample is taken. For best results, the blood sample
should be drawn over several breath cycles.
ventilation-perfusion; ventilation-volume; gas exchange
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INTRODUCTION |
INERT GAS TECHNIQUES, such as the multiple inert gas
elimination technique (MIGET), have long been used as a tool for
investigating the matching of ventilation and perfusion in the lungs.
In this technique a subject is infused intravenously with a selection of inert tracer gases, usually six, so that the mixed venous content of
these gases is constant. The gases are inert, and the blood content (C)
is related to the partial pressure (P) by C = 
P, where is the
Ostwald partition coefficient. The six gases are chosen so that their
partition coefficients are approximately evenly spaced on a logarithmic
scale. A typical choice of the gas of lowest solubility is sulfur
hexafluoride ( = 5.99 × 10
3); a typical
choice of the most soluble gas is acetone ( = 285). After a period
of time has elapsed, a "steady state" is reached, and the partial
pressure of each individual tracer gas in the mixed expired gases and
in the arterial blood is assumed to be constant. The ratio of mixed
expired partial pressure (P
) to
mixed venous partial pressure P
of each
tracer gas is known as the excretion; the ratio of arterial partial
pressure (P
to
(P
is known as the retention. From the
retention and excretion of each gas, predictions may be made about the
ventilation-perfusion (
/
)
distribution inside the lung. Initial work by Farhi and Yokoyama (2, 3)
used a simple lung model with one alveolar compartment. This was
developed by Wagner et al. (11) and Evans and Wagner (1) into a
50-compartment lung model.
Previous work on MIGET has made little reference to the effects of
tidal breathing. In particular, we demonstrate that tidal breathing
causes variations in P
within each breath.
These variations will also be dependent on the way in which the
ventilation is distributed within the lung, and so, theoretically,
MIGET should be dependent on the alveolar ventilation-volume
(A/VA) distribution as well
as the / distribution. We
investigate the magnitude of this effect in this study.
We apply the tidal breathing model described by Gavaghan and Hahn (4)
for the forced inspired sine-wave technique to MIGET. We begin by
describing the differences in the mathematical modeling. We then
proceed to demonstrate that the magnitude of the retentions vary over
the time course of each breath and investigate the effect of these
variations on a model with one alveolar compartment, as described here,
and that used by Evans and Wagner (1). We show that unless the
retentions are time averaged, considerable differences in both models
are caused by these time-varying retentions and that the
/ distributions recovered are very
dependent on the point of the breath at which the arterial blood is
sampled. For example, some retentions generated using a one-compartment tidal model may be recovered as a bimodal
/ distribution for a recovery
routine based on a continuous ventilation model. Finally, we
investigate the dependency of the retentions on the
A/VA, distribution. It is
shown that the width of the
A/VA distribution is only of
marginal significance but that the dependency on the total alveolar
volume (VAT) may not be neglected.
For clarity, we consider only unimodal
/ distributions, inasmuch as the
effects of the time-varying retentions can be demonstrated fully using
these distributions only.
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MODEL |
To study the effects of tidal breathing, it is necessary to give a
description of the inspiratory and expiratory processes by defining
VA as a function of time over each breath. We follow Gavaghan and Hahn (4) and assume a linear increase in volume on
inspiration and an exponential decrease in volume on expiration. If we
were to consider a lung with more than one mode of ventilation, volume,
and perfusion, then we would also have to consider sequential emptying
and filling of the compartments under consideration. It would then be
necessary to know the emptying pattern to calculate the
since the gas expired from an
alveolar compartment that empties last will be left in the series dead
space at the end of expiration and will not be included in the mixed
expired gases. For the same reason, the constitution of the dead space gas at the end of expiration is dependent on the emptying pattern of
the alveolar units, and similarly the distribution of this dead space
gas on inspiration is dependent on the filling pattern of the alveolar
compartments. Once these processes had been modeled, the conservation
of mass equations would still be dependent on the fraction of the tidal
volume associated with each compartment, and so the retentions and
excretions would now be dependent on the
A/VA distribution as well as
the / distribution. For these
reasons, we restrict ourselves to a lung model with unimodal / and
A/VA distributions.
One of the major differences between the tidal breathing model and the
continuous breathing model is that tidal breathing takes account of the
inspiration of gases left in the dead space at the end of each
expiration, and so the partial pressure (PD) of this
respired gas must be calculated. This is the first step in calculating
the retentions for the tidal model. After this has been done, the
conservation of mass equations for the tidal breathing model described
by Gavaghan and Hahn (4) may be solved.
Calculation of PD
Discrete compartment models.
Initially, we consider only unimodal distributions that are collections
of discrete compartments. We assume that there is no sequential
emptying and filling of compartments for such unimodal distributions.
This is generalized to continuous
A/VA and
/ distributions in Continuous
distributions of ventilation, perfusion, and volume. The
VA of these compartments
[VAi(t)] on an arbitrary
breath j is given by
|
(1)
|
on inspiration, where VTi is
the tidal volume of compartment i, TI is the
duration of inspiration, and tj is the time
at the beginning of breath j; for expiration
![V<SC>a</SC><SUB><IT>i</IT></SUB>(<IT>t</IT>) = <OVL>V</OVL><SC>a</SC><SUB><IT>i</IT></SUB> + V<SC>t</SC><SUB><IT>i</IT></SUB> <IT>e</IT><SUP>−&ggr;[<IT>t</IT>−(<IT>t</IT><SUB><IT>j</IT></SUB>+T<SC>i</SC>)]</SUP>](/content/vol87/issue1/fulltext/161/img008.gif)
|
(2)
|
where
Ai denotes
end-expiratory volume of compartment i and 
takes the same
value for each lung unit.
The total lung alveolar volume as a function of time
[VA(t)] and total tidal volume (VT)
are related by
|
(3)
|
|
(4)
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(5)
|
Let the VAT be
A + VD at time
TID after the beginning of inspiration, where
VD is the airway series dead space volume. Then, during the
time interval tj < t < tj + TID, the gas entering the alveolar compartment from the dead space volume
will be the gas that was left in the dead space at the end of the
previous expiration. At time t = tj + TID
This may be simplified using Eqs. 3-5 to give
In the time tj < t < tj + TID, the gas
left in the dead space will be inspired into the alveolar compartment.
In a similar manner, let the total alveolar volume be
A + VD at
TED after the beginning of expiration. Then,
for the remainder of expiration after this time, the gas leaving
the alveolar volume will be left in the dead space at the end of
expiration. We calculate TED by using an
argument similar to that used to derive TID to give
The gas leaving the alveolar compartment during the time
tj + TI + TED < t < tj+1 will be the gas that is left in the
dead space at the end of expiration. The partial pressure of this gas
(PD) may be calculated using
|
(6)
|
where t0 = tj + TI + TED and t1 = tj+1. There is a negative sign in
front of this integral, because the alveolar volume is
decreasing, and so dVAi/dt is negative.
Continuous distributions of ventilation, perfusion, and volume.
The underlying A/VA and
/ distributions are often assumed
to be log-normal distributions (6, 9). Assuming this to be the case, we
now derive an expression for PD for continuous / and
A/VA distributions. Later we
will use these continuous distributions to generate theoretical data
and compare the competing models. Defining
|
(7)
|
where Ad
is the alveolar volume distribution and VTd is
the tidal volume distribution at a point in the alveolar compartment. We may also write VTd as a function of
x and y
|
(8)
|
where 
and
are the mean
A/VA and mean
/, respectively. This log-normal
distribution has a peak at the point in the (x, y)
plane in the region of
(
) decaying to zero away from this point. The rate of decay is related to
the magnitude of 
x and y,
large values giving broad distributions and small values giving narrow
distributions. The constant A is found by noting that
and so
|
(9)
|
Equations 7 and 8 enable us to write
expressions for the alveolar volume and perfusions as functions of
x and y. Using Eq. 7, we write
|
(10)
|
In a similar way
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(11)
|
The total alveolar volume
(A) may be calculated as follows
Similarly, the pulmonary blood flow is given
by P = VTe
e
2y/2
TID and TED are
calculated in a manner analogous to that described above. We note that
the VAT at time t is given for
inspiration by
We may use the same argument that was used for discrete
compartments to show that the time taken for the dead space to be inspired is (VD/VT)TI. The same
method applied on expiration
gives TED = 1
ln(VT/VD)
We may now define PD for continuous
/ and
A/VA distributions to be the
triple integral
|
(12)
|
where t0 = tj + TI + TED and t1 = tj+1. We note that there is a minus sign
outside the integral to cancel the negative partial derivative of
VA with respect to time. This completes our description of
the calculation of PD.
Retentions and Excretions
Here we demonstrate how to calculate the retentions and excretions. We
begin by considering the classical three-compartment lung model, i.e.,
a lung model consisting of a dead space, a shunt, and a homogeneous
alveolar compartment. The governing conservation of mass equations are
described by Gavaghan and Hahn (4) for inspiration
|
(13)
|
and for expiration
|
(14)
|
where is the blood-gas partition coefficient,
P is the pulmonary blood flow,
P is the constant mixed venous partial pressure, and PIA(t) is the partial
pressure of the gas under consideration inspired from the dead space
volume, which is given by
where PD is the partial pressure of the gas in
the dead space at the end of expiration, as described in Eq. 6.
Equations 13 and 14 are solved numerically using library
computer subroutines (7), for N breaths, subject to the initial
condition PA(0) = 0. N is chosen large enough so
that a breath-by-breath steady state is reached: the partial pressures
at onset of inspiration and end of expiration on breath N are
equal and given by PA(tN). Equations 13 and 14 are then solved on breath N + 1 to show how PA(t) varies over the
course of one breath in this "pseudo"-steady state.
The retention is the ratio of P to
P. P is the
perfusion-weighted mean of the partial pressure from the alveolar and
shunt compartments, given by PA and
P, respectively. We see from Eqs. 13 and 14 that, over the course of each breath, PA is
a function of time and, therefore, so is the retention. We now define
the retention to be a function of time as well as solubility,
R(, t), given by
|
(15)
|
where S is the blood flow that
bypasses the lung alveolar compartment and
T is the total blood flow
(T = P + S).
In practice, the retentions are measured from a blood sample withdrawn
steadily over a period of time. If this sample is taken in the interval
0 < t < 1, then the retention
is given by
|
(16)
|
and we note the dependency of the retention on
0 and 1.
The excretion [E()] is calculated from the mixed expired gases and
so is calculated only once on each breath and is not a function of time
once the pseudo-steady state has been reached. The mixed expired gases
will contain a volume VD of gas that was left in the dead
space at the end of inspiration and so will contain no tracer gas,
together with a volume VT 
VD of alveolar
gas expired between times t = tj + TI and t = tj + TI + TED. We may therefore
write
|
(17)
|
where VA(t) is defined in Eq. 2.
Much previous work on MIGET has used only the retentions in the
recovery process (1, 5, 10). As a result, we consider the retentions in
much more detail in the remainder of the study.
We may now generalize the calculation of retentions and excretions to
continuous / and
A/VA distributions. For any
values of x and y, Eqs. 7 and 11 give
values of VT, VA, and .
Equations 13 and 14 together with the initial condition
P(x, y, 0) = 0 may then be solved in the same way as
for the one-compartment model to give a pseudo-steady-state solution.
The retentions as a function of t are now given by the double
integral
|
(18)
|
and the time-averaged retention over the time interval
0 < t < 1 is
|
(19)
|
The equation corresponding to Eq. 17 for the
excretion is
|
(20)
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Parameter Recovery
To evaluate the validity of the steady-state model, retentions will be
generated using the tidal breathing model, which we will term the
"given data." These data will then be substituted into the
parameter recovery routines for a 1- and a 50-compartment model, and
the parameters recovered will be compared with the parameters used to
generate the data. Changing to the more complex tidal breathing model
will affect only the given retentions. Because we use an identical
recovery procedure, the effect of experimental error will be identical
to those described by Ratner and Wagner (8). We do not therefore
consider the effect of experimental error.
One-compartment model.
We use the classical one-alveolar-compartment model together with a
dead space and shunt compartment, derived by using one alveolar
compartment in the model used by Wagner et al. (11). Given shunt
fraction
(S/T) and
/ for the alveolar compartments, we
may calculate the retentions
(Rci) for a given set of gases.
Given a set of retentions (Ri), we allow
S/T and
/ to vary and minimize
subject to
S/T and
/ being nonnegative.
Fifty-compartment model.
The recovery routine used for the 50-compartment model was identical to
that used by Evans and Wagner (1).
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RESULTS |
Variation of Retentions Within Each Breath
As previously pointed out, R(, t) varies over the course
of each breath, and a typical variation in R is illustrated in Fig. 1 for a single breath lasting 5 s
(inspiration of 1.5 s and expiration of 3.5 s). R(, t) is
calculated using the tidal breathing model with continuous
/ and
A/VA distributions by using
Eqs. 13, 14, and 18. VT was 700 ml,
VD was 150 ml, respiratory rate was 12 breaths/min, and
shunt was zero. The values of x and y, defined in Eq. 8, were both 0.5, and
the means of these distributions were chosen so that end-expiratory
VA was 2,380 ml and P was
5.95 l/min. R(, t) is plotted for each of the six values
of typically used in practice, as shown in Table 1. The retention of enflurane
( = 2.34) over the course of five individual breaths is shown with
an expanded y-axis in Fig. 2. There
are three distinct sections to the variation in retention over the time
course of each breath. First, the gas left in the dead space at the end
of the previous expiration is inspired: the partial pressure of the gas
under consideration in this dead space gas is very similar to that of
the gas inside the alveolar compartment, and so the partial pressure
inside the alveolar compartment is largely unchanged. In the next
section, gas inspired externally enters the alveolar compartment. This
gas contains no tracer gas, and so the partial pressure of the tracer
gas inside the alveolar compartment decreases through dilution.
Expiration is the third section. During this phase, the alveolar
partial pressure rises to the value at the start of inspiration.
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Fig. 1.
Variation in retentions in time over course of 1 breath. Lines
correspond to gases in Table 1: acetone (A), ether
(B), enflurane (C), cyclopropane (D),
ethane (E), and sulfur hexafluoride (F).
Inspiratory time is 1.5 s; expiratory time is 3.5 s, and respiratory
rate is, therefore, 12 breaths/min. Flat part of retention plot at
beginning of inspiration is due to dead space that is inspired before
fresh gas reaches alveoli.
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Fig. 2.
Magnification for data in Fig. 1, line C, showing variation in
retention of enflurane as a function of time over course of 5 breaths.
Trough in retention corresponds to end of inspiration; and peak
corresponds to time after start of inspiration when dead space gas
(from preceding breath) has been inspired.
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In Fig. 1, over the course of a breath, some of the retentions vary by
~20%. We would therefore expect any parameter recovery process to be
dependent on the exact time within the breath that the arterial blood
is sampled. We consider this further when we substitute the simulated
data into the parameter recovery procedure.
For the 50-compartment model, gases with low solubilities (and,
therefore, low retentions) give the most information about compartments
with low / ratios; gases with high
retentions give information about compartments with high
/ ratios. The second- and
third-highest retentions vary the most in Fig. 1. By recovering
distributions with use of the 50-compartment model (incorporating the
continuous ventilation assumption) by using retentions taken from
different stages of the breath, we would therefore expect compartments
with a high / ratio to be
altered most.
Effect of A/VA on
Retentions
As we stated earlier, the conservation of mass equations, Eqs.
13 and 14, are dependent on lung volume and
A/VA distribution as
well as the / distribution. Here we
demonstrate this numerically. First, we use a one-compartment model to
show the effect of alveolar volume on the retentions. Then we use
continuous / and
A/VA distributions to show
the effect of width of the distribution on retentions.
Effect of alveolar volume on retentions.
We consider six sets of retentions, all calculated using the tidal
breathing model, Eq. 18. These retentions are generated using
the solubilities of the gases typically used, as described by Kapitan
and Wagner (5), shown in Table 1. The first three sets are generated
using a lung with a unimodal distribution of ventilation, volume, and
perfusion. In the notation of Eq. 8, = ln 3, = ln 6, and
x = y = 0.5. VT was 500 ml, VD was 150 ml, shunt was 0, respiratory rate was 12 breaths/min, and breath
inspiratory-to-expiratory ratio was 1.5:3.5. This gave
A of 1,700 ml. By measuring the retentions at the point at which they are lowest for each gas in Fig.
1, we obtain one set of retentions (set R1,L), and
similarly by taking the highest retentions we obtain another set of
retentions (set R1,H). We may also time average the
retentions over the course of a whole breath to give us a third set,
1. Three more sets, R2,L, R2,H, and
2, are generated using the same
procedure, but this time = ln 6, giving A of 3,400 ml but
otherwise identical lung parameters and respiratory rate. These six
sets of retentions are shown in Table 2.
The retentions used in practice are time averaged because of the
physical necessity of taking a blood sample. Even so, the alveolar
volume at the end of expiration still makes a difference. There are
errors of >5% between 1 and
2. This error is larger than the
maximum experimental error expected in measurement of gas
concentrations in the blood (10), and so any model that does not take
into account A suffers from
modeling errors.
We plot the given distributions and the distributions recovered using
the 50-compartment model from the sets of retentions R1,H
and R2,H in Fig. 3A,
the distributions recovered from the sets of retentions
R1,L and R2,L in Fig. 3B, and the
distributions recovered from the sets of retentions
1 and
2 in Fig. 3C. When we
use the highest or lowest retentions over the course of a breath, the
distributions depend on alveolar volume. However, when the time-averaged retentions are used, as would be the case in practice, the distributions do not differ very much.
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Fig. 3.
Given ventilation-perfusion distribution (solid line) and distributions
recovered using 50-compartment model with use of retentions generated
with end-expiratory alveolar volume of 1,700 ml (dashed line) and
end-expiratory alveolar volume of 3,400 ml (dotted line). A:
distributions recovered when highest retentions are used; B:
distributions recovered when lowest retentions are used; C:
distributions recovered when time-averaged retentions are used.
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Effect of the width of the continuous distribution of volume,
ventilation, and perfusion.
We now consider the effect of the width of the
A/VA distribution on the
generated retentions. As with the one-compartment model, we generate
six sets of retentions using Eqs. 13, 14, and 18. The
first three sets, R1,L, R1,H, and
1, are generated using a narrow
A/VA distribution with
x = 0.0625. These retentions are the lowest,
highest, and time-averaged values of retention over the course of each
breath, respectively. The second three sets, R2,L,
R2,H, and 2, come
from a broad A/VA
distribution with x = 0.5. The retentions are
again the highest, lowest, and time-averaged retentions seen over the
course of a breath. Other lung parameters used to generate these
retentions were VT = 500 ml, VD = 150 ml,
= 4.0, = 6.0, and
S = 0; Respiratory rate was 12 breaths/min. The retentions are shown in Table
3.
For the retentions shown in Table 3, we see that, by comparing the
lowest retentions (sets R1,L and R2,L), the
highest retentions (sets R1,H and R2,H), and
the time-averaged retentions (sets
1 and
2), the width of the
A/VA distribution,
when varied from a very narrow to a very broad distribution, makes very
little difference in comparison, for example, to the effect of alveolar volume shown in Table 2. Therefore, the width of the
A/VA distribution is not a major factor in the retentions measured in practice.
Parameter Recovery
Here we investigate the effects of using tidally generated retentions
with the previously described parameter recovery methods. We generated
two sets of theoretical data using the continuous A/VA and
/ distributions described earlier
(Eqs. 13, 14, and 18. Both had VT = 700 ml,
x = y = 0.5, and = ln 3. This gave VA = 2,380 ml. The first case considered had
= ln 8 to give
P = 5.95 l/s and
S = 0. The second case had identical
parameters, except = ln 10 to
give P = 4.76 l/s, together with a 15%
shunt, S = 0.84 l/s. In each case, the
retentions over the course of a series of whole breaths were recorded.
We begin by considering the highest and lowest retentions over the
course of a breath. We then consider two time-averaged cases. First,
using Eq. 19, we time average the retentions over the course of
a whole breath. Second, we time average over an interval containing two
troughs of retentions (Fig. 1). Suppose a breath starts at time
tj. Then, using Eq. 19, we take a time
average of retentions over the interval tj + 0 < t < tj + 1, where 0 = 0.518 s and 1 = 7.317 s. This gives a time interval of 6-7 s, a feasible interval in which to take a blood sample. We justify taking this particular interval later.
One-compartment model.
Although the retentions were generated using a unimodal distribution,
because of the large fluctuations in retentions over the course of a
breath, as seen in Fig. 1, we do not assume that the recovered nontidal
model will be unimodal. Instead, we used a two-compartment model
recovery process. However, for all the retentions considered, the
recovered distribution had one compartment with zero perfusion, and so
the recovered distribution was unimodal, and an estimate for shunt
fraction and / ratio of the
ventilated compartment was given. These parameters, recovered using the
highest retentions, the lowest retentions, and the retentions time
averaged over a whole breath, are shown in Table
4.
The shunt fractions recovered in Table 4 are of acceptable clinical
accuracy. Shunt is mainly determined by the least soluble gas. We saw
in Fig. 1 that the retention of this gas
( = 5.99 × 103) hardly varied at all over the
course of each breath, and so the estimation of shunt is relatively
unaffected by the tidal nature of breathing.
The values of / ratios recovered
vary depending on when the arterial blood is sampled during the
respiratory cycle. If the lowest retentions are used, then in this
(standard) case, there is a 77% higher estimation of
/ ratio than would be estimated
using the highest retentions. Even in practice, where the time-averaged
retentions would be used, the value of the
/ ratio recovered is inaccurate.
Fifty-compartment models.
EFFECT OF ASSUMING CONTINUOUS VENTILATION IN THE PARAMETER RECOVERY
ROUTINE.
The retentions described above were given to the recovery process for
the 50-compartment model. The distributions recovered using the highest
and lowest retentions over the course of a breath are shown in Fig.
4A for zero shunt and Fig.
4B for 15% shunt. Also shown is the given
/ distribution. This allows us to consider the effect of substituting tidally generated data into a
parameter recovery routine that assumes continuous ventilation. The
recovered distributions in each plot are quite different. The highest
retentions give a narrow, unimodal distribution that is narrower and
higher than the given distribution and has a peak that is at a lower
/ ratio than the given
distribution; the lowest retentions give a bimodal distribution. This
bimodal distribution has one mode of approximately the correct width in
the same position as the unimodal distribution seen for the highest
retentions, together with a spurious, broad distribution of
compartments with high / ratios. As
a result, we become wary of any set of retentions that are time
averaged in such a way as to be weighted toward the troughs seen in
Fig. 1. This is why we have chosen our 0 and
1 to take the values chosen. The retentions are now
averaged over two troughs of retentions and the portion of the breath
between the troughs.
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Fig. 4.
Ventilation-perfusion distributions generated using retentions at
different stages of a single breath. A: data generated with no
shunt; B: data generated with 15% shunt. Solid lines, given
distribution; dashed lines, distribution recovered when highest
retentions measured are used; dotted lines, distribution recovered when
lowest retentions measured are used (see Fig. 1 for a whole breath).
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EFFECT OF DIFFERENT SAMPLING INTERVALS.
The distributions recovered using the time-averaged retentions are
shown in Fig. 5A for zero shunt and
Fig. 5B for 15% shunt. The given distributions, the
distributions recovered when the retentions are averaged over the
course of a whole breath, and the distributions recovered when the
retentions are averaged over the time interval
tj + 0 < t < tj + 1 described above are
shown. This allows us to investigate the effect of different experimental sampling intervals on the recovered distributions. Both
ways of time averaging the retentions lead to distributions skewed to
the right and a mode with a peak to the left of the peak of the given
distribution. However, the retentions that are time averaged over the
interval tj + 0 < t < tj + 1 give a
more highly skewed distribution that includes compartments with higher
/ ratios. Withdrawing an arterial
blood sample over a period of time that is an integer multiple of the
breath length will ensure that the retentions are representative of the whole breathing cycle and that there is no weighting of these retentions toward the troughs of the retentions shown in Fig. 1.
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Fig. 5.
Ventilation-perfusion distributions generated using retentions averaged
over different time intervals. A: data generated with no shunt;
B: data generated with 15% shunt. Solid lines, given
distribution; dashed lines, distribution recovered when retentions are
averaged over a whole breath (i.e., arterial blood samples are drawn
over a whole breath time interval, 5 s in this example); dotted lines,
distribution recovered when retentions are measured over a portion of 2 breaths (see RESULTS for details).
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DISCUSSION |
We have demonstrated using a mathematical model that the intrabreath
fluctuations in retention, caused by the nature of tidal breathing, are
significant in the parameter recovery process and that the simpler
continuous ventilation models previously used (2, 11) must be used with
care. Although the tidal breathing model is a more physically realistic
model than the continuous ventilation model, it must be remembered that
we have assumed a uniform filling and emptying of all compartments when
deriving the conservation of mass equations. It is unlikely that this
would be true in a patient with pulmonary dysfunction.
We generated retentions using a standard lung with unimodal
/ and
A/VA distributions. When a
tidal breathing model is used, it was shown that the retentions vary
with time. There is a variation of ~20% in some cases considered
here. We would normally expect that experimental data would be correct
to within 3% of the mean concentration of each gas (10), and so the
variations in the inert gas retentions within each breath should warn
us to expect differences in recovered parameters, depending on the time
at which the arterial blood is sampled during the respiratory cycle.
We have shown that we can indeed recover widely varying parameters when
using retentions at different points of the breath. Most notable is the
recovery of a bimodal distribution by the 50-compartment lung model
when the given distribution is truly unimodal when the lowest set of
retentions over the course of a breath is used. However, this problem
can be avoided to a large extent by the physical necessity of taking
time-averaged blood samples, and we have shown that the blood sample
used to measure the retentions should be taken over the course of a
series of whole breaths to avoid weighting toward the troughs in
retentions seen in Fig. 1. Even so, in the cases that we have
considered, this resulted in a distribution skewed to the right and
with the peak shifted to the left of the given distribution. We have
also shown that the one-compartment model, although it correctly
recovered a unimodal distribution, recovers parameters varying by up to 80%, so this model is not a practical alternative to the
50-compartment model.
Our final aim in this study was to investigate the effect of
A/VA distribution on the
retentions generated. We used log-normal functions of the
A/VA ratio and showed that
the width of the distribution had a negligible effect on the
retentions. However, the VAT could change the
retentions by an amount greater than that expected due to experimental
error. A higher alveolar volume had the effect of moving the
distribution toward higher / ratios.
We conclude by stating that if MIGET is to be used, then the retentions
should be time averaged over the course of a series of whole breaths.
This will minimize the effects of tidal breathing on the experimental data.
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ACKNOWLEDGEMENTS |
The authors acknowledge the financial support of The Wellcome Trust
through a Biomathematical Scholarship for J. P. Whiteley (Grant
041383/Z/94) and a Biomathematical Research Training Fellowship for D. J. Gavaghan (Grant 038286/Z/93/A) and the Medical Research Council of
the United Kingdom for a Career Development Fellowship for D. J. Gavaghan, which has allowed this research to be undertaken.
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FOOTNOTES |
Address for reprint requests and other correspondence: C. E. W. Hahn,
Nuffield Dept. of Anaesthetics, University of Oxford, Radcliffe
Infirmary, Oxford OX2 6HE, UK.
Received 23 September 1997; accepted in final form 18
February 1999.
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ABSTRACT
INTRODUCTION
