Abstract
The tidal breathing lung model described for the sinewave technique (D. J. Gavaghan and C. E. W. Hahn. Respir. Physiol. 106: 209–221, 1996) is generalized to continuous ventilationperfusion and ventilationvolume 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 ventilationvolume distribution. Simulated experimental data with a unimodal ventilationperfusion 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.
 ventilationperfusion
 ventilationvolume
 gas exchange
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 = λP, where λ 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 (λ = 5.99 × 10^{−3}); a typical choice of the most soluble gas is acetone (λ = 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
Previous work on MIGET has made little reference to the effects of tidal breathing. In particular, we demonstrate that tidal breathing causes variations in Pa̅ 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 ventilationvolume (V˙a/Va) distribution as well as the V˙/Q˙ 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 sinewave 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 timevarying retentions and that theV˙/Q˙ 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 onecompartment tidal model may be recovered as a bimodalV˙/Q˙ distribution for a recovery routine based on a continuous ventilation model. Finally, we investigate the dependency of the retentions on theV˙a/Va, distribution. It is shown that the width of theV˙a/Va distribution is only of marginal significance but that the dependency on the total alveolar volume (Va _{T}) may not be neglected.
For clarity, we consider only unimodalV˙/Q˙ distributions, inasmuch as the effects of the timevarying retentions can be demonstrated fully using these distributions only.
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
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 continuousV˙a/Va andV˙/Q˙ distributions in Continuous distributions of ventilation, perfusion, and volume. The Va of these compartments [Va
_{i}(t)] on an arbitrarybreath j is given by
The total lung alveolar volume as a function of time [Va(t)] and total tidal volume (Vt) are related by
Let the Va
_{T} beV̅
a + Vd at time Ti
_{D} after the beginning of inspiration, where Vd is the airway series dead space volume. Then, during the time interval t
_{j} < t< t
_{j} + Ti
_{D}, 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 = t
_{j}+ Ti
_{D}
In a similar manner, let the total alveolar volume beV̅
a + Vd at Te
_{D} 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 Te
_{D} by using an argument similar to that used to derive Ti
_{D} to give
Continuous distributions of ventilation, perfusion, and volume.
The underlying V˙a/Va andV˙/Q˙ distributions are often assumed to be lognormal distributions (6, 9). Assuming this to be the case, we now derive an expression for Pd for continuousV˙/Q˙ andV˙a/Va distributions. Later we will use these continuous distributions to generate theoretical data and compare the competing models. Defining
Ti
_{D} and Te
_{D} are calculated in a manner analogous to that described above. We note that the Va
_{T} at time t is given for inspiration by
We may now define Pd for continuousV˙/Q˙ andV˙a/Va distributions to be the triple integral
Retentions and Excretions
Here we demonstrate how to calculate the retentions and excretions. We begin by considering the classical threecompartment 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
The retention is the ratio of Pa̅ to Pv̅. Pa̅ is the perfusionweighted mean of the partial pressure from the alveolar and shunt compartments, given by Pa and Pv̅, 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(λ, t), given by
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 τ_{0} < t < τ_{1}, then the retention is given by
The excretion [E(λ)] 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 pseudosteady 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 = t
_{j} + Ti and t = t
_{j} + Ti + Te
_{D}. We may therefore write
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 V˙/Q˙ andV˙a/Va distributions. For any values of x and y, Eqs. 7
and
11
give values of Vt, Va, and Q˙.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 onecompartment model to give a pseudosteadystate solution. The retentions as a function of t are now given by the double integral
Parameter Recovery
To evaluate the validity of the steadystate 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 50compartment 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.
Onecompartment model.
We use the classical onealveolarcompartment 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 (Q˙s/Q˙t) andV˙/Q˙ for the alveolar compartments, we may calculate the retentions (
Fiftycompartment model.
The recovery routine used for the 50compartment model was identical to that used by Evans and Wagner (1).
RESULTS
Variation of Retentions Within Each Breath
As previously pointed out, R(λ, 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(λ, t) is calculated using the tidal breathing model with continuousV˙/Q˙ andV˙a/Va distributions by usingEqs. 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 ς_{x} and ς_{y}, defined in Eq. 8, were both 0.5, and the means of these distributions were chosen so that endexpiratory Va was 2,380 ml and Q˙p was 5.95 l/min. R(λ, t) is plotted for each of the six values of λ typically used in practice, as shown in Table1. The retention of enflurane (λ = 2.34) over the course of five individual breaths is shown with an expanded yaxis 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.
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 50compartment model, gases with low solubilities (and, therefore, low retentions) give the most information about compartments with low V˙/Q˙ ratios; gases with high retentions give information about compartments with highV˙/Q˙ ratios. The second and thirdhighest retentions vary the most in Fig. 1. By recovering distributions with use of the 50compartment model (incorporating the continuous ventilation assumption) by using retentions taken from different stages of the breath, we would therefore expect compartments with a high V˙/Q˙ ratio to be altered most.
Effect of V˙a/Va on Retentions
As we stated earlier, the conservation of mass equations, Eqs.13 and 14, are dependent on lung volume andV˙a/Va distribution as well as the V˙/Q˙ distribution. Here we demonstrate this numerically. First, we use a onecompartment model to show the effect of alveolar volume on the retentions. Then we use continuous V˙/Q˙ andV˙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,
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 R̅ _{1} andR̅ _{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 V̅ a suffers from modeling errors.
We plot the given distributions and the distributions recovered using the 50compartment model from the sets of retentions R_{1,H}and R_{2,H} in Fig. 3 A,the distributions recovered from the sets of retentions R_{1,L} and R_{2,L} in Fig. 3 B, and the distributions recovered from the sets of retentionsR̅ _{1} andR̅ _{2} in Fig. 3 C. When we use the highest or lowest retentions over the course of a breath, the distributions depend on alveolar volume. However, when the timeaveraged retentions are used, as would be the case in practice, the distributions do not differ very much.
Effect of the width of the continuous distribution of volume, ventilation, and perfusion.
We now consider the effect of the width of theV˙a/Va distribution on the generated retentions. As with the onecompartment model, we generate six sets of retentions using Eqs. 13, 14, and
18. The first three sets, R_{1,L}, R_{1,H}, andR̅
_{1}, are generated using a narrowV˙a/Va distribution with ς_{x} = 0.0625. These retentions are the lowest, highest, and timeaveraged values of retention over the course of each breath, respectively. The second three sets, R_{2,L}, R_{2,H}, and R̅
_{2}, come from a broad V˙a/Vadistribution with ς_{x} = 0.5. The retentions are again the highest, lowest, and timeaveraged retentions seen over the course of a breath. Other lung parameters used to generate these retentions were Vt = 500 ml, Vd = 150 ml,
For the retentions shown in Table 3, we see that, by comparing the lowest retentions (sets R_{1,L} and R_{2,L}), the highest retentions (sets R_{1,H} and R_{2,H}), and the timeaveraged retentions (setsR̅ _{1} andR̅ _{2}), the width of theV˙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 theV˙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 continuousV˙a/Va andV˙/Q˙ distributions described earlier (Eqs. 13, 14, and
18. Both had Vt = 700 ml, ς_{x} = ς_{y} = 0.5, and
Onecompartment 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 twocompartment 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 V˙/Q˙ 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 Table4.
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 (λ = 5.99 × 10^{−3}) 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 V˙/Q˙ 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 ofV˙/Q˙ ratio than would be estimated using the highest retentions. Even in practice, where the timeaveraged retentions would be used, the value of theV˙/Q˙ ratio recovered is inaccurate.
Fiftycompartment models.
EFFECT OF ASSUMING CONTINUOUS VENTILATION IN THE PARAMETER RECOVERY ROUTINE.
The retentions described above were given to the recovery process for the 50compartment model. The distributions recovered using the highest and lowest retentions over the course of a breath are shown in Fig.4 A for zero shunt and Fig.4 B for 15% shunt. Also shown is the givenV˙/Q˙ 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 lowerV˙/Q˙ 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 V˙/Q˙ 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 τ_{0} and τ_{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.
EFFECT OF DIFFERENT SAMPLING INTERVALS.
The distributions recovered using the timeaveraged retentions are shown in Fig. 5 A for zero shunt and Fig. 5 B 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 intervalt _{j} + τ_{0} < t <t _{j} + τ_{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 t _{j} + τ_{0} <t < t _{j} + τ_{1} give a more highly skewed distribution that includes compartments with higherV˙/Q˙ 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.
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 unimodalV˙/Q˙ andV˙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 50compartment 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 timeaveraged 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 onecompartment 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 50compartment model.
Our final aim in this study was to investigate the effect ofV˙a/Va distribution on the retentions generated. We used lognormal functions of theV˙a/Va ratio and showed that the width of the distribution had a negligible effect on the retentions. However, the Va _{T} 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 V˙/Q˙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.
Acknowledgments
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.
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.
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