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Generalized estimation of the ventilatory distribution from the multiplebreath washout: a bench evaluation study
BioMedical Engineering OnLinevolume 17, Article number: 3 (2018)
Abstract
Background
The multiplebreath washout (MBW) is able to provide information about the distribution of ventilationtovolume (v/V) ratios in the lungs. However, the classical, allparallel model may return skewed results due to the mixing effect of a common dead space. The aim of this work is to examine whether a novel mathematical model and algorithm is able to estimate v/V of a physical model, and to compare its results with those of the classical model. The novel model takes into account a dead space in series with the parallel ventilated compartments, allows for variable tidal volume (V_{T}) and endexpiratory lung volume (EELV), and does not require a ideal step change of the inert gas concentration.
Methods
Two physical models with preset v/V units and a common series dead space (v_{d}) were built and mechanically ventilated. The models underwent MBW with N_{2} as inert gas, throughout which flow and N_{2} concentration signals were acquired. Distribution of v/V was estimated—via nonnegative least squares, with Tikhonov regularization—with the classical, allparallel model (with and without correction for nonideal inspiratory N_{2} step) and with the new, generalized model including breathbybreath v_{d} estimates given by the Fowler method (with and without constrained V_{T} and EELV).
Results
The v/V distributions estimated with constrained EELV and V_{T} by the generalized model were practically coincident with the actual v/V distribution for both physical models. The v/V distributions calculated with the classical model were shifted leftwards and broader as compared to the reference.
Conclusions
The proposed model and algorithm provided better estimates of v/V than the classical model, particularly with constrained V_{T} and EELV.
Background
The estimation of the pulmonary ventilationtovolume (v/V) distribution may provide clinically useful information on intrapulmonary gasmixing but is an underused byproduct of the endexpiratory lung volume (EELV) measurements during mechanical ventilation. The v/V can be calculated with the multiplebreath washout (MBW) test, especially using N_{2} as the inert and low solubility gas (MBN_{2}W). The classical method [1,2,3] models the lungs as a set of allparallel units, including a dead space, whose contributions to the total lung ventilation are the unknowns. This approach has some limitations. For instance, it disregards the effects of the series dead space (v_{d}), whose volume may be estimated via the Fowler’s method [4] throughout the washout; not only the EELV but also the tidal volume (V_{T}) must remain constant during the MBN_{2}W; the inspired fraction of tracer gas should decrease instantaneously to zero. Recently, we [5] proposed a generalized multicompartmental model for MBN_{2}W that includes a series dead space and copes with a nonideal step change in gas concentration, variable V_{T} during the maneuver, and changes in EELV, as long as no compartment is completely emptied. Computational simulations showed that this model, together with an algorithm to estimate its parameters from measurements taken at the airway opening during MBN_{2}W, usually retrieved more correct estimations of the v/V distribution than previous proposals [5]. Furthermore, the alternative to impose a priori constraints determined along the MBN_{2}W limits the set of the v/V parameters estimates. However, since this same novel model drove the simulated MBN_{2}W, the results could have favored the algorithm in some form. It is arguable, hence, that bench tests with wellknown physical models would allow for a better, less biased assessment of the effects of modelling the series dead space in the estimates of v/V distributions.
The present work intends to compare the v/V distributions estimated by both the classical and generalized approaches employing experimental data obtained from physical models, under the conditions (constant V_{T} and EELV) required by the assumptions of the classical model. Similar estimation procedures were used for both models, employing nonnegative least squares and Tikhonov regularization plus a weighting matrix. The generalized approach adds a constrained least squares solver with imposed EELV, V_{T} and v_{d}. The results previously obtained by us [5], with numerically simulated experimental noise, directed the choice of the weighting matrix.
Methods
Mathematical model of the MBN_{2}W
The generalized mathematical model of the MBN_{2}W is as follows. The respiratory system comprises N parallel compartments, all connected through a single duct whereby the gases are exchanged with the ambient air. Each compartment J, whose volume is Vol_{J}, is an ideal mixer characterized by the fraction γ of V_{T} that enters and leaves it at each cycle, and its specific ventilation (S(J) = γV_{ T }/Vol_{ J }), the sum of all compartmental volumes being equal to EELVv_{d}. A series dead space is incorporated, considering that a compartment inspires a mixture of fresh gas from the inspiratory circuit and the content of the common duct. This also allows the model to be driven by a nonideal step in inspiratory concentration of the tracer gas. Variable V_{T} is admitted by defining S(J) with respect to a reference V_{T}, and variable EELV is achieved by tracking the differences between inspired and expired volumes, returning the distribution corresponding to EELV at the onset of maneuver [5].
In the experimental setup, where V_{T} and EELV were constant, the endtidal N_{2} concentration (\(F_{{N_{2} }}^{et}\)) at the kth cycle is modeled by
with the compartmental concentrations given by
where \(\alpha\) is the dead space to tidal volume ratio (v_{d}/V_{T}).
The classical approach to model multiple compartment MBN_{2}W considers an ideal step change of the inspired tracer gas at the onset of washout with the dead space as an additional parallel compartment. Under these assumptions, Eq. 2 simplifies to
and the combined compartmental concentrations are fitted to the measured mean expiratory N_{2}, by adjusting the respective weights. For a single compartment with a series dead space, it can be demonstrated, by using Eqs. 2 and 3, that this classical parallel model estimates a compartment with ventilation (\(1  \alpha\)) shifted leftwards (lower specific ventilation) from the real compartment. The estimated specific ventilation (S′) depends on the actual specific ventilation (S′ = (1 − α) · S/(αS + 1)), causing larger differences for faster compartments. Accordingly, the estimated compartmental volume is equal to EELV.
In case of a nonideal step at onset of washout, a further shift depending on the ratio of inspired to expired concentrations occurs. To distinguish partially between this effect of a nonideal step and the presence of a series dead space, an alternative classical model was tested. This is modeled by Eq. 2 with α = 0.
Experimental setup
To test the effect of a series dead space in the washout maneuver under controlled conditions, two physical models were assembled: one with four compartments of equal γ and different Vol_{J} (4C); and one with a single compartment (1C). The 4C allowed to examine the recovery of location, and the spread/breadth of the distribution, while with 1C the classical model distribution shift could be analytically predicted. Both models were ventilated by an Evita XL (Draeger Medical, Lübeck, Germany) and N_{2}, O_{2} and CO_{2} concentrations were measured by a fast mass spectrometer (AMIS 2000, Innovision, Glamsbjerg, Denmark). Pressure and flow signals were acquired directly from the ventilator and with a proximal pneumotachograph plus a pressure transducer. In order to synchronize the signals of gas concentration and flow, an uncalibrated flow signal was recorded from a pneumotachograph connected to the mass spectrometer, and the mainstream capnometer from the ventilator was placed close to the gas sampling port. All data were recorded simultaneously with a program written in LabView (National Instruments, Austin, USA).
The ventilated compartments were 1L anesthetic bags (VBM Medizintechnik GmbH, Sulz am Neckar, Germany) with endexpiratory volume maintained by application of a positive endexpiratory pressure (PEEP). A supersyringe inflation determined that at PEEP of 10 cmH_{2}O the volume of the bag was 1 L. CO_{2} production was simulated by a constant low flow of this gas into the compartment with the smallest v/V ratio. CO_{2} flow was titrated to achieve endtidal concentration between 0.5 and 1% to reduce effects in expired volume.
The series dead space comprised an anatomical and an instrumental dead space. The anatomical dead space was represented by a resistive piece and standard connectors used in mechanical ventilation, such as 22to15 mm reductions and Ypieces. The instrumental dead space was the connector for sidestream gas sampling and the pneumotachograph of the mass spectrometer, the mainstream capnometer of the ventilator, the proximal pneumotachograph and pressure outlet, a 90° connector to the resistance, and an HME filter (BB25, Pall Medical, Port Washington, USA) (Fig. 1). The total dead space volume (v_{d}), calculated from the geometry, were of 92 mL for 1C and of 152 mL for 4C.
Tidal volume and endexpiratory volume of compartments were selected to match, as nearly as possible, specific compartments from a logarithmic distribution of N = 50 ventilationtovolume ratios ranging from 0.01 to 100. The 1C model was ventilated with V_{T} = 250 mL, representing a compartment with S = 0.25. The respiratory frequency was of 15 breaths/min. A total of 5 washouts were performed with a N_{2} step change from 50% to zero. The 4C model compartments had 1.00, 0.83, 0.69 and 0.57 L and were ventilated with V_{T} = 560 mL, or 140 mL per compartment (S = 0.14, 0.83, 0.69 and 0.57, respectively). Where applicable, the compartmental endexpiratory volumes available for gas washout were reduced by inserting closed, impermeable plastic containers (Profissimo Gefrierbeutel, Germany) into the anesthetic bags, filled with appropriate volumes of air. The respiratory frequency was of 12 breaths/min (7 tests), 10 breaths/min (3 tests) or 15 breaths/min (2 tests). A total of 12 washouts were performed. In 9 tests, the N_{2} step change was from 50% to zero and in 3 tests the step change was limited from 10% to zero. Experiments were performed in ATPD conditions, disabling the ventilator’s BTPS compensation.
Signal processing
Before the data analysis, gas concentrations and flow were synchronized with a twostep procedure. First, flow curves from the ventilator and the mass spectrometer were aligned by maximizing their crosscorrelation. Second, the delay from gas sampling was compensated breathbybreath using the crosscorrelation between the CO_{2} signals from the mass spectrometer and the ventilator mainstream sensor.
The synchronized signals were processed to estimate v_{d}, EELV and the γ values of the compartments. The v_{d} was calculated from CO_{2} and volume curves using Fowler’s method [4]. The EELV was estimated from inspired and expired N_{2} volume during the washout (from onset until a N_{2} concentration ≤ 1/40th of initial value) [6]. Analogously, the distributions were estimated using the same number of cycles. The parameters of the multiple compartment model were estimated with nonnegative least squares and Tikhonov regularization with a fixed gain (4 × 10^{−3} for 1C and 3.3 × 10^{−2} for 4C) and a weighting matrix proportional to the compartmental washout ratio [2]. The generalized model was also estimated with a constrained least squares solver, imposing the sum of compartmental volume equal to the EELVv_{d} and unitary total ventilation [5]. Overall resistance and elastance were calculated from pressure and flow signals to ensure similar mechanical behaviors of the compartments. Data were analyzed in MatLab (Mathworks, USA).
Results
The time profile of inspiratory N_{2} was not that of an ideal step and, as expected, the washout of 4C was slower than that of 1C (Fig. 2a). EELV was estimated, from the MBN_{2}W inspired and expired N_{2} volumes, as 1.13 + 0.01 L for 1C and 3.24 ± 0.07 L for 4C. Typical expiratory capnogram curves were observed, despite the difference in magnitude (Fig. 2b). The estimated v_{d} were 73.8 ± 6.4 mL for 1C and 185.7 ± 4.5 mL for 4C (see the Additional file 1: Tables S1 and S2, for individual estimates of each experiment). The calculated overall resistance and elastance were R = 16.6 ± 0.3 cmH_{2}O/L/s and E = 78.5 ± 1.2 cmH_{2}O/L for 1C and R = 16.1 ± 0.6 cmH_{2}O/L/s and E = 20.8 ± 0.3 cmH_{2}O/L for 4C.
The distribution retrieved by the constrained generalized model for physical model 1C was located at the correct compartment, with a small contribution of an adjacent compartment, and corresponded to the smallest sum of squared errors between estimates and the real distribution of v/V (Fig. 3a and Table 1). In the case without constraints, the sum of compartmental volumes plus v_{d} underestimated EELV by 3%, and the total ventilation was overestimated by 5% (Fig. 3b). The classical model retrieved two or three compartments, located, however, leftwards from the actual compartment, as theoretically predicted (Fig. 3c). EELV was overestimated by 24%, and v_{d} (complement of total ventilation) was underestimated by 10%. The inclusion of the inspired N_{2} concentration partially corrected EELV estimations (mean error of 18%), but the distribution almost did not change (Fig. 3d). The estimated distribution of each test with each model is shown in the Additional file 1: Figures S1–S4.
For large N_{2} step changes (9 washouts, corresponding to cases A to I), the results for the model 4C were analogous to those for 1C. The constrained generalized model estimated v/V matching the expected specific ventilation, although narrower (Fig. 4a); the unconstrained generalized model underestimated EELV and overestimated the total ventilation by 13%, causing a rightwardshifted and broadened estimated v/V (Fig. 4b). The distribution estimated with the classical model was broader than expected and shifted leftwards from the actual distribution (Fig. 4c, d); the EELV was overestimated by 15% and v_{d} was underestimated by 6%. Again, including the measured inspired N_{2} in the classical model partially corrected EELV estimation reducing the errors to 7%. For N_{2} step changes limited to 10% (3 washouts, Fig. 4, corresponding to the cases J to L), all estimated distributions with the generalized as well as the classical approach resulted broadened (Fig. 4) and with larger sums of squared errors relative to the real distribution (Table 1), indicating the deleterious effect of a decreased signaltonoise ratio on the estimates. All individual estimated distributions are shown in the Additional file 1: Figures S5–S8.
Discussion
We proposed a bench comparison between a novel generalized mathematical model for the MBN_{2}W [5] and a classical allparallel model [1]. The tests were performed with a commercial intensive care unit ventilator and physical models mimicking lungs with one or four parallel compartments and a common series dead space. The main results are: (1) the retrieved v/V distribution with the constrained generalized approach was practically coincidental with the actual v/V distribution for both physical models for high N_{2} step changes; the unconstrained solution did not represent the expected distributions, missing the true values of EELV and V_{T}; (2) the v/V distribution retrieved with the classical approach was leftward shifted and broader, as compared to the actual, and its corresponding estimates of EELV were slightly favored when the nonideal step change of N_{2} at the washout onset was taken into account.
We used estimates of respiratory mechanics to provide a first assessment of the reproducibility of the tests and of the assumption of equal ventilation to each of the compartments in 4C. The small spread shows that the physical properties of the models may be considered constant along the washout repetitions, while the fourfold decrease in elastance in 4C compared to 1C suggests that all four anesthetic bags have similar compliances and, consequently, similar ventilations.
The anatomy of the airways consists of a network of ramifications where a strictly common dead space is restricted only to the trachea [7]. The set of subdivisions from the main bronchi to the deeper bronchioles results, during the expiration, in a mixture of alveolar gases originated from their respective airways. Thus, assuming the totality of the anatomical dead space simply as a common series duct is a considerable simplification, even though, as reported by Fortune and Wagner [8], most of the dead space lies proximal to the carina. Nevertheless, the lungs, as represented by the classical model (alveoli connected to the airways opening and the airways as one additional parallel compartment), is less corresponding to the reality. In the present experiments, the physical models agreed very well to the proposed mathematical model, since most of the tubings comprise the common dead space.
Because of the lack of correspondence between the classical model and the actual anatomy, two features arise: the retrieved distribution is shifted to the left as previously reported [8] and broadened as compared to the expected. The specific ventilation of the estimated 1C compartment was close to the theoretically predicted specific ventilation (see Fig. 3c). The spread of the distribution is influenced by factors inherent to the model, such as the difference in sensitivity to the common dead space for slow and fast compartments and the mixing of the contents of the compartments, which decreases the differences between the compartmental washout curves. The distribution curve is also sensitive to choices in data processing, for example the regularization gain used for the estimation of the parameters. The present gains were chosen on the basis of previously simulated experiments [5]. This may be a critical parameter in what concerns the shape of the estimated curve of v/V distribution, particularly its breadth and smoothness. Nevertheless, a tradeoff between accuracy and sensitivity to noise and artifacts is expected, hence this choice should be subjected to further investigations.
The distribution recovering technique applied to the classical model is essentially unconstrained. The solution includes the estimates of EELV and the parallel dead space of the distribution. This dead space does not necessarily correspond to v_{d}, representing the ventilation of a compartment with an infinite specific ventilation [3]. Regarding the EELV estimates, they were always overestimated with the classical model. EELV alone has been increasingly regarded as a useful parameter to evaluate the overall lung aeration [9], and it may be straightforwardly calculated by the breathbybreath summation of the net N_{2} (or other inert gas) volumes expired during the washout.
For the generalized model of MBN_{2}W, the EELV that serves as input to the constrained least squares estimation was calculated as above. The EELV estimates resulted accurate for both physical models. Gas exchange calculations based on measurements of gas concentrations and flow rate are very sensitive to the correction of the time delay between these signals [10]. A mainstream capnometer, currently a usual instrument in mechanical ventilation, was used as the time reference to synchronize the mass spectrometer measurements with the flow rate. This time correction, using just the maximal crosscorrelation between the CO_{2} concentration signals from the capnometer and the mass spectrometer, revealed feasible and reliable (EELV error < 5% and variability between repetitions < 10% [6]). Alternatively, an ultrasound flowmeter monitoring the washout of sulfur hexafluoride (SF_{6}), an inert and insoluble gas with a high molecular mass compared to the ambient air components, may be used. This device allows simultaneous and synchronous measures of flow rate and SF_{6} concentration and has been used for the estimation of ventilatory inhomogeneity [11, 12].
Breathbybreath estimates of the series dead space is a requirement for both the constrained and the unconstrained generalized v/V distribution. Instead of using prediction formulae, a direct measurement of that dead space is recommended, for example by applying Fowler’s technique [4] to the capnogram [5] as in the present work. Prediction formulae are scarce and inaccurate, especially for some conditions such as during mechanical ventilation, in which body position varies and EELV depends on the applied PEEP. For instance, there are conflicting reports as to the effect of the dead space on a vastly employed index to quantify ventilatory inhomogeneity, the lung clearance index (LCI). Despite Haidopoulou et al. [11] concluded that LCI is minimally affected by airway dead space, Neumann et al. [12] found an association between LCI and v_{d}/V_{T}. The LCI is an overall index of ventilatory inhomogeneity; in theory the increase of v_{d}/V_{T} should increase the magnitude of LCI. As an alternative, the alveolar lung clearance index (aLCI) [11] was proposed by considering the alveolar ventilation instead of the total ventilation as the bulk flow washing the alveolar units. The present generalized approach is based on the same assumption. Notably, an error in v_{d} estimation will result in a shifted distribution [5], as demonstrated with the extreme case of the classical model. Likewise, if v_{d} is overestimated the shift will be to the right (Additional file 1: Figure S9) due to slower washout (increased rebreathing) for each modeled v/V.
The v/V distribution of the respiratory system may be modeled by a continuous curve within a finite interval. The recovery of this distribution from the limited information present in a MBN_{2}W is an illposed problem and requires simplifying assumptions. The three assumptions relevant to the estimation method are smoothness, known bounds and discrete representativity. The first assumption was discussed above. The bounds used here are the same from [1, 13], and clearly will lead to wrong estimates if they don’t encompass all the v/V ratios of the real compartments. The a priori choice of 50 compartments is usual in the literature [1, 3, 13, 14]. In this study we tried to match every physical compartment to values present in the chosen 50 v/V ratios, favoring estimation: mismatch(es) between the physical v/V ratios and the set of chosen v/V ratios in the mathematical model will, in general, cause the true ratio to be represented by a combination of modeled compartments. This should affect mainly the amplitude and breadth of the distribution, and less its location. An example of the effects of such mismatch can be seen in Additional file 1: Figure S10.
Some limitations are addressed hereupon. To our best knowledge, this is the first report on multiplebreath washout of a multicompartmental physical model. Hence, we could not discuss our results against the literature as to possible comparative improvements. The physical models were limited to up to 4 units and this is far from the number of units found in experimental works with humans [3, 14,15,16]. Considering that the v/V is distributed on a log scale, a simulation with many more units would be difficult to perform in view of the present method of construction of v/V units. For the estimation of v/V distribution we used the same cycles selected for calculating the EELV [6]. For our combination of V_{T} and compartments’ volumes, this choice lead to a larger number of cycles than the commonly used of 17 [1, 2], which could have favored our results. Numerical simulations showed that, for the generalized model, both choices of cycles have similar estimations, although 17 cycles respected more the number of modes [5]. In the Additional file 1: Figures S1–S8, we show that this equivalence holds true in our experimental condition, including for the classical model. Lastly, one of the features of the generalized approach to estimate v/V distributions is that V_{T} and EELV are not necessarily constrained to be constant, as in the classical method. The present results did not include tests with variable ventilation [17] feasible at the laboratory since commercial mechanical ventilators currently feature this choice of strategy.
Conclusions
In conclusion, the present work compared the v/V distributions estimated by both the classical and generalized approaches employing experimental data obtained with in vitro models. The method that resulted in better coincidence with the actual distribution was the generalized approach with a constrained least squares solver with imposed EELV and V_{T}.
Abbreviations
 MBN_{2}W:

multiplebreath nitrogen washout
 EELV:

endexpiratory lung volume
 v_{d}:

series dead space
 v/V:

ventilationtovolume
 V_{T}:

tidal volume
 N:

number of modeled alveolar units
 Vol:

alveolar unit endexpiratory volume
 γ :

alveolar unit fraction of tidal volume
 J:

index of alveolar unit
 S :

specific ventilation
 k :

index of breath cycle
 \(F_{{N_{2} }}^{A}\) :

alveolar unit concentration of N_{2}
 \(F_{{N_{2} }}^{et}\) :

endtidal N_{2} concentration
 \(F_{{N_{2} }}^{I,A}\) :

alveolar unit inspired N_{2} concentration
 \(F_{{N_{2} }}^{I}\) :

ventilator delivered N_{2} concentration
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Authors’ contributions
GCMR and AGN conceived the work, developed the physical models and performed the tests. GCMR, FCJ, HW and AGN drafted, revised the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank Alessandro Beda for developing the time delay correction software.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The results from the experiments of the current study are available from the corresponding author on request.
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Funding
AGN was founded by CAPES, Ministério da Educação do Brasil (Fellowship BEX10876/138), AGN, FCJ and GMR were founded by FAPERJ—Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro and CNPq—Brazilian Research Council. To cover publishing costs, we acknowledge support from the German Research Foundation (DFG) and Leipzig University within the program of Open Access Publishing. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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Additional file
Additional file 1. The individual estimates of specific ventilation distributions are shown for each combination of physical and mathematical model, also considering estimates with 17 breath cycles. All estimates of endexpiratory lung volume, total ventilation and dead space are tabulated, together with the reference values. Sensitivity to error in estimated v_{d} and to the number N of modeled compartments.
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Keywords
 Pulmonary function tests
 Ventilatory distributions
 Multiplebreath washout
 Endexpiratory lung volume
 Functional residual capacity
 Dead space
 Nitrogen
 Ventilation to volume
 Tikhonov regularization
 Common dead space