British Journal of Anaesthesia

BJA Advance Access originally published online on August 21, 2006
British Journal of Anaesthesia 2006 97(5):718-731; doi:10.1093/bja/ael216

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A tidally breathing model of ventilation, perfusion and volume in normal and diseased lungs

J. S. Yem¹, M. J. Turner^1,*, A. B. Baker¹, I. H. Young² and A. B. H. Crawford³

¹ Department of Anaesthetics, The University of Sydney, Royal Prince Alfred Hospital Missenden Road, Camperdown, NSW 2050, Australia
² Department of Respiratory Medicine, The University of Sydney, Royal Prince Alfred Hospital Missenden Road, Camperdown, NSW 2050, Australia
³ Department of Respiratory Medicine, Westmead Hospital Westmead, NSW 2145, Australia

^*Corresponding author: Department of Anaesthetics, University of Sydney, Royal Prince Alfred Hospital, Building 89 Level 4, Missenden Road, Camperdown, NSW 2050, Australia. E-mail: mjturner{at}usyd.edu.au

Accepted for publication May 19, 2006.

	Abstract

Top Abstract Introduction Materials and methods Results Discussion Conclusions Supplementary data Appendix References

 
Background. To simulate the short-term dynamics of soluble gas exchange (e.g. CO₂ rebreathing), model structure, ventilation–perfusion () and ventilation–volume () parameters must be selected correctly. Some diseases affect mainly the distribution while others affect both and distributions. Results from the multiple inert gas elimination technique (MIGET) and multiple breath nitrogen washout (MBNW) can be used to select and parameters, but no method exists for combining and parameters in a multicompartment lung model.

Methods. We define a tidally breathing lung model containing shunt and up to eight alveolar compartments. Quantitative and qualitative understanding of the diseases is used to reduce the number of model compartments to achieve a unique solution. The reduced model is fitted simultaneously to inert gas retentions calculated from published distributions and normalized MBNWs obtained from similar subjects. Normal lungs and representative cases of emphysema and embolism are studied.

Results. The normal, emphysematous and embolism models simplify to one, three and two alveolar compartments, respectively.

Conclusions. The models reproduce their respective MIGET and MBNW patient results well, and predict disease-specific steady-state and dynamic soluble and insoluble gas responses.

Keywords: modelling; ventilation/perfusion distribution; ventilation inhomogeneity

	Introduction

Top Abstract Introduction Materials and methods Results Discussion Conclusions Supplementary data Appendix References

 
Simulation of respiratory exchange of soluble gases in diseased lungs under dynamic conditions requires that the model structure and parameters associated with the distributions of both ventilation–perfusion () and ventilation–volume () ratios are selected correctly. For example, the simulation of cardiac output measurement by short respiratory manoeuvres such as CO₂ rebreathing,¹ which is increasingly used in anaesthesia and intensive care for measurement and monitoring of cardiac output,^{2 3} requires models that predict well short-term changes in the transfer and storage of such a soluble gas. Steady-state exchange of soluble gases in diseased lungs depends strongly on the distribution of ratios but is independent of alveolar volumes. Exchange of soluble gases during transients, however, depends on the distributions of both and ratios. Some diseases, e.g. pulmonary embolism, affect mainly the distribution while others, e.g. emphysema, affect both and distributions. To simulate the transport and storage of soluble gases during dynamic manoeuvres in subjects with both and heterogeneity, the parameters associated with and distributions should be selected in a rational manner. Parameters of simple models are often selected arbitrarily to produce outputs that match clinical observations qualitatively.⁴–⁸ In more complex models arbitrary selection of parameters may lead to invalid or extreme predictions, particularly during dynamic changes in ventilation or perfusion.

At present there is no systematic approach for selecting mutually consistent sets of parameters for respiratory models that incorporate both and heterogeneity. In this study, we describe procedures for selecting alveolar compartment ventilation, volume and perfusion parameters for a tidally breathing respiratory model, based on multiple inert gas elimination technique (MIGET) and multiple breath nitrogen washout (MBNW) measurements. These models are developed for the purposes of simulating the exchange of soluble and insoluble gases during dynamic respiratory manoeuvres such as full, or partial rebreathing which is now commonly used in anaesthesia and intensive care.

	Materials and methods

Top Abstract Introduction Materials and methods Results Discussion Conclusions Supplementary data Appendix References

 
Background
Ventilation–perfusion ratio heterogeneity
The MIGET^{9 10} has long been used for investigating the matching of ventilation and perfusion in the lungs. Small quantities of six inert gases are dissolved in saline and i.v. infused until the mixed venous blood content of each inert gas is constant or changing at a slow uniform rate. The distributions are constructed from measurements of the steady-state concentrations of the inert gases in mixed venous blood, mixed arterial blood and mixed expired gas. Arterial and mixed expired PO₂ and PCO₂ calculated using the derived distribution compare well with corresponding measured values.¹¹–¹³

The shapes of MIGET-derived distributions for many respiratory conditions are known, and in many cases distinct patterns can be associated with specific respiratory conditions.¹⁴–¹⁷ As the distribution is a steady-state property of the lungs, predictions made by a respiratory model derived only from MIGET measurements are likely to be incorrect in non-steady-state conditions.¹⁵

The distributions recovered by MIGET have been shown to contain a limited amount of information.^{9 11 13 18}–²⁰ While the lungs contain a great number of gas exchange units, the MIGET has been shown to be able to discriminate only three distinct modes, or two modes in addition to shunt, and dead-space.^{9 11} Hence, in general, a model based on MIGET measurements needs to contain only three different lung compartments in addition to shunt and dead-space.

Ventilation–volume ratio heterogeneity
The MBNW technique is commonly used for investigating indices of ventilation inhomogeneity.²¹–²³ The subject breathes air before the procedure. At the start of the MBNW, the inspired gas is switched to a mixture containing no nitrogen, and end-tidal nitrogen concentration is monitored over a washout period that is typically 7 min.

Numerous studies have shown that the information in a washout curve is sufficient to describe only two or at most three compartments.^{22 24 25} Lewis and colleagues²² described a technique for recovering a continuous distribution of ventilation from a MBNW. This technique uses a smoothed least-squares fitting procedure similar to that used in the MIGET to recover distributions of ratios.¹³ Both normal and more complex distributions recovered from nitrogen washouts were shown to be reproducible within an individual.²² Therefore, significant changes in the shape of the distribution can be attributed to changes to the subject's lungs.²² Wagner²¹ examined the variability among compatible ventilation distributions, and found that in general, the achievable resolution depends on the specific underlying distribution, and physiologically significant features of the distribution can usually be specified, although the more complex a distribution is, the less resolution is possible. Other studies to assess the effects of experimental error on the resolution of the MBNW confirm that the information present in a MBNW is insufficient to allow confident resolution of more than two ventilation modes and an estimate of dead-space.^{25 26} Thus, a model based on MBNW measurements may contain lung compartments with only two different ratios.

Ventilation, perfusion and volume heterogeneity
We propose a subset of the three dimensional alveolar structure suggested by Whiteley and colleagues,²⁷ in a tidally breathing model, to simulate simultaneous ventilation, perfusion and volume heterogeneity. The eight alveolar compartments and shunt (Fig. 1) allow this model to exhibit steady-state gas exchange behaviour consistent with any measured distribution and dynamic characteristics consistent with any measured N₂ washout.

View larger version (16K): [in this window] [in a new window] [Download PowerPoint slide]   Fig 1 Alveolar compartment structure containing pure shunt, two alveolar dead-space compartments and six ventilated and perfused alveolar compartments.

 
Alveolar volume and the spread of the distribution have been shown to have negligible effect on distributions recovered using the MIGET⁵ if the retention ratios are averaged over a complete respiratory cycle. Although nitrogen has a low solubility, we anticipate that the N₂ washout and hence the measured distribution may be affected by the distribution by three mechanisms. First, the rate at which N₂ dissolved in body tissues is eliminated in expired gas²² may be affected by ratios of low compartments. Second, changes in the exchange of O₂ and CO₂ associated with changes in the distribution may affect the washout of N₂ by the second gas effect. Third, in a tidally breathing model with series dead-space, mixing of expired gases in common dead-spaces alters the effective ratio and the effective ventilation of each inert gas in each compartment,²⁸ and hence may affect the elimination of N₂.

In this study it is necessary to identify model parameters that enable our tidally breathing model to display characteristics that are consistent with both a measured distribution and a measured N₂ washout. Hence, it is necessary to select both the and the parameters in a single optimization procedure, so the interaction between and distributions is taken into account. In general, each ventilated compartment of our proposed alveolar model (Fig. 1) receives a fraction of the alveolar ventilation, and the volume of each compartment is selected so that compartments in the same row have approximately equal ratios. Two ventilated but unperfused compartments form an optional parallel dead-space. The remaining six ventilated compartments each receive a fraction of the pulmonary blood flow chosen so that each column has approximately equal ratio.

The tidal model
We modified an existing tidal model of the cardio-respiratory system of a healthy 70 kg adult male.²⁹–³³ Provision was made for the addition of up to four alveolar compartments to facilitate the simulation of simultaneous and inhomogeneity resulting in up to eight alveolar compartments and shunt (Fig. 1). The model is otherwise identical to that of Yem and colleagues,²⁹ and may be used to simulate dynamic responses of soluble and insoluble gases in the presence of and heterogeneity. The model simulates artificially controlled tidal breathing (as in anaesthesia or intensive care with a constant inspiratory flow and passive exponential expiration) through a branched respiratory tree and incorporates the effects on CO₂ dynamics of lung tissue mass, vascular transport delays, multiple body compartments and realistic blood–gas dissociation curves.³⁴ Nitrogen storage in blood and body tissues is simulated. The model is implemented using Matlab and Simulink (Mathworks, Natick, MA, USA).

Parameter estimation
Measured MIGET and MBNW data obtained in subjects who are representative of adults with normal lungs, emphysema and pulmonary embolism were selected from the literature. All model parameters other than compartmental V_A, and were obtained from the respective articles from which individual or data were obtained (Table 1). Parameters that were not available were selected using the standard mean values.³⁵–³⁷

View this table: [in this window] [in a new window]   Table 1 Model parameters. *Determined from 50-compartment MIGET predicitions. from MIGET distributions. . from MIGET distributions. #(). **(FRC–VInstDS)

 
The numbers of significant modes in each distribution and nitrogen washout for the respective individuals were estimated based on published patterns of ^{12 38}–⁴⁰ and ventilation distributions.^{22 41 42} The number of columns in the basic alveolar model (Fig. 1) was then reduced to match the number of estimated modes (), and similarly the number of rows was reduced to match the estimated number of modes (). This reduction procedure is necessary to allow a unique solution to be found from the simultaneous equations (A2), and to conform to the maximum number of alveolar compartments allowed by the respective MIGET and MBNW manoeuvres.^{9 11 18}–^{20 22 24}–²⁵ In each column, compartments that are known to exist in each disease were retained and compartments known to contribute little to gas exchange were excluded. These decisions were made according to current qualitative knowledge of subjects with normal lungs,^{12 22} patients with emphysematous lungs^{11 14 38 40} and subjects with pulmonary embolism.^{41 42} The number of alveolar compartments and the airway structure of the tidal model was modified appropriately and the tidal model was fitted to both inert gas retention/excretion ratios calculated from the measured distributions, and the normalized nitrogen washout data (see Appendix). After the tidal model was fitted to the data, the effective ventilation of each compartment was determined using a method that allows for the effects of gas mixing in common dead-spaces (see Appendix).

The subjects from whom the MIGET and MBNW measurements were obtained were of different sizes and had different lung volumes, ventilatory frequencies and tidal volumes. Analysing MBNW as a function of dilution number and alveolar dilution number has been shown to be insensitive to ventilatory frequencies⁴³ and the ratios of V_AnatDS/FRC and V_T/FRC.⁴⁴

The completed normal, emphysema and embolism models were evaluated in four ways. First, steady-state and were compared with values measured in the subjects from whom the MIGET data were obtained. Second, Bohr–Enghoff physiological dead-space fractions determined from the models with and without corrections for shunt⁴⁵ were compared with the respective CO₂ dead-space fractions calculated directly from the published distributions. CO₂ dead-space was calculated by substituting arterial and mixed expired PCO₂ values predicted by the 50-compartment continuous flow model⁹ into the Bohr–Enghoff dead-space equation. The O₂ and CO₂ dissociation curves of Olszowka and Farhi³⁴ were used. The acetone dead-space was calculated as for each respective model where E_6Model is the elimination ratio of acetone.

Third, simulated nitrogen washouts were compared with the published washouts. For the purpose of comparing the washouts, the independent variables of the measured washouts were transformed from total lung dilution numbers to alveolar dilution numbers by multiplying by

Fourth, the simulated arterial PCO₂ responses to step changes in ventilatory frequency were assessed, by increasing or decreasing ventilatory frequency by a factor of 1.5 at the start of an inspiration after the models had reached steady state.

	Results

Top Abstract Introduction Materials and methods Results Discussion Conclusions Supplementary data Appendix References

 
Representative distributions measured in subjects with normal lungs, and emphysema and embolism were obtained from Wagner and colleagues,⁴⁶ Melot and colleagues⁴⁷ and D'Alonzo and colleagues,⁴⁸ respectively (Fig. 2A–C). Representative MBNW curves measured in subjects with normal lungs and emphysema were obtained from Saidel and colleagues.⁴³

View larger version (15K): [in this window] [in a new window] [Download PowerPoint slide]   Fig 2 (A) The digitized normal MIGET distribution46 (black lines, left ordinate), and the recovered single compartment ratio (grey, right ordinate). (B) The digitized emphysema MIGET distribution47 (black, left ordinate) and the recovered three-compartment distribution (grey, right ordinate). (C) The digitized pulmonary embolism MIGET distribution56 (black, left ordinate), and the recovered two-compartment ratios (grey, right ordinate). Closed circles, perfusion; open circles, ventilation.

 
Normal model
parameters
The distribution of normal subjects is known to be unimodal with negligible shunt, and total dead-space approximates anatomical dead-space.¹⁴ The measured MIGET distribution⁴⁶ clearly contains a single mode at 1 with negligible shunt (Fig. 2A). We assumed that the normal subject had negligible alveolar dead-space, and therefore associated the dead-space ventilation in the measured distribution entirely with the series anatomical and instrument dead-space in the model. Therefore, only one compartment was used to simulate the normal lung (Fig. 2A).

parameters
The ventilation distribution of normal subjects is unimodal,²² which corresponds with a single compartment lung model.

Ventilation, perfusion and volume parameters of the normal model
To simulate a normal subject the airway structure of the tidal model is reduced to series anatomical dead-space and a single alveolar compartment that receives all the cardiac output and all the alveolar ventilation. The recovered series anatomical dead-space is 0.263 litre.

Emphysema model
parameters
In emphysema, the distribution is typically bimodal (Fig. 2B).^{14 47} The lower mode in Fig. 2B is assumed to be associated with the normal part of the lung in which the perfusion distribution is similar to that seen in normal subjects of equivalent age, and the ratio is slightly reduced as a result of reduced ventilation.¹⁴ The higher mode is assumed to be the abnormal emphysematous part of the lung, which has a high ratio as a result of the reduced alveolar surface area.^{11 14 38} Therefore, the emphysema model has two distinct compartments in addition to shunt and dead-space (Figs 2B and 3).

View larger version (14K): [in this window] [in a new window] [Download PowerPoint slide]   Fig 3 Allocation of active alveolar compartments for the emphysema model. Ventilation, perfusion and volume of each alveolar compartment are indicated. Alveolar ventilation values are effective ventilation values calculated using acetone as the indicator gas.

 
parameters
The ventilation distribution in emphysema is typically bimodal,²² thus requiring a model with two ratios (Fig. 3).

Ventilation, perfusion and volume parameters of the emphysema model
The emphysema model requires two and two compartments and therefore is potentially a four-compartment model. We reduced the model to three alveolar compartments by using qualitative information—we assumed that the lower mode and the faster compartment represent the normal part of the lungs. The lower mode is thus contained entirely within the faster compartment. The slow compartment is assumed to represent the diseased part of the lungs and is therefore entirely associated with the high mode. The parallel dead-space ventilation is assumed to be negligible.³⁸

The resulting alveolar structure is shown in Figure 3 and the tidal implementation of the airway of the emphysema model is shown in Figure 4. The three active alveolar compartments are connected at one ternary branching point so that the effects of common dead-space are similar between compartments.²⁸ The model parameters and 95% CIs estimated by fitting the emphysema model to the and MBNW measurements are shown in Table 2. The recovered and parameters reflect the initial qualitative selection of active compartments. There are two high compartments and one low compartment. In addition, the low compartment and one of the high compartments have fast gas turnovers, and the second high compartment has a distinctly slower gas turnover. The 95% CIs of the parameters (Table 2) do not include zero or one, indicating that the parameters are unique.

View larger version (23K): [in this window] [in a new window] [Download PowerPoint slide]   Fig 4 Respiratory tree of the emphysema model. The respiratory tree contains three perfused alveolar compartments and shunt. The grey areas indicate alveolar compartments that have similar ratios. WG, Weibel generations.

 

View this table: [in this window] [in a new window]   Table 2 Parameters that enable the emphysema and embolism tidal models to best fit MIGET and MBNW measurements. CI, confidence interval. *The numerical subscripts 1, 2, 3 and 4 refer to the low, mid and high , and alveolar dead-space compartments, respectively. The alphabetical subscripts a and b refer to compartments with normal and high ratios, respectively. (i) , , ; (ii) ,

 
Embolism model
parameters
In pulmonary embolism the distribution is typically bimodal with significant overlap (Fig. 2C).^{12 39} Therefore, the embolism model has two compartments in addition to shunt and dead-space (Figs 2C and 5).

View larger version (13K): [in this window] [in a new window] [Download PowerPoint slide]   Fig 5 Allocation of active alveolar compartments for the embolism model. Ventilation, perfusion and volume of each alveolar compartment are indicated. Alveolar ventilation values are effective ventilation values calculated using acetone as the indicator gas.

 
parameters
Pulmonary embolism does not result in significant redistribution of ventilation.^{41 42 49} Therefore, a normal ventilation distribution (single compartment) is assumed for pulmonary embolism (Fig. 5).

Ventilation, perfusion and volume parameters of the model
The alveolar model is reduced to shunt, anatomical dead-space and two compartments. The resulting alveolar structure is shown in Figure 5. The airway structure for the tidal embolism model is derived from the emphysema airway (Fig. 4) by setting to zero. Alveolar volumes are set proportional to ventilation so that the alveolar compartments have the same ratios. The model parameters and 95% CIs estimated by fitting the embolism model to the measurements are shown in Table 2. The 95% CIs of the embolism parameters (Table 2) do not include zero or one, indicating that the parameters are unique.

Model predictions
Steady-state and predicted by the normal, emphysema and embolism models, and Bohr–Enghoff dead-space ventilation fractions calculated from predicted arterial and mixed expired PCO₂ are shown in Tables 3 and 4. The largest differences between predicted and measured results in the normal case were in , which is underestimated by 4%. The simulated Bohr–Enghoff dead-space is two percentage points (or 4.5%) lower in the normal case than the value determined from Wagner's 50-compartment continuous flow model.⁹ The and predictions of the embolism and emphysema models both match measured values to within 3%. The Bohr–Enghoff dead-space of the emphysema model is three percentage points (or 4%) greater than that of the 50-compartment continuous model. The dead-space ventilation fractions of the tidal and continuous models of embolism do not differ by more than two percentage points (or 3%) without shunt correction, and one percentage point (or 2%) less with shunt correction. The acetone dead-space is similar to the Bohr–Enghoff dead-space in the normal case, but is substantially smaller in the emphysema and embolism models. V_D/V_T is the ratio of anatomical dead-space (including instrument dead-space) to tidal volume. The V_D/V_T prediction for the normal, emphysema and embolism models all match measured values to within four percentage points.

View this table: [in this window] [in a new window]   Table 3 Dead-space predictions. *Corrected for shunt45

 

View this table: [in this window] [in a new window]   Table 4 Steady-state model predictions

 
The MBNWs predicted by each model (normal, emphysema and embolism) are superimposed on the respective subject curves in Figure 6. In the normal case the standard error between the curves is 0.0125, and the maximum difference between the normalized curves is 0.040. The emphysema model produces a MBNW with a standard error of 0.0104 and a maximum deviation from the measured washout of 0.042. The embolism model produced a MBNW with a standard error of 0.0214, and a maximum deviation from the normal MBNW of 0.0718 of the measured value.

View larger version (13K): [in this window] [in a new window] [Download PowerPoint slide]   Fig 6 (A) The MBNW predicted by the normal and emphysema model, compared with the measured normal and emphysema subjects' MBNW.43 (B) The MBNW predicted by the embolism model, compared with the measured normal subject's MBNW.43

 
A step response is representative of a model's soluble gas response to dynamic changes. The models' responses to step changes in ventilation rates are shown in Figure 7. Over the 500 s period, the predicted by the normal and embolism models decreases by 18 and 24% in response to a 50% increase in ventilation, while the emphysema model shows a 10% decrease. Over the same period decreasing ventilation rates by a factor of 1.5 results in a 14% increase in in the emphysema and embolism models, while the normal model showed a 19% increase. The emphysema model exhibited the fastest change over the first 5–10 s.

View larger version (15K): [in this window] [in a new window] [Download PowerPoint slide]   Fig 7 The simulated arterial PCO2 responses to step changes in ventilatory frequency. The ventilatory frequency was increased or decreased by a factor of 1.5 for each model at t=0. Grey line, normal; black line, embolism; dashed line, emphysema. The open circles represent end-tidal points. Main figure –10 to 50 s. Inset figure 0–500 s.

 

	Discussion

Top Abstract Introduction Materials and methods Results Discussion Conclusions Supplementary data Appendix References

 
Dynamic models of respiration in health and disease should be capable of representing inequality and inhomogeneous ventilation, and be capable of simulating dynamic exchange of both soluble and insoluble gases for diseases such as emphysema and embolism. In addition, the cardiac output should be able to be varied easily and the recirculation time should vary as a function of the cardiac output. Tidal models tend to be ‘single path’ models⁵⁰ which do not simulate or parallel ventilation–volume heterogeneity, or high order ‘multiple path’ models ^{51 52} which simulate dynamic intrabreath gas exchange but require substantial computing power and so are not easily amenable for the study of practical dynamic situations such as the partial rebreathing measurement of cardiac output. In addition, there is no systematic approach for selecting mutually consistent sets of parameters for respiratory models that incorporate both and heterogeneity.

We present a tidally breathing respiratory model that can potentially simulate lungs that have up to three distinct compartments plus shunt and dead-space, and two distinct ratios. Information from MIGET measurements and from MBNW measurements are used together to estimate the parameters of one to three distinct modes, shunt and dead-space,^{9 11 13 18}–²⁰ and up to two distinct ratios.^{22 24 25} The resulting tidal model has an alveolar structure, which is a simplification of the model proposed by Whiteley and colleagues²⁷ The methods described in this study represent a general procedure that can potentially model any respiratory condition that can be characterized by MIGET and MBNW measurements, which allows modelling of dynamic respiratory manoeuvres including both soluble and insoluble gases.

Potentially any combination of the nine compartments may be used but in a specific situation the model can be reduced to a minimum of one compartment, as exemplified in this study by the model of the normal subject. A particular compartment is activated when it is clearly associated with one of the and modes or both. For rapid execution and to ensure unique solutions the model is kept as simple as possible. To represent other disease states such as ARDS^{14 53} more compartments than are used here might be required, in particular if the ARDS is superimposed on a pre-existing abnormality such as emphysema. Based on the assumptions that the MIGET provides a good measure of steady-state exchange of soluble gases and MBNW provides a good measure of the dynamic behaviour of gases with low solubility, the methods we describe produce models that are able to simulate the exchange of soluble gases in diseased lungs under both steady-state and dynamic conditions.

Normal model
In the normal subject, we assume that all the ventilation is to the mid compartment which has a single ratio. The steady-state and dynamic predictions of this very simple model are good approximations to the measured data, suggesting that the model is a valid representation of the normal lung. However, the lower predicted and lower predicted Bohr–Enghoff dead-space suggests that, for this particular subject, there may be some spread in the lung which the model ignores.

The model of the normal lung produces a washout that matches the measured subject's washout well. However, the early and late differences between the predicted and measured washouts indicates that the subject may also have some spread in ratios that is neglected in the model.

Emphysema model
This model contains three active alveolar compartments, reflecting the inhomogeneity of emphysematous lungs and the high degree of gas exchange impairment. The slow compartment receives about a quarter of the alveolar ventilation. This slow compartment is assumed to be associated with the diseased portion of the lungs, and to have a single high ratio. It represents the enlarged and putatively hypoperfused air spaces produced by emphysematous degradation of the associated alveolar walls.^{14 54 55} It should be noted that the well-ventilated high compartment has a gas turnover time constant approximately equal to that of the mid compartment. This result indicates that the emphysematous portion of the lung has faster washout initially, followed by very slow washout, which agrees well with the pathomorphology of this disease. The configuration is similarly bimodal. The mid compartment has a lower than normal ratio possibly because the pathology occurs mostly to high parts of the lung, shifting the blood flow to lower the average of the remaining lung. The model also contains two high compartments and no pure parallel dead-space. The recovered anatomical dead-space in the tidal model is higher than normal because all the dead-space ventilation is assumed to be as a result of anatomical dead-space.

The steady-state performance of the emphysema model is similar to that of the normal model. Arterial PO₂ and PCO₂ are slightly underestimated compared with the measured values, and the Bohr–Enghoff dead-space is slightly overestimated compared with the 50-compartment continuous flow model. These discrepancies may be partly atributable to the different origins (two different patients) of the and ventilation distribution data, as it is reasonable to expect variation in lung pathology between different manifestations of the disease. The discrepancies may also reflect the assumptions made in deriving the parameters of the model.

The emphysema model's responses to step changes in ventilation rate are distinctly different from the responses of the normal and embolism models and show clearly the effects of multiple ventilation time constants. The change in during the first 10 s is faster than the normal and embolism responses, which has not been described previously and may be of importance when for instance cardiac output is determined by a rebreathing technique. Beyond 10 s the change in of the emphysema model becomes much slower than that of the normal and embolism models, reflecting the dominance of the slower parts of the lung. In the range between 10 and 20 s, there is clearly a change in the shape of the emphysema curves, indicating recirculation of CO₂ through various body compartments appropriate for a cardiac output of 6 litre min^–1.

The emphysema model produces a nitrogen washout that approximates the emphysema subject's washout well.

Pulmonary embolism model
Vascular obstruction creates areas where alveoli are well ventilated but poorly perfused, resulting in high areas and increased dead-space ventilation.⁵⁶ Embolism also forces blood to flow through non-ventilated areas and creates areas of alveolar flooding that increase shunt. Because of the decrease in cardiac output, the ratios of all lung units tend to be increased.⁵⁶ Bohr–Enghoff dead-space is increased but there is minimal increase in series dead-space.⁵⁷ The recovery procedure yielded a mid compartment with a slightly greater than unity, a second compartment with a substantially increased ratio, and an approximately normal anatomical dead-space for the subject's weight of 70 kg. The Bohr–Enghoff dead-space determined from simulation results is increased as a result of the substantial high compartment. These results agree well with the available clinical data. It is assumed that there was no alteration in the ventilation distribution and that the subject had essentially normal lungs before the development of the pulmonary embolism.^{41 42 49}

The embolism model produces steady-state predictions very close to the measured data. The dynamic response as a result of ventilation rate changes demonstrates a few properties of the disease. First, there is no quasi-equilibrium observed within the first 50 s as seen in the emphysema model between 10 and 20 s. Second, the initial rate of change is greater than the normal model, as a result of higher minute volume and smaller FRC. Third, as a result of the predominantly high ratios which affect the rate of excretion, the rate of change of during increased ventilation is greater than during decreased ventilation over a longer period.

The embolism model produces a nitrogen washout that approximates the normal subject's washout well. The standard error is greater than the normal model, which may be caused by inefficiencies associated with the ratio spread or because of the data origins from two people.

Dead-space
Dead-space ventilation measured by the MIGET corresponds to ventilation of compartments that have ratios substantially larger than 100 (because acetone has a solubility of 300). The Bohr–Enghoff dead-space ventilation is measured using CO₂ and is the fraction of ventilation of lung units that have ratios substantially greater than approximately unity (because CO₂ has solubility of 4). A normal lung that has homogenous ventilation has minimal ventilation of units with ratios substantially greater than unity, and therefore exhibits similar acetone and Bohr–Enghoff dead-space ventilations. In diseased lungs any spread of ratios is likely to increase ventilation to units with ratios exceeding unity more than to units with ratios exceeding 100. Therefore, in diseased lungs the Bohr–Enghoff dead-space ventilation is expected to be larger than the acetone dead-space ventilation.⁵⁸ The dead-space ventilation values determined from our model results (Table 3) are consistent with this theory.

The effective ventilation (, see Appendix Table A1) in a tidal model with branching airway structure is affected by mixing in common dead-spaces. The composition of the gas in each common dead-space depends on the ratios of the alveolar compartments from which the expired gases originate and the solubilities of the inert gases.²⁸ These differences are demonstrated in Figure 8 for emphysema. The for each inert gas in the high compartments are more affected than the in the mid compartments, and as perfusion to each compartment is independent of the effect of common dead-space ventilation, the variation in results in variation in the ratio.

View this table: [in this window] [in a new window]   Table A1 List of symbols

 

View larger version (17K): [in this window] [in a new window] [Download PowerPoint slide]   Fig 8 The digitized emphysema MIGET ventilation distribution47 (grey, open circles, left ordinate), and the recovered three-compartment distributions showing for each inert gas with solubility increasing from left to right (SF6 to acetone). Open triangles, mid compartment; open squares, fast high compartment; open circles, slow high compartment.

 
Limitations of this study
An important limitation of this study is the lack of availability of and distributions measured in the same subjects. One of the assumptions used to produce the models in this study is that the alveoli of the lungs can be represented by compartments with different combinations of and ratios, which are obtained from measured distributions and distributions. We were not able to find published and distributions from the same individuals. Representative published and normalized distributions from different individuals were used, which must limit the realism of our results.

Diffusion limitation that sometimes accompanies emphysema⁵⁹ is not simulated in this study, although our model is able to simulate reduced diffusion. There are no published studies that report diffusion coefficients and distributions measured in the same subjects. The model in its present form is also able to simulate O₂ gradients in hypoxia.

Our models simulate tidal ventilation, which is a better representation of the respiratory systems of tidally breathing mammals than the conventional continuous ventilation models commonly used.⁶⁰ However, our model does not simulate sequential emptying which may occur in an inhomogeneous lung. This limitation is also acknowledged in the development of the MIGET model equations,⁶¹ and similarly all the lung units in our model empty at the same time.

The use of qualitative information to decide which of the nine alveolar compartments to retain may introduce uncertainties in the models, which may limit the application of this method for lungs that exhibit complex characteristics. Increasing the number of active compartments will improve the agreement between measurements and predictions, but is likely to widen the confidence intervals of the estimated parameters and may result in non-unique solutions. This relationship may depend on the information content of the MIGET and MBNW measurements.

For each situation, we have shown that the model adequately represents at least one person for each lung type, but may not necessarily represent adequately other patients or subjects. We have, however, shown the model to be robust across a variety of lung types. The model is very flexible and can be used to predict responses for other lung conditions such as ARDS, or be fine-tuned to any particular individual.

	Conclusions

Top Abstract Introduction Materials and methods Results Discussion Conclusions Supplementary data Appendix References

 
The disease-specific lung models produced in this study are able to predict most satisfactorily steady-state and dynamic exchange of soluble and insoluble gases with at worst very small systematic error, which does not inhibit the model representing changes accurately. In a companion paper, these models provide a means to investigate the effects of complex manoeuvres involving the dynamic exchange of soluble gases in the cardio-respiratory system.⁶²

	Supplementary data

Top Abstract Introduction Materials and methods Results Discussion Conclusions Supplementary data Appendix References

 
An Appendix showing details of the model parameter allocation procedures can be found as Supplementary data in British Journal of Anaesthesia online.

	Appendix

Top Abstract Introduction Materials and methods Results Discussion Conclusions Supplementary data Appendix References

 

	Acknowledgments

 
This study was funded by Australian Research Council ‘Strategic Partnership with Industry—Research and Training’ grant (ARC-SPIRT), Australia Pty Ltd, The Joseph Fellowship, The Jobson Foundation, The Woolcock Institute of Medical Research The University of Sydney and the Australian National Health & Medical Research Council (NHMRC). This work is attributed to Department of Anaesthetics, The University of Sydney.

	Footnotes

 
This article is accompanied by the Editorial.

	References

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