RESPIRATION and the AIRWAY a Tidally Breathing Model of Ventilation, Perfusion and Volume in Normal and Diseased Lungs†

British Journal of Anaesthesia 97 (5): 718–31 (2006) doi:10.1093/bja/ael216 Advance Access publication August 21, 2006

RESPIRATION AND THE AIRWAY A tidally breathing model of ventilation, perfusion and volume in normal and diseased lungs†

J. S. Yem1, M. J. Turner1*, A. B. Baker1, I. H. Young2 and A. B. H. Crawford3

1Department of Anaesthetics, and 2Department of Respiratory Medicine, The University of Sydney, Royal Prince Alfred Hospital, Missenden Road, Camperdown, NSW 2050, Australia. 3Department 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: [email protected]

Background. To simulate the short-term dynamics of soluble gas exchange (e.g. CO2 rebreath- _ ing), model structure, ventilation–perfusion (V_ A/Q) and ventilation–volume (V_ A/VA) parameters _ must be selected correctly. Some diseases affect mainly the V_ A/Q distribution while others _ affect both V_ A/Q and V_ A/VA distributions. Results from the multiple inert gas elimination technique (MIGET) and multiple breath nitrogen washout (MBNW) can be used to select _ _ V_ A/Q and V_ A/VA parameters, but no method exists for combining V_ A/Q and V_ A/VA 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 V_ A/Q 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. Br J Anaesth 2006; 97: 718–31 Keywords: modelling, ventilation/perfusion distribution, ventilation inhomogeneity Accepted for publication: May 19, 2006

Simulation of respiratory exchange of soluble gases in strongly on the distribution of V_ A/Q_ ratios but is indepen- diseased lungs under dynamic conditions requires that dent of alveolar volumes. Exchange of soluble gases during the model structure and parameters associated with the transients, however, depends on the distributions of both distributions of both ventilation–perfusion (V_ A/Q_ ) and V_ A/Q_ and V_ A/VA ratios. Some diseases, e.g. pulmonary ventilation–volume (V_ A/VA) ratios are selected correctly. embolism, affect mainly the V_ A/Q_ distribution while others, For example, the simulation of cardiac output measurement e.g. emphysema, affect both V_ A/Q_ and V_ A/VA distributions. 1 by short respiratory manoeuvres such as CO2 rebreathing, To simulate the transport and storage of soluble gases which is increasingly used in anaesthesia and intensive during dynamic manoeuvres in subjects with both V_ A/Q_ 23 care for measurement and monitoring of cardiac output, and V_ A/VA heterogeneity, the parameters associated with requires models that predict well short-term changes in the V_ A/Q_ and V_ A/VA distributions should be selected in a transfer and storage of such a soluble gas. Steady-state exchange of soluble gases in diseased lungs depends †This article is accompanied by the Editorial.

The Board of Management and Trustees of the British Journal of Anaesthesia 2006. All rights reserved. For Permissions, please e-mail: [email protected] A tidal model of diseased lungs rational manner. Parameters of simple models are often The subject breathes air before the procedure. At the selected arbitrarily to produce outputs that match clinical start of the MBNW, the inspired gas is switched to a observations qualitatively.4–8 In more complex models arbi- mixture containing no nitrogen, and end-tidal nitrogen trary selection of parameters may lead to invalid or extreme concentration is monitored over a washout period that predictions, particularly during dynamic changes in is typically 7 min. ventilation or perfusion. Numerous studies have shown that the information in a At present there is no systematic approach for selecting washout curve is sufficient to describe only two or at most mutually consistent sets of parameters for respiratory mod- three compartments.22 24 25 Lewis and colleagues22 des- els that incorporate both V_ A/Q_ and V_ A/VA heterogeneity. In cribed a technique for recovering a continuous distribution this study, we describe procedures for selecting alveolar of ventilation from a MBNW. This technique uses a compartment ventilation, volume and perfusion parameters smoothed least-squares fitting procedure similar to that used 13 for a tidally breathing respiratory model, based on multiple in the MIGET to recover distributions of V_ A/Q_ ratios. inert gas elimination technique (MIGET) and multiple breath Both normal and more complex distributions recovered nitrogen washout (MBNW) measurements. These models are from nitrogen washouts were shown to be reproducible developed for the purposes of simulating the exchange of within an individual.22 Therefore, significant changes in soluble and insoluble gases during dynamic respiratory the shape of the distribution can be attributed to changes manoeuvres such as full, or partial rebreathing which is to the subject’s lungs.22 Wagner21 examined the variability now commonly used in anaesthesia and intensive care. among compatible ventilation distributions, and found that in general, the achievable resolution depends on the specific underlying distribution, and physiologically significant Materials and methods features of the distribution can usually be specified, although the more complex a distribution is, the less resolution is Background possible. Other studies to assess the effects of experimental error on the resolution of the MBNW confirm that the Ventilation–perfusion ratio heterogeneity information present in a MBNW is insufficient to allow 910 The MIGET has long been used for investigating the confident resolution of more than two ventilation modes matching of ventilation and perfusion in the lungs. Small and an estimate of dead-space.25 26 Thus, a model based quantities of six inert gases are dissolved in saline and i.v. on MBNW measurements may contain lung compartments infused until the mixed venous blood content of each inert with only two different V_ A/VA ratios. gas is constant or changing at a slow uniform rate. The V_ A/Q_ distributions are constructed from measurements of the Ventilation, perfusion and volume heterogeneity steady-state concentrations of the inert gases in mixed We propose a subset of the three dimensional alveolar struc- venous blood, mixed arterial blood and mixed expired ture suggested by Whiteley and colleagues,27 in a tidally O CO gas. Arterial and mixed expired P 2 and P 2 calculated breathing model, to simulate simultaneous ventilation, using the derived V_ A/Q_ distribution compare well with 11–13 perfusion and volume heterogeneity. The eight alveolar corresponding measured values. compartments and shunt (Fig. 1) allow this model to exhibit _ _ The shapes of MIGET-derived V A/Q distributions steady-state gas exchange behaviour consistent with any for many respiratory conditions are known, and in many cases distinct patterns can be associated with specific res-

14–17 _ _ 2

V Q different V piratory conditions. As the A/ distribution is a · · · · Q Q Q Q steady-state property of the lungs, predictions made by a s · 1a · 2a · 3a · · respiratory model derived only from MIGET measurements V1a V2a V3a V4a Va 15 V1a V2a V3a V4a Va

are likely to be incorrect in non-steady-state conditions. · A

Shunt /V The V_ A/Q_ distributions recovered by MIGET have been · · · A 9111318–20 Q Q Q Compartments shown to contain a limited amount of information. · 1b · 2b · 3b · · While the lungs contain a great number of gas exchange V1b V2b V3b V4b Vb units, the MIGET has been shown to be able to discriminate V1b V2b V3b V4b Vb _ _ only three distinct V A/Q modes, or two modes in addition ·· ·· ·· 911 Shunt(V /Q ) (V /Q ) (V /Q ) Dead-space to shunt, and dead-space. Hence, in general, a model · A · 1 A · 2 A· 3 · Q Q Q Q VAlv, DS based on MIGET measurements needs to contain only s ·1 ·2 ·3 V V V three different V_ A/Q_ lung compartments in addition to 1 2 3 shunt and dead-space. ·· 3 different VA/Q compartments

Ventilation–volume ratio heterogeneity Fig 1 Alveolar compartment structure containing pure shunt, two The MBNW technique is commonly used for alveolar dead-space compartments and six ventilated and perfused investigating indices of ventilation inhomogeneity.21–23 alveolar compartments.

719 Yem et al. measured V_ /Q_ distribution and dynamic characteristics implemented using Matlab and Simulink (Mathworks, Nat- consistent with any measured N2 washout. ick, MA, USA). Alveolar volume and the spread of the V_ A/VA distribution have been shown to have negligible effect on V_ A/Q_ distributions recovered using the MIGET5 if the retention ratios Parameter estimation are averaged over a complete respiratory cycle. Although Measured MIGET and MBNW data obtained in subjects nitrogen has a low solubility, we anticipate that the N2 who are representative of adults with normal lungs, emphys- washout and hence the measured V_ A/Q_ distribution may ema and pulmonary embolism were selected from the be affected by the V_ A/Q_ distribution by three mechanisms. literature. All model parameters other than compartmental _ First, the rate at which N2 dissolved in body tissues is VA, V_ A and Q were obtained from the respective articles 22 eliminated in expired gas may be affected by V_ /Q_ ratios from which individual V_ A/Q_ or V_ A/VA data were obtained of low V_ /Q_ compartments. Second, changes in the exchange (Table 1). Parameters that were not available were selected _ 35–37 of O2 and CO2 associated with changes in the V_ A/Q distri- using the standard mean values. _ bution may affect the washout of N2 by the second gas The numbers of significant modes in each V_ A/Q distri- effect. Third, in a tidally breathing model with series bution and nitrogen washout for the respective individuals dead-space, mixing of expired gases in common dead- were estimated based on published patterns of V_ A/Q_ 12 38–40 22 41 42 spaces alters the effective V_ A/Q_ ratio and the effective vent- and ventilation distributions. The number ilation of each inert gas in each compartment,28 and hence of columns in the basic alveolar model (Fig. 1) was _ may affect the elimination of N2. then reduced to match the number of estimated V_ A/Q In this study it is necessary to identify model parameters modes (NVQ), and similarly the number of rows was that enable our tidally breathing model to display charac- reduced to match the estimated number of V_ A/VA teristics that are consistent with both a measured V_ A/Q_ modes (NVV). This reduction procedure is necessary to distribution and a measured N2 washout. Hence, it is nec- allow a unique solution to be found from the simultaneous essary to select both the V_ A/Q_ and the V_ A/VA parameters in equations (A2), and to conform to the maximum number of a single optimization procedure, so the interaction between alveolar compartments allowed by the respective MIGET 9 11 18–20 22 24–25 V_ /Q_ and V_ A/VA distributions is taken into account. In gen- and MBNW manoeuvres. In each V_ A/Q_ eral, each ventilated compartment of our proposed alveolar column, compartments that are known to exist in each model (Fig. 1) receives a fraction of the alveolar ventilation, disease were retained and compartments known to con- and the volume of each compartment is selected so that tribute little to gas exchange were excluded. These deci- compartments in the same row have approximately equal sions were made according to current qualitative 12 22 V_ A/VA ratios. Two ventilated but unperfused compartments knowledge of subjects with normal lungs, patients form an optional parallel dead-space. The remaining six with emphysematous lungs11 14 38 40 and subjects with pul- ventilated compartments each receive a fraction of the monary embolism.41 42 The number of alveolar compart- pulmonary blood flow chosen so that each column has ments and the airway structure of the tidal model was approximately equal V_ A/Q_ ratio. modified appropriately and the tidal model was fitted to both inert gas retention/excretion ratios calculated from the measured V_ A/Q_ distributions, and the normalized nitrogen washout data (see Appendix). After the tidal model was The tidal model fitted to the data, the effective ventilation of each com- We modified an existing tidal model of the cardio- partment was determined using a method that allows for respiratory system of a healthy 70 kg adult male.29–33 the effects of gas mixing in common dead-spaces (see Provision was made for the addition of up to four alveolar Appendix). compartments to facilitate the simulation of simultaneous The subjects from whom the MIGET and MBNW mea- V_ A/Q_ and V_ A/VA inhomogeneity resulting in up to eight surements were obtained were of different sizes and had alveolar compartments and shunt (Fig. 1). The model is different lung volumes, ventilatory frequencies and tidal otherwise identical to that of Yem and colleagues,29 and volumes. Analysing MBNW as a function of dilution may be used to simulate dynamic responses of soluble number and alveolar dilution number has been shown to 43 and insoluble gases in the presence of V_ A/Q_ and V_ A/VA be insensitive to ventilatory frequencies and the ratios 44 heterogeneity. The model simulates artificially controlled of VAnatDS/FRC and VT/FRC. tidal breathing (as in anaesthesia or intensive care with a The completed normal, emphysema and embolism mod- constant inspiratory flow and passive exponential expira- els were evaluated in four ways. First, steady-state Paco2 tion) through a branched respiratory tree and incorporates and PaO2 were compared with values measured in the the effects on CO2 dynamics of lung tissue mass, vascular subjects from whom the MIGET data were obtained. transport delays, multiple body compartments and realistic Second, Bohr–Enghoff physiological dead-space fractions blood–gas dissociation curves.34 Nitrogen storage in determined from the models with and without corrections 45 blood and body tissues is simulated. The model is for shunt were compared with the respective CO2

720 A tidal model of diseased lungs

P P † _ _ _ ‡ _ _ § _ _ _ Table 1 Model parameters. *Determined from 50-compartment MIGET predicitions. Q from MIGET V A/Q distributions. QAlv/Qs. V from MIGET V A/Q # _ distributions. (V total/f r). **(FRCVInstDS) Parameters Units Normal Emphysema Embolism

MIGET46 MBNW43 MIGET47 MBNW43 MIGET48

Subject data obtained directly from articles quoted above Age/sex yr/sex 22/M –/– 61/M –/– 71/F Weight/height kg/cm 91/193 –/– 70/175 –/– –/– 1 V_ total litre min 9.3 12.3 – 11.7 13.2 _ Qs % 0 – 0.3 – 20 MIGET V_ DS % 43 – 53 – 42 FRC litre 4.7 2.65 – 4.5 – VT litre – 0.98 – 0.54 – fr bpm 14 12.6 22 21.6 29 i F o2 % 2121212150

Paco2 kPa 5.21 – 6.8 – 3.2

Pao2 kPa 12.9 – 7.06 – 13.1 Calculated based on subject data VD/VT (Bohr–Enghoff)* % 45 – 72 – 61 _ † 1 QAlv litre min 5.7 – 6.0 – 2.9 _ ‡ 1 Qtotal litre min 5.7 – 6.0 – 3.7 § 1 V_ total litre min – – 15.8 – – # VT litre 0.66 – 0.72 – 0.46 FRC35 litre – – 4.05 – 3.18 VA** litre 4.39 – 3.68 – 2.97 1 O2 consumption* mmol s 163.0 163.0 186.6 186.6 114.9 1 Co2 production* mmol s 157.0 157.0 186.6 186.6 106.3 Model default values Weight kg – 70 – 70 – VInstDS litre 0.05 0.05 0.05 0.05 0.05 I:E36 Ratio 1:2.33 1:2.33 1:4 1:4 1:2.33 Hb g 100 ml1 wholeblood 15 15 15 15 15 BE mEq litre1 00000

dead-space fractions calculated directly from the published obtained from Wagner and colleagues,46 Melot and col- _ 47 48 V_ A/Q distributions. CO2 dead-space was calculated by sub- leagues and D’Alonzo and colleagues, respectively stituting arterial and mixed expired PCO2 values predicted by (Fig. 2A–C). Representative MBNW curves measured in the 50-compartment continuous ﬂow model9 into the Bohr– subjects with normal lungs and emphysema were obtained 43 Enghoff dead-space equation. The O2 and CO2 dissociation from Saidel and colleagues. curves of Olszowka and Farhi34 were used. The acetone dead-space was calculated as VDS/VT ¼ 1E6Model for each respective model where E6Model is the elimination ratio of Normal model acetone. V_ /Q_ parameters Third, simulated nitrogen washouts were compared with A The V_ /Q_ distribution of normal subjects is known to the published washouts. For the purpose of comparing A be unimodal with negligible shunt, and total dead-space the washouts, the independent variables of the measured approximates anatomical dead-space.14 The measured washouts were transformed from total lung dilution numbers 46 MIGET V_ A/Q_ distribution clearly contains a single mode to alveolar dilution numbers by multiplying by _ _ at V A/Q1 with negligible shunt (Fig. 2A). We assumed that ðVTVAnatDSÞ VFRC the normal subject had negligible alveolar dead-space, and ðVFRCVAnatDSÞ VT therefore associated the dead-space ventilation in the measured V_ A/Q_ distribution entirely with the series anatomical Fourth, the simulated arterial PCO2 responses to step and instrument dead-space in the model. Therefore, only one changes in ventilatory frequency were assessed, by increas- V_ A/Q_ compartment was used to simulate the normal lung ing or decreasing ventilatory frequency by a factor of 1.5 at (Fig. 2A). the start of an inspiration after the models had reached steady state.

V_ A/VA parameters Results The ventilation distribution of normal subjects is uni- 22 Representative V_ A/Q_ distributions measured in subjects modal, which corresponds with a single compartment with normal lungs, and emphysema and embolism were lung model.

721 Yem et al.

A 2 7 single alveolar compartment that receives all the cardiac Ventilation Normal DS43%

Ventilation/perfusion (litre min output and all the alveolar ventilation. The recovered series )

1 Perfusion 6 − anatomical dead-space is 0.263 litre.

5 Emphysema model 4 1 V_ A/Q_ parameters 3 In emphysema, the V_ A/Q_ distribution is typically bimodal 14 47 (Fig. 2B). The lower V_ A/Q_ mode in Fig. 2B is assumed to 2 be associated with the normal part of the lung in which the −

1 perfusion distribution is similar to that seen in normal

1 ) Ventilation/perfusion (litre min subjects of equivalent age, and the V_ A/Q_ ratio is slightly Shunt 0% 14 0 0 reduced as a result of reduced ventilation. The higher V_ A/Q_ mode is assumed to be the abnormal emphysematous _ _ B DS part of the lung, which has a high V A/Q ratio as a result of Emphysema 11 14 38

1 Ventilation/perfusion (litre min the reduced alveolar surface area. Therefore, the ) 53% 6 1 − emphysema model has two distinct V_ A/Q_ compartments 5 in addition to shunt and dead-space (Figs 2B and 3).

4 V_ A/VA parameters 3 The ventilation distribution in emphysema is typically 22 bimodal, thus requiring a model with two V_ A/VA ratios 2 (Fig. 3). − 1

1 ) Ventilation/perfusion (litre min Ventilation, perfusion and volume parameters of Shunt 0.3% 0 0 the emphysema model The emphysema model requires two V_ A/Q_ and two V_ A/VA C 1 Embolism DS 42% compartments and therefore is potentially a four-

Shunt 20% Ventilation/perfusion (litre min ) 6 1 compartment model. We reduced the model to three − alveolar compartments by using qualitative information— 5 we assumed that the lower V_ A/Q_ mode and the faster V_ A/VA 4 compartment represent the normal part of the lungs. The lower V_ A/Q_ mode is thus contained entirely within the faster 3 V_ A/VA compartment. The slow V_ A/VA compartment is assumed to represent the diseased part of the lungs and 2 is therefore entirely associated with the high V_ A/Q_ mode. The parallel dead-space ventilation is assumed to be −

1 38

1 )

Ventilation/perfusion (litre min negligible. The resulting alveolar structure is shown in Figure 3 and 0 0 0.001 0.01 0.1 1 10 100 the tidal implementation of the airway of the emphysema .. model is shown in Figure 4. The three active alveolar com- VA/Q ratio (non-dimensional) partments are connected at one ternary branching point so

A _ _ 46 that the effects of common dead-space are similar between Fig 2 ( ) The digitized normal MIGET V A/Q distribution (black 28 lines, left ordinate), and the recovered single compartment V_ /Q_ ratio compartments. The model parameters and 95% CIs A _ _ (grey, right ordinate). (B) The digitized emphysema MIGET V_ A/Q_ distri- estimated by ﬁtting the emphysema model to the V A/Q bution47 (black, left ordinate) and the recovered three-compartment and MBNW measurements are shown in Table 2. The recov- _ _ V A/Q distribution (grey, right ordinate). (C) The digitized pulmonary _ _ _ 56 ered V A/Q and V A/VA parameters reﬂect the initial quali- embolism MIGET V_ A/Q_ distribution (black, left ordinate), and the _ _ tative selection of active compartments. There are two high recovered two-compartment V A/Q ratios (grey, right ordinate). Closed _ _ _ _ circles, perfusion; open circles, ventilation. V A/Q compartments and one low V A/Q compartment. In addition, the low V_ A/Q_ compartment and one of the high V_ A/Q_ compartments have fast gas turnovers, and the second Ventilation, perfusion and volume parameters of high V_ A/Q_ compartment has a distinctly slower gas turn- the normal model over. The 95% CIs of the parameters (Table 2) do not To simulate a normal subject the airway structure of the tidal include zero or one, indicating that the parameters are model is reduced to series anatomical dead-space and a unique.

722 A tidal model of diseased lungs

·· ·· · − V /Q=0.32 V /Q =9.3 Q =0.002 litre min 1 A A s Normal speed · −1 · −1 Q2a=5.3 litre min Q3a=0.45 litre min compartments

· −1 · −1 V2a=1.7 litre min V3a=4.2 litre min

V2a=0.36 litre V3a=0.23 litre τ τ Shunt 2a=12.6 s 3a=10.0 s

·· VA/Q =11.5 Slow · −1 Q3b=0.16 litre min compartments

· −1 V3b=1.9 litre min

V3b=3.1 litre τ 3b=300 s

·· ·· Σ Dead-space Valv=3.7 litre Mid VA/Q High VA/Q

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.

Large Table 2 Parameters that enable the emphysema and embolism tidal models Airways airways to best ﬁt MIGET and MBNW measurements. CI, conﬁdence interval. *The _ _ WG 0–5 numerical subscripts 1, 2, 3 and 4 refer to the low, mid and high V A/Q, and alveolar dead-space compartments, respectively. The alphabetical subscripts a and b refer to compartments with normal and high V_ A/VA ratios, respectively. (i) F _ ¼1F _ F _ , F _ ¼ 1F _ F _ F _ , FV ¼ 1FV FV ; Airways V 3b V 2a V 3a Q 3b Q 2a Q 3a Q s 3b 2a 3a (ii) F _ ¼1F _ , F _ ¼ 1F _ F _ V 3a V 2a Q 3a Q 2a Q s

Parameters* Unit Optimized result 95% CI Airways Low High

Emphysemai F _ (fraction) 0.196 0.173 0.219 F · F · F · V 2a V V V F _ (fraction) 0.577 0.570 0.584 2a 3a 3b Medium V 3a F _ (fraction) 0.227 – – airways V 3b Airways Airways Airways F _ (fraction) 0.889 0.865 0.913 WG 6–17 Q 2a F _ (fraction) 0.072 0.066 0.078 Q 3a F _ (fraction) 0.039 – – Q 3b

FV2a (fraction) 0.102 0.074 0.129

Airways Airways Airways FV3a (fraction) 0.065 0.007 0.122

FV3b (fraction) 0.833 – – VAnatDS litre 0.335 0.310 0.359 Embolismii Small F (fraction) 0.334 0.259 0.409 V_ 2a airways F (fraction) 0.666 – – V_ 2a Airways Airways Airways WG 18–20 F _ (fraction) 0.507 0.488 0.526 Q 2a F _ (fraction) 0.493 – – Q 2a VAnatDS litre 0.157 0.072 0.241 Alv L Alv L T T Alveolar Embolism model volume _ _ Shunt WG 21–23 V A/Q parameters · · · F FQ Alv L FQ Q2a 3a In pulmonary embolism the V_ /Q_ distribution is typically s T A bimodal with signiﬁcant overlap (Fig. 2C).12 39 Therefore, the embolism model has two V_ A/Q_ compartments in F · Q3b addition to shunt and dead-space (Figs 2C and 5).

Fig 4 Respiratory tree of the emphysema model. The respiratory tree contains three perfused alveolar compartments and shunt. The grey areas V_ A/Q_ parameters _ _ indicate alveolar compartments that have similar V A/Q ratios. WG, Pulmonary embolism does not result in signiﬁcant Weibel generations. redistribution of ventilation.41 42 49 Therefore, a normal

723 Yem et al.

· ·· ·· −1 Qs=0.74 litre min VA/Q=1.4 VA/Q=4.5 · −1 · −1 Q2a=1.8 litre min Q3a=1.1 litre min · · V =2.6 litre min−1 V =5.1 litre min−1 Normal speed 2a 3a compartments

V2a=1.0 litre V3a=2.0 litre τ τ Shunt 2a=19.7 s 3a=19.7 s

·· ·· Σ Dead-space Valv=2.97 litre Mid VA/Q High VA/Q

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.

Table 3 Dead-space predictions. *Corrected for shunt45 Table 4 Steady-state model predictions

Parameters Unit Normal Emphysema Embolism Parameters Normal Emphysema Embolism

Bohr–Enghoff dead-space/ % 43 75 59 (57*) Pred Meas46 Diff Pred Meas47 Diff Pred Meas48 Diff tidal volume ratio calculated from Paco2 (kPa) 5.0 5.2 4% 6.7 6.8 1% 3.2 3.2 0% model predictions Pao2 (kPa) 12.8 12.9 1% 6.9 7.1 3% 12.7 13.1 3% Bohr–Enghoff dead-space/ % 45 72 61 (58*) tidal volume ratio calculated from 50- compartment continuous the embolism parameters (Table 2) do not include zero or model predictions one, indicating that the parameters are unique. Acetone dead-space/tidal %4452 42 volume ratio calculated from model predictions Acetone dead-space/tidal %4353 42 Model predictions volume ratio measured by Steady-state Paco and Paco predicted by the normal, MIGET46–48 2 2 VD/VT from model %4753 45 emphysema and embolism models, and Bohr–Enghoff predictions [includes an dead-space ventilation fractions calculated from predicted instrument dead-space of arterial and mixed expired PCO are shown in Tables 3 50 ml, i.e. (anatomical 2 dead-space+instrument and 4. The largest differences between predicted and mea- dead-space)/tidal volume] sured results in the normal case were in Paco2 , which is underestimated by 4%. The simulated Bohr–Enghoff dead-space is two percentage points (or 4.5%) lower in _ ventilation distribution (single V A/VA compartment) is the normal case than the value determined from Wagner’s assumed for pulmonary embolism (Fig. 5). 9 50-compartment continuous ﬂow model. The Paco2 and

Paco2 predictions of the embolism and emphysema models Ventilation, perfusion and volume parameters of both match measured values to within 3%. The Bohr– the model Enghoff dead-space of the emphysema model is three The alveolar model is reduced to shunt, anatomical dead- percentage points (or 4%) greater than that of the 50- space and two V_ A/Q_ compartments. The resulting alveolar compartment continuous model. The dead-space ventilation structure is shown in Figure 5. The airway structure for the fractions of the tidal and continuous models of embolism do tidal embolism model is derived from the emphysema air- not differ by more than two percentage points (or 3%) without way (Fig. 4) by setting F to zero. Alveolar volumes are shunt correction, and one percentage point (or 2%) less with V_ 3b set proportional to ventilation so that the alveolar compart- shunt correction. The acetone dead-space is similar to the ments have the same V_ A/VA ratios. The model parameters Bohr–Enghoff dead-space in the normal case, but is substan- and 95% CIs estimated by ﬁtting the embolism model to the tially smaller in the emphysema and embolism models. VD/VT V_ A/Q_ measurements are shown in Table 2. The 95% CIs of is the ratio of anatomical dead-space (including instrument

724 A tidal model of diseased lungs

A 1 24% in response to a 50% increase in ventilation, while the Normal 0.9 Emphysema emphysema model shows a 10% decrease. Over the same Normal model period decreasing ventilation rates by a factor of 1.5 results 0.8 Emph model in a 14% increase in Paco2 in the emphysema and 0.7 embolism models, while the normal model showed a 0.6 19% increase. The emphysema model exhibited the fastest 0.5 change over the ﬁrst 5–10 s. 0.4 0.3 Discussion 0.2 Dynamic models of respiration in health and disease should be capable of representing V_ /Q_ inequality and Normalised nitrogen concentration A 0.1 inhomogeneous ventilation, and be capable of simulating 0 dynamic exchange of both soluble and insoluble gases for 0246 8 diseases such as emphysema and embolism. In addition, the B 1 Normal cardiac output should be able to be varied easily and the 0.9 Emb model recirculation time should vary as a function of the cardiac 50 0.8 output. Tidal models tend to be ‘single path’ models which do not simulate V_ A/Q_ or parallel ventilation–volume het- 0.7 erogeneity, or high order ‘multiple path’ models 51 52 which 0.6 simulate dynamic intrabreath gas exchange but require sub- 0.5 stantial computing power and so are not easily amenable for the study of practical dynamic situations such as the partial 0.4 rebreathing measurement of cardiac output. In addition, 0.3 there is no systematic approach for selecting mutually 0.2 consistent sets of parameters for respiratory models that _ _ _

Normalised nitrogen concentration incorporate both V A/Q and V A/VA heterogeneity. 0.1 We present a tidally breathing respiratory model that can 0 potentially simulate lungs that have up to three distinct 02468 V_ A/Q_ compartments plus shunt and dead-space, and two Alv dilution no. distinct V_ A/VA ratios. Information from MIGET measure- Fig 6 (A) The MBNW predicted by the normal and emphysema model, ments and from MBNW measurements are used together compared with the measured normal and emphysema subjects’ to estimate the parameters of one to three distinct V_ A/Q_ MBNW.43 (B) The MBNW predicted by the embolism model, compared modes, shunt and dead-space,9 11 13 18–20 and up to two with the measured normal subject’s MBNW.43 22 24 25 distinct V_ A/VA ratios. The resulting tidal model has an alveolar structure, which is a simplification of the model proposed by Whiteley and colleagues27 The methods dead-space) to tidal volume. The VD/VT prediction for the described in this study represent a general procedure that normal, emphysema and embolism models all match can potentially model any respiratory condition that can be measured values to within four percentage points. characterized by MIGET and MBNW measurements, which The MBNWs predicted by each model (normal, emphys- allows modelling of dynamic respiratory manoeuvres ema and embolism) are superimposed on the respective including both soluble and insoluble gases. subject curves in Figure 6. In the normal case the standard Potentially any combination of the nine compartments error between the curves is 0.0125, and the maximum may be used but in a specific situation the model can be difference between the normalized curves is 0.040. The reduced to a minimum of one compartment, as exemplified emphysema model produces a MBNW with a standard in this study by the model of the normal subject. A particular error of 0.0104 and a maximum deviation from the compartment is activated when it is clearly associated with measured washout of 0.042. The embolism model produced one of the V_ A/Q_ and V_ A/VA modes or both. For rapid a MBNW with a standard error of 0.0214, and a maximum execution and to ensure unique solutions the model is deviation from the normal MBNW of 0.0718 of the kept as simple as possible. To represent other disease states measured value. such as ARDS14 53 more compartments than are used here

A Paco2 step response is representative of a model’s might be required, in particular if the ARDS is superim- soluble gas response to dynamic changes. The models’ posed on a pre-existing abnormality such as emphysema.

Paco2 responses to step changes in ventilation rates are Based on the assumptions that the MIGET provides a good shown in Figure 7. Over the 500 s period, the Paco2 predicted measure of steady-state exchange of soluble gases and by the normal and embolism models decreases by 18 and MBNW provides a good measure of the dynamic behaviour

725 Yem et al.

12% Normal 25% Embolism Emphysema 13%

–13%

–25% 0 100 200 300 400 500

0% change (%) 2 CO P

−12% −10020304050 10 Time (s )

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.

of gases with low solubility, the methods we describe single high V_ A/Q_ ratio. It represents the enlarged and puta- produce models that are able to simulate the exchange of tively hypoperfused air spaces produced by emphysematous soluble gases in diseased lungs under both steady-state and degradation of the associated alveolar walls.14 54 55 It should dynamic conditions. be noted that the well-ventilated high V_ A/Q_ compartment has a gas turnover time constant approximately equal to that _ _ Normal model of the mid V A/Q compartment. This result indicates that the emphysematous portion of the lung has faster washout ini- In the normal subject, we assume that all the ventilation is to tially, followed by very slow washout, which agrees well the mid V_ /Q_ compartment which has a single V_ /V ratio. A A A with the pathomorphology of this disease. The V_ /Q_ con- The steady-state and dynamic predictions of this very simple A figuration is similarly bimodal. The mid V_ /Q_ compartment model are good approximations to the measured data, sug- A has a lower than normal V_ /Q_ ratio possibly because the gesting that the model is a valid representation of the A pathology occurs mostly to high V_ A/Q_ parts of the lung, normal lung. However, the lower predicted Paco and 2 shifting the blood flow to lower the average V_ /Q_ of the lower predicted Bohr–Enghoff dead-space suggests that, A remaining lung. The model also contains two high V_ /Q_ for this particular subject, there may be some V_ /Q_ spread A A compartments and no pure parallel dead-space. The recov- in the lung which the model ignores. ered anatomical dead-space in the tidal model is higher than The model of the normal lung produces a washout normal because all the dead-space ventilation is assumed to that matches the measured subject’s washout well. How- be as a result of anatomical dead-space. ever, the early and late differences between the predicted The steady-state performance of the emphysema model and measured washouts indicates that the subject may is similar to that of the normal model. Arterial PO and PCO also have some spread in V_ /V ratios that is neglected 2 2 A A are slightly underestimated compared with the measured in the model. values, and the Bohr–Enghoff dead-space is slightly over- estimated compared with the 50-compartment continuous Emphysema model flow model. These discrepancies may be partly atributable This model contains three active alveolar compartments, to the different origins (two different patients) of the V_ A/Q_ reflecting the inhomogeneity of emphysematous lungs and ventilation distribution data, as it is reasonable to expect and the high degree of gas exchange impairment. The variation in lung pathology between different manifesta- slow compartment receives about a quarter of the alveolar tions of the disease. The discrepancies may also reflect ventilation. This slow compartment is assumed to be asso- the assumptions made in deriving the parameters of the ciated with the diseased portion of the lungs, and to have a model.

726 A tidal model of diseased lungs

The emphysema model’s responses to step changes in Emphysema DS ventilation rate are distinctly different from the responses 1 53% 8 of the normal and embolism models and show clearly the 7 Ventilation (litre min ) 1 effects of multiple ventilation time constants. The change in − 6 Paco2 during the ﬁrst 10 s is faster than the normal and embolism responses, which has not been described previ- 5 ously and may be of importance when for instance cardiac 4 output is determined by a rebreathing technique. Beyond 3

10 s the change in Paco2 of the emphysema model becomes − 1 ) much slower than that of the normal and embolism models, Ventilation (litre min 2 reflecting the dominance of the slower parts of the lung. 1 In the range between 10 and 20 s, there is clearly a change in the shape of the emphysema curves, indicating recirculation 0 0 of CO2 through various body compartments appropriate for 0.001 0.01 0.1 1 10 100 1 .. a cardiac output of 6 litre min . V/Q ratio (non-dimensional) The emphysema model produces a nitrogen washout that _ 47 approximates the emphysema subject’s washout well. Fig 8 The digitized emphysema MIGET V A ventilation distribution (grey, open circles, left ordinate), and the recovered three-compartment V_ A distributions showing V_ iEffective for each inert gas with solubility _ _ Pulmonary embolism model increasing from left to right (SF6 to acetone). Open triangles, mid V A/Q compartment; open squares, fast high V_ A/Q_ compartment; open circles, Vascular obstruction creates areas where alveoli are well slow high V_ A/Q_ compartment. ventilated but poorly perfused, resulting in high V_ A/Q_ areas and increased dead-space ventilation.56 Embolism also forces blood to flow through non-ventilated areas and cre- Dead-space ates areas of alveolar flooding that increase shunt. Because Dead-space ventilation measured by the MIGET corre- _ _ of the decrease in cardiac output, the V_ A/Q_ ratios of all lung sponds to ventilation of compartments that have V A/Q units tend to be increased.56 Bohr–Enghoff dead-space is ratios substantially larger than 100 (because acetone has increased but there is minimal increase in series dead- a solubility of 300). The Bohr–Enghoff dead-space vent- 57 space. The V_ A/Q_ recovery procedure yielded a mid ilation is measured using CO2 and is the fraction of vent- _ _ V_ A/Q_ compartment with a V_ A/Q_ slightly greater than ilation of lung units that have V A/Q ratios substantially unity, a second compartment with a substantially increased greater than approximately unity (because CO2 has solubil- V_ A/Q_ ratio, and an approximately normal anatomical dead- ity of 4). A normal lung that has homogenous ventilation _ space for the subject’s weight of 70 kg. The Bohr–Enghoff has minimal ventilation of units with V_ A/Q ratios substan- dead-space determined from simulation results is increased tially greater than unity, and therefore exhibits similar ace- as a result of the substantial high V_ A/Q_ compartment. These tone and Bohr–Enghoff dead-space ventilations. In diseased results agree well with the available clinical data. It is lungs any spread of V_ A/VA ratios is likely to increase vent- _ assumed that there was no alteration in the ventilation dis- ilation to units with V_ A/Q ratios exceeding unity more than _ tribution and that the subject had essentially normal lungs to units with V_ A/Q ratios exceeding 100. Therefore, in dis- before the development of the pulmonary embolism.41 42 49 eased lungs the Bohr–Enghoff dead-space ventilation The embolism model produces steady-state predictions is expected to be larger than the acetone dead-space very close to the measured data. The dynamic response ventilation.58 The dead-space ventilation values determined as a result of ventilation rate changes demonstrates a few from our model results (Table 3) are consistent with this properties of the disease. First, there is no quasi-equilibrium theory. observed within the first 50 s as seen in the emphysema The effective ventilation (V_ iEffective, see Appendix model between 10 and 20 s. Second, the initial rate of Table A1) in a tidal model with branching airway structure change is greater than the normal model, as a result of higher is affected by mixing in common dead-spaces. The com- minute volume and smaller FRC. Third, as a result of the position of the gas in each common dead-space depends on _ _ predominantly high V_ A/Q_ ratios which affect the rate of the V A/Q ratios of the alveolar compartments from which the expired gases originate and the solubilities of the inert Paco2 excretion, the rate of change of Paco2 during increased ventilation is greater than during decreased ventilation over gases.28 These differences are demonstrated in Figure 8 for a longer period. emphysema. The V_ iEffective for each inert gas in the high _ The embolism model produces a nitrogen washout that V_ A/Q compartments are more affected than the V_ iEffective _ approximates the normal subject’s washout well. The stan- in the mid V_ A/Q compartments, and as perfusion to each dard error is greater than the normal model, which may be compartment is independent of the effect of common dead- _ caused by inefficiencies associated with the V_ A/Q_ ratio space ventilation, the variation in V iEffective results in varia- _ spread or because of the data origins from two people. tion in the V_ A/Q ratio.

727 Yem et al.

Limitations of this study relationship may depend on the information content of An important limitation of this study is the lack of avail- the MIGET and MBNW measurements. ability of V_ A/Q_ and V_ A/VA distributions measured in the For each situation, we have shown that the model same subjects. One of the assumptions used to produce the adequately represents at least one person for each lung models in this study is that the alveoli of the lungs can be type, but may not necessarily represent adequately other represented by compartments with different combinations patients or subjects. We have, however, shown the model of V_ A/Q_ and V_ A/VA ratios, which are obtained from mea- to be robust across a variety of lung types. The model is very sured V_ A/Q_ distributions and V_ A/VA distributions. We were flexible and can be used to predict responses for other lung not able to find published V_ A/Q_ and V_ A/VA distributions conditions such as ARDS, or be fine-tuned to any particular from the same individuals. Representative published V_ A/Q_ individual. and normalized V_ A/VA distributions from different individuals were used, which must limit the realism of our results. Conclusions Diffusion limitation that sometimes accompanies The disease-specific lung models produced in this study are emphysema59 is not simulated in this study, although our able to predict most satisfactorily steady-state and dynamic model is able to simulate reduced diffusion. There are exchange of soluble and insoluble gases with at worst very no published studies that report diffusion coefficients and small systematic error, which does not inhibit the model V_ /Q_ distributions measured in the same subjects. The A representing changes accurately. In a companion paper, model in its present form is also able to simulate O 2 these models provide a means to investigate the effects gradients in hypoxia. of complex manoeuvres involving the dynamic exchange Our models simulate tidal ventilation, which is a better of soluble gases in the cardio-respiratory system.62 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 inho- Acknowledgements mogeneous lung. This limitation is also acknowledged in This study was funded by Australian Research Council ‘Strategic Partner- 61 ship with Industry—Research and Training’ grant (ARC-SPIRT), Drðger the development of the MIGET model equations, and Australia Pty Ltd, The Joseph Fellowship, The Jobson Foundation, similarly all the lung units in our model empty at the The Woolcock Institute of Medical Research The University of Sydney same time. and the Australian National Health & Medical Research Council (NHMRC). This work is attributed to Department of Anaesthetics, The use of qualitative information to decide which of the The University of Sydney. nine alveolar compartments to retain may introduce uncer- tainties 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 Supplementary data the agreement between measurements and predictions, but is An Appendix showing details of the model parameter likely to widen the confidence intervals of the estimated allocation procedures can be found as Supplementary parameters and may result in non-unique solutions. This data in British Journal of Anaesthesia online.

Appendix Table A1 List of symbols

Variables Unit Description

1 Cn mol litre Concentration of the nth inert gas EnMIGET – Elimination ratio of the nth inert gas calculated from published MIGET measurement EnModel – Elimination ratio of the nth inert gas calculated from model simulation Ei – Elimination ratio in the ith alveolar compartment FRC litre Functional residual capacity FM MIGET objective function FN MBNW objective function 1 fr min Respiration rate per minute FQi_ ði ¼ 1‚2‚3‚4Þ – Fraction of perfusion in the ith VAA/_ Q_ compartment FVi_ ði ¼ 1‚2‚3‚4Þ – Fraction of ventilation in the ith VAA/_ Q_ compartment FVj_ ðj ¼ a or bÞ – Fraction of ventilation in the jth VAA/_ VA compartment FVj ðj ¼ a or bÞ – Fraction of volume in the jth VAA/_ VA compartment _ NVQ – Number of VAA/Q_ modes _ NVV – Number of VAA/VA modes NAlv – Number of active alveolar compartments

728 A tidal model of diseased lungs

Table A1 Continued

Variables Unit Description

Pco2 kPa Partial pressure of carbon dioxide PN2 kPa Partial pressure of nitrogen PN2NBNWðnÞ kPa Partial pressure of expired nitrogen for the nth breath from MBNW measurement PN2ModelðnÞ kPa Partial pressure of expired nitrogen for the nth breath from model simulation PO2 kPa Partial pressure of oxygen Q_ litre min1 Perfusion Qi_ ði ¼ 1‚2‚3Þ litre min1 Perfusion in the ith VAA/_ Q_ compartment Qs_ litre min1 Shunted blood flow Qtotal_ litre min1 Total perfusion RnMIGET – Retention ratio of the nth inert gas calculated from published MIGET measurement RnModel – Retention ratio of the nth inert gas calculated from model simulation Ri – Retention ratio in the ith alveolar compartment V litre Volume VA litre Alveolar volume VDS litre Dead-space volume ViDSeffective litre Effective anatomical dead-space associated with ith alveolar compartment VFRC litre Volume of functional residual capacity VT litre Tidal volume _ VTi litre Tidal volume in the ith VAA/Q_ compartment VD/VT % Dead-space ratio V_ litre min1 Ventilation VA_ litre min1 Alveolar ventilation Vi_ ði ¼ 1‚2‚3‚4Þ litre min1 Ventilation in the ith VAA/_ Q_ compartment Vi_Effective ði ¼ 1‚2‚3‚4Þ litre min1 Effective ventilation in the ith VAA/_ Q_ compartment Vj_ ðj ¼ a‚bÞ litre min1 Ventilation in the jth VAA/_ VA compartment Viij_ ði ¼ 1‚2‚3‚4 j ¼ a or bÞ litre min1 Ventilation in the ith VAA/_ Q_ and the jth VAA/_ VA compartment of the nine-compartment model Vtotal_ litre min1 Total ventilation Voo2_ mmol s1 Oxygen consumption Vcoco2_ mmol s1 Carbon dioxide Flux VAA/_ Q_ – Ventilation–perfusion ratio ðVAA/_ Q_Þi – Ventilation–perfusion ratio for the ith VAA/_ Q_ compartment VA_ /VA min1 Ventilation–volume ratio ln – Blood–gas partition coefficient of the nth inert gas at 1 atm t s Time constant tj (j = a or b) s Time constant in the jth VAA/_ VA compartment tn – Dimensionless time constant tnj (j = a or b) – Dimensionless time constant in the jth VAA/_ VA compartment Subscript modifiers a Arterial Anat Anatomical A Alveolar E Mixed expired Inst Instrumental v Mixed venous

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