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Respiratory Physiology & Neurobiology 235 (2017) 8–17

Contents lists available at ScienceDirect

Respiratory Physiology & Neurobiology

journa l h om epa ge: www.elsev ier .com/ locate / resphys io l

odelling responses of the inert-gas washout and MRI toronchoconstriction

rody H. Foy ∗, David Kay, Rafel Bordasepartment of Computer Science, University of Oxford, Oxford, Oxfordshire, United Kingdom

r t i c l e i n f o

rticle history:eceived 13 June 2016eceived in revised form8 September 2016ccepted 20 September 2016vailable online 24 September 2016

a b s t r a c t

Many lung diseases lead to an increase in ventilation heterogeneity (VH). Two clinical practices for themeasurement of patient VH are in vivo imaging, and the inert gas multiple breath washout (MBW). Inthis study computational modelling was used to compare the responses of MBW indices LCI and scondand MRI measured global and local ventilation indices, �r and � local, to constriction of airways in theconducting zone of the lungs. The simulations show that scond, LCI and �r behave quite similarly to eachother, all being sensitive to increases in the severity of constriction, while exhibiting little sensitivity to

eywords:nert-gas washout

RIronchoconstrictionentilation heterogeneity

the depth at which constriction occurs. In contrast, the local MRI index � local shows strong sensitivity todepth of constriction, but lowered sensitivity to constriction severity. We finish with an analysis of thesensitivity of MRI indices to grid sizes, showing that results should be interpreted with reference to theimage resolution. Overall we conclude that the application of both local and global VH measures mayhelp to classify different types of bronchoconstriction.

. Introduction

Many diseases, such as asthma, emphysema and cystic fibrosisre characterised by ventilation heterogeneity (VH), which can beriven by constriction or closure of airways in the lungs. There-ore, accurate measurements of VH are of high utility in clinicalettings. Typically these measurements are inferred through sim-le lung function tests such as spirometry and plethysmography, orhrough more complex tests such as oscillometry (Oostveen et al.,003), inert-gas washouts (Robinson et al., 2013), or in vivo imagingde Lange et al., 2006; Simon, 2005; Tzeng et al., 2009).

A clinical test for VH that is becoming increasingly common ishe inert gas multiple breath washout (MBW) (Macleod et al., 2009;obinson et al., 2013; Verbanck et al., 2003). In short, the MBW iserformed in one of two ways; either air mixed with an inert-gassuch as SF6 or 3He) is washed-in (breathed in until it is assumed toave equilibrated in the lungs) by the patient and then washout ofhe gas is performed by switching the patient back to an air mixtureithout the gas; or in the case of N2 washouts, since N2 is alreadyresent in air, no wash-in is performed, and instead a washout is

erformed using pure oxygen. By monitoring the exhaled gas con-entrations during the washout process, a number of VH indicesan be derived. These include the Lung Clearance Index (LCI), scond,

∗ Corresponding author.E-mail address: brody.foy@new.ox.ac.uk (B.H. Foy).

ttp://dx.doi.org/10.1016/j.resp.2016.09.009569-9048/© 2016 Elsevier B.V. All rights reserved.

© 2016 Elsevier B.V. All rights reserved.

and sacin (Verbanck et al., 1998) (for exact definitions of each, thereader is referred to Appendix A). The uptake of the MBW as aclinical test of VH has in large part been driven by the correla-tion of MBW indices with clinician classifications of many lungdiseases, including cystic fibrosis (CF) (Aurora et al., 2004, 2005;Horsley et al., 2008) and asthma (Gustafsson, 2007; Verbanck et al.,1999). However, due to the complex nature of the test indices, pre-cise understanding of how the indices respond to VH has yet to beachieved (Robinson et al., 2013).

Since its initial development, understanding of the MBW hasbeen advanced by a combination of clinical and computationalstudies. Key to this was the concept of time constants (Otis et al.,1956), a measure of how effectively different pulmonary regionsof the lung ventilate. Typically, computational studies of the MBWhave used transport equations to simulate gas transfer through-out the breathing cycle. Early models (Cruz et al., 1997; Hanet al., 2001; Verbanck and Paiva, 1990) were simulated on ide-alised branching geometries based on the work of Weibel (1963).Later work used image processing to construct more realisti-cal simulation geometries from patient CT scans (Tawhai andHunter, 2001; Mitchell et al., 2012). Results from these studieshave suggested that indices from the MBW respond positivelyto increases in bronchoconstriction driven VH. The studies have

also strengthened the understanding that regional differences intime constants can significantly alter flow behaviour. However,there is still uncertainty about exactly what drives changes in theindices.

dx.doi.org/10.1016/j.resp.2016.09.009http://www.sciencedirect.com/science/journal/15699048http://www.elsevier.com/locate/resphysiolhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.resp.2016.09.009&domain=pdfmailto:brody.foy@new.ox.ac.ukdx.doi.org/10.1016/j.resp.2016.09.009

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B.H. Foy et al. / Respiratory Physio

A range of different imaging techniques have also been showno successfully resolve ventilation distribution heterogeneity,ncluding scintigraphy, computed tomography (CT), and positronmission tomography (Simon, 2005; Tajik et al., 2002; Tzeng et al.,009). However, while these techniques can produce high resolu-ion images, the use of ionising radiation limits application in manylinical settings, particularly paediatric, and longitudinal studies.

Due to the limitations of imaging techniques involving ionis-ng radiation, magnetic resonance imaging (MRI) techniques haveained traction as alternative tests for VH. In particular, proton MRIBauman et al., 2009; Biederer et al., 2012; Sá et al., 2010) and hyper-olarised gas MRI (Fain et al., 2007; Möller et al., 2002; van Beekt al., 2004) have exhibited increased uptake, and strong improve-ents in resolution and cost. From MRI data, measurements can beade of how well ventilated each voxel in an image is. From this, a

eries of different measures of VH can be derived, at either a globalBiederer et al., 2012; Horn et al., 2014) or local (Tzeng et al., 2009)cale.

MRI techniques have become more viable due to reductionsn time, alongside improvements in image resolution. However,he relative cost, and lack of standardised imaging protocols haveimited large scale uptake (Biederer et al., 2012). Equally, whilehere is a vast array of literature relating MBW indices to lungisease classification, there are fewer studies investigating theelationship of MRI outputs to disease classification. Some recentreliminary studies have investigated the relationships betweenRI and MBW within the same patient groups (Capaldi et al., 2015;orn et al., 2015; Horsley et al., 2014). However, to the best ofur knowledge no full studies in the literature have analysed thiselationship in significant detail, or confirmed preliminary results.

In this paper we aim to use computational investigations to bet-er understand the relationship between MBW and MRI derivedndices, and how they respond to bronchoconstriction. By simulat-ng the MBW and ventilation on a virtual lung structure, under a

ig. 1. Variation of scond, LCI, �r and �local against constriction severity. Positive responseBW indices as severity approaches airway closure. The variance of �local is also seen to b

Neurobiology 235 (2017) 8–17 9

variety of different bronchoconstriction scenarios, we investigatehow test indices respond to constriction of airways in the conduct-ing zone. Comparison of the MBW and MRI indices show that scond,LCI and global ventilation variance primarily respond to the severityof the bronchoconstriction, while local ventilation variance primar-ily responds to the degree of regionalisation in the constriction.Alongside this, we give insights to the sensitivity of MRI indices togrid resolution.

2. Methods

2.1. Ventilation model

Within our study, simulations were performed on a structuralrepresentation of the lungs, created through a combination of seg-mentation of a subject CT scan for the central airways (to generation6–10) and algorithmic airway generation (to an average genera-tion 16). The structural model used in this study was taken from aset presented by Bordas et al. (2015), with segmentation and algo-rithmic generation based on the work of Fetita et al. (2004) andTawhai et al. (2004). The airway tree structure consisted of 78,834branches, 39,420 of which were terminal bronchioles. Each ter-minal bronchiole was connected to a unique elastic acinar region,which expanded and contracted over time in accordance to pleuralpressure changes.

To remove any potential bias in ventilation heterogeneity fromthe base structure, the original CT based airway diameter data wasdiscarded, and regenerated using the idealised relationship pre-sented by Horsfield et al. (1976)

log d(x) = (x − N) log(RdS) + log(dn),

where d is the airway diameter, x is the Strahler order (Strahler,1957) of the branch, N is the maximum Strahler order of the tree,

s are initially seen from all four indices, followed by a decrease to baseline in thee significantly larger than that of the other indices.

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0 B.H. Foy et al. / Respiratory Physi

n is the maximum diameter, and RdS is the anti-log of the slopef the airway diameter plotted against Strahler order, taken as 1.4.or more details on the airway tree set, the reader is referred tohe work of Bordas et al. (2015). Airway radii in the structure wereaken as static, with all volume changes in the lungs occurring inhe acinar units that subtend the terminal bronchioles.

To infer the ventilation distribution from the lung structure,e assumed a pressure driven flow profile, the details of which

re presented in Appendix B. In short, the velocity (u) in eachranch is driven by the pressure gradient over the branch, with floweing conserved at each bifurcation. The flow model incorporatesoiseuille resistance with a correction term for energy dissipationroposed by Pedley et al. (1970). The expansion and contractionf each acinar region was driven by flow in the associated termi-al bronchiole, changes in pleural pressure, and the region’s tissueompliance. The pleural pressure was taken as sinusoidal, with areathing period of 4 s. The amplitude was chosen to enforce tidalreathing of 1 L per cycle, above a functional residual capacity of.5 L.

For total lung compliance, a standard value of 0.2 L cmH2O−1

Galetke et al., 2007; Mittman et al., 1965) was chosen. This valueas distributed unevenly across the lungs, to account for gravita-

ional effects seen in previous clinical studies (Hopkins et al., 2007;aneko et al., 1966; Musch et al., 2002). For both the MBW and MRI

he gravitational compliance gradient was applied in the dorso-entral axis, pertaining to the tests being performed in the supineosition. Due to computational constraints, a linear distributionf compliance was chosen over more complex models in the lit-rature, such as in Swan et al. (2012). Consistent with studies inhe literature (Kaneko et al., 1966; Musch et al., 2002) a gradienttrength of 1.5% cm−1 was applied.

.2. Model of the MBW

The MBW was simulated using a convection transport equation:

∂AC∂t

= −u ∂AC∂x

, (1)

here x is the axial direction of the flow, t is time, C(x, t) is theoncentration of the gas within air, A is the cross-sectional area ofhe airway, and u is the air velocity. The branch area A is taken asonstant within each branch, with differences between connectedranches accounted for in the finite volume formulation of theodel.Mass transfer between acinar regions and the conducting zone

as driven by flow rates from the terminal bronchioles. Withinach acinar region, gas mixing was assumed instantaneous, withach region being characterised by a single gas concentration. Forore details on the model, and the numerical scheme, the reader

s directed to the supplemental materials.Simulations were started from the washout phase, with a uni-

ormly distributed initial gas concentration of 4%. Initially theash-in phase was also simulated, however, even under high VH

cenarios, no significant difference in test indices was seen. Dur-ng washout an inspiratory tracheal boundary concentration of 0%

as applied. During expiration, gas was removed at the trachearoportional to the flow rate in the branch.

.3. Calculation of MRI indices

There are many competing standards for MRI indices withinhe literature. For simplicity, we chose two different indices; a

Neurobiology 235 (2017) 8–17

global ventilation measure, and a local ventilation measure. Wefirst define the ventilation ratio (Horn et al., 2014) of a region as

r = VfVf + Vr

,

where Vf is the fresh gas entering a region over an inhalation, andVr is the residual gas at the end of exhalation. Within our model,we can directly calculate an r value for each acinar region, based onvolume changes over the breathing cycle.

To approximate an MR image, we overlay a 3D grid to the lungstructure, and take the mean r value of all acinar regions withineach voxel (grid rectangular prism). From the set of voxel r values,the global ventilation variance (�2r ) was defined as the variance ofthe total set. The local variance (�2local) was defined as the variancerelative to a local mean, such that

�2local =1N

N∑i=1

(ri − ri,local)2,

where ri,local is the mean r value of all voxels sharing an edge withvoxel i, and N is the total number of voxels.

Fig. 2. Relationship between scond (left) and LCI (right) and �r . Data has been splitinto three groups based on constriction severity: less than 80% (blue), 80–90% (red)and greater than 90% (yellow). A positive relationship can be seen until constrictionlevels pass 80%.

B.H. Foy et al. / Respiratory Physiology & Neurobiology 235 (2017) 8–17 11

Fig. 3. Variation of scond, LCI, �r and �local against constriction depth (Strahler order). Results have been separated into six different constriction severity bands. For each band,the median output (line) is presented alongside an inner band representing the interquartile range (dark blue), and an outer band to the minimum and maximum values(light blue). The MBW indices show no consistent correlation with depth, but the MR indices respond more clearly, with �local having the strongest and most consistentresponse.

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Fig. 4. Ventilation maps. A series of 8 ventilation maps, corresponding to severeconstriction of 20% of airways at Strahler orders 1–4 (top, left to right), and 5–8

2 B.H. Foy et al. / Respiratory Physi

he literature (Horn et al., 2014; Qing et al., 2014; Stavngaard et al.,005).

.4. Protocol for selecting bronchoconstriction scenarios

A Monte Carlo random sampling approach was used to selectronchoconstriction scenarios. Each scenario was characterised by

constriction severity, and a constriction depth. Constriction sever-ty was the percentage that a chosen branch was constricted (hadts radius reduced) by. Strahler order, the minimum number ofranches between a branch and the nearest terminal bronchiole,as used as the marker of depth, with Strahler order 1 corre-

ponding to terminal bronchioles, and 10 corresponding to lobarronchi. Strahler order was used instead of branch generation, ashe mechanics of the model are more strongly driven by behaviourt the terminal bronchioles (and in the respiratory zone) than athe trachea. An investigation using branch generation as the depth

arker is presented in supplemental material.For each simulation, a constriction severity (0–99%) and con-

triction depth (Strahler order, 1–10) were uniformly randomlyhosen, and 20% of branches at that depth were constricted tohe chosen severity. Airway constriction was applied in a spatiallyoherent manner through adaptation of work by Leary et al. (2012),eaning constricted branches passed on some of their constriction

o the next generation. The diameter of a branch j with a parentranch i that has undergone constriction, inherited part of thisonstriction, such that

j = �di(

dbase,jdbase,i

)+ (1 − �)dbase,j,

here dbase denotes the diameter of the airway before constriction,nd � is known as the coherence parameter.

Simulations in this study were applied using coherence of 60%,eaning each child branch inherited 60% of the parent’s constric-

ion, each grandchild inherited 36% and so on. The value � = 0.6 washosen as it provided significant spatial coherence and smoothness,ut has been shown to not significantly affect the variance of airwayesistances (Leary et al., 2012).

Each simulation was performed until the mean exhaled gas con-entration fell below 1/40th of the mean exhaled concentration ofhe first breath for three consecutive exhalations, and at least sixurnovers had occurred (for definition of a turnover, see Appendix). Following this, the LCI, scond, �r and �local were calculated and aoxel map was generated.

. Results

As a reference, we give the baseline values for LCI, scond, �rnd �local produced with the model as 4.948 turnovers, 0.0012 L−1,.00016, and 2.19 × 10−5 respectively.

We note that within all simulations mass conservation waschieved to within 0.01% of original mass. Convergence of theumerical scheme in space and time was also checked (with detailsiven in the supplemental material).

.1. Response of MBW and MRI indices to constriction severity

Within Fig. 1 we see the response of the MBW and MRI indiceso increasing constriction severity. Initially we see all four indicesxhibit a positive response, however, as severity approaches 100%

he MBW indices monotonically decrease back towards baseline.his is representative of the inability of the MBW to detect air-ay closure, as seen previously in the literature (Mitchell et al.,

012). Considering the response of �local, we also see an increase as

(bottom, left to right). Each map has a similar �r value (0.08–0.11), but significantlydifferent �local values. Ventilation ratios for each voxel are indicated by brightness(lighter colours corresponding to higher ventilation).

constriction severity increases. However, the strength of thisresponse is weaker and more noisy than that of the other indices.

In Fig. 2 we show the correlation between �r and the MBWindices, with results separated into three severity bands: 90%. We see that initially the indices correlatestrongly, but that this correlation breaks down as airway closureis approached.

3.2. Response of MBW and MRI indices to constriction depth

Fig. 3 shows the change in the four indices as constrictiondepth becomes more proximal (Strahler order increases). To sepa-rate out the response from constriction severity, results have beengrouped into six different severity bands:

B.H. Foy et al. / Respiratory Physiology & Neurobiology 235 (2017) 8–17 13

Fig. 5. Variation of �local against constriction depth under various grid resolutions. Three different grid resolutions have been used: very coarse (top), coarse (middle), medium(bottom). For each grid resolution, results have been separated into six constriction severity bands. For each severity band the median output (line) is presented alongsidean inner band to the interquartile range (dark blue), and an outer band to the minimum and maximum values (light blue). Changes in grid resolution strongly affect �localchanging the depth at which the index maximises.

sreim(

ize of the neighbourhood. To address this we analyse theesponse of �local to changes in constriction depth, under differ-

nt grid resolutions. For comparison to the grid resolution usedn this study (80 × 80 × 80), three different resolutions were used:

edium (40 × 40 × 40); coarse (20 × 20 × 20); and very coarse10 × 10 × 10). Note that we present no results for finer resolutions

than 80 × 80 × 80, due to the number of acinar regions present inthe model, meaning higher resolutions resulted in too many empty

voxels to give accurate representations of ventilation.

In Fig. 5 we give the results of this analysis. As the figureshows, �local is strongly affected by image resolution. As resolu-tion decreases, the depth at which �local maximises becomes more

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roximal, peaking at Strahler orders 3, 5 and 7 for the medium,oarse and very coarse resolutions respectively. This suggests thathile a local VH parameter can help identify constriction depth,

nterpretation must be made with careful reference to the imagingesolution.

. Discussion

In this study, a mathematical model for gas transport within theungs was applied to the comparative analysis of how MBW and MRIndices respond to bronchoconstriction in the conducting zone. The

odel was simulated on a structural representation of the conduct-ng airways, constructed through image processing of a subject CTcan, with airway diameters being replaced by an idealised healthyodel, given in the work of Horsfield et al. (1976). By simulating

nd calculating MBW and imaging indices over a large range ofonstriction severities and depths, analysis could be made abouthe sensitivity of these indices as detectors of bronchoconstriction.

.1. Comparison of MBW and MRI indices

Comparative analysis of MBW and MRI indices produced inter-sting insights. As seen in Fig. 1, LCI, scond and �r all respondedositively to increases in constriction severity. Unlike the MBW

ndices though, �r remained unaffected by the development ofirway closure. This is logical, considering closed airways do notarticipate in the distribution of ventilation at the mouth, but willtill be resolved as unventilated by an MRI. The ability of MRI indiceso detect airway closure may offer explanation to outliers in corre-ations of the two measures, as seen in recent preliminary clinicaltudies (Capaldi et al., 2015; Horn et al., 2015; Horsley et al., 2014).

As noted in Fig. 3, despite strong dependence on constric-ion severity, the three indices showed only weak relationshipsith constriction depth. While previous studies in the literature

Robinson et al., 2010; Verbanck et al., 1998, 2003) have shownensitivity of the MBW to constriction in the small airways, theesults in this study suggest that the more dominant response maye severity of airway constriction.

.2. Local MRI indices

The local imaging index �local exhibits a reversal of thisehaviour. As shown in Figs. 1, 3 and 4 the index shows strongorrelations to constriction depth, despite only weak correlationso severity. However, the results in 5 suggest a high sensitivity tooxel size, meaning interpretation must be made with referenceo the imaging resolution. As noted in Section 2.3 there are manyompeting standards for MRI indices within the literature. Otherhoices in the literature have attempted to normalise the effect ofeighbourhood size that �local exhibited, through scaling, and these of covariance (Tzeng et al., 2009). Within this work we haveemonstrated the importance and utility of local MRI indices, how-ver, the comparative evaluation of local MRI indices is a clear areaor future research.

The results presented in this study show clear relationships andistinctions between how MBW and imaging indices respond toronchoconstriction. Some of the trends, such as the failure of theBW under airway closure have been exhibited in previous studies

Mitchell et al., 2012). However, to our knowledge, this is the firsttudy that has investigated the relationship between MBW and MRIerived indices in such detail.

.3. Limitations of the study and future work

In considering the limitations of the study, it should be notedhat the baseline values of scond and LCI produced by the model

Neurobiology 235 (2017) 8–17

before constriction were lower than standard healthy valuesreported in the literature (Verbanck et al., 1997). This can bepartially explained as resulting from the simplifying assumptionsmade in the model, which led to an underestimation of heterogene-ity. The development of more complex models, which incorporateairway elasticity, dynamic compliance, and more detailed acinarregion structures, will be undertaken in future work.

The exhibited maximum values of scond were also higher thanvalues seen in the literature for patients exhibiting bronchocon-striction (Verbanck et al., 1997). The constriction of a small numberof airways at a specific depth, by identical amounts will natu-rally lead to sharp distinctions between the time constants ofhealthy and unhealthy regions. These distinctions would be sharperthan what would be seen clinically, where the transition betweenhealthy and unhealthy regions occurs as a spectrum. This differenceexplains the higher index values exhibited in severe VH regimes,as well as the observation that most significant index changesoccurred once constriction severity exceeded 50%. This aligns withthe aim of our study, which was to investigate how test indicesrespond to structural changes in the lung, and not to specificallyrecreate patient scenarios.

Finally, this study investigated bronchoconstriction within theconducting zone of the lungs. This is most typical of lung diseasessuch as asthma. However, other common lung diseases such asemphysema and chronic obstructive pulmonary disease (COPD) aremore strongly characterised by regional compliance heterogeneity.While other studies have incorporated disease-driven heterogene-ity in respiratory zone compliance (Henry et al., 2012; Milic-Emiliet al., 2007; Tawhai and Hunter, 2001), there are still large ques-tions surrounding the use of the MBW in detection of this type ofrespiratory zone heterogeneity. The development of detailed mod-els of the acinar regions of the lungs, and the application of thesemodels to the investigation of disease-driven regional complianceheterogeneity, is a future area for research.

5. Summary

The simulations performed in this study suggest that LCI, scond,and �r are all quite sensitive to increases in constriction severity.Under severe constriction, �r appears to remain a strong predic-tor, while the MBW indices decrease back towards baseline, asexpected. This leads to a strong correlation between the MBWindices and �r except under airway closure. Despite strong depend-ence on constriction severity, all three indices exhibit only weakdependence on constriction depth. Local VH imaging indices suchas �local appear to relate more strongly to constriction depth.However, the nature of the relationship is dependent on imageresolution, meaning care must be taken in interpretation.

Acknowledgments

BF was funded by the Rhodes Trust, UK (Rhodes Scholarship,Queensland, 2015). RB was funded by the European Union Frame-work 7 AirPROM (Airway Disease Predicting Outcomes throughPatient Specific Computational Modelling) project (Grant Agree-ment No. 270194).

Appendix A. Multiple breath washout indices

To calculate the major MBW indices, the expired volume, and

expired gas concentration must be measured for the duration of thewashout phase. The expired gas concentration is typically plottedagainst time, and referred to as the washout curve. An examplewashout curve for 3He is given in Fig. 6.

B.H. Foy et al. / Respiratory Physiology & Neurobiology 235 (2017) 8–17 15

1009080706050403020100Time (s)

0

0.005

0.01

0.015

0.02

0.025

0.03

Exh

aled

3H

e (%

)

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Exhaled Volume (L)

0

0.5

1

1.5

2

2.5

3

Exh

aled

3H

e (%

)

Breath No. 1

1009080706050403020100Time (s)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Flo

w (

L . s

-1)

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0.01

0.012

0.014

0.016

0.018

0.02

0.022

Fig. 6. Multiple breath washout indices. A simulated 3He multiple breath washout curve (top left). The profile for the 1st breath (right, black line) is given, with the sIII slopebefore normalisation overlaid (red dashed line). Inhaled volume (i.e. volume above FRC) for the test is also given (bottom right). The total set of s-values is given (bottom left),with the slope scond overlaid (black dashed line), and the index sacin plotted (grey triangle). These figures were generated from simulation of the baseline (no constriction)s

Liu1ppb

ccFpit

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where subscripts upper and lower denote the proximal and distalbranches in the bifurcation.

Each terminal branch i, is connected to an acinar region withvolume Vacin,i, whose expansion is driven by the flow rate in theterminal branch

cenario of this study.

Three major indices are commonly taken from an MBW test: theung Clearance Index, scond and sacin. The Lung Clearance Index (LCI)s the amount of time taken from the start of the washout phasentil mean exhaled concentration over an exhalation drops below/40th of the initial exhaled concentration. To normalise betweenatients, this time is expressed as the number of lung turnoverserformed, turnovers being the cumulative expired volume dividedy the individual’s functional residual capacity (FRC).

The other two measures derived from the MBW are moreomplex in definition. For each exhalation, a plot of exhaled gas con-entration over cumulative expired volume is generated, as seen inig. 6. Typically this plot undergoes four distinct phases, an initiallateau, a steep rise, a second plateau, and a final rise. An sIII value

s defined as the slope of the third phase of each curve, divided byhe mean exhaled concentration over the third phase.

The statistic scond is defined as the mean gradient of the sIII val-es between turnovers 1.5–6, and sacin as the sIII value from therst exhalation, minus the contribution from the scond slope. Anxample of this is given in Fig. 6. We note that typically in the lit-rature, models of the MBW without diffusion produce sIII curveshat extrapolate to zero at turnover 0 (Hamid et al., 2005). As seen inig. 6, the sIII curve from the baseline scenario of this model extrap-lates to a small, but non-zero amount. We believe that this is due

o the nature of the ventilation model, allowing for variation inentilation of different acinar regions, even in the baseline scenario.

Appendix B. Ventilation model

The velocity (u) in each branch in the lungs was assumed tobe proportional to the pressure difference across the branch (�P),such that

�P = R(Q )Q,

where Q = Au is the flow rate, and A is the branch cross-sectionalarea. The resistance function R(Q) is taken as a Poiseuille resistance,with a correction term for energy dissipation proposed by Pedleyet al. (1970). At each branch bifurcation, flow is conserved, meaning

Qupper =∑

Qlower,

d

dtVacin,i = Qi.

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H

6 B.H. Foy et al. / Respiratory Physi

inally, each Vacin,i remains proportional to the difference betweenleural pressure (Ppl) and pressure at the distal end of each terminalranch (PT,i), meaning

T,i − Ppl =1Ci

Vacin,i,

here Ci is the compliance of acinar region i. The pleural pressureas specific as sinusoidal with period 4 s, and amplitude chosen

o enforce tidal breathing of 1 L per breath cycle, above a func-ion residual capacity of 2.5 L. To close the system an atmosphericressure boundary condition (P = 0) was applied at the trachea.

ppendix C. Supplementary data

Supplementary data associated with this article can be found, inhe online version, at http://dx.doi.org/10.1016/j.resp.2016.09.009.

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