RespirationPhysiology(1979)n, 16l-172
@ Elsevier/North-Holland Biomedical Press
A STATISTICAL DESCRIPTION OF THE HUMAN
TRACHEOBRONCHIAL TREE GEOMETRY
T.T. SOONGT,P. NICOLAIDEST,C.P. YUr and S.C. SOONG2
I Facuhyof Engineeringand Applied Scimces,Stote University of New York at Buffalo, Buffato, NY 14214
and2 WittiamsvitleEastHigh School,E. Amherst,NY 1405t,U.S.A.
Abstrect.Most physiologicalstudieswhich makeuseof lung geometryhave utilizedaveragedeterministic
modelsof the tracheobronchialtree geometry,suchas Weibel's Model A (1963).However, as shown
by morphometric studies, it is well known that there are significant inter-subject and intra-subject
variabilitiesin the structuralcomponentsof the human lung. Hence, inherent inaccuraciesexist when
deterministicdimensionsfor lung geometryare used.In this paper, a statisticaldescriptionof the lung
geometryispresented,
Using Weibel'sModel A asthe underlyingaveragemodel,probability distributions
for the lengthsand the diametersof ainrays and for the number and volume of alveoli are proposed
basedon morphometric data.
As a check for consistency,the probability distribution of the functional residualcapacity is derived
from thoseassociatedwith airways and alveoli and it is comparedwith reporteddata. Resultsof this
comparisonare favorable,suggestingthat the statisticaldescriptionpresentedherein representsa selfconsistentmodel for lung geometrywhich can be used for studiesof problemsrelated to pulmonary
physiology.
Airways
Lung
Alveolar volume
Functional residualcapacity
Lung volume
Morphometry
An accurateknowledge
of themorphometryof the respiratorytract is of fundamental
importancein the studyof a wide spectrumof problemsin pulmonary physiology,
rangingfrom generalanatomicinterestin the tracheobronchialtree itself to particle
depositionand gasdiffusionproblemsin the lung. The first carefulmorphometric
studyof the humanlung was caried out by Weibel (1963),which was followed
by numerousstudiesin order to establishan accurategeometricmodel for the
tree structure.More recently,Lfr extensivelung morphometry
tracheobronchial
studywasundertakenby Raabeet ol. (i976), which presentsthe most extensive
Acceptedfor publication 20 Febrwry 1979
l 6l
t62
T.T. SOONG er al
measurements
to date of the lung geometryfrom the tracheadown to and including
the tersrinalbronchiolesfiorhurnanand other rnammalianspecies.
It is clear from thesemorphometric studiesthat great inter-subjectvariabilities
exist in the geometric descipcion of the tracheobronchial tree. Furthermore,
variabilities of dimensionsand branchingsamong individual airway branchesand
alveoli in a gven zubjectprecludethe constructionof a simple yet accurate lung
model.Hence,to adcrpta simplemodelsuchaslVeibel's23-generation
model(1963),
variationsof geometriesof individual airwayswithin eachgenefirtionare significant
and a quantitativedescriptionof the variationsis helpful towardsconstructng a
more realistichacheobronchialtree structure.
Basedupon availablemeasurements,
this paper presentsa statisticaldescription
of thehumantrachrcbronchialtree.Probabilitydistributiors of airwaysand alveoli
are presented,which in turn iead to a probabilistic descriptionof the functional
residualcapacity (FRC). The FRC distribution is shown to be consistentwith
availableFRC measurements.
These distributions are formulatedon the basis
that Weibel's model of the human lung with dichotomy providesan adequate
averagestructure.
Weibel'sModel for fluman Tracheobronchial
Tree
The tracheobronchialtree(Model A) derivedby Weibel (1963)basedupon careful
anatomic measurements
has been widely used by hcalth physiciststo represent
the overalltree geometryof an averageadult human lung. It is also usedhere as
the basic averagemodel upon which the statistical descriptionis based. This
averagemodelassumesthat the respiratorytract is partitioned into 23 generations
of dichotomousbranchingsfrom thetracheato thelast order of terminalbronchioles.
ln orderto describetherespiratorytract in a statisticalframework,it is noteworthy
to discussthe principalcomponentsof the treegeometryas'givenby Weibel'smodel.
Beginningat the trachea(generation0) there are23 generationsof aimvayswhose
averagelengths,diameters,areas,and volumesare modelledby empiricalequations.
The modelalso predictsthe averagenumberof alveoli adheredto eachof the last
sevenairwaygenerations.
Weibel'sanatomic measurements
consistedof those from a singleplastic cast
of largerainrays and histologicalsectionsfrom smallerones.Thesesectionswere
inflated to the equivalentexpansionof 4800cm3lung volume of 314total lung
capacity(TLC) ascitedby Weibeland Gomez(1962)and Weibel(1963).The dead
spaceand alveolar ductsa@ountfor approximatelyll3 of the total lung volume
with the remainingvolume due to the alveolarair spaces.The linear dimensions
of the airways,the voluo6, and the numberof alveoli for eachgenerationas grven
by Weibel'sModel A are summarizedin table l.
163
HUMAN TRACHEOBRONCHIAL TREE GEOMETRY
TABLE I
Dimensions
of Weibel'sModelA
Generation
Number of
branches
Diameter
(cm)
Length
(cm)
Volume
(cm3)
0
I
2
3
4
5
6
7
I
9
t0
il
I
2
4
8
t6
32
64
t28
256
512
1024
2W
&96
8192
16384
32768
65536
13t072
262rU
524288
1M8576
2097t52
4t94n4
8388608
1.800
t.220
0,830
0.560
0.450
0.350
0.280
0.8a
0.186
0.154
0.t30
0.t09
0.095
0.082
o,o74
0.066
0.060
0.054
0.050
0.047
0.045
0,043
0.041
0.041
12.0w
4.760
1.900
0.760
r.270
t.070
0.900
0.760
0.640
0.5/m
0.460
0.390
30.500
n.250
3.970
t.520
3.&0
3.300
3.53
3.8s
4.45
5.t7
6.21
7.56
9.82
t2.45
t2
t3
t4
l5
l6
t7
t8
19
20
2l
22
23
o$a
0.270
0.230
0.200
0.165
0.141
0.n7
0.099
0.083
0.070
0.05e
0.050
rc.q
2t.70
29.70
4t.80
6t.10
93.20
r 39.50
224.30
350.00
59t.00
Number of
alveoti(105)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
2A
60
2lo
4t5
840
t430
Strrrtural Distributions
PROBABILITY
DISTRIBUTIONS
OFAIRWAYS
UsingWeibel'sModel A as the basicstructure,the first major task in establishing
a probabilisticmodelfor eachgenerationof ainrays is to estimatefrom available
morphometricmeasurements
sorneof the statisticalparametersof eachdistribution.
FiguresI and 2 presentestimatesof olp, the coefficientof variation (ratio of
standarddeviationto mean),of diametersand lengthsof ainvaysas functionsof
aiway generation
on thebasisof datasetsdueto Weibel(1963),Raabeet al. (1976),
Jesseph
andMerendino(1957),Parkeret al. (1971),andHansenandAmpaya(1975).
Oncethe meanvaluesfor the generationdiametersand lengthsare givenin table l,
the respective
standarddeviationsinay be easilye,alculated
from the ratios. In the
presentanalysis,the numberof airways per generationare not varied from the
dichotomyof Weibel'smodel.
t&
T.T. SOONG er aL
! llm.ctl€?s|
O &itlglr(l$7l
g
o-7
-9 0.6
€
L
^ct
>
05
b
C'
ItO or
C'
eoz
Fig. l- Coefftcientsof variation for 23 generationsof airway diameter: data and curve fit.
I
t
-
C'
b
s
rts
G'
.ll
b
o
G'
(,
+
Fig' 2. Coellicients of variation for 23 generationsof ainvay length: data
and curve fit.
In addition to the meanvaluesand standarddeviations,probability dis6ibutions
of airwaydiametersand lengthsfor eachgeneratnncan be postulated.On physical
as well as mathematicalground,natural candidatesfor their distributionsare lognormal and sammadistributionswhich preservepositivity of thesequantitiesand
'reflifot
the factiGiihe randomness
in thesedimensionsis a result of many random
sourcesof variabiiity. Thesedistributionswill be presentedfollowing a discussion
on the lung volumeadjustment.
\ - - - - a- - - *f u - 5
HUMAN TRACHEOBRONCHIAL TREE GEOMETRY
165
PROBABILITY DISTRIBUTIONS OF ALVEOLAR VOLUME AND NUMBER
In addition to airway variability, the variability of the alveolar volume was also
were usedto establishan average
investigatedby Weibel. Histological measurements
alveolar volume of 1.05x l0-t cm3.The male adult subject data of that study were
used to obtain a coefficient of variation of 0.2 which, together with the average
volume, yields the required standard deviation. Since random tissue sampleswere
usedin the lVeibel study, all alveoli in the presentmodel are assumedto have the
samecoefficientof variation irrespectiveof their correspondinggeneration.
A major obstacleencounteredin the counting of alveoli in a tissue sample with
two-dimensionaltechniquesis the relative inability to distinguish between alveoli
and their adjoining sacsand ducts as pointed out by Weibel (1963).Due to these
visual uncertainties,a coefficient of variation of 0.07 is used for the number of
alveoliper generation;a somewhathigher figure than that reported by Weibel.
The flusttwo central momentscan be calculatedfor the airway lengths, diameters,
number of alveoli and alveolar volume using the coefficients of variation shown
in figs. I and 2 and the averagevalues given in table l. These moments may be
verified by deriving an expression for a widely documented respiratory measure,
namely,the functional residual capacity (FRC).
FORFRC
DERIVED
DISTRIBUTION
The normal FRC is the volume of gps which is present in the respiratory system
at the beginningof quiet inhalation. In order that the FRC may be formulated,
a specificconfiguration for the ainvays is required to calculate the volume for each
generation.
For many flow calculations,a complicatedpathway may be treatedas if it were
composed of simple cylinders. Althougilr the airway branches are not cylinders,
the useof a cylindrical approximation alleviatesthe need for statistical parameters
for the airway volumes beyond those found for the linear dimensions.In addition,
the accuracyof this approximation may be assessedfrom Weibel's data.
Given dichotomy of Weibel's Model A, the calculated ainvay volume of the
ith generationmay be approximated as
2.
vft'= T (rEl4)d?u,ltu
(t)
k- l
whered* and l,* representthe kth ainuay diameterand length for the ith generation,
respectively.Given the cylindrical configuration, the FRC is defined as
21
23
FRC: I vfq+ I u"oo,
i= 0
i= 1 7
o
(2)
lffi
T.T. soo NG et at.
where vfo is given by eq- l, v" is the average
alveolar volume ancl n", is the number
of alveoli in generation i. since the alveolar
volume is assumed to be independent
of generation, q. 2 may be given by
FRC:
23
ufo v'\
,Io *
(3)
where \ denotes the total number of alveoli
in the lung. A substitution of
the
average model values and the average
atveotar volume into eq. 3 yields
a
FRc
which is about 99% of weibel's Model A
at il4 TLc. In the present analysis,
rhe
variables in eq' 3 are random with derived
statistical moments, and hence the
FRC
is a random variable.
From Appendix A, the expectedvalue
of FRc given by q. 3 is found to be
F,[FRq :
23
E
QtnH)Qt',,
+ o|)n, * I;".FN.
(4)
wherep and o denote the mean and
the standard deviation of the subscripted
randomvariables'It is noted that q. 4
is obtained by assuuingthat (. and
1,,
are independentrandomvariablesfor each
i. This assimption is in part justified
by weibel's observationthat the length-to-diameter
ratios of individual aimray
elementsvaryover averywiderange.An
examinationof the lengrh-diameter
scatter
diagnms for generations5 through t0 grvar
by weibel (1963)showsthat the
approximation
ElditiJ = E[dlJE[t,J
(s)
is adequate'calculationsusingtable I
and standarddeviationsfound from figs.
I
and2 yield an expectedvalueof 4Tg4.6
cm3.
The varianceof FRC lies betweenthe
two extremesas givenby eqs.A6 and Az.
Considerthe caseof qymmetricaldichotony
in which Ju Ja* _ di and lu = l* = li,
eq.A7 in AppendixA gves
var[FRQf
:
e'ttl4)'{@1,
+ 6pl,ol)Qrl+
o?,)
- Qrl,* o))rr?,lf
[,fl
+ lQrl.+oj.)?t'*.+ oi,.)- @!,yr*.)J
(6)
yielding,after substitutioninto
e9. 6, a standarddeviationof 9g6.3cm!.
Thedimensionsgivenin table i ure
basedupon a FRC at 3l4TLc as computed
previously'It is known that
expectedvalue of FRC is about l
12 TLc
-normally
(3200cm3usingweibel's model).
Thus,tne aimensionsand statisticsof the present
modelneedto be corrected.
If uniform expansionand contraction
are assumedto take placein the airways
andalveoliof the lung,thena linear
scalefactoris usedto scalethe 3l4TLc data.
Although uniform ainvay compliance
is iir*rr..t throughoutthe lung as reported
r67
HU MA}{ TRACHEOBRONCHIAL TREE GEOMETRY
by Marshall and Holden (1963), there is evidencethat volumetric changes are
proportionai to the cube of linear dimensionsas discussedby Hughes et al. (1972)
and Witson et al. fi974). We define the linear scalefactor, s' as
(7)
s : (320014784.6)rt3
such that the linear dimensions and alveolar volume along with their standard
deviationsare correctedto maintain the coefficientsof variation previously stated.
The results are summarizedin table 2. The present correction factor corresponds
to a uniform linear contraction of 12.5%which is quite close to the appropriate
i' ; ' ' - t ' {
2
x
b'8?b
of variation (olpl corresponding
Linearairway dimensions,volume calculationsand coelTicients
to a FRC of 3200cm3
Generation
Diameter (cm)
oltt
oltt
0
I
2
3
4
5
6
7
I
9
t0
ll
t2
l3
t4
l5
l6
t7
l8
l9
20
2l
22
23
t.574
1.067
0.726
0.490
0.394
0.306
0.245
0.201
0.163
0.t35
0.1l4
0.095
0.083
0.072
0.065
0.058
0.052
0.047
0.04
0.Ml
0.039
0.038
0.036
0.036
0.10
0.125
0.15
0.t75
0.20
0.23
0.275
0.325
0.35
0.42
0.50
0.575
0.66
0.675
0.60
0.50
0.40
0.30
o.22
0.15
0.13
0.10
0.10
0.098
t0.494
4.t6t
1.662
0.665
Llll
0.936
0.787
0.665
0.560
0,472
0.402
0.341
0.289
0.236
0.201
0.r75
0.r&
0.123
0.102
0.087
0.073
0.061
0.052
0.04
c
20.43
7.M3
2.750
1.002
2.t61
2.203
2.372
2.703
2.977
3.444
4.t82
4.985
6.fi8
7.812
10.839
14.995
20.48
28.308
40.277
60.227
92.575
t42.579
218.510
370.355
0.10
0.15
0.25
0.30
0.35
0,425
0.50
0.575
0.65
0.70
0.75
0.80
0.81
0.775
0.725
0.65
0.50
0.40
0.30
0.225
0.175
0.t5
0.rI
0.10
olp
Source
Alveolusvolume(cm3)
Number of alveoli
FRC (cm3)
Volume (cml)
Length (cm)
7.022x l0-6
2.981x lOt
3.200x 103
2.0 x t0-l
7.0 x lO-2
2.1 x lO-2
t68
T.T. SOONG et at.
0.4
o.t
2m no 320 3ro +n rrc sao $o
Funclimol
(cmtrlOl
Rctiduol
Copocrty
Fig' 3' c-omparisonof thecumulativedistriburion functions
for the postulatedFRC normal distribution,
N(3200.0,619.6ir,and subject dala,.
scalingcorrection
for thenonhomoBEneous
caseasreportedby Marshalland Holden
(1963)and Pedleyet al. (t970).
TheFRC dataforadurtmaresreportedby Heyder et (1975),
oI.
Bakeet ar. (rg74),
saidelet al' (l97ll and Palmeset al. (19i3) were
usedio **p are apostulated
normal distribution with the FRC statistics($ = 32w
cD!, o ='659.65lmr) after
the sourcedatais scaledto correspondto 32a0cmr,
The datastandarddeviationis
649'6cfr3, quite closeto that for the derivedFRc.
A plot of the cumulative
rr.fc o20
0
050 1.00 f.ro
J0
2n
2.n
AirroyDiometer
(cml
Fig- 4. log-normal density functions for airway
diametersby generation.
HUMAN TRACHEOBRONCHIAL TREE
GEOMETRY
0.800.t2 0J3
c
169
0.80 t.20 t.80
AirwoyLength(cml
Fig. 5. Log-normaldensityfunctionsfor airway lengths
by generation.
distributionsfor the postulatedFRc distribution
and the data are shown in fig. 3.
The fit appearsto be good despitethe small number
of subjectsin the study group.
Basedupon this preliminary fit the FRC appears
to have a normal distribution
for the adult male population. It is also noted
that the normality assumptionis
reasonablefrom the theoreticalpoint of view. Since
the FRC is given by eq. 3 as
a sumof a large number of independentrandom
variables,it is expectedto have
a normal distribution on the basisof the central
Limit Theorem.
As was discussedearlier, the distributionsfor
the airway lengthsand diameters
may be adequatelyapproximated by log-normal
or gammadistributions. Given
lung volume corrections,both tpm yirro similar
shapesfor the distributions.
Figures 4 and 5 give the probauitity density
funoion, ro, aimvaylengths and
diametersper generationusinglog-normaldistributions.
Conclusions
We have presenteda statistical descriptionof
the human tracheobronchial tree
that is compatiblewith measurementsreported
in the literature. using weibel,s
23'generationmodet as a basis for the irrrug
model, statistics associatedwith
airwaysand alveoliare obtained.Thesestatistics
are in turn used in the derivation
of a probability distribution for FRC, which is
shown to be consistenrwith FRc
measure'monts'
This verificationsuggests
that the resultsconstitutea self-consistent
statisticaldescriptionof the human tracheobronchial
tree.
It is noted that, since the statistical information
is presentedin the form of
standard deviaticn'to'rnean ratios, an average model
other than that of weibel
could also be used.The choiceof weibel's Model
A is made for conveniencesince
it has beenusedwidely in deterministicstudies.
na
T. T. SOONGer sl.
In lieht of inter-subject variabilities observed in studies
of lung geomerry,
statistiel approaches to many related subjects, such as gas
diffusi.rn and particle
deposition in the human lung becomeincreasinglyneces*ty. physiological
problems
which incorporate deterministic models suchas weibelt
moael may aliow additional
studies to quantify the effect of a random geometry on
various functions of the
human lung- This is espec:rrlly true in studies wnict rely
on the comparison of
predicted results with observed subject data in order
to
the acceptability of
a particular model.
"rr"r,
Acknowledgerents
This work was supported by Grant No. ES l23g from
the Nationar Institute of
Environmental Health Scienceand Grant No. ENG
7612269from the Nationar
ScienceFoundation.
AppendixA
From eq- 3 the functional residual epacity(FRc)
23
is defined as
2l
FRc=,Iop,t"nalt;1+v"\
(At)
where d1 and lil Nrc,respectivcly, the kt?r ainay
dbmeter and lengh in gencrationi and dichotomy
is assumedfor the arway structure.
For each i' the random varhbles dir and-liv aftas3unred
to be identiallydietriburcd with respective
meansFa, and Ft, a'ndvarianceso'0,
. some conelation is expccted to cxist bctween
d;; and d1
^\d.of
0*k)' betweenliiandl*6*k),
and
ueiiueendiiandt;1. Becauseof hck of informa:.ion
on thcse
correlations, two extreme casescan be assumed;.ihe
tini assumescomptete statistical independence
among all random variables d;1 and l;1 and, in
the secendcase,drj and d1 are perf*tly conelatd
with d;;: dik = di, similar assumntl.onrr3rds
for l;; and l;1, but sa'tistlcal inoepe' dence bewo,n
dii
and l;1. The secondcaseimplies random Uut
symm.irieal dichotomy.
Additional assumptions are nede.d on statistical
interdependenccamong airway geometries in
different generations' In what foltows, statistical
independenceis assumed among aiway diameters
and lengthsin ditferent generations,
Taking erpectationof both sidesof eq. Al and
usingeqs.Bl-84, we havc
23
2l
EtFRcl=,Io
lrrt"nntdit*l + E[v"\1
#{2)
Assuming that d;1 and l;1 are independent
and that the alveolar volume is indepurdent of the number
of alveoii, eq.A2 becomes
23
EtFRq:,I^
2l
.E.unetdAlEtt*l + E[v"]E[NJ
i= 0 k- l
E[FRql = .I^ Qh1+y1rt1,
+ o|,Jpt,*#v.rrN.
i= 0
(,{3)
(44)
HUMAN TRACHEOBRONCHIAL TREE GEOMETRY
I7I
wherep and d denote the meanand varianceof the corresponding
subscriptedvariables.
Assumingthat the total airway volume is independentof
the totar alveolar volume, the varianceof
FRC is given by
:
var[FRQI
+ varlva\l
"".[f, fi ,*,oroe,,-]
(A5)
using eq. 86.
For the lirst casewherestatisticalindependenceis assumedfor
airway diametersand lengths in a given
generation,eq. A5 gives,usingeqs. 86-89,
var[FRQf
:
23
,!o
6pj,o3,l@?,
F'hr4r2{Qtf,,+
+ oi,l- Qre,+o|)1i;)
+ [@3.+ o|.)@k-+ oir.)- p2u.pk,J
(A6)
In the secondcase'when d,i = dft: di, l;1= l;1= l;, oDd d; and
l; are independent,the varianceof
FRC takesthe form
var[FRQI=
23
,Io
l?hlq2 fttf,,+ 6p],o3,lta?,
t o2t,)
+ 62il2r1,1J
- QrA,
+ f@1.+o2".)@k.+
dft.)- p?Jrk.)
(A7)
AppendixB
kt X andY berandomvariables
andlet c bea constant.
Then[see,for example,
Meyer(lg7z))
l. E[c]= c
2. EtX+Yl= Etxl + E[y]
3. E[cX] = cE[X]
4. E[XYI = E[X]E[yJ; X,y independenr
5. var[cX] = cfuar[X]
6. var[X+ Y] = var[X] + var[y]; X,y independent
7. varlXl EfX2) - nztXl
8' lf x hasmeanp andvadanc'e
d andY=h(x), thenexpanding
h(X) aboutthemean
of X approximarcs
thernoments
of y as
EtYl =h($) + (U2)*G2h1dx\1*_o
var[Y] = o2(dhld)()2lx
_o
(Bl)
(821
(83)
(841
(Bs)
(86)
(87)
(88)
(Be)
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