RESEARCH ARTICLE
Modelling he dynamics o ube culosis
lesions in a i ual lung: Role o he b onchial
ee in endogenous ein ec ion
Ma ı
´Ca alà
1,2
, Jo di Bechini
3
, Mon se a Tenesa
3
, Rica do Pe
´ ez
3
, Ma iano Moya
3
,
C is ina VilaplanaID
4,5
, Joaquim VallsID
2
, Se gio AlonsoID
2
, Daniel Lo
´pez
2
, Pe e-
Joan Ca donaID
1,4,5
, Cla a P a sID
2
*
1Compa a i e Medicine and Bioimage Cen e o Ca alonia (CMCiB), Fundacio
´Ins i u d’In es igacio
´en
Ciències de la Salu Ge mans T ias i Pujol, Badalona, Ca alonia, Spain, 2Depa amen de Fı
´sica, Uni e si a
Poli ècnica de Ca alunya, Cas ellde els, Ba celona, Ca alonia, Spain, 3Se ei de Radiodiagnòs ic, Hospi al
Uni e si a i Ge mans T ias i Pujol, Badalona, Ca alonia, Spain, 4Expe imen al Tube culosis Uni , Fundacio
´
Ins i u d’In es igacio
´en Ciències de la Salu Ge mans T ias i Pujol, Uni e si a Au ònoma de Ba celona, Can
Ru i Campus, Edi ici Ma , Badalona, Ca alonia, Spain, 5Cen o de In es igacio
´n Biome
´dica en Red de
En e medades Respi a o ias, Mad id, Spain
*[email p o ec ed]
Abs ac
Tube culosis (TB) is an in ec ious disease ha s ill causes mo e han 1.5 million dea hs
annually. The Wo ld Heal h O ganiza ion es ima es ha a ound 30% o he wo ld’s popula-
ion is la en ly in ec ed. Howe e , he mechanisms esponsible o 10% o his ese e (i.e.,
o he la en ly in ec ed popula ion) de eloping an ac i e disease a e no ully unde s ood,
ye . The dynamic hypo hesis sugges s ha endogenous ein ec ion has an impo an ole in
main aining la en in ec ion. In o de o examine his hypo hesis o alsi iabili y, an agen -
based model o g ow h, me ging, and p oli e a ion o TB lesions was implemen ed in a
compu a ional b onchial ee, buil wi h an i e a i e algo i hm o he gene a ion o b onchial
bi u ca ions and ubes applied inside a i ual 3D pulmona y su ace. The compu a ional
model was ed and pa ame e ized wi h compu ed omog aphy (CT) expe imen al da a om
5 la en ly in ec ed minipigs. Fi s , we used CT images o econs uc he i ual pulmona y
su aces whe e b onchial ees a e buil . Then, CT da a abou TB lesion’ size and loca ion o
each minipig we e used in he pa ame e iza ion p ocess. The model’s ou come p o ides
spa ial and size dis ibu ions o TB lesions ha success ully ep oduced expe imen al da a,
hus ein o cing he ole o he b onchial ee as he spa ial s uc u e igge ing endogenous
ein ec ion. A sensi i i y analysis o he model shows ha he inal numbe o lesions is
s ongly ela ed wi h he endogenous ein ec ion equency and maximum g ow h a e o he
lesions, while hei mean diame e mainly depends on he spa ial sp eading o new lesions
and he maximum adius. Finally, he model was used as an in silico expe imen al pla o m
o explo e he ansi ion om la en in ec ion o ac i e disease, iden i ying wo main igge ing
ac o s: a high in lamma o y esponse and he combina ion o a mode a e in lamma o y
esponse wi h a small b ea hing ampli ude.
PLOS COMPUTATIONAL BIOLOGY
PLOS Compu a ional Biology | h ps://doi.o g/10.1371/jou nal.pcbi.1007772 May 20, 2020 1 / 25
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OPEN ACCESS
Ci a ion: Ca alàM, Bechini J, Tenesa M, Pe
´ ez R,
Moya M, Vilaplana C, e al. (2020) Modelling he
dynamics o ube culosis lesions in a i ual lung:
Role o he b onchial ee in endogenous
ein ec ion. PLoS Compu Biol 16(5): e1007772.
h ps://doi.o g/10.1371/jou nal.pcbi.1007772
Edi o : Dominik Woda z, Uni e si y o Cali o nia
I ine, UNITED STATES
Recei ed: June 5, 2019
Accep ed: Ma ch 4, 2020
Published: May 20, 2020
Copy igh : ©2020 Ca alàe al. This is an open
access a icle dis ibu ed unde he e ms o he
C ea i e Commons A ibu ion License, which
pe mi s un es ic ed use, dis ibu ion, and
ep oduc ion in any medium, p o ided he o iginal
au ho and sou ce a e c edi ed.
Da a A ailabili y S a emen : All ele an da a a e
wi hin he manusc ip and i s Suppo ing
In o ma ion iles.
Funding: CP, PJC and MC ecei ed unding om
“la Caixa” Founda ion (ID 100010434), unde
ag eemen LCF/PR/GN17/50300003; CV, PJC, MM,
JB, MT and RP ecei ed unding om Agència de
Ges io
´d’Aju s Uni e si a is i de Rece ca (AGAUR),
G up Uni a de Tube culosi Expe imen al, 2017-
SGR-500; CP, DL, SA, MC, JV ecei ed unding
om Minis e io de Ciencia, Inno acio
´n y
Au ho summa y
Tube culosis is, e en oday, among he 10 main causes o dea h in he wo ld. Despi e he
e ec i eness o cu en s a egies o igh he disease and hose ha a e unde de elop-
men , he huge ese oi o la en ly in ec ed indi iduals is a big hind ance in i s e adica-
ion. One o he challenges inhe en in his p oblem is ha he mechanisms ha cause
la en in ec ion o e ol e owa ds ac i e disease a e no ully unde s ood. Why will 90% o
in ec ed indi iduals ne e de elop an ac i e disease? In o he wo ds, wha a e he main
ac o s ha igge an ac i e disease in 10% o cases? We ha e ocused ou e o s on
unde s anding he mechanisms ha allow keeping in ec ion la en , especially hose ela ed
wi h endogenous ein ec ion. Since i is supposed o occu h ough he b onchial ee, we
ha e designed a 3D compu a ional model ha mimics his s uc u e, in which we ha e
implemen ed an agen -based model o lesion g ow h and p oli e a ion. Ou esul s we e
con as ed wi h compu ed omog aphy measu emen s in la en ly in ec ed minipigs, p o-
iding success ul esul s ha ein o ce he essen ial ole o endogenous ein ec ion
h ough he b onchial ee in keeping in ec ion la en .
In oduc ion
Tube culosis (TB) is an in ec ious disease ha in 2017 killed mo e han 1.6 million people.
Mycobac e ium ube culosis (M b) causes TB, and his bac e ium is he indi idual agen caus-
ing he highes mo ali y wo ldwide [1]. The Wo ld Heal h O ganiza ion (WHO) es ima es
ha 25 o 30% o he popula ion wo ldwide is in ec ed wi h M b, and ha a ound 10% o
in ec ed people will de elop ac i e ube culosis (ATB) in a ew yea s’ ime [2], al hough hese
pe cen ages a e being ques ioned and e- isi ed by ecen s udies [3]. WHO also es ima es ha
10 million humans de eloped ATB in 2017 [2].
TB in ec ion s a s a a pulmona y al eolus when M b is phagocy ed by an al eola mac o-
phage (AM). The M b esis s bac e icidal mechanisms induced by AM and eplica es inside
he mac ophage [4]. Unde p ope in i o condi ions M b eplica es once a day [5]. When he
in acellula bac e ial load o e comes he AM’s maximum ole abili y, mac ophage nec osis is
igge ed, he eby e u ning bacilli o he ex acellula milieu. These bacilli a e phagocy ed by
o he AMs and he cycle begins again gi ing ise o a u he inc ease in bacilli. The u he
inclusion o mo e AMs ails o con ol bacilla y g ow h. The dea h o AMs igge s a local
in lamma o y esponse i s , and hen a speci ic immune esponse, which inally con ols he
in ec ion. The end o he p og essi e in ec ion lea es an encapsula ed TB lesion [6]. Acco ding
o he dynamic hypo hesis o Ca dona [7], he e is a ce ain p obabili y ha a ew bacilli will
escape om he lesion, mos ly inside a oamy mac ophage, and s a a new in ec ion in ano he
al eolus. This p ocess is assumed o occu h ough he b onchial ee, and i is wha we deno e
as endogenous ein ec ion. This includes no only new in ec ions gene a ed om he ini ial
in ec ion si e, bu also hose o igina ed in successi e in ec ion oci (Fig 1). An M b in ec ion
may be comple ely clea ed by he o ganism [8], i may en e in o a la en s a e in which he
hos is in ec ed bu no sick and canno in ec o he people, called La en Tube culosis In ec-
ion (LTBI), o , i he immune and in lamma o y esponses a e no well balanced, he hos
may de elop Ac i e Tube culosis disease (ATB).
Sys ems biology and compu a ional models a e ui ul ools o inc easing unde s anding
o he p ocesses in ol ed in TB [1]. Recen ly, di e en models ha e been use ul in iden i ying
se e al TB key ac o s [9–13]. In pa icula , he Bubble model sugges s ha he coalescence o
closed lesions is he main mechanism o he g ow h o lesions in animals ha p og ess o
PLOS COMPUTATIONAL BIOLOGY
Dynamics o TB lesions in a i ual lung
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Uni e sidades and FEDER, wi h he p ojec
PGC2018-095456-B-I00; CVM has a Miguel Se e
II con ac unded by he Ins i u o Ca los III (ISCIII,
CPII18/00031). The unde s had no ole in s udy
design, da a collec ion and analysis, decision o
publish, o p epa a ion o he manusc ip .
Compe ing in e es s: The au ho s ha e decla ed
ha no compe ing in e es s exis .
ATB [14]. This model success ully explains expe imen al obse a ions in mice [15]. The Bub-
ble model assumes a gene alised logis ic (Richa d’s cu e) [16] g ow h o lesions, d i en by he
in lamma o y and immune esponses, wi h hei p oli e a ion acco ding o he endogenous
ein ec ion heo y, and a me ging be ween neighbou ing lesions when hey a e close enough.
The model success ully ep oduced ATB obse ed in C3HeB/FeJ mice, demons a ing he
impo ance o local in lamma ion, lesion p oli e a ion, and coalescence in he igge ing o
ac i e disease. These esul s a e ele an o mouse models; howe e , hey a e no easily ex ap-
ola ed o humans, because o he di e ences be ween he s uc u e o he lungs in he wo spe-
cies, in addi ion o he well-known di e ences in immune sys ems and encapsula ion capaci y.
Ac ually, he s uc u e o he lungs may play an impo an ole in he in ec ion dynamics o
TB. On he one hand, endogenous ein ec ion occu s mainly h ough he b onchial ee, and
mice ha e much simple pulmona y s uc u e han humans, as no seconda y lobula s uc u e
is ound in mice ( hey ha e li le o none in e lobula sep ae) [17]. On he o he hand, he
encapsula ion o lesions is d i en by ib oblas s and ib in om pulmona y memb anes like
in alobula sep ae. Ne e heless, mice do no possess in alobula sep ae and lesion encapsula-
ion is no possible excep close o he pleu a [18]. In addi ion, he immune esponse o
C3HeB/FeJ mice is much less e ec i e han ha o humans. In humans, a balanced Th1
immune esponse usually akes place a e TB in ec ion [19]. As mice’s immune esponse is
no s ong and encapsula ion is no possible, hese animal models canno de elop an LTBI si -
ua ion and all expe imen al obse a ions show ATB cases [15].
Al hough pigs and humans sha e a g ea deal o ana omy and physiology, esea che s a ely
employ pigs as in i o models o TB. Ye hei immune sys em and lung s uc u e a e pa icu-
la ly close o he co esponding sys em and s uc u e in humans. Thus, TB de elopmen in
Fig 1. Main ea u es o dynamic hypo hesis. Schema ic ep esen a ion o ini ial in ec ion (A, whi e a ow) lesions’ g ow h (A–
F), endogenous ein ec ion (C–E, yellow a ows) and coalescence o neighbou ing lesions (F).
h ps://doi.o g/10.1371/jou nal.pcbi.1007772.g001
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Dynamics o TB lesions in a i ual lung
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pigs is mo e simila o ha in humans han in mouse models [19,20]. Minipigs a e a gene i-
cally selec ed species, which is mo e con enien han o he pigs o expe imen s in a lab,
mainly o size easons. Expe imen al esul s in TB in minipigs esemble pa hological indings
desc ibed in human [21–23].
In his s udy we aim o adap and implemen he Bubble model in a i ual b onchial ee in
o de o unde s and he main enance o LTBI in minipigs. In pa icula , we wan o es he
alsi iabili y o he dynamic hypo hesis o Ca dona [7] ha explains his main enance, as well
as o ob ain some o de s o magni ude o i s dynamics. We use expe imen al minipig TB da a
o une he model [23]. Wi h he new model we pe o m se e al in silico expe imen s, which
success ully ep oduce expe imen al obse a ions, and, u he mo e, pe mi us o sys ema i-
cally explo e he ansi ion be ween ATB and LTBI.
In Ma e ials and Me hods we desc ibe he CT expe imen al da a, as well as he wo sub-
models used in simula ions, which co espond o he compu a ional lung and he e ised Bub-
ble model. We inish his sec ion p o iding de ails o he model’s implemen a ion and he
me hodology used o i s pa ame e iza ion and sensi i i y analysis. Resul s’ sec ion s a s wi h
an analysis o he compu a ional lung ob ained. Then, i p o ides he esul s o he model’s i -
ing o expe imen al da a and he sensi i i y analysis. Subsequen simula ion se ies a e used o
explo e he e ec o he ini ial con igu a ion and o es he ansi ion be ween LTBI and ATB
in minipigs. Finally, he conclusions o his s udy and hei implica ions in es ing he alsi i-
abili y o he dynamic hypo hesis a e d awn.
Ma e ials and me hods
Compu e omog aphy measu emen s o LTBI in minipigs
Six emale speci ic pa hogen- ee (sp ) minipigs we e in a acheally in ec ed by H37R Pas-
eu s ains o M b (10
3
CFU) unde seda ion [23]. They we e eu hanized wel e weeks pos -
in ec ion, wi hou ha ing ecei ed any TB ea men . None o hem had de eloped ATB
symp oms.
B oncho-pulmona y pieces we e ob ained om all minipigs and analysed wi h mul ide ec-
o compu ed omog aphy scan (CT), a GE Ligh Speed VCT wi h high image esolu ion
(64-slice). The mo phoana omical s udy was ca ied ou wi h olume ende ing so wa e. Fo
each o he minipigs, we eco ded he numbe o lesions, hei loca ion and size, Houns ield
uni s, and dis ance o closes pleu a. One o he analysed minipigs did no p esen any lesions;
i was conside ed non-in ec ed o echnical easons and excluded om he subsequen
analysis.
The 5 in ec ed animals showed 165 lesions in o al, 33 ±22 pe minipig. These lesions a e
shown in Fig 2. The mean diame e o he lesions was 1.3 ±0.2 mm. Lesions we e loca ed in
each minipig wi h a mean dispe sion o 16 ±4 mm. The numbe o lesions and hei posi ions
we e used o ain he compu a ional model [24].
E hics s a emen . All e hical equi emen s we e ollowed acco ding o Di ec i e 201/63/
EU, and he p o ocol and p ocedu es o he s udy we e app o ed by he co esponding e hical
commi ee on animal wel a e and he Ca alan Go e nmen (Pe mi numbe : 5796). All ani-
mals we e eu hanized a week 12 pos -in ec ion by in a enous injec ion o sodium
pen oba bi al.
Compu a ional b onchial ee
The main no el y o his modelling app oach is he use o an explici 3D space ha esembles a
pulmona y b onchial ee. The design o his explici space equi es he building o a compu a-
ional b onchial ee inside a ce ain pulmona y olume, limi ed by he ex e nal su ace. The
PLOS COMPUTATIONAL BIOLOGY
Dynamics o TB lesions in a i ual lung
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geome ical in o ma ion necessa y o his model can be ob ained om pulmona y CT images
o he s udied minipigs.
Acco dingly, he compu a ional b onchial ee consis s o wo models: (1) an empi ical
model o he ex e nal lung’ su ace ha limi s pulmona y olume, and (2) an a i icial i e a i e
model o bi u ca ions o build a b onchial ee inside his su ace. This model is de e minis ic,
since su aces a e ob ained om expe imen al CT measu emen s and he i e a i e model does
no inco po a e andomness.
Empi ical model o he lung’s ex e nal su ace. An empi ical su ace model is buil
using CT scan da a om one o he minipigs, andomly chosen. This ep esen a i e su ace is
subsequen ly e-scaled acco ding o he dimensions o o he s minipigs’ lungs, gi ing ise o 5
compu a ional su aces ha can be used o buildi he 5 di e en b onchial ees.
In o de o build he ep esen a i e pulmona y su ace, we use h ee images o he h ee
planes, i.e., co onal plane, sagi al plane, and axial plane. F om hese images, he con ou line
is ex ac ed, keeping he ca ina (i.e., he bi u ca ion poin o he achea whe e i di ides in o
he wo main b onchi) posi ion o pu poses o econs uc ion (Fig 3). The 3D econs uc ion
om con ou lines is ca ied ou wi h Ma lab. All con ou s a e no malized o 1 in o de o be
subsequen ly e-scaled wi h he speci ic dimensions o each o he 5 minipigs’ lungs on he
econs uc ion p ocess. Finally, he le lung is sligh ly o a ed (5˚) so ha he in e -pulmona y
space is educed.
All he lesions obse ed expe imen ally a e loca ed in he coo dina e sys em de ined by he
econs uc ions, aking he ca ina as he e e ence poin , in o de o check i hey a e loca ed
inside he ob ained compu a ional pulmona y su aces. As a esul , 95% o he de ec ed lesions
a e inside he compu ed su ace o in con ac wi h pleu a.
I e a i e model o he b onchial ee. A b onchial ee o he conduc i e zone is buil
inside he compu a ional pulmona y su ace wi h an algo i hm based on p e ious wo k on he
human b onchial ee [25,26]. S a ing a he achea, a se o i e a i e ules go e n he succes-
si e bi u ca ions. The b onchial ee o he minipigs is assumed o be mo phologically equi a-
len o he human one, bu wi h smalle dimensions [27].
Fig 2. Summa y o expe imen al esul s. Le : CT image econs uc ion o a minipig’s pulmona y su ace. Righ : 3D
ep esen a ion o loca ion and size o all minipig lesions; each colou is o a di e en minipig ( ed, magen a, blue,
yellow, and g een).
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Dynamics o TB lesions in a i ual lung
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Ou algo i hm assumes ha all he di isions a e bi u ca ions, i.e., hey occu in a dicho o-
mous way. The esul ing h ee b anches in ol ed in a bi u ca ion a e coplana , and he plane
ha con ains each bi u ca ion is called a bi u ca ion plane. The di isions a e assumed o occu
in successi e pe pendicula planes, i.e., igh -le , an e io -pos e io , and uppe -lowe . The e-
o e, each bi u ca ion plane is pe pendicula o he p e ious plane. The i s di ision s a s a
he ca ina and di ec s he new b anches in o he igh and le lungs.
When a ce ain conduc ing ai way 0 di ides in o conduc ing ai ways 1 and 2, he low con-
se a ion (Q
0
= Q
1
+ Q
2
) oge he wi h Mu ay’s law (Q = C �d
3
) [28] leads o he ollowing
ela ion be ween hei diame e s, d
i
:
d0
3¼d1
3þd2
3ð1Þ
Flo ens e al. [29] de i ed a a io o 3 o he leng h o a b anch (l
i
) and i s diame e (d
i
) o
mos o he b onchial ees:
li¼3dið2Þ
We also s udied his ela ion using Rozanek and Roubik’s expe imen al da a [30], ob aining
a p opo ionali y cons an o 3.07 and a goodness o i o R
2
= 0.98. This analysis is shown in
he Supplemen a y ma e ial sec ion 3 (Fig B in S1 File).
Fig 3. No malized con ou lines ob ained om CT-scan images. Con ou lines we e ob ained om CT-scan images and used o a 3D
compu a ional econs uc ion o he pulmona y su ace. T achea di ision poin (ca ina) is ma ked o pu poses o econs uc ion. (A)
Sagi al plane CT image. (B) Co onal plane CT image. (C) Axial plane CT image. (D) Sagi al plane ou line econs uc ion. (E) Co onal
plane ou line econs uc ion. (F) Axial plane ou line econs uc ion.
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Dynamics o TB lesions in a i ual lung
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The diame e s and angles o each bi u ca ion depend on he ela i e olume ha each new
b anch supplies. We de ine he q
i
ac o as:
qi¼Vi
V0
;i¼1;2ð3Þ
whe e V
1
and V
2
a e, espec i ely, he sub- olumes i iga ed by conduc ing ai ways 1 and 2
a e bi u ca ion, and V
0
is he olume supplied by he b anch 0 (be o e bi u ca ion). Taking
in o accoun Mu ay’s law, he diame e s a e he bi u ca ion a e:
di¼d0�qi
1=3;i¼1;2ð4Þ
Minimizing he wo k pe uni ime (associa ed wi h ic ion and o main ain he s uc u e)
in bi u ca ions, he ollowing ela ions be ween he angles o he bi u ca ion and he ac o q
i
a e ob ained [31]:
cos�i¼1
2qi2=31þqi
4=3 ð1qiÞ4=3
� �;i¼1;2ð5Þ
The calcula ion o he a io q
i
(Eq 3) canno be analy ically e alua ed. The e o e, a g id o
equispaced poin s is c ea ed so ha he numbe o poin s inside each conside ed olume, N
i
(i = 0,1,2), is assessed and he a io is e alua ed as:
qi¼Ni
N0
;i¼1;2ð6Þ
The dis ance be ween poin s is ini ially ixed a 1 mm [25] and hen educed o 0.2 mm o
inc ease p ecision and imp o e esul s.
In Fig 4 he e can be seen a diag am o a bi u ca ion example o q
1
= 0.6. Using Eq 4 i can
be de e mined ha he diame e o he daugh e b anches a e d
1
= 0.84�d
0
and d
2
= 0.74�d
0
,
espec i ely. Leng h is 3 imes he diame e o each b anch, hen: l
0
= 3�d
0
, l
1
= 3�d
1
= 2.53�d
0
=
0.84�l
0
and l
1
= 3�d
1
= 2.21�d
0
= 0.74�l
0
. Bi u ca ion angles can be compu ed using Eq 5 as: ϕ
1
=
32˚ and ϕ
2
= 43˚.
The ini ial es s wi h hese equa ions esul ed in a b onchial ee ha does no ully occupy
he uppe pa o he pulmona y su ace. The e o e, a co ec ion is added o he angle calcula-
ion [25]. I consis s o an e alua ion o he mass cen e o he olumes o be supplied,
!xMC;Vi, which is p ojec ed on he bi u ca ion plane. We will deno e wi h ψ
i
he angle
be ween he o iginal b anch and he p ojec ed mass cen e. Expe imen ally, bi u ca ions wi h
an angle g ea e han π/2 we e no obse ed [32]. The e o e, he co ec ed angle φ
i
is:
φi¼min �iþci
2;p
2
� �;i¼1;2ð7Þ
B anches wi h a diame e lowe han 0.5 mm a e conside ed e minals as a eas hose
b anches ha escape om he pulmona y su ace. The model o he building o successi e
bi u ca ions is summa ized in Table 1.
The Bubble model: an upda e
The Bubble model is an agen -based model in which he agen s a e he lesions. The dynamics
o he lesions a e d i en by h ee p ocesses: a gene alised logis ic g ow h, he ein ec ion p o-
cess ha pe mi s he gene a ion o new in ec ion ocuses, and he coalescence be ween neigh-
bou ing lesions.
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Dynamics o TB lesions in a i ual lung
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The Bubble model was o iginally designed and calib a ed o desc ibe he dynamics o ube -
culous lesions in mice wi h an ac i e disease [14]. The spa ial s uc u e was no ele an o
mice, due o he dimensions and he ela i ely simple s uc u e o mouse lungs, and aking
in o conside a ion he size o lesions o he ac i e disease. This model is upda ed and imple-
men ed as ollows.
Lesion g ow h. We model a lesion as a sphe e whose spa ial posi ion (3D coo dina es, in
mm), adius (in mm), and age (in days) a e a iables. Lesions a e i s ly de ec ed when hei
adius is
min
= 0.075 mm (smalle lesions canno be iden i ied). This occu s a e app oxi-
ma ely
min
= 14 days om he ini ial in ec ion. Then when a lesion is c ea ed i emains
“silen ” o 14 days be o e i is ini ialized wi h a
min
adius. The model employs a gene alised
logis ic g ow h o he adius o he lesions as ollows:
d ið Þ
d ¼ i� i ð Þ� 1 ið Þ
max
� �2
" # ð8Þ
whe e
i
is he adius o he lesion, ν
i
is he pa ame e ha se s he maximum g ow h a e, and
max
is he maximum adius. The pa ame e ν
i
is modelled as a Gaussian a iable wi h mean
alue νand s anda d de ia ion ν/3. The e o e, each lesion g ows a a sligh ly di e en eloci y
a each ime s ep. F om expe imen al da a i is known ha a ound he 28
h
day a 2 mm lesion
eaches i s limi [22]; he e o e, νis es ima ed as ν= 0.3 day
-1
(σ
ν
= 0.1 day
-1
).
Fig 4. Bi u ca ion diag am. Bi u ca ion o 0 b anch in o wo (1, 2) daugh e b anches. The cabal a io o b anch 1 is:
q
1
= 0.6. Leng h is 3 imes he diame e o each b anch as may be seen in Eq 2. Diame e ela ions a e ob ained om Eq
4, as d
1
= 0.84�d
0
and d
2
= 0.74�d
0
. Angula alues a e compu ed using Eq 5, as ϕ
1
= 32˚ and ϕ
2
= 43˚.
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Dynamics o TB lesions in a i ual lung
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Lesion p oli e a ion. The mul iplica ion o he numbe o lesions is caused by endoge-
nous ein ec ion. In his way, a mo he lesion gene a es new daugh e lesions om day 14 o
day 28. The o iginal ein ec ion p obabili y unc ion [14] includes wo e ms: (1) a linea ly
inc easing e m wi h he adius o he mo he , and (2) a linea ly dec easing p obabili y wi h
mo he lesion age [22]. The second e m is sligh ly modi ied in o de o allow he gene a ion o
new daugh e lesions om mo he lesions olde han 28 days, wi h a small non-ze o p obabil-
i y:
P ð Þd ¼ � ið Þ
min �eaðai14Þnd ;ai�14days ð9Þ
whe e ρ,α, and na e pa ame e s ha de ine he p obabili y p o ile, a
i
is he age o he lesion,
in days, and
min
is he minimum adius a which lesions a e iden i ied. Fig 5 shows o iginal
[14] and modi ied (Eq 9) models wi h α= 0.035 day
-n
and n= 1.63 wi h
max
= 1 mm and ρ=
0.10 day
-1
. The alues o αand na e ixed o ensu e ha he a ea unde he wo cu es a e
equal and o minimize he di e ence be ween he expe imen al esul s and he linea model.
The ein ec ion p obabili y ini ially g ows exponen ially wi h he lesion adius; howe e , o
longe imes he cu e decays exponen ially because o encapsula ion and calci ica ion.
In he o iginal model o mice, he loca ion o new lesions is selec ed om a p obabili y
which dec eases wi h he dis ance. In he cu en e sion o minipigs he dis ance is modelled
in he same way, bu conside ing he b onchial dis ance be ween wo e minals ins ead o he
geome ic dis ance be ween wo poin s. The model does no explici ly assume ha he dissemi-
na ion occu s exclusi ely h ough he ae ial pa , bu i could include a possible eci cula ion
h ough he adjacen ci cula o y o lympha ic sys ems as well. Then, possible loca ions a e
de e mined by he b onchial ee e minal posi ions:
P i !jð Þ ¼ ebdij
Pjebdij ð10Þ
whe e P(i!j) is he p obabili y ha a lesion appea s a a e minal jdue o a mo he lesion a
e minal i, and βis he dispe sion pa ame e ha de e mines he sp eading. Fig 6A shows he
mean dis ance o he appea ance o new lesions as a unc ion o he dispe sion pa ame e . Fig
Table 1. Summa y o he model o building he compu a ional b onchial ee o each minipig.
The no malized pulmona y su ace is speci ically e-scaled o each minipig, aking in o accoun he measu ed
dimensions.
A b onchial ee is buil inside each compu a ional pulmona y su ace.
The b onchial ee s a s a he end o he achea, aking he achea diame e and ca ina loca ion o each minipig as
e e ence.
The b onchial ee is a ubula s uc u e, and non- e minal b anches spli in a dicho omous way.
The h ee b anches implied in a di ision a e coplana (bi u ca ion plane).
In each di ision, he pulmona y e i o y is di ided in wo sub egions by he plane ha is pe pendicula o he
bi u ca ion plane, ollowing he di ec ion o he mo he b anch.
The bi u ca ion plane o he i s di ision is he e ical one ha sepa a es he igh and he le lungs.
The bi u ca ion planes a e pe pendicula om one gene a ion o he ollowing.
Once he bi u ca ion plane is de ined, he a io q
i
=V
i
/V
0
is nume ically e alua ed by a g id o equiespaced poin s
wi h a p ecision o 0.2 mm, so ha q
i
=N
i
/N
0
(N
i
and N
0
a e he numbe o poin s con ained in each sub olume).
The pa ame e s o each bi u ca ion a e e alua ed using Eqs 2,4,5and 7.
B anches ha escape he pulmona y su ace and hose wi h a diame e o less han 0.5 mm a e conside ed e minals.
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The lack o eliable in o ma ion abou he ini ial in ec ion en ails he need o a blind
assump ion. Following he law o pa simony, we assume he simples ini ial con igu a ion, and
al e na i e possibili ies will be explo ed la e on. We use as ini ial in ec ion o he simula ions
a single lesion loca ed a he nea es e minal o he mass cen e o all he lesions, gi en he
inal dis ibu ion shown by CT images. The se o pa ame e s is adjus ed o minimize he
e o s as explained in Ma e ials and Me hods. ollowed by he pe o mance o 500 indepen-
den simula ions o each minipig’s con igu a ion (i.e., a o al o 2500 simula ions). The se o
pa ame e s ob ained a e he minimiza ion o he objec i e unc ions (Eqs 17–19) is ρ= 0.13
day
-1
,β= 0.08 mm
-1
, and
max
= 0.68 mm, which co esponds o NLE = 0.089, DE = 0.283, and
SE = 0.299.
In Fig 9 we show he compa ison be ween he expe imen al and he simula ed ou come dis-
ibu ions o spa ial coo dina es X, Y, and Z, and o lesion diame e s. A wo-sample Kolmogo-
o -Smi no es o he ou pai s o dis ibu ions show ha he e a e no signi ica i e
di e ences in any o he cases wi h a signi ica ion le el o 0.05. The e o e, he dis ibu ions
esul ing om he nume ical simula ion success ully ep oduce he expe imen al obse a ions.
Simula ions show ha coalescence o lesions is nea ly non-exis en , on a e age less han
one coalescence pe minipig. This esul is in ag eemen wi h P a s e al. [9] and Ma zo e al.
[15], who p esen ed coalescence as a mechanism essen ial o he e olu ion owa ds an ac i e
Fig 9. Compa ison o expe imen al and compu a ionally ob ained dis ibu ions o lesion loca ion and size.
Compu a ional dis ibu ions we e ob ained conside ing as ini ial in ec ion one lesion in he mass cen e o he obse ed
lesions. The se o pa ame e s used is: ρ= 0.13 day
-1
,β= 0.08 mm
-1
, and
max
= 0.68 mm. (A) Coo dina e X (Le —
Righ ) his og am compa ison. (B) Coo dina e Y (An e io –Pos e io ) his og am compa ison. (C) Coo dina e Z
(Ve ical) his og am compa ison. (D) Diame e dis ibu ion his og am compa ison.
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Dynamics o TB lesions in a i ual lung
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disease om a la en in ec ion. A lack o coalescences, he e o e, would be a con ol indica o
o he la en in ec ion.
Fu he mo e, simula ion esul s show ha he p ocess o endogenous ein ec ion is c ucial
o unde s anding how lesions appea a di e en loca ions in he lungs. The use o a compu a-
ional b onchial ee o d i ing such ein ec ion p oduces spa ial dis ibu ions which esemble
he expe imen al cases. This p ocess has been shown o be a key ac o in main aining a la en
in ec ion inside big mammals like minipigs.
Sensi i i y analysis
Table 5 shows he esul s o he sensi i i y analysis, wi h he minimum alue o he ANOVA
es be ween he inc eased and he dec eased pa ame e s. This analysis e eals ha he numbe
o lesions is s ongly ela ed wi h ρ,ν, and T
max
. I was no ob ious ha he numbe o lesions
would be ela ed wi h he g ow h eloci y; howe e , when lesions g ow as e he e is an
inc ease in he likelihood ha he p ocess o endogenous ein ec ion will gene a e new lesions.
The mean diame e a ies wi h pa ame e s β,
max
, and T
max
. The in lamma o y esponse is
he cause o lesion g ow h, so ela ions wi h
max
and T
max
a e expec ed. The esul s also show
ha he dispe sion pa ame e , β, sligh ly a ec s he mean diame e . A smalle dispe sion
pa ame e causes lesions o be close , he eby inc easing he chance o a coalescence e en . In
ac , as seen in ex ended sensi i i y analysis o he explo ed pa ame e space,
max
and βa e
he wo pa ame e s ha a ec he mean diame e alue mos . An inc ease in one o hese
pa ame e s inc eases mean diame e alue.
Acco ding o his analysis, T
max
is he only pa ame e ha is no ela ed wi h he esul ing
numbe o coalescences. All o he pa ame e s a ec he coalescence p ocesses; ne e heless, a
coun e -in ui i e esul is ha he dispe sion pa ame e is no he mos s ongly ela ed. One
may expec he dispe sion pa ame e o be he pa ame e ha would a ec he coalescence
p ocess mos because i is he one ha de e mines coalescence sp eading, and hen de e mines
he dis ance a which new lesions appea .
The sensi i i y analysis e iden ly depends on he ini ial se o pa ame e s. Ne e heless, we
ha e employed o he se s o pa ame e s o ca y ou sensi i i y analyses, p o iding equi alen
esul s. These analyses a e shown in he Supplemen a y ma e ial sec ion 1 (Table A and
Table B).
Analysing he e ec o ini ial condi ions
A e con i ming ha he model wi h he simples assump ion o he ini ial condi ions is good
enough o explain he expe imen al esul s, we explo ed he possibili y o imp o ing he
Table 5. Sensi i i y analysis o he se o pa ame e s: S = {ρ,β,
max
} = {0.12 day
-1
, 0.08 mm
-1
, 0.68 mm}.
Inpu pa ame e s
β
max
ρνT
max
Ou come a iables Numbe o lesions 0.137 0.092 <0.001�� <0.001�� <0.001��
Mean diame e 0.003�� <0.001�� 0.475 0.106 <0.001��
Dispe sion <0.001�� 0.613 0.089 <0.001�� <0.001��
Coalescences 0.024�0.004�� <0.001�� <0.001�� 0.754
Each alue is he minimum o he p- alues om ANOVA es compa ing 500 uns o he o iginal se o pa ame e s and he se o pa ame e s whe e one pa ame e is
inc eased o dec eased by 10%. Numbe s ma ked wi h
�p esen s a is ically signi ican di e ences wi h p<0.05 and numbe s ma ked wi h
�� wi h p<0.01.
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ag eemen be ween he model and expe imen al measu emen s by es ing di e en ini ial dis-
ibu ions o lesions. Table 6 shows he 12 ini ial con igu a ions analysed, in addi ion o he
p e ious one. We explo e he choice o one o mo e lesions om CT da a as he ini ial in ec-
ion using loca ion, size, and densi y c i e ia, as well as di e en andom choices. Fo each ini-
ial dis ibu ion, he se o pa ame e s is adjus ed o minimize he objec i e unc ions as
explained in Ma e ials and Me hods. Table 6 shows he pa ame e alues ha minimize e o s
as well as he co esponding alues o each o he explo ed ini ial dis ibu ions.
The esul s o his analysis, shown in Table 6, do no p o ide a conclusi e c i e ion o dis-
inguishing hose lesions ha belonged o he ini ial in ec ion. Ne e heless, hey co obo a e
ha he inal lesion dis ibu ion is s ongly ela ed wi h he ini ial in ec ion dis ibu ion, since
he objec i e unc ion ha is mos a ec ed is ha o spa ial e o (SE). The dis ibu ions ha
assume as ini ial in ec ion one o wo andom expe imen al lesions p o ide be e spa ial e o
objec i e unc ion esul s; howe e , hey gi e ise o la ge alues o DE.
In some cases, he alue o he spa ial pa ame e ha minimizes he spa ial e o unc ion is
β= 0 mm
-1
. This alue pe mi s a mac ophage o a el o all o he e minals no aking in o
accoun he dis ance h ough he b onchial ee. In consequence, he inal dis ibu ion
becomes one ha ollows all e minal spa ial dis ibu ions and is no ela ed wi h he pa icula
ini ial dis ibu ion. The e o e, in hese cases he ini ial dis ibu ion is ela ed nei he wi h den-
si y no diame e .
F om la en o ac i e ube culosis: In silico expe imen s
Ma hema ically we de ine a case o ac i e disease as one wi h nume ical simula ions p o iding
a lesion la ge han 1 cm in diame e [35]. The model is designed o ep oduce expe imen al
esul s om la en ube culosis in minipigs. The e o e, no igge o disease is obse ed in any
o simula ions wi h he i ed pa ame e s. The ollowing se o simula ions is designed o
explo e he pa ame e space, looking o hose zones leading o ac i e disease.
Table 6. Ini ial con igu a ions explo ed wi h he model using expe imen al da a.
Ini ial in ec ion Pa ame e s se E o s
ρ
(day
-1
)
β
(mm
-1
)
max
(mm)
NLE DE SE SE/SE
con ol
Mass cen e (con ol) 0.134 0.071 0.67 0.015 0.25 0.30 1.00
Coo dina e cen e 0.134 0.070 0.66 0.026 0.23 0.34 1.14
Bigges lesion 0.129 0 0.68 0.006 0.18 0.74 2.45
Two bigges lesions 0.102 0 0.68 0.015 0.18 0.73 2.42
30% bigges lesions 0.084 0 0.69 0.009 0.17 0.77 2.58
Denses lesion 0.123 0 0.69 0.003 0.19 0.73 2.43
Two denses lesions 0.110 0 0.68 0.026 0.17 0.73 2.44
30% denses lesions 0.084 0 0.68 0.004 0.16 0.78 2.58
Densi y>150HU 0.078 0 0.68 0.020 0.17 0.79 2.62
One andom lesion 0.227 0.170 0.54 0.027 0.63 0.25 0.82
Two andom lesions 0.122 0.129 0.58 0.008 0.35 0.23 0.77
One andom e minal 0.212 0.160 0.55 0.011 0.59 0.71 2.36
Two andom e minals 0.129 0.150 0.59 0.016 0.40 0.71 2.37
The i s column shows he c i e ia o choosing which o he measu ed lesions we e assumed as ini ial in ec ion. The ollowing columns show he pa ame e alues ha
minimized e o s (ρ,β,
max
) and he alues o he h ee e o s ob ained (NLE,DE, and SE). The las column compa es he spa ial e o objec i e unc ion (SE) wi h he
one om he con ol simula ion. HU: Hounds ield uni s.
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The pa ame e space is delimi ed by β2[0, 0.2] mm
-1
,
max
2[1, 5] mm, and ρ2[0.02, 0.2]
day
-1
. We used equidis an poin s, 11 o β, 10 o
max
, and 4 o ρ. We explo ed a o al o 440
poin s. We an 2500 simula ions o each poin o his pa ame e s space, and 500 o each
minipig i ual lung. The ini ial in ec ion con igu a ion is se as he con ol (i.e., one lesion in
he mass cen e o he measu ed lesions’ dis ibu ion). Finally, we de ine an Ac i e Disease
Index as he equency o ac i e cases among he o al numbe o in silico expe imen s o each
poin o he pa ame e space.
We show in Fig 10 he ob ained Ac i e Disease Index o each o he explo ed poin s. This
index inc eases wi h any o he 3 pa ame e s,
max
,β, and ρ. Th ee di e en zones o he
pa ame e s space can be de ined: a la en zone, whe e mos o he simula ions esul in a la en
in ec ion (g een colou in Fig 10), an ac i e disease zone, whe e mos o he simula ions de i e
in o an ac i e disease ( ed colou in Fig 10), and a ansi ion zone, whe e bo h dynamics a e
possible ( om ligh g een o da k o ange in Fig 10).
The pa ame e
max
is ela ed wi h he e ec i e in lamma o y esponse in a b oad sense;
he g ea e he e ec o he in lamma o y esponse, he bigge he lesions and he g ea e
he likelihood o de eloping an ac i e disease. This esul is in ag eemen wi h expe imen al
obse a ions [15] and wi h o he biology sys ems app oaches [14]. O cou se, he e ec i e
dynamics o in lamma o y esponse can be modula ed by local p ope ies such as oxygen con-
cen a ion o mac ophages’ a ailabili y, among o he s, which a e no explici ly conside ed by
he model.
Pa ame e βis he dispe sion pa ame e ; a highe alue o βco esponds o lowe dispe -
sion inside he lung, which can be a consequence o a lowe b ea hing ampli ude. The e o e, a
Fig 10. Ac i e Disease Index o di e en se s o pa ame e s. Explo a ion o pa ame e s space (
max
,βand ρ) o see
he ac ion o in silico expe imen s ha p esen an ac i e TB disease. The colou is p opo ional o his equency;
g een colou means mos o he cases emained la en , ed colou means ha mos o he cases de i ed in o an ac i e
disease, and in e media e colou s mean ha bo h dynamics a e possible.
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low b ea hing ampli ude appea s again as a possible cause o he appea ance o big lesions, as
was p e iously desc ibed in he li e a u e [9]. I has o be aken in o accoun ha b ea hing
ampli ude changes om one lobe o ano he one, i.e., i is wide in lowe lobes. The e o e, he
condi ions ha acili a e he de elopmen o an ATB would a y om one lobe o ano he .
Ou model shows ha ac i e disease can be igge ed by a high in lamma o y esponse o due
o a mode a e in lamma o y esponse combined wi h a small b ea hing ampli ude. Ne e he-
less, a his poin he i ual lungs a e conside ed o be homogeneous, i.e., we a e no conside -
ing a ia ions o pa ame e s along he lungs’ s uc u e.
A e his explo a ion, he sensi i i y analysis is ex ended o check i sensi i i y o he model
depends on he se o pa ame e s used. In addi ion o he de aul pa ame e s ha belong o he
la en in ec ion zone, we chose wo combina ions o pa ame e s: one ep esen a i e o he
ac i e disease zone and he o he o he ansi ion zone; see Fig 10. A la en TB pa ame e se
shows ha he space pa ame e , β, de e mines mos ly he lesions dispe sion; howe e , a
pa ame e se o he ansi ion zone (
max
= 6.5 mm, β= 0.1 mm
-1
,ρ= 0.05 day
-1
) shows ha
he space pa ame e a ec s mos ly he numbe o lesions, mean diame e , and coalescences,
and does no a ec he dispe sion o he lesions. This can be seen in Supplemen a y ma e ial,
sec ion 1.
Discussion
Limi a ions and u he wo k
In his modelling app oach we ha e ollowed he law o pa simony (Occam’s azo ) [36], ying
o ind a simple solu ion o complex p oblems such as TB in ec ion dynamics in lungs. The
le el o complexi y was chosen acco ding o he ques ions o be add essed. This me hod was
de eloped speci ically o submi he main assump ions o he dynamic hypo hesis o alsi iabil-
i y es ing. The e o e, he cu en model includes he mos impo an s eps o TB in ec ion
e olu ion sugges ed by his hypo hesis: endogenous ein ec ion, lesion g ow h, and coales-
cence. These p ocesses a e supposed o cap u e he essence o TB dynamics in lungs, bu o
cou se hey a e no he only ones [6]. In ac , no model could be comple e [37]. This also allows
he use o a ewe numbe o pa ame e s, when compa ed wi h o he sys ems biology
app oaches o he same p oblem [1], and hus p o ides mo e obus ness o he i ing.
The p incipal no el y o his model is he implemen a ion o he bubble model in an explici
space like he b onchial ee in o de o simula e he endogenous ein ec ion p ocesses. Ne e -
heless, i s ill has a ew limi a ions ha should be men ioned, he mos impo an being he
ollowing:
- Exogenous ein ec ion is no ye conside ed in his model. I s inco po a ion may change
he ou come when simula ing an ATB in ec ion, as i ac s as a new mechanism o gene a e new
in ec ion ocuses. Ne e heless, he expe imen al da a used in his s udy we e ob ained unde
condi ions ha p e en ed exogenous ein ec ion. The e o e, he inclusion o his mechanism
should be suppo ed by expe imen al designs ha allow i .
- The b onchial ee model is absolu ely de e minis ic, o now. In he u u e we expec o
add some andom noise in his algo i hm in o de o ob ain di e en b onchial ees om a
single pulmona y su ace. This will be use ul o analysing he ole o speci ic b onchial ee
p ope ies in TB e olu ion as well as o accoun o he e ogenei y sou ces.
- In ec ion sp eading pa ame e s a e uni o m in each i ual lung. Ne e heless, b ea hing
ampli ude is no cons an , bu a ies om lowe and middle lobes (wide ampli ude) o uppe
ones (lowe ampli ude). B ea hing ampli ude is p obably ela ed wi h lesion sp eading; hen,
highe alues o βwould be e i he local beha iou o less dispe sion in he uppe lobe,
while lowe alues o βwould be app op ia e o desc ibing he sp eading in he lowe and
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Dynamics o TB lesions in a i ual lung
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middle lobes. This should be aken in o accoun by c ea ing a β a iable p o ile inside he
b onchial ee. Wi h ou model, we ha e seen ha a bigge dispe sion educes he p obabili y
o causing big lesions, which is ue locally. A he same ime, g ea e dispe sion may inc ease
he p obabili y o gene a ing new in ec ion oci in pa s o he lung wi h a smalle b ea hing
ampli ude, and he e o e wi h a highe p obabili y o e ol ing owa ds bigge g anulomas.
These dynamics will be ca e ully explo ed in he u u e.
Finally, we mus men ion he ou p incipal assump ions ha could be e ined and e en
e u ed in he u u e:
a. Ou model ollows dynamic hypo hesis assump ions [38], bu he e a e mo e hypo heses
ha can be submi ed o check easibili y such as [39,40]. In addi ion, o he impo an p o-
cesses such as he ole o oxygena ion [41,42] could be inco po a ed.
b. The model assumes ha he la ge he mo he , he mo e likely i is o gene a e new lesions.
This is based on he assump ion ha bigge lesions would ha e mo e oamy mac ophages
ha a e mo e likely o be d ained and, he e o e, o be able o cause a ein ec ion [43]. I is
also suppo ed by expe imen al obse a ions in macaques o a highe endency o bigge
lesions o dissemina e [44].
c. The ein ec ion model also conside s ha he olde a lesion is, he less likely an in ec ed
mac ophage is o escapes om i . This is in acco dance wi h expe imen al obse a ions
[22]. In he u u e, a e m ha depends on he ini ial in ec ion ime in addi ion o he age o
he lesion could be aken in o conside a ion because, as can be seen in [45], ini ial lesions
a e bigge han hei daugh e s in ATB.
d. An in ec ion is conside ed o co espond o an ac i e disease i a lesion g ows la ge han 1
cm. ATB de ini ion is mo e complica ed han simply ha ing a lesion bigge han 1 cm [35].
Then, o he indica o s should be explo ed o de ine an ATB. In ac , his h eshold is no a
s anda d accep ed bounda y. In supplemen a y ma e ial, sec ion 5 (Fig E in S1 File), an
explo a ion o di e en alues o his h eshold is shown. The s eng h o his model is ha
he same endencies ha we e obse ed in Fig 10 can be obse ed in he di e en explo a-
ion h eshold simula ions.
In gene al, model alsi iabili y could be success ully ca ied ou wi h da a om CT (o
equi alen ) o TB dynamics in big mammals, a di e en ime poin s. The ac ha only a inal
pho o o he sys em is a ailable is clea ly a d awback o he es ing o he p ocesses in ol ed.
Indeed, acking he g ow h o indi idual lesions a se e al imepoin s should be enough o
es ing he gene alized logis ic model. Ba coding echniques ha e also shown hei app op i-
a eness o dis inguish con ained om dissemina ed lesions in macaques [44], and hus could
p o ide a way o de e mine which he ini ial g anulomas a e and a pa e n o dissemina ion.
Bac e ia ba coding, oge he wi h a ime acking o he 3D cha ac e iza ion o loca ion and
size o lesions by means o CT, can p o ide key in o ma ion abou he ange and ela i e
impo ance o dissemina ion, as well as he possible geome ical cons ain s. Ne e heless, one
o he d awbacks o he equi ed expe imen al es s ha should be always kep in mind is ha
he n is usually small, while he in aspeci ic di e si y is high.
This app oach has consis ed o he es ing o a single model, which seems o go agains he
s ong in e ence in ma hema ical modelling [46]. Ne e heless, we ha e ocused on explo ing
and exploi ing all he possibili ies gi en by his model and he a ailable expe imen al da a. In
addi ion o he abo e-men ioned sea ch o new expe imen al measu emen s ha can e u e
he s a ed hypo heses, u u e wo k should include he es ing o al e na i e models whose
ejec ion would p o ide mo e clues on he na u al his o y o ube culosis.
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Dynamics o TB lesions in a i ual lung
PLOS Compu a ional Biology | h ps://doi.o g/10.1371/jou nal.pcbi.1007772 May 20, 2020 21 / 25

Conclusions
We buil a compu a ional model which includes he i ual lung and he upda ed Bubble
model. This model success ully i s CT expe imen al obse a ions o la en ube culosis in
minipigs. The model inco po a es he basis o he dynamic hypo hesis [7,37] o ep oduce
lesion p opaga ion h ough he b onchial ee on a la en ube culosis in ec ion. The ag ee-
men be ween expe imen al and nume ical esul s ein o ces he easibili y o he dynamic
hypo hesis, i.e., i is able o explain he expe imen al esul s obse ed in la en ly in ec ed mini-
pigs. In pa icula , we ha e obse ed he impo ance o he b onchial ee in he endogenous
ein ec ion (β>0).
Mos impo an pa ame e s o he model could be ela ed wi h he co esponding biophysi-
cal p ocesses. The e o e, he model is consis en wi h he da a. Pa ame e βcan be ela ed wi h
he b ea hing ampli ude as a ac o de e mining how a new lesions can appea ;
max
may be
ela ed wi h he e ec o in lamma o y esponse o he hos , as i is he main cause o lesion
g ow h; and ρde e mines he p obabili y o igge ing he endogenous ein ec ion p ocess. I
has been seen ha ν, he g ow h eloci y o he lesions, can play a simila ole in igge ing
endogenous ein ec ion p ocess, while slowing he g ow h eloci y o lesions may be a mecha-
nism o de aining endogenous ein ec ion.
Acco ding o he in silico expe imen s ca ied ou wi h he model, an ac i e TB in an immu-
nocompe en hos may be caused by high in lamma o y esponse o by mode a e in lamma-
o y esponse combined wi h small b ea hing ampli ude. The o me has al eady been
obse ed expe imen ally in se e al s udies [15,21,22]. The la e may be a clue o unde s and-
ing he usual p esence o ac i e TB in he uppe lobe, as sugges ed by Ca dona & P a s [9],
since i is in he lung zone ha he b ea hing ampli ude is smalle . In ac , u he e inemen s
and upda es o he model should include in e -lobula di e ences so ha his las possibili y
can be ca e ully explo ed.
Suppo ing in o ma ion
S1 Da ase . Expe imen al da ase o CT measu emen s. Shee “Lungs” con ains, o each
minipig, he pulmona y geome ical measu emen s and he global dis ibu ion o lesions in
each lung. Shee “Lesions” con ains he measu ed cha ac e is ics o each lesion.
(XLSX)
S1 File. Supplemen a y ma e ial. This ile includes he ollowing supplemen a y sec ions: (1)
Sensi i i y analysis o ansi ion and ac i e se ; (2) Ex ended sensi i i y analysis; (3) Diame-
e -leng h ela ion in b onchial ee; (4) De ails o he simpli ied p o ocol o model pa ame i-
za ion sensi i i y; and (5) Th esholds explo a ion.
(PDF)
Acknowledgmen s
We since ely hank Ma ina Vegue
´and Ca les Ba il o he building o he p elimina y human
b onchial ees, which was he basis o he cu en algo i hm. We hank Be na Puig o his
help on building he compu a ional pulmona y su ace.
Au ho Con ibu ions
Concep ualiza ion: Jo di Bechini, Mon se a Tenesa, Rica do Pe
´ ez, C is ina Vilaplana, Joa-
quim Valls, Daniel Lo
´pez, Pe e-Joan Ca dona, Cla a P a s.
PLOS COMPUTATIONAL BIOLOGY
Dynamics o TB lesions in a i ual lung
PLOS Compu a ional Biology | h ps://doi.o g/10.1371/jou nal.pcbi.1007772 May 20, 2020 22 / 25
Da a cu a ion: Ma ı
´Ca alà, Jo di Bechini, Mon se a Tenesa, Rica do Pe
´ ez, Ma iano Moya,
Pe e-Joan Ca dona.
Fo mal analysis: Ma ı
´Ca alà, Se gio Alonso, Daniel Lo
´pez, Cla a P a s.
Funding acquisi ion: Pe e-Joan Ca dona, Cla a P a s.
In es iga ion: Ma ı
´Ca alà, Jo di Bechini, Mon se a Tenesa, Rica do Pe
´ ez, Ma iano Moya,
C is ina Vilaplana, Pe e-Joan Ca dona, Cla a P a s.
Me hodology: Ma ı
´Ca alà, Jo di Bechini, Mon se a Tenesa, Rica do Pe
´ ez, C is ina Vila-
plana, Daniel Lo
´pez, Pe e-Joan Ca dona, Cla a P a s.
Resou ces: C is ina Vilaplana, Pe e-Joan Ca dona, Cla a P a s.
So wa e: Ma ı
´Ca alà, Joaquim Valls, Daniel Lo
´pez, Pe e-Joan Ca dona, Cla a P a s.
Supe ision: C is ina Vilaplana, Pe e-Joan Ca dona, Cla a P a s.
Valida ion: Ma ı
´Ca alà, Pe e-Joan Ca dona, Cla a P a s.
Visualiza ion: Ma ı
´Ca alà, Joaquim Valls, Se gio Alonso, Cla a P a s.
W i ing – o iginal d a : Ma ı
´Ca alà, Daniel Lo
´pez, Pe e-Joan Ca dona, Cla a P a s.
W i ing – e iew & edi ing: Ma ı
´Ca alà, Jo di Bechini, Mon se a Tenesa, Rica do Pe
´ ez,
C is ina Vilaplana, Joaquim Valls, Se gio Alonso, Daniel Lo
´pez, Pe e-Joan Ca dona, Cla a
P a s.
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