Modelling the effects of bacterial cell state and spatial location on tuberculosis treatment : insights from a hybrid multiscale cellular automaton model

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Contents lists available at ScienceDirect

Journal

of

Theoretical

Biology

journal homepage: www.elsevier.com/locate/jtbi

Modelling

the

effects

of

bacterial

cell

state

and

spatial

location

on

tuberculosis

treatment:

Insights

from

a

hybrid

multiscale

cellular

automaton

model

Ruth

Bowness

a ,∗

_,

_Mark

_A.J.

_Chaplain

b

_,

_Gibin

_G.

_Powathil

c

_,

_Stephen

_H.

_Gillespie

a

a_School_of_Medicine,_University_of_St_Andrews,_North_Haugh,_St_Andrews_KY16_9TF,_UK

b_School_of_Mathematics_and_Statistics,_University_of_St_Andrews,_North_Haugh,_St_Andrews_KY16_9SS,_UK c_Department_of_Mathematics,_Talbot_Building,_Swansea_University,_Singleton_Park,_Swansea,_SA2_8PP,_UK

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received17August2016 Revised1March2018 Accepted6March2018 Availableonline7March2018

Keywords:

Tuberculosis Cellularautomaton Hybridmultiscalemodel Antibiotics

Bacteria

a

b

s

t

r

a

c

t

If improvements are to be made in tuberculosis (TB) treatment, an increased understanding of disease in the lung is needed. Studies have shown that bacteria in a less metabolically active state, associated with the presence of lipid bodies, are less susceptible to antibiotics, and recent results have highlighted the disparity in concentration of different compounds into lesions. Treatment success therefore depends critically on the responses of the individual bacteria that constitute the infection.

We propose a hybrid, individual-based approach that analyses spatio-temporal dynamics at the cellular level, linking the behaviour of individual bacteria and host cells with the macroscopic behaviour of the microenvironment. The individual elements (bacteria, macrophages and T cells) are modelled using cellular automaton (CA) rules, and the evolution of oxygen, drugs and chemokine dynamics are incorpo- rated in order to study the effects of the microenvironment in the pathological lesion. We allow bacteria to switch states depending on oxygen concentration, which affects how they respond to treatment. This is the ﬁrst multiscale model of its type to consider both oxygen-driven phenotypic switching of the My-cobacteriumtuberculosis and antibiotic treatment. Using this model, we investigate the role of bacterial cell state and of initial bacterial location on treatment outcome. We demonstrate that when bacteria are located further away from blood vessels, less favourable outcomes are more likely, i.e. longer time before infection is contained/cleared, treatment failure or later relapse. We also show that in cases where bacteria remain at the end of simulations, the organisms tend to be slower-growing and are often located within granulomas, surrounded by caseous material.

1. Introduction

Althoughtuberculosis(TB)haslongbeenbothpreventableand curable,apersondiesfromtuberculosisapproximatelyevery eigh-teenseconds(WHOGlobalHealthReport2011).Currenttreatment requiresatleastsixmonthsofmultipleantibioticstoensure com-pletecureandmoreeffectivedrugsareurgentlyneededtoshorten treatment.Recentclinicaltrialshavenotresultedinashorteningof therapy andthere isa needto understandwhythesetrials were unsuccessfulandwhichnewregimenshouldbechosenfortesting inthecostlylong-termpivotaltrialstage(Gillespie et al., 2014 ).

The currentdrug developmentpathwayin tuberculosis is im-perfectas standardpreclinical methods maynot capturethe cor-rect pharmacodynamicsoftheantibiotics.Using invitromethods,

∗ _{Corresponding}_author.

E-mailaddress:rec9@st-andrews.ac.uk(R. Bowness).

itisdiﬃculttoaccuratelyreproducethenaturalphysiological envi-ronmentofMycobacteriumtuberculosis(M.tuberculosis)andthe re-liabilityofinvivomodelsmaybelimitedintheirabilitytomimic humanpathophysiology.

When M. tuberculosis bacteria enter the lungs,a complex im-muneresponseensues.Theoutcome ofinfectionisdependent on howeffectivethehost’simmunesystemisandonthe pathogenic-ityof the bacteria. The majority ofpatients will be able to con-trolinfectionandcontainit withinagranuloma, whichisa com-binationofimmunecellsthatsurroundthebacteria.Thecentreof the granuloma may exhibit caseous necrosis and havea cheese-likeappearance(Canetti et al., 1955 ).Mostcommonly,granulomas willundergoﬁbrosisorcalciﬁcationandtheinfectioniscontained andbecomeslatent (Canetti et al., 1955; Grosset, 1980 ). Inthese cases,however,theindividualsarestill atriskoffuturerelapse.If thegranulomasdonotcontainthediseaseandinfectioncontinues, thebacteriacangrowextracellularly.

https://doi.org/10.1016/j.jtbi.2018.03.006

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Fig.1. SchematicofpopulationpercentagesofoutcomesofTBdisease.Datatakenfrom(Ahmad,2010)andreferencestherein.

Although a minimum of six months of therapy is recom-mended, it haslong beenrecognised that manypatients require lessand are culture negative intwo months or less(Fox, 1981 ). Shortening treatment to four months or less results in unac-ceptably high relapse rates (Gillespie et al., 2014 ) and studies (Singapore, 1981; Study, 1981 )describedin(Fox et al., 1999 ).Ithas recentlybeen shownthat, forsomepatientswho becomeculture negativein only a week, there is still a non-zero risk of relapse (Phillips et al., 2016 ).

Fig. 1 describestheestimatesoftuberculosiscontainment ver-sus progression to active disease in the general population. In patientswith established disease, the outcome is perhaps deter-minedby the abilityofantibioticsto penetrateto thegranuloma (Prideaux et al., 2015 ).Inordertoimprovetuberculosistreatment, itisthereforevitaltoensuresuﬃcientconcentrationsofantibiotic reachthesitesofinfection(Via et al., 2015 ).

ItisincreasinglyrecognisedthatM.tuberculosisisabletoenter intoastate inwhichitis metabolicallylessactive.Dormant bac-teria havereducedsusceptibility to antibioticsof whichcell wall inhibitorsaremostaffectedbuttheactionoftheRNApolymerase inhibitor rifampicinand ﬂuroquinolones actingon DNA gyraseis also reduced (Wayne and Hayes, 1996 ). A signiﬁcant number of metabolic systems are down regulated in response to dormancy inducing stresses (Keren et al., 2004; 2011 ). This slower-growing state,associatedwiththepresenceoflipidbodiesinthe mycobac-teriacanincreaseresistanceby15fold(Hammond et al., 2015 ).It hasalsorecently beenshownthat around 60% ofbacteria in the lungare lipid rich(Baron et al., 2017 ).The reducedsusceptibility ofsome bacteria toantibioticsmeansthat itisveryimportantto studyand analyse the heterogeneity of the bacteria so more ef-fectivetreatmentprotocolscan bedeveloped.Thespatiallocation of the bacteria is also vitally important as the ability of antibi-oticstopenetratedifferentsitesofinfectioneffectivelyisvariable (Prideaux et al., 2015 ).

Multipleroutestodormancyhavebeenreportedandreviewed in detail (Lipworth et al., 2016 ). Oxygen concentration was one of the ﬁrst mechanisms demonstrated in an in vitro model to result in dormancy, and in vitro models have been developed to explore the antibiotic susceptibility andmetabolism of organ-isms in this slower-growing state (Wayne and Sramek, 1994 ). It hasbeenhypothesisedbymultipleauthorsthatlesionscontaining slower-growingbacteriaareresponsibleforrelapse(Grosset, 1980; Prideaux et al., 2015 ).

Cellular automaton modelling(and individual-based modelling ingeneral) hasbeen usedto model other diseases, mostnotably

tumour development and progression in cancer (Alarcón et al., 2003; Dormann and Deutsch, 2002; Gerlee and Anderson, 2007; Powathil et al., 2012; Swat et al., 2012; Zhang et al., 2009 ). Tu-berculosisgranulomashavebeensimulatedpreviouslythroughan agent-based model called‘GranSim’ (Cilfone et al., 2013; Marino et al., 2011; Pienaar et al., 2015; Segovia-Juarez et al., 2004 ),which aims to reconstruct the immunologicalprocesses involved in the developmentofagranuloma.

In(Pienaar et al., 2016 ) theauthorsmapmetaboliteand gene-scale perturbations, ﬁnding that slowly replicating phenotypes of M. tuberculosis preserve the bacterial population in vivo by continuously adapting to dynamicgranuloma microenvironments. (Sershen et al., 2016 ) alsocombinesaphysiologicalmodelof oxy-gendynamics,anagent-basedmodelofcellularimmune response and a systems-based model of M.tb metabolic dynamics. Their studysuggeststhat thedynamics ofgranuloma organisation me-diatesoxygenavailabilityandillustratestheimmunological contri-butionofthisstructuralhostresponsetoinfectionoutcome.

In this paper, we build on previous models such as (Cilfone et al., 2013; Marino et al., 2011; Pienaar et al., 2015; Segovia-Juarez et al., 2004 ),andreport thedevelopmentofa hybrid-cellular au-tomatonmodel.Ourmodelistheﬁrstmultiscalemodeltoconsider bothoxygendynamicsandantibiotictreatmenteffectswithina tu-berculosislesion, inorder to investigatethe role ofbacterial cell state heterogeneity andbacterialposition within the tuberculosis lesion onthe outcome of disease.This isa unique focus forthis typeofmodel.

2. Thehybridmultiscalemathematicalmodel

The model simulates the interaction between TB bacteria, T cells and macrophages. Immune responses to the bacterial in-fection can lead to an accumulation of dead TB bacteria and macrophages, creating caseum. Oxygen diffuses into the system from blood vessels: bacteria switch from a slow-growing (lipid rich) to a fast-growing (lipid poor) phenotype in an oxygen-rich environment (proximate to the blood vessels). Chemokine molecules are secreted by the macrophages, which direct the movementoftheimmunecells.Wetheninvestigatetheeffectthat antibioticshaveontheinfection.

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Fig.2. Schematicdescribingthebasicprocessesinthemodel.

Fig.3. Plotillustrating(a)aﬁxed,uniformdistributionofbloodvesselcross sec-tionsthroughoutthespatialdomainusedinthecellularautomatonsimulationsand (b)oneoutcomeofarandomdistributionofthebloodvessels.

beenchosensothateachautomatonelementisapproximatelythe same sizeasthe largestelement inthe system:the macrophage. Atpresentwealloweach gridcelltobeoccupiedbyamaximum ofoneelement.

Our modelis made up ofﬁve main components: (1)Discrete elements - the grid cell is occupied either by a TB bacterium, a macrophage, a T cell, caseum oris empty. If the grid cell is oc-cupied,automatonrulescontroltheoutcome;(2)thelocaloxygen concentration,whoseevolutionismodelledbyapartialdifferential equation;(3)chemokineconcentrations,modelledbyapartial dif-ferentialequation;(4)antibioticconcentrations,modelledbya par-tialdifferentialequationand(5)bloodvesselsfromwherethe oxy-gen andantibioticsare supplied within thedomain. A schematic overviewofthemodelisgivenin Fig. 2 .

2.1. Thebloodvesselnetwork

Atthetissuescale,weconsideroxygenanddrugdynamics.We introducea networkofbloodvessels inthemodel,whichisthen used asa sourceof oxygenandantibiotic within the model. Fol-lowing Powathil et al. (2012) ,we assume bloodvessel cross sec-tionsaredistributedthroughoutthetwodimensionaldomain,with density

φ

d=Nv/N2,where Nv is thenumber ofvessel cross sec-tions(Fig. 3 ).See Table 2 forvaluesofNandNv.Thisisreasonable ifweassumethatthebloodvesselsareperpendiculartothecross section ofinterestandthereare nobranchingpointsthroughthe

planeofinterest (Da ¸s u et al., 2003; Patel et al., 2001 ).We ignore anytemporaldynamicsorspatialchangesofthesevessels.

φ

Omr for resting macrophages,

φ

Oma for active

macrophages,

φ

O_mi for infected macrophages,

φ

O_mci for chronically infected macrophages and

φ

Ot for T cells (see Table 2 for these

values).m(x) denotes the vesselcross section atposition x,with m

(

x

)

=1forthe presenceofblood vesselatpositionx,andzero otherwise;the termrOm(x) thereforedescribesthe productionof oxygenatraterO.Weassumethattheoxygenissuppliedthrough the blood vessel network, and then diffuses throughout the tis-suewithin itsdiffusionlimit. Atthistime, weassume aconstant backgroundoxygen level arising from airways and focuson oxy-gendiffusionfromthevascularsystem.Sinceithasbeenobserved thatwhenavesselissurroundedbycaseousmaterial,itsperfusion anddiffusioncapabilitiesareimpaired(Datta et al., 2015; Pienaar et al., 2016 ),wehaveincorporatedthisbyconsideringalower dif-fusionandsupplyrateinthegranulomastructureascomparedto thenormalvessels,i.e.

DO=

⎧

⎪

⎨

⎪

⎩

DO

DOG , insideagranuloma,

DO, elsewherein thedomain,

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Fig.4. Plotshowingtheconcentrationproﬁleofoxygensuppliedfromtheblood vesselnetworkforthe(a)theﬁxed,uniformdistributionofbloodvesselsshown inFig.3(a),and(b)therandomdistributionofbloodvesselsshowninFig.3(b). Theredcolouredspheresrepresentthe bloodvesselcrosssectionsas shownin

Fig.3andthecolourmapshowsthepercentagesofoxygenconcentration.(For in-terpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferred tothewebversionofthisarticle.)

and

rO=

⎧

⎨

⎩

rO

r_OG, insideagranuloma,

rO, elsewherein thedomain.

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Theformulationofthemodelisthencompletedbyprescribing no-ﬂuxboundaryconditionsandan initialcondition(Powathil et al., 2012 ). Fig. 4 showsarepresentativeproﬁleofthespatial distribu-tionofoxygenconcentrationaftersolvingthe Eq. (1) withrelevant parametersasdiscussedin Section 2.5 .

x,t

)

,

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where DDrugi(x) is the diffusion coeﬃcient of the drug,

φ

Drugi is theuptakerateofthedrug, with

φ

_Drugi

b denotingtheuptakerate

by the bacteria and

φ

Drugim denoting the uptake rate by the

in-fected/chronicallyinfectedmacrophages. rDrugi isthe drug supply ratebythevascularnetworkand

η

_Drugiisthedrugdecayrate. In-sideagranulomastructure,thediffusionandsupplyratearelower to account for caseum impairingblood vessels and the fact that weknowthat antibiotic diffusioninto granulomatais lowerthan innormallungtissue (Kjellsson et al., 2012 ).Hence thetransport propertiesanddeliveryrateofthedrugareasfollows:

DDrugi=

⎧

⎪

⎨

⎪

⎩

DDrugi

D_DrugiG, insideagranuloma,

DDrugi, elsewhereinthedomain,

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Table1

ParametervaluesrelatingtoEqs.(2),(3),(5)and(6).

Parametervalue Reference

DOG 2.7 (Dattaetal.,2015)

rOG 4 (Dattaetal.,2015)

DDrugiG 7.28(R),3.85(H) (Pienaaretal.,2016)

rDrugiG 4 (Dattaetal.,2015)

Table2

Diffusion and decay parameters for antibiotics and Chemokine molecules.

Drug/Chemokine Diffusionrate(cm2_s−1₎ _Decay_rate_(hr−1₎

Rifampicin(R) 1.7×10−6 _0.17

Isoniazid(H) 1.5×10−5 _0.35

Pyrazinamide(Z) 1.6×10−5 _0.12

Ethambutol(E) 1.3×10−5 _0.2

Chemokine 10−6 _0.347

and

rDrugi=

_r_Drugi

r_DrugiG, insideagranuloma,

rDrugi, elsewherein thedomain.

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Thediffusionrateaboveiscurrentlybasedonrifampicin. Tostudythe eﬃcacyof thedrug, we have assumeda thresh-old drug concentration value, below which the drug has no ef-fect on the bacteria. If the drug reachesa bacterium orinfected macrophagewhen it’s concentration isabove thislevel (which is differentforfast-andslow-growingextracellular bacteriaandfor intracellular bacteria), then the bacterium will be killed and an empty space will be created (this will be described further in Section 3.1 ).Parametersarediscussedin Section 2.5 .

2.5. Parameterestimation

Inorder tosimulatethe modelwithbiologically relevant out-comes,itisimportantto useaccurateparameters values.Mostof theparametersarechosenfrompreviousmathematicaland exper-imentalpapers(see Tables 1 –Table 3 forasummaryofthe param-etervalues).

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Table3

Parameters.Whenthereisarangeforavalue,itissetrandomlybythemodel.

Parameter Description Value Source

N Sizeofgrid(N×N) 100 (Segovia-Juarezetal.,2004)

Nv Numberofbloodvessels 49 (Cilfoneetal.,2013)

φOb(Micromoles/cell/hr) Oxygenconsumptionrateofbacteria 20.8×10

−6 ₍_Sershen_et_al.,₂₀₁₆₎

φOmr(Micromoles/cell/hr) OxygenconsumptionrateofMr 1.15×10−

7 ₍_Sershen_et_al.,₂₀₁₆₎

φOma(Micromoles/cell/hr) OxygenconsumptionrateofMa 2.3×10−

φOmi (Micromoles/cell/hr) OxygenconsumptionrateofMi 3.45×10−

φOmci(Micromoles/cell/hr) OxygenconsumptionrateofMci 4.6×10−

φOt(Micromoles/cell/hr) OxygenconsumptionrateofTcells 0.14375×10−7 (Sershenetal.,2016)

φDrugb(Micromoles/cell/hr) Antibioticconsumptionrateofextracellularbacteria 2.1×10−

11 ₍_Pienaar_et_al.,₂₀₁₅₎

φDrugmi (Micromoles/cell/hr) Antibioticconsumptionrateofinfectedmacrophages 1.9×10

−6 ₍_Pienaar_et_al.,₂₀₁₅₎

Repf(hours) Replicationrateoffast-growingbacteria 15–32 (Shortenetal.,2013)

Reps(hours) Replicationrateofslow-growingbacteria 48–96 (Hendon-Dunnetal.,2016)

Olow(%) O2thresholdforfast→slow-growingbacteria 6 Estimated-seeSection4.1

Ohigh(%) O2thresholdforslow→fast-growingbacteria 65 Estimated-seeSection4.1

Mrinit InitialnumberofMrinthedomain 105 (Cilfoneetal.,2013)

MrMa ProbabilityofMr→Ma(multipliedbyno.ofTcellsinneighbourhood) 9 (Cilfoneetal.,2013)

Nici NumberofbacterianeededforMi→Mci 10 (Cilfoneetal.,2013)

Ncib NumberofbacterianeededforMcitoburst 20 (Cilfoneetal.,2013)

Mlife(days) LifespanofMr,MiandMci 0–100 (VanFurthetal.,1973)

Malife(days) LifespanofMa 10 (Segovia-Juarezetal.,2004)

tmoveMr(mins) TimeintervalforMrmovement 20 (Segovia-Juarezetal.,2004)

tmoveMa(hours) TimeintervalforMamovement 7.8 (Segovia-Juarezetal.,2004)

tmoveMi(hours) TimeintervalforMi/Mcimovement 24 (Segovia-Juarezetal.,2004)

Mrrecr ProbabilityofMrrecruitment 0.07 (Cilfoneetal.,2013)

Tenter BacterianeededforTcellstoenterthesystem 50 (Cilfoneetal.,2013)

Trecr ProbabilityofTcellrecruitment 0.02 (Cilfoneetal.,2013)

Tlife(days) LifespanofTcells 0–3 (Sprent,1993)

Tkill ProbabilityofTcellkillingMi/Mci 0.75 (Cilfoneetal.,2013)

tmoveT(mins) TimeintervalforTcellmovement 10 (Segovia-Juarezetal.,2004)

tdrug(hours) Timeatwhichdrugisadministered 168–2160 (AsefaandTeshome,2014;Oseietal.,2015)

DrugKillf(%) Drugneededtokillfast-growingbacteria 2 (Hammondetal.,2015)

DrugKills(%) Drugneededtokillslow-growingbacteria 10 (Hammondetal.,2015)

DrugKillMi(%) Drugneededtokillintracellularbacteria 20 (Aljayyoussietal.,2017)

assume a oxygen diffusion length scale of L=100μm and a dif-fusion constant of DO=2×10−5 cm2/s (Owen et al., 2004 ). Us-ingthesealongwiththerelationL=

D/

φ

,themeanoxygen up-takecanbeapproximatelyestimatedas0.2s−1_._The_oxygen_supply

throughthebloodvesselisapproximatelyrO=8.2×10−3molss−1 (Matzavinos et al., 2009 ).

Supplyanddiffusionratesinsideagranuloma areimpaired,as describedin Eqs. (2) ,(3 ),(5 )and(6 ).Parametervaluesareoutlined in Table 1 .

Nondimensionalisation gives T= 0.001 hours andhence each timestepissetto

t₌0_.001h,withonetimestepcorresponding to3.6s.

Becauseoxygendiffusionoccursonashorttimescale(ofthe or-derof10s),accuratelytrackingtransientvariationsinoxygen con-centration would requirea much smaller time step. With M. tu-berculosisreplicationratesofbetween15and96h(Hendon-Dunn et al., 2016; Shorten et al., 2013 ), associatedtimescalesare much fasterandhenceweassumeaquasi-steadyproﬁleforoxygen,with theoxygen concentration quicklycoming toequilibriumwithany changesinthemicroenvironment.

The simulations arecarried out withina two dimensional do-mainwithagridsizeN=100,whichsimulatesanareaoflung tis-sueapproximately2mm × 2mm.Thestepsizeusedisbasedon themodelproposedanddevelopedin(Cilfone et al., 2013; Marino et al., 2011; Segovia-Juarez et al., 2004 ), withone gridcell corre-spondingtotheapproximatediameterofthebiggest discrete ele-mentinoursystem, themacrophage(Krombach et al., 1997 ).The parameters that are used inthe equationsgoverningthe dynam-icsofantibioticsandchemokinemoleculesarechoseninasimilar way.

Oxygen is lighter incomparison tothe drugs, witha molecu-larweight of32amu (Hlatky and Alpen, 1985 ),andhenceit

dif-fusesfasterthanmostofthedrugsandthechemokinemolecules. Thechemokinemoleculesdiffuseslowerthantheantibiotics,being heavierthanmostdrugs.Oneofthedrugsunderthecurrentstudy, rifampicinhasa molecular weightof822.9amu (PubChem Com-poundDatabase).Toobtainorapproximateitsdiffusioncoeﬃcient, itsmolecularmasswascomparedagainstthemolecularmassesof knowncompounds,asin(Powathil et al., 2012 ),andconsequently takento be1.7×10−6 _cm2_s−1_._Similar_analyses _are_done _with

iso-niazid,pyrazinamideandethambutol,withparametervaluesgiven in Table 2 .Thedecayrateofthesedrugsare calculatedusingthe halflife values ofthe drugsobtained fromtheliterature and are alsooutlinedin Table 2 .

The threshold drug concentrations, DrugKillf, DrugKills and DrugKillMi, below which the drug has no effect on the TB bac-teria have been chosen to be the average densityof total drugs delivered through the vessels (total drug delivered/total number ofgrid points) and the total druggiven is kept the same forall drugstypes.ValuesforDrugKillf,DrugKillsandDrugKillMiarebased on data arising from in vitro experiments and are reported in (Aljayyoussi et al., 2017; Hammond et al., 2015 ). They are given in Table 3 .Arelativethresholdischosenhereinordertocompare the effects of bacterial cell state and location of bacteria, rather thanstudyinganyoptimisationprotocolsfordrugdosage.Asmore experimental databecomesavailable, thesethresholdscan be re-ﬁned.Valuesof10−6 _cm2_s−1 _to₁₀−7 _cm2_s−1 _have_been_reported

asdiffusion constantsforchemokinemolecules(Francis and Pals- son, 1997 ).Thehalf-lifeforIL-8,animportantchemokineinvolved intheimmuneresponseofM.tuberculoisis,hasbeenshowntobe 2–4h(Walz et al., 1996 ).Weusea diffusionrateof10−6 _cm2_s−1

andahalf-lifeof2hinoursimulations.

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3. Cellularautomatonrules

Theentiremultiscalemodelissimulatedoveraprescribedtime duration,currentlyset to12,000 h(500 days),anda vector con-tainingall grid cell positions is updated at every time step. The oxygen dynamics, chemokine dynamics and drug dynamics are simulatedusingﬁnitedifferenceschemes.

3.1.Rulesfortheextracellularbacteria

AminimalinfectiousdoseofM.tuberculosishasbeenshownto beoftheorderof10(Capuano et al., 2003 ).Forthisreason,we be-ginourCAsimulationswithoneclusterof12bacteriaonthegrid; 6fast-growingbacteriaand6slow-growingbacteria.Theseinitial bacteriareplicatefollowingasetofrulesandproduceaclusterof bacteriaona regular square latticewithno-ﬂux boundary condi-tions.The fast-andslow-growingbacteriaare assigneda replica-tionrate;Rep_fforthefast-growingandRepsfortheslow-growing. Whenabacteriumismarkedforreplication,itsneighbourhood of order3 is checked foran empty space.The neighbourhood type alternatesbetweena Moore neighbourhood anda VonNeumann neighbourhood to avoid square/diamond shaped clusters, respec-tively.Ifa spaceintheneighbourhood exists,anewbacterium is placedrandomlyinone oftheavailablegrid cells.Ifthereare no spacesintheneighbourhood oforder3,thebacteriumismarked as‘resting’.Ateachtimestep,theneighbourhoodofthese‘resting’ bacteriaarere-checkedsothatthey canstarttoreplicateagainas soonasspacebecomesavailable.

Asthismultiscalemodelevolvesovertime,theelementsmove andinteractwitheachother accordingtotheCAmodel.The bac-teriaandhost cellsalsoinﬂuence thespatialdistribution of oxy-gensince they consume oxygen fortheir essential metabolic ac-tivities. As the bacteria proliferate, the oxygen demand increases creatinganimbalancebetweenthesupplyanddemandwhichwill eventuallycreate astate where thebacteria are deprivedof oxy-gen.Bacteriacan changebetweenfast-growing andslow-growing states,depending on the oxygen concentration, scaled from0 to 100,attheirlocation.Fast-growingbacteriawheretheoxygen con-centration is below Olow will become slow-growing, and slow-growingbacteriacanturntofast-growinginareaswherethe oxy-genconcentration isaboveO_high (see Table 3 forthesevaluesand Section 3.5 formoredetails).

3.2.Rulesforthemacrophages

There are 4 typesof macrophagein oursystem: resting (Mr), active(Ma),infected(Mi)andchronicallyinfected(Mci).Thereare Mr_initrestingmacrophagesrandomlyplacedonthegridatthestart ofthe simulation. These resting macrophages can become active when T cells are in their Moore neighbourhood, with probabil-ityMrMa multiplied by the number of Tcells in the neighbour-hood.When active macrophages encounterextracellular bacteria, they kill the bacteria. If the resting macrophages encounter bac-teria, they become infected and can become chronically infected when they phagocytose more than N_ici bacteria. Chronically in-fectedmacrophagescanonlycontainNcibintracellularbacteria, af-terwhichtheyburst.Burstingmacrophagesdistributebacteria ran-domlyintotheirMooreneighbourhoodoforder3andthegridcell wherethemacrophagewaslocatedbecomescaseum.

While the oxygen and antibiotics enter the system via the bloodvesselnetwork,thechemokinesaresecretedbytheinfected, chronically infected and activated macrophages. Macrophages move in biasedrandom walks, with probabilities calculated asa functionofthe chemokineconcentration ofits Moore neighbour-hood.Resting, infectedand chronicallyinfected macrophages are randomlyassigned a lifespan, Mlife days, andactive macrophages

live for Malife days. Resting macrophages move every tmoveMr minutes, active macrophages move every tmoveMa hours and in-fected/chronicallyinfectedmacrophagesmoveeverytmoveMi hours. Resting macrophages arerecruited fromthe blood vessels witha probabilityofMrrecr.

3.3. RulesfortheTcells

The T cells enter the system once the extracellular bacterial loadreachesTenter andmovein abiasedrandom walk,similar to themacrophages.Tcellsarerecruitedfromthebloodvesselswith aprobabilityTrecr.TheyliveforTlife,andmoveeverytmoveTminutes. ActivatedTcellsareimmuneeffectorcellsthatcankillchronically infectedmacrophages.IfaTcellencountersaninfectedor chron-icallyinfectedmacrophage,itkillsthemacrophage(andall intra-cellular bacteria)withprobability T_kill andthat grid cellbecomes caseum.Tcellsalsoactivaterestingmacrophageswhentheyarein their Mooreneighbourhood, withprobability MrMamultiplied by thenumberofTcellsintheneighbourhood.

3.4. Rulesfortheantibiotics

Ifthe immune process does not clearthe infection,drugs are administeredattdrughours, arandomlychosentimebetweentwo values(see Table 3 ). This mimics the variabilityin time that pa-tientsseekmedicalattentionfortheirdisease(Asefa and Teshome, 2014; Osei et al., 2015 ). The antibiotics are administered for a period of 4380 h (six months). The antibiotic kills the bacteria whentheconcentration isoverDrugKillf orDrugKills,forthe fast-and slow-growing bacteria respectively. The antibiotics can also kill intracellular bacteria,by killingthe bacteriacontainedwithin the infected/chronically infected macrophages if the concentra-tion isover DrugKillMi.Thesemacrophages then returnto resting macrophages.

3.5. Oxygenthresholdsforbacterialcellstates

InordertochoosevaluesforOlowandOhigh,weconductedsome test simulations for 120 h, where nomacrophages were present. Thisallowedustoalterboththeoxygenswitchingthresholdsand seetheeffectithadonthebacteria.90simulationswererunin to-tal,witharangeofbothparametersbeingtested. Table 4 describes theoutcomeofthesesimulations:

Fig. 5 shows representative examples of these test simula-tions, where fast- and slow-growing bacteria are shown (by the blue/cyan lines, respectively), with O_high ﬁxed at 65 and O_low₌ 3,6,9. In (a) we see the effect of having a lower threshold for fast-growing bacteria to become slow-growing, with Olow=3, in (b)Olow=6andin(c),Olow hasa higherthresholdof9.In simu-lation(a),thebacteriadonotchangestate duringtheentire sim-ulation, Fig. 5 (a) supports thisas we do not seean increase in thecyanlinethatcorrespondswithadropintheblue.Simulation (b)hasO_low₌6andhereweseesometransferfromfast-to slow-growing during the 120 h. For simulation (c), however, the fast-growingbacteriatransfertoslow-growingalmostimmediately.

Similarly, Fig. 6 shows representative examples from varying Ohigh, where Olow is held at 6 and Ohigh=55,65,75. In simula-tion(a),whereOhigh=55,theslow-growingbacteriaallchangeto fast-growingveryneartothebeginningofthesimulation. Simula-tion(b)showssomebacteriachangingstatewhenO_high=65,and simulation(c)showsnotransferfromfast-toslow-growingwhen Ohigh=75.

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Table4

Oxygenswitch.

Olow=3 Olow=6 Olow=9

Ohigh=55 Allslow-growingbacteriaswitchedto

fast-growinginall10simulationsby

t=10h

In9ofthe10simulations,allslow-growing bacteriachangedtofast-growingbyt=10h

Inall10simulations,bacteriawasallthesamecell statebyt=10h:In4simulationsallwere slow-growing,in6simulationsallwere fast-growing

Ohigh=65 Nobacteriachangestateinanyofthe

10simulations

In7ofthesimulations,somebacteriachanged state:4fromfast-growingtoslow-growingand3 fromslow-growingtofast-growing(meantime forﬁrstswitchwast=21h).In3ofthe simulations,nobacteriachangedstate.

Allfast-growingbacteriabecomeslow-growingby

t=10hinall10simulations

Ohigh=75 Nobacteriachangestateinanyofthe

10simulations

Nobacteriachangedstateinanyofthe10 simulations

Allfast-growingbacteriabecomeslow-growingby

t=10hoursinall10simulations

Fig.5. Plotsofthefast-(blue)andslow-growing(cyan)extracellularbacteriafortheﬁrst120h,withOhighﬁxedat65and(a)Olow=3,(b)Olow=6and(c)Olow=9.(For

interpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig.6. Plotsofthefast-(blue)andslow-growing(cyan)extracellularbacteriafortheﬁrst120h,withOlowﬁxedat6and(a)Ohigh=55,(b)Ohigh=65and(c)Ohigh=75.(For

interpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

4. Simulationresults

Inordertostudytherelativeimportanceofbacterialcellstate andinitialspatiallocationofbacteria,westudytwoscenarios:one withaﬁxed,uniformbloodvesseldistributionandinitialbacterial locations(see fourexamples in Fig. 7 (a)), andanotherwherethe vesseldistributionandtheinitiallocationsoftheextracellular bac-teria aredetermined randomlyforeach simulation(see examples in Figs. 9 –11 (a)). Werunatotalof120simulationsfor500days: 20simulationsforthe‘ﬁxed’scenarioand100simulationsforthe ‘random’scenario.Thesesimulationswererunonserversthathave dual Intel Xeon E5-2640 CPUs and 128GB RAM. Each simulation tookapproximately8hourstorun.

4.1. Fixedbloodvesselsandinitialbacterialcluster

20 simulations were run withthe same initial distribution of Nvblood vessels andwithone bacterialcluster of6fast-growing

bacteriaand6slow-growingbacteria,locatedinthecentreofthe grid. Fig. 7 showsthisinitial setup. In this‘fixed’spatial config-uration,there was1blood vessel locatedwithin a 0.1mm radius ofthebacterialcluster,situated1gridcellaway(0.02mm).All20 simulationsresultedincontainmentofthedisease(whichwe de-fineasfewerthan 10extracellularbacteriaatthe endofthe 500 days), withthe macrophages andT cellspreventing disease pro-gression.Inall 20simulations nobacteriaremainedattheendof the500days.Asagranulomadevelopsinthesimulations,caseous cellsare created atthe centre. At 500 days,the simulations had a rangeof11–49caseous gridcells withamedian value of13.5. Inonlyone simulationthetotalbacterialloadexceededTenter and thereforeTcellsonly appeared inone simulation.See Table 1 in supplementary material for summary statistics forthese 20 sim-ulations. Supplementary material also contains a document with plotsshowingthedynamicsoftheTcells.

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eradi-Fig.7. Plotsshowingtheoutcomeoftworepresentativesimulations,(i)–(ii),withthefixedvesseldistributionandinitialbacteriallocation.(a)and(b)areplotsofthespatial distributionofallelements:Atthestartofthesimulation,withtheinitialbacterialclusterhighlightedbytheblackcircle(a)andattheendofthesimulation(b).Redcircles depictthebloodvessels,blackcirclesdepictthecaseum,bluecirclesshowthefast-growingextracellularbacteria,cyancirclesshowtheslow-growingextracellularbacteria, greendotsdepictmacrophages(withdarkergreenfortheinfected/chronicallyinfectedmacrophages)andyellowdotsdepicttheTcells.Plotsofbacterialnumbersareshown in(c),depictingfast-growingextracellularbacteria(darkblue),slow-growingextracellularbacteria(cyan)andintracellularbacteria(green).Notethatindividualfiguresare availableinSupplementarymaterialforcloserinspection.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionof thisarticle.)

cated.In the line plot, Fig. 7 (i)(c), we see that the macrophages effectivelyphagocytosethebacteriaearlyoninthesimulationand theinfected macrophagesgradually die out over 2500 h (around 100 days).In the model,asa macrophagereaches theend ofits lifespan,thegridcellbecomescaseumandtheintracellular bacte-riacontainedwithin isabletoburstout andgrowextracellularly. Thiscanonlyhappen,however,ifthereisspaceforthebacterium to move to. If instead, as in this case, the immune response is effectiveincontrollingthe diseasewithrestingmacrophages sur-roundingtheinfection,thereisnospacefortheseintracellular bac-teriatore-emergeasextracellularbacteria,andthebacteriumdies. Fig. 7 (ii) depicts the outcome of another example simulation witha starting ‘ﬁxed’ distribution. Fig. 7 (ii)(b)shows the end of thissimulation with49 caseousgrid cells. Ifwe look atthe line plot in Fig. 7 (ii)(c), we can see a more eventful simulation. The bacteriastart togrow atthebeginning ofthe simulationandwe see a more gradual immune containment of the infection. This isthe one simulation where Tcells entered the systemto assist inthis containment.Because of this, the Tcellsare also respon-siblefor killing infected macrophages and they also activatethe macrophages,which alsocontribute to thekilling ofthe infected macrophages.In Fig. 7 (ii)(c)wealsoseeincidencesofintracellular bacteriasuccessfully moving out ofthe macrophages asthey die naturally.Thesecan be seen assmallspikes inthe slow-growing line. In many of these cases, however, these newly escaped ex-tracellularbacteria arequicklyphagocytosed again.Eventually, by around3700h(around150days),thisinfectioniscompletely erad-icated.

Fig. 8 showssummary plots ofthe 20ﬁxed simulations, with themeanbacterialnumbersshownbythesolidlinesand95%

con-fidence intervals shown by the dashed lines. Fig. 8 (i) shows the entire 12,000 h and Fig. 8 (ii) shows only until 200 h, asthis is wheremostactivitytakesplace.Wecanseefromtheseplotsthat there isnot a great amount ofvariation betweenthe 20 simula-tions, withsmalldifferencesinbacterial numbers.It isclearalso that mostof the variability takes place close to the start of the simulations. This confirmsour conclusionsthat with a fixed dis-tributionof bloodvessels andinitialbacterialplacement, thereis verylittledifferenceintheoutcomebytheendofthesimulations.

4.2. Randombloodvesselsandinitialbacterialcluster

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Fig.8. Summaryplotsofall20fixedsimulations.Thesolidlinerepresentsthemeanbacterialnumbersandthedashedlinesshowthe95%confidenceintervals.(i)shows thebacterialnumbersfortheentire12,000hours,whereas(ii)showsuntiltime200hours,wheremostactivitytakesplace.Thefast-growingbacteriaareshownbythe bluelinesin(a),theslow-growingbacteriaareshownbythecyanlinesin(b)andtheintracellularbacteriabythegreenlinesin(c).(Forinterpretationofthereferencesto colourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

In Fig. 9 (i)(b), we seean example ofa simulation wherethe infection has been contained, with no extracellular bacteria and only 11 intracellular bacteria remaining. There is also a granu-loma visible at 500 days, containing 102 caseous grid cells. The line plot shownin Fig. 9 (i) (c)describes how the immune sys-tem hascontained thisinfection. In the500 days of the simula-tion, we seenumerousspikes in theextracellular bacteria,as in-fectedmacrophagesdieandtheintracellularbacteriaarereleased. In mostcircumstances, thisre-emergent infectionis quickly con-trolled.Atroughly9000h,however,theslow-growingbacteria be-ginto growagain.ItisatthispointinthesimulationthatTcells arerecruited,whichhelpstoregaincontroloftheinfection.

Fig. 9 (ii)showsanotherexampleofacontainedinfection,where notreatmentwasneeded.Inthissimulation,thereare6 extracel-lular bacteria remaining atthe endofthe 500 days.However, as these are surrounded by caseous material within the granuloma structure,theinfectioniscontrolledandthebacteriacannotgrow. In Fig. 9 (iii), we see a similar picture to that of shown in Fig. 9 (i). The difference hereisthat there wasno Tcell recruit-ment needed tocontrol theinfection. By theend of the simula-tion,weseein Fig. 9 (iii)(c)thatthereare16intracellularbacteria remaining.

Fig. 9 (iv)depicts a situation wherethe infection is controlled eﬃciently by thehost immunity. Fig. 9 (iv)(b) showsno bacteria remainingwithonly12caseousgridcells.

In 10simulations, the immune responsewasnot able to con-tain the disease within the granuloma and active disease devel-oped.Antibioticswerethenadministeredattdrughours.Insevenof thesesimulations,thetreatmentwas‘successful’,wherewedeﬁne successasfewerthantenextracellular bacteriaatthe endof500 days.Fourofthesesimulationshadbetween1and8slow-growing extracellular bacteria remaining at 500 days. These extracellular bacteriawereallsurroundedbycaseousgridcells.Althoughthese foursimulationsaredeemed‘successful’,asthereareasmall num-ber ofbacteriaremaining,thesecasesare capableofrelapsing.In the seven successfulsimulations, themajorityof thebacteria are

killedbytheantibiotics(meanvalueof75.1%).Fourrepresentative examplesofthese‘successfullytreatedcases’areshownin Fig. 10 . Fig. 10 (i)showsanexampleofasuccessfullytreatedsimulation. Bothtypesofextracellularbacteriagrowrapidlyatthestartofthe simulation,withtheimmune cellsunabletocontroltheinfection. Asthe bacteriagrow, theystart toconsumemoreandmore oxy-gen,reducingtheavailabilityandcausingthefast-growingbacteria toswitchstate. Weseethecleareffectsoftheantibioticsasthey enterthesystemat535hours,with85.8%ofthetotalbacteriain thesystembeingkilledbytheantibioticsatthistime.Nobacteria remainattheendofthissimulation.

Fig. 10 (ii)showsanotherexampleofasuccessfullytreated sim-ulation.Again,weseethesharpreductionofbacterialloadasthe antibioticsenterthesystemat708h,with80.6%ofthetotal bacte-riainthesystembeingkilledbytheantibiotics.Shortlyafterthis, thehostimmunitycontrolstheremaininginfection.Only1 slow-growingbacteriaremainsattheendofthesimulation,surrounded bycasousgridcells.

In Fig. 10 (iii),antibioticswereadministeredat215h,whichwe seein Fig. 10 (iii)(d)controlstheinfection,killing88.9%ofthetotal bacteria.

Fig. 10 (iv) shows an example where, although the antibiotics wereused andwereeffective,theimmune response wasactually responsible forthemajority ofthe killing(63.5%). Thisis inpart duetothat fact that antibioticswere givenearly, at 249h when thebacterialburdenwasnotparticularlyhigh.

The remaining three simulations, were deemed ‘unsuccessful’. Inthesesimulationstherewere16,241and354slow-growing bac-teriaremainingat500days.Thelattertwosimulationsareshown in Fig. 11 .

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Fig.9. Plotsshowingtheoutcomeoffourrepresentativesimulations,(i)–(iv),ofcontainedsimulations,with arandomly placedvesseldistributionand initialbacterial location.(a)–(b)areplotsofthespatialdistributionofallelements:atthestartofthesimulation,withtheinitialbacterialclusterhighlightedbytheblackcircle(a)and attheendofthesimulation(b).Redcirclesdepictthebloodvessels,blackcirclesdepictthecaseum,bluecirclesshowthefast-growingextracellularbacteria,cyancircles showtheslow-growingextracellularbacteria,greendotsdepictmacrophages(withdarkergreenfortheinfected/chronicallyinfectedmacrophages)andyellowdotsdepict theTcells.Plotsofbacterialnumbersareshownin(d),depictingfast-growingextracellularbacteria(darkblue),slow-growingextracellularbacteria(cyan)andintracellular bacteria(green).NotethatindividualﬁguresareavailableinSupplementarymaterialforcloserinspection.(Forinterpretationofthereferencestocolourinthisﬁgurelegend, thereaderisreferredtothewebversionofthisarticle.)

Finally, Fig. 11 (ii) shows an example of a simulation where treatmentwasreceivedandwasinitially‘successful’but,asitcan beseen in Fig. 11 (ii)(d), ataround8000h, extracellularbacteria begintogrowasdyingmacrophagesreleasetheirintracellular bac-teria.The slow-growing bacteria then continueto grow untilthe endofthesimulation.Thisisanexampleofarelapse.

Fig. 12 shows summary plots of the 100 random simulations, withthemeanbacterialnumbersshownbythesolidlinesand95% conﬁdenceintervals shown by the dashed lines. Fig. 12 (i) shows the entire 12,000 h and Fig. 12 (ii) shows only until 200 hours, asthisiswheremostactivitytakesplace.Incontrasttotheﬁxed summaryplots,weseemuchmorevariationinbacterialnumbers, especially for the slow-growing bacteria. We also note that this variabilityremains fortheentire12,000h.Thisfurtherhighlights

theincreasedvariabilityinoutcomewhenweinvestigatearandom placementofbloodvesselsandbacterialcluster.

Table 2 insupplementarymaterialshowssummarystatisticsfor these100 random simulations. Supplementary material also con-tainsadocumentwithplotsofthedynamicsoftheTcells.

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Fig.10. Plotsshowingthe outcomeoffourrepresentativesimulations,(i)–(iv),whichwere‘successfullytreated’,witharandomlyplacedvesseldistributionandinitial bacteriallocation.(a)–(c)areplotsofthespatialdistributionofallelements:atthestartofthesimulation,withtheinitialbacterialclusterhighlightedbytheblackcircle (a),justbeforethedrugentersthesystem(b)andattheendofthesimulation(c).Redcirclesdepictthebloodvessels,blackcirclesdepictthecaseum,bluecirclesshowthe fast-growingextracellularbacteria,cyancirclesshowtheslow-growingextracellularbacteria,greendotsdepictmacrophages(withdarkergreenfortheinfected/chronically infectedmacrophages)andyellowdotsdepicttheTcells.Plotsofbacterialnumbersareshownin(d),depictingfast-growingextracellularbacteria(darkblue),slow-growing extracellularbacteria(cyan)andintracellularbacteria(green).NotethatindividualﬁguresareavailableinSupplementarymaterialforcloserinspection.(Forinterpretation ofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

and p=4.6×10−5 _for_number _of _blood _vessels _within _a _0.1_mm

radius. Hence we have shown that these groups are statistically differentatthe 5% level.Thisseems tosuggest that, iftheinitial bacteria arelocatedfurther awayfromtheblood vessels,the less likelyit isthat thehostimmune responsewill containthe infec-tion.

5. Discussion

Individual-basedmodelshavealreadybeenshowntobe useful in understanding tuberculosis disease progression (Cilfone et al., 2013; Marino et al., 2011; Pienaar et al., 2015; 2016; Segovia-Juarez et al., 2004; Sershen et al., 2016 ).Herewehavebuiltahybrid cel-lular automatonmodelthat incorporatesoxygendynamics,which allowsbacteriatochangestate,andincludesantibiotictreatments.

Inaddition tofocusing onbacterial cellstate, we alsoinvestigate changesinspatiallocationofthebacteriaandtheirinﬂuences on diseaseoutcome.

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Fig.11. Plotsshowingtheoutcomeoftworepresentativesimulations,(i)–(ii),whichwere‘unsuccessfullytreated’,witharandomlyplacedvesseldistributionandinitial bacteriallocation.(a)–(c)areplotsofthespatialdistributionofallelements:atthestartofthesimulation,withtheinitialbacterialclusterhighlightedbytheblackcircle (a),justbeforethedrugentersthesystem(b)andattheendofthesimulation(c).Redcirclesdepictthebloodvessels,blackcirclesdepictthecaseum,bluecirclesshowthe fast-growingextracellularbacteria,cyancirclesshowtheslow-growingextracellularbacteria,greendotsdepictmacrophages(withdarkergreenfortheinfected/chronically infectedmacrophages)andyellowdotsdepicttheTcells.Plotsofbacterialnumbersareshownin(d),depictingfast-growingextracellularbacteria(darkblue),slow-growing extracellularbacteria(cyan)andintracellularbacteria(green).NotethatindividualﬁguresareavailableinSupplementarymaterialforcloserinspection.(Forinterpretation ofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig.12. Summaryplotsofall100randomsimulations.Thesolidlinerepresentsthemeanbacterialnumbersandthedashedlinesshowthe95%conﬁdenceintervals.(i) showsthebacterialnumbersfortheentire12,000h,whereas(ii)showsuntiltime200h,wheremostactivitytakesplace.Thefast-growingbacteriaareshownbytheblue linesin(a),theslow-growingbacteriaareshownbythecyanlinesin(b)andtheintracellularbacteriabythegreenlinesin(c).(Forinterpretationofthereferencesto colourinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

takealongertimefortheimmunecellstorespondtoandattempt tocontroltheinfection.Thisextratimecangive thebacteriatime to grow, making it harderfor the host immunity to contain the disease,thus in the majority ofcases, antibiotics are requiredto reduce the bacterial burden. We also note that in many of the

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thisisaﬂawinthemodeldesign,itdoesillustratetheimportance ofeffectivegranulomaformation.Infutureiterationsofthemodel wecouldconsideralargerdomainorlookatalternativeboundary conditionstocomparetheeffect.

Inthe100simulationsthatusedarandomdistributionofblood vessels andbacteria, we found that 90 (90%) ofthem were con-tainedbythehostimmunitybut32ofthese(36%)stillhadsome bacteria remaining, andare therefore latently infected, their dis-ease capableofreactivatingatalater date.10 ofthe100 simula-tions(10%)requiredtreatmentand7ofthese(70%)hadfavourable outcomes, withfewer thantenextracellularbacteriaremainingat the end of500 days.The remaining three simulations hadmore than tenextracellular bacteriaremaining attheend ofthe simu-lation.Twoofthesecasesweretreatedwithantibiotics,which re-ducedthebacterialloaddramatically.Inonecase,only16bacteria were remaining and these bacteria were situated within a gran-uloma.Theother casedepictsarelapse,wheretreatmentwas ini-tiallysuccessfulbutinfectionstartedtogrowagainposttreatment. Thelastcaseisanexampleofalatentlyinfectedindividualwhose diseasereactivatesjustbeforetheendofthesimulation.These per-centagesare comparablewiththose describedin Fig. 1 andfrom epidemiologicaldatafromtheWHOontheratesoflatentTB dis-ease(WorldHealthOrganisation,2013),with90% ofthecases re-sulting in containment of the disease but withmany capable of later relapse. We also show a case where an early clearance of the bacterialloadstill ended up withrelapseof thedisease, this is consistent withobservations fromthe REMoxTB trial,reported in(Phillips et al., 2016 ).Ourmodelsimulationsalsoshow thatin thecaseswherebacteriaremainattheendofthesimulation,they areslow-growing.Thisﬁndingofpersistentbacteriaisalso consis-tentwiththeliteraturedescribingpersistentTBinfectionininvivo models(Hu et al., 20 0 0; Manina et al., 2015 ).

An important feature of our model is that of caseation: when infected macrophages burst or die, or T cells kill in-fected macrophages, that grid cell becomes caseum. Hence, as macrophages movechemotacticallytowards theclustersof bacte-ria, a caseous granuloma starts toform andthis caseuminhibits drug diffusion.We have shownthat in simulations where bacte-riaaresurroundedbycaseum,theyoftenremainattheendofthe simulation.Thisemphasisestheimportanceofcaseousnecrosison the outcomeoftherapy.The implicationsofcaseumhave already been demonstrated (Grosset, 1980 ) and our simulations conﬁrm the importance of thistype ofnecrotic breakdown. Other hybrid models,such as(Pienaar et al., 2015 ), havealsoincludedcaseum, with non-replicating bacteria residing in the necrotic tissue and alsocallattentiontothisimportantfeatureintermsoftreatment outcome.Withrecentexperimentalstudiessuchas(Sarathy et al., 2018 ),wenowhavetheavailabledatatoexplorethedrug suscep-tibilityofbacteriaincaseum.

We have also shown that bacterial cell state has an impact on simulations, which is a characteristic that is only just start-ing to beunderstood. Allsimulations that haveextracellular bac-teria remaining at the end of 500 days are slow-growing bacte-ria. This is in part because any treatment received tends to kill off the faster-growing bacteria more quickly,perhaps leaving be-hind a slower-growing population. These bacteria are also often located inside a granuloma, where the oxygen supply and diffu-sion rate is impaired, which favours slower-growing phenotypes. There arerelativelyfew publicationsthat deﬁnethe susceptibility of slow-growingmycobacteriain relationto thestandard or new anti-tuberculosisdrugs andparticular emphasisshould be placed onsuchantibioticsthatcanpenetratewellintolesions.

Our preliminary simulations also highlight the importance of spatiallocationofthebacteria.Perhapsitisthoughtobviousthat spatial location of the bacteria is a key factor in treatment out-comebutprevious mathematical modelsto datehavenot

identi-fiedthisfact.Studieshavefocused onpharmacokinetic(PK)based on serum and simulations of epithelial lining fluid (ELF) bron-choalveolarlavage.Ourmodellinghasshownthatanatomical con-siderationsareimportantwhenchronicinfectioncreatesan anaer-obicenvironment andfibrosis aroundcavities. Treatment is com-poundedfurtherby bacterialcellstate,whichincreasesfunctional MICofbacteriathatcanbemoredifficulttokill duetopoor pen-etration.

Future models could address thiswith enhanced understand-ingofthe effectofdormancyorphenotypicresistance.This indi-catestheimportanceofworktodefinelesionalPK(Prideaux et al., 2015; Via et al., 2015 ). Infuture iterationsofthemodelwill also include more anatomical and immunological complexity, for ex-ample,airwayswill beaddedtothe domaintoexploreits effects andmore than one Tcell will be integrated intothe model. We could also explore the effect of fibrosis and cavity formation on outcome,buildingonrecentconcepts onlesionaldrug concentra-tions (Prideaux et al., 2015; Via et al., 2015 ).In addition,we will modelliquefaction, which will be importantto allow the release of‘trapped’bacteria,thusallowingustofurtherinvestigaterelapse cases.As ourunderstanding ofM. tuberculosiscell state increases wewillalsobeabletorefineourparameterestimatesofthis char-acteristicandbuilda bettermodel.Infuturemodelswe willalso investigatetheeffectofallowingmorethanoneelementtooccupy eachgridcell.Inthismodelthesimpledrugdiffusionthatis mod-elledissufficientforourinitialinvestigations.Infuturemodelswe willintegrateaPK/PDtreatmentmodelforcombinationtherapyso thatwecaninvestigatetherelativeimportanceoffront-linedrugs andtheirroleintargetingthevariousbacteria.

Sputum culture conversion during treatment for tuberculosis hasa limitedrole inpredictingtheoutcome oftreatment for in-dividualpatients(Phillips et al., 2016 ),so spatialmodelsthat ex-ploreTBinfectionandtreatmentinthelung areneededifwe are toincreaseourunderstandingofpatientoutcome.Inthisworkwe haveshown,usinganindividual-basedmodel,thataspatialmodel allowsustoexploremanyunansweredquestionsinTB.

Acknowledgements

This work was supported by the Medical Research Council [grantnumber MR/P014704/1 ]andthePreDiCT-TBconsortium(IMI Joint undertaking grant agreement number 115337 , resources of whichare composed of ﬁnancial contributionfrom theEuropean Union’s Seventh Framework Programme (FP7/2007-2013 ) and EF-PIAcompanies’inkindcontribution.

Supplementarymaterial

Supplementary material associated with this article can be found,intheonlineversion,atdoi:10.1016/j.jtbi.2018.03.006.

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