Delay effects in the response of low grade gliomas to radiotherapy: A mathematical model and its therapeutical implications

0.43 MB·

by  Victor M. Perez-Garcia

MathematicalMedicineandBiologyPage1of 26
doi:10.1093/imammb/dqnxxx
4
Delayeffects inthe response oflowgrade gliomasto radiotherapy: A
1
0 mathematical model andits therapeutical implications
2

n V´ICTOR M.PE´REZ-GARC´IA∗
a
J DepartamentodeMatema´ticas,UniversidaddeCastilla-LaMancha
ETSIIndustriales,Avda.CamiloJose´ Cela3,13071CiudadReal,Spain.
2
1 ∗Correspondingauthor:[email protected]

] MAGDALENA BOGDANSKA
M
MathematicsDepartment,TechnicalUniversityofLodz,
Q Lodz,Wolczanska214Street,Poland.
o. ALICIA MART´INEZ-GONZA´LEZ,JUAN BELMONTE-BEITIA
i DepartamentodeMatema´ticas,UniversidaddeCastilla-LaMancha
b
ETSIIndustriales,Avda.CamiloJose´ Cela3,13071CiudadReal,Spain.
-
q
[ PHILIPPESCHUCHT

Universita¨tsklinikfu¨rNeurochirurgie
1
v BernUniversityHospital,CH-3010Bern,Switzerland
3
LUIS A.PE´REZ-ROMASANTA
0
6 RadiotherapyUnit,UniversityHospitalofSalamanca,Salamanca,Spain
2
. [ReceivedonXXXXX]
1
0 Lowgradegliomas(LGGs)areagroup ofprimarybraintumorsusuallyencountered inyoung patient
4
populations.Thesetumorsrepresentadifficultchallengebecausemanypatientssurviveadecadeormore
1
andmaybeatahigherriskfortreatment-relatedcomplications. Specifically,radiationtherapyisknown
:
v to have a relevant effect on survival but in many cases it can be deferred to avoid side effects while
i maintaining its beneficial effect. However, a subset of low-grade gliomas manifests more aggressive
X
clinicalbehaviorandrequiresearlierintervention. Moreover,theeffectivenessofradiotherapydepends
r onthetumorcharacteristics.RecentlyPalludetal.,[Neuro-oncology,14(4):1-10,2012],studiedpatients
a
with LGGstreated with radiation therapy as a first line therapy. and found the counterintuitive result
thattumorswithafastresponsetothetherapyhadaworseprognosisthanthoserespondinglate. Inthis
paperweconstructamathematicalmodeldescribingthebasicfactsofgliomaprogressionandresponseto
radiotherapy.ThemodelprovidesalsoanexplanationtotheobservationsofPalludetal.Usingthemodel
weproposeradiationfractionationschemesthatmightbetherapeuticallyuseful byhelpingtoevaluate
thetumormalignancywhileatthesametimereducingthetoxicityassociatedtothetreatment.
Keywords:LowGradeGliomas,Radiotherapy,Mathematicalmodeloftumorresponse
1. Introduction
Lowgradeglioma(LGG)isatermusedtodescribeWHOgradeIIprimarybraintumorsofastrocytic
and/or oligodendroglialorigin. These tumors are highly infiltrative and generally incurable but have
mediansurvivaltimeof>5yearsbecauseoflowproliferation(Pignattietal.,2002;Pouratian&Schiff,
2010). Whilemostpatientsremainclinicallyasymptomaticbesidesseizures,thetumortransformation
(cid:13)c Theauthor2008.PublishedbyOxfordUniversityPressonbehalfoftheInstituteofMathematicsanditsApplications.Allrightsreserved.
2of26 V.M.PE´REZ-GARC´IAetal.
toaggressivehighgradegliomaiseventuallyseeninmostpatients.
ManagementofLGGhashistoricallybeencontroversialbecausethesepatientsaretypicallyyoung,
with few, if any, neurologicalsymptoms. Historically, a wait and see approach was often favored in
mostcases of LGG due to the lack of symptomsin these mostly youngand otherwisehealthyadults.
Thesupportforthispracticecamefromseveralretrospectivestudiesshowingnodifferenceinoutcome
(survival, qualityof life) if therapywas deferred(Olsonetal., 2000;Grier&Batchelor, 2006). Other
investigationshavesuggestedaprolongedsurvivalthroughsurgery(Smithetal.,2008). Inabsenceof
arandomizedcontrolledtrial,recentlypublishedstudiesmayprovidethemostconvincingevidencein
supportofanearlysurgerystrategy(Jakolaetal.,2012) andwaitingforthe useofothertherapeutical
options such as radiotherapy and chemotherapy. However, the decision on the individual treatment
strategy is based on a number of factors including patient preference, age, performance status, and
locationoftumor(Ruiz&Lesser,2009;Pouratian&Schiff,2010).
As to radiation therapy the clinical trial by Garciaetal (2004) showed the advantageof using ra-
diotherapy in addition to surgery. However, the timing of radiotherapy after biopsy or debulking is
debated. Itisnowwellknownthatimmediateradiotherapyaftersurgeryincreasesthetimeofresponse
(progression-free survival), but does not seem to improve overal survival while at the same time in-
ducingseriousneurologicaldeficitsasa resulttonormalbraindamage(VandenBent,2005). Overall
survivaldependsmoreonpatientandtumour-relatedcharacteristicssuchasage,tumourgrade,histol-
ogy and neurologicalfunction than the details of the plan of radiotherapytreatment. Radioteraphyis
usuallyofferedforpatientswithacombinationofpoorriskfactorssuchasage,sub-totalresection,and
diffuseastrocytomapathology(Higuchietal.,2004).
Mathematicalmodellinghasthepotentialtoselectpatientsthatmaybenefitfromearlyradiotherapy.
Also it may help in developingspecific optimalfractionationschemesfor selected patientsubgroups.
Howeverdespiteitsenormouspotential,mathematicalmodellinghashadalimitedusewithstrongfocus
onsomeaspectsofradiationtherapy(RT)forhigh-gradegliomas(Rockneetal.,2010;Bondiauetal.,
2011;Konukogluetal.,2010;Barazzuoletal.,2010). Uptonow,noideascomingfrommathematical
modellinghavebeenfoundusefulforclinicalapplication.
Thereisthusaneedformodelsaccountingforthefundamentalfeaturesoflow-gradegliomadynam-
icsandtheirresponseto radiationtherapywithoutinvolvingexcessivedetailsonthe -oftenunknown-
specificprocessesbutallowingthequalitativeunderstandingofthephenomenainvolved.Theavailabil-
ityofsystematicandquantitativemeasurementsofLGGgrowthratesprovideskeyinformationforthe
developmentandvalidationofsuchmodels(Palludetal.,2012,b).
Inthispaperwepresentasimplemathematicalmodelcapturingthekeyfeaturesseenintheresponse
of LGGsto radiation. Our modelincorporatesthebasic elementsoftumordynamics: infiltrationand
invasionofthenormalbrainbythetumorcells,proliferationandtumorcelldeathinresponsetother-
apy. Radiationtherapyis includedin analmostparameter-freeway thatcapturesthe essentials of the
dynamicsandexplainstherelationshipbetweenproliferation,responsetothetherapyandprognosisas
recentlyreportedbyPalludetal.(2012b).
In addition to explaining the counterintuitive observationsof Palludetal. (2012b) the model pre-
sentedinthispapercanbeusedtoexploredifferentradiationregimes. Theanalysistobepresentedin
thispapersuggeststhepossibilityofusingradiationtherapywithpalliativeintentandalsototestwhat
thetumorresponseisandhelptheoncologistsinmakingthebestpossibledecisionsonwhenandhow
toactonthetumor.
Ourplaninthispaperisasfollows. First,inSection2wepresentourmodelaccountingfortumor
celldynamicsandtheresponseofthetumorcellstoradiation. NextinSec. 3wepresenttheresultsof
thenumericalsimulationsofthemodelindifferentscenariosandstudythedependenceofthemodelon
DELAYEFFECTSINRADIOTHERAPYOFLGG 3of26
thedifferentparameters. InSec. 4wedisplaysomeanalyticalestimatesofthetypicaldynamicsofthe
tumorresponsetoradiation. InSec. 5wediscusssomehypothesissuggestedfromthemodelthatmay
beusefulfortherapy.Finally,inSec. 6wesummarizeourconclusions.
2. MathematicalmodelfortheresponseofLGGstoradiotherapy
2.1 Tumorcellcompartment
Inthelastyearstherehasbeenalotofactivityonmathematicalmodelsofgliomaprogression(Murray,
2007;Stamatakosetal.,2006,b;Frieboesetal.,2007;Swansonetal.,2008;Bondiauetal.,2008;Eikenberryetal.,
2009;Tanakaetal.,2009;Konukogluetal.,2010;Wangetal.,2009;Rockneetal.,2010;Pe´rez-Garc´ıaetal.,
2011;Pe´rez-Garc´ıa&Mart´ınez-Gonza´lez,2012;Hatzkirouetal.,2012;Badoualetal.,2012;Mart´ınez-Gonza´lezetal.,
2012;Guetal.,2012;PainterandHillen,2013).Inthispaperwewillconsideramodelforthecompart-
mentof tumorcells of the simplest possible type: a Fisher-Kolmogorov(FK) type equation(Murray,
2007). Morecomplicatedmodelssuchassingle-cellbasedmodelswouldallow,inprinciple,tofollow
theindividualmovementofthetransformedastrocytesthroughthebrainparenchyma.However,consid-
eringthatthebasicrulesbehindamodelaremoreimportantthanthemodeldetails,wehavediscarded
boththeuseofon-latticemodels,whicharenotrealisticwhencellmotionisconsidered,andoff-lattice
modelswhichassumefixedcellgeometriesand/orincorporateunknowncell-cellinteractions. Besides,
thesemodelsoftenrequiretheestimationofalargenumberofunknownparametersandthedetermina-
tionofinitialcellconfigurations,whichareextremelydifficulttovalidateininvivoexperimentsand/or
usingclinicaldata. Thus,tokeepourdescriptionassimpleasposiblewehaveoptedforacontinuous
modelasfollows
¶ C
= DD C+r (1 C)C, (2.1a)
¶ t −
C(x,0) = C (x), (2.1b)
0
whereC(x,t) is the tumor cell density as a function of time t and the spatial position x and it is
measuredinunitsofthemaximalcelldensityallowedinthe tissueC (typicallyaround103 cell/cm).
D =(cid:229) n ¶ 2/¶ x2 isthen dimensionalLaplacianoperator. ∗
j=1 j −
D is thediffusioncoefficientaccountingfortheaveragecellularmotilitymeasuredin mm2/dayto
beassumedinthispapertobeconstantandspatiallyuniform.Migrationingliomasisnotsimpleandin
factmanyauthorshaveproposedthatthehighlyinfiltrativenatureofhumangliomasrecapitulatesthe
migratorybehaviorofglialprogenitors(Suzukietal.,2002;Dirks,2001). Hereweassume,asinmost
models,thatgliomacellinvasionthroughoutthebrainisbasicallygovernedbyastandardFickiandiffu-
sionprocess.Morerealisticandcomplicateddiffusiontermsingliomasshouldprobablybegovernedby
fractional(anomalous)diffusion(Fedotovetal.,2011)orothermoreelaborateterms(Deroulersetal.,
2009) to accountfor the high infiltration observedin this type of tumors(Onishietal., 2011) and the
fact that cells do not behave like purely random walkers and may actually remain immobile for long
timesbeforecompelledtomigratetoamorefavorableplace. Inadditioninrealbraintherearespatial
inhomogeneities expected in the parameter values such as different propagation speeds in white and
graymatter,andanisotropies(e.g.onthediffusiontensorwithpreferentialpropagationdirectionsalong
white matter tracts). Many papers have incorporated these details (Jbadietal., 2005; Bondiauetal.,
2008;Konukogluetal.,2010;Clatz,2005;PainterandHillen,2013)mostlywiththeintentiontomake
patient-specificprogressionpredictions. However,themainlimitationisthelackofinformationonthe
(many) patientspecific unknowndetails what has limited progressesin that direction. Thus, in order
4of26 V.M.PE´REZ-GARC´IAetal.
to simplify the analysis and focus on the essentials of the phenomena we have chosen to study the
modelinonespatialdimensionandinisotropicmedia.Itisinterestingthatuptonowonlythesimplest
models such as those given by Eqs. (2.1) have been used to extract conclusionsuseful for clinicians
(Swansonetal.,2008;Wangetal.,2009).
The choiceof one-dimensionaldiffusionintendsto incorporatequalitativelydiffusionphenomena
inthesimplestpossibleway. Frontsolutionsofthe1DFKequationhavebeenextensivelystudiedand
areknowntopropagatewithaminimalspeedc=2√r Dwhenstartingfromstillinitialdata(Murray,
2007). It is interesting that the 1D model recapitulates the most relevant -for us-phenomenologyob-
servedinhigher-dimensionalscenarios. Firstitisobviousandwell-knownthatfront(invasionwaves)
solutions of the 1D FK equation also solve higher dimensional version of the equation (Xin, 2000;
BrazhnikandTyson,2000). Moreover,thosesolutionsareasymptoticallystable(Xin,2000;Sattinger,
1976)whatmeansthatunlikeothermorecomplicatednon-symmetricalsolutions(BrazhnikandTyson,
2000), they do arise as limits of non symmetric initial data. It is also well known that radially sym-
metric (in 2D) or spherically symmetric (in 3D) travelling wave solutions of FK do not exist in high
dimensions but that symmetric fronts also develop in those scenarios with a non-constant speed that
dependsonthelocalcurvatureR(BrazhnikandTyson,2000;Volpert&Petrovskii,2009). Asthefront
growswith time the now radius-dependentfrontspeed is givenby c(R)=c D/R(t). Thusgrowing
symmetricmultidimensionalsolutionswithlargecurvatureradii(R ¥ )gro−wwiththesamespeedas
→
1Dfronts(Volpert&Petrovskii,2009;GerleeandNelander,2012).
In this paper we are interested on the descriptionof low-gradegliomasthat typically are very ex-
tended when diagnosed, thus the initial data radius is large and fronts are well developed by then.
Althoughduring the initial stages of tumordevelopmentthe dimensionalitymay play a relevantrole,
for spatially extended tumors the effect of using higher-dimensional operators is not expected to be
substantial.
Moreover, some of the phenomena to be described later in this paper are found to be essentially
independentof diffusionand a verygood qualitativeagreementwill be foundbetween our simplified
analysisandthegrowthdynamicsofthemeantumordiameter. Takingintoaccountalltheseevidences
we will keep the system one-dimensional,since our intention is notto providea detailed quantitative
descriptionoftheprocesses-thatinanywaywouldbebeyondthereachofasimplemodelsuchasFK-
but instead to provide a qualitative description of the dynamics in the simplest possible way. As we
willdiscussindetaillater,thisapproachwillleadtoasimpleyetqualitativelycorrectdescriptionofthe
responseoflowgradegliomastoradiotherapy.
The parameterr in Eq. (2.1) is the proliferationrate (1/day)its inverse givingan estimate of the
typicalcelldoublingtimes. Wehavechosenalogistictypeofproliferationleadingtoamaximumcell
densityC(x,t)=1. Finally,thetumorevolvesfromaninitialcelldensitygivenbythefunctionC (x)
0
inanunboundeddomain,soweimplicitlyassumeittobelocatedinitiallysufficientlyfarfromthegrey
matter.
A very interesting feature of model Eqs. (2.1) is the well known fact that a tumor front arises
propagatingattheasymptotic(constant)speedofv =2√Dr whatisinverygoodagreementwiththe
∗
observed fact that the tumor mean diameter grows at an approximately constant speed (Palludetal.,
2012).
Whilemanyothermathematicalmodelsofgliomasincorporatedifferentcellphenotypes,e.g. nor-
moxic(proliferative)andhypoxic(migratory)phenotypes,suchasinMart´ınez-Gonza´lezetal.(2012);
Hatzkirouetal.(2012);Pe´rez-Garc´ıa&Mart´ınez-Gonza´lez(2012),herewefocusourattentiononLGGs
andassuchwillconsiderasingle(dominant)tumorcellphenotype.Inourmodelwedonotincludethe
possibleexistenceofdifferenttumorcellpopulationswithdifferentsensitivitiestotherapysuchasstem
DELAYEFFECTSINRADIOTHERAPYOFLGG 5of26
cellssincethefunctionandmechanismsofstemcellsinglioblastomaareyetunderdebate(Barrettetal.,
2012;Chenetal.,2012).
2.2 Responsetoradiation
Radiationtherapyhasbeenincorporatedindifferentformstomathematicalmodelsofhigh-gradeglioma
progression(Rockneetal.,2010;Bondiauetal.,2011;Konukogluetal.,2010;Barazzuoletal.,2010).
In thispaperwe wantto focusourattentiononLGGswhoseresponsetoradiationis verydifferentto
the oneobservedinHGGs. RadiationtherapyinLGGstypicallyinducesa responsethatprolongsfor
severalyearsaftertherapy.
Very interesting quantitative data on the response of LGGs to radiation have been reported in a
retrospectivestudybyPalludetal.(2012). TheauthorsstudiedpatientsdiagnosedwithgradeIILGGs
treatedwithfirst-lineradiotherapybeforeevidencesofmalignanttransformation. Patientswithapost-
RTvelocityofdiametricexpansion(VDE)(Palludetal.,2012)slowerthan-10mm/yearweretakenasa
subgroupofslowlygrowingLGGs.Patientswithapost-RTVDEof-10mm/yearorfasterwereincluded
inthegroupoffastgrowingLGGs. Theauthorsconcludedthatthepost-RTVDEwassignificantlyfaster
inthegroupwithhighproliferation.Alsointhepatientswithanavailablepre-RTVDE,thelowpre-RT
VDEsubgrouppresentedaslowerVDEatimagingprogression. Astothesurvivaltime(ST),thepost-
radiotherapyVDEcarriedaprognosticsignificanceonsurvivaltime,asthefastpost-radiotherapytumor
volumedecrease(VDE at-10 mm/yearor faster)were associated with a significantlyshortersurvival
thantheslowpost-radiotherapytumorvolumedecrease(VDEslowerthan-10mm/year).
Theveryslowresponsetoradiotherapy,leadingtotumorregressionlastingforseveralyearsisdiffi-
culttounderstandinthecontextofthestandardlinearquadraticmodelinwhichdamageisinstantaneous
andleadstocelldeathearlyafterradiationtherapy. Howeverakeyaspectofthecellresponsetoradia-
tionisthatirradiatedcellsusuallydiebecauseoftheso-calledmitoticcatastropheaftercompletingone
orseveralmitoses(VanderKogel&Joiner,2009). Thismeansthatslowlyproliferatingtumors,asitis
the caseofLGGswithtypicallow proliferationindexesbetween1%and5%inpathologicalanalyses
needaverylongtimetomanifesttheaccumulatedcelldamagethatcannotberepaired.
Thus,inordertocaptureinaminimalwaytheresponseofthetumortoradiationwewillcomplement
equation(2.1)forthedensityoffunctionallyalivetumorcellsC(x,t)withanequationfortheevolution
of the density of irreversibly damaged cells after irradiationC (x,t). Our model for the evolution of
d
bothtumorcelldensitieswillbegivenbytheequations
¶ C
= DD C+r (1 C C )C, (2.2a)
¶ t − − d
¶ C r
d = DD C (1 C C )C . (2.2b)
¶ t d− k − − d d
ThefirstequationisaFisher-KolomogorovtypeequationdescribingtheevolutionoftumorcellsC(x,t).
Thesaturationtermincludesthetotaltumorcelldensity,i.e. boththefunctionaltumorcellsandthose
damaged by radiationC (x,t). The evolution of cells irreversibly damaged by radiation is given by
d
Eq. (2.2b). Asitiswelldescribedintheliterature(VanderKogel&Joiner,2009),mostofthesecells
behavenormallyuntilacertainnumberofmitosiscycles,thuswewillconsiderthatafteranaverageof
k mitosis cyclesthese cells die resultingin a negativeproliferation. The longersurvivaltime kt with
t =1/r beingthetumorpopulationdoublingtime,resultsinareducedproliferationpotentialr /k,that
is the coefficientusedforthe negativeproliferationterm. Thus, theparameterk in Eq. (2.2b) hasthe
meaningoftheaveragenumberofmitosiscyclesthatdamagedcellsareabletocompletebeforedying.
6of26 V.M.PE´REZ-GARC´IAetal.
Aswiththenormalpopulationthenumberofcellsenteringmitosisdependsinanonlinearwayonboth
tumorcellpopulations(cf. lastterminEq.(2.2b)).
Wewillassumeaseriesofradiationdosesd ,d ,...,d givenattimest ,t ,...,t . Theinitialdatafor
1 2 n 1 2 n
thefirstsubintervalwillbegivenbytheequations
C(x,t ) = C (x), (2.3a)
0 0
C (x,t ) = 0. (2.3b)
d 0
The evolution of the tumor follows then Eqs. (2.2) until the first radiation dose d , given at time t .
1 1
Theirradiationresultsinafractionofthecells(survivingfraction)abletorepairtheradiation-induced
damagegivenbyS (d )andafraction1 S (d )ofcellsunabletorepairtheaccumulateddamagethus
f 1 f 1
−
feedingthecompartmentofirreversiblydamagedcells. Thesubsequentevolutionofthepopulationsis
againgivenbyEqs. (2.2)untilthenextRTdoseisgiven.Ingeneralaftereachirradiationeventweget
C(x,t+) = S (d )C(x,t ), (2.3c)
j f j −j
Cd(x,t+j ) = Cd(x,t−j )+ 1−Sf(dj) C(x,t−j ), (2.3d)
whereS (d )isthesurvivalfractionafteradoseofradi(cid:2)ationd ,i.e(cid:3).thefractionofdamagedtumorcells
f j j
after irradiation that are not able to repair lethal damage and are incorporatedto the compartmentof
damagedcells. Forthedosestobeconsideredindependenttheintervalbetweendoses(typically1day)
hastobelargerthanthetypicaldamagerepairtimes(oftheorderofhours). Theevolutionofbothcell
densitiesbetweenirradiationeventsisgivenbythePDEs(2.2).
2.3 Parameterestimation
Eqs.(2.2)togetherwiththeinitialconditionsforeachsubproblem(2.3)definecompletelythedynamics
ofalowgradegliomaintheframeworkofoursimplifiedtheoreticalapproach.
WehavechosentheparameterstodescribethetypicalgrowthpatternsofLGG.Fortheproliferation
rate we have chosen typical values to be small and around r =0.003 day 1 (see e.g. Badoualetal.
−
(2012)),thatgivedoublingtimesoftheorderofayear. Specificallywehaveconsideredvaluesranging
fromr =0.001day 1 forveryslowlygrowingLGGstor =0.01day 1. Forthecelldiffusioncoef-
− −
ficient we have taken values around D=0.0075mm2 /day (Jbadietal., 2005). This choice, together
withthepreviouslychosenr leadstoasymptotictumordiametergrowthspeedsgivenbyv=4√Dr of
the orderofseveralmillimetersperyear, inagreementwith typicaldiametricgrowthspeedsofLGGs
(Palludetal., 2012). Howeverthe factthat the asymptotic speed is only reachedwhen the tumorcell
densityisaroundonemayrequiretakinglargervaluesofDtomatchtherealgrowthspeeds.
Astotheradiobiologicalparameters,beinggliomasveryresistanttoradiationwehavetakenvalues
intherangeS (1.8Gy) SF 0.9consideringthemediansurvivalfractionvalue0.83afteronedose
f 1.8
≡ ∼
of2GygivenbyBarazzuoletal.(2010).
Finally, the average number of mitosis cycles completed before the mitotic catastrophe occurs is
difficult to estimate. This parameter intends to summarize in a single number a complex process in
which a cell hit by radiation and its progeny die after some more mitosis cycles leading to a final
extinctionafteravariabletime. Deathbymitoticcatastropheimpliesaminimalvalueofk=1andto
allowforsomemoretimewemaychoosevaluesintherangek=1 3(VanderKogel&Joiner,2009).
−
Wewillshowlaterthatthechoiceofthisparameterhasalimitedeffectonthemodeldynamicsandthat
instandardfractionationschemestheremaybebiologicalreasonstotakeitask=1.
OurtypicalchoicesforthefullsetofmodelparametersissummarizedinTable1.
DELAYEFFECTSINRADIOTHERAPYOFLGG 7of26
Variable Description Value(Units) References
C Maximumtumor 106cell/cm2 Swansonetal.(2008)
∗
celldensity
D Diffusioncoefficientfor 0.01mm2/day Jbadietal.(2005)
tumorcells
r Proliferation 0.00356day 1 Badoualetal.(2012)
−
rate
SF Survivalfraction 0.9 Barazzuoletal.(2010)
1.8
∼
for1.8Gy
k Averagenumberofmitosis 1-3 VanderKogel&Joiner(2009)
cyclescompletedbeforethe
mitoticcatastrophe
Table1.TypicalvaluesofthebiologicalparametersinthemodelofLGGevolution.
Inthispaperwewillfixthedoseperfractioninagreementwiththestandardfractionationschemes
forLGGstobe1.8Gy,theonlyrelevantparameteristhesurvivalfractionSF thatwillbetakentobe
1.8
aroundSF 0.9,asdiscussedabove. Inmanyexampleswewillchoosetheradiotherapyschemeas
1.8
∼
thestandardfractionationofatotalof54Gyin30fractionsof1.8Gyoveratimerangeof6weeks(5
sessionsperweekfrommondaytofriday).
3. Results
3.1 Computationaldetails
Wehavestudiedtheevolutionofthetumordiameterusingourmodelequations(2.2)and(2.3). Tosolve
thepartialdifferentialequationswehaveusedstandardsecondorderfinitedifferencesbothintimeand
space. Since the tumor diameterin the frameworkof this modeltendsto grow linearlyin any spatial
dimensionwe have focusedon the simplest one-dimensionalversionof the model. We havechecked
with simulationsin higherdimensionsthatthedynamicsisessentiallythesame andthushavesticked
to the simplest possible model. To avoid boundaryeffects and focus on the bulk dynamics, we have
assumedourcomputationaldomaintobemuchlargerthanthetumorsize.
In each simulation we have computed the tumor diameter as the distance between the points for
which the density reaches a critical detection value C that provides a signal in the T2 (or FLAIR)
th
MRI sequence. Althoughwhichisthatprecisevalueisa debatedquestionandin factdependsonthe
thresholdsusedintheimages,wehaveassumedthatC 0.05 0.07. Thisisinagreementwiththe
th
∼ −
reported value of cellular density about0.16 for detection (Swansonetal., 2008) and a normaltissue
densityofabout0.1. Inagreementwithpreviousstudieswetakeafataltumorburden(FTB)sizeof6
cmindiameter(Swansonetal.,2008;Wangetal.,2009). Asparameterscontainingusefulinformation
we have computed: the time in which the tumor starts regrowing after the therapy, usually called in
8of26 V.M.PE´REZ-GARC´IAetal.
60
(a)
½ = 0.00712 day-1
50
)
m
m
40
(

r
te ½ = 0.00356 day-1
e
m 30
a
i
D

r
o 20
m
u
T
10
0
0 1 2 3 4 5
Time (years)
0.5
(b)
0.4
A(t)
0.3
B(t)
0.2
0.1
0.0
0 40 80 120 160 200
Time (days)
FIG.1.(a)Tumordiameterevolutionfortwodifferentvaluesoftheproliferation:r =0.00356day−1,r =0.00712day−1.Other
parametervaluesare: D=0.0075mm2/day,criticaldetectionvalueCth=0.07,SF1.8=0.90andk=1. Theinitialtumorcell
densitiesaretakenasCd(x,0)=0,C(x,0)=0.4sech(x/6)withxmeasuredinmm,whatgivesaninitialtumordiameterof28.8
mm. Radiotherapy followsthestandardscheme(6weekswith1.8Gydosesfrommondaytofriday)andstartsattimet=0.
Circlesdenotemeasurementseverythreemonthsthatwouldcorrespondtoaclosefollow-upofthepatient.Theupperdashedline
(horizontal)showsthefataltumorburdensizetakenthroughthispapertobe6cmasdiscussedinthetext. (b)Evolutionofthe
tumorcellamplitudesA(t)=maxx C(x,t) andB(t)=maxx Cd(x,t) duringthefirst200daysshowingtheearlyresponsetothe
therapyforr =0.00356day−1. | | | |
DELAYEFFECTSINRADIOTHERAPYOFLGG 9of26
10
(a)
8
)
y
( 6

e
m
Ti
Survival time
4
2 Growth delay
Time to tumor progression
0
0.002 0.004 0.006 0.008 0.01
½ (day- 1 )
m) 0
m
(b)
(

n -4
o
ti
c
u
d -8
e
r

r
e
t -12
e
m
a
Di -16

x
a
M
-20
0.0 0.002 0.004 0.006 0.008 0.01
½ (day- 1 )
FIG.2. Dependenceofthe(a)survivaltime(solidline),growthdelay(dashedline),timetotumorprogression(dottedline)and
(b)maximalreductionindiameterasafunctionofr . Thecurvessummarizetheoutcomeofmanyindividualsimulationswith
initialdataCd(x,0)=0,C(x,0)=0.2exp x4/81920 withxmeasuredinmmthatgivesaninitialtumordiameterof33.80mm.
−
Radiotherapyfollowsthestandardscheme(6weekswith1.8Gydosesfrommondaytofriday)andstartsattimet=0. Other
(cid:0) (cid:1)
parametersusedinthesimulationsareasinFig.1.
10of26 V.M.PE´REZ-GARC´IAetal.
clinical practice time to tumor progression(TTP), the time for which the tumor size equals its initial
size-denotedasgrowthdelay(GD)-andthetimeforwhichthetumorsizeequalstheFTBorsurvival
time(ST).
Wehavestudiedabroadrangeofparametervaluescorrespondingtothepossiblerangeofrealistic
valuesintheframeworkofoursimpledescriptionofthetumordynamicsanditsresponsetoradiother-
apy. We have also taken differenttypesof initial data rangingfrom morelocalized (such as gaussian
initial profiles) to more infiltrative (such as sech type functions). In what follows we summarize our
results.
3.2 Tumorproliferationratedeterminestheresponsetoradiotherapy
Inafirstseriesofsimulationswehavestudiedthedependenceoftheevolutionofthetumordiameteron
theproliferationrate. Fig. 1(a)showstheevolutionofthetumordiameterfortwodifferentproliferation
rates r =0.00356 day 1 and r =0.00712 day 1 (Fig. 1). In the first case of low proliferation, the
− −
tumorrespondsmoreslowlytotherapyasmeasuredbythespeedoftumorregression(decreaseinsize)
butthetotalresponsetimeissignificantlylonger,beingthe timetotumorprogressionof16.9months
against8.2monthsinthelatercase. Alsothegrowthdelaysare14.7monthsforthefasterproliferating
tumor against29.9monthsfor the less proliferativeone. Finally the “virtualpatient” with the slowly
proliferating tumor survives much longer than that with the more proliferative one. This is just an
exampleofatendencyshowninallofoursimulationswheremoreaggressivetumorsrespondearlierto
thetherapy.
It is remarkablethat this fact is in fullagreementwith the results fromPalludetal. (2012b). Our
model based on reasonable biological assumptions leads to a long remission time (e.g. in Fig. 1 of
about 3 years) much larger than the treatment duration (6 weeks) and negatively correlated with the
tumor proliferationrate. As a second relevant finding, also seen in the results shown by Palludetal.
(2012b),weobservethattumorsrespondingfasterhavealsoshorterregrowthtime.
Fig 1(b) shows the dynamics of the maximum density of tumor cells (A(t)=max C(x,t)) and
x
damagedtumorcells(B(t)=max C (x,t)). Ascouldbeexpectedtheamplitudeoffunctionallyalive
x d
tumorcellsdecreasesduringthetherapywiththeexceptionofthebreaksintheweekendswereasmall
increaseisseenandcorrespondinglytheamplitudeofdamagedtumorcellsgrowsaftereveryirradiation
andforthefulltreatmentperiod(6weeks=42days). Aftert =42daysthepopulationoftumorcells
starts a slow recovery while the population of damaged cells declines in a much longer time scale.
However, the width of the total tumor population evolves only in the slow time scale and does not
displayanyeffectsduringthetreatmentperiod.
Itisimportanttoemphasizethatthisbehaviorisnottheresultofafortunatechoiceoftheparameters
but a generic behavior as we have confirmedthrougha large numberof simulations coveringthe full
clinically relevantparameterspace. As an example, in Fig. 2 we show how the variationof r overa
broadrangeofvaluesleadstothesameconclusion. Largerproliferationvaluesacceleratetheresponse
butleadtoearlierregrowthandassuch,shortergrowthdelaysandsurvivaltimes[Fig. 2(a)].Oursimu-
lationsalsopointoutthatthemaximumvolumereductionisonlyweaklydependentontheproliferation
rater [Fig. 2(b)],thesmallertheproliferationratesthelargerthemaximumreductionindiameter.This
factisalsoinverygoodagreementwiththeresultsofPalludetal.(2012b)(seee. g. Fig. 2bottomof
theirpaper).