Combination therapies and intra-tumoral compettion:insights from mathematical modeling

26 July 2021

Original Citation:

Combination therapies and intra-tumoral compettion:insights from mathematical modeling

Published version:

DOI:10.1016/j.jtbi.2018.03.014

Terms of use:

Open Access

(Article begins on next page)

Anyone can freely access the full text of works made available as "Open Access". Works made available under a

Creative Commons license can be used according to the terms and conditions of said license. Use of all other works

requires consent of the right holder (author or publisher) if not exempted from copyright protection by the applicable law.

Availability:

This is the author's manuscript

Please note that changes made in the online proofing system will

be added to the article before publication but are not reflected in

this PDF.

JID:YJTBI [m5G;March13,2018;16:54]

Journal of Theoretical Biology xxx (2018) xxx–xxx

ContentslistsavailableatScienceDirect

Journal

of

Theoretical

Biology

journalhomepage: www.elsevier.com/locate/jtbi

Combination

therapies

and

intra-tumoral

competition:

Insights

from

mathematical

modeling

Elena

Piretto

a,b

_,

_Marcello

_Delitala

b,∗

_,

_Mario

_Ferraro

c

Q1

a Department of Mathematics, Università di Torino, via Carlo Alberto, 10, Torino 10123, Italy

b Politecnico di Torino, Department of Mathematical Sciences, corso Duca degli Abruzzi 24, Torino 10129, Italy c Department of Physics, Università di Torino, via P. Giuria 1, Torino 10125, Italy

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 7 October 2017 Revised 31 January 2018 Accepted 12 March 2018 Available online xxx Keywords: Population dynamics Cancer modeling Evolutionary dynamics Chemotherapy Immunotherapy

a

b

s

t

r

a

c

t

Drugresistanceisoneofthemajorobstaclestoasuccessfultreatmentofcancerand,inturn,hasbeen recognizedtobelinkedtointratumoralheterogeneity,whichincreasestheprobabilityoftheemergence ofacancerclonesrefractorytotreatment.Combinationtherapieshavebeenintroducedtoovercome re-sistance,butthedesignofsuccessfulcombinedprotocolsisstill anopenproblem. Inordertoprovide someindicationsontheeffectivenessofmedical treatments,amathematical modelisproposed, com-prisingtwocancerpopulationscompetingforresourcesandwithdifferentsusceptibilitiestotheaction ofimmunesystemcells andtherapies: thefocus isontheeffects ofchemotherapy and immunother-apy,usedsingularlyorincombination.First,numericalpredictionsofthemodelhavebeentestedwith experimentaldatafromtheliteratureandnexttherapeuticprotocolswithdifferentdosesandtemporal orderhavebeensimulated.Finallytheroleofcompetitiveinteractionshasbeenalsoinvestigated,to pro-videsomeinsightsontheroleofcompetitiveinteractionsamongcancerclonesindeterminingtreatment outcomes.

© 2018PublishedbyElsevierLtd.

1. Introduction 1

Theideathatcancerisanevolutionarydiseaseandthatits

de-2

velopment occurs by the same processes through which animals

3

andhumans have emerged (Nowell, 1976) hasgained, nowadays,

4

a wideacceptance andhasdeeplyinfluencednot justour

under-5

standingofcancer butalsothedevelopmentofantitumoral

ther-6

apies(Basantaetal.,2012;GreavesandMaley,2012;Merloetal., 7

2006;Misaleetal.,2015).

8

The twin forces of evolution, mutation and selection, are at

9

work also in case of cancer: mutations at molecular level start

10

the process of tumorigenesis and ensure cancer heterogeneity,

11

whereas evolutionarypressures selectthe fittest species.The

re-12

sultisaprocessoccurringatmultiplescales(Bellomoetal.,2008);

13

for instance cancer heterogeneity arises at the microscopic level

14

via mutations and it is expressed at macroscopic level as a

va-15

riety ofclonal typesforming acommunity regulated bydifferent

16

types of interactions such as competition, cooperations,

mutual-17

ism(TabassumandPolyak,2015).

18

∗ _{Corresponding author.}

E-mail address: marcello.delitala@polito.it (M. Delitala).

Thegrowthofcancertypesisfurthershapedbytheinteractions 19

withdifferentenvironmentalfactors:inthehistoryoflifepowerful 20

protectivemechanisms haveevolvedtoensure thesurvival ofor- 21

ganismswithlargebodiesandlonglives.Inparticulartheimmune 22

systemensuresthatmanytumorsareroutinelyeradicated. 23

In conclusion, cancer can be considered an ecosystem (Hillen 24

andLewis,2014;Pacheco etal.,2014)formedbycoexistingpopu- 25

lations,embeddedinanenvironmentcomprisingnormalandim- 26

munecells (Marusyk etal., 2012). Growthof tumor speciesde- 27

pends on how effectively they are able to access resources and, 28

on the other hand, on how successfully they develop mecha- 29

nismsto prevent detection and eliminationby the immune sys- 30

tem(HanahanandWeinberg,2011). 31

The multiplicity of species in cancer populations has a clear 32

relevance forthe design oftherapies as heterogeneity is amajor 33

factor in cancer drug resistance, see e.g. Saunders et al. (2012); 34

eventhough atherapy candecimate a cancertype, one or more 35

variantsofthetumorpopulationexistwhichareresistant,driving 36

to the resurgence of treatment-refractory disease (Gerlinger and 37

Swanton, 2010). This observation has led to the idea of combi- 38

nation therapy, in which agents with different actions are com- 39

bined,thusincreasingthelikelihoodofsynergisticantitumoralef- 40

https://doi.org/10.1016/j.jtbi.2018.03.014

0022-5193/© 2018 Published by Elsevier Ltd.

fects,(DeVita andSchein,1973; Saundersetal.,2012).The design

41

ofcombinedprotocolsisachallengingproblemandspecificallythe

42

optimal dosing and timing in the combination of chemotherapy

43

andimmunotherapyisstillanopenissue,(Slovin,2012).

44

Not surprisingly, given the relevance of the problem, there

45

exist a large literature on mathematical models of cancer

dy-46

namics. For a review see for instance Eftimie et al. (2011),

47

Bellomoetal.(2008), Wilkie(2013)andreferencestherein.In

par-48

ticular several contributions can be found in the framework of

49

populationdynamics:amongothersontumorimmuneinteraction,

50

severalpopulations are considered in De Pillis et al. (2006) and

51

Wilson and Levy (2012) and spatial-temporal dynamics in

Al-52

Tameemietal.(2012).Theeffects oftherapies are studiedin the

53

context ofevolutionary dynamics (Gatenby etal., 2009a, 2009b),

54

whiletheimmunotherapyareconsideredin Eladdadietal.(2014),

55

Frascoliet al.(2014), Bunimovich-Mendrazitsky et al.(2008) and

56

theoptimizationoftherapeuticprotocolsin Ledzewiczand Schät-57

tler(2017)and Carrere(2017).

58

Population theory takes into account influences of the

envi-59

ronmentsuch aslimitedamountofresources,interactions among

60

speciesandpredation, seee.g. Murray(2002);thereforeitis able

61

to provide a suitable frame of reference to investigate the

ecol-62

ogyofcancer. In thispaper,the effectsof evolutionarypressures

63

ontumordevelopmentandonthe outcomesofthe therapiesare

64

investigatedwithamodel situatedinthistheoretical framework:

65

twopopulationsofcancercellscompeteforresourcesandare

sub-66

jectedtotheactionoftheimmunesystemandofthetherapies.

67

The restofthepaperis organizedasfollows.In Section2 the

68

mathematicalmodel is formalizedandits asymptotic behavioris

69

studied and next in Section 3, parameter estimation and

corre-70

spondingsensitivityanalysisareperformedtogetherwith

compar-71

isons withexperimental data; finally Section 4 is devoted tothe

72

analysisoftheroleoftemporalorderintheadministrationof

ther-73

apies,withan emphasistointra-tumoralcompetition.Specifically,

74

thefocus ison the effects ofchemotherapy andimmunotherapy,

75

usedsingularlyorincombination.Therapeuticprotocolsof

differ-76

ent time duration, intensity and order of administration are

ex-77

plored.

78

2. Mathematicalmodel

79

As mentioned earlier, the model has been developed in the

80

frameworkofpopulationdynamics(Murray,2002)todescribethe

81

evolution of cancer and immune system. Heterogeneity of

can-82

cer is taken into account by considering two cancer clones or

83

populations,withx1=x1

(

t

)

andx2=x2

(

t

)

denotingambiguously

84

both the cancer type and the corresponding number of tumor

85

cellsforeach clone. The numberof immune cells is represented

86

byz=z

(

t

)

.

87

Eachcancerspeciesisendowedwithdifferentphenotypictraits,

88

whichdeterminespecificcharacteristicswithrespect,forinstance,

89

totheabilitytoaccessresources andthe susceptibilityto the

ac-90

tion of the immune system or medical treatments. Cancer types

91

areassumedtocompetewitheachother,asarguably,competition

92

isthemoreimportanttypeofintra-tumoralinteractioninshaping

93

cancerdevelopment (Vivarellietal.,2012;Wagstaff etal.,2013).

94

Basic elements determining the evolution of the cell

popula-95

tions are proliferation, predation and competition for resources:

96

thegrowthofthecancerspeciesislimitedbythefinitenessof

re-97

sourcesandfurther constrainedby inter-specificcompetition and

98

bytheactionoftheimmune system.Inturn,theimmune system

99

growsbecauseoftwofactors: itisproduced bytheorganismand

100

itsnumerousisfurtherenhancedby clonalexpansioninpresence

101

ofcancer.

102

Fig. 1. Schematic representation of the model. The effects of therapies is to change

ri and c i . See text for explanation.

Themodel,sketchedin Fig.1,isformalizedasa systemofor- 103

dinarydifferentialequations: 104

dx1 dt =r1x1− r1 K1 x2 1







proli f eration −b12 K1 x1x2







competition − c1 K1 x1z







predation − g1

(

t

)

x1− h

(

t

)

K1 x1z







therapies , dx2 dt =r2x2− r2 K2 x2 2− b21 K2 x1x2− c2 K2 x2z− g2

(

t

)

x2− h

(

t

)

K2 x2z, dz dt =

β

z



1− z H









proli f eration +

α

1 Hx1z+

α

2 Hx2z







recognition . (1)

Consider tumor speciesx1: the first two terms in the RHSof 105 the equation represent the growth of x1 in isolation, i.e. in ab- 106 sence ofother cancer species,of theimmune system, andmedi- 107

cal treatment.Inthiscasex1 undergoesalogistic growthandthe 108 parameter r1 is thereproduction ratewhereas K1 corresponds to 109 the carrying capacity, i.e. the maximum value that x1 can take. 110 Development of x1 is constrained by the competition withclone 111 x2 (measuredby thecompetitionrateb12) andby theinteraction 112 withtheimmunesystem(ratec1).Furthermorethemedicaltreat- 113 ment (for instance chemotherapy) can act on x1 and its effects 114 are representedinthe modelby thetermg1(t)x1 whereg1 takes 115 intoaccount thedrugskineticsintheorganism, see DePillisand 116

Radunskaya(2001).This isequivalentto rewritethegrowthterm 117

of f1

(

t

)

x1=

(

r1− g

(

t

))

x1: wheng(t)>r1 thendx1/dt<0,meaning 118 thatthecancercanbeeradicatedbyagiventreatment. 119

Thecellsofimmunesystemspreyonthetumorcellsandtheir 120

action can be enhanced by immunotherapy. In particular, here 121

the focus is on a specific vaccination (DCstransduced withade- 122

noviruscontainingfull-lengthmousewild-typep53(Ad-p53))that 123

results in the generation of immune system cells (CTLs specific 124

for p53-derived peptide) inducing a specific antitumor immune 125

response (Nikitina et al., 2001). The effect of immunotherapy is 126

thenan increasedabilitybytheimmune systemtorecognizeand 127

kill cancer cellsasproposed in WilsonandLevy (2012) andit is 128

modeledby theterm h

(

t

)

K1

x1x2,h(t)>0,or,equivalently,bydefin- 129 inganewparameter

κ

1

(

t

)

=c1+h

(

t

)

. 130 The same considerations apply mutatis mutandis to tumor 131

species x2: in particular f2(t) and

κ

2(t) are the new growth and 132 predation parameters, defined in a way analogous to what has 133

beendoneforx1.Inthefollowing,forsimplicity’ssake, fi(t),

κ

i(t) 134

E. Piretto et al. / Journal of Theoretical Biology xxx (2018) xxx–xxx 3

JID:YJTBI [m5G;March13,2018;16:54]

Theequationsforthetumor speciesx1,x2 canthenbe

rewrit-136

teninmorecompactformas:

137 dx1 dt = f1x1− r1 K1 x2 1− b12 K1 x1x2−

κ

1 K1 x1z, dx2 dt = f2x2− r2 K2 x2 2− b21 K2 x1x2−

κ

2 K2 x2z. (2)

Weturnnowourattentiontotheimmunesystemz:inabsence

138

oftumor,itgrowswithanetrate

β

anditislimitedbythe

thresh-139

oldH,whichinthiscasecanbeinterpretedasacarryingcapacity.

140

Inpresenceofcancer,zcanexceedH,asitundergoesaclonal

ex-141

pansionweighted,respectively,bytherates

α

1,

α

2,whichmeasure

142

theabilityofimmunecellstodetectandrecognizecancercells.

143

This model is, admittedly, a drastic simplificationof the

pro-144

cesses underlying cancer evolution. In particular mutations are

145

not considered since this work focuses on the results of

thera-146

pies and then dealswith temporal windows, on average,shorter

147

than the onesinwhich significant mutationsoccur; thus we can

148

assume a fixed number ofclones typesbefore treatments, in

ac-149

cord to Greaves (2007). Also heterogeneity is taken into

ac-150

count by considering just two cancer populations, corresponding

151

to specific phenotypic traits. This assumption is not too

restric-152

tive as biological (clinical) data suggest that usually just a few

153

dominant types emerge (Ding et al., 2012); on the other hand,

154

selection to monomorphic or dimorphic scenarios is also

pre-155

dicted by complexmodels, seeforinstance Perthame (2006) and

156

Chisholmetal.(2015).

157

2.1. Stabilityanalysis

158

A completestability analysiscan be found inthe

Supplemen-159

taryInformation,herethemainresultsaresummarized.

160

In the positive orthant R3

+ the system (1) has 8 stationary

161

points, whose components willbe denoted by x₁∗, x∗₂,z∗, explicit

162

forms of these components can be found in the SI file. Four of

163

thesepointsareoftheformP=

(

x∗₁,x∗₂,0

)

withx1≥ 0,x2≥ 0and

164

they areclearlyunstable asdz/dt>0ifz<H: thusthey areof no

165

biologicalinterest.

166

The other points correspond to the case of tumor

eradica-167

tion Pte=

(

0,0,H

)

, survival of a singlecancer clone (competitive 168

exclusion) Pec1=

(

x∗1,0,z∗

)

, with x∗1>0 and z∗>0 (resp. Pec2= 169

(

0,x∗₂,z∗

)

, x₂∗>0, z∗>0) and coexistence of both cancer species

170

Pcoe=

(

x∗₁,x∗₂,z∗

)

,all componentsbeing greater than zero.In this 171

analysisweconsiderthetumorbeforetheapplicationofthe

ther-172

apysothat fi=ri,

κ

i=ci. 173

Tumoreradicationoccurswhen:

174

K1r1<c1H, K2r2<c2H, (3) in this case Pte is the only stationary point and it is (globally) 175

asymptotically stable, corresponding to the well known fact that

176

the immune systemroutinely suppresses tumoralcellsin the

or-177

ganism.

178

Onthecontrary,ifoneofconditions (3)doesnothold,the

cor-179

responding clone survives. For instance, if the first inequality of

180

(3)isnotsatisfiedandthesecondholds,x2tendstozeroandthe

181

systemreducestojusttwoequations(thefirstandthirdof (1))of

182

Lotka–Volterratype:thesolerestpointisPec1,whichisstable,but,

183

forcertainparametervalues,notnecessarilyasymptotically stable,

184

i.e. itmay be a center: thus in principle a continuum of

oscilla-185

tions mayoccur, but noisolated limit cycles (Hofbauer and Sig-186

mund,1998).

187

Obviously the sameconsiderations apply ifthe second

condi-188

tionin (3)isviolated andthefirstoneholds;thestablestationary

189

pointisnowPec2=

(

0,x∗2,z∗

)

.

190

If neither conditions (3) are satisfied, the eventual fate of

191

the system depends on the competition between x1 and x2 see

192

e.g. Murray(2002)and HofbauerandSigmund(1998).Whichcan- 193

certypesurvivesdependsontheparameters: 194

A1= r1+c1

α

1/

β

K1r1− c1H , A2= b12+c1

α

2/

β

K1r1− c1H , 195 B1= b21+c2

α

1/

β

K2r2− c2H , B2= r2+c2

α

2/

β

K2r2− c2H .

IfA1<B1 andA2<B2,thenx1 dominatesx2,anditistheonly 196 speciessurviving, in that Pec1 is stable, whereas Pec2 is unstable, 197 regardlessoftheinitialconditions,(obviously,theconverseistrue 198

iftheinequalities arereversed); thisisan exampleofsurvival of 199

thefittest. 200

If A1<B1 and A2>B2, there exist three stationary points, 201 namely,Pec1,Pec2,whicharelocallystable, andtheunstablePcoe∈ 202 intR3

+;whichspeciessurvivesdependsontheinitialconditions(an 203

exampleofsurvivalofthefirst). 204

Competitiveinteractions then may resultin the emergence of 205

dominantspecies,(Dingetal.,2012;Vivarellietal.,2012;Wagstaff 206

etal.,2013). 207

FinallyifA1>B1 andA2<B2,pointsPec1,Pec2 becomeunstable 208 whereasthecoexistencepointPcoe=

(

x∗_1,coe,x∗_2,coe,z∗coe

)

isstable. 209 Fig.2showstheprojectionsofthevectorfieldof Eq.(1)onthe 210

plane(x1,x2),whilethethirdcomponent(relativetotheimmune 211 systemz) is representedby an heat map. Fourcases are consid- 212

ered,namelytumoreradication,twotypesofcompetitiveexclusion 213

andcoexistence. 214

3. Parametersestimation 215

Theaimofthissection istopresentsimulations ofthemodel 216

underthe action of different therapies; therefore it is natural to 217

startfromasituationinwhichbothcancerclonesarepresent,see 218

e.g. Gerlinger andSwanton(2010).Thus, unless otherwisestated, 219

parametersvaluesareunderstoodto ensurecoexistenceofcancer 220

clones,i.e.Pcoe isstable. 221

Clonesarecharacterized bydifferentratesofgrowth,competi- 222

tiveefficiency(i.e.differentvaluesoftheparameters bij)andsus- 223 ceptibilitytotherapies. Inthe following,we shallassume that x1 224

hasalargerproliferationrate,r1>r2and,coherentlywiththeidea 225 thatafastproliferationinvolvesanevolutionarycost,wehaveset 226

b12>b21,i.e. x2 out-competes x1.Sensitivity to therapies will be 227

specifiedinthesequel. 228

As explained in Section 2, the effects of treatments on can- 229

cerdevelopment are modeled by alteringthe growthparameters 230

ri,andci,i=1,2,which measuresthe effectoftheimmune sys- 231 tem. Weconsider heretwotypes oftherapies: thefirst one,that 232

willbedenotedgenericallywiththetermchemotherapy,decreases 233

rto a newvalue f, whereas thesecond one, calledimmunother- 234

apy, increases the effect of the immune system by increasing c 235

to

κ

. We are aware that limiting the effect of chemotherapy to 236

cancer cells reproductionrate is a simplification in that, aswell 237

known(Zitvogeletal.,2008),chemotherapyhasatleastatwofold 238

effecton theimmune systemand onthe healthyhost tissue: on 239

onehanditweakenstheproliferationofimmunecellsbutonthe 240

otheritelicitsorreactivatesanticancerimmuneresponse,enhanc- 241

ingtheimmunogeniccharacteristics oftumor cells.Herewe have 242

focusedontheprimaryaspectofchemotherapy,namelythereduc- 243

tionofcancercellsproliferation. 244

Ideallyfi shouldbeloweredtosuch smallvaluesthat bothin- 245 equalities (3)aresatisfied;inthiscasebothcancertypeswilltend 246

todisappear,atleast aslong asthetherapy isapplied.The same 247

argument,obviously,applies to immunotherapy, whichshould be 248

strongenough tomake

κ

i so largeforboth (3)to hold.Unfortu- 249 nately,thissituationveryrarelyarisesandresistancetothetreat- 250

Fig. 2. All panels show the projections of the vector field of Eq. (1) on the plane ( x 1 , x 2 ), relative two tumor clones, while the third component (relative to the immune system z ) is represented by an heat map. Colored dots represent stationary points. Upper left panel: tumor eradication. Upper right panel: competitive exclusion. Bottom left panel: competitive exclusion with dependence on initial conditions. Bottom right panel: coexistence.

Table 1

Table of the parameters ranges.

Param. Unit Range/value Interpretation Source

ri days−1 0 . 02 − 0 . 2 Growth rate of clone i De Pillis et al. (2006) bij days−1 0 . 01 − 0 . 2 Intratumoral competition Est.

ci days−1 1 . 1 − 5 . 5 10 2 Immune killing rate of clone i Kuznetsov et al. (1994)

Ki cells 10 7 − 5 10 9 Carrying capacity of clone i De Pillis et al. (2006) ; Kuznetsov et al. (1994) H cells 5 10 4 − 5 10 5 _{Immune system threshold Kuznetsov et al. (1994)}

αi days−1 1 . 2 10 −5 _{− 3 10} −3 _{Immune recruitment due to clone i Kuznetsov et al. (1994)}

β days−1 _{0 . 5 − 55 Immune system growth rate Est.}

fective waysto battlecancer. Thus, in the followingwe shall

as-252

sumethat theclonex1 issensitivetochemotherapyandx2 is

re-253

sistant.

254

3.1.Sensitivityanalysis

255

Parameters rangesestimatedfrom theliterature arepresented

256

in Table1.Inparticular,forthenetratesoftumorgrowthriweuse 257

therangeproposedin DePillisetal.(2006)formice.Tumor

car-258

ryingcapacitiesKiare ina rangeconsistent withvaluesreported 259

in DePillis etal.(2006) and Kuznetsovetal.(1994); thewidth

260

ofthisrangeisduetodifferencesamongtumors.Thecompetition

261

parametersbij arefixedtoensurecoexistenceofthecancerclonal 262

populationasexplainedin Section2.1.Theimmune systemkilling

263

ratesc_iareestimatedfrom Kuznetsovetal.(1994).

264

The parameterH,whichrepresentsthenaturalgrowthlimitof

265

theimmunesysteminabsenceofcancercells,isinferredfromthe

266

stationarystate of the Kuznetsov model(Kuznetsovet al., 1994),

267

alsotakingintoaccountthefactthatwefocusonalocalimmune

268

response.The parameters

α

i, controllingthe immune cellsclonal 269

expansion,isevaluatedassumingasminimumandmaximumval- 270

ues, respectively, the corresponding values of the growth curve 271

used in Kuznetsov et al. (1994). The parameter

β

, the immune 272

systemrateofgrowth,isestimatedinanexploratoryway. 273

Aglobalsensitivityanalysisevaluatestheimpactofanestima- 274

tion error of the parameters range on the results of the model. 275

As described more in details in the Supplementary Information, 276

the analysisis carriedout with two techniques:the PartialRank 277

CorrelationCoefficient (PRCC),a sampling-based method,andex- 278

tendedFourierAmplitudeSensitivityTest(eFAST),avariance-based 279

method,see Marinoetal.(2008)and Saltellietal.(2004). 280

Thetwo methodsgive similar results,especiallyascriticalpa- 281

rameters areconcerned. As shownin Fig. 3,parameters withthe 282

greatestimpactonx1 andx2aretheproliferationratesri,thecar- 283 ryingcapacitiesKiandtheimmunethresholdH.Theimmunesys- 284 temappearstobesensitivetori,Ki,H,

α

iand

β

;howeveritshould 285 benotedthathereweareinterestedintheeffectsoftheimmune 286

system on tumoral dynamic ratherthan on the evolution of the 287

E. Piretto et al. / Journal of Theoretical Biology xxx (2018) xxx–xxx 5

JID:YJTBI [m5G;March13,2018;16:54]

Fig. 3. The panels show the sensitivity indexes of the parameters for variables x 1 (left panel), x 2 (center panel), z (right panel). Blue bar represent the first-order sensitivity index ( S i ) and the yellow bar the total-order sensitivity index ( S ti ). For more details see the Supplementary Information. (For interpretation of the references to colour in this

figure legend, the reader is referred to the web version of this article.)

Fig. 4. Comparison of the simulations with data from Misale et al. (2015) . The upper-left panel presents the case of control. In the upper-right panel the grey bar (green in the original) represents the administration of Pimasertib drug, and in the lower-left one the light grey bar (pink in the original) indicates injection of Cetuximab drug. In the lower right panel the combination therapy is represented by a dark grey bar (blue in the original). The piece-wise linear curve depicts the experimental data, while the dashed (red in the original) and dotted (blue in the original) curves are time courses of x 1 and x 2 respectively and the continuous curve (green in the original) is the total cancer load x . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3.2. Comparisonwithexperiments:firstdataset

289

Earlier inthis paperit hasbeen claimedthat interactions

be-290

tween clonesplaya relevantrole indeterminingtheevolutionof

291

cancer, andto validatethis assertionwe haveconsidered the

ex-292

periments presentedin Misaleetal.(2015),wherea combination

293

of molecular target-therapieswere used to treat colonrectal

can-294

cer (CRCs) in mice. The experiment consisted in the

administra-295

tionfor6weeksofthemoleculartarget-therapydrugs(aloneand

296

incombination): theanti-EGFR,Cetuximab,andtheanti-MEK,

Pi-297

masertib.ThedrugblockingMEKhadamarginal effectontumor

298

growthwhilecetuximabandcombinatorialtreatmentssignificantly

299

reducedtumorsizeinallanimals.In Misaleetal.(2015),datawere

300

then fitted witha probabilistic model assuming the existence of 301

twocancertypes,withdifferentsusceptibilitiestotherapies. 302

In our numerical computations some parameters are chosen 303

within the ranges reportedin Table 1: K1=K2=2.764· 109 cells 304 (as computed in Misale et al. (2015)), r1=0.11 days−1, r2= 305 0_.035 days−1_, b12=0.07 days−1, b21=0.01 days−1. Parameters 306 measuringthe effectofthe immune system, c1/K1=c2/K2=1.1· 307 10−7 days−1 cells−1 are derived from Kuznetsovetal.(1994).For 308

theimmunesystemvaluesparametersvaluesareH₌5_{· 10}4 _{cells 309} and

α

1/H=

α

2/H=6· 10−9 days−1 cells−1. Therate ofgrowthof 310 the immune system,

β

=55 days−1 is chosen in an exploratory 311

way. 312

Inaccordancewith Misaleetal.(2015),weassumethatonlyx1 313

Fig. 5. Comparison of the simulations with data from Ramakrishnan et al. (2010) . The code for curves is as in 4 . Grey bar (green in the original): application of chemotherapy. Light grey bar (pink in the original): application of immunotherapy. Upper panels: on the left the case of control, on the right the case of TAX therapy (chemotherapy); Initial conditions: x 1(0) = 1 . 4 · 10 5 cells, x 2(0) = 7 · 10 4 cells, z(0) = 5 · 10 2 cells. Lower panels: on the left the case of Ad-p53 therapy (immunotherapy), on the right results of combination therapy. Initial conditions: x 1(0) = 9 · 10 4 cells, x 2(0) = 4 · 10 4 cells, z(0) = 5 · 10 2 cells. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

f2=r2, whereas Pimasertib affects both clones andthis is

mod-315

eled by setting f1=0.06 days−1 and f2=−0.015 days−1; finally

316

thecombinationofthetwodrugsresultsin f1=−0.15days−1and

317

f2=−0.015days−1, theselatter valuesare obtained by a simple

318

linearsuperpositionoftheeffectsofthetwodrugs.

319

The initial conditions, x1

(

0

)

=1.2· 108 cells, x2

(

0

)

=0.024·

320

108_{cells,arethesameusedin Misaleetal.(2015)tofitthedata.}

321

Theimmunesystemissupposedtostartfromatumorfree

condi-322

tion,i.e.z

(

0

)

₌H cells. Theexperimental dataarecomparedwith

323

thecomputedtotalcancerloadx=x1+x2.

324

The simulations in Fig. 4 show good agreement with the

ex-325

perimentalfindings.Empiricaldatacannot beeasilyexplained

un-326

derthehypothesis ofa singleclonebutthey admit a

straightfor-327

wardinterpretationiftwo clonesaresupposed tobe present;

in-328

deed,asmentioned before,thisisalsothehypothesis formulated

329

in Misaleetal.(2015)tofitthedata.Inthecontrolcase,thetotal

330

cancerload ismade up by clonex1,which is moreproliferating,

331

whereasx2 ispresentata verylow level,since itis curtailedby

332

x1; thesametrendscan beobservedduringtheadministrationof

333

Pimasertibdrug, towhichbothclones aresensitive.The situation

334

changes when Cetuximab is used: clone x1 is susceptible to the

335

therapyandstartdecreasingbutthisallowx2togrowandindeed

336

thelarge rebound effectis dueto theincrease of x2,which now

337

almostcoincides withthe total cancerloadx. Finally,the

combi-338

nationtherapy results ina simultaneousdecrease ofboth clones,

339

againinagreementwiththedata.

340

3.3.Comparisonwithexperiments:seconddataset

341

Inordertoobtainsome reasonablevaluesofparameters range

342

fortheeffectsofchemo-andimmunotherapiesincombination,we

343

haveconsidereddataderivedfromanexperimentinwhichmurine

344

coloncarcinomatumors wereestablished inC57BL/6miceby s.c.

345

injectionof MC38 tumor cells (Ramakrishnan et al., 2010). Mice

346

were then treated withdifferent therapies: chemotherapy (pacli- 347

taxel (TAX),a widely used chemotherapeuticdrug), immunother- 348

apy(Ad-p53, a vaccinationwith DCs transduced withadenovirus 349

containingfull-lengthmousewild-typep53)andacombinationof 350

both. 351

Theresultspresentedin Ramakrishnan etal.(2010)show that 352

the immunotherapy agent, DCs transduced withadenovirus con- 353

taining full-lengthmouse wild-typep53 (Adp53), even though it 354

slowsdowncancerevolution,isnotenoughtoensuretumorsup- 355

pression. Similarly, treatment ofmice withthe chemotherapeutic 356

drugpaclitaxel(TAX)delayscancergrowth,buttumorprogression 357

resumes soon after the treatment is discontinued. On the other 358

hand,combinationofTAX andtheDC vaccineengendersa sharp 359

suppressionoftumorgrowth,whichcontinuesforatleast5weeks 360

afterstartofthetreatment.Thus,chemotherapyandimmunother- 361

apyenhanceeachother’seffects. 362

In our simulations parameters are in the ranges of Table 1: 363

in particular K1=K2=5· 107, H=3· 105. The proliferation rates 364 r_i are fixed in a way consistent with the tumor initial rate 365

of growth, as derived from the experiment; numerical val- 366

ues are r1=0.15 days−1 and r2=0.11 days−1. Interaction pa- 367 rameters are chosen in an exploratory way, assuming, as ex- 368

plained before, that the evolutionary cost of proliferation im- 369

plies a competitive disadvantage: b12=0.1 days−1 and b21= 370 0_.01 days−1. The parameters expressing the clonal expansion 371

α

1/H=

α

2/H=6· 10−9 days−1 cells−1 and the immune system 372 killing rates c1/K1=c2/K2=1.1· 10−7 days−1 cells−1 are taken 373 from Kuznetsov etal. (1994). Parameter

β

is chosen, within the 374

rangeof Table1,inanexploratoryway,

β

=0.8days−1. 375

Asbefore,onlyx1issensitivetochemotherapyandcorrespond- 376 inglyr1 isloweredto f1=−0.072 days−1,whereas r2 iskeptun- 377 changed. The effect of immunotherapy is to increase c_i to

κ

i= 378

E. Piretto et al. / Journal of Theoretical Biology xxx (2018) xxx–xxx 7

JID:YJTBI [m5G;March13,2018;16:54]

Fig. 6. Panels show the trajectories for a protocol with the immunotherapy first (left panel) and with the chemotherapy first (right panel). Code for treatments and time courses are as in the previous figures.

Fig. 7. Combination of therapies: dependence on the order and doses, in case of weak competition. Along the horizontal axis parameter the normalized dose of chemotherapy

ρis represented, along the vertical axis κfor different level s of immunotherapy is expressed in the normalized scale ξ(see text for explanation). Left Panels: immunotherapy first. Right Panels: chemotherapy first. Upper panels display, as a heat map, the total cancer load, normalized to the carrying capacity, at the end of the therapies, for different levels of treatment. Lower panels show the mean fitness. The checkerboard pattern represents the region where the fraction of cells are below the detection threshold of 10 6 cells ( _{Misale et al., 2015 ).}

Fig. 5 shows that the predictions of the model are in good

380

agreementwiththeexperimentalresults.Weareaware,ofcourse,

381

that these data can be fitted just considering a single cancer

382

species,hereweintended justto ensurethatnumericalvaluesof

383

theparametersleadtoresultsclosetotheexperimentaldata.

384

Examples shown so far, representthe situation in which just

385

one type of therapy cannot completely eradicate cancer, since a

386

clonal type isresistant totreatment. As mentionedbefore thisis

387

acommonoccurrenceinmostclinicalsituationandindeedoneof

388

the major problems inthe battle against cancer. A way to

over-389

come this problem is the adoption of combination therapies in

390

which differentcancerclones aretargetedby specific treatments.

391

Manyworkshavebeendonetoinvestigatetheeffectsof

combina-392

tiontherapies andthe processes that undergoes their success,or 393

otherwise. Inboth casesit is apparent that the efficacyof these 394

treatments dependson theduration andtemporalorderof drugs 395

administration,seeforinstance Ledford(2016)and Slovin(2012). 396

4. Relevanceoftemporalorder 397

Severalexperimental results,e.g. EmensandMiddleton(2015), 398

suggestthattheorderwithwhichtreatmentsareadministeredcan 399

lead to differentoutcomes. In order to studythis effect,pairs of 400

sequencesof chemotherapyfollowedby immunotherapy andvice 401

versaare simulated,with differentcombinationsof parameters fi 402

Fig. 8. Combination of therapies: dependence on the order and doses, in case of a dominating clone sensitive to chemotherapy. Competitive parameters are b 12 = 0 . 01 days −1 and b 21 = 0 . 8 days −1 . Left Panels: immunotherapy first. Right Panels: chemotherapy first. Upper panel show the fraction of total cancer cells surviving at the end of the treatments and lower ones the corresponding mean fitness. Color code is the same as in Fig. 7 .

andcorrespondingly f1 has replaced r1 while f2=r2; conversely

404

bothclones areequally affectedby immunotherapy, so

κ

1=

κ

2=

405

κ

.Parametersvaluesarethoseusedin Section3.3forthe

compar-406

isonwiththe seconddata set;inparticularbij valuesare such to 407

ensureaweak competitionandhence,beforetreatments, the

co-408

existence ofcancer types.The initial conditions are chosen as in

409

thelower panels of Fig.5. Therapiesare supposed to be applied

410

consecutivelyandwiththesameduration.

411

In Fig. 6, examples of cancer development are shown, under

412

chemotherapy and immunotherapy applied sequentially: the left

413

panel refers to immunotherapy followed by chemotherapy while

414

intherightpaneltheorderisreversed.Itisclearfromthefigure

415

that theeffects are different: in both casesthe total cancer load

416

xisreducedbut, ifchemotherapy isapplied firstboth x1,x2,are

417

decreasingwhereas, withthe opposite sequence,the totalcancer

418

loadrebounds atthe endof the cure, mainlybecause of the

in-419

creaseofresistant clonex2,nomoreconstrainedby x1.This

sug-420

geststhat not just xneeds to be takeninto account but alsoits

421

rateofchangeattheendofthetreatment.

422

An obvious measure of x rate of change is the logarithmic

423 derivative, 424 w=1 x dx dt = dlgx dt ; (4)

theuseofthelogarithmicderivativemakesthismeasuresensitive

425

tochangesofxatlowlevelsandthustothegrowthofthetumor

426

afteranapparentlysuccessfultreatment.

427

The parameter whas also a different characterization: define 428

thefitnessofcancerspeciesx1,x2 as 429 wi=ri− ri Ki− bi j Ki xj− ciz Ki, (5)

wherewirepresentstheability oftheithcancer speciesto grow, 430 giventheintra-andinter-specificinteractionsandpredationbythe 431

immunesystem.Then,itistrivialtoshowthat 432

w=w1x1+w2x2 x1+x2 ,

(6)

thatiswisthemeanfitnessofthecancerpopulation. 433

Inordertogeneralizetheexampleof Fig.6,wehavecomputed 434

xandwfordifferentlevelsoftreatment,representedbydifferent 435

valuesofg1,andhenceoff1,(measuringtheeffectofchemother- 436 apy), and

κ

(representing effects of immunotherapy), see upper 437

panelsof Fig.7.Findingarerepresentedasan heatmapwithval- 438

uesofxnormalizedtothecarryingcapacities,forvaryingstrength 439

ofchemotherapyonthehorizontalaxisversustheimmunotherapy 440

ontheverticalaxis. 441

Forclarity’s sake, f1 and

κ

have beenreplaced by normalized 442

parameters.Forchemotherapywedefine 443

ρ

= g1 g1,u

where g1, as explained earlier, is defined by g1=r1− f1, and 4 4 4 g1,u=0.25is the chosen maximumvalue ofg1 corresponding to 445 f1=−0.1.Incaseofthe immunotherapythenormalizedparame- 446

E. Piretto et al. / Journal of Theoretical Biology xxx (2018) xxx–xxx 9

JID:YJTBI [m5G;March13,2018;16:54]

Fig. 9. Combination of therapies: dependence on the order and doses, in case of a dominating clone resistant to chemotherapy. Results for total cancer load and mean fitness when the competitive parameters are b 12 = 0 . 8 days −1 and b 21 = 0 . 01 days −1 . Left Panels: immunotherapy first. Right Panels: chemotherapy first. Upper panels show the fraction of total cancer cells surviving at the end of the treatments and lower ones the corresponding mean fitness. Color code is the same as in Fig. 7 .

teris

447

ξ

=

κ

κ

u

where

κ

u=30 isthemaximum valueof

κ

.The zeroofthescale 448

correspondstothesituationofnotherapy.

449

In the heat map values go from black (low) to white (high)

450

whilethecheckerboardpatternindicatesthatcancerloadisbelow

451

detectability threshold (corresponding to 106 _{cells, Misale et al.,}

452 2015).

453

Comparison of figures inupper panels clearlyshows that the

454

sequence withchemotherapyfirst(at therightofthepanel)

per-455

forms better than the opposite order (at the left of panel): a

456

chemotherapy first strategy leads to a decrease of cancer loadx

457

belowdetectability,andalsotheblackarea(correspondingtolow

458

xvalues)islargerinthiscase.Thedifferencesbetweenthetwo

se-459

quencesareevenclearerwhenoneconsidersw_,thatistheability

460

ofcancertorecoverfromthetherapy: notethat,with

chemother-461

apyfirst,to low cancerlevelthere correspondsmallvalues ofw,

462

whereasthat isnottrueforimmuno-first,meaningthatwiththis

463

latterstrategyonemustexpectamuchfasterregrowthofcancer.

464

It should be notedthat hereclonex1 ismore proliferatingin

465

absenceoftreatment(r1>r2),butsensitivetochemotherapy,while

466

x2 has a competitive advantage (b12>b21) and is resistant: both

467

clones are equally affected by immunotherapy. Then, the better

468

performance ofthechemo-firststrategy meansthatitismore

ef-469

fective totargetfirst clonex1,beforeit growstoomuch,whereas

470

x2,whichhasalowerrateofgrowth,canbedealtwithlater.

471

4.1. Competitiveinteractions 472

In this subsection we extend our study to different configu- 473

rations involvingcompetition amongcancer clones,moving away 474

theparticularexperimental data.Thuswe presentnew”insilico” 475

experiments withthe same parameters asbefore but withvalue 476

ofthe competition parameters b12, b21 takenin a wayto ensure 477 that exclusive competition applies. This is, admittedly, a some- 478

how extreme case, but it may correspond to the appearance of 479

a particularly competitive mutant in a population. We have set 480

first b12=0.01 days−1 and b21=0.8 days−1.Now the morepro- 481 liferating clone is also more competitive, so this case x1 can be 482 considered thestrongclone: inan evolutionarytime the popula- 483

tionx1 dominatesx2 (i.e.Pce1 is theonlystablepoint). Resultsof 484 thesimulations,presented in Fig.8,showthat nowthe sequence 485

withimmunotherapyfirstisthemostefficient.Ifthevaluesofbij 486

areswitched,b12=0.8,b21=0.01sothatthestrongcloneisalso 487 resistant to chemotherapy, the chemo-first sequence is again the 488

moreefficient,compare Fig.9. 489

Apossibleinterpretationofthe resultsisthat,ifthedominat- 490

ingcloneissusceptibletoachemicaltreatment,itisbettertohave 491

a cure (in this caseimmunotherapy) which controlsboth cancer 492

typeswhilelettingx1eliminatex2 andnextusechemotherapyon 493 x1.Onthecontrary,ifx2 isthestrongclonebetterresultsareob- 494 tainedfirsttargetingthemoreproliferatingtype,to avoidarapid 495

tumoralgrowth,andnextusingatherapyeffectiveonbothclones. 496

Thus competitive interactions play a role in determining the 497

best sequence of therapies and this may explain why there ex- 498

ist apparently contradictory experimental findings about opti- 499

a chemotherapy first approach (Antonia et al., 2006; Basanta 501

et al., 2012) and others suggesting to use immunotherapy first

502

(Hodietal.,2008).

503

In conclusion, the effects of the therapy on a cancer clone

504

depend on how the others tumors types respond to treatment

505

and this underlines the relevance of the interactions among

506

cancer species in designing effective therapies (Tabassum and 507

Polyak,2015).

508

5. Discussion 509

Basic premise of this paper is that growth of cancer cells is

510

ruledbythesameselectiveforcesshapingtheevolutionofspecies

511

and that population-theoretical perspective can provide a useful

512

frameworkformodelingcancerdevelopment:iffollowsthat

com-513

petitionamongcancertypesplayarelevantroleevenwhen

med-514

icaltreatmentsareapplied.

515

Along the lines of this hypothesis, a model of two

compet-516

ingcancerspeciessubjectedtotheactionofimmune systemcells

517

andtherapies has been proposed andformalized by a system of

518

three ordinary differential equations, whose asymptotic behavior

519

hasbeenanalyzed.

520

Rangeofparametershavebeenestimatedbyexperimentaldata

521

availableinthe literatureanda sensitivityanalysishasbeen

per-522

formed.Preliminarily,themodelhasbeentestedwithdatacoming

523

fromtwo experiments,(Misale et al., 2015; Ramakrishnan et al., 524

2010).

525

Combinations of immuno- and chemo-therapies of different

526

strengthanddifferentorderofapplicationhavebeensimulated.

527

The resultsofthemodelsconfirmthat temporalorderof

ther-528

apies administration is crucial for the effectiveness of cures, in

529

agreementwith findings fromclinical trials, (Slovin, 2012); more

530

interestingly, they show that the competitive interaction among

531

cancertypes determines whichsequence ismore effective.Ifthe

532

competition between cancer clones is weak, the sequence with

533

chemotherapy first appears to be more efficient: it is conducive

534

to a greater reduction of the total cancer load and to a slower

535

regrowthofthe tumor.However, whenthe competitionis strong

536

the situation is lessclear-cut andsomehow counter-intuitive

re-537

sultsoccur. If thedominating clone is susceptibleto

chemother-538

apy,animmuno-firsttherapyyieldsbetterresults,whereas,onthe

539

contrary,achemo-firsttreatmentisbetterifthedominantcloneis

540

resistanttochemicaltreatment.Anexplanationhasbeenprovided

541

whichtakesintoaccountthedifferenteffectsofclonesinteraction

542

inthecoexistenceversuscompetitive exclusionconfigurations.As

543

notedearlier, these findings may explain whythere exist

appar-544

entlycontradictory experimental findings aboutoptimal order of

545

curesadministration.

546

Ingeneral,modelresultsemphasizetheneedofcombining

dif-547

ferenttreatments tolimittheeffectofdrugresistanceand

under-548

liestherelevanceoftheinteractionsamongdifferentclones,when

549

determiningthetemporalorderoftherapies.

550

In conclusion,analysisandnumericalsimulations confirmthat

551

thismodel,albeitsimple,isabletoreproducesomeofthe

experi-552

mentalfindingandemergingbehaviorsaboutcancerdevelopment

553

undertherapy,and,also,tomakesometestablepredictions.

554

Supplementarymaterial

555

Supplementary material associated with this article can be

556

found,intheonlineversion,atdoi:10.1016/j.jtbi.2018.03.014.

557

References 558

Al-Tameemi, M., Chaplain, M., d’Onofrio, A., 2012. Evasion of tumours from the con- 559

trol of the immune system: consequences of brief encounters. Biol. Direct 7 (1), 560

31. doi: 10.1186/1745-6150-7-31 . 561

Antonia, S.J., Mirza, N., Fricke, I., Chiappori, A., Thompson, P., Williams, N., Bepler, G., 562 Simon, G., Janssen, W., Lee, J.-H., et al., 2006. Combination of p53 cancer vaccine 563 with chemotherapy in patients with extensive stage small cell lung cancer. Clin. 564 Cancer Res . 12 (3), 878–887. doi: 10.1158/1078- 0432.CCR- 05- 2013 . 565 Basanta, D., Gatenby, R.A., Anderson, A.R., 2012. Exploiting evolution to treat drug 566 resistance: combination therapy and the double bind. Mol. Pharm. 9 (4), 914– 567

921. doi: 10.1021/mp200458e . 568

Bellomo, N., Li, N., Maini, P.K., 2008. On the foundations of cancer modelling: se- 569 lected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 570 18 (04), 593–646. doi: 10.1142/S0218202508002796 . 571 Bunimovich-Mendrazitsky, S., Byrne, H., Stone, L., 2008. Mathematical model of 572 pulsed immunotherapy for superficial bladder cancer. Bull. Math. Biol. 70 (7), 573 2055–2076. doi: 10.1007/s11538- 008- 9344- z . 574 Carrere, C., 2017. Optimization of an in vitro chemotherapy to avoid resistant tu- 575 mours. J. Theor. Biol. 413, 24–33. doi: 10.1016/j.jtbi.2016.11.009 . 576 Chisholm, R.H., Lorenzi, T., Lorz, A., Larsen, A.K., De Almeida, L.N., Escargueil, A., 577 Clairambault, J., 2015. Emergence of drug tolerance in cancer cell populations: 578 an evolutionary outcome of selection, nongenetic instability, and stress-induced 579 adaptation. Cancer Res. 75 (6), 930–939. doi: 10.1158/0 0 08- 5472.CAN- 14- 2103 . 580 DeVita, V.T., Schein, P.S., 1973. The use of drugs in combination for the treatment of 581 cancer: rationale and results. N. Engl. J. Med. 288 (19), 998–1006. doi: 10.1056/ 582

NEJM197305102881905 . 583 De Pillis, L.G., Radunskaya, A., 2001. A mathematical tumor model with immune re- 584 sistance and drug therapy: an optimal control approach. Comput. Math. Meth- 585

ods Med. 3 (2), 79–100. 586

De Pillis, L.G., Gu, W., Radunskaya, A.E., 2006. Mixed immunotherapy and 587 chemotherapy of tumors: modeling, applications and biological interpretations. 588 J. Theor. Biol. 238 (4), 841–862. doi: 10.1016/j.jtbi.2005.06.037 . 589 Ding, L., Ley, T.J., Larson, D.E., Miller, C.A., Koboldt, D.C., Welch, J.S., Ritchey, J.K., 590 Young, M.A., Lamprecht, T., McLellan, M.D., et al., 2012. Clonal evolution in re- 591 lapsed acute myeloid leukaemia revealed by whole-genome sequencing. Nature 592 481 (7382), 506–510. doi: 10.1038/nature10738 . 593 Eftimie, R., Bramson, J.L., Earn, D.J., 2011. Interactions between the immune system 594 and cancer: a brief review of non-spatial mathematical models. Bull. Math. Biol. 595 73 (1), 2–32. doi: 10.1007/s11538-010-9526-3 . 596 Eladdadi, A., Radunskaya, A., et al., 2014. Modeling cancer-immune responses 597 to therapy. J. Pharmacokinet. Pharmacodyn. 41 (5), 461–478. doi: 10.1007/ 598

s10928- 014- 9386- 9 . 599 Emens, L.A., Middleton, G., 2015. The interplay of immunotherapy and chemother- 600 apy: harnessing potential synergies. Cancer Immunol. Res. 3 (5), 436–443. 601

doi: 10.1158/2326-6066.CIR-15-0064 . 602

Frascoli, F., Kim, P.S., Hughes, B.D., Landman, K.A., 2014. A dynamical model of tu- 603 mour immunotherapy. Math. Biosci. 253, 50–62. doi: 10.1016/j.mbs.2014.04.003 . 604 Gatenby, R.A., Brown, J., Vincent, T., 2009. Lessons from applied ecology: cancer 605 control using an evolutionary double bind. Cancer Res. 69 (19), 7499–7502. 606

doi: 10.1158/0 0 08- 5472.CAN- 09- 1354 . 607

Gatenby, R.A ., Silva, A .S., Gillies, R.J., Frieden, B.R., 2009. Adaptive therapy. Cancer 608 Res. 69 (11), 4 894–4 903. doi: 10.1158/0 0 08- 5472.CAN- 08- 3658 . 609 Gerlinger, M., Swanton, C., 2010. How darwinian models inform therapeutic failure 610 initiated by clonal heterogeneity in cancer medicine. Br. J. Cancer 103 (8), 1139– 611

1143. doi: 10.1038/sj.bjc.6605912 . 612

Greaves, M., 2007. Darwinian medicine: a case for cancer. Nat. Rev. Cancer 7 (3), 613

213–221. doi: 10.1038/nrc2071 . 614

Greaves, M., Maley, C.C., 2012. Clonal evolution in cancer. Nature 481 (7381), 306– 615

313. doi: 10.1038/nature10762 . 616

Hanahan, D., Weinberg, R.A., 2011. Hallmarks of cancer: the next generation. Cell 617 144 (5), 646–674. doi: 10.1016/j.cell.2011.02.013 . 618 Hillen, T., Lewis, M.A., 2014. Mathematical ecology of cancer. In: Managing Complex- 619 ity, Reducing Perplexity. Springer, pp. 1–13. doi: 10.1007/978- 3- 319- 03759- 2 _ 1 . 620 Hodi, F.S., Butler, M., Oble, D.A., Seiden, M.V., Haluska, F.G., Kruse, A., MacRae, S., 621 Nelson, M., Canning, C., Lowy, I., et al., 2008. Immunologic and clinical effects 622 of antibody blockade of cytotoxic t lymphocyte-associated antigen 4 in pre- 623 viously vaccinated cancer patients. Proc . Nat l. Acad . Sci . 105 (8), 3005–3010. 624

doi: 10.1073/pnas.0712237105 . 625

Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics. Cam- 626

bridge university press. 627

Kuznetsov, V.A ., Makalkin, I.A ., Taylor, M.A., Perelson, A.S., 1994. Nonlinear dynamics 628 of immunogenic tumors: parameter estimation and global bifurcation analysis. 629 Bull. Math. Biol. 56 (2), 295–321. doi: 10.1016/S0092- 8240(05)80260- 5 . 630 Ledford, H., 2016. The perfect blend. Nature 532 (7598), 162–164. 631 Ledzewicz, U., Schättler, H., 2017. Application of mathematical models to metro- 632 nomic chemotherapy: what can be inferred from minimal parameterized mod- 633 els? . Cancer Lett. doi: 10.1016/j.canlet.2017.03.021 . 634 Marino, S., Hogue, I.B., Ray, C.J., Kirschner, D.E., 2008. A methodology for performing 635 global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol. 254 636 (1), 178–196. doi: 10.1016/j.jtbi.2008.04.011 . 637 Marusyk, A., Almendro, V., Polyak, K., 2012. Intra-tumour heterogeneity: a looking 638 glass for cancer? Nat. Rev. Cancer 12 (5), 323–334. doi: 10.1038/nrc3261 . 639 Merlo, L.M., Pepper, J.W., Reid, B.J., Maley, C.C., 2006. Cancer as an evolutionary and 640 ecological process. Nat. Rev. Cancer 6 (12), 924–935. doi: 10.1038/nrc2013 . 641 Misale, S., Bozic, I., Tong, J., Peraza-Penton, A., Lallo, A., Baldi, F., Lin, K.H., Truini, M., 642 Trusolino, L., Bertotti, A., et al., 2015. Vertical suppression of the egfr pathway 643 prevents onset of resistance in colorectal cancers. Nat. Commun. 6. doi: 10.1038/ 644

ncomms9305 . 645

Murray, J.D., 2002. Mathematical Biology. Springer-Verlag. 646 Nikitina, E.Y., Clark, J.I., van Beynen, J., Chada, S., Virmani, A.K., Carbone, D.P., 647

E. Piretto et al. / Journal of Theoretical Biology xxx (2018) xxx–xxx 11

JID:YJTBI [m5G;March13,2018;16:54]

Gabrilovich, D.I., 2001. Dendritic cells transduced with full-length wild-type p53 648

generate antitumor cytotoxic t lymphocytes from peripheral blood of cancer pa- 649

tients. Clin . Cancer Res . 7 (1), 127–135. 650

Nowell, P.C., 1976. The clonal evolution of tumor cell populations. Science 194 651

(4260), 23–28. doi: 10.1126/science.959840 . 652

Pacheco, J.M., Santos, F.C., Dingli, D., 2014. The ecology of cancer from an evolution- 653

ary game theory perspective. Interface Focus 4 (4), 20140019. doi: 10.1098/rsfs.

654

2014.0019 . 655

Perthame, B., 2006. Transport Equations in Biology. Springer Science & Business Me- 656

dia. 657

Ramakrishnan, R., Assudani, D., Nagaraj, S., Hunter, T., Cho, H.-I., Antonia, S., Al- 658

tiok, S., Celis, E., Gabrilovich, D.I., 2010. Chemotherapy enhances tumor cell sus- 659

ceptibility to ctl-mediated killing during cancer immunotherapy in mice. J. Clin. 660

Invest. 120 (4), 1111. doi: 10.1172/JCI40269 . 661

Saltelli, A., Tarantola, S., Campolongo, F., Ratto, M., 2004. Sensitivity Analysis in Prac- 662

tice: A Guide to Assessing Scientific Models . John Wiley & Sons. 663

Saunders, N.A., Simpson, F., Thompson, E.W., Hill, M.M., Endo-Munoz, L., Leggatt, G., 664

Minchin, R.F., Guminski, A., 2012. Role of intratumoural heterogeneity in can- 665

cer drug resistance: molecular and clinical perspectives. EMBO Mol. Med. 4 (8), 666

675–684. doi: 10.1002/emmm.201101131 . 667

Slovin, S., 2012. Chemotherapy and immunotherapy combination in advanced 668 prostate cancer. Clin. Adv. Hematol. Oncol. 10 (2), 90–100. 669 Tabassum, D.P., Polyak, K., 2015. Tumorigenesis: it takes a village. Nat. Rev. Cancer 670

15 (8), 473–483. doi: 10.1038/nrc3971 . 671

Vivarelli, S., Wagstaff, L., Piddini, E., 2012. Cell wars: regulation of cell survival 672 and proliferation by cell competition. Essays Biochem. 53, 69–82. doi: 10.1042/ 673

bse0530069 . 674

Wagstaff, L., Kolahgar, G., Piddini, E., 2013. Competitive cell interactions in cancer: 675 a cellular tug of war. Trends Cell Biol. 23 (4), 160–167. doi: 10.1016/j.tcb.2012.11. 676

002 . 677

Wilkie, K.P., 2013. A review of mathematical models of cancer–immune interactions 678 in the context of tumor dormancy. In: Systems Biology of Tumor Dormancy. 679 Springer, pp. 201–234. doi: 10.1007/978- 1- 4614- 1445- 2 _ 10 . 680 Wilson, S., Levy, D., 2012. A mathematical model of the enhancement of tumor 681 vaccine efficacy by immunotherapy. Bull. Math. Biol. 74 (7), 1485–1500. doi: 10. 682

1007/s11538- 012- 9722- 4 . 683 Zitvogel, L., Apetoh, L., Ghiringhelli, F., Kroemer, G., 2008. Immunological aspects of 684 cancer chemotherapy. Nat. Rev. Immunol. 8 (1), 59–73. doi: 10.1038/nri2216 . 685