A survey of adaptive cell population dynamics models of

www.biomathforum.org/biomath/index.php/biomathREVIEW ARTICLE

A survey of adaptive cell population dynamicsmodels of emergence of drug resistance in

cancer, and open questions about evolutionand cancer

Jean Clairambault∗, Camille Pouchol†∗INRIA Paris & Sorbonne Université,

UMR 7598, LJLL, BC 187, 75252 Paris Cedex 05, [email protected]†Department of Mathematics

KTH, Brinellvägen 8, 114 28 Stockholm, [email protected]

Received: 9 March 2019, accepted: 14 May 2019, published: 24 May 2019

Abstract—This article is a proceeding sur-vey (deepening a talk given by the first authorat the BioMath 2019 International Conferenceon Mathematical Models and Methods, heldin Będlewo, Poland) of mathematical modelsof cancer and healthy cell population adaptivedynamics exposed to anticancer drugs, todescribe how cancer cell populations evolvetoward drug resistance.

Such mathematical models consist of par-tial differential equations (PDEs) structuredin continuous phenotypes coding for the ex-pression of drug resistance genes; they in-volve different functions representing targetsfor different drugs, cytotoxic and cytostatic,with complementary effects in limiting tu-mour growth. These phenotypes evolve con-tinuously under drug exposure, and their fate

governs the evolution of the cell populationunder treatment. Methods of optimal controlare used, taking inevitable emergence of drugresistance into account, to achieve the beststrategies to contain the expansion of a tu-mour.This evolutionary point of view, which re-

lies on biological observations and resultingmodelling assumptions, naturally extends toquestioning the very nature of cancer as evo-lutionary disease, seen not only at the shorttime scale of a human life, but also at thebillion year-long time scale of Darwinian evo-lution, from unicellular organisms to evolvedmulticellular organs such as animals and man.Such questioning, not so recent, but recentlyrevived, in cancer studies, may have conse-quences for understanding and treating can-cer.

Copyright: c© 2019 Clairambault et al. This article is distributed under the terms of the Creative CommonsAttribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium,provided the original author and source are credited.Citation: Jean Clairambault, Camille Pouchol, A survey of adaptive cell population dynamicsmodels of emergence of drug resistance in cancer, and open questions about evolution andcancer, Biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 Page 1 of 23

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J. Clairambault, C. Pouchol, A survey of adaptive cell population dynamics models of emergence ...

Some open and challenging questions maythus be (non exhaustively) listed as:

• May cancer be defined as a spatially lo-calised loss of coherence between tissuesin the same multicellular organism, ‘spa-tially localised’ meaning initially startingfrom a given organ in the body, but alsopossibly due to flaws in an individual’sepigenetic landscape such as imperfectcontrol of differentiation genes?

• If one assumes that “The genes of cellularcooperation that evolved with multicel-lularity about a billion years ago arethe same genes that malfunction in can-cer.” (Davies and Lineweaver, 2011), howcan these genes be systematically investi-gated, looking for zones of fragility - thatdepend on individuals - in the ‘tinker-ing’ (F. Jacob, 1977) evolution is made of,tracking local defaults of coherence?

• What is such coherence made of andto what extent is the immune systemresponsible for it (the self and differen-tiation within the self)? Related to thisquestion of self, what parallelism can beestablished between the development ofmulticellularity in different species pro-ceeding from the same origin and thedevelopment of the immune system inthese different species?

Keywords-Cell population dynamics; struc-tured models; Darwinian evolution; drug-induced drug resistance; cancer therapeutics;optimal control

I. Introduction: Motivation from andfocus on drug resistance in cancerSlow genetic mechanisms of ‘the great evolu-

tion’ that has designed multicellular organisms,together with fast reverse evolution on smallertime windows, at the scale of a human disease,may explain transient or established drug re-sistance. This will be developed around the so-called atavistic hypothesis of cancer.

Plasticity in cancer cells, i.e., epigenetic [30],[64] (much faster than genetic mutations, andreversible) propension to reversal to a stem-like, de-differentiated phenotypic status, result-

ing in fast adaptability of cancer cell popula-tions, makes them amenable to resist abruptdrug insult (high doses of cytotoxic drugs, ion-ising radiations, very low oxygen concentrationsin the cellular medium) as response to cellularstress.

Intra-tumour heterogeneity with respect todrug resistance potential, meant here to modelbetween-cell phenotypic variability within can-cer cell populations, is a good setting to rep-resent continuous evolution towards drug re-sistance in tumours. This is precisely what iscaptured by mathematical (PDE) models struc-turing cell populations in relevant phenotypes,relevant here meaning adapted to describe anenvironmental situation that is susceptible toabrupt changes, such as introduction of a deadlymolecule in the environment. Beyond classical(in ecology) viability and fecundity, reversibleplasticity for cancer cell populations may alsobe set as one of such phenotypes.

Such structured PDE models have the advan-tage of being compatible with optimal controlmethods for the theoretical design of optimisedtherapeutic protocols involving combinations ofcytotoxic and cytostatic (and later possibly epi-genetic [89]) treatments. The objective functionof such optimisation procedure being chosen asminimising a cancer cell population number, theconstraints will consist of minimising unwantedtoxicity to healthy cell populations. The innova-tion in this point of view is that success or failureof therapeutic strategies may be evaluated by amathematical looking glass into the hidden coreof the cancer cell population, in its potential ofadaptation to cellular stress.

The poor understanding of the determinantsof drug resistance in cancer at the epigeneticlevel thus far, and the unexplained failure - orpartial failure - of initially promising treatmentssuch as targeted therapies and immunotherapiesmake it mandatory, from our point of view, toexamine cancer, its evolution and its treatmentat the level of a whole multicellular organismthat - locally, to begin with - progressively lacks

Biomath 8 (2019), 1905147, http://dx.doi.org/10.11145/j.biomath.2019.05.147 Page 2 of 23



its within- and between-tissue cohesion.II. Biological background

A. The many facets of drug resistance in cancerDrug resistance is a phenomenon common

to various therapeutic situations in which anexternal pathogenic agent is proliferating at theexpense of the resources of an organism: an-tibiotherapy, virology, parasitology, target pop-ulations are able to develop drug resistancemechanisms (e.g., expression of β-lactamase inbacteria exposed to amoxicillin). In cancer, thereis in general no external pathogenic agent (eventhough one may have favoured the disease) andthe target cell populations share much of theirgenome with the host healthy cell population,making overexpression of natural defence phe-nomena easy (e.g., ABC transporters in cancercells). Note that drug resistance (and resistanceto radiotherapy) is one of the many forms offast resistance to cellular stress, possibly codedin ‘cold’, i.e., strongly preserved throughout evo-lution, rather than in ‘hot’, i.e., mutation-prone,genes [107].

At the molecular level in a single cell (that isper se insufficient to explain the emergence ofdrug resistance), overexpression of ABC trans-porters, of drug processing enzymes, decreaseof drug cellular influx, etc. [38] are relevant todescribe endpoint molecular resistance mecha-nisms. At the cell population level, representingdrug resistance by an abstract continuous vari-able x standing for the level of expression of aresistance phenotype (in evolutionary game the-ory [4]: a strategy of the population) is adaptedto describe continuous evolution from total sen-sitivity (x = 0) towards total resistance (x = 1).Is such evolution towards drug resistance dueto sheer Darwinian selection of the fittest bymutations in differentiation at cell division or,at least partially, due to phenotype adaptationin individual cells? This is by no means clearfrom biological experiments. In particular, it hasbeen shown in [88] that emergence of drug re-sistance may be totally reversible, and, further-more, that it may be completely dependent on

the expression and activity of epigenetic controldrugs (DNA methyltransferases). This has beencompleted by molecular studies of the role ofrepeated sequences in drug tolerance in [42].

B. Ecology, evolution and cancer in cell popula-tions“Nothing in biology makes senses except in

the light of evolution.” (Theodosius Dobzhansky,1964 [27])

The animal genome (of the host to cancer) isrich and amenable to adaptation scenarios that,especially under deadly environmental stress,may recapitulate salvaging developmental sce-narios - in particular blockade of differentiationor dedifferentiation, allowing better adaptabil-ity but resulting in insufficient cohesion of theensemble - that have been abandoned in theprocess of the great evolution [45], [46] fromunicellular organisms (aka Protozoa) to coher-ent Metazoa [23] (aka multicellular organisms).In cancer populations, enhanced heterogeneitywith enhanced proliferation and poor differenti-ation results in a high phenotypic or genetic di-versity of immature proliferating clonal subpop-ulations, so that drug therapy may be followed,after initial success, by relapse due to selectionof one or more resistant clones [25].

As regards ecology and evolution, geneticsand epigenetics: ecology is concerned with thriv-ing or dying of living organisms in populationsin the context of their trophic environment.Evolution is concerned by the somatic changes,either inscribed by genetic mutations of basepairs in the marble of their DNA, or only -and sometimes geneticists in that case dismissthe term evolution, preferring adaptation - re-versibly (however transmissible to the next gen-erations) modified by silencing or re-expressinggenes by means of grafting methyl or acetylradicals on on the base pairs of the DNA oron the aminoacids that constitute the chromatin(i.e., histones) around which the DNA is wound.In the latter case, such evolution is determinedby epigenetic mechanisms, i.e., mechanisms that




do not change the sequence of the base pairs,but only locally modify their transcriptionalfunction. At the level of a same multicellular or-ganism, such so-called epimutations govern thesuccession of events that physiologically resultin cell differentiations.

At this stage, one must clearly distinguishthree different meanings of the words evolutionand differentiation/maturation: 1) at the short-term time scale of a cancer cell population,evolution by (plastic) adaptation of phenotypeor by mutations - both phenomena may beencountered - to a changing environment suchas infusion of an anti-cancer drug [32]; 2) at themid-term time scale of a developing animal orhuman organism, programmed succession of celldifferentiations, leading from one original cellto the circa 200-400 different cell types (in thehuman case) that constitute us as multicellularorganisms (development): no mutations, onlydifferentiations (aka maturations), i.e., epige-netic changes within the same genome untileach cell reaches its completely physiologicallymature state (an example of an ODE model ofdifferentiation may be found, e.g., in [44]); 3) atthe very long-term (billions of years) time scaleof Darwinian evolution of species, succession ofmutations from protozoa until evolved metazoa.

Epigenetic changes in a cell population arenot rare events and may be fast, operating underenvironmental pressure by means of epigeneticcontrol enzymes (methylases and acetylases,DNA methyltransferases, etc.), and they arereversible, however likely less quickly than theyhave occurred [88]. It is also likely, and indeedthis has been shown in some cases, that follow-ing such reversible epigenetic events, rare eventsthat are mutations (not so rare in the contextof genetic instability that often characterisescancer cell divisions) stochastically happen, fix-ing in the DNA an acquired advantage in thecontext of a changing environment. Conversely,mutations in parts of the genome that code forepigenetic control enzymes may determine epi-genetic changes, metaphorically representable in

the Waddington epigenetic landscape [45], [46],[106]. Such genetic to epigenetic modificationsand vice versa are discussed in [14], [36], [37].Also note that the relationships between ecol-ogy, evolution and cancer are extensively devel-oped in the book [102].

C. The atavistic theory of cancerThere has been some debate in the past 20

years about two opposed views of cancer, theclassic one, advocated by P. Nowell [71] statesthat cancer starts from a single “renegade”cell (a cheater), that by a succession of muta-tions initiates cancer (Somatic Mutation The-ory, SMT), being followed by strict Darwinianselection of the fittest, while the less admittedTissue Organisational Field Theory (TOFT),advocated in particular by A. Soto and C. Son-nenschein [90], contends that cancer is a matterof deregulated ecosystem, amenable to eradica-tion by changing the tumour ecosystem. Theatavistic theory of cancer is completely differentfrom those two, in as much as it relates cancerto a regression in the billion year-long evolu-tion of multicellularity. The idea that cancer isa form of backward evolution from organisedmulticellularity toward unicellularity, stalled atthe poorly organised forms of multicellularitytumours consist of (as cancer cell populations,escaping the collective control present in delicateorganismic organisations that constitute coher-ent multicellularity, continuously reinvents thewheel of multicellularity, starting from scratchfor their own sake) is not new and it has atleast been proposed by T. Boveri in 1929 [7]and L. Israel in 1996 [47]. However, it hasregained visibility thanks to the documentedand simultaneous studies by physicists P.C.W.Davies and C.H. Lineweaver [23], [55] and oncol-ogist M. Vincent [103], [104], followed by varioussubsequent studies, constituting a new bodyof knowledge [11], [19], [97], [98], [108] underthe name of atavistic theory of cancer. Thistheory, or hypothesis, postulates that, althoughcancer reinvents the wheel of multicellularity,it has at its disposal for this task “an ancient




toolkit of pre-existing adaptations” that makesit fundamentally differ from classical Darwinianevolution [23]. In this respect, cancer is clearly“more an archeoplasm than a neoplasm” (MarkVincent [103]).

What is relatively new in this theory, com-pared with the previously cited ones, is theidea that an intermediate set of coarse formsof multicellularity (which they call “Metazoa1.0”), lacking coherent control of intercellular- and between cell populations - cooperativ-ity and proliferation, qualified a “robust toolkitfor the survival , maintenance and propagationof non-differentiated or weakly-differented cells”is a safety state to which “Metazoa 2.0” (us,in particular) revert when our sophisticatedform of multicellularity goes astray in cancer.Such events are due to failures in control ofevolved cooperativity genes, and this incoher-ent, chaotic, poorly organised “Metazoa 1.0”system endows the tissues in which it is installedwith high phenotype adaptability, aka cancerplasticity, on which tumour development relies.Such plasticity makes tumour cells in particularable to exploit for their own sake, plasticityresulting in resistance to cytotoxic drugs, epige-netic enzymes that were physiologically designedto control finely tuned cell differentiations, inacquired resistance to cancer treatments. Thisilluminating view of cancer, according to whichthe genes that malfunction are precisely “thegenes of cellular cooperation that evolved withmulticellularity about a billion years ago” (PaulDavies and Charles Lineweaver [23]), and thereason of resistance is to be found in ancient,well preserved, genes of our DNA, is howeverquite often not admitted by many biologistsof cancer who strongly believe in the strictlyDarwinian nature of evolution in cancer cellpopulations [33], [34], [39], [40], without anykind of such “genomic memory”.

Compatible with the atavistic hypothesis thatpostulates such backward evolution, a possiblescenario suggests that cancer may start witha local deconstruction of the epigenetic con-

trol of cell differentiation (that is an essentialpiece of the coherence of multicellularity, e.g.,in haematopoiesis, by genes TET2, DNMT3A,ASXL1), followed by deregulation of cooper-ativity between cell populations (essential todivision of work in a multicellular organism)initiated by disruption of transcription factorsresponsible for differentiation (e.g., by genesRUNX1, CEBPα, NPM1) and finally deregula-tion of the determinants of the strongest andmost ancient bases of multicellularity, prolif-eration and apoptosis (e.g., by genes FLT3,KIT, and genes of the RAS pathway). Eventhough many cancer biologists are reluctant toendorse this scenario, biological observations ex-ist, showing that a scenario of successions ofmutations may be found in fresh blood sam-ples of patients with acute myeloid leukaemia,phylogenetically recapitulating such hierarchi-cally ordered deconstruction of the multicellu-lar haematopoietic structure, from the finest(epigenetic) to the coarsest (proliferation andapoptosis) elements of the construction of mul-ticellularity [43].

From the point of view of therapeutic appli-cations, the atavistic theory of cancer has theconsequence that, even though those genes ofcooperativity that are altered in cancer (the“multicellularity gene toolkit of Metazoa 1.0”)have taken one billion years of Darwinian evo-lution to achieve (by ‘tinkering’ [49]) coher-ent evolved multicellular organisms, they arenevertheless in finite number and can be sys-tematically investigated, as has been initiatedin phylostratigraphic studies led by TomislavDomazet-Lošo and Diethard Tautz [28], [29].Such systematic between-species phylogeneticbiocomputer studies should open observationwindows onto altered genes in patients and theirpossible correction in the future.

From the point of view of mathematical mod-elling, the fact that ancient genes of survivalhave been developed in the course of evolutionto make individual cells, and later coherentlyheterogeneous and nevertheless communicating




together (failures in intercellular communica-tions [99], [100], [101] incidentally being alsoa possible source of default of cooperativityin cell populations), and may be conserved assilent capacities in our genome, only waiting tobe unmasked by epigenetic enzymes [88] put onthe service of survival in highly plastic cancercells [107], gives reasonable biological supportto the notion of cell populations structured inphenotype of survival and of drug resistance.How such ancient genes (‘cold genes’ [107]) havebeen preserved in our genome while servingin rare and extreme environmental conditionsonly is not clear (in principle, genes that arenot expressed are prone to disappear), howeverobservations reported in [107] propose that an-cient genes evolve more slowly than youngerones. Hence, preserved in the genomic memoryas survival genes, revivable in plastic cancercell populations (plastic here meaning that theyhave easy access to epigenetic enzymes to changetheir phenotype under environmental pressure),their level of expression may offer a basis forevolvability and reversibility, under environmen-tal pressure, of continuous phenotypes structur-ing the heterogeneity (aka biological variability)of cancer cell populations that is developed inmathematical models of adaptive cell populationdynamics.

III. Models of adaptive dynamics

A. Models structured in resistance phenotypeThe simplest model of a resistance phenotype-

structured cell population may be described by anonlocal Lotka-Volterra-like integro-differentialequation, here x ∈ [0, 1] representing a contin-uous resistance phenotype, from x = 0, totalsensitivity to the drug, until x = 1, total insen-sitivity:

∂n

∂t(t, x) =

(r(x)− d(x)ρ(t)

)n(t, x),

with

ρ(t) :=∫ 1

0n(t, x) dx and n(0, x) = n0(x).

Note that this simple integro-differential equationmay, when this makes biological sense, be gen-eralised to a reaction-diffusion-advection (RDA)one written as

∂n

∂t(t, x) + ∂

∂x{v(x)n(t, x)}

= β∂2

∂x2n(t, x) + {r(x)− d(x)ρ(t)}n(t, x).

We assume reasonable (L∞) hypotheses onr and d, and n0 ∈ L1([0, 1]). Phenotype-dependent functions r and d stand for intrinsicproliferation rate and intrinsic death rate dueto within-population competition for space andnutrients, respectively. Note that space is repre-sented here only in the abstract nonlocal logisticterm d(x)ρ(t). It is nevertheless possible to mixphenotype and actual Cartesian space variablesto structure the population, as will be shownlater.

One can then prove for the simple integro-differential model the asymptotic behaviour the-orem:

Theorem 1. [24], [48], [74](i) ρ converges to ρ∞, the smallest value ρ suchthat r(x) − d(x)ρ ≤ 0 on [0, 1] (i.e., ρ∞ =max[0,1]

rd).

(ii) The population n(t, ·) concentrates on thephenotype set

{x ∈ [0, 1], r(x)− d(x)ρ∞ = 0

}.

(iii) Furthermore, if this set is reduced to asingleton x∞, then n(t, ·) ⇀ ρ∞δx∞ in M1(0, 1).

(the measure space M1(0, 1) being the dualfor the supremum norm of the space of continu-ous real-valued fonctions on [0, 1]; note the Diracmass on the RHS, convergence is here meant inthe sense of measures.)

Although in the one-population case, asstated above, a direct proof of convergence basedon proving that ρ(t) is BV on the half-line,from which concentration easily follows fromexponential growth, it is interesting to note,as this argument can be used in the case oftwo interacting populations, that a global proofbased on the design of a Lyapunov function givesat the same time convergence and concentration:




choosing any measure n∞ on [0, 1] with supportin argmax r

d such that∫ 1

0n∞(x) dx = ρ∞ = max

[0,1]

r

d,

and for an appropriate weight w(x) (in fact 1d(x))setting

V(t)=∫ 1

0w(x){n(t,x)−n∞(x)−n∞(x) lnn(t,x)}dx,

one can show thatdV

dt= −(ρ(t)− ρ∞)2

+∫ 1

0w(x) {r(x)−d(x)ρ∞}n(t, x) dx,

which is always nonpositive, tends to zero fort → ∞, thus making V a Lyapunov function,and showing at the same time convergence andconcentration.

Indeed, in this expression, the two terms arenonpositive and their sum tends to zero; the zerolimit of the first one accounts for convergenceof ρ(t), and the zero limit of the second oneaccounts for concentration in x (on a zero-measure set) of lim

t→+∞n(t, x).

Starting from this simple model, one can gen-eralise it to the case of two interacting cell pop-ulations, cancer (nC) and healthy (nH), againusing a nonlocal Lotka-Volterra setting, withtwo different drugs, u1, cytotoxic (= cell-killingdrug, towards which resistance evolves accordingto the continuous phenotype x ∈ [0, 1]) and u1,cytostatic (only thwarting proliferation withoutkilling cells):

∂

∂tnH(t, x) =[rH(x)

1+kHu2(t)−dH(x)IH(t)−u1(t)µH(x)]nH(t,x),

∂

∂tnC(t, x) =[rC(x)

1+kCu2(t)−dC(x)IC(t)−u1(t)µC(x)]nC(t,x).

(1)

The environment in the logistic terms is definedby:

IH(t) = aHH .ρH(t) + aHC .ρC(t),IC(t) = aCH .ρH(t) + aCC .ρC(t),

with aHH > aHC , aCC > aCH (higher within-species than between-species competition) and

ρH(t)=∫ 1

0nH(t, x) dx, ρC(t)=

∫ 1

0nC(t, x) dx.

The cytotoxic drug terms, tuned by drug sen-sitivity functions µC and µH , act as added deathterms to the logistic term, whereas the cyto-static drug terms act by inhibiting the intrinsicproliferation rates rC and rH . Functions µC andµH obviously have to be decreasing functionsof x, and so, less obviously, but representinga trade-off between survival and proliferation(“cost of resistance”), have to be rC and rH . Asregards dC and dH , no modelling choice imposesitself; however, in order to make the functionrd globally decreasing and thus, in the absenceof drug, obtain its maximum around zero, itwas assumed in this study that it is a non-decreasing function of x. Biologically, this meansthat the more resistant a cell is, the strongeropposition to its proliferation it encounters in itsown species, cancer or healthy, which is anotherway, coherent with the modelling choice madeon rC and rH , to express a cost of resistance.

In this 2-population case, following an ar-gument by Pierre-Emmanuel Jabin and GaëlRaoul [48], one can prove, as in the 1-populationcase, at the same time convergence and concen-tration by using a Lyapunov functional of theform∫w(x) {n(t, x)− n∞(x)− n∞(x) lnn(t, x)} dx.

We have also in this case the asymptoticbehaviour theorem:

Theorem 2. [81], [83] Assume that u1 and u2are constant: u1 ≡ u1, and u2 ≡ u2. Then,for any positive initial population of healthy andof tumour cells, (ρH(t), ρC(t)) converges to the




equilibrium point (ρ∞H , ρ∞C ), which can be exactlycomputed as follows:Let a1 ≥ 0 and a2 ≥ 0 be the smallest nonnega-tive real numbers such that

rH(x)1 + αH u2

− u1µH(x) ≤ dH(x)a1 and

rC(x)1 + αC u2

− u1µC(x) ≤ dC(x)a2.

Then (ρ∞H , ρ∞C ) is the unique solution of theinvertible (aHH .aCC > aCH .aHC) system

I∞H = aHHρ∞H + aHCρ

∞C = a1,

I∞C = aCHρ∞H + aCCρ

∞C = a2.

Let AH ⊂ [0, 1] (resp., AC ⊂ [0, 1]) be the setof all points x ∈ [0, 1] such that equality holds inthe inequalities above. Then the supports of theprobability measures

νH(t) = nH(t, x)ρH(t) dx and νC(t) = nC(t, x)

ρC(t) dx

converge respectively to AH and AC as t tendsto +∞.

In [81], this result is complemented with nu-merical simulations which show the failure ofconstant administration of high doses of bothdrugs. The theorem explains the phenomenon:such a strategy makes the cancer cell densityconcentration on a very resistant phenotypenear x = 1. Once most of the mass is close tox = 1, further treatment is hopeless as the tu-mour has become mostly resistant, and it startsincreasing again after having first decreased.This can be interpreted as relapse.

Note that numerical studies based on a sim-ilar model of adaptive dynamics in a reaction-diffusion version, dealing with the question ofrelapse, can be found in [16], [17], [56], [57],see also [74], [75] for more theoretical consid-erations.

This result extends to two competitively in-teracting populations the result of convergenceand concentration for nonlocal Lotka-Volterraphenotype-structured models previously pub-lished in [24], [48], [74]. Note that it assumes

the invertibility of the square matrix [aij ] wherei, j ∈ {H,C}.

A natural question then arises: is it possibleto extend this result to N > 2 interactingpopulations? This is the object of the study [82],in which the following N -dimensional nonlocalLotka-Volterra system is set, for which one canlook for coexistence of positive steady states(i.e., persistence of all species):

∂

∂tni(t, x) =

ri(x) + di(x)N∑j=1

aijρj(t)

ni(t, x),

in which x stands for all xi ∈ Xi for simplicity,each Xi being a compact subset of someRpi , ri, di smooth enough, and as usualρi(t) =

∫Xi

ni(t, x) dx.

This system generalises to a nonlocal settingclassical Lotka-Volterra models (for 2 popula-tions in an ODE setting, see, e.g., Britton [8] orMurray [69]) with ecological cases: mutualisticif aij > 0 and aji > 0, competitive if aij < 0and aji < 0 , predator-prey-like if aijaji < 0, forthe interaction matrix A = [aij ]

In [82], to which the reader is sent for moredetails, it is proved that a coexistent positivesteady state

ρ∞ = [ρ∞1 , . . . , ρ∞i , . . . , ρ∞N ]t

exists in RN if and only if, settingI∞ = [I∞1 , . . . , I∞i , . . . , I

∞N ]t,

where each I∞i = maxx∈Xi

ri(x)di(x) , the equation

Aρ + I∞ = 0 has a solution ρ∞ ∈ RN . Then,under some precise conditions on A, it can beproved, again using the same kind of Lyapunovfunction as in [81], that the solution to theN -dimensional nonlocal Lotka-Volterra systemexists and is globally defined; furthermore, thesolution ρ∞ to the equation Aρ + I∞ = 0 inRN is then unique and globally asymptoticallystable. As in the 1- and 2-dimensional cases,a result of concentration in phenotype follows,with moreover an estimation of the speeds ofconvergence and of concentration.




B. Modelling mutualistic tumour-stroma inter-actions

Noting that breast and prostate tumoursare accompanied in their stroma by so-calledcancer-associated adipocytes (CAAs) or cancer-associated fibroblasts (CAFs) [18], [26], [60],which favour cancer growth, likely by exchang-ing bidirectional messenger molecules, one canmodel such mutualistic interactions in a nonlo-cal Lotka-Volterra way:

∂

∂tnA(t,x)=[rA(x)−dAρA(t)+sA(x)ϕC(t)]nA(t,x)

∂

∂tnC(t,y)=[rC(y)−dCρC(t)+sC(y)ϕA(t)]nC(t,y)

withρA(t) =

∫nA(t, x) dx, ρC(t) =

∫nC(t, y) dy,

ϕA(t) =∫ψA(x)nA(t, x) dx and

ϕC(t) =∫ψC(y)nC(t, y) dy,

for some weight functions ψA and ψC (that inthe absence of known data may be chosen assimply affine), and some given initial conditionsnA(0, x) = n0

A(x), nC(0, y) = n0C(y) for all (x, y)

in [0, 1]2, x standing for transformation towardsa CAA or CAF state in the adipocyte popula-tion and y standing for strength of malignancyin the cancer cell population. This model isstudied, theoretically and numerically, in [80] inits generalised reaction-diffusion-advection form(see above) with explicit functions and initialfunctions n0

A, n0C assumed to be Gaussian.

In a setting in which mutualistic interactionsbetween two cell species, one of them being ini-tially healthy, but susceptible to become cancer-ous, namely proliferating haematopoietic stemcells and early progenitors nh, in the mandatorypresence of the other species ns, supportingstromal cells, a model closely related to theprevious one is presented [70]. It writes∂tnh(t,x)=

[rh(x)−ρh(t)−ρs(t)+α(x)Σs(t)

]nh,

∂tns(t,y)=[rs(y)−ρh(t)−ρs(t)+β(y)Σh(t)

]ns.

This system is completed with initial data

nh(0, x) = nh0(x) ≥ 0, ns(0, y) = ns0(y) ≥ 0.

Here the assumptions and notations are

• ρh(t) :=∫ b

anh(t, x)dx, ρs(t) :=

∫ d

cns(t, y)dy

are the total populations of HSCs andtheir supporting stromal cells MSCs, re-spectively, x representing a malignancy po-tential in haematopoietic cells and y atrophic potential in stromal cells.

• The functions

Σh(t) :=∫ b

aψh(x)nh(t, x) dx, and

Σs(t) :=∫ d

cψs(y)ns(t, y) dy

denote an assumed chemical signal (Σh)from the hematopoietic immature stemcells (haematopoietic stem cells, HSCs)to their supporting stroma (mesenchymalstem cells, MSCs), i.e., “call for support”and conversely, a trophic message (Σs) fromMSCs to HSCs. The weight functions ψh, ψsare nonnegative and defined on (a, b) and(c, d), intervals of the real line.

• The function rh ≥ 0 represents the intrin-sic (i.e., without contribution from trophicmessages from MSCs) proliferation rate ofHSCs. Assume that rh is non-decreasing,rh(a) = 0 and rh(b) > 0.

• The function α ≥ 0, satisfying α′ ≤ 0 andα(b) = 0, is the sensitivity of HSCs to thetrophic messages from supporting cells.

• For the function rs ≥ 0, it is assumed thatr′s(y) ≤ 0. The function β(y) ≥ 0 withβ′(y) ≥ 0 represents the sensitivity of thestromal cells MSCs to the (call for support)message coming from HSCs.

Some examples for rh, α are given by rh =r∗h(x − a) or rh = r∗h(x − a)2, α(x) = α∗(b − x)with positive constants r∗h, α∗, ψs(y) = y andψh(x) = x.

The reader is sent to [70] for a detailed studyof this model. In particular, theoretical condi-tions for extinction, invasion or possible stable




coexistence of a leukaemic clone emerging in aninitial healthy HSC population together with amaintained healthy fraction of it, with numericalsimulations, are given in this study. They arerelated to convexity or concavity properties offunctions of the model describing proliferationof the population, rh and α, the same kindof evolution being possible in the stromal cellpopulation.C. Models structured in phenotype and space

Although purely space-structured models lackthe necessary heterogeneity in phenotype totake into account continuous evolution towardsdrug-induced drug resistance, purely phenotype-structured models lack the possibility to exam-ine possible heterogeneities due to extension oftumours in Cartesian space, in particular dueto diffusion of molecules (anticancer drugs andnutrients) in the medium. Hence, provided thatsomething is known of the geometry of the spaceoccupied by a cancer cell population, and thisis indeed the case with initial tumours thatspontaneously thrive in spheroids, mixing spacewith phenotype to structure a model of a cancercell population under drug exposure, to study itsbehaviour with respect to drug resistance, is anatural way to proceed.

Relying on modelling principles developedin [52], [58], integrated in spheroid-like space,such a model is studied in [59]:

∂tn(t, r, x) =[

p(x)1+µ2c2(t,r)s(t,r)

−d(x)%(t,r)−µ1(x)c1(t,r)]n(t,r,x),

−σs∆s(t, r)+[γs+

∫ 1

0p(x)n(t,r,x)dx

]s(t,r) = 0,

−σc∆c1(t,r)+[γc+

∫ 1

0µ1(x)n(t,r,x)dx

]c1(t,r)=0,

−σc∆c2(t,r)+[γc+µ2

∫ 1

0n(t,r,x)dx

]c2(t,r) = 0,

with zero Neumann conditions at r = 0(spheroid centre) coming from radial symmetry

and Dirichlet boundary conditions at r = 1(spheroid rim):s(t, r = 1) = s1, ∂rs(t, r = 0) = 0,c1,2(t, r = 1) = C1,2(t), ∂rc1,2(t, r = 0) = 0,where:• The function p(x) is the intrinsic (i.e., in-dependently of cell death) proliferation rate ofcells expressing resistance level x due to theconsumption of resources. The factor

11 + µ2c2(t, r)

mimics the effects of cytostatic drugs, which actby slowing down cellular proliferation, ratherthan by killing cells. The parameter µ2 modelsthe average cell sensitivity to these drugs.• The function d(x) models the death rate ofcells with resistance level x due to the competi-tion for space and resources with the other cells.• The function µ1(x) denotes the destructionrate of cells due to the consumption of cytotoxicdrugs, whose effects are here summed up directlyon mortality.• Parameters σs and σc model, respectively,the diffusion constants of nutrients and cyto-toxic/cytostatic drugs.• Parameters γs and γc represent the decayrate of nutrients and cytotoxic/cytostatic drugs,respectively.

The model can be recast in the equivalentform

∂tn(t, r, x)=R(x, %(t, r), c1,2(t, r), s(t, r)

)n(t,r,x),

in order to highlight the role played by the netgrowth rate of cancer cells, which is describedbyR(x, %(t, r), c1,2(t, r), s(t, r)

):=

p(x)1 + µ2c2(t, r)s(t, r)− d(x)%(t, r)− µ1(x)c1(t, r).

The following considerations and hypotheses areassumed to hold:•With the aim of translating into mathematicalterms the idea that expressing cytotoxic re-sistant phenotype implies resource reallocation




(‘cost of resistance’, i.e., redistribution of ener-getic resources from proliferation-oriented taskstoward development and maintenance of drugresistance mechanism, such as higher expressionor activity of ABC transporters in individualcells), p is assumed to be decreasing

p(·) > 0, p′(·) < 0.

As regards function d, one can note that inthis study [59], the advocated modelling choice(d′(·)< 0) is the opposite of the one that wasmade in [58], a study nevertheless publishedby the same authors. In [58], the underlyingbiological reason is possibly that ‘the more re-sistant a cell is, the stronger opposition to itsproliferation it encounters in its own species,cancer or healthy, which is another way, coherentwith the modelling choice made on rC and rH ,to express a cost of resistance’ (see this argu-ment developed above in Subsection III-A). Asa matter of fact, the simulations shown in thisstudy [59] always use a constant value for d, and,contrary to [58], no theorem is proposed to thereader of [59], which should induce to actuallychoose d′(·) ≥ 0.• The effects of resistance to cytotoxic therapiesare modeled by the obvious condition that thedrug sensitivity function µ1 is non-increasing:

µ1(·) > 0, µ′1(·) ≤ 0.

The interesting results of this model consist ofsimulations, illustrated by figures to which theinterested reader is referred.

D. Models structured in cell-functional variablesA puzzling observation on an in-vitro ag-

gressive cancer cell culture (PC9, a variant ofNSCLC cells) exposed to high doses of anti-cancer drug, experiment reported in [88], is that:1) even though 99.7% of cells quickly die whenexposed to the drug, sparse and tiny subpopula-tions (0.3%) survive, named drug-tolerant per-sisters (DTPs), and for some time just survive,exposed to the same very high concentration ofdrug; 2) after some time (not precisely definedin the paper), these surviving cells change their

phenotype as expressed by membrane markers,and proliferate again, then named drug-tolerantexpanded persisters (DTEPs), unabashed in themaintained high drug dose; 3) when the drugis washed out from the cell culture, the cellpopulation reverts to initial drug sensitivity, andsuch resensitisation occurs ten times more slowlyat the DTEP stage than at the DTP stage; 4)if the cell culture is exposed to an inhibitor ofthe epigenetic enzyme KDM5A together withthe drug, be it at the DTP or DTEP stage,DTPs - or DTEPs - die. Such clearly epigeneticand completely reversible mode of resistance,developed in two stages, called for designing acell population dynamic model structured, notas previously, monotonically in drug resistancegene expression level, but in phenotypes linkedto the cell fate, which in cell populations al-ways may be reduced to proliferation, deathor differentiation (senescence being a version ofdelayed death). In the modelling and numericalstudy [13], 2 phenotypes are thus chosen to takeinto account the cell population heterogeneityrelevant for the experiment: survival potentialunder extreme environmental conditions (calledby ecology theoreticians viability), x, and pro-liferation potential (called fecundity), y. Theresulting model is described by the reaction-diffusion-advection equation, that describes thebehaviour of a very plastic cell population underexposure to a high dose of anticancer drug:

∂n

∂t(x, y, t) + ∂

∂y

(v(x, c(t); v)n(x, y, t)

)︸︷︷︸

Stress-induced adaptationof the proliferation level

=

β∆n(x,y,t)︸︷︷︸Non-genetic

phenotype instability

+[p(x,y,%(t))−d(x,c(t))

]n(x,y,t)︸︷︷︸

Non local Lotka-Volterra selection

• %(t)=∫ 1

0

∫ 1

0n(x, y, t) dx dy,

p(x, y, %(t))=(a1+a2y+a3(1−x))(

1− %(t)K

)and d(x, c) = c(b1 + b2(1− x)) + b3




• The global population term

%(t) =∫ 1

0

∫ 1

0n(x, y, t) dx dy

occurs in p as a logistic environment lim-iting term (availability of space and nutri-ents).

• The drift term w.r.t. proliferation po-tential y represents possible (if v 6=0) ‘Lamarckian-like’, epigenetic and re-versible, adaptation from PC9s to DTPs;switching from v ≥ 0 to v = 0 heremeans switching from a possible adaptationscenario to a strictly Darwinian one (it isbiologically impossible to decide betweenthe two scenarios).

• v(x, c(t); v) = −vc(t)H(x∗ − x) where t 7→c(t) is the drug infusion function, and x∗ isa fixed viability threshold.

• No-flux boundary conditions.Of note, another, individual-based, model

(IBM) yielding the same simulation results (notheorem) is proposed in a complementary wayto the interested reader, sent to [13].

The simulation results firstly show totalreversibility to drug sensitivity when the drugis withdrawn, and also allow to study theevolution of the two phenotypes in the absenceof drug, under drug exposure, and when thedrug is withdrawn. Furthermore, the model wasput at stake by asking 3 questions:

Q1. Is non-genetic instability (Laplacian term)crucial for the emergence of DTEPs?

Q2. What can we expect if the drug dose islow?

Q3. Could genetic mutations, i.e., an integralterm involving a kernel with smallsupport, to replace both adapted drift(advection) and non-genetic instability(diffusion), yield similar dynamics?

Consider c(·) = constant and two scenarios:

(i) (‘Lamarckian’ scenario (A): the outlaw)Only PC9s initially, adaptation present(v 6= 0)

(ii) (‘Darwinian’ scenario (B): the dogma)PC9s and few DTPs initially, no adaptation(v = 0)

To make a long story short [13],

• Q1. Always yes! Whatever the scenario.

• Q2. Low doses result in DTEPs, but noDTPs.

• Q3. Never! Whatever the scenario.

Can such cell-functional models be used toactually manage drug resistance in the clinic?An idea would be to counter the plastic adap-tation that cancer cell populations show in thepresence of high doses of drugs by infusing atthe same time as cytotoxic drugs inhibitors ofepigenetic enzymes such as KDM5A in [88].However, even though epigenetic drugs are theobject of active research in the pharmaceutic in-dustry [89], the importance of epigenetic controlof physiological processes (all differentiation isepigenetic!) and the role of impaired epigeneticcontrols in impaired cell differentiation, which isa characteristic of cancer, has been stressed [31]makes them delicate to manage in the clinic sofar.

IV. Optimisation and optimal controlA. ODE models and their optimal control incancer

To go beyond the administration of constantdoses, one is led to let drug infusion rates varyin time and try to find the best such rates tominimise a given criterion, such as the numberof cancer cells at the end of a given time-window.This is the purpose of the mathematical field ofoptimisation (see, e.g., in another framework [5],[20]) and optimal control, with all its availabletheoretical and numerical tools.




At this stage, it is noteworthy that the dis-cretisation of the phenotype-structured PDEmodels introduced so far leads to ODEs, usuallyof Lotka-Volterra type. To illustrate the idea, letus go back to the prototype integro-differentialmodel

∂n

∂t(t, x) =

(r(x)− d(x)ρ(t)

)n(t, x).

If one discretises the phenotype space intoNx + 1 equidistant phenotypes through xi =i∆x, ∆x = 1

Nxthe above equation is approxi-

mated by the ODE system

y′

i(t) =(r(xi)− d(xi)ρ(t)

)yi(t), i = 0, . . . , N

where yi(t) ≈ ∆xn(t, xi) and ρ(t) =∑Nj=0 yi(t).

This remark is general and applies to thenumerical simulation of phenotype-structuredPDE models (this is nothing but a semi-discretisation of the corresponding PDE). Thispoint of view also makes the link betweenODE models where resistance is representedby a binary variable, or more generally, a dis-crete variable. With a coarser discretisation,the ODE model has few equations and is moreamenable to parameter identification, quick nu-merical simulation, but is also less accurate inrepresenting resistance.

When it comes to optimal control, ODE mod-els with a discrete representation of resistancehave long been studied, either theoretically ornumerically [91], [22], [50]. This is one aspect ofthe rich literature on optimal control for cancermodelling, see the reference book [86]. Note thatthese ODE models can be made richer, as theymay additionally model healthy cells, cells indifferent compartments of the cell cycle, immunecells, etc.

Independently of the number of equations,the investigation of the optimal control problemtypically leads to optimal strategies being theconcatenation of bang-bang and singular arcs.Bang-bang arcs correspond to drugs being ei-ther given at the maximum tolerated dose ornot at all, whereas singular arcs correspond tointermediate doses which can be computed in

feedback form from the yi’s. These results areobtained either numerically, or theoretically byapplying the Pontryagin Maximum Principle,possibly with higher order criteria (such as theLegendre-Clebsch criterion) and/or with geo-metric optimal control techniques.

The usual clinical practice is to use maximumtolerated doses, a strategy which has been calledinto question as it can lead to an initial dropin tumour size before regrowth due to acquiredresistance [25]. This corresponds to bang-bangcontrols. Instead, alternative and more recentstrategies advocate for the infusion of interme-diate doses [35], [73], [93].

Thus, as explained in [53], understandingwhether the optimal controls do contain sin-gular arcs is of paramount importance, andmight depend from the parameters governingthe cost. The recent work [12], where resistanceis modelled to be binary, also features parameterregions leading to singular arcs which follow afirst arc with maximum tolerated doses.

This naturally poses the question of opti-mal scheduling for PDE models of resistance.The corresponding optimal control problems arethen significantly harder to solve. Numerically,this is because a fine discretisation leads tocomputationally demanding algorithms. Theo-retically, as is already the case for a high dimen-sional ODE, it becomes more difficult to obtainprecise results on the optimal control strategy,even from an (infinite-dimensional) PontryaginMaximum Principle.

B. Optimal control of phenotype-structured PDEmodels

These difficulties might explain why thereare up to date few optimal control results onphenotype-structured PDEs for resistance. Moststudies are restricted to constant doses and theoptimisation is then performed on the resultingscalar parameters. This can be done by nu-merical investigation of the parameter space asin [16], [58] or theoretically for cases in whichexplicit solutions are available [3]. Other non-constant infusion strategies mimicking popular




protocols are sometimes also tested in the afore-mentioned works. Finally, in [3], restriction toGaussian solutions allows the authors to reducethe PDE to a system of three ODEs (for thetotal mass, the mean and the standard devia-tion). All these studies conclude that the con-tinuous administration of maximum tolerateddoses might lead to relapse, and that alternativestrategies with lower infusion of drugs might bepreferable.

There is, up to our knowledge, very littlework in the direction of tackling a full optimalcontrol problem for the phenotype-structuredPDEs models, without such simplifications asabove. The two works we are aware of are [72]and [81], both concerned with the model (1)(see Section III-A). In [72], the model is morecomplex since genetic instability is introduced inthe PDE, modelled by diffusion terms. The goalis to minimise the total number of cancer cellsρC(T ), and the overall model is complementedwith the constraints• maximum tolerated doses:

0 ≤ u1(t) ≤ umax1 , 0 ≤ u2(t) ≤ umax2 ,

• control of the tumour size:ρH(t)

ρH(t) + ρC(t) ≥ θHC , (2)

• control of the toxic side-effects:

ρH(t) ≥ θHρH(0), (3)

where 0 < θHC , θH < 1.In order to solve the problem numerically,

the approach consists in discretising the wholeproblem in phenotype and time, thus using aso-called direct method in numerical optimalcontrol [96]. This is equivalent to discretising intime an ODE system which has as many equa-tions as there are discretised phenotypes. Theoptimal control problem then becomes a highfinite-dimensional optimisation problem, whichcan be handled, for example, by interior pointmethods.

As is common to most numerical optimisationproblems, the biggest difficulty lies in choosing

the initial guess for the algorithm. The approachof [81] is to solve the optimisation problemwith a very coarse discretisation (few unknowns)before scaling the problem up progressively toa fine discretisation. For the generalised modelwith mutations, such a strategy fails because ofthe computational cost of Laplacians.

To circumvent this, the numerical strategyintroduced in [72] is to simplify the PDEs bysetting some coefficients to zero, so that theresulting optimal control problem can be solvedby a Pontryagin Maximum Principle. Althoughthis problem is non-realistic from the applica-tive a point of view, it provides an excellentstarting point for a homotopy procedure whichallows to go all the way back to the originalmore complicated problem, with a very accuratediscretisation.

An optimal strategy clearly emerges fromthese two works, when the initial tumour isheterogeneous (as a result of a first standardadministration of cytotoxic drugs). The ideais to let the tumour density evolve to a sen-sitive phenotype by using no cytotoxic drugsand intermediate (constant) doses of cytostaticdrugs for a long phase, during which the con-straint on tumour size saturates. Only then onetakes profit of a sensitive tumour by using themaximum tolerated doses, up until the side-effects constraint saturates. It is then possibleto further reduce the tumour size by loweringthe cytotoxic dosage.

The asymptotic analysis comes in handy inunderstanding the optimality of such a strategy:the first long phase leads to the convergence ofthe cancer cell density onto a Dirac mass locatedon a sensitive phenotype (or a smoothed versionof such a Dirac mass when there is a diffusionterm). This property allows the authors of [81]to perform a theoretical study of the optimalcontrol problem in a reduced control set wherethe controls are forced to take constant valuesduring a first long phase. The strategy obtainednumerically is then proved to be optimal in atheorem, informally given below.




Theorem 3. [81]When the final time T is large, the optimalsolution is such that1) at the end of the first phase, the densityof cancer cells has concentrated on a sensitivephenotype,2) the optimal strategy is then the concatenationof three arcs• an arc with saturation of the constraint on

ρH

ρH+ρC.

• a free arc with maximum tolerated doses,namely u1 = umax1 and u2 = umax2 ,

• an arc with saturation of the constraint onρH and u2 = umax2 .

We insist that this result is proved only inthe absence of diffusion. The proof relies on thefact that Dirac mass concentration at the endof the first phase allows to replace the PDEsystem by an 2x2 ODE system, up to an errorbecoming arbitrarily small as the length of thephase increases. The resulting optimal controlproblem can then be handled with a PontryaginMaximum Principle with state constraints.

C. Future prospects in optimal controlApplying the strategy advocated in [72], [81]

requires thinking it in a quasi-periodic manner,and as a strategy relevant after the traditionaladmnistration of the first dose, which usuallyinduces resistance. The idea would then be toalternate between:1. a long phase with cytostatic doses and no

cytotoxic doses (a drug holiday) to resensi-tise the tumour,

2. a short phase with maximum tolerateddoses until the toxicity is considered to havereached its limit, with a possible subsequentswitch in dose for the cytotoxic drugs tokeep diminishing the tumour size.

Such a protocol requires to assess the level ofresistance in order to decide when to switch from1. to 2. and determine when damage to healthytissue justifies switching back to 1. A majordifficulty is of course the scarce availability ofbiological markers, which critically depends on

each particular cancer. For instance, in prostatecancer, a regrowth of the plasmatic level of PSA,routinely available to clinical measures for quitea long time, after some stagnation time undertreatment may indicate the emergence of resis-tance. In the same way, for colorectal cancer,it has been advocated that circulating tumoralDNA detection may be used for clinical manage-ment [54], and the same is true of circulating tu-mour cells [10]; however these techniques are farfrom being clinical routine. As regards damagesto healthy tissue, they are numerous (e.g., for5-FU and other cytotoxic drugs, classical hand-foot syndrome, mouth sores, neutropenia, thatoften lead to treatment interruption), dependingon each molecule and most of all on the evalu-ation of their severity by the oncologist in theclinic, given the health status of the patient un-der treatment (in the case of laboratory animals,weight loss is a common indicator of toxicity).

Taking advantage of the models introducedin Section III, there are several directions foranalysing such types of optimal control but ina slighly different or generalised setting. Thiswould both test the robustness of the strategypresented above, and possibly lead to alternativeones depending on the context.

The addition of an advection term wouldhelp modulate the speed at which emergenceand resensitisation occur. Modelling how suchterms would depend (or not) on a given drugis already an issue. However, it is likely thatthe addition of such a term will not jeopardisethe numerical computation of optimal controlswith direct methods refined with homotopies.Of course, considering higher-dimensional phe-notype variables or adding a space variable willinevitably lead to an explosion in complexity.

This is why the numerical optimal controlof phenotype-structured PDEs will benefit fromstate-of-the-art methods in that field, which inturn highly rests on the quality of optimisa-tion solvers. In other words, expert numericalmethods will undoubtedly be at the core ofany attempt at solving these complex infinite-




dimensional problems.The theoretical aspects are more exploratory.

Even for the integro-differential system of [81],the optimal control had to be solved in a re-stricted class of controls, and a complete under-standing of the interplay between concentrationphenomena and optimal control is yet to emerge.With few tools available for the control and op-timal control of non-local PDEs (an active areaof research), a theoretical analysis of optimalcontrols of PDE structured models is at thisstage a real hurdle.

V. Open and challenging questions

A. Conflicting phenotypes and multicellularityThe question of emergence of drug-resistant

clones in a possibly totally genetically homoge-neous cancer cell population (as is likely the caseof the observations reported in [76], [77], [88])under environmental pressure, here drug expo-sure, is related to the emergence of multicellu-larity in unicellular organisms. This question hasbeen the object of many studies by evolutionarybiologists [1], [63], [65], [66], [67], and theyhypothesise that, confronted with a challeng-ing, possibly deadly, environmental pressure, analready existing, without specialisation, multi-cellular aggregate (this had to occur after thebeginning of massive oxygenation of the oceanand atmosphere, about one billion years ago, as,to stick together, cells need some glue of collagenfamily, which is synthesised only in the presenceof free oxygen [94], [95]) had to specialise tosurvive. The proposed paradigmatic scenario isin [63], [65] the conflict between proliferation - orfecundity, with adhesivity, to maintain againstpredators a colony of replicating cells on a goodenvironmental trophic niche - and motility - tomake the aggregate able to change its location,to leave for a more favourable one when re-sources are become scarce or when predatorsare threatening. The solution of such conflictis found in specialisation in two phenotypes,later to be refined, likely by bifurcations inmore than two if the environmental pressure is

diversified. This question has been tackled byYannick Viossat together with Richard Michodin a simple setting [67], from which one findsthat according to the convexity or concavity ofa level set on which an optimum of fitness isto be found, there may be coexistence of twophenotypes or predominance of a single one.Such a situation is encountered in an adaptivedynamics framework in [70] (see Section III-B)for the possible invasion, or coexistence withhealthy cells, of a leukaemic cell clone. However,how to model such specialisation and cooper-ativity in general, or in the particular case ofviability vs. fecundity for cancer cells under drugpressure, still escapes our efforts.

B. Coherence in an organism and its controlThe question of coherence - and of within-

and between-tissue cohesion - of a whole mul-ticellular organism with so many diverse andspecialised subpopulations is seldom posed, andexcept in [78], it is generally ignored. MatejPlankar and co-authors ask the main questionthat is so often dodged when speaking of canceras a developmental disorder: ‘What exactly isdisorganised that was previously organised?’ andthey propose that the biophysical base of suchcoherence resides in the coherence, in the phys-ical sense, of oscillatory signals between cellsthat might be of electromagnetic or quantumnature, transporting energy and information,that could originate from, or be transmitted by,microtubules working like antennas, to - likelytoo vaguely and unfaithfully - sum up theirhypotheses. How are such signals synchronised?Is there a forcing signal originating from anorganisational centre, or is it based on some sortof multilevel system of phase-locked loops?

A possible candidate for such an organisingsystem is the circadian system, that is made ofcircadian clocks [84], [87], existing in all nucle-ated cells (and even, so it seems, in red bloodcells by different mechanisms), and that consistof oscillators based on gene networks that existin all, at least, animal cells (but have also beenindividuated in some plants). Such oscillators




have firstly been evidenced in fruitflies [51], andlater in all mammals [105] where they have beensearched for. They have the general property tobe daily reset by the sun (or by routine socialactivities when the sun does not shine its rays),and they date back to a very ancient past of ourplanet. There exists a central control centre, thecircadian pacemaker located in the suprachias-matic nuclei of the hypothalamus in mammals,that receives synchronising electric signals fromexternal light via the retinohypothalamic tractand physiologically sends synchronising mes-sages to all peripheral cells via hormones and theautonomic nervous system. The activity of thecentral circadian pacemaker, that is in particularreflected by body temperature oscillations andby oscillations of corticosteroids in the surrenalgland, is known to be disrupted in cancer [85],and this all the more so as cancer is moreadvanced. Although the authors of [78] do notmention this synchronising system, it is coherentwith their view. Is the circadian the synchronis-ing system? or is it a dubbing system, underthe dependence of an electromagnetic or quan-tum signalling system advocated in [78]? Morehidden than the circadian system, could someorganising coded plan, progressively establishedtogether with the immune system in the devel-opment of multicellularity, be the glue and con-trol on which all Metazoan between- and within-tissue coherence relies? In other words, couldthere exist a set of genes, already present in earlyMetazoa, likely related to epigenetic control,that would on the one hand define, in a MHC(major histocompatibility complex)-like way, inits fixed part a species and, at the individuallevel, an individual within a species, and on theother hand a variable part within a species thatwould give rise to the different cell phenotypesthat make a coherent multicellular organism(in limited number, 200 to 400 cell types orso, from enterocytes to neurons in a Human).Would this be the case, then one could imaginethat tumours - as Metazoa 1.0, according tothe atavistic hypothesis of cancer - might have

developed a sort of primitive, failed, immune re-sponse system whose main failure and differencewith respect to the host normal immune systemwould be a strong tolerance to plasticity, i.e.,to lack of belonging to a well-differentiated cellclass. In the metaphoric Waddingtonian view,this would imply an ablated, flattened epigeneticlandscape, with plenty of room for dediffer-entiation and transdifferentiation between cellphenotypes. Could such flattened Waddington’sepigenetic barriers in return be interpreted interms of undecided, empty, spins borne on cellantigens that should normally, to avoid recogni-tion as foe by antigen presenting cells, be codedas either 1 (differentiated) or 0 (open to furtherdifferentiation in a well-defined cell fate), butnot blank? Some support to these speculationsmay be found in a study dedicated to the originof the Metazoan immune system [68].

C. Intra-tumour cooperativity, plasticity, bethedging

If tumours, as Metazoa 1.0, have developedsome internal cooperativity that makes themable to survive as a whole to cytotoxic stressand to friend-or-foe recognition by the immunesystem, what does such cooperativity consist of?Experimental evidence exists that such cooper-ativity exists [21], [79], [92] and is necessary fora tumour to thrive, while some studies focuson evidencing tumour genetic or only pheno-typic heterogeneity [61], [62] without necessarilyproposing a rationale for such heterogeneity.However, as pointed by Mark Vincent, “Hetero-geneity, even though it might in some superficialway, ‘explain’ differential drug sensitivity, is notin itself an explanation of cancer; rather, itis the heterogeneity itself that requires explana-tion.” [104]. Indeed, focusing only on drug resis-tance, we might satisfy ourselves with the plainobservation of tumour heterogeneity to explainthe variety of drug resistance mechanisms andwhy it is so uneasy to eradicate cancer. But try-ing to understand its determinants is more diffi-cult. Leaving aside the obvious fact that spatialisolation of cells inside a tumour may lead under




different forms of environmental pressure to dif-ferent (phenotypically or genetically) clones thatmay have nothing to do with cooperativity, wemay wonder why different phenotypes may befound in the same spatial niche. To begin with,does cooperativity exist with a fixed repartitionof phenotypes inducing some division of labourin a tumour, or is it not a transient state thatis observed only when a cancer cell populationis put at stake under cellular stress?... Or inartificial lab conditions [21], [79], [92]?

Can we consider that no actual cooperativity,in the sense of division of labour in an integratedstructure, exists within a cancer cell population,all the more so as cell differentiations are im-paired in cancer, but that plasticity of cancercells (and not only of cancer populations) isso high - within a preestablished Metazoa 1.0plan, i.e., it is not infinite, but takes advan-tage of many, but finitely many, stress responsemechanisms inscribed in their genome and easilyreactivatable - that tumours can react to deadlyinsults by different resistance mechanisms, thesimplest one being enhanced proliferation outof control, making them winners in all (knownto their genome) cases? The sole idea of cancercooperativity should be examined with care, ifone admits that cancer cells are fundamentallycheaters, or otherwise said, defectors in theevolutionary game of multicellularity [2]. How-ever, primitive Metazoa, such as sponges [68]or algae show some cooperativity, in particularas regards immunity to invasion by pathogens.Have successful tumours regressed in evolutionat an earlier stage than sponges? Likely yes, asmulticellularity in sponges is well controlled.

Following the theme of cancer cell plastic-ity, an interesting notion has recently emerged,the so-called bet hedging fail-safe strategy ofcancer cell populations [9], [41]. According tothis hypothesis, cancer cells - or cancer cellpopulations - are so plastic that they can adapttheir phenotypes to sustain different insults in-volving critical cell stress by developing differ-ent adapted subpopulations. It has also been

observed that some cancer cells may expressvery ancient genes (so-called ‘cold genes’, i.e.,that are conserved throughout evolution, beingprotected from evolution due to their essentialrole in facing unpredictable, but already met in aremote past of evolution, deadly insults) in caseof exposure to chemotherapies [107]. One couldspeculate that some sentinel cells, expressingthese ‘cold genes’, might send various resistancemessages to other cells, or that they could them-selves, being extremely plastic, differentiate intodiverse categories of cell subpopulations, eachone developing one of the resistance mechanismselaborated in the course of evolution from aprotozoan state, and then sheer darwinian se-lection would prevail. Whether cells themselvesare plastic and can adapt in a sort of Lamar-ckian (necessarily epigenetic) way or only cellpopulations are plastic, constituted by preexist-ing (prior to any insult) genetically well-definedsubpopulations is not easy to decide, and in [13],both scenarios were challenged by a reaction-diffusion-advection model (see Section III-D).However, the very fact that cell differentiationsare always - to some extent - impaired in can-cer cells, the fact that inhibitors of epigeneticenzymes have been shown in some cases to an-nihilate drug resistance in very aggressive formsof cancer (NSCLC cells in culture in [88]) induceus to propose plasticity as a distinctive andcontinuous trait of cancer cells. With respect tothe class of cell-functional models proposed inSection III-D, i.e., structured in the conflictingcontinuous traits named viability (x, potential ofsurvival in extreme conditions, opposed to apop-tosis) and fecundity (y, proliferation potential),one could then add a plasticity trait (θ, opposedto differentiation), characterising, together withthe first two, each cell in its relevant variabilityin a heterogeneous cancer cell population (whichwould not give an explanation of such hetero-geneity, but might help understand its evolutionunder cellular stress).

A general class of cell population adaptivedynamics models that would be structured in




(x, y, θ) could then, following an idea popu-larised in [6] for the so-called cane toad equation(that describes the invasion of cane toads inAustralia by using an equation than cannot beof the classical reaction-diffusion type yieldingtravelling waves), be described by

nt +∇ · {V (x, y, θ,D)n}

= α(θ)nxx + β(θ)nyy + γ(θ)nθθ

+n{r(x, y, θ)− µ(x, y, θ,D)− ρ(t)

C(x, y)

},

where r(x, y, θ) is the intrinsic (i.e., in theabsence of any limitation) growth rate of thepopulation, µ is an added death term due to thedrug dose D, condition C(x, y) ≤ K representsan environmental constraint, V an optional ad-vection function standing for abrupt modifica-tions of the environment (such as the major cellstress-inducing delivery of high doses of drugsas in [13]) and

ρ(t) =∫

[0,1]3n(x, y, θ, t) dx dy dθ

is the total cell population at time t, put asusual in Lotka-Volterra settings in a logisticposition to represent competition, e.g., for nu-trients, hence growth limitation, within the cellpopulation.

How such class of models might lead to repre-sent the emergence of different cell subpopula-tions under environmental pressure is still workunderway.

VI. Conclusion

In this review of models of adaptive dynam-ics dedicated to represent, analyse and controldrug-induced drug resistance in cancer, we havefirstly proposed a brief description of the biolog-ical background of cancer evolution, describingin particular the atavistic hypothesis of cancer,which to our meaning illuminates the scenery ofthe cancer disease, with more and more observa-tion facts to support it. Then (neglecting clas-sical compartmental ODE models that cannot

claim to represent adaptive phenomena), we pre-sented different cell population dynamics mod-els that belong to the mathematical categoryof adaptive dynamics, i.e., integro-differentialor partial differential equations structured bycontinuous traits describing the heterogeneityof cancer cell populations and their evolutionunder drug exposure. In a third part, we showedhow optimal control methods can be applied tosuch equations of adaptive dynamics and usedto design theoretical optimal therapeutic controlstrategies. Such strategies, even though methodsfor the identification of the parameters andfunctions of the models still remain to be found,are amenable to predict qualitative behaviour ofcancer cell populations under optimised time-scheduled drug exposure. Finally, we presentedsome challenging questions, addressed to evolu-tionary biologists and ecologists of cancer, to on-cologists, and to mathematicians to accuratelyrepresent, analyse and control the behaviour ofcancer cell populations.

Acknowledgments

The authors are gratefully indebted to-wards their colleagues Luis Almeida, RebeccaChisholm, Tommaso Lorenzi, Alexander Lorz,Benoît Perthame, and Emmanuel Trélat, co-authors with them of the mathematical pa-pers on structured cell population models ofevolution towards drug-induced drug resistance,whose main results have been presented in thisreview. C.P. acknowledges support from theSwedish Foundation of Strategic Research grantAM13-004.

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