É D Fr

Billy Érique
email: frederique.billy@inria.fr

Jean And

Clairambault
email: jean.clairambault@inria.fr

DESIGNING PROLIFERATING CELL POPULATION MODELS WITH FUNCTIONAL TARGETS FOR CONTROL BY ANTI-CANCER DRUGS

Keywords: 2000 Mathematics Subject Classification. Primary: 92-02; Secondary: 92D25 92C50 35Q80 Cell population dynamics, Proliferation, Cell division cycle, Adaptive dynamics, Cancer drugs, Control targets

We review the main types of mathematical models that have been designed to represent and predict the evolution of a cell population under the action of anti-cancer drugs that are in use in the clinic, with effects on healthy and cancer tissue growth, which from a cell functional point of view are classically divided between "proliferation, death and differentiation". We focus here on the choices of the drug targets in these models, aiming at showing that they must be linked in each case to a given therapeutic application. We recall some analytical results that have been obtained in using models of proliferation in cell populations with control in recent years. We present some simulations performed when no theoretical result is available and we state some open problems. In view of clinical applications, we propose possible ways to design optimal therapeutic strategies by using combinations of drugs, cytotoxic, cytostatic, or redifferentiating agents, depending on the type of cancer considered, acting on different targets at the level of cell populations.

1. Introduction. Mathematical models have been called for some time already by cancer biologists and clinical oncologists to help improve the efficacy of anti-cancer treatments. Indeed, understanding better the evolution of cancers and how to treat them in an optimal way is still an open question, that might benefit from contributions of mathematics to represent cell proliferation control by drugs.

Knowing that most anti-cancer treatments use combinations of drugs with different molecular targets and different functional effects on proliferating cell populations, we advocate considering these effects not just as on inhibition of a global 'birth minus death' rate, but rather with a refined point of view, considering multiple targets, representing differential drug effects on birth, death or differentiation, by different control targets in mathematical models. The closer these model representations are to actual clinical questions, the better.

The main two pitfalls of clinical oncology, that limit increasing drug doses, are unwanted toxic side effects on healthy cell populations and occurrence of resistance to drugs in cancer cell populations. According to the medical questions at stake, one may consider different models to represent the underlying biological phenomena that are the object of control by drugs. Spatial representations may be partly helpful, in particular when tumour neo-angiogenesis and anti-angiogenic drugs are present. However, in as much as drug effects are the tools of control considered here, and since they act mostly by modifying the physiology of cells, physiologically structured models should always be used, with or without added spatial structure. In particular, focus will be set here on models structured 2 FR ÉD ÉRIQUE BILLY AND JEAN CLAIRAMBAULT according to age in the cell cycle, or according to an evolutionary phenotype if evolution towards drug resistance is the main issue.

The paper is organised in the following way: firstly, we review the main difficulties inherent to cell population modelling for drug delivery control, and categories of drugs used in the clinic, from a pharmacological point of view. We then present in an abstract way, but with concrete instances, how anti-cancer drug targets may actually be represented in cell proliferation dynamic models. Finally, we briefly review the main types of models lately used in the scientific literature on drug control in healthy and cancer cell populations, sketching some results recently obtained, either theorems, or hints to possible future results coming from mere simulations, when proper mathematical analyses still seem out of reach.

2. Anticancer drug effects and their representation in mathematical models. The types of models used to predict cell population behaviour under control by drugs range from cellular automata to ordinary and partial differential equations (ODEs and PDEs), PDEs being amenable to transformation into delay differential equations (DDEs) by integration of PDEs along characteristics [START_REF] Foley | Dynamic hematological disease: a review[END_REF][START_REF] Perthame | Transport Equations in Biology[END_REF]. Biological variability (physiological differences between cells) is easily taken into account by stochastic models, and so-called individual-based models (IBMs) are amenable to include any kind of rule one puts in the individual agents. These models are hardly amenable to mathematical analysis, in particular to the study of their asymptotics, given the intrinsically finite number of cells they take into account, but their simulation may give hints to possible properties of their behaviour, that need to be further explored by mathematical analysis [START_REF] Byrne | Individual-based and continuum models of growing cell populations: a comparison[END_REF]. As regards physiologically structured PDEs, their structure variables (e.g., age in cell cycle models) electively represent biological variabilities considered as the most relevant for the question at stake, and their asymptotics can be studied when the model is tractable, sometimes resulting in theorems -which is hardly possible with agent-based models.

A model, what for?

Guidelines to design a model for an application. It should be stressed that designing cell population models under the action of drugs puts from the beginning modellers in a perspective completely different from the one used to represent the "natural history" of tumour growth under the influence of mechanical or physiological factors, but without built-in drug control. A model for anticancer therapeutics should be thought of toward a precise aim, with the idea to control by a given means the biological system under study (the growth of a cell population, or of different cell populations), or to analyse a precise aspect of it. Moreover, given the complexity of biological phenomena underlying tissue growth, i.e., proliferation of histologically homogeneous cell populations (notwithstanding some biological variability between individual cells), choosing well delineated questions of therapeutics arising in the clinic of cancers as a source of inspiration helps designing practical models adapted to represent actual treatments, with the aim to answer questions (about prediction of unwanted toxic side effects, drug resistance, and optimal combinations of drugs) asked by clinicians.

Optimisation of drug delivery is always a concern for clinicians. This for mathematicians implies defining an objective function, usually the number or density of cancer cells to be minimised, under constraints that may be limitation of toxicity to healthy cells or avoidance of the thriving of a drug-resistant subpopulation in cancer cells, or both. Oncologists seldom use only one drug, but rather, combinations of drugs acting on different molecular targets with the aim to potentiate their effects. If one wants to accurately represent and study such combinations, it is necessary to design models with built-in specific targets for the drugs in use [START_REF] Clairambault | Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments[END_REF].

Anticancer drugs are most often known for their molecular effects, i.e., effects that involve blockade or re-establishment of an intracellular signalling pathway, for instance targeting a specific enzyme or a chain of molecular reactions starting with a membrane or cytosol receptor. Nevertheless, such effects are not always specific, and even when a drug has been designed to block a given pathway by conformational analysis of known molecules, unexpected other effects may be unravelled by preclinical and clinical trials, and these effects may be deleterious on other pathways or to other cells. Furthermore, anticancer drugs show their therapeutic effects in a measurable way only at the cell population level by actual cell functional effects, i.e., on cell death, birth or differentiation.

This points out the multiscale nature of drug delivery problems: drugs are given at the whole body level (collection of communicating cell populations), exert their effects at the single cell level, but these effects, as far proliferation is concerned, are actually measurable at the cell population level only [START_REF] Clairambault | Optimising cancer pharmacotherapeutics using mathematical modelling and a systems biology approach[END_REF]. The most relevant level to describe proliferation and its control by drugs is clearly the cell population level, but the single cell (molecular, at which drugs chemically act) and whole body (at which drugs at delivered) levels must also be considered, at least if one aims at designing models for practical applications, i.e., for therapeutic optimisation. Drug effects must be different in cancer and in healthy cell populations if a therapeutic benefit is searched for, which stresses the fact that identical functional targets should be represented as behaving differently in cancer and in healthy tissues.

Note that we will consider here only proliferating cell populations. In an extended representation including applications to fast invading tumours (e.g., gliomas) and their treatments, a fourth fate for tumours beyond cell proliferation, death and differentiation that should be considered is cell motion, which ("go or grow" alternative) is, in a given time lapse, incompatible with proliferation [START_REF] Chauvière | A model of cell migration within the extracellular matrix based on a phenotypic switching mechanism[END_REF].

2.2.

A gap between molecular and functional targets. Anticancer drugs that are designed for their effects at the molecular level are most often known to block a specific intracellular signalling pathway (a chain of molecular reactions) that is assumed to end up by being involved in "apoptosis, proliferation, differentiation" all together, with seldom further precisions, because of the entanglement of the various molecular pathways participating in these physiological cell functions. Molecules designed at the single cell level thus need to be further studied at upper levels, by cell population dynamic studies in cell cultures and in experimental animal models, to investigate in a differential way all these physiological functions, plus drug resistance in diseased cell populations.

Otherwise said, in the same way as silencing a gene may have effects on other phenotypes than the targeted one, blocking a molecular pathway has seldom specific effects on a functional target in a cell population, hence the interest of investigating combinations of drugs by their functional more than by their molecular effects, and at the cell population rather than at the molecular level, from an experimental point of view in cell cultures. Such process obviously ought to be completed by further studies using whole body animal experiments, to unravel possible unexpected toxic side effects on other cell populations, possibly resulting in prohibitive toxicities to healthy tissues.

The gap to bridge between these three levels of observation (single cell, cell population, whole body) of this multiscale perspective is an intrinsic difficulty for which no miracle solution is known in general. Specific ways to integrate them depend on the type of model used at the cell population level, which is central in proliferation. A common suggestion [START_REF] Clairambault | Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments[END_REF][START_REF] Clairambault | Optimising cancer pharmacotherapeutics using mathematical modelling and a systems biology approach[END_REF] is to use compartmental ODEs for pharmacology (pharmacokineticspharmacodynamics, PK-PD for short) of anti-cancer drugs, structured PDEs for cell populations in each compartment, and a simultaneous management of the cell populations attacked by drugs by optimisation algorithms at the whole body level.

2.3. Drugs used in cancer treatments considered at the single cell level. Tissue renewal, which at the cell population level is made of proliferation, cell death and differentiation, relies on physiological phenomena that are built of elementary molecular reaction chains, and it is on these cell biochemical reactions that anti-cancer drugs act. In particular, proliferation of cell populations relies at the single cell level on the cell division cycle [START_REF] Morgan | The Cell Cycle: Principles of Control[END_REF].

2.3.1. DNA damaging agents. Also known as alkylating agents, these drugs act by directly binding to the DNA and creating damages in it, such as double strand breaks, that are hard to repair by the cell. They are in principle not cell cycle phase-specific. However, the fact that DNA is less protected in S phase, when it is duplicated, makes them more active in this phase. Commonly used alkylating agents are for instance cisplatin, oxaliplatin and cyclophosphamide. DNA damaging agents are cytotoxic drugs (see below).

2.3.2.

Cell cycle phase-specific agents. S phase-specific drugs block DNA replication either by acting as substrate substitutes in metabolic reactions (antimetabolites) or by inhibiting enzymes of these metabolic reactions, or both. Such are 5-Fluorouracil, a substitute for normal uracil, that also acts by blocking the enzyme thymidilate synthase, and irinotecan, that blocks topoisomerase II, an enzyme that is essential to DNA replication. M phase specific drugs block mitosis either by destroying the mitotic spindle (spindle poisons, such as vinca alkaloids, e.g., vincristine) or preventing its dissociation, such as taxanes (e.g., docetaxel and taxotere). These drugs secondarily create damages to the DNA (and thus are cytotoxic) since it cannot be properly duplicated (S phase-specific drugs) or they produce cells that are unable to divide; in both cases, cells under attack are short-lived due to cell control mechanisms occurring at so-called checkpoints (see below Cell death).

Molecular targeted therapies.

Although in principle all drugs have molecular targets, that are defined as specifically as possible, this recently denominated category of drugs was firstly restricted to chemicals shown to very specifically re-establish the normal functioning of a molecular signalling pathway perturbed in cancer. One of the first used molecular targeted therapies was in 1986 all-transretinoic acid or ATRA, a molecule that corrects the normal granulocyte differentiation process blocked in acute promyelocytic leukaemia (APL, also known as type 3 acute myeloblastic leukaemia) by the chimeric protein PML-RARα, ATRA destroying this protein [START_REF] Haferlach | Molecular genetic pathways as therapeutic targets in acute myeloid leukemia[END_REF]. The term 'molecular targeted therapy' now includes mostly monoclonal antibodies (associations of a growth factor receptor antagonist with an antibody that is specific of the receptor, with a name usually ending in -mab) and tyrosine kinase inhibitors (TKIs, that directly block growth factor receptors in kinases at a tyrosine site, with a name usually ending in -nib). The best known success story in molecular targeted therapies is that of imatinib mesylate, that has completely transformed the prognosis of chronic myelogenous leukaemia (CML) [START_REF] Druker | Efficacy and safety of a specific inhibitor of the BCR-ABL tyrosine kinase in chronic myeloid leukemia[END_REF]. ATRA and imatinib actually cure most of APL (over 80%, by using it in combination with a cytotoxic drug) and CML (over 95%, in monotherapy) patients, respectively, likely because they are directed against well identified chimeric (abnormal) proteins, PML-RARα and BCR-Abl, respectively. However, in cases where other protein targets are normal but only abnormally overexpressed, molecular targeted therapies are not so effective, and sometimes highly toxic. Other recent molecular targeted therapies include histone deacetylase (HDAC) inhibitors, drugs that block the effects of HDACs, electively in cancer cells, since healthy cells seem to be more protected against their effects. Since these enzymes are involved in most cell processes (proliferation, cell differentiation and death), they may be molecule-specific, but not functionally specific, as are for instance imatinib on proliferation or ATRA on differentiation; however, tentative functional distinctions between them have been proposed [START_REF] Witt | HDAC family: What are the cancer relevant targets?[END_REF]. Molecular targeted therapies are usually cytostatic (see below) but may become cytotoxic at high doses.

2.4. Drug effects and targets at the cell population level. Indeed, since the aimed-at and clinically measurable effects of drugs are functional (on proliferation, cell differentiation or death) at the cell population level, it is at least as interesting to consider such drug effects as acting on functional terms in model equations representing tissue growth. As mentioned above, bridging the gap between the molecular and functional effects of anti-cancer drugs remains in general an open question. Another issue in the measurement of drug effects in cell populations comes from technological limitations due to the tools used to identify model parameters, that seldom can be all highlighted, unless the model is very simplified. Such identification relies in particular on flow cytometer measurements [START_REF] Ubezio | Unraveling the complexity of cell cycle effects of anticancer drugs in cell populations[END_REF][START_REF] Ubezio | Quantitative assessment of the complex dynamics of G1, S, and G2-M checkpoint activities[END_REF] and, more recently, on fluorescence-based methods investigating cell by cell a whole population [START_REF] Billy | Synchronisation and control of proliferation in cycling cell population models with age structure[END_REF][START_REF] Sakaue-Sawano | Visualizing spatiotemporal dynamics of multicellular cell-cycle progression[END_REF][START_REF] Sakaue-Sawano | Tracing the silhouette of individual cells in S/G2/M phases with fluorescence[END_REF].

2.4.1. Cytotoxics and cytostatics. Cytotoxic drugs are those that are aimed at killing cells -usually killing not only cancer cells -, sending them to death either by directly launching apoptosis (i.e., 'clean' cell death), or blocking them in an irreversible phase of the division cycle where long-term survival is impossible. On the contrary, cytostatic drugs may kill not even a single cell, being not directed at creating damages to the cells, but rather slow down the growth of the cell population as a whole, which may be experimentally evidenced by a lengthening of the cell population doubling time. It is known however that, depending on the drug dose, various cytostatics may become cytotoxic, and this has been represented in a model dealing with lapatinib [START_REF] Hinow | A mathematical model separates quantitatively the cytostatic and cytotoxic effects of a HER2 tyrosine kinase inhibitor[END_REF].

Cell death.

A direct enhancement of death rates by drugs is the simplest way to represent the effect of cytotoxics, and it is the only one available in the simplest ODE models where a "birth minus death" term is the only possible target in the equations (i.e., dn dt = r(n).n -d.n). Although apoptosis pathways, involving at the molecular level members of the Bcl2 family, have been explored with the aim to find targets for drugs that would be specific of apoptosis launching, it does not seem that specific proapoptotic drugs are already routinely available in the clinic. However, if one wants to oppose drug effects that are clearly cytotoxic to others that are only cytostatic, it is licit, provided that the model allows separate identifiability of cytotoxic and cytostatic effects, to represent cytotoxic effects on death rates, as opposed to cytostatic ones on proliferation rates. In age-structured models of the cell division cycle, where cell cycle phases are distinct, separated by transition rates between them, it is also possible, knowing that these transitions are under the control of protein p53 -"the guardian of the genome" -, itself triggered by DNA damage, to represent cytotoxic effects, rather than directly on death rates, by p53-mediated blockade of the cell cycle at these transitions (so-called checkpoints, mainly between G 1 and S, and between G 2 and M phases). We allude here at McKendrick-like PDE models for the density n i (t, x) ≥ 0 of cells with age x in cell cycle phase i = 1, . . . , I at time t, in which inputs of drug may be considered as impacting death terms d i (t, x) in phases or boundary terms, p53-controlled transition rates K i→i+1 (t, x) between phases in transport equations for cell division cycle phases

∂ ∂t n i (t, x) + ∂ ∂x {v i (x)n i (t, x)} + {d i (t, x) + K i→i+1 (t, x)}n i (t, x) = 0 (1)
with boundary terms

n i (t, x = 0) = ξ≥0 K i-1→i (t, ξ) n i-1 (t, ξ) dξ. (See below Eq. ( 5
)
for a more complete description.) But then one would have to describe in equations the actual sequencing of physiological mechanisms by which, as classically stated in biology tutorials, p53 arrests the cell cycle and launches DNA repair or apoptosis [START_REF] Vogelstein | Surfing the p53 network[END_REF], subsequently relating by complementary equations the d i to the K i→i+1 , i.e., how should cell death launch be represented when cells have spent "too much time" blocked at checkpoints? This is a task that to our knowledge has not been done thus far, essentially because of lacking physiological knowledge on the timing in this sequence of events. Likely, cell energetic considerations (e.g., on ATP consumption) could be helpful. Nevertheless, cell death at check points can be mimicked in a coarse way in numeric simulations by imposing an arbitrary maximum number of runs of the division cycle for a given cell until it passes to next phase.

Furthermore, enzymatic repair mechanisms (nucleotide excision repair, NER, after damage to the DNA) are important to consider since resistance to anti-cancer drugs may be due to their over-expression (and it is also the only way to explain resistance to radiotherapy). Thus they should also be included in a model of control of proliferation to complete the representation of this p53-related missing link between DNA damage induced by cytotoxic drugs and its consequences on death rates at the cell population level. This also remains to be done, to our knowledge.

2.4.3.

Proliferation terms: birth rate or ageing speed. Effects of cytostatic drugs, that slow down proliferation without destroying cells, may be represented in simple models by a decrease in a multiplicative growth factor affecting the cell population variable, e.g., by introducing a drug effect f in an equation of the form

dn dt = r(n).n 1 + f (t) -d.n, (2)
which obviously does not distinguish them, in this simple form, from additive effects on death rates. But they may be represented in different and richer ways in models of the cell division cycle, such as constituted of copies of the McKendrick equation presented above, and also below in a more detailed way, see Eq. ( 5), by an inhibiting action on the speed v i with which phase i (mostly G 1 or G 2 ) is scrolled through, or by introducing an inhibiting factor before the boundary terms ξ≥0 K i→i+1 (t, ξ) n i (t, ξ) dξ, both ways to mean a negative effect on the influence of growth factors, which usually is the result of the action of cytostatic drugs.

2.4.4.

Sending cells to quiescence. Another way to represent the effect of cytostatic drugs is to use age-structured models in which cell cycle phases are not necessarily detailed, keeping only one proliferative phase, but introducing exchanges of proliferating cells with a quiescent phase in which cells do not grow. One may thus represent cytostatic effects by a contrasted fate at mitosis, sending proliferating cells with density p(t, x) either back into the division cycle, or to a storage siding Q representing a quiescent phase, as in [START_REF] Gabriel | The contribution of age structure to cell population responses to targeted therapeutics[END_REF], depending on the cytostatic drug-controlled factor f (see below Eq. ( 6)). Such a model is still linear and thus amenable to asymptotic analysis by investigating its first eigenvalue, or Malthus exponent [START_REF] Perthame | Transport Equations in Biology[END_REF]. But it would be also possible to perform the same representation of drug effects on exchanges between proliferation and quiescence in non linear models involving furthermore feedback from quiescence onto proliferating cell populations, following models for stem cell populations as proposed by Mackey in a delay-differential form in 1978 [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF], and later by many others, in particular [START_REF] Brikci | Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle[END_REF][START_REF] Brikci | An age-and-cyclin-structured cell population model for healthy and tumoral tissues[END_REF][START_REF] Doumic | Analysis of a population model structured by the cells molecular content[END_REF] in a PDE form with both age and molecular structure, further studied with reduction to the only molecular (cyclin D) structure in [START_REF] Borges | Equilibria of a cyclin structured cell population model[END_REF][START_REF] Borges | Oscillations in a molecular structured cell population model[END_REF]. It is also possible in these models to linearise the system around stationary points (zero or infinite total cell population) to perform asymptotic analyses.

2.4.5. Differentiation terms. The same kind of models, applied to haematopoietic cell populations, for which cell differentiation is relatively well known and in which it is completely blocked at different maturation stages in acute myelobloblastic leukaemia (AML), in particular at the promyelocytic stage mentioned above about APL, has been studied by Adimy et al. They proposed a model with continuous age and discrete maturity structure [START_REF] Adimy | Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF], for which a stability analysis was performed and stability conditions involving both proliferation and differentiation, are given [START_REF] Ozbay | Stability analysis of cell dynamics in leukemia[END_REF][START_REF] Ozbay | Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics[END_REF]. In its delay-differential version (obtained by integration in age along characteristics [START_REF] Foley | Dynamic hematological disease: a review[END_REF][START_REF] Perthame | Transport Equations in Biology[END_REF]), the model describes at each maturation stage i the dynamics of both quiescent and proliferating cells, as in Mackey's models [START_REF] Foley | Dynamic hematological disease: a review[END_REF][START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF], see below 3.3.2.

Effects of anti-angiogenic drugs.

There are a lot of models dedicated to specifically represent the action of these anticancer agents, that do not act directly on the cancer cell populations themselves, but on their vascular environment. They will not be considered as such here, but in as much as their effects on cell populations are by limiting their proliferation, not by directly killing them, they may be considered as belonging to the class of cytostatic drugs. The representation of their effects depends on the prior choice of a model. Angiogenic drugs have been considered in particular in ODE models [START_REF] Hahnfeldt | Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy[END_REF][START_REF] Hahnfeldt | Minimizing long-term tumor burden: the logic for metronomic chemotherapeutic dosing and its antiangiogenic basis[END_REF] and in PDE models, physiologically structured or not [START_REF] Billy | A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy[END_REF][START_REF] Bresch | Computational modeling of solid tumor growth: the avascular stage[END_REF][START_REF] Frieboes | Prediction of drug response in breast cancer using integrative experimental/computational modeling[END_REF][START_REF] Ribba | A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents[END_REF][START_REF] Ribba | Chemotherapy may be delivered based on an integrated view of tumour dynamics[END_REF]. In these models, either they act by decreasing the "carrying capacity" of the tumour, or they choke progression in the cell cycle at the G 1 /S transition.

2.4.7. Other models. In a more abstract way, it is also possible to consider a very general model for the action of drugs on cell populations, without neither molecular nor functional targets, aiming at optimising the sequence of drug delivery times, as proposed in [START_REF] Agur | Optimizing chemotherapy scheduling using local search heuristics[END_REF]; such models lying not exactly within the scope of this study, which deals molecular and functional targets for anticancer drugs in proliferating cell populations, we limit ourselves to only mention this possibility.

3. Short review of cell population models with targets for drug effects. In this section, we present a brief review of various types of models that have been designed to investigate drug efficacy on cancer cell populations. A lot of work has been performed in this domain since the end of the 20th century, and we do not claim here to be exhaustive, but only sketch the scenery by choosing examples. More can be found in [START_REF] Billy | Optimisation of cancer drug treatments using cell population dynamics[END_REF], where the presentation is focused on drug delivery optimisation, and in the synoptic [START_REF] Swierniak | Mathematical modeling as a tool for planning anticancer therapy[END_REF] and in the references in these articles. We focus here on the representation of drug targets in these models.

Cellular Automata.

A popular way among physicists, chemists and biologists to represent the cell division cycle is to consider this cycle as a set of prescribed biological rules that govern cell evolution. In this way, cellular automata enable to describe individual cell evolution within a cell population and to investigate drug efficacy. Alarcón et al. [START_REF] Alarcón | A cellular automaton model for tumour growth in inhomogeneous environment[END_REF] used a cellular automaton model to represent tumour growth in a vascular environment, opening the way to the possible representation of anti-angiogenic therapies. Tumour growth at the vascular stage was further studied by means of an ODE model [START_REF] Alarcón | A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells[END_REF] and later on with the addition of a PDE model [START_REF] Alarcón | A multiple scale model for tumor growth[END_REF] (see below).

Altinok and Goldbeter developed a cellular automaton for the cell cycle [START_REF] Altinok | An automaton model for the cell cycle[END_REF][START_REF] Altinok | A cell cycle automaton model for probing circadian patterns of anticancer drug delivery[END_REF][START_REF] Altinok | Optimizing temporal patterns of anticancer drug delivery by simulations of a cell cycle automaton[END_REF][START_REF] Altinok | Identifying mechanisms of chronotolerance and chronoefficacy for the anticancer drugs 5-fluorouracil and oxaliplatin by computational modeling[END_REF]. Transition between two states of the automaton, that represent phases of the cell cycle, correspond to cell progression through, or exit from, the cell cycle, and are assumed to respect some prescribed rules. For instance each phase of the cell cycle is assumed to be characterised by a mean duration and a variability in order to take into account inter-cell variability that can appear within a population. This cellular automaton was coupled with a model of the circadian clock in order to investigate the cytotoxic effects of time-scheduled (delivered according to a periodic schedule) infusions of 5-fluorouracil (5-FU) [START_REF] Altinok | A cell cycle automaton model for probing circadian patterns of anticancer drug delivery[END_REF][START_REF] Altinok | Optimizing temporal patterns of anticancer drug delivery by simulations of a cell cycle automaton[END_REF]. The authors modelled the effects of 5-FU on the cell cycle by increasing the probability that cells submitted to this drug while in S phase exit from the cell cycle at the next G 2 /M transition. Altinok et al. also investigated the effects of oxaliplatin time-scheduled therapies on cancer cells [START_REF] Altinok | Identifying mechanisms of chronotolerance and chronoefficacy for the anticancer drugs 5-fluorouracil and oxaliplatin by computational modeling[END_REF]. Contrary to 5-FU, oxaliplatin is an anti-cancer agent that is not phase-specific. Therefore the authors modelled the effects of oxaliplatin on the cell cycle progression by increasing the probability for exposed cells of exiting the cycle at the next checkpoint (G 1 /S or G 2 /M transitions).

Ordinary Differential Equations.

The most popular models that formed the basis of the development of models to investigate drug efficacy are primarily the exponential model ( dn dt = λn), the logistic (

dn dt = λn 1 - n K
, where K is the maximum tumour size, or "carrying capacity" of the environment), and the Gompertz (

dn dt = λn ln K n ,
where again K is the carrying capacity). A lot of studies on drug control are based on these models [START_REF] Barbolosi | Optimizing drug regimens in cancer chemotherapy: a simulation study using a PK-PD model[END_REF][START_REF] Basdevant | Optimisation of time-scheduled regimen for anti-cancer drug infusion[END_REF][START_REF] Clairambault | Modelling oxaliplatin drug delivery to circadian rhythm in drug metabolism and host tolerance[END_REF][START_REF] Martin | Optimal control drug scheduling of cancer chemotherapy[END_REF][START_REF] Martin | Low-intensity combination chemotherapy maximizes host survival time for tumors containing drug-resistant cells[END_REF][START_REF] Martin | Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy[END_REF][START_REF] Murray | Optimal control for a cancer chemotherapy problem with general growth and loss functions[END_REF][START_REF] Murray | Some optimal control problems in cancer chemotherapy with a toxicity limit[END_REF][START_REF] Murray | The optimal scheduling of two drugs with simple resistance for a problem in cancer chemotherapy[END_REF]. To model the action of cytotoxic drugs, these models integrate a cell loss term that depends on the drug concentration and that can be generically written as:

dn dt = λn ln K n -L(n, D) (3)
where L : R 2 → R is a function, not necessarily linear, of the density of cells N and of the drug concentration D. The drug concentration D is usually given as the solution of an ODE that depends on the drug infusion rate and that can be seen as the output of a more or less complicated PK-PD model. For instance, Martin [START_REF] Martin | Optimal control drug scheduling of cancer chemotherapy[END_REF] developed such a model with a function L linear in n and D to optimise chemotherapy schedules under constraints of maximal tolerated doses. In a more mechanistic way, Barbolosi and Illiadis [START_REF] Barbolosi | Optimizing drug regimens in cancer chemotherapy: a simulation study using a PK-PD model[END_REF] defined the drug concentration thanks to a two-compartment model of the chemotherapy PK. These models consider only one cell population, whereas others integrate several kinds of cell populations. Distinguishing between tumour cell and healthy cell populations enables to take into account possible side-effects of the treatment on the population of normal cells [START_REF] Basdevant | Optimisation of time-scheduled regimen for anti-cancer drug infusion[END_REF][START_REF] Clairambault | Modelling oxaliplatin drug delivery to circadian rhythm in drug metabolism and host tolerance[END_REF][START_REF] Murray | Optimal control for a cancer chemotherapy problem with general growth and loss functions[END_REF][START_REF] Murray | Some optimal control problems in cancer chemotherapy with a toxicity limit[END_REF]. For instance, in [START_REF] Basdevant | Optimisation of time-scheduled regimen for anti-cancer drug infusion[END_REF] Basdevant et al. proposed two optimisation problems. The first one consisted in determining the drug infusion scheme that would minimise the number of tumour cells while kipping the number of healthy cells above a given threshold. The second one consisted in finding a quasi-periodic drug infusion scheme that would maintain the tumour cell population at the lowest possible level while preserving the healthy cell population. As tumour cell resistance can be responsible for the failure of chemotherapy, Martin et al. [START_REF] Martin | Optimal control of tumor size used to maximize survival time when cells are resistant to chemotherapy[END_REF] considered two tumour cell subpopulations: one sensitive to treatment and the other one insensitive to treatment. They assumed that sensitive tumour cells could spontaneously become insensitive and that chemotherapy had a cytotoxic effect only on sensitive cells. In another work [START_REF] Martin | Low-intensity combination chemotherapy maximizes host survival time for tumors containing drug-resistant cells[END_REF], the same authors studied the effect of a combination of two chemotherapies and thus considered three cell subpopulations, differentiating cells being insensitive to one of the two chemotherapies or to both chemotherapies.

To design a more realistic model of tumour growth, in particular to study the effects of anti-angiogenic agents on tumour growth, Hahnfeldt et al. [START_REF] Hahnfeldt | Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy[END_REF] considered, in a Gompertz model, the carrying capacity K as a variable. The variations of this carrying capacity were given by an ODE integrating the spontaneous, the tumour-induced and the anti-angiogenic drug-induced vasculature loss. More recently, some authors based themselves on this approach to study other kinds of variations for the carrying capacity K and to analyse the effect of anti-angiogenic therapies combined with chemotherapies or radiotherapies [START_REF] Ergun | Optimal scheduling of radiotherapy and angiogenic inhibitors[END_REF][START_REF] Ledzewicz | Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy[END_REF][START_REF] Onofrio | Rapidly acting antitumoral antiangiogenic therapies[END_REF][START_REF] Onofrio | Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al[END_REF][START_REF] Onofrio | A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy[END_REF]. For instance, in [START_REF] Ledzewicz | Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy[END_REF], Ledzewicz et al. analysed an optimisation problem that consisted in minimising the final volume of a tumour submitted to a combination of an anti-angiogenic and a cytotoxic anti-cancer treatment, under constraints on the total amounts of the two drugs. This study enabled the authors to propose optimal infusion schemes for such combination of anti-cancer agents.

Most anti-cancer agents are phase-specific, which means that they target only cells that are in a specific phase of the cell division cycle. To analyse the effect of such chemotherapies on cancer cell populations, models that integrate two or more compartments representing the phases or additions of the phases of the cell cycle have been developed. The simplest ones distinguish between cycling and non-cycling cells and suppose that only cycling cells are sensitive to chemotherapy through a death term [START_REF] Panetta | A mathematical model of cell-specific chemotherapy[END_REF][START_REF] Panetta | A mathematical model of breast and ovarian cancer treated with paclitaxel[END_REF][START_REF] Webb | A cell population model of periodic chemotherapy treatment[END_REF][START_REF] Webb | A non linear cell population model of periodic chemotherapy treatment[END_REF]. Other more detailed models combine for instance G 1 and S phases in one compartment and G 2 and M in another one [START_REF] Kozusko | A mathematical model of in vitro cancer cell growth and treatment with the antimitotic agent curacin A[END_REF][START_REF] Swierniak | Optimal control for a class of compartmental models in cancer chemotherapy[END_REF], or consider the phases G 0 and/or S in separate compartments [START_REF] Alarcón | A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells[END_REF][START_REF] Panetta | Mechanistic mathematical modelling of mercaptopurine effects on cell cycle of human acute lymphoblastic leukaemia cells[END_REF]. These models aim at accounting for the effects on cancer cell populations of phase-specific chemotherapies, that can be cytotoxic or cytostatic. Alarcón et al. [START_REF] Alarcón | A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells[END_REF] for instance established by a linear stability analysis that a minimal oxygen concentration is necessary for the tumour to actually grow, instead of being stabilised at a maximum size level. Independently, Kozusko et al. [START_REF] Kozusko | A mathematical model of in vitro cancer cell growth and treatment with the antimitotic agent curacin A[END_REF] analysed the effects on cell cycle progression and cell viability of several doses of curacin A by representing its effects on the transition rates between phases G 1 and S, and between G 2 and M of the cell division cycle, and on cell apoptosis rate, cells being distinguished between sensitive and resistant cells. Panetta et al. [START_REF] Panetta | Mechanistic mathematical modelling of mercaptopurine effects on cell cycle of human acute lymphoblastic leukaemia cells[END_REF] then completed this model by separating the phases G 1 and S in order to investigate the effects of 6-mercaptopurine, an S phase-specific drug, on the dynamics of the cell cycle in populations of cells that were more or less resistant to the treatment. Swierniak et al. [START_REF] Swierniak | Optimal control for a class of compartmental models in cancer chemotherapy[END_REF][START_REF] Swierniak | Optimal control problems arising in cell-cycle-specific cancer chemotherapy[END_REF] distinguished between G 0 , G 1 , and S/G 2 /M to analyse the effects of a cytotoxic chemotherapy combined with 'recruiting' agents such as cytokines, that enable the global cell population to recruit cells from the quiescent phase G 0 into the proliferating phase, which is assumed to make then sensitive to the cytotoxic treatment, i.e., to subsequently kill them when they are definitely committed in the cell cycle.

Partial Differential Equations: physiologically structured transport equations.

Cell population growth also depends on the physiological properties of cells. Such physiological properties can be age of the cells (i.e., the time elapsed since the last cell division), mass or volume of the cells, their degree of resistance to treatment, their DNA content, the size of the induced metastases, etc. To take into account in a population of cells betweenindividual variability linked to these physiological parameters, the McKendrick (or Von FR ÉD ÉRIQUE BILLY AND JEAN CLAIRAMBAULT Foerster-McKendrick) PDE framework is particularly well suited:

           ∂n ∂t (x, t) + ∂ ∂x {v(x)n(x, t)} + d(x)n(x, t) = 0 (t > 0, x > x min ) , n(x min , t) = ∞ xmin β(ξ)n(ξ, t) dξ (t > 0) , n(x, 0) = n 0 (x) (x > x min ) , (4)
where n(x, t) is the density of proliferating cells with characteristic x (age, mass, volume, DNA content, etc.) at time t, v is the cell growth rate (a velocity relating physiological characteristic x to time t), d is the death rate, β is the birth rate, x min ≥ 0 is the minimum value of x for a cell to actually proceed in the cycle. Note that v, d, β depend on x.

The McKendrick model is positive and linear, and as such its asymptotics is governed by a first eigenvalue λ, also called Malthus exponent. It can be proved indeed (using the Krein-Rutman theorem and a generalised relative entropy -GRE -principle, see, e.g., [START_REF] Perthame | Transport Equations in Biology[END_REF]) that its solution may be represented for large times t by a bounded function times exp(-λt).

Physiologically structured cell population dynamics models have been extensively studied in the last 25 years, see e.g., [START_REF] Arino | A survey of structured cell population dynamics[END_REF][START_REF] Arino | Comparison of approaches to modeling of cell population dynamics[END_REF][START_REF] Arino | A survey of cell population dynamics[END_REF][START_REF] Barbolosi | Mathematical and numerical analysis for a model of growing metastatic tumors[END_REF][START_REF] Basse | Modelling the flow cytometric data obtained from unperturbed human tumour cell lines: parameter fitting and comparison[END_REF][START_REF] Basse | A generalised age-and phase-structured model of human tumour cell populations both unperturbed and exposed to a range of cancer therapies[END_REF][START_REF] Benzekry | Modeling the impact of anticancer agents on metastatic spreading[END_REF][START_REF] Billy | Synchronisation and control of proliferation in cycling cell population models with age structure[END_REF][START_REF] Clairambault | Comparison of Perron and Floquet eigenvalues in age structured cell division models[END_REF][START_REF] Clairambault | Circadian rhythm and cell population growth[END_REF][START_REF] Gyllenberg | A nonlinear structured population model of tumor growth with quiescence[END_REF][START_REF] Hinow | A mathematical model separates quantitatively the cytostatic and cytotoxic effects of a HER2 tyrosine kinase inhibitor[END_REF][START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF][START_REF] Kheifetz | Long-range predictability in models of cell populations subjected to phase-specific drugs: growth-rate approximation using properties of positive compact operators[END_REF][START_REF] Metz | The dynamics of physiologically structured populations[END_REF][START_REF] Ribba | A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents[END_REF][START_REF] Webb | Resonance phenomena in cell population chemotherapy models[END_REF]. We focus here, as mentioned earlier, on those that explicitly include a target for the representation of drug effects to control their dynamics.

Age-structured models for the cell division cycle. Age of cells in the cell cycle is

one of the the most used physiological characteristics in the literature on physiologically structured models. The main interest of considering an age-structured model is in distinguishing, in a representation of the cell division cycle, between physiological time (age) and external time, in the perspective of controlling the cycle by drugs that act on it, as do most anti-cancer drugs.

In the McKendrick model, the variable a corresponds to the age of the cells in the cell division cycle for a one-compartmental model (cf Eq. ( 4)), or more generally to the age of the cells in each of the I phases (or addition of phases, e.g., S and G 2 , or G 2 and M ) of the cell cycle:

               ∂ ∂t n i (t, x) + ∂ ∂x {v i (x) n i (t, x)} + {d i (t, x) + K i→i+1 (t, x)} n i (t, x) = 0 , n i (t, x = 0) = ξ≥0 K i-1→i (t, ξ) n i-1 (t, ξ) dξ 2 ≤ i ≤ I , n 1 (t, x = 0) = 2 ξ≥0 K I→1 (t, ξ) n I (t, ξ) dξ , (5)
in which protein p53 is assumed to control phase transition kernels K i→i+1 between phases i and i + 1, e.g., G 1 and S.

As in the case of ODE models, the simplest way to represent the action of a cytotoxic treatment is to enhance the death rate of the cell population according to the treatment activity. Thus Kheifetz et al. [START_REF] Kheifetz | Long-range predictability in models of cell populations subjected to phase-specific drugs: growth-rate approximation using properties of positive compact operators[END_REF] used a one-compartment age-structured cell cycle model where the death rate integrates the drug activity through an exponentially decreasing multiplicative function.

As mentioned earlier, age-structured models have also been used to represent the action of cytotoxic agents by their effects, not directly on death rates, but primarily on phase transition rates, death occurring only secondarily, when cells have been blocked long enough at phase transition checkpoints. In [START_REF] Billy | Synchronisation and control of proliferation in cycling cell population models with age structure[END_REF][START_REF] Clairambault | Comparison of Perron and Floquet eigenvalues in age structured cell division models[END_REF][START_REF] Clairambault | Circadian rhythm and cell population growth[END_REF], the authors consider a multiphase agestructured PDE model of the cell cycle in which is introduced a time dependency of the parameters (death rate, transition rate from one phase of the cell cycle to the next one) to analyse the effects of a circadian control (a biologically known physiological control with period 24 hours) on tissue proliferation, in particular tumour growth, with or without therapy. Thus in [START_REF] Clairambault | Comparison of Perron and Floquet eigenvalues in age structured cell division models[END_REF][START_REF] Clairambault | Circadian rhythm and cell population growth[END_REF], the authors compare the growth rate (Malthus exponent) of a cell population submitted to different time-periodic controls between them, including the case of a no control (i.e., a control function being replaced by a multiplicative constant set to an average of the control function it is compared to).

In [START_REF] Billy | Synchronisation and control of proliferation in cycling cell population models with age structure[END_REF], it is proved that when transition rates K i→i+1 (t, x) are time-independent, the stiffer these rates behave as a function of age x, the lower is the Malthus exponent λ. More precisely, if the cycle phase duration probability density function (p.d.f.) K(x)e -x 0 K(ξ) dξ (which is indeed a p.d.f., on R + provided that +∞ 0 K(ξ) dξ = +∞) is taken in a family of laws with fixed mean µ and varying variance σ 2 , then λ is an increasing function of σ 2 . In other words, the higher the incertitude on the phase duration, the higher the growth exponent. This result is not completely original, and it corresponds to the intuitive notion that healthy cell populations are well synchronised with respect to cell cycle timing, passing from one phase to the following one in good order, whereas cancer cells are more loosely coordinated, resulting in a higher growth rate for a cancer cell population.

This being settled, the question of a target for periodic control by physiological or pharmacological inputs may be assessed in the McKendrick model. A rather unexpected result proved in [START_REF] Clairambault | Circadian rhythm and tumour growth[END_REF] (Theorem 2.1) is that when a time-periodic control is exerted only on death rates, then the Malthus exponent λ is always higher than its counterpart for an uncontrolled time-stationary model designed with the same time-averaged coefficients (death rates). Otherwise said, periodic gating control exerted on death rates enhances proliferation. But if the same periodic control is exerted only on transition rates instead of death rates, then no clear hierarchy can be found between the periodic and the stationary λs [START_REF] Clairambault | Comparison of Perron and Floquet eigenvalues in age structured cell division models[END_REF][START_REF] Clairambault | Circadian rhythm and cell population growth[END_REF][START_REF] Clairambault | An inequality for the Perron and Floquet eigenvalues of monotone differential systems and age-structured equations[END_REF]. This theoretical finding, combined with the fact that protein p53 is known to exhibit periodic oscillations in case of DNA damage exerted by a cytotoxic drug ( [START_REF] Lahav | Dynamics of the p53-mdm2 feedback loop in individual cells[END_REF][START_REF] Bar-Or | Generation of oscillations by the p53-mdm2 feedback loop: A theoretical and experimental study[END_REF], see also recent reviews and new physiologically based models for p53 oscillations in [START_REF] Dimitrio | A spatial physiological model for p53 intracellular dynamics[END_REF][START_REF] Sturrock | Spatio-temporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways[END_REF][START_REF] Sturrock | Influence of the nuclear membrane, active transport, and cell shape on the hes1 and p53-mdm2 pathways: insights from spatio-temporal modelling[END_REF]), together with the fact that cytotoxic drugs themselves often show variations in their concentrations that are under the dependence of periodically controlled enzymes of the cell metabolism, induces to consider as more likely an action of cytotoxic drugs on cell cycle phase transition rates rather than only directly on death rates.

In [START_REF] Billy | Synchronisation and control of proliferation in cycling cell population models with age structure[END_REF], an optimisation problem is considered, consisting in minimising the exponential growth rate λ C of a population of tumour cells submitted to a phase-specific cytotoxic therapy under a toxicity constraint on the population of healthy cells. This constraint is also represented by an exponential growth λ H to be maintained over a (tunable) constant value Λ. The phase-specific chronotherapy results in the blocking of cells at the G 2 /M checkpoint, and subsequent cell death when too much time has been spent in such blocked status, for cells that are committed in the division cycle. As mentioned earlier, in the absence of physiological knowledge on how cell death occurs after cell cycle arrest, subsequent cell death is coarsely represented in numeric simulations in [START_REF] Billy | Synchronisation and control of proliferation in cycling cell population models with age structure[END_REF] by a maximum number of runs of the division cycle, i.e., a limited number of loops on age x for transition rates. It results after this number is passed in a null transition rate K i-1→i (t, a) and null boundary term ξ≥0 K i-1→i (t, ξ) n i-1 (t, ξ) dξ for all phases i. The model in a 2-phase form has been partly (i.e., without drug control) identified on biological data (an NIH-3T3 cell line, mouse embryonic fibroblasts in culture) by using the FUCCI fluorescence method [START_REF] Sakaue-Sawano | Visualizing spatiotemporal dynamics of multicellular cell-cycle progression[END_REF][START_REF] Sakaue-Sawano | Tracing the silhouette of individual cells in S/G2/M phases with fluorescence[END_REF]. Only simulations have been performed, and the model has been only partly identified, but optimal control strategies for a periodic drug delivery are proposed in [START_REF] Billy | Synchronisation and control of proliferation in cycling cell population models with age structure[END_REF] , that successfully solve the problem, decreasing the cancer growth rate while maintaining the healthy cell population growth rate over a given threshold.

Other works model cytostatic, rather than cytotoxic, effects by an action on the ageing velocity v in Eq. ( 4). Thus, Hinow et al. [START_REF] Hinow | A mathematical model separates quantitatively the cytostatic and cytotoxic effects of a HER2 tyrosine kinase inhibitor[END_REF] investigated the effect of lapatinib (a cytostatic drug that is known to become cytotoxic at high doses) on proliferating and non-proliferating cells. To be consistent with experimental data the authors considered a slowing (cytostatic) effect of the drug on the velocity of ageing in the proliferative cell population in G 1 (i.e., with the notations of Eq. ( 4)), they took v(a) = 1 -δ(a, t) where the function δ also depended on the drug dose), and simultaneously, at high drug doses, a cytotoxic effect (death term) on the two cell populations.

Another way to represent cell population growth control by cytostatics in age-structured models is to send (and to maintain) proliferating cells in a quiescent phase where they do not proliferate. Gabriel et al. [START_REF] Gabriel | The contribution of age structure to cell population responses to targeted therapeutics[END_REF] investigated the effect of erlotinib, another cytostatic drug, on cancer cells, using an age-structured model that includes a proliferative compartment and a quiescent one. They assumed that the rate of proliferating cells that become quiescent is an increasing function of the cytostatic drug dose. This simple linear McKendrick model is written as

                           ∂ ∂t p(t, x) + ∂ ∂x p(t, x) + {µ + K(x)} p(t, x) = 0 , p(t, x = 0) = 2(1 -f ) ξ≥0 K(ξ) p(t, ξ) dξ , p(t, x = 0) = p 0 (x) , d dt Q(t) = 2f ξ≥0 K(ξ) p(t, ξ) dξ -νQ(t) , Q(0) = Q 0 , (6)
and the drug target here is f , rate of escape at mitosis towards the siding phase Q, f to be enhanced by a cytostatic drug. The model was identified on the human Non Small Cell Lung Cancer (NSCLC) cell line PC-9 submitted to erlotinib.

3.3.2.

Extension to delay differential models. Following a 30 year-old tradition of models for haematopoiesis that date back to [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF], using a distinction between proliferating and nonproliferating (i.e., quiescent) cell compartments, Adimy et al. designed a model with continuous age and discrete maturity structure [START_REF] Adimy | Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia[END_REF]. In its delay-differential version (obtained by integration of an age-structured PDE model along characteristics [START_REF] Foley | Dynamic hematological disease: a review[END_REF][START_REF] Perthame | Transport Equations in Biology[END_REF]), the model describes at each maturation stage i the dynamics of both quiescent cells with density x i and proliferating cells y i . It may be written as

               ẋi (t) = -δ i x i (t) -w i (t) + 2(1 -K i ) τi 0 e -γia f i (a)w i (t -a) da + 2K i-1 τi-1 0 e -γi-1a f i-1 (a)w i-1 (t -a) da , ẏi (t) = -γ i y i (t) + w i (t) - τi 0 e -γia f i (a)w i (t -a) da , (7)
where K i , with K 0 = 0 (i = 0 representing the stem cell state), is the rate of cells that differentiate to the next maturation stage (i + 1), γ i and δ i are death rates at stage i in the proliferating and quiescent states respectively, w i (t) := β i (x i (t))x i (t), where β i is a nonlinear feedback ("reintroduction function", often taken as a decreasing Hill function with limit zero at infinity, following [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF]) from the ith quiescent to the ith proliferating phase. The introduction of the discrete maturity state (i = 1 . . . I) allows to represent the action of redifferentiating agents such as ATRA on the differentiation rates K i , that in AML are zero at some stage i, e.g., at the promyelocyte stage in APL. In this setting, one can also represent the action of cytotoxic drugs by an increase in the death rates γ i in the proliferating phase, and of cytostatic drugs by a decrease in the feedback functions β i , considered here as representing (formerly in the boundary terms in the original PDE) the dependence on growth factor receptors that is negatively impacted by cytostatic drugs. Stability analyses were performed and theoretical drug targets, involving both proliferation and differentiation, were proposed on β functions and rates K (see [START_REF] Adimy | Asymptotic behavior and stability switch for a mature-immature model of cell differentiation[END_REF][START_REF] Ozbay | Stability analysis of cell dynamics in leukemia[END_REF][START_REF] Ozbay | Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics[END_REF] and references therein). They involve an inequality on model parameters at a non trivial equiibrium point, the existence of which is proven under conditions in [START_REF] Ozbay | Stability analysis of cell dynamics in leukemia[END_REF]. Such stability conditions can be guidelines in the future to use such models as a rationale for the delivery of drugs in combined therapies mixing cytotoxic (acting on death rates γ and δ), cytostatic (acting on the feedback function β) and redifferentiating (acting on differentiation rate K) molecules, aiming at re-establishing a lost equilibrium. For the time being, the focus in modelling has been set on combinations between cytotoxics and cytostatics. This is in particular the case of AML with a mutation of the Flt-3 growth factor receptor (Flt3-ITD gene duplication, resulting in an abnormal tyrosine kinase, that does not need its normal ligand to be activated), which is present in about 30% of all AMLs, generally resulting in poor prognosis [START_REF] Haferlach | Molecular genetic pathways as therapeutic targets in acute myeloid leukemia[END_REF].

3.3.3.

Other physiologically structured transport equations. Physiologically structured models have also been used to study the dynamics of metastatic cell population. Instead of considering age of cells in the cell cycle as the main structure variable, these models consider the size of the metastatic colony [START_REF] Barbolosi | Mathematical and numerical analysis for a model of growing metastatic tumors[END_REF][START_REF] Benzekry | Modeling the impact of anticancer agents on metastatic spreading[END_REF][START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF]]. Thus, denoting by x the tumour size, Iwata et al. [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF] assumed that a primary tumour was generated from a single cell at time t = 0, grew at rate v(x) and emitted at rate β(x) metastatic single cells that developed as the primary tumour. The growth rate v(x) was assumed to follow a Gompertz model:

v(x) = αx log b x (8)
and the evolution of the colony size distribution of metastatic tumours with cell number x at time t was assumed to be governed by a McKendrick model (cf Eq. ( 4)). So that this model is a combination of a Gompertz model and of an age-structured one. In [START_REF] Barbolosi | Mathematical and numerical analysis for a model of growing metastatic tumors[END_REF] Barbolosi et al. proposed a mathematical analysis of this model. Later, Benzekry et al. [START_REF] Benzekry | Modeling the impact of anticancer agents on metastatic spreading[END_REF] introduced in this model the effect of docetaxel, an M phase-specific cytotoxic drug, using a PK-PD three-compartmental model and an interface model to determine the drug exposure. The effect of this drug was modeled via a death term in the growth velocity. In the same work, they also analysed the effect of several infusion schedules of an anti-angiogenic agent. To do this, they considered an additional equation (ODE) for the time evolution of the carrying capacity, just as done by Hahnfeldt et al. in [START_REF] Hahnfeldt | Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy[END_REF] (cf Section 3.2), in which they introduced a death term as a function of the concentration in the anti-angiogenic agent. These authors accounted for the experimentally observed phenomenon of metastatic acceleration after anti-angiogenic therapy (using endostatin) and analysed the influence of several infusion schedules on this phenomenon. Finally, they compared the effect both on the primary tumour and on the number of metastases of several infusion schemes of combination of etoposide (cytotoxic drug) and bevacizumab (anti-angiogenic agent).

Transitions between the phases of the cell cycle are accompanied by changes in DNA content, from 2n to 4n DNA content during S phase, so that models structured by DNA content have also been developed. Basse et al. [START_REF] Basse | A mathematical model for analysis of the cell cycle in cell lines derived from human tumors[END_REF] proposed a model for the four phases of the cell cycle in which obviously only cells in S phase undergo DNA changes. To account for DNA variability induced by flow cytometry, the authors also introduced in the equation on the density of cells in S phase with DNA content x at time t a dispersion term. This model enabled them to compare calculated flow cytometry profiles with experimental ones. Later, Basse et al. [START_REF] Basse | Modelling cell death in human tumour cell lines exposed to the anticancer drug paclitaxel[END_REF][START_REF] Basse | Modelling cell population growth with applications to cancer therapy in human tumour cell lines[END_REF] extended this model by adding an apoptosis phase that could be reached via transition from the mitotic phase. The authors also introduced an age structure in the M phase of the cell cycle, so that this phase is included in this model. They use a model structured in both age and DNA content. This extended model enabled them to analyse the effects on tumour cell lines of paclitaxel, an M phase-specific anti-cancer agent known to induce mitotic cell cycle arrest and cell death via a transition from the mitotic phase to the apoptotic one blocks mitotic spindle dissociation, thus preventing cytokinesis, the last part of mitotic phase, during which a dividing cell actually becomes two). The model parameters were determined by fitting them to experimental profiles obtained by cytometry.

Other models consider several physiological variables as structure variables. Thus Bekkal Brikci et al. [START_REF] Brikci | Analysis of a molecular structured population model with possible polynomial growth for the cell division cycle[END_REF][START_REF] Brikci | An age-and-cyclin-structured cell population model for healthy and tumoral tissues[END_REF] developed an age-structured model that was also structured by the amount of the complex cyclin D/(Cdk4 or 6), known to be the main regulator of the cell cycle at the restriction point R in late G 1 phase, an age after which cells are irreversibly committed to proceed towards division [START_REF] Morgan | The Cell Cycle: Principles of Control[END_REF]. The model considers a global proliferative phase that is age-and cyclin D/(Cdk4 or 6)-structured and a quiescent one that is not physiologically structured. The exchanges of cells between the proliferative compartment and the quiescent one depend on Hill functions of the total cell population, as in Mackey's models [START_REF] Mackey | Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis[END_REF]. Although the inclusion of drug control has been in [START_REF] Brikci | An age-and-cyclin-structured cell population model for healthy and tumoral tissues[END_REF] postponed to 'the future', the model is adapted to represent targets for cytostatics at the level of these exchanges and for cytotoxics either on death rates or on boundary terms for the proliferating population. More recently, Borges et al. [START_REF] Borges | Equilibria of a cyclin structured cell population model[END_REF][START_REF] Borges | Oscillations in a molecular structured cell population model[END_REF] considered the age-independent version of this model, i.e., a cyclin-structured model, still without drug targets. Also Frieboes et al. [START_REF] Frieboes | Prediction of drug response in breast cancer using integrative experimental/computational modeling[END_REF] developed a spatial transport equation model of tumour growth to predict the response to a drug that also takes into account cell phenotype, i.e., that distinguished between drug sensitive and drug resistant viable tumour cells. In this model, cell proliferation and death depend on the intratumoral concentrations of oxygen, nutrients and a cytotoxic drug (additionally represented by spatial reaction-diffusion equations).

3.3.4.

Models integrating space and age-structure. As mentioned earlier, the most relevant level to describe proliferation and its control by drugs is clearly the cell population level, but the level of local surrounding cell populations must also be considered, in particular to account for the interactions between the tumour and its environment. Bresch et al. [START_REF] Bresch | Computational modeling of solid tumor growth: the avascular stage[END_REF] developed a tumour growth model in which cell cycle regulation also depends on the tumour environment, typically cell density and oxygen level. The authors considered two proliferative phases P 1 and P 2 , representing respectively G 1 before the restriction point [START_REF] Morgan | The Cell Cycle: Principles of Control[END_REF], and the rest of the cell cycle (remainder of G 1 and S/G 2 /M ), two proliferative phases to which a quiescent phase Q is added. They assumed that only proliferative phases were age-structured and they added an advection term to model the passive transport of tumour cells induced by cell division:

                                 ∂P 1 ∂t + ∂P 1 ∂a + ∇ • (v P1 P 1 ) = 0 , ∂P 2 ∂t + ∂P 2 ∂a + ∇ • (v P2 P 2 ) = 0 , ∂Q ∂t + ∇ • (v Q Q) = (1 -f )P 1 (a = a max,P1 ) - ∂f ∂t + Q(t -) , P 1 (a = 0) = 2 P 2 (a = a max,P2 ) , P 2 (a = 0) = f P 1 (a = a max,P1 ) + ∂f ∂t + Q(t -) , (9)
where P i = P i (t, x, y, z, a), (i = 1, 2), Q = Q(t, x, y, z), a representing the age of cells in each proliferative phase, ∇ denotes the derivative with respect to the 3d-space variable x = (x, y, z), v P1 , v P2 , v Q are the velocities (in 3d-space with respect to time t) in the phases P 1 , P 2 , Q respectively, a max,P1 and a max,P2 the maximal age a cell can spend in P 1 and P 2 respectively, [•] + the positive part. The function f is a boolean function that depends on time and space, equal to one if there is no overpopulation and no hypoxia at space location x = (x, y, z) and time t, equal to zero otherwise. Assuming that the three velocities were equal and that the total number of tumour and healthy cells was constant yielded an additional equation to compute the velocity. Bresch et al. [START_REF] Bresch | Computational modeling of solid tumor growth: the avascular stage[END_REF] analysed the influence on tumour growth of a membrane surrounding this tumour. Ribba et al. [START_REF] Ribba | A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents[END_REF] considered a simplified version of this model as they did not account for the oxygen but they considered the action of matrix metalloproteinases (MMPs). MMPs are enzymes secreted by tumour cells, known to digest the extracellular matrix so as to facilitate local invasion by tumour cells. Using this model, Ribba et al. analysed the effect of MMP inhibitors (MMPIs), anti-cancer agents known to reduce cancer growth in animal models but whose clinical development was not as successful as expected (they were too toxic). In this model, MMPIs were assumed to promote the passage of proliferative cells into the quiescent phase at the restriction point. They assessed the therapeutic benefit of MMPIs via a parameter that compared the proportion of quiescent cells in a population submitted to treatment with the one in the same population without treatment. Later, Billy et al. [START_REF] Billy | A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy[END_REF], based on the model developed by Bresch et al. [START_REF] Bresch | Computational modeling of solid tumor growth: the avascular stage[END_REF], investigated the effects on tumour growth of an anti-angiogenic therapy. They coupled, via oxygen concentration, the model developed by Bresch et al. with a continuous PDE model of angiogenesis that accounts for the density of endothelial cells (ECs, cells that constitute blood vessels), oxygen and some pro-and antiangiogenic substances. The authors investigated the effect of an anti-VEGF therapy (VEGF being the main pro-angiogenic factor) that consisted in increasing the local concentration of endostatin, an anti-angiogenic factor that competes with VEGF for binding to EC receptors. This therapy was modelled through an endostatin oversecretion since endostatin was known to be endogenously secreted by tumour cells. Thus, in this model, endostatin is assumed to directly target VEGF binding to ECs since an increase of endostatin concentration induces a decrease of the VEGF binding rate to ECs, which leads to lower activation of ECs by VEGF, i.e., lower proliferation and migration rates of ECs and even EC death for high endostatin concentrations (more details can be found in [START_REF] Billy | A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy[END_REF] Sections 3 and 4). Such therapy can lead to regression of the vascular network and thus induce hypoxia or even death of tumour cells. Several infusion schedules were studied and their efficacies on the tumour volume were compared. The authors highlighted the existence of a critical local concentration of endostatin below which it was more efficacious on tumour growth to increase the rate of oversecretion of endostatin instead of its duration and above which the opposite was true. Following the line of previous articles [START_REF] Alarcón | A cellular automaton model for tumour growth in inhomogeneous environment[END_REF][START_REF] Alarcón | A mathematical model of the effects of hypoxia on the cell-cycle of normal and cancer cells[END_REF], Alarcón et al. proposed in [START_REF] Alarcón | A multiple scale model for tumor growth[END_REF] a multiscale model using a cellular automaton (for the vascular network), ODEs (for the cell cycle) and PDEs (for oxygen and VEGF diffusion). Ribba et al., following the line of their 2006 article [START_REF] Ribba | A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents[END_REF] published in 2009 another multiscale model to study the effects of cell cycle specific drugs by introducing pharmacokinetic-pharmacodynamic representation of the drug fate [START_REF] Ribba | Chemotherapy may be delivered based on an integrated view of tumour dynamics[END_REF]. In a more recent model, Powathil et al. [START_REF] Powathil | Modelling the effects of cell-cycle heterogeneity on the response of a solid tumour to chemotherapy: biological insights from a hybrid multiscale cellular automaton model[END_REF] used in a way close to the Alarcón models a multiscale model of a hybrid nature, integrating the cell cycle and the vascular environment by an ODE system and a cellular automaton, and the diffusion of both oxygen and anticancer drugs by spatial PDEs. This allowed these last authors to study and optimise theoretical combinations of various cell cycle phase specific drugs acting on a tumour cell population.

3.4. Cell Darwinism, adaptive dynamics, phenotype-structured population models.

3.4.1.

Cancer as an evolutionary disease. The idea that cancer is an evolutionary disease is not new, but strangely enough, not many articles on this theme, let alone on mathematical models on Darwinism in cancer cell populations, have been published so far, according to a recent review [START_REF] Aktipis | Overlooking evolution: A systematic analysis of cancer relapse and therapeutic resistance research[END_REF], even though the necessity to consider cancer evolution as amenable to a Darwinian selection principle is becoming popular [START_REF] Gatenby | A change of strategy in the war on cancer[END_REF][START_REF] Gatenby | Adaptive therapy[END_REF]. The idea promoted by Gatenby et al. in these recent articles is that tumour eradication is a difficult goal to achieve, and that it can even have adverse consequences, inducing resistant cell subpopulations that will be impossible to control, so that stabilising the tumour (obtaing "dormancy", i.e., a nonproliferative state) rather than trying to eradicate it should be more reasonable and more successful in terms of life expectancy for cancer patients. Towards this aim, metronomic chemotherapy that consists in chronic administration of low chemotherapeutic drug doses, as advocated for instance in [START_REF] Pasquier | Metronomic chemotherapy: new rationale for new directions[END_REF], and studied from a modelling point of view in [START_REF] Hahnfeldt | Minimizing long-term tumor burden: the logic for metronomic chemotherapeutic dosing and its antiangiogenic basis[END_REF], might contribute to some extent.

Note that it is a resistance phenotype, not a genotype, that is taken into account in such models. Whereas the genetic paradigm, i.e., cancer due to a single "renegade" mutated cell developing in a malignant clone, has most often been considered as the only explanation of evolution towards cancer, with little attention brought so far to (possibly reversible) environmental effects exerted at the level of cell populations, and this may partly explain the present state of publications regarding cancer as an evolutionary disease. In the perspective of cell Darwinism, which always concerns populations of individual cells, considering drug effects as part of an environmental pressure resulting in selection and possibly speciation (emergence of a genetically resistant species) is a rather new idea in mathematical modelling to account for evolution towards drug resistance in a cancer cell population.

3.4.2.

Phenotype-structured population models. These models, that are widespread in the ecological modelling community, are only beginning to diffuse in the cancer biological modelling world. They can be used to describe the dynamics of different cell populations in interaction, healthy or tumour, with the addition of environmental variables such as nutrients, natural molecules linked to the influence of energetic and metabolic settings, that are assumed to play a role in pharmacotherapeutics, of the immune system and drugs. Physiologically structured, usually without space variables -but nothing can be opposed in principle to the introduction of a spatial variable if it is relevant to the question at stake -, they consist of integro-differential equations to take into account the existence of nonlocal interactions, leading for instance to mutations in cell populations. An exemple is [START_REF] Delitala | A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions[END_REF],

and another one, focusing on the problem of drug resistance is [START_REF] Lorz | Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies[END_REF], presented below. Finally, it is worth noting that a natural extension of these models are the kinetic-like models presented in a general setting in [START_REF] Bellomo | Modelling Complex Living Systems -A Kinetic Theory and Stochastic Game Approach[END_REF][START_REF] Bellomo | From the mathematical kinetic, and stochastic game theory to modeling mutations, onset, progression and immune competition of cancer cells[END_REF][START_REF] Bellouquid | Modelling Complex Multicellular Systems -A Kinetic Theory Approach[END_REF], which are concerned with the description of cancer-immune competition.

3.4.3. Phenotype-structured population models for drug resistance, general form. This phenotype-structured population model relies on equations with a continuous structure variable representing, rather than age or any relevant molecule, a resistant phenotype (x = 0, no resistance; x = 1, completely resistant population), which may be reversible (if resistance is due to an epigenetic rather than genetic phenomenon) or not. In models of the type presented below (Eq. ( 10)), evolution towards acquired drug resistance, i.e., the development of a subpopulation bearing a resistant phenotype, close to x = 1, may be the result of mutations, but it may also be the result of exchanges with the environment, without mutations, representing in the latter case possibly reversible acquired resistance [START_REF] Lorz | Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies[END_REF]:

∂ ∂t n(x, t) = mutations and renewal θ 1 + αc 2 (t) r(y)M (y, x)n(y, t)dy -r(x)n(x, t) + r(x) 1 + αc 2 (t) -d(x)I(t) n(x, t)
growth with cytostatic therapies and death

- c 1 (t)µ(x)n(x, t).
effect of cytotoxic therapies [START_REF] Altinok | Optimizing temporal patterns of anticancer drug delivery by simulations of a cell cycle automaton[END_REF] Here M is a mutation kernel, with radius a priori larger in the cancer than in the healthy cell population, θ is the proportion of cells that undergo mutations at mitosis, and, introducing subscripts H and C for healthy and cancer cells n H and n C , competition in the environment is represented by variables I H and I C that may be only fixed linear combinations of the total healthy and cancer cell populations n C (x, t) dx, but may also be variables in interaction with the environment, e.g., with cytokines.

Mutations only.

The main interest of including a drug target on mutation rates is to represent genomic instability in cancer cells by a higher probability of mutation under the influence of a drug. This is presented in the frame of model equation ( 10) when θ = 0. Nevertheless, at least in this setting, mutations do not seem to play the main part in the establishment of a resistant phenotype, yielding only diffusion around a Dirac δ distribution for the dominant phenotype. This rules out the possibility of representing in this model acquired drug resistances only due to mutations of the target, as for instance imatinib resistance, for which mutations of the target, BCR-Abl protein, have been evidenced. Likely, other versions, focusing on mutations, of the same model, should be used in this case.

3.4.5. Competition for resources and exchanges with the environment. In this type of model, it is easy to obtain evolution towards resistance without mutations (i.e., setting θ = 0). In fact, the theorems demonstrated in [START_REF] Lorz | Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies[END_REF] (Theorems 3.1, 3.2, 3.3) state asymptotic convergence towards a single Dirac mass concentrated around a unique value of the phenotypic trait, and these theorems hold with or without mutations. The healthy cell population case is characterised by a homeostatic factor of the form 1 1 + I(t) before the proliferation term r(x), where I(t) represents the total (healthy and cancer) cell population, preventing the healthy cell population from exploding, whereas in the cancer population case, no such homeostasis has been put in the evolution equation.The theorems in [START_REF] Lorz | Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies[END_REF] show that under a cytotoxic therapy, the convergence occurs towards a single Dirac mass (monomorphism of the fittest population) , concentrated around x = 0 in the healthy case, and concentrated around a fittest trait x C = 0, i.e., a resistant phenotype, in the cancer case.

3.4.6. Combining cytotoxic and cytostatic effects. With settings close to those shown in Eq. ( 10), neglecting mutations, but representing the action of two different drugs and a 2-dimensional resistance phenotype (x, y), one corresponding to a cytostatic drug that acts on proliferation, and another one corresponding to a cytotoxic drug that acts on a death term, it is possible to obtain numerically dimorphism of asymptotic traits in the cancer cell population, one fittest subpopulation concentrated around (1, 0) and the other around (0, 1) (Fig. 2), i.e., asymptotic coexistence of two different subpopulations, one resistant to the cytostatic, the other resistant to the cytotoxic drug. The 2-drug model for tumour cells runs ∂ t n(t, x, y) = r(x, y) 1 + µ 2 (x, y)c 2 (t) -d(x, y)I(t) -µ 1 (x, y)c 1 (t) n(t, x, y),

where c 1 and c 2 represent the effects of a cytotoxic and of a cytostatic drug, respectively, and I(t) is again the total (weighted) population of all cells, healthy and tumour, the term d(x, y) thus representing competition for space and nutrients, while µ 1 and µ 2 represent the targets for cytotoxic and cytostatic drugs, respectively. Simulation results, with particular choices for the target functions (not shown), taking into account the fact that developing resistance hinders proliferation capacities, and with initial conditions centered around mean values (x, y) = (0.5, 0.5) show the possibility of evolution towards a dimorphic cell population under the simultaneous infusion of cytotoxic and cytostatic drugs, whereas, in the absence of therapeutic agents, the cell population remains monomorphic and evolves towards a totally sensitive (but highly proliferative) state, as illustrated on Fig. 1 and2. It is also possible, as numerically shown in [START_REF] Lorz | Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies[END_REF], Fig. 6.6, to obtain different asymptotic behaviours, in particular evolution towards resistance or extinction of the cancer cell population while keeping healthy cells alive, only by varying constant doses of the two drugs. This may be considered as a step towards drug delivery optimisation with respect to the question of avoiding drug resistance, by combining cytotoxic and cytostatic drugs.

Note that one can also (work underway in the line of [START_REF] Lorz | Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies[END_REF]) represent simultaneously space and an evolutionary phenotype in integro-differential models accounting for the evolution of tumour spheroids submitted to an externally delivered therapy. In this case, a 1d radial spatial coordinate is a relevant complementary variable that must be considered together with the cell phenotype responsible for drug resistance.

Conclusion.

We have reviewed in this article some old and more recent models of proliferating cell population dynamics designed to theoretically solve problems of therapeutic optimisation encountered in the clinic of cancers, always considering them from the point of view of drug target representation. Polychemotherapy is usually the rule in oncologywith a few known exceptions such as imatinib in chronic myelogenous leukaemia -, so that designing a rationale to optimally combine treatments acting on different functional targets in the physiological mechanisms that control proliferation in cell populations should eventually be a help in the clinic. This involves taking into account in multiscale mathematical models the moving physiological knowledge of these mechanisms, of how old and new drugs can modify them, and of how they can be investigated at the different levels of observation: single cell, cell population, and whole body as a collection of interacting cell populations. We stress that this program implies frequent interactions, going far beyond an attitude of sort of 'scientific service providers', between teams of mathematicians, biologists and clinicians, involving in a much more committed way mutual efforts to understand how each discipline can benefit of each others' representations and findings. We hope that this review of models and results, focusing on drug targets, can be helpful to this aim.

x≥0nH

  (x, t) dx and x≥0

FIGURE 1 .

 1 FIGURE 1. Cancer cells without therapies. Starting from an average 2d phenotype [(x, y) = (0.5, 0.5), left], time evolution towards total sensitivity: no resistance. Model, simulation and figure adapted from[START_REF] Lorz | Dimorphism in cancer cell populations evolving under drug pressure[END_REF].

16 FIGURE 2 .

 162 FIGURE 2. Cancer cells with therapies. Starting from the same average 2d phenotype [(x, y) = (0.5, 0.5), left], time evolution towards two separate resistant clones, one resistant to the cytotoxic drug, the other one to the cytostatic drug. Model, simulation and figure adapted from [65].

Acknowledgments. The authors are gratefully indebted to Alexandre Escargueil, Tommaso Lorenzi, Alexander Lorz, and Benoît Perthame for fruitful discussions and phenotypestructured population models for cell Darwinism in healthy and cancer cell populations.