Benoît Perthame
email: benoit.perthame@upmc.fr

Some mathematical aspects of tumor growth and therapy

Keywords: Tumor growth, Hele-Shaw equation, Free boundary problems, Structured population dynamics, Resistance to therapy Mathematics Subject Classification (2010): 35K55, 35B25, 76D27, 92C50, 92D25

Mathematical models of tumor growth, written as partial differential equations or free boundary problems, are now in the toolbox for predicting the evolution of some cancers, using model based image analysis for example. These models serve not only to predict the evolution of cancers in medical treatments but also to understand the biological and mechanical effects that are involved in the tissue growth, the optimal therapy and, in some cases, in their implication in therapeutic failures.

The models under consideration contain several levels of complexity, both in terms of the biological and mechanical effects, and therefore in their mathematical description. The number of scales, from the molecules, to the cell, to the organ and the entire body, explains partly the complexity of the problem. This paper focusses on two aspects of the problem which can be described with mathematical models keeping some simplicity. They have been chosen so as to cover mathematical questions which stem from both mechanical laws and biological considerations. I shall first present an asymptotic problem describing some mechanical properties of tumor growth and secondly, models of resistance to therapy and cell adaptation again using asymptotic analysis.

Introduction

Since the paper of H. P. Greenspan [START_REF] Greenspan | Models for the growth of a solid tumor by diffusion[END_REF] in 1972, an increasing mathematical activity has been developing, that creates new models, new numerical methods, new analysis of partial differential equations representing various aspects of tumor growth and therapy. This activity follows the National Cancer Act, usually called 'war on cancer', signed in 1971 and the awareness that the disease becomes a major health problem in many countries. Despite several decisive progresses (more than 1500 americans are declared recovered from cancer every day), the many faces of the problem, and their complexity in terms of scales, agents and scientific background, explain that cancer remains a challenge for XXIst century medicine.

Interestingly enough, many aspects have lead to mathematical modeling and I would like to mention some of them. The molecular basis of tumors are mutations of cells, which are modeled by random processes [START_REF] Durrett | Probability Models for DNA Sequence Evolution[END_REF], and which opens the route of molecular targets for drug design. The number of scales, from the molecule, to the cell , to the tumor itself and to the organ, also explains the complexity of the phenomena. Considering an assembly of cells, bridging gene activity to cell behavior, are possible with agent based methods, and related discrete methods, giving a detailed account of tissue growth and organization [START_REF] Anderson | Single-cell-based models in biology and medicine[END_REF][START_REF] Byrne | Individual-based and continuum models of growing cell populations: a comparison[END_REF][START_REF] Hoehme | A cell-based simulation software for multi-cellular systems[END_REF][START_REF] Moreira | Cellular automata models of tumour developmenta critical review[END_REF]. However, in these notes, I will consider continuous models, used for large populations of cells, and only solid tumors even if liquid tumors (blood or lymphatic cancers) have also led to a core of mathematical literature, see [START_REF] Colijn | Bifurcation and bistability in a model of hematopoietic regulation[END_REF][START_REF] Adimy | On the stability of a maturity structured model of cellular proliferation[END_REF] and the references therein.

Ordinary differential equations are however the first modeling tool. They are efficient because parameter identification is simpler than in the more elaborate partial differential equations. They can can also provide direct qualitative behaviours in complex situations when several effects combine. This is the case for several examples when modeling angiogenesis (see below also) and supply both of nutrients and therapy to the tumor by neovasculature which can be of low quality [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF][START_REF] Benzekry | A new mathematical model for optimizing the combination between antiangiogenic and cytotoxic drugs in oncology[END_REF], tumorimmune system interaction [START_REF] De Angelis | Qualitative Analysis of a mean field model of tumor-immune system competition[END_REF][START_REF] Bellomo | Modelling and mathematical problems related to tumor evolution and its interaction with the immune system[END_REF][START_REF] Kirschner | On the global dynamics of a model for tumor immunotherapy[END_REF], metastases development [START_REF] Benzekry | Maximum Tolerated Dose Versus Metronomic Scheduling in the Treatment of Metastatic Cancers[END_REF], drug optimization during therapy [START_REF] Ledzewicz | Singular controls and chattering arcs in optimal control problems arising in biomedicine[END_REF], interactions between cell cycle and circadian cycle, [START_REF] Clairambault | Comparison of the Perron and Floquet eigenvalues in monotone differential systems and age structured equations[END_REF][START_REF] Clairambault | Circadian rhythm and cell population growth[END_REF]. Note that, in these works, the model description based on ODEs can be complemented with physiological variables, thus leading to integrodifferential equations, the so-called 'structured population' models. Continuous models allow for numerical simulations at the scale of the organ and are used for predicting tumor progression in combination with medical imaging [START_REF] Swanson | Quantifying the role of angiogenesis in malignant progression of gliomas: in silico modeling integrates imaging and histology[END_REF][START_REF] Lowengrub | Nonlinear modelling of cancer: bridging the gap between cells and tumours[END_REF][START_REF] Cornelis | In vivo mathematical modeling of tumor growth from imaging date: Soon to come in the future?[END_REF][START_REF] Colin | System identification in tumor growth modeling using semi-empirical eigenfunctions[END_REF]. These models can incorporate several features as nutrients availability, angiogenesis (the process by which necrotic cells in the core of the tumor emit molecular signals attracting new vasculature), adhesion to the extracellular matrix and its degradation, interaction with the healthy cells, proliferative or quiescent or necrotic states of the cells; these features and many others are described in the many papers and in several surveys available in the literature [START_REF] Byrne | Free boundary value problems associated with the growth and development of multicellular spheroids[END_REF][START_REF] Sherratt | A new mathematical model for avascular tumour growth[END_REF][START_REF] Araujo | A history of the study of solid tumour growth: the contribution of mathematical models[END_REF][START_REF] Bellomo | On the foundations of cancer modelling: selected topics, speculations, and perspectives[END_REF][START_REF] Bellomo | Modelling and mathematical problems related to tumor evolution and its interaction with the immune system[END_REF][START_REF] Friedman | A hierarchy of cancer models and their mathematical challenges[END_REF][START_REF] Greenspan | Models for the growth of a solid tumor by diffusion[END_REF][START_REF] Lowengrub | Nonlinear modelling of cancer: bridging the gap between cells and tumours[END_REF][START_REF] Roose | Mathematical models of avascular tumour growth: a review[END_REF].

In order to present both the impact of physical laws and biological aspects, these notes address two different aspects of tumor growth. Considering fluid mechanical aspects, section 2 describes one of the simplest models in the area and is followed, in section 3 by the derivation of a free boundary problem in the 'stiff law-of-state' limit. Then, we turn to an approach, based on asymptotic analysis, to a question related to therapy and resistance to drugs; this is section 4.

Mechanical aspects of tissue growth

Solid tumors grow under the effect of cell proliferation limited by several factors. Space availability, and the pressure induced by higher cell population, appears to be the first cause of growth limitation by contact inhibition [START_REF] Byrne | Individual-based and continuum models of growing cell populations: a comparison[END_REF][START_REF] Ranft | Fluidization of tissues by cell division and apaptosis[END_REF]. This can be included in the simplest models for a cell population density n(x, t) where pressure generates both movement and growth limitation, leading to write

       ∂ ∂t n + div nv = nG(p), x ∈ R d , t ≥ 0, n(x, t = 0) = n 0 (x) ≥ 0, v(x, t) = -∇p(x, t), p(x, t) ≡ Π γ n(x, t) := n(x, t) γ , γ > 1.
(

) 1
The rule v(x, t) = -∇p(x, t) is a simplified version of Darcy's law expressing isotropic and homogeneous friction with the surrounding environment. This expression for the velocity field means that cells are only pushed by mechanical forces (variants are mentioned later). The particular choice for the law-ofstate Π γ (n) := n γ is made for simplicity, see considerations on this issue in [START_REF] Ciarletta | The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis[END_REF]. Finally the growth term, the right hand side in [START_REF] Alikakos | Convergence of the Cahn-Hilliard Equation to the Hele-Shaw Model[END_REF], is of Lotka-Volterra type, and takes into birth and death of cells.

Because pressure generates contact inhibition, we assume that the

C 1 function G(•) satisfies G(0) = G M > 0, G ′ (•) < 0, G(P h ) = 0, for some G M > 0, P h > 0. ( 2
)
The name 'homeostatic pressure' has been proposed for P h ( [START_REF] Ranft | Fluidization of tissues by cell division and apaptosis[END_REF]). At this stage it might also be useful to mention that dimensions d = 2 is relevant for in vitro experiments on a dish and d = 3 is relevant both in vitro and in vivo. As well known for the porous medium equation, one property of such partial differential equations is to describe solutions with compact support than expand [START_REF] Vázquez | The porous medium equation. Mathematical theory. Oxford Mathematical Monographs[END_REF]. For our purpose here, this is enough and we do not bother with a bounded domain and associated boundary conditions. This feature is however relevant both for realistic models and numerics.

As far as existence is concerned, this equation is standard and is a semi-linear version of the 'porous medium equation', [START_REF] Vázquez | The porous medium equation. Mathematical theory. Oxford Mathematical Monographs[END_REF]. Therefore, several bounds are known under some assumptions on the initial data. Now, we follow closely [START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF]. Because we are interested in the dependence on the parameter γ (and large values of it), we consider a family of initial data n 0 γ such that for some constant K 0 ,

R d n 0 γ dx ≤ K 0 , p 0 γ := Π γ (n 0 γ ) ≤ P h , R d |∇n 0 γ |dx ≤ K 0 . (3)
Proposition 2.1 With assumptions (2)-( 3), the solution of equation (1) satisfies the followinf a priori estimates n(x, t) ≥ 0, p(x, t) ≤ P h ,

R d n(x, t)dx ≤ K 0 e G M t , R d |∇n(x, t)|dx ≤ K 0 e G M t , T 0 R d |∇p(x, t)|dxdt ≤ C(T, P h , K 0 ), R d p(x, t)dx ≤ P (γ-1)/γ h K 0 , T 0 R d |∇p(x, t)| 2 dxdt ≤ 1 + γG M T γ -1 P (γ-1)/γ h K 0 .
Proof. The estimates for n are straightforward. For the TV bound, we just notice that, the equation for n can also be written

∂ ∂t n -∆Φ(n) = nG p(x, t) , with Φ ′ (n) = nΠ ′ γ (n).
Therefore, the equation for

w i = ∂n(x,t) ∂x i is ∂ ∂t w i -div[Φ ′ (n)∇w i ] = w i G p(x, t) + nG ′ p(x, t) ∂p(x, t) ∂x i ,
and finally

∂ ∂t |w i | -div[Φ ′ (n)∇|w i |] = |w i |G p(x, t) -n G ′ p(x, t) ∂p(x, t) ∂x i ≤ |w i |G M .
After integration and use of the Gronwall lemma, this gives the L 1 estimate on the gradient of n and keeping the term with ∂p ∂x i gives the bound on the gradient of p (see [START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF] for details).

The second line of bounds in Proposition 2.1 follows from the equation on the pressure. Namely, we compute

∂ ∂t p -nΠ ′ (n)∆p -|∇p| 2 = nΠ ′ (n)G p(x, t) . ( 4
)
This equation is in the strong form, the maximum principle applies and gives the bound p ≤ P h . It gives the L 1 control on p because

p = n γ = nn γ-1 = np (γ-1)/γ ≤ nP (γ-1)/γ h ,
and it remains to apply the L 1 control on n.

The L 2 estimate on the gradient is better seen when identifying the pressure, as p = n γ in (4), to find

∂ ∂t p -γp∆p -|∇p| 2 = γpG(p). (5)
Integrating by parts, we obtain, for T > 0,

R d [p(x, T ) -p 0 (x)]dx + (γ -1) T 0 R d |∇p| 2 dxdt ≤ γG M T 0 R d p(x, t)dxdt.
which, combined with the L 1 estimate for p gives the last inequality. The bounds in Proposition 2.1 are fine to ensure compactness in space. It remains to prove estimates implying time compactness. An easy way is to notice that under the assumption that n 0 is a subsolution, that is

-div n 0 ∇Π(n 0 ) ≤ n 0 G p 0 (x) ,
we have ∂ ∂t n 0 ≥ 0. We may apply the same argument as for space derivatives and w = ∂ ∂t n satisfies

∂ ∂t w -div[Φ ′ (n)∇w] = wG p(x, t) + nG ′ p(x, t) γn γ-1 w,
an equation which gives us the property

∂ ∂t n 0 ≥ 0 =⇒ ∂ ∂t n ≥ 0. ( 6
)
This property is very strong and shows one limitation of the model at hand. It is incompatible with the observations that the cell population decreases in the center of the tumor, the necrotic core. This effect, which typically occurs at the size of 1mm 3 , can be obtained when the effects of nutrients are included in the equation, see [START_REF] Bénilan | The mesa problem for Neumann boundary value problem[END_REF] below.

In this situation, which we call 'well prepared initial data', we conclude

d dt R d |w(x, t)|dx ≤ G M R d |w(x, t)|dx,
and thus

R d ∂ ∂t n(x, t) dx ≤ R d div n 0 ∇Π(n 0 ) + n 0 G p 0 (x) dx. (7)
It is possible to improve these estimates and avoid the restrictive assumption that the initial data is a subsolution. We recall from [START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF] the Proposition 2.2 For a constant r G depending only on G(•), the estimates hold, for all t > 0,

∂ ∂t p(x, t) ≥ -γ r G p(x, t) e -γr G t 1 -e -γr G t , ∂ ∂t n(x, t) ≥ -r G n(x, t) e -γr G t 1 -e -γr G t .
These inequalities express a regularizing effect with a fast transition at t = 0 (the right hand side is singular then). They extend a family of similar inequalities initiated in [START_REF] Crandall | Regularizing effects for u t = ∆φ(u)[END_REF]. They are stronger than those in (6) because they do not assume any further assumption on the initial data than those in Proposition 2.1 (no nedd that n 0 is a subsolution of the stationary equation). A remarkable feature here, is that the semi-linear source term improves the usual inequalities for the porous medium equations, which are recovered for r G → 0.

To conclude this section, we present some additional effects which are used in more realistic models of tumor growth. A possible additional ingredient is to take into account nutrients. Then, we arrive to the model, also treated in details in [START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF] 

        ∂ t n -div(n∇p) = n Φ(p, c), ∂ t c -∆c = -n Ψ(p, c), c(x, t) = c B > 0 as |x| → ∞, (8)
where c denotes the density of nutrients, and c B the far field supply of nutrients (from blood vessels). The coupling functions Φ, Ψ are assumed to be smooth and to satisfy the intuitive hypotheses

   ∂ p Φ < 0, ∂ c Φ ≥ 0, Φ(P h , c B ) = 0, ∂ p Ψ ≤ 0, ∂ c Ψ ≥ 0, Ψ(p, 0) = 0. (9)
Variants are possible; for instance, we could assume that nutrients are released continuously from a vasculature or an other source [START_REF] Ciarletta | The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis[END_REF], several nutrients (oxygen, glucose) can be considered. Traveling wave profiles are special one-diemensional solutions under the form n(x, t) = n(x -σt), c(x, t) = c(x -σt), which connect the healthy to the cancer states for y = ±∞ and y = x -σt; they give an insight of the local shape of solutions when a permanent regime is established. The determinatination of such profiles is usual in this field [START_REF] Sherratt | A new mathematical model for avascular tumour growth[END_REF][START_REF] Perthame | Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient[END_REF].

Another ingredient is to take into account active movement of cells and not only their passive movement under pressure forces. This leads to write the model, which is analyzed in [START_REF] Perthame | Derivation of a Hele-Shaw type system from a cell model with active motion. Interfaces and Free Boundaries[END_REF],

∂ t n -div n∇p -ν∆n = nG p . ( 10
)
The effect of the diffusion term -ν∆n is drastic and progression is much faster with smoother profile (but [START_REF] Perthame | Derivation of a Hele-Shaw type system from a cell model with active motion. Interfaces and Free Boundaries[END_REF] show that a free boundary problem can still be defined as we do it in the next section).

Finally, Darcy's law relating velocity and pressure can also extended to a visco-elastic fluid and gives

   ∂ t n -div n∇W -ν∆n = nG p , -ν∆W + W = p,
see [START_REF] Bresch | A viscoelastic model for avascular tumor growth[END_REF].

More generally, the formalism of multiphase fluids can be used in the present context [START_REF] Byrne | A two-phase model of solid tumor growth[END_REF][START_REF] Preziosi | Multiphase modeling of tumor growth and extracellular matrix interaction: mathematical tools and applications[END_REF] in order to represent the complexity of cell surrounding. One can also add many additional biological features, which have led to mathematical models, and which we do not mention here.

The Hele-Shaw asymptotic and free boundary formulation

As long as cells are well separated, the pressure forces are negligible. When the population density increases, there is a maximum possible compaction which cannot be exceeded. To represent this effect with a fast transition, the simplest formalism is to consider the limit as γ → ∞ in the equation of state, see [START_REF] Alikakos | Convergence of the Cahn-Hilliard Equation to the Hele-Shaw Model[END_REF], and which we call the stiff pressure asymptotic. This type of modeling is mostly used in practical use of cancer models and software development [START_REF] Friedman | A hierarchy of cancer models and their mathematical challenges[END_REF][START_REF] Lowengrub | Nonlinear modelling of cancer: bridging the gap between cells and tumours[END_REF][START_REF] Roose | Mathematical models of avascular tumour growth: a review[END_REF][START_REF] Colin | Computational modeling of solid tumor growth: the avascular stage[END_REF][START_REF] Colin | System identification in tumor growth modeling using semi-empirical eigenfunctions[END_REF][START_REF] Cristini | Nonlinear simulations of tumor growth[END_REF].

Free boundary problem

This limit results in a model that generalizes the Hele-Shaw equation of fluid mechanics and which is usually seen as a free boundary problem. The tumor occupies a domain Ω(t), healthy cells fill the space outside Ω(t). The boundary ∂Ω(t) of the domain Ω(t) is moving with the velocity

v ∞ (x, t) = -∇p ∞ (x, t) (11)
where the pressure field is computed thanks to the equation

-∆p ∞ = G(p ∞ ) x ∈ Ω(t), p ∞ = 0 on ∂Ω(t). (12)
In order to define this dynamic, some smoothness of the free boundary is necessary. Such a property has been widely studied, see [START_REF] Friedman | A hierarchy of cancer models and their mathematical challenges[END_REF][START_REF] Escher | Classical solutions for Hele-Shaw models with surface tension[END_REF] and the references therein. An alternative is to set this problem in the general framework of viscosity solutions with a correct viscosity condition on the interface, see [START_REF] Kim | Uniqueness and existence results on viscosity solutions of the Hele-Shaw and the Stefan problems[END_REF][START_REF] Kim | Homogenization of a Hele-Shaw problem in periodic and random media[END_REF]. Surface tension may also be included [START_REF] Escher | Classical solutions for Hele-Shaw models with surface tension[END_REF][START_REF] Friedman | A hierarchy of cancer models and their mathematical challenges[END_REF][START_REF] Alikakos | Convergence of the Cahn-Hilliard Equation to the Hele-Shaw Model[END_REF], then the Dirichlet boundary condition has to be changed to p ∞ = aκ(x, t) with a a parameter and κ the mean curvature.

As we mentioned earlier, the biophysical modeling gives growth terms G that depend on p, and not on n as in [START_REF] Bénilan | The mesa problem for Neumann boundary value problem[END_REF] for instance. Remarkably, this property allows us to extend nicely the usual Hele-Shaw theory and recover the semi-linear elliptic equation [START_REF] Bouin | A Hamilton-Jacobi approach for a model of population structured by space and trait[END_REF]. A recent interest for the Hele-Shaw equation also arises in other fields of mathematics with the stochastic Loewner evolutions, Laplacian growth, diffusion limited aggregation, etc

Weak formulation

Besides the free boundary formulation, there is also a weak formulation of the limit γ → ∞ in the equation ( 1). This limit gives a more general setting allowing a 'pretumor zone' where healthy and tumor cells are present in a mixed state. This weak formulation was derived in [START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF] and leads to the equation

       ∂ ∂t n ∞ -div n ∞ ∇p ∞ = n ∞ G p ∞ (x, t) , x ∈ R d , t ≥ 0, n ∞ (x, t = 0) = n 0 ∞ (x) ≥ 0, p ∞ (1 -n ∞ ) = 0, 0 ≤ n ∞ ≤ 1. (13)
In other words, when n ∞ < 1 then p ∞ = 0. Consequently, n ∞ and p ∞ are so weakly related that their dynamics can be somewhat independent. Nevertheless, a remarkable property is that the weak solution of ( 13) is unique (see [START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF]).

To present the result, we now insert the index γ to the notations n and p for the solutions of (1). The following result holds Theorem 3.1 (Hele-Shaw limit, [START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF]) With the assumptions of Proposition 2.1, as γ → ∞, we have

n γ → n ∞ ≤ 1, p → p ∞ ≤ P h a.e. in R d × (0, ∞), ∇p γ ⇀ ∇p ∞ in L 2 R d × (0, T ) -weak, ∀T > 0, ∂ ∂t n ∞ ≥ 0, ∂ ∂t p ∞ ≥ 0.
The limit of equation (1) is equation [START_REF] Bresch | A viscoelastic model for avascular tumor growth[END_REF].

Notice that, from the BV (bounded variation) properties of n γ and p γ in Proposition 2.1, we derive strong compactness. We also conclude that

n ∞ ∈ L ∞ (0, T ); L 1 ∩ L ∞ (R d ) , p ∞ ∈ L ∞ (0, T ) × R d ∩ L 1 (0, T ) × R d
and that, as measures although we use the notation of

L 1 functions, |∇n ∞ (x, t)| and |∇p ∞ (x, t)| are bounded with R d |∇n ∞ (x, t)|dx ≤ K 0 e G M t , T 0 R d |∇p ∞ (x, t)|dxdt ≤ C(T, P h , K 0 ).
The other results follow immediately. For example, because , we find the relation

n γ p γ = n γ+1 = p
n ∞ ∇p ∞ = ∇p ∞ , /quada.e..
In other words, the equation on n ∞ , in [START_REF] Bresch | A viscoelastic model for avascular tumor growth[END_REF], can also be written

∂ ∂t n ∞ -∆p ∞ = n ∞ G p ∞ (x, t) .
This is the form used in [START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF] to prove uniqueness of weak solutions.

A more difficult result is the derivation of the 'complementary relation', ( 14) below, which is equivalent to the strong convergence of ∇p γ .

∇p γ → ∇p ∞ in L 2 loc R d × (0, ∞) -strong,
The 'complementary relation' also holds

p ∞ ∆p ∞ + G(p ∞ ) = 0 in D R d × (0, ∞) . ( 14
)
The complementary relation [START_REF] Byrne | Free boundary value problems associated with the growth and development of multicellular spheroids[END_REF] is not an obstacle problem (a sign is incompatible) and the solution is not unique. It is a weak version of the equation [START_REF] Bouin | A Hamilton-Jacobi approach for a model of population structured by space and trait[END_REF] with

Ω(t) = p ∞ (x, t) > 0 , (15)
as set which evolution cannot be deduced from ( 14), but from the weak formulation [START_REF] Bresch | A viscoelastic model for avascular tumor growth[END_REF]. the meaning, in distributions, of ( 14) is that for all smooth test functions ϕ with compact support, it holds

R d ×(0,∞) ϕ(x, t) -|∇p ∞ | 2 + p ∞ G(p ∞ ) - R d ×(0,∞) p ∞ ∇ϕ.∇p ∞ = 0
which makes sense with the available regularity for p ∞ in Proposition 2.1.

The proof of Theorem 3.2 relies on a functional analysis argument which uses the L ∞ control from below for ∂ ∂t n γ ≥ 0 as given in Proposition 2.2.

From the weak formulation to the free boundary statement

To begin withn notice that 1 I {Ω(t)} = 1 I {n∞(x,t)=1} . Indeed, on the one hand, 1 I {Ω(t)} ⊂ 1 I {n∞(x,t)=1} . On the other hand, when p ∞ = 0, then from (13), we conclude that ∂ ∂t n ∞ = n ∞ G M , which means that we cannot have n ∞ (x, t) = 1 otherwise n ∞ would continue to grow thus contradicting the bound n ∞ (x, t) ≤ 1.

Therefore, when n ∞ (x, t) takes the values 0 or 1 only, then we have

n ∞ (x, t) = 1 I {Ω(t)} . (16)
In this situation and assuming some smoothness for Ω(t), it is easy to derive the Hele-Shaw free boundary formulation mentioned in Section 3.1. This is written in details (and in more generality in the sense below) when Ω(t) is a ball in [START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF], then one can establish precisely the speed of the free boundary given by [START_REF] Bouin | Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration[END_REF].

However, the weak formulation contains more than the free boundary statements [START_REF] Bouin | Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration[END_REF], ( 12) which only holds true when initially n 0 = 1 I {Ω(t=0)} so as to ensure [START_REF] Byrne | A two-phase model of solid tumor growth[END_REF]. One can formally see this, because in the interior of Ω(t), we can write ∂ ∂t n ∞ = 0 and thus the weak formulation [START_REF] Bresch | A viscoelastic model for avascular tumor growth[END_REF] gives immediately the elliptic equation [START_REF] Bouin | A Hamilton-Jacobi approach for a model of population structured by space and trait[END_REF]. But, if there is a zone where n 0 < 1, then we still have n ∞ (x, t) < 1 for some time. In this space-time zone, we have p ∞ = 0 and ( 13) is reduced to the simple differential equation

∂ ∂t n ∞ = n ∞ G M .
A numerical simulation, illustrating this interpretation is displayed in Figure 1.

A similar, but less complete, theory can be carried out for the case with active motion [START_REF] Benzekry | Maximum Tolerated Dose Versus Metronomic Scheduling in the Treatment of Metastatic Cancers[END_REF], see [START_REF] Perthame | Derivation of a Hele-Shaw type system from a cell model with active motion. Interfaces and Free Boundaries[END_REF], and for the system with nutrient (8) and furthermore, the permanent shape, given by a traveling wave can be written exactly [START_REF] Perthame | Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient[END_REF].

Adaptation and resistance to drugs

Besides mechanical aspects which we have presented so far, mathematical models of tumor growth also deal with questions which are more connected to biology than mechanics, and resistance to treatment is a typical example. The subject of resistance is considered presently as one of the challenges is medical treatment (see [START_REF] Kimmel | Control theory approach to cancer chemotherapy: benefiting from phase dependence and overcoming drug resistance[END_REF][START_REF] Tomasetti | An elementary approach to modeling drug resistance in cancer[END_REF][START_REF] Lorz | Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors[END_REF][START_REF] Lorz | Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies[END_REF] and the references therein).

A possible modeling of this phenomena is related to Darwinian evolution and to selection of the fittest traits. A subject that bridges probability [START_REF] Champagnat | From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution[END_REF] for finite populations, game theory as introduced by J. Maynard Smith and PDEs, the formalism we use below.

Population adaptive dynamic

In the view of [START_REF] Lorz | Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors[END_REF][START_REF] Lorz | Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies[END_REF], cells are assumed to carry a resistance phenotype y ∈ [0, 1]. In the simplest description, one considers the population density n(y, t), this is usually called a structured population, [START_REF] Perthame | Transport equations in biology[END_REF]. One can postulate an equation for the dynamic of n(y, t), expressing birth and death of cells. A general, yet simple, formalism is, following [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF][START_REF] Mirrahimi | A singular Hamilton-Jacobi equation modeling the tail problem[END_REF], to write a type of Lotka-Volterra equation

∂ ∂t n(y, t) = n(y, t)R(y, ρ(t)) + µ∆n(y, t), ρ(t) = 1 0 n(y, t)dy,
with Neuman boundary conditions (these are somewhat artificial but simplify the presentation). The diffusion term stands for mutations; several other forms are possible as integral operators [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF] and, as well as diffusion, can de derived from stochastic individual models [START_REF] Champagnat | From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution[END_REF]. Again the choice of diffusion is made for simplicity. The term R(y, ρ) represents the growth rate (death and birth), an example being

R(y, ρ) = b(y) -ρ k(y) -d(y) c th , (17)
with b(•) the intrinsic division rate, d(•) the death rate induced by the therapeutic drug given with the concentration c th . Finally, k(•) represents the death rate due to competition, for space and nutrients, with all the cells whatever is their resistance level. Therefore, in the general setting, we assume that, for some constant α > 0, ∂ ∂ρ R(y, ρ) ≤ -α < 0.

Then, according to the interpretation of y as a resistance gene expression, we can assume some kind of resource allocation. When a cell uses energy to generate resistance, there is less energy for the cell division cycle, therefore we have

b ′ (•) < 0, d ′ (•) < 0, k ′ (y) < 0,
the last assumption means that resistant cells are also better competitors (an assumption that could be released by introducing another phenotypic trait).

The main qualitative property of solutions is better stated with a renormalization of time according to the scale µ = ε 2 µ 0 , t new = εt old , with t old the generation time, t new the evolution time. This renormalization leads to re-write the equation on n(y, t) as

ε ∂ ∂t n ε (y, t) = n ε (y, t)R(y, ρ ε (t)) + ε 2 µ 0 ∆n ε (y, t), ρ ε (t) = 1 0 n ε (y, t)dy. ( 18
)
This rescaling is standard in parabolic equation, in particular because it is the basis for deriving various front motions, see [START_REF] Evans | A PDE approach to geometric optics for certain semilinear parabolic equations[END_REF][START_REF] Souganidis | Front propagation: theory and applications, CIME course on 'viscosity solutions[END_REF] for instance.

The analysis carried out in [START_REF] Diekmann | The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach[END_REF][START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF][START_REF] Mirrahimi | A singular Hamilton-Jacobi equation modeling the tail problem[END_REF] leads to use two main tools. The first one is a uniform Total Variation bound (TV in short) on ρ ε (t)

0 < c ≤ ρ ε (t) ≤ C, T 0 ρε (t) dt ≤ C. (19)
The lower bound expresses non-extinction and can be recovered a posteriori, it is however convenient to have it proved directly when this is possible. The BV bound is needed for nonlinear dependence on ρ in R(x, ρ); it is not fundamental for the case [START_REF] Champagnat | The evolutionary limit for models of populations interacting competitively with many resources[END_REF] for instance. The second tool is the WKB change of unknown

u ε (y, t) = ε ln n ε (y, t)
and according to the observation of natural selection, as is standard in adaptive dynamics [START_REF] Diekmann | A beginner's guide to adaptive dynamics[END_REF], the population should be highly concentrated around the fittest trait (think of a Gaussian). Then, initially one assumes that for some y 0 ∈ (0, 1),

     n 0 ε (y) -→ ε→0 δ(y -y 0 ) (weakly), u 0 ε is bounded in Lip(0, 1), u 0 ε -→ ε→0 u 0 , max 0<y<1 u 0 (y) = u 0 (y 0 ) (strict maximum). (20)
This initial concentration effect remains true for all times under structural assumptions on (e.g. assuming that R is monotonic in y as in [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF], or that R is concave in y as in [START_REF] Lorz | Dirac mass dynamics in multidimensional nonlocal parabolic equations[END_REF]). Then, it is established that

     ρ ε (t) -→ ε→0 ρ(t) ∈ L ∞ ∩ T V (0, +∞), a.e. n ε (y, t) -→ ε→0 ρ(t)δ y -y(t) . (21)
Figure 2: No therapy. Solution of system ( 18) with µ 0 = 0 for R(y, ρ) = 3 2 -y + ρ 1.5+y , and departing from a distribution concentrated near y = .95 as a Gaussian with parameter ε = 0.02. Left: the isovalues of n(y, t), abscissae are y and ordinates are t. Center: the function t → ρ(t). Right: the distribution n(y, t final ) at t final = 20, which concentrates at the point y = y ∞ = 0.

The next question is to characterize the dynamic of the two unknowns ρ(t) and y(t). The answer is expressed through the limiting behavior of u ε (t). Still under technical assumptions depending on the case at hand (monotonic or concave function R), one has u ε (y, t) -→ ε→0 u(y, t) uniformly, locally in time, and the functions u(y, t) and ρ(t) satisfy the constrained Hamilton-Jacobi equation

           ∂ ∂t u(y, t) = R y, ρ(t) + µ 0 |∇u| 2 , 0 < x < 1, t ≥ 0, max 0≤y≤1 u(y, t) = 0 = u y(t), t , u(y, t = 0) = u 0 (y), (22)
with Neuman boundary conditions (note that only cases in the full line have been studied so far). The interpretation is as follows: ρ(t) is a Lagrange multiplier associated with the algebraic constraint that max y u(y, t) = 0. For this reason, the usual property of contraction in L ∞ of Hamilton-Jacobi equations is lost in the case with a constraint. However Lipschitz bounds for u(y, t) are still available (and motivate the corresponding assumption in [START_REF] Clairambault | Circadian rhythm and cell population growth[END_REF]) and are enough to prove existence of a viscosity solution. Uniqueness is only known in the particular case when R has a specific form as in [START_REF] Champagnat | The evolutionary limit for models of populations interacting competitively with many resources[END_REF], see [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF].

Canonical equation and evolutionary stable distribution

One can go further (to the expense of more regularity on u, a condition that can be proved with concavity assumptions on R) and establish the form of canonical equation ( [START_REF] Diekmann | A beginner's guide to adaptive dynamics[END_REF]) as follows:

   R y(t), ρ(t) = 0, ẏ(t) = -D 2 u(y(t), t) -1 • D y R(y(t), ρ(t)).
Because R is invertible in ρ, the first equation gives ρ(t) as a function of y(t), and then the ordinary differential equation for y(t) is in closed form when u is known. This form of a canonical equation is not explicit as long as u is not computed, but can however give some information on the sign of ẏ(t) and on the long term dynamics of y(t). When a steady state is attained, it is called the evolutionary stable distribution [START_REF] Jabin | On selection dynamics for competitive interactions[END_REF] (ESD in short), a notion closely related the evolutionary stable strategy in adaptive dynamics [START_REF] Diekmann | A beginner's guide to adaptive dynamics[END_REF].

For instance, for the case of [START_REF] Champagnat | The evolutionary limit for models of populations interacting competitively with many resources[END_REF] with no therapy, c th = 0, and weak competition compared to proliferation (|b ′ | large compared to |k ′ |, then we find ẏ(t) ≤ 0 as b ′ < 0, because D 2 u(y(t), t) ≤ 0 at a maximum point. And we conclude that less resistant cells are selected. The dynamics will stop at an ESD (ρ ∞ , y ∞ ) which achieves both conditions

R y ∞ , ρ ∞ = 0, D y R(y ∞ , ρ ∞ ) = 0, (23)
and this value y ∞ corresponds to a maximum of R in y. Here we should emphasize that because µ 0 = 0, the restriction that y ∈ (0, 1) is only useful for the biological interpretation. Mathematically, the dynamics might lead y(t) to become negative. This ESD is illustrated by Figure 2; with the rate function R(y, ρ) used in this figure, one can readily check that y ∞ = 0. For a strong drug concentration, c th large, then one finds on the contrary ẏ(t) ≥ 0 as -d ′ > 0 and thus resistant cells are selected which escape therapy if ρ(t) does not vanish. Here, it might occur that, for c th large enough, then ρ(t) vanishes and the lower bound in [START_REF] Ciarletta | The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis[END_REF] fails; this can be interpreted as recovery. In such a case, the constraint in the constrained Hamilton-Jacobi equation [START_REF] Clairambault | Optimal control of cancer chemotherapies to overcome drug resistance[END_REF] does not hold because the Lagrange multiplier is fixed at ρ(t) = 0. The Figure 3 illustrates a case where resistance occurs and the ESD, characterized by [START_REF] Colijn | Bifurcation and bistability in a model of hematopoietic regulation[END_REF], is for y ∞ = 1.

The effect of a multi-therapy to prevent resistance can be included in the model and gives rise to the following extension

∂ ∂t n(y, t) = n(y, t) b(y) 1 + c Stat -ρ(t) k(y) -d(y) c Tox , ρ(t) = 1 0 n(y, t)dy. (24)
Here two types of effects are taken into account; c Tox represents the cytotoxic drugs which induce apoptosis (usually by DNA damage during the Synthesis phase of the cell cycle), and c Stat represents cytostatic effects which slow down the cell cycle (for instance using molecules that inhibit cyclines). Then the question is to determine the optimal scheduling c Stat (t), c Tox (t) with constraints on total toxicity. This is studied in [START_REF] Clairambault | Optimal control of cancer chemotherapies to overcome drug resistance[END_REF] with the constraints to keep a high enough population of healthy cells.

Space structure and heterogeneity

Selection of a monomorphic population (that means a single Dirac mass) as derived before is compatible with the Gause competitive exclusion principle which is used in ecology; with N environmental variables, a bioreactor can sustain N interacting species, [START_REF] Champagnat | The evolutionary limit for models of populations interacting competitively with many resources[END_REF]. Here N = 1 and the environmental variable is just measured by the total population. However, genetic tests show a wide heterogeneity in tumor cells. Several explanations are possible as random mutations which generate a peaked distribution n(y, t) but not exactly a Dirac mass; the parameter ε is small but not zero. Another possible explanation is spacial heterogeneity within tumor environment due to local availability of nutrients.

In order to write corresponding equations, one should describe population densities n(x, y, t) where x stands for the position and y for the phenotypical trait. With space and trait, to derive statements similar to those in ( 21)-( 22) is a much more recent topic, see [START_REF] Bouin | Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration[END_REF][START_REF] Mirrahimi | Dynamics of sexual populations structured by a space variable and a phenotypical trait[END_REF][START_REF] Bouin | A Hamilton-Jacobi approach for a model of population structured by space and trait[END_REF], with unexpected outcomes and difficulties. New phenomena as accelerating waves occur and mathematically, a priori bounds are more complicated because they should reflect the L 1 theory in the trait and the L ∞ theory in space. The model proposed in [START_REF] Lorz | Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors[END_REF] contains aspects coming both from the spatial model with nutrient (8) and the evolutionary aspects introduced in Section 4.1. To begin with, we present a simpler version, taken from [START_REF] Mirrahimi | Asymptotic analysis of a selection model with space[END_REF], which explains the expected behavior of the solutions. We denote by n ε (y, x, t) the population density of cells which are located at the position x, with the trait y. To simplify we choose y ∈ (0, 1) and x ∈ R to simplify the statements. The spatial dependence determines local conditions for trait adaptation, according to available nutrient concentration c(x, t). Following the rescaling proposed in [START_REF] Champagnat | From Individual Stochastic Processes to Macroscopic Models in Adaptive Evolution[END_REF], we write

ε∂ t n ε (y, x, t) = [r(y)c ε (t, x) -d(y)(1 + ̺ ε (x, t))] n ε (y, x, t), x ∈ R, 0 < y < 1, t ≥ 0, (25)
∂ ∂t c ε -∆ x c ε (x, t) + [̺ ε (x, t) + λ] c ε (x, t) = λc B , x ∈ R, t ≥ 0, (26)
̺ ε (x, t) = n ε (y, x, t)dx, x ∈ R, t ≥ 0. ( 27
)
In other words, we have chosen k(y) = d(y) in ( 17), neglected mutations and added a parameter x which dependency is ruled by a parabolic PDE. To handle the asymptotic behavior in [START_REF] Colin | System identification in tumor growth modeling using semi-empirical eigenfunctions[END_REF], the main difficulty is to find strong estimates for ̺ ε (x, t). Uniform L ∞ bounds are immediate but strong compactness, as derived from [START_REF] Ciarletta | The radial growth phase of malignant melanoma: multi-phase modelling, numerical simulations and linear stability analysis[END_REF] in the x-independent case, are not available. With technical assumptions that we skip here, it is proved in [START_REF] Mirrahimi | Asymptotic analysis of a selection model with space[END_REF], that there is ̺(y, t), X(y, t) such that A qualitative consequence is heterogeneity which is expressed by the phenotypes Y (x, t), x ∈ R, which are represented at a time t.

To be closer to the case of tumor treatment, the system used in [START_REF] Lorz | Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors[END_REF] includes additional matter. The space variable represents the distance to the center, effect of therapeutic drugs are included (following the ideas leading to the equation ( 24)), and both nutrients and therapy are delivered from a vasculature on the boundary of the tumor.

Conclusion

One should keep in mind that mathematical biology is not a recent subject. It has a long record of success as the Lotka-Volterra equations in ecology, statistics and random processes in genetics, the Turing instability for pattern formation and developmental biology, the Hodgkin-Huxley system for electric pulse propagation along nerves, the Keller-Segel system for cell chemotaxis, and many others. Subjects as epidemiology, population genetics, neuroscience use mathematical models for a long time. Biofluids, biomechanics are now well established subjects with applications to medicine and medical industry. Even though more recent, mathematics motivated by questions around tumor growth are now numerous and a search on publications data basis shows a fast growing activity in the field. This fast development, can be observed in many other fields of life sciences under two effects. Biologists have now access to new experimental devices giving enormous quantities of data as images; data analysis is needed to handle them and mathematical modeling is needed to give sense to them. Physicists have entered the field massively and have now access to simplified living systems; it might be simpler for mathematicians to speak with them. However, because of the specificities of the living matter, classical models must be revisited with new variants. But new questions, on new models, also appear which require to develop new mathematical tools. This paper is an attempt to show these two faces.

Figure 1 :

 1 Figure 1: Numerical simulations at three different times of the system (1) with γ = 40 and G(p) = 5(2 -p). The density n is plotted in solid line whereas the pressure p is represented with -+ -(smoother curve).

  the strong limits, we find in the limit the relation p ∞ (1 -n ∞ ) = 0. Another property follows immediately from the same argument; because n γ ∇p γ = ∇p γ+1 γ γ

Theorem 3 . 2 (

 32 Complementary relation) Additionally to Theorem 3.1, one also has

Figure 3 :

 3 Figure 3: With therapy. Solution of system (18) with µ 0 = 0 for R(y, ρ) = 3 2 -y + ρ 1.5+y -C Tox (1-x), and departing from a distribution concentrated near y = .05 as a Gaussian with parameter ε = 0.02. Left: the isovalues of n(y, t), abscissae are y and ordinates are t. Center: the function t → ρ(t). Right: the distribution n(y, t final ) at t final = 180, which concentrates at the point y = y ∞ = 1.

c

  ε (x, t) -→ ε→0 c(x, t) locally uniformly, ̺ ε (x, t) -→ ε→0 ̺(x, t) pointwise, n ε (y, x, t) -→ ε→0 ̺(x, t)δ y -Y (x, t) weakly in measures.