Hamid Ladjal
email: hamid.ladjal@liris.cnrs.fr

Jean-Luc Hanus
email: jean-luc.hanus@ensi-bourges.fr

Antoine Ferreira
email: antoine.ferreira@ensi-bourges.fr

Microrobotic Simulator for Assisted Biological Cell Injection

Keywords: micro-injection ICSI, finite element, modeling, real-time interaction, haptic feedback

This paper presents a complete computer-based biological training system to simulate intracytoplasmic sperm injection (ICSI) procedures in virtual environments (VEs). Using the proposed system, the user can practice and improve its control by developing a gesture similar to that performed in reality. In order to minimize the risk of damaging membranes during the injection, the needle must be removed quickly and safely. The human-machine user's interface provides an efficient haptic-based guidance tool. The key challenge of deformable simulation is to satisfy the conflicting requirements of real-time interactivity and physical realism. The compromise between the realism of bio-mechanical models used, the accuracy and stability of computational algorithms for real-time haptic rendering are fullfilled with the use of a mechanical modeling based on an hyperelastic model (St-Venant-Kirchhoff) and the writing of a specific dynamic finite element code. Simulated responses are compared to experimental data. These comparisons show the effectiveness of the proposed physically-based model.

INTRODUCTION

Cell manipulation is a prevalent process in the field of molecular biology. This process plays an important role in intracytoplasmic sperm injection (ICSI), pronuclei deoxyribonucleic acid (DNA) injection, therapeutic and regenerative medicine, and other biomedical areas. ICSI has progressively replaced all other micro-injection procedures for overcoming intractable male-factor infertility and has emerged in a relatively short time as a routine procedure for many in-vitro procedures. Nevertheless, injections are currently performed manually and technicians are required to be skillful enough to not destruct the cell structure of the ovum during the ICSI process. As a consequence, the success rate remains relatively low and strongly dependant on the experience and skills of the operator [START_REF] Kastrop | Comparison between intracytoplasmic sperm injection and in-vitro fertilization (IVF) with high insemination concentration after total fertilization failure in a previous IVF attempt[END_REF], [START_REF] Dumoulin | Embryo development and chromosomal anomalies after ICSI effect of the injection procedure[END_REF], [START_REF] Ebner | A prospective study on oocyte survival rate after ICSI influence of injection technique and morphological features[END_REF]. Various cell injection systems have been developed to provide more controllable manipulation of biological cells [START_REF] Yu | Microrobotic Cell Injection[END_REF], [START_REF] Liu | Vision-based cellular force measurement using an elastic microfabricated device[END_REF]. If visual servoing is a necessary condition for precise micro-injection operations, the knowledge of interactive forces acting during pipette insertion plays an important role too since it can be used to provide force feedback for precise regulation of needle penetration speed and stength [START_REF] Pillarisetti | Evaluating the Effect of Force Feedback in Cell Injection[END_REF]. Reducing cell deformation and needle deflection will also lower the risk of cell damage. However, when manipulating deformable biological objects, force sensor measurement will provide only the local forces at the pipette puncture point which limits strongly the operator haptic rendering [START_REF] Pillarisetti | Evaluating the Effect of Force Feedback in Cell Injection[END_REF]. In order to learn, train and analyze basic cell injection procedures, most operators are aiming to use artificial cells because practicing on human oocyte has ethical concerns and potential risks [START_REF] Ladjal | Interactive Cell Injection Simulation Based on 3D bio-mechanical Tensegrity Model[END_REF]. The development of computer-based systems using visual and force-feedback for both injection training and injection assistance will help to overcome some of these problems. Similar research on virtual reality environments platforms has been carried out at a macroscopic scale in surgery simulation and modeling of soft tissues [START_REF] Kim | Virtual surgery simulation for medical training using multi-resolution organ models[END_REF], [START_REF] Satava | Surgical education and surgical simulation[END_REF], [START_REF] Meier | Real-time deformable models for surgery simulation: a survey[END_REF]. Needle punctures and passes through different tissue layers such as skin, muscle, fatty and connective tissue have been modeled [START_REF] Alterovitz | Planning for steerable bevel-tip needle insertion through 2D soft tissues with obstacles[END_REF], [START_REF] Nienhuys | A computational technique for interactive needle insertions in 3D nonlinear material[END_REF], [START_REF] Goksel | 3D simulation of needletissue interaction with application to prostate brachytherapy[END_REF], [START_REF] Goksel | Needle insertion modeling and simulation[END_REF]. By analogy to in vivo intracytoplasmic sperm injection, similar boundary conditions occur. Within the framework of microinjection, the cell geometry and bio-mechanical subcomponents properties (biomembrane, cytoskeleton, cytoplasm, nucleus) are of major importance in simulation and modeling because these factors affect the amount of cell membrane deformation, needle deflection and interaction forces [START_REF] Bolhassani | Needle insertion into soft tissue : A survey[END_REF], [START_REF] Pillarisetti | Evaluating the Effect of Force Feedback in Cell Injection[END_REF]. A nonlinear bio-mechanical cell model is required to render in a realistic way the interactions between the needle and the cell for accurate insertion [START_REF] Ammi | Evaluation of 3D Pseudo-Haptic Rendering using Vision for Cell Micromanipulation[END_REF]. The objective of this paper is to develop and implement a bio-mechanical finite element approach dedicated to real-time cell injection to facilitate training of ICSI operations. We developed two models based either on an explicit linear finite element model or on a non-linear finite element Saint-Venant-Kirchooff material. These models include the topological information of the living cells (shape and dimensions), and the biological structure (cytoplasm layers, cytoskeletons and nucleus) [START_REF] Ladjal | Interactive Cell Injection Simulation Based on 3D bio-mechanical Tensegrity Model[END_REF], [START_REF] Lulevich | Cell Mechanics using Atomic Force Microscopy-based Single-Cell Compression[END_REF], [START_REF] Ladjal | Reality-based real-time Cell Indentation Simulator[END_REF]. This paper is organized as follows: in Section II, we introduce the main requirements for ICSI trainers. In Section III, we present the bio-mechanical finite element approach dedicated to real-time injection. In Section IV, we present the methodology of the virtual environment system for cell injection. In Section V, we make a comparative study between our real-time cell injection simulator and experimental data. Finally, we give some concluding remarks and the directions for future work.

INTRACYTOPLASMIC SPERM INJECTION TRAINERS

The intracytoplasmic sperm injection (ICSI) is one of the most important advances in assisted reproductive technology in recent years. ICSI is considered a breakthrough in the treatment of male infertility issues. Conventional methods of ICSI injection require the operator to undergo long training (over a period of one year), and the success rates are low (around 10% -15%) due to poor reproducibility. The fragile nature of a biological cell requires the operator to be efficient; otherwise the patrician may damage the cell. There are also chances of contamination due to direct human involvement. The drawbacks involved in conventional methods have motivated the research community to develop robotic-based tools to perform cell injection tasks. One of the main features of these systems is to have better control on the movement of the micro-pipette used for injecting cells. In order to facilitate training of biological cell injection operations, we have developed and implemented an interactive virtual tool for clinical and ethical formation. The operator improves his gesture in a similar way to that carried out in reality without involving ovocyte tests coming from animals or humans. The design of such an environment of simulation requires to satisfy the conflicting requirements of realtime interactivity and physical realism. A key challenge of deformable simulation is to find a compromise between the complexity of the realistic models and the computing speed. The proposed physics-based simulator aims to accurately restore bio-mechanical characteristics of the micro-injection task in a three-dimensional virtual environment.

BIOMECHANICAL MODELS FOR CELL INJECTION

We implemented and compared two models: the linear explicite finite element model and the hyperelastic model. In this section, we present the numerical development of the linear elastic mass-tensor model [START_REF] Delingette | Soft tissue modelling for surgery simulation[END_REF], [START_REF] Cotin | A Hybrid Elastic Model for Real-Time Cutting, Deformations, and Force Feedback for Surgery Training and Simulation[END_REF], and its extension to non-linear elasticity [START_REF] Picinbono | Non-linear and Anisotropic Elastic Soft Tissue Models for Medical Simulation[END_REF] with tetrahedral linear elements. For an isotropic elastic or St Venant-Kirchhoff hyperelastic material the elastic energy, noted W , can be written as:

W (E) = λ 2 ( trE ) 2 + µ tr ( E 2 ) ( 1
)
where E is the Green-Lagrange strain tensor. A displacementbased finite element solution is obtained with the use of the principe of virtual works [START_REF] Bathe | Finite Element Procedure[END_REF]. Using finite element method (FEM) notations, inside each tetrahedron T k , the displacement field is defined by a linear interpolation

[ N k ] of the nodal displacement vector { u k
} of the four vertices of tetrahedron:

{ U(x) k } = [ N k (x) ] { u k } (2)

A. Linear elastic finite element model

In this section, we introduce the fundamental concepts for description and measurement of the deformation using finite element. For small deformations, the Green-Lagrange strain tensor is linearized into the infinitesimal strain tensor. The elastic energy is only a quadratic function of the displacement field. The elementary submatrices connecting the elementary force acting on the node i, to the displacement of the node j are:

[

K k ij ] = 1 36 V k ( λ { m i } { m j } T + (3) µ { m j } { m i } T + µ { m i } T { m j } [ I ] )
where { m } are unit outward-pointing normals to triangular faces and V k is the volume of the tetrahedron T k . Taking into account the contribution of all adjacent tetrahedra, the global internal force acting on a node l can be expressed as follows:

{ F l int } = ∑ k∈V l ( 4 ∑ j=1 [ K k ij ] { u j } ) (4)
where V l is the neighborhood of vertex l (i.e. the tetrahedra containing node l).

B. Hyperelastic finite element model

The simplest hyperelastic material model is the Saint Venan-tKirchhoff model which is just an extension of the linear elastic material model to the nonlinear regime. The algorithm is built in the same way. Since the strain is now quadratic in displacement, the elastic energy is a fourth-order polynomial in displacement. The relation between nodal forces and nodal displacement is non-linear and can be written as:

{ F τ i } = 4 ∑ j=1 [ B τ ij ] { u j } F τ 1 + 4 ∑ j,k=1 { u k } { u j } T { C jki } + 1 2 ( { u j } T { u k } ) { C ijk } F τ 2 +2 3 ∑ j,k,l=0 D jkli { u l } { u k } T { u j } F τ 3
(5) where the global force can be written as:

F T i = F T 1 Linearf orce + F T 2 + F T 3 non-linearf orce (6)
where

[ B ij ] , { C ijk }
, D ijkl are respectively a matrix, a vector and a scalar. These tensors, depending on the remaining geometry and Lame's coefficients are constant. They can be pre-computed in an off-line phase. This is the essential advantage of the mass-tensor approach which makes it useful for real-time application.

C. Dynamic Model

The deformation of the deformable object is given by the displacement of the nodes according to the acting external and internal forces. In an interactive simulation the applied forces change in time and the virtual objects have to react to them in real time. Therefore, the FEM has to be simulated dynamically. The equation of motion of a vertex l of the cell mesh can be written:

M l { ül } + γ l { ul } + ∑ k∈V l ( 4 ∑ j=1 [ K k ij ] { u j } ) = { F l ext } (7)
where M l and γ l are respectively the mass and damping coefficients of each vertex. To solve the dynamic system, we tested different integration schemes (implicit and explicit) taking into account the tradeoff between real-time simulation and haptic stability requirements. We choose the explicit centered finite-difference scheme. 4. 3D REAL-TIME VIRTUAL REALITY BASED ICSI SIMULA-TOR Real time virtual reality based cell simulation system, the operator would be able to interact with the threedimensional model of cell using his sense of vision as well as actively manipulate using his sense of touch. Fig. 1 shows the architecture of the real-time virtual reality based cell ICSI simulator. This includes the computer generated mesh of oocyte, the needle, collision detection algorithm, physical-based models of deformable cell modeling and haptic interaction controller. The operator is able to interact with the three-dimensional (3-D) model of the cell using his sense of vision as well as actively manipulate using his sense of touch. Usually, realtime graphics translates to an update rate of 25 Hz, stable haptic interaction in virtual environments (VE's) requires much higher update rate of around 1 kHz. A significant difficulty of using the finite element technique for real-time simulation is most computationally costly, these mesh-based schemes require expensive numerical integration operation for the computation of the system stiffness matrices. We adopted a computational architecture with two simulation stages: (i) an off-line computation of stiffness matrices for each triangular element and (ii) visual and haptic interaction for real-time ICSI simulation.

A. Off-line computation

The proposed computational methodology is composed of an off-line pre-calculation step. The most costly and timeconsuming operations are realized during this step. The database contains geometric properties of oocyte cells using the micro-injection setup image analysis. The design geometric of oocyte cell models are based on a commercial computeraided design (CAD) package (MARC-ADAMS). The meshing of the 3D internal structure is then carried out through a dedicated 3D meshing software (GID software) modeled in exact dimensions. Although they were displayed as 3D texturemapped objects to the user, they were modeled as connected line segments to reduce the number of collision computations during real-time interactions. The oocyte cell was approximated as an assembly of discrete tetrahedral elements interconnected to each other through a fixed number of nodes (see Fig. 2). The displacements of these nodal points for applied external injection forces were the basic unknowns of our FEM analysis. The coordinates of vertices, the tetrahedron indexing, and the connectivity of vertices were derived from the geometric model. Then, the mechanical (young modulus, density) and geometrical (diameter, volume) properties of the cell structure are determined by a real tests carried out on a mouse oocyte [START_REF] Sun | Mechanical property characterization of mouse zona pellucida[END_REF]. Finally, these properties are injected in the FEM algorithm for calculating the physically-based behavior of biological soft tissues. The off-line calculation of all tensors K ij allows precalculation stiffness values for preselected cells. The stiffness tensors K ij should be calculated for a set of oocyte parameters.

B. Real-time haptics-enable simulator

Since haptics and graphics have different update frequencies, we implemented separate threads to update the loops. During the injection task, the contact between the tip and the cell must occur at a special set of points (nodal points). The collision detection module specifies the type of tool-tissue interaction. The tool in a virtual environment can be modeled as a complex 3D object, composed of numerous surfaces, edges and vertices. However, due its structure complexity, realtime interaction is very challenging. In our implementation, we used a point based representation of the pipette injector and employed a simple ray-triangle intersection with local search technique [START_REF] Moller | Fast, Minimum Storage Ray-Triangle Intersection[END_REF]. To optimize the collision detection algorithm and cell deformation, we refine our mesh only in a puncture area. The mesh refinement is carried out in the precomputational process. The visual feedback is updated after each displacement calculation step and the haptic feedback is returned after each force calculation step.

1) Virtual coupling for stability of the haptic rendering:

The internal operating loop of the haptic interface requires an update frequency around 1 KHz. However, the finite element model update frequency is in the range of 25 -30 Hz. This frequency difference threatens the coherence between both systems leading to instabilities of the user haptic rendering. We adopted the solution to use a virtual coupling model defined in [START_REF] Adams | Stability and performance of haptic displays : theory and experiments[END_REF]. This approach introduces a virtual passive link between the simulation model and the haptic interface in order to ensure the stability and the performance of the system Fig. (4.a). When we combine the impedance display implementation with an appropriate virtual coupling network, we get the admittance matrix for the combined interface. The linear two-port is said to be absolutely stable if exists no set of passive terminating one-port impedances for which the system is unstable. Llewellyn's stability criteria [START_REF] Llewellyn | Some fundamental properties of transmission systems[END_REF] provides both necessary and sufficient conditions for absolute stability of linear two-ports. We get the conditions for absolute stability of the haptic interface:

Re(Z di (z)) ≥ 0, 1 Z cvi (z) ≥ 0 (8) cos(∠ZOH(z)) + 2Re(Z di (z))Re( 1 Zcvi(z) ) |ZOH(z)| ≥ 1 (9)
Where Z cvi (z) is the virtual coupling impedance (k c , b c ), ZOH(z) is a zero order holder, and Z di (z) is the PHANToM impedance. The inequality (9) can be rewritten to get an explicit expression of absolute stability of the haptic interface: Re(

1 Z cvi (z) ) ≥ 1 -cos(∠ZOH(z)) 2Re(Z di (z)) |ZOH(z)| (10)
For the virtual coupling, the impedance display induces a limit on the maximum impedance which can be rendered. We use [START_REF] Bolhassani | Needle insertion into soft tissue : A survey[END_REF] to find the virtual coupling which makes the haptic interface absolutely stable, please refer to the authors previous work [START_REF] Ladjal | Reality-based real-time Cell Indentation Simulator[END_REF].

EXPERIMENTS AND VALIDATION

In order to test the accuracy and reliability of the proposed user interface system with haptics enabled simulation, we used experimental real data. The geometrical dimensions and the mechanical properties are determined through a microinjection setup [START_REF] Sun | Mechanical property characterization of mouse zona pellucida[END_REF]. The experimental data (shown in Fig. 5) depicts the force and deformation measurement process on a mouse oocyte ZP. The forces increase nonlinearly as deformation increase. When the deformation reaches about 45µm, the ZP and the plasma membrane are punctured, the puncturing forces are approximately 7.5µN [START_REF] Sun | Mechanical property characterization of mouse zona pellucida[END_REF]. The experimental data are plotted with 95 % prediction bounds (error bounds 5 %) for more precision (Fig. 5). The Fig. 6 presents the error between experimental data and regression curve. In order to demonstrate the validity of the proposed real-time finite element method, we compared the simulation results of a needle insertion task using the physics-based FEM model with the experimental data provided by a real test carried out on a mouse oocyte [START_REF] Sun | Mechanical property characterization of mouse zona pellucida[END_REF]. The mechanical and geometrical properties used in the finite element simulations are settled in Table .I.

The Fig. 7 presents the experimental data, the linear and non-linear finite element relationship between the needle insertion and the reaction force for mouse oocyte cells. The simulations show that the non-linear FE St-Venant-Kirchhoff model is in good agreement with experimental data. It can be clearly seen that the linear finite element model is valid only for small displacement (less than 10 % of the mesh size). A. Evaluation An experimental system was implemented on a personal computer with a Intel Core Duo 3.2 GHz CPU with 4 GB memory, and GeForce 8600 GTS with 512Mo memory. In this experiment, we investigated the time required for precomputation (off-line computation) and the real-time deformation process by changing the number of vertices of a cell model (Table.II). The relationship between the number of vertices and the off-line computation was identified by a nonlinear law or this relation between the number of vertices and realtime simulation was described by a linear law (see Fig. 9). However, The accuracy and computational time of the finite element analysis depends on the number of the vertices and tetrahedron included in the mesh. The problem should be solved by using graphics processing unit (GPU) to perform finer mesh simulations.

CONCLUSION

We have developed a computer-based training system to simulate real-time cell micro-injection procedures in virtual environments for training biologist residents. The simulator provides the user with visual and haptic feedback. The simulation of this procedure involves the real-time rendering, physically-based modeling of soft biological tissue, and display of touch and force sensations to the training through the simulation of haptic interactions. We first investigated the challenging issues in the real-time modeling of the biomechanical properties of the cell micro-injection through finite element models. Compared to experimental data performed on oocyte cells, we can see clearly that the proposed physicallybased FEM model is able to simulate the cell deformation through real-time simulation constraints. Currently, we are working on integrating others effects such as friction, viscosity, and adhesion forces, and the simulator is prepared to doing an experimental training evaluation from trained and non-trained candidates. All modalities will be merged in an ergonomic and intelligent biological simulator to support learning microinjection and training tasks.

Fig. 1 .

 1 Fig. 1. Computational architecture for simulating force-reflecting deformable cell micro-injection in a virtual environment. The figure shows the two simulation phases used for the real-time micro-injection of the cell: (i) off-line pre-calculations of stiffness matrices and (ii) simulation of visual and haptic interaction.

Fig. 2 .

 2 Fig. 2. Virtual graphic of the ICSI environnement: the 3D tetrahedral mesh of the mouse oocyte cell.

Fig. 3 .

 3 Fig. 3. Scheme illustrating the collision detection algorithm using a simple triangle points intersection problem: needle intersection with triangle surface of the cell, and the refining of the mesh.

Fig. 4 .

 4 Fig. 4. Stability conditions using the virtual coupling model. (a); The model of virtual coupling, (b); Impedance display virtual coupling

Fig. 5 .

 5 Fig. 5. Experimental data: force versus needle insertion deformation of the mouse oocyte cell with 95 % prediction bounds.

Fig. 6 .

 6 Fig. 6. Error between experimental data and regression curve.

Fig. 7 .

 7 Fig. 7. Visual comparison of the various stages of deformation of needle insertion between FEM simulations and experimental data.

Fig. 8 .

 8 Fig. 8. Force versus deformation: comparison between experimental data, non-linear St Venant Kirchhoff and linear finite element simulations.

Fig. 9 .

 9 Fig. 9. Computation time versus vertices number: Pre-computation time and real-time simulation.

TABLE I MECHANICAL

 I AND GEOMETRICAL PROPERTIES OF MOUSE OOCYTE.

		Mechanical and Geometrical Properties
	Young modulus	17.9KP a
	Poisson coefficient	0.49
	Diameter	56µ m
	The mass of the cell	9.1952 10 -5 mg
	Vertex mass	0.07995 10 -5 mg
	Cell volume	91952 µ m 3
	Mesh of the cell	115 vertices and 332 Tetra
	Maximum deformation	40 ∼ 44µ m


S Benzekry

N André

A Benabdallah

J Ciccolini

C Faivre

F Hubert
email: fhubert@cmi.univ-mrs.fr

D Barbolosi

Modeling the impact of anticancer agents on metastatic spreading

Keywords: modeling, metastases, anti-angiogenic therapy, metronomic chemotherapy AMS subject classification: 35F16, 92C50

Treating cancer patients with metastatic disease remains an ultimate challenge in clinical oncology. Because invasive cancer precludes or limits the use of surgery, metastatic setting is often associated with (poor) survival, rather than sustained remission, in patients with common cancers like lung, digestive or breast carcinomas. Mathematical modeling may help us better identify non detectable metastatic status to in turn optimize treatment for patients with metastatic disease. In this paper we present a family of models for the metastatic growth. They are based on four principles : to be as simple as possible, involving the least possible number of parameters, the main informations are obtained from the primary tumor and being able to recover the variety of phenomena observed by the clinicians. Several simulations of therapeutic strategies are presented illustrating possible applications of modeling to the clinic.

Introduction

Classification of cancer as localized or metastatic disease remains the mainstay for determining the best therapeutic strategy to be undertaken at bedside. Although simple, this classification may underestimate the risk of undetectable, early metastatic stages [START_REF] Comen | Clinical implications of cancer self-seeding[END_REF]. Many studies have been per-formed to identify prognostic biomarkers for evaluating the risk of metastatic relapse in patients with apparently localized disease. We have developed original mathematical models designed to evaluate the metastatic state of the patient. These models including the efficacy of some anticancerous drugs, are designed to help clinicians to define the best therapeutic strategy.

Our approach consists in deriving a transport equation for the metastatic colony distribution ranged by some physiological traits denoted by X ∈ R n of the tumor growth, for example X = x the size of the tumor ( = Volume, either expressed in number of cells or mm 3 , using the conversion rule 1 mm 3 ≃ 10 6 cells). We suppose that the primary tumor is generated by a single cell at t = 0 and grows with the rate G(t, X) per time unit. The growing tumor emits metastatic cells with the rate β(X). Each metastatic cell develops into a new tumor, which also grows at rate G(t, X) and emits new nuclei of metastasis just as the primary tumor does. Let ρ(t, X) represent the colony distribution structured by the trait X at time t. The dynamics of the colony distribution is then given by the following Mac-Kendrick Von Foerster equation:

   ∂ ∂t ρ(t, X) + div[G(t, X)ρ(t, X)] = 0, X ∈ Ω, t ≥ 0 -G(t, σ) • νρ(t, σ) = N(σ) Ω β(X)ρ(t, X)dX + β(X p (t)) , σ ∈ ∂Ω, t ≥ 0 ρ(0, X) = 0, X ∈ Ω (1.1)
with X p (t), representing evolution of the primary tumor in the trait space satisfies Ẋp (t) = G(t, X p (t)) and N is the distribution of metastases at birth. From the solution ρ of this partial differential equation problem, various quantities of interest can be derived :

Total number of metastases = Ω ρ(t, X)dX, Visible metastases = Ω 1 x≥x vis (X)ρ(t, X)dX

Metastatic mass =

Ω xρ(t, X)dX with x vis being the threshold for metastases to be visible (typically, 10 7 cells). The first model in that direction was derived in [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF] where the tumor growth follows some Gompertz law. The variable X ∈ R is the size of the tumor and for untreated tumor G(t, X) = aX ln(θ/X). The potentialities of this model have been widely discussed in [START_REF] Barbolosi | Mathematical and numerical analysis for a model of growing metastatic tumors[END_REF][START_REF] Devys | A model describing the growth and the size distribution of multiple metastatic tumors[END_REF][START_REF] Verga | Modélisation mathématique de processus métastatiques[END_REF][START_REF] Barbolosi | Modélisation du rique d'évolution métastatique chez les patients supposés avoir une maladie localisée[END_REF]. The section 2 is devoted to introduction of a chemotherapy in this previous model. We exhibit how the model can give some hints on the choice of the number of cycles for a cytotoxic drug in a post-surgery context (adjuvant chemotherapy). Because most anticancer drugs are highly toxic, tolerance is an issue because treatment has often to be discontinued in patients with severe toxicities (eg, hematologic). In addition to standard cytotoxic agents all displaying a poor efficacy/toxicity balance, targeted therapies (eg, anti-angiogenic drugs as monoclonal antibodies or tyrosin kinase inhibitors) have emerged as a new paradigm to treat patients with cancer in a safer manner.

In the section 3. we explain how to integrate the action of these drugs within our approach. In this model, developed in [START_REF] Benzekry | Mathematical analysis of a two-dimensional population model of metastatic growth including angiogenesis[END_REF][START_REF] Benzekry | Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers[END_REF][START_REF] Benzekry | Passing to the limit 2D-1D in a model for metastatic growth[END_REF], the tumor growth rate is bi-dimensional and integrates dynamics of the vascular capacity following the model of [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF]. We first study and compare effect of the scheduling of anti-angiogenic drugs on both primary tumor growth and metastases. Then, we propose a model to take into account the metastatic acceleration phenomenon reported in [START_REF] Ebos | Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis[END_REF] after Sunitinib therapy.

Metronomic chemotherapy [START_REF] Pasquier | Metronomic chemotherapy: new rationale for new directions[END_REF] -the chronic administration of chemotherapy at relatively low, minimally toxic doses on a frequent schedule of administration, at close regular intervals, with no prolonged drug-free breaks -have been recently developed to reduce the toxic-risk in patients. This novel approach is an alternative way for fragile patients as children and old patients [START_REF] André | Metronomic etoposide/cyclophosphamide/celecoxib regimen to children and adolescents with refractory cancer : a preliminary monocentric study[END_REF][START_REF] Baruchel | Safety and pharmacokinetics of temozolomide using a doseescalation, metronomic schedule in recurrent paediatric brain tumours[END_REF][START_REF] Casanova | Vinorelbine and low-dose cyclophosphamide in the treatment of pediatric sarcomas: pilot study for the upcoming European Rhabdomyosarcoma Protocol[END_REF][START_REF] Stempak | A pilot pharmacokinetic and antiangiogenic biomarker study of celecoxib and low-dose metronomic vinblastine or cyclophosphamide in pediatric recurrent solid tumors[END_REF][START_REF] Choi | Feasibility of metronomic maintenance chemotherapy following high-dose chemotherapy for malignant central nervous system tumors[END_REF][START_REF] Kieran | A feasibility trial of antiangiogenic (metronomic) chemotherapy in pediatric patients with recurrent or progressive cancer[END_REF][START_REF] Sterba | Combined biodifferentiating and antiangiogenic oral metronomic therapy is feasible and effective in relapsed solid tumors in children: single-center pilot study[END_REF][START_REF] Fontana | Metronomic chemotherapy for metastatic prostate cancer: a 'young' concept for old patients?[END_REF]. These therapies seem to exhibit an anti-angiogenic effect on the tumor [START_REF] Kerbel | The anti-angiogenic basis of metronomic chemotherapy[END_REF][START_REF] Hahnfeldt | Minimizing long-term tumor burden : the logic for metronomic chemotherapeutic dosing and its antiangiogenic basis[END_REF] and reveal to be more efficient than classical cytotoxic therapies on large times. For such new therapeutical strategies, modeling should be of great help to give some prediction of the efficacy of the treatment. In section 5., we propose a model for such therapies. Numerical experiments reinforce constations of the clinicians, that is long time gain of metronomic chemotherapy.

Adjuvant chemotherapy

Model

We assume here that the tumor growth follows some Gompertz law with parameter a (proliferation rate, related to the tumor doubling time, taken constant) and θ > 1 (carrying capacity = maximum tumor size). The chemotherapy treatment (CT) then acts as a reduction of the growth speed via a log-kill term, that is

G(t, x) = ax ln θ x -C(t)(x -x min ) + (2.1)
where t → C(t) stands for the exposure of the cancer cells to the drug which is given by the pharmacokinetics (PK)/pharmacodynamics (PD) model, and x min is the minimal size for the treatment to be active (size of one cell). The PK model for Docetaxel is a three-compartmental model, coming from [START_REF] Bruno | A population pharmacokinetic model for docetaxel (Taxotere): model building and validation[END_REF]. We use the interface model from [START_REF] Meille | An interface model for dosage adjustment connects hematotoxicity to pharmacokinetics[END_REF] as PD model. All together, the equations are

       ċ1 (t) = -k e c 1 (t) + k 12 (c 1 (t) -c 2 (t)) -k 13 (c 1 (t) -c 3 (t)) + I(t) V ċ2 (t) = k 21 (c 1 (t) -c 2 (t)) ċ3 (t) = k 31 (c 1 (t) -c 3 (t)) Ċ(t) = -α I e -β I C(t) C(t) + c 1 (t) (2.2)
The last equation is the interface model introduced in [START_REF] Meille | An interface model for dosage adjustment connects hematotoxicity to pharmacokinetics[END_REF] to model the effect (exposure) of the drug. It is intended to have more flexibility than just considering the area under the curve (which is obtained by taking α I = 0) or an effect compartment (β I = 0). The term I stands for the drug input and writes

I(t) = M i=1 D i 1 t i ≤t≤t i +τ (2.3)
where {t i ; i = 1, .., M} are the administration times of the drug, {D i ; i = 1, .., M} the administrated doses and τ the injection duration. The parameter values used for the PK model can be found in Table 1.

The density of metastases ρ(t, x) structured in size x of the tumor as in [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF][START_REF] Barbolosi | Mathematical and numerical analysis for a model of growing metastatic tumors[END_REF] [START_REF] Bruno | A population pharmacokinetic model for docetaxel (Taxotere): model building and validation[END_REF].

∂ ∂t ρ(t, x) + ∂ ∂x [G(t, x)ρ(t, x)] = 0, x ∈ [1, θ), t ≥ 0 (2.4-a) g(1)ρ(1, t) = b 1 β(x)ρ(x, t)dx + β(x p (t)), (2.4-b) ρ(x, 0) = 0. ( 2
β(x) = mx α , (2.5)
where m is the colonization coefficient and α is the fractal dimension of blood vessels infiltrating the tumor. The parameter α expresses how blood vessels geometrically distribute in or on a tumor. If vasculature is superficial the fractal dimension α is assigned to be 2/3 because assuming that the tumor has the shape of a sphere implies that the surface area is proportional to 2/3. Otherwise, in the case of an homogeneously distributed vasculature in the whole tumor, α is taken to be equal to 1.

Note that for each patient, we have to identify the parameters involved in the model. A calibration of the model for non treated tumors has been done in [START_REF] Barbolosi | Modélisation du rique d'évolution métastatique chez les patients supposés avoir une maladie localisée[END_REF] thanks to a clinical study performed in Institut Gustave Roussy between 1958 and 1972 (see [START_REF] Koscielny | Breast cancer: relationship between the size of the primary tumour and the probability of metastatic dissemination[END_REF]).

Some features on adjuvant chemotherapy

The following section illustrates how such a model could help the oncologist to decide which best strategy should be undertaken (eg, starting an adjuvant therapy), in the case of chemotherapy. Consider some virtual patients with a breast cancer treated during 6 cycles of 21 days with a cocktail of Epirubicine (100mg) and Doxetaxel (75mg) as proposed in [START_REF] Viens | Epirubicindocetaxel combination in first-line chemotherapy for patients with metastatic breast cancer: final results of a dose-finding and efficacy study[END_REF]. We use meanvalues of parameters for the drug as reported in [START_REF] Meille | An interface model for dosage adjustment connects hematotoxicity to pharmacokinetics[END_REF], and represent a virtual patient by a set of parameters (a, θ, α, m). We fix the values of a = 4.71 × 10 -4 , θ = 10 12 , α = 2/3 and let the value of m

Patients m # Metastases Patients m # Metastases n • 1 1.7 × 10 -8 0 n • 6 7.0 × 10 -8 0 n • 2 1.9 × 10 -8 0 n • 7 1.3 × 10 -7 1 n • 3 2.7 × 10 -8 0 n • 8 2.7 × 10 -7 2 n • 4 5.0 × 10 -8 0 n • 9 4.0 × 10 -7 3 n • 5 6.1 × 10 -8 0 n • 10 6.1 × 10 -7 5
Table 2: Metatstatic state of the patients, five years after the end of the treatment suggested in [START_REF] Viens | Epirubicindocetaxel combination in first-line chemotherapy for patients with metastatic breast cancer: final results of a dose-finding and efficacy study[END_REF] vary from m = 1.7 × 10 -8 to m = 6.1 × 10 -7 . Table 2 shows that the last four patients present a highly metastatic risk five years after the end of the treatment. We suggest for the last four patients to increase the number of cycles of therapy or to use optimal protocol as described in the clinical essay MODEL I [START_REF] Meille | New adaptive method for phase I trials in oncology[END_REF]. Table 3 shows adaptation of the number of cycles of the treatment or the use of an adapted therapy may reduce the metastatic risk for the patients.

Patients m Viens's protocol Optimized protocol 6 cycles 9 cycles 12 cycles 9 cycles 13 cycles 18 cycles 126 days 189 days 252 days 126 days 182 days 252 days n

• 7 1.3 × 10 -7 1 0 ⋆ 0 ⋆ ⋆ n • 8 2.7 × 10 -7 2 1 0 2 0 ⋆ n • 9 4.0 × 10 -7 3 2 1 3 1 0 n • 10 6.1 × 10 -7 5 4 3 4 3 1
Table 3: Metastatic state of the patients n • 7 -10 for various protocols Identification of the parameter m would thus be crucial in view of clinical application. Our convinction is that the main information on metastatic aggressiveness lies in the primary tumor (as suggested in [START_REF] Comen | Clinical implications of cancer self-seeding[END_REF]) so that even if the patient is in a micro-metastatic state in an adjuvant setting we can hope to identify this parameter from histological data from the primary tumor.

Anti-angiogenic therapy

Another class of treatment consists in targeting the tumoral vasculature [START_REF] Folkman | Antiangiogenesis : new concept for therapy of solid tumors[END_REF] since neo-angiogenesis is a fundamental process in cancer evolution, consisting for the tumor in developing its own vascular network. This process ensures nutrient's supply to the tumor and allows it to grow beyond 2-3 mm diameter. Anti-angiogenic (AA) therapies have emerged in the early 2000's as a new class of anticancer agents with limited toxicities. They are classified as small molecules (eg, tyrosine kinase inhibitors) and monoclonal antibodies. The rise of targeted therapies has generated a great hope in clinical oncology. However, despite marked achievements in increasing response rates, most of these new drugs failed in prolonging disease-free survival or overall survival in pa-tients with cancer (see [START_REF] Gasparini | Combination of antiangiogenic therapy with other anticancer therapies: Results, challenges, and open questions[END_REF] for a synthesis of results of AA drugs). Determining the best temporal administration protocol for these agents is still a clinical open question [START_REF] Reynolds | Potential relevance of bell-shaped and u-shaped dose-responses for the therapeutic targeting of angiogenesis in cancer[END_REF][START_REF] Gasparini | Combination of antiangiogenic therapy with other anticancer therapies: Results, challenges, and open questions[END_REF].

Model

In order to take into account for the effect of anti-angiogenic (AA) therapy, we consider the model of Hahnfeldt -Folkman [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF] for vascular tumoral evolution, which has been biologically validated in [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF] by comparison with mice data. It consists in taking the parameter θ from the Gompertz equation (2.1) as being related to the vasculature of the tumor and dynamically evolving in time, the dynamics resulting from the competition between endogenous stimulation and inhibition from the tumor in the following way θ(t) = cx -dx 2/3 θ.

The two terms of this equation result from a ingenious analysis of the diffusion of inhibitory and stimulatory agents within the tumor which concludes to a constant rate of production for stimulating molecules and a rate of production proportional to the surface of the tumor for inhibitory ones, thus explaining the term in x 2/3 . Anti-angiogenic therapy is then modeled by adding a log-kill term in this equation. Since we will consider situations with the combination of AA and CT, we also add a log-kill term in the Gompertz equation on the tumoral evolution. The growth rate is now two-dimensional and writes

G(t, X) = G(t, x, θ) = ax ln θ x -C(t)(x -x min ) + cx -dx 2/3 -eA(t)(θ -θ min ) + (3.1)
where C(t) and A(t) stand respectively for the CT and AA drug effective concentrations resulting from a PK/PD model and x min = θ min is the size of one cell, taken as minimal value for the drugs to be active. We assume that the metastases have size x and vascular capacity θ (which has same unit than x) bigger than x min and less than b = c d 3/2 which is the maximal reachable size and vascular capacity by a tumor following the Hahnfeldt-Folkman model. Hence they evolve in the square Ω =]x min , b[×]x min , b[. The metastatic density is structured by a two-dimensional trait X = (x, θ) and solves the problem (1.1), with G given by (3.1). Considering that all metastases are born with size x min = 1 cell = 10 -6 mm 3 and vascular capacity θ 0 we take N(σ) = δ σ=(1,θ 0 ) , the Dirac mass at point (1, θ 0 ). We refer to [START_REF] Benzekry | Mathematical analysis of a two-dimensional population model of metastatic growth including angiogenesis[END_REF][START_REF] Benzekry | Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers[END_REF][START_REF] Benzekry | Passing to the limit 2D-1D in a model for metastatic growth[END_REF] for mathematical and numerical analysis of the resulting two-dimensional, measure-valued renewal equation.

Mice parameters

We use the parameters values obtained in [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF] from fits to data on mice bearing Lewis Lung carcinoma. The metastatic parameters are fixed arbitrarily. The values are given in the Table 4.

Human parameters

In the paper of Iwata et al. [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF] where the growth rate was a Gompertz, parameters where identified from data on a hepatocellular carcinoma. To determine realistic parameters for human situations, we fix values of the parameters for the model of Hahnfeldt et al.

a c d θ 0 m α 0.192 5.85 0.00873 625 0.001 2/3 day -1 day -1 day -1 mm -2 mm 3 Nb of meta•day -1 • mm -3α
Table 4: Values of the growth and metastatic parameters for mice. Parameters a, c and d where fitted on mice data in [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF].

reproducing the gompertzian growth curve of Iwata et al., keeping the carrying capacity from [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF] equal to c d 3/2 and fixing θ 0 as being rescaled from the value of [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF] by the ratio of the maximal reachable sizes from the two papers. We use the same value for α and adapt the value of m to a size unit in mm 3 : in [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF]

m = 5.3 • 10 -8 (Number of metastases)•(cell -α )•day -1 gives m = 5.3 • 10 -8 • 10 6α •(Number of metastases)•(mm -3α )•day -1 . a c d θ 0 m α 0.0042 1 5.7251• 10 -4 2630.14 5.3 •10 -4 2/3 day -1 day -1 day -1 mm -2 mm 3 Nb of meta•day -1 • mm -3α
Table 5: Values of the growth and metastatic parameters for human. a, m and α are from [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF].

Results : Primary Tumor VS Metastases

We recall here various simulations of AA monotherapy that can be found in [START_REF] Benzekry | Mathematical and numerical analysis of a model for anti-angiogenic therapy in metastatic cancers[END_REF] and investigate the difference in effectiveness of various drugs regarding to their pharmacokinetic/pharmacodynamic parameters. The first result shown in Figure 1 takes the three drugs which were used in [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF] where only the effect on tumor growth was investigated, and simulates the effect on the metastases. The three drugs are TNP-470, endostatin and angiostatin and each drug is characterized by two parameters in the model : its efficacy e and its clearance rate clr A . The first one appears in the second component of the growth rate (see expression 3.1) and the second one in the one-compartmental pharmacokinetics model for the concentration A(t) :

A(t) = D N i=1 e -clr(t-t i ) 1 t≥t i , (3.2)
where D is the administrated dose which is given as bolus at times t i . These parameters were retrieved in [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF] by fitting the model to mice data. The administration protocols are the same for endostatin and angiostatin (20 mg every day) but for TNP-470 the drug is administrated with a dose of 30 mg every two days. We observe that TNP-470 seems to have the poorest efficacy, both on tumoral growth, vascular capacity and total number of metastases, due to its large clearance. As noticed in [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF], the ratio e/clr A should govern the efficacy of the drug and its value is 0.13 for TNP-470 and 0.39 for both endostatin and angiostatin. The model we developed is now able to simulate efficacy of the drugs on the metastatic evolution (figure 1.C). Interestingly, the drug which seems to be more efficient regarding to the tumor size at the end of the simulation (day 15), namely angiostatin, is not the one which gives the best result on the metastases. Indeed, the lower efficacy of endostatin regarding to ultimate size is due to a relatively high clearance provoking a quite fast rebound of the angiogenic capacity once the treatment stops. But since the tumor size was lower for longer time, the metastatic evolution was better contained. This shows that the model could be a helpful tool for the clinician since the response to a treatment can differ from the primary tumor to metastases, but the clinician has no data about micro-metastases which are not visible with imagery techniques.

In the figure 2, we investigate the influence of the AA dose (parameter D) on tumoral, vascular and metastatic evolution. We observe that the model is consistent since it exhibits a monotonous response to variation of the dose.

Influence of the scheduling One of our main postulate in the treatment of cancer is that for a given drug, the effect can vary regarding to the temporal administration protocol of the drug, due to the combination of the pharmacokinetic of the drug and the intrinsic dynamic of tumoral and metastatic growth. To investigate the effect of varying the administration schedule of the drug, we simulated various administration protocols for the same drug (endostatin). The results are presented in figure 3. We gave the same dose and the same number of administrations of the drug but either uniformly distributed during 10 days (endostatin 2), concentrated in 5 days (endostatin 1) or in 2 days and a half (endostatin 3).

We observe that the tumor growth is better stabilized with a uniform administration of the drug (endostatin 2) but the number of metastases is better reduced with the intermediate protocol (endostatin 1). It is interesting to notice that again if we look at the effects at the end of the simulation, the results are different for the tumor size and for the metastases. The two protocols endostatin 1 and endostatin 2 give the same size at the end, but not the same number of metastases. Moreover, the best protocol regarding to minimization of the final number of metastases (endostatin 1) is neither the one which provoked the largest regression of the tumor during the treatment (endostatin 3) nor the one with the most stable tumor dynamic (endostatin 2).

Metastatic acceleration

In a recent paper [START_REF] Ebos | Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis[END_REF], Ebos et al. obtained surprising results after AA therapy with Sunitinib, a tyrosine kinase inhibitor of VEGFR, a Vascular Endothelial Growth Factor (VEGF) receptor. The treatment could induce metastatic acceleration in mice, while substantially inhibiting primary tumor growth. They used two different experimental protocols to evidence this phenomenon, on mice : by intravenous injection of cancerous cells or by orthotopic implantation of a tumor in the mammary fat pad and then removal of the primary tumor. In both cases, they obtained acceleration of the metastatic mass in groups treated by the AA drug, compared to an untreated group. However, on the primary tumor, the effect of the treatment was beneficial, as shown in Figure 4A of [START_REF] Ebos | Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis[END_REF]. Moreover, sustained therapy at 60 mg/kg/day exhibited better primary tumor slowdown than a temporal protocol of 120 mg/kg/day during 7 days, starting the day after tumor implantation. These results were corroborated by another paper by Paez-Ribes et al. [START_REF] Paez-Ribes | Antiangiogenic therapy elicits malignant progression of tumors to increased local invasion and distant metastasis[END_REF] in the very same issue of the journal.

Model

In this subsection, we modify our model in order to qualitatively reproduce these results, in a first attempt to have a theoretical tool aiming at controlling this paradoxical metastatic acceleration effect. In [START_REF] Ebos | Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis[END_REF], the authors propose as possible explanation for the phenomenon an upregulation of proangiogenic factors. To integrate this feature in the model, we propose to modify the angiogenesis stimulation parameter c of the tumoral growth model of Hahnfeldt -Folkman, only for the growth of metastases, by making it dependent of the AA drug concentration A(t), with an increased value c M ≫ c when the AA concentration is above a threshold A τ :

c(A(t)) = c M if A(t) ≥ A τ c if A(t) < A τ
See Figure 4 for an illustration. Our model is thus designed to take into account for the possible counter-attack of tumor cells towards AA therapy, as suggested by [START_REF] Ebos | Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis[END_REF] and shown in [START_REF] Chefrour | Erlotinib in combination with capecitabine (5'dFUR) in resistant pancreatic cancer cell lines[END_REF]. Indeed, in this last paper the authors demonstrate an increase of VEGF tumoral expression after erlotinib therapy in an in vitro experiment.

The equation for the dynamics of the vasculature of the metastases during the therapy becomes

dθ dt = c(A(t))x -dx 2/3 θ -eA(t)(θ -θ min ) + = G 2 (t, x, θ). (3.3)
and we see that the effect of the treatment is balanced between this "boost" effect and the natural anti-angiogenic effect of the drug.

We justify the fact that this angiogenic boost effect only occurs on the metastases and not on the primary tumor in view of the following arguments :

1. The primary tumor is big and relatively stable concerning angiogenic dynamics, having already established a vascular network and thus it is less active regarding to this process. On the contrary, metastases are small and fully active, passing through the angiogenic switch.

Hence they are more reactive to an external stress. We could be more accurate by making c M depend on the size of the tumor. In first approximation though, we don't do so.

2. Metastases are known to be genetically more aggressive, as the detaching cells which give rise to a malignant secondary tumor must have survived to various adverse events (intravasation into blood vessels, extravasation, settling in a new environment...). Thus they could have a stronger reactive phenotype towards AA therapy.

In absence of pharmacokinetics data for the Sunitinib, we perform the simulations in the case of a one-compartmental PK model (equation (3.2)), with clearance equal to 1.7 (endostatin in [START_REF] Hahnfeldt | Tumor development under angiogenic signaling : a dynamical theory of tumor growth, treatment, response and postvascular dormancy[END_REF]).

Simulations

Metastatic acceleration To perform the simulations, we use the tumoral growth parameters of Table 4. For the other parameters, we use

e = 0.2, clr A = 1.7, c M = 50, m = 0.001, α = 2 3 , (x 0 , θ 0 ) = (10 -6 , 10 -5 ). (3.4)
Of course, to obtain metastatic acceleration, we have to consider small value of the threshold A τ , which can be approximated by taking A τ = 0 (immediate boost effect). In the Figure 5 are presented simulation results in this situation, which qualitatively reproduce the results of Figure 2A from [START_REF] Ebos | Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis[END_REF]. We simulated the model without therapy, with initial primary tumor values (x 0,p , θ 0,p ) = (1, 10 -5 ), virtually performed resection of the primary tumor on day 14.5 (which consists in removing the source term in the PDE for the metastatic density) and then administrated the AA drug starting day 15, during 6 days. The temporal administration protocol that we used for the drug is 20mg/day. This result shows that for the parameter values (3.4), the model is able to reproduce metastatic acceleration, the balance between the anti-angiogenic effect and the boost effect in equation (3.3) being in favor of the second one.

In Figure 6, we reproduced in silico the situation of Figure 4A from [START_REF] Ebos | Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis[END_REF] by not performing resection of the primary tumor and testing two different temporal protocols for the AA drug on primary tumor evolution : protocol 1 consists in giving the drug at 20mg/day during 7 days and protocol 2 administrates half of the dose, 10mg/day, but during a larger time, from day 13 (which is the time for which the tumor reaches 200mm 3 in the model) until the end. We also used A τ = 0 in this simulation. We observe good qualitative agreement between Figure 6.A, showing the primary tumor growth, and the Figure 4A from [START_REF] Ebos | Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis[END_REF]. Indeed, since the primary tumor is not subjected to the boost effect in the model, the AA drug induces inhibition of the growth. We also retrieve the fact that better results are obtained with sustained therapy (protocol 2), on the primary tumor. In Figures 6.B, C and D we show some metastatic quantities, respectively the total number of metastases, the metastatic mass and the number of visible metastases (size exceeding 10 8 cells = 100 mm 3 ) . The following fact is interesting to notice : for both protocol, the total number of metastases is reduced compared to the situation without treatment; however, for the metastatic mass and the visible metastases, the effect depends on the protocol. While protocol 1 induces increased metastatic mass and number of visible metastases, protocol 2 has a positive effect, being even able to avoid apparition of visible metastases whereas protocol 1 provokes the presence of almost two visible metastases at the end of the simulation. It would be interesting to compare these in silico predictions to the metastatic data corresponding to Figure 4A from [START_REF] Ebos | Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis[END_REF] (unfortunately not available in the paper).

We could imagine that the threshold value A τ is a parameter which depends on the AA drug considered, or on the patient. Identification of its value would then be of fundamental importance since metastatic acceleration can occur or not depending on this value, as illustrated in Figure 7 where we performed the same numerical experiment as in Figure 5, but with A τ = 20, and observe reduction of the total number of metastases and of the metastatic mass, that is, the opposite of Figure 5.

Influence of the scheduling

In the Figure 8 we compare two protocols regarding to the metastatic acceleration phenomenon. Protocol 1 consists in giving the drug every day at dose 20 mg and Protocol 2 administrates the double dose every two days, each one being administrated from day 15 to 42 and with A τ = 7.

The two protocols have different implications : protocol 1 implies metastatic acceleration while protocol 2 results in deceleration of metastatic growth. These results are to be compared with the Figure 5 from the publication [START_REF] Onofrio | The dynamics of tumour-vasculature interaction suggests low-dose, time-dense anti-angiogenic schedulings[END_REF] concerning primary tumor evolution, where the equivalent protocols give the opposite qualitative results, namely a better effect of protocol 1. Of course, what we observe here in the metastatic acceleration context (i.e., using equation (3.3) for the vascular dynamics of the metastases) is totally related to the PK of the drug. In comparing two schedulings, the one resulting in larger boost effect will be the one for which the total time above A τ in the drug concentration time profile will be the largest (see Figure 4).

Again, the result depends on the value of the parameter A τ , as shown in Figure 9 where we performed the same simulation with A τ = 20 and obtained the opposite result : protocol 1 is better than protocol 2. It would be interesting to perform biological experiments to identify the values of the threshold A τ and of the overexpression of VEGF c M . In particular they should probably depend on the drug under consideration. Estimating inter-individual variability of these parameters could reveal very helpful for designing better temporal administration protocols for AA agents.

Cytotoxic and anti-angiogenic drugs combination

Due to their poor anti-tumor efficacy, AA drugs are neither administered alone but rather in combination with classical chemotherapies. An important problem in clinical oncology is to determine how to combine a cytotoxic drug that kills the proliferative cells and an anti-angiogenic drug which acts on the angiogenic process. For instance, in [START_REF] Riely | Randomized phase ii study of pulse erlotinib before or after carboplatin and paclitaxel in current or former smokers with advanced non-small-cell lung cancer[END_REF], the authors obtained different response rates and overall survival depending on the scheduling of a CT composed of carboplatin and paclitaxel and erlotinib (tyrosin kinase inhibitor), in the case of non-small-cell lung cancer. Two questions are still open : which drug should come before the other and then what is the best temporal repartition for each drug? Here, we perform a brief in silico study of the first question.

The treatments impact by reducing the tumoral growth rate. Following the log-kill assumption for chemotherapy which says that cytotoxic drugs kill a constant fraction of the tumoral cells (and not a constant number of them), the growth rate has the following expression

G(t, x, θ) = ax ln θ x -f C(t)(x -x min ) + cx -dx 2/3 -eA(t)(θ -θ min ) +
and we will use the parameters from Table 5, except m = 1.

We place ourselves in the human case and consider combination of Etoposide, Considering a clinical setting, we evaluate the combination between Etoposide, which is a CT agent used in wide variety of cancers (lung, testicle cancers, lymphoma, leukemia...) and Bevacizumab (monoclonal antibody targeting Vascular Endothelial Growth Factor, used mainly in colorectal and breast cancers). Several recent clinical trials have been evaluating the combination of both drugs in lung cancers and glioblastoma, with mixed results [START_REF] Spigel | Randomized Phase II Study of Bevacizumab in Combination With Chemotherapy in Previously Untreated Extensive-Stage Small-Cell Lung Cancer: Results From the SALUTE Trial[END_REF][START_REF] Reardon | Phase II study of metronomic chemotherapy with bevacizumab for recurrent glioblastoma after progression on bevacizumab therapy[END_REF][START_REF] Jordan | Bevacizumab in combination with sequential high-dose chemotherapy in solid cancer, a feasibility study[END_REF][START_REF] Francesconi | Carboplatin and etoposide combined with bevacizumab for the treatment of recurrent glioblastoma multiforme[END_REF]. The PK model and parameters for the CT drug are from [START_REF] Barbolosi | Optimisation de la posologie et des modalités d'administration des agents cytotoxiques à l'aide d'un modèle mathématique[END_REF] and for Bevacizumab from [START_REF] Lu | Clinical pharmacokinetics of bevacizumab in patients with solid tumors[END_REF]. Both publications show that the PK of the two drugs can be described by a two-compartmental model :

dc 1 (t) dt = -k 12 c 1 (t) + k 21 c 2 (t) + I(t) V dc 2 (t) dt = -k 21 c 2 (t) + k 12 c 1 (t).
and then A(t), C(t) = c 2 (t) (for the respective parameters). The dose injection I(t) can be expressed as in (2.3) with injection durations τ C and τ A and respective administration doses. The parameter values can be found in Tables 6 and7. We take as non-zero initial condition the traits

Parameter V 1 V 2 k 10 k 20 k 12 k 21 τ A e
L L day -1 day -1 day -1 day -1 min L•mg -1 • day -1
Table 6: Parameter values for the PK model for Bevacizumab. All parameters except e are from [START_REF] Lu | Clinical pharmacokinetics of bevacizumab in patients with solid tumors[END_REF].

Parameter V 1 V 2 k 10 k 20 k 12 k 21 τ C f Value 25 15 1.6 9.36 0.4 0 24 25 Unit L L day -1 day -1 day -1 day -1 h L•g -1 • day -1
Table 7: Parameter values for the PK model for Etoposide. All parameters except f are from [START_REF] Barbolosi | Optimisation de la posologie et des modalités d'administration des agents cytotoxiques à l'aide d'un modèle mathématique[END_REF].

(x 0,p , θ 0,p ) = (17112, 44849) corresponding to the values reached by the primary tumor after 1000 days when starting with one cell and the parameters from Table 5, except m = 1. We also take the corresponding value of ρ(1000) as ρ 0 .

In the Figure 10 we compare each monotherapy case to the combined treatment with the two drugs. The administration protocol for Bevacizumab is 5 mg/kg/2 weeks [START_REF] Lu | Clinical pharmacokinetics of bevacizumab in patients with solid tumors[END_REF] (with a virtual patient of 70 kg) and the Etoposide one is 0.5 g/m 2 at day 1 of the cycle [START_REF] Barbolosi | Optimisation de la posologie et des modalités d'administration des agents cytotoxiques à l'aide d'un modèle mathématique[END_REF] (virtual patient with a Body Surface Area of 1.75 m 2 ). We observe that, with the efficacy parameter e and f that we chose, the effect of the AA drug is to stabilize tumoral growth as well as the number of visible metastases. The CT has an important reduction effect and combination of both is better than the two monotherapy cases since it allows to reduce the tumor burden and then stabilize it to a low level.

We investigate now the effect of changing the order of administration of the two drugs. In the Figure 11 we simulated two situations with one administration of each drug : either Bevacizumab at day 0 and Etoposide at day 8 or the opposite. An interesting fact to observe is that the best way of combining both regarding to the tumor size, namely AA first and then CT is not the best for the total number of metastases, which rather suggests that the best is the CT first and then the AA. Although the difference between both is very low we still have qualitatively different answers for primary tumor size and metastases, for the problem of the order of administration.

Metronomic chemotherapy

During the last decade, a novel therapeutic approach called metronomic chemotherapy (also named low dose anti-angiogenic therapy [START_REF] Browder | Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer[END_REF]) appeared. Various Phase I studies have been performed [START_REF] André | Metronomic etoposide/cyclophosphamide/celecoxib regimen to children and adolescents with refractory cancer : a preliminary monocentric study[END_REF][START_REF] Baruchel | Safety and pharmacokinetics of temozolomide using a doseescalation, metronomic schedule in recurrent paediatric brain tumours[END_REF][START_REF] Casanova | Vinorelbine and low-dose cyclophosphamide in the treatment of pediatric sarcomas: pilot study for the upcoming European Rhabdomyosarcoma Protocol[END_REF][START_REF] Stempak | A pilot pharmacokinetic and antiangiogenic biomarker study of celecoxib and low-dose metronomic vinblastine or cyclophosphamide in pediatric recurrent solid tumors[END_REF][START_REF] Choi | Feasibility of metronomic maintenance chemotherapy following high-dose chemotherapy for malignant central nervous system tumors[END_REF][START_REF] Kieran | A feasibility trial of antiangiogenic (metronomic) chemotherapy in pediatric patients with recurrent or progressive cancer[END_REF][START_REF] Sterba | Combined biodifferentiating and antiangiogenic oral metronomic therapy is feasible and effective in relapsed solid tumors in children: single-center pilot study[END_REF] using this new way of administrating cytotoxic agents which consists in giving the drug at low dose but as continuously as possible, whereas standard protocols administrate drugs at their respective MTD (maximal tolerated doses) as frequently as possible, with respect to the patient's hematological tolerance and the management of the drug-induced side effects. Indeed, this scheduling of the drug is believed to have better efficacy, one argument being that it would generate less resistances in the cancerous cells population in some circumstances. However, metronomic schedules are now thought to be potentially more efficient while reducing the toxicities according to the paradigm that this low dose/time dense scheduling of the drug would have important anti-angiogenic effect [START_REF] Kerbel | The anti-angiogenic basis of metronomic chemotherapy[END_REF][START_REF] Hahnfeldt | Minimizing long-term tumor burden : the logic for metronomic chemotherapeutic dosing and its antiangiogenic basis[END_REF]. Indeed, the endothelial cells which are proliferating during tumoral neo-angiogenesis are also targeted by the cytotoxic agent, while developping less resistances as they are genetically more stable than malignant cells. Moreover, the dynamical effect of metronomic schedules would be higher than the one of MTD protocols, for large times. In this context, determining what is the best metronomic scheduling for chemotherapeutic agents remains a major issue.

In this section, we use the Hahnfeldt -Folkman model to give insights on this topic, and observe the effect of metronomic schedules on the metastases population. As expressed above, one of the main ingredients explaining the benefit of metronomic schedules compared to MTD ones are the resistances, which we shall thus introduce in the model. We place ourselves in the context of breast cancer and will first consider a monotherapy situation with Docetaxel as the CT drug, and then combination of Docetaxel with Bevacizumab.

Model

The main assumptions underlying our modeling of the metronomic paradigm are the following :

1. The CT has an anti-angiogenic effect by killing proliferative endothelial cells.

2. Cancerous cells develop resistances to the CT whereas endothelial cells don't.

3. The killing action of the drug is stronger on the endothelial compartment than on the tumoral one (endothelial cells are more fragile).

We start from the Hahnfeldt -Folkman, in which we integrate anti-angiogenic effect of a chemotherapy (assumption 1). We consider that the CT drug acts in both compartment : on the tumoral compartment x (classical cytotoxic effect) and on the vasculature θ (anti-angiogenic effect due to the killing of proliferative endothelial cells). The expression of the growth rate G is

G(t, X) = ax ln θ x -C 1 (t)(x -x min ) + cx -dx 2/3 θ -(eA(t) + C 2 (t))(θ -θ min ) + (5.1)
where C 1 and C 2 are the exposures of the CT drug respectively on the tumoral cells and on the vascular compartment. They are defined from the output C(t) of the PK model for Docetaxel (2.2), expressed in mg• L -1 . We take into account for the resistances (assumption 2) by assuming that for each time t, each cell has probability RC(t) to become resistant. The probability of being resistant at time t is then exponentially distributed and we set

C 1 (t) = α 1 e -R R t 0 C(s)ds C(t), C 2 (t) = α 2 C(t).
L•mg -1 L•mg -1 •day -1 L•mg -1 •day -1 L•mg -1 •day -1 min
Table 8: Parameter values for the PD model for Docetaxel. Values α I and β I come from [START_REF] Meille | An interface model for dosage adjustment connects hematotoxicity to pharmacokinetics[END_REF]. Parameters α 1 , α 2 and R were fixed arbitrarily.

being given by (5.1). The tumoral growth and metastatic parameters used in the simulations are those in Table 5 coming from [START_REF] Iwata | A dynamical model for the growth and size distribution of multiple metastatic tumors[END_REF] where they were fitted to data of a hepatocellular carcinoma. We take as non-zero initial condition the traits (x 0,p , θ 0,p ) = (902.28, 15401) corresponding to the values reached by the primary tumor after 600 days when starting with one cell and the parameters from Table 5. We also take the corresponding value of ρ(600) as initial metastatic density ρ 0 .

Simulation results

Metronomic Docetaxel In the publication [START_REF] Baruchel | Safety and pharmacokinetics of temozolomide using a doseescalation, metronomic schedule in recurrent paediatric brain tumours[END_REF], for pediatric brain cancers, the authors compared a classical and a metronomic protocol. During 49-days therapy cycles, the classical protocol delivers 200 mg/m 2 of Temozolomide per day during the first 5 days whereas the metronomic protocol gives 85 mg/m 2 per day during 42 days followed by a 7-day rest period. The total amounts of drug delivered during a cycle are respectively 1000 mg/m 2 and 3570 mg/m 2 , the second one being thus able to give more than 3.5 fold the total dose of the first one. We wish now to mimic this situation in the case of breast cancer with Docetaxel as CT and compare the two following schedules, based on a 21-day long therapy cycle :

1. MTD schedule : 100 mg at day 0, as considered in [START_REF] Meille | An interface model for dosage adjustment connects hematotoxicity to pharmacokinetics[END_REF].

2. Metronomic schedule : 10 mg/day every day, without resting period.

The total dose administrated during one cycle are respectively 100 mg and 210 mg. The simulation results for the tumoral evolution and the metastases are presented in Figure 12. We observe in Figure 12 that, at the beginning, the MTD schedule induces better tumoral reduction than the metronomic one, which exhibits only limited regression and even regrowth of the tumor. This could be misleading in practical situations since one could decide to stop the therapy when observing the regrowth. However, due to resistances the MTD schedule gets worse and eventually doesn't contain regrowth of the primary tumor. On the opposite, over long periods the metronomic schedule gives better results, being able to overcome the resistance phenomenon, and eventually leading to stable tumoral reduction, which still persists for times bigger than 220 days (simulation not shown). The explanation of this fact can be understood by looking at Figure 12.B representing the effect of the drug on the vascular capacity. While the MTD schedule does not provoke overall decrease of the vascular capacity on large time scale, the metronomic one ensures deeper and more stable effect on the vasculature, which eventually explains the large time tumor decrease. Hence, the overall superiority of the metronomic schedule is explained in the model by the anti-angiogenic action of the drug. The MTD schedule does also exhibit anti-angiogenic effect but, combined with the intrinsic dynamic of the vasculature, it is not asymptotically efficient because the large rest period lets time for the vasculature to recover. The metronomic schedule on the opposite does not let time for vascular recovery and, since endothelial cells are not subjected to resistances to the drug, is more efficient on large time scale.

Concerning metastases, we observe the same behavior as described above on the number of visible metastases (Figure 12.D). While the untreated curve leads to apparition of one visible metastasis at the end of the simulation, both protocols don't. But the MTD schedule asymptotically has growing number of visible tumors whereas the metronomic one is able to decrease it and keep it under control. On the total number of metastases, the MTD schedule is slightly better but we can suspect that for larger times both curves will cross since, according to Figure 12.A, all tumors will eventually decrease, thus leading to less emission of neo-metastases.

If the dose used in the metronomic schedule is too low, then it is not efficient, as shown in Figure 13 where the same metronomic schedule is simulated, but using a dose of 8 mg/day. This result suggests that there is an optimal dose to use for metronomic schedules, and a mathematical model can be helpful in determining this optimum (see [START_REF] Benzekry | An optimal control problem for anti-cancer therapies in a model for metastatic evolution[END_REF] for an optimal control problem for the scheduling). Integrating the toxicity issue in the model should help to further optimize the optimization of metronomic scheduling.

Metronomic Docetaxel + Bevacizumab

We perform now the same comparison between the metronomic schedule and the classical one, but we add action of Bevacizumab. The scheduling that we use for this last drug comes from [START_REF] Lu | Clinical pharmacokinetics of bevacizumab in patients with solid tumors[END_REF] and is 7.5 mg/kg every three weeks (Bevacizumab has a large half-life). We consider a virtual patient of 70 kg. The simulation results are plotted in Figure 14. We observe that the two scheduling efficiently reduce the tumor size and control explosion of the total number of metastases and of the visible ones. If we look at evolution during the whole simulation, the classical schedule is better than the metronomic one. Indeed, as shown in Figure 14.B when comparing it with Figure 12.B, addition of the AA drug ensures large reduction of the vasculature leading to tumor suffocation. This happens independently from the temporal administration protocol of the CT drug. Our model suggests thus no benefit of the metronomic schedule on the classical one in the case of combination therapy with AA drugs.

Conclusion

One of the main issue in cancer therapy is to evaluate the metastatic risk of the patient. Determining the best regimen and the optimal dosing schedule is still an unsolved question in various clinical settings. Oncogenetics, pharmacogenomics and pharmacogenetics are new tools increasingly used that are dedicated to provide biomarkers for treatment efficacy and tolerance. Similarly, mathematical modeling could help and give valuable information to the physician to optimize treatments. The models that we presented in this paper can give informations on the (micro)metastatic state of the patient and the impact of therapy on its evolution. They deliberately involve a few number of parameters in view of clinical applications. We illustrated through numerical simulations how our models could be used to give insights on concrete clinical issues.

The models are flexible enough to integrate action of the two main chemical therapies available to clinicians : cyototoxic and anti-angiogenic. We were able to qualitatively reproduce experimental observations such as metastatic acceleration after AA therapy and to test biological hypotheses susceptible to provide rationale for the use of metronomic schedules in chemotherapy. The models were then used to run predictive in silico tests which have now to be confronted to biological data. We are currently performing mice experiments to validate the model in the ANR project MEMOREX-PK.

All along the study, we tried to use as much as possible parameter values coming from biological data available in the literature. However, all parameters were not available, in particular for metastatic emission and efficacy of the drugs. A possible way to estimate these last ones is to use results of clinical studies giving various evolution percentages of the disease (Complete Remission, Partial Remission, Stable Disease, Progressive Disease). The parameters can be calibrated using statistical methods to reproduce these results, as done for example in [START_REF] Barbolosi | Modélisation du rique d'évolution métastatique chez les patients supposés avoir une maladie localisée[END_REF]. About the metastatic emission parameters m and α, we are currently working on methods to identify them from data on the primary tumor.

We are aware that, since our ultimate goal is to provide effective results in the clinic, it is mandatory to take into account for the toxicities induced by the considered therapies (hematological toxicities, mucositis, cutaneous rushes,...). Models dealing with these issues like [START_REF] Meille | New adaptive method for phase I trials in oncology[END_REF][START_REF] Barbolosi | Optimizing drug regimens in cancer chemotherapy: a simulation study using a pk-pd model[END_REF][START_REF] Iliadis | Optimizing drug regimens in cancer chemotherapy by an efficacytoxicity mathematical model[END_REF] could be integrated in future versions. Furthermore, in the model we presented for combination of CT and AA therapies no nonlinear interactions between the drugs are taken into account. But since delivery of the CT depends on the amount of blood vessels reaching the tumor, a negative effect of AA therapy on the CT could be expected. On the opposite, a "normalization" effect [START_REF] Jain | Normalizing tumor vasculature with anti-angiogenic therapy: A new paradigm for combination therapy[END_REF] could also temporarily happen following administration of AA agents, thus improving CT transportation. We are currently working on a refinement of the Hahnfeldt -Folkman model which deals with these features (See [START_REF] Benzekry | A new mathematical model for optimizing the combination between anti-angiogenic and cytotoxic drugs in oncology[END_REF]).

The actual therapies are still experiencing various problems. This work shows that mathematical modeling could help to provide new leads to overcome the failures faced up to now.

CFigure 1 :

 1 Figure 1: Effect of the three drugs from [26]. The treatment is administrated from days 5 to 10. Endostatin (e = 0.66, clr A = 1.7) 20 mg every day, TNP-470 (e = 1.3, clr A = 10.1) 30 mg every two days and Angiostatine (e = 0.15, clr A = 0.38) 20 mg every day. A : tumor size. B : Angiogenic capacity. C : Number of metastases.

Figure 2 :

 2 Figure 2: Effect of the variation of the dose for endostatin. A : tumor size. B : Angiogenic capacity. C : Number of metastases.

CFigure 3 :

 3 Figure 3: Three different temporal administration protocols for the same drug (Endostatin). Same dose (20 mg) and number of administrations (6) but more or less concentrated at the beginning of the treatment. Endostatin 1 : each day from day 5 to 10. Endostatin 2 : every two days from day 5 to 15. Endostatin 3 : twice a day from day 5 to 7.5. A : tumor size. B : Angiogenic capacity. C : Number of metastases.

Figure 4 :

 4 Figure 4: Illustration of activation of the boost effect.

Figure 5 :

 5 Figure 5: Metastatic acceleration for A τ = 0. A. Total number of metastases, only from day 15 to day 30. B. Metastatic mass

DFigure 6 :Figure 7 :CFigure 8 :

 678 Figure 6: Without resection. Protocol 1 : 20mg/day from day 15 to day 21. 10mg/day from day 13 until the end A. Primary tumor. B. Total number of metastases, from day 15 until the end. C. Metastatic mass (log scale). D. Visible metastases

BFigure 9 :

 9 Figure 9: Influence of the scheduling. A τ = 20. Protocol 1 : 20 mg/day. Protocol 2 : 40 mg/2 days. A. Total number of metastases. B. Metastatic mass.

Figure 10 :

 10 Figure 10: Comparison between the two monotherapy cases and the combined therapy. A : Primary tumor size. B : Visible metastases. C : Angiogenic capacity. D : Total number of metastases.

DFigure 11 :

 11 Figure 11: Administer the CT before or after the AA? A : Primary tumor size. B : Visible metastases. C : Angiogenic capacity. D : Total number of metastases.

We transpose assumption 3 by taking α 2 > α 1 .

 21 The parameter values are gathered in Tables 1 for the PK model and 8 for the PD one. For the metastatic evolution we use the model (1.1) with G

Figure 12 :

 12 Figure 12: Comparison between MTD and metronomic schedules for Docetaxel. A. Primary tumor size. B. Primary tumor vascular capacity. C. Total number of metastases. D. Number of visible metastases (log scale).

Figure 13 :

 13 Figure 13: Metronomic schedule for Docetaxel with dose 8 mg/day. A. Tumor size. B. Vascular capacity

Figure 14 :

 14 Figure 14: Comparison between MTD and metronomic schedules for Docetaxel in combination with Bevacizumab. A. Primary tumor size. B. Primary tumor vascular capacity. C. Total number of metastases D. Number of visible metastases (log scale).

Table 1 :

 1 Parameter values of the PK model for Docetaxel

Acknowlegments

The authors were partially supported by l'Agence Nationale de la Recherche under grant ANR-09-BLAN-0217-01 and a Grant # 50009 from the Association pour la Recherche contre le Cancer (ARC).