Section: New Results

Stochastic modeling

Tumor growth modeling

Participants: P. Vallois, S. Wantz-Mézières

External collaborator: J-S. Giet (IECL,Université de Lorraine)

A cancer tumor can be represented for simplicity as an aggregate of cancer cells, each cell behaving according to the same discrete model and independently of the others. Therefore to measure its size evolution, it seems natural to use tools coming from dynamics of population, for instance the logistic model. This deterministic framework is well-known but the stochastic one is worthy of interest. We work with a model in which we suppose that the size $V_t$ at time $t$ of the tumor is a diffusion process of the type :

where ${\left(B_t\right)}_{t\ge 0}$ is a standard brownian motion starting from zero. Then (i) We define a family of time continuous Markov chains which models the evolution of the rate of malignant cells and approximate (under some conditions) the diffusion process $\left(V_t\right)$. (ii) We study in depth the solution to equation (1 ). This diffusion process lives in a domain delimited by two boundaries: 0 and $\kappa >0$. In this stochastic setting, the role of $\kappa$ is not so clear and we contribute to understand it. We describe the asymptotic behavior of the diffusion according to the values of the parameters. The tools we resort to are boundary classification criteria and Laplace transform of the hitting time to biological worthwhile level. We are able in particular to express the mean of the hitting time. We have an accepted paper in the journal Theory of Stochastic Processes [70] .

A Multitype Branching Process Model of Heterogeneous Damages in vitro Cancer Cell Populations Treated by Radiotherapy

Participants: T. Bastogne, P. Vallois

External collaborator: S. Pinel (CRAN, Université de Lorraine)

Cancer is the result of inter-dependent multi-scale phenomena and this is mainly why the understanding of its spread is still an unsolved problem. In integrative biology, mathematical models play a central role; they help biologists and clinicians to answer complex questions through numerical simulations and statistical analyses. The main issue here is to better understand and describe the role of cell damage heterogeneity and associated mutant cell phenotypes in the therapeutic responses of cancer cell populations submitted to a radiotherapy sessions during in vitro experiments. The cell heterogeneity is often described as randomness in mathematical modeling and different representations, such as Markov chains, branching processes and even stochastic differential equations, have been recently used. Conversely to these previous studies, which only focused on the steady-state responses of cell populations, we are interested by modeling the transient behavior after treatment and to identify the role of mutation heterogeneity in the global dynamic response of the cell populations. We propose to describe the survival response of anin vitro cancer cell culture treated by radiotherapy as a superposition of independent dynamics. Each cell is represented by a finite collection of cell mutation states with possible transitions between them. The population dynamics is given by an age-dependent multi-type branching process. From this representation, we obtain equations satisfied by the average size of the global survival population as well as the one of subpopulations associated with 10 mutation phenotypes. This work was presented via a poster communication in a international congress [40] .

Modeling of response to chemotherapy in gliomas

Participant: S. Wantz-Mézières

External collaborators: M. Ben Abdallah, Yann Gaudeau, J.-M. Moureaux (CRAN, Université de Lorraine) and M. Blonski, L. Taillandier (CHU Nancy)

In the framework of a collaboration with neurologists (Luc Taillandier, Marie Blonski, CHU Nancy) and automaticians (Jean-Marie Moureaux, Yann Gaudeau, CRAN), around the thesis supervision of M. Ben Abdallah, our aim is to work out personalized therapeutic strategy in the monitoring of diffuse low-grade glioma patients. Regular monitoring with MRI are used to estimate the tumour volume ; we proposed a method by manual segmentation and statistically assessed its reproducibility by a subjective test. In order to design a decision-aid tool for the response to chemotherapy, our approach is phenomelogical and we used simple regression tools to model and predict the cinetics of the tumour growth. We identified two different models. These results open up many perspectives, the main one being the modeling by multi-factor models, including biological and anatomopathological factors. This work is currently in progress.

Photodynamic therapy

Participant: C. Lacaux

External collaborators: T. Obara and M. Thomassin (CRAN, Université de Lorraine), L. Vinckenboch (Fribourg)

Our project focuses on an innovative application: the interstitial PDT for the treatment of high-grade brain tumors. This strategy requires the installation of optical fibers to deliver the light directly into the tumor tissue to be treated, while nanoparticles are used to carry the photosensitizer into the cancer cells. In order to optimize the intra-cerebral position of our optical fiber, two fundamental questions have to be answered: (1) What is the optimal shape and position of the light source in order to optimize the damage on malignant cells? (2) Is there a way to identify the physical parameters of the tissue which drive the light propagation?

Notice that we are obviously not the first ones to address these issues, and there is nowadays a consensus in favor of the algorithm proposed by L. Wang and S. L. Jacques for the simulation of light transport in biological tissues. However, our starting point is the observation that the usual methods slightly lack of formalism and miss formal representations that answer the questions of identifiability. In [16] , in the framework of homogeneous biological tissues, we propose an alternative MC method to Wang’s algorithm. Then we also propose a variance reduction method. Interestingly enough, our formulation also allows us to design quite easily a Markov chain Monte Carlo (MCMC) method based on Metropolis-Hastings algorithm and to handle the inverse problem (of crucial importance for practitioners), consisting in estimating the optical coefficients of the tissue according to a series of measurements. We have compared the proposed MC and MCMC method and Wang’s algorithm: we see that our MC method is much more consistent. However, MCMC methods induce quick mutations, which paves the way to very promising algorithms in the inhomogenous case. To handle the inverse problem, we derive a probabilistic representation of the variation of the fluence with respect to the absorption and scattering coefficients. This leads us to the implementation of a Levenberg-Marquardt type algorithm that gives an approximate solution to the inverse problem. Our results open the way for new improvements of Monte-Carlo methods in the context of light propagation. They should rather be seen as a starting point for new methods, including in inhomogeneous tissue. This work has been presented in several french seminars (Lille, Avignon, Paris Descartes, Orléans).

Time-changed extremal process as a random sup measure

Participant: C. Lacaux

External collaborator: G. Samorodnitsky (Cornell, USA)

In extreme value theory, one of the major topics is the study of the limiting behavior of the partial maxima of a stationary sequence. When this sequence is i.i.d., the unique limiting process is well-known and called the extremal process. Considering a long memory stable sequence, the limiting process is obtained as a simple power time change extremal process. Céline Lacaux and Gennady Samorodnistky have proved in [38] that this limiting process can also be interpreted as a restriction of a self-affine random sup measure. In addition, they have established that this random measure arises as a limit of the partial maxima of the same long memory stable sequence, but in a different space. Their results open the way to propose new self-similar processes with stationary max-increments. Céline Lacaux has presented this work in an invited session of the international conference Extreme Value Analysis at Ann Arbor (June 2015).

Modulus of continuity of some conditionally sub-Gaussian fields, application to stable random fields

Participant: C. Lacaux

External collaborator: H. Biermé (Poitiers)

Hermine Biermé and Céline Lacaux maintain their collaboration on the study of anisotropic random fields. They have extended their previous work in the framework of conditionally sub-Gaussian random series. For such anisotropic fields, they have obtained a modulus of continuity and a rate of uniform convergence. Their framework enables the study of study e.g., Gaussian fields, stable random fields and multi-stable random fields. As invited speaker, Céline Lacaux has presented this work in the international conference Adventure in Self-similarity at Cornell University (June 2015) [17] . Another of their works in progress deals with the simulation of anisotropic Gaussian random fields and the estimation of their parameters using quadratic variations.

DNA sequences analysis

Participants: P. Vallois

External collaborators: A. Lagnoux and S. Mercier (Toulouse)

Here we want to determine the sequences that are biologically interesting and compare the results using the single local score Hn and using the pair (Hn;Ln) where Ln is the length of the segment that realizes the best score. In that view, we work on the p-values associated to the observed samples.

Multicriteria Agregation for Health Economic Assessment

Participants: T. Bastogne, Y. Petot, P. Vallois

The framework of this work is the PhD thesis of Yann Petit. The first chapter of the thesis is a state of the art identifying the current challenges in medico-economic analyses. A review article should be submitted in spring 2016. We are currently working on the aggregation operators, based on fuzzy measures and the Choquet integral. Theoretical results have been obtained and a publication is planned to be submitted in the second half of 2016. Work continues by introducing probabilities. The next step will be to apply our theoretical results to real clinical cases.

Spatial and spatio-temporal modeling

Participant : A. Gégout-Petit

External collaborators: S. Li, L. Guerin-Dubrana (Inra Bordeaux)

In the framework of a collaboration with INRA Bordeaux about the esca-illness of vines, Anne Gégout-Petit with Shuxian Li developed different spatial models and spatio-temporal models for different purposes: (1) study the distribution and the dynamics of esca vines in order to tackle the aggregation and the potential spread of the illness (2) propose a spatio-temporal model in order to capture the dynamics of cases and measure the effects of environmental covariates. For this, we propose different hierarchic models with latent process associated with a bayesian inference. A part of the research has been submitted in a journal of biology [39] . Shuxian Li defended his PhD on December the 15th.

Stochastic modeling of fatigue crack propagation

Participants: R. Azaïs, A. Gégout-Petit

External collaborators: A.B. Abdessalem, M. Puiggali, M. Touzet (Bordeaux)

Fatigue crack propagation is a stochastic phenomenon due to the inherent uncertainties originating from material properties and environmental conditions. In a recent preprint [35] , we propose to model and to predict the fatigue crack growth by a piecewise-deterministic Markov process associated with deterministic crack laws of the literature, namely the Paris-Erdogan equation defined by $\text{d}a/\text{d}N=C{\left(\Delta K\right)}^m$ and the Forman equation given by $\text{d}a/\text{d}N=C{\left(\Delta K\right)}^m/(K_c(1-R)-\Delta K)$, where $a$ is the crack length, $N$ denotes the number of cyclic mechanical loads, $\Delta K$ is the range of the stress intensity factor and $C$, $m$, $K_c$ and $R$ are different parameters. We introduce a regime-switching model to express the transition between Paris' regime and rapid propagation which occurs just before failure. We also investigate the prediction of the fatigue crack path and its variability based on measurements taken at the beginning of the propagation. This work has also been presented in an international conference [25] .