Guillemette Chapuisat

Mathematical Modeling of Tumor Growth and Metastatic Spreading: Validation in Tumor-Bearing Mice

Defining tumor stage at diagnosis is a pivotal point for clinical decisions about patient treatment strategies. In this respect, early detection of occult metastasis invisible to current imaging methods would have a major impact on best care and long-term survival. Mathematical models that describe metastatic spreading might estimate the risk of metastasis when no clinical evidence is available. In this study, we adapted a top-down model to make such estimates. The model was constituted by a transport equation describing metastatic growth and endowed with a boundary condition for metastatic emission. Model predictions were compared with experimental results from orthotopic breast tumor xenograft experiments conducted in Nod/Scidγ mice. Primary tumor growth, metastatic spread and growth were monitored by 3D bioluminescence tomography. A tailored computational approach allowed the use of Monolix software for mixed-effects modeling with a partial differential equation model. Primary tumor growth was described best by Bertalanffy, West, and Gompertz models, which involve an initial exponential growth phase. All other tested models were rejected. The best metastatic model involved two parameters describing metastatic spreading and growth, respectively. Visual predictive check, analysis of residuals, and a bootstrap study validated the model. Coefficients of determination were [Formula: see text] for primary tumor growth and [Formula: see text] for metastatic growth. The data-based model development revealed several biologically significant findings. First, information on both growth and spreading can be obtained from measures of total metastatic burden. Second, the postulated link between primary tumor size and emission rate is validated. Finally, fast growing peritoneal metastases can only be described by such a complex partial differential equation model and not by ordinary differential equation models. This work advances efforts to predict metastatic spreading during the earliest stages of cancer.

Modelling methodology in physiopathology.

Diseases are complex systems. Modelling them, i.e. systems physiopathology, is a quite demanding, complicated, multidimensional, multiscale process. As such, in order to achieve the goal of the model and further to optimise a rather-time and resource-consuming process, a relevant and easy to practice methodology is required. It includes guidance for validation. Also, the model development should be managed as a complicated process, along a strategy which has been elaborated in the beginning. It should be flexible enough to meet every case. A model is a representation of the available knowledge. All available knowledge does not have the same level of evidence and, further, there is a large variability of the values of all parameters (e.g. affinity constant or ionic current) across the literature. In addition, in a complex biological system there are always values lacking for a few or sometimes many parameters. All these three aspects are sources of uncertainty on the range of validity of the models and raise unsolved problems for designing a relevant model. Tools and techniques for integrating the parameter range of experimental values, level of evidence and missing data are needed.

Exploration of beneficial and deleterious effects of inflammation in stroke: dynamics of inflammation cells

The inflammatory process during stroke consists of activation of resident brain microglia and recruitment of leucocytes, namely neutrophils and monocytes/macrophages. During inflammation, microglial cells, neutrophils and macrophages secrete inflammatory cytokines and chemokines, and phagocytize dead cells. The recruitment of blood cells (neutrophils and macrophages) is mediated by the leucocyte–endothelium interactions and more specifically by cell adhesion molecules. A mathematical model is proposed to represent the dynamics of various brain cells and of immune cells (neutrophils and macrophages). This model is based on a set of six ordinary differential equations and explores the beneficial and deleterious effects of inflammation, respectively phagocytosis by immune cells and the release of pro-inflammatory mediators and nitric oxide (NO). The results of our simulations are qualitatively consistent with those observed in experiments in vivo and would suggest that the increase of phagocytosis could contribute to the increase of the percentage of living cells. The inhibition of the production of cytokines and NO and the blocking of neutrophil and macrophage infiltration into the brain parenchyma led also to the improvement of brain cell survival. This approach may help to explore the respective contributions of the beneficial and deleterious roles of the inflammatory process in stroke, and to study various therapeutic strategies in order to reduce stroke damage.

Traveling fronts guided by the environment for reaction-diffusion equations

This paper deals with the existence of traveling fronts guided by the medium for a KPP reaction-diffusion equation coming from a model in population dynamics in which there is spatial spreading as well as genetic mutation of a quantitative genetic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of a threshold value on the existence of a nonzero asymptotic profile (a stationary limiting solution). When a nonzero asymptotic profile exists, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case. We also study here the bistable case. The equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if the area where theere is reaction is large enough and the non-existence if it is too small.

Existence and Nonexistence of Traveling Wave Solutions for a Bistable Reaction-Diffusion Equation in an Infinite Cylinder Whose Diameter is Suddenly Increased

ABSTRACT We consider a reaction-diffusion equation of bistable type in a square cylinder whose diameter varies with Neumann boundary conditions in dimension 2 and 3. We prove the nonexistence of generalized traveling wave solution of this equation when the diameter is suddenly strongly increased. At the same time, we prove that the solution of the equation with an exponentially decreasing initial condition cannot pass over a certain threshold far enough in the direction of propagation. The proof is divided in two steps. First, we extend the solution in the cylinder to a solution of the same equation in the half space. Then we overestimate it using Greens functions.

Asymptotic profiles for a travelling front solution of a biological equation

We are interested in the existence of depolarization waves in the human brain. These waves propagate in the grey matter and are absorbed in the white matter. We consider a two-dimensional model ut = �u + f(u) |y|≤R − �u |y|>R, with f a bistable nonlinearity taking effect only on the domain R × [−R,R], which represents the grey matter layer. We study the existence, the stability and the energy of non-trivial asymptotic profiles of the possible travelling fronts. For this purpose, we present dynamical systems techniques and graphic criteria based on Sturm-Liouville theory and apply them to the above equation. This yields three different behaviours of the solution u after stimulation, depending of the thickness R of the grey matter. This may partly explain the difficulties to observe depolarization waves in the human brain and the failure of several therapeutic trials.

In Silico Study of the Influence of Intensity and Duration of Blood Flow Reduction on Cell Death Through Necrosis or Apoptosis During Acute Ischemic Stroke

Ischemic stroke involves numerous and complex pathophysiological mechanisms including blood flow reduction, ionic exchanges, spreading depressions and cell death through necrosis or apoptosis. We used a mathematical model based on these phenomena to study the influences of intensity and duration of ischemia on the final size of the infarcted area. This model relies on a set of ordinary and partial differential equations. After a sensibility study, the model was used to carry out in silico experiments in various ischemic conditions. The simulation results show that the proportion of apoptotic cells increases when the intensity of ischemia decreases, which contributes to the model validation. The simulation results also show that the influence of ischemia duration on the infarct size is more complicated. They suggest that reperfusion is beneficial when performed in the early stroke but may be either inefficacious or even deleterious when performed later after the stroke onset. This aggravation could be explained by the depolarisation waves which might continue to spread ischemic damage and by the speeding up of the apoptotic process leading to cell death. The effect of reperfusion on cell death through these two phenomena needs to be further studied in order to develop new therapeutic strategies for stroke patients.

Abstract 402: A new mathematical model for describing metastatic spreading: Validation in tumor-bearing mice, confrontation with clinical data and in silico simulations to optimize treatment modalities.

Proceedings: AACR 104th Annual Meeting 2013; Apr 6-10, 2013; Washington, DC Occult metastatic disease is a major concern in clinical oncology because optimal therapy can not been undertaken in a timely manner. Developing mathematical tools for describing and anticipating early metastatic stages would help clinicians to choose the most adequate therapeutic strategy, even if no metastasis is yet detectable at bedside. We have developed a mathematical model that provides a Metastatic Index (MI). We used a phenomenological approach based on a structured transport equation with non local boundary condition for the colony size distribution of metastases. The velocity of this transport was related to a Gompertz laws growth. The colonization rate of the tumors reflects not only the metastatic diffusion but also some fractal dimension of the blood vessels infiltrating the tumors. This model is mostly based upon few parameters and can integrate the impact of surgery or chemotherapies on tumor growth and spreading. Model structure and parameters were first adjusted by fitting the predictions with observed data from several experiments performed in mice bearing the MDA-231LUC+ breast orthotopic xenograft. Three-dimmensional bioluminescence monitoring was carried out so as to detect early metastases as small as 10ˆ5 cells. A pre-validation step was carried out to check the ability of bioluminescence imaging to discriminate small metastatic sites from artifactual signals. Various molecular markers (A-cadherine, alcohol dehydrogenase, metaloproteases) were studied from tumor biopsies to refine some of the model parameters. Data were finally compared using Monolix software and Visual Predictive Check confirmed the ability of the model to predict accuratelly both tumor growth-rate and invasiveness. Additionally, we compared the in silico predictions of the model with clinical data from 2648 breast cancer patients (a.k.a. the Koscielny cohort) with a follow-up of metastatic reccurence depending on the initial tumor size measured upon surgery. Results showed that the models predictions matched the clinical observations (rˆ2=0.98), thus suggesting that our MI could be a useful tool indeed to forecast metastatic spreading in patients with cancer. Finally, in silico simulations were performed to study the impact of surgery or treatment with cytotoxics alone or combined with antiangiogenics. Marked variations in efficacy were observed depending on the treatment modalities. The resulting simulations suggest therefore that our mathematical model could be used as well to determine in silico the best scheduling and dosing of cancer chemotherapy, especially when anti-angiogenic and cytotoxic drugs are associated. Citation Format: Severine Mollard, Joseph Ciccolini, Niklas Hartung, Christian Faivre, Sebastien Benzekri, Guillemette Chapuisat, Assia Benabdallah, Florence Hubert, Dominique Barbolosi. A new mathematical model for describing metastatic spreading: Validation in tumor-bearing mice, confrontation with clinical data and in silico simulations to optimize treatment modalities. [abstract]. In: Proceedings of the 104th Annual Meeting of the American Association for Cancer Research; 2013 Apr 6-10; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2013;73(8 Suppl):Abstract nr 402. doi:10.1158/1538-7445.AM2013-402