Section: Application Domains

Estimation for complex and biological systems

Our main application for this line of investigation is the photodynamic therapy developed by T. Bastogne. We shall also focus on bacteriophage therapies and subdiffusion within molecules.

(i) Photodynamic therapy. One of the main application we have in mind for our identification problems is to model photodynamic therapy. This promising cancer treatment involves selective uptake and retention of a photosensitive drug in a tumor, followed by irradiation with light at an appropriate wavelength. Photosensitizers are photoactive compounds such as for instance porphyrins and chlorins. The activated photosensitizer is thought to produce singlet oxygen at high doses and thereby to initiate apoptotic and necrotic death of tumor. Due to the lack of response reproducibility, the complexity of interactions between physical, chemical and biological aspects and the high cost of experiments, there is a real demand in good mathematical and physical models which might help to better control and understand PDT responses. We are particularly concerned with modeling the drug uptake into cancer cells, the photoreactions induced by light exposition and tumor growth kinetics.

(ii) Bacteriophage systems. A collaboration between our team, the Mathematics and the Genetics and Microbiology Departments at the Universitat Autònoma de Barcelona (UAB) is being set up, focusing on probabilistic aspects of bacteriophage therapies for animal diseases like hemorrhagic septicemia in cattle or atrophic rhinitis in swine. This kind of therapy consists in inoculating a (benign) virus to animals in order to kill the bacteria known to be responsible of the disease. It was in use in the Soviet Union until the 80s, and is now re-emerging, still at an experimental level, due to the progressive slowdown in antibiotic efficiency.

Within this context, our analysis of a noisy predator-prey competition modeling the treatment helps to calibrate and to understand better the behavior of the system in terms of fluctuations around an equilibrium. Note that our preliminary contacts with the Genetics and Microbiology Departments at UAB also open the way to a particle model in order to represent the couple bacteria/virus living on a surface.

(iii) Subdiffusion into molecules. Our purpose here is a better understanding of the phenomena observed in nanoscale Biophysics, as explained in the series of papers [52] . The technological advances in nanoscale technologies allow the observation of single molecules, and thus the description of newly observed phenomenon. A typical example of this new kind of observation is given by the fluctuations in the folding of a protein-enzyme compound called Fre, which is involved in the DNA synthesis of the (canonical) bacterium E. Coli.

More specifically, the paper [52] advocates for modeling this folding fluctuations by means of a Volterra type equation driven by a fractional Brownian motion. This convincing model is based on some experimental and physical evidences, and have also been observed in a wide number of recent biological experiments. However, the model exhibited in [52] also raises some unsolved questions: some stochastic equations appearing in the models are not properly defined and their long time behavior is still mysterious. The lack of a method in order to simulate and estimate coefficients of these equations on a solid mathematical ground should also be mentioned. This is the kind of topic we wish to address, for which a preliminary contact with S. Kou and N. Pillai (Princeton University, USA) has been established.

(iv) Osteoporosis. During the year 2011-2012, C. Lacaux has been visiting the MAP 5 (Paris Descartes University) laboratory and joined the ANR Project MATAIM (Modèles Anisotropes de Textures. Applications à l'Imagerie Médicale). This project, which involves both mathematicians and practitioners, is in particular interested in the osteoporosis diagnostic. The paper [34] is a first step in the direction of modeling trabecular bone x-ray images by some operator scaling fields. Actually the estimation of the matrix, which characterizes the anisotropy of the model, is crucial for practical purposes. Hermine Biermé (Paris Descartes Univesity) and Céline Lacaux are working on this problem using quadratic variations. Once the problem of estimation is solved, they plan a comparison of the theoretical model with real data provided by our Biologist colleagues of the MATAIM project. If the model corresponds to real data (as suggested in [34] ), this approach may help for the diagnostic of osteoporosis: a numerical study has to be performed in order to find the parameter value which characterizes osteoporosis.