Mots-Clés

deep learning, Monte Carlo, partial differential equations, melanoma, resistance to treatment

Description

Internship: Simulating high dimensional models of tumor metabolic plasticity by probabilistic methods and deep learning

Start and duration: February to April 2023 for at least 4 months.  

Location: Computational Systems Biology team, LPHI, University of Montpellier, Montpellier.

https://systems-biology-lphi.cnrs.fr/

Candidates: M1 or M2 students / and or engineering students with excellent background in numerical analysis and machine learning and a taste for interdisciplinary research. The intern will receive a salary (gratification) at the standard rate used by French public research institutions.

Context: Tumor cell dynamics is described by high dimensional partial differential equations. Indeed, tumor cells are characterized by their position in the physical space, but also by many other dimensions characterizing the functioning of the intracellular signaling and metabolic pathways [1]. This raises important challenges concerning the computational cost, the precision and stability of traditional numerical simulation schemes based on finite differences or elements. In this internship we will investigate two alternative approaches. The first approach is probabilistic and based on Monte Carlo simulations. By the Feynman-Kac formula, solutions of high dimensional reaction-diffusion PDEs can be related to solutions of high dimensional stochastic differential equations [2]. The advantage of the probabilistic approach over finite differences or elements is that the convergence rate does not depend on the dimension of the problem and suffers less from the curse of dimensionality. The second approach exploits the universal approximation properties of artificial neural networks (ANN)2. ANNs have the capacity to overcome the curse of dimensionality: they can approximate solutions of PDEs arbitrarily well with parametric complexity growing polynomially in the PDE dimension [3]. High dimensional PDEs have numerous other applications in optimal control theory of large industrial systems (Hamilton-Jacobi equation), pricing financial derivatives in finance (Black-Scholes model), dynamics of large many-particle systems in physics (Schrödinger equation).  

Problem: A model of tumor adaptation to treatment extending our previously published model1 will be provided as high dimensional reaction-diffusion PDEs. The PDE model will have three spatial dimensions and up to 20 structural dimensions corresponding to metabolic markers. The intern will develop appropriate numerical schemes allowing the effective simulation of the model (either Monte Carlo or deep learning). We are interested in testing how the 3D distribution of blood vessels in the tumor influences the metabolic heterogeneity of the tumor and its resistance to treatment. The model’s predictions will be validated experimentally using multimodal imaging mass cytometry in the team of Laurent LeCam (INSERM and IRCM Montpellier). The numerical methods will be implemented using Julia or Python programming.

References

1 A Hodgkinson, D Trucu, M Lacroix, L Le Cam, O Radulescu. Computational Model of Heterogeneity in Melanoma : Designing Therapies and Predicting Outcomes. Frontiers in Oncology, 12 (2022).

2 E Weinan, J Han, A Jentzen. Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. Nonlinearity 35, 278 (2022).

3 M. Hutzenthaler, A. Jentzen, T.Kruse, TA Nguyen. A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical equation of semilinear heat equations. SN Partial Differential Equations and Applications, 1, 10 (2020), arXiv:13333v2.