PhD Position F/M PhD in applied mathematics: Stochastic modeling and statistics for quantifying and predicting the evolution of tumor heterogeneity in chronic lymphocytic leukemia

Tasks

In this PhD project, we propose to tackle the problem of clonal reconstruction, first from data collected at a single time (already available), and second from longitudinal data. Data will be collected throughout the duration of the PhD thesis.

The main problem consists in reconstructing the phylogenetic tree of mutations and the dynamics of frequencies of each clone. The originality comes from the fact that data are heterogeneous: we will have the full profile of VDJ mutations of clones with frequencies and each cancer genes variants with allele frequencies. From the mathematical modeling perspective, VDJ data share common features with single-cell data since full sequences can be reconstructed using tools like MiXCR (https://mixcr.com). Existing packages for clonal heterogeneity analysis are B-SCITE (Malikic et al., 2017) and ddClone (Salehi et al., 2017). They are able to deal with both types of data (bulk and single-cell) and could in principle be used here. However, there are specificities of CLL that do not fit into these methods.

The PhD student will first construct a probabilistic model accounting for all the data. This model will contain the phylogenetic tree as latent variable, where each node in the tree corresponds either to a VDJ mutation, a mutation of cancer genes, or a chromosomic alteration, where each mutation occurs only once in the tree. The observations will then be obtained, following the classical rules of the infinitely many sites model, as linear combi- nations of the frequency of every clone in the sample (which are other latent variables), possibly with some noise.

Treating latent variables as parameters, we could use the maximum likelihood method, but maximization is a difficult problem in practice due to the very large number of possible trees. We will test genetic algorithms (Metropolis-Hastings, MCMC...), but we expect better results using a Bayesian approach, combined with a variational method to maximize the a posteriori likelihood.

Second, the PhD student will validate the method from data simulated from our model, then using the benchmark simulation tool proposed by Foglierini et al. (2020), adapting them to our double heterogeneity context, and finally comparing with single-cell sequencing data of 3D in vitro cultures of proliferating cells that will be collected all along the project. Prediction performances will also be tested.

Finally, we will try to detect if groups of patients have similar mutational patterns (such as phylogenetic tree topology), which could correspond to a similar tumorigenesis, or a similar stage of progression, or a similar response to treatments. This is a clustering problem that can be addressed by model-free artificial intelligence tools (such as latent Dirichlet allocation: Pritchard et al., 2000; Falush et al., 2003), or using models like those developed by Beerenwinkel et al. (2004, 2005). This will allow us to build a predictive model of treatment efficiency given the clonal heterogeneity of a patient, that can be used by clinicians in a context of personalized medicine.

Skills

The candidate should have skills in statistics and/or stochastic modeling. R, Python or Matlab programming skills are also required. An affinity or experience with medical applications will be highly appreciated.

Keywords: Applied probability, stochastic modeling, statistical modeling for medicine, variational Bayesian methods, clonal heterogeneity, chronic lymphocytic leukemia