Low-rank tensor methods for Markov chains with applications to tumor progression models

Journal of Mathematical Biology, vol. 86, pp. 7

Abstract

Cancer progression can be described by continuous-time Markov chains whose state space grows exponentially in the number of somatic mutations. The age of a tumor at diagnosis is typically unknown. Therefore, the quantity of interest is the time-marginal distribution over all possible genotypes of tumors, defined as the transient distribution integrated over an exponentially distributed observation time. It can be obtained as the solution of a large linear system. However, the sheer size of this system renders classical solvers infeasible. We consider Markov chains whose transition rates are separable functions, allowing for an efficient low-rank tensor representation of the linear system's operator. Thus we can reduce the computational complexity from exponential to linear. We derive a convergent iterative method using low-rank formats whose result satisfies the normalization constraint of a distribution. We also perform numerical experiments illustrating that the marginal distribution is well approximated with low rank.

Authors 6

1. Peter Georg

   University of Regensburg

   Affiliation as printed

   Department of Physics, University of Regensburg, 93040, Regensburg, Germany

2. Lars Grasedyck Aachen Institute for Geometry and Applied Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Geometry and Applied Mathematics, RWTH Aachen University, 52062, Aachen, Germany

3. Maren Klever corresponding Aachen Institute for Geometry and Applied Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Geometry and Applied Mathematics, RWTH Aachen University, 52062, Aachen, Germany. klever@igpm.rwth-aachen.de

   Institute for Geometry and Applied Mathematics, RWTH Aachen University, 52062, Aachen, Germany

4. Rudolf Schill

   University of Regensburg

   Affiliation as printed

   Department of Statistical Bioinformatics, Institute of Functional Genomics, University of Regensburg, 93040, Regensburg, Germany

5. Rainer Spang

   University of Regensburg

   Affiliation as printed

   Department of Statistical Bioinformatics, Institute of Functional Genomics, University of Regensburg, 93040, Regensburg, Germany

6. Tilo Wettig

   University of Regensburg

   Affiliation as printed

   Department of Physics, University of Regensburg, 93040, Regensburg, Germany