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Low-rank tensor methods for Markov chains with applications to tumor progression models
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 tu...
Autores principales: | , , , , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer Berlin Heidelberg
2022
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9718722/ https://www.ncbi.nlm.nih.gov/pubmed/36460900 http://dx.doi.org/10.1007/s00285-022-01846-9 |
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author | Georg, Peter Grasedyck, Lars Klever, Maren Schill, Rudolf Spang, Rainer Wettig, Tilo |
author_facet | Georg, Peter Grasedyck, Lars Klever, Maren Schill, Rudolf Spang, Rainer Wettig, Tilo |
author_sort | Georg, Peter |
collection | PubMed |
description | 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. |
format | Online Article Text |
id | pubmed-9718722 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2022 |
publisher | Springer Berlin Heidelberg |
record_format | MEDLINE/PubMed |
spelling | pubmed-97187222022-12-04 Low-rank tensor methods for Markov chains with applications to tumor progression models Georg, Peter Grasedyck, Lars Klever, Maren Schill, Rudolf Spang, Rainer Wettig, Tilo J Math Biol Article 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. Springer Berlin Heidelberg 2022-12-02 2023 /pmc/articles/PMC9718722/ /pubmed/36460900 http://dx.doi.org/10.1007/s00285-022-01846-9 Text en © The Author(s) 2022 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) . |
spellingShingle | Article Georg, Peter Grasedyck, Lars Klever, Maren Schill, Rudolf Spang, Rainer Wettig, Tilo Low-rank tensor methods for Markov chains with applications to tumor progression models |
title | Low-rank tensor methods for Markov chains with applications to tumor progression models |
title_full | Low-rank tensor methods for Markov chains with applications to tumor progression models |
title_fullStr | Low-rank tensor methods for Markov chains with applications to tumor progression models |
title_full_unstemmed | Low-rank tensor methods for Markov chains with applications to tumor progression models |
title_short | Low-rank tensor methods for Markov chains with applications to tumor progression models |
title_sort | low-rank tensor methods for markov chains with applications to tumor progression models |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9718722/ https://www.ncbi.nlm.nih.gov/pubmed/36460900 http://dx.doi.org/10.1007/s00285-022-01846-9 |
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