Distributed delays in a hybrid model of tumor-Immune system interplay
A tumor is kinetically characterized by the presence of multiple spatio-temporal scales in which its cells interplay with, for instance, endothelial cells or Immune system effectors, exchanging various chemical signals. By its nature, tumor growth is an ideal object of hybrid modeling where dis...
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| Language: | English |
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AIMS Press
2012-11-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.37 |
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| author | Giulio Caravagna Alex Graudenzi Alberto d’Onofrio |
| author_facet | Giulio Caravagna Alex Graudenzi Alberto d’Onofrio |
| author_sort | Giulio Caravagna |
| collection | DOAJ |
| description | A tumor is kinetically characterized by the presence of multiple spatio-temporal scales in which its cells interplay with, for instance, endothelial cells or Immune system effectors, exchanging various chemical signals. By its nature, tumor growth is an ideal object of hybrid modeling where discrete stochastic processes model low-numbers entities, and mean-field equations model abundant chemical signals. Thus, we follow this approach to model tumor cells, effector cells and Interleukin-2, in order to capture the Immune surveillance effect. We here present a hybrid model with a generic delay kernel accounting that,due to many complex phenomena such as chemical transportation and cellular differentiation,the tumor-induced recruitment of effectors exhibits a lag period. This model is a Stochastic Hybrid Automata and its semantics is a Piecewise Deterministic Markov process where a two-dimensional stochastic process is interlinked to a multi-dimensional mean-field system. We instantiate the model withtwo well-known weak and strong delay kernels and perform simulations by using an algorithm to generate trajectories of this process. Via simulations and parametric sensitivity analysis techniques we $(i)$ relate tumor mass growth with the two kernels, we $(ii)$ measure the strength of the Immune surveillance in terms of probability distribution of the eradication times, and $(iii)$ we prove, in the oscillatory regime, the existence of a stochastic bifurcation resulting in delay-induced tumor eradication. |
| format | Article |
| id | doaj-art-af1ebafd907048dca99cd6eb969601e5 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2012-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-af1ebafd907048dca99cd6eb969601e52025-01-24T02:25:25ZengAIMS PressMathematical Biosciences and Engineering1551-00182012-11-01101375710.3934/mbe.2013.10.37Distributed delays in a hybrid model of tumor-Immune system interplayGiulio Caravagna0Alex Graudenzi1Alberto d’Onofrio2Department of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I-20126 MilanDepartment of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I-20126 MilanDepartment of Informatics, Systems and Communication, University of Milan Bicocca, Viale Sarca 336, I-20126 MilanA tumor is kinetically characterized by the presence of multiple spatio-temporal scales in which its cells interplay with, for instance, endothelial cells or Immune system effectors, exchanging various chemical signals. By its nature, tumor growth is an ideal object of hybrid modeling where discrete stochastic processes model low-numbers entities, and mean-field equations model abundant chemical signals. Thus, we follow this approach to model tumor cells, effector cells and Interleukin-2, in order to capture the Immune surveillance effect. We here present a hybrid model with a generic delay kernel accounting that,due to many complex phenomena such as chemical transportation and cellular differentiation,the tumor-induced recruitment of effectors exhibits a lag period. This model is a Stochastic Hybrid Automata and its semantics is a Piecewise Deterministic Markov process where a two-dimensional stochastic process is interlinked to a multi-dimensional mean-field system. We instantiate the model withtwo well-known weak and strong delay kernels and perform simulations by using an algorithm to generate trajectories of this process. Via simulations and parametric sensitivity analysis techniques we $(i)$ relate tumor mass growth with the two kernels, we $(ii)$ measure the strength of the Immune surveillance in terms of probability distribution of the eradication times, and $(iii)$ we prove, in the oscillatory regime, the existence of a stochastic bifurcation resulting in delay-induced tumor eradication.https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.37piecewise deterministic markov processtumorstochastic hybrid automataimmune systemdelay differential equationdistributed delays. |
| spellingShingle | Giulio Caravagna Alex Graudenzi Alberto d’Onofrio Distributed delays in a hybrid model of tumor-Immune system interplay Mathematical Biosciences and Engineering piecewise deterministic markov process tumor stochastic hybrid automata immune system delay differential equation distributed delays. |
| title | Distributed delays in a hybrid model of tumor-Immune system interplay |
| title_full | Distributed delays in a hybrid model of tumor-Immune system interplay |
| title_fullStr | Distributed delays in a hybrid model of tumor-Immune system interplay |
| title_full_unstemmed | Distributed delays in a hybrid model of tumor-Immune system interplay |
| title_short | Distributed delays in a hybrid model of tumor-Immune system interplay |
| title_sort | distributed delays in a hybrid model of tumor immune system interplay |
| topic | piecewise deterministic markov process tumor stochastic hybrid automata immune system delay differential equation distributed delays. |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.37 |
| work_keys_str_mv | AT giuliocaravagna distributeddelaysinahybridmodeloftumorimmunesysteminterplay AT alexgraudenzi distributeddelaysinahybridmodeloftumorimmunesysteminterplay AT albertodonofrio distributeddelaysinahybridmodeloftumorimmunesysteminterplay |