Abstract
Although macrophages are part of the human immune system, it has been remarkably observed in laboratory experiments that decreasing its number can slow down the tumor progression. We analyze through a recently mathematical model proposed in the literature, necessary conditions for aggregation of tumor cells and macrophages. In order to do so, we prove the possibility of having blow-up in finite time. Next, we study if the aggregation of macrophages can occur when having a low density of tumor cells, and vice versa. With this purpose, we consider the problem of analyzing the existence or not of a simultaneous blow-up. We achieve this goal thanks to a novel process that allows us to compare the entropy functional associated with the density of each population, which turns out to be also a method to find enough conditions for having a simultaneous blow-up.
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Acknowledgements
The author gratefully acknowledges to Ph.D. Matthias Piesche, Facultad de Medicina, Universidad Católica del Maule, for his important comments for contextualize adequately in cancer progression the mathematical system studied in this work. The author gratefully acknowledges the comments on the first and the second drafts made by reviewers, which have greatly enriched this subsequent manuscript.
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For K. Vilches this work was funded by CONICYT PAI/ACADEMIA 79150021 2016–2018. For C. Conca this work is partially supported by PFBasal-001 and AFB170001 Projects.
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Espejo, E., Vilches, K. & Conca, C. A simultaneous blow-up problem arising in tumor modeling. J. Math. Biol. 79, 1357–1399 (2019). https://doi.org/10.1007/s00285-019-01397-6
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DOI: https://doi.org/10.1007/s00285-019-01397-6