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Speed Switch in Glioblastoma Growth Rate due to Enhanced Hypoxia-Induced Migration

Curtin, Lee; Hawkins-Daarud, Andrea; van der Zee, Kristoffer G.; Swanson, Kristin R.; Owen, Markus R.


Lee Curtin

Andrea Hawkins-Daarud

Kristin R. Swanson


We analyze the wave-speed of the Proliferation Invasion Hypoxia Necro-sis Angiogenesis (PIHNA) model that was previously created and applied to simulate the growth and spread of glioblastoma (GBM), a particularly aggressive primary brain tumor. We extend the PIHNA model by allowing for different hy-poxic and normoxic cell migration rates and study the impact of these differences on the wave-speed dynamics. Through this analysis, we find key variables that drive the outward growth of the simulated GBM. We find a minimum tumor wave-speed for the model; this depends on the migration and proliferation rates of the normoxic cells and is achieved under certain conditions on the migration rates of the normoxic and hypoxic cells. If the hypoxic cell migration rate is greater than the normoxic cell migration rate above a threshold, the wave-speed increases above the predicted minimum. This increase in wave-speed is explored through an eigenvalue and eigenvector analysis of the linearized PIHNA model, which yields an expression for this threshold. The PIHNA model suggests that an inherently faster-diffusing hypoxic cell population can drive the outward growth of a GBM as a whole, and that this effect is more prominent for faster proliferating tumors that recover relatively slowly from a hypoxic phenotype. The findings presented here act as a first step in enabling patient-specific calibration of the PIHNA model.


Swanson, K. R., Curtin, L., Hawkins-Daarud, A., van der Zee, K. G., Swanson, K. R., & Owen, M. R. (2020). Speed Switch in Glioblastoma Growth Rate due to Enhanced Hypoxia-Induced Migration. Bulletin of Mathematical Biology, 82(3),

Journal Article Type Article
Acceptance Date Mar 4, 2020
Online Publication Date Mar 16, 2020
Publication Date Mar 16, 2020
Deposit Date Mar 6, 2020
Publicly Available Date Mar 17, 2021
Journal Bulletin of Mathematical Biology
Print ISSN 0092-8240
Electronic ISSN 1522-9602
Publisher Springer Verlag
Peer Reviewed Peer Reviewed
Volume 82
Issue 3
Article Number 43
Keywords Immunology; General Biochemistry, Genetics and Molecular Biology; Computational Theory and Mathematics; General Neuroscience; Pharmacology; General Agricultural and Biological Sciences; General Mathematics; General Environmental Science
Public URL
Publisher URL
Additional Information Received: 4 September 2019; Accepted: 4 March 2020; First Online: 16 March 2020


BMB Wavespeed Changes Unmarked - Curtin Hawkins-daarud VanderZee Swanson Owen 2020 (1.5 Mb)

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