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A Mathematical Model of Glioblastoma Tumor Spheroid Invasion in a Three-Dimensional In Vitro Experiment

Andrew M. Stein ^*, Tim Demuth , David Mobley , Michael Berens  and Leonard M. Sander 

^* Department of Mathematics, University of Michigan, Ann Arbor, Michigan; Cancer and Cell Biology Division, TGen, Phoenix, Arizona; Department of Pharmaceutical Chemistry, University of California at San Francisco, San Francisco, California; and Department of Physics, University of Michigan, Ann Arbor, Michigan

Correspondence: Address reprint requests to A. M. Stein, Dept. of Mathematics, University of Michigan, 2074 East Hall, 525 Church St., Ann Arbor, MI 48109. E-mail: amstein{at}umich.edu.

Glioblastoma, the most malignant form of brain cancer, is responsible for 23% of primary brain tumors and has extremely poor outcome. Confounding the clinical management of glioblastomas is the extreme local invasiveness of these cancer cells. The mechanisms that govern invasion are poorly understood. To gain insight into glioblastoma invasion, we conducted experiments on the patterns of growth and dispersion of U87 glioblastoma tumor spheroids in a three-dimensional collagen gel. We studied two different cell lines, one with a mutation to the EGFR (U87EGFR) that is associated with increased malignancy, and one with an endogenous (wild-type) receptor (U87WT). We developed a continuum mathematical model of the dispersion behaviors with the aim of identifying and characterizing discrete cellular mechanisms underlying invasive cell motility. The mathematical model quantitatively reproduces the experimental data, and indicates that the U87WT invasive cells have a stronger directional motility bias away from the spheroid center as well as a faster rate of cell shedding compared to the U87EGFR cells. The model suggests that differences in tumor cell dispersion may be due to differences in the chemical factors produced by cells, differences in how the two cell lines remodel the gel, or different cell-cell adhesion characteristics.

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