S. Sadhukhan, D. Santra, S. Dey
On the cellular level, glioma cells seem to be either in a "go" state or in a "grow" state. It is observed that the outer edge of the tumor grows over time. A general rule c [≤] 1/2c_F was proven by Thiessen, Conte, Stepien, and Hillen via a substitution argument; however, they weren't able to derive the exact formula for the tumor's speed or determine exactly what conditions would cause a tumor to reach its maximum speed. In this article, it is assumed that the cells switch back and forth between moving and growing very fast, and the exact equation for the speed of the tumor expansion is c* = 2{surd}{phi}i_m {phi}_p rD. It is found that the same rate of maximum possible growth occurs when cells move for 50\% of the time and grow for 50% of the time. If the cells spend more of their time doing one than the other, it grows even more slowly. Moreover, a dimensionless parameter is introduced: {chi} = c/2{surd}rD, to determine the exact hidden phenotype balance through plugin the speed at which one individual cell moves (D), the cell division rate (r), and the tumor expansion rate (c). The outcome will be near 1 if the old standard tumor model is correct. If it is 0.5 or below, our model is correct; it is either a "go" or "grow" state. In addition, the exact number indicates the percentage of tumour cells that are moving. In the human brain, tumors do not spread evenly in a circle; they spread much faster along with the white-matter tracts apparent through diffusion tensor imaging (DTI). It is shown that our 50% speed limit remains valid along these "brain highways" a result that has not been previously shown in a mathematical study. Lastly, we address a theoretical question that was left unanswered by past studies using computer simulations and discuss the implications for clinical interpretation of MRI growth rates from real patients using our results.