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February  2004, 4(1): 289-295. doi: 10.3934/dcdsb.2004.4.289

Dynamics of a model for brain tumors reveals a small window for therapeutic intervention

Kristin R. Swanson 1, , Ellsworth C. Alvord, Jr 2, and  J. D. Murray 3,

1. 

Department of Pathology, Harborview Medical Center, and Department of Applied Mathematics, University of Washington, Seattle, WA, United States
2. 

Department of Pathology, Harborview Medical Center, Seattle, WA, United States
3. 

Department of Applied Mathematics, University of Washington, Seattle, WA, United States

Received  December 2002 Revised  June 2003 Published  November 2003

Glioblastomas are the most malignant and most common glioma, a type of primary brain tumor with the unfortunate ability to recur despite extensive treatment. Even with the advent of medical imaging technology during the last two decades, successful treatment of glioblastomas has remained elusive. It has become increasingly clear that, along with the proliferative potential of these neoplasms, it is the subclinically diffuse invasion of glioblastomas that primarily contributes to their resistance to treatment. In other words, the inevitable recurrence of these tumors is the result of diffusely invaded but invisible tumor cells peripheral to the abnormal signal on medical imaging and to the current limits of surgical, radiological and chemical treatments.
Mathematical modeling has presented itself as a viable tool for studying complex biological processes (Murray, 1993, 2002). We have developed a mathematical model that portrays the growth and extension of theoretical glioblastoma cells in a matrix that accurately describes the brain's anatomy to a resolution of 1 cu mm (Swanson, et al, 1999, 2000, 2002, 2003a, 2003b). The model assumes that only two factors need be considered for such predictions: net growth rate and infiltrative ability. The model has already provided illustrations of theoretical glioblastomas that not only closely resemble the MRIs (magnetic resonance imaging) of actual patients, but also show the distribution of the diffusely infiltrating cells.
Keywords: invasion., brain tumor, Mathematical model.
Mathematics Subject Classification: 93A3.
Citation: Kristin R. Swanson, Ellsworth C. Alvord, Jr, J. D. Murray. Dynamics of a model for brain tumors reveals a small window for therapeutic intervention. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 289-295. doi: 10.3934/dcdsb.2004.4.289
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