Journal of Mathematical Biology

, Volume 58, Issue 4, pp 561-578

First online:

A mathematical model for brain tumor response to radiation therapy

  • R. RockneAffiliated withDepartment of Pathology, University of Washington
  • , E. C. AlvordJr.Affiliated withDepartment of Pathology, University of Washington
  • , J. K. RockhillAffiliated withDepartment of Pathology, University of Washington
  • , K. R. SwansonAffiliated withDepartment of Pathology, University of Washington Email author 

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Gliomas are highly invasive primary brain tumors, accounting for nearly 50% of all brain tumors (Alvord and Shaw in The pathology of the aging human nervous system. Lea & Febiger, Philadelphia, pp 210–281, 1991). Their aggressive growth leads to short life expectancies, as well as a fairly algorithmic approach to treatment: diagnostic magnetic resonance image (MRI) followed by biopsy or surgical resection with accompanying second MRI, external beam radiation therapy concurrent with and followed by chemotherapy, with MRIs conducted at various times during treatment as prescribed by the physician. Swanson et al. (Harpold et al. in J Neuropathol Exp Neurol 66:1–9, 2007) have shown that the defining and essential characteristics of gliomas in terms of net rates of proliferation (ρ) and invasion (D) can be determined from serial MRIs of individual patients. We present an extension to Swanson’s reaction-diffusion model to include the effects of radiation therapy using the classic linear-quadratic radiobiological model (Hall in Radiobiology for the radiologist. Lippincott, Philadelphia, pp 478–480, 1994) for radiation efficacy, along with an investigation of response to various therapy schedules and dose distributions on a virtual tumor (Swanson et al. in AACR annual meeting, Los Angeles, 2007).


Modeling Radiation therapy Glioma Linear-quadratic Tumor response Treatment fractionation

Mathematics Subject Classification (2000)