A mathematical model for brain tumor response to radiation therapy

Abstract

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 (rho) 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).

Fig. 1

T1Gd and T2-weighted MRI images of a patient diagnosed with glioblastoma multiforme (GBM)

Fig. 2

a Two-dimensional cross section of actual dose distribution for a patient with a GBM. Enhancement margin is nonuniform in order to avoid high doses of radiation to the brain stem. b One-dimensional dose distribution created with a spline interpolation of nine reference points. c Cumulative dose distribution showing enhancement region plus margin for initial and boost phases of dose delivery for the standard treatment scheme

Fig. 2

a Two-dimensional cross section of actual dose distribution for a patient with a GBM. Enhancement margin is nonuniform in order to avoid high doses of radiation to the brain stem. b One-dimensional dose distribution created with a spline interpolation of nine reference points. c Cumulative dose distribution showing enhancement region plus margin for initial and boost phases of dose delivery for the standard treatment scheme

Fig. 2

a Two-dimensional cross section of actual dose distribution for a patient with a GBM. Enhancement margin is nonuniform in order to avoid high doses of radiation to the brain stem. b One-dimensional dose distribution created with a spline interpolation of nine reference points. c Cumulative dose distribution showing enhancement region plus margin for initial and boost phases of dose delivery for the standard treatment scheme

Fig. 3

a Simulated tumor growth and response prior to, during and after XRT as measured on T1Gd. Solid black line shows no visible tumor for ~30 days. Time t = 0 days corresponds to the beginning of standard XRT simulated with range of sensitivity values 0.025 Gy⁻¹ ≤ α ≤ 0.036 Gy⁻¹. Dotted black line shows projected untreated tumor growth using Fisher’s approximation: D = 1.43mm²/year, ρ = 16.52 /year. b Detail of a with typical response window (TRW) of −30 to +20% of pre-treatment radius

Fig. 3

a Simulated tumor growth and response prior to, during and after XRT as measured on T1Gd. Solid black line shows no visible tumor for ~30 days. Time t = 0 days corresponds to the beginning of standard XRT simulated with range of sensitivity values 0.025 Gy⁻¹ ≤ α ≤ 0.036 Gy⁻¹. Dotted black line shows projected untreated tumor growth using Fisher’s approximation: D = 1.43mm²/year, ρ = 16.52 /year. b Detail of a with typical response window (TRW) of −30 to +20% of pre-treatment radius

Fig. 4

Simulated tumor response to XRT using a one per day dose delivery scheme for several days of therapy with the total dose held constant at 61.2Gy

Fig. 5

Response to therapy as measured by percent reduction in pre-treatment radius at the end of therapy as visible on T1Gd and T2 MRI for 1, 5, 10, 15, …, 40DOT at each of 1–4 dose fractions per day using two different dose distribution schemes: equal fractions and boost. Total dose held constant at 61.2 Gy

Fig. 6

Recovery time, measured as the minimum time for which the tumor attains its pre-treatment radius for each of four fractions per day and 1, 5, 10, 15, …, 40DOT. Recovery time was set to one day for treatments that yielded no response