Radiation Oncology/Radiobiology/Equations

Radiobiology Equations

• Tumor Growth
  • Mitotic Index: MI = # mitoses / # cells = λ * T_m / T_c
  • Labeling Index: LI = # labeled / # cells = λ * (T_s / T_c)
  • Growth fraction: GF = LI / MI
  • Tumor volume doubling time: T_d
  • Potential doubling time: T_pot = T_c / GF = λ * (T_s / LI)
  • Cell loss faction: CLF = 1 - T_pot / T_d
  • Gompertzian Growth (progressively slowing: V = V₀ * exp[A/B * (1 - exp-Bt)]
    • Small t (early): V = V₀ * exp(At)
    • Large t (late): V = V₀ * exp (A/B)
• Cell survival curves
  • Plating efficiency: # colonies counted / # number cells seeded
  • Surviving fraction: SF = # colonies counted / (# cells seeded * plating efficiency)
  • Do not distinguish mode of death (mitotic vs apoptotic)
• Target theory
  • SF(single hit) = exp(-D/D₀)
  • SF(multi-hit) = 1 - (1 - exp(-D/D₀))ⁿ
  • D_q = D₀ * ln(n)
  • D₁₀ = 2.3 * D₀
  • SF(single fx)ⁿ = SF(multifraction), where n= number of fractions
• Linear Quadratic model
  • SF_D = exp -(αD+βD^2)
  • SF_n = (exp -(αD+βD^2))ⁿ
  • BED (same RBE) = n * d * (1 + d/(α/β))
  • BED (RBE adjusted) = n * d * (RBE_max + d/(α/β))
  • BED (time adjusted) = n* d * (1 + d/(α/β)) – (0.693 * t) / (α * Tpot)
  • Isoeffect dose: D₂ = D₁ * (d1 + α/β) / (d2 + α/β)
  • Equivalent 2Gy dose: EQD₂ = n * d * (d + α/β) / (2 + α/β)
• Dose-response
  • TCP = exp(-λ)
  • TCP = exp(-N₀ * exp -(αD+βdD))
  • TCP = (SF₂)^N, where N = number of fractions
• LET
  • LET = dE / dl, where dE is average energy locally imparted to medium, and dl is track length
  • Co-60 photon 0.2 keV/μm
  • 250 kVp photon 2.0
  • 150 MeV proton 0.5
  • 10 MeV proton 4.7
  • 14 MeV neutron 100
  • 2.5 MeV alpha 166
  • 2 GeV Fe 1000
  • Optimal RBE as a function of LET at 100 keV/μm
• RBE
  • RBE = dose(250 kV) / dose (test)
• Hypoxia
  • Oxygen enhancement ratio: OER = dose in hypoxic cells / dose in aerated cells
  • Initial proportion of hypoxic cells = SF aerated / SF hypoxic