Radiobiology Equations

Tumor Growth

• Mitotic Index:^[1]:252
  ${\displaystyle MI={{\text{Number of mitoses}} \over {\text{Number of cells}}}={\lambda \times {T_{m} \over T_{c}}}}$
• Labeling Index:
  ${\displaystyle LI={{\text{Number labeled}} \over {\text{Number of cells}}}={\lambda \times {T_{s} \over T_{c}}}}$
• Growth fraction:
  ${\displaystyle GF={LI \over MI}}$
• Tumor volume doubling time:
  ${\displaystyle T_{d}}$
• Potential doubling time:
  ${\displaystyle T_{pot}={T_{c} \over GF}={\lambda \times {T_{s} \over LI}}}$
• Cell loss factor:
  ${\displaystyle CLF={1-{T_{pot} \over T_{d}}}}$
• Gompertzian Growth^[2]:475
  • Progressively slowing:
    ${\displaystyle V=V_{0}\times \exp {\left[{{A \over B}\times {\left(1-\exp {\left[-Bt\right]}\right)}}\right]}}$
  • Small t (early):
    ${\displaystyle V=V_{0}\times \exp \left(At\right)}$
  • Large t (late):
    ${\displaystyle V=V_{0}\times \exp \left({A \over B}\right)}$

Cell survival curves

• Plating efficiency:
  ${\displaystyle PE={{\text{Number of colonies counted}} \over {\text{Number of cells seeded}}}}$
• Surviving fraction:
  ${\displaystyle SF={{\text{Number of colonies counted}} \over {\text{Number of cells seeded}}\times {PE}}}$

Do not distinguish mode of death (mitotic vs apoptotic)

Target theory

• Surviving fraction (single target-single hit):^[3]
  ${\displaystyle SF=\exp \left({-D \over D_{0}}\right)}$
• Surviving fraction (multiple target-single hit):
  ${\displaystyle SF=1-{\left(1-\exp \left[{-D \over D_{0}}\right]\right)}^{n}}$
• Quasi-threshold dose:^[4]
  ${\displaystyle D_{q}=D_{0}\times \ln n}$
• ${\displaystyle D_{10}=2.3\times D_{0}}$
• ${\displaystyle {SF_{\text{single-hit}}}^{N}=SF_{\text{multi-hit}}}$

Linear Quadratic model

• Fraction of cells surviving single dose ${\displaystyle d}$:^{[1]:228[5]:31}
  ${\displaystyle SF_{d}=\exp \left(-\alpha d-\beta d^{2}\right)}$
• Fraction of cells surviving fractions ${\displaystyle N}$:^[5]:31
  ${\displaystyle SF_{N}={\left(\exp \left[-\alpha d-\beta d^{2}\right]\right)}^{N}=\exp \left[-\alpha D-\beta Dd\right]}$
• Biologically Effective Dose (same RBE):^[1]:230
  ${\displaystyle BED_{\alpha \over \beta }=N\times d\times \left[1+{d \over {\left({\alpha \over \beta }\right)}}\right]}$
• BED for high LET radiation (RBE adjusted):^[4]:268
  ${\displaystyle BED_{H}=N\times d\times \left[RBE_{max}+{d \over {\left({\alpha \over \beta }\right)}}\right]}$
• BED (time adjusted):^[6]
  ${\displaystyle BED_{time}=N\times d\times \left[1+{d \over {\left({\alpha \over \beta }\right)}}\right]-{0.693 \over \alpha \times T_{p}}\times {\left[T-T_{k}\right]}}$
• Isoeffective dose:^[7][8]
  ${\displaystyle D_{\text{IsoE}}=D\times W_{\text{IsoE}}}$
  ${\displaystyle D_{2}=D_{1}\times \left[{{d_{1}+{\alpha \over \beta }} \over {d_{2}+{\alpha \over \beta }}}\right]}$
• Equivalent Dose in 2 Gy Fractions:
  ${\displaystyle EQD_{2}=N\times d\times {{d+{\alpha \over \beta }} \over {2+{\alpha \over \beta }}}}$

Dose-response

• Tumor control probability (TCP)
  ${\displaystyle TCP={{\text{Number of colonies counted}} \over {\text{Number of cells seeded}}\times {PE}}}$
  ${\displaystyle TCP=\exp \left[-\lambda \right]}$
  ${\displaystyle TCP=\exp \left[-N_{0}\times \exp \left(-\alpha D-\beta dD\right)\right]}$
  ${\displaystyle TCP={SF_{2}}^{N}}$

Linear Energy Transfer

• Linear Energy Transfer (LET):^[9]:106
  ${\displaystyle LET={\operatorname {d} E \over \operatorname {d} l}}$

Optimal RBE as a function of LET at 100 keV/μm

Relative Biological Effectiveness

• Relative Biological Effectiveness (RBE):^[9]:115
  ${\displaystyle RBE={D_{250} \over D_{r}}}$

Hypoxia

• Oxygen enhancement ratio:^[1]:237
  ${\displaystyle OER={{\text{dose in hypoxic cells}} \over {\text{dose in aerated cells to cause same effect}}}}$
• OER Values:
  • photon 3
  • proton 3
  • neutron 1.6
  • energized ion 1
  • alpha 1
• ${\displaystyle {\text{Initial proportion of hypoxic cells}}={{\text{SF aerated}} \over {\text{SF hypoxic}}}}$

References