Radiation Oncology/Radiobiology/Equations

Radiobiology Equations

Tumor Growth

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

Definitions

• $T_{m}$=M phase duration
• $T_{c}$=cell cycle duration (total duration of all phases)
• $\lambda$=correction factor for uneven distribution of cells
• $T_{s}$=S phase duration
• $V$= tumor volume
• $V_{0}$= original tumor volume
• $t$= time
• $A$,$B$= constants

Cell survival curves

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

Do not distinguish mode of death (mitotic vs apoptotic)

Target theory

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

Definitions

• $D$=dose
• $D_{0}$=dose that decreases surviving fraction to 37%
• $n$=extrapolation number, $D_{0}$ doses required to kill all cells
• $D_{10}$=dose that decreases SF to 10%
• $N$=number of fractions

Linear Quadratic model

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

Definitions

• $N$=number of fractions
• $d$=dose
• $\alpha$=linear coefficient, reflects cell radiosensitivity
• $\beta$=quadratic coefficient, reflects cell repair mechanisms
• $T_{k}$=kick-off or onset time
• $T_{p}$=average cell-number doubling time
• $D$=total absorbed dose
• $W_{\text{IsoE}}$=weighting factor

Dose-response

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

Definitions

• $N$=number of fractions

Linear Energy Transfer

• Linear Energy Transfer (LET):
  $LET={\operatorname {d} E \over \operatorname {d} l}$

Radiation type	LET (keV/μm)
Co-60 photon	0.2
250 kVp photon	2.0
150 MeV proton	0.5
10 MeV proton	4.7
14 MeV neutron	100
18 MeV carbon	108
2.5 MeV alpha	166
75 MeV argon	250
2 GeV iron	1000

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

Definitions

• $\operatorname {d} E$=average energy locally imparted to medium
• $\operatorname {d} l$=track length

Relative Biological Effectiveness

• Relative Biological Effectiveness (RBE):
  $RBE={D_{250} \over D_{r}}$

Definitions

• $D_{250}$=dose of 250 kVp x-rays
• $D_{r}$=dose of test radiation required to produce equal biological effect to $D_{250}$

Hypoxia

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