Radiation Oncology/Radiobiology/Equations

Radiobiology Equations

Tumor Growth

• Mitotic Index:Template:Rp
  $MI=\frac{\text{Number of mitoses}}{\text{Number of cells}}=\lambda \times \frac{T_m}{T_c}$
• Labeling Index:
  $LI=\frac{\text{Number labeled}}{\text{Number of cells}}=\lambda \times \frac{T_s}{T_c}$
• Growth fraction:
  $GF=\frac{LI}{MI}$
• Tumor volume doubling time:
  $T_d$
• Potential doubling time:
  $T_{pot}=\frac{T_c}{GF}=\lambda \times \frac{T_s}{LI}$
• Cell loss factor:
  $CLF=1-\frac{T_{pot}}{T_d}$
• Gompertzian GrowthTemplate:Rp
  • Progressively slowing:
    $V=V_0\times \exp \left[\frac{A}{B}\times (1-\exp [-Bt])\right]$
  • Small t (early):
    $V=V_0\times \exp \left(At\right)$
  • Large t (late):
    $V=V_0\times \exp \left(\frac{A}{B}\right)$

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

Do not distinguish mode of death (mitotic vs apoptotic)

Target theory

• Surviving fraction (single target-single hit):
  $SF=\exp \left(\frac{-D}{D_0}\right)$
• Surviving fraction (multiple target-single hit):
  $SF=1-{(1-\exp \left[\frac{-D}{D_0}\right])}^n$
• Quasi-threshold dose:
  $D_q=D_0\times \ln n$
• $D_{10}=2.3\times D_0$
• ${S{F}_{\text{single-hit}}}^N=S{F}_{\text{multi-hit}}$

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 $d$:Template:RpTemplate:Rp
  $S{F}_d=\exp (-\alpha d-\beta d^2)$
• Fraction of cells surviving fractions $N$:Template:Rp
  $S{F}_N={(\exp [-\alpha d-\beta d^2])}^N=\exp [-\alpha D-\beta Dd]$
• Biologically Effective Dose (same RBE):Template:Rp
  $BE{D}_{\frac{\alpha }{\beta }}=N\times d\times [1+\frac{d}{\left(\frac{\alpha }{\beta }\right)}]$
• BED for high LET radiation (RBE adjusted):Template:Rp
  $BE{D}_H=N\times d\times [RB{E}_{max}+\frac{d}{\left(\frac{\alpha }{\beta }\right)}]$
• BED (time adjusted):
  $BE{D}_{time}=N\times d\times [1+\frac{d}{\left(\frac{\alpha }{\beta }\right)}]-\frac{0.693}{\alpha \times T_p}\times [T-T_k]$
• Isoeffective dose:
  $D_{\text{IsoE}}=D\times W_{\text{IsoE}}$
  $D_2=D_1\times \left[\frac{d_1+\frac{\alpha }{\beta }}{d_2+\frac{\alpha }{\beta }}\right]$
• Equivalent Dose in 2 Gy Fractions:
  $EQ{D}_2=N\times d\times \frac{d+\frac{\alpha }{\beta }}{2+\frac{\alpha }{\beta }}$

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)
  $TCP=\frac{\text{Number of colonies counted}}{\text{Number of cells seeded}\times PE}$
  $TCP=\exp [-\lambda ]$
  $TCP=\exp [-N_0\times \exp (-\alpha D-\beta dD)]$
  $TCP={S{F}_2}^N$

Definitions

• $N$=number of fractions

Linear Energy Transfer

• Linear Energy Transfer (LET):Template:Rp
  $LET=\frac{\mathrm{d}E}{\mathrm{d}l}$

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

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

Relative Biological Effectiveness

• Relative Biological Effectiveness (RBE):Template:Rp
  $RBE=\frac{D_{250}}{D_r}$

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