Front Page: Radiation Oncology | RTOG Trials | Randomized Trials

### 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):
${\displaystyle V=V_{0}\times \exp \left({A \over B}\right)}$

#### Definitions

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

### 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]
${\displaystyle D_{q}=D_{0}\times \ln n}$
• ${\displaystyle D_{10}=2.3\times D_{0}}$
• ${\displaystyle {SF_{\text{single-hit}}}^{N}=SF_{\text{multi-hit}}}$

#### Definitions

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

### 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:
${\displaystyle EQD_{2}=N\times d\times {{d+{\alpha \over \beta }} \over {2+{\alpha \over \beta }}}}$

#### Definitions

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

### Dose-response

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

#### Definitions

• ${\displaystyle N}$=number of fractions

### Linear Energy Transfer

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

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

### Relative Biological Effectiveness

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

#### Definitions

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

### 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

1. a b c d David S. Chang, Foster D. Lasley, Indra J. Das, Marc S. Mendonca, Joseph R. Dynlacht (2014). Basic Radiotherapy Physics and Biology. Springer. ISBN 9783319068411.{{cite book}}: CS1 maint: uses authors parameter (link)
2. H. Awwad (2013). Radiation Oncology: Radiobiological and Physiological Perspectives. Springer. ISBN 9789401578653.{{cite book}}: CS1 maint: uses authors parameter (link)
3. Beyzadeoglu, Murat, Ozyigit, Gokhan, Ebruli, Cüneyt (2010). Basic Radiation Oncology. Springer. ISBN 978-3-642-11665-0.{{cite book}}: CS1 maint: uses authors parameter (link)
4. a b Roger G. Dale, Bleddyn Jones (2007). Radiobiological Modelling in Radiation Oncology. British Institute of Radiology. ISBN 9780905749600.{{cite book}}: CS1 maint: uses authors parameter (link)
5. a b Lemoigne, Yves; Caner, Alessandra (2011). Radiation Protection in Medical Physics. Dordrecht: Springer. ISBN 9789400702479.
6. Levitt, S.H. (2006). Technical basis of radiation therapy : practical clinical applications ; with 146 tables (4th ed.). Berlin: Springer. p. 8. ISBN 978-3-540-21338-3.
7. Wambersie, A.; Menzel, H. G.; Andreo, P.; DeLuca, P. M.; Gahbauer, R.; Hendry, J. H.; Jones, D. T. L. (7 December 2010). "Isoeffective dose: a concept for biological weighting of absorbed dose in proton and heavier-ion therapies". Radiation Protection Dosimetry. 143 (2–4): 481–486. doi:10.1093/rpd/ncq410.
8. Brahme, Anders (2014). Comprehensive Biomedical Physics. Newnes. p. 137. ISBN 9780444536334.
9. a b Hall, Eric J.; Giaccia, Amato J. (2006). Radiobiology for the radiologist (6th ed.). Philadelphia: Lippincott Williams & Wilkins. ISBN 9780781741514.