Radiation Oncology/Radiobiology/Equations

< Radiation Oncology < Radiobiology

Front Page: Radiation Oncology | RTOG Trials | Randomized Trials

Radiobiology Equations

Tumor Growth

Mitotic Index:^[1]^:252
$MI={{\text{Number of mitoses}} \over {\text{Number of cells}}}={\lambda \times {T_{m} \over T_{c}}}$
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^[2]^:475
- 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):^[3]
$SF=\exp \left({-D \over D_{0}}\right)$
Surviving fraction (multiple target-single hit):
$SF=1-{\left(1-\exp \left[{-D \over D_{0}}\right]\right)}^{n}$
Quasi-threshold dose:^[4]
$D_{q}=D_{0}\times \ln n$
$D_{10}=2.3\times D_{0}$
${SF_{\text{single-hit}}}^{N}=SF_{\text{multi-hit}}$

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 $d$ :^[1]^:228^[5]^:31
$SF_{d}=\exp \left(-\alpha d-\beta d^{2}\right)$
Fraction of cells surviving fractions $N$ :^[5]^:31
$SF_{N}={\left(\exp \left[-\alpha d-\beta d^{2}\right]\right)}^{N}=\exp \left[-\alpha D-\beta Dd\right]$
Biologically Effective Dose (same RBE):^[1]^:230
$BED_{\alpha \over \beta }=N\times d\times \left[1+{d \over {\left({\alpha \over \beta }\right)}}\right]$
BED for high LET radiation (RBE adjusted):^[4]^:268
$BED_{H}=N\times d\times \left[RBE_{max}+{d \over {\left({\alpha \over \beta }\right)}}\right]$
BED (time adjusted):^[6]
$BED_{time}=N\times d\times \left[1+{d \over {\left({\alpha \over \beta }\right)}}\right]-{0.693 \over \alpha \times T_{p}}\times {\left[T-T_{k}\right]}$
Isoeffective dose:^[7]^[8]
$D_{\text{IsoE}}=D\times W_{\text{IsoE}}$
$D_{2}=D_{1}\times \left[{{d_{1}+{\alpha \over \beta }} \over {d_{2}+{\alpha \over \beta }}}\right]$
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):^[9]^:106
$LET={\operatorname {d} E \over \operatorname {d} l}$

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):^[9]^:115
$RBE={D_{250} \over D_{r}}$

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

References

1 2 3 4 David S. Chang, Foster D. Lasley, Indra J. Das, Marc S. Mendonca, Joseph R. Dynlacht (2014). Basic Radiotherapy Physics and Biology. Springer. ISBN 9783319068411.
↑ H. Awwad (2013). Radiation Oncology: Radiobiological and Physiological Perspectives. Springer. ISBN 9789401578653.
↑ Beyzadeoglu, Murat, Ozyigit, Gokhan, Ebruli, Cüneyt (2010). Basic Radiation Oncology. Springer. ISBN 978-3-642-11665-0.
1 2 Roger G. Dale, Bleddyn Jones (2007). Radiobiological Modelling in Radiation Oncology. British Institute of Radiology. ISBN 9780905749600.
1 2 Lemoigne, Yves; Caner, Alessandra (2011). Radiation Protection in Medical Physics. Dordrecht: Springer. ISBN 9789400702479.
↑ 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.
↑ 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.
↑ Brahme, Anders (2014) (in en). Comprehensive Biomedical Physics. Newnes. p. 137. ISBN 9780444536334. https://books.google.co.uk/books?id=9RR0AwAAQBAJ&dq=isoeffective+dose+d1+d2&source=gbs_navlinks_s.
1 2 Hall, Eric J.; Giaccia, Amato J. (2006). Radiobiology for the radiologist (6th ed.). Philadelphia: Lippincott Williams & Wilkins. ISBN 9780781741514.

This article is issued from Wikibooks. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.