# Radiation Oncology/Radiobiology/Equations

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Radiobiology Equations

### Tumor Growth

* Mitotic Index:$^{:252}$ $MI={{\text{Number of mitoses}} \over {\text{Number of cells}}}={\lambda \times {T_{m} \over T_{c}}}$
* 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$^{:475}$
** 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): $SF=\exp \left({-D \over D_{0}}\right)$
* 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 $d$:$^{:228}$$^{:31}$ $SF_{d}=\exp \left(-\alpha d-\beta d^{2}\right)$
* Fraction of cells surviving fractions $N$:$^{:31}$ $SF_{N}={\left(\exp \left[-\alpha d-\beta d^{2}\right]\right)}^{N}=\exp \left[-\alpha D-\beta Dd\right]$
* Biologically Effective Dose (same RBE):$^{:230}$ $BED_{\alpha \over \beta }=N\times d\times \left[1+{d \over {\left({\alpha \over \beta }\right)}}\right]$
* BED for high LET radiation (RBE adjusted):$^{:268}$ $BED_{H}=N\times d\times \left[RBE_{max}+{d \over {\left({\alpha \over \beta }\right)}}\right]$
* 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):$^{:106}$ $LET={\operatorname {d} E \over \operatorname {d} l}$
| 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):$^{:115}$ $RBE={D_{250} \over D_{r}}$

#### 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:$^{:237}$ $OER={{\text{dose in hypoxic cells}} \over {\text{dose in aerated cells to cause same effect}}}$
* 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.{{cite book}}: CS1 maint: uses authors parameter (link)
* ↑ H. Awwad (2013). Radiation Oncology: Radiobiological and Physiological Perspectives. Springer. ISBN 9789401578653.{{cite book}}: CS1 maint: uses authors parameter (link)
* ↑ 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)
* 1 2 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)
* 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). Comprehensive Biomedical Physics. Newnes. p. 137. ISBN 9780444536334.
* 1 2 Hall, Eric J.; Giaccia, Amato J. (2006). Radiobiology for the radiologist (6th ed.). Philadelphia: Lippincott Williams & Wilkins. ISBN 9780781741514.