放射生物学方程式
有丝分裂指数:[ 1] :252 M I = Number of mitoses Number of cells = λ × T m T c {\displaystyle MI={{\text{Number of mitoses}} \over {\text{Number of cells}}}={\lambda \times {T_{m} \over T_{c}}}}
标记指数 L I = Number labeled Number of cells = λ × T s T c {\displaystyle LI={{\text{Number labeled}} \over {\text{Number of cells}}}={\lambda \times {T_{s} \over T_{c}}}}
生长分数 G F = L I M I {\displaystyle GF={LI \over MI}}
肿瘤体积倍增时间 T d {\displaystyle T_{d}}
潜在倍增时间 T p o t = T c G F = λ × T s L I {\displaystyle T_{pot}={T_{c} \over GF}={\lambda \times {T_{s} \over LI}}}
细胞丢失因子 C L F = 1 − T p o t T d {\displaystyle CLF={1-{T_{pot} \over T_{d}}}}
Gompertzian 生长[ 2] :475 逐渐减缓 V = V 0 × exp [ A B × ( 1 − exp [ − B t ] ) ] {\displaystyle V=V_{0}\times \exp {\left[{{A \over B}\times {\left(1-\exp {\left[-Bt\right]}\right)}}\right]}}
小 t（早期） V = V 0 × exp ( A t ) {\displaystyle V=V_{0}\times \exp \left(At\right)}
大 t（晚期） V = V 0 × exp ( A B ) {\displaystyle V=V_{0}\times \exp \left({A \over B}\right)} T m {\displaystyle T_{m}} T c {\displaystyle T_{c}} λ {\displaystyle \lambda } T s {\displaystyle T_{s}} V {\displaystyle V} V 0 {\displaystyle V_{0}} t {\displaystyle t} A {\displaystyle A} B {\displaystyle B} 铺板效率 P E = Number of colonies counted Number of cells seeded {\displaystyle PE={{\text{Number of colonies counted}} \over {\text{Number of cells seeded}}}}
存活分数 S F = Number of colonies counted Number of cells seeded × P E {\displaystyle SF={{\text{Number of colonies counted}} \over {\text{Number of cells seeded}}\times {PE}}} 不区分死亡方式（有丝分裂 vs 凋亡）
存活分数（单靶点 - 单击中）:[ 3] S F = exp ( − D D 0 ) {\displaystyle SF=\exp \left({-D \over D_{0}}\right)}
存活分数（多靶点 - 单击中） S F = 1 − ( 1 − exp [ − D D 0 ] ) n {\displaystyle SF=1-{\left(1-\exp \left[{-D \over D_{0}}\right]\right)}^{n}}
准阈值剂量:[ 4] D q = D 0 × ln n {\displaystyle D_{q}=D_{0}\times \ln n}
D 10 = 2.3 × D 0 {\displaystyle D_{10}=2.3\times D_{0}} S F single-hit N = S F multi-hit {\displaystyle {SF_{\text{single-hit}}}^{N}=SF_{\text{multi-hit}}} D {\displaystyle D} D 0 {\displaystyle D_{0}} n {\displaystyle n} D 0 {\displaystyle D_{0}} D 10 {\displaystyle D_{10}} N {\displaystyle N} 单次剂量 d {\displaystyle d} [ 1] :228 [ 5] :31 S F d = exp ( − α d − β d 2 ) {\displaystyle SF_{d}=\exp \left(-\alpha d-\beta d^{2}\right)}
分数 N {\displaystyle N} [ 5] :31 S F N = ( exp [ − α d − β d 2 ] ) N = exp [ − α D − β D d ] {\displaystyle SF_{N}={\left(\exp \left[-\alpha d-\beta d^{2}\right]\right)}^{N}=\exp \left[-\alpha D-\beta Dd\right]}
生物学有效剂量（相同 RBE）：[ 1] :230 B E D α β = N × d × [ 1 + d ( α β ) ] {\displaystyle BED_{\alpha \over \beta }=N\times d\times \left[1+{d \over {\left({\alpha \over \beta }\right)}}\right]}
高 LET 辐射的 BED（RBE 调整）：[ 4] :268 B E D H = N × d × [ R B E m a x + d ( α β ) ] {\displaystyle BED_{H}=N\times d\times \left[RBE_{max}+{d \over {\left({\alpha \over \beta }\right)}}\right]}
BED（时间调整）：[ 6] B E D t i m e = N × d × [ 1 + d ( α β ) ] − 0.693 α × T p × [ T − T k ] {\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]}}
等效剂量：[ 7] [ 8] D IsoE = D × W IsoE {\displaystyle D_{\text{IsoE}}=D\times W_{\text{IsoE}}} D 2 = D 1 × [ d 1 + α β d 2 + α β ] {\displaystyle D_{2}=D_{1}\times \left[{{d_{1}+{\alpha \over \beta }} \over {d_{2}+{\alpha \over \beta }}}\right]}
2 Gy 分数等效剂量 E Q D 2 = N × d × d + α β 2 + α β {\displaystyle EQD_{2}=N\times d\times {{d+{\alpha \over \beta }} \over {2+{\alpha \over \beta }}}} N {\displaystyle N} d {\displaystyle d} α {\displaystyle \alpha } β {\displaystyle \beta } T k {\displaystyle T_{k}} T p {\displaystyle T_{p}} D {\displaystyle D} W IsoE {\displaystyle W_{\text{IsoE}}} 肿瘤控制概率 (TCP) T C P = Number of colonies counted Number of cells seeded × P E {\displaystyle TCP={{\text{Number of colonies counted}} \over {\text{Number of cells seeded}}\times {PE}}} T C P = exp [ − λ ] {\displaystyle TCP=\exp \left[-\lambda \right]} T C P = exp [ − N 0 × exp ( − α D − β d D ) ] {\displaystyle TCP=\exp \left[-N_{0}\times \exp \left(-\alpha D-\beta dD\right)\right]} T C P = S F 2 N {\displaystyle TCP={SF_{2}}^{N}} N {\displaystyle N} 线性能量转移 (LET):[ 9] :106 L E T = d E d l {\displaystyle LET={\operatorname {d} E \over \operatorname {d} l}}
辐射类型
LET (keV/μm)
钴-60 光子
0.2
250 kVp 光子
2.0
150 MeV 质子
0.5
10 MeV 质子
4.7
14 MeV 中子
100
18 MeV 碳离子
108
2.5 MeV α粒子
166
75 MeV 氩离子
250
2 GeV 铁离子
1000
最佳 RBE 作为 LET 的函数，在 100 keV/μm 处
d E {\displaystyle \operatorname {d} E} d l {\displaystyle \operatorname {d} l} 相对生物效应 (RBE):[ 9] :115 R B E = D 250 D r {\displaystyle RBE={D_{250} \over D_{r}}} D 250 {\displaystyle D_{250}} D r {\displaystyle D_{r}} D 250 {\displaystyle D_{250}} 氧增强比:[ 1] :237 O E R = dose in hypoxic cells dose in aerated cells to cause same effect {\displaystyle OER={{\text{dose in hypoxic cells}} \over {\text{dose in aerated cells to cause same effect}}}}
OER 值光子 3
质子 3
中子 1.6
带电离子 1
α 粒子 1
Initial proportion of hypoxic cells = SF aerated SF hypoxic {\displaystyle {\text{Initial proportion of hypoxic cells}}={{\text{SF aerated}} \over {\text{SF hypoxic}}}}
↑ a b c d David S. Chang, Foster D. Lasley, Indra J. Das, Marc S. Mendonca, Joseph R. Dynlacht (2014). 基础放射治疗物理学和生物学 . Springer. ISBN 9783319068411 {{cite book }}: CS1 维护: 使用 authors 参数 (link )↑ H. Awwad (2013). 放射肿瘤学：放射生物学和生理学视角 . Springer. ISBN 9789401578653 {{cite book }}: CS1 维护: 使用 authors 参数 (link )↑ Beyzadeoglu, Murat, Ozyigit, Gokhan, Ebruli, Cüneyt (2010). 基础放射肿瘤学 . Springer. ISBN 978-3-642-11665-0 {{cite book }}: CS1 维护: 使用 authors 参数 (link )↑ a b Roger G. Dale, Bleddyn Jones (2007). 放射肿瘤学中的放射生物学模型 . 英国放射学会. ISBN 9780905749600 {{cite book }}: CS1 维护: 使用 authors 参数 (link )↑ a b Lemoigne, Yves; Caner, Alessandra (2011). 医学物理学中的辐射防护 . 多德雷赫特: Springer. ISBN 9789400702479 ↑ Levitt, S.H. (2006). 放射治疗的技术基础：实际临床应用；包含 146 个表格 (第 4 版). 柏林: Springer. 第 8 页. 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. (2010 年 12 月 7 日). "等效剂量：质子治疗和重离子治疗中吸收剂量生物加权的概念". 辐射防护剂量学 . 143 (2–4): 481–486. doi :10.1093/rpd/ncq410 . ↑ Brahme, Anders (2014). 综合生物医学物理学 ISBN 9780444536334 ↑ a b Hall, Eric J.; Giaccia, Amato J. (2006). 放射生物学 for 放射科医生 (第 6 版). 费城: Lippincott Williams & Wilkins. ISBN 9780781741514 
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![{\displaystyle SF=1-{\left(1-\exp \left[{-D \over D_{0}}\right]\right)}^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0d79df23469db6bc2bbd83c5bac66fb5a7201d)
![{\displaystyle SF_{N}={\left(\exp \left[-\alpha d-\beta d^{2}\right]\right)}^{N}=\exp \left[-\alpha D-\beta Dd\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53d895155df2668e30d0ef5df0aaf7ccf1b3fce7)
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![{\displaystyle BED_{H}=N\times d\times \left[RBE_{max}+{d \over {\left({\alpha \over \beta }\right)}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4647822331f668eafe85e73376167a40ef097945)
![{\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]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f65907e58a274fcd611cf45497000c7593c1be38)
![{\displaystyle D_{2}=D_{1}\times \left[{{d_{1}+{\alpha \over \beta }} \over {d_{2}+{\alpha \over \beta }}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99ab27028162c7adf6dca1683be60529608d1b2e)
![{\displaystyle TCP=\exp \left[-\lambda \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf3da6e4ae150fc5239289e5a00588c46ee98f59)
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