放射生物學方程式
有絲分裂指數:[ 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}}}}
標記指數 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}}}}
生長分數 G F = L I M I {\displaystyle GF={LI \over MI}}
腫瘤體積倍增時間 T d {\displaystyle T_{d}}
潛在倍增時間 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}}}
細胞丟失因子 C L F = 1 − T p o t T d {\displaystyle CLF={1-{T_{pot} \over T_{d}}}}
Gompertzian 生長[ 2] :475 逐漸減緩 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} 鋪板效率 P E = Number of colonies counted Number of cells seeded {\displaystyle PE={{\text{Number of colonies counted}} \over {\text{Number of cells seeded}}}}
存活分數 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}}} 不區分死亡方式（有絲分裂 vs 凋亡）
存活分數（單靶點 - 單擊中）:[ 3] S F = exp ( − D D 0 ) {\displaystyle SF=\exp \left({-D \over D_{0}}\right)}
存活分數（多靶點 - 單擊中） S F = 1 − ( 1 − exp [ − D D 0 ] ) n {\displaystyle SF=1-{\left(1-\exp \left[{-D \over D_{0}}\right]\right)}^{n}}
準閾值劑量:[ 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} 單次劑量 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)}
分數 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]}
生物學有效劑量（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 輻射的 BED（RBE 調整）：[ 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（時間調整）：[ 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]}}
等效劑量：[ 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 分數等效劑量 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}}} 腫瘤控制機率 (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} 線效能量轉移 (LET):[ 9] :106 L E T = d E d l {\displaystyle LET={\operatorname {d} E \over \operatorname {d} l}}
輻射型別
LET (keV/μm)
鈷-60 光子
0.2
250 kVp 光子
2.0
150 MeV 質子
0.5
10 MeV 質子
4.7
14 MeV 中子
100
18 MeV 碳離子
108
2.5 MeV α粒子
166
75 MeV 氬離子
250
2 GeV 鐵離子
1000
最佳 RBE 作為 LET 的函式，在 100 keV/μm 處
d E {\displaystyle \operatorname {d} E} d l {\displaystyle \operatorname {d} l} 相對生物效應 (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}} 氧增強比:[ 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
質子 3
中子 1.6
帶電離子 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). 基礎放射治療物理學和生物學 . Springer. ISBN 9783319068411 {{cite book }}: CS1 維護: 使用 authors 引數 (link )↑ H. Awwad (2013). 放射腫瘤學：放射生物學和生理學視角 . Springer. ISBN 9789401578653 {{cite book }}: CS1 維護: 使用 authors 引數 (link )↑ Beyzadeoglu, Murat, Ozyigit, Gokhan, Ebruli, Cüneyt (2010). 基礎放射腫瘤學 . Springer. ISBN 978-3-642-11665-0 {{cite book }}: CS1 維護: 使用 authors 引數 (link )↑ a b Roger G. Dale, Bleddyn Jones (2007). 放射腫瘤學中的放射生物學模型 . 英國放射學會. ISBN 9780905749600 {{cite book }}: CS1 維護: 使用 authors 引數 (link )↑ a b Lemoigne, Yves; Caner, Alessandra (2011). 醫學物理學中的輻射防護 . 多德雷赫特: Springer. ISBN 9789400702479 ↑ Levitt, S.H. (2006). 放射治療的技術基礎：實際臨床應用；包含 146 個表格 (第 4 版). 柏林: Springer. 第 8 頁. 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). "等效劑量：質子治療和重離子治療中吸收劑量生物加權的概念". 輻射防護劑量學 . 143 (2–4): 481–486. doi :10.1093/rpd/ncq410 . ↑ Brahme, Anders (2014). 綜合生物醫學物理學 ISBN 9780444536334 ↑ a b Hall, Eric J.; Giaccia, Amato J. (2006). 放射生物學 for 放射科醫生 (第 6 版). 費城: Lippincott Williams & Wilkins. ISBN 9780781741514 
![{\displaystyle V=V_{0}\times \exp {\left[{{A \over B}\times {\left(1-\exp {\left[-Bt\right]}\right)}}\right]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a01ceeca6a3082220e2d004d019988763c4cb56d)
![{\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)
![{\displaystyle BED_{\alpha \over \beta }=N\times d\times \left[1+{d \over {\left({\alpha \over \beta }\right)}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92359c150e3a7647aca773c8d92e26d9ec9c3641)
![{\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)
![{\displaystyle TCP=\exp \left[-N_{0}\times \exp \left(-\alpha D-\beta dD\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2716ee26534150b0327933736f321a25519c142)
