Radiation Oncology/Radiobiology/Equations

Radiobiology Equations

Tumor Growth
$$MI={\text{Number of mitoses} \over \text{Number of cells}}=$$ $$LI={\text{Number labeled} \over \text{Number of cells}}={\lambda \times {T_s \over T_c}}$$ $$GF={LI \over MI}$$ $$T_d$$ $$T_{pot}={T_c \over GF}={\lambda \times {T_s \over LI}}$$ $$CLF={1-{T_{pot} \over T_d}}$$ $$V=V_0 \times \exp {\left [ {{A \over B} \times {\left ( 1- \exp {\left [-Bt \right ]} \right )}} \right ]}$$ $$V=V_0 \times \exp \left ( At \right )$$ $$V=V_0 \times \exp \left ( {A \over B} \right )$$
 * Mitotic Index:
 * Labeling Index:
 * Growth fraction:
 * Tumor volume doubling time:
 * Potential doubling time:
 * Cell loss factor:
 * Gompertzian Growth
 * Progressively slowing:
 * Small t (early):
 * Large t (late):

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
$$PE={\text{Number of colonies counted} \over \text{Number of cells seeded}}$$ $$SF={\text{Number of colonies counted} \over {\text{Number of cells seeded}} \times {PE}}$$ Do not distinguish mode of death (mitotic vs apoptotic)
 * Plating efficiency:
 * Surviving fraction:

Target theory
$$SF= \exp \left ( {-D \over D_0} \right )$$ $$SF=1- {\left ( 1- \exp \left [ {-D \over D_0} \right ] \right )}^n$$ $$D_q=D_0 \times \ln n$$
 * Surviving fraction (single target-single hit):
 * Surviving fraction (multiple target-single hit):
 * Quasi-threshold dose:
 * $$D_{10} = 2.3 \times D_0$$
 * $${SF_{\text{single-hit}}}^N=SF_{\text{multi-hit}}$$

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
$$SF_d=\exp \left( - \alpha d - \beta d^2 \right )$$ $$BED_{\alpha \over \beta}=N \times d \times \left [ 1 + {d \over {\left ( {\alpha \over \beta} \right )}} \right ]$$ $$BED_H=N \times d \times \left [ RBE_{max} + {d \over {\left ( {\alpha \over \beta} \right )}} \right ]$$ $$BED_{time}= N \times d \times \left [ 1 + {d \over {\left ( {\alpha \over \beta} \right )}} \right ] - {0.693 \over \alpha \times T_p} \times { \left [ T-T_k \right ]}$$ $$D_{\text{IsoE}}=D \times W_{\text{IsoE}}$$ $$D_2=D_1 \times \left [ { {d_1 + {\alpha \over \beta}} \over {d_2 + {\alpha \over \beta}}} \right ]$$ $$EQD_2 = N \times d \times { {d + { \alpha \over \beta } } \over {2 + {\alpha \over \beta } } }$$
 * Fraction of cells surviving single dose $$d$$:
 * Fraction of cells surviving fractions $$N$$: $$SF_N= {\left ( \exp \left [ - \alpha d - \beta d^2 \right ] \right )}^N = \exp \left [ - \alpha D - \beta D d \right ] $$
 * Biologically Effective Dose (same RBE):
 * BED for high LET radiation (RBE adjusted):
 * BED (time adjusted):
 * Isoeffective dose:
 * Equivalent Dose in 2 Gy Fractions:

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
$$TCP={\text{Number of colonies counted} \over {\text{Number of cells seeded}} \times {PE}}$$ $$TCP= \exp \left [ - \lambda \right ]$$ $$TCP= \exp \left [ - N_0 \times \exp \left (-\alpha D - \beta d D \right ) \right ]$$ $$TCP={SF_2}^N$$
 * Tumor control probability (TCP)

Definitions

 * $$N$$=number of fractions

Linear Energy Transfer
$$LET = {\operatorname{d}E\over\operatorname{d}l}$$
 * Linear Energy Transfer (LET):

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
$$RBE = {D_{250} \over D_r}$$
 * Relative Biological Effectiveness (RBE):

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
$$OER= {\text{dose in hypoxic cells} \over \text{dose in aerated cells to cause same effect}}$$
 * Oxygen enhancement ratio:
 * OER Values:
 * photon 3
 * proton 3
 * neutron 1.6
 * energized ion 1
 * alpha 1
 * $$\text{Initial proportion of hypoxic cells} = {\text{SF aerated} \over \text{SF hypoxic}}$$