Theory

The BED equation is based on the linear-quadratic model (LQ model), following the original Equation (2)Reference Hall and Giacca ^{6 ,} Reference Bruzzaniti, Abate, Pedrini, Benassi and Strigari ⁷ as

(2) $$S{\equals}\exp {\minus}\left( {\alpha D{\plus}\beta D^{2} } \right)$$

S is the surviving fraction, α and β the radiobiological parameters and D the radiation dose (Gy). Equation (3) is the conversion of Equation (2) into a natural logarithm:

(3) $$\ln (s){\equals}{\minus}\!\left( {\alpha D{\plus}\beta D^{2} } \right)$$

(4) $$E{\equals}{\minus}\!\ln (s){\equals}\alpha D{\plus}\beta D^{2} $$

E in Equation (4) is the effect of a single fraction of radiation treatment. For fractionation treatment of n fractions and dose per fraction d, the E can be expressed as

(5) $$E{\equals}{\minus}\!\ln (s)^{n} {\equals}{\minus}\!n\ln (s)$$

(6) $$E{\equals}{\minus}\!n(\!{\minus}\alpha d{\minus}\beta d^{2} ){\equals}(nd\alpha )\left( {\!1{\plus}d{\beta \over \alpha }} \right)$$

Thus, as shown in Equation (1), ${\rm BED}{\equals}{E \over \alpha }{\equals}(nd)\left( {1{\plus}d{\beta \over \alpha }} \right)$

If the two treatment plans have equal BED or IsoBED, the equation can be expressed as

(7) $$n_{1} d_{1} \left[ {1{\plus}{{d_{1} } \over {\alpha /\beta }}} \right]{\equals}n_{2} d_{2} \left[ {1{\plus}{{d_{2} } \over {\alpha /\beta }}} \right]$$

Where n ₁ is number of fractions of the radiation dose d ₁ in the original schedule and n ₂ is the number of fractions of radiation dose d ₂ for the new schedule.

However, the LQ model has some limitations, as it does not take overall treatment time into account, and for the bi-fractionated treatments and at high dose per fraction the predicted effect of radiation is not accurate.Reference Voyant, Julian, Roustit, Biffi and Lantieri ⁸ Therefore, this new software program was based on the linear-quadratic-linear (LQ-L) model so the limitations of the LQ model could be elucidated. The BED equation based on LQ-L model, following the software by Cyril Vovant et.al.Reference Voyant, Julian, Roustit, Biffi and Lantieri ⁸ can be expressed as

1. 1. BED for target volume

   1. 1.1. case d>d _t

      (8) $$\eqalignno{BED&{\equals}n\left[ {d_{t} \left( {1{\plus}{{d_{t} } \over {\alpha \,/\,\beta }}} \right){\plus}{\gamma \over a}\left( {d{\minus}d_{t} } \right)} \right]\cr \quad{\minus}\left[ {\theta \left( {T{\minus}T_{k} } \right)\left( {{{\ln (2)} \over {\alpha T_{{pot}} }}} \right)\left( {T{\minus}T_{k} } \right)} \right]$$
      where d is dose per fraction, dt is LQ-L threshold $\left( {d_{t} \,\sim\,2{\alpha \over \beta }} \right)$ , n is number of fractions, θ(x)is the Heaviside function, $${\gamma \over a}$$ is the parameter of the LQ-L model, T is overall treatment time, T _k is the time at which repopulation begins after the start of treatment (day) and T _pot is potential doubling time (day).

   2. 1.2. case d≤d _t

   (9) $$\eqalignno{{\rm BED}&{\equals}nd\left[ {1{\plus}\left( {1{\plus}H_{m} } \right){d \over {\alpha \,/\,\beta }}} \right]\cr\quad{\minus}\theta \left( {T{\minus}T_{k} } \right)\left( {{{\ln (2)} \over {\alpha T_{{{\rm pot}}} }}} \right)\left( {T{\minus}T_{k} } \right)$$
   (10) $$H_{m} {\equals}\left( {{2 \over m}} \right)\left( {{\phi \over {1{\minus}\phi }}} \right)\left( {m{\minus}{{1{\minus}\phi ^{m} } \over {1{\minus}\phi }}} \right)$$
   (11) $$\phi {\equals}e^{{{\minus}(\mu & \,\Delta T)}} $$
   (12) $$\mu {\equals}{{\log _{e} 2} \over {T_{{1\,/\,2}} }}$$
   Where H _m is the LQ model correction taking the poly-fractionation into consideration, m the number of fractions per day, ϕ the incomplete repair, ∆T the duration between two irradiations and $T_{{{1 \over 2}}} $ the time taken to repair the damage to half of the original level.

2. 2. BED for organs at risk

   1. 2.1. case d>d _t

      (13) $$\scale95%{{\rm BED}{\equals}n\left[ {d_{t} \left( {1{\plus}{{d_{t} } \over {\alpha /\beta }}} \right){\plus}{\gamma \over a}\left( {d{\minus}d_{t} } \right)} \right]{\minus}D_{{{\rm rec}}} T}$$
      (14) $$D_{{{\rm rec}}} {\equals}{{\ln (2)} \over {\alpha T_{{{\rm pot}}} }}$$
      Where D _rec is the recovered dose

   2. 2.2. case d≤d _t

(15) $$\scale96%{{\rm BED}{\equals}nd\left[ {1{\plus}\left( {1{\plus}H_{m} } \right){d \over {\alpha /\beta }}} \right]{\minus}D_{{{\rm rec}}} T$$}

Tumour Control Probability (TCP) and Normal Tissue Complication Probability (NTCP) can be presented as a function of BED as shown in Equations (16)Reference Hedman, Bjork-Eriksson, Brodin and Dasu ⁹ and (17):Reference Kehwar ¹⁰

(16) $${\rm TCP}_{{{\rm BED}}} {\equals}\exp [{\!\minus\!}M.\exp (\!{\minus}\!\alpha {\rm BED})]$$

Where M is all clonogenic cells

(17) $${\rm NTCP}(D,v){\equals}\exp \left[ \!{{\minus}\!N_{0} v^{{{\minus}k}} \exp (\!{\minus}\!\alpha {\rm BED})} \right]$$

Where N ₀ is density of specific tissue

k is tissue-specific parameter

v is uniformly irradiated partial volume of the normal tissue/organ