Isoeffective Formula And Biological Equivalent Dose

Isoeffective Formulae are designed to allow different treatment regimes to be compared. The now out-of-date NSD and related models were described in the effect of overall treatment time topic.

## Biologically Effective Dose

The biologically effective dose is a relatively modern method of accounting for the effects of fractionation on the observed effects, taking into account the different behaviour of early and late reacting tissues.
Summarising the process in Hall:
The biological effect E of a dose of D Gy is:

(1)
\begin{align} \text{E} = \alpha D + \beta D^2 \end{align}

If the dose D is split into n fractions of d dose each, then:

(2)
\begin{align} \text{E} = n(\alpha d + \beta d^2) \end{align}

This can be rewritten as:

(3)
\begin{align} \text{E} = n.d. (\alpha + \beta d) = \alpha. n.d. (1 + \frac{d}{\frac{\alpha}{\beta}}) \end{align}

Since the total dose D = n.d,

(4)
\begin{align} \text{E}=\alpha.(\text{Total Dose}).(\text{Relative Dose Effectiveness}) \end{align}

Where the 'Relative Dose Effectiveness' is:

(5)
\begin{align} \text{Relative Dose Effectiveness}=1 + \frac{d}{\frac{\alpha}{\beta}} \end{align}

The final step is to divide equation 4 by $\alpha$, giving

(6)
\begin{align} \frac{\text{E}}{\alpha} = (n.d) \times (1 + \frac{d}{\frac{\alpha}{\beta}}) = (\text{Total Dose})\times (\text{Relative Dose Effectiveness}) \end{align}

The value $\frac{\text{E}}{\alpha}$ is known as the biologically effective dose (BED). This value can be used to compare different fractionation regimes with regards to the the effects on individual tissues.

(7)
\begin{align} \text{BED}= (\text{Total Dose})\times (\text{Relative Dose Effectiveness}) \end{align}

## Effective dose in 2 Gy Fractions

Many tissue toxicities are presented with a total dose with conventional fractionation. Conventional fraction is 2 Gy per fraction. It is therefore useful to compare altered fractionation regimes with a 2 Gy per fraction regime.

The formula for this is:

(8)
\begin{align} \text{EQD}_{2 \text{ Gy}}=\frac{D_{(x\text{ Gy})}(x + \frac{\alpha}{\beta})}{(2+\frac{\alpha}{\beta})} \end{align}

For instance, if equivalent dose in spinal cord ($\frac{\alpha}{\beta}=1$) in 2 Gy fractions for a treatment of 20 Gy in 5 fractions was to be calculated:

(9)
\begin{align} \text{EQD}_{2 \text{ Gy}}=\frac{20_{(4\text{ Gy})}(4 + 1)}{(2+1)}=\frac{20 \times 5}{3}=\frac{100}{3} = 33.\dot{3}\text{ Gy} \end{align}