Clinical Use of the Linear Quadratic Equation

The linear quadratic equation has already been discussed.

The linear quadratic equation models the number of cells killed following exposure to a varying amount of radiation. The shape of the curve is determined by:

\begin{align} \text{SF}(D)=e^{-(\alpha D + \beta D^2)} \end{align}

The α component describes the linear component of the curve; the β component describes the quadratic (curving) portion of the curve. The $\frac{\alpha}{\beta}$ ratio is the point at which linear cell kill is equivalent to quadratic cell kill. Beyond this point, the surviving fraction drops rapidly as the quadratic cell kill takes over. In general:

  • Early responding tissues have a high $\frac{\alpha}{\beta}$ ratio, leading to a linear increase in cell kill at therapeutic doses. The average $\frac{\alpha}{\beta}$ of early responding tissues is 10.
  • Late responding tissues have a low $\frac{\alpha}{\beta}$ ratio, leading to less cell kill at lower doses and greater cell kill at higher doses. The average $\frac{\alpha}{\beta}$ of late responding tissues is 3.
  • Most tumours have a high $\frac{\alpha}{\beta}$ ratio (10 or above)

Some tumours (eg: prostate) may have an $\frac{\alpha}{\beta}$ similar to late tissues (about 3).
Some tumours may have a very low $\frac{\alpha}{\beta}$ ratio (under 1), such as melanoma and some sarcomas. These tumours are resistant to low doses of radiation.

Use of the linear quadratic formula to compare fractionation schedules

The most modern method of comparing fractionation schedules is to use the linear quadratic equation to calculated the equivalent dose in 2 Gy fractions. The underlying equation which relates two fractionation schedules (with total dose D1 with fractions of d1 Gy per fraction, versus total dose D2 with d2 Gy per fraction) is:

\begin{align} D_1(d_1 + \frac{\alpha}{\beta})=D_2(d_2 + \frac{\alpha}{\beta}) \end{align}

If one of the schedules is using 2 Gy fractions with a total dose of EQD2 Gy, and other has a total dose of Dx Gy Gy with x Gy per fraction, then:

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

This is then divided by $(2 + \frac{\alpha}{\beta})$ to give:

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

For accurate calculation on EQD2 Gy it is important to have:

  • Accurate $\frac{\alpha}{\beta}$ ratios for the tissues in question
  • Doses that are usable with the linear quadratic model
    • There is evidence that at very low doses (under 0.5 Gy per fraction) and very high doses (over 8 Gy per fraction) that the linear quadratic model is not applicable.

Example of equivalent dose calculation

A common palliative treatment is for painful bony metastases in the spine. The usual treatment is 20 Gy, delivered in 5 fractions of 4 Gy per fraction. Substituting these values into the above equation, and assuming an $\frac{\alpha}{\beta}$ ratio of 1 for spinal cord:

\begin{align} EQD_{2 \text{Gy}}= \frac{20(4 + 1)}{(2 + 1)}=\frac{100}{3}=33.3 \text{ Gy (in 2 Gy fractions, for spinal cord late effects)} \end{align}

If the tumour is a prostate cancer with an $\frac{\alpha}{\beta}$ ratio of 3, then the effective dose on the tumour would be:

\begin{align} EQD_{2 \text{Gy}}= \frac{20(4 + 3)}{(2 + 3)}=\frac{140}{5}=28 \text{ Gy (in 2 Gy fractions, for effect on prostate cancer cells)} \end{align}

This calculation is useful as many tolerance doses for normal tissues are provided with conventional fractionation (ie. 2 Gy per fraction). If the patient represents in one year with recurrence at the site of treatment, it can be seen that repeating the treatment would give a total dose of 66.6 Gy if the previous treatment is remembered by the tissue. At doses of 66.6 Gy there is a nearly 50% chance of developing radiation myelopathy.

Fortunately the spinal cord seems to forget a significant amount of dose after 1 - 3 years. Therefore this dose may be repeatable, if 50% of the dose is forgotten after one year. Our department would change fractionation, and deliver 20 Gy in 10 fractions (2 Gy per fraction). This would limit the possibility of late effects developing if significant remembered dose remains.