For a given number of days to compensate there are two extremes:

### Offensive compensation

The compensating treatment is defined in such a way as to achieve the same effect in the tumor tissue: i.e. 100% of the original dose.

### Defensive compensation

The other extreme is defining the compensatory treatment in such a way as to avoid increasing risk to normal tissues, i.e. the late normal tissues complication probability remains the same as in the initially planned treatment. This might be of interest in plans that are designed based on the dose constraints of critical organs, such as the brainstem or optic chiasm for example.

The aggressivity of a compensated treatment plan will be judged differently in palliative or curative intention. One always needs to keep in mind that there are limitations in the values used, which are based on estimates.

### Aggressivity

We term the axis from defensive to offensive compensation *aggressivity* (see Figure1).

Simple calculators, including sliders, may be used to find a clinically acceptable compromise between offensive and defensive treatment compensation[13], an example of a smartphone based program is shown in Figure2.

In the example in Figure2 the *aggressivity* of treatment can be chosen based on the relative effect of the new dose per fraction on tumour as well as late effects. For typical calculations the alpha/beta ratio for tumour tissue is 10, and for normal tissue late effects: 3. The optimal compensation treatment depends on dose per fraction and number of additional treatments.

The optimal compensation treatment needs to be searched for in two dimensions: *aggressivity* as well as the number of days used for compensation (Figure3).

We developed a java based compensation treatment calculator calculating tumour as well as normal tissue doses based mostly on the calculations proposed by the Royal College of Radiologists[9].

Input into the model includes: Parameters of the planned treatment:

The planned number of fractions

The planned dose per fraction

The planned duration of treatment (the planned number of fractions + e.g. planned weekend brakes)

Parameters of the planned treatment:

The applied number of fractions

The applied dose per fraction

The duration of the applied treatment (up to the treatment brake)

Duration of the treatment brake.

Radiobiological parameters specific for tumour and normal tissue

The factor in Gy by which repopulation is accounted for (starting with day 29)

The alpha/beta value for the tumour

The alpha/beta value for late normal tissue effects

A sample calculation is shown in Figure4. A head&neck cancer patient received only 30 of 35 planned fractions of 2Gy in the originally planned time period (46days). In this example five days are available to compensate the treatment interruption. For each day beyond the 4^{th} week of treatment, the effective tumour dose is reduced by 0.9 Gy per day. The BED_{10}planned represents the biologically equivalent dose (BED) that was intended to be delivered to the tumour. The BED_{3}planned represents the effect on normal tissue. The applied dose is represented by BED_{10}applied and BED_{3}applied respectively. To achieve the same tumour effect a total tumour BED of 67.8Gy is required (the same value as the BED_{10}planned). A loss of 0.9Gy per day after the 4^{th} week is subtracted (including the 5 days of treatment prolongation due to compensatory treatment). When the formula is solved a dose per fraction of 2.62Gy is obtained. The BED_{3}new is calculated using this dose per fraction and a new BED value for normal tissue is determined. The BED_{3}ratio represents the ratio between BED_{3}new and BED_{3}original, in this case 106.7%. In this example the compensation treatment was calculated to maintain the originally planned tumour dose leading to a BED_{10}ratio of 100%.

Every proposed compensation treatment results in two important values: the new tumour dose as well as the new late effects dose, each expressed in percent compared to the initially planned treatment.

In the example in Figure4 the tumour dose would be 100%, the late effects dose would be 106.7%. Due to the nature of the sigmoid shapes of the dose response curves larger deviations from the planned treatment will probably result in over-proportional deviations in results.

We assumed it would be logical to search for a compromise between offensive and defensive treatments where both values are close to 100%. The goal should be to keep the sum of the squares of the deviations as low as possible. This value we termed the *compensability index:*

\begin{array}{c}\mathit{Compensability}\phantom{\rule{0.5em}{0ex}}\mathit{Index}={\left(1-\frac{\mathit{new}\phantom{\rule{0.5em}{0ex}}\mathit{tumor}\phantom{\rule{0.5em}{0ex}}\mathit{effect}}{\mathit{planned}\phantom{\rule{0.5em}{0ex}}\mathit{tumor}\phantom{\rule{0.5em}{0ex}}\mathit{effect}}\right)}^{2}\\ \phantom{\rule{7.5em}{0ex}}+{\left(\frac{\mathit{new}\phantom{\rule{0.5em}{0ex}}\mathit{normal}\phantom{\rule{0.5em}{0ex}}\mathit{tissue}\phantom{\rule{0.5em}{0ex}}\mathit{effect}}{\mathit{planned}\phantom{\rule{0.5em}{0ex}}\mathit{normal}\phantom{\rule{0.5em}{0ex}}\mathit{tissue}\phantom{\rule{0.5em}{0ex}}\mathit{effect}}-1\right)}^{2}\end{array}

When the treatment can be compensated for with the same tumour and late effect the compensability index is 0, with increasing deviations the compensability index increases with the sum of square powers of the deviations.

The square power was chosen for weighting the deviations as the effect would be expected to be exponential. This too is a simplification as the change in effect is also dependent on the position within the sigmoid curve. However as these details are not available for individual patients we believe the square power represents a useful practical solution.