A radiotherapy treatment margin model proposed by van Herk was based on a two-Gaussian-process model comprising systematic positioning errors and random positioning errors. The coefficients of the model parameters were calculated in spherical coordinate system, leading to the following planning target volume (PTV) margin approximation, Mptv   :
where S is a standard deviation (SD) of the systematic errors and σ is an SD of the random errors. The coefficient k1 depends on a given coverage probability of the patient population in a facility, whilst k2 varies with biological penumbra of the treatment fields. For example, k1 was set to 2.5, where 90% of the patients receive at least 95% of the prescribed dose in the clinical target volume (CTV)   . The coefficient k2 was given as 0.7 for prostate treatment   and 0.5 for typical lung treatment because the beam penumbra for lung treatment is broader than that for abdominal treatment  . The margin formula (1) was originally proposed for normal fractionation for body radiotherapy, when systematic errors were relatively large compared to random errors   . After on-board cone-beam computer tomography (CBCT) was employed for tumor localization, the systematic errors were much reduced down to the level of the random errors  .
Meanwhile, a different formula was proposed for intracranial stereotactic radiotherapy, where a head fixation system comprising an integrated mask- mouthpiece was employed under CBCT image guidance  . In this case, the SD of the systematic errors was practically zero and thus S = 0. Looking back to the formula (1), the PTV margin giving the treatment coverage probability of 90% resulted from the systematic error distribution. When the systematic error is practically zero, the PTV margin needs to be calculated by the remaining random error distribution that follows a Gaussian distribution with an SD of σ. In other words, the following formula may hold  :
In this context, the coverage probability of 90% may have a different meaning. For the original van Herk formula, it means a patient population coverage, but for hypofractionated intracranial SRT, it can mean coverage probability of the number of fractions out of all the fractions delivered in the facility, where the random error distribution is assumed a single Gaussian distribution without inter-patient variability. The formula (2) was also graphically supported by Stroom et al.  , where a target dose volume histogram with S = 0 appeared favorable when a PTV margin was increased to 2σ. A natural interest is how to bridge the above two formulas (1) and (2) for an intermediate value of S. The purpose of this study was to derive this formula.
2. Derivation of the Formula
Assuming that formula (1) is valid when S goes down to σ  , Mptv for 0 ≤ S ≤ σ may be approximated by a first-order interpolation as described below. The formula (1) can be written as follows:
The formula (3) shows that Mptv/σ is a linear function of S/σ with a slope of k1 and an intercept of k2. Then, a linear interpolation function of Mptv/σ for 0 ≤ S ≤ σ may be assumed as
where a and b are unknown coefficients. The coefficients a and b can be solved
Figure 1. Calculated plot of piecewise-linearly-approximated normalized treatment margins MPTV as a function of normalized standard deviations of systematic positioning errors S, where each normalization was performed relative to the standard deviation of random positioning errors σ.
by using two boundary conditions of Mptv/σ = k1 with S = 0 from (2), and Mptv/σ = k1 + k2 with S = σ from (3), which results in a = k2 and b = k1. Consequently, we obtain the following formula:
Figure 1 shows the resulting PTV margin plot as a function of S/σ with k1 = 2.5 and k2 = 0.7, where the PTV margin is also normalized relative to σ. The plot suggests that a use of (1) may be discouraged for 0 ≤ S ≤ σ because it may underestimate the PTV margins. The maximum amount of the underestimation may be (k1 − k2)σ when S = 0. To the author’s knowledge, this is the first report that provides a linearly approximated PTV margin formula for 0 ≤ S ≤ σ.
It is interesting to note that the two variables S and σ in the formulas (1) and (6) were swapped each other. This may be interpreted as follows: a coverage probability (either patient population or the number of fractions) of 90% is governed by a spatially-broader probability distribution between the systematic errors and the random errors, whilst a spatially-narrower probability distribution serves as an additive margin correction term.
A linearly approximated PTV margin formula of k1σ + k2S has been given when the systematic errors are relatively small compared to the random errors, which in turn leads to a more general form of k1 max(S, σ)+ k2 min(S, σ), a piecewise linear approximation for any values of S and σ.