_{1}

A radiotherapy treatment margin formula has been analytically derived when a standard deviation (SD) of systematic positioning errors Ʃ is relatively small compared to an SD of random positioning errors σ. The margin formula for 0 ≤ Ʃ ≤ σ was calculated by linearly interpolating two boundaries at Ʃ = 0 and Ʃ = σ, assuming that the van Herk margin approximation of <i>k<sub>1</sub></i>Ʃ + <i>k<sub>2</sub></i>σ is valid at Ʃ = σ. It was shown that a margin formula for 0 ≤ Ʃ ≤ σ may be approximated by <i>k<sub>1</sub></i>σ + <i>k<sub>2</sub></i>Ʃ, leading to a more general form of <i>k<sub>1</sub></i> max(Ʃ,σ) + <i>k<sub>2</sub></i> min(Ʃ,σ) which is a piecewise linear approximation for any values of Ʃ and σ.

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, M_{ptv} [

where S is a standard deviation (SD) of the systematic errors and σ is an SD of the random errors. The coefficient k_{1} depends on a given coverage probability of the patient population in a facility, whilst k_{2} varies with biological penumbra of the treatment fields. For example, k_{1} was set to 2.5, where 90% of the patients receive at least 95% of the prescribed dose in the clinical target volume (CTV) [_{2} was given as 0.7 for prostate treatment [

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 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. [

Assuming that formula (1) is valid when S goes down to σ [_{ptv} 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 M_{ptv}/σ is a linear function of S/σ with a slope of k_{1} and an intercept of k_{2}. Then, a linear interpolation function of M_{ptv}/σ for 0 ≤ S ≤ σ may be assumed as

where a and b are unknown coefficients. The coefficients a and b can be solved

by using two boundary conditions of M_{ptv}/σ = k_{1} with S = 0 from (2), and M_{ptv}/σ = k_{1} + k_{2} with S = σ from (3), which results in a = k_{2} and b = k_{1}. Consequently, we obtain the following formula:

Equivalently,

_{1} = 2.5 and k_{2} = 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 (k_{1} − k_{2})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 k_{1}σ + k_{2}S 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 k_{1} max(S, σ)+ k_{2} min(S, σ), a piecewise linear approximation for any values of S and σ.

Yoda, K. (2017) A Radiotherapy Treatment Margin Formula When Systematic Positioning Errors are Relatively Small Compared to Random Po- sitioning Errors: A First-Order Approxima- tion. International Journal of Medical Phy- sics, Clinical Engineering and Radiation Oncology, 6, 193-196. https://doi.org/10.4236/ijmpcero.2017.62017