An Empirical Model for Describing the Small Field Penumbra in Radiation Therapy

We developed a mathematic empirical model for describing the small field penumbra in order to analyze the potential dose perturbation caused by overlapping field to avoid the dose calculation errors in linear accelerator-based radiosurgery.

A ball phantom was fabricated for measuring penumbra at 4 different gantry angles in the coplanar plane. A least square root estimation (LSRE) Model was created to fit the measured penumbra dose profile and to predict the penumbra dose profile at any gantry angles. The Sum of Squared Errors (SSE) was used for finding the parameters n and t for the best fitting of the LSRE model. Geometric and mathematical methods were used to derive the model parameters.

The results showed that the larger the gantry angle of the field, the more the expansion of the penumbra dose profile. The least square root estimation model for describing small field penumbra is as follows: $$ where Penumbra denotes the dose profile D(š) at the penumbra region, T is the penumbra height (usually in scalar 100), n is the parameter for curvature, š = x - W /2 (x and š are the values in cm on x-axis), and t is the radiation transmission of the collimator. Geometric analysis establishes the correlation between the penetration depth of the exposure and its effect on the penumbra region in ball phantom. The penumbra caused by an exposure at any arbitrary angles can be geometrically derived by using a one-variable quadratic equation.

The dose distribution in penumbra region of small field can be created by the LSRE model and the potential overdosage or underdosage owing to overlapping field perturbation can be estimated.