The Multivariate Gaussian: a new approach to multichannel radiochromic film dosimetry Ignasi Méndez and Aljoša Polšak Department of Medical Physics, Institute of Oncology Ljubljana, Zaloška cesta 2, Ljubljana 1000, Slovenia E-mail:
[email protected] Introduction Radiochromic films in combination with a flatbed scanner is the dosimetry system of choice for many applications in radiotherapy and radiology. Flatbed scanners can deliver three simultaneous measures of the dose distribution: one for each color channel (red, green and blue). Several methods have been proposed in the literature that combine all three measures into a single more accurate dose distribution. State-of-the-art methods for multichannel film dosimetry are perturbation models [1]. These models consider that differences between each channel dose distribution and the real dose distribution absorbed by the film are caused by small local perturbations of the dosimeter response. Perturbations are correlated between the channels. Different models postulate different correlations and, more generally, different properties of the perturbations. The purpose of this study was to propose a new approach to multichannel radiochromic film dosimetry: the Multivariate Gaussian method; and to compare it against perturbation methods. The Multivariate Gaussian method considers that, for any dose, the probability density function (pdf) of the channel responses follows a multivariate Gaussian distribution. Materials and Methods Five Gafchromic EBT3 films (Ashland Inc., Wayne, NJ) from lot 06061401 where situated on top of the IBA (IBA Dosimetry GmbH, Germany) MatriXX detector inside the IBA MULTICube phantom. The phantom was set up at source-axis distance from a Novalis Tx accelerator (Varian, Palo Alto, CA). The beam energy was 6 MV and the field size was 20 cm x 20 cm. Films were irradiated with doses 1, 2, 4, 8 and 16 Gy, respectively. Doses were simultaneously measured with the MatriXX detector. Films were scanned prior to and 24 h following irradiation. The scanner was warmed up before readings. A 4.5 cm x 20.3 cm strip of an unexposed film was fixed on the scanner bed and scanned beside the films to correct deviations of the scanner's repeatability. A 3 mm thick glass sheet was placed on top of the films to keep constant the distance between film and light source. Ten scans were taken each time for each film. In order to ensure the stability of the scanner lamp, the average scan was calculated with the last five images of each film, the first five were discarded. The lateral artifact was corrected using the model proposed by Lewis and Chan [2], the doses measured by the MatriXX detector, and the sensitometric curves for all three channels. The sensitometric curves were calculated using regions of interest centered on the field with dimensions 3 cm x 3 cm and cubic spline interpolation. In this way, associated with each pixel of each film were: the measured dose, the pixel values (PVs) of each color channel before irradiation, and the PVs after irradiation, all PVs corrected by scanner's repeatability and lateral artifact.
Six algorithms were employed to calculate the dose from the PVs and compare it with the measured dose: the method proposed by Mayer et al.[3] using PVs (after irradiation) and net optical densities (NOD), a channel independent perturbation (CHIP) [1] method with normal perturbation's pdf using PVs and NOD, the Multivariate Gaussian (MG) method using PVs after irradiation, and the MG method using PVs before and after irradiation. For a given dose, the conditional probability of the PVs according to the MG method can be expressed as: P(v | D) ~ N(μ , Σ) , where μ represents the mean vector and Σ the covariance matrix. Following the Bayes's theorem, the conditional probability of the dose given the PVs is: P(D | v) P(v) = P(D) P(v | D) . Considering the doses equiprobable, the dose pdf given the PVs is proportional to the PVs pdf given the dose. Mean vectors and covariance matrix can be obtained during the calibration for a set of doses, we interpolated them for the rest of doses using cubic splines. Thus, given the PVs measured in a pixel, the MG yields the pdf of the dose. It is then straightforward to obtain the most probable dose and its uncertainty. Results Fig 1. and Fig.2 show 2D probability density functions for two different combinations of channel responses. For every combination of two channels, similar (approximately bivariate Gaussian) distributions were found. Fig 3. Shows relative dose differences between measured and calculated dose distributions using the Multivariate Gaussian method with irradiated and non irradiated channels. To avoid uncertainties derived from steep dose gradients, only pixels with measured doses between ± 5% of the prescribed dose were evaluated. The dose uncertainty, i.e., the standard deviation of the dose difference between measured and calculated doses, obtained for each method were: 1.6, 1.6, 1.6, 1.4, 1.3 and 1.1 % for the Mayer PV, CHIP PV, Mayer NOD, MG irradiated channels, CHIP NOD, and MG irradiated and non irradiated channels, respectively. Conclusion This work presents a novel method for multichannel radiochromic film dosimetry: the Multivariate Gaussian method. For the lot under study, the Multivariate Gaussian method was found to provide more accurate doses than the Mayer and the CHIP methods. References [1]. Méndez, I., Peterlin, P., Hudej, R., Strojnik, A. and Casar, B., 2014. On multichannel film dosimetry with channel‐independent perturbations. Medical physics, 41(1). [2]. Lewis, D. and Chan, M.F., 2015. Correcting lateral response artifacts from flatbed scanners for radiochromic film dosimetry. Medical physics, 42(1), pp.416-429.
[3]. Mayer, R.R., Ma, F., Chen, Y., Miller, R.I., Belard, A., McDonough, J. and O'Connell, J.J., 2012. Enhanced dosimetry procedures and assessment for EBT2 radiochromic film. Medical physics, 39(4), pp.2147-2155.
Figures
Fig 1. Probability density function 2D of PVs for each dose value. Irradiated red and green PVs are represented in the X and the Y axis, respectively. PVs are shown as differences between PVs and the mean PV for those channel and dose.
Fig 2. Probability density function 2D of PVs for each dose value. Irradiated red and non irradiated blue PVs are represented in the X and the Y axis, respectively. PVs are shown as differences between PVs and the mean PV for those channel and dose.
Fig 3. Relative dose differences between measured and calculated dose distributions using the Multivariate Gaussian method with irradiated and non irradiated channels.
