Dose uncertainty

1. Polynomial function

1.1 Film response

Net optical density represents the film response [1] as:

\[ netOD = -log_{10}\frac{I}{I_0} \]

where \(I_0\) and \(I\) are the reading for the unexposed and exposed film piece, respectively.

Using error propagation, the associated standard deviation is given by [2,3]:

\[ SD(netOD) = \frac{1}{ln10} \sqrt{ \left(\frac{SD(I_0)}{I_0} \right)^2 + \left(\frac{SD(I)}{I} \right)^2 } \]

1.2 Calibration

Film response to dose relationship can be fitted to a power function of the form: $\( D = a\cdot netOD + b\cdot (netOD)^n \)$

where a, b, and n are fitting parameters and D is the measured dose in Gy.

Using propagation error analysis [3], the dose uncertainty is given by:

\[SD(D) = \sqrt{ SD_{exp}^2 + SD_{fit}^2 }\]

where \(SD_{exp}\) is the experimental uncertainty associated with the film irradiation and scanning procedures [2]. \(SD_{fit}\) represents the fitting uncertainty.

\[SD_{exp} = (a + n \cdot b \cdot netOD^{n-1}) \cdot SD(netOD) \]

\[SD_{fit} = \sqrt{netOD^2\cdot SD_a^2 + netOD^{2n} \cdot SD_b^2} \]

where \(SD_a\) and \(SD_b\) are the fitting parameter uncertainties.

2. Rational function

2.1 Film response

The normalized pixel value represents the film response to dose [1], \(x\) as: $\(x = \frac{I}{I_0}\)\( where \)I\( and \)I_0$ are the reading for the exposed and unexposed film, respectively. Using error propagation expression and ignoring cross-correlation, the associated standar deviation for film response is given by:

\[ SD(x) = \frac{I}{I_0}\sqrt{\left(\frac{SD(I)}{I}\right)^2 + \left(\frac{SD(I_0)}{I_0}\right)^2} \]

where \(SD(I)\) and \(SD(I_0)\) are the associated standard deviations for \(I\) and \(I_0\), respectively.

2.2 Calibration

The response to dose relationship is constructed by plotting the dose \(D\), as a function of film response \(x\):

\[D = -c + \frac{b}{x-a}\]

Using propagation error analysis [3], the dose uncertainty is given by:

\[SD(D) = \sqrt{ SD_{exp}^2 + SD_{fit}^2 }\]

where \(SD_{exp}\) is the experimental uncertainty associated with the film irradiation and scanning procedures [2]. \(SD_{fit}\) represents the fitting uncertainty.

\[SD_{exp} = \frac{b}{(x-a)^2}SD(x)\]

\[SD_{fit} = \sqrt{ \left(\frac{b}{(x-a)^2}SD(a) \right)^2 + \left(\frac{SD(b)}{x-a} \right)^2 }\]

where \(SD_a\) and \(SD_b\) are the fitting parameter uncertainties.

Bibliography

[1] Azam Niroomand-Rad, et al, Report of AAPM Task Group 235 Radiochromic Film Dosimetry: An Update to TG-55, Medical physics, 47(12), (2020), 5986-6025.

[2] L.A. Olivares-J., Distribución de dosis en radioterapia de intensidad modulada usando películas de tinte radiocrómico. Tesis de maestría, UNAM, (2019).

[3] E.Y. León Marroquin, et al, Evaluation of the uncertainty in an EBT3 film dosimetry system utilizing net optical density, J. Appl. Clin. Med. Phys., 17 (2016), pp. 466-481.

[4] Devic S, Seuntjens J, Hegyi G, et al. Dosimetric properties of improved GafChromic films for seven different digitizers. Med Phys. (2004);31(9):2392–401.