Methodology for correcting the nonlinear response of radionuclide calibrators

Due to constructive and operational issues, the response of a radionuclide calibrator used in Nuclear Medicine can behave in a non-linear manner, especially in the transition of scales. Although the deviations from linearity are small, they may be important for standard secondary or reference radionuclide calibrators used in calibration laboratories. In the document TRS 454 IAEA it is proposed that the deviation from linearity for these instruments should be in the range of ± 2%. However, calibration laboratories may have some difficulty meeting the requirement. This article proposes an easy-to-implement methodology to correct the response of reference radionuclide calibrators from calibration laboratories, in order to meet the requirement of ± 2% for the linearity parameter recommended by the IAEA.


INTRODUCTION
In nuclear medicine, it is important to measure the activity of radiopharmaceuticals before they are injected into patients to certify that the activity is the same as the one prescribed [1]. The instruments used to perform these measurements are radionuclide calibrator. The range of activities employed for diagnostic and therapeutic purposes is quite wide, ranging from a few MBq to the order of GBq. These activities generate currents in ionization chambers [2] ranging from fA to μA. Thus, an electrometer associated with an ionization chamber must be capable of processing eight orders of magnitude of current. Thus, it is necessary for the electrometer to adjust its scales automatically.
Discontinuities in the responses of the electrometers may occur in the transitions between scales [1].
Considering that the range of activities measured is wide and the radionuclide calibrator are calibrated with only one activity value [3,4], it is important that the electrometer response be linear (the ratio between the measured activity and the conventional true activity must be constant over the entire range of currents provided by the activity-meter manufacturer). The International Atomic Energy Agency -IAEA [5] -recommends performance criteria for various metrological parameters. Different tolerance levels are proposed for field radionuclide calibrator, reference radionuclide calibrator and secondary-standard radionuclide calibrator. Regarding linearity, it is proposed for reference radionuclide calibrator and secondary-standard radionuclide calibrator that non-linearity be within the range of ±2%. This variable may represent an important contribution to measurement uncertainty. A methodology is proposed to correct the nonlinearity of the radionuclide calibrator' responses to meet the requirements proposed by the IAEA and to reduce the estimate of measurement uncertainty.

MATERIALS AND METHODS
The present study investigated the responses of CRC-15R and CRC-25R radionuclide calibrator manufactured by Capintec as a function of the activity of a 99m Tc source. All measurements were done at radionuclide calibrator 99m Tc setting. The activities were measured and recorded every two minutes. For the CRC-25R radionuclide calibrator, a source with an initial activity of 8.92 GBq was used and measured for 79 hours. For the CRC-15R radionuclide calibrator, the source initial activity was 3 GBq, measured for 60 hours.
The result of each measurement Mt at time t was corrected for radioactive decay corresponding to a reference time t0. For the CRC-25R radionuclide calibrator, tref corresponds to the time at which 800 MBq activity was recorded. This procedure has the objective of avoiding a possible effect due to ion recombination for higher activity values [2]. For the CRC-15R radionuclide calibrator, tref corresponds to the beginning of the measurements. Thus: The analysis of the values of Mtc(t) allows the identification of the discontinuity points that represent the transition between the various electrometer scales of the radionuclide calibrator and the nonlinearity regions. Thus, it is possible to section the interval of measurements of the activity into p virtual scales. Each scale will be treated individually.
For each of the identified scales, the factor fi (A) is calculated, which will correct the discrepancy that occurred in the automatic scale changes of the electrometer, as well as the observed nonlinearity. The function that best describes the data obtained by eq. 2 is: f i (A) = a n x n + a n-1 x n-1 +…a 2 x 2 + a 1 x 1 +a 0 Where n is the degree of the polynomial. For the same data set, the degree n can assume several values. Due to the statistical characteristics of the data, the value of the correlation coefficient R 2 of the fit may not be the most appropriate estimator for the selection of the optimal value of n. The residual sum of squares (SQE) has the largest amplitude of variation and is the most sensitive estimator to detect the appropriate degree of polynomial for fitting [6].
Thus, the criterion used was to determine n from which the SQE value shows no significant variations.
Where Yj is the value of the correction factor of the original data in measurement j; Ŷ j is the value of the factor predicted by the fitted polynomial in the j-th measurement.       For the CRC-25R and CRC-15R radionuclide calibrator, Figures 5 and 6 show, respectively, the original 99m Tc source measurement values corrected by radioactive decay and corrected by the function fij. The range of variation of the residuals obtained after the nonlinearity correction is shown in Table   3. Table 3: Range of variation of the residuals.

RESULTS AND DISCUSSION
The methodology presented improves the linearity of the radionuclide calibrator responses as a function of the measured activity. At first, the better the proposition of the number p, the better the correction. Figures 5 and 6 indicate that the methodology and the definition of the number p of scales were able to significantly reduce the non-linearity of the responses of the radionuclide calibrator studied.
The range of variation of residuals relative to the difference between the experimental and theoretical data is significantly reduced after the correction of nonlinearity (  [5]. The methodology can additionally be evaluated by comparing the experimental values of the decay constant λ before and after data correction with an internationally accepted theoretical value [7]. The corrected experimental data were fit by the least squares method for the function LN(A) = -λt + LN(A0). The results of the fit are shown in Table 4. Table 4: Statistics of linear regression of the response curves of the radionuclide calibrator before and after correction of the non-linearity.
The value of the decay constant λ of 99m Tc calculated from the corrected experimental data obtained by the both radionuclide calibrators studied coincides with the value adopted by LNHB [7] ( Table 5) to the fifth significant digit. However, the difference between the values of λ obtained before and after nonlinearity correction has little relevance for most practical situations. The difference between the activity values corrected by radioactive decay for t = 10 hours is only 0.2%.