Investigation of Uncertainty Sources in the Determination of Gamma Emitting Radionuclides in the UAL A. Specification Gamma-spectrometry method is used to identify and determine the activity concentration in Bq/dm 3 of the gamma-emitting radionuclides in urine samples. Measurement procedure The detectors are calibrated for energy and efficiency using certified calibration sources. The sources were prepared dispersing radioactive material in silicon resin. The volume of the resin is 1000±10 cm 3 Cert. No: 9031-OL-83/05 and 9031-OL-85/05 Type: CBSS. The received urine samples are weighted for the volume determination. The end-volume of the bottles must be 1000 cm 3 to have the counting geometry of 1L Bottle. The specific information on how to calibrate and measure is given in the WI-3-UA-04 and WI-3-UA-06. The measure consists of the following stages: Volume determination Preparation of Measuring Geometry Background Determination Gammaspectrometric Determination Calibration RESULT Figure 1: Gamma-spectrometry Measurement 1
Calculation The measurand is the nuclide activity concentration in Bq/dm 3, based on the peak at energy, E, is given by: A i N = ε t γ K d [Bq] where: A = the activity of nuclide I i N = the background corrected net peak area ε = the detector efficiency t = the live time γ d C Ai = = the branching ratio for this nuclide the activity concentration of this nuclide V = the volume of the urine sample K = the decay correction T 1/ = the half-life of this nuclide t e K ln t e T1/ = e Ai CA i = [Bq/L] V for this nuclide = the elapsed time between the start of acquisition and the time of sample collection
B. Identifying and Analysing Uncertainty Sources The estimation of uncertainties contains the following steps: Exploration of all factors influencing the measurement, Quantification of the uncertainties connected to each factors by measurement results or by expert judgment if applicable, Estimation of the overall uncertainty of the measurement. Uncertainty Sources Net Peak Area Detector Efficiency Branching Ratio Sample Volume Decay Correction Measuring time Temperature The different effects and their influences are shown as a cause and effect diagram in Figure. Quantifying the uncertainty components Net Peak Area, Detector Efficiency, Branching Ratio and Decay Correction The GammaVision evaluation software analyse spectrum files and produce a list of the background, net area, counting uncertainty and net count rate for all peaks in the spectrum. It also gives a list of average activity of the nuclides in the sample, the activity of each nuclide based on each gamma-ray energy in the library. The uncertainties of these components are combined and reported by the software during spectrum analysis. Assuming that the density of the Calibration Standard and the Sample is similar the internal absorption (attenuation) contribution cancels itself out (see Urine Density Determination Study). 3
Net Peak Area Determined by the spectrum evaluation software Background correction N = G B where: G = Gross sum of the counts within the peak region (ROI), B = background area counts for the peak. The uncertainty in background area counts. where: n B = 1 + n 1 ( B B ) n B1 B u ( B) = + n1 n n = number of channels in the ROI, BB1 = Area of left background region, width n1 channels, B B = Area of right background region, width n channels. If the background regions are of equal width (n 1 = n ), the uncertainty in background area counts reduces to: u n + ( B ) ( B) = 1 B n 1 In the spectrometry practice the statistical uncertainty of the net peak area, in connection with the Poisson distribution, is commonly expressed as: u ( N) + B = G u( ) (1.a) Subtraction of Peak Environmental Background N = ( G B) I The environmental background interference peak is calculated as I = t t B I B where: I = number of count due to the environmental background 4
t B B = the live time of background spectrum I B B = the net peak area of the peak in the background The variance of the background interference can be calculated as: u I = ( ) u( I B tb u(i B ) B = the uncertainty of the net peak area for the interference peak, it can be calculated by equation (1.a). t In case of the presence of an interference peak, i.e. a residual area in the ROI due to environmental background radiation or detector contamination, etc. determined by a separate background measurement, the following equation shall be used instead of (1.a): ( G + u( B) u( ) ) ) u ( N) = + I (1.b) Non-statistical errors connected with the calculation of net peak area and uncertainty of net peak area Determined by the spectrum evaluation software Expert comments: The calculation of net area and net area uncertainty is also associated with no pure statistical errors, as: - Set of ROI, i.e. peak width, - Set of width of background regions, - Peak fitting. Detector Efficiency Determined by the spectrum evaluation software The efficiency calibration consists of the following steps: collection of spectrum of the calibration standard, evaluation of the spectrum, calculation of photo peak efficiencies and fit of the energy-efficiency curve. ε E = A Ei N E t γ K d i Where: K i = the decay correction for radionuclide I, K ln t e T1/ = e t e = the elapsed time between the start of acquisition and the stated reference time in the certificate T 1/ = the half-life of this nuclide 5
Uncertainty Sources in the Efficiency Calibration: Activity in the reference standard, Preparation of calibration standard, Peak areas in calibration spectrum, Decay correction, Fitting of energy-efficiency curve, Difference in calibration and measurement geometry, Shift of the peak energy. Branching ratio Activity and preparation of the calibration standard The combined uncertainty of the activity of radionuclides in a reference standard and preparation of the calibration geometry shall be given in the certificate attached with it. The value of relative uncertainties does not exceed 1- percent, generally. If the reported uncertainty is given with confidence level, it has to be divided by the appropriate percentage point on the normal distribution for the level of confidence given to calculate the standard deviation. Without confidence level, it is normally appropriate to assume a rectangular distribution. The uncertainty should be divided by 3 1/ to calculate the standard deviation. Net Peak Area Determined by the spectrum evaluation software Expert comments: The statistical uncertainty of net peak areas of calibration spectrum shall be reduced by use of high activity standards and long measuring time. However, the use of very high activities should also be avoided, because of the possible uncertainty from pile-up in the measuring equipment. In practice, a dead time up to 3-5 percent is quite acceptable. Due to the summation effect the use of radionuclides with cascade decay scheme needs careful investigation for some measuring geometries. It can be stated, however that for the commonly used geometries at the IAEA UAL (bottle) the relative uncertainty from the summation effect is well below 1- percent. The relative statistical uncertainty of the net peak areas in the calibration spectrum can be kept below 5 percent, usually. The variation of background plays only role in case of interference peaks, when the correction of the peak net area is based on a separate background measurement. Therefore, the frequency of background measurement shall be high enough to exclude the deterministic or systematic uncertainty and the measuring time long enough to reduce the relative statistical uncertainty to less than 5-10 percent. For the calculation of it the equation (1.a) or (1.b) can be used. 6
Decay correction Determined by the spectrum evaluation software The uncertainty of the decay correction factor is calculated as K ln() t u( K) = e u( T1/ ) ( T ) 1/ In all cases, if the u(t 1/ ) is not available, it is set to zero. Fitting of energy-efficiency curve Determined by the spectrum evaluation software Expert Comments: Most of the spectrum evaluation software offer the possibility to fit different energy-efficiency curves on the measured values. To decide the goodness of fit of the efficiency curve the following criteria shall be evaluated: chi-square value of fit and systematic difference between the fitted curve and several measuring points in any energy regions. The relative uncertainty associated with the efficiency curve fit shall not exceed -3 percent. Difference in calibration and measurement geometry Determined by expert judgment. The equality of calibration and measurement geometries shall refer to the following factors in general: distribution of radionuclide, volume, shape or dimensions, mass, density, atomic number or effective atomic number, mass number. The identity of geometries for measurement of urine by gamma-spectrometry can be achieved to be a great extent, i.e. the relative uncertainty shall be less than 1 percent. Shift of the peak energy Determined by expert judgment. The shift of the photopeak energy has only minor influence on the efficiency, usually. The maximum of the derivative of the efficiency curve, except of again the energies below 50-60 kev, is about 10-7 /kev, which is negligible comparing the efficiency of approx. 10-4. 7
Combined standard uncertainty for the efficiency calibration The standard uncertainty u(ε) of the detector efficiency can be calculated as: u( N) u( A) u( γ ) u( K) u( ε ) = ε + + + N A γ K Branching ratio Determined by the spectrum evaluation software Expert comments: There are many nuclide libraries, data sets containing gamma photon emission probabilities (yields) for radionuclides. The spectrum evaluation software contains also editable nuclide library, usually. There could be no remarkable differences of yields given in different data sources, at least for string peaks of commonly used radionuclides. However, it is worth to mention, that the gamma emission probabilities are given for the decay of the radionuclide and not the parent nuclide, generally. A simple example: 137 C is a pure beta-emitting radionuclide, the gamma photons of 661.6 kev are emitted by the radioactive daughter, 137m Ba. The yield of 137m 137 the gamma line of the above energy is 0.898 for Ba, however the Cs transition to 137m Ba has a branching fraction of 0.946. It means, that the gamma emission probability for the decay of 137 Cs is 0.898x0.946 = 0.85, instead of 0.898, even if the parent and daughter radionuclides are in radioactive equilibrium. Finally, it can be concluded, that following a careful check of nuclide library of the spectrum evaluation software, the uncertainty associated with the gamma emission probability is usually negligible. Measuring time Determined by expert judgment. Although, the dead time of the ADC is compensated for each measuring equipment (live-time correction) the very high count rates, resulting dead time higher than about 5-10 percent without pile up rejection shall be avoided. In normal spectrometry practice the count rates are very low usually, therefore the uncertainty of the measuring time is negligible. Environmental effects Determined by expert judgment. The most important environmental factor, which might influence the measurement result, is temperature. However, the change of temperature can cause mainly the change of amplification, which results in peak shift. As it was stated above the peak shift has only minor effect on efficiency. Otherwise, the temperature stability of the spectrometers are high enough to ensure that there is no observable peak shift for the temperature range of normal laboratory conditions. 8
Volume of the Sample The volume of the sample in the bottle is subject to three major sources of uncertainty: The mass of the sample m-v calibration of the Balance Model equation: V V water sample = msample [L] mwater Sample Mass (m gross) The gross mass of the sample is obtained by weighing the urine together with the bottle. The manufacturer identifies three uncertainty sources for the tare weighing: The repeatability The readability (digital resolution) of the balance scale The contribution due to the uncertainty in the calibration function of the scale (linearity and sensitivity). Model equation: m = m + m + m sample rep lin read [g] The uncertainty associated with the mass of the sample is estimated using data from the manufacturer s recommendations. Repeatability : the standard deviation is quoted as ±0.01 g. Normal distribution. Readability : the digital resolution of the balance is 0.01 g. A rectangular distribution is assumed: Linearity 0.01g = 0.006g 3 : the difference from the actual weigh on the scale pan and the reading of the scale is within the limits of ±0.0 g. A rectangular distribution is assumed. 0.0g = 0.01g 3 The three components are combined to give the standard uncertainty u(m gross ). u( msample) = 0.01 + 0.006 + 0.01 = 0. 017g 9
m-v Calibration of the Balance To obtain the volume of the sample the balance has to be calibrated. Water Volume The volume of the water contained in the volumetric flask, used for the volume measurement, is subject to three major sources of uncertainty: The uncertainty in the certified internal volume of the flask. Variation in filling the flask to the mark The actual temperature differs from the temperature at which the volume of the flask was calibrated Calibration: The manufacturer quotes a volume for the Grade A volumetric flask of 1000±0.4mL measured a temperature of 0 o C. A triangular distribution is assumed. 0.4mL = 0.163mL 6 Repeatability: The uncertainty due to the variations of filling was estimated from a series of ten fill and weight experiment and gave a standard uncertainty of 0.167mL. Temperature: According to the manufacturer the volumetric flask has been calibrated at temperature of 0 o o C, whereas the laboratory temperature varies between ±5 C. The uncertainty from this effect can be calculated from the estimation of temperature range and the coefficient of volume expansion for water. During the balance calibration the room temperature was 5 o C. o o 1000mL ± 5 C.1E 4 C 1 = 1. 05mL The standard uncertainty is calculated using the assumption of a rectangular distribution for the temperature variation: 1.05mL = 0.606mL 3 The three contribution are combined to the standard uncertainty u(v) of the water volume. u( V water ) = 0.163 + 0.606 + 0.167 = 0. 649mL 10
Water Mass The gross mass is obtained by weighing the water together with the bottle. As the same balance is used the standard uncertainty of the water mass is the same like the standard uncertainty of the sample mass. u( mwater ) = 0.01 + 0.006 + 0.01 = 0. 017g The maximum bias (Bρ) due to the difference between urine and water density is 1%. The bias of the bottle (B ) is assumed to be 0.5% bottle The combine uncertainty for the slope is: 1. u( slope) = 8.96E 04 water ) water ) = slope + Bρ water water u( V V 6.49E 04 1 u( m + m + 0.017 1115.95 + B bottle + 0.01 + 0.005 = = 1.417E 05L / g Using the internal counting function of the balance, the program allows automatic conversion of weights into piece counts based on a reference sample weight (1L of Water). The pieces readout is converted to volume in L divided it by the factor of 100. It makes possible the quickly determination of the sample volume in the UAL, avoiding cross contamination. Calibration data Parameters Description x y m 0.0896 SLOPE(y,x) 0 0.00 b 0.000 INTERCEPT(y,x) 1115.95 g 100 pcs f 0.01 Conversion factor to L Table 1- Regression Calibration of the Balance Derived data Derived values x y Slope, m m 8.96E-04 SLOPE(y,x) 0.0 0.00 Intercept, b b 8.88E-16 INTERCEPT(y,x) 1000.0 0.90 Observations, n n 1.000 COUNT(x) 100.0 0.91 Std error in estimate, Syx SYX 1.09E-08 STEYX(y,x) 1040.0 0.93 Average x XAVG 1008.333 AVERAGE(x) 1060.0 0.95 SSX SSX 1.15E+06 DEVSQ(x) 1080.0 0.97 t(a,df) t.8 TINV(0.05,n-) 1100.0 0.99 110.0 1.00 1140.0 1.0 1160.0 1.04 1180.0 1.06 100.0 1.08 Table - Regression Line Confidence Interval 11
. Combining with the regression standard deviation u ( slope, c) = (1.417E 05) + (1.09E 08) = 1.417E 05L / g The combine uncertainty for the sample volume is: u( V sample ) = V sample u( m m sample sample ) u( slope, c) + slope [ L] The relative uncertainty of sample weighing is less than 0.1 %. Therefore it can be neglected. The result for the relative standard uncertainty for the sample volume is assumed to be 1.581%. Calculating the combined standard uncertainty N Ai = [Bq] ε t γ K Ai CA i = [Bq/L] V u( A u V i ) ( sample ) u( CA ) = C A + + Ej i i A [Bq/L] i V sample Ej is the expert judgment regarding to the difference between the calibration and counting geometry. Its value is 1% with is entered as systematic error into the evaluation software. The expanded uncertainty is obtained by multiplying the combined standard uncertainty with a coverage factor of. d For the determination of the combined standard uncertainty the GUM Workbench will be used. Spectrum/Nuclide Evaluation Software Report Combining Relative Standard Uncertainty GUM Workbench QAF0A4.An1/Am-41 1.59 %.3.9 Attached Table 7 Attached Table 3 Quantified Uncertainty by different methods 1
Example Calculations Example 1: 137 Cs in mix gamma standard source (file QAFoA4.An1) Description Value x u(x) u(x)/x Time corrected activity and sigma total uncertainty 1.99E+03 Bq 30.47 Bq 0.0157 Certified Volume 0.9999 L 0.01 L 0.0100 Table 4: Values and uncertainties u( C Ai u( A ) u( V ) i sample ) = C A + = 1.99E + 03 0.0157 + 0.01 = 3.591E + 01Bq/L i Ai Vsample 1 A V Value 1.99E+03 0.9999 3 Uncertainty 30.47 0.01 4 5 A 1.99E+03 1.959E+03 1.99E+03 6 V 0.9999 0.9999 1.01 7 8 CA 1.99E+03 1.959E+03 1.910E+03 9 u(y,xi) 30.47-19 10 u(y), u(y,xi) 189.4 98.4 361 11 1 u(c A ) 3.591E+01 Table 5: Spreadsheet calculation of uncertainty V A CA 0.00E+00 1.00E+01.00E+01 3.00E+01 4.00E+01 u(y,xi) Figure Uncertainty contributions in Measurement of Activity Concentration Reporting expanded uncertainty (@ σ) Cs-137 Activity concentration: 1.99E+03 ± 7.18E+01 Bq/L 13
Example : Using 15 Eu standard source in the file in the file QA4D4.An1as urine sample to simulate the procedure A B C D E F Slope, 1 A msample Vsample [L] regression Value 3.33E+04 1104.91 8.96E-04 0.99 3 Uncertainty 104.0 0.017 1.4E-05 1.57E-0 4 5 A 3.33E+04 3.43E+04 3.33E+04 3.33E+04 3.33E+04 6 m sample 1104.91 1104.91 1104.93 1104.91 1104.91 7 Slope,reg. [L/g] 8.96E-04 8.96E-04 8.96E-04 9.10E-04 8.96E-04 8 V sample [L] 0.99 0.99 0.99 1.01 1.01 9 0.00 0.0.45E-04 10 11 C A 3.36E+04 3.47E+04 3.36E+04 3.31E+04 3.31E+04 1 u(y,xi) [Bq/L] 1.03E+03-5.17E-01-5.4E+0-5.4E+0 13 u(y), u(y,xi) 1.6E+06 1.07E+06.68E-01.74E+05.74E+05 14 15 u(c A ) [Bq/L] 1.7E+03 3.78% Table 6 - Spreadsheet calculation of uncertainty The values of the parameters are entered in the second row into C, E and F. The D value is calculated from F and E. Their standard uncertainties are in the row bellow (C3-E3). The standard uncertainty of the sample volume (F3) is calculated combining the standard uncertainties entered in D3 and E3. The spreadsheet copies the values from C-F into the second column from B5 to B8. The result (C A ) using B5 and B8 is given in B11. The C5 shows the value of A from C plus its uncertainty given in C3. The result of the calculation using values C5 and C8 is given in C11. The columns D-F follow a similar procedure. The values shown in row 1 (C1-F1) are the differences of the row (C11-F11) minus the value given in B11. In row 13 (C13-F13) the values of row 1 (C1-F1) are squared and summed to give value shown in B13. B15 gives the combined standard uncertainty, which is the square root of B13. The value shown in C15 is the relative standard uncertainty. 14
msample Vsample [L] A CA 0.00E+00.00E+0 4.00E+0 6.00E+0 8.00E+0 1.00E+03 1.0E+03 1.40E+03 u(y,xi) [Bq/L] Figure 3 Uncertainty contributions in Measurement of Activity Concentration The graph shows that the major contribution is coming from the calculation of the activity by the software. The reported extended uncertainty will be the reported during spectrum analysis, which is already multiply by the coverage factor of. 15
Practical Example Intercomparison 004 Sample B File Cs-137 activity [Bq/L] Total reported uncertainty [%] Reference Date Measuring date: Method Spectrum Evaluation Software GammaVision 1464.An1 5.4.38 004-0-01 08:00 004-0-4 1:54 Description Uncertainty in Percent Net Peak Area (includes *) 3.38 statistical (Poisson distribution) and * Non-statistical (set of ROI, peak fitting) * Efficiency (includes *).37 Activity in the reference standard (certificate) * Curve fitting * Decay correction * Net Peak Area (statistical and non-statistical) * Branching ratio * Nuclide data (includes *) < 0.1 Decay correction * Branching ratio * Square of combined uncertainty of the above factors 17.04 difference in calibration and counting geometry 1 Expert Judgment Cascade summing (calibration) - shift of photopeak energy < 0.1 gamma emission probability < 0.1 measuring time( dead time) < 0.1 stability against environmental effects < 0.1 Square of combined uncertainty of the above factors 1 Laboratory Sample Volume 1.581 The overall combined uncertainty for the activity in this example is: Square of combined uncertainty of the above factors.5 u( A) = A 17.04 + 1+.5 = 4.53% @ 1sigma 16
Am-41 activity [Bq/L] 6371.4 Bq/L Uncertainty @ 1sigma in the report 1.59 % Reference Date 003-01-01 08:00 Measuring date: 005-09-1 1:4 Method Description Uncertainty in Percent Spectrum Evaluation Software Net Peak Area (includes *) 0.9 GammaVision statistical (Poisson distribution) and * Non-statistical (set of ROI, peak fitting) * Efficiency (includes *) 1.73 Activity in the reference standard (certificate) * Curve fitting * Decay correction * Net Peak Area (statistical and non-statistical) * Branching ratio * Nuclide data (includes *) < 0.1 Decay correction * Branching ratio * Square of combined uncertainty of the above factors 3.084 Expert Judgment difference in calibration and counting geometry 1 Cascade summing (calibration) - shift of photopeak energy < 0.1 gamma emission probability < 0.1 measuring time( dead time) < 0.1 stability against environmental effects < 0.1 Square of combined uncertainty of the above factors 1 Laboratory Sample Volume 1.581 Square of combined uncertainty of the above factors.5 Total Combined Relative Standard Uncertainty.6 Table 7 Combined Relative Standard Uncertainty for high activity 17
N Sample γ d Sample Expert judgment K Sample Background Correction Attenuation T 1/i Attenuation N Ei Standard K Standard T 1/ ε γ Ei Standard A Ei Standard m gross Sample Repeatability Calibration Balance Linearity Readability V Sample Repeatability Calibration Flask Linearity m gross Water V Water C Ai Calibration Balance Readability m-v Calibration Temperature Repeatability Figure 4: Uncertainties in Gamma Measurement 18