Tutorial to evaluate measurement uncertainty as applied in 21NRM05 STASIS

1 Tutorial to evaluate measurement uncertainty as applied in 21NRM05 STASIS Standartisation for safe implant scanning in MRI Theoretical introduction When evaluating measurement uncertainty, the very basic source to rely on is the document JCGM 100:2008 Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM 1995 with minor corrections). This Guide establishes general rules for evaluating and expressing uncertainty in measurement that are intended to be applicable to a broad spectrum of measurements. It provides general rules for evaluating and expressing uncertainty in measurement. Following are very few excerpts from the Guide, used as a basic example for evaluating the measurement data. 1. Estimate of the output quantity In most cases, a measurand Y is not measured directly, but is determined from N input quantities X1, X2, ....XN through a functional relationship Y = f (X1, X2, ..., XN ) (1) An estimate y of the measurand Y, is obtained from Equation (1) using input estimates x1, x2, ..., xN for the values of the N input quantities X1, X2, ..., XN. Thus the output estimate y, which is the result of the measurement, is given by 𝑦=𝑌=1𝑛∑𝑌𝑘 𝑛𝑘=1 =1𝑛∑𝑓(𝑋1,𝑘,𝑋2,𝑘,…,𝑋𝑁,𝑘,) 𝑛𝑘=1 (2) That is, y is taken as the arithmetic mean or average of n independent determinations Yk of Y, each determination having the same uncertainty and each being based on a complete set of observed values of the N input quantities Xi obtained at the same time. 2. Measurement uncertainty of the input quantities Evaluation of the measurement uncertainty of input quantitites can be carried out in two ways: a. Type A evaluation of measurement uncertainty, b. Type B evaluation of measurement uncertainty. Type A evaluation of measurement uncertainty (VIM 2.28) It is evaluation of a component of measurement uncertainty by a statistical analysis of measured quantity values obtained under defined measurement conditions Features: 1. Employed for input quantities which are based on replicate measurements (i.e. measurements under repeatability conditions) 2. At least 4 replicate measurements are necessary for type A evaluation 3. The uncertainty is related to the estimate of input quantity, i.e. to the arithmetic mean in this case 𝑢A=√1 𝑛(𝑛−1)∑ (𝑥𝑖−𝑥)2 𝑛𝑖=1 (3) where n number of replicate measurements 21NRM05 STASIS 2 xi the ith measurement result of replicate measurements 𝑥 estimate of the overall measurement result – arithmetic mean Type B evaluation of measurement uncertainty (VIM 2.29) It is evaluation of a component of measurement uncertainty determined by means other than a Type A evaluation of measurement uncertainty. Features: 1. employed for input quantities obtained from other sources 2. if boundaries a (sometimes designated as zmax) of the approximated values are given, the uncertainty evaluated by the type B method can be calculated as 𝑢𝐵=𝑎𝑘 (4) where k is the value belonging to the selected approximation of the probability distribution: EXAMPLES. Type B evaluation of measurement uncertainty is based on information: - associated with authoritative published quantity values, - associated with the quantity value of a certified reference material, - obtained from a calibration certificate, - about drift, - obtained from the accuracy class of a verified measuring instrument, - obtained from limits deduced through personal experience. 3. Evaluation of measurement uncertainty uy of the output quantity y 1. For non-correlated input quantities (no common influence on pairs of input quantities): 𝑢𝑦2=∑𝐴𝑖2𝑢𝑥𝑖 2 𝑚 𝑖=1 (5) where Ai (Aj respectively) are sensitivity coefficients, which can be calculated as 𝐴𝑖=𝜕𝑓(𝑋1,𝑋2,...𝑋𝑚) 𝜕𝑋𝑖|𝑋1=𝑥1,...𝑋𝑚=𝑥𝑚 (6) 2. For correlated input quantities (common influence on pairs of input quantities exists): 𝑢𝑦2=∑𝐴𝑖2𝑢𝑥𝑖 2 𝑚 𝑖=1 +2∑ ∑ 𝐴𝑖𝐴𝑗𝑢𝑥𝑖,𝑗 𝑚−1 𝑗<𝑖 𝑚 𝑖=2 (7) where 𝑢𝑥𝑖,𝑗 is a covariance among estimates x1, x2,... xm of correlated input quantities X1, X2,... Xm 3. If certain correlation between the two input quantities Xi and Xj exists, i.e. if one quantity somehow depends on the other one, their covariance must be considered as a part of the overall uncertainty of measurement. Covariance can increase or decrease the overall uncertainty of measurement. 3.a Evaluation of covariance by the type A method If two input quantities Xi and Xj with estimates xi and xj are corelated, covariance evaluated by the type A method is 𝑢A 𝑥𝑖,𝑗=1 𝑛(𝑛−1)∑ (𝑥𝑖𝑘−𝑥𝑖) 𝑛𝑘=1 (𝑥𝑗𝑘−𝑥𝑗) (8) 3.b Evaluation of covariance by the type B method 21NRM05 STASIS 3 If two input quantities Xi and Xj with estimates xi and xj are correlated, the covariance evaluated by the type B method is 𝑢B 𝑥𝑖,𝑗=𝑟𝑥𝑖,𝑗𝑢𝑥𝑖𝑢𝑥𝑗 (9) where 𝑟𝑥𝑖,𝑗is a correlation coefficient between estimates xi and xj. 𝑢𝑥𝑖 resp. 𝑢𝑥𝑗 are uncertainties of estimates xi and xj. Finding the correlation between the power deposited into the implant and the following temperature increase 1. Scope Determination of the relationship between the induced power and the temperature increase of the implant. 2. Technical specification ISO/TS 10974 In the chapter 8, the point 8.4.4.4 states that the local temperature rise T or SAR at a point location in a hot spot produced by the AIMD can be related to the toral power deposition using a calibrated RF power injection method. For each AIMD hot spot, a conversion factor m between T or SAR at the hot spot and injected power is experimentally determined (i.e. T = mPinject or SAR = mPinject). 3. The task For each implant, define a coefficient, with associated uncertainty/confidence level, that allows estimating the temperature increase after a specified time instant given the power deposited into the implant. The input data are represented by a pair – deposited power and a corresponding temperature rise. 4. Theoretical background It is assumed that the relationship between the deposited power (denoted as X), and the temperature difference after a given time (denoted as Y), can be approximated by linear regression. The two forms of linear regression can be employed: a. the linear regression (line) which crosses the zero point (intersection of x and y axes), i.e. 𝑌=𝛽 𝑋 b. the linear regression (line) which is shifted from the zero point (intersection of x and y axes), i.e. 𝑌=𝛼+ 𝛽 𝑋 For the linear regression in the form of 𝒀=𝜷 𝑿 1. Employing the least squares method, the unknown coefficient b can be determined as follows: 𝑏= 1 ∑𝑥𝑖2∑𝑥𝑖𝑦𝑖 where xi is the measured power input value (from the supplied data), yi is the measured temperature rise (from the supplied data). 2. The uncertainty ub of the coefficient b can be determined as follows: 𝑢𝑏=1 √∑𝑥𝑖2𝑠, 21NRM05 STASIS 4 where s can be estimated by a formula for sample residual variance 𝑠=√1 𝑛−1∑[𝑦𝑖−(𝑏𝑥𝑖)]2 𝑛 𝑖=1 3. The uncertainty uy of the calculated value y = bx can be determined as follows: 𝑢𝑦=𝑥 𝑢𝑏=𝑥 1 √∑𝑥𝑖2𝑠 4. Coefficient of determination R2 is a statistical measure of how well the regression predictions approximate the real data points 𝑅2=1−∑ (𝑦𝑖−𝑏𝑥𝑖)2 𝑛𝑖=1 ∑ (𝑦𝑖−𝑦)2 𝑛𝑖=1 For the linear regression in the form of 𝒀=𝜶 + 𝜷 𝑿 1. Employing the least squares method, the unknown coefficients a, and b can be determined as follows: 𝑎=∑𝑥𝑖2∑𝑦𝑖−∑𝑥𝑖∑𝑥𝑖𝑦𝑖 𝑛∑𝑥𝑖2−(∑𝑥𝑖)2 𝑏=𝑛∑𝑥𝑖𝑦𝑖−∑𝑥𝑖∑𝑦𝑖 𝑛∑𝑥𝑖2−(∑𝑥𝑖)2 2. The uncertainties ua, ub of both coefficients a, and b can be determined as follows: 𝑢𝑎2=∑𝑥𝑖2 𝑛∑𝑥𝑖2−(∑𝑥𝑖)2𝑠2 𝑢𝑏2=𝑛 𝑛∑𝑥𝑖2−(∑𝑥𝑖)2𝑠2 where 𝑠2=1 𝑛−2∑(𝑦𝑖−𝑦)2=1 𝑛−2∑(𝑦𝑖−𝑎−𝑏𝑥𝑖)2 3. The covariance ua,b between the two coefficients a, and b can be determined as follows: 𝑢𝑎,𝑏=−∑𝑥𝑖 𝑛∑𝑥𝑖2−(∑𝑥𝑖)2𝑠2 4. The uncertainty uy of the calculated value y can be determined as follows (the formula for uy2 is stated here): uy2 = ua2 + x2ub2 + 2xua,b 5. Coefficient of determination R2 is a statistical measure of how well the regression predictions approximate the real data points 𝑅2=1−∑ (𝑦𝑖−𝑎−𝑏𝑥𝑖)2 𝑛𝑖=1 ∑ (𝑦𝑖−𝑦)2 𝑛𝑖=1 21NRM05 STASIS 5 5. Graphic example Ankle plate - linear regression in the form of 𝒀=𝜷 𝑿 Ankle plate - linear regression in the form of Y = a + b X