Nuclear Medicine Physics 02 Oct. 2007

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1 Nuclea Medicine Physics Oct. 7 Counting Statistics and Eo Popagation Nuclea Medicine Physics Lectues Imaging Reseach Laboatoy, Radiology Dept. Lay MacDonald 1//7 Statistics (Summaized in One Slide) Type of analysis which includes the planning, summaizing and intepeting of obsevations of a system followed by pedicting o foecasting futue events. Desciptive Statistics-Descibe o summaize obseved measuements of a system Infeential Statistics-Infe, pedict, o foecast futue outcomes, tendencies, behavios of a system Types of Eos Systematic Eos uncetainties in the bias of the data, such as an unknown constant offset implies that all measuements ae shifted the same (but unknown) amount fom the tuth measuements with a low level of systematic eo, o bias, have a high accuacy. Random Eos aise fom instument impecision (e.g. electonic noise) and/o the inheent natue of the phenomena (e.g. biological vaiability, counting statistics) each measuement fluctuates independently of pevious measuements, i.e. no constant offset o bias measuements with a low level of andom eo have a high pecision. 1 3 Systematic Eos Random Eos Eo Eamples Systematic eos typically cannot be chaacteized with statistical methods but athe must be analyzed case-by-case. Measuement standads should be used to avoid systematic eos as much as possible. double-check equipment against known values established by standads. If thee is a fatal flaw in a study, it is usually fom an ovelooked systematic eo (i.e. bias). Attention to epeimental detail is the only defense! Even if we had no instumentation andom eos, andom eos will esult fom biological and/o patient vaiability Random eos can be analyzed with statistical methods Tue Y Tue Y Tue X Tue Y Tue Y Tue X Tue X Tue X 5

2 Nuclea Medicine Physics Oct. 7 Illustation: Hypothetical tace uptake fom a PET scan SUV measued data points tue uptake uptake cuve estimated fom measued data points SUV estimated uptake cuve bias Chaacteizing Random Phenomena (and Eos) Measues of Cental Tendency: Mode - Most Fequent Measuements (not necessaily unique) Median - Cental Value dividing data set into equal pats (unique tem) Mean (Aithmetic Mean) Chaacteizing Random Phenomena (and Eos) Measues of Dispesion: Range - Diffeence of lagest and smallest values Vaiance - Measues dispesion aound mean:! = 1 n #( i " ) n "1 i=1 Standad Deviation:! =! minutes measuements with: Random Eo: High Systematic Eo: Low --> Low Pecision, High Accuacy --> Low Bias, High Noise minutes measuements with: Random Eo: Low Systematic Eo: High -->High Pecision, Low Accuacy --> High Bias, Low Noise 7 = 1 n n! i=1 i Standad Eo of the mean:!( ) =! n 9 Simple Eample Following numbes ae maimum CT values (in HU) of the same tumo measued in the same individual What is the best estimate of the ma CT value of this tumo? 3 Scenaios: mean= 3. vaiance= 91.7 standad deviation= 17.7 standad eo of the mean =.5353 median= 37.1 Mean ± standad eo of the mean 3. ±.5 HU 1. Measued on diffeent scans fom diffeent days at eactly same location?. Measued on same image volume by individuals? Repoting Statistics Estimated value of (e.g. mean), and its eo (e.g. standad deviation): ± δ Significant digits in calculated values should be the same as measued values; measue = 1. cm, 1.5 cm, 1.1 cm, 1.3 cm mean = ( ) / = 1.75 cm = 1.3 cm σ =.177 cm =. cm = 1.3 ±. cm In geneal, epoted values and eos should agee in significant digits, and be dictated by the pecision of the measuements Chaacteizing Random Eos With a Distibution Statistical Models fo Random Tials 1. Binomial Distibution Random independent pocesses with two possible outcomes. Poisson Distibution Simplification of binomial distibution with cetain constaints 3. Gaussian o Nomal Distibution Futhe simplification if aveage numbe of successes is lage (e.g., >) 3. Measued on diffeent scans fom same day at eactly same location?

3 Nuclea Medicine Physics Oct Binomial Distibution Independent tials with two possible outcomes Binomial Density Function: n! P binomial () =!(n! )! p (1! p) n! Pobability of successes in n ties when p is pobability of success in single tial Binomial pocess Tial can have only two outcomes define a success (anything else is a failue ) Binomial pobability density function mean and vaiance N is total numbe of tials p is pobability of success is mean, σ is standad deviation Eample: What is the pobability of olling a 1 on a si sided die eactly 1 times when the die is olled fo a total of times. = 1, n =, p = 1/, P binom (=1) =.5 ~ 1 in If p is vey small and a constant then: Eample: What is the pobability that a clinical tial will include 1 smokes in a andom cohot of 1, when the pobability a peson is a smoke is X%. = 1, n = 1,, p = X% Poisson Distibution Limiting fom of binomial distibution as p and N As in nuclea decay. Have many, many nuclei, pobability of decay and obsevation of decay vey, vey small Poisson Distibution vs. Binomial Poisson: only fo positive values; µ > Asymmetic µ = () = µ ep(!µ)! µ = 1. () = µ ep(!µ)! µ = 1. Only one paamete, µ In a Poisson Pocess Mean = Vaiance 1 = pobability of measuing when mean µ = 1 Binomial: mean = pn µ =

4 Nuclea Medicine Physics Oct Gaussian (Nomal) Distibution 1 $ ( # µ) ' P Gaussian () = ep &# )! " %! ( Symmetic about the mean Useful in counting statistics because distibutions ae appoimately nomal when N > Vaiance and mean not necessaily equal mean µ = 1 19 Gaussian (Nomal) Distibution Confidence Intevals Inteval about measuement ±.7σ ± 1.σ ±1.σ ± 1.9σ ±.5σ ± 3.σ Pobability that mean is within inteval (%) How accuate is a single measuement? Take a single measuement of adioactive souce and detect 1, events. Radioactive decay => Poisson, 1k events => Gaussian Estimate that the eal numbe of detectable events is 1,; mean µ = 1k vaiance σ = 1k std. dev. σ = 1 How close is this to the tuth? % Pobability: 9,9 < tuth < 1,1 95% Pobability: 9, < tuth < 1, 99.7% Pobability: 9,7 < tuth < 1,3 1 Simple Popagation of Eo Quantities of inteest ae often detemined fom seveal measuements pone to andom eo. If the quantities ae independent, then add independent contibutions to eo in quadatue as follows: The simplest eamples ae addition, subtaction, and multiplication by a constant. If the quantities a and b ae measued with known eo δ a and δ b, then the eo in the quantities, y, z when = a + b y = a - b z = k*a, k = constant (no eo) ae:! =! y =! a +! b Geneal Popagation of Eo Still assuming the individual measuements (a,b,c, ) ae independent of each othe; and the desied quantity is a function of a,b,c, : = (a,b,c, ) The contibution of measuement a to the eo in, δ a is given by:! a = " "a! a contibutions add in quadatue:! =! a +! b +! c +... δ z = k*δ a 3 Geneal Popagation of Eo Eample 1: Addition & subtaction (a,b) = a ± b! a = " "a! a =! a Same fo b. Note absolute value of patial deivative ---> cumulative eo will be individual eos.! =! a +! b!! =1,!a!b =1 =! a +! b

5 Nuclea Medicine Physics Oct. 7 Geneal Popagation of Eo Eample : Multiplication by constant No eo in the constant k (a) = ka!! a = k! a!a = k! =! a = k! a Geneal Popagation of Eo Eample 3: Multiplication of eo-pone vaiables:! =! a +! b (a,b) = a * b! a = " "a! a = b! a = b! a + a! b Simple Eamples A adioactive souce is found to have a count ate of 5 counts/second. What is pobability of obseving no counts in a peiod of seconds? Five counts in seconds? Mean count ate: 5 cnts/sec. --> 1 cnts/ sec. mean = µ, obseved cnts = : By etension of E.1 & E.: (a,b) = ka ± b! = k! a +! b 5 Eample : Division of eo-pone vaiables: (a,b) = a /b! a = " "a! a = 1 b! a! = " 1% " $ '! a % # b& a + $ ' # b &! b = " "b! b = a b! b! b ( = ) = (µ =1)= ep(!(µ =1)) =.5 *1!5 ( = )! ( = 5) = (1)5 ep(!1) =.3 (5)! 7 Simple Eamples The following ae measuements of counts pe minute fom a Na souce. What is the decay ate and its uncetainty? 1 ˆ µ = =. 15!(ˆ µ ) =! n =. = 1 5 Count Rate = ( ± 1)counts/min 31 O, could view mean as sum of values (each with eo) * constant: f = (a + b + c + d + e) *1/5 Chi Squae Test Chi squae test is often used to assess the "goodness of fit" between an obtained set of fequencies in a andom sample and what is epected unde a given statistical hypothesis E. Detemine if andom vaiations obseved ae consistent with Poisson distibution! f =! a +! b +! c +! d +! e *1/5 = 1 9 (Some Infeential Statistics) The Maimum Likelihood Estimato Suppose we have a set of N measuements ( 1,, N ) fom a theoetical distibution f( θ), whee θ is the paamete to be estimated (e.g. 1 is obseved/detected counts fom piel θ ). We fist calculate the likelihood function, L(!) = f ( 1,!) f (,!)K f ( N,!) which can be seen as the pobability of measuing the sequence of values ( 1,, N ) fo a value θ. Maimum likelihood estimato is the value of θ that povides the maimum value fo L( θ) E.g. ML Estimate of mean of a Gaussian distibution is just mean of measuements 3

6 Nuclea Medicine Physics Oct. 7 Raphe Question D7. How many counts must be collected in an instument with zeo backgound to obtain an eo limit of 1% with a confidence inteval of 95%? Raphe Answe Raphe question G7. A adioactive sample is counted fo 1 minute and poduces 9 counts. The backgound is counted fo 1 minutes and poduces 1 counts. The net count ate and net standad deviation ae about and counts. A. 1 B. 31 C. 1, D., E. 1, CI = 95% --> measue within σ % Eo = (eo/measue) 1%;! N = "1% (! = N ) N D7. How many counts must be collected in an instument with zeo backgound to obtain an eo limit of 1% with a confidence inteval of 95%? D. A 95% confidence inteval means the counts must fall within two standad deviations (SD) of the mean (N). Eo limit = 1% = SD/N, but SD = N 1/. Thus.1 = (N 1/ )/N = / N 1/. Whee [.1] = /N and N =,. A., B., 3 C. 9, D. 9, 3 E. 99, 3 Measued value is best guess of the mean, std. dev. equals sqt(mean): N = measued counts, R = count ates (counts pe min.) Goss counts N g = 9 ± (9) 1/ --> 1 minute --> R g = 9±3 cpm Backgound N b = 1 ± (1) 1/ --> 1 min. --> R b = 1±1 cpm N > ( / 1%) = () N >, 31 3 Net count ate = goss ate - backgound ate: R n = R g - R b R n = 9 cpm - 1 cpm = 9 cpm! n =! g +! b = 3 +1 " 3 33 Raphe answe G7. A adioactive sample is counted fo 1 minute and poduces 9 c ounts. The backgound is cou nted fo 1 minutes and p oduces 1 c ounts. The net count ate and net standad deviation ae about and counts/min. D. The net count ate is: [ ( N s/t s) - (N b/t b)] = [(9/1) - (1/1)] = 9. The net standad deviation, is: [(N s/t s) + (N b/t b)] 1/ = [(9) + (1)] = 3. Summay of Topics Types of Eo systematic vs. andom; accuacy vs. pecision Repoting Eo consistency in significant figues Statistical Desciptions of Random Eo pobability distibution functions Popagation of Eo cumulative eos fom independent souces 3 Net week: Gamma Cameas Robet Miyaoka 35