sorenson chpt 6

8/12/2019 Sorenson Chpt 6

1/10

Physics nNuclear MedicineJames A. Sorenson, Ph.D.Professor of RadiologyDepartment of RadiologyUniversity of Utah Medical CenterSalt Lake City, UtahMichael E. Phelps, Ph.D.Professor of Radiological SciencesDepartment of Radiological SciencesCenter for Health SciencesUniversity of CaliforniaLos Angeles, California

~JGrune & StrattonA Subsidiary of Harcourt BraceJovanovich, PublishersNew York LondonParis San Diego San Francisco Sao PauloSydney Tokyo Toronto


2/10

6Nuclear Counting Statistics

All nuclearmedicineprocedures re basedon radiation countingmeasurements.Like any other measurements f physical quantities, hey are subject o meas-urementerrors. This chapterdiscusseshe types of errors that occur, how theyare analyzed,and how, in somecases, hey can be minimized.

A. TYPESOFMEASUREMENTRRORMeasurement rrors are of three general ypes:Blunders are errors that are adequatelydescribedby their name. Usuallythey produce grossly naccurate esults and their occurrences easily detected.Examples n radiation measurementsnclude the use of incorrect instrumentsettings, ncorrect abeling of samplecontainers, njecting the wrong radiophar-maceutical nto the patient, etc. There is no way to "analyze" errors of thistype, only to avoid them by careful work.Systematicerrors produceresults hat differ consistently rom the correctresult by some fixed amount. The same result may be obtained in repeatedmeasurements, ut it is the wrong result. For example, ength measurementswith a "warped" ruler, or radiation counting measurements ith a "sticky"timer or other persistent nstrument malfunction, could contain systematicer-rors. Observer"bias" in the subjective nterpretationof data (e.g., scan ead-ing) is anotherexample of systematicerror. Measurement esults having sys-tematic errors are said to be inaccurate.It is not always easy o detect he presence f systematicerror. Measure-ment results affected by systeinaticerror may be very repeatableand not too

100


3/10

Nuclear Counting Statistics 101different from the expected results, which may lead to a mistaken sense ofconfidence. One way to detect systematic error is by the use of measurementstandards, which are known from previous measurements with a properly op-erating system to give a certain measurement result. For example, radionuclidestandards, containing a known quantity of radioactivity, are used in various"quality assurance" procedures to test for systematic error in radiation countingsystems. Some of these procedures are described in Chapter 12, Section D.

Random errors are variations in results from one measurement to the next,arising from physical limitations of the measurement system or from actualrandom variations of the measured quantity itself. For example, length meas-urements with an ordinary ruler are subject to random error because of inexactrepositioning of the ruler and limitations of the human eye. Random error isalways present in radiation counting measurements because the quantity that isbeing measured-namely, the rate of emission from the radiation source-isitself a randomly varying quantity.

Random error affects measurement reproducibility. Measurements that arevery reproducible-in that nearly the same result is obtained in repeated meas-urements-are said to be precise. It is possible to minimize random error byusing careful measurement technique, refined instrumentation, etc; however, itis impossible to eliminate it completely. There is always some limit to theprecision of a m~asurement system. The amount of random error present issometimes called the uncertainty in the measurement.

It is also possible for a measurement to be precise (small random error)but inaccurate (large systematic error), or vice versa. For example, length meas-urements with a warped ruler may be very reproducible (precise); nevertheless,they are still inaccurate. On the other hand, radiation counting measurementsmay be imprecise (because of inevitable variations in radiation emission rates)but still they can be accurate, at least in an average sense.

Because random errors are always P esent in radiation measurements, it isnecessary o be able to analyze them and to obtain estimates of their magnitude.This is done using methods of statistical analysis. (For this reason, they are alsosometimes called statistical errors.) The remainder of this chapter describesthese methods of analysis.

B. RANDOM ERRORS INRADIATION COUNTING MEASUREMENTS

1. The Poisson DistributionSuppose that a long-lived radioactive sample is counted repeatedly under

supposedly dentical conditions with a properly operating counting system. Be-cause he disintegration rate of the radioactive sample undergoes random vari-ations from one moment to the next, the numbers of counts recorded in suc-cessive measurements (N1, Nz, N3, etc.) are not the same. Given that different


4/10

102 Physics in Nuclear Medicineresults are obtained from one measurement to the next, one might question ifa "true value" for the measurement actually exists. One possible solution is tomake a large number of measurements and use the average N as an estimate forthe "true value":

True Value = N (6-1)N = (Nt + N2 + . . . + Nn)/n (6-2)

n N= ~ -1 (6-3)i= n

where n is the number of measurements taken. The notation ~. indicates thatIa sum is taken over values of the parameter with the subscript i.Unfortunately, multiple measurements are impractical in routine practice,

and one must usually be satisfied with taking only one measurement. The ques-tion then is, how good is the result of a single measurement as an estimate ofthe "true value," i.e., what is the uncertainty in this result? The answer to thisdepends on the frequency distribution of the measurement results. Figure 6-1shows a typical frequency distribution curve for a series of radiation countingmeasurements. It is a graph showing the different possible results, (i.e., numberof counts recorded) versus the probability of getting each result. The curve ispeaked at a mean value m, which is the "true value" for the measurement.Thus if a large number of measurements were made and their results averaged,one would obtain

N = m (6-4)The curve in Figure 6-1 is described mathematically by the Poisson dis-

tribution. The probability of getting a certain result N when the true value is mis

P(N;m) = e-m ~/N (6-5)where e (= 2.718 . . .) is the base of natural logarithms and N (N factorial)is the product of all integers up to N (i.e., 1 X 2 X 3 X . . . X N). FromFigure 6-1 it is apparent that the probability of getting the exact result N = mis rather small; however, one could hope that the result would at least be "closeto" m.The probability that a measurement result will be "close to" m dependson the relative width, or dispersion, of the frequency distribution curve. Thisis related to a parameter called the variance, (T2,of the distribution. The varianceis a number such that 68.3 percent (-2/3) of the measurement results fall within


5/10

Nuclear Counting Statistics 103

0.13 . .0.12.. I .

0.10z .IIf- 0.09 .~:::>~ 0.08a: .>- 0.07f-- ..= 0.06CD


6/10

104 Physics in Nuclear MedicineThe range :t -.IiI is the uncertainty in N. The percentage uncertainty in

N isV = (-.IiI/N) x 100% (6-7)= 100%/-.IiI (6-8)

Example 6-1.Comparehe percentagencertaintiesn the measurements = 100counts and Nz = 10,000 counts.Answer.

For N1 = 100 counts, VI = 100%/VlOO = 10% (Equation 6-8). ForNz = 10,000 counts, Vz == 100%/~ = 1%. Thus the percentageuncertainty in 10,000 counts is only VIO he percentage uncertainty in100 c~unts.

Equation 6-8 and Example 6-1 indicate that large numbers of counts havesmaller percentage uncertainties and are statistically more reliable than smallnumbers of counts.

Other confidence intervals can be defined in terms of (1 or -.IiI.Thex aresummarized in Table 6-1. The 50 percent confidence interval (0.675VN) issometimes called the probable error (PE) in N.

2. The Standard DeviationThe variance (1z s related to a statistical index called the standard deviation

(SD). The standard deviation is a number that is calculated for a series ofmeasurements. If n counting measurements are made, with results NI, Nz, N3,. . ., N n' and a mean value N for those results is found, the standard deviationis

SD = (i (Nj - N)Z)Iz (6-9);=1 n - 1

(recall that raising a quantity to the Vz power is the same as taking its squareroot). The standard deviation is a measure of the dispersion of measurementresults about the mean and is in fact an estimate of (1, the square root of thevariance. For radiation counting measurements one should therefore obtain

SD = -.IiI (6-10)This can be used as a test to determine if the random error observed in a seriesof counting measurements is consistent with that predicted from random vari-ations in source decay rate, or if there are additional random errors present,e.g., from faulty instrument performance. This is discussed further in Section E. i


7/10

Nuclear Counting Statistics 105Table 6-1Confidenceimits n Radiation ountingMeasurements ;-/

Range ConfidenceLimit for m(True Value) (%) , 'c '

N: : 0.675


8/10

106 Physics n Nuclear MedicineC. PROPAGATIONOF ERRORS

The precedingsection describedmethods or estimating he random erroror uncertainty n a single counting measurement; owever, most nuclear med-icine procedures nvolve multiple counting measurements,rom which ratios,differences, and so on are computed o determinea final resQlt. This sectiondescribesmethods or estimatinguncertaintiesn mathematical :ombinations fmultiple counting measurements.

1. Sums and DifferencesIf two quantitiesA and B are subject o randomerrors 0" and 0"B' then theuncertainty n either their sum or difference s given by

O"(A t B) = y~~ (6-1Applying this to radiation counting measurementsne obtains

. O"(Ni t NJ = VN~~ (6-14)The rule is readily extended o longer sequences

O"(Ni t N2 :t N3 :t . . .) = YNi + N2 + N3+ . ;~ (6-15)2. Constant MultipliersIf a quantity A having random error 0" is multiplied by a constantk (i.e.,

a numberhaving no random error), the uncertainty n the result, kA isO"(kA)= kO"A (6-16)

Thus for a radiation counting measurement multiplied by a constantk,0"(kN) = kVN (6-17)

The percentage ncertainty n the product kN isV(kN) = [0"(kN)/kN] x 100% (6-18)

= 100%/VN (6-19)which is the same esult as Equation6-8. Thus there s no statisticaladvantageto be gained in multiplying the number of counts recordedby a constant omake he number arger. The percentage ncertaintystill depends n the actualnumberof counts ecorded.


9/10

NuclearCounting Statistics 1073. Products and RatiosIf two quantitiesA and B have percentage ncertaintiesV and VB, then

the percentage uncertainty in their product or ratio isXV(A + B) = VV~ +V~ (6-20)For radiation counting measurement J and N2, this implies

V(NJ~NJ = VV~I +V~2 (6-21)~ ViIN; + llN2 x 100% (6-22)

The ule canbe extendedo longersequences:x x x ~ . ". .. ,.. . . "Y.V (NJ + N2 + N3 + . . .) ~ V llNJ + llN2 + llN3 + ' . . x 100% (6-23)

4. More Complicated CombinationsMany nuclear medicine procedures nvolve both differencesand ratios ofcounting measurementse.g., thyroid uptakes, blood volume determinations,etc.). A general orm of thesecalculations s

Y = k(NJ - NJ/(N3 - N4) (6-24)whereNJ, N2, N3, and N4 are measured ounts, k is a constant, and Y is thecalculatedquantity (thyroid uptake, blood volume, etc.). The percentageun-certainty in Y is obtained by first applying the rule for ratios and products(Equation6-20),

Vy = VV~1-N2 + V~3-N4 (6-25)and hen the rule for sums and differences Equation6-14),

V~1-N2 [(NJ+NJ/(NJ-NJ1 x 100% (6-26)V~3-N4 = [(N3+N4)/(N3-N4Y] x 100% (6-27)

The final result isVy ~ V(NJ+NJ/(NJ-NJ2 + (N3+N4)/(N3-N4Y 100% (6-28)

The uncertainty T can then be obtained rom(Ty ~ Vy x YIlOO% (6-29)


10/10

108 Physics n Nuclear Medicine

Example6-3.A patient s injected with a radionuclide.At some ater time a bloodsample is withdrawn for counting in a well counter and Np = 1200counts are recorded.A blood samplewithdrawn prior to injectiongives a blood backgroundofNpb = 400 counts. A standardpreparedfrom the injection preparation ecordsN. = 2000 counts, and a"blank" sample records an instrument background ofNb = 200counts. Calculate he ratio of net patient samplecounts o net standardcounts, and the uncertainty n this ratio.

Answer.The atio s Y = (Np- Npb)/(N. Nb)= (1200 - 400)/(2000 - 200)= 800/1800 = 0.44

The percentage ncertainty n the ratio is (Equation6-28)Vy = \/(1200 + 400)/(800)2 (2000 + 200)/(1800)2 100%

= 5.6%The uncertainty s 5.6% x 0.44 = 0.02; thus the ratio and itsuncertaintyare Y = 0.44 j: 0.02.

~-'"

";:rc,,, lti' "hi, "; ;h;; "P

.7 u.., ;;;,((';

sorenson chpt 6

Documents