LLD for personnel dosimetry systems 0 P. L. ROBERSON AND R. D. CARLSON
3
Table 1. Notation for mathematical derivation. Dosimeter Control Total Net reader dosimeter dosimeter dosimeter noise (background) reading reading ~
True value Observed value Standard deviation Observed SD Critical level Detection level
~~
~~
~
PN
PB
PT
PH
N
HB
T
H
uN
UB
UT
-
SB
-
OH
SH
Lc LD 12
Signal
Critical Level (L,)-the signal that provides a confidence level of 1-a that the result is not due to a fluctuation of the background. All results with values less than L, should not be reported as detection of a positive signal. Results with values greater than L, should be reported as positive values and may be accompanied by a confidence interval. Detection Level (LD)-the delivered exposure for which the confidence level is 1-/3 that the result will be detected and properly reported as a (qualitative) positive result. This value may be identified as the LLD when expressed in units of dose equivalent.
Fig. 1. Schematic representation of the normal distribution for the background (or noise) signals and the signals representing a delivered dose (equivalent) of LD,the detection level.
For one dosimeter reading, these values are replaced by observed values: H = T - WB,
(2) where the background is determined by many ( n ) dosimeters and the bar denotes mean value. The variance of the net signal is (rH2
THEORY Some interpretation is required to convert a dosimeter reading to a dose-equivalent value. Necessary steps in the interpretation typically included subtraction of a background signal and application of an interpretation algorithm that may use multiple element readings from the same dosimeter. The initial development of the formalism for the LLD will assume a simple signal. More complicated systems will be discussed in the Applications section. The background signal to be subtracted from the dosimeter reading should include all signal sources not due to the radiation exposure to be determined. This may include reader noise, incidental signals, and background radiation. Examples of incidental signals are stray luminescence (e.g., residual deep traps) for thermoluminescence dosimeters or fog level for film dosimeters. The signal due to reader noise only is denoted pN,with associated standard deviation of uN. The background can be characterized using a mathematical mean (true value, p B ) and a standard deviation (uB). A background determination using n independent measurements (dosimeters) will result in an estimate of the mean with accompanying standard deviation of the mean (uB/&). The net dosimeter reading (true value, pH) is the difference between the total value (or raw signal) and the mean background value:
(arbitrary units)
=
UT2
+ uB2/n.
(3)
Note that uT is never smaller than uB. For a zero net signal (pH = 0, UH = (TO, UT = uB), uo2 = ug2 (1
+ l/n).
(4)
The critical level (L,) provides a confidence level of 1-a that the determined reading is not due solely to a fluctuation of the background (see Fig. l), Lc = k&O,
(5)
where k, is the abscissa of the standard normal distribution corresponding to a single-sided probability of 1a. Values greater than L, due solely to fluctuations in the background are errors of the first kind, or the incorrect rejection of the null hypothesis (false-positive) (USNBS 1963). Dosimeter readings below L, should not be reported as an indication of the detection of a positive signal. Not reporting a positive value for a true exposure is an error of the second kind, or the incorrect acceptance of the null hypothesis (false-negative). The exposure level received for which the probability of reporting a zero value (below L,) is no more than /3 is termed the detection level (LD), LD = L,
+
kpbD,
(6)
where kp is the abscissa of the standard normal distribution corresponding to a single-sided probability of 1/3, and where uDis the standard deviation of the readings at the exposure level LD.
Health Physics
4
The total signal (pT) is a combination of a signal due to the effect being measured ( p ~ )a,signal attributed to the dosimeter but not part of the effect being measured ( p a - pN), and a signal not associated with the dosimeter, labeled reader noise ( p ~ ) : PT
= pH
f PB
= (PH
f pB
- pN) + YN,
(7)
where the value in brackets is the signal from the dosimeter irrespective of origin. The standard deviation of the dosimeter signal is assumed to be a constant fraction of the signal level5(Piesch 1981). The standard deviation of the total signal, uT, may be split into two components, one dependent on the dosimeter signal level (pT - pN) and the other identified with the reader noise level ( p ~ ) , aT2
=
2 6,
x
(pT
- pN)2
+ cN2,
(8)
where a, is the constant value of the relative standard deviation for large signals. For the background readings, aB2
2
= cfi
x
(PB
- pN)2
+ cN2,
January 1992, Volume 62, Number 1 a,pg’ >> UN (low noise) and large n, ~ , p g ‘ and the LDsimplifies to 2 k 3 / ( 1- ka,). For finite sample sizes, = Hg - N replaces p g ’ , Student’s t values replace the k values, and sample standard deviations (S) replace a values. Usingn background dosimeter readings to calculate SOand HB’, and rn dosimeter readings of a known large dose (- 10 mGy) compared to background to calculate S,,
Ilk. For
ug
=
UO,
nB’
where tn (tm)is Student’s t statistic for n - 1 (rn - 1) degrees of freedom, and S, is the relative sample standard deviation for rn - 1 degrees of freedom. A typical confidence level for t is 95%. Necessary formulas include the following (USNBS 1963): m
s,2= (H) - I C (rn - 1) i=l
=
‘T,
2
x
(pT
- pN)2
+
(TN2
n=’.Z Hi rn i=1
+ cB2/n.
(10)
For p~ = p~ + pg from eqn (l), expanding and collecting terms and using eqn (9), aH2
6; x [pH2 + 2pH x
(16)
(9)
and, from eqns (3) and (8), for general readings, aH2
(Hi - W)2
1
n
=
(pB
- pN)] +
aB2
(1
+ 1/n)* (1 1)
The variance for the detection level (pH = LD) from eqn (1 1) is ~ T D =~0;
X
(LD2+ 2 LD p g ’ )
+ 0o2,
(12)
where pB’ pB - p ~Solving . for CTDin eqn (6), inserting eqn ( 5 ) for L,, squaring, setting the result equal to eqn (12), and collecting terms yields (1 - k g):a
x LD2- [2 (k, a0 + kp2:a
pB’)]
x LD + (km2- k t ) x go2 = 0.
(13)
where the (Hg)i represents background readings and the Hi represents known dose readings. The reader noise may be measured (or adjusted to zero when level possible) before dosimeter readout. The n appearing in the equation for So represents the number of background dosimeters routinely used to determine the background subtraction level and may be different from the number of dosimeters used to calculate SB.
(w)
For k, z kDz k, LD
= 2 (k
UO
+ k2
pg’)
1 - k2 c;
In nearly all cases, the dominant term in the expression for LD is 2kao. Since k at 95% confidence for a singlesided distribution is 1.7, LD (at first approximation) is 3.4 times the standard deviation of the background. The denominator factor contributes significantly for a large relative standard deviation. The second term in the numerator is significant for large values of the background. The second term and denominator factor represent small corrections to 2kao unless a, approaches § The dominant error is in the readout process, which affects the full signal in the same manner irrespective of signal magnitude.
APPLICATIONS Applying the LD formula, eqn (15), may require special handling of the data. Implicit in its derivation is the assumption that the units for the background readings are the same as are quoted for the positive readings, presumably in units of dose equivalent. Alternatively, the readings may be expressed in other units (e.g., reader units) and the LD level derived for the measurement system in those units. Then the LLD may be defined as the LD multiplied by the conversion factor to dose equivalent. For example, with an energy-dependent dosimeter, one value of LDmay correspond to various values of LLD, depending on the energy calibration factor.
LLD for personnel dosimetry systems 0 P. L. ROBERSON AND R. D. CARLSON
Multiple-element dosimeters An analysis algorithm is typically applied to multiple-element dosimeters. Several components may be analyzed to ultimately result in interpretations of the penetrating and superficial dose equivalent values. A typical analysis may separate the photon, beta, and neutron dosimetry in order to apply different calibration factors before performing the final sum. In this situation, application of the LD formula in the simple form requires that the analysis algorithm be a function of the element readings without discontinuities and that the background dosimeters be evaluated with the full algorithm. The background dosimers should be analyzed as though they were field dosimeters. For example, even if the background dosimeters are known to receive (at most) photon irradiation from the environment, they should be analyzed for beta and neutron exposure as well. The LLD will be greater if there is a possibility of radiations other than photons. Algorithms for most beta and neutron dosimeters routinely subtract the signal due to photon exposures before applying the calibration factor. When evaluating the background dosimeters for the mean (RB) and standard deviation (SB)values, both negative and positive values must be retained. They represent fluctuations, present at positive exposure levels, which are due to the uncertainty in the background. (Note: while a value, if present, is technically required negative by the LD formula, its presence probably indicates a miscalibrated dosimetry system. The importance of retaining negative values is for the calculation of SB.) Here, the LLD represents the beta or neutron dose equivalent that results in a positive report if photon exposures do not exceed background levels. This condition will not be satisfied under mixed-field conditions (see below). Another consideration is the relationship between the LLD and the energy dependence of the dosimeter. For example, the value of the LD (in reader units) may be independent of the exposure energy. However, application of an energy-dependent calibration factor, applied to calculate the LLD in dose equivalent, results in an energy-dependent LLD. For most dosimetry systems, the LLD associated with each type and energy (or energy range) of radiation should be determined.
wB’
Mixed fields Many radiation field environments are composed of more than one type of radiation. For example, neutron sources are rarely present without significant associated gamma and/or x-ray sources. Likewise, it is unusual for a significant beta-radiation source to be present without some gamma radiation. It may be more important to determine the LLD in real mixed fields than to record the LLD for optimal conditions of photon-only sources. The above formulation for the lower level of detection cannot be directly applied to mixed-field conditions where the ratios of the components of the total
5
dose equivalent (e.g., beta/gamma ratio) are unknown. Given a calculational algorithm for the interpretation of the signal levels, one can define an LD expressed in terms of signal level. If the calibration factor is dependent on the ratio of the components of the total dose equivalent, the LLD will similarly be component ratio dependent. The result is an LLD dependent on workplace conditions. There are several dosimeter design strategies used for the determination of dose equivalent in mixed fields. These include dosimetry systems in which: 1) the components are measured separately and summed after the conversion to dose equivalent (e.g., separate gamma and neutron dosimeters, where the neutron dosimeter is insensitive to gamma exposure and vice versa); 2) the components are measured separately but the dosimeter element is sensitive to both radiation types so that the background due to a mixture component must be subtracted before the dose equivalent calibration conversion can be performed (e.g., a thermoluminescence albedo neutron dosimeter); 3) the same dosimeter elements are used to detect both components and a calculational algorithm is used to separate the components prior to the conversion to dose equivalent (e.g., a beta/ photon dosimeter using shallow and deep elements to separate beta, x-ray, and gamma-ray components). These configurations will be discussed in turn. In the first case above, an LLD can be determined using the LD equation for each component separately as illustrated in Fig. 2. However, a combined LLD would (at best) be dependent on the mixture ratio. The usefulness of the concept is questionable in cases where the separate LLDs are significantly different and the mixture ratio is not unique. In the second case, the LLDs may also be determined for the components separately, but since the measured background consists of signal from both types of radiation, the large-dose standard deviation, ( T ~ , is increased. This implies that the LLD level is dependent on the mixture ratio present, but in a predictable manner for each dosimeter design. An LLD for the combined signal remains questionable. For the dosimeter system in the third case, the LLD for each mixture component is a function of the component ratio. For the example of a beta/x-ray/ gamma dosimeter, the deep-dose component may be handled as a single-component dosimeter. The betadose LLD may be a function of both the beta/photon mixing ratio and the beta energy. The LD, expressed in “reader units,” is typically independent of beta energy because the reading of only the shallow-depth element is used. All of the beta energy dependence is in the beta calibration factor, whether determined by site measurement or dosimeter element ratios. Thus, the LD can be expressed as a function of the beta/photon signal ratio. Assuming that the background is not varying, the go value is independent of the ratio present in the work environment. The CT,, value is a function of the ratio because the higher the ratio, the greater the relative
Health Physics
6
1.2
January 1992, Volume 62, Number 1
I
30 (a)
0
10
Signal
20
30
(arbitrary units)
Fig. 2. Schematic representation of the normal distributions
for a two-component dosimeter, each component with a different Lo when measured alone.
8-
r 0
6-
=
Q)
0-
E
U.
difference of measured values for the dosimeter elements. Thus, the high-dose limit of the standard deviation must be determined for a range of beta/photon ratios. Note that as the mixture approaches an exclusively photon dose, determination of the beta dose is increasingly difficult because it depends on the difference of large numbers. The zero-dose standard deviation becomes large and the beta-dose LDbecomes large. The beta-dose LLD may be expressed as a function of beta energy by multiplying by the energy-dependent calibration factor (if applicable). The LLD for the combined beta plus photon shallow dose equivalent is not easily derived due to the expected difference in LLD for the separate components. However, for a given radiation field composition (mixture ratio and energies) the LLD is calculable. Performance testing using a limited number of dosimeters An estimate of the LLD achievable under test conditions can be obtained from an analysis of background dosimeters and the results of NVLAP or DOELAP performance testing. At least 15 background dosimeters handled similarly to those used for routine dosimetry should be used for the calculation of R'Band SO.It is important to calculate the values for the background dosimeters using the assumptions for each field condition or test category considered. The S, value can be estimated from the test results for each test category using the relative bias (B) and the relative standard deviation (S) (ANSI 1983): S s, = B + 1'
4-
2-
n , . . . . .
" ? " Y ? ? " ? ? ' " ? Y 9 " ? U
P
u
o
P
P
P
l
m
Reading
m
m
m
m
m
m
m
m
m
(mGy)
Fig. 3. Dosimeter readings for 154 unirradiated dosimeters (a) and 30 I3'Cs-irradiated dosimeters (b). Histograms of frequency of reading vs. reading are presented.
Values of the LLD for single-field categories may be derived using eqn ( 1 5). The calculation of LLD for mixed-field categories is dependent on dosimeter design, and use of eqn (1 5) may not be appropriate. It should be the goal of dosimetrists to maintain the LLD well below the lower limit of the testing ranges in ANSI N13.11 or the DOE testing standard. An LLD value less than half the lower limit is reasonable. For example, for an LLD value of half the lower limit of the testing range and a U, value near 0.1, eqns (1 1) and (14) predict a relative standard deviation at the lower limit of the testing range of approximately twice the ufl value. A factor of 2 increase in the standard deviation is acceptable considering the low number of test dosimeters assigned dose equivalents near the lower limit of the testing range. Example The calculation of the LLD is illustrated using data submitted by a DOELAP participant. Readings from 154 unirradiated dosimeters and 30 dosimeters irradiated to a large dose (Fig. 3) were used to estimate the LLD.
LLD for personnel dosimetry systems 0 P. L. ROBERSON AND R. D. CARLSON
I
Table 2. Critical value and detection level for sample data. (a1
*
Calculation method
-
Data points Normal curve Log normal curve
2o
Data Normal distribution Lognormal distribution
C
Lt (mGy) 0.066 0.057 0.071
LO( ~ G Y ) 0.1 12 0.115 0.1 14
a
U
!?
10
0 0.0
0.1
Reading
LL 2
J
0.2 (mGy)
0.3
0 1
A
o ! 4
5
Reading
6
(mGy)
Fig. 4. Calculated normal and lognormal curves compared to dosimeter readings. (a) Unirradiated dosimeters; (b) '37Cs-
irradiated dosimeters.
The distribution of readings from the unirradiated dosimeters does not obey a normal distribution. Fig. 4a shows the data and fitted curves using normal and lognormal distributions. The lognormal distribution provides a better fit to the data. Values for LC were derived for the data set, the normal curve, and the lognormal curve (Table 2). The value derived from the data set is the difference between the dose that exceeds 95% of the data points and the average dose (RB), multiplied by the (1 1/12) factor. For purposes of this analysis, it was assumed that 30 background dosimeters are used for the routine background subtraction (i.e., a 1/12] factor of 1.03). The LC value from the [l lognormal distribution gives better agreement with the value derived from the data set than with the value derived from the normal distribution, reflecting the better fit to the data set for the lognormal curve. Some of the discrepancy between the normal-distribution value and the data set value observed for LC is compensated during the calculation of L D by the
+
+
asymmetric distribution of the data. The 2tSo term represents the upper half of the background distribution plus the lower half of the LD distribution. This concept is developed more fully in the Appendix. The data extend above the mean value much more than below. The last column in Table 2 contains the LD values calculated using the upper and lower parts of the curves for the calculation of the 2tS0 term, compared to the calculation using the normal and lognormal distribution. Differences are relatively small compared to the uncertainty of the LDvalue calculated from the data set (an uncertainty of 14% from distribution-free tolerance limits at 95% confidence). Thus, the assumption of normal statistics does not greatly influence the result. The data set for the high-dose irradiation is consistent with the fitted normal curve (Fig. 4b). The derived relative standard deviation is reliably represented. The input values for eqn (1 5 ) and the derived LD value are given in Table 3. In this case, as in most, all but the leading term (2tS0) are small.
DISCUSSION The lower limit of detection has not received much attention as applied to personnel dosimetry. The lower dose levels were indirectly emphasized by the specifications in ANSI N13.11 (ANSI 1983)through the lower limit of the testing range. The lower ranges were determined with the goal of achieving a reasonably low dose measurement capability so that multiple dosimeters summed to provide annual and lifetime cumulative doses would retain sufficient precision. The exception was for the more difficult categories (beta, neutron, and mixtures) where higher levels were chosen consistent with prevailing technology. The DOE performance testing standard (USDOE 1986) included a requirement for the calculation of the LLD without a performance requirement. For ease of use, a formula similar to eqn ( 14) is specified. More recently, Spacher et al. (1990) used increasing exposure levels and normal statistics to determine the LLD, while Hirning" applied a formula derived from the approach used in the DOE standard to describe the determination of the LLD for operating dosimetry systems. The ANSI N13.11 specification of a performance criteria constant with exposure level contrasts with the recommendations of national and international organizations. Recommended accuracies are within a factor )J Hirning, C . R. Detection and determination limits for thermoluminescence dosimetry. Submitted to Health Physics.
Health Physics
8
Table 3. Derived Darameter values for samde case. Inout
Value
0.165 mGy 0.033 mGy 5.07 mGy
R E
SE
R
0.044
S“ ~
Outout
Lc LDW D )
Value 0.057 mGy 0.115 mGy
of 3 at levels less than one-tenth the maximum permissible levels (ICRU 1971; NCRP 1978) or within a factor of 2 at 95% confidence at levels below 1 rem (ICRP 1982). The ANSI N13.11 specification was determined from the desire to keep the test as simple as possible and the observation that the relative precision did not vary greatly with dose level (ANSI 1983). This effect is easily understood from eqn (15). With the standard deviation of the background sufficiently low for the LLD to be well below (less than one-half) the lower limit of the testing range, the signal-dependent term surpasses the background term at two to three times the lower limit of the test range (0.5 to i .O mGy for the photon categories). The relatively small increase in standard deviation at the lower levels would be difficult to observe. Alternatively, if the LLD is at or above the lower limit of the testing range, there is an unacceptably large risk of reporting a test dosimeter as a zero or minimal reading (a catastrophic test failure). The test range specifications in the ANSI N13.11 test are the dominant requirement at lower dose levels. Another issue is the conflicting goals of reporting only those exposure levels that can be identified as nonzero (i.e., above Lc) while retaining all data for future statistical analysis. Exposure levels below the LLD have a high probability of being identified as minimal (below Lc). Summing zero or minimal reported values provides no advantage for future evaluation. However, when multiple dosimeter readings are summed, the LLD of the net signal may decrease, improving the precision of the result. The sum could be applied for a class of workers over a short time period or for one worker over a long time period. Minimal readings for large numbers of workers may be valuable for future epidemiological studies. The ideal solution is to report “minimal” in individual exposure records while retaining the actual readings and quality assurance records separately.
CONCLUSIONS Estimates of the lower level of detection for a personnel dosimetry system may be easily calculated and monitored using routinely collected data and formulas derived assuming random variables and normal statistics. Performance testing results may be used to
January 1992, Volume 62, Number 1
estimate LLD values for the various testing categories. Under unusual circumstances, it may be advisable to test the results using multiple, escalating low-level exposures. The concept of the critical level may be used to control the number of zero (or minimal) dose-equivalent readings reported. A more rigorous definition of a positive reported reading may improve knowledge of exposures for radiation protection managers. This procedure implies that the dose-equivalent level at which a minimal report is made will vary with field conditions. A typical dosimetry system has a varying LLD as a function of the radiation environment. Reported LLD values should include specifications of particle type and energy, and mixture ratio under mixed field conditions. Acknowledgement(s)-The authors thank Dr. L. A. Cume and Dr. C. R. Hirning for their critical review of the manuscript and Ms. Gem Roberts for the preparation of the manuscript.
REFERENCES American National Standards Institute. Criteria for testing personnel dosimetry performance. New York ANSI; ANSI N13.11-1983; 1983. Bevington, P. R. Data reduction and error analysis for the physical sciences. New York: McGraw-Hill Book Company; 1969. Currie, L. A. Limits for qualitative detection and quantitative determination. Anal. Chem. 40: 586-593; 1968. International Commission on Radiation Units and Measurements. Radiation protection instrumentation and its applications. Bethesda: ICRU; ICRU Report No. 20; 1971. International Commission on Radiological Protection. General principles of monitoring for radiation protection of workers. New York: Pergamon Press; ICRP Publication No. 35; 1982. National Council on Radiation Protection and Measurements. Instrumentation and monitoring methods for radiation protection. Bethesda: NCRP; NCRP Report No. 57; 1978. Piesch, E. Application of TLD systems for environmental monitoring. In: Oberhofer, M.; Scharman, A., eds. Applied thermoluminescence dosimetry. Bristol: Adam Hilgar Ltd; 1981:197-228. Spacher, P. J.; Mis, F. J.; Klueber, M. R. Lower limits of detection for thermal luminescent dosimeters. Radiat. Prot. Manag. 7:44-50; 1990. U. S. Department of Energy. Department of Energy standard for the performance testing of personnel dosimetry systems. Oak Ridge, TN: National Technical Information Service; DOE/EH-0027; 1986. U. S . National Bureau of Standards. Experimental statistics, NBS handbook 9 1. Washington, DC: US. Government Printing Ofice; 1963.
APPENDIX The origin of the terms in eqn (14) is made clear by following the derivation in more detail. Starting
LLD for personnel dosimetry systems 0 P.L. ROBERSON AND R. D. CARLSON
k,ao LD =
+ 6pB‘ + [2kau06pLg’
+ (6PB’)* + k t a z + ( k 2 - k t ) C726]1’2 1-6
3
(2A)
where 6 = k,2 6;. The 6 value is typically much less than 1. Expanding the terms under the square root in powers of 6 yields
9
This formula simplifies to eqn (14) for k, = kp. The leading terms are due to the minimization of falsepositive readings (ko)and of false-negativereadings (kp). The remainder of the formula accounts for the usually small increase in the standard deviation at L D over the standard deviation of the background, represented here by the 6 parameter. From eqn (1 2),
3
Table 1. Notation for mathematical derivation. Dosimeter Control Total Net reader dosimeter dosimeter dosimeter noise (background) reading reading ~
True value Observed value Standard deviation Observed SD Critical level Detection level
~~
~~
~
PN
PB
PT
PH
N
HB
T
H
uN
UB
UT
-
SB
-
OH
SH
Lc LD 12
Signal
Critical Level (L,)-the signal that provides a confidence level of 1-a that the result is not due to a fluctuation of the background. All results with values less than L, should not be reported as detection of a positive signal. Results with values greater than L, should be reported as positive values and may be accompanied by a confidence interval. Detection Level (LD)-the delivered exposure for which the confidence level is 1-/3 that the result will be detected and properly reported as a (qualitative) positive result. This value may be identified as the LLD when expressed in units of dose equivalent.
Fig. 1. Schematic representation of the normal distribution for the background (or noise) signals and the signals representing a delivered dose (equivalent) of LD,the detection level.
For one dosimeter reading, these values are replaced by observed values: H = T - WB,
(2) where the background is determined by many ( n ) dosimeters and the bar denotes mean value. The variance of the net signal is (rH2
THEORY Some interpretation is required to convert a dosimeter reading to a dose-equivalent value. Necessary steps in the interpretation typically included subtraction of a background signal and application of an interpretation algorithm that may use multiple element readings from the same dosimeter. The initial development of the formalism for the LLD will assume a simple signal. More complicated systems will be discussed in the Applications section. The background signal to be subtracted from the dosimeter reading should include all signal sources not due to the radiation exposure to be determined. This may include reader noise, incidental signals, and background radiation. Examples of incidental signals are stray luminescence (e.g., residual deep traps) for thermoluminescence dosimeters or fog level for film dosimeters. The signal due to reader noise only is denoted pN,with associated standard deviation of uN. The background can be characterized using a mathematical mean (true value, p B ) and a standard deviation (uB). A background determination using n independent measurements (dosimeters) will result in an estimate of the mean with accompanying standard deviation of the mean (uB/&). The net dosimeter reading (true value, pH) is the difference between the total value (or raw signal) and the mean background value:
(arbitrary units)
=
UT2
+ uB2/n.
(3)
Note that uT is never smaller than uB. For a zero net signal (pH = 0, UH = (TO, UT = uB), uo2 = ug2 (1
+ l/n).
(4)
The critical level (L,) provides a confidence level of 1-a that the determined reading is not due solely to a fluctuation of the background (see Fig. l), Lc = k&O,
(5)
where k, is the abscissa of the standard normal distribution corresponding to a single-sided probability of 1a. Values greater than L, due solely to fluctuations in the background are errors of the first kind, or the incorrect rejection of the null hypothesis (false-positive) (USNBS 1963). Dosimeter readings below L, should not be reported as an indication of the detection of a positive signal. Not reporting a positive value for a true exposure is an error of the second kind, or the incorrect acceptance of the null hypothesis (false-negative). The exposure level received for which the probability of reporting a zero value (below L,) is no more than /3 is termed the detection level (LD), LD = L,
+
kpbD,
(6)
where kp is the abscissa of the standard normal distribution corresponding to a single-sided probability of 1/3, and where uDis the standard deviation of the readings at the exposure level LD.
Health Physics
4
The total signal (pT) is a combination of a signal due to the effect being measured ( p ~ )a,signal attributed to the dosimeter but not part of the effect being measured ( p a - pN), and a signal not associated with the dosimeter, labeled reader noise ( p ~ ) : PT
= pH
f PB
= (PH
f pB
- pN) + YN,
(7)
where the value in brackets is the signal from the dosimeter irrespective of origin. The standard deviation of the dosimeter signal is assumed to be a constant fraction of the signal level5(Piesch 1981). The standard deviation of the total signal, uT, may be split into two components, one dependent on the dosimeter signal level (pT - pN) and the other identified with the reader noise level ( p ~ ) , aT2
=
2 6,
x
(pT
- pN)2
+ cN2,
(8)
where a, is the constant value of the relative standard deviation for large signals. For the background readings, aB2
2
= cfi
x
(PB
- pN)2
+ cN2,
January 1992, Volume 62, Number 1 a,pg’ >> UN (low noise) and large n, ~ , p g ‘ and the LDsimplifies to 2 k 3 / ( 1- ka,). For finite sample sizes, = Hg - N replaces p g ’ , Student’s t values replace the k values, and sample standard deviations (S) replace a values. Usingn background dosimeter readings to calculate SOand HB’, and rn dosimeter readings of a known large dose (- 10 mGy) compared to background to calculate S,,
Ilk. For
ug
=
UO,
nB’
where tn (tm)is Student’s t statistic for n - 1 (rn - 1) degrees of freedom, and S, is the relative sample standard deviation for rn - 1 degrees of freedom. A typical confidence level for t is 95%. Necessary formulas include the following (USNBS 1963): m
s,2= (H) - I C (rn - 1) i=l
=
‘T,
2
x
(pT
- pN)2
+
(TN2
n=’.Z Hi rn i=1
+ cB2/n.
(10)
For p~ = p~ + pg from eqn (l), expanding and collecting terms and using eqn (9), aH2
6; x [pH2 + 2pH x
(16)
(9)
and, from eqns (3) and (8), for general readings, aH2
(Hi - W)2
1
n
=
(pB
- pN)] +
aB2
(1
+ 1/n)* (1 1)
The variance for the detection level (pH = LD) from eqn (1 1) is ~ T D =~0;
X
(LD2+ 2 LD p g ’ )
+ 0o2,
(12)
where pB’ pB - p ~Solving . for CTDin eqn (6), inserting eqn ( 5 ) for L,, squaring, setting the result equal to eqn (12), and collecting terms yields (1 - k g):a
x LD2- [2 (k, a0 + kp2:a
pB’)]
x LD + (km2- k t ) x go2 = 0.
(13)
where the (Hg)i represents background readings and the Hi represents known dose readings. The reader noise may be measured (or adjusted to zero when level possible) before dosimeter readout. The n appearing in the equation for So represents the number of background dosimeters routinely used to determine the background subtraction level and may be different from the number of dosimeters used to calculate SB.
(w)
For k, z kDz k, LD
= 2 (k
UO
+ k2
pg’)
1 - k2 c;
In nearly all cases, the dominant term in the expression for LD is 2kao. Since k at 95% confidence for a singlesided distribution is 1.7, LD (at first approximation) is 3.4 times the standard deviation of the background. The denominator factor contributes significantly for a large relative standard deviation. The second term in the numerator is significant for large values of the background. The second term and denominator factor represent small corrections to 2kao unless a, approaches § The dominant error is in the readout process, which affects the full signal in the same manner irrespective of signal magnitude.
APPLICATIONS Applying the LD formula, eqn (15), may require special handling of the data. Implicit in its derivation is the assumption that the units for the background readings are the same as are quoted for the positive readings, presumably in units of dose equivalent. Alternatively, the readings may be expressed in other units (e.g., reader units) and the LD level derived for the measurement system in those units. Then the LLD may be defined as the LD multiplied by the conversion factor to dose equivalent. For example, with an energy-dependent dosimeter, one value of LDmay correspond to various values of LLD, depending on the energy calibration factor.
LLD for personnel dosimetry systems 0 P. L. ROBERSON AND R. D. CARLSON
Multiple-element dosimeters An analysis algorithm is typically applied to multiple-element dosimeters. Several components may be analyzed to ultimately result in interpretations of the penetrating and superficial dose equivalent values. A typical analysis may separate the photon, beta, and neutron dosimetry in order to apply different calibration factors before performing the final sum. In this situation, application of the LD formula in the simple form requires that the analysis algorithm be a function of the element readings without discontinuities and that the background dosimeters be evaluated with the full algorithm. The background dosimers should be analyzed as though they were field dosimeters. For example, even if the background dosimeters are known to receive (at most) photon irradiation from the environment, they should be analyzed for beta and neutron exposure as well. The LLD will be greater if there is a possibility of radiations other than photons. Algorithms for most beta and neutron dosimeters routinely subtract the signal due to photon exposures before applying the calibration factor. When evaluating the background dosimeters for the mean (RB) and standard deviation (SB)values, both negative and positive values must be retained. They represent fluctuations, present at positive exposure levels, which are due to the uncertainty in the background. (Note: while a value, if present, is technically required negative by the LD formula, its presence probably indicates a miscalibrated dosimetry system. The importance of retaining negative values is for the calculation of SB.) Here, the LLD represents the beta or neutron dose equivalent that results in a positive report if photon exposures do not exceed background levels. This condition will not be satisfied under mixed-field conditions (see below). Another consideration is the relationship between the LLD and the energy dependence of the dosimeter. For example, the value of the LD (in reader units) may be independent of the exposure energy. However, application of an energy-dependent calibration factor, applied to calculate the LLD in dose equivalent, results in an energy-dependent LLD. For most dosimetry systems, the LLD associated with each type and energy (or energy range) of radiation should be determined.
wB’
Mixed fields Many radiation field environments are composed of more than one type of radiation. For example, neutron sources are rarely present without significant associated gamma and/or x-ray sources. Likewise, it is unusual for a significant beta-radiation source to be present without some gamma radiation. It may be more important to determine the LLD in real mixed fields than to record the LLD for optimal conditions of photon-only sources. The above formulation for the lower level of detection cannot be directly applied to mixed-field conditions where the ratios of the components of the total
5
dose equivalent (e.g., beta/gamma ratio) are unknown. Given a calculational algorithm for the interpretation of the signal levels, one can define an LD expressed in terms of signal level. If the calibration factor is dependent on the ratio of the components of the total dose equivalent, the LLD will similarly be component ratio dependent. The result is an LLD dependent on workplace conditions. There are several dosimeter design strategies used for the determination of dose equivalent in mixed fields. These include dosimetry systems in which: 1) the components are measured separately and summed after the conversion to dose equivalent (e.g., separate gamma and neutron dosimeters, where the neutron dosimeter is insensitive to gamma exposure and vice versa); 2) the components are measured separately but the dosimeter element is sensitive to both radiation types so that the background due to a mixture component must be subtracted before the dose equivalent calibration conversion can be performed (e.g., a thermoluminescence albedo neutron dosimeter); 3) the same dosimeter elements are used to detect both components and a calculational algorithm is used to separate the components prior to the conversion to dose equivalent (e.g., a beta/ photon dosimeter using shallow and deep elements to separate beta, x-ray, and gamma-ray components). These configurations will be discussed in turn. In the first case above, an LLD can be determined using the LD equation for each component separately as illustrated in Fig. 2. However, a combined LLD would (at best) be dependent on the mixture ratio. The usefulness of the concept is questionable in cases where the separate LLDs are significantly different and the mixture ratio is not unique. In the second case, the LLDs may also be determined for the components separately, but since the measured background consists of signal from both types of radiation, the large-dose standard deviation, ( T ~ , is increased. This implies that the LLD level is dependent on the mixture ratio present, but in a predictable manner for each dosimeter design. An LLD for the combined signal remains questionable. For the dosimeter system in the third case, the LLD for each mixture component is a function of the component ratio. For the example of a beta/x-ray/ gamma dosimeter, the deep-dose component may be handled as a single-component dosimeter. The betadose LLD may be a function of both the beta/photon mixing ratio and the beta energy. The LD, expressed in “reader units,” is typically independent of beta energy because the reading of only the shallow-depth element is used. All of the beta energy dependence is in the beta calibration factor, whether determined by site measurement or dosimeter element ratios. Thus, the LD can be expressed as a function of the beta/photon signal ratio. Assuming that the background is not varying, the go value is independent of the ratio present in the work environment. The CT,, value is a function of the ratio because the higher the ratio, the greater the relative
Health Physics
6
1.2
January 1992, Volume 62, Number 1
I
30 (a)
0
10
Signal
20
30
(arbitrary units)
Fig. 2. Schematic representation of the normal distributions
for a two-component dosimeter, each component with a different Lo when measured alone.
8-
r 0
6-
=
Q)
0-
E
U.
difference of measured values for the dosimeter elements. Thus, the high-dose limit of the standard deviation must be determined for a range of beta/photon ratios. Note that as the mixture approaches an exclusively photon dose, determination of the beta dose is increasingly difficult because it depends on the difference of large numbers. The zero-dose standard deviation becomes large and the beta-dose LDbecomes large. The beta-dose LLD may be expressed as a function of beta energy by multiplying by the energy-dependent calibration factor (if applicable). The LLD for the combined beta plus photon shallow dose equivalent is not easily derived due to the expected difference in LLD for the separate components. However, for a given radiation field composition (mixture ratio and energies) the LLD is calculable. Performance testing using a limited number of dosimeters An estimate of the LLD achievable under test conditions can be obtained from an analysis of background dosimeters and the results of NVLAP or DOELAP performance testing. At least 15 background dosimeters handled similarly to those used for routine dosimetry should be used for the calculation of R'Band SO.It is important to calculate the values for the background dosimeters using the assumptions for each field condition or test category considered. The S, value can be estimated from the test results for each test category using the relative bias (B) and the relative standard deviation (S) (ANSI 1983): S s, = B + 1'
4-
2-
n , . . . . .
" ? " Y ? ? " ? ? ' " ? Y 9 " ? U
P
u
o
P
P
P
l
m
Reading
m
m
m
m
m
m
m
m
m
(mGy)
Fig. 3. Dosimeter readings for 154 unirradiated dosimeters (a) and 30 I3'Cs-irradiated dosimeters (b). Histograms of frequency of reading vs. reading are presented.
Values of the LLD for single-field categories may be derived using eqn ( 1 5). The calculation of LLD for mixed-field categories is dependent on dosimeter design, and use of eqn (1 5) may not be appropriate. It should be the goal of dosimetrists to maintain the LLD well below the lower limit of the testing ranges in ANSI N13.11 or the DOE testing standard. An LLD value less than half the lower limit is reasonable. For example, for an LLD value of half the lower limit of the testing range and a U, value near 0.1, eqns (1 1) and (14) predict a relative standard deviation at the lower limit of the testing range of approximately twice the ufl value. A factor of 2 increase in the standard deviation is acceptable considering the low number of test dosimeters assigned dose equivalents near the lower limit of the testing range. Example The calculation of the LLD is illustrated using data submitted by a DOELAP participant. Readings from 154 unirradiated dosimeters and 30 dosimeters irradiated to a large dose (Fig. 3) were used to estimate the LLD.
LLD for personnel dosimetry systems 0 P. L. ROBERSON AND R. D. CARLSON
I
Table 2. Critical value and detection level for sample data. (a1
*
Calculation method
-
Data points Normal curve Log normal curve
2o
Data Normal distribution Lognormal distribution
C
Lt (mGy) 0.066 0.057 0.071
LO( ~ G Y ) 0.1 12 0.115 0.1 14
a
U
!?
10
0 0.0
0.1
Reading
LL 2
J
0.2 (mGy)
0.3
0 1
A
o ! 4
5
Reading
6
(mGy)
Fig. 4. Calculated normal and lognormal curves compared to dosimeter readings. (a) Unirradiated dosimeters; (b) '37Cs-
irradiated dosimeters.
The distribution of readings from the unirradiated dosimeters does not obey a normal distribution. Fig. 4a shows the data and fitted curves using normal and lognormal distributions. The lognormal distribution provides a better fit to the data. Values for LC were derived for the data set, the normal curve, and the lognormal curve (Table 2). The value derived from the data set is the difference between the dose that exceeds 95% of the data points and the average dose (RB), multiplied by the (1 1/12) factor. For purposes of this analysis, it was assumed that 30 background dosimeters are used for the routine background subtraction (i.e., a 1/12] factor of 1.03). The LC value from the [l lognormal distribution gives better agreement with the value derived from the data set than with the value derived from the normal distribution, reflecting the better fit to the data set for the lognormal curve. Some of the discrepancy between the normal-distribution value and the data set value observed for LC is compensated during the calculation of L D by the
+
+
asymmetric distribution of the data. The 2tSo term represents the upper half of the background distribution plus the lower half of the LD distribution. This concept is developed more fully in the Appendix. The data extend above the mean value much more than below. The last column in Table 2 contains the LD values calculated using the upper and lower parts of the curves for the calculation of the 2tS0 term, compared to the calculation using the normal and lognormal distribution. Differences are relatively small compared to the uncertainty of the LDvalue calculated from the data set (an uncertainty of 14% from distribution-free tolerance limits at 95% confidence). Thus, the assumption of normal statistics does not greatly influence the result. The data set for the high-dose irradiation is consistent with the fitted normal curve (Fig. 4b). The derived relative standard deviation is reliably represented. The input values for eqn (1 5 ) and the derived LD value are given in Table 3. In this case, as in most, all but the leading term (2tS0) are small.
DISCUSSION The lower limit of detection has not received much attention as applied to personnel dosimetry. The lower dose levels were indirectly emphasized by the specifications in ANSI N13.11 (ANSI 1983)through the lower limit of the testing range. The lower ranges were determined with the goal of achieving a reasonably low dose measurement capability so that multiple dosimeters summed to provide annual and lifetime cumulative doses would retain sufficient precision. The exception was for the more difficult categories (beta, neutron, and mixtures) where higher levels were chosen consistent with prevailing technology. The DOE performance testing standard (USDOE 1986) included a requirement for the calculation of the LLD without a performance requirement. For ease of use, a formula similar to eqn ( 14) is specified. More recently, Spacher et al. (1990) used increasing exposure levels and normal statistics to determine the LLD, while Hirning" applied a formula derived from the approach used in the DOE standard to describe the determination of the LLD for operating dosimetry systems. The ANSI N13.11 specification of a performance criteria constant with exposure level contrasts with the recommendations of national and international organizations. Recommended accuracies are within a factor )J Hirning, C . R. Detection and determination limits for thermoluminescence dosimetry. Submitted to Health Physics.
Health Physics
8
Table 3. Derived Darameter values for samde case. Inout
Value
0.165 mGy 0.033 mGy 5.07 mGy
R E
SE
R
0.044
S“ ~
Outout
Lc LDW D )
Value 0.057 mGy 0.115 mGy
of 3 at levels less than one-tenth the maximum permissible levels (ICRU 1971; NCRP 1978) or within a factor of 2 at 95% confidence at levels below 1 rem (ICRP 1982). The ANSI N13.11 specification was determined from the desire to keep the test as simple as possible and the observation that the relative precision did not vary greatly with dose level (ANSI 1983). This effect is easily understood from eqn (15). With the standard deviation of the background sufficiently low for the LLD to be well below (less than one-half) the lower limit of the testing range, the signal-dependent term surpasses the background term at two to three times the lower limit of the test range (0.5 to i .O mGy for the photon categories). The relatively small increase in standard deviation at the lower levels would be difficult to observe. Alternatively, if the LLD is at or above the lower limit of the testing range, there is an unacceptably large risk of reporting a test dosimeter as a zero or minimal reading (a catastrophic test failure). The test range specifications in the ANSI N13.11 test are the dominant requirement at lower dose levels. Another issue is the conflicting goals of reporting only those exposure levels that can be identified as nonzero (i.e., above Lc) while retaining all data for future statistical analysis. Exposure levels below the LLD have a high probability of being identified as minimal (below Lc). Summing zero or minimal reported values provides no advantage for future evaluation. However, when multiple dosimeter readings are summed, the LLD of the net signal may decrease, improving the precision of the result. The sum could be applied for a class of workers over a short time period or for one worker over a long time period. Minimal readings for large numbers of workers may be valuable for future epidemiological studies. The ideal solution is to report “minimal” in individual exposure records while retaining the actual readings and quality assurance records separately.
CONCLUSIONS Estimates of the lower level of detection for a personnel dosimetry system may be easily calculated and monitored using routinely collected data and formulas derived assuming random variables and normal statistics. Performance testing results may be used to
January 1992, Volume 62, Number 1
estimate LLD values for the various testing categories. Under unusual circumstances, it may be advisable to test the results using multiple, escalating low-level exposures. The concept of the critical level may be used to control the number of zero (or minimal) dose-equivalent readings reported. A more rigorous definition of a positive reported reading may improve knowledge of exposures for radiation protection managers. This procedure implies that the dose-equivalent level at which a minimal report is made will vary with field conditions. A typical dosimetry system has a varying LLD as a function of the radiation environment. Reported LLD values should include specifications of particle type and energy, and mixture ratio under mixed field conditions. Acknowledgement(s)-The authors thank Dr. L. A. Cume and Dr. C. R. Hirning for their critical review of the manuscript and Ms. Gem Roberts for the preparation of the manuscript.
REFERENCES American National Standards Institute. Criteria for testing personnel dosimetry performance. New York ANSI; ANSI N13.11-1983; 1983. Bevington, P. R. Data reduction and error analysis for the physical sciences. New York: McGraw-Hill Book Company; 1969. Currie, L. A. Limits for qualitative detection and quantitative determination. Anal. Chem. 40: 586-593; 1968. International Commission on Radiation Units and Measurements. Radiation protection instrumentation and its applications. Bethesda: ICRU; ICRU Report No. 20; 1971. International Commission on Radiological Protection. General principles of monitoring for radiation protection of workers. New York: Pergamon Press; ICRP Publication No. 35; 1982. National Council on Radiation Protection and Measurements. Instrumentation and monitoring methods for radiation protection. Bethesda: NCRP; NCRP Report No. 57; 1978. Piesch, E. Application of TLD systems for environmental monitoring. In: Oberhofer, M.; Scharman, A., eds. Applied thermoluminescence dosimetry. Bristol: Adam Hilgar Ltd; 1981:197-228. Spacher, P. J.; Mis, F. J.; Klueber, M. R. Lower limits of detection for thermal luminescent dosimeters. Radiat. Prot. Manag. 7:44-50; 1990. U. S. Department of Energy. Department of Energy standard for the performance testing of personnel dosimetry systems. Oak Ridge, TN: National Technical Information Service; DOE/EH-0027; 1986. U. S . National Bureau of Standards. Experimental statistics, NBS handbook 9 1. Washington, DC: US. Government Printing Ofice; 1963.
APPENDIX The origin of the terms in eqn (14) is made clear by following the derivation in more detail. Starting
LLD for personnel dosimetry systems 0 P.L. ROBERSON AND R. D. CARLSON
k,ao LD =
+ 6pB‘ + [2kau06pLg’
+ (6PB’)* + k t a z + ( k 2 - k t ) C726]1’2 1-6
3
(2A)
where 6 = k,2 6;. The 6 value is typically much less than 1. Expanding the terms under the square root in powers of 6 yields
9
This formula simplifies to eqn (14) for k, = kp. The leading terms are due to the minimization of falsepositive readings (ko)and of false-negativereadings (kp). The remainder of the formula accounts for the usually small increase in the standard deviation at L D over the standard deviation of the background, represented here by the 6 parameter. From eqn (1 2),
