Detection Sensitivity - MDA and LLD - Ludlum Measurements Inc.

Ludlum Measurements, Inc. frequently receives calls from people who perform environmental monitoring with questions regarding release limits and the method for determining them. While Ludlum is not the source authority on regulatory limits or instructions on how to perform environmental monitoring, calculation of detection limits does depend upon the inherent characteristics of the instrument used to perform the monitoring. Two commonly used detection limits are MDA (minimum detectable activity) and LLD (lower limit of detection).

Introduction

The most prevalent detection limit is the MDA. A good definition of MDA as used in the field of nuclear measurements is the smallest amount of activity distinguishable from background which can be quantified at a given confidence level (which in the nuclear industry is nearly always 95%). MDA is an a priori (before the fact) calculation designed to give an indication of the basic capabilities of a counting system. It has been referred to as an "advertising level". Another way of thinking about it is a "minimum quantifiable activity". Just because a result is less than the calculated MDA does not mean activity distinguishable from background is not present; it just means that there is not enough to quantify an activity with great precision. MDA is a characteristic of a counting system and is not specific to an individual sample or measurement.

Another closely related detection limit is the LLD. There are many different (and conflicting) definitions for LLD, just as there are for MDA. Possibly the best way of distinguishing the two is to think of LLD as the MDA with sample-specific parameters added. For example, MDA would be the appropriate term in performing measurements of surfaces or wipe samples since the detection limit will remain the same for all of them. LLD would apply to counting air samples since it can be different for each sample due to differences in flow rate or collection time.

Equations for Calculating MDA

In this section on MDA, a 95% confidence level (which is standard in the nuclear industry) is used in all equations and calculations. The equations are given in several similar forms for what we hope will be clarity. If you are still not sure which one is appropriate, several examples are provided at the end to help you find the equation that applies to your specific application.

If you've made it this far into the article, you've probably seen at least a couple of different equations for calculating MDA. In truth they all will give nearly the same results and are in most cases mathematically identical. The best all-encompassing form of the MDA equation is:

\[\mathsf{MDA=\frac{2.71+3.29\sqrt{R_b t_s[1+\frac{t_s}{t_b}]}}{(t_s)(E)}\tag{Equation 1}}\]

where:

MDA = minimum detectable activity in dpm
R_b = background count rate in cpm
t_s = sample counting time in minutes
t_b = background counting time in minutes
E = detector efficiency in counts per disintegration

In the case where the sample counting time is the same as background counting time, the equation can be simplified to:

\[\mathsf{MDA=\frac{2.71+4.65\sqrt{R_b t}}{(t)(E)}\tag{Equation 2}}\]

where:

MDA, R_b, and E are as defined above
t = sample and background counting time in minutes

In many instances we want the MDA to be in terms of a specific area. For samples such as smears for removable contamination, the above equations are understood to be in terms of dpm per smear. For surface measurements, we usually want the MDA in terms of dpm per 100 cm². For this, an additional factor can be included in the denominator to adjust the MDA for the area of the detector you are using:

\[\mathsf{MDA=\frac{2.71+3.29\sqrt{R_b t_s[1+\frac{t_s}{t_b}]}}{(t_s)(E)(\frac{A}{100})}\tag{Equation 3}}\]

where:

MDA, R_b, t_s, t_b and E are defined as above
A = detector area in cm²

and

\[\mathsf{MDA=\frac{2.71+4.65\sqrt{R_b t}}{(t)(E)(\frac{A}{100})}\tag{Equation 4}}\]

where:

MDA, R_b, E, and A are as defined above
t = sample and background counting time in minutes

Using MDA Equations with Ratemeters

Determining MDA for instruments that count a sample or surface for a preset time is relatively straightforward using the above equations. Sometimes people get stuck, however, in calculating MDA for a ratemeter because there is no clear sample and background counting "time". Fortunately, it has been accepted that in ratemeter applications, twice the instrument's time constant can be substituted for the sample and background counting times.

An instrument's time constant is not the same as an instrument's response time, although they are related. The response time is the value we are most familiar with, and it is defined as "the time interval required for the instrument reading to change from 10% to 90% of the final reading (or vice versa) following a sudden significant change in the radiation field at the detector." Another way of thinking about response time is the amount of time over which the currently indicated value is averaged.

The time constant, on the other hand, is defined as "the time involved in the charging or discharging of an inductor or capacitor; one time constant is the length of time required to reach 63% of the full charge or discharge." As described our three-part series “Determining Ratemeter Time Constants for MDA Equations”, the time constant can be approximated by multiplying the response time by 0.44. The calculated time constant, however, will be in seconds, so you must divide that number by 60. Alternately, to approximate from response time (in seconds) directly to time constant (in minutes) in a single step, divide the response time by 0.00733.

The table below contains the relevant values for Ludlum's analog ratemeter instruments. The response time (and thus the time constant) of digital ratemeters varies with the current count rate. Refer to the articles listed above to determine the response time for digital ratemeters.

Instrument Model	Response Time	Time Constant	Value to Use as "t" in MDA Calculations (twice the time constant)
177	Fast = 2.2 s	0.0161 min.	0.0323 min.
177	Slow = 22 s	0.161 min.	0.323 min.
3 12 14C 16 18 2221	Fast = 4 s	0.0293 min.	0.0587 min.
3 12 14C 16 18 2221	Slow = 22 s	0.161 min.	0.323 min.

MDA Examples

We offer the following four examples as typical of actual conditions, illustrating calculation of the MDA with the equations published in Part I. (The equation numbers referenced refer to the equations as numbered in Part I.)

Example 1

You are using a Model 1000 with a 43-32 alpha detector to count wipes for removable ²³²Th contamination. You count background for one hour and count each wipe for one minute. The background count over the hour is 57 counts (or 0.95 cpm) and the efficiency for ²³²Th is 28% (or 0.28 counts per disintegration). Using Equation 1,

\[\mathsf{MDA=\frac{2.71+3.29\sqrt{(0.95\;cpm)(1\;min)[1+\frac{1\;min}{60\;min}]}}{(1\;min)(0.28\;counts/disintegration)}=21.2\;dpm\;per\;wipe}\]

Example 2a

Using the same conditions as in Example 1, except that both the background and the wipes are counted for one minute. Using Equation 2,

\[\mathsf{MDA=\frac{2.71+4.65\sqrt{(0.95\;cpm)(1\;min)}}{(1\;min)(0.28\;counts/disintegration)}=25.9\;dpm\;per\;wipe}\]

Example 2b

You are using a Model 3 with a 44-9 GM pancake detector to screen wipes for ⁶⁰Co. The Model 3 does not have a scaler options, so you are "counting" the wipes using slow response. The background in the area is 60 cpm, and the ⁶⁰Co efficiency you are using is 18% (0.18 counts per disintegration). From the table published in Part I, the value to use for sample and background counting time is 0.323 minutes. Therefore, using Equation 2,

\[\mathsf{MDA=\frac{2.71+4.65\sqrt{(60\;cpm)(0.323\;min)}}{(0.323\;min)(0.18\;counts/disintegration)}=399\;dpm\;per\;wipe}\]

Example 3

You are using a Model 3 (which has had the scaler option installed) to quantify ¹⁴C contamination on a bench top. You use the scaler to count the background for 10 minutes and get 422 counts (or 42.2 cpm). You use the ratemeter function on fast response to measure activity on the bench top. Efficiency for ¹⁴C is 4.3% and the area of the detector is 15 cm². From the table published in Part I, the value to use for sample counting time is 0.0587 min. Using equation 3,

\[\mathsf{MDA=\frac{2.71+3.29\sqrt{(42.2\;cpm)(0.0587\;min)[1+\frac{0.0587\;min}{10\;min}]}}{(0.0587\;min)(0.43\;counts/disintegration)(\frac{15\;cm^2}{100})}=20,874\;dpm\;per\;100\;cm^2}\]

Example 4a

Using the same conditions as in Example 3, except the background and the surface will be counted for two minutes using the scaler option. Using Equation 4,

\[\mathsf{MDA=\frac{2.71+4.65\sqrt{(42.2\;cpm)(2\;min)}}{(2\;min)(0.043\;counts/disintegration)(\frac{15\;cm^2}{100})}=3,522\;dpm\;per\;100\;cm^2}\]

Example 4b

Using the same conditions as in Example 3, except the background and the surface will be determined using only the ratemeter function on slow response. From the table published in Part I, the value to use for time is 0.323 min. Using Equation 4,

\[\mathsf{MDA=\frac{2.71+4.65\sqrt{(42.2\;cpm)(0.323\;min)}}{(0.323\;min)(0.043\;counts/disintegration)(\frac{15\;cm^2}{100})}=9,541\;dpm\;per\;100\;cm^2}\]

Equation for Calculating LLD

The equation for calculating LLD for air samples is similar to the MDA equations, but includes additional factors in the denominator:

\[\mathsf{LLD=\frac{2.71+3.29\sqrt{R_b t_s[1+\frac{t_s}{t_b}]}}{(t_s)(E_d)(E_f)(FF)(SAF)(Vol_{cc})(2.22 x 10^6)}}\]

where:

LLD = lower limit of detection in µCi/ml
R_b = background count rate in cpm
t_s = sample counting time in minutes
t_b = background counting time in minutes
E_d = detector efficiency in counts per disintegration
E_f = filter efficiency
FF = fraction of filter counted
SAF = self absorption factor
Vol_cc = air sample volume in cc (or ml)
2.22 x 10⁶ = factor to convert dpm to µCi

Consider the following example for use of the LLD equation:

Example LLD

You are going to be counting air samples for ²³²Th using a Model 2224 with a 43-1 alpha scintillation detector. The entire air sample filter fits exactly underneath the detector so the filter fraction will be 1. The air sample will be drawn at 20 lpm (20,000 ml per minute) for 2 hours (120 minutes) making the total sample volume 2.4 x 10⁶ ml. The background was counted for 24 hours (1,440 minutes), and the air samples will be counted for one hour (60 minutes) each. The alpha self-absorption factor for these glass-fiber filters is 0.9, and the filter efficiency is 99.97% (0.9997). The detector efficiency is 23.4% (0.234). Using the equation above:

\[\mathsf{LLD=\frac{2.71+3.29\sqrt{(0.43\;cpm)(120\;min)[1+\frac{120\;min}{1,400\;min}]}}{(120\;min)(0.234\;counts/dis)(0.9997)(1)(0.9)(2.4\;x\;10^6 \;ml)(2.22\;x\;10^6)}=2.03\;x\;10^{-13}\;µCi\;per\;ml}\]

References

Gollnick, D.A., Basic Radiation Protection Technology, 3rd Edition. Altadena, CA; Pacific Radiation Corporation; April 1994.

American National Standard Performance Specifications for Health Physics Instrumentation - Portable Instrumentation For Use In Normal Environmental Conditions. New York: Institute of Electrical and Electronic Engineers: ANSI N42.17A-1989.

Berger, J.D., Manual for Conducting Radiological Surveys in Support of License Termination (NUREG/CR-5849) June 1997.

Strom, D.J. and Stansbury, P.S., Minimum Detectable Activity when Background is Counted Longer Than the Sample, Journal of the Health Phys. 63(3):360-361; 1992.