Radon (Rn-222) is a rare gas with radioactivity. It is derived from uranium decay series and is widely present in nature. In particular, radon is enriched in basements and mines, where air circulation is commonly poor and the environment is surrounded by soils or rocks that contain high concentrations of uranium (∼thousands of Bq/m³) (Cevik et al., 2010; Cevik et al., 2011; Somlai et al., 2011; Meng et al., 2015; Fijałkowska Lichwa, 2016; Sahu et al., 2016; Sun et al., 2020; Kojo et al., 2021). Once radon is inhaled, the alpha particles emanating from the radon and its daughters inevitably deposit energy in the lungs, thereby causing radiation damage to the lung tissues (Black et al., 1968; Blanchard et al., 1969; Wahl et al., 2000; Momcilovic et al., 2001; Pillalamarri et al., 2013). The cumulative effects of such radiation damage from long-term living or working in a high radon environment greatly increase the risk of lung cancer (WHO, 2009). This is arguably the leading reason why the probability of lung cancer among miners is much higher than that of the general public. Accordingly, the World Health Organization (WHO) has listed radon as a class I carcinogen and identified it as the second leading cause of lung cancer, next only to smoking.

It is thus desirable to determine the potential alpha energy exposure (PAEE) of individuals with a history of substantial radon exposure, e.g., miners, so that retrospective assessments of their health conditions can be performed in a timely manner. Currently, the dominant method of reconstructing PAEE relies on two factors, namely radon concentration onsite and the estimated duration of exposure (UNSCEAR, 2019; Tomasek, 2020). For example, the working level month (WLM) represents the cumulative exposure of an individual working 170 h in total in an environment with an emission of 1.3 × 10⁵ MeV potential alpha energy per liter from short-lived daughters of radon (Black et al., 1968; Blanchard et al., 1969). However, this method suffers from large uncertainties because radon activity in the environment often strongly fluctuates over time and locations, the human respiration rate may differ significantly among individuals, and residence time in a high radon environment often depends on subjective recollection, which may considerably deviate from reality (Laurer et al., 1993).

In contrast, reconstructing the cumulative radon intake of a tested person based on the direct measurement of Pb-210 activity in the living skull is a very promising, cutting-edge method. Its scientific basis can be briefly described as follows: Pb-210 is a relatively stable nuclide formed after short-term decay of Rn-222, and its half-life is ∼22 years, as shown in Figure 1. Since lead is a chemical congener of calcium, its biodynamic characteristics in the human body are very similar to those of Ca (ICRP, 1993; Dantas et al., 2006); hence, the Pb-210 formed from the decay of radon that has been inhaled shall ultimately transport to the bones and distribute evenly among them. In other words, the bones can act as a good “dosimeter” (Laurer et al., 1999), and the activity of Pb-210 contained therein directly reflect individual’s cumulative exposure to radon. Among the many bones, the skull has the largest surface area (Estrada et al., 1993), accounting for approximately 13–15% of the total bone mass (ICRP, 1975; Wahl et al., 2000). Moreover, its upper epidermis is relatively thin, and the difference between individuals is small. Therefore, it is the best choice for Pb-210 activity measurement in vivo (Cohen et al., 1977).

Admittedly, there are indeed potentially large uncertainties underlying this approach, because there can be assorted sources contributing to the Pb-210 activity in the skull other than radon (Laurer et al., 1999). However, we are strong confident that this approach promises great potential for the following three reasons: Firstly, the associated uncertainty can be reasonably suppressed through carefully setting up the control group for measurements; secondly, the Pb-210 activity originating from other sources becomes trivial comparing to radon when the person of interest had been subject to substantial radon exposure (see Discussion for more details). Thirdly, in many cases, the Pb-210 in vivo measurement could be the only viable measure to reconstruct one’s radon intake, as the radon concentration may not be available.

In 1968, Eisenbud et al. first verified the feasibility of this idea by directly measuring the characteristic γ-rays emitted by Pb-210 in the living skull of miners (Eisenbud et al., 1969). Following this pioneering study, the technique of Pb-210 in vivo measurements has attracted considerable interests from researchers all around the world and enjoyed continued advancement in the past 50 years (Cohen et al., 1973; Cohen et al., 1977; Zheng et al., 1992; Estrada et al., 1993; Laurer et al., 1993; Scheler et al., 1998; Wahl et al., 2000; Muikku et al., 2003; Dantas et al., 2006; Dantas et al., 2007; Pillalamarri et al., 2013). It is generally consent that the detection of Pb-210 in vivo is extremely subject to the background interference because the activity of Pb-210 in the skull is often very low (∼10 Bq) (Wahl et al., 2000; Dantas et al., 2007), the branching ratio of γ decay is as low as 4.25%, and the energy of released γ-rays is only 46.5 keV, thereby characterizing poor penetrating capability. Consequently, researchers often adopt the limit of detection to assess the sensitivity of their measuring devices. Nonetheless, throughout the literature, researchers have defined the limit of detection in a variety of ways (Altshuler et al., 1963; Isenhour et al., 1964; Kirkpatrick et al., 2013; Done et al., 2016), which severely compound the interstudy comparison and obscure how various factors make their impacts. To unravel this challenge, we resort to the concept of minimum detectable activity (MDA) defined by Currie (Currie, 1968) and reanalyze the data reported in the literature. The results presented in this work clearly demonstrate how key factors, such as detector’s active area and energy resolution, background level, and measurement time, influence MDA and thus shed light on the future direction for further improvement.

The rest of this article is organized as follows: in MDA Derivation, we demonstrate the derivation of MDA and identify the critical factors therein; in Results and Discussion, we present the key figures from the literature and reanalyze the reported data; we make concluding remarks in Conclusion.

The derivation of MDA was described in detail in Currie’s classical article (Currie, 1968), which we briefly recapitulate below. We note that the MDA_ISO defined by the ISO standard has also been widely used in many studies (ISO, 2010). Comparing with Currie’s MDA, the ISO standard takes the uncertainty in the calibration factor into account. In this regard, MDA_ISO is admittedly more rigorous. Nonetheless, there are two reasons for adopting MDA_Currie in our study: First, among the studies focusing on Pb-210 in vivo measurement, the uncertainty in the calibration factor was mostly not considered and thus not documented in the publication. Consequently, it become impractical to employ MDA_ISO in our study. Second, it has been pointed out by Kirkpatrrick that two definitions of MDA yield quite similar results when the uncertainty in the calibration factor is smaller than 10% (Kirkpatrrick et al., 2013). Because it is common to employ an anthropomorphic phantom enclosing Pb-210 with activity up to 5,000 Bq for calibration, the resultant uncertainty in the calibration factor should be fairly small. Hence, we consider two definitions of MDA should lead to essentially the same results.

Let’s imagine a general scenario, where a surveyor needs to rely on a certain measurement system to detect radiation from a sample with unknown radioactivity. First, the surveyor needs to determine the background level under the measurement conditions. Because the processes of nuclear decay, particle transportation, and radiation detection all characterize random features, the results of each measurement differ statistically. By measuring the background several times with equal measurement duration of t _b , the arithmetic average of multiple measurement results and their variance can be computed, which are denoted μ ¯ b and σ μ ¯ b , respectively. With these statistical samplings, one can consider the average background level is approximately μ ¯ b and calculate its standard deviation σ b as follows: σ b = μ ¯ b + σ μ ¯ b 2 (1) where it is assumed that the background statistics obey the Poisson distribution, and the principle of error propagation is used.

In the next step, the surveyor needs to perform measurements under the presence of the sample with measurement duration of t _s and analyze if it contains radioactivity. Two types of errors may occur. The first type of error is misjudging that the sample is radioactive whereas in reality it isn’t. The critical level L _C is a threshold set to avoid such errors under a certain confidence level. To be more specific, it equals to the upper limit of the background count under the confidence of 1−α, as shown in Figure 2, and can be expressed as below: L C = μ ¯ b + k 1 − α σ b (2) where k _1-α represents the abscissa corresponding to the confidence interval 1−α on the standard normal distribution. The second type of error is misjudging that the sample is not radioactive whereas in reality it is. The detection limit L _D is a threshold set to avoid such errors under a certain confidence level. In the similar manner, it equals to the lower limit of the total count (including background and sample) under the 1−β confidence level, as shown in Figure 2, and can be expressed as follows: L D = ( μ S + μ ¯ b ) − k 1 − β μ S + σ b 2 (3) where μ _s represents the actual net count resulted from the sample, and k _1-β has the same meaning as k _1-α . Setting μ _s = L _D and α = β leads to the following equation: L D = k 1 − α 2 + 2 k 1 − α μ ¯ b + σ μ ¯ b 2 (4)

Finally, MDA is defined by converting the decision limit L _D through the calibration factor ν, namely: M D A = L D t s ⋅ ν = L D t s η ε = k 1 − α 2 + 2 k 1 − α μ ¯ b + σ μ ¯ b 2 t s η ε (5) where η is the probability of the sample emitting the characteristic γ-ray to be measured in one decay (i.e., branching ratio), and ε is the detection efficiency of the γ-rays by the measuring system. Note that the calibration factor ν equals to the product of η and ε. MDA thus represents the minimum activity of the sample that can be recognized by the measuring device with confidence level of k _1−α through counting the characteristic γ-rays emanating from the sample under the measurement conditions including duration, detection efficiency, and background.

It should be noted that t _b can be different from t _s , but then the average and standard deviation of the background count need to be normalized according to t _s . In addition, researchers often use single measurement to estimate the background and set the confidence level to be 95%. Under these circumstances, σ μ ¯ b 2 = μ ¯ b , and k _1−α = 1.64; finally, the following simplified expression can be obtained: M D A = k 1 − α 2 + 2 k 1 − α 2 μ ¯ b t s η ε = 2.71 + 4.65 μ ¯ b t s η ε (6)

Equation 6 is arguably the most commonly used expression in the field of radiation detection.

In addition to the MDA listed above, researchers also employed other quantities to evaluate the limit of detection including L C ′ and LLD (Altshuler et al., 1963), which are shown below: L C ′ = k 1 − α 2 μ ¯ b t s v (7) L L D = 2 k 1 − α 2 μ ¯ b t s v (8)

The application of these various measures for limit of detection apparently causes great difficulties for comparisons among studies as shown next.

Table 1 summarizes the key features from the studies focusing on Pb-210 in vivo measurements (Eisenbud et al., 1969; Cohen et al., 1973; Cohen et al., 1977; Zheng et al., 1992; Estrada et al., 1993; Scheler et al., 1998; Wahl et al., 2000; Haninger et al., 2002; Dantas et al., 2007), including characteristics of detectors, measurement time t _s , calibration factor v, and background count rate N _b .

It is first noticed that two types of detectors were commonly employed, namely NaI-CsI scintillator detectors and high-purity germanium (HPGe) semiconductor detectors. Because they characterize drastically distinct energy resolution, the region-of-interest (ROI) and its corresponding width in the measurement spectrum are dramatically different. To be more precise, the full width at the half maximum (FWHM) of the scintillator detector is ∼9 keV at 46.5 keV, and that of the semiconductor detector is ∼0.6 keV at 46.5 keV. As a result, the width of ROI for NaI-CsI detector is approximately 40 keV, whereas that for HPGe detector is roughly 2 keV. Besides energy resolution, two types of detectors also differ considerably in active area A; it is common that scintillator detectors often characterize much larger active area than HPGe detectors. On average, the active area for the studies employing NaI-CsI detectors is 4.4 × 10⁴ mm² while that for HPGe detectors is 2.8 × 10⁴ mm².

The width of ROI and active area are two critical factors that affect the limit of detection. Under a fixed level of background, it can be understood that a wider ROI would necessarily lead to a higher background count rate N _b and equivalently the background count μ ¯ b within the measurement duration. In this sense, a smaller ROI resulted from a finer energy resolution is more advantageous for lowering MDA. In contrast, the active area influences the limit of detection in a more complex way. On the one hand, a larger active area would improve the efficiency in detecting the targeted γ-rays, because the solid angle spanned by detector towards the skull is wider; on the other hand, it can also increase the counts on the background in the same way. We analyze these factors in more detail in Analysis of Background Count Rate N _b and Analysis of the Calibration Factor v.

In addition to the detector factors, the duration of measurement also plays a considerable role. We see the measurement time varies widely among studies and is commonly set to more than 1 h. Furthermore, as shown in Table 1, researchers also employed various quantities to measure the limit of detection. In Comparison of MDA Among Studies, we rely on the reported data and unify these measures according to the MDA derived in the prior section. Moreover, we demonstrate the evolution of MDA under the variation of detector’s active area and the length of measurement.

In this work, we refer to the studies focusing on the Pb-210 in vivo measurements and reanalyze the reported data therein to understand how several key factors influence the limit of detection.

Firstly, we focus on the reported background count rates and analyze their dependence on detector’s energy resolution and active area. It is found that the background level inside the shielding room of respective studies is almost the same, which can be rationalized by the saturating capability of metallic walls in suppressing the background through attenuating environmental γ-rays. It is largely due to difference in detector's energy resolution and active area that leads to the considerable differences in background count rates across various studies.

Secondly, we turn to the reported calibration factors and conduct analysis in the same manner. It is found that the detection efficiency is linearly dependent on the active area and is independent on the type of detectors.

Thirdly, we calculate MDA for each study and demonstrate its dependence on the active area of detector and measurement duration. In the limit of the largest achievable active area (∼75,000 mm²), it is found that the asymptotic MDA is approximately 6 (4) Bq and 15 (11) Bq under 30 (60) min measurement using NaI-CsI scintillator and HPGe semiconductor detectors, respectively.

Finally, we discuss these asymptotic MDA in the context of model-estimated Pb-210 activity in the skull resulted from a hypothetical history of radon exposure. It is suggested that a lower MDA would be beneficial to recognize an individual’s radon exposure in an earlier stage, which is critical for retrospective assessment on lung cancer risks in a timely manner.