Applicability Limits of the TDCR Method for Non-Pure β Emitters: Case Studies of ²¹⁰Pb and ⁹⁰Sr

Applicability Limits of the TDCR Method for Non-Pure β Emitters: Case Studies of ²¹⁰Pb and ⁹⁰Sr | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Applicability Limits of the TDCR Method for Non-Pure β Emitters: Case Studies of ²¹⁰Pb and ⁹⁰Sr Camilla S. Sampaio, Wanderson O. Sousa, Denison Souza-Santos, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9161859/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The triple-to-double coincidence ratio (TDCR) method is a well-established primary technique for the absolute standardization of pure β-emitting radionuclides in liquid scintillation counting (LSC). Its potential application to radionuclides of dosimetric and environmental relevance characterized by time-dependent decay schemes remains, however, insufficiently explored. This work presents a systematic experimental investigation of the applicability limits and correction strategies required to extend TDCR to non-pure β-emitters, using lead-210 (²¹⁰Pb) and strontium-90 (⁹⁰Sr) as representative and complementary case studies. Measurements were performed using a Hidex 300 SL TDCR system following chemical separation of parent and daughter radionuclides. Direct TDCR-based activity determination yielded statistically consistent results only within a restricted temporal window, corresponding to daughter contributions below approximately 20–22% of the total counting rate. Beyond this interval, TDCR values increased under constant quench conditions, demonstrating sensitivity to changes in the effective β-energy spectrum rather than to quenching effects alone. Activities derived from TDCR measurements were found to follow the temporal evolution predicted by the Bateman formalism, providing a physical basis for implementing a Bateman-based correction. After correction, time-independent parent activities were obtained at any measurement time, with z-scores consistently within statistical acceptance limits. The proposed framework preserves the calibration-standard-free, primary character of TDCR and enables reliable low-level activity determination for radionuclides with complex decay schemes. These results demonstrate that Bateman-corrected TDCR is well suited for early post-intake measurements, routine internal dosimetry, and emergency situations where access to calibration standards is limited or unavailable. TDCR non-pure beta-emitters ²¹⁰Pb ⁹⁰Sr liquid scintillation counting decay-chain correction Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Highlights Applicability limits of TDCR for non-pure β-emitters were evaluated using ²¹⁰Pb and ⁹⁰Sr Direct TDCR application is statistically valid only for daughter contributions below ~20–22% TDCR variations reflect changes in the effective β-energy spectrum under constant quenching Bateman-based correction restores time-independent parent activities TDCR enables activity determination without external calibration standards 1. Introduction The accurate quantification of long-lived β-emitting radionuclides is a central requirement in internal dosimetry and radiation protection, particularly in scenarios involving early post-intake measurements, emergency response, and retrospective dose assessment. Among these radionuclides, strontium-90 (⁹⁰Sr) and lead-210 (²¹⁰Pb) are of particular relevance due to their long physical half-lives, radiological significance, and strong affinity for bone tissue, making them key contributors to long-term committed doses following internal contamination (IAEA, 2014 ; ICRP 130, 2015). Liquid Scintillation Counting (LSC) is widely used for the determination of β-emitting radionuclides. When combined with the Triple-to-Double Coincidence Ratio (TDCR) method, LSC enables activity determination as a primary measurement technique, eliminating the need for external efficiency calibration standards (Broda et al., 2007 ). This characteristic is particularly advantageous in emergency or field situations, where certified reference materials may be unavailable, delayed, or impractical to obtain. TDCR relies on a physical model of the detection process to derive efficiency directly from measured coincidence ratios. While the method is well established for pure β-emitters, its applicability to non-pure β-emitters relevant to internal dosimetry and environmental exposure remains insufficiently characterized, especially under non-equilibrium conditions typical of real samples (Kossert et al., 2014 ). The main challenge arises from the time-dependent nature of parent-daughter systems. In the case of 90 Sr (T 1/2 = 28.8 y), the ingrowth of its high-energy daughter, 90 Y (E max ≈ 2.28 MeV) rapidly modifies the detected β spectrum. Similarly, 210 Pb (T 1/2 = 22.3 y) decays to 210 Bi (E max ≈ 1.16 MeV), while also emitting a characteristic low-energy γ-ray at 46.5 keV. These additional radiative components alter the balance between triple and double coincidence, potentially biasing TDCR-based activity determination if not properly accounted for (Bobin et al., 2023 ). The applicability of TDCR-based liquid scintillation counting to radionuclides with complex decay schemes has been previously demonstrated for environmental matrices using the HIDEX 300 SL system, including detailed investigations of parent–daughter ingrowth relationships and β-spectral interferences for ²¹⁰Pb and radium isotopes (Eikenberg et al. 2014 ). Although the focus of that work was primarily environmental and included α-emitting radionuclides, it provides important experimental evidence that TDCR-based methodologies implemented on the HIDEX 300 SL can handle complex decay schemes and overlapping spectral components under realistic measurement conditions. Previous approaches addressing decay-chain effects in LSC measurements have often relied on the Bateman formalism. In routine laboratory contexts, Bateman-based corrections have been successfully applied to accelerate the quantification of ⁹⁰Sr, albeit typically within the framework of secondary or relative measurements (Wiatr et al. 2022 ). From a primary metrological perspective, extensions of the TDCR model to complex decay chains have been demonstrated under conditions requiring detailed spectral modeling and strict experimental control (Kossert 2019). In parallel, applied LSC studies have established detection limit benchmarks for ²¹⁰Pb relevant to dosimetric and environmental applications (Sampaio et al., 2020 ). Despite these advances, a quantitative definition of the applicability limits of direct TDCR measurements in non-equilibrium decay systems remains lacking, particularly in the context of internal dosimetry and emergency bioassay, where measurements are often performed shortly after intake or chemical separation. By experimentally elucidating the sensitivity of the TDCR value to spectral changes under constant quench conditions, this work defines practical applicability limits for direct TDCR use and introduces a physically motivated correction based on the Bateman formalism. This approach preserves the calibration-standard-free character of TDCR while enabling statistically consistent activity determination beyond the early post-separation regime. 2. Materials and Methods 2.1 Instrumentation and Measurement Conditions Measurements were performed using a Hidex 300 SL liquid scintillation counter operated in TDCR mode, equipped with three photomultiplier tubes arranged at 120°. Samples were prepared in polyethylene vials containing 20 mL of HiSafe scintillation cocktail. Regions of interest were defined to include the full β spectra of parent and daughter radionuclides. Background corrections were applied using blank samples prepared under identical conditions. 210 Pb samples were counted for 100 min while 90 Sr samples were counted for 20 min each. 2.2 Sample Preparation and Chemical Separation Reference materials of ²¹⁰Pb and ⁹⁰Sr provided within the framework of the Brazilian National Intercomparison Program (PNI) were used in this study. These materials are routinely employed for performance evaluation and interlaboratory comparison, providing independent reference activity values without serving as calibration standards. Chemical separation procedures were applied to isolate the parent radionuclides from their respective daughters (²¹⁰Bi and ⁹⁰Y). The reference time (t = 0) was defined as the end of the separation or precipitation step. Samples were measured repeatedly over time to monitor daughter ingrowth. 2.3 TDCR Determination and Quench Monitoring Double and triple coincidence events were recorded by the acquisition system, which directly provided the TDCR values for each measurement. The quench parameter equivalent (QPE) was monitored throughout the experiments in order to track possible variations in scintillation or optical conditions during the measurement period. 2.4 Bateman Formalism and Activity Correction Activities were initially determined using the TDCR method under the simplifying assumption of a single β-emitting radionuclide, applying TDCR-derived detection efficiencies and neglecting explicit contributions from radioactive progeny. Under this operational assumption, the activity obtained directly from TDCR represents an effective activity associated with the composite detection response of the decay system at a given time. To account for the presence of radioactive progeny in parent–daughter decay systems, an activity correction based on the Bateman formalism was applied. The corrected parent activity, A corr (t), was calculated as: $$\:{A}_{corr}\left(t\right)=\:\frac{{A}_{TDCR}\left(t\right)}{1+{f}_{d}\left(t\right)}=\frac{{R}_{s}-{R}_{bg}}{{\epsilon\:}_{TDCR}\left(t\right)\bullet\:60\bullet\:m\bullet\:[1+{f}_{d}\left(t\right)]}\:\:\:\:\left(1\right)$$ Where A TDCR (t) is the activity obtained directly from TDCR assuming a single β-emitter, R s and R bg are the sample and background count rates, respectively, e TDCR (t) is the TDCR-derived detection efficiency, m is the sample mass, and f d (t) is the the relative contribution of the daughter activity with respect to the parent activity, calculated from the Bateman equations. At the reference time immediately after chemical separation (t = 0), the theoretical value of f d (0) is zero, and the correction factor reduces to unity, such that the corrected activity equals the directly determined TDCR activity. As daughter ingrowth proceeds, f d (t) increases continuously. Under secular equilibrium, where the parent and daughter activities are equal, \(\:{f}_{\text{d}}=1\) and the corrected activity becomes half of the activity obtained by direct TDCR application without correction. This correction approach is physically grounded, as it directly reflects the decay kinetics of the parent–daughter system and explicitly separates spectral evolution effects from the activity determination. Importantly, it enables reliable determination of the parent activity at any measurement time after separation while fully preserving the primary, calibration-standard-free character of the TDCR method. 2.5 Detection Limits and Uncertainty Considerations Detection limits were evaluated in accordance with ISO 11929 (2019), using the same measurement model adopted for activity determination. The activity was expressed as a function of the background count rate, the TDCR-derived detection efficiency, the sample mass, and the relative daughter contribution obtained from the Bateman formalism. Uncertainty components associated with each of these quantities were considered, including background counting statistics, TDCR efficiency stability, sample mass, and uncertainty in the daughter fraction. The combined uncertainty of the activity estimate was then used to derive the detection limit for symmetric decision and detection risks (α = β = 5%). 3. Results and Discussion 3.1. Spectral interference and daughter ingrowth effects The determination of Sr-90 and Pb-210 by LSC is inherently complicated by continuous β spectra and spectral overlap with their daughters, Y-90 and Bi-210. As shown in Fig. 1 , β spectra recorded shortly after separation differ substantially from those at secular equilibrium, reflecting the increasing contribution of the higher-energy daughter radionuclides. For the Sr-90/Y-90 system, secular equilibrium is reached approximately 10 days after separation, while for Pb-210/Bi-210 equilibrium is established after about 15 days. During this period, the increasing contribution of the higher-energy β emitters (Y-90 and Bi-210) significantly modifies the spectral shape observed by the liquid scintillation detector. 3.2. Temporal behavior of TDCR and ingrowth dynamics Figure 2 shows the evolution of the normalized TDCR as a function of the relative daughter contribution predicted by the Bateman equations. For both radionuclide systems investigated, TDCR increases systematically as the contribution of the high-energy β-emitting daughter grows. Although the magnitude of this increase is small—ranging from 73% to 79% for Pb-210 and from 91% to 94% for Sr-90—it remained consistent throughout daily monitoring and therefore cannot be attributed to statistical fluctuations in isolated measurements. Despite this systematic trend, the temporal evolution of TDCR does not follow the growth kinetics of the daughter activity. Instead, TDCR exhibits a bounded and gradually saturating behavior inherent to its definition as a ratio of coincidence probabilities. As a consequence, TDCR cannot reproduce the rapid increase in total counting rate associated with daughter ingrowth once the daughter contribution becomes significant, even though a measurable effect on the coincidence balance is already present. The stability of the Quench Parameter Equivalent (QPE), shown in Fig. 3 , confirms that the scintillation yield and optical attenuation properties of the detection medium remained unchanged throughout the measurement period (u QPE < 0.5%). This observation excludes quenching variations as the origin of the observed TDCR increase and demonstrates that the effect arises under constant quench conditions. Under these conditions, the increase in TDCR must therefore originate from changes in the effective detected β-energy spectrum. As daughter radionuclides with higher β end-point energies progressively contribute to the signal, the relative balance between triple and double coincidences is modified, leading to an increase in TDCR even in the absence of quenching variations. This behavior demonstrates that TDCR is intrinsically sensitive to additional spectral contributions arising from time-dependent decay-chain evolution. Accordingly, a systematic increase in TDCR without a corresponding variation in QPE may be interpreted as an experimental indicator of evolving spectral composition. For radionuclides with time-dependent decay schemes, TDCR therefore reflects changes in the effective β-energy distribution rather than acting solely as a quench monitor. This observation provides the physical basis for defining applicability limits for direct TDCR use and for implementing correction strategies when additional spectral contributions become significant. 3.3. Activity bias and applicability limits of direct TDCR Table 1 summarizes the evolution of activity bias and z-scores obtained using direct TDCR values for ²¹⁰Pb and ⁹⁰Sr as a function of time after chemical separation. For both radionuclides, a pronounced increase in bias is observed after the first 1–2 days, while z-scores during the earliest measurements remain within the statistical compatibility criterion (|z| ≤ 2), despite the presence of positive bias. Table 1 – Bias and z-score of 210 Pb and 90 Sr obtained when using direct TDCR value instead of efficiency for different time intervals of water sample from intercomparison program. Time after separation Bias (%) 210 Pb z-score 210 Pb Bias (%) 90 Sr z-score 90 Sr 1 day 8–18 0.4–0.9 18–28 1.2–1.9 2 days 16–28 0.8–1.4 35–45 2.4–3.0 ≥ 3 days 30–85 ≥ 1.5–4.2 45–88 ≥ 3.0–6.2 Note: Individual datasets are provided in the Supplementary Material. Each time point corresponds to independent replicate measurements (n = 3), allowing statistical evaluation of bias and z-score. During this initial post-separation interval, the contribution of the daughter radionuclides remains limited and the composite β spectrum is still dominated by the parent emission. Under these conditions, the decay system behaves approximately as a quasi-pure β-emitter, allowing the application of TDCR without explicit correction when statistical acceptance criteria are adopted. Figure 4 presents the activity bias obtained by direct TDCR application as a function of the relative daughter contribution predicted by the Bateman equations for the ⁹⁰Sr/⁹⁰Y and ²¹⁰Pb/²¹⁰Bi decay systems. A strong correlation is observed between the increase in activity bias and the progression of daughter ingrowth. The green line identifies the temporal window in which direct TDCR application remains feasible without correction. Within this window, corresponding to additional contributions below approximately 20%, activity bias remains moderate and the associated z-scores satisfy the statistical compatibility criterion. This defines a practical regime in which TDCR may be applied without explicit correction for routine or screening purposes. It must be emphasized, however, that this criterion reflects statistical consistency rather than strict metrological validity. In primary standardization and high-accuracy applications, substantially tighter bias limits are typically required, and corrections accounting for additional contributions to the detected signal—beyond the parent β-emission—become necessary. Accordingly, even within the early post-separation interval, the use of direct TDCR should be considered conditional and dependent on the intended application. Beyond approximately 1–2 days after separation, both Table 1 and Fig. 4 show a rapid increase in activity bias, accompanied by z-scores exceeding acceptance limits typically adopted in metrology. This behavior reflects the increasing contribution of the high-energy β-emitting daughters (⁹⁰Y and ²¹⁰Bi), which significantly modifies the effective detected β spectrum. Because TDCR is a bounded ratio, it cannot reproduce the corresponding increase in effective detection efficiency associated with the evolving decay scheme. The close agreement between experimental activity bias and Bateman-predicted evolution provides a clear physical basis for introducing a correction framework, which is presented in the following section. 3.4 Corrected versus uncorrected activity determination On the basis of the systematic bias observed for direct TDCR application and its clear correlation with the Bateman-predicted evolution of the decay chain, a correction framework was implemented in which the TDCR-derived activity is explicitly adjusted according to the relative contribution of additional decay components. This approach allows the parent activity to be decoupled from the time-dependent composite β spectrum. Figure 5 presents the activity determination for ²¹⁰Pb as a function of time after chemical separation, comparing results obtained by direct TDCR application with those corrected using the Bateman formalism. Activities derived from direct TDCR application already show a systematic positive bias from the first 1–2 days after separation, consistent with the bias trends identified in Section 3.3 . This early overestimation reflects the rapid spectral influence of ²¹⁰Bi, whose higher β endpoint energy significantly affects the detected spectrum even at low relative activity contributions. As ingrowth progresses, the deviation between direct TDCR results and the reference value increases steadily, leading to pronounced overestimation well outside the associated uncertainty band. In contrast, activities obtained using Bateman-corrected TDCR remain stable throughout the entire measurement period and are statistically consistent with the reference value within its uncertainty. The corrected results exhibit no systematic temporal trend, demonstrating that the correction effectively compensates for the evolving decay scheme. Figure 6 shows the corresponding results for the ⁹⁰Sr/⁹⁰Y system. Direct TDCR application leads to measurable overestimation of the activity already within the first day after separation, with the bias increasing as the contribution of ⁹⁰Y grows. Although secular equilibrium in the ⁹⁰Sr/⁹⁰Y system is established only several days after separation, the spectral impact of the high-energy β-emission of ⁹⁰Y becomes detectable much earlier, resulting in a progressive divergence from the reference activity. Once the Bateman-based correction is applied, the TDCR-derived activities for ⁹⁰Sr remain stable over time and fully compatible with the reference value. The absence of systematic temporal dependence confirms that the correction framework is effective for both radionuclides, despite their markedly different β-energy distributions and ingrowth kinetics. Taken together, Figs. 5 and 6 demonstrate that, while direct TDCR application is intrinsically sensitive to changes in the decay scheme and leads to early overestimation of activity, the Bateman-corrected TDCR approach provides a robust and time-independent determination of parent activity, extending the applicability of TDCR beyond the initial post-separation regime. Although demonstrated here for two-member decay chains, the approach can, in principle, be extended to more complex decay schemes, provided that individual spectral contributions are well characterized. 3.5 Z-score analysis and statistical consistency of Bateman-corrected TDCR approach The performance of the Bateman-corrected TDCR approach was further assessed through z-score analysis, which provides a standardized metric for evaluating statistical compatibility between measured activity values and reference results while accounting for their combined uncertainties. This analysis is particularly relevant in the context of intercomparison exercises, where acceptance criteria are commonly expressed in terms of z-scores. Figures 7 and 8 show the temporal evolution of z-score values obtained for ²¹⁰Pb and ⁹⁰Sr, respectively, using TDCR-derived activities corrected according to the Bateman formalism. For comparison, direct TDCR-based activity results discussed in previous sections exhibit a systematic increase in z-score with time after chemical separation, exceeding the commonly adopted acceptance criterion (|z| ≤ 2) once the contribution of additional spectral components becomes significant. In contrast, the Bateman-corrected TDCR results yield z-scores predominantly within the interval |z| ≤ 2 over the entire measurement period for both radionuclides, including times approaching secular equilibrium. No systematic temporal trend is observed in the corrected z-score distributions, indicating that the correction effectively compensates for the evolving decay scheme and stabilizes the activity determination despite substantial changes in the composite β spectrum Importantly, this improvement reflects not only a reduction in systematic bias but also the restoration of full uncertainty-consistent agreement with the reference values. The z-score analysis therefore provides an independent statistical validation of the corrected TDCR methodology, complementing the physical interpretation based on activity evolution and bias trends presented in the preceding sections. From an experimental perspective, the statistical consistency demonstrated here establishes the necessary condition for extending TDCR-based activity determination beyond the early post-separation interval. Once bias and uncertainty propagation are properly controlled through Bateman-based correction, the remaining limitation on practical applicability is governed by counting statistics and background contributions, which directly determine the achievable detection limits. These aspects are addressed quantitatively in the following section. 3.6 Detection Limits The practical applicability of TDCR-based activity determination is ultimately constrained by the achievable detection limits under the prevailing measurement conditions. Having established in Section 3.5 that the Bateman-corrected TDCR approach provides statistically consistent and time-independent activity results, the present section focuses on the quantitative detection capability of the method as a function of time after chemical separation. Detection limits were evaluated in accordance with ISO 11929 using the same measurement model adopted for activity determination. Activities derived directly from TDCR measurements performed shortly after chemical separation were compared with activities corrected for daughter ingrowth using the Bateman formalism. This approach enables a direct assessment of detection capability under early post-separation conditions—where direct TDCR application remains statistically acceptable—and at later times, when correction becomes necessary. The evaluation explicitly accounts for experimental background count rates, counting time, TDCR-derived detection efficiency, sample mass, and the uncertainty associated with the relative daughter contribution predicted by the Bateman equations. In this framework, detection limits reflect not only instrumental performance but also the physical evolution of the decay system, as spectral changes directly influence the effective detection efficiency. Table 2 summarizes the detection limits obtained for ⁹⁰Sr and ²¹⁰Pb under representative early post-separation and secular equilibrium conditions. The highest detection limits are observed on the first day after chemical separation for both radionuclides. At these early measurement times, the detected spectra are dominated by the low-energy β emissions of the parent radionuclides, resulting in relatively low signal-to-background ratios over the selected regions of interest. Under these conditions, background count rates—approximately 40 counts per minute for ⁹⁰Sr (20 min counting time) and 37 counts per minute for ²¹⁰Pb (100 min counting time)—constitute the dominant contribution to the combined uncertainty and therefore govern the resulting detection limits. Table 2 - Detection limits obtained using direct TDCR and Bateman-corrected TDCR activity determination Radionuclide Time after separation Activity model Detection limit (Bq L⁻¹) ⁹⁰Sr Day 1 Direct TDCR 0.23 ⁹⁰Sr Day 1 Bateman-corrected TDCR 0.19 ⁹⁰Sr Secular equilibrium Bateman-corrected TDCR 0.12 ²¹⁰Pb Day 1 Direct TDCR 0.21 ²¹⁰Pb Day 1 Bateman-corrected TDCR 0.19 ²¹⁰Pb Secular equilibrium Bateman-corrected TDCR 0.10 As daughter ingrowth progresses and secular equilibrium is approached, detection limits decrease systematically for both radionuclides. This reduction reflects the increasing contribution of higher-energy β emissions from the daughter radionuclides, which enhances the effective detection efficiency and improves counting statistics while background conditions remain unchanged. The observed improvement in detection capability is therefore driven by physical changes in the detected β-energy spectrum rather than by modifications to the experimental setup or measurement protocol. The results in Table 2 further indicate that the relatively elevated detection limits observed at early post-separation times are primarily governed by background and counting statistics, rather than by intrinsic limitations of the TDCR method itself. From an experimental perspective, these detection limits could be further reduced by increasing counting times, thereby decreasing the statistical contribution of the background to the combined uncertainty, without requiring any modification to efficiency modeling or calibration procedures. Overall, the detection limit analysis confirms that, once statistical consistency is ensured through Bateman-based correction, the practical sensitivity of TDCR-based measurements is determined by conventional counting statistics and background considerations. 4. Conclusions This work demonstrates that the Triple-to-Double Coincidence Ratio (TDCR) method can be reliably extended to radionuclides characterized by non-pure β-decay schemes that are of direct relevance to internal dosimetry and radiological emergency response, using ²¹⁰Pb and ⁹⁰Sr as representative and complementary case studies. Despite the intrinsic complexity introduced by daughter ingrowth and time-dependent spectral evolution, TDCR provides statistically robust and physically interpretable activity determinations when decay-chain effects are explicitly accounted for. A key outcome of this study is the confirmation that TDCR enables accurate low-level activity determination without reliance on external calibration standards. This feature is particularly critical in internal dosimetry applications and emergency situations, where rapid assessments of intake are required and suitable reference materials may be unavailable or impractical to obtain. This constitutes a fundamental advantage over conventional liquid scintillation and Cherenkov counting approaches, which typically require certified standards and additional empirical corrections to address decay-chain effects. By relying exclusively on measured coincidence ratios and well-established decay kinetics, TDCR preserves its character as a primary measurement method, even for radionuclides traditionally considered challenging due to spectral non-stationarity. The observed temporal evolution of TDCR parameter for both radionuclides was shown to be driven primarily by changes in the effective detected β-energy spectrum associated with daughter ingrowth, rather than by variations in quenching conditions. This finding provides a clear physical basis for defining applicability limits for direct TDCR use in early post-separation or early post-intake scenarios, as well as for implementing correction strategies when measurements extend beyond this regime. While direct TDCR application remains statistically acceptable during the early measurement period, the Bateman-based correction introduced in this work extends the validity of the method over the full ingrowth interval, including conditions approaching secular equilibrium, yielding time-independent parent activities with z-scores consistently within acceptance criteria. Regarding detection capability, the achievable detection limits are mainly governed by the background count rate and counting time, particularly in the early post-separation period. The higher detection limits observed on day 1 result from the dominance of background over the low-energy β emissions of the parent radionuclides, while daughter ingrowth progressively reduces detection limits by enhancing the effective detection efficiency and counting statistics. Although the resulting detection limits are higher than those obtained with ultra low-level systems, the ability to derive statistically consistent activities without external calibration standards represents a clear practical advantage for internal dosimetry and emergency assessments, especially in early post-intake scenarios where rapid measurements are required and reference materials may be unavailable. Overall, this study expands the practical and metrological applicability of the TDCR method beyond pure β-emitters and demonstrates that, when combined with physically grounded Bateman-based corrections, TDCR constitutes a robust and flexible tool for activity determination in internal dosimetry, emergency response, and related environmental assessments, without compromising its calibration-standard-free nature. Declarations No funding was received for conducting this study. Conflict of Interest/Competing Interests The authors have no conflicts of interest to declare that are relevant to the content of this article. Author Contribution C.S.S. and W.O.S. wrote the main manuscript text. All authors reviewed the manuscript Acknowledgement The authors gratefully acknowledge Antonio Copote-Cuellar for insightful discussions and valuable comments that contributed to the interpretation of the TDCR behavior and the physical consistency of the proposed correction framework. 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Appl Radiat Isot 156:108972. https://doi.org/10.1016/j.apradiso.2019.108972 Wiatr K, Rubel B, Kardas M (2022) Rapid 90 Sr quantification method based on the Bateman equation for routine laboratory work. Nukleonika 67(4):6772 https://doi.org/10.2478/nuka-2022-0006 Additional Declarations No competing interests reported. Supplementary Files SupplementaryMaterial.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Sampaio","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABC0lEQVRIiWNgGAWjYJCCAw8KYEwDBjk2EAlh49GSAJc1YDAmSgtDApJsYgMhLfztpxOBttgxyLefTnvwo8AmvU/68AbGLxWHGcylD2DVInEmdwNQSzKDwZnc7YY9Bmm5bXxpBcwyZw4zWPYlYNViwADWwgxibJPgMTic28bDY8As2ZYGNAS7wwz434K01DPI97/dJvnH4HA6G0EtEmBbDjMw3MjdJg20JQGkhfFjmw1OLRI3wLYc5zG48Xa7sYxBmmEbD1vBYYYzNjyWPThCrD9384cPFdVy8v252x6++WMjL9/DvPHhjwoJOXMe7FpgACTNBucd5oGIEAQILYw/iFE/CkbBKBgFIwUAAA9MWLJS2iHpAAAAAElFTkSuQmCC","orcid":"","institution":"Institute of Radioprotection and Dosimetry","correspondingAuthor":true,"prefix":"","firstName":"Camilla","middleName":"S.","lastName":"Sampaio","suffix":""},{"id":614351124,"identity":"d46bdcd8-0ecc-47a3-9067-227e239b8749","order_by":1,"name":"Wanderson O. Sousa","email":"","orcid":"","institution":"Institute of Radioprotection and Dosimetry","correspondingAuthor":false,"prefix":"","firstName":"Wanderson","middleName":"O.","lastName":"Sousa","suffix":""},{"id":614351125,"identity":"2eb33432-9f1a-4469-af47-9833d62532cd","order_by":2,"name":"Denison Souza-Santos","email":"","orcid":"","institution":"Institute of Radioprotection and Dosimetry","correspondingAuthor":false,"prefix":"","firstName":"Denison","middleName":"","lastName":"Souza-Santos","suffix":""},{"id":614351126,"identity":"9d1d98b5-c5f9-488a-8470-eed1a2ecc59b","order_by":3,"name":"Ademir Xavier Silva","email":"","orcid":"","institution":"COPPE, Federal University of Rio de Janeiro (UFRJ)","correspondingAuthor":false,"prefix":"","firstName":"Ademir","middleName":"Xavier","lastName":"Silva","suffix":""}],"badges":[],"createdAt":"2026-03-18 17:09:13","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9161859/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9161859/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106092105,"identity":"68d8b1b3-2298-46d3-ad66-f0a63b599bab","added_by":"auto","created_at":"2026-04-03 11:17:53","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":122443,"visible":true,"origin":"","legend":"\u003cp\u003eExperimental β spectra of ²¹⁰Pb/²¹⁰Bi and ⁹⁰Sr/⁹⁰Y measured shortly after chemical separation (top) and after secular equilibrium (bottom), showing the increasing contribution of the high-energy β-emitting daughter radionuclides\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-9161859/v1/ae6e1e45738e09bfa9643293.png"},{"id":106095120,"identity":"7f758e16-d269-4a9b-bc45-75bb45d44f4a","added_by":"auto","created_at":"2026-04-03 11:44:37","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":68277,"visible":true,"origin":"","legend":"\u003cp\u003eTemporal evolution of the TDCR following chemical separation, illustrating a systematic increase under constant quench conditions driven by spectral changes associated with daughter ingrowth\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-9161859/v1/29059af1c6d9a84d57a06701.png"},{"id":106092107,"identity":"a0457519-b1e3-4bdf-8816-5110eb76cca0","added_by":"auto","created_at":"2026-04-03 11:17:53","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":24197,"visible":true,"origin":"","legend":"\u003cp\u003eRelationship between TDCR and quench parameter equivalent (QPE) for measurements performed before and after chemical separation. The observed stability of QPE indicates that variations in TDCR are not associated with changes in quenching conditions\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-9161859/v1/ded6f346a219e61196179bd4.png"},{"id":106092108,"identity":"d64d6312-fbb8-452f-b730-424b86f3d6d4","added_by":"auto","created_at":"2026-04-03 11:17:53","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":53119,"visible":true,"origin":"","legend":"\u003cp\u003eRelative contribution of the daughter radionuclide as a function of time after chemical separation for the ⁹⁰Sr/⁹⁰Y and ²¹⁰Pb/²¹⁰Bi decay systems, versus Activity bias calculated using TDCR. The green line highlights the quasi-pure β-emitter regime, in which direct TDCR application remains valid\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-9161859/v1/64e53ac564ec1e4343dc3e08.png"},{"id":106092109,"identity":"b8ab57ff-c4b3-4ec7-b45b-e2d059f3d236","added_by":"auto","created_at":"2026-04-03 11:17:53","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":99829,"visible":true,"origin":"","legend":"\u003cp\u003eTime-dependent determination of ²¹⁰Pb activity following chemical separation, showing direct (uncorrected) TDCR-based activities and activities corrected for daughter ingrowth using the Bateman formalism\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-9161859/v1/c65ee56fa5596e73fd985656.png"},{"id":106095119,"identity":"c398d784-d18f-4a48-b7b1-7632835ff4a9","added_by":"auto","created_at":"2026-04-03 11:44:37","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":105335,"visible":true,"origin":"","legend":"\u003cp\u003eTime-dependent determination of \u003csup\u003e90\u003c/sup\u003eSr activity following chemical separation, showing direct (uncorrected) TDCR-based activities and activities corrected for daughter ingrowth using the Bateman formalism\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-9161859/v1/650e7eef67b5f0087978305d.png"},{"id":106094876,"identity":"26da3c92-b6fb-4f87-a0be-b270f558db02","added_by":"auto","created_at":"2026-04-03 11:43:35","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":28297,"visible":true,"origin":"","legend":"\u003cp\u003eZ-score values for the temporal activity determination of ²¹⁰Pb using Bateman-corrected TDCR-based activity, demonstrating statistical consistency over the full measurement period\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-9161859/v1/bc4c7401ec3eae00819deda0.png"},{"id":106095937,"identity":"f2629c3c-3643-486c-aa4e-309bb9d76611","added_by":"auto","created_at":"2026-04-03 11:51:54","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":29197,"visible":true,"origin":"","legend":"\u003cp\u003eZ-score values for the temporal activity determination of ⁹⁰Sr using Bateman-corrected TDCR-based activity\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-9161859/v1/0a279d17a592b637c04b46e7.png"},{"id":109240085,"identity":"047040be-0519-43de-ba60-c0392880ad47","added_by":"auto","created_at":"2026-05-14 06:25:48","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":631972,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9161859/v1/af5dbeea-2911-4e89-b2a5-9bcdf1826a55.pdf"},{"id":106092104,"identity":"c18f7d38-f616-4d18-a870-dddf3a785e8a","added_by":"auto","created_at":"2026-04-03 11:17:53","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":24112,"visible":true,"origin":"","legend":"","description":"","filename":"SupplementaryMaterial.docx","url":"https://assets-eu.researchsquare.com/files/rs-9161859/v1/d54c954bb7fef7d1393795bf.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Applicability Limits of the TDCR Method for Non-Pure β Emitters: Case Studies of ²¹⁰Pb and ⁹⁰Sr","fulltext":[{"header":"Highlights","content":"\u003cul\u003e\n\u003cli\u003eApplicability limits of TDCR for non-pure \u0026beta;-emitters were evaluated using \u0026sup2;\u0026sup1;⁰Pb and ⁹⁰Sr\u003c/li\u003e\n\u003cli\u003eDirect TDCR application is statistically valid only for daughter contributions below ~20\u0026ndash;22%\u003c/li\u003e\n\u003cli\u003eTDCR variations reflect changes in the effective \u0026beta;-energy spectrum under constant quenching\u003c/li\u003e\n\u003cli\u003eBateman-based correction restores time-independent parent activities\u003c/li\u003e\n\u003cli\u003eTDCR enables activity determination without external calibration standards\u003c/li\u003e\n\u003c/ul\u003e"},{"header":"1. Introduction","content":"\u003cp\u003eThe accurate quantification of long-lived β-emitting radionuclides is a central requirement in internal dosimetry and radiation protection, particularly in scenarios involving early post-intake measurements, emergency response, and retrospective dose assessment. Among these radionuclides, strontium-90 (⁹⁰Sr) and lead-210 (\u0026sup2;\u0026sup1;⁰Pb) are of particular relevance due to their long physical half-lives, radiological significance, and strong affinity for bone tissue, making them key contributors to long-term committed doses following internal contamination (IAEA, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; ICRP 130, 2015).\u003c/p\u003e \u003cp\u003eLiquid Scintillation Counting (LSC) is widely used for the determination of β-emitting radionuclides. When combined with the Triple-to-Double Coincidence Ratio (TDCR) method, LSC enables activity determination as a primary measurement technique, eliminating the need for external efficiency calibration standards (Broda et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). This characteristic is particularly advantageous in emergency or field situations, where certified reference materials may be unavailable, delayed, or impractical to obtain. TDCR relies on a physical model of the detection process to derive efficiency directly from measured coincidence ratios. While the method is well established for pure β-emitters, its applicability to non-pure β-emitters relevant to internal dosimetry and environmental exposure remains insufficiently characterized, especially under non-equilibrium conditions typical of real samples (Kossert et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2014\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe main challenge arises from the time-dependent nature of parent-daughter systems. In the case of \u003csup\u003e90\u003c/sup\u003eSr (T\u003csub\u003e1/2\u003c/sub\u003e = 28.8 y), the ingrowth of its high-energy daughter, \u003csup\u003e90\u003c/sup\u003eY (E\u003csub\u003emax\u003c/sub\u003e \u0026asymp; 2.28 MeV) rapidly modifies the detected β spectrum. Similarly, \u003csup\u003e210\u003c/sup\u003ePb (T\u003csub\u003e1/2\u003c/sub\u003e = 22.3 y) decays to \u003csup\u003e210\u003c/sup\u003eBi (E\u003csub\u003emax\u003c/sub\u003e \u0026asymp; 1.16 MeV), while also emitting a characteristic low-energy γ-ray at 46.5 keV. These additional radiative components alter the balance between triple and double coincidence, potentially biasing TDCR-based activity determination if not properly accounted for (Bobin et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2023\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe applicability of TDCR-based liquid scintillation counting to radionuclides with complex decay schemes has been previously demonstrated for environmental matrices using the HIDEX 300 SL system, including detailed investigations of parent\u0026ndash;daughter ingrowth relationships and β-spectral interferences for \u0026sup2;\u0026sup1;⁰Pb and radium isotopes (Eikenberg et al. \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). Although the focus of that work was primarily environmental and included α-emitting radionuclides, it provides important experimental evidence that TDCR-based methodologies implemented on the HIDEX 300 SL can handle complex decay schemes and overlapping spectral components under realistic measurement conditions.\u003c/p\u003e \u003cp\u003ePrevious approaches addressing decay-chain effects in LSC measurements have often relied on the Bateman formalism. In routine laboratory contexts, Bateman-based corrections have been successfully applied to accelerate the quantification of ⁹⁰Sr, albeit typically within the framework of secondary or relative measurements (Wiatr et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). From a primary metrological perspective, extensions of the TDCR model to complex decay chains have been demonstrated under conditions requiring detailed spectral modeling and strict experimental control (Kossert 2019). In parallel, applied LSC studies have established detection limit benchmarks for \u0026sup2;\u0026sup1;⁰Pb relevant to dosimetric and environmental applications (Sampaio et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eDespite these advances, a quantitative definition of the applicability limits of direct TDCR measurements in non-equilibrium decay systems remains lacking, particularly in the context of internal dosimetry and emergency bioassay, where measurements are often performed shortly after intake or chemical separation. By experimentally elucidating the sensitivity of the TDCR value to spectral changes under constant quench conditions, this work defines practical applicability limits for direct TDCR use and introduces a physically motivated correction based on the Bateman formalism. This approach preserves the calibration-standard-free character of TDCR while enabling statistically consistent activity determination beyond the early post-separation regime.\u003c/p\u003e"},{"header":"2. Materials and Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Instrumentation and Measurement Conditions\u003c/h2\u003e \u003cp\u003eMeasurements were performed using a Hidex 300 SL liquid scintillation counter operated in TDCR mode, equipped with three photomultiplier tubes arranged at 120\u0026deg;. Samples were prepared in polyethylene vials containing 20 mL of HiSafe scintillation cocktail. Regions of interest were defined to include the full β spectra of parent and daughter radionuclides. Background corrections were applied using blank samples prepared under identical conditions. \u003csup\u003e210\u003c/sup\u003ePb samples were counted for 100 min while \u003csup\u003e90\u003c/sup\u003eSr samples were counted for 20 min each.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Sample Preparation and Chemical Separation\u003c/h2\u003e \u003cp\u003eReference materials of \u0026sup2;\u0026sup1;⁰Pb and ⁹⁰Sr provided within the framework of the Brazilian National Intercomparison Program (PNI) were used in this study. These materials are routinely employed for performance evaluation and interlaboratory comparison, providing independent reference activity values without serving as calibration standards.\u003c/p\u003e \u003cp\u003eChemical separation procedures were applied to isolate the parent radionuclides from their respective daughters (\u0026sup2;\u0026sup1;⁰Bi and ⁹⁰Y). The reference time (t\u0026thinsp;=\u0026thinsp;0) was defined as the end of the separation or precipitation step. Samples were measured repeatedly over time to monitor daughter ingrowth.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 TDCR Determination and Quench Monitoring\u003c/h2\u003e \u003cp\u003eDouble and triple coincidence events were recorded by the acquisition system, which directly provided the TDCR values for each measurement. The quench parameter equivalent (QPE) was monitored throughout the experiments in order to track possible variations in scintillation or optical conditions during the measurement period.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Bateman Formalism and Activity Correction\u003c/h2\u003e \u003cp\u003eActivities were initially determined using the TDCR method under the simplifying assumption of a single β-emitting radionuclide, applying TDCR-derived detection efficiencies and neglecting explicit contributions from radioactive progeny. Under this operational assumption, the activity obtained directly from TDCR represents an effective activity associated with the composite detection response of the decay system at a given time.\u003c/p\u003e \u003cp\u003eTo account for the presence of radioactive progeny in parent\u0026ndash;daughter decay systems, an activity correction based on the Bateman formalism was applied. The corrected parent activity, A\u003csub\u003ecorr\u003c/sub\u003e(t), was calculated as:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:{A}_{corr}\\left(t\\right)=\\:\\frac{{A}_{TDCR}\\left(t\\right)}{1+{f}_{d}\\left(t\\right)}=\\frac{{R}_{s}-{R}_{bg}}{{\\epsilon\\:}_{TDCR}\\left(t\\right)\\bullet\\:60\\bullet\\:m\\bullet\\:[1+{f}_{d}\\left(t\\right)]}\\:\\:\\:\\:\\left(1\\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere A\u003csub\u003eTDCR\u003c/sub\u003e(t) is the activity obtained directly from TDCR assuming a single β-emitter, R\u003csub\u003es\u003c/sub\u003e and R\u003csub\u003ebg\u003c/sub\u003e are the sample and background count rates, respectively, e\u003csub\u003eTDCR\u003c/sub\u003e(t) is the TDCR-derived detection efficiency, m is the sample mass, and f\u003csub\u003ed\u003c/sub\u003e(t) is the the relative contribution of the daughter activity with respect to the parent activity, calculated from the Bateman equations.\u003c/p\u003e \u003cp\u003eAt the reference time immediately after chemical separation (t\u0026thinsp;=\u0026thinsp;0), the theoretical value of f\u003csub\u003ed\u003c/sub\u003e(0) is zero, and the correction factor reduces to unity, such that the corrected activity equals the directly determined TDCR activity. As daughter ingrowth proceeds, f\u003csub\u003ed\u003c/sub\u003e(t) increases continuously. Under secular equilibrium, where the parent and daughter activities are equal, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{f}_{\\text{d}}=1\\)\u003c/span\u003e\u003c/span\u003e and the corrected activity becomes half of the activity obtained by direct TDCR application without correction.\u003c/p\u003e \u003cp\u003eThis correction approach is physically grounded, as it directly reflects the decay kinetics of the parent\u0026ndash;daughter system and explicitly separates spectral evolution effects from the activity determination. Importantly, it enables reliable determination of the parent activity at any measurement time after separation while fully preserving the primary, calibration-standard-free character of the TDCR method.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.5 Detection Limits and Uncertainty Considerations\u003c/h2\u003e \u003cp\u003eDetection limits were evaluated in accordance with ISO 11929 (2019), using the same measurement model adopted for activity determination. The activity was expressed as a function of the background count rate, the TDCR-derived detection efficiency, the sample mass, and the relative daughter contribution obtained from the Bateman formalism.\u003c/p\u003e \u003cp\u003eUncertainty components associated with each of these quantities were considered, including background counting statistics, TDCR efficiency stability, sample mass, and uncertainty in the daughter fraction. The combined uncertainty of the activity estimate was then used to derive the detection limit for symmetric decision and detection risks (α\u0026thinsp;=\u0026thinsp;β\u0026thinsp;=\u0026thinsp;5%).\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results and Discussion","content":"\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n\u003ch2\u003e3.1. Spectral interference and daughter ingrowth effects\u003c/h2\u003e\n\u003cp\u003eThe determination of Sr-90 and Pb-210 by LSC is inherently complicated by continuous \u0026beta; spectra and spectral overlap with their daughters, Y-90 and Bi-210. As shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, \u0026beta; spectra recorded shortly after separation differ substantially from those at secular equilibrium, reflecting the increasing contribution of the higher-energy daughter radionuclides.\u003c/p\u003e\n\u003cp\u003eFor the Sr-90/Y-90 system, secular equilibrium is reached approximately 10 days after separation, while for Pb-210/Bi-210 equilibrium is established after about 15 days. During this period, the increasing contribution of the higher-energy \u0026beta; emitters (Y-90 and Bi-210) significantly modifies the spectral shape observed by the liquid scintillation detector.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\n\u003ch2\u003e3.2. Temporal behavior of TDCR and ingrowth dynamics\u003c/h2\u003e\n\u003cp\u003eFigure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e shows the evolution of the normalized TDCR as a function of the relative daughter contribution predicted by the Bateman equations. For both radionuclide systems investigated, TDCR increases systematically as the contribution of the high-energy \u0026beta;-emitting daughter grows. Although the magnitude of this increase is small\u0026mdash;ranging from 73% to 79% for Pb-210 and from 91% to 94% for Sr-90\u0026mdash;it remained consistent throughout daily monitoring and therefore cannot be attributed to statistical fluctuations in isolated measurements.\u003c/p\u003e\n\u003cp\u003eDespite this systematic trend, the temporal evolution of TDCR does not follow the growth kinetics of the daughter activity. Instead, TDCR exhibits a bounded and gradually saturating behavior inherent to its definition as a ratio of coincidence probabilities. As a consequence, TDCR cannot reproduce the rapid increase in total counting rate associated with daughter ingrowth once the daughter contribution becomes significant, even though a measurable effect on the coincidence balance is already present.\u003c/p\u003e\n\u003cp\u003eThe stability of the Quench Parameter Equivalent (QPE), shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e, confirms that the scintillation yield and optical attenuation properties of the detection medium remained unchanged throughout the measurement period (u\u003csub\u003eQPE\u003c/sub\u003e \u0026lt; 0.5%). This observation excludes quenching variations as the origin of the observed TDCR increase and demonstrates that the effect arises under constant quench conditions.\u003c/p\u003e\n\u003cp\u003eUnder these conditions, the increase in TDCR must therefore originate from changes in the effective detected \u0026beta;-energy spectrum. As daughter radionuclides with higher \u0026beta; end-point energies progressively contribute to the signal, the relative balance between triple and double coincidences is modified, leading to an increase in TDCR even in the absence of quenching variations. This behavior demonstrates that TDCR is intrinsically sensitive to additional spectral contributions arising from time-dependent decay-chain evolution.\u003c/p\u003e\n\u003cp\u003eAccordingly, a systematic increase in TDCR without a corresponding variation in QPE may be interpreted as an experimental indicator of evolving spectral composition. For radionuclides with time-dependent decay schemes, TDCR therefore reflects changes in the effective \u0026beta;-energy distribution rather than acting solely as a quench monitor. This observation provides the physical basis for defining applicability limits for direct TDCR use and for implementing correction strategies when additional spectral contributions become significant.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\n\u003ch2\u003e3.3. Activity bias and applicability limits of direct TDCR\u003c/h2\u003e\n\u003cp\u003eTable 1 summarizes the evolution of activity bias and z-scores obtained using direct TDCR values for \u0026sup2;\u0026sup1;⁰Pb and ⁹⁰Sr as a function of time after chemical separation. For both radionuclides, a pronounced increase in bias is observed after the first 1\u0026ndash;2 days, while z-scores during the earliest measurements remain within the statistical compatibility criterion (|z| \u0026le; 2), despite the presence of positive bias.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 1 \u0026ndash;\u003c/strong\u003e Bias and z-score of \u003csup\u003e210\u003c/sup\u003ePb and \u003csup\u003e90\u003c/sup\u003eSr obtained when using direct TDCR value instead of efficiency for different time intervals of water sample from intercomparison program.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Taba\" border=\"1\"\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTime after separation\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eBias (%)\u003c/p\u003e\n\u003cp\u003e\u003csup\u003e210\u003c/sup\u003ePb\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003ez-score\u003c/p\u003e\n\u003cp\u003e\u003csup\u003e210\u003c/sup\u003ePb\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eBias (%)\u003c/p\u003e\n\u003cp\u003e\u003csup\u003e90\u003c/sup\u003eSr\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003ez-score\u003c/p\u003e\n\u003cp\u003e\u003csup\u003e90\u003c/sup\u003eSr\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1 day\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e8\u0026ndash;18\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.4\u0026ndash;0.9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e18\u0026ndash;28\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.2\u0026ndash;1.9\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2 days\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e16\u0026ndash;28\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.8\u0026ndash;1.4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e35\u0026ndash;45\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2.4\u0026ndash;3.0\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026ge;\u0026thinsp;3 days\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e30\u0026ndash;85\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e\u0026ge;\u0026thinsp;1.5\u0026ndash;4.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e45\u0026ndash;88\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e\u0026ge;\u0026thinsp;3.0\u0026ndash;6.2\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003ctfoot\u003e\n\u003ctr\u003e\n\u003ctd colspan=\"5\"\u003eNote: Individual datasets are provided in the Supplementary Material.\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tfoot\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eEach time point corresponds to independent replicate measurements (n\u0026thinsp;=\u0026thinsp;3), allowing statistical evaluation of bias and z-score. During this initial post-separation interval, the contribution of the daughter radionuclides remains limited and the composite \u0026beta; spectrum is still dominated by the parent emission. Under these conditions, the decay system behaves approximately as a quasi-pure \u0026beta;-emitter, allowing the application of TDCR without explicit correction when statistical acceptance criteria are adopted.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e presents the activity bias obtained by direct TDCR application as a function of the relative daughter contribution predicted by the Bateman equations for the ⁹⁰Sr/⁹⁰Y and \u0026sup2;\u0026sup1;⁰Pb/\u0026sup2;\u0026sup1;⁰Bi decay systems. A strong correlation is observed between the increase in activity bias and the progression of daughter ingrowth. The green line identifies the temporal window in which direct TDCR application remains feasible without correction.\u003c/p\u003e\n\u003cp\u003eWithin this window, corresponding to additional contributions below approximately 20%, activity bias remains moderate and the associated z-scores satisfy the statistical compatibility criterion. This defines a practical regime in which TDCR may be applied without explicit correction for routine or screening purposes.\u003c/p\u003e\n\u003cp\u003eIt must be emphasized, however, that this criterion reflects statistical consistency rather than strict metrological validity. In primary standardization and high-accuracy applications, substantially tighter bias limits are typically required, and corrections accounting for additional contributions to the detected signal\u0026mdash;beyond the parent \u0026beta;-emission\u0026mdash;become necessary. Accordingly, even within the early post-separation interval, the use of direct TDCR should be considered conditional and dependent on the intended application.\u003c/p\u003e\n\u003cp\u003eBeyond approximately 1\u0026ndash;2 days after separation, both Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e show a rapid increase in activity bias, accompanied by z-scores exceeding acceptance limits typically adopted in metrology. This behavior reflects the increasing contribution of the high-energy \u0026beta;-emitting daughters (⁹⁰Y and \u0026sup2;\u0026sup1;⁰Bi), which significantly modifies the effective detected \u0026beta; spectrum. Because TDCR is a bounded ratio, it cannot reproduce the corresponding increase in effective detection efficiency associated with the evolving decay scheme.\u003c/p\u003e\n\u003cp\u003eThe close agreement between experimental activity bias and Bateman-predicted evolution provides a clear physical basis for introducing a correction framework, which is presented in the following section.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\n\u003ch2\u003e3.4 Corrected versus uncorrected activity determination\u003c/h2\u003e\n\u003cp\u003eOn the basis of the systematic bias observed for direct TDCR application and its clear correlation with the Bateman-predicted evolution of the decay chain, a correction framework was implemented in which the TDCR-derived activity is explicitly adjusted according to the relative contribution of additional decay components. This approach allows the parent activity to be decoupled from the time-dependent composite \u0026beta; spectrum.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e presents the activity determination for \u0026sup2;\u0026sup1;⁰Pb as a function of time after chemical separation, comparing results obtained by direct TDCR application with those corrected using the Bateman formalism. Activities derived from direct TDCR application already show a systematic positive bias from the first 1\u0026ndash;2 days after separation, consistent with the bias trends identified in Section \u003cspan class=\"InternalRef\"\u003e3.3\u003c/span\u003e. This early overestimation reflects the rapid spectral influence of \u0026sup2;\u0026sup1;⁰Bi, whose higher \u0026beta; endpoint energy significantly affects the detected spectrum even at low relative activity contributions.\u003c/p\u003e\n\u003cp\u003eAs ingrowth progresses, the deviation between direct TDCR results and the reference value increases steadily, leading to pronounced overestimation well outside the associated uncertainty band. In contrast, activities obtained using Bateman-corrected TDCR remain stable throughout the entire measurement period and are statistically consistent with the reference value within its uncertainty. The corrected results exhibit no systematic temporal trend, demonstrating that the correction effectively compensates for the evolving decay scheme.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e shows the corresponding results for the ⁹⁰Sr/⁹⁰Y system. Direct TDCR application leads to measurable overestimation of the activity already within the first day after separation, with the bias increasing as the contribution of ⁹⁰Y grows. Although secular equilibrium in the ⁹⁰Sr/⁹⁰Y system is established only several days after separation, the spectral impact of the high-energy \u0026beta;-emission of ⁹⁰Y becomes detectable much earlier, resulting in a progressive divergence from the reference activity.\u003c/p\u003e\n\u003cp\u003eOnce the Bateman-based correction is applied, the TDCR-derived activities for ⁹⁰Sr remain stable over time and fully compatible with the reference value. The absence of systematic temporal dependence confirms that the correction framework is effective for both radionuclides, despite their markedly different \u0026beta;-energy distributions and ingrowth kinetics.\u003c/p\u003e\n\u003cp\u003eTaken together, Figs.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e demonstrate that, while direct TDCR application is intrinsically sensitive to changes in the decay scheme and leads to early overestimation of activity, the Bateman-corrected TDCR approach provides a robust and time-independent determination of parent activity, extending the applicability of TDCR beyond the initial post-separation regime. Although demonstrated here for two-member decay chains, the approach can, in principle, be extended to more complex decay schemes, provided that individual spectral contributions are well characterized.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\n\u003ch2\u003e3.5 Z-score analysis and statistical consistency of Bateman-corrected TDCR approach\u003c/h2\u003e\n\u003cp\u003eThe performance of the Bateman-corrected TDCR approach was further assessed through z-score analysis, which provides a standardized metric for evaluating statistical compatibility between measured activity values and reference results while accounting for their combined uncertainties. This analysis is particularly relevant in the context of intercomparison exercises, where acceptance criteria are commonly expressed in terms of z-scores.\u003c/p\u003e\n\u003cp\u003eFigures \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e show the temporal evolution of z-score values obtained for \u0026sup2;\u0026sup1;⁰Pb and ⁹⁰Sr, respectively, using TDCR-derived activities corrected according to the Bateman formalism. For comparison, direct TDCR-based activity results discussed in previous sections exhibit a systematic increase in z-score with time after chemical separation, exceeding the commonly adopted acceptance criterion (|z| \u0026le; 2) once the contribution of additional spectral components becomes significant.\u003c/p\u003e\n\u003cp\u003eIn contrast, the Bateman-corrected TDCR results yield z-scores predominantly within the interval |z| \u0026le; 2 over the entire measurement period for both radionuclides, including times approaching secular equilibrium. No systematic temporal trend is observed in the corrected z-score distributions, indicating that the correction effectively compensates for the evolving decay scheme and stabilizes the activity determination despite substantial changes in the composite \u0026beta; spectrum\u003c/p\u003e\n\u003cp\u003eImportantly, this improvement reflects not only a reduction in systematic bias but also the restoration of full uncertainty-consistent agreement with the reference values. The z-score analysis therefore provides an independent statistical validation of the corrected TDCR methodology, complementing the physical interpretation based on activity evolution and bias trends presented in the preceding sections.\u003c/p\u003e\n\u003cp\u003eFrom an experimental perspective, the statistical consistency demonstrated here establishes the necessary condition for extending TDCR-based activity determination beyond the early post-separation interval. Once bias and uncertainty propagation are properly controlled through Bateman-based correction, the remaining limitation on practical applicability is governed by counting statistics and background contributions, which directly determine the achievable detection limits. These aspects are addressed quantitatively in the following section.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e\n\u003ch2\u003e3.6 Detection Limits\u003c/h2\u003e\n\u003cp\u003eThe practical applicability of TDCR-based activity determination is ultimately constrained by the achievable detection limits under the prevailing measurement conditions. Having established in Section \u003cspan class=\"InternalRef\"\u003e3.5\u003c/span\u003e that the Bateman-corrected TDCR approach provides statistically consistent and time-independent activity results, the present section focuses on the quantitative detection capability of the method as a function of time after chemical separation.\u003c/p\u003e\n\u003cp\u003eDetection limits were evaluated in accordance with ISO 11929 using the same measurement model adopted for activity determination. Activities derived directly from TDCR measurements performed shortly after chemical separation were compared with activities corrected for daughter ingrowth using the Bateman formalism. This approach enables a direct assessment of detection capability under early post-separation conditions\u0026mdash;where direct TDCR application remains statistically acceptable\u0026mdash;and at later times, when correction becomes necessary.\u003c/p\u003e\n\u003cp\u003eThe evaluation explicitly accounts for experimental background count rates, counting time, TDCR-derived detection efficiency, sample mass, and the uncertainty associated with the relative daughter contribution predicted by the Bateman equations. In this framework, detection limits reflect not only instrumental performance but also the physical evolution of the decay system, as spectral changes directly influence the effective detection efficiency.\u003c/p\u003e\n\u003cp\u003eTable 2 summarizes the detection limits obtained for ⁹⁰Sr and \u0026sup2;\u0026sup1;⁰Pb under representative early post-separation and secular equilibrium conditions. The highest detection limits are observed on the first day after chemical separation for both radionuclides. At these early measurement times, the detected spectra are dominated by the low-energy \u0026beta; emissions of the parent radionuclides, resulting in relatively low signal-to-background ratios over the selected regions of interest. Under these conditions, background count rates\u0026mdash;approximately 40 counts per minute for ⁹⁰Sr (20 min counting time) and 37 counts per minute for \u0026sup2;\u0026sup1;⁰Pb (100 min counting time)\u0026mdash;constitute the dominant contribution to the combined uncertainty and therefore govern the resulting detection limits.\u0026nbsp;\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\n\u003cp\u003e\u003cstrong\u003eTable 2\u003c/strong\u003e - Detection limits obtained using direct TDCR and Bateman-corrected TDCR activity determination\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tabb\" border=\"1\"\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eRadionuclide\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eTime after separation\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eActivity model\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eDetection limit (Bq L⁻\u0026sup1;)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e⁹⁰Sr\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDay 1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDirect TDCR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.23\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e⁹⁰Sr\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDay 1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBateman-corrected TDCR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.19\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e⁹⁰Sr\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eSecular equilibrium\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBateman-corrected TDCR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.12\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026sup2;\u0026sup1;⁰Pb\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDay 1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDirect TDCR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.21\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026sup2;\u0026sup1;⁰Pb\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eDay 1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBateman-corrected TDCR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.19\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u0026sup2;\u0026sup1;⁰Pb\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eSecular equilibrium\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBateman-corrected TDCR\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.10\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eAs daughter ingrowth progresses and secular equilibrium is approached, detection limits decrease systematically for both radionuclides. This reduction reflects the increasing contribution of higher-energy \u0026beta; emissions from the daughter radionuclides, which enhances the effective detection efficiency and improves counting statistics while background conditions remain unchanged. The observed improvement in detection capability is therefore driven by physical changes in the detected \u0026beta;-energy spectrum rather than by modifications to the experimental setup or measurement protocol.\u003c/p\u003e\n\u003cp\u003eThe results in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e further indicate that the relatively elevated detection limits observed at early post-separation times are primarily governed by background and counting statistics, rather than by intrinsic limitations of the TDCR method itself. From an experimental perspective, these detection limits could be further reduced by increasing counting times, thereby decreasing the statistical contribution of the background to the combined uncertainty, without requiring any modification to efficiency modeling or calibration procedures.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eOverall, the detection limit analysis confirms that, once statistical consistency is ensured through Bateman-based correction, the practical sensitivity of TDCR-based measurements is determined by conventional counting statistics and background considerations.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4. Conclusions","content":"\u003cp\u003eThis work demonstrates that the Triple-to-Double Coincidence Ratio (TDCR) method can be reliably extended to radionuclides characterized by non-pure β-decay schemes that are of direct relevance to internal dosimetry and radiological emergency response, using \u0026sup2;\u0026sup1;⁰Pb and ⁹⁰Sr as representative and complementary case studies. Despite the intrinsic complexity introduced by daughter ingrowth and time-dependent spectral evolution, TDCR provides statistically robust and physically interpretable activity determinations when decay-chain effects are explicitly accounted for.\u003c/p\u003e \u003cp\u003eA key outcome of this study is the confirmation that TDCR enables accurate low-level activity determination without reliance on external calibration standards. This feature is particularly critical in internal dosimetry applications and emergency situations, where rapid assessments of intake are required and suitable reference materials may be unavailable or impractical to obtain. This constitutes a fundamental advantage over conventional liquid scintillation and Cherenkov counting approaches, which typically require certified standards and additional empirical corrections to address decay-chain effects. By relying exclusively on measured coincidence ratios and well-established decay kinetics, TDCR preserves its character as a primary measurement method, even for radionuclides traditionally considered challenging due to spectral non-stationarity.\u003c/p\u003e \u003cp\u003eThe observed temporal evolution of TDCR parameter for both radionuclides was shown to be driven primarily by changes in the effective detected β-energy spectrum associated with daughter ingrowth, rather than by variations in quenching conditions. This finding provides a clear physical basis for defining applicability limits for direct TDCR use in early post-separation or early post-intake scenarios, as well as for implementing correction strategies when measurements extend beyond this regime. While direct TDCR application remains statistically acceptable during the early measurement period, the Bateman-based correction introduced in this work extends the validity of the method over the full ingrowth interval, including conditions approaching secular equilibrium, yielding time-independent parent activities with z-scores consistently within acceptance criteria.\u003c/p\u003e \u003cp\u003eRegarding detection capability, the achievable detection limits are mainly governed by the background count rate and counting time, particularly in the early post-separation period. The higher detection limits observed on day 1 result from the dominance of background over the low-energy β emissions of the parent radionuclides, while daughter ingrowth progressively reduces detection limits by enhancing the effective detection efficiency and counting statistics. Although the resulting detection limits are higher than those obtained with ultra low-level systems, the ability to derive statistically consistent activities without external calibration standards represents a clear practical advantage for internal dosimetry and emergency assessments, especially in early post-intake scenarios where rapid measurements are required and reference materials may be unavailable.\u003c/p\u003e \u003cp\u003eOverall, this study expands the practical and metrological applicability of the TDCR method beyond pure β-emitters and demonstrates that, when combined with physically grounded Bateman-based corrections, TDCR constitutes a robust and flexible tool for activity determination in internal dosimetry, emergency response, and related environmental assessments, without compromising its calibration-standard-free nature.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eNo funding was received for conducting this study.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflict of Interest/Competing Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors have no conflicts of interest to declare that are relevant to the content of this article.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eC.S.S. and W.O.S. wrote the main manuscript text. All authors reviewed the manuscript\u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e\u003cp\u003eThe authors gratefully acknowledge Antonio Copote-Cuellar for insightful discussions and valuable comments that contributed to the interpretation of the TDCR behavior and the physical consistency of the proposed correction framework.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eBobin C, Thiam C, M\u0026rsquo;Hayham M-D, Mougeot X (2023) Activity standardization of \u003csup\u003e60\u003c/sup\u003eCo and \u003csup\u003e106\u003c/sup\u003eRu/\u003csup\u003e106\u003c/sup\u003eRh by means of the TDCR method and the importance of the beta spectrum. 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Nukleonika 67(4):6772\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003e\u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.2478/nuka-2022-0006\u003c/span\u003e\u003cspan address=\"10.2478/nuka-2022-0006\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"TDCR, non-pure beta-emitters, ²¹⁰Pb, ⁹⁰Sr, liquid scintillation counting, decay-chain correction","lastPublishedDoi":"10.21203/rs.3.rs-9161859/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9161859/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe triple-to-double coincidence ratio (TDCR) method is a well-established primary technique for the absolute standardization of pure β-emitting radionuclides in liquid scintillation counting (LSC). Its potential application to radionuclides of dosimetric and environmental relevance characterized by time-dependent decay schemes remains, however, insufficiently explored. This work presents a systematic experimental investigation of the applicability limits and correction strategies required to extend TDCR to non-pure β-emitters, using lead-210 (\u0026sup2;\u0026sup1;⁰Pb) and strontium-90 (⁹⁰Sr) as representative and complementary case studies.\u003c/p\u003e \u003cp\u003eMeasurements were performed using a Hidex 300 SL TDCR system following chemical separation of parent and daughter radionuclides. Direct TDCR-based activity determination yielded statistically consistent results only within a restricted temporal window, corresponding to daughter contributions below approximately 20\u0026ndash;22% of the total counting rate. Beyond this interval, TDCR values increased under constant quench conditions, demonstrating sensitivity to changes in the effective β-energy spectrum rather than to quenching effects alone.\u003c/p\u003e \u003cp\u003eActivities derived from TDCR measurements were found to follow the temporal evolution predicted by the Bateman formalism, providing a physical basis for implementing a Bateman-based correction. After correction, time-independent parent activities were obtained at any measurement time, with z-scores consistently within statistical acceptance limits. The proposed framework preserves the calibration-standard-free, primary character of TDCR and enables reliable low-level activity determination for radionuclides with complex decay schemes.\u003c/p\u003e \u003cp\u003eThese results demonstrate that Bateman-corrected TDCR is well suited for early post-intake measurements, routine internal dosimetry, and emergency situations where access to calibration standards is limited or unavailable.\u003c/p\u003e","manuscriptTitle":"Applicability Limits of the TDCR Method for Non-Pure β Emitters: Case Studies of ²¹⁰Pb and ⁹⁰Sr","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-04-03 11:17:48","doi":"10.21203/rs.3.rs-9161859/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"871a4b79-9a1e-495d-b0cc-b1bea44075d3","owner":[],"postedDate":"April 3rd, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Rejected","date":"2026-05-14T06:20:11+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-11T12:41:16+00:00","index":13,"fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-06T07:59:28+00:00","index":12,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-05-14T06:25:37+00:00","versionOfRecord":[],"versionCreatedAt":"2026-04-03 11:17:48","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9161859","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9161859","identity":"rs-9161859","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}