Rapid ⁹⁰Sr quantification method based on the Bateman equation for routine laboratory work

The main sources of radiochemical contamination are nuclear weapon tests [1] and accidents at nuclear power plants (e.g., Chernobyl in 1986) [2]. It is worth noting that the accident at the Fukushima Daiichi power plant released very low levels of ⁹⁰Sr [3]. The radioisotopes released in such accidents enter the biosphere and food chain, ultimately reaching humans. Among many fission products, ⁹⁰Sr is considered to be long-lived (half-life = 28.8 years [4]) and has high radiotoxicity because of the released β radiation. The maximum radiation energies and transition probabilities of ⁹⁰Sr and ⁹⁰Y decay are shown in Fig. 1.

Fig. 1

Because strontium and calcium exhibit similar chemical properties, these two elements compete with one another in human bone chemistry. However, their behaviours in organisms differ because strontium is absorbed relatively less effectively in the intestines, excreted more effectively by the kidneys and removed more rapidly from the bone surface [5]. Following acute ⁹⁰Sr absorption, an adult person (i.e., 25–45 years old) eliminates only 2.8–3.2% of the initial ⁹⁰Sr amount each year [5]. The β radiation released by ⁹⁰Sr and its daughter isotope ⁹⁰Y can cause leukosis, as well as other cancers that affect the bones and adjacent soft tissues. For this reason, environmental monitoring of ⁹⁰Sr and analysis of foodstuffs for ⁹⁰Sr content are crucial, and special emphasis should be placed on milk and dairy products [6, 7, 8, 9].

Typical analysis begins with sample mineralization by ashing in a muffle furnace, followed by dissolution of that ash in nitric acid. Further steps enable ⁹⁰Sr or ⁹⁰Y separation from the matrix and isolation from one another. Yttrium can be separated from strontium via liquid–liquid extraction with tributyl phosphate, which forms a complex with yttrium in the presence of nitric acid [10]. Thus, yttrium is transferred to the organic phase, while strontium remains in the aqueous phase. Yttrium is then back-extracted with 25% ammonia solution to form Y(OH)₃ [9, 11]. Finally, Y(OH)₃ dissolves in the nitric acid, and yttrium is precipitated as an oxalate [11, 12] and then further purified on ion exchange resin [11]. An alternative technique is to co-precipitate interfering radioisotopes with barium chromate, while ⁹⁰Y is precipitated as Y(OH)₃ [9, 13]. In this approach, strontium is left in the solution to promote ⁹⁰Y ingrowth. Then, yttrium is precipitated as an oxalate, and only the activity of ⁹⁰Y is the subject of subsequent measurements. There are also other potential procedures involving ion-exchange and extraction chromatography [8, 9, 14, 15, 16]. One strategy was developed based on forming a strontium complex with crown ether (4,4′(5′)-di-t-butyl-cyclohexano-18-crown-6), which is very selective for strontium [8, 9, 14, 15, 16]. Commonly employed detectors mostly use proportional gas flow and liquid scintillation counters [9, 10, 16].

Challenges related to ⁹⁰Sr analysis include the presence of calcium in the matrix, overlapping ⁹⁰Sr, ⁹⁰Y and ⁸⁹Sr β-spectra, and the required separation of other radionuclides [7, 8, 11]. These difficulties can be resolved by focusing on either ⁹⁰Y separation and immediate radioactivity measurements, or ⁹⁰Sr separation and waiting for secular equilibrium, at which point, half of the measured radioactivity originates from ⁹⁰Sr. Some methods rely on the separation of strontium from yttrium at this stage, and the measurement of ⁹⁰Y activity, which is then equal to the ⁹⁰Sr activity. The equilibrium between ⁹⁰Sr and ⁹⁰Y is reached after approximately 18 days, which makes for a long analytical process [9]. Another method is measurement of Cherenkov effect by liquid scintillation counter when scintillation cocktail is not added to the sample. This approach is preferred for samples where ⁸⁹Sr is present. If ⁹⁰Y is freshly separated from strontium, then the Cherenkov effect caused by ⁸⁹Sr high energy β radiation (E_max = 1495.1 keV) can be measured. Then subsequent measurement can detect the same effect caused by ⁹⁰Y β radiation for further calculation of ingrowth [9]. Measuring the activity immediately after ⁹⁰Sr separation is considered as a last-resort strategy for emergency situations because the accuracy of such analysis is compromised. However, there are mathematical approaches that can provide accurate results before secular equilibrium is reached [7, 17]. Therefore, it is not necessary to wait a long time for the equilibrium state. The aim of the present work was to develop a ⁹⁰Sr determination method based on the Bateman equation to enable calculations of daughter radionuclide activities according to the initial activity of the parent radionuclide and the decay rate of all considered radionuclides in absence of ⁸⁹Sr. The Bateman equation for the ⁹⁰Sr/⁹⁰Y isotope pair is shown in Eq. (1), where A_Y is ⁹⁰Y activity (Bq), A_Sr0 is initial ⁹⁰Sr activity (Bq), λ_Y and λ_Sr are the decay constants of ⁹⁰Y and ⁹⁰Sr (day⁻¹), t is time (days) and e is the Euler number: (1) AY=λYλY−λSrASr0(e−λSrt−e−λYt) {A_{\rm{Y}}} = {{{\lambda _{\rm{Y}}}} \over {{\lambda _{\rm{Y}}} - {\lambda _{{\rm{Sr}}}}}}{A_{{\rm{S}}{{\rm{r}}_0}}}\left({{e^{- {\lambda _{{\rm{Sr}}}}t}} - {e^{- {\lambda _{\rm{Y}}}t}}} \right)

The proposed calculation method also applies the concept of measuring the total activity and calculating the ⁹⁰Sr fraction in a modified manner [17].

In general, the total activity of the sample (A_tot) (in Bq) is equal to the sum of ⁹⁰Sr (A_Sr), ⁹⁰Y (A_Y) activities, as expressed in Eq. (2). Those individual activities change along with the decay of the corresponding radionuclides. However, since A_Sr and A_Y are unknown, they must be expressed as fractions of A_tot. Equation (3) defines A_tot as a sum of fractions changing with time (t; days), where λ_Sr represent the ⁹⁰Sr decay constant (day⁻¹) (2) Atot=ASr+AY {A_{{\rm{tot}}}} = {A_{{\rm{Sr}}}} + {A_{\rm{Y}}} (3) Atot=xAtote−λSrt+(1−x)Atote−λSrt {A_{{\rm{tot}}}} = x\,{A_{{\rm{tot}}}}{e^{- {\lambda _{{\rm{Sr}}}}t}} + (1 - x){A_{{\rm{tot}}}}{e^{- {\lambda _{{\rm{Sr}}}}t}}

The fraction of ⁹⁰Sr (i.e., x) is defined as shown in Eq. (4): (4) x=ASrAtot−ASrASr+AY x = {{{A_{{\rm{Sr}}}}} \over {{A_{{\rm{tot}}}}}} - {{{A_{{\rm{Sr}}}}} \over {{A_{{\rm{Sr}}}} + {A_{\rm{Y}}}}}

Because A_Y is unknown, it must be calculated using the Bateman equation. It is important to note that A_Sr is assumed to be equal to A_Sr0 because the half-life of ⁹⁰Sr is much longer than the time elapsed during the preparation and measurement of the sample. Thus, x can be expressed as shown in Eq. (5). Decay constant of ⁹⁰Y is denoted as λ_Y (day⁻¹): (5) x=ASrASr+λYλY−λSrASr(e−λSrt−e−λYt) x = {{{A_{{\rm{Sr}}}}} \over {{A_{{\rm{Sr}}}} + {{{\lambda _{\rm{Y}}}} \over {{\lambda _{\rm{Y}}} - {\lambda _{{\rm{Sr}}}}}}{A_{{\rm{Sr}}}}\left({{e^{- {\lambda _{{\rm{Sr}}}}t}} - {e^{- {\lambda _{\rm{Y}}}t}}} \right)}}

Because λ_Y is much greater than λ_Sr, the latter can be assumed to be equal to zero. Therefore, Eq. (5) is simplified to Eq. (6) (6) x=12−eλYt x = {1 \over {2 - {e^{{\lambda _{\rm{Y}}}t}}}}

To calculate A_Sr, Eqs. (2) and (3) are combined to give the expressions in Eqs. (7) and (8) as (7) ASr=xAtote−λSrt {A_{{\rm{Sr}}}} = x\,{A_{{\rm{tot}}}}{e^{- {\lambda _{{\rm{Sr}}}}t}} (8) AY=(1−x)Atote−λSrt {A_{\rm{Y}}} = (1 - x)\,{A_{{\rm{tot}}}}{e^{- {\lambda _{{\rm{Sr}}}}t}}

In the case of ⁸⁹Sr presence the approach presented above cannot be applied because ⁸⁹Sr and ⁹⁰Sr activities are independent of each other. In such instance, the total activity of the sample can be expressed as shown in Eq. (9), where A₈₉ is initial ⁸⁹Sr activity (Bq) and λ₈₉ its decay constant (day⁻¹) as (9) Atot=ASr0e−λSrt+λYλY−λSrASr0(e−λSrt−e−λYt)+A89e−λ89t \matrix{{{A_{{\rm{tot}}}}} \hfill & {= {A_{{\rm{S}}{{\rm{r}}_0}}}{e^{- {\lambda _{{\rm{Sr}}}}t}} + {{{\lambda _{\rm{Y}}}} \over {{\lambda _{\rm{Y}}} - {\lambda _{{\rm{Sr}}}}}}{A_{{\rm{S}}{{\rm{r}}_0}}}\left({{e^{- {\lambda _{{\rm{Sr}}}}t}} - {e^{- {\lambda _{\rm{Y}}}t}}} \right)} \hfill \cr {} \hfill & {+ {A_{89}}{e^{- {\lambda _{89}}t}}} \hfill \cr}

For calculation of ⁸⁹Sr and ⁹⁰Sr activities, separate measurement of ⁹⁰Y activity is desired after its ingrowth. In an emergency situation one cannot presume that ⁹⁰Sr and ⁹⁰Y are in equilibrium and also cannot wait for it. The activity of ⁹⁰Sr is shown in Eq. (10), where t_i is time of ⁹⁰Y ingrowth and t_m is time from ⁹⁰Y purification to the midpoint of the measurement as (10) ASr=AY(1−e−λYti)e−λYtm {A_{{\rm{Sr}}}} = {{{A_{\rm{Y}}}} \over {(1 - {e^{- {\lambda _{\rm{Y}}}{t_i}}}){e^{- {\lambda _{\rm{Y}}}{t_m}}}}}

In order to calculate the ⁸⁹Sr activity the sum of ⁹⁰Sr and ⁹⁰Y activities must be subtracted from the total activity. It is described in Eq. (11) where t₈₉ is the time from start to the midpoint of the activity measurement and t_Y is the time from yttrium separation to the midpoint of activity measurement, A_Sr+Y is the sum of ⁹⁰Sr and ⁹⁰Y activities and A_Y1 is ⁹⁰Y activity measured after its ingrowth. This approach is described in detail in [18]: (11) A89=Atot+ASr+Y−[AY1(1−e−λYtY)]e−λ89t89 {A_{89}} = {{{A_{{\rm{tot}}}} + {A_{{\rm{Sr}} + {\rm{Y}}}} - \left[ {{A_{{\rm{Y}}1}}\left({1 - {e^{- {\lambda _{\rm{Y}}}{t_{\rm{Y}}}}}} \right)} \right]} \over {{e^{- {\lambda _{89}}{t_{89}}}}}}

All reagents used in these experiments were of analytical grade or higher. The strontium carrier was prepared from 15.21 g of SrCl₂·6H₂O (Sigma-Aldrich) dissolved in 500 mL of distilled water. Nitric acid, aqueous ammonia solution (25%), oxalic acid, ammonium carbonate, sodium hydroxide, and disodium ethylenediaminetetraacetate (Na₂EDTA) were supplied by Chempur, Sr-resin (SR-B100-A; particle size = 100–150 μm) was supplied by Triskem, the scintillation cocktail Insta-Gel Plus was purchased from PerkinElmer, and ⁹⁰Sr and ⁸⁵Sr standard solutions were obtained from Polatom, Świerk (Poland).

Radiometry was performed on the investigated radionuclides using a low-background scintillation counter (Quantulus 6220, PerkinElmer). The counting protocol was set to a normal counting mode, with the transformed spectral index of the external standard (tSIE) as the quench indicator and the static control switched on (coincidence time = 18 ns; delay before burst = 500 ns; pulse amplitude comparison strength = 50). Guard compensation technology (GCT) optimization was conducted during the procedure development stage and was stored in the equipment memory; the GCT mode was set to ‘low’. The counting process proceeded for 60 min over a counting range of 0.5–1200 keV in low-diffusion plastic vials (PerkinElmer). Background subtraction ‘1st vial’ mode was used, where a ‘blank’ vial containing a sample without the analyte was placed in the first position on the rack. Then, its spectrum was considered as the background and was subtracted from the subsequent spiked sample spectrum. The Quantulus instrument was calibrated by measuring a ⁹⁰Sr/⁹⁰Y source in secular equilibrium. The counting efficiency was determined to be >98% and assumed to be 100% in all sample measurements.

In addition, samples spiked with ⁸⁵Sr were analysed to verify the strontium recovery independently from the proposed computational method. These analyses were performed using a high-purity germanium gamma spectrometer (Tennelec CPVDS-30 20190), which had 18% counting efficiency. The measurements were conducted in 100-mL cylindrical containers that are routinely used for ⁸⁵Sr determinations.

The linearity was determined by measuring four ⁹⁰Sr/⁹⁰Y sources in secular equilibrium at different activity levels. Then the counts were plotted as a function of the source activity, which enabled calculation of the determination coefficients.

The reliability of the applied procedure was determined based on ⁹⁰Sr recovery. Each subsample's radioactivity was measured immediately after preparation, and also later at a random time. This was necessary to evaluate the accuracy at any given moment.

Milk powder samples were analysed in terms of their ⁹⁰Sr content with an accredited method [13]. Specifically, the activity concentration was confirmed to be below the minimal detectable activity (MDA), and then a large sample of the material was ashed in a muffle furnace. The ash was divided into subsamples representing about 50 g of the original material. Sub-samples were placed in 250-mL beakers and spiked with ⁹⁰Sr standard solution by weight; there were two fortification levels (5 Bq·kg⁻¹ and 50 Bq·kg⁻¹) and 12 subsamples in each level. Finally, the analytical results were compared with previously calculated activity concentrations as a recovery measure.

The procedure began with the addition of the strontium carrier solution to the ash. In the next step, the ashed sample was dissolved in 60 mL (1:1 v/v) nitric acid in water and filtered. The solution volume was increased to 200 mL with distilled water, and then, 200 mL of 8% oxalic acid was added, with five drops of bromocresol green. Strontium oxalate precipitation was achieved upon adding ammonia solution to reach pH 4–4.5. The precipitate was ashed in a muffle furnace at 830°C for approximately 4 h. The subsequent steps are presented in Fig. 2. It is worth to point that elutes which here are considered as waste contain ⁹⁰Y which could be used for ⁸⁹Sr and ⁹⁰Sr calculation as described in Eqs. (10) and (11). This allows to calculate the results much quicker in comparison to methods which require ⁹⁰Sr/⁹⁰Y equilibrium.

Fig. 2

Prior to the analysis, 3 g of Sr-Resin was conditioned in distilled water for 12 h and then poured into a glass chromatographic column (inner diameter = 10 mm) and further conditioned with 20 mL of 65% nitric acid. The column was cleaned and refilled with Sr-Resin after each third analysis. After each first and second analyses, the resin was regenerated with 0.1 M Na₂EDTA/0.1 M NaOH solution, as advised by the manufacturer.

The ⁹⁰Sr concentrations (C_Sr-90; Bq) were calculated using Eq. (12) as (12) Csr−90=xCPM60e−λSrtrSrm {C_{{\rm{sr}} - 90}} = {{x{{{\rm{CPM}}} \over {60}}{e^{- {\lambda _{{\rm{Sr}}}}t}}} \over {{r_{{\rm{Sr}}}}m}} where x is determined from Eq. (6), CPM represents the count rate per minute, r_Sr is the strontium recovery, m is the sample mass (kg) and the time (t) was measured from the 0.05 M nitric acid addition until the end of the liquid scintillation counting (LSC) measurement. The chemical recovery of strontium was calculated with Eq. (13), where m_carb is the mass of the dry carbonate (g), C_car is the concentration of the carrier solution (g·g_sol⁻¹) and m_car is the mass of the added carrier solution (g): (13) rSr=mcarb1.685Ccarmcar {r_{{\rm{Sr}}}} = {{{m_{{\rm{carb}}}}} \over {1.685\,{C_{{\rm{car}}}}{m_{{\rm{car}}}}}}

The matrix effect was assessed by analysing ⁸⁵Sr-spiked samples. The analytical procedure was the same as that used for ⁹⁰Sr determination until the end of the chromatography step. After collecting the strontium fraction, the solution was diluted to 100 mL and measured using a gamma spectrometer to determine the ⁸⁵Sr recovery.

The MDA (Bq·kg⁻¹) [19] was calculated using Eq. (14) for each individual sample (14) MDA=2.71+4.65CPM603600m {\rm{MDA}} = {{2.71 + 4.65\sqrt {{{{\rm{CPM}}} \over {60}}}} \over {3600m}}

The combined standard uncertainty was estimated using the Nordtest method; this approach has previously been described in [20].

Table 1 presents the recovery statistics as the standard deviation (SD) for both fortification levels. The testing revealed no significant difference between the immediate and delayed results.

The mean ⁹⁰Sr recovery range indicates the negative bias of the procedure. The matrix effect was evaluated based on ⁸⁵Sr recovery measurements, and the results also indicate 83% mean recovery of strontium. Neither LSC detection nor calculations based on the Bateman equation were used in this experiment. The recovery results from both experiments are consistent, which indicates that the bias originates from the chemical treatment of the sample. Further optimization should improve this procedure and minimize the bias.

Table 2 presents the performance characteristics of the applied method.

Another aspect of the validation involved proficiency tests, which were organized by Procorad (an experienced proficiency test provider; http://procorad.org/). To test the method, two spiked (21SRB and 21SRC) and one blank sample of urine were analysed. Table 3 shows the obtained test results alongside reference values for comparison. The developed method detected a ⁹⁰Sr concentration in the blank sample below the MDA. Analysis of sample 21SRB resulted in a z-score of 0.5 and a normalized error of 0.3; acceptable values for z-score range from −2 to 2, and for normalized error, from −1 to 1. The result for the 21SRC sample obtained from analysis via the proposed method was not submitted for evaluation.

To evaluate radiological hazards due to the accidental intake of radionuclides, it is necessary to employ analytical methods that can rapidly determine the levels of specific radionuclides, especially pure β-emitters, with sufficient accuracy. This report confirms that the proposed approach satisfies this requirement and achieves accuracy comparable with slower methods currently being used in laboratory practice. However, it must be stressed that the proposed method where ⁹⁰Sr fraction of total activity is calculated should not be used immediately after a nuclear accident, because it does not take ⁸⁹Sr into account.

Overall, the time required for analysis was shortened significantly thanks to the Bateman equation, which is beneficial for commercial services and scientific research regarding ⁹⁰Sr in any matrix. Whole analysis takes 8 days and chromatographical separation with LSC sample preparation can be done in 8 h. The analytical precision was not influenced by implementing these calculations based on the Bateman equation. However, further optimizations of the sample treatment and chromatography stages are needed for matrices that are notorious for high Ca concentrations and those artificially fortified with mineral salts.