Bone marrow dosimetry for mice: exposure from bone-seeking 89,90Sr

Studies of radiobiological effects in murine rodents exposed to internal radiation in the wild or in laboratory experiments require dosimetric support. The main problem of bone marrow (BM) dosimetry for bone-seeking β-emitters is dosimetric modeling, because the bone is a heterogeneous structure with complex microarchitecture. To date, there are several approaches to calculating the absorbed dose in BM, which mostly use rough geometric approximations. Recently, in the framework of studies of people exposed to 90Sr in the Urals, a new approach (SPSD) has been developed. The aim of the current study was to test for the first time the possibility of extension of the SPSD approach elaborated for humans to mice. For this, computational phantoms of femur bones of laboratory animals (C57BL/6, C57BL/6 J, BALB/c, BALB/cJ) aged 5–8 weeks (growing) and > 8 weeks (adults) were created. The dose factors DFSr-90(BM ← TBV + CBV) to convert the Sr isotope activity concentration in a bone tissue into units of dose rate absorbed in the bone marrow were 1.75 ± 0.42 and 2.57 ± 0.93 μGy day−1 per Bq g−1 for growing and adult animals, respectively, while corresponding values for DFSr-89(BM ← TBV + CBV) were 1.08 ± 0.27 and 1.66 ± 0.67 μGy day−1 per Bq g−1, respectively. These results are about 2.5 times lower than skeleton-average DFs calculated assuming homogenous bone, where source and target coincide. The results of the present study demonstrate the possibility of application of the SPSD approach elaborated for humans to non-human mammals. It is concluded that the study demonstrates the feasibility and appropriateness of application of the SPSD approach elaborated for humans to non-human mammals. This approach opens up new prospects for studying the radiobiological consequences of red bone marrow exposure for both laboratory and wildlife mammals.


Introduction
Beta-emitting strontium isotopes ( 90 Sr and 89 Sr) are present in the environment as a result of anthropogenic radiation events that may lead to significant consequences for humans and biota. For example, large amounts of Sr isotopes were released and globally dispersed due to atmospheric weapons testing in the mid of the last century (Povinec et al. 2005). Some local territories were also contaminated accidentally. For example, vast territories of Russia, Belorussia and Ukraine were contaminated with long-lived 90 Sr (half-life: ~ 29 years) due to releases of the Mayak Production Association facility into the Techa River (Degteva et al. 2016), and due to the Kyshtym (Avramenko et al. 1997;Izrael 2013) and Chernobyl accidents (Askbrant et al. 1996). Solubility of Sr results in a high bio-accessibility through food chains. Stable strontium is a low toxicity element. However, bone-seeking beta emitters may be adverse for hematopoietic and mesenchymal stem cells surrounded by bone structures. 90 Sr may irradiate the active bone marrow over a long period of time. Consequently, adverse health effects of chronic radiation exposure for small mammals are a highly debated topic (Fesenko 2019;Dahl et al. 2021;Shishkina et al. 2021a).
Internal radiation dosimetry for animals is an important issue of radiation research. There are several approaches to dosimetric modeling. The first one was elaborated in the framework of the system of environmental protection developed by the International Commission on Radiological Protection (ICRP) (ICRP 2008). This system is quite conservative and based on a set of principles, viz.: (1) a simplified representation of exposure geometry (elliptic body shapes); (2) a uniform radionuclide distribution within homogeneous body media; and (3) an absorbed dose averaged over the whole body. The second approach is used to support animal experiments of chronic radionuclide intakes (Bitar et al. 2007;Keenan et al. 2010;Locatelli et al. 2017). In this approach, realistic voxel-based three-dimensional computer models of mice and rats of different sex and age allow calculating doses from selected source or target pairs (cross-organ doses). The second approach has the great advantage that it can be combined with biokinetic models, and that it thus allows obtaining accurate organ-specific doses (Bolch et al. 1999(Bolch et al. , 2009. Existing realistic animal models describe the skeleton as a homogeneous medium and do not take into account any bone microstructure. However, doses in active marrow exposed to bone-seeking beta emitters, such as 89 Sr or 90 Sr/ 90 Y (energies of electrons emitted by these radioisotopes are 0-1.5 MeV and 0-2.4 MeV, respectively), should be calculated considering both macroand microstructures of bones (Shishkina et al. 2021b). The description of spongy bone microarchitecture as well as separation of the source and target (bone and bone marrow) are important because the fraction of energy absorbed in spongiosa is deposited within the bone outside the bone marrow volume.
Advanced methods for human bone dosimetry are based on combinations of ex vivo computed tomography (CT) images and micro-CT (μ-CT) or nuclear magnetic resonance micro-images of trabecular bone (Kramer et al. 2012;Zankl et al. 2018). Such methods provide a rather realistic model of a scanned bone. However, the dimensions of an individual bone may differ from those of a bone typical of population-average dimensions. Moreover, an image-based computational phantom is non-parametric and does not allow for the estimation of uncertainties associated with individual variability of bone morphology. The stochastic parametric skeletal dosimetry (SPSD) approach , which was originally proposed by Shishkina et al. (2018) and Zalyapin et al. (2018), is an alternative, which allows for the generation of models based on population averaged, sex-and age-specific morphometric data. This parametric approach allows creating a set of random models reflecting individual variability of bone micro-and macrodimensions. In the present study, computational bone phantoms were generated by the «Trabecula» software (Shishkina et al. 2020) as figures of simple geometric shape (Sharagin et al. 2018). Bone phantoms consist of a spongiosa, where rod-like bone trabeculae penetrate the bone marrow, and a cortical bone layer covering the spongiosa. The parameters to generate such a phantom are as follows: (1) macroparameters include linear bone dimensions and cortical thickness (Ct.Th); (2) microparameters of spongiosa include trabecular thickness (Tb.Th), trabecular separation (Tb.Sp) and bone volume fraction (BV/TV). Human-specific parameters were evaluated based on literature analyses as described in Sharagin et al. (2021) and Tolstykh et al. (2021). Similarly, a vast amount of morphometric information on bone-specific micro-and macro-dimensions for mice is available in the published literature and could be used in the present study for parameterization of murine-specific computational phantoms for bone dosimetry.
The aim of the current study was to test for the first time the possibility of extension of the SPSD approach elaborated for humans to non-human mammals. Mice, which belong to the reference animals typical of terrestrial ecosystems and which can be used as a convenient model for understanding any "exposure-dose" and "dose-effect" dependences, were chosen as the object of the study. The main questions to ask were as follows: (1) is there enough literature data on microarchitecture available to estimate the parameters of computational phantoms and (2) is consideration of bone microarchitecture (which would complicate bone dosimetry significantly) important for mice bone marrow dosimetry? One of the main hematopoietic sites of the adult murine skeleton are limb bones comprising from 20 to 35% of total active marrow (Shaposhnikov 1979;Boggs 1984;Epp et al. 1959;Taketa et al. 1970). Since the most of the published bone microarchitecture measurements are available for experimental animals, in the present study published results on femur micro-and macro-dimensions for laboratory mice of different strains were collected. Based on the parameters derived from the literature, a set of computational bone phantoms was generated and used for estimation of dose factors, converting the specific activity of 90 Sr and 89 Sr incorporated in cortical and trabecular bone volumes to dose rate in active bone marrow.

Creation of a computational phantom of murine femur
In the present study, the shape of the femur bone was simplified with cylindrical approximation (Fig. 1). The height (l) and diameter (d) of the cylinder were assumed to be equal to the most commonly used morphometric indices, viz., the maximum length of the femur (L), which is the distance from the greatest trochanter to the medial condyle, and the diameter (D), which is the width of bone in the middle of diaphysis. The thickness of the cortical layer ( Ĉ t.Th ) (which was assumed to cover the lateral cylindrical surface only) was assumed to be uniform and equal to the average Ct.Th of the bone. Parameters of the bone microstructure, viz., trabecular thickness ( T b.Th ), trabecular separation ( T b.S p) and bone volume fraction ( B V∕TV ), correspond to the histomorphometry parameters (Tb.Th, Tb.Sp and BV/TV) of standard nomenclature (Bouxsein et al. 2010).
Collection of published data on morphometric studies was based on the following criteria: − Only original papers were considered. − Only studies of "healthy" bones were accepted (without fractures, diseases or involvement of drugs that lead to damage to bone structures). − Only studies including a well-described experiment (strains, number of animals, sex and age, animal housing conditions and diet) were considered. − Only results obtained under similar conditions were included (no extreme diet or temperature or any other factors of influence). − All measurements on macro-dimensions (L, D) obtained with CT, microscopy or caliper were accepted. − Histomorphometry results (Ct.Th, Tb.Th, Tb.Sp and BV/TV) obtained with microscopes or μ-CT with method resolution < 70 μm were accepted.
The collected results comprise both male and female data and are related to mice of two wild types, 16 inbred strains of laboratory animals, as well as one publication summarizing bone microarchitecture data on 62 strains of inbred mice (Chanpaisaeng et.al. 2019). All data derived from the literature are presented in the electronic Supplementary Information file (Supplementary Information. xlsx). Bone dimensions of mice of different sex and strains were compared (Spearman correlation with α = 0.05, Mann-Whitney U test with α = 0.05) to decide whether it was possible to pool the data together for combined analyses.
The weighted averaging of morphometric dimensions (x i ) of different sample sizes (n i ) derived from publication i was done according to Eq. (1): where x is a a model parameter; x represents the uncertainty of x ; and N is the total number of data available for parameter x.
The individual variability of parameter x was calculated as a propagation of weighted average individual variabilities ( x i ) and the non-excluded systematic error of x equal to x (Eq. 2).
The obtained estimates of the parameters and their individual variability were used as an input for the generation of a computational phantom with the «Trabecula» software (Shishkina et al. 2020). Another input to create the SPSD model is the intra-specimen variability of the trabecular thickness and the trabecular separation. It is noted, however, that no description of the intra-specimen variability of these parameters in mouse bones could be found in the available literature. Therefore, values of the variation coefficients typical of human bones (as it was presented in Supporting materials S1 to Degteva et al. 2021) were used as a surrogate, namely, 22% and 23% for T b.Th and T b.Sp , respectively. Computational phantoms with average dimensions ( x) and 12 random variative models (within ± ̂ x ) were generated using the software «Trabecula». The software automatically calculates the volumes of source and target tissues for each of the generated phantoms.

Monte Carlo simulations of radiation transport
Each set of phantoms was used for Monte Carlo simulation of electron and photon transport within the bone and bone marrow media to calculate the energy deposition ( E r (BM ← S) ) in bone marrow (BM) per decay of Sr isotopes (r-radionuclide considered) uniformly distributed in cortical or trabecular bone media (S-the source tissues). Probability of electron emission of different energies for 89 Sr, 90 Sr and 90 Y decays were taken from the Atomic Data Information System-«Janis 4.1» (Javabased Nuclear Data Information System) (Soppera 2014) available in the public domain. Mean energy of 89 Sr electrons per decay is equal to 0.585 MeV (Q = 1.495 MeV). The decays of 90 Sr (Q = 0.546 MeV) and its progeny 90 Y (2.28 MeV) were computed with equal probability (assuming secular equilibrium). The mean 90 Sr + 90 Y energy per 90 Sr disintegration is equal to 0.570 MeV (similar to that for 89 Sr). The overall energy deposition was calculated per mother's radionuclide decay. Elemental composition of simulated media was assumed to be the same as for humans (Table. 1) (ICRP89 2002). In contrast to the chemical composition, the murine bone density is lower than that typical of humans (1.64-1.89 г/cm 3 ) (Broulík et al. 2013), and a value of 1.57 g/cm 3 is used. Radiation transport was done using «MCNP 6.2» (Monte Carlo N-Particle Transport Code). The number of histories was at least 4,000,000, and the statistical error less than 1%.

Calculation of absorbed dose rate in the bone marrow of murine femur
Dose factors ( DF r (BM ← S) ) to convert the radionuclide specific activity in a source tissue into units of dose rate absorbed in the BM were calculated by normalization of the energy deposition per decay according to Eq. (3): where m S is the mass of a source tissue; m BM is the mass of the target tissue Total dose rate in BM can be expressed by Eq. (4) where Ḋ r is the dose rate in BM; A r (CBV) and A r (TBV) are the radionuclide (r) activity concentrations in the cortical (CBV) and trabecular (TBV) bone volumes, respectively. Assuming uniform radionuclide distribution in the whole bone media ( A r (CBV) ≈ A r (TBV) = A r ), the overall dose rate per 1 Bq g −1 was calculated with Eq. (5).

Computational phantom of a murine femur
Because macro-dimensions (L and D) did not correlate with murine strains, all literature-derived macro-dimensions could be combined (14 papers describing > 500 animals). In contrast, Spearman correlation analysis showed a statistically significant relationship (P < 0.05) between the mouse strain and all microarchitecture measurement results (BV/TV, Tb.Th, Tb.Sp, Ct.Th). In other words, the macro-dimensions are not different for different strains and all bone linear dimensions available can be pooled together for analysis. However, the micro-dimensions are different in different strains. Several strains only have no statistically significant difference in micro-dimensions, viz., C57BL/6, C57BL/6 J, BALB/c, BALB/ cJ. Consequently, the laboratory animals of these strains were selected as modeling subjects. It should be noted that the greatest number of studies were devoted exactly to these strains. In total, 15 papers describing different morphometric parameters of more than 400 mice were considered for spongiosa microarchitecture modeling (Table 2). No statistically significant relationships between microarchitecture dimensions and sex were found. Therefore, the computational phantoms were constructed without sex differentiation. However, age-related changes in bone dimensions were considered. Data on animals of different age were divided into three groups: juvenile (≤ 5 weeks), growing (5-8 weeks) and adult (> 8 weeks). As it can be seen from Table 2, almost no papers provided information on microparameters for juveniles. Data on femur length (L) were mainly from growing and adult animals. For example, the minimal age of animals with measured L, which were studied in parallel with microparameters (Table 2), was 4 weeks. Data on femur diameters were found for adults only (Dubrovsky et al. 2020). Consequently, femur computational phantoms were created for growing and adult mice only, and the femur diameter of the growing mouse was assumed to be equal to that of an adult mouse as a first approximation.
Mean values of Tb.Th, Tb.Sp and BV/TV were almost equal across age groups. However, the corresponding individual variability was about twice higher in adult than in growing mice. Ct.Th increased up to an age of 24 weeks (Papageorgiou et al. 2020). Therefore, both Ct.Th and corresponding individual variability were two times higher in adults. Table 3 shows the parameters estimated based on morphometric data for femur phantoms of growing and adult laboratory animals.
As a result, two sets of computational bone phantoms were generated for two age groups. Each one includes a femur model with average group parameters (basic phantom) and 12 randomly generated femur models to simulate the individual variability of bone dimensions (supplementary phantoms). These phantoms were used for dose factor calculations. Table 4 presents the dose factors calculated for 90 Sr (in equilibrium with 90 Y) and 89 Sr considering TBV and CBV as source organs, to convert the radionuclide specific activity into the dose rate in the BM. The results are shown a central estimates (calculated with the basic phantom) ± root mean square deviations (rmsd) of 12 DF r (BM ← S) values obtained using the supplementary phantoms. The rmsd reflects the effect of individual variability of bone dimensions.

Dose factors
According to Table 4, the dose rate due to 1 Bq g −1 of strontium isotopes in TBV of adults is about 20% lower than Table 2 The literature sources (Dubrovsky et al. 2020Doucette et al. 2015Xiang et al. 2007;Cao et al. 2010;Martín-Badosa et al. 2003a, b;Glatt et al. 2007;Voide et al. 2008;Chiang and Pan 2011;Verdelis et al. 2011;Ma et al. 2011;Wu et al. 2013;Bagi et al. 2011;Tamasi et al. 2013;Turner et al. 2000) providing useful information for the present study: gray cells indicate the presence of information on the parameter studied in the paper (m male, f female) that of growing mice. This is a combined effect of a slightly lower bone fraction of spongiosa (BV/TV) for adult as compared to growing mice and a greater spongiosa surface (66 mm 2 versus 57 mm 2 ), which increases the probability of energy loss from the spongiosa volume. In contrast, DF r (BM ← CBV) of adult mice is about two times higher than that of growing mice. This is the effect of a twice as high Ĉ t.Th value in the adult phantom. The difference in cortical thickness led to the difference in source tissue volume. As a result, this changed the m S m BM ratio (Eq. 3). The variability of DFs did not exceed 64% for S = TBV and 22% for S = CBV. The individual variability of BV/TV (proportional to spongiosa density) is quite high (Table 3): 38% and 75% for growing and adult mice, respectively. As a result, the uncertainties associated with DF r (BM ← TBV) were about the same. In addition to BV/TV, the individual variability of Ct.Th is also an important factor that influences DF r (BM ← CBV) uncertainty. The variability of Ct.Th for growing and adult mice was 22% and 42%, respectively. The combined effect results in an uncertainty for DF r (BM ← CBV) 38% and 66%, respectively.
Separation of the trabecular and cortical bones as source tissues can be useful when combining a dosimetric model with a biokinetic one that takes into account the difference in bone remodeling with time of these two types of bones. However, currently available biokinetic models for rodents do not distinguish between cortical and trabecular bones (Malinovsky et al. 2013) and, consequently, in the present study doses were calculated assuming a uniform radionuclide distribution across all bone structures. Therefore, Table 4 includes the dose factors of a combined CVB + TBV source. The overall uncertainties of the estimates were about 25% and 40% for growing and adult mice, respectively.

Dose factors calculated with different approaches
Dose calculations can be made with different accuracy and precision depending on the specific task. For example, for the purpose of radiological protection conservative estimates are usually made for a reference animal. In contrast, radiobiological and medical studies often need accurate individual radiation doses. To estimate the differences in dose factors calculated with different approaches, SPSDbased dose factors were compared in the present study with dose factors calculated with the conservative approach implemented in the ERICA Tool 2.0 (Brown et al. 2016) as well as with dose factors calculated using the RODES VI software developed by IRSN (Locatelli et al. 2017). The Erica Tool was elaborated to support decision-making on environmental issues related to the effects of ionizing radiation on non-human biota with emphasis on ensuring the structure and functioning of ecosystems. On the contrary, the RODES VI software was created to support dosimetry for animal experiments including chronic radionuclides intake. Within the ERICA Tool 2.0, mouse body is assumed to have a spheric/ellipsoid shape and to be filled with a homogeneous medium containing uniformly distributed radioactivity. In other words, both body heterogeneity and the bone-seeking nature of strontium are ignored. In contrast, the RODES VI software uses more realistic voxelbased three-dimensional computer models of mice and rats, which were developed based on magnetic resonance imaging. However, RODES VI also uses some simplifications, such as the assumption of bone homogeneity. Consequently, in this software mean absorbed energy is calculated for the whole uniformly-contaminated skeleton. In other words, source (bone) and target (BM) are not separated. Table 5 compares dose factors calculated with different approaches for adult mice due to incorporated 89,90 Sr. The source tissue is indicated as S; the target is indicated as T. Erica Tool 2.0 calculates the body-average dose (for a body with dimensions 8.9 × 2.5 × 2.5 cm and m = 20.5 g). Using RODES VI the skeleton-average dose for mice with a body mass m = 20 g was calculated.
As it can be seen from the table, the conservative ERICA approach is not appropriate for internal BM dosimetry: estimating the body-average radionuclide activity concentration based on data on 90 Sr contamination of bones needs additional information on strontium intake and biodistribution. A rough Table 4 Dose factors DF r (BM ← S) for 90 Sr and 89 Sr considering TBV and CBV as source organs (S = TBV or S = CBV) to convert the radionuclide specific activity into the dose rate in the BM-target organ; rmsd-root mean square deviation a Assuming uniform radionuclide distribution within all bone structures S source organ, BM bone marrow, TBV trabecular bone volume, CBV cortical bone volume Age DF r (BM ← S) ± rmsd, (μGy day −1 ) per (Bq g −1 ) extrapolation of radionuclide bone burden to the whole body results in a body-average dose five times higher than that calculated with SPSD for bone marrow directly. Neglecting the bone microstructure and calculating average energy absorption in the skeleton (RODES VI) leads to an overestimation of the dose to the bone marrow by a factor of 2.5 ( Table 5). The skeleton dose rates calculated using RODES VI cannot be propagated to BM either; they do not fall within the 90% confidence intervals of SPSDbased results. This example illustrates the importance of taking into account bone microstructure in the evaluation of internal exposure of BM to bone-seeking beta-emitting radionuclides. It should be noted, however, that the authors of the RODES software explicitly mentioned this issue as a further possibility to improve their software (Locatelli et al. 2017).

Dosimetry of rodents exposed at the East Urals radioactive trace
Vast territories of the Urals were contaminated due to long-lived 90 Sr released as a result of a thermal explosion of a storage tank of radioactive waste in the territory of the Mayak Production Association, Russia (the so-called Kyshtym accident) (Avramenko et al. 1997;Izrael 2013) on 29 September 1957. In the course of the accident, radionuclides were deposited on the soil surface forming the socalled East Urals radioactive trace (EURT), which led to the radiation exposure of the inhabitants of the contaminated territories, rodents among them. Despite the fact that more than 60 years (about two half-lives of 90 Sr) have passed since the Kyshtym accident, a large part of the EURT territory is still heavily contaminated. For example, in 2001-2012 the concentration of 90 Sr in the first 10 cm layer of soil from that area reached 20 Bq g −1 (Molchanova et al. 2009(Molchanova et al. , 2014Mikhailovskaya et al. 2019). Many studies of radiation effects in EURT rodents have already been published (Ilyenko 1974;Gileva 2002;Orekhova and Modorov 2017;Orekhova et al. 2019) and continue to be carried out. These studies on radiation effects should be supported by radiation dosimetry studies. In principle, internal BM exposure due to incorporated 90 Sr can be estimated based on the information about skeleton contamination (Starichenko 2004;Starichenko et al. 2014). In Shishkina et al.'s (2021a) study, results on 90 Sr skeleton contamination of herb field mouse (Sylvaemus uralensis) obtained during a long-term (2003-2012 years) study are summarized. In that study, three sampling areas were selected for the purpose with the following initial (just after the explosion) 90 Sr deposition density: (1) 3.7-37 MBq m −2 ; (2) 74-3,700 kBq m −2 and (3) 37 kBq m −2 . Animals were classified according to four functional-age groups: (1) juveniles; (2) subadults-nonbreeding animals; (3) breeding underyearlings-not wintered adult animals; (4) breeding overwintered adult animals. The data on subadults can be roughly used for growing mice; in contrast, two groups of breeding individuals (underyearlings and overwintered animals) can be associated with adults in the current study. Table 6 presents the group-average 90 Sr activity concentrations in the murine bones (per wet mass) estimated according to Shishkina et al. (2021a) and the corresponding BM dose rates. The 90% confidence intervals (CIs) for the dose rate reflects the overall individual 1 3 variability of the activity concentrations and dose factors. The overall variation coefficients were calculated with the uncertainty propagation law; 90% CIs were estimated using the lognormal approach.
In the wild, the mean life span of a mouse-like rodent is about 1.5 years. The cumulative internal dose in the rodent bone marrow estimated with the SPSD approach is about 70 mGy and does not exceed 120 mGy in the EURT territories with 3.7-37 MBq m −2 of surface contamination.
Taking into account the non-uniform distribution of boneseeking 90 Sr in an organism, the maximum organ dose is formed in bone tissue and bone marrow. The remaining tissues are exposed to a lesser extent (Malinovsky et al. 2013(Malinovsky et al. , 2014. Therefore, the dose rates obtained for bone tissue and bone marrow could be considered as a conservative estimate of whole-body dose rates to be compared with a screening value (the dose rate threshold that does not lead to an unacceptably high effect on the structure and function of an ecosystem). A screening value of 240 μGy day −1 for chronic exposure was established in the Erica Assessment Tool (Brown et al. 2016) based on the analysis of chronic exposure data from more than 26,000 data entries in the original Fine-Root Ecology Database (FRED) (Iversen et al. 2017). Dose rates estimated with the SPSD-based approach are notably lower than the screening level.
To compare this result with the commonly accepted approach implemented in ERICA Tool 2, the bone-specific activity concentration of 90 Sr had to be converted to a body-average activity concentration value. For this, the biokinetic model of 90 Sr (Malinovsky et al. 2013) for adult mice has been used here to calculate the soft tissue exposure assuming chronic intake and steady state between the organism and environment. As a result it was found that the soft tissue activity concentration should be 0.162 times the bone activity concentration. Assuming the mass fraction of skeleton as 13% of total body mass (Malinovsky et al. 2013), 1 Bq g −1 of 90 Sr in bone tissue corresponds to 0.27 Bq g −1 of body-average activity concentration. Therefore, to calculate the body-average dose rate based on a given 90 Sr activity concentration in bones the dose factors from ERICA Tool2 (Table 5) should be corrected. For example, 0.27 × DF Sr−90 (wholebody ← wholebody) = 3.48 (μGy day −1 )/(Bq g −1 of bone wet mass). This value is 1.8 times higher than the corresponding SPSD dose factor (Table 5). Even with this conservative approach, the mean dose rates in the sampling areas do not exceed 200 µGy day −1 . The maximum dose calculated for the area with maximum contamination level can reach 290 μGy day −1 (calculated with the maximum 90 Sr activity concentration ~ 110 Bq g −1 in mice bones reported by Starichenko et al. (2004); Starichenko et al. (2014)) In other words, both the conventional approach developed for the purpose of radiation protection and the SPSD-based bone marrow dose estimates do not, on the average, exceed the screening level of 240 μGy day −1 from ERICA Tool2.
In contrast, considering homogenous spongiosa (RODES VI) for dose prediction provides values comparable with the screening value in the most contaminated area (3.7-37 MBq m −2 ), i.e., a mean dose rate of about the screening level ~ 240 μGy day −1 (up to 400 μGy day −1 ). This approach has been used for murine-specific internal dosimetry in a number of EURT studies (Malinovsky et al. 2014;Modorov 2014) using dose factors estimated by Chesser et al. (2000) and calculations based on Stabin et al. (2006); these dose factors are quite similar to those from RODES VI (6.5 (μGy day −1 ) per (Bq g −1 ) for 90 Sr and 3.6 (μGy day −1 ) per (Bq g −1 ) for 89 Sr). It should be noted that no changes in murine population size, reproductive activity and morphophysiological characteristics, which may affect any EURT inhabiting population as a structural unit of the local community and ecosystem, were found in the studies (Tarasov 2000;Lyubashevsky et al. 2002a, b;Olenev and Pasechnik 2003;Orekhova and Modorov 2017). The absence of any pronounced radiation-induced changes at the population level is not consistent with the risks expected from dose predictions based on the homogenous spongiosa approach. In other words, in case of bone-seeking beta emitters, doses to the bone calculated assuming a uniform radionuclide distribution in a homogenous bone medium don't reflect real bone marrow exposure and overestimate body-average doses considerably. Thus, the fact that the observed radioecological effects are not consistent with accumulated doses estimated for the EURT by means of RODES VI, which shows a 2.5 times dose overestimation due to the applied approach of homogenous spongiosa caused considerable scientific discussions (Shishkina et al. 2021a), may be solved with the help of the adequate dosimetry. This example highlights the importance of taking bone microarchitecture into account, as was done in the present study by applying the SPDS approach.

Further direction of bone marrow dosimetry for rodents
Limb bones, including femur and bone marrow therein, present only a part of the murine skeleton. The skeleton includes additional regions where parts of the hematopoietic system are located such as the spine (20-32% of active marrow), skull (11-20% of active marrow) and pelvic bones (9-13% of active marrow) (Shaposhnikov 1979). Therefore, to get an accurate estimate of the skeleton-average dose factor for bone marrow, it is necessary to include all these sites in the dose modeling. Consequently, any next step of dosimetric modeling of bone marrow exposure should include development/elaboration of computation phantoms of the sites mentioned above.
According to Shishkina et al. (2021b), the mean free path of electrons (in continuous slowdown approximation) emitted during the decay of 90 Sr + 90 Y and 89 Sr in spongiosa is between 0.15 and 0.22 cm (calculated with Eq. 6). This range of values is comparable with typical mice bone sizes.
According to Volchkova et al. (2022), the main factors that influence DF Sr−90 (BM ← S) for such small bones are: (1) the surface area (depending on bone dimensions) where radiation losses are possible; and (2) the source-totarget mass ratio (proportional to BV/TV). Therefore, it is important to specify both bone geometry and microarchitecture parameters for computational phantoms of different bone sites with hematopoiesis. Taking into account the smaller dimensions of pelvic and skull bones (as compared to the femur), which are very thin, one may expect greater energy losses from the volume of such small bones. And, as a result, the skeleton-average DF r (BM ← S) may be smaller than that calculated for the femur.
Another unsolved issue is the lack of sufficient data to create adequate phantoms of juvenile animals (younger than 5 weeks). In this context, the problem is not in the absence of morphometric data, but in the daily morphometric changes of bones during the early period of development. One solution could be to create a phantom of a newborn mouse, and then interpolate the values of dose coefficients obtained for the newborn between 0 and 5 weeks.
Finally, the assumption of equal radionuclide activity concentrations in TBV and CBV used in the calculations is not ideal. Actually, immediately after intake Ca-like elements are predominantly incorporated in TBV. Sometime later, however, they will be concentrated in CBV, due to continuous resorption and remodeling of trabecular bone. For humans, for example, it takes about ten years until the concentrations of long-lived 90 Sr become equal in trabecular and cortical bone . For mice, this may take less than a year (rough human-to-mice extrapolation based on life span). In the case of chronic intake of longlived 90 Sr (as it is typical for populations living in the EURT region) one can expect a steady-state radionuclide activity concentration distribution between various body compartments. However, for the scenarios of a single 90 Sr intake and of single and chronic intakes of 89 Sr (T 1/2 ~ 50 days) one needs an accurate biokinetic description of radionuclide accumulation in TBV and CBV separately. Dosimetric modeling with the SPSD approach is suitable for this purpose (see Table 4). However, the Sr biokinetic models for mice should be improved to take TBV and CBV compartments separately into account.
These results are about 2.5 times lower than those calculated for the whole skeleton assuming homogenous bone, where the source and target region coincide. The lower doses estimated in the present paper might explain that no significant ecological effects on a population level has been observed among mouse populations that live in the EURT region. In the future, the present dosimetric models for growing and adult mice should be improved. The improvement should include creation of computational phantoms of all hematopoietic bone sites of mice skeleton taking into consideration bone size and shape as well as a site-specific bone microarchitecture. An additional challenge is the dosimetric modeling for the first 5 weeks of the life of a mouse.
Besides the dosimetric modeling, bone dosimetry would benefit from an improvement of the 90 Sr biokinetic model to describe the age-dependent dynamics of strontium biodistribution in trabecular and cortical bone regions included as separate compartments.

Supplementary Information
The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s00411-022-01010-3. Funding The authors have no relevant financial or non-financial interests to disclose. The authors did not receive support from any organization for the submitted work.

Author contributions
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The authors have no competing interests to declare that are relevant to the content of this article. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.