1 Introduction

A generalized model of a mouse skeleton was developed with the purpose of generating absorbed fractions and dose factors for ⁹⁰Sr and ⁹⁰Y in the skeleton. The starting point was a model developed by Muthuswamy et al. [1]. In their model, four types of bone were described (1) Ribs, clavicle, sternum, pelvis; (2) Limb bones; (3) Vertebrae; and (4) Skull. They used simplified geometric constructs, namely (1) a 300 μm thick slab; (2) a 900 μm diameter cylinder; (3) a 200 μm diameter sphere; and (4) a 170 μm diameter sphere, to represent the four structures. They then estimated electron absorption in these structures for ¹³¹I, ¹⁸⁶Re, and ⁹⁰Y using a point kernel approach, numerically integrating absorption over the length of the particles’ path.

We adopted their geometric model and performed similar calculations using the Monte Carlo transport code MCNP [2]. This simple study was performed to answer a specific question related to a study involving the injection of strontium chloride into mice. Xie et al. [3] developed a skeletal dosimetry model for a rat model and provided absorbed fractions for photons and electrons at discrete starting energies. Xie and Zaidi [4] developed dose factors for a series of mouse models. They treated the skeleton as a uniform mixture of bone and marrow. Keenan et al. [5] developed absorbed fractions and dose factors for several mouse and rat models, and also modeled skeletal regions as a uniform mixture of bone and marrow, and the provided factors were averaged over all regions. We chose to adopt the region-specific descriptions developed by Muthuswamy et al. [1], using Monte Carlo method and employing the full beta spectra of ⁹⁰Sr and ⁹⁰Y to address the question posed.

2 Methods

Muthuswamy et al. [1] suggested that 47% of the marrow is in the first bone type, 20% in the second, 21% in the third, and 12% in the fourth. The geometric models were:

1) Ribs, clavicles, sternum, and pelvis (47% of total marrow): this region was modeled as a uniform bone/marrow mixture in a slab with a 300 μm thickness, a length and width of 2 cm, and surrounded by a sphere of tissue-equivalent material with a 10 cm radius. The composition of the mixture will be described below.

2) Limb (20% of total marrow): the ’limb’ region was modeled as an inner cylinder with a radius of 450 μm, containing marrow and surrounded by a cylinder of thickness of 350 μm, containing bone, length: 2 cm. The source was assumed to be in the bone and dose factors were calculated for the bone and marrow components separately. The length of the cylinder was set to an arbitrary value known to be longer than the range of the electrons, and was surrounded by a sphere of tissue-equivalent material with a 10 cm radius.

3) Vertebrae (21% of total marrow): this region was modeled as a uniform bone/marrow mixture, in a sphere with a 200 μm diameter, surrounded by a sphere of tissue-equivalent material with a 10 cm radius.

4) Skull (12% of total marrow): this region was modeled as a uniform bone/marrow mixture, in a sphere with a 170 μm diameter, surrounded by a sphere of tissue-equivalent material with a 10 cm radius.

In cases 1, 3, and 4, as noted, bone and marrow regions were not treated separately. The bone was considered to be a uniform mixture of bone and marrow. The densities of the bone and marrow were 2.02 g/cm^-3 and 1.04 g/cm^-3, respectively. We used the values established in humans, as we did not find specific values for various animal models in the literature. The compositions and fractions [6] used in this simulation are shown in Table 1.

Fractions of marrow in individual bones and fractions of bone in the whole skeleton of humans (detailed data for mice are not available) were taken from the MIRD mathematical phantom [6] and ICRP Publication 89 [7], to calculate the compositions of the mixtures, which are shown in Table 2 and Table 3. It’s assumed that the mouse has the same composition for skeleton as humans, the composition data were used for the simulation.

Densities for the bone-marrow mixture of the three regions of mixtures were calculated to be 1.38, 1.44 and 1.67 g/cm³, respectively.

The elemental compositions of the four bone types, as derived from humans [6, 7], are given in Table 4.

MCNP input files were prepared to represent various geometries, using the available appropriate combinatorial geometries. Beta spectra for ⁹⁰Sr and ⁹⁰Y were taken from the decay data compendium of Stabin and da Luz [8]. The material compositions from Tables 1 and 4 were coded into the MCNP materials cards, using the weight fraction option. An estep value of 6 was used, with the F8 tally. From 25000–150000 starting particles were employed. Reported tally uncertainties were under 1%.

3 Results

The MCNP simulation results are shown in Table 5. Shown are the fractions of the whole skeleton assumed to be comprised by the components, and the fractions of the total electron energy that was absorbed in that component, with comparisons to the values reported by Muthuswamy et al. [1] where possible.

Using the weight fractions from the human skeleton for the fraction of the total skeleton comprised by each bone type, and calclating the total fraction of Sr-90 and Y-90 energy absorbed in these bones, we obtain dose factors for the whole skeleton, which are 1.56×10^-11 Gy/disintegration(dis) for Sr-90 and 1.74×10^-11 Gy/dis for Y-90.The calculated individual dose factors are shown in Table 6.

4 Discussion

Bone and marrow dose models are some of the most difficult to characterize. Models for human bone and marrow have been evolving for decades [9] and are still under development by several groups. Anatomic models for rodents, and accompanying dose factors have been reported by Keenan et al. [5]. Dose factors for the whole skeleton were given, based on a simple model of the skeleton as a uniform bone/marrow mixture, which is widely used by researchers [3-5] and may lead to an error of 4% at most for the results, with scoring of photon and electron energy in individual bones, and no characterization of bone microstructure. Their approximate values for the Sr-90 or Y-90 sources in the skeleton of a 30 g mouse were 1.0×10^-11 Gy/dis and 2.1×10^-11 Gy/dis, which agree reasonably well with our values. Our study extended the results of Muthuswamy et al. [1], using modern Monte Carlo method and the full beta spectra of the nuclides instead of mean values. Ours was a fairly simplified treatment of bone and marrow dosimetry. Advanced techniques, such as small animal modeling and three dimensional modeling of electron transport in individual bone cavities can extend or confirm these results, but this is a very big undertaking.