**Department of Mathematics and Statistics, Auburn University

Ellipsoid Model Development

We assume that a radionuclide is uniformly distributed within the volume of an ellipsoid. The ellipsoid shape was chosen because it permits a variety of geometric combinations to represent a particular tissue, body mass or anatomical region. Ellipsoids may be spherical, linear, or practically any shape in between. By representing tissue masses as ellipsoids, we can improve our estimates of energy dissipation affecting specific organs, organelles, or collections of tissue types. A particle or photon emitted by nuclear decay may traverse the ellipsoid in any direction and thus, different distances may be required for the energy to be expended outside of the ellipsoid.

A Cartesian coordinate system was chosen so that an ellipsoid e is defined as

for positive constants of a, b and c. Given , we define the inner product

and denote the norm induced by this inner product by

The Euclidean norm of x is denoted by , and with this notation the definition of the ellipsoid becomes

We denote the location at which radionuclide decay occurs by the random vector X and assume that X is uniformly distributed on e. When radioactive decay occurs, a particle or photon is ejected into a random direction. A unit vector U describes this direction, where U is uniformly distributed on the unit sphere . Since the surface area of the unit sphere is equal to 4p, the probability density function of U is given by

We assume that the random vectors X and U are independent vectors. Furthermore, we assume that the 

Ellipsoids have many possible shapesmaking it possible to mimic various sizes and geometries of animals, tissues, and media. Importantly, by using ellipsoids the geometric properties are estimable and the mathematical procedures remain the same for all shapes and sizes.

The methods developed in this work estimate the percent of particles or photons that are absorbed within the volume of any ellipsoid shape or size. This will enable researchers to calculate radiation dose imparted by radionuclides deposited within the body of many different kinds of animals and tissues.

The models developed here permit ellipsoids to be flattened, such as coins or leaves. These properties permit many more possibilities for modeling animal/tissue geometries and radiation dose.

The daily absorbed dose conveyed by a gamma photon or beta particle measured in milliGray per day (mGyd^-1) can be estimated by the following formula

where T is the concentration of a radionuclide measured in Becquerels (Bq) per gram, m is the mass (g) of the volume, and E is the average energy measured in MeV per disintegration (equation , c.f., Chesser et al. 2000). Of course, radioactive decay may impart dose via a variety of radionuclides present in the tissue and by a combination of beta and gamma energy contributions of a single radionuclide (total of R types). Therefore, the daily absorbed dose in the tissue comprising the ellipsoid e  

particle ejected by radioactive decay will travel some distance within the harboring tissue before interacting with another particle, at which time it conveys its total energy to that interaction. The rate of absorption of the particle or photon within the tissue is assigned the parameter l For a gamma photon, the constant l_g is equal to the "effective" tissue absorption coefficient (average)

For a beta particle, the constant l_b can be determined from the half-value layer (HVL) which is estimated by

(where E denotes the average beta particle energy) and the absorption coefficient is 

We define the Euclidean distance from the point X to the point on the boundary of the ellipsoid e as L=L(X,U). This also defines the ray emanating at X into the direction of U. Note that which is the probability that a particle is absorbed within the ellipsoid. Using the law of total probability and the independence of the random vectors X and U, one obtains

Using the appropriate rules for the change of variables and the calculation of a surface integral, one obtains for

the integrated Riemann integral

 is then given by

where f_i is the average fraction of energy from a single disintegration of the ith particle or photon that is absorbed within the ellipsoid e. Because T, m, and E are determined by direct empirical measurements, it remains to determine the absorbed fraction, f, for each particle type and energy. The equations (Riemann Integrals) on the left estimate this value.

The fraction of energy absorbed from beta particles released by

radioactive decay quickly reaches an asymptotic function as body radius exceeds one centimeter (mouse).

The rate of attainment of asymptotic energy absorption is fastest when the b energy is low (eg, ⁹⁰Strontium) and slowest with

high energy betas (eg, ⁹⁰Yttrium).

Gamma energy absorbed within the volume remains low even when

the body size exceeds that of a dinosaur and blue whale (~82%).

About 98% of gamma photon energy escapes the body of a

mouse (1 cm radius).

The methods for dose calculations for non-human species recently

released by the United States Department of Energy assume an

infinite body size. The dose rates from b particles will be far

overestimated when body sizes are small (mouse or less). Dose

rates from gamma photons will be substantially overestimated

for all species.

NUMERICAL SOLUTIONS OF THIS RIEMANN INTEGRAL PROVIDE ESTIMATES OF THE ABSORBED FRACTION (f) OF ENERGY FROM EACH NUCLEAR DECAY.

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