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

RESULTS
Absorbed
Fractions (f)
Particles
and g
Photons
Man
is approximately 10 cm
in
radius. Mouse is 1 cm.

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. 


Dose Rate Estimates
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 halfvalue 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.
CONCLUSIONS
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, ^{90}Strontium) and slowest with
high
energy betas (eg, ^{90}Yttrium).
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 nonhuman 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.
