For a given treatment plan P consisting of specified beam angles and
individual beam doses, let 
be the represent the
predicted dose per voxel as a function of P and voxel location x.
We wish to measure the utility or ``goodness'' of
D -- denoted by 
-- in a manner that is both clinically
useful and computationally feasible.
We propose to assume 
is of the following form:
where each 
is a function of the values of 
restricted to some subset 
of all possible voxels. Each voxel
will be contained in exactly one subset 
. Each subset 
corresponds to a single tissue type, such as normal brain tissue,
tumor, or a particular type of critical normal tissue.
might, for example, combine 
through 
in some simple fashion such as a weighted arithmetic
or geometric mean.
Every 
is assumed to be a function of a histogram of
the values that 
takes on over the set of 
.
Formally,
where
is the count of voxels 
such that 
;
and 
are the dose values corresponding to the 
histogram's 
bin's lower and upper bounds, respectively.
For example, Figure 1 shows a crude representation of a
cross-section of a possible brain scan. The tissue voxels have been
partitioned into five different subsets: 
for normal brain
tissue, 
for tumor tissue, 
for the eyes,
for the brain stem, and 
for the skull.
Figure 2 shows a possible histogram for the radiation
per voxel over the 
voxel subset; Figure 3 shows
a possible histogram for the 
subset.
One factor of possible clinical significance that the above
assumptions may rule out modeling is the fact that it may be worse to
have a few large ``hot spots'' within the tissue rather than many
smaller ``hot spots'' with the same total volume and radiation
intensity. However, such factors would be computationally expensive
to model, and the assumption that each 
is histogram-based still
allows for much flexibility in terms of what objective functions can
be modeled. For example:
