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\title{Histogram-Based Objective Function for Radiotherapy Planning}
\author{}
\date{}

\begin{document}

\maketitle

\section*{Assumed functional form}

For a given treatment plan $P$ consisting of specified beam angles and
individual beam doses, let $D(P, x)$ 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 $U(D)$ --- in a manner that is both clinically
useful and computationally feasible.

We propose to assume $U(D)$ is of the following form:

\[U(D) = U(H_1(D), \ldots, H_N(D))\]

where each $H_r(D)$ is a function of the values of $D(P, x)$ 
restricted to some subset $S_r$ of all possible voxels.  Each voxel
will be contained in exactly one subset $S_r$.  Each subset $S_r$ 
corresponds to a single tissue type, such as normal brain tissue,
tumor, or a particular type of critical normal tissue.

$U(H_1(D), \ldots, H_N(D))$ might, for example, combine $H_1(D)$
through $H_N(D)$ in some simple fashion such as a weighted arithmetic
or geometric mean.

Every $H_r(D)$ is assumed to be a function of a {\em histogram} of 
the values that $D(x)$ takes on over the set of $x \in S_r$.
Formally,

\[H_r(D) = F(c^r_1, \ldots, c^r_M) \]

where

\begin{itemize}
\item the histogram has $M$ bins;
\item each $c^r_i$ is the count of voxels $x \in S_r$ such that $l^r_i \leq D(x) < u^r_i$;
\item $l^r_i$ and $u^r_i$ are the dose values corresponding to the $r^{th}$
histogram's $i^{th}$ bin's lower and upper bounds, respectively.
\end{itemize}

\normalfig{head}{Example tissue partitions $S_r$}
\normalfig{tumorhist}{Example histogram for tumor tissue}
\normalfig{eyehist}{Example histogram for critical normal tissue (e.g., brain stem)}

For example, Figure~\ref{head} shows a crude representation of a
cross-section of a possible brain scan.  The tissue voxels have been
partitioned into five different subsets: $S_{norm}$ for normal brain
tissue, $S_{tumor}$ for tumor tissue, $S_{eyes}$ for the eyes,
$S_{stem}$ for the brain stem, and $S_{skull}$ for the skull.
Figure~\ref{tumorhist} shows a possible histogram for the radiation
per voxel over the $S_{tumor}$ voxel subset; Figure~\ref{eyehist} shows
a possible histogram for the $S_{stem}$ 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 $H_r$ is histogram-based still
allows for much flexibility in terms of what objective functions can
be modeled.  For example:

\subsection*{Penalties for Deviations from Prescribed Dose}

The histogram-based approach may easily be used to implement an
objective function that penalizes deviations from prescribed
doses on a voxel-by-voxel basis.  A weighted-sum-of-squared-deviations 
objective function could be implemented simply by specifying:

\[H_r(D) = w_r * \sum_{i=0}^{M} [c^r_i * (P_r - (\frac{u^r_i - l^r_i}{2})^2)] \]

where $P_r$ is the prescripted $w_r$ is a weight indicating the
relative importance of getting as close to the prescribed dose as
possible for the particular tissue type $r$.  (Note that we don't
actually need to use a finite number $M$ of bins for this objective
function; we could simply iterate over all the voxels in $S_r$
instead.)

Such an objective function should cause the optimizer to:
\begin{itemize}
\item Keep the doses close to the prescribed doses, particularly
in critical tissue or other tissue with high weights $w_r$.
\item This should naturally lead to high dose gradients at tissue
boundaries, particularly boundaries between tumor tissue and
normal critical tissue.
\end{itemize}

Alternatively, we could formulate even simpler objective functions
with penalties proportional to the amount of normal tissue dosed over
a certain threshold, plus additional penalties proportional to the
amount of tumor tissue receiving doses under another threshold.  Or,
we could allow the practitioner to graphically specify more
complicated penalty functions for specific deviations from the
prescribed dose via the use of splines.

Any of the above possible objective functions could be supplemented
with an additional term that explicitly penalizes nonuniform dose
distributions throughout the tumor.  For example, some multiple of the
standard deviation of the dose values found in the tumor tissue's
histogram could be subtracted from the rest of the objective function.

\subsection*{Cell Response Models}

Instead of an objective function based on penalizing deviations from a
prescribed dose, one might wish for an objective function that
attempted to explicitly model the probability of controlling the tumor
without inducing serious complications in normal tissue.  Most 
current theoretical models of tumor control probability (TCP) and
normal tissue complication probability (NTCP) are based on dose
histograms, largely since more complicated models would require more
data than currently available.  Thus, a histogram-based approach
such as the one above would be sufficient to implement a 
biological-response-based objective function.

\section*{Computational advantages}

A histogram-based objective function of the form described above has
the computational advantage that if an single beam
weight or position is varied, one only need look at the voxels
intersecting the old and new beam positions in order to update
the value of the objective function.  This may provide a significant
computational advantage when using optimization algorithms that
adjust only a few beams at a time while exploring the space of
possible solutions.

\end{document}