Approximation and Segment Count Reduction in Intensity Modulated Radiation Therapy

For a given mtimesn nonnegative real matrix A, a binary segmentation with 1-norm relative error e is a set of pairs (alpha,S)={(alpha
₁
, S
₁
), (alpha
₂
, S
₂
), ..., (alpha
_k
, S
_k
)}, where each alpha
_i
is a positive number and S
_i
is an mtimesn binary matrix, and e=|A-Sigma
_i

^k
=alpha
_i
S
_i
|1/|A|
₁
where |A|
₁
is the 1-norm of a vector which consists of exactly all the entries of the matrix A. Given A and positive scalars gamma, delta, we consider the optimization problem of finding a binary segmentation alphaS that minimizes z=Sigma
_i

^k
=alpha
_i
+gammak+deltae subject to certain constraints on S
_i
. This problem provides a major challenge in preparing a clinically acceptable treatment plan for intensity modulated radiation therapy (IMRT) and is known to be NP-hard. In this paper, we present an effective heuristic algorithm for the above problem with an additional constraint klesK, where K is a given positive integer. Reducing k subject to achieving an acceptable value of e is very desirable in both IMRT and intensity modulated arc therapy (IMAT). Our algorithm can be used as a valuable tool to find a clinically deliverable solution with a tradeoff between these two competing metrics