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The Genetic Algorithm-Based Optimization Approach for
Gamma Unit Treatment Planning System
1 Restatement of the Problem
Stereotactic radiosurgery delivers a single high dose of ionizing radiation to a
radiographically well-defined, small intracranial 3D brain tumor without delivering
any significant fraction of the prescribed dose to the surrounding brain tissue. One of
the three often-used modalities is the gamma knife unit. The gamma knife unit
delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit
sources through a heavy helmet. All 201 beams simultaneously intersect at the
isocenter, resulting in a spherical (approximately) dose distribution at the effective
dose levels. Irradiating the isocenter to deliver dose is termed a “shot”. Shots can be
represented as different spheres. Four interchangeable outer collimator helmets with
beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different
size volumes. For a target volume larger than one shot, multiple shots can be used to
cover the entire target. By combining multiple shots of radiation, the treatment plan
can be customized to treat lesions of varying sizes and shapes. In practice, most target
volumes are treated with 1 to 15 shots.
Gamma knife treatment plans are conventionally produced using a manual
iterative approach. In each iteration, the planner attempts to determine: (1) the number
of shots, (2) the shot sizes, (3) the shot locations, and (4) the shot exposed times
(weights) that would adequately cover the target and spare critical structures. For
large or irregularly shaped treatment volumes, this process becomes rather tedious and
time consuming. Also, the quality of the plan produced often depends upon both the
patience and the experience of the user. Consequently, a number of researchers have
studied techniques for automating the gamma knife treatment planning process, e.g.,
[1, 2]. The algorithms that have been tested include simulated annealing [3, 4], and
mixed integer programming, and a nonlinear programming [5-8], etc.
In treatment planning problems, the objective is to deliver a homogeneous
(uniform) dose of radiation to the tumor (typically called the target) area while
avoiding unnecessary damage to the surrounding tissue and organs. One approach
approximates each radiation shot as a sphere [9], thus reducing the problem to one of
geometric coverage. Then, a sphere-packing approach [10] can be used to determine
the shot locations and sizes. In this paper, considering the physical limitations and
biological uncertainties involved in the gamma knife therapy process, we will also
formulate the optimal treatment planning for a gamma knife unit as a sphere-packing
problem, and propose a reasonable algorithm to find a solution.
2 Assumptions
To account for all physical limitations and biological uncertainties involved in the
gamma knife therapy process, we make several assumptions as follows:
(A1) The shape of the target is not too irregular, and the target volume is a
bounded. As a rule of thumb, the target to be treated should be less than 3.5 cm in all
dimensions. Its three-dimensional (3D) digital image, usually consisting of millions of
points, can be obtained from a CT or MRI.
(A2) Considering the target volume as a 3D grid of points, we divide this grid into
two subsets, the subset of points in and out of the target, is denoted as
T
and ,
respectively.
N
(A3) Four interchangeable outer collimator helmets with beam channel diameters
={4, 8, 14, 18} mm are available for irradiating different size volumes. We use
to denote the coordinates of the center location of the shot, to denote
the time (weight) that each shot is exposed. The total dose delivered is a linear
function of the . For a target volume larger than one shot, multiple shots can be
used to cover the entire target. In the gamma knife case, there is a bound of the
number, , of the shots. The typical range of values for
w
),,(
sss
zyx
ws
t
,
ws
t
,
n n
is:
151
≤
≤
n
. (1)
(A4) Neurosurgeons commonly use isodose curves as a means of judging the
quality of a treatment plan, and may wish to impose a requirement that the entire
target is surrounded by an isodose line of , i.e., 30~70%. In our model, we will
use an isodose line of 50%, which means that a constraint that the 50% line must
surround the target.
%x
(A5) The dose cloud was approximated as a spherically symmetric distribution by
averaging the profiles along the
x
, , and axes. Other effects will be ignored in
the following optimization formulations.
y
z
(A6) The total dose deposited in the target and critical organ should be more than
a fraction,
P
, of the total dose delivered by a plan. Typically, we choose the values
for
P
:
%40%25
≤
≤
P
.
3 Optimization Models
3.1 Analysis of the Problem
The goal of radiosurgery is to deplete tumor cells while preserving normal
structures. In general, an optimal treatment plan is designed to meet the following
requirements:
(R1) Match specified isodose contours to the target volumes.
(R2) Match specified dose-volume constraints of the target and critical organ.
(R3) Constrain dose to specified normal tissue points below tolerance doses.
(R4) Minimize the integral dose to the entire volume of normal tissues or organs.
(R5) Minimize the dose gradient across the target volume.
(R6) Minimize the maximum dose to critical volumes.
Also, we have the following constraints:
(C1) Prohibit shots from protruding outside the target.
(C2) Prohibit shots from overlapping (to avoid hot spots).
(C3) Cover the target volume with effective dosage as much as possible. But at
least 90% of the target volume must be covered by shots.
(C4) Use as few shots as possible.
In order to satisfy the above requirements and constraints, we can formulate the
optimal treatment planning for a gamma knife unit as a sphere-packing problem. The
optimization is performed in two steps. First, for the sphere-packing problem, a
geometry-based heuristic is used to produce a reasonable configuration of shot
number, sizes and locations. Next, a dose-based optimization is used to produce the
final treatment plan.
3.2 Geometry-Based Heuristic for the Sphere-Packing Problem
The geometry-based heuristic is designed to quickly produce a reasonable
configuration of shot number, sizes and locations. In our geometry-based heuristic,
each shot of radiation is modeled as a sphere, and the medial axis transform, known as
the “skeleton”, of the target volume is used to guide the placement of the shots. The
skeleton is frequently used in shape analysis and other related areas [11-13]. In our
case, we will just use the skeleton to quickly find good locations of the shots.
Our geometry-based heuristic is in three stages. First, we generate the skeleton
using a 3D skeleton algorithm. Then, we place shots and choose their sizes along the
skeleton to maximize a measure of our objective. This process is done by a genetic
algorithm (GA)-based shot placement approach. After the number of focusing helmets
to be included in the treatment plan, the planning produces a list of the possible
helmet combinations and a suggested number of shots to use for the dose-based
optimization outlined in Section 3.3.
3.2.1 Skeleton Generation
In our geometry-based heuristic, a 3D skeleton algorithm that follows similar
procedures to that in [5] is adopted. Here, we use a morphologic thinning approach
[14] to create the skeleton as opposed to the Euclidean distance technique. The first
step in the skeleton generation is to compute the contour map containing distance
information from the point to a nearest target boundary. Then, based on the contour
map, several known skeleton extraction methods in the literature [5, 11-14] can be
used. Since the method in [5] is simple and fast, we use it for our skeleton generation.
3.2.2 Genetic Algorithm-Based Shot Placement
In the shot placement algorithm, we restrict our attention to points on the skeleton.
Our approach is to search a good location of the point to place a shot from all points
of the skeleton. We start from a special type of skeleton points, an end point, that help
determining the shot size and the location, as shown in Fig. 1, while in [5] uses two
special types of skeleton points, i.e., an end point and a cross point. Then, based on
the end points of the skeleton, we randomly find a best location to place a shot by
using the GA-based shot placement algorithm.
Fig. 1: Examples of the end-points.
We should first give the definitions of an end point [5].
Definition 1. A point is an end-point if: (1) It is in the skeleton; (2) It has only one
V-neighbor in the skeleton.
Starting from an end-point, we then look for the best point to place a shot and
determine the shot size by using GA [15, 16]. In the GA-based shot placement
algorithm, we must solve the following problems:
(1) The encoding method. In general, the bit string (0-1’s) encoding is the most
common method adopted by GA researchers because of its simplicity and tractability.
However, in this case, if we directly encode the point coordinates into a
bit string, after the crossover and mutation operation, it will result in generating some
points that are not in the skeleton. In order to solve this problem, we build a
corresponding table to the relationship of the point number and the point coordinates
in the skeleton, as shown in Table 1. As such, we just encode the point
number into a bit string. We just select m points from all points of the skeleton to
form a population. One point is a chromosome.
),,(
sss
zyx
),,(
sss
zyx
Table 1. The relationship of the point number and the point coordinates
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资源评论
- limurui2014-02-08终于找到了!2003年伽马刀的O奖论文。应该是东华大学的。用到了抽骨架模型,遗传算法和目标规划。虽然是较早的论文,但还是很有参考价值
袁飞远
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