美国数学建模优秀论文03年B题

美国数学建模优秀论文03年B题

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.

美国数学建模优秀论文03年B题

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 N, respectively.

(A3) Four interchangeable outer collimator helmets with beam channel diameters w={4, 8, 14, 18} mm are available for irradiating different size volumes. We use (xs,ys,zs) to denote the coordinates of the center location of the shot, ts,w to denote the time (weight) that each shot is exposed. The total dose delivered is a linear function of the ts,w. 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, n, of the shots. The typical range of values for n is:

1≤n≤15. (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 x%, 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.

(A5) The dose cloud was approximated as a spherically symmetric distribution by averaging the profiles along the x, y, and z axes. Other effects will be ignored in the following optimization formulations.

(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:

25%≤P≤40%.

3 Optimization Models

3.1 Analysis of the Problem

美国数学建模优秀论文03年B题

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 …… 此处隐藏：25695字，全部文档内容请下载后查看。喜欢就下载吧 ……