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This application is based on, incorporates herein by reference, and claims the benefit of U. This invention relates generally to systems and methods for developing control plans for radiation dosage and, more particularly, to systems and methods to determine a desired plan for applying a radiation dose. The application of radiation to an individual is commonplace in a number of industries, such as the medical industry, where radiation is used in imaging applications and therapy applications, and the security industry, where radiation is used to perform inspection processes.
Regardless of the industry, whenever radiation is applied to a living subject, a balance is sought between the ability to achieve the best results of the process utilizing the radiation, which typically leans toward increased radiation dosage, and the safety of the subject, which typically leans toward decreased radiation dosage.
To this end, a constant difficulty that arises in radiotherapy treatment planning is the patient-specific tradeoff between providing an appropriate radiation dose to a tumor and keeping the healthy tissue dose low. This is traditionally handled by forming a joint objective function that includes an objective that rewards high, uniform dose to the tumor, as well as separate objectives that penalize dose to various healthy organs or tissue.
The problem with this approach is that a good set of relative weighting factors on the different objectives is not a priori known, and must be found by the treatment planner using a, time-consuming, iterative, process, often based at least partially on trial and error. Furthermore, a set of weights found for one patient will likely not work well for another patient. Indeed, it has been shown that for a concave tumor phantom and a nearby critical structure, the geometry parameterized by the separation distance and the relative weights that give comparable plans are different for each instance.
Therefore, it would be desirable to have a system and method that allows treatment planners and physicians to understand the tradeoffs for individual patients, while simultaneously avoiding the time-consuming human-iteration loop of searching for a good set of objective function weights. The present invention overcomes the aforementioned drawbacks by providing a system and method for calculating well-placed points on a Pareto surface PS and applying it to an intensity modulated radiation therapy IMRT planning problem.
The invention includes an algorithm that uses a feasible set and objective functions that are convex and this condition is satisfied by staying within a linear programming environment. The algorithm specifies how to add a single point to the surface to best reduce the surface position uncertainty. These benefits are not available in traditional constraint methods, such as the normal boundary intersection method, the epsilon constraint method, or the uniform weights scalarization approach.
In accordance with the present invention, a method for selecting a desired portion of a subject to receive a radiation dose is disclosed. The method includes selecting a plurality of points on a Pareto surface PS using a plurality of weights to identify the plurality of points, iteratively choosing additional weights to run to build up a model of the PS, and examining each point that has been found on the PS along with the weights used to produce the point to determine if the point is indicated a Pareto optimal.
The method further includes identifying new weights and repeating the previous steps using the new weights until a Pareto optimal stop tolerance is met.
In accordance with another aspect of the present invention, a method for selecting a desired portion of a subject to receive a radiation dose is disclosed. The method includes selecting a lower bound and an upper bound to sandwich a PS therebetween, performing a convex hulling to obtain a facet representation of the PS, and determining lower distal point LDP for each facet.
The method also includes calculating a distance from each facet to each LDP, comparing the distance to a threshold value, and repeating the above steps until the distance is below the threshold value for all sides of the PS.
In accordance with yet another aspect of the present invention, a system for selecting a desired portion of a subject to receive a radiation dose is disclosed. The system includes a computer having a computer program that when executed causes the computer to select a plurality of points believed to be on a PS about the desired portion of the subject using a plurality of weights to identify the plurality of points and iteratively choose additional weights to run to build up a model of the PS.
The computer is also caused to examine each point that has been selected along with the weights used to produce the point; and identify new weights. Also, the computer is caused to repeat the above steps using the new weights until a geometric stop tolerance is met. Various other features of the present invention will be made apparent from the following detailed description and the drawings.
The invention will hereafter be described with reference to the accompanying drawings, wherein like reference numerals denote like elements, and:.
As will be described, the present invention approaches the multiple objective intensity modulated radiation therapy IMRT optimization problem by computing a database of Pareto optimal plans for subsequent exploration.
N, a beamlet solution x is Pareto optimal if there does not exist a strictly better feasible solution. The algorithm is suitable for moderate dimensional multiobjective problems, for example, N up to 5 or 6. The computation of a representative set of Pareto optimal solutions is not the only approach to multiobjective programming MOP.
Two other common approaches for handling multiple objectives are goal programming GP and lexicographic ordering LO. These methods require that the decision maker specifies his or her preferences before optimization begins, and based on these preferences, a single Pareto optimal plan is computed, which makes these techniques attractive from a computational standpoint.
In GP, objectives are re-expressed as goals. Goals are achieved by minimizing a weighted sum of the deviations from these goals. Based on the set of weights chosen, a single balanced plan is computed. In LO, also called preemptive goal programming, the objectives are prioritized. The first optimization problem minimizes the objective with the highest priority.
The result of this serves as a constraint for the next problem, where the objective with the next highest priority is minimized with a constraint that the first objective cannot go above its optimal value in the first solution. This is repeated for subsequent prioritization levels. LO leads to extreme solutions, based on the priority ordering, which may not be desirable by decision makers. Therefore, in accordance with the present invention, the computation of a representative set of Pareto optimal solutions is preferred since it is the only approach that actually shows the tradeoffs to the planners, allowing them to choose a plan with an understanding of the different options available.
The general Pareto MOP can be represented as: Whereas single optimization problems have a single solution, MOPs have a set of optimal solutions called the Pareto set. Therefore, a discrete set of points on the PS that approximates the surface well is sought. It should be noted that PS is used herein to refer to the surface in objective function space. That is, PS is the space with one coordinate axis for each objective function F i. Put another way, it could be thought of as the surface of Pareto optimal points in the underlying decision variable x space.
In settings where the calculation of an individual point on the PS is computationally expensive, there is a motivation to compute a good representation of the surface in as few points as possible. In two dimensions 2D , for example, if the Pareto tradeoff curve is kinked, two end points and a point at the kink produce a sufficient and economical representation of the surface. However, a priori, the shape of the PS is unknown. Nevertheless, as will be explained below, convexity properties of the problem lead to geometric lower and upper bounds on the PS, which can be used to guide to a well-distributed placement of points on the PS.
Scalarization algorithms are a common method for handling equation 1 when the constraint set X is convex and the functions F i are convex. Under these convexity assumptions, Pareto optimal solutions to equation 1 can be found by solving the following:.
However, this is highly impractical, and even a fine discrete sampling is impractical for any number of objectives above 3.
Furthermore, there is no guarantee that a pre-specified set of weight vectors to run will lead to a good distribution of points on the PS. Despite these apparent weaknesses, there are several advantages to using a scalarization method. One is that in IMRT planning, many treatment planning systems TPS are designed where such scalarization is done by hand, with planners iteratively changing the weights to find a good plan.
Therefore, by developing an algorithm that works on this principle, the algorithm can be hooked up to these solvers, provided they are using convex formulations, with minimal effort.
Another advantage is the empirical observation that the solution time for a scalarized problem is much faster than other techniques, which get individual points on the PS of a MOP. In particular, the normal boundary intersection NBI method, the normal constraint NC method, and the epsilon constraint method, such as described, for example, in S. It has been observed that these constraints can make the solution process significantly slower.
These techniques also clip edge portions off of the PS in dimensions over 2, and thus should be used only when this is acceptable.
On the other hand, the algorithm of the present invention overcomes these problems by sandwiching the PS between a lower and an upper convex approximation. Others, such as K. However, the approach of Klamroth et al. The approach of Solanki et al. One is that the present invention provides a closed form algebraic solution to the problem of finding the distances between the lower approximation and the upper approximation.
On the other hand Solanki et al. The second key difference is in how the boundary of the PS is handled. Since each run is expensive, the present invention does not use negative weight vectors, and instead attempts to find the proper weighting vector based on the weight vectors used to create the boundary region being targeting. In certain applications, the user will opt to not flush out the boundary points of the Pareto set, in which case the issue of dealing with the boundary is irrelevant.
In general though, the user typically has formed a set of constraints such that any solution that satisfies those constraints is a potentially viable solution, and hence, the present invention seeks to determine the entire PS given that constraint set.
The algorithm of the present invention relies heavily on the following result from multiobjective programming: This means that any hyperplane plane tangent to the PS is a supporting hyperplane, hence, the PS is entirely on one side of the hyperplane.
The method of the present invention is applicable to general convex optimization MOPs. As will be explained with respect to FIG. By examining the current points that have been found on the PS along with the weights Used in equation 2 that produced those points, PGEN produces a new weight vector to run, and repeats this process until a geometric stop tolerance is met.
That is, it is a lower bound of the PS. This gives, with each new point calculated, a new lower bound on the PS. This is illustrated in FIG. An upper bound is obtained by the convex combination of all of the points thus far found on the PS. The lower and upper bounds are then combined and, at places where the bounds have the largest gap, new points are added to the surface.
Referring now to FIGS. The concepts extend to N dimensions, but nontrivially, particularly when dealing with the boundary of the PS. For simplicity it is assumed that each function is to be minimized with positive weights. Maximization problems are turned into minimization problems with negative weights, but this detail is unnecessary and only complicates the exposition.
These points bound the lower extent of the PS in each respective dimension. After the N anchor points are computed, a single balanced point is computed by either using equal weights on each objective function, or by using the outward-facing normal of the hyperplane through the N anchor points, provided all of the components are non-negative. The intersection of the lower bound tangent planes leads to the points I 1 and I 2 which are called lower distal points LDPs. As show in FIG.
The convex hull of a set of points in N-dimensional ND space is the set of all points that can be expressed as convex combinations of the points in the original set. This infinite set is compactly represented as a set of bounding facets, such as line segments in 2D, triangles in 3D, and the like, and their normals. Since it is known that for convex optimization MOPs the PS points lie on the boundary of a convex set, a convex hulling algorithm can be used to obtain a facet representation of the PS.
For example, a freely available ND convex hull algorithm called convhulln and available in software such as Qhull can be used. Convhulln takes as input the points and returns the connectivity of those points.
That is, the software returns the facets of the convex hull, and the face normals. Only the lower facets are used as the upper approximation to the PS, so in this case, facets [P 1 , P 3 ] and [P 3 , P 2 ] are used, and facet [P 1 , P 2 ] is thrown away. Facet rejection in this 2D example is straightforward and could be done by looking at the normal vector of each facet.
Using the convention that normals point into the convex hull polygon, a facet the normal to which consists of all negative components could be removed.