Optimization of intensity modulated beams in inverse planning by norm minimization
Abstract
There are two general paradigms for treatment planning: forward planning and inverse planning. In forward planning, an initial set of treatment parameter values are refined by trial and error until a desired dose distribution is produced. In inverse planning the problem is to determine the appropriate parameter values directly from the desired dose distribution. In this thesis the inverse planning problem (IPP) is posed mathematically as an inverse problem and as an optimization problem. In the case where scatter and attenuation are ignored (the NS-NA model) the inverse problem reduces to the inversion of the Dual Radon operator; if rotational symmetry is assumed, the problem simplifies to the solution of the Abel integral equation. The inverse solutions exhibit negative components and are therefore nonphysical. By formulating the IPP as a constrained optimization problem, the occurrence of negative components can be eliminated using non-negativity constraints. Additional constraints, such as dose limits, can be easily incorporated. The optimality criteria are based on minimizing the L1 and the L2 norms: the former is formulated as a linear program and the latter is formulated as a quadratic program. The numerical solutions of these mathematical programs are obtained in the NS-NA model and in the case where attenuation and scatter are present (SCAT models). In the case of the symmetric NS-NA model, analytical solutions are found using the variational calculus. Since linear and quadratic programs are convex programs, the solutions are global solutions (but not necessarily unique). Comparisons are made between the L1 and the L2 solutions in both the SCAT and the NS-NA dose models for various phantom geometries. The results indicate that norm minimization is a viable approach in the resolution of the IPP; furthermore, the underlying algorithms are sufficiently fast for use in an interactive environment. On a PC running Window NT 4.0 with 192MB RAM, solving the L1 linear program with 800 to 1190 constraints and 2776 variables takes 145.9 seconds on average, with a median of 66.3 seconds; solving the L2 quadratic program with 1128 to 1510 constraints and 324 variables takes 648.1 seconds on average, with a median of 281.2 seconds.
Degree
Ph.D.
Advisors
Landolt, Purdue University.
Subject Area
Radiation|Radiology
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