The inverse problem of depth dose curve estimation

Abstract

The inverse problem of the depth dose curve is formulated and a proposition for its solution is presented. The solution is based on the approximation of the observation equation with a numerical quadrature operator and the regularization of this inverse problem with a smoothness side constraint. The problem formulation is applicable for both the electron and the photon depth dose curve estimation. Moreover, the method is equivalent for, for example, all energies, field sizes and source-to-phantom distances. Simulations show that the estimation error is smaller with the proposed method than with direct linear interpolation. The main result of the paper, however, is the formulation of the problem that allows feasible extensions and modifications for different measurement situations.

A strategy for selecting measurement points in the determination of depth-dose curves and dose profiles

“…. , N can be expressed as an integral of the true depth-dose curve D(z) over the interval [z k − h/2, z k + h/2] which is occupied by the measurement probe [5], namely…”

“…Equation ( 6) is called the observation equation. In the case in which the observations were carried out with an equally spaced grid [5] the matrix K was a block Toeplitz matrix (with nonsquare blocks), which is no longer the case here.…”

“…Another alternative is to use nonparametric interpolation, which also has its own disadvantages [1]. On the other hand, the measurements are actually not samples from the true depth-dose curve [5] due to the physical size of the measurement probe. Thus interpolation is not the correct approach even though it would otherwise be applicable.…”

“…In [5] the inverse problem of estimation of the depth-dose curve was formulated and a regularization method for its solution was proposed. The main result of this formulation is that is makes extensions and modifications for different measurement situations feasible.…”

“…In this paper we study the applicability of the regularization method proposed in [5] to sparse unequally spaced measurements and propose a method for selecting the measurement points. The proposed method is based on the curvatures of a set of prototype depth-dose curves and profiles.…”