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Table 1 Condition functions of typical objectives

From: Applying pytorch toolkit to plan optimization for circular cone based robotic radiotherapy

Objectives g
Max dose \(H(D_{i} - D_{i}^{0} )\)
Min dose \(H(D_{0} - D_{i}^{0} )\)
Uniform dose 1
Max DV \(\left\{ {\begin{array}{*{20}c} {H(D_{i} - D_{i}^{0} )} & {{\text{head}}} \\ 0 & {\text{others }} \\ \end{array} } \right.\)
Min DV \(\left\{ {\begin{array}{*{20}c} {H(D_{i}^{0} - D_{i} )} & {{\text{tail}}} \\ 0 & {{\text{others}}} \\ \end{array} } \right.\)
  1. H is the Heaviside function. For dose volume (DV) objectives, head and tail refer to the voxels at the top and end position sorted by dose in descending order