^{1,a)}

### Abstract

The processes of extraction and sweep are basic segmentation steps that are used in leaf sequencing algorithms. A modified version of a commercial leaf sequencer changed the way that the extracts are selected and expanded the search space, but the modification maintained the basic search paradigm of evaluating multiple solutions, each one consisting of up to 12 extracts and a sweep sequence. While it generated the best solutions compared to other published algorithms, it used more computation time. A new, faster algorithm selects one extract at a time but calls itself as an evaluation function a user-specified number of times, after which it uses the bidirectional sweeping window algorithm as the final evaluation function. To achieve a performance comparable to that of the modified commercial leaf sequencer, 2–3 calls were needed, and in all test cases, there were only slight improvements beyond two calls. For the 13 clinical test maps, computation speeds improved by a factor between 12 and 43, depending on the constraints, namely the ability to interdigitate and the avoidance of the tongue-and-groove under dose. The new algorithm was compared to the original and modified versions of the commercial leaf sequencer. It was also compared to other published algorithms for 1400, random, , test maps with 3–16 intensity levels. In every single case the new algorithm provided the best solution.

The clinical intensity maps came from a variety of sources over the past few years, and the author would like to thank St. Jude Children’s Research Hospital, DKFZ (German Cancer Research Center), and the University of California in San Francisco for providing them.

I. INTRODUCTION

II. MATERIAL AND METHODS

II.A. Basic processes

II.B. Reduction algorithm

II.C. Performance evaluation

III. RESULTS AND DISCUSSION

IV. CONCLUSION

### Key Topics

- Multileaf collimators
- 13.0
- Sequence analysis
- 8.0
- Dosimetry
- 3.0
- Iteration theory
- 3.0
- Optimization
- 3.0

## Figures

This is a flow chart of the variable depth recursion algorithm. The bold “T” and “F” around the decision diamonds stand for “True” and “False,” respectively. The subscript “*j*” is the number of levels being tested and is also the iteration index for a segment. The inner loop, responsible for testing segments with various levels, is indicated by the dashed rectangle. The process in the thick line rectangle is where the recursion takes place as described in the text.

This is a flow chart of the variable depth recursion algorithm. The bold “T” and “F” around the decision diamonds stand for “True” and “False,” respectively. The subscript “*j*” is the number of levels being tested and is also the iteration index for a segment. The inner loop, responsible for testing segments with various levels, is indicated by the dashed rectangle. The process in the thick line rectangle is where the recursion takes place as described in the text.

Comparison of the number of segments (top) and relative fluence (bottom) resulting from the various IMFAST related algorithms when there is no sequencing constraint. The line is for the fully modified IMFAST algorithm (“modified”), the “x” mark is for the original IMFAST algorithm (“original”), and the square is for the variable depth recursion (“vdr”) algorithm. The values are normalized to those for the algorithm of Bortfeld *et al.* The square lies on top of the line in the top graph, indicating that VDR is tied with the fully modified IMFAST algorithm. The increases in the relative fluence are minimal, with VDR having a slightly higher result than the minimum in some cases.

Comparison of the number of segments (top) and relative fluence (bottom) resulting from the various IMFAST related algorithms when there is no sequencing constraint. The line is for the fully modified IMFAST algorithm (“modified”), the “x” mark is for the original IMFAST algorithm (“original”), and the square is for the variable depth recursion (“vdr”) algorithm. The values are normalized to those for the algorithm of Bortfeld *et al.* The square lies on top of the line in the top graph, indicating that VDR is tied with the fully modified IMFAST algorithm. The increases in the relative fluence are minimal, with VDR having a slightly higher result than the minimum in some cases.

Comparison of the number of segments (top) and relative fluence (bottom) resulting from the various IMFAST related algorithms with the collision constraint. The symbols have the same meaning as those in Fig. 2. Again, the squares lie on top of the line, indicating that VDR and the fully modified IMFAST algorithm perform equally well.

Comparison of the number of segments (top) and relative fluence (bottom) resulting from the various IMFAST related algorithms with the collision constraint. The symbols have the same meaning as those in Fig. 2. Again, the squares lie on top of the line, indicating that VDR and the fully modified IMFAST algorithm perform equally well.

Comparison of the number of segments (top) and relative fluence (bottom) resulting from the various IMFAST related algorithms with the tongue-and-groove constraint. The symbols have the same meaning as those in Fig. 2. Again, the squares lie on top of the line, indicating that VDR and the fully modified IMFAST algorithm perform equally well.

Comparison of the number of segments (top) and relative fluence (bottom) resulting from the various IMFAST related algorithms with the tongue-and-groove constraint. The symbols have the same meaning as those in Fig. 2. Again, the squares lie on top of the line, indicating that VDR and the fully modified IMFAST algorithm perform equally well.

Comparison of the number of segments (top) and relative fluence (bottom) resulting from the various IMFAST related algorithms with the tongue-and-groove and collision constraint. The symbols have the same meaning as those in Fig. 2. Again, the squares lie on top of the line, or appear slightly below it as many times as they appear slightly above it, indicating that VDR and the fully modified IMFAST algorithm perform equally well.

Comparison of the number of segments (top) and relative fluence (bottom) resulting from the various IMFAST related algorithms with the tongue-and-groove and collision constraint. The symbols have the same meaning as those in Fig. 2. Again, the squares lie on top of the line, or appear slightly below it as many times as they appear slightly above it, indicating that VDR and the fully modified IMFAST algorithm perform equally well.

Relative speeds of the various IMFAST related algorithms. The fully modified IMFAST algorithm has a value of 1. VDR is about 10–40 times faster than the fully modified IMFAST algorithm.

Relative speeds of the various IMFAST related algorithms. The fully modified IMFAST algorithm has a value of 1. VDR is about 10–40 times faster than the fully modified IMFAST algorithm.

## Tables

(A) The average of the number of segments and the average of the relative fluence for the various algorithms for the 100 random maps with various maximum map levels, for the case where no constraint was applied. ‘B’ is the algorithm of Bortfeld *et al.* (Ref. 9), ‘G’ is the algorithm of Galvin, Chen, and Smith (Ref. 4), ‘X’ is the algorithm of Xia and Verhey (Ref. 5), ‘Q’ is the algorithm of Que (Ref. 6), and ‘C’ is the algorithm of Crooks *et al.* (Ref. 8). ‘V0’ and ‘V15’ stand for the variable depth recursion algorithm with and , respectively. The last two columns give the percent decrease in the number of segments for V15 relative to V0 and B. The top three results for each level are in bold.

(A) The average of the number of segments and the average of the relative fluence for the various algorithms for the 100 random maps with various maximum map levels, for the case where no constraint was applied. ‘B’ is the algorithm of Bortfeld *et al.* (Ref. 9), ‘G’ is the algorithm of Galvin, Chen, and Smith (Ref. 4), ‘X’ is the algorithm of Xia and Verhey (Ref. 5), ‘Q’ is the algorithm of Que (Ref. 6), and ‘C’ is the algorithm of Crooks *et al.* (Ref. 8). ‘V0’ and ‘V15’ stand for the variable depth recursion algorithm with and , respectively. The last two columns give the percent decrease in the number of segments for V15 relative to V0 and B. The top three results for each level are in bold.

(A) The average of the number of segments and the average of the relative fluence for the various algorithms for the 100 random maps with various maximum map levels, for the case where no interdigitation was allowed. The meanings of the labels are the same as those in table I.

(A) The average of the number of segments and the average of the relative fluence for the various algorithms for the 100 random maps with various maximum map levels, for the case where no interdigitation was allowed. The meanings of the labels are the same as those in table I.

The number of segments and relative fluence resulting from the application of various recursion depths to 100 random maps with maximum map levels in the typical clinical range. “D0” stands for no recursion, where as “D1,” “D2,” and “D3” stand for recursion depths of 1, 2, and 3, respectively. The collision and tongue and groove constraints were applied.

The number of segments and relative fluence resulting from the application of various recursion depths to 100 random maps with maximum map levels in the typical clinical range. “D0” stands for no recursion, where as “D1,” “D2,” and “D3” stand for recursion depths of 1, 2, and 3, respectively. The collision and tongue and groove constraints were applied.

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