### Abstract

The gamma index is a tool for dose distribution comparison. It combines both dose difference (DD) and distance to agreement (DTA) into a single quantity. Though it is an effective measure, making up for the inadequacy of DD or DTA alone, its calculation can be very time-consuming. For a k-D space with N quantization levels in each dimension, the complexity of the exhaustive search is O(N^{2k}). In this work, we proposed an efficient method that reduces the complexity from O(N^{2k}) to O(N^{k}M), where M is the number of discretized dose values and is comparable to N. More precisely, by embedding the reference dose distribution in a (k+1)-D spatial-dose space, we can use fast Euclidean distance transform with linear complexity to obtain a table of gamma indices evaluated over a range of the (k+1)-D spatial-dose space. Then, to obtain gamma indices for the test dose distribution, it requires only table lookup with complexity O(N^{k}). Such a table can also be used for other test dose distributions as long as the reference dose distribution is the same. Simulations demonstrated the efficiency of our proposed method. The speedup for 3D gamma index calculation is expected to be on the order of tens of thousands (from O(N^{6}) to O(N^{3}M)) if N is a few hundreds, which makes clinical usage of the 3D gamma index feasible. A byproduct of the gamma index table is that the gradient of the gamma index with respect to either the spatial or dose dimension can be easily derived. The gradient can be used to identify the main causes of the discrepancy from the reference distribution at any dose point in the test distribution or incorporated in treatment planning and machine parameter optimization.

Original language | English (US) |
---|---|

Pages (from-to) | 2037-2047 |

Number of pages | 11 |

Journal | Physics in Medicine and Biology |

Volume | 54 |

Issue number | 7 |

DOIs | |

State | Published - 2009 |

### ASJC Scopus subject areas

- Radiology Nuclear Medicine and imaging
- Radiological and Ultrasound Technology

### Cite this

*Physics in Medicine and Biology*,

*54*(7), 2037-2047. https://doi.org/10.1088/0031-9155/54/7/012

**Efficient gamma index calculation using fast Euclidean distance transform.** / Chen, Mingli; Lu, Weiguo; Chen, Quan; Ruchala, Kenneth; Olivera, Gustavo.

Research output: Contribution to journal › Article

*Physics in Medicine and Biology*, vol. 54, no. 7, pp. 2037-2047. https://doi.org/10.1088/0031-9155/54/7/012

}

TY - JOUR

T1 - Efficient gamma index calculation using fast Euclidean distance transform

AU - Chen, Mingli

AU - Lu, Weiguo

AU - Chen, Quan

AU - Ruchala, Kenneth

AU - Olivera, Gustavo

PY - 2009

Y1 - 2009

N2 - The gamma index is a tool for dose distribution comparison. It combines both dose difference (DD) and distance to agreement (DTA) into a single quantity. Though it is an effective measure, making up for the inadequacy of DD or DTA alone, its calculation can be very time-consuming. For a k-D space with N quantization levels in each dimension, the complexity of the exhaustive search is O(N2k). In this work, we proposed an efficient method that reduces the complexity from O(N2k) to O(NkM), where M is the number of discretized dose values and is comparable to N. More precisely, by embedding the reference dose distribution in a (k+1)-D spatial-dose space, we can use fast Euclidean distance transform with linear complexity to obtain a table of gamma indices evaluated over a range of the (k+1)-D spatial-dose space. Then, to obtain gamma indices for the test dose distribution, it requires only table lookup with complexity O(Nk). Such a table can also be used for other test dose distributions as long as the reference dose distribution is the same. Simulations demonstrated the efficiency of our proposed method. The speedup for 3D gamma index calculation is expected to be on the order of tens of thousands (from O(N6) to O(N3M)) if N is a few hundreds, which makes clinical usage of the 3D gamma index feasible. A byproduct of the gamma index table is that the gradient of the gamma index with respect to either the spatial or dose dimension can be easily derived. The gradient can be used to identify the main causes of the discrepancy from the reference distribution at any dose point in the test distribution or incorporated in treatment planning and machine parameter optimization.

AB - The gamma index is a tool for dose distribution comparison. It combines both dose difference (DD) and distance to agreement (DTA) into a single quantity. Though it is an effective measure, making up for the inadequacy of DD or DTA alone, its calculation can be very time-consuming. For a k-D space with N quantization levels in each dimension, the complexity of the exhaustive search is O(N2k). In this work, we proposed an efficient method that reduces the complexity from O(N2k) to O(NkM), where M is the number of discretized dose values and is comparable to N. More precisely, by embedding the reference dose distribution in a (k+1)-D spatial-dose space, we can use fast Euclidean distance transform with linear complexity to obtain a table of gamma indices evaluated over a range of the (k+1)-D spatial-dose space. Then, to obtain gamma indices for the test dose distribution, it requires only table lookup with complexity O(Nk). Such a table can also be used for other test dose distributions as long as the reference dose distribution is the same. Simulations demonstrated the efficiency of our proposed method. The speedup for 3D gamma index calculation is expected to be on the order of tens of thousands (from O(N6) to O(N3M)) if N is a few hundreds, which makes clinical usage of the 3D gamma index feasible. A byproduct of the gamma index table is that the gradient of the gamma index with respect to either the spatial or dose dimension can be easily derived. The gradient can be used to identify the main causes of the discrepancy from the reference distribution at any dose point in the test distribution or incorporated in treatment planning and machine parameter optimization.

UR - http://www.scopus.com/inward/record.url?scp=63649102975&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=63649102975&partnerID=8YFLogxK

U2 - 10.1088/0031-9155/54/7/012

DO - 10.1088/0031-9155/54/7/012

M3 - Article

C2 - 19287084

AN - SCOPUS:63649102975

VL - 54

SP - 2037

EP - 2047

JO - Physics in Medicine and Biology

JF - Physics in Medicine and Biology

SN - 0031-9155

IS - 7

ER -