# Arc Duplication/PTV Splitting

## Terminology

*Convex*: Curving out or extending outward`①`*Concave*: Curving in or hollowed inward`②`

## Motivation

The VMAT algorithm does not perform well on concave objects. Concave objects are objects containing a volume or volumes smaller than their own convex hull `①`. Therefore, Spine SRS uses a PTV splitting algorithm to better treat concave objects such as vertebrae.

The PTV splitting algorithm automatically determines the optimal number and shape of sub-PTVs that the VMAT algorithm uses for optimization `②`.

## Algorithm Description

For each sub-PTV, the arc setup is copied and the fields of these arcs are initialized with IMRT segments covering only this sub-PTV. Once the VMAT optimization is started, MLC shapes of an arc initialized to one of the sub-PTVs are allowed to also cover other sub-PTVs.

The only visible effect is the duplication of the arc setup. Arc duplication/PTV splitting can be switched on or off by setting Arc Duplication is enabled to either Yes or No `①`.

If the arc setup contains more than one arc, it might be sensible to limit the total number of arcs. This directly limits the number of sub-PTVs that are automatically created. Set the maximum number of arcs under Maximum number of arcs.

For example, if there are two arcs in the setup and Maximum number of arcs is set to 6 `②`, PTV splitting may generate up to three sub-PTVs (2 x 3 arcs). However, if Maximum number of arcs = 4, only up to two sub-PTVs are allowed to be generated (2 x 2 arcs).

## Limitations

Currently, it is *not* possible to apply the algorithm to PTVs that have several disconnected components (i.e., several parts that are not adjacent to each other). For example, imagine a single PTV that covers two vertebrae but spares the vertebra between them. Two adjacent vertebrae could also be contoured so that the respective volumes do not overlap.

## Algorithm Details

The PTV splitting algorithm is very specific to the complicated shapes of vertebrae. Therefore, it is only available in the Spine SRS workflow.

When treating spine metastases, either the complete vertebra or only parts (or sectors) of a vertebra may be included in the PTV. The algorithm is able to distinguish these two cases based on the skeleton of the PTV. The skeleton of a complete vertebra is dominated by a circular branch around the hole where the spinal cord passes through the vertebra. Skeletons of partial vertebrae do not contain such a prominent circular branch.

## Complete Vertebrae

The PTV is divided into 8 parts by the k-means algorithm (using k++ initialization). These parts are subsequently merged until 4 parts are left. At each merging step, the two parts are chosen so that the sum of the convex hull volumes of all the parts left after merging is minimal. If the Maximum number of arcs is smaller than 4 times the number of arc plane definitions, the complete vertebra is accordingly split into less than 4 parts. This is achieved by merging multiple parts.

## Partial Vertebrae

The PTV is divided into 2 parts by trial-and-error. At each voxel on the skeleton of the PTV, a plane is determined that minimizes the intersection area of the plane with the PTV. The planes are parallel to a predetermined rotation axis that is aligned with the mass centroid axis of the local part of the spinal cord. If neither spinal cord, spinal canal, spine or cauda equina is available as OAR in the application, the Z-axis of the reference image set is taken as the rotation axis. Candidate planes are created by rotating an initial plane going through the skeleton voxel around this rotation axis.

A naïve, geometric way by a plane

`①`. However, this is not optimal because the bottom part is not connected.By flood-filling starting from the skeleton voxel

`②`. The flood-filling algorithm is not allowed to cross the plane in the local neighborhood of the skeleton voxel`③`, but is allowed to cross the plane in other parts of the object`④`.

If the summed convex hull volume of the parts was significantly reduced by splitting into two parts, it is possible to split one of the parts again. Additionally, the PTV is split twice at the same time, which means that all possible combinations of two skeleton voxels are selected and the PTV divided into three parts with two planes. The result with the lower summed convex hull volume is selected. Certain steps are not executed depending on the Maximum number of arcs setting. For example, if there are 2 arc plane definitions and Maximum number of arcs = 4, then splitting twice at the same time is not performed because this will lead to 3 parts and 6 arcs, respectively.