## Definitions

For the purposes of this chapter, only a few definitions are required. Vascular tree morphology and morphometry have already been defined as the shape or structure of the tree, and methods (in this context particularly imaging-based methods) for quantifying it, respectively. The repeated subunit as the tree progresses from trunk to periphery is the bifurcation. A bifurcation is a branch point in the tree where a parent vessel divides to give rise to two daughter vessels as shown in Fig. 1 [4]. The unbranching tube between consecutive bifurcations is called a vessel segment. At a bifurcation, the parent vessel diameter is denoted by d0, and the larger and smaller daughter diameters by d1 and d2, respectively. The 3D midline of the vessel lumen is called the vessel axis or the medial axis of the vessel segment. The branching angles, 61 and 02, measured in the planes containing the parent and each daughter's medial axes, are the angles between the parent's medial axis and the

larger and smaller daughter's medial axes, respectively. Zamir has claimed that the three vessel axes for all coronary bifurcations lie approximately in a plane, rendering the three-dimensionality of bifurcations negligible [4]. The space-filling characteristics of the entire tree are achieved by translations and rotations of the numerous bifurcation planes relative to each other. The bifurcation index is defined as the ratio d2/d1 and varies between 0.0 and 1.0. A symmetrical bifurcation, in which the two daughters have approximately equal diameters, would have a bifurcation index close to unity, whereas for a side branch where a very small vessel branches off a main trunk — a highly asymmetric bifurcation — this quantity would approach zero. Thus, the primary fundamental quantities of interest in arterial tree morphometry are vessel segment lengths and diameters and the branching angles at bifurcations. Other parameters reported on in the radiological literature, such as the branching coefficient, defined as the ratio of the cross-sectional area of the daughters to that of the parent [5], can be derived from this base set of metrics assuming circular vessel cross-sections. Of course, there are thousands or even millions of these elementary measurements available from a single 3D image of a complete arterial tree, so the necessity arises to summarize their functional or hemodynamic significance in some meaningful and intelligible or interpretable way. There are also some higher order statistics such as measures of connectivity that have been applied to vascular (the connectivity matrix) [6-9] and other network-type structures (the Euler number or Euler-Poincare index) [10-12], but these are not emphasized in this chapter.

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