Geometric Structure of 3D Spinal Curves: Plane Regions and Connecting Zones

Abstract

This paper presents a new study of the geometric structure of 3D spinal curves. The spine is considered as an heterogeneous beam, compound of vertebrae and intervertebral discs. The spine is modeled as a deformable wire along which vertebrae are beads rotating about the wire. 3D spinal curves are compound of plane regions connected together by zones of transition. The 3D spinal curve is uniquely flexed along the plane regions. The angular offsets between adjacent regions are concentrated at level of the middle zones of transition, so illustrating the heterogeneity of the spinal geometric structure. The plane regions along the 3D spinal curve must satisfy two criteria: (i) a criterion of minimum distance between the curve and the regional plane and (ii) a criterion controlling that the curve is continuously plane at the level of the region. The geometric structure of each 3D spinal curve is characterized by the sizes and orientations of regional planes, by the parameters representing flexed regions and by the sizes and functions of zones of transition. Spinal curves of asymptomatic subjects show three plane regions corresponding to spinal curvatures: lumbar, thoracic and cervical curvatures. In some scoliotic spines, four plane regions may be detected.

Figure 1

The techniques involved in the calculation of the 3D spinal curve from biplanar radiography coupled with photogrammetric reconstruction.

Figure 2

The technique allowing the access to the geometric structure of the spinal curve is illustrated by the results obtained with an asymptomatic subject (male 35 years). Are presented: the sagittal and frontal radiographic images, the sagittal, frontal, and horizontal global projections of the 3D spinal curve and the pelvic modeled as a triangle, the projections of the three plane regions, and the short zones of transition. A table gives numerical values: pelvic incidence PI and pelvic tilt PT, main parameters of the regional plane zones 2, 4, 6: components of the normal vector to the regional plane (ψ axial rotation about Z ₀, θ lateral flexion about X ₀), vertebral levels of regional limits, magnitude of the flexion angle bending the regional curve, main parameters of the zones of transition 1, 3, 5: vertebral levels of the zone, offsets of angular displacements (Δψ and Δθ).

Figure 3

Cases of two asymptomatic adults (40 and 30 years). The comparison between modeling the spinal curve structure using a unique plane of greatest curvature PGC and the model involving regional planes and zones of transition. For each asymptomatic subject, are drawn: the PGC and the three regional planes and the zones of transition. The two PGCs have similar positions, but the regional structures are strongly different.

Figure 4

Patient suffering from a low back pain. The geometric structure is characterized by a strong rotation of the spinal lumbar plane plane region and by a correlative increasement of the sizes of zones of transition.

Figure 5

The scoliosis alters significantly the geometric structure of the spine.

Figure 6

The different steps of scoliotic evolutions. The new technique allows medical doctors to quantify local effects of deforming pathologies.

Figure 7

Illustration of the effects of a surgical operation upon a deformed spine.