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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.

Sagittal balance of asymptomatic subjects and patients has been studied from sagittal radiographic images [

The mathematical equation of the sagittal spinal projection has been obtained from using a new technique [

The photogrammetric technique reconstructs points in space from their two images in projection plane. The photogrammetry applied to radiographic images has been described by Suh [

The biplanar radiography involves the setting of specific devices in case of simultaneous exposures [

The goal of this paper is to study the geometric structure of spinal curves defined from a series of points reconstructed from photogrammetric techniques. Stagnara et al. [

The principle of local planes fitted to regions of the 3D rough spinal curve has been retained. Two criteria must be satisfied locating these planes: (i) the maximum distance between the 3D curve facing the regional plane and the unknown plane must be low (2 mm) and (ii) the regional curve must be continuously plane without local anomalies. A great number of spinal curves have an heterogeneous structure. They show plane regions, where the spine curve is purely flexed, and zones of transition between adjacent plane regions. The plane regions often correspond to lumbar, thoracic, and cervical curvatures. In asymptomatic subjects the regional planes are oriented closely. They may be strongly spreaded in patients with spinal deformities. Examples are shown for illustrating the method. Results are presented and discussed.

Frontal and sagittal images of subjects and patients have been obtained from two different radiographic systems.

New set-up shooting simultaneously from frontal and sagittal incidences. 125 members of paramedical staffs of a local hospital accepted a biplanar radiographic examination in the frame of a clinical research studying relations between spinal features and chronic low back pain.

A standard radiographic system completed by a rotating plateau has been set in an hospital of Lyon specialized in back pathologies. All patients having back deformities are submitted to a biplanar radiographic examination with successive X-rays with the objective of following the 3D spinal feature evolutions versus time. The access to the geometric structure of spinal curves at different steps of their evolution brings new information for clinicians.

The detection is based upon two criteria: the criterion of proximity and the criterion of continuous regional planeity.

The criterion of proximity to a plane is satisfied when all points of a local section of the rough spinal curve are close to the unknown regional plane. The maximum distance between rough curve and the facing plane is very low (2 mm). Accuracy reconstructing points using biplanar radiography coupled with photogrammetry is 1 mm.

The criterion of continuous regional planeity controls that the plane region is continuously plane. Series of three points, located close together along the spinal curve are successively considered. The normal vector to the local plane passing through the three points is calculated. The bundle of normal vectors to the regional curve is defined. This bundle must be strongly concentrated. Aberrant points along the spinal curve introduce aberrant normal vector directions. These points are eliminated. The direction of the median normal vector is calculated from the bundle of normal vectors. Limit points bounding the plane region are then determined.

Limit points bounding each plane region are located along a diagram representing the unrolled spine from the L_{5} lower plate center to the C_{3} upper plate center. Each vertebra and each intervertebral disc are located along the unrolled spine using normalized values defined statistically [

The axes of the fixed frame

This technique introduces rotations about fixed axes. The direct comparison between orientations of the different plane regions does not depend on the position of the moving axes involved by the Euler calculation.

Spinal curves show generally three (or more) plane regions. The planes

The geometric structure of the spinal curve is heterogeneous. It has several plane regions

The plane curves

Positions of limit points starting points

Length

Position of

Angles

Global flexion angle

All parameters characterizing the shape of each flexed region are calculated. But main parameters representing the geometric structure of spine, pelvis in upstanding patients, are presented.

The pelvis in erected patient is represented by two parameters.

One parameter represents its morphology: the pelvic incidence.

One parameter describes its orientation in erected posture: the pelvic tilt.

The spinal structure is first characterized by the number of its plane flexed regions. Each regional plane is located using the two angles (

The goal of this paper is to describe the geometric structure of 3D spinal curves assumed to be previously calculated. Figure

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

The results of two different treatments are then compared. The first one assumes that the global spinal curve entirely belongs to a unique plane: the plane of maximum curvature. The second one detects several strictly plane regional sections along the spinal curve. Local geometric discontinuities are then exhibited. In asymptomatic subjects, the spinal structure is compound of three plane regions corresponding to lumbar, thoracic, and cervical curvatures. Short zones of transition are interposed between adjacent regions oriented differently. The modeling technique has been applied to patients suffering from low back pain. Local structural discontinuities of the spinal curve are observed. Scoliosis modifies significantly the spinal structure. An example of evolutive scoliosis is presented. Effects of implant on the spinal structure are shown.

25 asymptomatic adults accepted a biplanar global examination (sagittal and frontal radiographic incidences). Photogrammetric reconstructions allowed to represent 3D models of pelvis and spinal curve. The pelvis is represented as a triangle based on femoral head centers and sacral plate center. 3D spinal curve and pelvic triangle are projected on sagittal plane, frontal plane, and horizontal plane. The spinal curve is compound of three plane regions and three zones of transition.

A case (male 35 years) is presented in Figure

zone 2 from

zone 4 from

zone 6 from

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 (

Three zones of transition are interposed between adjacent flexed regions:

zone 1:

zone 3:

zone 5:

The plane regions are characterized by the orientation of the normal vector to the regional plane (axial rotation

Angular offsets between normal vectors to adjacent regional planes: Δ

These techniques locate the unique plane of maximum curvature and the heterogeneous structure of spine compound of flexed regions and zones of transition.

Two examples of asymptomatic subjects are presented (Figure

the plane of greatest curvature (PGC) is characterized by the orientation of the global normal vector to the plane (angles _{eg} from 3D curve to PGC,

the regional planes are represented by the orientations of their normal vector (_{lr }and angular _{ar} coefficients. They represent linear and angular maximum offsets between 3D curve and regional planes.

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.

The presented asymptomatic subjects have different spinal curve features: their PGC is not significantly rotated versus the sagittal plane

90 patients suffering from low back pain accepted a biplanar radiographic examination. Figure

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.

The scoliosis alters significantly the geometric structure of the spine.

The radiographic file of an adult patient is presented for illustrating different effects of the pathology on the structure of the spinal curve. The whole spinal structure is strongly rotated versus the patient sagittal plane, main part of the rotation concerning the lumbar plane region. The thoracic curvature (from

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

A scoliotic patient accepted four biplanar radiographic examinations after orthopaedic treatment along a period of eight years (before, 2 years, 5 years, 8 years). The spinal structure keeps comparable global features versus time, but spinal deformations are related to pelvic tilting. After eight years, the set pelvis spine recovers its initial feature. It can be observed that regional planes slightly move between two successive radiographic examinations but corresponding flexion angles keep roughly constant. The regional curvatures are submitted to small rotations versus time, but flexed shapes are kept.

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

Three radiographic files describe the structural changes brought by a surgical operation: initial state, 6 months (before the surgical operation), and one year (after the operation).

The two first radiographic files show the structural evolution of the spinal curve before surgical operation: the pelvic tilt is the same, the lumbar regional plane slightly increases its axial rotation versus the patient sagittal plane. The angular offset between lumbar and thoracic regions is increased.

The surgical operation consisted in implanting a rod between

The surgical operation reduces significantly the axial rotation value of the lumbar region (28.4° versus 68.9°). Axial rotation offsets between adjacent plane regions strongly decrease: from −24.1° to −3.9° for the lumbar/thoracic transition zone and from −24.5° to −6.9° for the zone of transition between thoracic and cervical regions.

The surgical operation acts directly upon the orientation of the lumbar region, however the lumbar flexion angle is roughly constant. This operation has indirect effects upon the global spinal structure: initial discontinuities between plane regions are strongly decreased.

In perfectly balanced subjects, spinal curves belong entirely to the anatomical sagittal plane. Lumbar, thoracic, and cervical curvatures are bounded by points of inflexion of the plane spinal curve. Our objective was to describe the spinal geometric structure of spines weakly or strongly deformed.

The mathematical representation of spinal curves is a prerequisite for accessing this goal. In the present study, it involved the access to radiographic systems giving images of the motionless patient shot under different incidences. The radiographic technique is coupled with photogrammetric reconstruction of points from their two images. Spinal curves are defined from series of isolated points.

Plane regions are detected along the rough spinal curve. This detection requires two successive steps, the first one locates its position, the second one eliminates aberrant points and restrict the regional extent so that two criteria would be satisfied. The first one checks that linear distances between points of the spinal curve and facing plane are lower than a maximum value. The second criterion verifies that planes passing through three adjacent points along the curve are very closed to the regional plane. The regional normal vector to the regional plane is calculated. The rough spinal curve facing each regional plane is projected on this plane. The regional plane projection of the 3D rough curve is uniquely flexed. The flexion angle bending this region is then obtained.

Zones of transition between two adjacent plane regions must transmit the angular offsets between the two plane orientations.

Directions of normal vectors to regional planes are related to anatomical axes (fixed axes) using two angles. The normal vector direction is obtained from two successive rotations moving the transverse axis

Geometric structures of spinal curves have been first estimated using different techniques. Experimental testing demonstrated the existence of an optimal radiographic incidence: the spinal curve shows its greatest curvatures when projected on the plane perpendicular to this specific incidence [

The results show that asymptomatic subjects have closely oriented regional planes. The drawing of a unique plane of greatest curvature PGC may be licitly proposed. In other asymptomatic subjects, the proposal of a unique plane of greatest curvature is not obvious. The 3D location of unknown plane PGC is always possible, but linear distances between spinal curve and PGC may reach high values, not consistent with an accurate geometric modeling. The proposed geometric model of the spinal structure displays its heterogeneity. Even in asymptomatic subject, some regional sections of the spinal curve belong to planes oriented differently. In these regions the spinal curve is purely flexed. Axial rotation and abduction components due to the different orientations of adjacent regional planes are concentrated at level of zones of transition. These zones are restricted to a unique intervertebral disc when the adjacent regional planes are close together. They are longer in patients showing high regional discontinuities.

The proposed model has been applied to asymptomatic subjects. Local unbalances are displayed. It has also been tested with patients suffering from back pathologies, chronic low back pain and scoliosis. These pathologies entail spinal structure deformations. Geometric description of spinal curve features may allow clinical people to propose well-adapted correcting techniques.

A large diffusion of this technique involves the access to the mathematical representation of the spinal curve. Local solutions have been found from using biplanar radiography with successive or simultaneous exposures coupled with photogrammetric reconstructions. But this modeling technique may be adapted to any imaging system giving the accurate representation of spinal curves.

E. Berthonnaud, R. Hilmi, and J. Dimnet disclose any financial and personal relationships with other people or organisations that could inappropriately influence their work