The modeling of the cervical spine is a design and use of an anticipated mathematical model, allowing a simplified understandable representation, a form of prediction of the functioning of the cervical spine, thanks to these biomechanical properties. Thanks to this modeling it will be possible to parameterize, to calibrate or adjust the functioning of the spine in order to draw practical conclusions applicable in humans, during treatment of cervicospondyloarthrosis by cervical traction Cervical spondyloarthrosis clinically manifesting itself in various forms, the most common of which is radiculopathy on vertebral disco impingement or cervicobrachial neuralgia (CBN) is one of three clinical syndromes of cervical spondyloarthrosis(1) (CS). CBN is caused by compression or irritation of nerve roots (2,3). The World Health Organization recommends that degenerative joint conditions should be managed by noninvasive, nonpharmacological therapeutic measures. This can be done manually or mechanically by applying a traction force to stretch the muscles, lengthen the soft tissues and create the space between the cervical vertebrae (Fig. 1,2), [4–5]. The litterature suggests that for a good clinician, the traction force applied should neither be excessive nor insufficient. Thus, an appropriate pulling force must be fixed in advance.
The problem justifying this study is in the fact that previous research has shown that a traction force of 11 to 16 kg and a duration of 20 to 25 minutes was necessary to obtain a measurable change in the structures of the cervical spine and a good muscle relaxation [6,7]. However, there is not yet clear evidence on the impact of loading time rise (the evolution of the increase in intervertebral spaces) and on the results. Therefore, the present study aimed to evaluate the influence of the intensity of the traction force and the time of the rise on the progressive evolution of the intervertebral spaces, for a controlled, effective and safe cervical traction.
This study will provide the necessary information on the evaluation of the evolution of the spacing of the intervertebral spaces during a brutal loading, or rectangular impulse and during a progressive loading following a slope or progressive impulse. was put on the impact of the time of the rise or the loading on the gain of intervertebral spaces observed. Finally, a comparison will be made in the discussion part with the experiment carried out in vitro in goats in previous study.
MODELING TRACTION OF THE HUMAN CERVICAL RACHIS
Structure of the human cervical spine
As shown in the images (Fig. 1), the human cervical spine is made up of seven vertebrae ( \({C}_{1}{C}_{7}\)) connected by soft tissues which are the intervertebral discs, ligaments, neck muscles and facet joints. The intervertebral discs act as a shock absorber during axial loading and carry the load from one vertebra to the other; the articular facets only support the load of one vertebra on the other; the ligaments and muscles of the neck stabilize the cervical spine.
Physical model of the human cervical spine
To facilitate the analysis, a few assumptions were made on the human cervical spine model:

The vertebrae are considered as rigid and dimensionless, identical bodies whose masses in grams per unit of vertebra are 6.3. [8];

Discs and ligaments are treated as in parallel combination of linear parts, springs providing longitudinal stiffness.
Using the previous assumptions, the eight degrees of freedom model is established as shown in Fig. 7
, where \({k}_{ij}\)( \(i=OC,\text{1,2},\text{3,4},\text{5,6},7;j=\text{1,2},\text{3,4},\text{5,6},7,T1)\)represent different stiffnesses.
Stiffness values are given in Table 1 and Table 2 for, respectively, various intervertebral ligaments and discs.
Table 1
spinal level

Ligaments

Stiffness (N/mm)

CO–C1

JC (joint capsules)

32.6


AAOM (anterior atlanto – occipital membrane)

16.9


PAOM ( posterior atlanto – occipital membrane)

5.7

C1–C2

ALL ( anterior longitudinal ligament)

24.0


JC

32.3


LF ( ligament flavum )

11.6

OCC2

TM ( tectorial membrane )

7.1


Apical

28.6


Alar

21.2


CLV ( cruciate ligament, vertical part)

19.0

C2–C5

ALL

16.0


PLL ( posterior longitudinal ligament)

25.4


LF

25.0


SIL

7.74

C5–T1

ALL

17.9


PLL

23.0


LF

21.6


ISL ( ligament interspinous )

6.4

Table 2
Level

Stiffness (N/mm)

C2–C3

63.5

C3–C4

69.8

C4–C5

66.8

C5–C6

22.0

C6–C7

69.0

C7–T1

82.2

Modeling of the axial tensile force
To model the pulling force, we first consider a rectangular pulse of magnitude\({F}_{0}\) (Fig. 4 ), which represents a pulling force suddenly applied to the occiput.
We then apply a force gradually following a linear transition in time \({T}_{0}\)(rise time) to a constant load of amplitude \({ F}_{0}\), as shown in Fig. 5, which represents a pulling force applied gradually to the occiput (OC).
Rectangular pulse

On a slope with ascent time

\(F\left(t\right)=\left\{\begin{array}{c}{F}_{0} pour t\le {t}_{end} \\ 0 pour t>{t}_{end} \\ \end{array}\right.\)

\(F\left(t\right)=\left\{\begin{array}{c}{F}_{0}\left(\frac{t}{{T}_{0}}\right) pour t\le {T}_{0} \\ {F}_{0} pour {T}_{0}<t\le {t}_{end} \\ 0 pour t>{t}_{end} \\ \end{array}\right.\)

ASSESSMENT OF INCREASE IN INTERVERTEBRAL SPACE MEASUREMENT
Equations of Motion/Governing Equations
The cervical spine was modeled as an eightdegreeoffreedom system as shown in Fig. 3, so the motion of the system can be completely described by the coordinates\({x}_{OC}\left(t\right),\) \({x}_{1}\left(t\right)\), \({x}_{2}\left(t\right)\), \({x}_{3}\left(t\right)\), \({x}_{4}\left(t\right)\), \({x}_{5}\left(t\right)\), \({x}_{6}\left(t\right)\), \({x}_{7}\left(t\right)\)which define the vertical displacements of the vertebra represented by the masses\({m}_{OC},\) \({m}_{1}\), \({m}_{2}\), \({m}_{3}\), \({m}_{4}\), \({m}_{5}\), \({m}_{6}\), \({m}_{7}\)at any time \(t\)from equilibrium positions.
Applying Lagrange's principle, principle of derivative of the equations of motion to the model leads to the system of equations which can be written in general matrix–vector form as:
$$M\left\{\ddot{x}\left(t\right)\right\}+K\left\{x\left(t\right)\right\}=\left\{F\left(t\right)\right\}$$
Or
$$M=\left(\begin{array}{cccccccc}{m}_{OC}& 0& 0& 0& 0& 0& 0& 0\\ 0& {m}_{1}& 0& 0& 0& 0& 0& 0\\ 0& 0& {m}_{2}& 0& 0& 0& 0& 0\\ 0& 0& 0& {m}_{3}& 0& 0& 0& 0\\ 0& 0& 0& 0& {m}_{4}& 0& 0& 0\\ 0& 0& 0& 0& 0& {m}_{5}& 0& 0\\ 0& 0& 0& 0& 0& 0& {m}_{6}& 0\\ 0& 0& 0& 0& 0& 0& 0& {m}_{7}\end{array}\right)$$
$$K=\left(\begin{array}{cccccccc}{k}_{OC1}+{k}_{OC2}& {k}_{OC1}& {k}_{OC2}& 0& 0& 0& 0& 0\\ {k}_{OC1}& {k}_{OC1}+{k}_{12}& {k}_{12}& 0& 0& 0& 0& 0\\ {k}_{OC2}& {k}_{12}& {k}_{OC2}+{k}_{12}+{k}_{23}+{k}_{25}& {k}_{23}& 0& {k}_{25}& 0& 0\\ 0& 0& {k}_{23}& {k}_{23}+{k}_{34}& {k}_{34}& 0& 0& 0\\ 0& 0& 0& {k}_{34}& {k}_{34}+{k}_{45}& {k}_{45}& 0& 0\\ 0& 0& {k}_{25}& 0& {k}_{45}& {k}_{25}+{k}_{45}+{k}_{56}+{k}_{5T1}& {k}_{56}& 0\\ 0& 0& 0& 0& {k}_{56}& {k}_{67}& {k}_{56}+{k}_{67}& 0\\ 0& 0& 0& 0& 0& 0& {k}_{67}& {k}_{67}+{k}_{7T1}\end{array}\right)$$
$$\left\{F\left(t\right)\right\}=\left(\begin{array}{c}F\left(t\right)\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right)$$
Resolution
To solve the problem of governance equation, we applied the method of modal analysis exploiting the properties of orthogonality properties, vibration modes to transform the eight equations of motion from a physical system where the equations are coupled to a modal coordinate system where the equations of motion are decoupled. Each of the decoupled equations can be solved independently. The flowchart displayed in the figure below summarizes the modal analysis method (Fig. 6):