The Method Exploration for Measuring the Strain of the Intervertebral Disc

Many researches point out that intervertebral pressure and transformation are key parameters for evaluating intervertebral disc degeneration. Aiming at avoiding the damage caused by direct and indirect measuring methods, this research proposes a cylindroid hypothesis and measuring method, which can monitor the strain condition of the intervertebral disc in vivo and real-time without being damaged.


Introduction
According to incomplete statistical analysis, around 7% -10% of the 1.3 billion people in China have some kind of cervical disease.It shows a growing trend.After some kind of treatment, the motion function of the fused segments is lost forever and the adjacent level becomes degenerate easily [1].Both biochemical and mechanical factors are thought to deal with the cascade of events caused by degeneration [2,3].Many researches have pointed out that intervertebral pressure and transformation were the key parameters for evaluating degeneration.Monitoring the intervertebral strain in vivo was very important for evaluating the long-term result of surgery and checking the degeneration of the adjacent levels [4,5].Many analytic or geometric and finite element models have been developed to describe the structural response of intervertebral discs [6].At present, direct measurement can't be done nondestructively [7], indirect measurement and simulation can't reach an agreement [8][9][10].
Based on the physiology structure and properties of the intervertebral disc, this study raised a solution that could monitor the intervertebral strain in vivo without being damaged.It can solve the intervertebral disc strain measure dilemma that direct measure methods can only be executed on a few healthy volunteers and other simulation or indirect measures are too hard to achieve an agreement.
Aiming at getting the strain condition of the intervertebral disc, a mathematic model and a measurement system based on strain gauges was built.In our hypothesis, the complete strain condition and the dimension of the disc can be calculated by only fixing 6 strain gauges on the front side of an intervertebral disc.

Description of the Intervertebral Disc
Strain Measurement Model

Definition of Initial Condition
In some imaging measurement about intervertebral disc, we can easily find out that the transverse section of the disc is an axialsymmetrypolygonwith a smooth border.At first, the study assumes the geometric shape of the intervertebral disc as a cylindroid, sets the origin at the geometric center and establishes a coordinate in the vertical view.As Figure 1 shown, the right side is set as the X axis, the front side is set as the Y axis, and the top side is set as the Z axis.According to the result of actual measurement, the length in the coronal plane is longer than the length in the sagittal plane, the expression of the side face of the disc in Cartesian coordinates is: A is the long axis of the ellipse, B is the short axis of the ellipse, and A > B. As the dimension of the ellipse can't be measured in vivo, A and B are both unknown.With the help of the transfer equation between Cartesian coordinates and polar coordinates: We can rewrite (1) as: Link a certain point on the ellipse's angle with the origin, the angle between the X axis and it is  ,  is the distance between from the point to the origin.Any point on the ellipse can be described as   ,    .

Equation of Geometric Condition
In order to realize the nondestructive measurement, only a bit of arc symmetric with Y axis is available.
As   a, for th in b, c, d, e are tensile strain gauges and sensitive e tension in Z direction.f is a shear strain gauge and sensitive for the in torsion Z direction.The measuring result of f is independent to others' equation, we focus on the geometric equation and strain equation of a, b, c, d, e.
After fixing the flexible PCB right in the front of the tervertebral disc, there are only 4 independent parameters: , , , .
We can get 2 independent equations: Add the arc length formula of plane curve: We can get: In the equation above: and are known quantity,     are unknow uantity n q .

Equation of Physical Condition
In the normal stress analysis theory, the entire deformapatibility, the st tion of the intervertebral disc can be resolved into 4 kinds: 1) Bending deformation in X direction; 2) Bending deformation in Y direction; 3) Bending deformation in Z direction; 4) Torsional deformation in Z direction.
According to the equation of strain com rain at a, b, c, d, e can be described as below: sin cos sin cos After solving the equation system above, we B

Solution
uation system can be described as For convenience, the eq below: •sin 2 •sin 2 •cos 2 •cos 2 Because the elliptic integral exists in Equations (18) an sed.d (19), the equation system is hard to be solved directly.So we assume the long axis radius, short axis radius and angle of the measure point at first and then select the fitting answer.The specific steps were presented in Table 3.

Model Modification and Improvement
The model above is theoretical and hypothesis ba Some modification and improvement will make the model more effective and accurate.
Firstly, the error the cylindroid simply introduced into the model needs to be calculated and modificated.Some imaging research tells that the front height of the intervertebral disc is longer than the back height, so an angle about 3˚ -6˚ exists between the top and bottom surface of the intervertebral disc naturally.
Secondly, the strain gauge should stick on the surface directly.But confined by the biocompatibility and operation time, the strain gauge had to be packed in a biocompatibility flexible board and sticked at the same time.The  Based on the range of rough m lect A and B at a certain step length and get a two dimension easuring result, se al matrix.5 7 Using the numerical method (function "EllipticE") in Matlab, the arc length can be calculated a certain step length.Combining the known quantity and  3) and ( 4), α, β can be solved.
Bringing α, β into Equations ( 1) and ( 2), ρ α and ρ can be solved.(5)-( 7) compose a over-determined system, some solutions of ε N got with the linear fit method.0 0 (8) and ( 9) compose a over-determined system some solutions of ρ y can be got with the linear fit method.
Compare the solutions and determined the optimal one.board's properties especially the thic ess will bri me error to the result.
en specificity, installation error, m ral pressure or strain measurement method will m e strain condition of the intervertebral model and a measurement system nd can be operated on th iment stage and further modify and adjust

Figure 1 .
Figure 1.Coordinate built on the intervertebral disc.

Figures 2 and 3
show, 6 strain gauges were packaged in flexible PCB and fixed on the front surface of the intervertebral disc, symmetrical with YZ plane.ab = de and bc = cd.

Figure 2 .
Figure 2. Front view of the strain gauges' location.

Figure 3 .
Figure 3. Top view of the strain gauges' location.

14) 2 . 4 .
Model Summary including 7 known quantities: In summary, the model , , , , , , l      , details are shown in Table 1.ntities: , , , , , , , , y        ., details are shown in Table 2 disc's radi of curvature in plane XZ 9 independent equations: ngelo and K. T. Foley, "Motion Compensation Associated with Single-Level Cervical Fusion: Where Does the Lost Motion Go," Spine, Vol.31, No. 21, 2006, kn ng the model accordingly.so Thirdly, it is hard to modify the model in vivo because of the unavoidable specim

Table 2 . Description of unknown quantity.
A radThe angle between point a and the X axis