Determination of Stresses in the Apparatus of External Fixation of the Human Spine
Piven Valeriy Vasilievich1
1 Department of Machinery and Equipment of the Oil and Gas Industry, Industrial University of Tyumen, Tyumen, Russia.
Email: pivenvv @ yandex.ru
ABSTRACT
This present work aims to increase the effectiveness of the spine external fixation device by optimizing its stressstrain state. Insufficient studies of the problems of the stressstrain state of the pedicle screw closure vertebra column, as well as the occurrence of dangerous stresses in the pedicle screw with various combinations of control efforts, are made to correct spinal deformity with a large factor of safety. In many cases, it is not possible to create the required correction action on the spine. A high degree of statistical determination of the structure, the ambiguity of the impact of the device on the vertebrae, and the change in the internal control efforts in adjusting the distraction rods limit the use of external fixation for complex spine deformities. Determination of stresses in dangerous sections of rodsscrews depending on the control actions is the basis for mathematical modeling of the process of correction. An algorithm for mathematical modeling of the stressstrain state of the external fixation apparatus of the spine is proposed. By calculationand experimentation, we were able to determine the mechanical stresses in the dangerous section of the structural elements of the device. To prevent dangerous stresses when turning the vertebrae, it is necessary to apply asymmetrical multidirectional load. With complex deformities of the spine, an additional impact of a couple of forces is necessary to turn the vertebrae.
Key words: Distraction force, External fixation, Spine, Scoliosis
INTRODUCTION
Overload of vertebral column segments caused by various factors and in some cases, it leads to kyphoscoliosis  a deformity of the spine [16]. When choosing the optimal tactics of surgical treatment [716] it is possible to install various implants and frame structures mounting directly on the vertebrae, with a strong degree of scoliosis with the use of anexternal fixation device of the spine [1719]. In some cases, it is advisable combining the use of these two ways: in the first stage of correction of the deformation of the spine for external fixation, then the subcutaneous installation of implants or other structures.
Applied as the optimization process of correcting the scoliotic deformity of a person, the external fixation device is a spatial core structure fixed with the help of rodscrews in the vertebral bodies [20]. Similar constructive schemes and details can also be used in the treatment of various spinal injuries.
The present work aims to increase the effectiveness of the external fixation device by optimizing its stressstrain state. Determination of stresses in dangerous crosssections of rodsscrews depending on the control actions acts as the basis of the mathematical modeling correction process.
Mathematical design and optimal loading conditions of the device to ensure the necessary corrective actions on the vertebrae can be realized using a developed schemewith the original rheological properties of an elasticviscousplastic base (vertebral bodies), and ways of embedding the supporting elements, the stressstrain state of spatial rod system and its geometric and mechanical characteristics [21].
There are technological and technical methods of reducing stress in a dangerous section of parts of the external fixation device, but in all cases, it is necessary to take into account the stress in the dangerous sections of the parts of the device and the vertebral bodies [22].
MATERIALS AND METHODS
To achieve a positive result for the correction of spinal deformity, the parts of the device must create a stressdeformed state to provide the necessary torque on the vertebrae in the sagittal plane. This should carry out strength limitations on the parts of the apparatus and the body of the individual as well. A dangerous section of the shaft of the screw is its point embedded in the vertebra. With the destruction of the rodscrews, the problem arises out of the threaded portion, in which case the surgery for the correction of spinal deformity will have to be postponed.
Optimization of the above options can be achieved by a mathematical design of the process of correction. The decomposition of the optimization process (Figure 1) should include the formation of a block of input data, a mathematical design block with the search for optimal values of the adjustable parameters of internal forces in the details of the apparatus. With a phased implementation of the algorithm, it is necessary to provide feedback to the initial data generation unit to take into account the actual strength state of the vertebral body.
Figure 1. Decomposition scheme of correction of spinal deformity 
During mathematical modeling for finding the unknown power factors, the force method was used [23, 24]. The external fixation device of the spine is a statistically indeterminate spaceframe design. A high degree of static in determination leads to an additional internal effort in the details of the apparatus, to high stress, and reduces the efficiency of the apparatus.
To estimate the internal forces in the dangerous sections of the rodscrews of the external fixation device, we considered the scheme of the device with two lateral plates and the sealing of shafts of screws in the vertebral bodies (Figures 2 and 3). The computational model adopted is perpendicular to the installation rod screws to the plate.
Considering the case when one of the adjusting rods creates alongitudinal tensile force (P) and the other longitudinal compressive stress (P). The specified force system is a skewsymmetric load with a plane ofsymmetry passing through the ZZ, Z'Z'axes. To disclose static indeterminacy, as the main system [23], we take the force scheme shown in Figure 4.
To solve this problem by the method offorcesto find the four unknown values of internal forces, it is necessary to solve the following system of canonical equations [23]:
(1) 
where δĳ– move in the direction of the i – th power factor under the action of j – th individual factor; Xi is the internal force factors; Δip  move in the direction of the i – th power factor under the action of the force P.
In the experiment, the internal forces and moments were measured using force sensors.

Figure 2. Diagram of apparatus: 1  rodscrew; 2  plate; 3  adjusting rod 
Figure 3. Scheme fixation rodscrews to the vertebra 
Figure 4. Basic system in the calculation of forces 
Plot bending and torsional moments from the action of load P are shown in Figure 5.
Figure 5. Plot bending (Mben) and torsional (Mtor) moments of force P 
Calculations of canonical coefficients carried out in a known mannerwhen the value of the force P = 200 N, the parameters a = 0.06 m; b = 0.04 m; s = 0.03 m; d = 0.02 m. as a result we get the following system of equations [23]:
(2) 
RESULTS AND DISCUSSION
When solving the system of equations (2) in the program "MATCAD" we obtained the following values of unknowns: X1= 61.0; Х2 = 2.1; Х3 = 6.8; Х4= 0.9.
Bending moment in a dangerous section of rodscrews (A)
(3) 
Torque T.
(4) 
Stress in p. A
(5) 
To turn the vertebra, the sum of the effects of power factors should be aimed at creating maximum torque MtorZ.
(6) 
The discrepancy between the theoretical and experimental data was not more than 12%.
The obtained results may be used in the optimization of the stressstrain state of the apparatus for external fixation of the spine in various layout solutions, including a large number of elements. The choice of specific vertebrae, which should influence the maximum torque should be produced taking into account the initial deformation of the spine.
The determination of the maximum diameter of the rodscrew is limited by the mechanical properties of the bone tissue. This diameter is chosen depending on the type of vertebra, its anatomical structure. For example, for the lumbar vertebra L1, the length of the threaded part of the screw rod should be 4042 mm, and its diameter4 mm.
Optimal values of control forces in the distraction rods must be performed using mathematical modeling, the decomposition diagram of which is shown in Figure 1. The limiting parameters in the simulation are stresses in dangerous sections, determined from the dependences obtained in the work.
Further studies on the correction of spinal deformities should be continued in the direction of developing guidelines for physician staff on controlling the device for external fixation of the spine, developed based on mathematical modeling, taking into account the results obtained in this study.
A series of experiments were carried out to experimentally determine the stresses arising in the rodscrew. Mechanical stresses were evaluated using strain gauge equipment. Strain gauges were mounted at the base of the rodscrew, which was screwed into a wooden bar (Figure 6). The rodscrewwas rigidly connected by a plate, to which they were attached with Lshaped clamps.
The distance from the sealing point of the rodscrew to the Lshaped clamp was 73 mm. The force P was created using weights of 10 H and 15 H. The points of application of the load P are indicated by the numbers 19 (Figure 7). Securing loads through holes 1, 2, 3, 5, 7, 8, 9 (Figure 7) corresponds to the possible options for the mounting location of the adjustment rods 3 (Figure 2).
According to the obtained calculated and experimental data, a graph of the dependence of the bending stresses in the rodscrew, depending on the point of application of the force P, is constructed (Figure 7). The presented results show the percentage of stress in the dangerous crosssection of the rodscrewin comparison with the case when the entire load falls on one screw rod without fixing it to the plate.
The analysis of the results showed the following. With an asymmetric load of the rodscrew (Figure 6), one of the rodscrew carries a significant load. This can lead to dangerous stress in the sealing of the rodscrew or the body of the vertebra. For example, when a load is applied at point 1 (Figure 7), the first screw rod accounts for up to 70% of the total stress, while the second one accounts for up to 30 %. The symmetrical and equal stress of the rodscrewis achieved when a load is applied at point 5. When moving away from this point, the stress difference in the rodscrew may differ by 2.5 times. The discrepancy between the experimental and theoretical data was 911%.
Figure 6. Diagram of the load application (P) on the rodscrew I and II connected by a plate 
For the stresses in any rodscrew not to reach the limit value, it is necessary to apply asymmetrical load to plate 2 (Figure 2). At the same time, the maximum torque of the MtorZ will also be provided (Formula 6). Forces P must be applied at both ends of the plate in opposite directions.
Analysis of formula 8 allows us to conclude: to create the maximum torque MtorZ on the vertebra, it is advisable to add the impact of a pair of forces. This pair of forces is denoted as X2 (Figure 4). The direction of action of the pair of forces should be opposite to the direction shown in the figure:  X2.
The need for an additional impact of a pair of forces on the plates for turning the vertebrae arises in cases where, due to the peculiarities of the curvature of the spine, it is impossible to provide a symmetrical impact with the forces of P. The need for a pair of forces appears when installing the external fixation device of the spine, consisting of three or more plates with complex deformities of the spine. In this case, it is advisable to apply a couple of forces for additional rotation of individual vertebrae.
Figure 7. Graph of the dependence of stresses (σ) on bending in the rodscrew (I and II) depending on the point of application of the force P and the diagram of the plate with holes: 1, 2, 3, ... 9 – holes through which loads were attached 
CONCLUSION
This study presents the results of the study in the simulation of mechanical stresses in the details of the external fixation apparatus of the spine. The simplest case of using an external fixation device is considered when only two vertebrae are used to correct spinal deformity. In more complex forms of scoliosis, at least three vertebrae are involved. The external fixation device must be equipped with a set of simple elements. This arrangement is considered in this paper. The calculation of more complex structures of devices should be carried out based on the results obtained.
ACKNOWLEDGMENTS : None
CONFLICT OF INTEREST : None
FINANCIAL SUPPORT : None
ETHICS STATEMENT : None
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