Spatial deformation of osteosynthesis systems. Message 1. Criteria of functional reliability

Annotation. Based on experimental studies, an analytical method is proposed for assessing the functional reliability of osteosynthesis systems. As criteria for the rigidity of osteosynthesis systems, displacements and mutual rotations of parts of bone fractures with fixation tools were used. The characteristics of the “bone with a fracture fixation system”, necessary for calculations, determined experimentally from individual loads by compression, bending, and torsion of bone samples with simulated fractures and fixation tools. The characteristics of the plates for fixation of tibial fractures are determined, the level of allowable loads is calculated according to the combined action of compression, bending and torsion, and the areas of valid loads are determined. The proposed criteria for functional reliability make it possible, using simple tests and analytical calculations, to determine permissible load levels that do not lead to the occurrence of dangerous movements in the fracture area.


Introduction
During the knitting of bone fractures fixed by means of fixation, in most cases, disposable or reusable, mainly cyclic, loads occur. At the same time, it is impossible to completely eliminate the dangerous displacements of individual points in the fracture area. These displacements can lead to improper reposition of fractures or slow down the process of fusion [1][2][3][4][5].
The practice of using various means of fixation of fractures (OS systems) has shown that one of the main functional indicators of these systems is the mutual linear displacements (Λ) of adjacent fracture points and the mutual rotation angles (G) of the fracture parts. Based on many years of research in the field of practical surgery, the established allowable linear displacements (displacement) of fracture points [Λ] = 1 mm, and the mutual angles of rotation of the fracture parts [G] = 3° [6,7].
Thus, the requirements for deformation indicators of the systems "bone with a fracture -a means of fixation of a fracture", which are determined by the physiological properties and type of fracture, have the form [8]: where i is the designation of the fracture points. It can be [9]: M -medial (located in the middle, closer to the longitudinal axis of the human body), L -lateral (located on the side, away from the longitudinal axis), V -ventral (located in front), D -dorsal (located behind) or other fracture points.

Accounting for the mutual displacement of the fracture points
Displacements and rotation angles in the fracture area are determined by the influence of such forces ( Fig. 1, a): Р1 -force acting along the longitudinal axis of the bone (for the lower extremities it is mainly human gravity); P 2 -transverse force (muscle strength, limb weight in the supine position of the victim); P 3 -the force acting on the foot when walking (muscle action during rotational movements of the body during walking, the weight of the foot in the supine position of the victim) and causes torsion of the bones of the limbs. Due to the action of these forces, bending moments and torsion moment arise: where e k -the distances from the line of action of the corresponding force to the place of fixation of the fracture; 1, 2, 3 k  -designation of forces and corresponding moments (Fig. 2). We take into account that most OS tools are located on the lateral (external to the center of the body) or medial (internal) side of the bone. In this case, the axis of the longitudinal force P1 passes through the frontal plane (the one that divides the body into the front and back) of the bone. In the future, we consider the case when the line of action of the most dangerous force P 2 also lies in the front plane. Therefore, the displacement arising from these forces and the corresponding moments must be added algebraically.
Note that the direction (sign) of external forces is not known in advance. Therefore, in what follows we will consider the most dangerous arrangement of the forces P 1 , P 2 , and P 3 when the directions of the projections of the movement of all these forces coincide. In this regard, when constructing the criteria of functional reliability, consider that all components of displacements have the same sign.
If the components of the displacement are known (the projections of the total vector Λ on the coordinate axis ( Fig. 1, b) at the most dangerous point, then condition (1) can be written as: We denote by λ the displacements as the ratio of the absolute displacements to the acting forces, and by  the displacements as the ratio of the absolute values of the displacements to the acting moments: where i jk  -displacement projections of the і-th points caused by forces Р k on the j-th axis.
The reduced displacements are determined experimentally by measuring the displacements under the influence of the corresponding forces and moments, for example, by digital recording using a camera [9,10,11].
If the displacements are known, the components of the absolute displacements can be determined for any values of the components of the force vector P k and the corresponding components of the moment vector M k . So, the projection of the total displacement Λ i on the axis X is determined as follows: The projections of the displacement Λ i on the Y and Z axes are determined similarly. In general terms, we have: The stiffness condition (3) using, as a criterion, the maximum linear mutual displacement in compressed form is written as follows:

Accounting for the angles of mutual rotation of the parts of the fracture
The angle of rotation of the line connecting two opposite fracture points M and L.  Choosing the maximum mutual angle of rotation of the fracture parts as the rigidity criterion and taking into account the relation (5), condition (2) is obtained in the following form:

Accounting for cyclic loads
The process of treating victims is accompanied by exposure to limbs cyclically acting loads (walking, exercise). In this case, the mutual displacements of the fracture points can significantly increase, and the conditions of stiffness (4) and (6) must take into account the deformations (displacements) that accumulate in the OS system during the application of cyclic loads: In (7) and (8) it was taken into account that the components of the point displacement consist of instantly elastic displacement This ratio can be expressed through the reduced displacements which are related to the action of forces   i jk  and Index 0 means instant-elastic component, index Naccumulated displacements: From a practical point of view, the greatest interest is the displacement of the points of fracture, which accumulate when exposed to a sufficiently significant number of load cycles. In this case, conditions (7), (8) show displacements that occur on the N-th cycle, it is expedient to replace them with limiting ones (with the maximum number of cycles): is the coefficient of influence of cyclic creep, which shows an increase in the total displacement occurring at a significant number of cycles compared to the displacement under a single load.

Analytical description of cyclic creep curves
Previous experiments have shown that the curve that connects the upper points of the cyclic creep diagram (dashed in Fig. 4) it is possible to approximate with a sufficient degree of accuracy the sum of exponential functions. The number of members of this sum is sufficient to limit by three so that the error of approximation does not exceed the instrumental error. This is, in fact, the use of a generalized Kelvin-Voigt model by formally replacing the period of the force's application by the number of load cycles.
Imagine fracture displacements under the action of long cyclic loads as follows: Similarly for the reduced displacement ( ) i jk N  :

Generalized forms of conditions of functional reliability
Taking into account (13) and (14), the stiffness conditions (7) and (8) at repeated cyclic loads will have the following form: ISSN 2521-1943. Mechanics and Advanced Technologies #3 (87), 2019

Determination of the allowable loads at fractures of tibia bone, fixed with plates
As an example, consider the application of the above criteria to estimate the allowable loads of the system "tibia with fracture -fixing plate" [8,12,13]. The following points of fracture were considered: point M (medial, point near the attachment of the plates) and point L (lateral, the point furthest from the plates). We assume that the external forces are applied in such a way that the directions of the projections of the displacement vectors of the point L, which arise due to individual forces and moments of forces, coincide (the most dangerous case).
According to formula (4), in order to accurately evaluate the functional reliability of the fracture fixation and to determine the allowable loads, it is necessary to know nine values of the reduced displacements i jk  and nine values of i jk  . Obviously, identifying all the components, even using the digital method of recording movements [9], is quite difficult. However, it turned out that the number of displacements that influence the results of calculations can be significantly smaller.
The above displacements were determined from the results of experiments for angled medial plates with angular stability (hereinafter 2M), medial blocked plates (3M), and X-shaped medial blocked plates for open correction osteotomy (4M).
Tibial bones with simulated fractures fixed by these plates have been tested for compression, bending and torsion by the method [9,14]. In table 1 shows the distances from the points of application of loads to the locking plates. Table 1 The location of the point of application of loads to the locking plate (see Fig. 2 The most dangerous area is the area of fracture in which, under the action of loads, maximum mutual displacements of adjacent points occur. It is obvious (and this is confirmed experimentally) that in most cases maximum displacements will occur at the point of fracture farthest from the fixation means (plate or rod apparatus of external fixation). Therefore, in our opinion, as the point і in the calculations of maximum displacements, we need to take the point itself from the clamp (point L in Fig. 2).
Analyzing the experimental material [12,14] we can conclude that the main displacement vector Λ is determined mainly by the components The first four components are the longitudinal (along the X-axis) and transverse (along the Z-axis) displacements by forces P 1 and P 2 . The fifth component -is the displacement in the plane perpendicular to the longitudinal axis of the bone due to the moment of rotation M 3 .
The results of calculations of the reduced displacements, which are included in formulas (4) and (6), are given in Table 2. The other components of the displacement of the breakpoints are close to zero and are within the measurement error. Table 2 The reduced displacements of the point farthest from the fixing point by the plate L and closest to the plate M Using the criterion of maximum linear displacement recorded in the form (17) and the value of the reduced displacements, are given in Table 2, the values of the allowable loads [P 1 ], [P 2 ] and [P 3 ] were calculated with their simultaneous action. The results of the calculations are shown in Table 3. It is permissible for the reciprocal movement of the breakpoints to be taken   1 mm   .   (Table 3) calculated through the allowable linear displacements under condition (17) were selected as valid ones.
For clarity, the data in table 3 is presented in the form of three-dimensional diagrams (limit regions) that limit the areas of allowable loads. Any point under each of the surfaces corresponds to three values of loads P 1 , P 2 , and P 3 , the joint action of which leads to the occurrence of displacements and angles of rotation of fracture parts exceeding allowable: [Λ] < 1mm and [G] < 3°.
If the point belongs to the surface, then this combination of P 1 , P 2 , and P 3 causes a displacement of [Λ] = 1mm. In the case where the point is beyond the limiting surface, such a combination of loads is dangerous, with the displacement of the fracture point Λ > 1 mm away from the plate. It should be noted that the reduced displacements (see Table 2) were determined experimentally on full-scale bone specimens with simulated diastase fractures not filled with bone regenerate. This fracture state corresponds only to the initial stages of fusion, when the regenerate is absent or the mechanical properties do not affect the deformation (displacement) of the fracture points under the influence of physiological loads.
Therefore, the values of allowable loads (see Table 3), calculated using these reduced displacements, also correspond only to the initial condition of the bone regenerate. To determine the allowable loads for later stages of fracture fusion, it is necessary to know the kinetics of changes in the deformation properties of bone tissue in the fracture region during its regeneration.

Conclusions
1. The criterion ratios obtained for calculating load levels that do not cause unacceptable displacements and mutual rotations of fracture parts. The coefficients required for the calculations (displacements are given) are determined by simple tests at separate loads by compressing, bending, and torsion of bone samples with simulated fractures and fixation means.
2. Using displacement data obtained for simple types of loads (compression, bending, torsion), displacements for three types of plates for fixing tibial fractures are calculated.