Comments on the Effect of an Orthogonal Locking Plate and Primary Plate Working Length on Construct Stiffness and Plate Strain in an In vitro Fracture-Gap Model

Among other tests, the authors of this paper studied the effect of working length on plate strain between the two innermost plate holes (area of interest) of a single LCP implant.¹ In Table 3, the authors report no significant difference in the strain between SWL and MWL, but the strain in LWL was significantly higher than that of both SWL and MWL. Nevertheless, in the abstract, the authors state that “single plate construct strain was significantly, incrementally, higher as working length was extended.” This statement gives the wrong impression as it creates a pattern of a direct relationship between plate strain and working length, which does not reflect the findings.

In the last paragraph of the Discussion, the authors state that they used four-point bending “to produce a constant bending moment across the entire construct to allow comparison between constructs of varied stiffness.” This statement correctly suggests that the bending moment in the area of interest was the same at SWL, MWL, and LWL. [Fig. 1] of this letter illustrates a model and its graphical representation of the four-point bending used in the paper. Considering that the area of interest is composed of a homogenous material and isotropic, with a constant cross-sectional area and a longitudinal plane of symmetry within which the bending moment lies, the stress at various points of this area can be calculated using the flexure formula ([Fig. 1]). From the formula, it is apparent that the stress at any point in that area depends on parameters that were constant among SWL, MWL, and LWL. This suggests that the flexural stresses should also be constant at any point within the area. The flexural stresses would be the only normal stresses applied in the area of interest as the area was under pure bending. The same stress will produce the same strain within the elastic deformity boundaries where the experiment took place. Contrary to that, the authors found a significant difference between strains in SWL and MWL and LWL. It would be acceptable for the authors not to reject the null hypothesis for this test, as, despite the small test value, the result does not apply in real life since it is against the mathematical theory of pure bending.[2] The results could be attributed to inaccuracies in measurement or because the strain at 300 N was estimated by interpolation from a linear model they created using a small number of measurements, or other reasons.

Fig. 1 Left: Mechanical model of the four-point bending as described in the paper and the diagrams of the shear force (V) and bending moment (M) across the structure. Ft: Peak force, F: Force on each roller, N: Newton, Nm: Newton metre, m: metre, A,B,C,D: positions of rollers. The distance between the inner and outer rollers is d = 0.025 m. The area delineated as blue is the area of interest where strain was measured. The shear force from the right of point A to the left of point B is constant V =  240 N. The bending moment increases linearly across this area, reaching a maximum of M = V x d = 240 N x 0.025 m = 6 Nm. The maximum moment only depends on V and d and is independent of the inner roller's span, the plate's working length, the construct's shape, and its composition (simple or composite). Notice that the maximum moment (6 Nm) is the area under the curve of the shear diagram from A to B and that the rate of change of the bending moment from A to B is the shear force across A to B. That is because the moment is the integral of the shear force, and the shear force is the derivative of the moment. Between the right of point B and the left of point C, the shear force is zero. This area of the construct is under pure bending. In this area, the moment remains constant. This means that the bending moment at every cross-section of the area of interest is 6 Nm, irrespective of the working length. From the right of point C to the left of point D the shear force remains constant and in opposite direction than between A and B, and the bending moment decreases linearly until it reaches 0 at the left of point D. Right: A transverse section of the area of interest under pure bending (the only normal stresses are flexural, and there are no shear stresses). The arrows represent the stresses at points of the beam across the cross-section. The stresses are maximum at the surface and zero at the neutral plane. The points above the neutral plane are under compression, and the ones below are under tension. The transverse distance of a point from the neutral plane is y. The flexural stress at that point can be calculated by the flexure formula , where I is the area moment of inertia of the beam's cross-section about the neutral axis. The researchers measured the strain at the surface of the beam, so y will be the distance from the neutral axis to the surface. Notice that M, y and I are constant between SWL, MWL and LWL, suggesting that the stress in the area of interest should also be constant.

As the authors chose to accept the importance of this test in real life, it is important that they provide a model that can explain how the same beam area under the same bending moment experienced different strains when subjected to deformations within the proportional limit.

In the Discussion section, paragraph 7, the authors compare their results with previous studies that found opposite results, which, according to the current study's authors, were explained by strain distribution over a shorter working length. Nevertheless, the previous studies examined different loading conditions (axial compression), and different results would be predictable ([Fig. 2]). More specifically, in one of these studies,[3] the authors discuss in detail how different loading conditions can produce different relationships between stress and working length. Moreover, in their Discussion section,[3] they state that “the present authors also disagree with the statement that the stress was lower in the condition with long working length because the load distributed in the longer working length and vice versa…and mechanically under the same bending load, long plates deformed more than short ones, but the stress should be the same.” This statement is in accordance with the flexural formula. The authors of this study provided a mechanical model to explain their findings, which is lacking in the study.[1]

Fig. 2 Different mechanical models will produce different results. F: Load applied on the bone fragment, d: moment arm, M: Bending moment, Fs: Shear force, Fn: Normal force, C: Centre of rotation Left: A load is applied to the bone fragment. Middle: The load generates a bending moment bending the implant. The more elastic the implant is (e.g. long working length), the more it will rotate, and the more the lever arm will increase (d2 >  d1). The larger level arms will result in larger bending moments which will result in larger flexural stresses between the innermost screws. Right: Similar scenario, but the direction of the load keeps changing to remain parallel to the bone fragment. The more elastic the implant, the more it will bend, but this will not result in an increase in the lever arm and, subsequently, in the bending moment. Notice that the line of the force is always a tangent of the trajectory of rotation. Hence, its perpendicular distance from C (lever arm) is always the radius d1 of the circle. Nevertheless, due to the direction of the load, a shear load will be generated between the innermost screws, and the normal force will be smaller. Notice how small differences in the loading conditions can result in differences in the implant stress. Also, notice that increasing the working length will increase the von Mises stresses in the plate in the middle model but may decrease (small shear and smaller normal load) or leave the von Mises stresses unchanged in the right model.

Over the last decade, there has been extensive discussion within the veterinary community about the relationship between strain and working length. The debate has always been based not on mechanical modeling but on arbitrary interpretation of mechanical concepts and intuitive assumptions, which formed the basis of laboratory experimentation. This has led to the prevalence of the belief that long working lengths will result in less strain in any loading condition, a statement that made its way into the book.[4] If we continue to not base our assumptions on mechanical models and not interpret our findings using the rigorous mathematical reasoning such models require, we will keep producing contradictory results and pass the wrong statement in the next book.

For all the above reasons, I would kindly ask the authors to provide a more comprehensive explanation of their findings and their implications in clinical practice. This should start with a detailed explanation of why, in their opinion, the flexure formula does not apply to their experiment.

Publication History

Article published online:
27 December 2024

• References

• 1 de Bruyn BW, Glyde M, Day R, Hosgood G. Effect of an orthogonal locking plate and primary plate working length on construct stiffness and plate strain in an in vitro fracture-gap model. Vet Comp Orthop Traumatol 2024; 37 (04) 173-180
• 2 Petrie A, Watson P. An introduction to hypothesis testing, Distinction between statistical and biological significance. In: Petrie A, Watson P. eds. Statistics for Veterinary and Animal Science. Chichester: John Wiley & Sons, Ltd; 2013: 75-84
• 3 Chao CK, Chen YL, Wu JM, Lin CH, Chuang TY, Lin J. Contradictory working length effects in locked plating of the distal and middle femoral fractures—a biomechanical study. Clin Biomech (Bristol) 2020; 80 (07) 105198-105198
• 4 Moreno MR, Zambrano S, Dejardin LM, Saunders WB. Bone biomechanics and fracture biology. In: Johnston SA, Tobias KM. eds. Veterinary Surgery: Small Animal. 2nd ed.. St. Louis, MO: Elsevier; 2018: 646