Biomechanical studies that appear in the literature typically use fixtures that constrain some components of relative 3D motion between ends of the test specimen, other than in the direction of loading, and/or do not apply a uniform component of load over the instrumented length of the specimen [1–8], and use test machine ram displacement to determine stiffness without correcting for potting-fixture-test machine (PFT) stiffness error [1, 2, 4, 6, 7, 9]. Direct measurement of relative 3D interfragmentary motions during in vitro testing has been performed [5, 7, 10–12], including those that have eliminated PFT stiffness [11, 12]. In vivo 3D interfragmentary motions have been measured using a transducer telemetry system [13], a motion capture system [6, 7, 9], and a 3D optical tracking system [14]. To the authors' knowledge, combined use of 3D unconstrained fixtures with 3D optical tracking of test segment ends to eliminate PFT stiffness error has not appeared in human or veterinary literature.

The purpose of the investigation described in this paper was to design and verify the capabilities of an in vitro loading-measurement system that: 1) mimics in vivo unconstrained relative 3D motion between fracture-model (herein after referred to as "fracture") segments of long bone ends, 2) applies uniform compression, torsion, or bending moment loads over the entire length of the test specimen, 3) measures 3D relative motion between test segment ends to directly determine 3D stiffness components of intact and instrumented test segments, and 4) identifies the weakest aspects of an instrumented specimen during increased uniform loading over its entire length. The ultimate goal was to use this loading-measurement system to measure and compare the relative 3D biomechanical characteristics of intact and two Association for the Study of Internal Fixation (ASIF) techniques for repair of an oblique fracture involving the distal (inferior) diaphysis of an equine radius.

Major periosteal and endosteal cross section diameters measured 49.8 (± 4.4) mm and 30.5 (± 4.8) mm, respectively, whereas the minor periosteal and endosteal cross section diameters measured 32.1 (± 1.4) mm and 18.4 (± 1.7) mm, respectively. The mean intact measured K_3D values were within the average elliptical to rectangular cross section theoretical range for axial compression, torsion and ML bend; and were greater than the highest theoretically predicted by a factor of 1.1 for LM bend, 1.3 for CaCr bend, and 1.5 for CrCa bend (Table 1). The length normalized axial and torsion K_3DN results were a factor of 19.6 and 7.5 higher, respectively, than the corresponding K_H results (Table 2). Likewise, the 101 mm CrCa and LM bending K_3DN results were a factor of 8.3 and 10.6 higher than the corresponding K_H, respectively; whereas, the 11.5 mm CrCa and LM bending K_3DN results were a factor of 72 and 92 higher (Table 2). The fixtures allowed components of 3D motion other than in the direction of loading to occur. For axial and torsional loading, unconstrained relative 3D motion across the fracture site was most noticeable for the ostectomy due to definite lack of load sharing with bone at the fracture site and the unsymmetrical double plate construct. The least amount of 3D motion between ends of the test segment occurred for the intact radius. This is illustrated in the torsion loading example which shows detectable CrCa and LM bending rotation for the ostectomy, and negligible CrCa and LM bending rotation for the intact radius and the reduced osteotomy (Figure 6). The loading fixtures during axial and torsion testing were also observed to have CrCa and LM bending rotation, which was most prevalent for the ostectomy model. Relative 3D motion between ends of the test segment was detected by the optical tracking system for bending loads which in some cases was visible by the relative motion across the fracture; however, 3D motion of the loading fixtures was not readily visible.

Construct failure configurations were consistent with theoretical failure modes for brittle material (bone) and reproducible for both the torsion and bending load to failure tests. Torsion failure was consistently attributable to spiral fracture propagation that initiated perpendicular to the osteotomy or ostectomy at cranial screw holes. LM and CrCa bend failure initiated by cis-cortex fracturing through either the proximal or distal screw holes adjacent to the osteotomy or ostectomy, and progressed perpendicular to the longitudinal axis of the radius; with the exception of one ostectomy undergoing LM bend to failure experiencing substantial bone-plate interface separation due to lateral screw pull out of bone without any bone fracture. LM and CrCa bends produced visible osteotomy and ostectomy gap opening on the non-plated side due to it being in tension, and visible proximal and distal screw bending adjacent to the osteotomy or ostectomy. ML and CaCr bending produced visible ostectomy gap closure on the non-plated side due to it being in compression.

No failures occurred within the potted ends or of the fixtures themselves during torsional and bending loads to failure. No failures occurred through rigid body screw holes.

Measured stiffness (K_sm) of a long bone specimen is a function of parameters associated with morphologic/anatomical characteristics (material properties, cross section size and shape, length) which in turn are influenced by animal (age, sex, size, breed, diet, and training history), and parameters associated with testing protocol (specimen history/test conditions that effects its material properties [wet/dry, temperature, time from death to removal to storage, time in storage], test loading [strain] rate, relative stiffness of the potting-fixtures-testing machine [K_pft] to actual stiffness of the specimen [K_sa], fixture constraints on the relative 3D motion between specimen ends, and load range of data used for stiffness calculation). Hanson et al. [1] considered all but the later three when discussing why stiffness values from all modes of testing (compression, torsion, bending) intact equine tibia by McDuffee et al. [15] were 2.5 to 6 times greater, and contributed this to McDuffee's greater loading rates. McDuffee et al. [2, 15], Hanson et al. [1], and our compression testing was conducted in position control at constant ram velocity, with corresponding constant strain rate (ram velocity/exposed length of tibia or radius) = (0.58, 0.54), 0.00011, and 0.000018 (1/sec). McDuffee's compression strain rates were 527 and 491 times higher than Hanson's while ours' was 0.16 times lower. Similarly for torsion, (ram angular velocity/exposed length of tibia or radius) = (0.058, 0.054), 0.0017, and 0.00114 (deg/[mm*sec]); with McDuffee's being 34 and 32 times higher than Hanson's while ours' was 0.67 times lower. A quantitative comparison of strain rate for bending is not realistic due to load [2, 15] versus position control [1] used, and that applied bending moment (strain) rate is not uniform over the length of a specimen for 3-point bending, ranging from 0 at the outer supports to a maximum at the central load application point.

Panjabi et al. [16] reported that cortical bone longitudinal modulus of elasticity increases with increase in strain rate by a factor of 1.5 for strain rate of 1500 compared to 1.0 (1/sec), thus explaining in part the greater stiffness observed by McDuffee et al. [2, 15] compared to Hanson et al. [1]. If strain rate were the predominant factor, our length normalized measured stiffness values should be lower than or similar to those reported by Hanson et al. [1], since our strain rates were lower.

The order of magnitude difference in our K_3DN results compared to those reported by Hanson et al. [1] is likely due to their K_pft being relatively low compared to the K_sa of the specimen. When using test machine ram displacement to calculate K_sm of a specimen, ram displacement is the sum of that due to specimen deflection and PFT machine deflection, which includes bone crushing when applying bending loads directly to bone [1]. The theoretical model is two springs in series [17], one with stiffness K_sa the other with K_pft, resulting in a measured stiffness K_sm = K_sa/(1 + K_sa/K_pft). Ideally K_pft is infinite (rigid PFT), thus K_sm = K_sa. The general rule-of-thumb when using ram displacement is to have K_pft ≥ 10*K_sa so that K_sa ≥ K_sm ≥ (1/1.1) K_sa and thus error is ≤ 9% when reporting K_sm for K_sa. Assuming K_3DN = K_sa and K_H = K_sm produces K_sa/K_sm ratios ranging from 7.5 to 19.6, which would require that Hanson's K_pft = 0.15 K_sa to 0.05 K_sa. This is low but possible; however, Cowin stated that with many testing machines, the stiffness of the bone specimen is greater than that of the load frame [17]. Difference in K_pft may also have contributed to the higher stiffness reported by McDuffee et al. [2, 15] compared to Hanson et al. [1]. Use of direct measurement of test segment end motion to determine its stiffness is not subject to PFT stiffness error, which is one reason we selected to use 3D optical tracking and is reason for using an extensometer when testing for material modulus of elasticity [17]. The holes drilled in the radius to mount the LED rigid bodies were at the end cross sections of the test segment, thus having no effect on its stiffness.

Theoretically, in the absence of friction, the loading fixtures presented in this investigation allow 3D unconstrained components, of proximal relative to distal end motion of the radius, other than associated with the applied load component. Thus during loading, the ends of the specimen move into a relative 3D position that is dictated only by the 3D resistance of the (unsymmetrical or not) instrumented segment to the known (only) component of externally applied load such that mechanical equilibrium is achieved. The bending fixtures are the most likely to be subject to constraint by friction since the hardened cross bars were observed to indent the unhardened extension pipe surface during bending load to failure; the effect of this was reduced by periodically smoothing the pipe contact surfaces with a file.

Improvements could be made to reduce joint friction by using rollers incorporated into the 4-point bending load applicators [18]. Use of a commercially available universal joint has been reported [4] and was considered but not used since its pivot center would be several centimeters above or below the proximal or distal end of the bone being tested, respectively. Since the applied axial load remains coincident with the axis through the center of the universal joints, this would create, as load was increased, an ever increasing non-physiological lateral displacement of the plate from the axis of loading and thus bending moment on the specimen. Our goal was to keep the center of the spheres as close as possible to the ends of the radius being tested, so that the line of action of the axial applied load would remain close to the central point on each end of the radius. The authors' feel that the loading fixtures used and described in this manuscript, particularly axial and torsion, are a significant improvement towards achieving clinically relevant 3D unconstrained "worst case" in vivo loading simulation, compared to the fixtures reported in the literature [1, 2, 4].

Another advantage of this loading fixture design was that the same potting of the ends could be used to test a specimen in all three modes of loading: axial compression, torsion and bending. In addition the bending fixtures were capable of applying uniform 4-point bending moment in any transverse plane over the entire length of the instrumented specimen with no point of load application applied directly to bone or instrumentation hardware, and is independent of specimen and fixture cross section size and material since the center pivot assures equal forces at the inner application points. Thus the weakest aspect of an instrumented segment can be located, in contrast to it being predisposed to be at the location of the highest applied bending moment [1, 2, 15]. Also it avoids the problem of applying loads to non-circular bone cross sections and failure due to bone crushing at the point of transverse force application [1].

Stiffness is typically determined as the slope of the "linear portion" of each load versus displacement curve [1, 2, 15] leaving the range of data used being subjectively determined and variable, which can have a notable effect on the numerical value obtained. To reduce the subjectivity, we used data over a consistent load range to determine stiffness for each loading modality. The load range was selected so that all tests would have a linear load versus displacement characteristic within the range. Our load versus test machine ram displacement curves typically had lower slope during the first loading cycle compared to 2nd and 3rd loading cycles (the later two being similar), due to settling of bone-potting-fixture interconnections during the first cycle [17]. McDuffee et al. [2, 15] and Hanson et al. [1] obtained stiffness from single cycle to failure tests, which could be a factor in the variability of and lower stiffness values observed.

The theoretical stiffness equations are based on the assumption that cross sections remain plane as load is applied, and thus corresponding experimental stiffness determination depends upon accurate 3D measurement of the position of these planes. For axial and torsional testing the initial position of the LEDs was pointing directly towards the 3 camera optical measuring head (Figures 2, 3) but were oriented at approximately 45 degrees for bending (Figures 4, 5). Accuracy in detecting the location of individual LEDs is known to deteriorate as their angle deviates from pointing directly at the measuring head [19]. This may explain why the average measured axial and torsional stiffness values were within the predicted range, while the average measured bending stiffness values were 0.97 to 1.5 times greater than the largest theoretically predicted stiffness for the four bending modes. This also might be an explanation for the occasional saw tooth type characteristic in some bending moment versus relative angular displacement graphs to which we used regression to fit a straight line to get the general trend and thus stiffness. Another possibility is that the single point of attachment to the radius of each rigid body did not produce an accurate representation of the 3D motion of the cross section plane. A recommended option would be to attach two rigid bodies to a C-ring with three pointed machine screws to make cortical bone contact at three points on the cross section plane. This would also eliminate the LED rigid body mounting hole's stress riser and potential for fracture initiation at this site, which did not occur in our tests to failure.

Long bone is known to be anisotropic and heterogeneous (differing in diaphyseal and epiphyseal properties), and is primarily elastic at low deformation rates [16, 17]. The principal stress is longitudinal for compression and bending load modes. Published low strain rate longitudinal modulus of elasticity E determined by machine testing ranged from 17,000 to 22,600 (N/mm²) for human and bovine femur and tibia [17]. Torsional stiffness is a function of the longitudinal-circumferential shear modulus G, with published machine testing determined values ranging from 3300 to 5000 (N/mm²) for human and bovine femur and tibia [17]. Thus use of E = 18,000 and G = 4,615 for the short 50 mm long intact diaphyseal test segment produced representative theoretical stiffness values for use as a magnitude accuracy comparison reference.

In conclusion, assuming negligible friction, the 3D loading-measurement system described in this manuscript: a) mimics unconstrained relative 3D motion between radius ends that occurs in clinical situations, b) applies uniform compression, torsion, and 4-point bending loads over the entire length of the test specimen, c) measures interfragmentary 3D relative motion between test segment ends to directly determine stiffness thus being void of PFT machine stiffness error, and d) has the resolution to detect differences in the 3D motion and stiffness of intact as well osteotomized-instrumented and ostectomized-instrumented equine radii. It is the authors' opinion that the 3D loading-measuring system described in this manuscript is capable of creating "worst case" clinically relevant loading conditions, and accordingly has the capacity to detect location and type of instrumented segment modes of failure and has the ability to produce more accurate stiffness results that are independent of PFT machine stiffness error.

Axial compression stiffness: K_C = AE/L(N/mm)   (1)

Torsional stiffness:   K_T = (1.745 * 10^-5)*GK/L   (Nm/deg)   (2)

LM & ML bending stiffness:   K_LM = (1.745 * 10^-5)*EI_yy/L   (Nm/deg)   (3)

CrCa & CaCr bending stiffness:   K_CC = (1.745 * 10^-5)*EI_xx/L   (Nm/deg)   (4)

where:

D_p, D_e = Major diameter periosteal, endosteal respectively   (mm)

d_p, d_e = Minor diameter periosteal, endosteal respectively   (mm)

A = Cross section area   (mm²)

= (π/4) * (D_pd_p - D_ed_e)   for hollow ellipse    (5a)

= (D_pd_p - D_ed_e)   for hollow rectangle    (5b)

I_yy = Area moment about y axis   (mm⁴)

= (π/64) * (D_p ³d_p - D_e ³d_e)   for hollow ellipse    (6a)

= (1/12) * (D_p ³d_p - D_e ³d_e)   for hollow rectangle    (6b)

I_xx = Area moment about x axis   (mm⁴)

= (π/64) * (D_pd_p ³ - D_ed_e ³)   for hollow ellipse    (7a)

= (1/12) * (D_pd_p ³ - D_ed_e ³)   for hollow rectangle    (7b)

K = Polar moment type term   (mm⁴)

= (π/16)* [(D_p ³d_p)(1-q⁴)]/[D_p ² + d_p ²]   for hollow ellipse    (8a)

= (1/12)* [(D_pd_p*(D_p ²+d_p ²)) - (D_ed_e*(D_e ²+d_e ²))]   for hollow rectangle    (8b)

q = ratio of major endosteal/periosteal diameters = (D_e/D_p)

L = Length of the test segment = 50 mm

E = Cortical bone modulus of elasticity = 18,000   (N/mm²)

Range in literature (8,900 ≤ E ≤ 42,000)

G = Cortical bone shear modulus = 4,615   (N/mm²)

Range in literature (3,500 ≤ G ≤ 8,710)