Measured stiffness (Ksm) 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 [Kpft] to actual stiffness of the specimen [Ksa], 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 K3DN results compared to those reported by Hanson et al. [1] is likely due to their Kpft being relatively low compared to the Ksa of the specimen. When using test machine ram displacement to calculate Ksm 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 Ksa the other with Kpft, resulting in a measured stiffness Ksm = Ksa/(1 + Ksa/Kpft). Ideally Kpft is infinite (rigid PFT), thus Ksm = Ksa. The general rule-of-thumb when using ram displacement is to have Kpft ≥ 10*Ksa so that Ksa ≥ Ksm ≥ (1/1.1) Ksa and thus error is ≤ 9% when reporting Ksm for Ksa. Assuming K3DN = Ksa and KH = Ksm produces Ksa/Ksm ratios ranging from 7.5 to 19.6, which would require that Hanson's Kpft = 0.15 Ksa to 0.05 Ksa. 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 Kpft 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.