Numerous studies have been conducted to determine the structural fracture tolerance of the human leg in bending (Table 18.2). The studies report a significant variation in fracture tolerance that is likely due to both the wide variety of test conditions applied in the tests as well as variation in the human tissue specimens used. Injury risk functions that account for the applicable covariates are ideal for FE model validation of fracture/failure tolerance since they permit the application of conservative estimates, with statistical bases, to model simulations.

Loading rate is used as a primary evaluation criterion for previous studies. Some of these studies report results from tests conducted at quasi-static loading rates, which may significantly underestimate the bending tolerance of the femur under impact loading [17, 18]. Carter and Hayes [17] estimated that both strength and modulus of cortical bone were proportional to , where is the strain rate. Additionally, Mather [8] and Martens [19] suggest that bone exhibits an increase in strength and stiffness at higher strain rates during dynamic loading, and Schreiber et al. [12] showed a dramatic difference in bending tolerance of the human leg between quasi-static tests and dynamic tests. While this suggests bone may experience a strain rate sensitivity of fracture tolerance, this effect may be smaller than inter-specimen variation when strain rates vary by less than an order of magnitude. For an order of magnitude estimate, unpublished data from vehicle-pedestrian impact experiments performed using strain gage instrumented PMHS at the University of Virginia suggest that the tibia can experience strain rates from 100 %/s to as high as 700 %/s depending on pedestrian stance and vehicle geometry. Thus, tests where specimens were loaded only quasi-statically, will not be considered in the following review.

Mather [8] tested a large number of tibiae in dynamic drop testing with different impact energies to determine which proportion of specimens failed at each level of impact energy. No moment or force calculations were made and thus the results of the study cannot be used in the development of injury risk functions.

In Rabl et al. [11], the authors’ goals were to gain basic information on the fracture behavior and fracture morphology of the tibia and to determine the relevance of the results for general practice. Thirty-two human tibiae (16 matched pairs) were loaded in bending, with 8 specimens each loaded on each of the 4 anatomical faces of the bone (posterior, anterior, lateral, and medial). The authors constrained the ends of the bone, which likely influenced the wide range of fracture tolerances presented (2,475–12,206 N). The authors do not report span length or moment data, nor do they report any anthropometrical measurements of the specimens (age, gender, stature, and weight were provided). The specimens were loaded at 0.3 m/s and data was sampled at 600 Hz from an accelerating load cell that was not inertially compensated. Additionally, only 20 cm long sections of tibiae were used, harvested from the specimen 6 cm distal to the tibial plateau regardless of specimen length. Thus, results from their study cannot be used to develop injury risk functions for bending loading of the tibia/fibula/leg. However, interestingly, despite mid-shaft loading of all specimens, the study reports distal third fractures in almost 60 % of the anterior loading cases, as well as in 20 % of cases with medial loading and 20 % of cases with lateral loading.

Almost all of the remaining studies (other than Nyquist et al. [10]) were performed by the University of Virginia [12–16]. Among these studies, only Schreiber et al. [12] presents data for leg bending in the posterior-anterior (P-A) direction, while the rest present data for bending in the lateral-medial (L-M) direction. Studies presenting data on the tolerance of the leg/tibia/fibula under L-M loading will be discussed first, followed by studies discussing the tolerance in P-A direction.

In Kerrigan et al. [13], four matched pairs (all male) of leg specimens were loaded in L-M three point bending (Fig. 18.4). The epiphyses of each specimen were potted in rigid potting cups and attached to rollers. The rollers rested on greased support plates to create a simple support boundary condition at each end. The central loading was provided by a padded aluminum ram attached to the actuator of a servohydraulic materials testing machine able to deliver downward displacement pulses at controllable rates as high as 1.6 m/s. Each of the four matched pair bending tests was used to investigate the influence of a particular parameter on the bending strength of the leg/tibia (Table 18.3). In the first two pair of specimens, the influence of location of the central loading point was investigated by comparing the fracture tolerance of the tibia loaded at the mid-shaft versus loading at the proximal third location (first pair), and mid-shaft versus loading at the distal third location (second pair). The distal third and proximal third loading locations were defined by measuring to a point located 33 % of the bone’s length measured from the distal tip of the medial malleolus and from the proximal tip of the tibial spine. The fracture tolerance in bending was slightly higher in proximal third loading than in mid-shaft loading in the first pair, and slightly higher in mid-shaft loading than in distal third loading in the second pair. The effect of rate sensitivity was investigated using the third pair of specimens, both loaded at the proximal third location; however, one was loaded at 0.001 m/s while the other was loaded at the dynamic rate. A larger difference in fracture tolerance was reported in the pair loaded at rate than in either of the loading location pairs. In the fourth pair, one denuded tibia was loaded to failure at the mid-shaft and the other specimen with tissue intact (fibula and surrounding flesh were intact, including proximal and distal tibio-fibular joints). The greatest difference in fracture tolerance was noted in this pair: the denuded tibia failed at a bending moment less than 75 % of the failure moment in the leg segment case. This result highlights the importance of not only validating whole bone bending response of the tibia and fibula, but that the entire leg specimen (tibia, fibula and surrounding flesh) bending tolerance should be as well.

A second study was performed by the University of Virginia [14], where 10 whole leg specimens (tibia, fibula, and surrounding flesh) were loaded in L-M three point bending (Fig. 18.5). This study was performed to highlight the rate sensitivity of the bending tolerance and response of whole leg specimens: six specimens were loaded in dynamic bending at 1.6 m/s, and four specimens were loaded in quasi-static bending. The tests were performed using identical hardware and equipment to those performed in the other University of Virginia study [13]. The bending responses were scaled and used to develop a quasi-static and a dynamic response corridor.

In the third University of Virginia study [15], previously published data were combined with data from new experiments in an effort to create an injury risk function for the dynamic L-M bending tolerance of the leg. The new tests presented in this study included four (no matched pairs) leg specimen bending tests performed identically to those presented in the other two Kerrigan et al. [13, 14] studies. The previously published data that were included in the development of the injury risk function were the single leg specimens from Kerrigan et al. [13], the six dynamically loaded leg specimens from Kerrigan et al. [14], and nine tests published by Nyquist et al. [10].

Nyquist et al. [10] performed a series of leg/tibia bending experiments that included nine dynamic (2.1–4.7 m/s) L-M impacts to simply supported leg specimens (with tibia, fibula and surrounding flesh). Nyquist et al. [10] reported that force data were filtered at 100 Hz to “smooth the traces and slightly attenuate the short duration spikes”. Upon examination of the filtered and unfiltered data, it was concluded that the filtered data attenuated the peak forces by approximately 10 %. Nyquist also reported that there was no evidence to suggest that tibial fracture occurs at the peak force, so only filtered values are presented in the paper.

Thus for the purpose of the injury risk function development in Kerrigan et al. [15], all of the data from Nyquist et al. [10] were considered to be right-censored (fracture happens at a force or moment greater than the published values) and data from the other previous University of Virginia studies were to be considered exact, or uncensored (for more information on data censoring or injury risk function development see Kent and Funk [20]). The (moment-based) fracture tolerance of each of the 20 specimens was geometrically scaled to represent the 50th percentile male fracture tolerance using a whole bone length scale factor and 379 mm as the 50th percentile male tibia length. After scaling, 19 of the 20 available data points were used to construct the injury risk function for the leg. One of the Nyquist et al. [10] data points was not used because after scaling the fracture tolerance (634 Nm) was 2.5 standard deviations from the mean of the Nyquist et al. [10] data, 3.2 standard deviations away from the mean of all of the data in Table 18.2, and lastly because it was a right-censored point.

A parametric injury risk curve was developed from a Weibull univariant survival model that was fit to the scaled fracture moment data (8 right censored points and 11 exact points) to determine the relationship between the probability of fracture and peak applied moment. A Weibull model was chosen because it is not necessarily symmetric, permits a zero risk value at zero moment, and has been shown to be no worse than other distributions for modeling biomechanical injury data [20]. The injury risk function for mid shaft L-M bending of the leg is given with 95 % confidence intervals in Fig. 18.6.

In the fourth University of Virginia study [16], the results of eight more dynamic L-M leg (tibia, fibula and surrounding flesh) bending experiments were presented. Four of the specimens were loaded at the distal third location and the remaining four specimens were loaded at the proximal third location. While the scaled (to 50th percentile male) bending tolerance data are presented in this study, no statistical analyses were performed due to the low number of specimens tested in each bending scenario.

In terms of bare bone (tibia or fibula) bending experiments the only available tolerance data for specimens loaded dynamically in the L-M direction were performed by the University of Virginia. However, none of the data have large enough sample sizes to determine injury risk functions with reasonable confidence intervals. Dynamic tolerance for seven bare tibiae loaded in the L-M direction were presented in Kerrigan et al. [13], and the University of Virginia has unpublished data for bending tests performed on the fibulae separated from the tibiae tested for the Kerrigan et al. [13] study. The fibula bending experiments were performed in the L-M direction where matched pairs were used in a manner similar to the tibia tests from Kerrigan et al. [13] to evaluate the influence of loading location and loading rate.

Nyquist et al. [10] conducted A-P bending tests on leg specimens in addition to the L-M tests. Nyquist concluded that the tibia failure moment does not depend on the loading direction when comparing A-P bending (avgs. 320 Nm male, 280 Nm female) to L-M bending (avgs. 330 Nm male and 264 Nm female). Despite the shortcomings of the data (see previous section), Nyquist confirmed the earlier finding by Yamada [9] that bending strength was independent of loading direction.

In one study, Schreiber et al. [12] conducted dynamic (5.55 m/s impact velocity) P-A three-point bending tests of the leg complex for 12 intact leg specimens (Table 18.4). For comparison, six additional tests were conducted in A-P direction using identical test conditions (see Ref. [21] and Table 18.4). A Student’s t-test of the P-A and A-P loading scenarios showed no statistically significant differences with a 95 % level of confidence. Crandall et al. [21] concluded that by combining the University of Virginia data with the results of Nyquist et al. [10], the data indicated that the tibia has approximately the same bending moment to failure for all loading directions.

The concept behind the tibia index comes from classical beam theory that involves a prismatic beam subjected to a combination of bending and compression. The form of the tibia index is closely related to the formula for the response of a beam-column:
where P is the axial compression force, M is the applied bending moment across the beam, C_m is an empirically derived coefficient, M_P is the plastic moment capacity (M_p = σ_yZ_P) where σ_y is the yield strength of that material and Z_P is the plastic section modulus. P_cr is the compression capacity of the beam, or Euler load if the slenderness ratio is sufficiently large.

Following the form of this equation, Mertz [22] developed the tibia index as an injury criterion to aid in the design of injury mitigation strategies for lower limb injuries. In this case, the beam-column relation was reworked as the tibia index relation:
where F is the axial force in the tibia, M is the bending moment in the tibia, and F_cr and M_cr are the critical axial force and bending moment. If the tibia index exceeds the value of 1.0, a tibia injury is predicted.

Using tibia properties derived by Yamada [9] and scaling them to different anthropometries, Mertz determined the critical compression force and moments for various occupants (Table 18.5).

Since the introduction of the tibia index to the biomechanics community, one study [12] has evaluated the validity of the criterion for combined axial and bending loads. This study combined a dynamic bending load and a static compression load and produced a decrease in bending moment by 19 % with an averaged scaled failure moment for a 505 male of 372 Nm. This appears to confirm that the interaction of the bending and compression forces combine to affect the fracture tolerance of the tibia.

Crandall et al. [21] reviewed all subsequent failure tests involving the tibia and offered revised tolerance values for the tibia index. Specifically, this review found that axial and bending tolerance for quasi-static loads should be approximately 10.3 kN and 240 Nm, respectively. Assuming that the fibula can handle 10 % of the static axial load of the tibia, the critical static force for the tibia index increases to 11.3 kN. Review of dynamic bending tests suggest a dynamic bending tolerance of 450 Nm. Using the static-to-dynamic scaling presented by Schreiber et al., the dynamic threshold of 16.6 kN is obtained without the fibula, and 18.3 kN with the fibula.

For simplification purposes, the ankle and subtalar joints have been linked in this chapter for the major motions of the hindfoot. Between the talus and calcaneus lies the subtalar joint, which has the primary function of enabling xversion rotation of the hindfoot. A summary of the response and injury data in both quasi-static and dynamic testing is provided.

The more controlled environment of quasi-static tests permitted evaluation of not only the moment-angle characteristics, but also the joint centers of rotation, which were shown to vary slightly with ankle position [24, 25]. Without inertial effects, the results were found to be less variable overall. Petit et al. [25] tested 25 limbs quasi-statically in dorsiflexion with simulated muscle tension applied through the Achilles tendon. The legs were amputated mid-shaft, and the proximal ends of the tibia and fibula were potted together in a cup with cement and Steinmann pins. The calcaneus was potted similarly. In order to apply muscle tension, the Achilles tendon was cut and secured to a cable, which was routed through the test fixture to a counterweight. Dorsiflexion tests were performed with 900 N of tension applied to the Achilles. The prepared specimens were placed in a test fixture which applied rotation to the joint with an electric motor while allowing the specimen to translate in two directions. Of seven legs tested in dorsiflexion, the center of rotation was found to be 76 ± 8 mm proximal from the sole of the foot and 61 ± 6 mm anterior from the posterior face of the heel. The dorsiflexion tests had an average 47 ± 17 N-m and 49° ± 5° failure point. Typical injuries included medial malleolus fractures and calcaneofibular ligament tears.

The quasi-static tests by Parenteau et al. [24] were performed in a fixture similar to the one used by Petit et al. [25], but without the applied muscle tension. Specimen preparation involved cutting the leg mid-shaft and potting the tibia and fibula together in a cup with cement and Steinmann pins. The calcaneus was held with pins and cement in a cup, as well. Torque was applied to the calcaneus cup, which induced rotation in the ankle or subtalar joint, depending on the orientation of the specimen. A motor applied torque at a constant rate of rotation until failure was detected by a drop in moment. Nine tests were performed in dorsiflexion, five of which sustained an injury. Injuries included two lateral malleolus fractures, two calcaneofibular ligament tears and one interosseous talocalcaneal ligament tear. Failure, defined by a drop in moment with increased rotation, occurred at an average moment of 33.1 ± 16.5 N-m and dorsiflexion of 44.0° ± 10.9°, while the average injury moment was 36.2 ± 14.8 N-m with dorsiflexion of 47.0° ± 5.3°. The center of rotation in dorsiflexion was found to be located at the ankle joint, near the center of the talar dome.

Schreiber et al. [34] conducted quasi-static tests in a six-degree of freedom test fixture that was capable of both constrained (rotation about on the dorsiflexion axis) and unconstrained motion. Six cadaveric lower limbs were amputated above the knee and subjected to complete cycles of dorsiflexion and plantarflexion at non-injurious loading levels. For each lower limb, four boundary conditions were used to determine the influence of various below-knee anatomical structures on the ankle response. The ankle test apparatus consisted of a gimbal assembly, with two links centered at the ankle’s theoretical center of rotation. A sprocket, driven by a hydraulic piston at a constant rate, produced a moment in the outermost gimbal assembly aligned with the flexion axis of rotation. As the ankle was forced into flexion, the other axes were unconstrained. The tibia was mounted rigidly to a fixed frame while the foot was mounted to a foot plate fixed to the inner gimbal link. Translation was accommodated by roller bearings in the inner gimbal link. In all, six degrees of freedom (three rotational and three translational), were permitted in the ankle’s motion. The results suggest that dorsiflexion of the ankle is highly dependent on the triceps surae muscle group, the superior tibio-fibular joint, and the relative spacing of the ankle mortise. The insensitivity of plantarflexion motion to varying anatomical conditions suggests that the boney stops, ligaments, and other local anatomical structures determine the extension response.

The overall utility of the tests by Schreiber et al. [34] for FE model validation may lie in the sequential identification of anatomical components to the overall response of the lower limb. Using a range of motion (ROM) defined as the angle achieved when 6 Nm of torque was applied in any direction, Schreiber et al. produced a stiffness mapping with different boundary conditions (Table 18.7).