Variations in humeral and femoral strains across body sizes and limb posture in American alligators

Knowledge of limb bone loading can contribute significantly to the understanding of limb function during locomotion. By comparing limb bone loads throughout ontogeny and across taxa, we can gain insights into how animal locomotion is influenced by factors ranging from limb shape to limb posture and body size. Size-dependent variation in limb loading has been a primary focus of past studies. Despite expectations from geometric and dynamic similarity, which suggest that skeletal stress would scale with mass^1/3, peak limb bone stresses during terrestrial locomotion remain consistent in mammals across more than a 10,000-fold increase in body mass (Biewener, 1990; Biewener and Taylor, 1986; Biewener et al., 1988; Rubin and Lanyon, 1982). This pattern was primarily attributed to the use of a more upright limb posture (resulting in reduced mass-specific limb muscle forces), along with enhanced bone robustness in larger mammals (Bertram and Biewener, 1990; Biewener, 1989, 1990, 2005). As a result of the fairly uniform bending strengths of their limb bones, eutherian mammals maintain safety factors for compression above 1.4 based on yield and within 2–4 based on failure strain/stress during strenuous activities, such as running and jumping (Biewener, 1990, 1993; Blob et al., 2014; Copploe et al., 2015; Currey, 1984, 1987, 1990). Similarly, small to medium-sized ground-dwelling birds exhibit limb bone safety factors for compression within 2.9–3.7 based on expected yield strains during running (Biewener, 1993; Biewener et al., 1986; Blob et al., 2014; Rubin and Lanyon, 1984).

Limb bone strains, stresses and safety factors during terrestrial locomotion have also been examined in tetrapods with non-parasagittally moving limbs, such as salamanders, frogs, lizards, turtles and alligators (Blob and Biewener, 1999; Blob et al., 2014; Butcher et al., 2008; Kawano et al., 2016; Sheffield et al., 2011; Young et al., 2017). These tetrapods generally exhibit higher limb bone safety factors for bending and torsion compared with mammals and birds, owing to smaller bone loads experienced during locomotion and higher yield as well as failure strains/stresses in limb bones (Blob and Biewener, 1999; Blob et al., 2014). Their elevated bone safety factors have been ascribed to various factors, including low activity levels negating the costs of carrying bone materials (Alexander, 1981; Diamond and Hammond, 1992), low activity metabolism and small capacities for bone remodeling (Burr et al., 1985; Seymour et al., 2011), and high variability of loads (Blob and Biewener, 1999; Lowell, 1985). Despite these insights, the body size range of the amphibians and reptiles explored represents only a small fraction of that studied in mammals, and it remains uncertain whether their bone loads and safety factors are maintained across a broader size range. This limitation hinders our ability to understand the relationship between body size and limb bone loading during the limb posture transition (Blob, 2001; Blob and Biewener, 1999, 2001).

In addition to considerations of loading magnitudes and safety factors, previous studies have also revealed variation in loading regimes across taxa and limb postures. Axial compression and bending have been noted as primary loading regimes in most mammals with parasagittally moving limbs (Biewener and Taylor, 1986; Biewener et al., 1983, 1988; Goodship et al., 1979; Lanyon et al., 1975; Main and Biewener, 2004; Rubin and Lanyon, 1982), whereas shear predominates in the femora of rats and Virginia opossums with a crouched limb posture (Butcher et al., 2011; Keller and Spengler, 1989). In contrast, hindlimb bones of frogs, lizards, turtles, alligators and birds experience strong shear loads (Biewener and Bertram, 1993; Biewener et al., 1986; Blob and Biewener, 1999; Blob et al., 2014; Butcher et al., 2008; Carrano, 1998; Main and Biewener, 2007). Theoretically, greater shear strains would develop with more horizontally oriented limb bones (Butcher et al., 2011; Carrano, 1998) along with strong long-axis rotator muscles (Blob and Biewener, 1999, 2001), which increase torsional moments from the ground reaction force and muscle forces, respectively. Therefore, with the use of a more upright limb posture, primary loading regimes are expected to transition from shear to axial compression and bending across the crouched-to-upright posture spectrum in mammals with parasagittally moving limbs. However, it remains uncertain whether a comparable transition in bone loading regimes persists across the sprawling-to-erect posture spectrum among tetrapods with non-parasagittally moving limbs. Previous research on American alligators revealed that, within an individual, the use of more erect limb posture unexpectedly increased principal and shear strains of the femur, possibly because of greater knee extensor muscle forces acting in the hindlimb during locomotion that used an erect stance (Blob and Biewener, 1999, 2001; Reilly and Blob, 2003). This finding underscores the need for further investigation into the relative contribution of axial and off-axial loads across limb postures.

Historically, transitions from sprawling to erect limb posture occurred in multiple independent lineages, including archosaurs (Bonaparte, 1984; Carrano, 2000; Charig, 1972; Hutchinson, 2001a,b; Parrish, 1986) and synapsids (Blob, 2001; Bishop and Pierce, 2024a,b; Fröbisch, 2006; Kemp, 1978, 1982). Although the musculoskeletal changes associated with these postural shifts appear to be complex and gradual, trackway evidence indicates that these shifts occurred abruptly across tetrapods during the Early Triassic, implying an arms race for enhanced locomotor stamina (Benton, 2021; Benton and Wu, 2022; Kubo and Benton, 2009). As limb bone loading is crucial in limiting the locomotor capacities of tetrapods, comparing locomotor bone strains and stresses in living animals across different limb postures can offer insights into the biomechanical requirements and consequences of evolutionary changes in limb posture that helped shape vertebrate locomotor diversity.

In this study, we evaluated in vivo strains of the humerus and femur in juvenile to subadult American alligators, as they walked with various limb postures. Our objective was to test for variation in strain magnitudes and loading regimes across different sizes and limb postures. We predicted that: (1) limb bone strain is either maintained across sizes by adopting a more erect limb posture in larger sizes or increases if limb posture remains constant across sizes, and (2) axial loads become more dominant over shear loads when a more erect limb posture is adopted. By testing these predictions on size- and posture-dependent changes in limb bone strains, we aimed to clarify the relationships between body size, limb posture and limb bone loading in tetrapods, providing a new context for understanding evolutionary limb posture transitions.

Four juvenile and subadult American alligators, Alligator mississippiensis (Daudin, 1802), identified as al10, al07, al09 and al05, with total lengths of 0.53, 0.82, 0.9 and 1.28 m, and mass of 0.37, 1.53, 1.82 and 4.98 kg, respectively, were used for locomotor experiments. The sex and age of the animals are unknown. These animals were captured by staff biologists of the Louisiana Department of Wildlife and Fisheries alligator program and housed at the Rockefeller Wildlife Refuge in Baton Rouge, LA, USA, until transport to Clemson University in Clemson, SC, USA. Individuals were housed in separate enclosures with water access within a vivarium facility featuring a glass roof and walls, for 18–22 months after transport. Daily temperature inside the vivarium was maintained within the range 22–38°C. The smaller individuals were fed pellets (Mazuri crocodilian diet, small) and the largest individual received live feeder fish or pieces of boneless pork twice a week. Experimental procedures were approved by Clemson University IACUC (protocol 2019-037).

The procedures for surgery and preparation of strain gauges followed those outlined by previous studies (Biewener, 1992; Blob and Biewener, 1999; Blob et al., 2014). Before surgery, individuals were sedated via intramuscular injection of butorphanol (0.4–1.0 mg kg^–1), ketamine (30–100 mg kg^–1) and xylazine (0.5–1.0 mg kg^–1), and subsequently maintained under isoflurane (0.5–1.0% at an oxygen supply of 1 liter min^–1) using an intubated inhalation machine throughout surgery. Following intubation, an incision was made over the medial surface of the right upper arm and thigh, and muscles were separated to expose the humeral and femoral midshaft. A window of the periosteum was removed using a periosteal elevator and the bone surface was defatted and cleaned with ether.

Strain gauges were attached to the midshaft surface of the right humerus and femur of each individual, using self-catalyzing cyanoacrylate adhesive (Duro Super Glue, Henkel Corp., Avon, OH, USA). In each individual, a three-element rosette – with elements placed at 0, 45 and 90 deg (FRA-1-11 or EFRA-05-11-002LE; Tokyo Measuring Instruments Laboratory Co., Ltd) – was attached to the medial cortex of the humerus and ventromedial cortex of the femur, while two single-element gauges (FLK-1-11 or EFLK-02-11) were attached to the ventral cortex of the humerus and dorsomedial cortex of the femur. Efforts were made to align the central element of the three-element rosette and single-element gauges with the long axis of the bone. The lead wire (336-FTE, etched Teflon insulation; Micro-Measurements, Vishay Precision Group, Inc.) from the gauges, which extended towards the proximal end of the bone, was passed subcutaneously through a small incision at the posterior shoulder and anterior hip, allowing for some slack to accommodate limb movement. The exposed lead wires were soldered into micro-connectors, and the soldered junctions were coated with epoxy resin. All incisions were sutured, and wires and micro-connectors were secured around the shoulder and hip with self-adhesive Vet Wrap bandages.

Data collection sessions for each individual were conducted over two consecutive days, following a 24 h recovery period. The animals exhibited various degrees of postsurgical lameness, rendering the limb kinematics observed in these individuals not directly comparable to those that have not undergone surgery (Iijima et al., 2021, 2023). Among the four individuals used in this study, one (al09: 1.82 kg mass) typically used a high walk with its torso lifted off the ground, while the other three (al10, al07 and al05: 0.37, 1.53 and 4.98 kg mass, respectively) typically used more abducted limb postures with their torsos partially in contact with the ground. During trials, animals were encouraged to walk on a treadmill (model DC5; JOG A DOG, LLC, Ottawa Lake, MI, USA) fitted with custom Plexiglas walls. Animal movement was facilitated by gently tapping the tail as needed. Treadmill speed was adjusted to match the natural walking pace of each animal, and strain and video recording commenced while animals maintained this steady speed for 16–20 s. Two digital high-speed cameras (Phantom v.5.1; Vision Research Inc., Wayne, NJ, USA) captured the locomotion from lateral and dorsolateral angles at 100 Hz. 3D camera calibration was performed using the direct linear transformation method with a static object of known dimensions in DLTdv8 and DLTcal8 apps (Hedrick, 2008) within MATLAB R2022B (MathWorks, Inc., Natick, MA, USA). Strain and video recordings were synchronized by a 1.5 V square-wave pulse sent to the strain signals and a simultaneous light pulse captured in the video. Absolute walking speed for each trial was determined from the videos using white paint marks on the treadmill belt spaced at 10 cm intervals; these were then converted to dimensionless speed=absolute speed/(gh)^1/2, where g is gravitational acceleration (9.81 m s^–2) and h is the length from the hip to the ankle with the knee joint fully extended. h was measured using a tape measure after the experiment.

To analyze limb kinematics, the shoulder, elbow, wrist and metacarpophalangeal joints in the forelimb and the hip, knee, ankle, and metatarsophalangeal joints in the hindlimb were marked with non-toxic white paint during trials. Joint angles were calculated based on the joint coordinate system described in Iijima et al. (2021, 2023). In this study, we exclusively calculated the shoulder and hip abduction–adduction angles. These angles are formed between the horizontal plane and either the shoulder–elbow line or hip–knee line, respectively (Iijima et al., 2021, 2023). Positive and negative angles indicate the elevation and depression of the upper arm and thigh relative to the horizontal plane, respectively. The shoulder and hip abduction–adduction angles were computed at midstance during each stride, coinciding with the vertical alignment of the shoulder and metacarpophalangeal joints in the forelimb, as well as the hip and ankle joints in the hindlimb, as observed from the lateral angle in the video footage. We adopted the midstance definition from the inverted pendulum model, corresponding to the moment where the shoulder or hip height is at its peak and potential energy of an animal's center of mass is maximized. This typically occurs when shoulder–metacarpophalangeal joints and hip–ankle joints are vertically aligned, as the center of pressure of the hand and foot would be closer to the metacarpophalangeal and ankle joints, respectively.

To collect strain recordings, micro-connectors on the shoulder and hip were plugged into shielded cables connected to Vishay conditioning bridge amplifiers (model 2120B; Micro-Measurements, Vishay Precision Group, Inc.). Raw voltage signals were converted from analog to digital format using a multifunction data acquisition board (model PCI-6031E; National Instruments Corp., Austin, TX, USA), and sampled at 2000 or 2500 Hz using LabVIEW software (v.6.1, National Instruments Corp.). Before each recording session, signal output in volts was calibrated to microstrain using 1000 με toggle switches on the amplifier. Upon completion of strain recording trials, animals were euthanized (intraperitoneal injection of Beuthanasia-D Special, Merck Animal Health, Millsboro, DE, USA; 120 mg kg^–1) and frozen for later dissection and CT scanning.

To verify gauge positions and capture the shape of the humerus and femur, thawed cadaveric specimens were CT scanned at the Veterinary Teaching Hospital of the University of Georgia in Athens, GA, USA, using a Siemens Sensation 64 with a peak tube voltage of 120 kV, tube current of 152 mA, exposure time of 1000 ms, in-plane pixel resolution of 0.576–0.742 mm and slice thickness of 0.6 mm. Gauge connectivity, positions and orientations were subsequently confirmed through manual dissection. Malfunctional gauges were identified through visual confirmation of their connectivity, combined with signal characteristics, such as absence or intermittent recording of data, as well as drifting baselines. Data from these gauges were removed from the analyses.

To estimate the maximum tensile and minimum compressive strains normal to the cross-sectional plane of the bone midshaft, as well as the neutral axis of bending, planar strain theory was employed for the humerus using strains recorded from the central rosette element and the two single-element gauges (Biewener, 1992; Carter et al., 1981) (Fig. 1C). First, the right humerus of each individual was segmented from the CT data using Avizo 8.1 (ThermoFisher Scientific Inc., Waltham, MA, USA), and the midshaft was cut orthogonal to the bone's long axis using Blender v.4.0 (https://www.blender.org/). The cross-sectional outline was traced in Adobe Illustrator and then exported as a 500×500 pixel JPG file. A set of coordinates corresponding to the cross-sectional outline was extracted using the import_jpg1 function in the R package Momocs (Bonhomme et al., 2014; http://www.R-project.org/), and the strain gauge coordinates were determined by referring to the CT data. The equation for the line defining the neutral axis of bending was derived by solving a set of three equations, ε_n=ax_n+by_n+c, where ε is the recorded strain, x and y are the xy-coordinates of the nth gauge site, and a, b and c are unknown variables, for each of the three axial gauges (Biewener, 1992; Carter et al., 1981). Strain per unit distance normal to the neutral axis was computed, enabling the calculation of strain at any xy-coordinate based on its distance from the neutral axis. By sweeping through all xy-coordinates corresponding to the cross-sectional outline, the maximum tensile and minimum compressive strains were estimated. The neutral axis slope (–90 to 90 deg) was calculated relative to the horizontal orientation, established by the line connecting the medial and lateral condyles of the distal humerus. Positive slopes indicate counterclockwise rotation relative to the horizontal orientation, while negative slopes indicate clockwise rotation, when observing the cross-section from the distal perspective. To visualize the distribution of strains through the cross-section, strain isoclines were drawn at 100 με intervals at 20%, 40%, 60% and 80% of selected strides (Fig. 1C). Owing to malfunctions in one of the three axial gauges in each individual, planar strain analysis was not conducted for the femur.

The relationship between strain characteristics and limb posture was assessed for both the humerus and femur in each individual. Multiple linear regressions were performed with strain variable as the response, humerus or femur adduction angle and dimensionless speed as predictors, and their interaction, using the R package pequod (https://CRAN.R-project.org/package=pequod; http://www.R-project.org/). The interaction term was included because the effect of the limb joint angle on a strain variable might depend on dimensionless speed. Both predictors were centered on their means to reduce multicollinearity. Peak maximum and minimum principal strain, peak absolute shear strain, mean shear index and peak axial tensile and compressive strains during stance were used as response variables. Regressions were conducted for each strain variable from each individual, with strain variables from a single trial lacking variation in walking speed excluded.

Safety factors of the humerus and femur were calculated as the ratio of the yield strain for compression or torsion to the bone strain recorded during locomotion. Peak axial compressive strains estimated from three axial gauges and peak torsional strains calculated from the three-element rosette during stance were utilized as bone strains during locomotion. Unlike previous work, peak torsional strains were not corrected for gauge location based on planar strain analysis (Biewener, 1992; Carter et al., 1981), given the observed discrepancies between experimentally measured and predicted peak torsional strains using this method (Verner et al., 2016). Tensile and torsional yield strains of the humerus and femur were obtained from earlier research (Blob and Biewener, 1999; Blob et al., 2014), and the compressive yield strain was considered to be 33.3% higher than the tensile yield strain (Biewener, 1993; Currey, 1984). Safety factors for compression and shearing were calculated using both mean and single greatest peak values for axial compressive and absolute shear strains during stance across available strides. We did not consider size-dependent changes in limb bone strength, as such data are currently unavailable for alligator limb bones. Data manipulation and visualization were performed using R base, tidyverse and ggplot2 packages (http://www.R-project.org/; Wickham, 2016; Wickham et al., 2019).

Strain signals were recorded for 40 strides in al07 and 35 strides in al09 for the humerus (1.53 and 1.82 kg mass for al07 and al09, respectively) and 29 strides in al10, 48 strides in al07, 160 strides in al09 and 18 strides in al05 for the femur (0.37, 1.53, 1.82 and 4.98 kg mass for al10, al07, al09 and al05, respectively), using a three-element rosette and two single-element gauges in each individual (Fig. S1). The humeri of al10 and al05 were excluded from the analysis because of malfunction of both the three-element rosette and two single-element gauges, thereby preventing the application of rosette analysis (Dally and Riley, 1978) and planar strain analysis (Biewener, 1992; Carter et al., 1981). Throughout the trials, the three smaller individuals (al10, al07 and al09) walked at a mean dimensionless speed ranging from 0.211 to 0.253, while the largest alligator (al05) walked slower, with a mean dimensionless speed of 0.117 (Fig. 2A). Our four individuals with implanted strain gauges displayed various humeral and femoral postures that differed from the size-dependent limb postures previously observed in alligators without implantations (Iijima et al., 2021, 2023) (Fig. 2B,C). It is noteworthy that limb postures varied even among individuals of similar sizes, such as al07 and al09 (1.53 and 1.82 kg mass, respectively) (Fig. 2B).

Rosette analysis was conducted for 40 strides in al07 and 4 strides in al09 for the humerus, and 29 strides in al10, 48 strides in al07, 160 strides in al09 and 18 strides in al05 for the femur. For the humerus, larger peak maximum principal strains and smaller (i.e. larger negative) peak minimum principal strains were observed in al07 compared with al09 (Table 1, Fig. 3A,B). Peak absolute shear strains were greater in al07 than in al09, and the mean shear index was high (>0.6) in both individuals (Table 1, Fig. 3C,D). Shearing occurred in the direction of the external rotation of the humerus during the first half of stance in both al07 and al09, while the direction changed to the opposite during the last half of stance in al09 (Fig. 3C). For the femur, significantly larger peak maximum principal strains and smaller (larger negative) peak minimum principal strains were recorded in al05, with the absolute peak principal strains more than twice as large as those in smaller individuals (Table 1, Fig. 4A,B). Moreover, although our four individuals exhibited different femur adduction angles (Fig. 2C), the three individuals with the most similar hindlimb postures (al10, al07 and al05) showed an increase in their principal strain magnitudes with larger body sizes (Table 1, Fig. 4A,B). Peak absolute shear strains were largest in al07, followed by al05, al10 and al09 (Table 1, Fig. 4C). The mean shear index was high (>0.7) in al10 and al07, moderate in al05 (∼0.3) and very small in al09 (∼0.1) (Fig. 4D). Shearing predominantly occurred in the direction of external rotation of the femur during stance in al10 and al07, while it was in the opposite direction in al05. No significant shearing was observed in al09, which is likely associated with its more adducted limb posture (Fig. 4C; see Discussion).

Planar strain analysis was performed for 40 strides in al07 and 35 strides in al09 for the humerus (Table 2). The neutral axis of bending was primarily negatively rotated throughout stance in both individuals, with compression dominating on the dorsolateral cortex and tension on the ventromedial cortex (Fig. 3G). The estimated mean peak axial tensile strains were approximately 500–600 με, while the mean peak axial compressive strains were more than twice as large in absolute value in both al07 and al09 (Table 2, Fig. 3E,F).

Multiple linear regressions revealed the effect of limb posture on strain characteristics within each individual (Table 3). Interactions between the humeral or femoral adduction angle and dimensionless speed were non-significant (P>0.05) in most models (Table 3). Thus, the effect of limb posture on strain variables is largely independent of dimensionless speed. The humerus adduction angle had a significant (P<0.05) or nearly significant (P<0.1) effect on peak minimum principal strain and peak absolute shear strain in al07, and peak axial compressive strain in al09 (Table 3, Fig. 5). Typically, smaller (larger negative) peak minimum principal strains, larger peak absolute shear strains and smaller (larger negative) peak axial compressive strains were recorded when the humerus was more abducted. The femoral adduction angle had a significant or nearly significant effect on peak minimum or maximum principal strain in al10 and al09, peak absolute shear strain in al09 and al05, and mean shear index in al10 and al07 (Table 3, Fig. 5). Peak minimum principal strains were larger (smaller negative) in al10, while peak maximum principal strains were larger in al09 with a more abducted femur. Peak absolute shear strains showed opposite posture-dependent trends in al09 and al05: they were higher in al09 and lower in al05 with a more abducted femur. Mean shear index was higher with a more abducted femur in both al10 and al07.

Safety factors for axial compression were lower in the humerus of al09, which exhibited a more adducted humerus compared with al07. In al09, they were calculated as 4.8 based on the mean peak strain across all strides, and 3.4 based on the single greatest (largest negative) peak compressive strain (Table 4). Safety factors for torsion were lower for individuals that used a more abducted humerus and femur. In the humerus and femur of al07, they were 11.5 and 9.3 based on the mean peak strain and 5.0 and 6.2 based on the single greatest peak strain, respectively (Table 4).

The three-element rosette attached on the ventromedial cortex of the femoral midshaft recorded strains in four individuals, enabling comparisons of principal strains across sizes. The magnitudes of the principal tensile and compressive strains increased with body size among individuals that adopted similar limb postures (al10, al07 and al05), supporting size-dependent changes in femoral bone strains (Table 1, Fig. 4A,B, summarized in Fig. 6).

An increase in limb bone strains with larger sizes is expected from geometric and dynamic similarity (Alexander and Jayes, 1983; Biewener and Patek, 2018). If body and limb shape are geometrically similar and bone material properties remain constant in animals moving in a dynamically similar manner, limb bone strain and stress would scale with mass^1/3 (Biewener and Patek, 2018; McMahon, 1975). In American alligators, the femur length scales with mass^0.286, as determined by the reduced major axis regression of log-transformed variables in Farlow et al. (2005). Additionally, the femur midshaft circumference scales with the femur length^1.133 (Iijima and Kubo, 2019). As the result, the midshaft cross-sectional area, disregarding the medullary cavity, scales with mass to the power of 0.65 (=0.286×1.133×2). This implies that femoral midshaft stress would scale with mass^0.35 if external, muscle and inertial forces applied to the bone scale proportional to body mass, which closely aligns with the scaling exponent expected from geometric similarity.

Larger American alligators are known to address a size-dependent increase in bone and soft tissue strain and stress, as well as mass-specific muscle force, by adopting a more erect limb posture. Previous research on limb kinematics in juvenile to adult alligators (0.2–223 kg mass) revealed that larger individuals employ a more adducted humerus and femur and possibly a more extended elbow and knee, with the most significant changes occurring at a body mass of 1 kg (Iijima et al., 2021, 2023). By adopting a more erect limb posture, larger alligators mitigate an increase in mass-specific muscle forces by reducing size-normalized limb joint moments (Iijima et al., 2021). This interpretation aligns with the experimentally measured muscle activation patterns of m. pectoralis, the largest forelimb muscle and primary shoulder adductor (Iijima et al., 2024). Reducing the required limb muscle forces would help mitigate limb bone strains and stresses in larger alligators, as a significant portion of bone stress derives from forces exerted by muscles (Aiello et al., 2013; Biewener, 1989, 1990, 2005). In this study, three of our implanted animals (al10, al07 and al05: 0.37–4.98 kg mass) did not modulate limb posture and adopted more abducted femora, with their torsos partially in contact with the ground. This might contribute to increased femoral strains in larger individuals.

An ontogenetic increase in limb bone strains has been documented in a few other taxa. In juvenile to adult goats during walking, trotting and galloping, bending strains of the radius increased towards larger sizes, primarily as a result of the strong negative growth allometry of the midshaft cross-sectional area of the radius (⋑mass^0.53, where ⋑mass^0.67 is the expectation under isometry) (Main and Biewener, 2004, 2006). This negative growth allometry might be attributed to the precociality of goats, which allows juveniles to move at the same absolute speed as adults in herds (Main and Biewener, 2004, 2006). Similarly, in hatchling to adult emu during walking and running, bending and torsional strains of the femur and tibiotarsus increased towards larger sizes, possibly as a result of near-isometric growth of bone cross-sectional area (Main and Biewener, 2007). For alligators, it remains unclear whether the observed changes in limb posture in individuals that have not undergone surgery can compensate for expected increases in limb bone strains. Integrated data on limb kinematics, ground reaction force, limb bone cross-sectional geometry and material properties would be necessary to fully understand the factors responsible for the observed loading patterns (Main et al., 2021; Reilly et al., 2005).

Interpretation of the changes in femoral loading regimes across sizes is complicated for our animals because of potential postsurgical lameness shown by some individuals; however, differences in their shear directions are noteworthy. Among the three individuals that adopted more abducted limb postures with their torsos partially in contact with the ground (al10, al07 and al05), the femur experienced torsion in the direction of external rotation during early to mid-stance and then in the direction of internal rotation during late stance in al10 and al07 (Fig. 4C). This pattern aligns roughly with the external moment about the long axis of the femur during stance (Blob and Biewener, 2001; Iijima et al., 2021), suggesting that the external moment is a major contributor of femoral torsion. In contrast, the femur underwent torsion in the direction of internal rotation throughout stance in al05 (Fig. 4C). This may indicate a more lateral foot placement during stance in al05, causing the ground reaction force vector to pass lateral to the femur, which results in a persistent internal rotation moment about the bone's long axis. Additionally, there could be an increased contribution from an internal long axis rotator muscle, m. caudofemoralis (Blob, 2000; Blob and Biewener, 1999, 2001), during stance in al05.

Overall, the observed effects of limb posture on humeral and femoral strain variables support our predictions that axial loads predominate over shear loads with the use of a more erect limb posture (summarized in Fig. 6). Lesser peak shear strains, as well as a smaller shear index, are associated with a more adducted humerus in al07 and femur in al07, al09 and al10 (Table 3, Fig. 5B,F,H,I). Although the femur in al05 exhibited the opposite trend, with greater peak shear strains towards a more adducted femur (Table 3, Fig. 5G), the sample size is smallest, and the range of limb postures covered is narrower for this comparison, requiring careful interpretation. The comparison of loading regimes between individuals of similar body sizes yielded comparable results. In al07 and al09, with a body mass of 1.53 and 1.82 kg, respectively, lesser magnitudes of peak shear strains and a smaller shear index were observed in the humerus and femur of al09, which exhibited a more adducted humerus and femur (Table 1, Figs 3 and 4). It is noteworthy that the femur of al09, which used a more upright posture than other individuals (Fig. 2), was subjected almost exclusively to axial loadings (Table 1, Fig. 4). Interpreting the distinct loading patterns in the femur of al09 is challenging, given that its limb posture overlaps to some extent with that of other individuals (Fig. 2). Nonetheless, the loading pattern in this individual contrasts with the common perception that the limb bones of alligators experience significant torsion (Blob and Biewener, 1999; Blob et al., 2014). Conceivably, in American alligators that employ a range of limb postures, bone loading regimes may not be uniform and may vary depending on limb posture.

These findings suggest that primary loading regimes shift from shear to axial compression and bending across the sprawling-to-erect posture spectrum in animals whose limbs move non-parasagittally. This is comparable to the shift in loading regimes observed in mammals across the crouched-to-upright posture spectrum, where limbs move in a parasagittal plane (Biewener and Taylor, 1986; Biewener et al., 1983, 1988; Butcher et al., 2011; Goodship et al., 1979; Keller and Spengler, 1989; Lanyon et al., 1975; Main and Biewener, 2004; Rubin and Lanyon, 1982). The shift in bone loading regimes associated with the transition from sprawling to erect posture has been examined in extinct tetrapods by comparing the cross-sectional geometry of long bones. Theoretically, a circular shape maximizes the polar moment of inertia to resist torsion, while an ellipsoid shape increases the second moment of area along a specific axis to resist bending (Main et al., 2021). Across synapsid taxa inferred to experience this posture transition, the cross-sectional shapes of the limb bones shifted from circular to ellipsoid, consistent with the expectation (Blob, 2001).

In addition to the magnitudes of shear strains, those of axial strains and combined axial and shear strains may vary with limb posture. Observations of within-individual trends in limb bone strains revealed smaller magnitudes of peak axial compressive strains with the adoption of a more erect limb posture in the humerus of al09 (Table 3, Fig. 5C). Moreover, smaller peak absolute principal strains were recorded when adopting a more erect limb posture in the humerus of al07 and the femur of al09, while a weak opposite trend was found in the femur of al10 (Table 3, Fig. 5A,D,E). Overall, these results seem to support the decrease in magnitude for axial and combined axial and shear strains with the adoption of a more erect posture, particularly for the humerus, in which consistent trends were observed. The use of a more erect posture may help mitigate limb bone strains by decreasing joint moments caused by both external forces and antigravity muscle forces (Iijima et al., 2024). As larger alligators that have not undergone surgery use a more erect limb posture (Iijima et al., 2021, 2023), the transition from sprawling to erect limb posture in the Triassic (Bonaparte, 1984; Carrano, 2000; Charig, 1972; Kubo and Benton, 2009) might lead to or result from the evolution of larger body sizes in archosaurs. Such correlated evolution of locomotor biomechanics and body size has been proposed for tetrapods from the Carboniferous to early Permian based on trackway data (Buchwitz et al., 2021).

The current findings concerning posture-dependent changes in limb bone strain magnitudes show distinct patterns from those observed previously in the femora of American alligators. A previous study demonstrated that in the alligator femur, principal and shear strains on the dorsal and ventral cortex along the midshaft perimeter tend to be greater, while axial strains on the anterior cortex tend to be smaller when the bone is more adducted (Blob and Biewener, 1999). Peak axial compressive strains estimated from the planar strain analysis also appeared to be greater with a more adducted femur (Blob and Biewener, 1999). To interpret these counterintuitive results, it was previously hypothesized that during a more erect walk in alligators, the center of pressure of the foot shifts anteriorly, requiring larger ankle extensor forces from m. gastrocnemius to counteract the increased ankle dorsiflexion moment (Blob and Biewener, 1999, 2001; Reilly and Blob, 2003). As m. gastrocnemius is a biarticular muscle that also flexes the knee joint, larger knee extensor forces from m. femorotibialis and m. iliotibialis are required, which increase the compressive strains on the dorsal cortex of the femur (Blob and Biewener, 1999, 2001; Reilly and Blob, 2003). As the current results on posture-dependent changes in humeral strains differ from previous data on femoral strains (Blob and Biewener, 1999), an increase in axial strains with a more erect limb posture could be characteristic only of the hindlimb with a long pes and may not apply to the forelimb with a short manus (Iijima et al., 2024). Moreover, during an erect walk, the lifted proximal tail could potentially cause a disproportionate increase in loading on the hindlimb compared with the forelimb, leading to elevated femoral strains.

The digitization of the cross-sectional shape of the limb bone midshaft, along with the application of planar strain analysis to all coordinates along the cross-sectional outline throughout stance, enabled the rigorous estimation of peak axial strains. The estimated safety factor for axial compression was 4.8 based on the mean peak strain and 3.4 based on the single greatest (largest negative) peak compressive strain in the humerus of al09 (1.82 kg mass) (Table 4). These values are much lower than the safety factor of 8.4 for axial tension in the humerus of similarly sized alligators (1.7–2.0 kg mass) (Blob et al., 2014). In their study, the safety factor was calculated based on the mean peak axial strain on a three-element rosette at a specific time frame. This was then corrected for a proportional increase in strain along the cross-sectional outline using planar strain analysis from a single trial (Blob and Biewener, 1999; Butcher et al., 2008).

The safety factors for torsion were as low as 11.5 and 9.3 based on the mean peak strain in the humerus and femur, respectively, while these values decreased by 43–67% when based on the single greatest peak strain (Table 4). Although the mean peak torsional strains are higher than previously reported torsional safety factors of 7.8 in the humerus and 5.0 in the femur, we did not correct for a proportional increase in shear strain using planar strain distribution (Blob and Biewener, 1999; Blob et al., 2014). Because such corrections may not accurately predict peak shear strains along the cross-sectional outline, corrected calculations could be underestimates (Verner et al., 2016). Assessing whether alligator limb bones face a greater risk from axial or off-axial strains is challenging based on the current data, as it depends on the primary loading regimes and thus limb posture adopted by each individual. Bending safety factors were usually lower for individuals using a more erect posture, while torsional safety factors were lower for those using a less erect posture (Table 4).

Among tetrapods, the lowest safety factor for bending in the humerus of alligators, as computed from the mean peak strain in this study (4.8), is still higher than those based on yield strains in eutherian mammals (∼2.5: Biewener, 1993; Blob et al., 2014) and birds (2.9–3.7: Biewener, 1993; Biewener et al., 1986; Blob et al., 2014; Rubin and Lanyon, 1984). Meanwhile, the bending safety factor in the alligator humerus is comparable to those recorded in Virginia opossums and turtles (5.1 and 4.4, respectively: Butcher et al., 2008, 2011) and smaller than those in frogs (7.7–8.5: Blob et al., 2014). It is possible that the origin of the low limb bone safety factor in birds could be traced back to slightly decreased safety factors in archosaurs including crocodylians (Nesbitt, 2011), or even archelosaurians including turtles (Crawford et al., 2015). As limb bone safety factors in alligators were based on bone strains during walking, data recorded during strenuous activities, such as running and lunging, would further narrow the gap in safety factors between crocodylians and birds (Blob and Biewener, 1999). Such data across tetrapods other than mammals and birds (e.g. Blob and Biewener, 1999; Munteanu et al., 2023) would contribute to a better understanding of the evolution of limb bone safety factors and their underlying mechanisms.

We acknowledge several limitations to our current study. Although we initially implanted strain gauges on the humerus and femur of six individuals, data from four humeri and two femora were unusable because of malfunctions in both the three-element rosette and single-element gauges. Consequently, only two humeri and four femora were used for the current analysis. The small number of samples precluded the use of statistical tests to assess the effect of body size on bone strains, so caution is required when interpreting the results. Additionally, for four femora, peak axial strains were not estimated using planar strain analysis because of missing strain signals from either of the single-element gauges. Thus, comparisons of femoral strains across sizes relied solely on the principal and shear strains determined through rosette analysis. Although three-element rosettes were consistently placed on the ventromedial cortex of the femoral midshaft in all individuals, potential changes in the neutral axis of bending would complicate comparisons of principal strain magnitudes across individuals. The small number of samples also limited our ability to identify the causes of distinct strain patterns, such as the principal and shear strains in the femur of al09, and shear direction in the femur of al05. Lastly, postsurgical lameness and potential alterations in muscle function due to surgery might contribute to the differences in limb kinematics between individuals in this study and those in previous studies that had not undergone surgery (Iijima et al., 2021, 2023). Notably, the highly abducted humeri and femora in al07 and al05 (1.53 and 4.98 kg mass, respectively) are significantly different from those in previous studies involving individuals that had not undergone surgery (Iijima et al., 2021, 2023). Despite these limitations, this study is the first to describe the bone loading in relation to body size and limb posture in tetrapods with non-parasagittal limb posture, providing a foundation for future investigations.

We express our gratitude to R. Elsey, D. LeJeune and M. Miller (Louisiana Department of Wildlife and Fisheries) for providing alligators, and M. Hart (South Carolina Department of Natural Resources), R. Flynt (Mississippi Department of Wildlife, Parks and Fisheries), C. Threadgill and T. Ancelet (Alabama Department of Conservation and Natural Resources), and J. Hawkins (Georgia Department of Natural Resources) for granting permission to transport and house the alligators. We also thank D. Knight and T. Pruitt (Clemson University) for their help with animal surgeries, and J. Jones (Clemson University), N. Northrup, M. Perlini and C. Romine (University of Georgia) for facilitating the CT scanning of animal cadavers. Animal care was assisted by K. Diamond, C. Kinsey, A. Palecek and D. Adams (Clemson University).