Estimating ankle joint angle from skeletal geometry: a mechanical model of the calcaneal lever in terrestrial mammals

Introduction

The hindlimbs perform some essential functions for terrestrial life, such as supporting body mass, generating propulsion, and lifting the body. During walking, in particular, the hindlimbs contribute to forming an inverted pendulum (Cavagna, Heglund & Taylor, 1977; Alexander & Jayes, 1978a, 1978b; Hildebrand, 1984; Hildebrand & Hurley, 1985; Alexander, 1991; Griffin, Main & Farley, 2004) or spring-loaded inverted pendulum (SLIP) model (Geyer, Seyfarth & Blickhan, 2006), both of which facilitate efficient locomotion by generating propulsive energy. The SLIP model proposes that the limbs act as springs, while also emphasizing the importance of optimized stiffness (Blickhan & Full, 1993; Geyer, Seyfarth & Blickhan, 2006). In these models, the rod of the inverted pendulum is defined as the line between the grounding point of the metatarsus and the center of mass (Cavagna, Heglund & Taylor, 1977; Alexander & Jayes, 1978a, 1978b; Blickhan & Full, 1993; Griffin, Main & Farley, 2004; Geyer, Seyfarth & Blickhan, 2006); therefore, the rod length is governed by the joint angles, particularly those of the knee joint and the ankle joint in the hindlimbs. The previous studies showed that the knee joint angles among mammalian species do not change drastically—remain relatively constant with small changes—during the stance phase, and differ between species (Manter, 1938; Gray, 1944; Goslow, Reinking & Stuart, 1973; Goslow et al., 1981; Alexander & Jayes, 1983; Inuzuka, 1996; Fischer et al., 2002; McGowan, Baudinette & Biewener, 2005; Dutto et al., 2006; Day & Jayne, 2007; Ren et al., 2008; Patel et al., 2013). The variations in the joint angles contribute to differences in body height, locomotion, and biology between species. However, Mizuno & Kohno (2023) showed that the angle between the line of the action of the musculus semimembranosus and the tibia resembles across mammals, even when these taxa belong to different orders. Similarly, Fujiwara (2009) and Fujiwara & Hutchinson (2012) showed that the angle between the olecranon process and the m. triceps brachii remains similar despite differences in the elbow joint angle typically measured between the humerus and ulna. These studies indicate that while joint angles observable from outside the body vary across species, may be conserved across mammals due to skeletal morphology. Therefore, identifying the mechanisms that determine such skeletal angles enables the reconstruction of limb postures in extinct animals, whose joint angles cannot be directly observed.

The foot is a part of the inverted pendulum of the entire hindlimb, and also functions as a lever system driven by extensor muscles to generate greater ground reaction force (Gregory, 1912; Manter, 1938; Biewener, 1989, 1990; McGowan, Baudinette & Biewener, 2005; Deforth et al., 2019). The upward rotation of the calcaneal tuber, caused by contraction of these muscles, induces the downward rotation of the metatarsus, thereby generating ground reaction force. Anatomically, the extensor muscles—specifically the m. gastrocnemius, m. soleus, and m. flexor digitorum superficialis—contribute to raising the calcaneal tuber. These extensor muscles form the tendo calcaneus (Achilles tendon), although the m. soleus is absent or weakened in some taxa. This muscle originates from the flexor side of the knee joint—specifically, the distal femur and the proximal tibia and fibula—and inserts into the distal end of the calcaneal tuber via the Achilles tendon, running nearly parallel to the tibia (Kato & Yamauchi, 2003; König, Liebich & Bragulla, 2007). Therefore, the net force vector produced by the extensor muscles can be assumed to run parallel to the axis of the tibia. The functional role of the ankle extensors during the stance phase to support body mass and generate propulsive force against gravity has been well established (Engberg & Lundberg, 1969; Goslow, Reinking & Stuart, 1973; Rasmussen, Chan & Goslow, 1978; Walmsley, Hodgson & Burke, 1978; Goslow et al., 1981; Nicolopoulos-Stournaras & Iles, 1984; Whelan, Hiebert & Pearson, 1995). In terrestrial mammals, particularly in unguligrade and digitigrade species, the ankle joint is actively extended to counteract gravitational flexion forces during both standing and stance phase. Accordingly, an extensor torque must be generated at the ankle joint, with the calcaneal tuber playing a central role in its propagation. This extensor torque arises from the force applied via the calcaneal tuber and the internal moment arm, defined as the perpendicular distance between the joint’s center of rotation and the line of action of the muscle force (Fujiwara, 2009). Given this biomechanical context, it is reasonable to hypothesize that mammals adopt joint angles that maximize this moment arm during the stance phase, thereby enhancing mechanical advantage and stability.

We hypothesize that (i) the functional ankle extension angle (FAEA), defined as the angle between the line of action of the extensor muscles and calcaneal tuber, is maintained within a range that optimizes muscle force production and mechanical advantage during the stance phase of walking in terrestrial mammals; (ii) FAEA is similar among species with different observable ankle joint angle (θ) which is typically measured; and (iii) θ can be estimated from a mechanical model of the ankle extensor apparatus. This study aims to test these hypotheses using a mechanical model based on the skeletal geometry of each target species, and to provide insight into the fundamental constraints shaping limb posture, which may aid in the functional interpretation of extinct taxa.

Materials and Methods

Mechanical model

To examine the hypotheses, we constructed a simplified mechanical model of joint rotation. For instance, the trochlea of the talus was approximated as a circular arc, and the articulations between the foot bones were assumed to be rigid. This simplification was necessary to allow general comparison across species, as skeletal morphology varies not only between species but also among individuals. Assuming the foot as a simple body of rotation about the ankle joint (Fig. 1) during extension and flexion, the ankle extensor torque ( $\tau$) is generated by the extensor muscles, which inserts at the most distal point on the calcaneus (C) (Fig. 1). $\tau$ can be expressed by the following equation:

(1) $$\tau =l\times F$$where F is a contractile force of the extensor muscles, and l is the moment arm length. The moment arm l is defined as the perpendicular distance from the center of the ankle joint rotation (O) to the line of action of F (Fig. 1). If F is constant, τ is directly proportional to l. The moment arm l can be calculated using the following equation:

(2) $$l=\mathrm{O}\mathrm{C}\cdot \mathrm{s}\mathrm{i}\mathrm{n}\alpha$$where OC is the distance between O and C, and α represents the value of FAEA. Since the length OC is fixed for a given specimen, the value of l depends entirely on the angle α (Fig. 1). The function sin α reaches its maximum when α is 90° (Fig. 1). Therefore, l is maximized (sin α = 1) when the line OC is perpendicular to the tibia and vector F. The value of sin α can thus be regarded as the mechanical advantage of the calcaneal lever.

Assuming that the F runs parallel to the shaft of the tibia and that the articulations between the tarsal and metatarsal bones are rigid, we defined the estimated ankle joint angle (θ_est) as the angle at which α = 90° (Fig. 1). At this angle, the mechanical advantage of the calcaneal lever (sin α) is maximized. The mechanical advantage of the calcaneal lever decreases when the ankle joint angle θ deviates above or below this value θ_est. To investigate the directional relationship between the tibia and the Achilles tendon, anatomical observations were performed on several cadaveric specimens. Details of these observations are provided below. All angles (e.g., θ and α) are expressed in degrees unless otherwise noted. θ varies depending on the specimen (Fig. 2).

Collection from skeletal specimens

The ankle joint angles at which the extensor moment arm is maximized were measured using skeletal specimens. The data were collected for the same 26 species used in the observations of θ_obs (Table 2). Disarticulated hindlimb bones stored at the National Museum of Nature and Science (NMNS: Tsukuba, Japan), Toyohashi Museum of Natural History (Toyohashi, Japan), Kanagawa Prefectural Museum of Natural History (Odawara, Japan), and Osaka Museum of Natural History (Osaka, Japan) (Table 2) were used. A total of 42 skeletal specimens were measured. No specimens showed significant pathologies or malformations. Furthermore, we avoided using juvenile specimens.

We reassembled the metatarsals and tarsal bones (calcaneus, talus, cuneiforms, navicular, and cuboid) using a soft, reusable adhesive (Hittsukimushi, Kokuyo, Osaka, Japan), and then the flexibility of the joint connections was confirmed to be negligibly small. The Mongolian gerbil foot skeleton was stored with the bones connected by tendons and ligaments. The reassembled bones were photographed in lateral view using a digital camera (D3400; Nikon, Tokyo, Japan).

Two lines were then drawn on each image, and the angle between them was measured following procedure described below. First, a circle centered at O, which best fits the ridge of the trochlea of the talus, was drawn. Next, a perpendicular line (line A) to the line segment OC, originating from point O, was drawn. Following this, a line connecting the dorsal tips of the proximal and distal epiphyses on the metatarsal III (line B) was drawn (see description in Mechanical Model section). Finally, the angle between lines A and B was defined as the ankle joint angle where the extensor moment arm maximized for the specimen (Fig. 3). These figures were also created using Inkscape. The average of the angles for each species was defined as θ_est.

Results

The directions of the Achilles tendon and the tibia were nearly parallel in the species dissected in this study (Fig. 4). This parallel orientation of the Achilles tendon relative to the tibia was consistently observed over a wide range of joint angles. Therefore, the direction of the line of action of the ankle joint extensor muscle can be approximated by the tibia. Additionally, the transitions of the ankle joint angle during the stance phase were checked to ensure the θ_obs represented the angle of the target species. According to the line graph of the ankle joint angle transitions, these were gradual and the maximum differences between the average angle (θ_obs) and the maximum or minimum angle were small in the many studied species (Fig. 5): in 22 species of the 27 studied species, the maximum differences were less than 20 degrees. Thus, θ_obs was considered representative for each species and was suitable for comparison with θ_est. The Mongolian gerbil which had the smallest body mass, 0.06 kg, had both the minimum θ_obs and θ_est (64° and 95°, respectively). The white rhinoceros, which has the largest body mass, 3,400 kg, had the second maximum θ_est (141°). Thoroughbred, with the fourth largest body mass (518 kg), had the maximum θ_obs (157°), whereas the maximum θ_est was 146° for the Indian rhinoceros, which had the second largest body mass (2,100 kg) (Table 3). The Japanese serow, reindeer, and Japanese macaque had the smallest absolute difference between θ_est and θ_obs (0°), and the thoroughbred had the largest difference, 38° (Table 3).

In 24 species (including the Kiso horse), the value of the mechanical advantage of the calcaneal lever exceeded 0.9; among these, 18 species had values greater than 0.95. Three species had the mechanical advantage of the calcaneal lever between 0.8 and 0.9. The thoroughbred had the lowest mechanical advantage of the calcaneal lever at 0.79 (Table 3 and Fig. 6). PhyANOVA and pairwise comparisons were conducted to assess significant differences in the mechanical advantage of the calcaneal lever among locomotor modes. The phyANOVA showed no significant differences (p = 0.602), and all pairwise Holm-adjusted p-values were equal to 1.00; therefore no significant differences in the mechanical advantage of the calcaneal lever among locomotor modes (Table 4). Additional phyANOVA and pairwise comparisons were also conducted among orders. However, since Scandentia was represented by only one species, it was excluded from the analysis. The phyANOVA among orders also showed no significant differences (p = 0.665), and all pairwise Holm-adjusted p-values were equal to 1.00 (Table 4). Thus, neither locomotor mode nor order showed significant differences in the mechanical advantage of the calcaneal lever.

Since the mechanical advantage of the calcaneal lever was not normally distributed, the Spearman’s rank order correlation was used to assess the correlation coefficient. The correlation coefficient between the θ_obs and the θ_est was 0.75, with the p-value less than 0.05 (p = 1.21 × 10⁻⁵: Table 4 and Fig. 7). This indicated a strong positive correlation between θ_obs and θ_est, and supporting a significant correlation in the population.

We also performed multiple linear regression analysis via PGLS using three data sets: (1) Kiso horse data representing horse; (2) thoroughbred data representing horse; and (3) excluding horse to examine the relationships between θ_obs, θ_est, and body mass. This approach was taken because the mechanical advantage of the calcaneal lever values differed notably between the Kiso horse and the thoroughbred, even though they belong to same species (Table 3). The regression equation for the θ_obs can be expressed as:

(3) $${\theta }_{obs}={\beta }_0+{\beta }_1\cdot {\theta }_{est}+{\beta }_2\cdot \mathrm{b}\mathrm{o}\mathrm{d}\mathrm{y}\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}$$

In the Kiso horse data set, the regression coefficients were β₀ = 4.90 (95% confidence interval (CI) [−34.25 to 43.05], intercept), β₁ = 0.48 (95% CI [0.12–0.83], coefficient for θ_est, p = 0.009), and β₂ = 4.87 (95% CI [3.10–6.25], coefficient for body mass, p = 2.9 × 10⁻⁶), with a significant overall model (p = 6.1 × 10⁻⁸, adjusted R² = 0.74) (Table 5). In the Thoroughbred data set, the coefficients were β₀ = 8.42 (95% CI [−37.85 to 54.69]), β₁ = 0.43 (95% CI [0.03–0.83], p = 0.032), and β₂ = 4.87 (95% CI [3.07–6.67], p = 8.7 × 10⁻⁶) (overall model p = 6.7 × 10⁻⁷, adjusted R² = 0.68) (Table 5). In the data set excluding Equus caballus, the coefficients were β₀ = 5.46 (95% CI [−33.30 to 44.22]), β₁ = 0.47 (95% CI [0.12–0.82], p = 0.009), and β₂ = 4.61 (95% CI [2.97–6.25], p = 4.0 × 10⁻⁶) (overall model p = 9.5 × 10⁻⁸, adjusted R² = 0.75) (Table 5). Pagel’s λ values indicated a strong phylogenetic signal across all data sets (0.943, 0.977, and 0.938, respectively) (Table 5).

Discussion

A total of 85.2% (23 out of 27) of the studied species showed a mechanical advantage of the calcaneal lever greater than 0.9 (sin α > 0.9). This species included individuals with a wide range of body masses, from 0.18 kg (common tree shrew) to 3,400 kg (white rhinoceros), as well as species representing three locomotor modes and six orders (Table 3). The phyANOVA and pairwise comparison tests for mechanical advantage of the calcaneal lever across locomotor modes and taxonomic orders detected no significant differences. Additionally, the correlation coefficient between mechanical advantage of the calcaneal lever and body mass was not statistically significant (p = 0.12), suggesting that the mechanical advantage of the calcaneal lever is not strongly influenced by these factors. These results suggest that FAEA of terrestrial mammals is maintained within a range that optimizes mechanical advantage of the calcaneal lever during the stance phase under steady-speed walking conditions, across a wide range of body masses and taxa. A higher mechanical advantage of the calcaneal lever enhances torque generation at the ankle joint for a given muscle force, contributing to more efficient support and propulsion during walking (Biewener, 1989; Biewener & Roberts, 2000; Ray & Takahashi, 2020). This mechanical advantage is particularly important during the stance phase, when joints must resist gravitational flexion forces.

Based on the multiple linear regression analyses using PGLS, the Kiso horse data set more closely resembled the excluding E. caballus data set (Table 5); therefore, the Kiso horse data set was treated as representative of E. caballus. The regression equation showed that the effect of body mass on θ_obs is notably stronger than that of θ_est, where each unit increase in θ_est leads to an increase in θ_obs by about 0.48 degrees. Both factors significantly influence θ_obs, with body mass having a more substantial impact. However, in species exceeding 10 kg in body mass, the recalculated θ_obs values using the regression equation (hereafter referred to as θ_r-obs) showed larger differences from θ_obs than differences of θ_est values. These mean values were 11.1 and 6.0 degrees, respectively. The θ_r-obs values showed smaller differences than θ_est in 11 out of 26 species, but five of these 11 species had a body mass below 10 kg (Mongolian gerbil, common tree shrew, yellow-toothed cavy, meerkat, and cat) (Table 6). Notably, only seven out of the 26 species had a body mass below 10 kg (Table 6). Among the 19 species exceeding 10 kg, three species (pygmy hippopotamus, tiger, and Kiso horse) showed identical differences from the θ_obs for both θ_r-obs and θ_est. In the remaining 16 species, θ_est showed smaller differences than θ_r-obs in 11 cases. On average differences of θ_r-obs were 3.8 degrees in species below 10 kg, and 11.1 degrees in species exceeding 10 kg. In contrast, the differences in θ_est were −16.3 degrees in species below 10 kg, and 6.0 degrees in species exceeding 10 kg (Table 6). In addition, the correlation coefficient between the θ_obs and the θ_est was significantly positive and strong (r = 0.74). Therefore, θ_est can be reliably used to reconstruct θ_obs in species exceeding 10 kg, even if the skeletal specimens have been stored disarticulated.

Notably, kangaroos (Macropus spp.) showed negative values for θ_obs minus θ_est (−23 and −21, respectively) (Table 6), indicating that the θ_obs were more flexed than the θ_est. Kangaroos employ a unique form of locomotion known as pentapedal, which uses the tail to support body mass during quadrupedal walking (Windsor & Dagg, 1971; O’Connor et al., 2014). However, because the tail is used during the swing phase of the hindlimbs, other factors may contribute to this result. One possible explanation is that quadrupedal walking is not the typical mode of locomotion for them; they primarily use hopping as their main mode of locomotion. The hopping gait is most commonly employed during regular movement, escape responses, and long-distance travel (Windsor & Dagg, 1971; O’Connor et al., 2014). Their hindlimbs are highly specialized for hopping, e.g., elongated metatarsals and shortened forelimbs (Alexander & Vernon, 1975; Polly, 2007; McGowan & Collins, 2018; Jones, Travouillon & Janis, 2024). The elongated metatarsals of hopping species increase the distance between the ground and the ankle joint, whereas short forelimbs reduce the distance between the ground and the trunk. It is possible that flexion of the hindlimb joints during quadrupedal walking, as observed in this study, serves to lower the posterior trunk, which might partly account for the more flexed θ_obs relative to θ_est in kangaroos. However, because this study includes two kangaroos species, additional data from other hopping species such as wallaby and kangaroo rat are needed to confirm the idea.

The data from two breeds of horses, the Kiso horse and the thoroughbred, implied that artificial selection had influenced on their morphology. These two breeds are classified in the same species, Equus caballus (horse), but their sin α differed by 0.16, with the Kiso horse showing higher value (0.95) than the Thoroughbred (0.75) (Table 3). This difference was greater than between the highest and the second lowest sin α of this study, 0.13 (Table 3). This difference appeared to result from more extended hindlimb posture in thoroughbred, where θ_obs exceeded θ_est, reducing the mechanical advantage of the calcaneal lever. While both breeds are domesticated, their morphology reflects fundamentally different breeding purposes. The Kiso horse is a medium-sized Japanese native horse that had been raised since around fifth century and used for agriculture and transportation in mountain regions until the middle of the 20th century (Nakagawa, 2014; Yamashita, 2014). Today, it is maintained primarily for conservation. In addition, the Japanese native horses have remained independent from modern breeding (Nozawa, 1992). Thoroughbred, on the other hand, was developed in the late 18th century and has been strongly selected for racing abilities and characteristics until today (Cunningham, 1991; Bower et al., 2012). It is possible that artificial selection for enhanced galloping ability in Thoroughbreds has favored morphologies that maximize propulsion during high-speed locomotion. This may include joint configurations that enhance the rotational torque around the ankle joint (Gnagey, Clayton & Lanovaz, 2006). Additionally, efficient elastic energy storage requires significant tendon stretch, which is facilitated by joint flexion (Dimery, Alexander & Ker, 1986; Biewener, 1998). Therefore, more extended posture during walking of thoroughbred may reflect an adaptation for maximize both elastic energy recovery and mechanical advantage of the calcaneal lever during running. However, this study had only two breeds of horses because we aimed to collect data from wide range of taxa. To test this idea, it will be necessary to collect data from broad range of species and breeds within Equidae.

There were some possibilities to lead errors in this study. In vivo measurement was limited by animal behavior and shooting environment. In addition, the measurements were not performed with any markers directly put on the living specimens. When measuring skeletal specimens, although the articulation flexibility is small, thickness of the adhesive material could cause errors, and the effect of measurement tools accuracy also coursed errors. The ideas to avoid these errors, such as using motion capture, making a strait walkway for recording video, and taking video with x-ray cinematography. Furthermore, no speed-related parameters, such as actual velocity, Froude number, or duty factor, were measured in this study. Although all animals were observed walking at self-selected speeds, variation in walking velocity among species may have affected joint postures. Therefore, the lack of speed standardization represents a limitation when comparing θ_obs values across taxa. Future studies may benefit from quantifying stride parameters to better control for interspecific variation in locomotor dynamics.

Conclusion

In this study, we proposed a mechanical model of the ankle extensor apparatus of terrestrial mammals that approximates the ankle joint as a simple hinge and the force of the ankle extensors as acting parallel to the tibia. Our model predicts an estimated joint angle (θ_est) at which the moment arm of the ankle extensors is maximized. By comparing this predicted angle with observed ankle joint angles (θ_obs) during walking, we found that despite large variations in θ_obs among species, mechanical advantage of the calcaneal lever (sin a) remains within a relatively narrow range, supporting the notion that, across taxa, skeletal morphology at the ankle joint has been selected for optimization of mechanical advantage of the calcaneal lever for the ankle plantar flexors. This hypothesis is supported even in species with varying ankle joint postures and ranges of motion. This insight provides a new perspective for reconstructing locomotor postures in extinct terrestrial mammals and enhances understanding of the biomechanical constraints shaping limb function.

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