## Introduction

As the point of contact between bones of the skeleton, articular facets of interosseous joints (primarily those of the synovial type, but also certain fibrous joints such as the intercentral joints, pubic symphysis, and iliosacral joints) transmit forces throughout the skeletal system. Thus, these articulations are of particular interest in studies of locomotion (Jungers, 1984; Ruff, 1988; Godfrey et al. 1995; Hamrick, 1996; Parr et al. 2011) and mastication (Smith et al. 1983; Bouvier, 1986; Taylor, 2006; Terhune, 2013), activities that may demand the transmission of large forces between bones. To prevent overloading and to maintain relatively low stresses, facet areas should scale in proportion to the forces they transmit (i.e. large facets should experience higher forces than small ones). More specifically, Alexander (1980:96) suggested that if animals are geometrically similar, then ‘maximum joint stresses may have the same order of magnitude in mammals of all sizes’.

Assuming this proposition is correct, two primary models have been proposed to explain the variance in force magnitude that drives synovial joint construction. Alexander (1980) postulated a model in which muscle-induced forces dominated joint construction. The ultimate prediction of this model is that facet area will scale isometrically with muscle physiological cross-sectional area (PCSA), the variable that most directly correlates with potential muscle force (Bodine et al. 1982; Perry & Wall, 2008; Myatt et al. 2011). Unfortunately, data on PCSAs are extremely rare. However, assuming (i) an isometric relationship between PCSA and muscle mass and, in turn (ii), an isometric relationship between muscle mass and body mass, a more easily testable prediction avails: the areas of articular surfaces and the ability to resist joint forces should scale isometrically with body mass (i.e. in an area–volume relationship, the scaling coefficient should be equal to 0.66). Despite the required assumptions, these two variables (body mass and facet area) have been most frequently utilized to address this hypothesis by previous authors (Jungers, 1988; Ruff, 1988; Swartz, 1989; Godfrey et al. 1991; Jungers, 1991; Ruff & Runestad, 1992; Williams et al. 1992; Godfrey et al. 1995; Hamrick, 1996; Parr et al. 2011).

The major alternative to Alexander's hypothesis of muscle-induced forces dominating joint formation is the hypothesis by Biewener (1982; Biewener & Taylor, 1986) that mass-induced forces dictate joint construction. Given two objects of the same shape and density but different absolute sizes, the larger one will have a higher mass/surface area ratio due to the greater dimensionality of mass. Increasing the size of the object will result in increasing stress, as mass scales to area with an exponent of 1.5. Because bone has the same physical properties regardless of body size (Biewener, 1982; Currey, 1984), larger animals operate closer to the safety limit of their skeletal elements. To compensate for this reduced safety margin, facet areas in large animals may be greater than predicted by isometric scaling. Thus, if mass-induced forces dominate joint construction, articular surface areas should scale with positive allometry (i.e. scaling coefficient greater than 0.66). Due to the different predictions for the relationship between facet area and body mass, Alexander's (1980) hypothesis of muscle-dominated forces is an isometric or geometrically similar model, whereas Biewener's (1982) hypothesis of mass-induced forces is an allometric model.

Several previous studies have examined the relationship between facet areas and body mass with conflicting results (Jungers, 1988; Ruff, 1988; Swartz, 1989; Godfrey et al. 1991; Jungers, 1991; Ruff & Runestad, 1992; Williams et al. 1992; Godfrey et al. 1995; Hamrick, 1996; Parr et al. 2011). Swartz (1989) suggested positive allometric scaling for the facet areas in 11 of 12 limb joints among anthropoids, highlighting the role of mass-induced forces and requisite compensation for a reduced safety factors among larger animals. These results were contested by Godfrey et al. (1991), who noted that Swartz's sample contained two distinct groups at opposite ends of the body mass spectrum: small-bodied monkeys and large-bodied apes. Although a regression line generated for both groups indicates positive allometry, isometry is maintained if both groups are analyzed separately (as shown by Ruff & Runestad, 1992). The larger-bodied suspensory hominoids have the same scaling coefficient, but the higher intercept for the group (interpreted as indicating a greater mobility at the shoulder) inflates the slope for the total sample.

Jungers (1991) examined linear dimensions of articular surfaces in hominoid limb joints, finding that five of 16 joints displayed positive allometry, and although isometry could not be ruled out for the other joints, there was a positive allometric trend among them as well. Jungers suggested these results may require ‘qualifications to Alexander's model’ (Jungers 1991:397), but did not invoke Biewener's model as a possible explanation.

Recently, Parr et al. (2011) examined scaling relationships for talar facet areas among hominoids. They expected to confirm the generality of isometric scaling, and allowed for the possibility that convex (male-type) facets would scale allometrically depending on locomotor roles. However, their results show only the ectal (or posterior calcaneal facet), a concave (female-type) facet, scaled with positive allometry. Among the other facets, although confidence intervals do not rule out isometry, there is a trend toward positive allometry in the relationship between facet surface area and body mass (Parr et al. 2011).

It is important to reiterate that the relationship between body mass and facet area is only a proxy for what Alexander's model actually posits. Isometric scaling between body mass and articular surface area is not necessarily expected if isometry does not hold for the scaling relationships of body mass, muscle mass, and muscle PCSA. For instance, if muscle mass increases with positive allometry relative to body mass, then we might expect the capacity for muscle force also to scale allometrically. Under this scenario, positive allometry between facet area and body mass would not refute Alexander's hypothesis because facet area might still be scaling isometrically with respect to muscle force. Alexander et al. (1981) addressed this particular assumption by showing an isometric relationship between body mass and muscle mass of distal leg muscles in a wide sample of mammals, including primates. More recently, Muchlinski et al. (2012) reported that primate muscle mass exhibits slight (but significant) positive allometry with body mass, bringing into question the validity of testing joint construction hypotheses by looking at scaling between total body mass and facet area. However, neither study utilized phylogenetic regression methods in their analyses, introducing the possibility of data distortion similar to Swartz (1989).

Nonetheless, the relevance of allometry between muscle to body mass depends on the scaling relationship between body mass and PCSA: the most pertinent variable for determining the capacity of a muscle to generate force. PCSA is a function of the mass of the muscle belly, the muscle's density and the length of muscle fascicles, and can be calculated with the equation:

PCSA = [*cos* (pennation angle) × muscle mass]/(muscle density × muscle fascicle length)

The maximum force a muscle can produce reflects the maximum number of sarcomeres contained within the muscle cross-sectional area (Bodine et al. 1982; Perry & Wall, 2008; Myatt et al. 2011); both a larger absolute area and an increased pennation angle can accommodate a greater number of sarcomeres. Therefore, a larger PCSA indicates a muscle's ability to generate larger absolute force.

If muscle-induced forces dominate joint construction, then PCSA should scale isometrically with articular facet area (area–area relationship, scaling coefficient = 1). Biewener's model for mass-induced forces influencing joint construction makes no predictions regarding the relationship between PCSA and facet area; neither positive nor negative allometry will refute the model. Comparing a broad range of mammalian species, both Alexander et al. (1979) and Pollock & Shadwick (1994) found positive allometric scaling in muscle cross-sectional area relative to body mass. However, analyzing a hominoid-only sample, Myatt et al. (2011) reported that scaling patterns of PCSA in most hindlimb muscles conformed to isometry. Once again, phylogenetically and functionally diverse groups have generated positively allometric slopes, whereas more restricted samples conform to isometry (Godfrey et al. 1991).

Here, we examine the scaling relationships between facets of the talus and body mass for euarchontans, the clade composed of primates, treeshrews, and colugos, as well as several smaller taxonomic and functional groups. The talus is well suited for this purpose, as it contains both convex (male-type) and concave (female-type) facets. Additionally, as no muscles attach directly to the bone, there is little chance that joint expansion has occurred to accommodate increased ligament attachment, as argued by Currey (1984). Recognizing that scaling between facet area and body mass is an indirect assessment of the competing hypotheses being evaluated, we also examine scaling relationships between muscle mass and body mass using phylogenetic generalized least squares regression, and determine the relationship between PCSA of the extrinsic muscles of the foot and talar facet areas.

We designed our approach to address key issues in assessing the alternative hypotheses of Alexander and Biewener on joint construction. Key elements of our approach were (i) to consider articular surfaces of distal limb elements; (ii) to measure actual facet areas rather than interpolating from linear measurements; (iii) to utilize phylogenetic generalized least squares regression to account for clade-level differences; and (iv) directly to test the relationship between muscle PCSA and facet areas. Previous studies have focused on proximal limb elements (Swartz, 1989; Godfrey et al. 1991; Jungers, 1991; Ruff & Runestad, 1992; Godfrey et al. 1995), particularly the expansion of the femoral head and its relation to increased bipedality (Jungers, 1988; Ruff, 1988). There are fewer analyses of more distal limb bones (Hamrick, 1996; Parr et al. 2011). As more distal elements have higher load-bearing requirements [i.e. they must support the axial mass and the mass of the limb of which they are a component (Demes & Gunther, 1989)], it is important to evaluate articular surfaces at interosseous joints throughout the body.

Additionally, as noted by Godfrey et al. (1991), most studies of articular surfaces have been limited to linear measurements to approximate true areas (Jungers, 1988, 1991; Ruff, 1988). This source of imprecision and random error can be somewhat alleviated by molding articular facets and flattening the three-dimensional mold to two dimensions (Swartz, 1989; Godfrey et al. 1995) but this method is difficult, time-consuming, and not ground-truthed. However, digital surface files, generated from surface laser scans (Parr et al. 2011) or microCT scans (this study), are highly amenable to surface area manipulations and measurements. These advances allow us to quantify accurately more complex shapes than previously afforded.

To address concerns highlighted by Godfrey et al. (1991, 1995), we employ phylogenetic generalized least squares regression (PGLS) for taxonomically diverse samples and reduced major axis regression (RMA) when phylogeny no longer provides meaningful information (i.e. λ is not significantly different from zero).

We recognize two hypotheses for joint construction: a strict isometric model of muscle-induced force (Alexander, 1980; Godfrey et al. 1991) and a more relaxed allometric model of mass-induced force (Biewener, 1983). Expectations for these models are presented graphically in Fig. 1. Under the strict model of geometric similarity, if muscle-induced forces dominate joint construction, then articular facet areas should scale isometrically with body mass (scaling coefficient = 0.66). Under the relaxed model, if mass-induced forces influence joint construction, then articular facet areas should scale with positive allometry relative to body mass (scaling coefficient > 0.66).

In this study, we accept Alexander's (1980) suggestion that facets must maintain similar stresses in animals of different size. Given this, if Alexander's (1980) hypothesis that muscle-induced forces dominate joint construction is correct, the following relationships can be expected to hold true when phylogenetic affinity and/or functional specializations are accounted for:

- body mass and muscle mass scale with isometry (scaling coefficient = 1),
- body mass and PCSA scale with isometry (scaling coefficient = 0.66),
- body mass and facet areas scale with isometry (scaling coefficient = 0.66),
- facet areas and PCSA scale with isometry (scaling coefficient = 1).

As listed, these relationships are progressively better (more direct) tests for evaluating the two models of joint construction we have presented here, culminating in Condition 4, which directly tests Alexander's postulate that muscle-induced forces dominate joint construction. Therefore, if taxonomically equivalent samples of all variables were available with equal sample sizes, it would be pointless to look at any relationship except that between PCSA and facet area. However, the rarity of PCSA data among extant taxa and its non-existence for fossils makes proxy methods critical for gaining knowledge. Furthermore, data on scaling of body mass and facet area can be applied to other important evolutionary questions as detailed in the discussion.