Feeding in billfishes: inferring the role of the rostrum from a biomechanical standpoint

The billfishes (marlins, spearfishes, sailfishes and swordfish) are large and fast pelagic predators characterized by the extreme elongation of their upper jaw bones to form an elongated rostrum, or bill (Nakamura, 1985; Davie, 1990; Fierstine, 1990; Fierstine and Voigt, 1996). Surprisingly, the function of this structure is still controversial (Nakamura, 1983; Frazier et al., 1994). The rostrum has been hypothesized to improve hydrodynamics by reducing drag (Wisner, 1958; Ovchinnikov, 1970; Aleyev, 1977; but see Sagong et al., 2013), to be used for defense (as evidenced by rostral fragments found embedded in large predators; Fierstine, 1997; Fierstine et al., 1997) and to be used to strike, immobilize or dismember prey before ingestion, thereby facilitating feeding (Gudger, 1940; Talbot and Penrith, 1964; Scott and Tibbo, 1968). Although it is certainly possible that the elongated rostrum may have been selected for multiple roles, substantial evidence from stomach contents and recent field observations strongly support the latter feeding-related hypothesis (Scott and Tibbo, 1968; Stillwell and Kohler, 1985; Frazier et al., 1994; Shimose et al., 2007; Domenici et al., 2014).

The geometries and relative sizes of the rostra of different billfish species are in some cases strikingly different, suggesting that the use of the rostrum in feeding is species specific. The rostrum of swordfish (the only species within the Xiphiidae), which can exceed 50% of body length, is wide, distally tapered and dorsoventrally flattened with a lenticular cross-section, having sharp edges and lacking teeth (Gregory and Conrad, 1937; Gudger, 1940; Poplin et al., 1976; Nakamura, 1983; Fierstine, 1990; Fierstine and Voigt, 1996; Nohara et al., 2003). Conversely, in istiophorids (marlins, spearfishes and sailfishes), the rostrum is more circular in cross-section, has no sharp edges, and is surrounded by small villiform teeth. The length of the rostrum varies among species but is proportionally shorter than in swordfishes, being approximately 24% of body length in blue marlin (Gudger, 1940; Nakamura, 1983). These structural differences between swordfish and marlin are indeed reflected in anecdotal accounts of feeding events, where swordfish are suggested to slash their prey with only lateral movements of the rostrum (Scott and Tibbo, 1968; McGowan, 1988) and istiophorids stun and/or spear their prey, thereby using a wider range of rostral movements (Talbot and Penrith, 1964; Fierstine, 1997; Fierstine et al., 1997; Shimose et al., 2007). Correlations of these differences in rostral form and behavior between species, and their mechanical implications at the tissue level have, however, been nearly impossible to study because of the logistical issues associated with the study of large, fast, pelagic predators such as billfishes (but see Domenici et al., 2014).

Biomechanical approaches offer feasible ways to investigate the potentially differing roles of the rostrum in the various billfish species, enabling quantification of mechanical performance and facilitating the inference of behavior from morphology. The utility of these approaches has been demonstrated for experimentally intractable taxa, via investigations of feeding mechanics in white sharks, bull sharks and rorqual whales (Wroe et al., 2008; Field et al., 2011; Habegger et al., 2012). Significant advances in biomechanical studies have been accomplished through the utilization of beam theory, an engineering approach that describes the mechanical behavior of a beam under loading. This approach is applicable because many biological structures can be treated as beams (e.g. trees, echinoderms, vertebral columns; Koehl, 1977; Baumiller, 1993; Ennos, 1993; Huber et al., 2013), and beam theory also facilitates better comprehension of form and function relationships by teasing apart factors that contribute to a biological beam's performance under loading. A beam's resistance to bending is given by the flexural stiffness EI, which is a function of both its material properties (via Young's modulus, E) and its geometry (via second moment of area, I) (Koehl, 1976, 1977; Wainwright et al., 1976; Biewener and Full, 1992). Analyses of a structure's cross-sectional geometry and the response of its material to load (e.g. the force per unit area, or stress, σ, that builds in a material under load in response to deformation, or strain, ε) can therefore provide an understanding of the overall structure's mechanical capabilities and limits, with stress distributions and yield behavior pointing to performance boundaries, all of which can offer clues to loading regimes likely to be experienced in vivo. Because EI accounts for both material and structural properties, it is a useful metric for characterizing mechanical function in comparative studies, particularly when linking organismal function with evolutionary and ecological pressures (Koehl, 1976, 1977; Macleod, 1980; Etnier, 2003). Flexural stiffness has been shown to correlate with loading regime and direction for a range of biological body support systems, from the limb skeletons of batoids and dogs, to the jaws of whales and pelicans, to the exoskeletons of crabs (Kemp et al., 2005; Taylor et al., 2007; Field et al., 2011; Macesic and Summers, 2012); these taxonomic comparisons also illustrate that higher levels of bending resistance can be attained evolutionarily by increases in either E or I, or in both.

The goal of this study is to use biomechanical models and material testing to infer the possible biological role(s) of the rostrum in two billfish species with differing rostrum morphologies, the blue marlin (Makaira nigricans Lacepede 1802) and the swordfish (Xiphias gladius Linnaeus 1758). Utilizing beam theory as our primary model, we estimate several biomechanical variables and describe the material properties and geometric attributes of the rostrum along its length in order to acquire a better understanding of its function and mechanical capabilities. Inferences of the putative feeding behaviors of these two apex predators are then made based on these analyses, paying special attention to suggestions of how mechanical and growth demands may be balanced for these biological cantilevers. Since failure or large deflections could be detrimental during feeding we hypothesize that stress will be equally distributed along the length of the rostrum, preventing any localized region of higher stress (‘weak spots’) that could lead to failure during feeding. Correspondingly, we expect flexural stiffness to be uniform along the length of the rostrum and sufficiently large to avoid appreciable deflections while striking prey. Based on previous descriptions of cross-sectional geometry, we expect that the swordfish rostrum will be better suited for lateral striking of prey, whereas that of the blue marlin will be suited for a larger range of motions.

The rostra of both species are composed of similar tissue types that appear to be arranged in similar patterns along the length of the rostra (Figs 1 and 2). Three main tissues were observed: acellular bone (a bone lacking osteocytes, Cohen et al., 2012), adipose tissue, and hyaline cartilage, with proportions of the tissues changing in distinct ways from tip to base. The primary component of distal cross-sections (Fig. 1A–D and Fig. 2A–D) was acellular bone. Moving proximally towards the base of the rostrum, adipose tissue and hyaline cartilage become more dominant, particularly in area 4, where they comprise most of the center of the cross-section (Fig. 1E,F and Fig. 2E,F) and acellular bone is relegated to the periphery. Although the majority of the adipose tissue was found in area 4, this tissue was also observed intermittently in smaller quantities within canals present throughout the acellular bone of the rostrum.

Swordfish and blue marlin exhibited diagnostic differences for most of the variables in our CT-scan-based analysis of cross-sectional geometry, indicating that the observed species-specific morphologies have mechanical consequences. Second moment of area (I, a descriptor of the cross-sectional distribution of material in the direction of loading and a proxy for bending resistance) increased proximally in both species and in both principle loading directions, particularly at ∼75–80% bill length (BL), where the rostrum begins to merge with the head region. However, the ratio of I for loading in the dorsoventral and lateral directions (I_maj/I_min) differed considerably between species, as a function of the dissimilar cross-sectional shapes of the rostra. Swordfish rostra are extremely asymmetric in morphology and in the contribution of rostrum shape to bending resistance, with I in lateral bending ∼10–16 times greater than that for dorsoventral bending. This ratio is greatest distally and decreases proximally as the rostrum deepens closer to the head (Fig. 3A). In blue marlin, I was only slightly greater for loading in the lateral direction, and the ratio of I_maj/I_min was fairly constant along the length of the rostrum (∼2–3), indicating a more uniform distribution of material and isotropic bending resistance (Fig. 3B; Table 1).

In proximal regions of the rostrum, bone forms only a thin peripheral layer but is arranged far from the neutral axis, resulting in a high I for a comparatively small amount of bone tissue. This is reflected in the abrupt proximal (∼75–80% BL) decrease in the proportion of mineralized tissue in each cross-section (Fig. 3C,D), as well as in our histological data, which show a proximal decrease in bone and increase in adipose tissue and cartilage (Figs 1,2). Whereas in blue marlin the proportion of mineralized tissue in cross-sections is approximately 100% in distal sections (<80% BL), representing a solid bony rostrum, in swordfish the proportion of mineralized tissue shows jagged oscillation distally, reflecting the periodic bony chambers (Fierstine, 2006) within the distal rostrum (Fig. 3C). Both species show a proximal-to-distal increase in the mean relative bone mineral density of the rostra, whereas the pattern of decrease in swordfish is more gradual, and that of blue marlin is consistent until approximately 70% BL and then decreases more sharply (Fig. 3C,D).

Stresses along the rostrum were calculated as a function of rostral geometry and load during bending experiments where rostra tips were displaced ±1 cm. Although the magnitude of stresses in the rostra of the two species were considerably different, with blue marlin stresses being two to four times higher, both species showed an approximately similar distal-to-proximal trend of slight increase in stress from area 1 to 2 and then a gradual decrease to area 4. During lateral loading, the average maximum stress distribution did not vary much along the swordfish rostrum, ranging from 1.4 to 1.6 N mm⁻² (areas 1–4, respectively). During dorsoventral loading, the stress distribution was higher along the rostrum ranging from 2.1 to 1.3 N mm⁻² (areas 1-4, respectively, Fig. 4A, supplementary material Table S1). For the blue marlin, the average maximum stress distribution during lateral loading ranged from 7.8 to 3.9 N mm⁻² (areas 1–4, respectively) with stress in the three most distal areas being considerably greater than that in the most proximal section (area 4). Similarly, during dorsoventral loading, the stress distribution in the three distal areas ranged from 8.7 to 9.2 N mm⁻², and area 4 decreased to 4.9 N mm⁻² (Fig. 4B, supplementary material Table S1).

In the swordfish, Young's modulus, E, decreased proximally from 12.2 to 5.3 GPa when calculated in lateral loading (areas 1–4, respectively) and from 12.9 to 3.1 GPa when calculated in dorsoventral loading (Fig. 4C, supplementary material Table S1). Although similar trends in E were observed for the blue marlin, values tended to be larger, ranging from 14.5 to 3.5 GPa in lateral loading and from 16.4 to 5.7 GPa in dorsoventral loading (Fig. 4D, supplementary material Table S1).

Flexural stiffness, which accounts for both material and geometric contributions to bending resistance, increased proximally in the swordfish during lateral (3.7×10⁵ to 9.6×10⁵ GPa mm⁴) and dorsoventral loading (2.9×10⁴ to 1.2×10⁵ GPa mm⁴) (Fig. 4E). Flexural stiffness also increased proximally in the blue marlin, although values between the two loading planes were more similar than those of the swordfish, especially for the distal half of the rostrum. During lateral loading, EI ranged from 4.2×10⁵ to 2.0×10⁶ GPa mm⁴ and dorsoventrally from 2.4×10⁵ to 1.3×10⁶ GPa mm⁴ (Fig. 4F). (For the full range of values, see supplementary material Table S1.)

It took considerably larger loads to displace the blue marlin rostrum the same fixed distance (±1 cm) as that of the swordfish (maximum loads applied to the blue marlin rostrum were 173.0 N and 123.9 N compared with the swordfish 19.3 N and 7.1 N, lateral and dorsoventral loading, respectively). Therefore, rather than comparing stresses resulting from the maximum loads applied to rostra of both species, we determined the largest moment value during the swordfish experiments, and then for each strain gauge, determined the corresponding stresses resulting from a similar moment value in both blue marlin specimens. This ‘standardized’ stress allowed us to illustrate the difference in the tissue stress responses to the same moment, as opposed to the responses to the same maximum displacement. The use of the swordfish maximum moment as a reference value for both species was preferred over load standardizations (standardizing the values to similar loads for each species) to account for small differences in the position of the strain gauges between species because bending moment calculations take into account both load and distance from load. Results from standardized moments showed similar stress values for both species during lateral loading with the exception of area 4 for blue marlin, which was considerably lower. However, during dorsoventral loading, swordfish rostra showed almost three times greater stresses (Table 2, Fig. 5A,B).

For a hypothetical point loading scenario, swordfish had higher stress values along the rostrum in both loading planes compared with blue marlin (Fig. 6A). Stresses calculated for the swordfish rostrum during dorsoventral loading were considerably larger than those for lateral loading, diminishing towards the base (Fig. 6A). Stress values for blue marlin were lower and very similar for both loading planes. Simulations of hydrodynamic loading during head swinging indicated that swordfish experience the highest maximum stress and drag during dorsoventral loading and lowest maximum stress and drag during lateral loading, with blue marlin values being intermediate for both variables in both loading directions (Fig. 6B,C).

Our data suggest that the morphological differences observed between swordfish and blue marlin rostra have performance implications, and that these morphological and mechanical differences may be consistent with different feeding behaviors. Considering the possible role of the rostrum during feeding, the blue marlin rostrum is built like a short, solid javelin with cross-sectional anatomy suggesting multi-axial use; in contrast, the swordfish rostrum appears suited for use as a light-weight, low drag, uniaxial weapon.

Despite the cross-sectional shape differences between the two species, our data show several material and structural commonalities between blue marlin and swordfish rostra, providing strong evidence that common evolutionary pressures (e.g. growth, use in feeding) may have shaped rostral anatomy in both species. In general, both rostra show opposing trends in I and E, with the former increasing rapidly proximally, but the latter increasing distally (Fig. 3, Fig. 4C,D). The decrease in E proximally means that the increase in EI must solely result from disproportionate changes in I. The distribution of grayscale intensity in rostrum CT scans suggests that bone mineral density increases distally in the rostrum in both species (Fig. 3C,D), perhaps explaining the observed stiffness gradient, because E and mineral density are positively associated in a variety of mineralized tissues (Currey, 2004; Horton and Summers, 2009; Magwene and Socha, 2013).

Stress values in the tissue resulting from the bending experiments were shown to be approximately similar among the distal regions (areas 1–3) for both species (Fig. 4A,B). A strategy to keep stress constant along the length of a cantilever beam is to increase the cross-sectional area towards the fixed end, compensating for the increase in bending moment (Currey, 1984). Billfish rostral cross-sectional area and I increase proximally (Figs 1,Fig. 2,–3), probably explaining the consistent stresses in the distal regions. Stress values tended to be comparatively low in area 4, achievable because of the comparatively large I values at the base (>60% bill length). This increase in I is a function of the change in geometry and the large cross-sectional areas in proximal regions, but also the peripheral arrangement of bone in this area. This arrangement allows for a high I without necessitating increased weight, because the core of the base of the rostrum is filled with low density materials (adipose tissue and cartilage) that are lighter than a comparable rostrum of solid bone (Figs 1 and 2, area 4). Biological structures that have a similar conformation (solid exterior and a softer, light-weight interior) such as the toucan beak, porcupine quills and feather shafts (Bonser, 2001; Seki, et al., 2005; Meyers et al., 2013) have been suggested to not only increase I in a structure but also to decrease the probability of local buckling by dissipating fracture energy and preventing the collapse of the thin cortical walls, while also minimizing weight (Ma et al., 2008; Meyers et al., 2013). Furthermore, Gudger (1940) proposed this filling in billfishes may also act as shock absorber, dampening vibrations during striking of prey.

The ecological implications of the distal increase in material stiffness may be associated with feeding, because this change minimizes deflection of the distal portion of the rostrum and provides a hard surface for striking prey. Billfish rostra are subject to periodic loadings over long periods of time, which may result in fatigue damage and favor the formation of microcracks in this stiffer area, as in bone and nacre (Currey, 1984). Similar gradients of stiffness have been reported in other biological structures including bird feathers (Macleod, 1980; Bonser and Purslow, 1995), squid beaks (Miserez et al., 2008) and spider fangs (Politi et al., 2012); the latter two systems also acting as predatory weapons, similar to billfish rostra. In addition, the higher incidence of remodeled bone morphology in the distal region of the rostrum suggest that the tissue is under regular loading and therefore more prone to damage and consequently, remodeling (Atkins et al., 2014). Distal stiffening of rostral material could also counteract structural tapering relating to hydrodynamic demands. Previous work has shown that the drag experienced during lateral swiping of the elongate rostrum in alligators is proportional to the height of the jaws (McHenry et al., 2006). Minimizing element height (via distal tapering) may reduce drag in billfish rostra; however, it also comes at the cost of potentially reducing cross-sectional area and resistance to bending (McHenry et al., 2006); distal stiffening of the rostrum may ultimately be a strategy for maintaining adequate flexural stiffness in areas where I is selected to be comparatively low for hydrodynamic reasons. Additionally, since the bow wave generated by a predator can be detected by prey (Visser, 2001), the relatively smooth and elongated rostrum of billfishes may reduce water disturbance, thereby delaying detection of approaching billfish by prey, as suggested for other vertebrates with long jaws including water snakes and alligators (Alfaro, 2002; McHenry et al., 2006).

Generalized patterns of stress, material stiffness, second moment of area, and flexural stiffness in rostra of both billfish seem to follow a common pattern that may aid in overcoming conflicting demands imposed by the rostrum's function and shape. Evidence from the histology and mineral density gradient of the rostrum found in this study, in addition to that in an ongoing investigation of billfish tissue morphology (Atkins et al., 2014), indicate that distal regions of the billfish rostrum are less porous, more mineral dense and more remodeled. This suggests that distal areas may contain older tissue than basal regions and therefore that the rostrum in billfishes may grow from its base by ‘pushing’ older material distally (although rostrum development is not fully understood and other factors such as mechanical demands may play a role in its development). We have shown that basal regions of the rostrum (i.e. suggested growth zones) are prone to less stress and, because of the proximal increase in I, EI was also increased. The lower values of stress at the base of rostrum also coincide with a region of multiple sutures, where multiple cranial bones converge and where the rostrum connects to the rest of the head (Fig. 7). Sutures have been found to influence patterns of stress and strain by resisting or allowing deformations depending on load orientation and/or magnitude (Rafferty et al., 2003; Markey et al., 2006; Krauss et al., 2009) but can also act as bone deposition sites in mammals (Opperman, 2000; Sun et al., 2004) and so may provide free surfaces for growth and bone deposition in billfish rostra. All of these factors may decrease the likelihood of failure at this region, preventing the compromise of vital organs such as the nares, and also preventing damage to a region of higher vascularization and growth. Coincidentally, rostra in several different billfish species have been more often documented to show breaks in their distal halves (Frazier et al., 1994; Fierstine and Crimmen, 1996; Fierstine et al., 1997).

Although broad structural similarities clearly exist between rostra of these species, their morphological differences revealed mechanical disparities that suggest different feeding behaviors. While I was approximately equal for both species during lateral loading (Fig. 3A,B), the blue marlin rostrum showed higher I during dorsoventral loading and greater equivalence of bending resistance between lateral and dorsoventral loading (I_maj/I_min∼1.0) as a result of its almost circular cross-section, compared with the swordfish (I_maj/I_min>8.0) (Fig. 3A,B). Additionally, blue marlin rostra showed E values higher than those for swordfish for most of the regions along the rostrum (Fig. 4C,D) and E varied with respect to loading regime in swordfish, suggesting material heterogeneity in this species. Overall, stiffness values calculated from distal strain gauge data (area 1, 12.9 GPa swordfish and 16.4 GPa blue marlin) agree with those determined in a different study for five billfish species (including those studied here) using material testing techniques (E ranged between 12 and 20 GPa; Atkins et al., 2014) but also exceed the values found in other acellular boned fishes, such as great sculpin, Myoxocephalus polyacanthocephalus (6.48 GPa; Horton and Summers, 2009) and tilapia Oreochromis aureus (7.20 GPa; Cohen et al., 2012) as well as those with cellular bone found in the common carp Cyprinus carpio (8.51 GPa; Cohen et al., 2012). Moreover, values of stiffness in billfish are not only shown to be higher than most values reported for other fish, but also to be closer to values reported for mammals (Atkins et al., 2014).

Mechanical demands associated with feeding may not be the only factors dictating shape differences between species. Drag forces, for example, are influenced by a structure's shape, surface characteristics, speed of movement and the surrounding fluid's density and viscosity (Vogel, 2003), and therefore drag forces acting on the rostra can conceivably influence feeding. Our results from modeled biological loading scenarios indicate that drag force and stress during dorsoventral loading are greater in swordfish than blue marlin. Swordfish rostra, however, would exhibit the lowest drag force during lateral loading (Fig. 6C) because of the depressed lenticular form of the rostrum. Consequently, swordfish may be limited in the extent to which they can use dorsoventral striking, which incurs 40-times higher drag forces than lateral striking (Fig. 6C). Interestingly, this support for differences in rostrum motion during striking are also suggested by differences in vertebral morphology between the two species. The vertebral column of the blue marlin is reinforced with zygapophyses, which provide reinforcement and stability but decrease flexibility (Hebrank et al., 1990). In contrast, the lack of zygapophyses in swordfish may allow extensive lateral flexibility of their body, because Hebrank et al. (1990) found that the flexibility of the blue marlin vertebral column increased when the zygapophyses were removed. The potentially greater lateral flexibility of the swordfish body may be related to the dorsoventral depression of the rostrum and its reduced hydrodynamic drag in the horizontal plane, facilitating low-resistance, lateral swiping at its prey.

Patterns of σ, E, I, EI and theoretical drag calculations suggest that blue marlin may have a rostrum better suited to strike its prey in a wider range of motions, similar to the multi-plane striking behavior observed during feeding in sailfish (Domenici et al., 2014), a species with a rostrum similar in shape to that of the blue marlin. Rod-like structures are suited to withstand loading in multiple directions, whereas elliptical cross-sectional beams, such as the swordfish rostrum, are better designed to withstand loading in a single plane (Etnier and Vogel, 2000; Currey, 2002). Injuries to prey provide insight into billfish feeding behavior (Scott and Tibbo, 1968; Nakamura, 1985; Stillwell and Kohler, 1985). Prey fishes in the stomach contents of blue marlin showing lacerations to the dorsal, ventral and caudal body regions (Shimose et al., 2007) and images of sailfish prey bearing injury on differing regions of the body (Fig. 8A) support our theory of multi-directional striking in istiophorids. In contrast, gut contents of swordfish showed prey being severed, which may be due to the lateral swiping of the sharp, lenticular rostrum (Goode, 1883; Stillwell and Kohler, 1985) (Fig. 8B).

In conclusion, blue marlin rostra are stiffer and better suited to strike prey in multiple planes, whereas swordfish rostra seem to be highly specialized for lateral striking as a result of the arrangement of skeletal material in a way that maximizes lateral bending resistance while decreasing drag. However, despite the differences found between these species, there are still many variables converging into similar patterns that may explain how to build an efficient biological beam that can act as a weapon in multiple ways. Whether this structure evolved as an adaptation for feeding remains uncertain, but its design, tissue architecture, and material properties favor its role in prey capture.

Four swordfish (Xiphias gladius) and two blue marlin heads (Makaira nigricans) (Table 3) with intact rostra were obtained from fishing tournaments in the Gulf of Mexico, USA. Heads were transported to the University of South Florida and frozen with the rostrum wrapped in plastic to reduce dehydration. After being thawed, all heads were CT scanned with a 64-slice Aquilon Toshiba scanner (Toshiba American Medical Systems Inc., Tustin, CA, USA) with slice thickness ranging from 0.75 to 1.0 mm (image size, 512×512 pixels; pixel size, 0.625 mm). After scans were completed, samples were wrapped in plastic bags and kept frozen until material testing. Whole-skull scans were rendered for figure purposes using Mimics software (Materialise HQ, Belgium). The use of all tissue samples was approved under the University of South Florida Institutional Animal Care and Use Committee, IACUC protocol number T 3884.

In order to investigate the tissues comprising the rostra, transverse sections were prepared for both swordfish (n=1) and blue marlin (n=1) at four regions along each rostrum, mirroring the locations of strain gauge placement (see below). Four transverse sections (areas 1–4, distal to proximal) of approximately 0.5 cm thickness were cut from one representative rostrum of each species. Transverse sections were preserved in 10% buffered formalin followed by decalcification in formic acid (50% HCOOH, 50% H₂O) sodium citrate (500 g NaH₂C₆H₅O₇, 2500 ml H₂O). Sections of 4 µm were made with a Bioacut microtome (Leica/Reichert Jung model 2030, Wetzlar, Germany) and stained using hematoxylin and eosin. Samples were observed under compound microscope (Leica CTR 6500 Wetzlar, Germany). Pictures were taken at two magnifications, ×50 and ×100, with a digital camera (Leica DFC 420c Wetzlar, Germany). Although this tissue had been subjected to previous freeze–thaw cycles, which may compromise bone tissue and cellular ultrastructure (e.g. nuclei and osteocytes; Andrade et al., 2008), these changes are irrelevant for the scale of our observations and osteocytes are lacking from the bone of these species.

In order to understand geometric and structural contributions to the mechanical performance of billfish rostra, we applied beam theory analyses to transverse sections of the rostra of four swordfish and two blue marlin (Table 3) using a custom MATLAB script described below. Our analyses were restricted to the contribution of hard tissue (bone) because CT scanning captures only bone structure accurately and bone occupies the majority of the rostra.

Rostral bone was isolated within each dataset using the Segmentation Editor in Amira (Mercury Computer Systems, Berlin, Germany) by first digitally isolating the rostrum (by cropping out all tissue caudal to the nares). The rostrum includes a range of tissue types (e.g. bone, cartilage, adipose tissue); to extract the bony tissue, we selected the largest peak from the scan's histogram distribution of grayscale values, and then made small-scale adjustments to the selection range while noting the effect on the morphological accuracy of the resultant volume (i.e. whether the volume represented more or less bone than was known from dissection to be there). This ‘bone selection’ method resulted in more morphologically accurate volumes than an alternative method, where the grayscale thresholding range was set at the start of the MATLAB script through an iterative series of threshold steps on individual original (full grayscale range) CT slices. However, when thresholding methods were compared for the same CT datasets, analyses of cross-sectional geometry in MATLAB (see below) produced nearly identical trends, indicating the technique is robust, regardless of thresholding method.

Each resultant thresholded (i.e. ‘bone-only’) volume was sectioned orthogonal to the longitudinal axis of the rostrum and an image stack of cross-sections exported to the MATLAB script. The script then performed a slice-wise analysis of cross-sectional geometry, beginning by normalizing rostrocaudal positions to percentage lengths to facilitate comparison across scans (i.e. the slice containing the rostral tip represented 0% length, the slice containing the caudal end of the rostrum represented 100% length). Slices were then selected at intervals of 2% element length; this selection interval was determined via sensitivity analysis to provide a balance of resolution of data trends and script efficiency.

For each of the 51 slices per rostrum, the script converted the grayscale ‘bone-only’ images to binary images (white, bone; black, soft tissue and voids) and analyzed the distribution of white pixels around the cross-section centroid in order to determine a series of variables describing rostrum cross-section geometry: second moment of area (I); cross-sectional area (A); and filled-CSA (A_f; the area of the element cross-section with any holes or internal black pixels converted to white pixels) (Fig. 9).

Second moment of area describes the distribution of material around a neutral axis (NA) in a cross-section and perpendicular to the line of applied force, and is therefore indicative of resistance to bending. Billfish rostra contain multiple tissues with different material properties, which can be accounted for by scaling the contribution of each tissue according to its Young's modulus (Gere, 2002). However, as the non-bony tissues are located closer to the NA and their moduli are as of yet unknown (although likely orders of magnitude less), we simplify the analysis by focusing only on the bone tissue and assuming it has a consistent modulus down the length of the rostrum and between species.

where n is the number of individual pixels; I_pix is the second moment of area of each pixel relative to its own centroid (a⁴)/12; A_pix is the pixel area, a², where a=0.625 mm; y is the perpendicular distance from the NA (NA was determined as the axis perpendicular to the applied force passing through the centroid of each cross section). Apart from the term (I_pix) in the summation, this equation is identical to the standard integral; the I_pix term, however, is important when there are few pixels in an examined cross-section and/or y is small compared with the pixel size (i.e. at the extreme proximal or distal ends of the rostra), because the area of each pixel is proportionately more important to the cross-section second moment of area. For large and/or thick-walled cross-sections, when y is large compared with the pixel size, the corrective term I_pix would be negligible and our equation and the standard integral would effectively yield the same results.

However, as the second moment of area of asymmetrical sections can vary with the chosen bending axis – evidenced by the example of an architectural I-beam with a higher second moment of area when bent in line with its longer cross-sectional axis – we calculated second moment of area for bending in line with the major axis (I_maj) and the bending in line with the minor axis (I_min) of each cross-section. The major and minor axes are defined as the longest and shortest diameters of the cross-section; in both species, these structural axes correspond to the lateromedial and dorsoventral axes of the rostra, respectively (see cross-sections in Fig. 9). The ratio of the two moments (I_maj/I_min) is a mechanical aspect ratio, describing the anisotropy of bending resistance for the cross-section. Whereas an ellipse would exhibit a higher ratio, indicating a greater resistance to bending along one axis, a circle's moment ratio would be 1.0, indicating isotropy in bending resistance.

In addition, the relative mineral content per cross-section was estimated by calculating the average grayscale value of mineralized tissue. As the scans were not calibrated to materials of known mineral density (e.g. using imaging phantoms of varying hydroxyapatite composition), these values are only meaningful as relative measurements and so are presented as percentages of each scan's maximum value, from 0 to 100%.

Billfish heads were thawed and prepared for material testing by removing all soft tissues from the skull. Once cleaned, each skull (neurocranium) was potted in fast-curing cement (Rockite, Hartline Products, Cleveland, OH, USA), firmly encasing the head while leaving the entire rostrum exposed, simulating a cantilever beam. The embedded skull was then anchored to the floor via a holding device (Fig. 10) and the tip of the rostrum fixed to the crosshead of a materials testing device by a custom-made wooden clamp that enveloped the rostrum cross section allowing vertical motion and load dispersal along the cross section. The attachment point of the MTS was located approximately 15% of the rostrum length from the tip, representing a loading point (LP) that might be expected during prey contact. To determine the reaction of tissue along the length of the rostrum to distal loads, four transverse regions were demarcated (areas 1–4, distal to proximal), starting at 30% rostrum length and spaced approximately 20% apart. The criteria for the selection of these points were based on the relative positions that represented the best locations to survey strain along the majority of the rostrum accounting for differences in length in both species. Each area was fitted with six strain gauges oriented parallel to the longitudinal axes of the rostra (C2A-13-125LW-120, Vishay Measurements Group, Inc., Raleigh, NC, USA): four on the dorsal surface (two lateral and two medial) and two on the medial ventral surface (Fig. 10). For proper strain gauge attachment, each region was smoothed with 400-grade sandpaper, and excess oil removed with acetone. Strain gauges were glued to the underlying bone with Mbond glue (Vishay Measurements Group, Inc., Raleigh, NC, USA). Because some of the skeletal elements at the base of the most proximal portion of the rostrum in blue marlin (area 4) are supported medially by soft tissue, and removing them could greatly affect the structural integrity of the rostrum, strain gauges could not be attached here, decreasing the total number of strain gauges for this species to 22.

Because the location of the NA depends on load orientation, different strain gauges were selected for the two different loading regimes utilized (lateral and dorsoventral). For example, only strain gauges located on the lateral edges of the rostrum were analyzed during lateral loading (Fig. 10). Because of the lack of knowledge of rostral bending during normal feeding behaviors in billfish and the limitations of our MTS system, displacements were fixed to ±1 cm for all bending tests. Each rostrum was loaded under displacement control at the same loading point with a 50 lb (∼23 kg) load cell (JP 50, Honeywell, Golden Valley, MN, USA) at a frequency of 2 Hz in a material testing system, this was the maximum frequency allowed by our system, because higher values resulted in vibration interference (MTS System Corporation, Minneapolis, MN, USA). Loading tests were conducted in dorsoventral and lateral planes to simulate likely striking directions, with data (strains, cross-head displacement and measured force) recorded at 1000 Hz in a Megadac data collection system (Optim Electronics Corporation, Germantown, MD, USA).

where M is the bending moment at each area calculated by multiplying the maximum measured load from each bending regime by the distance from the loading point to each position where most strain gauges were aligned for each area; at a given position, d, y is the in-plane distance from the NA to each strain gauge at that location (Fig. 10); and I is the second moment of area associated to each NA (mm⁴). For each area's cross-section, I and y were calculated in MATLAB; for the latter, physical measurements of the distances of strain gauges from the lateral edge of the rostrum were used to determine their dorsoventral positions in each digital cross-section, and the dorsoventral distance y of those points from the neutral axis. Since we were interested in estimating the maximum stress generated by the rostrum during a set displacement, stress was calculated utilizing the largest bending moment at each area for each experiment (made by the product of the largest load and the distance from the strain gauges at each area to the loading point). Although the stress formula has been formulated based on simple structures in engineering, it has been widely used in biological systems, however, caution should be taken at interpreting its results since some deviation from its original assumptions may apply (i.e. the rostra do not have a constant cross-sectional area).

where E is Young's modulus and I is the second moment of area previously calculated. This parameter estimates the resistance to bending of the rostrum at all areas, allowing an integrated understanding of material properties and responses to load along the length of the rostrum.

In addition to the geometric analyses above, we used a second MATLAB script to examine the effects of cross-sectional form on rostral stresses for both species for two possible loading conditions: point-loading (a single/point load, equivalent to the billfish striking a prey item with the tip of the rostrum) and hydrodynamic loading (a load along the full length of the rostrum, resulting from the rostrum being swiped through the water) (Fig. 11).

For each loading scenario (tip-loading vs hydrodynamic loading), stresses were calculated along the length of the rostrum for two loading directions: simulating the rostrum moving laterally (i.e. in line with the rostrum's cross-sectional major axis) and moving dorsoventrally (i.e. in line with the rostrum's cross-sectional minor axis). Stresses were calculated using equations for beam bending with the assumption of small deformations. Rostrum cross-section dimensions were determined from CT scan data (see above), with both species’ rostra scaled to 700 mm tip-to-base length to allow a length-independent, interspecies comparison.

Maximum normal stress is a function of the applied load, F; the observed cross-section's rostrocaudal distance from the loading point (i.e. distance from the tip of the rostrum), x; a linear dimension of the cross-section in line with the applied load: b=one-half major axis length for lateral loading or a=one-half minor axis length for dorsoventral loading; and the second moment of area for that loading direction (I_maj or I_min, respectively; determined as described above and rescaled to a rostrum length of 700 mm). Maximum stresses resulting from tip-loading were calculated at 2% distance-increments from the tip to the base of the rostrum for both species and for both lateral and dorsoventral loading.

As the following calculations of hydrodynamic loading stresses are simplified considerably by using symmetrical geometric cross-sections, we selected a cross-sectional shape that could act as a proxy for actual rostral morphology, by comparing the maximum stress profiles for tip-loading for actual rostrum morphologies (see previous paragraph) with maximum stress profiles for hypothetical rostra with elliptical or diamond-shaped cross-sections. This allowed determination of the simplified cross-sectional shape that most accurately approximated stresses calculated for actual rostra, and therefore the shape which, for these purposes, was most mechanically similar to actual rostra. The aspect ratios for these idealized cross-sections were the same as those for the biological sections (i.e. the elliptical and diamond cross-sections were as wide and tall as the rostrum itself at any given tip-to-base location); these dimensions were used to calculate I along the length of the hypothetical bill using available equations for ellipses and diamonds (Gere, 2012) and these values were used in Eqn 6 to calculate maximum stresses per cross-section. The elliptical cross-section proved to be the more accurate approximation of both species’ cross-sectional profiles and so was used for the subsequent hydrodynamic loading calculations.

for dorsoventral motion the same equation applies, with b(x) replacing a(x).

As with the tip-loading regime, hydrodynamic stresses were calculated at 2% increments along the bill, allowing determination of a maximum hydrodynamic stress profile for both species and in two loading directions; for dorsoventral rotation of the rostrum (i.e. motion parallel to the minor axis of the cross-sections), a(x) was exchanged for b(x) and vice versa in Eqns 7–11.

This study could not have been done without Anna Avrigian and Jim Frank's generous help during specimen collection. Eric Prince allowed us to collect in fishing tournaments. Rick Stanczyk, J. Gardiner, A. Ferguson, J. Morris, R. Hueter and Mote Marine laboratories provided samples. Tony Villicana and V. Melnichuk assisted during experiments. Susan Herring provided input on strain gauge selection. John Long provided feedback on backbone biomechanics. Candy Miranda performed histology work and T. Crisman and C. Mitraki provided access to microscopes. Summer Decker and J. Ford made it possible to scan specimens and render 3D images. Many thanks to M. Montocchio, R. de la Parra and E. Walker for sharing their beautiful artwork. Thank you to S. Mulvany for helping to collect specimens and editing this manuscript. Special thanks go to M. Berdugo for his continuous assistance in every step of the process. We are extremely grateful to H. Fierstine for providing insightful answers to all our questions and for his endless support and encouragement.