Variation among specimens referred to a single fossil taxon can originate from several biological sources, such as ontogeny, sexual dimorphism, and individual variation, but taphonomy can also be a source of morphological variation in fossils. During fossilization and diagenesis, bones can become deformed, and this deformation can lead to difficulties in understanding taxonomic variation, phylogenetic relationships, and functional morphology [1], [2], [3], [4]. Understanding the effects of taphonomic deformation on bones is therefore important for interpreting morphological variation.

Fossils can become distorted from the effects of brittle or plastic deformation (or both). In geological terms, brittle deformation results in fractures, joints, and faults, and plastic deformation results in folds. Whether or not a fossil undergoes brittle or plastic deformation is dependent on the temperature, confining pressure, and strain rate it experiences. Brittle deformation occurs at low temperatures, low confining pressures, and high strain rates; plastic deformation occurs at high temperatures, high confining pressures, and low strain rates. Many fossils undergo brittle deformation prior to burial, cracking and fracturing during transport, and brittle deformation can occur during diagenesis as well, such as if a fossil is faulted. Plastic deformation of a fossil is more likely to occur during fossilization and diagenesis, during which time bone can act like a ductile material. Fossils rarely survive more than a single phase of plastic deformation, and as such, identifiable but plastically distorted fossils typically have a simple deformation history [5]. Not all fossils in a single bedding plane may deform homogeneously, and not all elements within a single specimen will necessarily deform homogeneously [5]. The orientation of a specimen within the sediment will also affect how the specimen deforms [6].

The goal of this study is to introduce some techniques for understanding three-dimensional (3D) plastic deformation in ankylosaurid dinosaur skulls. First, skulls of extant vertebrates were examined to determine if the shape of the orbit can be used as an indicator for whether or not plastic deformation has occurred. If the periorbital rims of a variety of extant vertebrates are generally circular, then fossil skulls with elliptical orbits have probably undergone some amount of plastic deformation. Retrodeformation and finite element analysis were then used as tools for understanding what parts of an ankylosaur skull are most likely to undergo deformation and therefore least likely to be phylogenetically useful. This information can then be used to enhance the quality of cranial characters used in phylogenetic analyses. No attempt was made to undistort taphonomically distorted skulls into their original shape, as there are few features on the skull to act as constraints guiding the decisions in retrodeforming ankylosaur skulls. Retrodeforming an ankylosaur skull with the goal of restoring its true shape would be highly subjective. Instead, the focus of this study is on understanding which morphological features on an ankylosaur skull are most likely to become taphonomically deformed.

The software program Geomagic [7] is used to investigate potential effects of deformation by modifying digital models of ankylosaur skulls. It can be used to restore symmetry to a skull, and to measure the amount of shape change in various models of the same structure. Finite element analysis (FEA) can be used to investigate the way in which we might expect a fossil to have deformed under a variety of geological forces. FEA has been used to investigate the effects of biologically-induced forces in extant and extinct vertebrates [8], [9], [10]. However, FEA has not been used to investigate the effects of geological forces on vertebrate fossils, such as sediment compaction and diagenesis. In this paper, the retrodeformation analyses represent the subtraction of deformation from a skull, and the finite element analyses represent the addition of deformation to a skull. If the same parts of the skull undergo shape change during both retrodeformation and FEA, then these parts of the skull are most likely to experience deformation during fossilization and diagenesis.

This study examines two cases where understanding deformation can be used to better interpret ankylosaur cranial morphology: 1) intraspecific variation in Euoplocephalus tutus Lambe, 1910 [11], and 2) the taxonomic validity of Minotaurasaurus ramachandrani Miles and Miles, 2009 [12]. Euoplocephalus (Fig. 1) is the best represented ankylosaurid from the Late Cretaceous of North America, and more than 15 skulls have been referred to this genus. Coombs [13] synonymized four taxa within Euoplocephalus tutus: Dyoplosaurus acutosquameus Parks, 1924 [14], Scolosaurus cutleri Nopcsa, 1928 [15], and Anodontosaurus lambei Sternberg, 1929 [16]. Arbour et al. [17] recognized Dyoplosaurus as a distinct taxon, a result supported by an ankylosaur phylogenetic analysis by Thompson et al. [18]. Penkalski [19] noted a great deal of variation among skulls referred to Euoplocephalus, and identified sexual dimorphism, ontogeny, and individual differences as the sources for much of this variation, in addition to potential taxonomic differences. Many of the distinctive features of individual specimens noted by Penkalski [19] are unlikely to change during deformation, because they represent quantities or surface texture (e.g. surface texture of cranial sculpturing, number of osteoderms in the nuchal crest). However, some, such as the erectness of the squamosal horns, may be affected by dorsoventral compaction. Retrodeformation of two Euoplocephalus skulls (AMNH 5405 and UALVP 31) will highlight the changes that can occur during crushing, and can be used to identify areas of the skull that are most likely to change and therefore be less taxonomically informative. For example, if the erectness of the squamosal horns changes with retrodeformation, or if there is high strain in this area after FEA, then the erectness of the squamosal horn may be affected by dorsoventral compaction.

The second case study examines the taxonomic validity of Minotaurasaurus (Fig. 1), known from a single specimen of unknown provenance, but likely from the Gobi Desert of Mongolia or China [12]. This taxon bears a strong overall resemblance to Saichania chulsanensis Maryańska, 1977 [20], Tarchia gigantea Maryańska, 1977 [20], and Tianzhenosaurus youngi Pang and Cheng, 1998 [21], although the most recent phylogenetic analysis of the Ankylosauria [18] found a close relationship between Minotaurasaurus ramachandrani and Pinacosaurus grangeri Gilmore, 1933 [22] (but not Pinacosaurus mephistocephalus Godefroit, Pereda Suberbiola, Li, and Dong, 1999 [23]). Although the holotype of Minotaurasaurus does not appear obviously taphonomically distorted, it has a much lower, flatter profile compared to ankylosaurs such as Euoplocephalus. Additionally, several features are described by Miles and Miles [12] as being flatter or more dorsoventrally compressed compared to other taxa, such as the orientation of the pterygoid, the articular surface of the quadrate, the pterygoid-quadrate contact, and the angle of projection of the quadratojugal horn. If the pterygoid, quadrate, and quadratojugal horn undergo more shape change than other portions of the skull during retrodeformation and FEA, then these features are most likely the result of dorsoventral compaction and the diagnosis of Minotaurasaurus should be revised.

Taphonomic distortion of some ankylosaur skulls is immediately easy to identify if there are obvious and extreme asymmetries, such as those seen in the holotypes of Crichtonsaurus benxiensis Lü, Ji, Gao, and Li, 2007 [31] (BXGM V0012) and Nodocephalosaurus kirtlandensis Sullivan, 1999 [32] (SMP VP900). Prieto-Márquez [33] noted that bending ridges and unusual bulges can also be signs of dorsoventral crushing in fossil skulls. However, Boyd and Motani [34] have shown that a symmetrical model does not indicate that plastic deformation from overburden compaction has been removed, and it can be easy to reconstruct a skull into an incorrect shape if there is no knowledge of accurate skull morphology. As such, symmetry alone may be insufficient for identifying deformation.

Measurements of the ellipticity of extant, undeformed vertebrate orbits suggest that orbits are not perfectly circular, but that the length:height ratio is generally between 1.00 and 1.28. As such, elliptical orbits in fossil specimens may not necessarily indicate that dorsoventral compaction has occurred. However, an orbit shape ratio greater than 1.28 in fossil skulls may indicate that some amount of dorsoventral crushing has occurred.

The higher orbit ratios in the few crocodilian and avian taxa in this study (representing the extant phylogenetic bracket for ankylosaurs) may suggest that archosaurian orbits are less circular than those of mammals, and that undeformed orbit ratios from 1.3–1.7 could be expected for dinosaurs. However, many of the ankylosaurid skulls had orbit ratios well above the maximum undeformed ratio recorded in this study (1.66 for Varanus sp.), and the range of orbit ratios was much greater for ankylosaurs than for all extant taxa combined. A plot of ankylosaur orbit ratios (Fig. 3) shows that few specimens have a ratio below 1.28. This suggests that either ankylosaurid orbits were not generally circular, or that many skulls have undergone some dorsoventral crushing during fossilization and diagenesis. AMNH 5405 has surprisingly high orbit ratios, given that the arched profile of the skull suggests little crushing took place. In contrast, Crichtonsaurus has a relatively low orbit ratio, despite the fact that this skull is highly asymmetrical and has certainly been flattened and distorted. Several specimens (AMNH 5403, MOR 433) have noticeably different orbit ratios for the left and right orbits, which suggests that the skulls underwent shearing or uneven dorsoventral compaction. Orbit ratios may be most useful when compared across multiple specimens of the same taxon, and very high ratios above 2 (in specimens where the orbit is completely encircled by the periorbital rim) are likely to indicate that dorsoventral crushing has occurred. The orbit ratio can serve as a general indicator if an ankylosaurid skull has been dorsoventrally compacted, but cannot be used to definitely indicate how much compaction has occurred. The true orbit ratio may not be known for a given fossil taxon, but high orbit ratios relative to the mean for a given sample of fossil specimens could also be used to identify if dorsoventral compaction has occurred. The orbit ratio could be a useful indicator of compaction for skulls that are symmetrical and which may not be obviously deformed.

Geomagic is a useful tool for investigating potential shape changes resulting from dorsoventral compression. The results of these tests can be independently assessed using finite element analysis to investigate which areas of the skull are most likely to experience strain (and therefore shape change). The FEA tests (Figs. 5, 6) showed high strain on the jugals, quadratojugals, and squamosals, which correspond to areas of change in the Geomagic models (Fig. 4). Strain was also present on the quadrates, pterygoids, and vomers, which did not change much in the Geomagic models. This indicates that retrodeforming a flattened skull in Geomagic will provide a good approximation for which features have been most affected, but may not reveal changes in all regions of the skull. Finite element analysis of several taphonomic scenarios is useful for determining which forces a skull may have been subjected to during deformation.

Taphonomic distortion may be responsible for some of the variation in skulls referred to Euoplocephalus. For example, Penkalski [19] suggested that the more upright squamosal horns of MOR 433 (in comparison to USNM 11892) may have been a result of crushing. This is supported by results from this study, where dorsoventrally compressing AMNH 5405 in Geomagic resulted in more upright squamosal horns similar to those of UALVP 31 (Fig. 4). The most noticeable change to AMNH 5405 was the flattening of the skull in lateral view. Skulls referred to Euoplocephalus have a range of morphologies in lateral view, from arched (AMNH 5405, ROM 1930), to flat (CMN 8530, USNM 11892). It is possible that the arching of the skull may be related to ontogeny, in which case a correlation between flatness and size would be expected. It is also possible that the relative flatness may be a true taxonomic difference. However, many of the skulls that are flat also have subcircular orbits, which suggests that the skulls have undergone crushing and in life were more arched.

Miles and Miles [12] identify several features of Minotaurasaurus as being flatter or more horizontal than their equivalents in other ankylosaurids: the angle of projection of the jugal horns, the articular surface of the quadrate, the pterygoid-quadrate contact, and the orientation of the pterygoid body. Additionally, the ‘flaring’ narial osteoderms may be a product of dorsoventral crushing. Retrodeformation of INBR21004 in Geomagic resulted in more ventrally projecting quadratojugal horns, but did not affect the quadrates or pterygoids (Fig. 4). However, finite element analyses simulating crushing in INBR21004 showed increased strain (and therefore shape change) in the quadrates and the caudal portion of the pterygoids (Fig. 6). This suggests that the retrodeformation techniques outlined in this study do not necessarily capture all of the shape changes on the ventral side of the skull, and emphasizes the need for multiple approaches when attempting to understand deformation in fossils. The dorsoventral angle of projection of the quadratojugal horn can be easily affected by taphonomic distortion, and should not be used as a diagnostic character for ankylosaur taxa. It is less clear if the articular surface of the quadrate, pterygoid-quadrate contact and horizontal pterygoid body in Minotaurasaurus are a result of deformation or represent true taxonomic differences. The flaring appearance of the narial osteoderms did not change during retrodeformation (Fig. 4), and dorsoventral compaction of AMNH 5405 did not result in more flaring narial osteoderms. UALVP 31, which is probably dorsoventrally compacted, also lacks flaring narial osteoderms (Fig. 4). In the finite element analyses of INBR21004, the narial osteoderms never experienced increased strain under any of the load regimes (Fig. 6). This suggests that the wide, flaring nares of Minotaurasaurus are real, and not an artifact of preservation.

Although Geomagic contains tools that could be used to correct plastic deformation in a fossil, there are many challenges associated with reconstructing a distorted fossil into its true, original shape. It is difficult to determine the accuracy of the retrodeformed skull in which there is no extant, undeformed analog. Simply restoring symmetry is insufficient to determine if a retrodeformed skull represents an accurate shape. Boyd and Motani [34] demonstrated that a digitally fragmented and distorted skull could be pieced back together into a symmetrical, but incorrect shape. As such, the results presented in this paper should not be taken to indicate that dorsoventrally compacted ankylosaur skulls can be retrodeformed into their true shape, but that retrodeformation tools can be used to understand which parts of the skull were most likely to be deformed. Three-dimensional retrodeformation techniques are useful for understanding potential sources of morphological variation in ankylosaur skulls, but it is not possible to confidently retrodeform an ankylosaur skull to its original shape.

Retrodeformation of a specimen may result in new taxonomic interpretations because of changes in shape. The accuracy of 3D retrodeformation techniques is still being investigated; retrodeformation is more likely to be successful when morphological constraints, based on features of extant taxa, can be used [3]. Although the FEA results differed somewhat from the retrodeformation results, some morphological features consistently changed (or did not change), and this provides information on which ankylosaur cranial characters may or may not be taxonomically informative. Overall skull morphology was easily changed with minimal retrodeformation, but features of the palate and braincase were less likely to be affected. The dorsoventral angle of projection of the quadratojugal horn is easily altered by dorsoventral compaction and should not be used to support taxonomic distinctions among ankylosaurs. Many of the diagnostic features of Minotaurasaurus did not change during retrodeformation, which suggests that these features are either unique to this genus or represent intraspecific or ontogenetic variation within a different taxon. Much of the variation in skull morphology in specimens referred to Euoplocephalus may also be a result of taphonomic distortion, although again intraspecific and ontogenetic variation cannot be ruled out.