Finite-element model construction for the virtual synthesis of the skulls in vertebrates: Case study of Diplodocus

Authors

  • Ulrich Witzel,

    Corresponding author
    1. Forschungsbereich Biomechanik, Institut für Konstruktionstechnik, Fakultät Maschinenbau, Ruhr-Universität Bochum, Bochum, Germany
    • Forschungsbereich Biomechanik, Institut für Konstruktionstechnik, Fakultät Maschinenbau, Ruhr-Universität Bochum, Bochum, Germany
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  • Holger Preuschoft

    1. Formerly Subdepartment of Functional Anatomy, Anatomisches Institut, Medizinische Fakultät, Ruhr-Universität Bochum, Bochum, Germany
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Abstract

The measurement of strains in real skulls is an inductive method that yields information about the stresses occurring in the a priori existing shape. In contrast, the approach taken here to determine the relationship between skull function and skull shape applies Wolff's law through a deductive technique of structure synthesis. This article describes the application of this method in the exact virtual synthesis of a sauropod skull, e.g., Diplodocus longus Marsh from Wyoming. An unspecific homogeneous solid is first constructed, giving the stresses ample volume to spread between points of force application and constraint. ANSYS 7.0 is used to form 10-noded tetrahedral finite elements with a maximum of 130,000 nodes. The initial conditions are the functional spaces for the eye openings, muscle forces, and the placement of the dental arcade, including assumed bite forces. Enforcing equilibrium of forces, the primary 3D stress flows in each load case are summarized by a physiological superposition, which accumulates the highest value of stress in each finite element. If the stress free parts are eliminated and the summarized stress flows are maintained, a reduced model appears, which is very similar to the real skull. This reduction of shape can be repeated iteratively and leads to a more exact form. Changes in the form of the dental arcade, its position relative to the braincase, the origins of muscles, or the height of the face lead to models that clearly resemble morphological differences between genera. The synthesis of a skull in this way demonstrates the direct correlation between functional loading and the biological structure and shape and can be used to test hypotheses regarding the relationship between structure and function during skull evolution. © 2005 Wiley-Liss, Inc.

The analysis of in vivo strains and stresses in skulls using strain gauge measurements and finite-element analysis (FEA) of CT-based models demonstrates that the skull is adapted to resist biting and other functional forces (Rayfield, 1998; Fastnacht et al., 2002; Dumont et al., this issue; Ross et al., this issue). These inductive approaches facilitate determination of, for example, the combination of muscle forces needed to produce specified bite forces or the ability of specially formed sliding sutures to transfer compressive stress, but not high shear stress (Rayfield, 2004). They also verify the general strategy of skull evolution, namely, to minimize mass and energy. One of us (U.W.) has also used the inductive approach in the field of orthopedic biomechanics to calculate the 3D stress distribution in the bony embedding of implants (Witzel, 1996, Witzel, 2000; Witzel et al., 2004) and in a special case to analyze an individual CT-based model of a human lower jaw under unilateral biting forces to get biomechanical information about carrying capacity from the working side to the balancing side (data not shown).

Deducing Skull Form Using FEA

Despite their advantages, these inductive approaches subordinate the theory of morphogenesis and do not provide precise explanations for the existence of, for example, a suture or a strut of bone in a specific position in the first place. In sum, they do not allow the explanation of skull shape as an absolute necessity and a mechanical answer to mechanical loading (Wolff, 1892). Rather, they are analyses of shapes that exist a priori.

In contrast, the deductive technique of structure synthesis used in this study is based on Wolff's law (Wolff, 1892). This method was established in 1985 by synthesizing a cross-section of the diaphysis of a human femur (Witzel, 1985). A similar approach was later developed by Carter and Beaupre (2001). In general, the method starts from an unspecific homogeneous body that offers the stresses ample volume for spreading between points of force application and constraint. External forces are applied to the model, low stress areas are iteratively removed, and the resulting shapes are compared with those observed in nature. In this way, the method is used to deduce skull form from a few initial and boundary conditions.

This method has been used to elucidate the mechanical reasons for the pneumatized spaces in the skulls of primates (Preuschoft et al., 2002) and for the prominent bony nose in humans and other primates (Witzel and Preuschoft, 1999), as well as a virtual synthesis of the facial part of a human and a gorilla skull (Witzel and Preuschoft, 2002; Preuschoft and Witzel, 2004). The present article extends this method to the deductive virtual synthesis of the skull of Diplodocus longus Marsh, 1878.

MATERIALS AND METHODS

The modeling procedure begins with a 3D Bauraum (Fig. 1), a wedge-shaped solid structure that envelops the total structure of the objects of interest. This Bauraum (in a free scale; in this case, 1:4.23) is created using ANSYS 7.0 and consists of 10-noded tetrahedral finite elements (FEs) with 36,491 nodes. Young's modulus is 17 Mpa and the Poisson ratio is 0.3.

Figure 1.

Wedge-shaped solid: 3D Bauraum.

To synthesize the skull of Diplodocus, a model is constructed that makes only five a priori assumptions: the position of the two eye openings, the shape and position of the dental arcade, the relative bite forces (Table 1), the observed muscle insertions, and the estimated activity of the muscles under considerations of the equilibrium of the whole system. The stable three-point support, which is a basic necessity for FE calculations, is achieved by constraining the positions of the quadratomandibular joint and the occipital condyle (Fig. 6A) with their bearing points where reaction forces are applied.

Table 1. Study Data
 NumberLoad/components
[N][N]
Frontal teeth (left, right)1, 1′6020
 2, 2′6020
 3, 3′3020
 4, 4′3020
 5, 5′3020
Side teeth (left, right)6, 6′3020
 7, 7′2020
 8, 8′2020
 9, 9′2020
 10, 10′2020

The primary 3D stress flows found for each load case—biting (Fig. 6A), lateral pulling to either side (Fig. 6B and C), gravity (Fig. 6D), and opening the snout (Fig. 6E)—are summarized by a load case technique. This is not an arithmetic addition, but a physiological superposition because every finite element accumulates the highest value that occurs in one of the load cases.

For example, consider four load cases acting on a simple beam (Fig. 2), which is fixed on his right side and loaded on the other in four different ways: from top (Fig. 2A), at the front (Fig. 2B), from the bottom (Fig. 2C), and from behind (Fig. 2D). An arithmetic addition of all stress flows generates a stress-free beam (Fig. 3); the physiological superposition (Fig. 4A) shows a pipe-like stress distribution (Fig. 4B). The center is nearly free of stresses.

Figure 2.

Demonstration of load cases. Beam model, fixed on the right side. A: Load from top. B: Load from front. C: Load from bottom. D: Load from behind.

Figure 3.

Beam model. Loading of all load cases and arithmetic addition of all stress flows. Result: beam without stresses.

Figure 4.

Beam model. A: Physiological superposition of all load cases. Every finite-element accumulates the highest value of compressive stresses that occurs in one of the load cases. B: View on the right side of the beam model. Pipe-like stress distribution. The center is nearly free of stresses.

If the stress-free elements are eliminated (during the synthesis of a skull) and the summarized stress flows are maintained, a reduced model appears that is similar to the real skull. This reduction of shape can be repeated iteratively and leads stepwise to more exact forms. The final iteration is dictated by reaching a physiological stress distribution. Changes in the form of the dental arcade, its position relative to the braincase or the origins of muscles, or the length or height of the skull all lead to models that clearly resemble morphological differences between genera.

For the Diplodocus model, the geometry of the Bauraum (Fig. 1) was derived from the outer shape, the dental arcade and the positions of joints taken from the fossil Diplodocus skeleton in Senckenberg Museum, Frankfurt am Main, Germany. The insertions of m. mandibulae and m. pterygoideus anterior (the active tension chord in the maxilla and premaxilla) (Rossmann et al., 2001; Preuschoft and Witzel, 2002), the m. pseudotemporalis and m. intramandibularis (the active tension chord in the mandible), the m. depressor mandibulae and longitudinal musculature (dorsal neck muscles; Fig. 5) were used to locate the places of application of the muscle forces.

Figure 5.

Analysis of a fossil skull: Diplodocus (Senckenberg Museum, Frankfurt am Main, Germany). A: Side view. l.m., longitudinal musculature (dorsal neck muscles); m.dep.m., m. depressor mandibulae; m.intrm., m. intramandibularis; m.m., m. mandibulae; m.pst., m. pseudotemporalis; m.pt.a., m. pterygoideus anterior. B: Rear view.

Before starting the FE calculation, bite forces and muscle forces were brought into equilibrium for each load case (Fig. 6) according to mechanical laws as follows: ΣFi x,y,z = 0, with Fi x = all forces in the direction of x-coordinate, and so forth; and ΣMi x,y,z = 0, with Mi x = all moments about x-coordinate, and so forth.

Figure 6.

5 load cases. A: Biting. b.f., bite forces (see Table 1); m.m., m. mandibulae; m.pst., m. pseudotemporalis; m.pt.a., m. pterygoideus anterior; occ.co, occipital condyle with x,y,z bearing; qu.m.j., quadratomandibular joint with y,z bearing. B: Lateral pulling to the right side. l.f., lateral forces; lat.m., lateral neck muscle, x,y bearing and y slide bearing. C: Lateral pulling to the left side. D: Gravity. l.m., longitudinal musculature; s.occ., supraoccipital with x slide bearing; qu.m.j., quadratomandibular joint with y slide bearing. E: Opening the snout. m.d.m., m. depressor mandibulae; qu.m.j.: quadratomandibular joint with x,y bearing.

The models were constructed on a 2.4 GHz portal and storage computer with 2 GB of RAM and a storage capacity of 240 GB and all calculations were performed on the central computer of Ruhr-University Bochum: an HP Superdome with 27 processors and 56 GB of RAM. After the models were loaded, the postprocessor of the FE program was used to extract all principal stresses, including the von Mises stresses. Wolff's theory and Pauwels' law of functional adaptation (Pauwels, 1965, 1973) emphasize compressive stress. This was also the case in this study because tensile stresses are usually taken by muscle forces and in physiological cases the bending moments are zero.

RESULTS

Figure 7B shows the distribution of compressive stress on the surface of the model. Areas under high load are the quadratomandibular joint, basioccipital and supraoccipital (green to blue); the unloaded or little-stressed regions are the corners marked yellow to red.

Figure 7.

First model: Bauraum. A: With finite elements, forces, and bearings, location of cross-sections. B: Distribution of compressive stresses at the surface of the model. Areas under high load are the quadratomandibular joint (qu.m.j.), basioccipital (b.occ.) and supraoccipital (s.occ.; green to blue). Unloaded or little-stressed regions are the corners (unl.c.; yellow to red).

The stress distribution in the interior of the volume can be shown by cross-sections through the model. For example, cross-section 14 (see Fig. 7A for section number) gives information about stress distributions in the palate (Fig. 8A) with three openings (low stress areas correspond to the positions of the openings in the real skull), the frontal and the antorbital fenestra. The 20th cross-section through the eye holes (Fig. 8B) shows the quadratomandibular joint, the infratemporal opening, the empty space, and the end of the nose opening in the upper frontal. The basioccipital, supraoccipital, and foramen magnum are presented by different values of compressive stresses in cross-section 26 (Fig. 8C).

Figure 8.

Cross-sections with compressive stresses. A: Cross-section 14 through the snout with details of the palate (p.) with three openings (o.), the dorsal plate, and the antorbital fenestrae (ao.f.). B: Cross-section 20 through the eye holes (e.h.) showing the quadratomandibular joint (qu.m.j.), the infratemporal openings (it.o.), the empty space (e.sp.), and the end of the nose opening (n.o.) in the upper frontal. C: Cross-section 26 near to the occipital end of the skull shows the basioccipital (b.occ.), supraoccipital (s.occ.), and the foramen magnum (f.m.).

If the lightly stressed parts in these cross-sections using an arbitrarily selected threshold of −0.8 N/mm2 are eliminated, the equivalents of the remaining bony cross-sections are shown Figure 9. Cross-section 4 presented in Figure 9A through the middle of the dental arcade shows a low-stressed region between the palate and premaxilla corresponding to a pneumatized space in Diplodocus, which contains the m. pterygoideus anterior as the active tension chord to resist bending (Figs. 5A and 6A). The force in this active tension chord is considered in our calculation and is an important factor to shape the bony structure.

Figure 9.

Reduced cross-sections. Stress-free parts are eliminated. A: Cross-section 4 through the middle of the dental arcade shows the palate, the premaxilla, and a pneumatized space (pn.sp.). B: Cross-section 18 through the beginning of the eye openings (e.h.), the middle of the nasal aperture (n.o.) in the frontal and infratemporal fenestrae (it.o.). C: Last cross-section through the occipital condyle (occ.co.).

Cross-section 18 cuts through the beginning of the eye openings (Fig. 9B) and the middle of the nasal aperture in the frontal. A bony opening was not expected at this location. Below the orbits, an infratemporal fenestra extends downward and forward to separate the postorbital from the quadratojugal and the quadrate, which forms a column bracing the mandibular joint. The occipital segment of the model (cross-section 28), shown in Figure 9C, shows the equivalent of the occipital condyle.

The merging together of all cross-sections leads to a 3D reconstruction of the reduced model as an “incarnation” of compressive stresses (Preuschoft and Witzel, 2004). The result, realized using CAD software Solid Edge, is presented in Figure 10. In Figure 10A, the reduced model is seen from top left with its characteristic silhouette of the snout. In front of the orbit (white circle), there are the antorbital and preantorbital fenestrae. Below the orbit is the infratemporal fenestra above the quadratomandibular joint. The frontal is characterized by a wide opening, which can be interpreted as a nasal aperture and the massive plate for the insertions of the longitudinal musculature (neck muscle). Figure 10B and C show the anterior view from above and from the bottom; in the latter view, there is a free passage through the brain case and the foramen magnum. Above the occipital condyle, and on both sides of the foramen magnum, the skull of Diplodocus shows smooth surfaces in contact with the atlas. These joint surfaces are visible in Figure 10D.

Figure 10.

Complete reduced model in a CAD presentation. A: Seen from top left. B: Anterior view from above. C: Anterior view from bottom. D: skull from behind. E: Sagittal section. The white sectional plane is circumscribing the pneumatized space and the braincase. Below the premaxilla the vault of palate and, in the background, the short dental arcade are visible. Lacrimal (lac.,), frontal wall of the braincase (f.w.b.), and parasphenoid)p.sp.) are not synthesized.

In a sagittal section (Fig. 10E), the white section plane circumscribes the pneumatized space and the braincase. Below the premaxilla the vault of palate and, in the background, the short dental arcade are visible.

For the next step of reduction, this 3D model was meshed with 10-noded tetrahedral finite elements (Fig. 11). The subsequent calculation with the same loading regime (load case technique) as in the first case yields the surprising results shown in Figure 12. All cortical shells change to a more concentrated walled construction. If the lightly stressed parts (red areas) in all cross-sections with a threshold of −2 N/mm2 (1% of the ultimate strength of Harversian bone: 172–220 N/mm2) (Wainwright et al., 1976) are eliminated or transformed into a lower stiffness, the synthesized structure very close resembles the skull of Diplodocus.

Figure 11.

Preparation of the reduced model for the next step of reduction by load case technique using the same loading regime as in the beginning.

Figure 12.

Result of the second FE calculation with the thinning effect of cortical walls. A: Cross-section 4 through the snout. B: Cross-section 9. The wall of the premaxilla is reduced to a cortical-trabecular bone sandwich.

In Figures 12A, 14A, the thinning effect is demonstrated in the snout. Other thick walls, such as the wall of the premaxilla (Fig. 12B), are reduced into a cortical-trabecular bone sandwich like a technical light-weight construction. Highly stressed areas (marked in blue and gray) are artifacts due to the notch (stress raiser) effect, or there is a deficit of material.

Figure 13.

The fossil skull of Diplodocus. b., bar between the openings; lac., lacrimal; f.w.b., frontal wall of the braincase; p.sp., parasphenoid; o.prm., two small openings in the premaxilla.

Figure 14.

Three-dimensional reconstruction of the first 11 cross-sections of the second iteration. A: Seen from top left. o.pr.m., three small openings in the premaxilla. The middle one is an artifact. B: Lateral view. b., bar between the openings is not synthesized.

DISCUSSION

The results of the described synthesis of the Diplodocus skull demonstrate the direct correlation between functional loading and the biological structure and shape. This synthesis suggests that the evolution of skull form in Diplodocus reflects natural selection for optimal skull construction, where optimality is defined as maximum strength with minimum material.

In this study, the lacrimal, the parasphenoid, and the frontal wall of the braincase were not synthesized (Fig. 13). This is because of practical limits to the FEA software, which could not model the small structures such as very thin shells and bars. The model in Figure 10 is also not identical to Diplodocus (Fig. 13) because the bar between the series of openings in the side wall anterior to the orbits is missing (Fig. 14B), as well as the sharp rim that continues the dental arcade toward the mandibular joint. All these traits, however, do exist in the fossil Rapetosaurus (Curry Rogers and Forster, 2004: Fig. 1): one continuous long opening anterior to the orbits and shrinking of the maxillary contour behind the tooth row. If our models are not in full agreement with the shape in one fossil but in another, the most probable explanation is that our assumptions about loading do not fit perfectly to Diplodocus. However, they do fit Rapetosaurus.

This study demonstrates the utility of FEA for the virtual synthesis of vertebrate skulls in order to test assumptions and hypotheses regarding the relationship between skull function and structure.

Acknowledgements

The authors are greatly obliged to the Rechenzentrum der Ruhr-Universität Bochum and the Institut für Konstruktionstechnik under personal supervision of its director Professor E.G. Welp for providing the FE software package ANSYS 7.0 and sufficient computer capacity for many repetitions of the calculations necessary to find those solutions faithful to their biological structures. Their special thanks go to Mr. Rainer Goessling, Mr. Danijel Gagovic, and Mr. Timm Hoffmann, who executed numerous calculations, and to the collection of Senckenberg Museum with its director Dr. G. Plodowski, where it was possible to study the Diplodocus skeleton with the helpful assistance of his coworker. They also thank the editor of this volume, Dr. C. Ross, for his careful reading and constructive criticism of their article. This study is contribution number 5 to the DFG Research Unit “Biology of Sauropod Dinosaurs.”

Ancillary