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Keywords:

  • biomechanics;
  • finite element analysis;
  • geometric morphometrics;
  • multibody dynamics analysis

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Building interesting models
  5. Analysis and interpretation of the results of biomechanical modelling
  6. Concluding remarks
  7. Acknowledgements
  8. Author contributions
  9. References

The development of virtual methods for anatomical reconstruction and functional simulation of skeletal structures offers great promise in evolutionary and ontogenetic investigations of form–function relationships. Key developments reviewed here include geometric morphometric methods for the analysis and visualization of variations in form (size and shape), finite element methods for the prediction of mechanical performance of skeletal structures under load and multibody dynamics methods for the simulation and prediction of musculoskeletal function. These techniques are all used in studies of form and function in biology, but only recently have they been combined in novel ways to facilitate biomechanical modelling that takes account of variations in form, can statistically compare performance, and relate performance to form and its covariates. Here we provide several examples that illustrate how these approaches can be combined and we highlight areas that require further investigation and development before we can claim a mature theory and toolkit for a statistical biomechanical framework that unites these methods.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Building interesting models
  5. Analysis and interpretation of the results of biomechanical modelling
  6. Concluding remarks
  7. Acknowledgements
  8. Author contributions
  9. References

Functional morphologists apply a variety of methods to explore the workings of and constraints on skeletal form and function. These include building physical models (e.g. Demes, 1984; Demes et al. 1984); simplified graphical and mathematical models (e.g. Hylander, 1975; Greaves, 1978, 2000; Throckmorton et al. 1980; Demes & Creel, 1988; Spencer & Demes, 1993); direct experimentation (e.g. Hylander, 1979a,b; Hylander & Crompton, 1986; Kay et al. 1986; Hylander et al. 1987, 2000; Hylander & Johnson, 1997; Ravosa et al. 2000; Daegling & Hotzman, 2003; Wang et al. 2006) and, more recently, complex computer-based simulations (e.g. Koolstra & van Eijden, 1997a,b, 1999, 2005; Langenbach & Hannam, 1999; Langenbach et al. 2002, 2006; Sellers & Crompton, 2004; Rayfield, 2005; Ross et al. 2005; Curtis et al. 2008, 2010; Moazen et al. 2008; Dumont et al. 2009; Strait et al. 2009). These models allow the investigator to ask questions regarding how the system works, how variations in the geometry of the complex affect function and what functional-architectural constraints there may be on a structure. Such investigations also allow insight into developmental and evolutionary pathways and have a direct application to biomedical engineering.

Two engineering approaches are frequently used for musculoskeletal simulation and are increasingly impacting on the biological literature. The first, finite elements analysis (FEA) is commonly applied in studies of skeletal function to estimate the effects of loading, computing the resulting deformation and usually expressing this in terms of stress and strain. FEA offers the possibility of interpreting skeletal structure in terms of functional performance even in extinct species. Consequently, it has excited a generation of comparative and functional morphologists, resulting in its application to a wide range of living and extinct species during the last decade (e.g. Gupta et al. 1973; Hart et al. 1992; Korioth & Hannam, 1994; Vollmer et al. 2000; Koolstra & van Eijden, 2005; Moazen et al. 2009; Rayfield, 2005; Richmond et al. 2005; Ichim et al. 2007; Strait et al. 2007; Wroe et al. 2007; Kupczik et al. 2007; Dumont et al. 2009; Strait et al. 2009). The second engineering approach which is becoming increasingly popular is multibody dynamics analysis (MDA). MDA is a computer modelling approach which analyses the motion and dynamic behaviour of interconnecting bodies. This technique has been applied to studies of craniofacial form to simulate musculoskeletal function and facilitates in-depth exploration of the relationships between musculoskeletal geometry, muscle parameters, forces and motion in fish (Westneat, 2003), pigs (Langenbach et al. 2002, 2006), lizards (Curtis et al. 2010), humans (Koolstra & van Eijden, 1997a,b, 1999, 2005; Langenbach & Hannam, 1999; Sellers & Crompton, 2004) and non-human primates (Curtis et al. 2008). A typical application might involve, for example, estimating masticatory muscle forces based on known motion, bite forces and skull morphology. Thus MDA has a clear link with FEA. MDA can be used to simulate and predict forces which can then be applied to finite element (FE) models for analysis of how these biological structures respond to the predicted loading regimen (Koolstra & van Eijden, 2005; Curtis et al. 2008, 2010; Moazen et al. 2008).

With the advent of sophisticated computer modelling, we have the ability to produce large numbers of high-resolution virtual models which can be exploited (both in terms of geometry and operation) for deep exploration of form–function relationships. However, dealing with large biological sample sizes and complex geometric structures poses problems for investigators, including how to represent the specimen (do you use an exemplar, the mean form or create models for extremes within a sample?), how to identify which aspects of shape/form appear to be different and/or related to function within your population, and how to alter accurately the geometry of the model. Following the functional analysis, how does one quantify and compare how specimens vary in terms of deformation and other aspects of performance? Further, where there is uncertainty regarding input parameters (e.g. muscle vectors, force magnitudes, constraints) careful sensitivity analyses are required to provide insight into the consequences of errors in modelling and the results of such analyses need appropriate statistical treatment to reach appropriate conclusions. Such problems require quantification and comparison of deformations; changes in form.

In recent years geometric morphometric (GMM) methods have proved to be a significant advance in relation to the quantitative description and comparison of form using landmarks. They are applicable to the analysis of changes in form such as results from FEA but also in many other biologically interesting circumstances such as motion analysis and studies of development and evolution. These methods comprise manipulations of landmark data that facilitate the quantification of variations in form (size and shape) among landmark configurations taken on different specimens using multivariate methods. They have been employed by morphologists interested in describing and comparing complex shapes for two decades and have impacted enormously on comparative anatomical studies (Bookstein, 1991; O’Higgins, 1997; Dryden & Mardia, 1998; Rohlf, 2000; Slice, 2005, 2007). GMM allows for the study of covariations between forms or shapes or between these and extrinsic variables (e.g. bite force, phylogenetic history, climate, etc.) using for example the methods of regression and partial least squares (PLS; Bookstein, 1986; Rohlf & Corti, 2000). Despite obvious applicability to direct form–function investigations GMM has yet to be fully utilized by functional morphologists (but see Pierce et al. 2008; O’Higgins et al. 2009).

This review details the application of this suite of techniques to model building, functional studies of biological form as well as analysis and interpretation of the results of biomechanical modelling studies (in particular those using MDA and FEA). Our purpose here is to review these strands of methodological development and consider how they might be brought together in a novel framework for multivariate form–function.

Building interesting models

  1. Top of page
  2. Abstract
  3. Introduction
  4. Building interesting models
  5. Analysis and interpretation of the results of biomechanical modelling
  6. Concluding remarks
  7. Acknowledgements
  8. Author contributions
  9. References

Reconstructing fragmentary material

Perhaps one of the most developed applications of GMM in relation to subsequent functional modelling of skeletal form is in reconstruction of damaged or fragmentary specimens. Gunz et al. (2009) provide an extensive review of how methods of virtual reconstruction and applications of GMM can be used to reconstruct damaged, distorted and fragmentary hominin fossil crania. The key GMM approaches include: statistical reconstruction, in which regression or two block partial least squares (PLS) is used (iteratively if necessary) to estimate missing landmark coordinates; geometric reconstruction, in which missing parts are reconstructed by warping parts from other material into the specimen using thin plate splines (TPS); and, the application of reflected relabelling (Gunz et al. 2009) to produce a mirror image specimen which is then used with the original to estimate a symmetric reconstruction. TPS is used in GMM to generate deformed Cartesian transformation grids, a triplet of splines providing a smooth function for the interpolation of 3D points from the vicinity of a reference specimen to that of a target. The same approach can be used to warp curves or surfaces between landmarked reference and target forms. In Fig. 1 a skull of Australopithecus boisei is reconstructed using where possible mirror imaging of complete anatomy from one side to the opposite incomplete side and warping from fossil near relatives to estimate the architecture of missing parts. This includes the whole of the midcranial region (consisting of parts of the sphenoid, vomer, ethmoid, maxilla, temporal, parietal and frontal bones). This reconstruction provides an estimate of the form and a virtual cranium suitable for subsequent FEA. Note that while the warping of missing anatomy is subject to considerable error, the extent of this error can be assessed by carrying out sensitivity studies in which the effects of different choices of landmarks, specimens from which missing anatomy is warped and other aspects of reconstruction can be quantified and evaluated.

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Figure 1.  The surface of a reconstructed Australopithecus africanus skull (A; Sts5) is used to reconstruct Australopithecus boisei (B; OH5) via warping. (A) 325 reference landmarks (yellow) are placed on the reference surface. (B) The reference surface is then warped to the target configuration (blue landmarks) placed on the surface of OH5. (C) The fossil is then reconstructed using the warped surface to estimate the architecture of missing parts. The grey zygoma in (B) is reconstructed via mirror imaging of the opposite side.

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Creating and experimenting with functional models

Another potential application of GMM in relation to virtual functional simulation is in the rapid creation of variants of a single original model and experimentation with form to investigate form–function relationships through warping of models. While this is not necessarily straightforward in relation to FEA, as we discuss later, warping of form offers a route to the production of smoothly deformed modifications of the original model. This approach has been demonstrated by Sigal et al. (2008) who warped a reference rat caudal vertebra surface into the form of a target vertebra. One of the warping methods they used with good effect was TPS. A similar approach using TPS was applied by Stayton (2009) to the warping of an FE model of a turtle shell between species. This technique allows one model to be warped into a target form (described by landmark coordinates) to create a new model. Such warping potentially provides the investigator with the opportunity to create large numbers of specimens without the necessity to virtually reconstruct each 3D model from CT data, a time-consuming task. Furthermore, in certain situations CTs may not be available, thus the ability to create a 3D model from a sample of landmark coordinates (collected for example by 3D Microscribe, or a digitized surface collected from a surface scanner) provides the investigator with a way of increasing their access to a wider range and larger sample of study specimens.

As well as the potential to generate globally deformed models, warping can be conducted such that only localized deformations are produced. Such isolated changes in geometric form offer the opportunity to test form–function and functional integration hypotheses in a controlled way. It is possible to alter the size or shape of a feature of biomechanical interest while keeping all other aspects of morphology constant, for example the degree of prognathism in the skull.

To produce an isolated warp, control landmarks are placed all over the original virtual reconstruction (reference form), including a subset selected to represent the region of interest (e.g. those over the area you wish to alter). A second landmark dataset (target form) is then produced containing the same control landmarks held constant in location but with an alternative subset represented by landmarks which have been moved to the desired position. TPS is then used to warp the reference smoothly into the target. Such an application is illustrated in Fig. 2, where a virtual reconstruction of the gracile australopith Australopithecus africanus (Sts5) (Fig. 2A) is modified such that it comes to have zygomatico-maxillary regions like those of the robust australopith A. boisei (OH5) (Fitton et al. 2009). This is achieved by densely landmarking the relevant regions of both specimens. The regions that are not to be warped are landmarked in the gracile specimen (yellow landmarks) and these landmarks are used to protect these regions from unwanted warping (anchor landmarks). Other landmarks are placed densely around the region which is to be altered (red landmarks), in this case in the zygomatico-maxillary region, and combined with the anchor landmarks they define the reference form. The robust specimen’s zygomatico-maxillary landmarks (blue) are scaled and brought into the required register with the gracile specimen, in this case such that the lower orbital borders, roots of zygomas and posterior aspects of zygomatic arch best match. Next, the landmarks from A. boisei are combined with the anchor landmarks in the gracile specimen and this new configuration represents the target form into which the gracile reconstruction is warped using a triplet of thin plate splines. The end result is a hypothetical variant of gracile australopith morphology. This can be used in subsequent analyses in which everything except form is held constant in order to examine the effects of this variation in morphology on muscle action and loading through MDA or on skull deformation through FEA (Fitton et al. 2009). This basic approach can be extended to other anatomical features and anatomical systems and so provides a novel way of investigating the relationships between animal form and function.

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Figure 2.  Warping the zygomatic region of a virtual reconstruction of a gracile australopith. (A) The red landmarks (region of interest) are combined with the yellow (control landmarks) to create a reference landmark configuration representing the original specimen. The blue landmarks in (A) taken from Australopithecus boisei (OH5) are combined with the yellow landmarks to define a target configuration (see text). The original surface (A) is then warped to the target configuration to create a modified form that comes to have zygomatico-maxillary regions like those of a robust australopith (B) (see Fig. 1C for robust morphology).

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Making models of hypothetical forms

As well as using TPS to warp models, GMM includes other approaches useful for exploring form–function relationships. Geometric morphometric analyses often yield results that can be visualized as deformed landmark configurations and which could well be used to define the geometry of a subsequent model of interest. Thus, while most FEA or MDA analyses of cranial function in animals are based on a single individual (and so ignore variation), GMM can readily generate a sample mean and extreme landmark configurations within a population that can then be used as targets for the warping of 2D and 3D models. Using such approaches, the prospect of statistical-biomechanical understanding of how morphological and functional variations intertwine becomes a reality. In turn this means that we can control error due to sampling and address evolutionary functional questions using well established and powerful statistical approaches. Landmark configurations from GMM analyses have been used to generate FE models by Pierce et al. (2008), who carried out a geometric morphometric analysis of skull form in crocodilian skulls using 2D landmark data. The results of this analysis were presented as a principal components analysis (PCA) plot and the forms of the 2D configurations represented by the extremes of the first few PCs were used to generate 2D FE models that were then used to relate form variability to function, ecology and feeding.

An obvious extension is to use such GMM analyses to generate warpings of detailed 3D models for FEA and MDA by using thin plate splines as described in the previous section. Furthermore, in functional studies there is the potential of using GMM to estimate the relationship between biomechanically interesting variables and form. This can be achieved directly using either multivariate regression in which form is regressed on the variable(s) of interest, or partial least squares (PLS) in which the association (rather than dependency as in regression) between form and biomechanical variable(s) is modelled. Subsequently, the result of the regression can be used to warp a functional model and so facilitate direct mechanical assessment of this relationship. Such an analysis is illustrated in Fig. 3, where multivariate regression has been used to characterize the relationship between the square root of post-canine dental area (as defined in McHenry, 1994) and cranial shape in a sample of hominin fossils including representatives of most australopith and early Homo species. Of the total variance in the sample, 23% is accounted for by this regression, i.e. is explained by the square root of post-canine area. The mean form of these hominins is shown in the middle row of Fig. 3 as a landmark configuration with a triangulated surface drawn between landmarks to facilitate visual interpretation and as a warping of a reconstruction of A. africanus to this mean. Likewise the shapes predicted for small and large post-canine areas are shown in bottom and top rows of Fig. 3, respectively. The implication is that if post-canine dental area relates to masticatory effort or post-canine loading, then these warps represent forms adapted to the extremes of this. This can be further assessed by functional modelling.

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Figure 3.  Results of a multivariate regression of cranial shape on the square root of post-canine dental area (McHenry, 1994) in a sample of hominin fossils in which 23% of the total variance is explained. (A) Extremes (top and bottom rows) and mean shapes (B; middle row) are represented by a surface triangulated among 56 cranial landmarks. Using these landmark configurations a virtual reconstruction of Australopithecus africanus (Sts5) (see Fig. 2) is warped using TPS. These resulting surfaces (B) represent hypothetical forms which possess small (B; bottom) mean (B; middle) and large (B; top) post-canine areas.

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The mechanical significance of the differences in shape between these extremes is amenable to detailed functional analysis. By modelling functional performance, the extent to which the morphometric association is indeed functionally important with regard to performance can be assessed. Such analyses can potentially clarify how form and function interact. MDA can be used to assess the effects of cranial warpings on variables such as biting force, jaw kinematics, muscle forces and joint reaction forces. Additionally, the warped models can be subjected to FEA. This is illustrated in Fig. 4 where these cranial shapes associated with small (Fig. 4, column A) or large (Fig. 4, column B) post-canine dental areas are subjected to FEA simulating molar (m2, Fig. 4, top row) or incisor (Fig. 4, bottom row) bites. Comparing molar bites between the shapes associated with small and large post-canine area it is clear that they differ mainly in the level of strain experienced on the anterior aspect of the zygomatic, extending on to the maxilla on the working (right) side. The strain is much less in the predicted cranial form for large post-canine area. This is an expected finding as it confirms that the facial morphology, principally enlarged and flared zygomatics, associated with large post-canine area in these Plio-Pleistocene hominins is better able to withstand molar loads than that associated with small post-canine area. There is concordance between the morphometric and functional findings that suggests that the morphometric findings are indeed functionally significant. In contrast, when the incisors are loaded in both models the situation is reversed, in that the small post-canine area model appears better able to resist this load, with strains much reduced over the subnasal region, when compared to the large post-canine area model. This is an unexpected finding in that incisor size was not included in the regression analysis. It may point to further functionally significant differences between large and small cheek-toothed hominins although the effects of general size and differences in muscle forces also need to be considered.

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Figure 4.  Von Mises strain maps from FEA of the hypothetical hominin models produced in this figure (see text). Cranial shapes associated with small (A) or large (B) post-canine dental areas are subjected to FEA simulating molar loading (top images) and incisor loading (bottom images). Young’s modulus 17GPa, Poisson’s ratio 0.3; muscle forces (calculated using OH5 cross-section estimates; Demes & Creel, 1988): Anterior temporalis 659.2 N, medial pterygoid 316.4 N, masseter 527.4 N; constraints: occipital condyles (all directions), incisors (I1 + I2) (vertical) simulating incisor bites or m2 right (vertical) to simulate molar bite.

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This preliminary study serves principally as an illustration of how GMM and FEA can be usefully combined in novel ways. There are, however, a number of difficulties that need addressing in this example study, such as the fact that these models are warpings of a composite reconstruction of Sts5 and Sts52, representing a gracile australopith. In warping this model, its internal architecture is also warped and this may be driving some of the functional findings (see below). A further issue arises with regard to scale, in that the landmarks used to warp the specimen are much lower density than the nodes of the eventual FE model. This means that while strains are eventually computed at a very fine scale they should not be over-interpreted at this scale. Additionally, the fossil sample is heavily dependent on availability and different samplings may give different results. Similarly, the relationship between post-canine area and cranial morphology is clearly heavily influenced by the robust australopiths in the sample. As such this approach needs to be applied to different groupings of hominins to fully understand the potential mechanical links between dental morphology and facial form in early hominins. Finally, the significance of enlarged post-canine area with its associated facial form is unclear; it may relate to large but occasional bite forces or, equally, it may be a correlate of chewing effort (Strait et al. 2009; Grine et al. 2010). This study can be extended in useful ways, for instance, it would be interesting to use other variables (e.g. muscle cross-sectional areas) as the independent predictor of cranial form and to examine how different biomechanically informative variables and size interact by combining them as predictors in multiple multivariate regressions. Indeed, there are many avenues to explore in advancing statistical biomechanical modelling approaches as these are in their infancy.

Issues with warping

There are a number of issues to consider and explore when working with warped models, particularly for FEA. In such warping experiments it is inevitable that the internal as well as the external morphology of bone is warped. Variations in cortical thickness and in internal architecture (e.g. trabecular organization and density) potentially have a large effect on the ways in which skeletal elements deform under load. It is therefore important to consider the impact warping has on analytical outcomes. Sigal et al. (2008, 2010) avoided this issue by warping only the surface model and later allocating material properties to the warped model based on the CT grey levels of the target. After allocating material properties to the target form using the CT data from the target specimen they found good correspondence between the FEA results of the true target vertebra and its warped model. This approach offers some considerable saving of time in segmentation and model building but could not be applied where the CT of the target specimen is not available. This is the case when working with hypothetical forms such as the population mean and it is often the case when working with fossils where matrix and mineralization can have a major effect on sample density and CT grey scale levels.

An example of the effect landmark-based warping can have on an FEA result is illustrated in Fig. 5. Von Mises strains arising from a second molar bite are mapped in a series of male macaques. Two adult male macaques (Fig. 5A,B; an adult male with M3 missing) are both loaded in the same way. The resulting strain maps show differences particularly over the zygomatic region. Interestingly, when the male skull of Fig. 5B is warped into that of Fig. 5A the resulting strain map (Fig. 5C) does not match that of Fig. 5A; rather it shows some features in common with both. This is a consequence of the internal and cortical architecture (here meaning; the arrangement of solid and trabecular bone and of spaces within the structure as well as thickenings of the cortical layers) of Fig. 5B being warped as well as the external shape. The differences between this warped internal architecture and the actual architecture in the original macaque of Fig. 5A appear to exert a significant influence on the FEA results.

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Figure 5.  Von Mises strains mapped in male macaques subsequent to simulated second molar biting using simplified muscle forces (50 N applied directly inferiorly at the attachment area of the superior and deep masseter muscles). (A) Adult male and (B) another adult male lacking M3 are both loaded in the same way. When the skull (B) is warped using 56 landmarks into the configuration of (A), the resulting strain map (C), while similar, does not match perfectly the FE results of the target form (A), rather it shows some features in common with both (B) and (A).

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This problem can be approached by working with models that have the same or standardized ‘internal’ morphology (e.g. equal cortical bone thickness throughout (Gröning et al., in press) or solid models with no internal spaces and trabeculae). In these cases careful thought must be given to the question that is being addressed. In comparing models where the interior anatomy has been replaced with solid material, or the cortex is rendered as having even thickness, the magnitudes and distributions of stresses and strains will inevitably differ from what would be found in a more natural model. However, such models can be usefully applied to exploration of the consequences of variations in external form on load bearing in skeletal structures. Extensive studies of the applicability of such models to specific questions are needed, but one clear role is in the investigation of the interaction between external form and internal architecture in resisting applied loads.

While warping may have unwanted or unanticipated effects if not controlled for correctly, the applicability of the GMM toolkit to the creation of forms for functional analyses is potentially very exciting and clearly needs further development and assessment.

Multibody dynamic analysis (MDA)

GMM analyses also have potential in MDA through their application to motion analysis. For example, using MDA, muscle activation patterns in the head of Sphenodon were predicted from motion and musculoskeletal anatomy by Curtis et al. (2009). Of necessity, this study used CT data from one individual to build the model and motion from another. However, just as GMM offers a straightforward way of estimating mean or other interesting forms for functional analyses, it can also estimate motions such as the mean motion in an individual or sample and this could then be used to estimate the functioning of, for example, a model representing the mean form generated using GMM approaches as described above.

Motion analysis is achieved by breaking the motion into tiny steps, and representing it as a series of landmark configurations that vary over time. The analysis is then reduced to estimating and comparing trajectories of form change over time. This will be a repeated closed loop within form space for cyclical motions or an open curve for other motions. Such an approach (Slice, 1999; Adams & Cerney, 2007) allows the detailed quantification and analysis of complex motions involving many joints or complex motion at few joints (e.g. the jaw in chewing) and is especially useful in analysing the motions of structures where there are few ‘joints’ but rather a deformable surface with landmarks moving in complex ways such as the face during expression or speech (O’Higgins et al. 2002). There are subtleties to such an analysis that concern registration, its meaning and effect, and how motions are captured, equally spaced in time or in the form space (Slice, 2003). Of further interest, possibilities arise in examining covariations between motions and other variables such as forces or aspects of anatomy. This approach has received little attention to date, but it seems likely that, together with GMM approaches for skeletal form (above) it may well offer useful approaches to investigating form and function relationships through MDA at a level beyond the individual.

Analysis and interpretation of the results of biomechanical modelling

  1. Top of page
  2. Abstract
  3. Introduction
  4. Building interesting models
  5. Analysis and interpretation of the results of biomechanical modelling
  6. Concluding remarks
  7. Acknowledgements
  8. Author contributions
  9. References

Geometric and multivariate morphometric methods are potentially useful in comparing the results of functional modelling. With regard to MDA, motion analysis via GMM might be useful in comparing motions resulting from simulations with each other or with in vivo data, although as far as we are aware this has never been tried. With regard to FEA statistical approaches to assessment of results are more established, although in general the focus has been on comparing strains at comparable points rather than examining deformations as a whole. In this section we focus on the issues that arise in comparing FEA results using strains and consider the potential application of GMM to comparing deformations arising from FEA.

FEA: the comparison of strains

Applied loads deform a continuous body. The classic engineering approach to quantifying such deformation (defined as a change in form, i.e. size and shape, but not location or orientation) relies on the consideration of relative displacements and rotations, and computation of strains and other biomechanical parameters derived from these (Fagan, 1992). Strains are key measures in an engineering assessment of the response of a structure to loads and are commonly used in studies of skeletal biomechanics. Knowledge of local deformation (expressed as strains) is useful in assessing the risk of fracture or failure and has formed the basis of attempts to model and predict the adaptive response of bones to loads (e.g. Frost, 1964, 1987, 2003; Oxnard, 1993; Huiskes, 2000; Currey, 2002). Strains are usually measured at ‘points’ using strain gauges (e.g. Hylander, 1979a,b; Hylander & Crompton, 1986; Kay et al. 1986; Hylander et al. 1987, 2000; Hylander & Johnson, 1997; Ravosa et al. 2000; Kupczik et al. 2007), although a full-field assessment is possible using coats of photoelastic material (e.g. Evans & Lissner, 1948) and, more recently, laser interferometry, which measures and maps strain magnitudes and directions over moderately large regions of bone surfaces (Gröning et al. 2009; Panagiotopoulou et al. 2010a,b).

The strain field is a continuum and the accuracy with which it is represented depends (amongst other things) on the resolution of the FE model. The size (or number) and mathematical complexity of the finite elements determine how well the model predicts deformations throughout the body. Small elements give higher resolution, whereas large elements lead to a coarser approximation of the predicted deformation. However, most craniofacial FEAs have used large numbers of small elements and so issues related to detail of modelling and lack of convergence on a stable solution do not arise. Using principal strain magnitudes or directions, or strain quantities derived from these (e.g. von Mises strain), provides a way of focusing on a particular aspect (parameter) of the deformation that has meaning in terms of engineering principles (e.g. fracture risk) or has special significance in relation to the control or prediction of a biological event, such as bone remodelling. Ideally, the choice of parameter should be explicit, related directly to the question at hand.

Strain maps arising from FEA are not only used to examine local strains and strain gradients in relation to fracture risk or remodelling, but are also frequently interpreted as a whole, e.g. to compare general aspects of deformation under load as in Figs 4 and 5. That general deformations are of interest to the functional morphologist is illustrated by the paper in this volume by Ross et al. (in press) which concerns the response of the orbit to chewing and by extensive functional studies of human and primate mandibular form, function and evolution (e.g. Demes et al. 1984; Hylander, 1984; Daegling, 1990, 1993; Hylander & Johnson, 1994). These studies have examined the link between morphological adaptations and stresses and strains resulting from large scale bending and twisting of craniofacial skeletal structures during masticatory function. Quantitative methods for examining local and global deformations and the relationships between these could advance understanding of mechanical performance at different scales. Further, as bone generally adapts to functional loads by minimizing material while maintaining sufficient structural stiffness in relation to normally encountered loads, studies of large-scale deformation under different loads could potentially lead to better understanding of the relationships between internal and external bone architecture animal function, behaviour and ecology. This was extremely difficult to investigate prior to the advent of virtual modelling because deformations are typically very small and could only reasonably be measured for a few exemplar real bones in vivo using a small number of strain gauges that sample only what is going on in a tiny area underneath them.

The performance of a single object under different loads can be assessed quantitatively by comparing strain parameters at key points on the object. There are no judgements to be made regarding the equivalence of location of these sampling points; they are known absolutely with respect to the same unloaded object. However, such analyses consider strains at relatively few points. This means that to compare the performance of objects as a whole, strains from throughout the model have to be considered simultaneously. Multivariate approaches could feasibly be applied in such analyses but there are issues that arise from a statistical perspective. These include the distributional properties of strain magnitudes and directions (angles) and the shape space for such data (Rohlf, 2000). Additionally other subtle issues need full consideration and further work; these include the effects of different meshings of objects or scalings of objects and forces (Dumont et al. 2009; Herrel et al. 2010) on eventual results.

However, in biology an issue of tremendous importance arises that is not common in engineering applications of FEA; variation. The biologist, is concerned with samples that vary in form and so there is a need to compare FE results between models and between groups of models representing biological groups. Additional issues arise when comparing strain magnitudes and directions between different objects (or different meshings of the same object) subject, for example, to similar loadings. These include the need for satisfactory registration and for the identification of anatomical equivalences between them. The matching of equivalent points between objects is not trivial (Oxnard & O’Higgins, 2009); at some points, equivalence is more certain than at others (e.g. prosthion is readily identified, whereas points over the central portion of the parietal bone are more ambiguous).

To compare the absolute strain response throughout two or more objects, a mapping of the objects into each other is required and the comparison is dependent on this mapping. One simple approach to mapping between objects is to calculate and compare strains at equivalent anatomical landmarks. An analysis using 56 anatomical landmarks in two crania is presented in Fig. 6. Each is loaded in two ways and the minimum principal strains at the landmark locations are submitted to PCA. In this analysis, the first PC differentiates the robust from the gracile australopith principally because of the overall difference in strain magnitudes; the robust australopith shows generally absolutely smaller (less negative) minimum principal strains than do gracile. The second PC differentiates incisor from molar bites, indicating that there are aspects of the anatomical distribution and values of minimum principal strains that are shared between bites. This analysis is based only on the strains at the equivalent points between specimens and uses only the ‘compressive’ strains. As such it relates to specific, local aspects of the deformation resulting from FEA, not to the whole deformation; a point that must be kept in mind when considering whether the analysis is appropriate to the question at hand.

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Figure 6.  PCA of minimum principal strains (A) recorded at 56 landmark locations in two different species (Australopithecus africanus and Australopithecus boisei) during two different load cases (incisor and molar bites). (A) Landmarks are shown in white; (B) wireframe connecting these.

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Over most regions, point equivalences cannot be unequivocally defined on anatomical grounds and so comparison is not possible without a mathematical mapping of equivalences between objects (Oxnard & O’Higgins, 2009). Where necessary, a mapping of strains between objects could be carried out using a convenient function such as TPS grounded in matched landmarks and, possibly, semilandmarks. These matched strains between objects could then be compared using univariate or multivariate approaches. Such comparisons would suffer the same practical and statistical issues outlined above for comparisons between different load cases applied to the same objects in addition to issues related to the effects of choice landmarks and mapping function. There is an additional problem in that the two different objects will most often have no ‘true’ registration between them and this causes problems for comparing the directions of principal strains; they cannot be referred back to a common unloaded model or a shared reference frame. This means that the orientations of principal strains can only be compared between objects relative to equivalent structures on each or after declaring a common registration. Different registrations will, however, lead to different impressions of the differences in deformation, although the impact of this source of error is likely minimal where objects are very similar in shape. The problems of using strains to understand how skeletal structures deform globally and to statistically compare local and global deformations between models are significant and merit consideration of alternative approaches.

FEA: alternative approaches to comparing deformations using GMM

There are many other ways in which changes in form or differences in form can be assessed (O’Higgins & Johnson, 1988; O’Higgins, 1997) and the best understood and most appropriate in many circumstances is GMM (Bookstein, 1978, 1991; Dryden & Mardia, 1998; Rohlf, 2000; Zelditch et al. 2004; Slice, 2007). GMM was developed specifically to characterize form differences and covariances with form using landmark (and semilandmark) data. The methods are directly applicable to the task of characterizing the changes in form that arise during loading of a bone. Because the geometry of the landmark configuration is preserved at all stages of analysis, it is feasible to actualize results as images. An example might be the mean change in form of a sample of bones under ‘identical’ loads which could be shown as a pair of diagrams (mean unloaded and mean loaded forms) or as a single diagram using a transformation grid. An alternative that has special mechanical significance is to represent this as a strain map. The preservation of geometry in GMM analyses means that at any stage, strain magnitudes and directions can be computed from a pair of objects (reference and target, e.g. loaded and unloaded). Nothing is lost as long as the landmarks and semilandmarks are of sufficient density to allow deformations at the scale of interest to be characterized. This is analogous to the issue of finite element size raised earlier in the discussion of strains, but differs in one key aspect; the nodes of finite elements are not necessarily equivalent between objects, whereas landmarks and semilandmarks are intended to be so. However, with current FE models now typically defined with millions of nodes and elements, in practice GMM landmark datasets will rarely if ever have a similar density of sampling points, except where small regions are being compared.

The limit on density of matched points is not only a practical one in terms of computer software and anatomical knowledge but also a philosophical one; there is often no such matching in reality – an issue that becomes very apparent at finer scales of bone architecture. Thus, for example the concept of matched or equivalent points on trabeculae (and of course parts of trabeculae as in FEA) is elusive. Particular trabeculae may have similar anatomical relations, functions or developmental histories but they are unlikely to share all equivalences and so subjective decisions have to be made regarding matchings. The decisions will affect the outcome of comparison and so the matchings need justifying in terms of the question at hand. This is an area that requires further investigation and experiment. It is important to realize that this is not a problem that affects GMM alone. The limits imposed by the reliability and basis of matchings between objects equally affect our ability to compare the outcomes of FEA between different objects when using strain or landmark-based methods such as GMM.

In comparing deformations of bones or regions of bones, GMM approaches have one great advantage over strain-based ones; the landmark data are treated as a whole. Thus, while the density of landmarks relates to level of detail (resolution) in the comparison, it is always a comparison of deformation of the object or region as a whole. In contrast, strain-based analyses that sample local deformations at disparate points inevitably lose, at the outset, the integration of the whole. The strains relate only to deformation in the immediate vicinity of the point at which they are sampled. Where all strain magnitudes and directions computed for all elements are used, then full information about the deformation is preserved. In contrast, the strain vectors at disparate points do not tell us about how the configuration as a whole is deforming but rather what is happening in the immediate vicinity of each of those points.

The simple example of Fig. 7 serves to illustrate how GMM methods open up novel ways of relating loading to deformation. FEA is used to load a gracile australopith cranial reconstruction, simulating incisor or second molar bites. The deformation resulting from such loading is characterized using the same 56 landmarks as were used in Fig. 6. The configurations of landmarks from the unloaded and two loaded cases are then submitted to generalized Procrustes analysis (GPA) followed by PCA of form. The plot in Fig. 7 accounts for 100% of the total variance and so completely describes how these landmark configurations deform between unloaded and loaded states. The deformations due to molar and incisor bites are visualized in Fig. 7 by warping a surface model and transformation grids using triplets of thin plate splines. The deformations of the grids are useful in understanding how the skull deforms in each bite and the deformed surface models could potentially be used in conjunction with the undeformed geometry to compute strains if so desired, although this needs further investigation. Clearly there will be significant error in this because so few landmarks have been used but this can be reduced to some degree by adding further landmarks. While this simple example is limited, the real value of this approach is that it is readily extensible to comparisons between different specimens and has powerful statistical underpinnings that, as noted above, open up the possibility of relating deformations between specimens and species to covariates of interest.

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Figure 7.  Form space PCA and visualization of the deformations resulting from FEAs simulating molar (m2) and incisor bites in the two australopiths (see legend to Fig. 4 for boundary conditions). The same 56 landmarks used in Fig. 6 were included in this analysis. 100% of the total variance is explained by PC1 and PC2. Vectors connect the unloaded and loaded forms and these are visualized as warped surface models and transformation grids computed using triplets of thin plate splines. Deformations are shown ×100 to facilitate interpretation.

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In comparing deformations of different objects, the same problems apply as to the comparison of the same object under load in that there has to be a mapping of equivalent points between the two that is justifiable in terms of the question being investigated. This requires careful thought in planning landmark sets (Oxnard & O’Higgins, 2009). A further issue arises in that the vectors of deformation will occupy different locations in the form space. Thus, in the analysis of Fig. 7 a standard GMM analysis of form differences was carried out. The deformations are represented by the vectors connecting the undeformed specimen and the deformed ones. If other crania had also been deformed through FEA, then they would occupy different regions of form space to the extent that they differ in form and deformation. In this circumstance, one approach to comparing deformations is to work with the vectors themselves, comparing their lengths and directions and, if working with samples, to use a permutation test to assess significance. Visualizations of the differences in deformation could be readily achieved by, for example, adding the relevant Procrustes shape variables (residuals) to the mean.

Finally, the biologist is also concerned to understand, from a functional perspective, why structures are as they are; in other words, how form and function covary. GMM analyses of the deformation of FE models facilitate such analyses in that they offer potential in assessing covariances between global or local bone deformations under load and ecological and other interesting factors while taking account of important variables like phylogeny or geography.

Concluding remarks

  1. Top of page
  2. Abstract
  3. Introduction
  4. Building interesting models
  5. Analysis and interpretation of the results of biomechanical modelling
  6. Concluding remarks
  7. Acknowledgements
  8. Author contributions
  9. References

The combination of GMM, MDA and FEA methods opens the door to a statistical biomechanics of form that can potentially transform comparative, evolutionary or ontogenetic studies of function. All methods depend on virtual reconstruction of the structures under analysis and generate informative visual as well as quantitative data regarding form and function. GMM offers great potential in both the preparation of models for functional simulation and analysis of results. Much of what has been presented here by way of example studies is tentative, they are small and are intended to illustrate possibilities rather than to provide definitive insights. They illustrate what we know and serve to underline the many issues that require further mathematical, statistical, biomechanical and anatomical development. An important endeavour in the immediate future will be the clarification of these issues at fundamental theoretical (mathematical and statistical) and at more practical (biomechanical and anatomical modelling) levels. Sensitivity analyses will be important in guiding the development and application of these combined approaches. If the initial promise holds, then we are at an exciting juncture in the historical development of statistical mechanical studies of skeletal form and function.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Building interesting models
  5. Analysis and interpretation of the results of biomechanical modelling
  6. Concluding remarks
  7. Acknowledgements
  8. Author contributions
  9. References

We are grateful to the Anatomical Society of Great Britain and Ireland and to the Hull York Medical School for supporting the Craniofacial Biomechanics Symposium where this work was first presented and for commissioning this special volume. Many of the participants there contributed greatly to discussions that led to this paper and we should like to thank them all. We are also grateful to colleagues who have contributed in various ways through discussion, critique or practical experimentation and trials of analyses in our laboratories; R. McNeill Alexander, Fred Bookstein, Janine Chalk, John Currey, Neil Curtis, Kornelius Kupczik, Charles Oxnard, Junfen Shi, Lee Page, Dennis Slice, Amanda Smith, Jack Wharton and Ulrich Witzel. The work is supported by research grants from The Leverhulme Trust (F/00224), BBSRC (BB/E013805; BB/E009204), and by EVAN (MRTN CT-2005-019564) and PALAEO (MEST-CT-2005-020601) EU: Marie Curie Initiatives, Vienna, Hull and York.

Author contributions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Building interesting models
  5. Analysis and interpretation of the results of biomechanical modelling
  6. Concluding remarks
  7. Acknowledgements
  8. Author contributions
  9. References

This review arises from joint research projects that involve the authors in different combinations. P.O.H. is the lead author and with S.C., L.F., F.G. and M.J.F. is responsible for the biomechanical analyses presented here. R.P. and J.L. developed the software we have used to carry out our analyses. L.F. and F.G. were responsible for most of the virtual reconstructions and warpings. The key ideas in this paper arose during our frequent exchanges in the course of our joint work and all authors contributed to the drafting of this review.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Building interesting models
  5. Analysis and interpretation of the results of biomechanical modelling
  6. Concluding remarks
  7. Acknowledgements
  8. Author contributions
  9. References