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.

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.

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.