New methods drive to new ideas

​In recent years, FEA models and the respective results have been compared in a predominantly qualitatively way, by comparing the stress or strain distributions. Visual comparison using only the stress or strain distribution of contour plots can be a good option when the number of models to is small and the results are distinctive enough between the analysed models (See Rayfield, 2007 for a review of FEA models in vertebrates). When we are solving a model using Finite Element Analysis, the results are obtained via a distribution map. These internal distributions of the forces -called stress- and ratios of deformation -strain- appear in the inner regions of the models due to the action of external forces. Computational biomechanics allow to model how different forces act on a biological structure like bones. The models of the bony structures are subdivided in small pieces called "elements" using a mesh where the equations are solved. Then, theoretical forces are applied to the model and the stress and strain values of each element are recorded and mapped in a coloured map called stress or strain distribution which enables a qualitative comparison between different models. Of course, this "coloured map" is related with numerical values. Specifically, with the values obtained from the solution in each element of the mesh representing stress or strain.

​​We published a work in 2015 “Finite Element Analysis of the Cingulata Jaw: An Ecomorphological Approach to Armadillo’s Diets” with several FEA models of armadillo mandibles (figure 1) in order to study the biomechanical performance during chewing and trying to relate it with the dietary preferences of each specie (Serrano-Fochs et al., 2015). Although quantifying stress or strain data at specific points of the model has been useful in previous ecomorphological analyses of the mammal mandible, in our work we could not find interesting patterns to be considered for analysing differences between models.

Figure 1 Distribution of von Mises stress in the eleven armadillos’ mandibles separated by diet: Omnivore/carnivore and specialist or generalist insectivore. Modified from Serrano-Fochs et al (2015).

At this point, we decided to start developing new methods for quantifying the stress state from FEA of the armadillo mandibles. To analyse the stress values in a quantitative framework could be complicated, as these elements have different size in the same mesh. for this reason, we published recently the work "Accounting for differences in element size and homogeneity when comparing Finite Element models: Armadillos as a case study" (Marcé-Nogué et al., 2016). In this work we proposed a method to obtain the average mean and median of the distribution of these stresses in a Finite Element model weighting for the differences in elements size (mesh-weighted values). On the other hand, we proposed a procedure to check whether the meshes used to generate the elements provide accurate results to be used later in statistical analysis (quasi-ideal mesh). In other words, a consistent way to be sure that the stress values can be used as a proxy of the relative strength of vertebrate structures in a comparative framework for comparing the obtained mechanical results of different models between them. Figure 2A is an example of good practice. Boxplots of Von Mises stress distributions can be displayed only when Quasi-Ideal Meshes are assumed for all the different Cingulata mandibles analysed. It enables a qualitative comparison between species and, in this case, diets. The computation of average values or the display of stress is wrong unless all the elements of the mesh have the same size or have been corrected with the weight-meshed values. Mathematically speaking, the heterogeneity (or uniformity) in the size of the elements of the mesh can conditionate the results of the average value.

Figure 2 (A) Boxplots of Von Mises stress distributions when Quasi-Ideal Meshes (QIM) are assumed for the cingulate mandibles analysed (B) PCA based on the correlation matrix. The loadings for each variable are coloured according with the range of stress they represent, with reddish colours for high level of stress, and bluish for low levels. X-axis: PC1. Y-axis: PC2. Modified from Marcé-Nogué et al. (2016, 2017).

Recently in another work, we published "The intervals method: a new approach to analyse finite element outputs using multivariate statistics" (Marcé-Nogué et al., 2017a). This paper was a continuation of our previous works with armadillo mandibles where we were developing new methods to help the interpretation of FEA results. In this paper, we proposed a new method, named the intervals’ method, to analyse data from finite element models in a comparative multivariate framework. The method incorporates element volumes to provide an additional way of quantifying and comparing FEA results, thus avoiding the problems with the mesh. This method could allow for considerably more effective comparisons of finite element models, and maybe more precise distinction between dietary traits. As a case study once again, several armadillo mandibles were analysed, showing that the proposed method is useful to distinguish and characterise biomechanical differences related to diet/ecomorphology. The intervals’ method consists of generating a set of variables, each one defined by an interval of stress values. Each variable is expressed as a percentage of the area of the mandible occupied by those stress values. Afterwards these newly generated variables can be analysed using multivariate methods (see the biplot in figure 2B). Applying this novel method to the biological case study of whether armadillo mandibles differ according to dietary groups, we showed that the intervals’ method is a powerful tool to characterize biomechanical performance and how this relates to different diets. This allowed us to positively discriminate between specialist and generalist species and, moreover, something that it is very important when working with FEA: We show that the proposed approach is a useful methodology not affected by the characteristics of the finite element mesh. 

​Summarizing, these post-process methods are specially interesting because 1) open a new and imaginative way to compare large number of FEA models combined also with phylogenetic data (See Marcé-Nogué et al., 2017b in primates or Zhou et al., 2019 in ungulates) and 2) they can be used as an input data for the use of both supervised and non-supervised machine-learning (ML) algorithms. The use of ML is a powerful way that allow to classify sets of data as well as a predictor and has an incredible potential in inferring behaviours of extinct taxa. See Püschel et al., 2018 as an example of using the intervals method combined with ML algorithms.

​References

Marcé-Nogué, J., De Esteban-Trivigno, S., Escrig, C., Gil, L., 2016. Accounting for differences in element size and homogeneity when comparing Finite Element models: Armadillos as a case study. Palaeontol. Electron. 19, 1–22. https://doi.org/10.26879/609
Marcé-Nogué, J., De Esteban-Trivigno, S., Püschel, T.A., Fortuny, J., 2017a. The intervals method: a new approach to analyse finite element outputs using multivariate statistics. PeerJ 5, e3793. https://doi.org/10.7717/peerj.3793
Marcé-Nogué, J., Püschel, T.A., Kaiser, T.M., 2017b. A biomechanical approach to understand the ecomorphological relationship between primate mandibles and diet. Sci. Rep. 7, 8364. https://doi.org/10.1038/s41598-017-08161-0
Püschel, T.A., Marcé-Nogué, J., Gladman, J.T., Bobe, R., Sellers, W.I., 2018. Inferring locomotor behaviours in Miocene New World monkeys using finite element analysis, geometric morphometrics and machine-learning classification techniques applied to talar morphology. J. R. Soc. Interface 15, 20180520. https://doi.org/10.1098/rsif.2018.0520
Rayfield, E.J., 2007. Finite Element Analysis and Understanding the Biomechanics and Evolution of Living and Fossil Organisms. Annu. Rev. Earth Planet. Sci. 35, 541–576. https://doi.org/10.1146/annurev.earth.35.031306.140104
Serrano-Fochs, S., De Esteban-Trivigno, S., Marcé-Nogué, J., Fortuny, J., Fariña, R.A., 2015. Finite Element Analysis of the Cingulata Jaw: An Ecomorphological Approach to Armadillo’s Diets. PLoS One 10, e0120653. https://doi.org/10.1371/journal.pone.0120653
Zhou, Z., Winkler, D.E., Fortuny, J., Kaiser, T.M., Marcé-Nogué, J., 2019. Why ruminating ungulates chew sloppily: Biomechanics discern a phylogenetic pattern. PLoS One 14, e0214510. https://doi.org/10.1371/journal.pone.0214510