The morphology of the mandible does not provide a large number of distinctly identifiable landmarks to attain comprehensive coverage of its surface, potentially leading to loss of important phenotypic information. However, even if achievable, dense landmarking would not be sufficient unless complemented with reasonable confidence of landmark homology (correspondence) across specimens. We recognize two main approaches for establishing correspondences and landmarking of 3D surface meshes; both use a reference template (atlas) with the landmarks of interest already identified on it. The first approach performs a rigid alignment of the template mesh to match the target, followed by deformable registration to refine the match, and then transfers the landmarks from the atlas to the target mesh. Examples are ALPACA35,36 and MeshMonk37,38, which mainly differ in their non-rigid registration algorithm. Variants of this approach abolish landmarks altogether and achieve correspondences between the vertices of the meshes directly14,39.
The second approach needs digitization of a (relatively small) subset of the landmarks on both meshes, then performs a TPS warping of the template, driven by this subset of landmarks, and transfers the remaining points to the target mesh, usually by projection on the mesh surface. Thus, the first approach is essentially a mesh-to-mesh registration, whereas the second is a TPS warping of point configurations. This method belongs to the geometric morphometric toolbox17,28,40 and is usually followed by sliding of semilandmarks to enhance geometric correspondence25,26 and mapping of the specimens in a shape space via generalized Procrustes alignment41 and PCA.
The shape space is the final goal of all methods, as it represents a statistical shape model13,14 which describes population variability and can be used both as a reference for testing novel shapes and as a generative source for creating plausible anatomy. We consider GM methods preferable, due to their solid statistical foundation and excellent visualization tools. Although highly dense models are produced from methods that establish correspondences at the mesh vertex level13,14,39, the anatomical correspondence (homology) is not guaranteed26,42.
Studies of the mandible with dense landmarking are scarce. Most studies limit the landmarks on the anterior and posterior ramus ridges, the mandibular notch, the inferior outline, and the symphysis outline on the midsagittal plane, in addition to ubiquitous conventional landmarks, such as Gonion, Gnathion, Coronoid and the condylar poles5,6,7,9,43,44,45,46. The total number of landmarks ranges from below 20 (e.g.45) to above 100 (e.g. 11344, 30146 or 10005); however, seldom is the mandibular surface between ridges landmarked5,8,23,36.
In creating an atlas of shape variability, the design of the template is a key factor24. The base mesh does not need to be detailed or be one of the meshes of the sample; indeed, simpler geometries sometimes work better24. We used a simplified symmetric mandibular mesh, expanded outwards by 1.5 mm, to avoid the common problem of the landmarks projecting on the wrong surface24, especially in the area of the gonial angle, where the ramus can be thin and the gonial angle everted. We aimed for a large number of landmarks, dispersed evenly over the whole surface, to capture both the shape of the main ridges and the smooth areas in-between. The number of fixed landmarks was small, limited to the condyle poles, the coronoid processes, Gonion and Gnathion. Gonion is a problematic landmark, showing high identification error, both in 2D and 3D digitizations47,48,49. Although we could set this as a sliding semilandmark, or remove it altogether, we opted to retain it as a sentinel between the ramus and the corpus, to avoid the curve and surface semilandmarks from invading the wrong area. However, we located it algorithmically using a clearly defined geometric procedure, to reduce identification error (Supplementary Table S1). Gnathion was similarly located, as the farthest point from the condyle on the midsagittal plane. Locating landmarks algorithmically avoids subjectivity, and ensures repeatability and validity, although biological homology is debatable42. The curve semilandmarks were few and sparsely dispersed, to allow them to adjust by sliding, since a very high density effectively prohibits sliding and reverts to equidistant sampling. The surface semilandmarks were dispersed on the template mesh via a diffusion algorithm, to ensure an initially even distribution.
Our sample was a convenience sample, imaged for various reasons, most commonly for dental implant planning and third molar pre-extraction evaluation. Although there is no assurance that the average coincides with the average of the population, research has shown minimal effect of including even extreme cases50. Sample size was adequate for assessing average shape and shape variability18,19. Sex grouping was based on birth-assignment. Although several factors can affect the phenotype, such as genetic variations, sex chromosomes, epigenetic variation, hormones, environmental factors, and others51, it was not possible to investigate them in the present sample.
Landmarking was accurate, as shown by the repeated digitizations, because the curves were placed on well-defined ridges and the remaining surface points were located by a TPS warping of the template followed by sliding. Fixed landmarks were placed by automated heuristics (e.g. Gnathion, Gonion) further minimizing subjective identification52. The only curves that required full human intervention were the alveolar buccal and lingual curves.
Almost 80% of the shape variance was described by the first 12 PCs, the first 3 of these describing 49%, so most of the shape variability was included in only a few PCs, comparable to previous work7,36. Van der Wel et al.11 report a larger spread of shape variance among the PCs; this can be attributed to their sample comprising patients treated by orthognathic surgery, therefore potentially more extreme cases, and to segmentation artifacts in the area of the teeth, due to metallic orthodontic appliances. The number of landmarks is a significant factor affecting the percentage of shape variance distributed between the PCs. Studies with a few landmarks report a large fraction of variance in the first few PCs because shape is more coarsely measured (e.g. 14 landmarks: 67% shape variance in the first 2 PCs53, 13 landmarks: 61% variance in the first 2 PCs9,45).
The shape patterns were similar to those reported elsewhere, mainly related to mandibular width, angulation between the ramus and corpus, inclination of the symphysis and prominence of the gonial angle. PC1 described mandibular width variation, in relation to ramus height and corpus length (Fig. 4) whereas PC2 mainly described the ramus-corpus angulation. The same primary patterns are seen in the work of van der Wel et al.11, and potentially Fournier et al.36 and Kim et al.39, although the visualizations in those publications do not facilitate a direct comparison. The shape patterns obviously depend on several factors, including ethnicity, age, sex, and degree of edentulism. Our sample was mono-ethnic, equally divided by sex, and of low edentulism prevalence (average number of teeth missing: 2.2, Table 1), so the results need to be assessed under these conditions.
A significant effect of edentulism on mandibular shape has been observed9,45,54, which we detected here, but only in the male group and the pooled sample. In addition to reduction in the height of the alveolar process, loss of teeth was associated with retraction of the anterior alveolar area with relative prominence of menton, increase of the gonial angle and intercondylar distance, and posterior inclination of the ramus (Supplementary Fig. S6). We mention these associations with great caution, even though they agree with previous reports overall9,45,54, since our sample contained very few patients with many (> 5) missing teeth, did not contain completely edentulous mandibles, and alveolar resorption had not progressed significantly in several patients.
A clear sex difference was evident, both regarding size and shape, as expected for an adult sample10. Centroid size, computed from the mesh vertices, was 8.9% larger in males; however, mesh volume was 25% larger. A discrepancy between the two is expected because, with scaling, volume increases to the third power, whereas centroid size to the first power. However, the expected volume change would be larger, at 29% (1.0893 = 1.291). This can be explained by the mandible’s shape and the position of the centroid, which lies in empty space, on the midsagittal plane, at the level of the molars. The distance of the landmarks relative to the centroid is affected mostly by variation in mandibular width, and not so much by variation in ramus height, ramus anteroposterior width, or ramus and corpus thickness, factors that significantly affect volume. The superimposition of the size-adjusted averages (Fig. 5) shows differences in shape that explain this discrepancy between volume and centroid size ratios, e.g. a higher ramus and a more pronounced gonial and symphyseal area in the males.
The mesh-based size and volume measurements included the tooth regions. Any sex differences assessed by these variables could therefore be confounded by an unequal number of missing teeth between the groups, or by sex differences in tooth size. The first factor was not pertinent here because the degree of edentulism was similar between the groups (Table 1), but the second factor could be relevant, as males tend to have larger teeth than females55,56. However, the dentition area is small relative to the whole mesh, and tooth-size sex differences are also too small for them to be of concern here. In any case, these measurements are of low reliability, additionally because volume segmentation was often uncertain due to streaking artefacts.
The landmark-based centroid size difference was similar to the mesh-based difference (8.3% vs. 8.9%, respectively) and is considered more reliable as it is not affected by inclusion of teeth. Size differences have been noted in all previous studies of adult samples. Franklin et al.7 reported almost the same centroid size difference (7.8%) for their sample of 30 mandibles. Vallabh et al.57 list various linear measurements, of which ramus height shows the largest relative difference between sexes (14%) whereas width measurements are comparable to our centroid size ratio (intercondylar width: 5.6% and intergonial width 8.7%). This difference in intercondylar and intergonial widths is also reflected in the shape difference we detected (Fig. 5). Kranioti et al.58, in a sample of the same ethnic origin as ours, report a comparable inter-gonial width difference of 7.6%, giving a sex classification accuracy of 71%.
Shape differences were less pronounced than size differences, as seen when comparing Fig. 3 and Supplementary Fig. S5. In addition to a higher ramus, more pronounced gonial and mental areas, males showed a wider inter-gonial distance. Such sex differences have been noted by other investigators as well7,8,53.
One of the traits considered a male characteristic is gonial eversion, presumably arising from a strong masseteric attachment. However, evidence suggests that this difference is lower than initially assumed29,59. To overcome the qualitative nature of this trait, Oettlé et al.59 applied GM methods confined to the posterior ramal and gonial areas and obtained quantitative data. Although they detected differences between males and females, mainly in the extent and location of the eversion, the accuracy of sexing was below 75%. Our sample showed a larger inter-gonial width in males (Fig. 5), but there was no clear gonial eversion when examining the gonial area. On the contrary, the female outline curved towards the lingual and the male outline was straight (Fig. 7). A difference in condylar axis angulation was also observed.
Another alleged dimorphic trait is ramus flexure, an “angulation of the posterior border of the mandibular ramus at the level of the occlusal surface of the molars”30. Although the initial results for this trait were positive, later evidence is conflicting10,29,60,61. Unfortunately, this trait is qualitative and could not be incorporated into the GM analysis. Visual inspection of the posterior ramus border did not show flexure in our sample (Fig. 7).
Discriminant analysis based on the first 2–4 PCs in form space was very successful in assigning subjects to their correct group (91–93% accuracy). However, this was achieved mainly due to size differences, similarly to previous research using linear measurements between anatomical landmarks10,58. Classification accuracy was higher than that reported by previous studies using conventional linear and angular measurements. Kranioti et al.58 report an accuracy of 80% based on 2 linear variables in a Greek population. Other reports vary, from around 75% to almost 90%, using univariate or multivariate models62,63,64,65,66,67,68,69. Franklin et al.70 report an exceptionally high accuracy of 95%, but this needs to be interpreted with caution, as 10 variables were applied on a sample of 40 mandibles, suggesting a danger of overfitting.
We did not detect an age-related shape change, even though the degree of edentulism was related to both age and shape. This is in contrast to some previous reports, e.g., Costa-Mendes et al.53. As noted above, our sample size was relatively small in the higher and lower age bins; however, we had a similar degree of edentulism in both sex groups, whereas this was not recorded in53 and could have biased the results.
The atlas (average form and variability patterns) can be used clinically as a guide for planning surgical mandibular reconstruction. In cases of missing or deformed parts, the intact mandible can be used to fit the model and obtain a plausible anatomical form of the remaining. Virtual reconstruction based on statistical shape models has been demonstrated with good results12,15,16. It is evident that dense landmarking, for detailed representation of the anatomy, and matching of the population from which the model was constructed to the patient characteristics, are essential factors of success.
Limitations
Future research on mandibular shape variability and sex differences could improve on this study in several ways. First, the template could be augmented with more points on the condylar head, so variability in condylar form could be investigated in detail. However, a GM analysis, focused on that region, instead of encompassing the whole mandible, would probably be more appropriate. Such an analysis could be performed on the mental region, as well, as this has also been reported to show sex differences. A notable challenge with regional analyses lies in delineating the region to be studied, particularly when clear morphological boundaries are lacking.
A second limitation was related to sample composition. Ours was of Greek ethnicity, and contained few patients in the tail ends of the age distribution, making it difficult to evaluate age effects. Also, time of tooth loss was not available. The effect of tooth loss on alveolar bone resorption, and subsequent functional issues that depend on prosthetic rehabilitation are expected to affect mandibular shape, but could not be evaluated here. Our analysis showed that tooth loss and sex were the main factors related to mandibular shape; since tooth loss is heavily tied to age, it is not easy to uncover a potential age-related effect of sex. A sample with minimal edentulism over a wide age range would be valuable in this respect.
