Of all 100 LCRs analysed in this study, not one had all four of Sassouni’s described planes converge precisely at point O. This contrasts with Sassouni’s original study where 16 of his sample had all four planes converge at point O. This result may be explained by a difference in the samples selected between Sassouni’s study and this study. Sassouni’s original sample were 7–15-year-old Mediterranean Caucasians. The sample used in this study were aged between 13 – 48 years of age and of any ethnicity. Therefore, point O which is required as a centre of which to draw either the posterior arc or the anterior arc was a theoretical point used in this study, calculated automatically by Dolphin Imaging Plus™ Software by using an averaged angle of the four marked horizontal planes.
The results show that the mean difference for the estimated position of Go (x) was − 5.72 mm while Go (y) was − 2.64 mm relative to the traced position of Go (x,y). In context, these differences can be considered small, and therefore, the application of the Sassouni+ analysis is reasonably accurate in estimating the position of Go. The (y) co-ordinate estimation was more accurate than the (x) co-ordinate and this is to be expected as the intersection between the posterior arc and the MdP (Go-Me) was used to determine the estimated position of Go. As Go forms one of the posterior landmarks on the MdP, it is not surprising the vertical (y) co-ordinate was more accurate than the horizontal (x) co-ordinate.
The results also show that the mean difference for the estimated position of Pog (x) was − 2.35 mm while Pog (y) was − 8.52 mm. The difference in Pog (x) can be considered small; however, the difference in Pog (y) is relatively large; therefore, the application of the Sassouni+ analysis is reasonably accurate in estimating the horizontal position of Pog but inaccurate in estimating the vertical position of Pog. In contrast to the estimated position of Go, the (y) co-ordinate was far less accurate than the (x) co-ordinate, with the analysis consistently estimating the position of Pog (y) to be more inferior than the traced position of Pog (y) was. This is also to be expected, as the intersection between the anterior arc and the MdP (Go-Me) was used to determine the position of Pog. The traced position of Pog would always be expected to be above the MdP as it is not a landmark that is used in determining the MdP. Pog should always lie superior to menton (Me), which was one of the points used in determining the MdP in this study.
The Euclidean straight-line distance analyses show a 7.89-mm mean difference between the estimated position of Go and a 11.15-mm mean difference for Pog. The difference in accuracy between the two can be explained by the consistently inferior estimation of Pog (y) by the methodology of this study. Overall, it can be concluded that the application of Sassouni’s analysis can be reasonably accurate in determining the position of both Go and Pog when the position of the MdP (Go-Me) is already known.
There was a positive correlation coefficient of r = 0.381 between the anterior cranial base length (SN) and the length of the mandibular body (Go-Me). This result is also to be expected, as the length of the anterior cranial base increases, it would be expected that the length of the mandibular body (Go-Me) also increases to maintain facial proportion. Further regression analysis showed the length of the anterior cranial base (SN) could be used as a predictive variable for the length of the mandibular body using the equation 22.65 + 0.5426x, where x = length of the anterior cranial base (SN) in millimetres. This has application as a potential starting point in determining a predictive length of mandibular body (Go-Me) where the mandible is missing, however, caution has to be taken in solely using the cranial base as a predictor of mandibular length as it has been found that cranial base length correlated strongly with maxillary length but weakly with mandibular length .
The results also show that there is a significant correlation between convex-shaped palates and oblique-shaped mandibles. The oblique mandibular shape described by Sassouni shows typical features of a backward clockwise growth rotation pattern as described by Björk . However, if the palate was concave or horizontal, then there was no association found with the shape of the mandible. This is in part likely due to the method of the study. With the method of the study, the author AO had to use ‘best fit’ by comparing the shape of the palate on the LCR with the diagrams described by Sassouni (, which introduced subjectivity in the assessment and potential bias. Furthermore, the concave and horizontal shapes of the palate as described by Sassouni were relatively similar to each other when compared with the convex shape of the palate, which had a much more characteristic shape that was readily identifiable. This meant that there was greater uncertainty when classifying a two-dimensional radiographic image of the palate on a LCR as either concave or horizontal.
Overall, the accuracy of the Sassouni analysis in estimating the positions of Go and Pog is surprisingly good when considering the sample selected. The sample consists of 100 LCRs taken in a secondary care orthodontic department and therefore an assumption could be made that most of the selected sample would likely have a greater degree of malocclusion with facial skeletal disproportion relative to the general population. This is because those who are considered to be in great need of orthodontic treatment  with more complex malocclusions requiring interdisciplinary care  are more likely to be referred to orthodontic secondary care in the UK. Bearing this in mind, the expectation would be that the Sassouni analysis would be inaccurate in estimating the positions of Go and Pog as most of the sample population would not be expected to have balanced facial proportions or a normal occlusion.
Interestingly, an exploratory finding is that in all 8 cases where the estimated Go (x) co-ordinates were exactly coincident with the traced Go (x) co-ordinates, every single case was identified as skeletal class I which suggests the analysis is most accurate in skeletal class I cases. This is an important consideration for future investigation.
The implications of these results are that there are some useful indicators on a LCR that can be used to help predict the two-dimensional shape of the mandible. The length of the anterior cranial base (SN) can be used to give an estimate of the length of the mandible at (Go-Me) using the given regression analysis equation 22.65 + 0.5426x, where x = length of the anterior cranial base (SN) in millimetres. Where the palate is a convex shape, it is likely that the shape of the mandible is oblique and that the individual is hyperdivergent. The Sassouni+ analysis can be used with a theoretical point O and posterior arcs and anterior arcs used to estimate the positions of Go and Pog respectively.
The validity of this approach can be questioned, considering the Sassouni analysis was originally designed for use when the reference planes converge exactly at the point of origin O. In this study, this did not occur in any of the selected sample, and therefore, a theoretical average O had to be used instead. In this way, the original analysis had to be adapted for use in this study and therefore the analysis ‘transformed’ the individual into one with average vertical proportions even if they were not. This likely explains why the adapted Sassouni’s analysis was more accurate in estimating the positions of Go and Pog than expected based on the study sample population characteristics.
A limitation of the method used to estimate the positions of Go and Pog in this study is that the method used the known MdP as determined by Go-Me as the inferior limit of intersection with the posterior and anterior arcs. Where the mandible may be missing, such as in forensic science or facial reconstruction, then a different inferior limit or plane of intersection with the posterior arc or the anterior arc will be required to estimate the y co-ordinates of both Go and Pog. This would be required before the Sassouni analysis is carried out as it forms one of the planes which may or may not converge with the other planes at point O. A method would have to be devised to extrapolate this, possibly by using the other angular measurements between the marked horizontal planes in order to determine the vertical position of the mandibular reference points. Orthlieb et al.  studied correlations between mandibular shape and lower facial height and found that the mandibular (gonial) angle (Articulare-go-chin point) had the strongest coefficient of correlation with lower facial height (r = 0.691) but with large dispersion, highlighting the difficulties with predicting vertical facial proportions from other measurements.
A further limitation of this study is that by design, this method is limited to two-dimensional shape prediction based on two-dimensional radiographic imaging, whereas the mandible is a complex three-dimensional object . Transverse measurements such as bigonial width or any asymmetries  could not be assessed using this methodology. Three-dimensional imaging with further landmarks, measurements and volumetric data would be required in order to assess more complex differences in shape, size and position.
In this study, the focus was mainly on the Cartesian co-ordinates of landmarks Go and Pog. To more accurately predict the shape of the mandible, a greater number of cephalometric landmarks would be required in order to predict other parts of the mandible such as the mandibular ramus, condyle and coronoid process. A common shape analysis used in anthropology employs the use of geometrical morphometrics which uses Cartesian landmark co-ordinates that can capture shape variables. Linear measurements and angular measurements on their own are unreliable in predicting complex shapes as they combine shape and size together  and do not always include details of the more subtle aspects of the mandibular form .
A weakness of the Sassouni+ analysis and a number of other cephalometric analyses [16,17,18] is that they can be time consuming to carry out and like many cephalometric studies, are subject to error of the method with differences between examiners in landmark identification, and errors [19, 20] in both angular and linear measurements [21, 22]. This can affect the reliability and reproducibility of the data collected.
Predicting the shape of the mandible presents numerous difficulties as there can be significant differences in the shape, size and position of the mandible between individuals. Furthermore, the size and shape of the mandible is subject to changes with growth over time. Interestingly, between the ages of 16-99, Parr et al.  found that few mandibular measurements exhibited age-related changes, and most were affected by antemortem tooth loss. Chen et al.  found no significant changes in mandibular shape between the ages of 9 and 11 years old, and some significant changes between 11 and 15 years old which coincided with the onset of the pubertal growth spurt.
Different populations may show differences in mandibular size or shape based on evolutionary or genetic factors . There may also be trends linked to environmental factors such as diet  which influences functional demands and the shape or size of the mandible over different time periods [27, 28]. These differences must be considered when trying to predict the shape of the mandible for any one individual from any one time period as there can be morphological plasticity in the shape of the mandible through time .
In context with other research into the shape of the mandible, Lavelle  looked at the mandibular shape of a sample of 90 female patients aged 12–15 years, with 30 class I, 30 class II and 30 class III by using LCRs. He used the technique of medial axis transformation of the mandibular outline form originally described by de Souza and Houghton . The results showed consistency between the overall mandibular shape outline in all three groups. The medial axis lengths were all shorter (average 9%) in class II cases and all longer (average 11%) in class III cases . These findings suggest while the size and position of the mandible may vary based on skeletal pattern, there may be less variation in the outline of the shape of the mandible.
Šidlauskas et al.  investigated the genetic and environmental impact on mandibular morphology using twin based studies in both monozygotic and dizygotic twins who had completed mandibular growth. The results showed that the saddle angle (NSBa) showed high genetic determination as well as the sagittal relationship of the mandible to the cranial base. In addition to this, the gonial angle was also under high heritability but linear measurements such as mandibular body length, ramus width and ramus height were more likely due to environmental factors or non-additive genetic factors . Manfredi et al.  similarly found that mandibular structure seemed more genetically determined than mandibular size.
Other researchers have assessed different landmarks and measurements and related them to the shape of the mandible. Neha et al.  looked at the size and dimensions of the ST and whether there was any correlation with the size of the mandible or maxilla. Their results showed a correlation between the S length and area with both mandibular ramus height and mandibular body length and that the ratio between these measurements and S area was found to be nearly constant . The area of S could potentially be used with the length of the anterior cranial base (SN) in order to more accurately predict the length of the mandibular body.
As the mandibular condyle articulates with the glenoid fossae of the temporal bone of the skull at the TMJ, attention has been paid to the shape of the glenoid fossae and the shape of the mandible. Kantomaa  investigated the correlation between the shape of the glenoid fossae and the morphology of the mandible and found that the inclination of the glenoid fossae in relation to the SN line correlated strongly with the configuration of the mandible. A vertically orientated articulating surface of the glenoid fossae seemed to direct condylar growth more vertically than a more horizontal articulating surface and therefore the inclination of the glenoid fossae may be a useful predictor for assessing skeletal divergency or vertical position of the MdP.
Halozenetis et al.  investigated the shape of the mandible from Art to Gn and used a Fourier analysis to analyse the shape of the mandible at circumpubertal timepoints. The results showed that the angles GoGn-SN, Frankfort horizontal-mandibular line (FH-ML) and Pal-GoGn were moderately to highly correlated to the shape of the mandible at all time points. These findings are of interesting relevance, as both SN and Pal (ANS-PNS) were measured in our study design. When used in conjunction with Go-Gn, which can be used as a method to determine the MdP, it is unsurprising these angles had a high correlation to the shape of the mandible around the circumpubertal period.
One of the key challenges is determining the size, position and shape of the inferior border of the mandible and delineating the MdP. This is commonly assessed by cephalometric analysis using the Maxillary Mandibular Plane Angle or Frankfort Mandibular Planes Angle. Where this angle is increased, this suggests that the individual is ‘high angle or hyperdivergent’ and where the angle is decreased, the individual is classified ‘low angle or hypodivergent’. Oh et al.  used mandibular landmarks to predict adult facial vertical types as described above. They found that Go, Me and Art were the best discriminating landmarks for predicting vertical facial pattern. As Go-Me can be used to determine MdP, it is expected that those would be useful discriminating landmarks for predicting vertical facial pattern. It is surprising that Art was one of the best discriminating landmarks for predicting adult facial vertical type and therefore this could have some application in conjunction with the inclination of the glenoid fossae  in predicting skeletal divergency.
Ayoub et al.  looked at whether the angle of the mandible could be used to differentiate between males and females (sexual dimorphism) in a Lebanese population with normocclusion and balanced facial proportions. They measured the mandibular angle using the ramal plane (Pr) and three different methods of determining the MdP: Down’s analysis using Go-Me , Steiner’s analysis using Go-Gn  and the Sassouni analysis using Go-Me . The results showed that the female mandibular angles were slightly smaller than the males but that the mandibular angle could not be used as a differentiator for gender in a Lebanese population . These findings suggest that neither gender nor the size of the angle of the mandible is particularly useful characteristics which may help predict the shape of the mandible in a Lebanese population. Alarcón et al.  also found that sex-specific mandibular traits behave differently across vertical facial patterns. In contrast to these findings, Schmittbuhl et al.  found that 84.1% of males and 81.2% of females presented significant sexual dimorphism of the shape of the mandible after size normalisation, while Franklin et al.  found the subadult mandible is not dimorphic. These differences in findings are likely down to differences in methodology between the studies, and therefore, while males tend to have larger sized mandibles than females, it is unclear from the current available evidence whether there are distinct differences in the shape of the mandible between males and females at different ages.
It is important to state at this point how accurate two-dimensional cephalometric measurements translate to anthropometric measurements of the face in reality. This gives an idea to how useful this prediction would be for anthropometric facial reconstruction. Budai et al.  investigated the relationship between anthropometric and cephalometric measurements of the face of human adult Caucasian participants. The results showed 97.4% of facial surface measurements and 96.7% of cephalometric measurements were considered normal. The cephalometric upper face, nose and upper lip heights did not differ significantly from the anthropometric counterparts but the cephalometric face height, lower face height and lower alveolar height were all lower in the cephalometric measurements than the anthropometric measurements. This suggests that two-dimensional cephalometric mandibular prediction may underestimate the vertical position or size of the mandible relative to the anthropometric measurements, although these differences are likely to be small.