The mandible is the strongest, largest and lowest bone in the human face (Nirmale et al., 2012), affecting orofacial functions, occlusion, and jaw relationships (Drevenšek et al., 2005). Knowledge of mandibular growth and development is essential for malformation diagnosis and early interception (So, 1997; Efstratiadis et al., 1999). Data on monthly morphological changes during mandibular growth will provide useful information for the establishment of such knowledge. However, bone growth is a very complex process during which the shape and structure of the bone are modified via a dynamic process involving both bone resorption and deposition (Enlow and Harris, 1964; Gu and McNamara Jr, 2007), indicating that mandibular growth may not be a simple homogeneous (uniform) expansion of the bone (Pal, 2014; Kelly et al., 2017). Therefore, techniques for accurate modeling and analysis of regular 3D imaging data over the growth period are essential for the quantitative description of the non-uniform morphological changes of the mandible.

Cephalograms are often used as the basis for diagnostic imaging and have been used to image mandible shape changes at limited time instances in a limited number of in vivo studies (Nanda, 1955; Liu et al., 2010). Bjork (1963) was the first to use overlapping lateral cephalometric radiographs to observe the growth of the mandible implanted with tantalum metal markers in children in vivo (Björk, 1963). However, using cephalograms for long-term, in vivo measurement of human mandible growth is not feasible for ethical reasons because it requires multiple exposures to ionizing radiation over the monitoring period. On the other hand, the use of cephalograms in monitoring the growth of the maxillofacial regions of miniature pigs has been well documented because they are similar to humans in the morphology of the mandible (size and shape) and their bone metabolism rate (Ström et al., 1986; Langenbach et al., 2002; Huh et al., 2006) (Ide et al., 2013; Holton et al., 2015). More recent development in computed tomography (CT) enables accurate three-dimensional (3D) reconstruction of the bone and the measurement of its morphology (Adams et al., 2004; Cavalcanti et al., 2004) (Broadbent, 1981; Mah and Hatcher, 2003; Adams et al., 2004). Three-dimensional CT has been used to study human mandibular growth using cadaveric specimens of different growth stages (Kelly et al., 2017; Schipper et al., 2021), or in patients with diseases that require follow-up with CT imaging (Cavalcanti et al., 2004; Carvalho et al., 2010). However, the adverse effects of ionizing radiation on living bodies preclude the long-term monitoring using CT in vivo. While low-dose, dental Cone-Beam-Computed-Tomography (CBCT) allowed relatively straightforward and rapid to obtain 3-D images of the mandible (Hildebolt et al., 1990; Cavalcanti and Vannier, 1998; DeCoster et al., 2012), the radiation dose is still not low enough for long-term monitoring of the human mandibular growth. Therefore, miniature pigs remain a good alternative to humans in mandibular growth studies.

Measurement of the mandibular growth in miniature pigs using CBCT has been reported by Lin H.-S. et al. (2014). The development of the mandible was quantified by measuring the magnitudes and average rates of monthly changes of the three-dimensional distances between selected anatomical landmarks over the growth period. While this approach revealed the length changes of a limited number of bone segments of the mandible in a selection of positions and directions during growth, it does not provide the quantitative, continuous patterns of morphological changes of the mandibular bone during growth. Methods are needed for a complete description of the continuous changes in the 3D morphology of the mandible from measurements at a series of discrete-time instants during growth.

The purpose of the current study was to develop a new approach based on 3D time-dependent, non-linear osteometric modeling of the bones to describe the 3D quantitative, continuous patterns of the morphological changes of the mandible during growth from the CBCT data of miniature pigs taken monthly from the age of one month onwards over 12 months.

The RMSE of the positions between the new method and the gold standard for the positions of the 17 anatomical landmarks over the 12 months of monitoring were all less than 0.86 mm (Table 2). The RMSE were all less than 0.44 mm over the first eight months of monitoring and 0.86 mm over months 9–12 (Table 2). The average of the RMSE of 17 anatomical landmarks ranged between 0.17 and 0.66. The average of the RMSE of 17 anatomical landmarks was less than 0.29 over the first eight months of monitoring and 0.66 mm over months 9–12 (Table 2). The RMSE of the rates of inter-landmark distance changes between the new method and the gold standard over the 12 months of monitoring were mostly less than 0.05%/month (Table 3). The average of the RMSE of the rates of inter-landmark distance changes was less than 0.06%/month over the first eight months of monitoring and 0.08%/month over months 9–12 (Table 3).

As shown in the lateral and superior views, the mandible appeared to change its size over the 12-month growing period but with largely similar shapes (Figures 2, 3). The bone thickness of the anterior and posterior ramus regions and the mediolateral width of the condylar head also increased over time (Figure 4). As seen from the lateral view, the growth rate of the mandible showed an anteriorly decreasing trend from the posterior part of the ramus to the anterior corpus edge (Figure 5). The condyle region showed the fastest growth rate during the initial growth period (T1-T4), with an average growth rate of about 1.19%/month, 1.03%/month, and 0.85%/month in the anterior/posterior, superior/inferior, and medial/lateral direction respectively (Table 4). The second-fastest growth rate occurred in the posterior ramus with average growth rates of about 0.75, 0.62, and 0.53%/month in the anterior/posterior, superior/inferior, and medial/lateral directions, respectively, over T1-T12 (Table 4). The slowest growth rates occurred in the middle corpus region with average values of about 0.39, 0.39, and 0.32%/month in the anterior/posterior, superior/inferior, and medial/lateral directions, respectively; and in the anterior corpus region with corresponding values of 0.53, 0.45, and 0.37%/month (Table 4). In the anterior/posterior direction, the fastest growth of the condyle occurred at T1-T2, while the most rapid growth of the other regions occurred at T2-T3 (Table 4). In the superior/inferior direction, the fastest growth of the condyle and ramus regions occurred at T1-T2, while the most rapid growth of the corpus regions occurred at T2-T3 (Table 4).

In the frontal and posterior views, the largest growth rate (a total rate of about 2.66%/month) also occurred in the condyle, both medial and lateral sides (Figures 5–8). The second-fastest growth rate (a total rate of about 2.40%/month) occurred in the posterior ramus, while the lowest growth rates were in the middle and anterior corpus regions, with total rates of about 0.46 and 0.64%/month, respectively (Figures 5–8 and Table 4). The condyle had a similar growth rate as the anterior and posterior ramus regions from T1 to T12 (Figure 9 and Table 4). In the medial/lateral direction, the fastest growth of the posterior ramus occurred at T1-T2, and the fastest growth of the condyle, anterior ramus, middle, and anterior corpus occurred at T2-T3 (Table 4). The most rapid growth of all the regions occurred in the anterior/posterior direction, except the middle corpus (Table 4). The results of the superior view further confirmed the patterns seen in the other views (Figure 8).

The current study aimed to develop a new approach based on 3D time-dependent, non-linear osteometric modeling of the bones to describe the 3D quantitative, continuous patterns of the morphological changes of the mandible during growth from the CBCT data of miniature pigs during the growth over the first 12 months. The continuous growth patterns of the mandible during the 12-month monitoring period were obtained, averaged over six selected anatomical parts, and displayed as continuous color maps over the surface of the mandible. Generally, the growth rate of the mandible was found to decrease anteriorly, with the condyle region and the posterior part of the ramus growing faster than the rest of the mandible. The condyle region showed the fastest growth rate and the posterior ramus the second during the growth period, while the middle and anterior corpus regions showed the slowest growth rates. The current results suggest that the new approach will be useful for future studies on the growth of the mandible in different animals.

The new approach developed in the current study enabled the monitoring and description of the 3D mandibular growth for each vertex on the surface and within the body of the mandible at high accuracy over a long period of time. This is in contrast to previous studies that described the growth of the mandible in terms of the changes in the distances between two landmarks (Lin H.-S. et al., 2014). Line segments defined by two landmarks only revealed the gross patterns of the growth. More detailed information for a small area that cannot be defined by two landmarks could not be obtained. For example, the condyle or coronoid process could not be measured with two apparent landmarks. Another limitation of previous approaches is that the descriptions were mostly in two dimensions. A planar description cannot reveal the growth changes of the overall mandible because it is three-dimensional in shape and nature.

The CBCT-based measurement method and the new analysis technique based on 3D osteometric modeling used in the current study enabled the quantification of the 3D growth in terms of the monthly changes of the vertex points on and within the mandible. This new approach involved between-model registration and time-fitting to simulate the non-linear growth of the mandible. The method searched for the corresponding points on the bone models at two consecutive instances in time with the highest match between anatomical and morphological features. The RMSE of the positions between the new method and the gold standard for the positions of the 17 anatomical landmarks over the 12 months of monitoring were small (less than 0.86 mm), especially over the first eight months of monitoring (less than 0.44 mm). These validation results indicate that the new approach developed in the current study could be used for measuring the quantitative, continuous patterns of growth of the mandibular bone. While quantitative values of the average growth rates were reported over the six selective parts of the mandible, continuous growth maps over the mandibular bone surface were also helpful for a general picture of the growth patterns. Color maps could not be generated for the growth rates for points within bones; further development of display methods for these points will be needed for the ease of clinical applications.

With the current three-dimensional modeling approach, the overall mandible was described by hundreds of triangular elements, the rates of morphological changes of which were indicated by color-codes. This was in contrast to previous representations of mandible growth using length changes between landmarks. For example, landmarks Go and GT have been used to represent the length of the mandible, and its distance changes used to describe mandibular growth. It is noted that bony growth between Go and GT was assumed to have a linear trend. However, the current results showed that the bony growth rates were non-linear between Go and GT, decreasing gradually from Go to GT (Figure 5). Similar to Go-GT, growth rates along the line joining landmarks Cd-lat and Go, indicating the mandible height, were also found to be non-linear, increasing from the lateral pole of the condylar head (Cd-lat) to the ramus body (Go) (Figure 6). These results suggest that the inter-marker distances may affect the representation of bony growth using line segments. The current new approach to finding correspondence between any two models over time enabled us to describe the growth and growth rates (or morphological changes) for each vertex on the bone model. Since the vertices were very close to each other in terms of the accuracy of the CBCT, the growth rates could be described in much greater detail than traditional line segment representations.

The growth patterns of the current pigs’ mandibles during the first four months appeared to be related to the germination of the initial mandibular dentition, which begins from the third lower deciduous incisors and the deciduous canines immediately after birth, followed by the first deciduous incisors (Cheng et al., 2010). To provide space for the germination of the anterior dentition, the anterior corpus region in four-week-old pigs appeared to have grown to a size suitable for the growth of the incisors and canines, as shown in the first CT scan and the reducing rate of changes in sizes for the rest of the monitoring period. Subsequent to the first CT scan, the condyle and posterior ramus regions were found to have the fastest growth rate during T1-T4, with the largest monthly change in the condyle part (Table 3, 4 and Figures 5–8). The mandible became taller and longer as a result of the continuous growth of the condyle and the middle and posterior corpus regions, which was accompanied by germination and growth of the posterior teeth (Figure 5). The current results suggest that the size of the condylar region at the time of the first CT scan and the subsequent morphological changes of the mandible in the early period of growth are helpful for the development of the temporomandibular joint for mandibular movement and the germination of the mandibular dentition.

The current study was limited to quantifying the surface morphological changes and the average rates of such changes over six selected anatomical parts of the mandible during the 12-month monitoring period, showing the non-uniform change patterns of different regions of the mandible to describe the overall growth. The proposed approach can be further extended to include volumetric information of the mandible to provide further information on the morphological changes at any point within the mandible bone during growth. It is noted that without implanting markers within the bone, it was challenging to indicate the locations and amount of bone apposition and resorption that resulted in the morphological changes observed in the current study and to identify any bone rotation within the mandible. On the other hand, the mandibular growth was monitored over 48 weeks until the pig became too big to be imaged by the current CBCT machine, so the growth patterns reported were limited to the period from eruption up to 2/3 of the mixed dentition stage. In the future, advances in imaging techniques may reduce harmful ionizing radiation to such low levels that the proposed approach could safely be validated in human subjects.

The current study presented for the first time in the literature the 3D continuous morphological changes of the mandible in miniature pigs during the growth of the first 12 months as average morphological changes over six selected anatomical parts and continuous changes over the surface of the mandible using color maps. This was achieved by integrating CBCT and a new osteometric modeling approach, which quantified the non-linear patterns and non-linear rate of morphological changes in different growth regions and the whole mandible. The current approach will be useful for future studies on the growth of the mandible in other animals and the exploration of the growth of different dentition patterns, such as protruded or retruded mandibles, which may contribute to a complete understanding of the underlying growth factor of dental occlusion.