Geometric morphometric analysis of growth patterns among facial types

  • Ryan P. Knigge
    Affiliations
    Department of Integrative Biology and Physiology, University of Minnesota Medical School, Minneapolis, Minn

    Department of Pathology and Anatomical Sciences, University of Missouri School of Medicine, Columbia, Mo

    Department of Orthopaedic Surgery, University of Missouri School of Medicine, Columbia, Mo
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  • Kieran P. McNulty
    Affiliations
    Evolutionary Anthropology Lab, Department of Anthropology, University of Minnesota, Minneapolis, Minn
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  • Heesoo Oh
    Affiliations
    Department of Orthodontics, Arthur A. Dugoni School of Dentistry, University of the Pacific, San Francisco, Calif
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  • Anna M. Hardin
    Affiliations
    Department of Pathology and Anatomical Sciences, University of Missouri School of Medicine, Columbia, Mo

    Biology Department, Western Oregon University, Monmouth, Ore
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  • Emily V. Leary
    Affiliations
    Department of Orthopaedic Surgery, University of Missouri School of Medicine, Columbia, Mo
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  • Dana L. Duren
    Affiliations
    Department of Pathology and Anatomical Sciences, University of Missouri School of Medicine, Columbia, Mo

    Department of Orthopaedic Surgery, University of Missouri School of Medicine, Columbia, Mo
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  • Manish Valiathan
    Affiliations
    Department of Orthodontics, School of Dental Medicine, Case Western Reserve University, Cleveland, Ohio
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  • Richard J. Sherwood
    Correspondence
    Address correspondence to: Richard J. Sherwood, Department of Pathology and Anatomical Sciences, University of Missouri School of Medicine, M263 Medical Sciences Building, One Hospital Dr, Columbia, MO 65212.
    Affiliations
    Department of Pathology and Anatomical Sciences, University of Missouri School of Medicine, Columbia, Mo

    Department of Orthopaedic Surgery, University of Missouri School of Medicine, Columbia, Mo

    Department of Orthodontics, School of Dental Medicine, Case Western Reserve University, Cleveland, Ohio
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Open AccessPublished:June 23, 2021DOI:https://doi.org/10.1016/j.ajodo.2020.04.038

      Highlights

      • Understanding vertical facial divergence is critical to treatment optimization.
      • Growth in 3 vertical facial types is quantified with geometric morphometrics.
      • Normodivergent and hypodivergent growth are distinct from hyperdivergent growth.
      • Anteroposterior maxillomandibular growth is restricted in hyperdivergent patients.
      • Hyperdivergent growth patterns indicate a greater risk for Class II malocclusion.

      Introduction

      Extreme patterns of vertical facial divergence are of great importance to clinicians because of their association with dental malocclusion and functional problems of the orofacial complex. Understanding the growth patterns associated with vertical facial divergence is critical for clinicians to provide optimal treatment. This study evaluates and compares growth patterns from childhood to adulthood among 3 classifications of vertical facial divergence using longitudinal, lateral cephalograms from the Craniofacial Growth Consortium Study.

      Methods

      Participants (183 females, 188 males) were classified into 1 of 3 facial types on the basis of their adult mandibular plane angle (MPA): hyperdivergent (MPA >39°; n = 40), normodivergent (28° ≤ MPA ≤ 39°; n = 216), and hypodivergent (MPA <28°; n = 115). Each individual had 5 cephalograms between ages 6 and 20 years. A set of 36 cephalometric landmarks were digitized on each cephalogram. Landmark configurations were superimposed to align 5 homologous landmarks of the anterior cranial base and scaled to unit centroid size. Growth trajectories were calculated using multivariate regression for each facial type and sex combination.

      Results

      Divergent growth trajectories were identified among facial types, finding more similarities in normodivergent and hypodivergent growth patterns than either share with the hyperdivergent group. Through the use of geometric morphometric methods, new patterns of facial growth related to vertical facial divergence were identified. Hyperdivergent growth exhibits a downward rotation of the maxillomandibular complex relative to the anterior cranial base, in addition to the increased relative growth of the lower anterior face. Conversely, normodivergent and hypodivergent groups exhibit stable positioning of the maxilla relative to the anterior cranial base, with the forward rotation of the mandible. Furthermore, the hyperdivergent maxilla and mandible become relatively shorter and posteriorly positioned with age compared with the other groups.

      Conclusions

      This study demonstrates how hyperdivergent growth, particularly restricted growth and positioning of the maxilla, results in a higher potential risk for Class II malocclusion. Future work will investigate growth patterns within each classification of facial divergence.
      Classifications of facial and dental morphology are frequently used in orthodontic practice for diagnosis, treatment planning, and prognosis. Schudy
      • Schudy F.
      Vertical growth versus anteroposterior growth as related to function and treatment.
      first characterized the interaction of vertical and anteroposterior growth in the face as patterns of facial divergence (ie, hyperdivergent, normodivergent, and hypodivergent), and since then, additional terminology has been used to describe similar morphologic patterns. The hyperdivergent skeletal pattern has been referred to as an open bite
      • Nanda S.K.
      Patterns of vertical growth in the face.
      • Nanda S.K.
      Growth patterns in subjects with long and short faces.
      • Moon S.C.
      • Kim H.K.
      • Kwon T.K.
      • Han S.H.
      • An C.H.
      • Park Y.S.
      Patterns of vertical facial growth in Korean adolescents analyzed with mixed-effects regression analysis.
      or long face syndrome,
      • Schendel S.A.
      • Eisenfeld J.
      • Bell W.H.
      • Epker B.N.
      • Mishelevich D.J.
      The long face syndrome: vertical maxillary excess.
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      • Björk A.
      • Skieller V.
      Contrasting mandibular growth and facial development in long face syndrome, juvenile rheumatoid polyarthritis, and mandibulofacial dysostosis.
      and the hypodivergent pattern as a deep bite
      • Nanda S.K.
      Patterns of vertical growth in the face.
      • Nanda S.K.
      Growth patterns in subjects with long and short faces.
      • Moon S.C.
      • Kim H.K.
      • Kwon T.K.
      • Han S.H.
      • An C.H.
      • Park Y.S.
      Patterns of vertical facial growth in Korean adolescents analyzed with mixed-effects regression analysis.
      or short face syndrome.
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      ,
      • Opdebeeck H.
      • Bell W.H.
      The short face syndrome.
      Although Schudy’s
      • Schudy F.
      Vertical growth versus anteroposterior growth as related to function and treatment.
      classification system was based on the mandibular plane angle (MPA), other measurements have been used to classify patterns of vertical facial divergence, including the anterior to posterior facial height ratio,
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      the lower anterior to total anterior facial height ratio,
      • Nanda S.K.
      Patterns of vertical growth in the face.
      • Nanda S.K.
      Growth patterns in subjects with long and short faces.
      • Moon S.C.
      • Kim H.K.
      • Kwon T.K.
      • Han S.H.
      • An C.H.
      • Park Y.S.
      Patterns of vertical facial growth in Korean adolescents analyzed with mixed-effects regression analysis.
      and the angle between mandibular and Frankfort horizontal planes.
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      The clinical significance of these classifications relates to their associations with dental malocclusion and other functional or esthetic issues. The hyperdivergent facial type is commonly associated with an anterior open bite referring to a lack of vertical overlap in maxillary and mandibular incisors.
      • Proffit W.
      • Fields Jr., H.
      • Sarver D.
      Contemporary Orthodontics.
      Furthermore, skeletal hyperdivergence is related to reduced masticatory performance
      • Proffit W.R.
      • Fields H.W.
      • Nixon W.L.
      Occlusal forces in normal- and long-face adults.
      • Proffit W.R.
      • Fields H.W.
      Occlusal forces in normal- and long-face children.
      • Ingervall B.
      • Minder C.
      Correlation between maximum bite force and facial morphology in children.
      • Cha B.K.
      • Kim C.H.
      • Baek S.H.
      Skeletal sagittal and vertical facial types and electromyographic activity of the masticatory muscle.
      and airway constriction.
      • Joseph A.A.
      • Elbaum J.
      • Cisneros G.J.
      • Eisig S.B.
      A cephalometric comparative study of the soft tissue airway dimensions in persons with hyperdivergent and normodivergent facial patterns.
      The hypodivergent facial type has been linked to a deep or excessive vertical overbite of the maxillary and mandibular incisors, which can interfere with lateral and anterior mandibular movements and temporomandibular joint dysfunction.
      • Riolo M.L.
      • Brandt D.
      • TenHave T.R.
      Associations between occlusal characteristics and signs and symptoms of TMJ dysfunction in children and young adults.
      ,
      • Bergersen E.O.
      A longitudinal study of anterior vertical overbite from eight to twenty years of age.
      Understanding differences in morphology and growth associated with vertical facial divergence patterns is critical for providing appropriate treatment and prognoses for participants. Morphologic differences among skeletal facial types are apparent around 5 or 6 years of age.
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      ,
      • Karlsen A.T.
      Craniofacial growth differences between low and high MP-SN angle males: a longitudinal study.
      • Karlsen A.T.
      Association between facial height development and mandibular growth rotation in low and high MP-SN angle faces: a longitudinal study.
      • Oh H.
      • Knigge R.
      • Hardin A.
      • Sherwood R.
      • Duren D.
      • Valiathan M.
      • et al.
      Predicting adult facial type from mandibular landmark data at young ages.
      • Hardin A.M.
      • Valiathan M.
      • Oh H.
      • Knigge R.P.
      • McNulty K.P.
      • Leary E.V.
      • et al.
      Clinical implications of age-related change of the mandibular plane angle.
      Most studies agree that facial type differences are predominantly found in the lower face and mandible, but the specific measurements that differ among facial types vary by study.
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      ,
      • Karlsen A.T.
      Craniofacial growth differences between low and high MP-SN angle males: a longitudinal study.
      • Karlsen A.T.
      Association between facial height development and mandibular growth rotation in low and high MP-SN angle faces: a longitudinal study.
      • Oh H.
      • Knigge R.
      • Hardin A.
      • Sherwood R.
      • Duren D.
      • Valiathan M.
      • et al.
      Predicting adult facial type from mandibular landmark data at young ages.
      ,
      • Bishara S.E.
      • Augspurger E.F.
      The role of mandibular plane inclination in orthodontic diagnosis.
      ,
      • Björk A.
      • Skieller V.
      Normal and abnormal growth of the mandible. A synthesis of longitudinal cephalometric implant studies over a period of 25 years.
      Previous cephalometric analyses of linear distance and angular measurements describing childhood and/or adolescent growth patterns across facial types have also found contrasting results. For example, Bishara and Jakobsen
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      note parallel incremental growth curves for many craniofacial measurements from 5 to 25 years of age, indicating measurement differences among facial types were already present at 5 years of age, with similar subsequent growth patterns. Conversely, Karlsen
      • Karlsen A.T.
      Craniofacial growth differences between low and high MP-SN angle males: a longitudinal study.
      ,
      • Karlsen A.T.
      Association between facial height development and mandibular growth rotation in low and high MP-SN angle faces: a longitudinal study.
      reported differences in growth up to 12 years of age among facial types, whereas Nanda
      • Nanda S.K.
      Patterns of vertical growth in the face.
      found variation in the adolescent growth spurt among facial types leading to increased measurement differences during adolescence. These disagreements likely result from the use of different study samples, classification methods, or the age at classification, requiring further work to harmonize previous studies.
      In this analysis, we use geometric morphometric methods (GMM), a multivariate, landmark-based approach to evaluate differences in craniofacial morphology and growth patterns among 3 skeletal facial types. The GMM approach allows for the evaluation of coordinated patterns of vertical and anteroposterior growth in craniofacial structures by preserving the geometric or spatial relationships of landmarks throughout the analysis, which traditional linear cephalometrics are poorly equipped to quantify. We used the strengths of this approach to clarify inconsistencies among previous studies of craniofacial growth patterns of hyperdivergent, normodivergent, and hypodivergent participants. The goals of this study are to (1) describe and compare growth trajectories for hyperdivergent, normodivergent, and hypodivergent facial types and (2) determine shape differences among facial types at different ages.

      Material and methods

      The Craniofacial Growth Consortium Study houses longitudinal, lateral cephalograms from 6 historical growth studies in the United States, which sampled populations of primarily European ancestry from 1930-1982.
      • Oh H.
      • Knigge R.
      • Hardin A.
      • Sherwood R.
      • Duren D.
      • Valiathan M.
      • et al.
      Predicting adult facial type from mandibular landmark data at young ages.
      ,
      • Hardin A.M.
      • Valiathan M.
      • Oh H.
      • Knigge R.P.
      • McNulty K.P.
      • Leary E.V.
      • et al.
      Clinical implications of age-related change of the mandibular plane angle.
      ,
      • Sherwood R.J.
      • Oh H.S.
      • Valiathan M.
      • et al.
      Bayesian approach to longitudinal craniofacial growth: the craniofacial growth consortium study.
      A total of 1855 cephalograms from 371 participants (183 females, 188 males) in the Craniofacial Growth Consortium Study were included in this analysis. Each individual had 1 cephalogram taken within each of the following age categories: 6-8, 9-11, 12-14, 15-17, and 18-20 years, for a total of 5 cephalograms per participant. If a participant had more than 1 cephalogram per age category, the cephalogram taken closest to the median age for that category was selected.
      The age range of 6-20 years, and associated age categories, were chosen to maximize the available sample, which includes the primary period for clinical treatment related to facial growth. Some previous studies have considered the importance of postadolescent changes in facial dimensions,
      • Bishara S.E.
      • Treder J.E.
      • Jakobsen J.R.
      Facial and dental changes in adulthood.
      • Bondevik O.
      Growth changes in the cranial base and the face: a longitudinal cephalometric study of linear and angular changes in adult Norwegians.
      • Forsberg C.M.
      Facial morphology and ageing: a longitudinal cephalometric investigation of young adults.
      • Forsberg C.M.
      • Eliasson S.
      • Westergren H.
      Face height and tooth eruption in adults--a 20-year follow-up investigation.
      • Fudalej P.
      • Kokich V.G.
      • Leroux B.
      Determining the cessation of vertical growth of the craniofacial structures to facilitate placement of single-tooth implants.
      • Thilander B.
      • Persson M.
      • Adolfsson U.
      Roentgen-cephalometric standards for a Swedish population. A longitudinal study between the ages of 5 and 31 years.
      • Behrents R.G.
      Growth in the Aging Craniofacial Skeleton.
      • Tallgren A.
      The effect of denture wearing on facial morphology. A 7-year longitudinal study.
      although the amount and clinical significance of this growth have been debated.
      • Bishara S.E.
      • Treder J.E.
      • Jakobsen J.R.
      Facial and dental changes in adulthood.
      ,
      • Bondevik O.
      Growth changes in the cranial base and the face: a longitudinal cephalometric study of linear and angular changes in adult Norwegians.
      ,
      • Fudalej P.
      • Kokich V.G.
      • Leroux B.
      Determining the cessation of vertical growth of the craniofacial structures to facilitate placement of single-tooth implants.
      The greatest magnitude of postadolescent growth is often observed in anterior facial height but rarely exceeds 1 mm over 10 years, a suggested threshold for clinical significance.
      • Bondevik O.
      Growth changes in the cranial base and the face: a longitudinal cephalometric study of linear and angular changes in adult Norwegians.
      Facial type classification was based on adult morphology for all participants using their cephalograms in the 18-20 year age interval; these classifications were applied to juvenile cephalograms from the same participants. MPA, the angle between sella-nasion and gonion-menton, was calculated and used for facial type classification: hypodivergent (MPA <28°), normodivergent (28° ≤ MPA ≤ 39°), and hyperdivergent (MPA >39°) (Table I). A set of 36 cephalometric landmarks (Table II; Fig 1) was collected on each cephalogram, with no missing data, to quantify the morphology of the craniofacial skeleton and anterior dentition. This study was approved by the University of Missouri Institutional Review Board.
      Table ISummary of sample by sex and adult facial type classification
      ClassificationFemalesMales
      ParticipantsCephalogramsParticipantsCephalograms
      Hyperdivergent211051995
      Normodivergent113565103515
      Hypodivergent4924566330
      Total183915188940
      Table IIDescription of landmarks collected for this study
      No.LandmarkDefinition
      1SELLAThe midpoint of the pituitary fossa
      2NASIONThe most anteroinferior point on the frontal bone at the nasofrontal suture
      3ORBITALEInferiormost point of the orbital margin of the more anterior orbital image
      4U I APEXPoint of intersection between the long axis of the more anteriorly positioned maxillary central incisor and the contour of that tooth’s root-end curvature
      5POINT AThe deepest point on the curvature of the surface of the maxillary bone between ANS and the alveolar crest of the maxillary central incisor
      6U I EDGETip of the incisal edge of the more anteriorly placed maxillary central incisor
      7L I EDGETip of the incisal edge of the more anteriorly placed mandibular central incisor
      8POINT BThe deepest point on the curvature of the anterior border of the mandible between pogonion and the alveolar crest of the mandibular central incisor
      9L I APEXPoint of intersection between the long axis of the most anteriorly positioned mandibular incisor and the contour of that tooth’s root-end curvature
      10POGONIONAnterior-most point of the bony chin at the midline
      11MENTONInferiormost point on the mandible at the symphysis
      12CONDYLEThe point on the posterior-superior contour of the condyle that is the longest distance from pogonion
      13PNSThe intersection between the posterior extension of the superior surface of the palate and the downward extension of the pterygomaxillary fissure
      14BASIONInferiormost point on the anterior margin of the foramen magnum in the midsagittal plane
      15PORIONMost superior point of the external auditory meatus of the right ear
      16ANSAnterior-most point of the anatomic anterior nasal spine
      17GLABELLAThe most anterior point on the osseous forehead
      18SE
      Anterior cranial base landmarks used for superimposition.
      The midpoint between the intersections of the 2 great wings of the sphenoid bone with the sphenoid plane
      19POINT P
      Anterior cranial base landmarks used for superimposition.
      Point of greatest convexity between the anterior contour of sella turcica and the sphenoid plane
      20PTMPterygomaxillary fissure point: the intersection of the inferior border of the foramen rotundum with the posterior wall of the pterygomaxillary fissure
      21SIGMOIDThe deepest point of the sigmoid notch
      22P RAMUSPosterior ramus point where the inflection starts
      23CORONOIDTip of the coronoid process
      24A RAMUSThe deepest point in the anterior ramus
      25INFRADENTALEMost superior and anterior point of the mandibular alveolar process
      26LINGUAL L1Most superior and anterior point of alveolar bone on the lingual surface
      27LINGUAL POINT BIntersecting point of a posterior extension of point B parallel to the mandibular plane connecting gonion and menton on the lingual cortical plate of the symphysis
      28L SYMPHYIntersecting point of a posterior extension of pogonion parallel to the mandibular plane connecting gonion and menton on the lingual cortical plate of the symphysis
      29ANTEGONIONThe deepest point of the antegonial notch
      30PROSTHIONMost inferior and anterior point of the maxillary alveolar process
      31FRONTPoint of intersection between the roof of orbit line and the internal cortical plate of the frontal bone
      32ROOF
      Anterior cranial base landmarks used for superimposition.
      Most superior point of the roof of the orbit, the middle point between FRONT and SE-S
      33SE-S
      Anterior cranial base landmarks used for superimposition.
      Point of intersection of the greater wing of the sphenoid bone and the roof of the orbit (midpoint between 2 greater wings of the sphenoid bone)
      34ETHMOID
      Anterior cranial base landmarks used for superimposition.
      Point of intersection between the lateral border of the ethmoid bone and the vertical extension of the most anteriorly positioned Key ridge
      35ZYGOMAMost inferior point of the most anteriorly positioned key ridge
      36GONIONAverage of upper and lower gonion points
      Anterior cranial base landmarks used for superimposition.
      Figure thumbnail gr1
      Fig 1Depiction of cephalometric landmarks used in this study. Landmark numbers correspond to descriptions in . Landmarks 18, 19, 32, 33, and 34 represent homologous points of the anterior cranial base used in superimposition.
      For each sex, a modified generalized Procrustes analysis was used to align landmark configurations by removing variation in location, rotation, and scale so that only differences in shape remained.
      • Gower J.C.
      Generalized Procrustes analysis.
      ,
      • Rohlf F.J.
      • Slice D.
      Extensions of the Procrustes method for the optimal superimposition of landmarks.
      Each landmark configuration was superimposed to align 5 homologous landmarks of the anterior cranial base but scaled to the unit centroid size of the entire landmark configuration (see Table II; Fig 1). The resulting aligned landmark coordinates were used as shape variables in subsequent analyses.
      To model growth trajectories, multivariate regressions of the shape variables on age
      • Mitteroecker P.
      • Gunz P.
      • Bookstein F.L.
      Heterochrony and geometric morphometrics: a comparison of cranial growth in Pan paniscus versus Pan troglodytes.
      ,
      • Bulygina E.
      • Mitteroecker P.
      • Aiello L.
      Ontogeny of facial dimorphism and patterns of individual development within one human population.
      were performed for each adult facial type, separately by sex. In this context, we specifically refer to growth trajectories as changes in landmark configurations (ie, shape change) with age. For each of the 6 facial type and sex combinations, 4 different models were fit (linear to fourth-order polynomial). Leave-one-out cross-validation and mean squared error values were used to select the best fit models. Predicted shapes were calculated for the 6 selected models at 1-year age intervals, from 6-20 years, to represent annual mean shape estimates for each trajectory. Pairwise permutation tests, described by Mitteroecker et al,
      • Mitteroecker P.
      • Gunz P.
      • Bookstein F.L.
      Heterochrony and geometric morphometrics: a comparison of cranial growth in Pan paniscus versus Pan troglodytes.
      were used to test identical trajectories among the 3 facial types using the residual sum of squares as a test statistic with 10,000 iterations.
      Facial type growth trajectories were compared, by sex, using a principal component analysis of age-shape space, which is a principal component analysis of the matrix of predicted shape coordinates with an additional column for age (cf, Mitteroecker et al
      • Mitteroecker P.
      • Gunz P.
      • Bernhard M.
      • Schaefer K.
      • Bookstein F.L.
      Comparison of cranial ontogenetic trajectories among great apes and humans.
      ). Growth patterns for each facial type were visualized using wireframe plots to compare the shape change within each trajectory at ages 6, 10, 14, and 18 years.
      • Bookstein F.L.
      Morphometric Tools for Landmark Data: Geometry and Biology.
      Comparisons were also made between the hyperdivergent and hypodivergent landmark configurations at these ages to visualize morphologic differences between the most extreme facial types.
      Three angular measurements, SNA (sella-nasion-point A), SNB (sella-nasion-point B), and ANB (point A-nasion-point B), were calculated for each cephalogram to demonstrate the change in anteroposterior (A-P) maxillomandibular relationships with age, across facial types. Although relative maxillomandibular relationships are visualized using GMM, the SNA, SNB, and ANB angles are provided for comparison as clinically important cephalometric measurements for identifying interactions with anteroposterior relationships (ie, Class I, II, and III). One-way analysis of variance with post-hoc Tukey’s Honest Significant Difference test was used to test significant differences in each angle among facial types by sex and age group. All analyses were performed in R using the Morpho,
      • Schlager S.
      Morpho and Rvcg - shape analysis in R.
      geomorph,
      • Adams D.
      • Collyer M.
      • Kaliontzopoulou A.
      • Baken E.
      Geomorph: geometric morphometric analyses of 2D/3D landmark data.
      and stats packages.
      R Core Team
      R: a language and environment for statistical computing.

      Results

      For each of the 6 facial type and sex combinations, quadratic models were selected for multivariate regressions of shape variables on age (Table III). In females and males, the first principal component axis of age-shape space describes variation in age and patterns of age-correlated shape. When plotted against age, the second principal component axis distinguishes hyperdivergent and hypodivergent trajectories, with the normodivergent trajectory between them (Fig 2). The differences in growth trajectories represent unique patterns of average craniofacial growth for each facial type. In females, the 3 trajectories become increasingly parallel at older ages, whereas, in males, the hyperdivergent and hypodivergent trajectories continue to diverge from the normodivergent trajectory. Raw data for each individual are projected onto the age-shape axes to visualize the individual variation within each facial type (Fig 2). Individual trajectories overlap among facial types, but permutation tests indicate significantly different trajectories for each facial type (P ≤0.0003 for all comparisons) (Table IV).
      Table IIIModel comparison of the mean squared error using leave-one-out cross-validation
      ClassificationLinearSecond-orderThird-orderFourth-order
      Females
       Hyperdivergent0.010480.010450.010470.01045
       Normodivergent0.012040.011980.011980.01199
       Hypodivergent0.012520.012410.012410.01247
      Males
       Hyperdivergent0.011070.011070.011090.01119
       Normodivergent0.010470.010390.010390.01039
       Hypodivergent0.010460.010410.010430.01045
      Figure thumbnail gr2
      Fig 2A plot of second principal component (PC2) of age-shape space against chronological age for females (top) and males (bottom). Each line represents a growth trajectory (ie, shape change of a landmark configuration) with age. The thick lines represent the average trajectories for each facial type, with thin lines representing participants (blue, hypodivergent; yellow, normodivergent; and green, hyperdivergent).
      Table IVP values from pairwise permutation tests with 10,000 iterations
      MalesfemalesHyperdivergentNormodivergentHypodivergent
      Hyperdivergent0.00030.0001
      Normodivergent0.00010.0002
      Hypodivergent0.00010.0001
      Note. Females, upper triangle; males, lower triangle.
      Growth patterns within each facial type trajectory are presented in Figure 3 by visualizing predicted landmark configurations at ages 6, 10, 14, and 18 years by sex. In both females and males, the hyperdivergent type exhibits slight anteriorly downward rotation of the maxilla from ages 6 to 10 years, with little rotation occurring after age 10 years. The maxilla also becomes relatively shorter with age, and the subnasal region (ANS-point A-prosthion) becomes increasingly concave. At each subsequent age, the mandibular ramus becomes more vertically oriented, resulting in a more anterior position of the condyle and coronoid process with corresponding backward rotation of the corpus. The backward rotation of the corpus also coincides with a relatively taller mandibular symphysis.
      Figure thumbnail gr3
      Fig 3Visualization of the shape change along each facial type and sex growth trajectory shown in as represented by landmark configurations estimated at ages 6 (gray), 10 (green), 14 (yellow), and 18 (blue) years.
      Slight downward rotation of the anterior maxilla from ages 6 to 10 years is also present in the normodivergent and hypodivergent types. Similar to growth in the hyperdivergent type, the normodivergent type exhibits a more vertical orientation of the mandibular ramus but with no corresponding backward rotation of the corpus. As the normodivergent corpus becomes relatively longer with age, the mandibular symphysis projects more anteriorly with no change in relative height. The hypodivergent growth pattern does not depict as much change in the orientation of the ramus, but the corpus rotates forward with a greater projection of the chin. In both normodivergent and hypodivergent trajectories, the antegonial notch flattens from ages 6 to 18 years.
      In males, growth patterns within each facial type are very similar to those in females, but the magnitude of shape change and degree of rotation is greater (Fig 3). In females, there is less change in facial shape between ages 14 and 18 years, whereas there is still considerable shape change between ages 14 and 18 years in males, likely an indication of an earlier cessation of growth in females compared with males.
      Comparisons of hypodivergent and hyperdivergent face shapes at different ages (Fig 4) show that distinct morphologies are present at 6 years of age and persist to 18 years of age in both sexes. At each age interval, the hyperdivergent face differs from the hypodivergent face by a larger gonial angle with a more distinct antegonial notch, smaller posterior facial height, shorter length of the posterior cranial base, a narrower mandibular ramus, anterior (or forward) rotation of the ramus, backward rotation of the mandibular corpus, downward rotation of the anterior maxilla, greater subnasal height, and greater upper and lower anterior dentoalveolar heights (Fig 4). The more vertical orientation of the mandibular ramus in the hyperdivergent face results in a higher positioned condyle and coronoid process. The lengths of the maxilla and mandible are relatively shorter and posteriorly positioned in the hyperdivergent face compared with the hypodivergent face. Furthermore, the hyperdivergent maxillary and mandibular incisors exhibit reduced overlap compared with hypodivergent maxillary and mandibular incisors.
      Figure thumbnail gr4
      Fig 4Differences between the average hyperdivergent (green) and hypodivergent (blue) landmark configurations at ages 6, 10, 14, and 18 years. The landmark configurations represented by wireframe plots are the predicted configurations from the average growth trajectories depicted in .
      The smaller SNA and SNB angles in the hyperdivergent type demonstrate the retrognathic positioning of both the maxilla and mandible compared with other facial types (Fig 5; Table V). SNA and SNB angles increase with age to a lesser degree in the hyperdivergent group than the hypodivergent and normodivergent groups. Furthermore, the hyperdivergent ANB angle is, on average, larger compared with other facial types indicating a more retrognathic mandible relative to the maxilla, but this difference is statistically significant at all ages only in females.
      Figure thumbnail gr5
      Fig 5Plots of SNA, SNB, and ANB angles by facial type for each age group. Points are medians and whiskers show the range for the 25th to 75th percent quantiles. Left column, females; right column, males.
      Table VResults from the post-hoc Tukey's HSD tests
      FemalesMales
      Hyper-NormoHyper-HypoNormo-HypoHyper-NormoHyper-HypoNormo-Hypo
      6-8 y
       SNA0.4130.012
      Significant at 0.05 level.
      0.025
      Significant at 0.05 level.
      0.008
      Significant at 0.05 level.
      0.004
      Significant at 0.05 level.
      0.868
       SNB0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.003
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.028
      Significant at 0.05 level.
       ANB0.008
      Significant at 0.05 level.
      0.004
      Significant at 0.05 level.
      0.7570.9990.2130.017
      Significant at 0.05 level.
      9-11 y
       SNA0.1480.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.008
      Significant at 0.05 level.
      0.001
      Significant at 0.05 level.
      0.369
       SNB0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.002
      Significant at 0.05 level.
       ANB0.036
      Significant at 0.05 level.
      0.049
      Significant at 0.05 level.
      0.9880.6140.047
      Significant at 0.05 level.
      0.039
      Significant at 0.05 level.
      12-14 y
       SNA0.1420.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.005
      Significant at 0.05 level.
      0.002
      Significant at 0.05 level.
      0.796
       SNB0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.035
      Significant at 0.05 level.
       ANB0.000
      Significant at 0.05 level.
      0.002
      Significant at 0.05 level.
      0.9820.9510.1900.045
      Significant at 0.05 level.
      15-17 y
       SNA0.039
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.001
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.312
       SNB0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.003
      Significant at 0.05 level.
       ANB0.009
      Significant at 0.05 level.
      0.003
      Significant at 0.05 level.
      0.6130.9680.2210.049
      Significant at 0.05 level.
      18-20 y
       SNA0.011
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.001
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.528
       SNB0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.000
      Significant at 0.05 level.
      0.003
      Significant at 0.05 level.
       ANB0.005
      Significant at 0.05 level.
      0.001
      Significant at 0.05 level.
      0.4230.7700.0510.016
      Significant at 0.05 level.
      Significant at 0.05 level.

      Discussion

      In this study, we used GMM to evaluate and compare growth trajectories among different clinical classifications of vertical facial divergence. The strength of GMM is the preservation of geometric relationships and the orientation of structures as configurations of landmarks allowing for the evaluation of coordinated patterns of vertical and anteroposterior growth in the craniofacial complex. This landmark-based approach is novel to studies of craniofacial growth related to clinical classifications of facial morphology as previous work has focused on modeling the growth of linear distance measurements or isolated interlandmark angles of the face. In contrast, GMM uses information about the spatial relationships among all of the landmarks simultaneously.
      Growth models based on distance measurements are very useful for evaluating the timing and changes in growth velocity among individual measurements; however, traditional linear cephalometrics are poorly equipped to quantify the spatial relationships among variables or measurements. For example, it is possible for landmark positions or orientations to change from one configuration to another even if the linear distance itself does not change. In this situation, shape change would be very difficult to detect by studying linear measurements and would require a complicated simultaneous interpretation of multiple such measurements; GMM is designed to overcome this challenge.
      This study builds on previous analyses of growth related to vertical facial classifications by identifying nuanced differences in the growth and orientation of facial components among vertical skeletal facial types. The identification here of novel correlative anatomic change provides a better understanding of structural relationships that can ultimately lead to improved growth modification strategies. Differences in results among previous studies have often been attributed to variation among sample populations and classification procedures. Some studies have included only participants with the most extreme hyperdivergent and hypodivergent conditions
      • Nanda S.K.
      Patterns of vertical growth in the face.
      ,
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      or have restricted the sample to only those with Class I malocclusion.
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      In this study, we included all participants within the classification ranges to fully represent the variation within each facial type. Furthermore, we include all participants with Class I, II, and III malocclusions to examine the full range of vertical and anteroposterior relationships across facial types. We also determined facial type classifications using cephalograms in the oldest age group (18-20 years). This approach allows for maximum morphologic distinction among facial types and assessment of how the adult facial morphology was achieved during growth and development. Other studies have classified participants during the adolescent growth spurt,
      • Nanda S.K.
      Patterns of vertical growth in the face.
      ,
      • Moon S.C.
      • Kim H.K.
      • Kwon T.K.
      • Han S.H.
      • An C.H.
      • Park Y.S.
      Patterns of vertical facial growth in Korean adolescents analyzed with mixed-effects regression analysis.
      ,
      • Karlsen A.T.
      Craniofacial growth differences between low and high MP-SN angle males: a longitudinal study.
      ,
      • Karlsen A.T.
      Association between facial height development and mandibular growth rotation in low and high MP-SN angle faces: a longitudinal study.
      ,
      • Han S.H.
      • Park Y.S.
      Growth patterns and overbite depth indicators of long and short faces in Korean adolescents: revisited through mixed-effects analysis.
      or during childhood.
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      This analysis indicates that some differences in average morphology among the 3 facial types are established by 6 years of age, as previously noted.
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      ,
      • Oh H.
      • Knigge R.
      • Hardin A.
      • Sherwood R.
      • Duren D.
      • Valiathan M.
      • et al.
      Predicting adult facial type from mandibular landmark data at young ages.
      Nevertheless, here we show that each type follows a unique average growth trajectory until adulthood, indicating different adolescent growth patterns,
      • Nanda S.K.
      Patterns of vertical growth in the face.
      ,
      • Moon S.C.
      • Kim H.K.
      • Kwon T.K.
      • Han S.H.
      • An C.H.
      • Park Y.S.
      Patterns of vertical facial growth in Korean adolescents analyzed with mixed-effects regression analysis.
      ,
      • Karlsen A.T.
      Craniofacial growth differences between low and high MP-SN angle males: a longitudinal study.
      ,
      • Han S.H.
      • Park Y.S.
      Growth patterns and overbite depth indicators of long and short faces in Korean adolescents: revisited through mixed-effects analysis.
      with hypodivergent and normodivergent growth patterns more similar to each other than either is to the hyperdivergent growth pattern (Fig 2). These findings suggest the need for a nuanced interpretation of growth when attempting to estimate future face shape that accounts not only for early established differences in morphology but also the distinct growth trajectories among facial types that will further contribute to differences in face shape at later ages.
      The differences in growth trajectories among facial types also vary by sex. Male trajectories continue to diverge through 20 years of age, whereas the female trajectories become nearly parallel by age 18 years (Fig 2). The near-parallel trend in females compared with males is likely the result of the earlier cessation of adolescent craniofacial growth in females than males, with minor changes in later facial shape from postadolescent growth.
      • Nahhas R.W.
      • Valiathan M.
      • Sherwood R.J.
      Variation in timing, duration, intensity, and direction of adolescent growth in the mandible, maxilla, and cranial base: the Fels longitudinal study.
      The diverging male growth trajectories reflect the continued differential growth of craniofacial structures and would be expected to approach a near-parallel configuration, similar to females, during postadolescent growth.
      The results of this study suggest that morphologic differences between hyperdivergent and hypodivergent types result from coordinated changes in the orientation of the maxilla and mandible relative to the anterior cranial base, in conjunction with differences in anterior facial growth. In comparison with the average hypodivergent configurations, the hyperdivergent mandible and maxilla are rotated downward and backward, relative to the anterior cranial base, resulting in superior and anterior positioning of the mandibular condyle, downward rotation of the palatal plane anteriorly, and backward rotation of the mandibular corpus (Fig 4). The degree of rotation is greatest in the corpus. Backward rotation of the mandibular corpus in the hyperdivergent configuration also corresponds with an increase in relative lower anterior facial height, in contrast to the forward mandibular rotation and reduction in relative lower anterior facial height exhibited in hypodivergent growth. Differences in both posterior and anterior facial heights among facial types, manifest in part by coordinated maxillomandibular orientation, are supported by previous work.
      • Karlsen A.T.
      Craniofacial growth differences between low and high MP-SN angle males: a longitudinal study.
      ,
      • Björk A.
      • Skieller V.
      Normal and abnormal growth of the mandible. A synthesis of longitudinal cephalometric implant studies over a period of 25 years.
      Our results contrast with Nanda
      • Nanda S.K.
      Patterns of vertical growth in the face.
      and others
      • Moon S.C.
      • Kim H.K.
      • Kwon T.K.
      • Han S.H.
      • An C.H.
      • Park Y.S.
      Patterns of vertical facial growth in Korean adolescents analyzed with mixed-effects regression analysis.
      ,
      • Han S.H.
      • Park Y.S.
      Growth patterns and overbite depth indicators of long and short faces in Korean adolescents: revisited through mixed-effects analysis.
      ,
      • Enoki C.
      • Telles Cde S.
      • Matsumoto M.A.N.
      Dental-skeletal dimensions in growing individuals with variations in the lower facial height.
      who have argued that these morphologic differences are associated primarily with differential growth in the vertical anterior dimensions of the face with little difference in posterior facial height among classifications.
      When facial type was determined using the ratio of lower anterior facial height to total anterior facial height, rather than MPA, Enoki et al
      • Enoki C.
      • Telles Cde S.
      • Matsumoto M.A.N.
      Dental-skeletal dimensions in growing individuals with variations in the lower facial height.
      found that participants with smaller lower anterior facial heights (ANS-Me) have palatal planes that are rotated downward compared with participants with greater lower anterior facial height. Despite the noted difference in the angle from the mandibular plane to the palatal plane, Enoki et al
      • Enoki C.
      • Telles Cde S.
      • Matsumoto M.A.N.
      Dental-skeletal dimensions in growing individuals with variations in the lower facial height.
      found there was no difference in MPA by facial type. This contrasts with the findings of the current study that identify downward palatal plane rotation in the hyperdivergent face with a corresponding increase in lower anterior facial height. These contrasting results may stem from the use of different classification systems; the MPA in the current study and the ratio of lower anterior facial height to total anterior facial height by Enoki et al.
      • Enoki C.
      • Telles Cde S.
      • Matsumoto M.A.N.
      Dental-skeletal dimensions in growing individuals with variations in the lower facial height.
      As demonstrated by both the geometric and traditional morphometric analyses in this study (Figs 3 and 5), the hyperdivergent maxilla and mandible are relatively shorter and posteriorly positioned than the hypodivergent facial type, with this difference becoming more pronounced with age. In the hyperdivergent face, the relative anterior positioning of the mandible does not change substantially, but the maxilla becomes relatively shorter with age. Conversely, the hypodivergent face exhibits an increase in relative mandibular length with age, but relative maxillary length remains unchanged. These results are contrary to those found by Bishara and Jakobsen
      • Bishara S.E.
      • Jakobsen J.R.
      Longitudinal changes in three normal facial types.
      and Opdebeeck et al,
      • Opdebeeck H.
      • Bell W.H.
      The short face syndrome.
      who identified no difference in mandibular or maxillary length among facial types. Our results are supported by the A-P relationships of the maxilla and mandible found by Joseph et al
      • Joseph A.A.
      • Elbaum J.
      • Cisneros G.J.
      • Eisig S.B.
      A cephalometric comparative study of the soft tissue airway dimensions in persons with hyperdivergent and normodivergent facial patterns.
      using SNA and SNB angles, and Opdebeeck et al
      • Opdebeeck H.
      • Bell W.H.
      The short face syndrome.
      also find a more retrusive position of the maxilla in the hyperdivergent group.
      The observations reported in this study have clinical implications for understanding how and when features associated with vertical facial divergence arise. The identification of morphologic features indicative of facial divergence at younger ages can help improve the capacity of clinicians to diagnose and treat participants. Gonial angle and dentoalveolar height are mandibular features previously identified as good predictors of adult facial type at young ages.
      • Oh H.
      • Knigge R.
      • Hardin A.
      • Sherwood R.
      • Duren D.
      • Valiathan M.
      • et al.
      Predicting adult facial type from mandibular landmark data at young ages.
      In this study, these features, in addition to the depth of the antegonial notch and height of the subnasal maxillary region as described above, were also found to distinguish hyperdivergent and hypodivergent facial patterns at 6 years of age.
      Furthermore, patterns of facial divergence are often associated with different types of malocclusion. This analysis demonstrates how the A-P relationships of the maxilla and mandible change with age across facial types. In the normodivergent and hypodivergent types, forward rotation and elongation of the mandible, while maintaining proportional growth of the maxilla, effectively reduces the ANB angle over time. However, the A-P positioning of the mandible relative to the maxilla in the hyperdivergent face does not change substantially, with slightly greater anterior growth of the mandible than the maxilla. This suggests a greater propensity for maintaining a higher ANB angle and Class II malocclusion in the hyperdivergent group.

      Conclusions

      In this study, we find distinct patterns of craniofacial growth among the 3 classifications of vertical facial morphology (hyperdivergent, normodivergent, and hypodivergent). Patterns of growth are more similar among normodivergent and hypodivergent groups, with the magnitude of change being greater in the latter. Key morphologic differences among facial types were already present by age 6 years and intensified through adolescence. Much of the variation is concentrated in the shape and growth of the mandible, but differences were also present in the orientation and relative A-P dimension of the maxilla, indicating different patterns of coordinated maxillary and mandibular growth. Specifically, these results demonstrate that the downward and backward rotation of both the maxilla and mandible in conjunction with restricted relative maxillary growth in the hyperdivergent face may lead to increased risk for Class II malocclusions. Further analysis is needed to identify under what conditions vertical hyperdivergence may result in satisfactory or deficient A-P growth. This study will serve as a foundation for future work evaluating growth patterns within facial types and in conjunction with other vertical and A-P malocclusions.

      Author credit statement

      Ryan P. Knigge contributed to conceptualization, methodology, formal analysis, visualization, original and draft preparation, and review and editing; Kieran P. McNulty contributed to supervision, methodology, formal analysis, and review and editing; Heesoo Oh contributed to conceptualization, methodology, and review and editing; Anna M. Hardin contributed to review and editing, conceptualization, data curation, and methodology; Emily V. Leary contributed to conceptualization, methodology, and review and editing; Dana L. Duren contributed to conceptualization and review and editing; Manish Valiathan contributed to conceptualization and review and editing; and Richard J. Sherwood contributed to review and editing, project administration, funding acquisition, resources, conceptualization, and methodology.

      Acknowledgments

      The authors are grateful for the long-term commitment of the almost 2000 study participants who make up the overall study sample.

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