Bayesian approach to longitudinal craniofacial growth: The Craniofacial Growth Consortium Study

Abstract

Early in the 20th century, a series of studies were initiated across North America to investigate and characterize childhood growth. The Craniofacial Growth Consortium Study (CGCS) combines craniofacial records from six of those growth studies (15,407 lateral cephalograms from 1,913 individuals; 956 females, 957 males, primarily European descent). Standard cephalometric points collected from the six studies in the CGCS allows direct comparison of craniofacial growth patterns across six North American locations. Three assessors collected all cephalometric points and the coordinates were averaged for each point. Twelve measures were calculated from the averaged coordinates. We implemented a multilevel double logistic equation to estimate growth trajectories fitting each trait separately by sex. Using Bayesian inference, we fit three models for each trait with different random effects structures to compare differences in growth patterns among studies. The models successfully identified important growth milestones (e.g., age at peak growth velocity, age at cessation of growth) for most traits. In a small number of cases, these milestones could not be determined due to truncated age ranges for some studies and slow, steady growth in some measurements. Results demonstrate great similarity among the six growth studies regarding craniofacial growth milestone estimates and the overall shape of the growth curve. These similarities suggest minor variation among studies resulting from differences in protocol, sample, or possible geographic variation. The analyses presented support combining the studies into the CGCS without substantial concerns of bias. The CGCS, therefore, provides an unparalleled opportunity to examine craniofacial growth from childhood into adulthood.

1 ∣. INTRODUCTION

The longitudinal study of growth has a long history in North America. Beginning in 1913, a number of studies were initiated with the simple intention of charting growth from infancy to adulthood. Even with the simple technology available at the time, it became clear that characterization of a child's growth status provided important insight into the health of the individual. As the relationship between health and growth became increasingly apparent, the number of growth studies increased rapidly in the 1940s (see Garn, 1981 for a broader review). Individual studies had different goals and protocols, but several notable studies included craniofacial growth and dental maturation as primary areas of interest. Craniofacial records in these studies often included multiple radiographic images, most commonly lateral and postero-anterior (PA) cephalograms, and dental casts. Standards of craniofacial growth derived from these studies soon became important clinical tools across multiple disciplines. The longitudinal nature of these growth records provides an unparalleled opportunity to examine individual growth trajectories, and to examine how perturbations in timing of specific milestones may result in clinical conditions.

In 2008, curators of the largest craniofacial growth studies in North America received funding from the American Association of Orthodontists Foundation (AAOF) to assemble a virtual collection of longitudinal cephalograms and make them available to researchers via a web interface (http://www.aaoflegacycollection.org/; Baumrind & Curry, 2015). Each study in this collection provides a snapshot of normal craniofacial growth in a relatively discrete geographic region and a particular point in history. Individuals in these studies are predominantly of European descent. While these studies have been used for decades to quantify and describe growth, there has never been a detailed comparison of the growth trajectories among the studies. In 2015, we initiated a large-scale evaluation of craniofacial growth using cephalograms from the AAOF collection as well as additional cephalograms from three of the studies: Fels Longitudinal Study, Michigan Growth Study, and the Bolton Brush Growth Study. Together, this sample is collectively known as the Craniofacial Growth Consortium Study (CGCS).

1.1 ∣. Goals of the Craniofacial Growth Consortium Study

Data from the individual collections within the CGCS have served, for decades, as the focus of numerous studies/publications describing and detailing aspects of craniofacial growth. Rarely have investigators compared the ontogenetic trajectories from multiple studies, and even less often, have multiple studies been combined into a single study (Al-Jewair, Stellrecht, Lewandowski, & Chakaki, 2018; Baumrind & Curry, 2015; Oh et al., 2019) Among the primary goals of the CGCS is the development of new growth standards and individualized predictive models of craniofacial growth. Initial work by our group has shown the value and utility of this combined dataset (Hardin et al., 2019, 2020; Knigge et al., 2018, 2019; Sherwood et al., 2018; Sherwood, Hardin, et al., 2019; Sherwood, McNulty, et al., 2019). Prior to large-scale analyses making use of the CGCS sample, however, it is important to characterize and compare the ontogenetic trajectories from the separate studies to identify differences that may arise from factors ranging from geographic sampling to protocol variation that could potentially introduce systematic bias.

The goals of the current article, therefore, are to provide a brief history of the originating studies, describe the original protocols, and discuss the challenges of using historical radiographic archives that are up to 90 years old. In this article, we also identify and discuss many issues related to the use of historical radiographic archives and the steps we have taken to minimize systematic error and bias. Further, we model the ontogenetic trajectories of a set of commonly used cephalometric traits to assess the value and validity of combining multiple studies into one. To begin this process, we have identified a set of measurements that regularly play a role in clinical or anatomical assessment. Growth models for each trait, by study, are developed, for the purposes of evaluating any differences in ontogenetic trajectory in preparation for the combination of the individual studies into a single database.

1.2 ∣. Historical background of growth studies

In general, the early decades of the 20th century saw a growing interest in understanding human growth. This was due, in part, to the burgeoning field of physical anthropology (Sherwood & Duren, 2013b), as well as a growing clinical understanding of the lasting effects of childhood disease on lifetime health. In 1909, the first White House Conference on Children was held. Occurring each decade thereafter, these conferences were seen as “advisory to the American people as a whole and to the representatives in local, State, and national legislative bodies, to professional and citizens' groups, to public and private agencies” (U.S. Department of Health, 1967, p. 1). The 1930 conference focused on child health and protection and was held at a time when the effects of the Great Depression were being felt hardest by children (White House, 1930). While a few of the earliest growth studies in the U.S. predate the 1930 conference, it is not surprising that the number of longitudinal studies increased dramatically in the following years (Figure 1).

FIGURE 1.

Timeline showing activity in each of the six growth studies comprising the CGCS. For each study, the period when cephalographs were collected is highlighted

Accurate information regarding the history of each of the six studies comprising the CGCS is of critical importance. For example, given the variation in radiographic techniques, the historic details of the radiographic protocols of each study are particularly significant for interpreting results based on these studies, especially when combined into a single dataset. The degree of radiographic enlargement, for example, can vary both within and between studies, but correction of radiographic enlargement is a simple matter if the protocol is known. Over the course of the current investigation, it became clear that the information available in publications and elsewhere was incorrect for some studies (details are provided below) potentially biasing inferences drawn from those studies. Because including biased data in a combined database would be problematic, we prioritized the verification of protocol details, including correction factors, for each study. Below, we provide information on the active dates of each study, the primary motivation for the study, the number of participants, the types of data collected, and the number of radiographs.

1.2.1 ∣. Iowa

The Iowa Growth Study began in 1917 when Bird T. Baldwin, an educational psychologist with a long-standing interest in the study of growth, became the director of the newly funded Iowa Child Welfare Research Station. Baldwin had published multiple studies on growth and intellect in school children before coming to Iowa (e.g., Baldwin, 1911, 1914). Until his death in 1928, the study collected 1,300 radiographs of the hands and wrists of children from birth to 17 years of age (Tanner, 1981).

Charles McCloy and Howard Meredith became directors after Baldwin's passing. In 1946, Meredith began collaborating with the University of Iowa College of Dentistry to start the Iowa Facial Growth Study. A total of 183 children (92 males and 91 females) participated in this part of the study. Although some were examined as early as 3 years of age, children were typically examined every 6 months from age 5–12 years and then annually until 18 years of age. In addition to lateral cephalograms, dental casts and anthropometric measurements were taken at each visit. Most participants were of Northwestern European ancestry and from families of above-average socio-economic status, though approximately 27% of the fathers were considered semi-skilled or unskilled workers (Sturdivant, Knott, & Meredith, 1962). After gathering approximately 3,600 lateral cephalograms, the study concluded in 1960 (AAOF Legacy Collection, 2013). Twenty-six (16 males, 10 females) participants over 23 years of age returned for examination in 1968, and additional follow-up records were collected for 31 participants in mid-adulthood (AAOF Legacy Collection, 2013).

In 1964, the Iowa Child Welfare Research Station was renamed the Institute of Child Behavior and Development (Cravens, 1993) which closed in 1974.

1.2.2 ∣. Denver

The Denver Growth Study was initiated in 1923 by clinician Dr. W.W. Wasson. Wasson began the longitudinal study at the University of Colorado to study disease detection in high risk children. The study was incorporated into the Child Research Council in 1927, but it lasted in this form for only 3 years. When Al Washburn, a pediatrician, became the director of the council in 1930, he steered the research toward preventative pediatrics.

Participants in the Denver Growth Study were examined every month until 1 year of age, every 3 months until the end of the adolescent growth spurt, then every year until adulthood (Tanner, 1981). Lateral cephalograms were collected from 1931 to 1966 (Tanner, 1981). Along with those films, additional data were collected, including: anthropometry, skull measurements, skeletal maturation, and others (McCammon, 1970). Psychological data were also collected but were inconsistent due to the changes in the field. In total, the study enrolled 334 people of Northwest European ancestry from 215 families. A total of 968 cephalograms, from individuals ranging from 4 to 26 years of age, were collected (Tanner, 1981). Of the 334 study participants, 256 were enrolled before they were born and stayed in the study until they reached full maturity. Enrollment ended in 1966 and the study was concluded in 1971.

Major scientific contributions produced by the Denver Growth Study include Pritchett's growth charts (Bortel & Pritchett, 1993; Pritchett, 1988, 1992, 1997) and one of the first manuscripts to fit a curve to longitudinal human growth data (Deming & Washburn, 1963).

1.2.3 ∣. Fels Longitudinal Study

Samuel S. Fels, a businessman and philanthropist, and Arthur Morgan, President of Antioch College in Yellow Springs, Ohio, founded the Fels Longitudinal Study in 1929. It is said that Morgan wanted to answer the question, “what makes people different?” and Fels was eager to fund the study to answer this question (Roche, 1992). Recruitment of participants began in 1930 from the surrounding community. The Fels Longitudinal Study was unique among growth studies in that it actively recruited family members, this would provide an opportunity to study genetic influence on traits in the future (Duren et al., 2011; Duren, Seselj, Froehle, Nahhas, & Sherwood, 2013; Šešelj, Duren, & Sherwood, 2015; Sherwood et al., 2011; Sherwood & Duren, 2008). The Fels Longitudinal Study was also unique in that it continued to follow participants well past childhood; indeed, by the end of the study in 2018, original participants seen in the early years were still coming in for visits.

The basic plan for Fels Longitudinal Study participant visits was to be seen shortly after birth, then at 3, 6, 9 months followed by biannual visits at their half- and full birthday until they reached skeletal maturity. As adults, local Fels participants were brought in every 2 years and those who lived out of state were seen at least once every 5 years. Visits included collection of standard anthropometrics (e.g., height, weight, skin-fold thickness, etc.) and measurements of body mass via underwater weighing later transitioning to dual energy x-ray absorptiometry (DXA). Improving on prior skeletal maturity methods, an objective method for assessing skeletal maturity based on hand-wrist radiographs from children in the Fels Longitudinal Study was published in 1988 (Roche, Chumlea, & Guo, 1988; Roche, Chumlea, & Thissen, 1988). As personnel and technology changed, participant visits intermittently included other phenotypes such as psychological testing, gait analysis, and cardiac ultrasound.

Dental and craniofacial research first became an important part of the Fels Institute when investigator and pediatrician, Meinhard Robinow, published on deciduous tooth emergence (Robinow, Richards, & Anderson, 1942). Lateral cephalograms were of primary interest, but additional views including P-A cephalograms, oblique, or dental radiographs, were also collected. Starting in 1940, Arthur Lewis, a local dentist (and future editor of The Angle Orthodontist) began spending one day a week supervising dental casting and head radiographs. In 1952, Stanley Garn, an anthropologist and notable figure in the field of growth and development, came to the Fels Research Institute as chair of the Physical Growth Department. Lewis and Garn published several classic articles on craniofacial and dental growth and development (e.g., Garn, Lewis, & Kerewsky, 1965, 1967; Garn, Lewis, & Polacheck, 1958; Lewis, 1991; Lewis & Roche, 1972, 1974, 1977, 1988; Lewis, Roche, & Wagner, 1985). In the mid-1970s, the Fels Longitudinal Study became part of the newly formed Wright State University School of Medicine and moved to Dayton, Ohio in the early 2000s. The research focus of the Fels Longitudinal Study had shifted away from craniofacial growth in the 1980s and craniodental radiography was discontinued in 1984. Interest in the craniofacial program was revitalized by Richard Sherwood and Dana Duren in 2003 (Sherwood et al., 2008, 2011; Sherwood & Duren, 2008, 2011, 2012, 2013a, 2013b; Sherwood, Duren, Czerwinski, Blangero, & Towne, 2003; Sherwood, Duren, Subedi, & Williams-Blangero, 2013). Sherwood became Director of the Lifespan Health Research Center housing the Fels Longitudinal Study in 2008.

1.2.4 ∣. Bolton-Brush

The Bolton-Brush Growth Study consists of the combined data of the Brush Inquiry and the Broadbent-Bolton Growth Study. The Brush Inquiry, originally called the Health Inquiry, was a longitudinal growth study initiated in 1928 by T. Wingate Todd at Western Reserve University in Cleveland, OH, with financial support from Charles Brush and the Brush Foundation (Behrents & Broadbent Jr., 1984; Nelson, Hans, Broadbent Jr., & Dean, 2000). It monitored the growth and development of healthy white children from well-educated, upper-class families. In the early years, health contests were held to determine who would be enrolled in the study. Local teachers were asked to pick the “boy and girl who best exemplified ideal health,” and teachers and students were asked to pick the students who were “the scholastic and social bests” (Behrents & Broadbent Jr., 1984, p. 9). These contests were the basis for what would become the measurements and records for the Brush Inquiry (Behrents & Broadbent Jr., 1984).

In addition to recruitment through schools, many Brush Inquiry participants were recruited as babies and seen at 3, 6, 9, and 12 months of age and then every 6 months until 5 years of age, at which point participants were seen annually (Behrents & Broadbent Jr., 1984; Nelson et al., 2000; Tanner, 1981). In total, the Brush Inquiry recruited 4,435 subjects (Nelson et al., 2000). Radiographs of the regions of the skeleton and health, nutrition, anthropometric, and psychologic information were collected at each visit. During Todd's tenure as director, an atlas was created whereby a child's hand radiograph could be compared to a set of standard images from the same age and sex in order to determine skeletal maturity (Todd, 1937). When William W. Greulich became the director of the inquiry in 1940 Todd's atlas was updated with the help of S. Idell Pyle, to create the Greulich-Pyle Atlas (Gruelich & Pyle, 1950). Other skeletal maturity standards were devised from the Brush population, for other regions of the body including the knee, foot, and ankle (Hoerr, Pyle, & Francis, 1962; Pyle & Hoerr, 1955). Other major publications include a monograph regarding the early anthropometric data (Simmons & Greulich, 1944). The Brush Inquiry stopped recruiting and closed in 1942 due to the socioeconomic effects of the Second World War (Behrents & Broadbent Jr., 1984).

The companion Bolton Study began in 1929 when B. Holly Broadbent, Sr., an orthodontist who worked in Todd's research lab, became a professor of dentofacial morphology in the Department of Anatomy and initiated a study on craniofacial growth (Behrents & Broadbent Jr., 1984). Funded by the Charles Bingham Bolton fund, the Bolton study began as “an independent but coordinated study in conjunction with the Brush Inquiry” (Behrents & Broadbent Jr., 1984, p. 6). Along with lateral and frontal cephalograms, hand-wrist x-rays were taken (Tanner, 1981). Nutritional, dental and medical status, and disease history were also recorded in handwritten notes (Hans, Broadbent Jr., & Nelson, 1994). Enrollment stopped in 1959 after 4,309 participants had been enrolled. Of the Bolton participants, 2,432 individuals were also enrolled in the Brush inquiry. Forty to fifty thousand plaster casts of children's teeth were gathered, and 409,000 radiographs were taken, with an average of 15 timepoints per participant. Of the 4,309 participants, 53.1% were female, 46.9% male, 90.7% were white, 9.2% black, and 0.1% were labeled “other” (Behrents & Broadbent Jr., 1984). They were seen at 3, 6, 9, and 12 months and then every 6 months until 5 years of age. After 5 years of age, participants were seen annually.

In 1948, Case Western Reserve University (CWRU) incorporated the Bolton Study into the School of Dentistry as its own center. The Bolton Study took over the Brush Inquiry and became the Bolton-Brush Growth Study Center at CWRU around 1960. A set of serial outlines representing ideal craniofacial form and growth, known as the Bolton Standards, were created in 1975 from 32 subjects: 16 boys and 16 girls (Standerwick, Roberts, Hartsfield Jr., Babler, & Katona, 2009), and are still in use today. In 1982, 113 of the original participants returned for a recall study by Rolf Behrents (Behrents, 1985).

1.2.5 ∣. Michigan

Students enrolled in the University Elementary and High School, approximately 3–18 years of age, were recruited for the study (Riolo, Moyers, McNamara, & Hunter, 1974). No original sources describing the ancestry of the participants can be found. Dental casts, lateral jaw radiographs, and occlusal plane radiographs were the only dental information gathered in the first years of the study (Riolo et al., 1974). In addition, hand/wrist X-rays were taken as well as psychological, behavioral, educational, and physical growth data. In 1953, annual lateral cephalograms were initiated, and they discontinued dental casts, lateral jaw radiographs, and occlusal plane radiographs. Dr. Robert E. Moyers was chairman of the Orthodontics department at the time and, with funding from NIH, the study thrived. In 1966 Donald Enlow joined Moyers with additional NIH funding (HD 02272 continued later as DE 43120) to continue the work. The AAOF Craniofacial Growth Legacy collection reports 3,266 lateral cephalograms of 721 individuals from 150 families. The collection of lateral cephalograms ended when the University School closed in 1970 (AAOF Legacy Collection, 2013).

In 1974, a symposium series honoring Moyers began that continues to this day. This symposium has generated several classic monographs on Craniofacial Growth (https://deepblue.lib.umich.edu/handle/2027.42/146667/recent-submissions?offset=0), including a comprehensive analysis of craniofacial traits in “An Atlas of Craniofacial Growth” (Riolo et al., 1974).

1.2.6 ∣. Oregon

The Child Clinic Study conducted at the Oregon Health and Science University Dental School in Portland, Oregon, commonly known as the Oregon Growth Study, was started by orthodontist Bhim Savara in 1952 and concluded in 1974. Although the impetus for the study is not explicitly described, one can surmise a primary interest in longitudinal growth was the principal motivator.

Records regarding total number of participants enrolled and the age ranges for data collection are conflicting. Hunter, Baumrind, and Moyers (1993) reports that 409 individuals (221 female and 188 male) were enrolled in the study. Tanner (1981) report participants ranged between 3 and 18 years of age while Savara and Steen (1978) write that participants were between 3 and 28 years of age. Cephalograms in the AAOF collection include individuals aged 2–30 years. The number of twins enrolled in the study is also unclear, with the number of sets ranging from 20 (AAOF Legacy Collection, 2013) to 45 (University of Oregon Bulletin, 1964).

All participants lived in or near Portland, Oregon, self-reported Northwest European ancestry, and had parents of middle and upper socioeconomic status (Savara & Steen, 1978). Study data included lateral cephalograms, P-A cephalograms, hand/wrist radiographs, and intraoral radiographs (AAOF Legacy Collection, 2013). All cephalograms were collected annually using the Bolton-Broadbent method (Savara, 1965a, 1965b). These images were collected along with medical histories (Hunter et al., 1993). The study also collected subjects' physical measurements, dental study casts, calf-and-knee roentgenograms, photographs (University of Oregon, 1964), dietary records, and health histories (AAOF Legacy Collection, 2013). There are 1,948 images in the AAOF Legacy Collection.

Though others published data from the Child Clinic Study, Bhim Savara was the most prolific and published numerous articles on craniofacial growth and dental eruption. Other notable researchers include G. Dave Singh (Singh & Savara, 1966), Norifumi Nakamura (Nakamura, Savara, & Thomas, 1972), and Douglas L. Buck (Buck & Church, 1972), who also served as director of the study.

2 ∣. METHODOLOGY

2.1 ∣. Sample

Table 1 provides the details of the overall sample available in each study of origin and in the CGCS by sex. The total number of cephalograms examined in the CGCS is 15,407. The median number of cephalograms is 9–12 per individual in five of the studies and four cephalograms per individual in the Michigan Growth study, with an overall median of 9. For these analyses, individuals with a single cephalogram were excluded; the range, therefore is 2–22 images per person. In general, the age ranges for each study are dense in the target period between six and 18 years. As described, several studies either stopped seeing participants in mid to late adolescence or reduced visit frequency such that the number of images past 18 years is notably less than in the earlier periods. Figure 2 provides the distribution of images by participant and by age for the CGCS and each study contained within.

TABLE 1.

Sample size of CGCS by study

	Total # of individuals in CGCS
Set	Females	Males	Total
Bolton Brush	193	209	402
Fels	267	281	548
Michigan	349	326	675
Oregon	56	47	103
Iowa	47	44	91
Denver	44	50	94
Total	956	957	1913

FIGURE 2.

Number of images comprising the CGCS. (a) number of images per participant by study. Median number of images = 9–12 in each study but Michigan with a median of four images per person. (b) number of images per age group (years) by study. Total number of images in the CGCS = 15,407

The University of Missouri Institutional Review Board approved all procedures used in this study.

2.2 ∣. Cephalogram quality from historical collections

When dealing with large historical radiographic collections, such as those that make up the CGCS, there is concern that image deterioration will reduce the available sample for analyses. During the course of curation of the Legacy Collection, or as part of other studies (Šešelj et al., 2015; Sherwood et al., 2011; Sherwood & Duren, 2008), more than 50,000 cranial radiographs (multiple views) from the historical collections described here were individually examined prior to digital archiving. Very few of these radiographs suffered from age-related deterioration (e.g., cracks or peeling emulsion), and these few damaged radiographs are excluded from the study.

Excluding radiographs with obvious physical deterioration, the overall quality of images was variable due to changes in radiographic technology occurring during the period of image collection. In general, growth studies in North America began using plain film radiography in the 1920s to capture changes of anatomical regions in the growing child. Over the years, there were numerous advancements in film technology, including the use of fluorescent screens (i.e., intensifying screens) and the improvement of these screens over time, which greatly improved the image and reduced radiation exposure.

Complete imaging of cranial structures is complicated by variation in hard and soft tissue thickness and density of the anatomical regions of the skull, such that exposure settings to maximize anterior facial or nasal regions will result in underexposed intracranial structures such as the sella turcica. Correcting exposure for proper imaging of sella in turn overexposes anterior regions, and so on. These regional image inequalities increase with age as bones thicken and increase density and the overall cranial volume increases. To compensate for this difference in tissue thickness and density, an aluminum wedge filter was used to selectively attenuate the beam in the region of the face. When used, this sufficiently improved the image such that soft-tissue structures (e.g., the nose and lips) are visible. Use of a wedge filter is variable between and within studies of the CGCS. Modern improvements include digital detectors and easy manipulation of image properties via software.

Variation in image quality was a primary challenge with the radiographic collection for the current study. In some cases (The Fels Longitudinal Study in particular), the collection had been copied at one point in an effort for long-term preservation. As the current study uses computer-based data collection, all cephalograms were scanned at their home institutions. Scanners vary in quality due to differences in hardware and software with the effect that image quality may potentially suffer in the transition from hard copy to digital image. The ability to manipulate brightness and contrast or spontaneously apply filters during assessment frequently compensates for minor issues.

In cases where the digital versions of the cephalograms had substantial exposure issues, we found that image quality could be improved through the application of histogram equalization algorithms (Gonzalez, Woods, & Eddins, 2004; Hum, Lai, & Salim, 2014). Through this, we often found that images previously thought largely unusable could be significantly improved such that data loss was minimized and accuracy improved. An example of the application of these algorithms is seen in Figure 3; the python code used to apply the algorithms is available via NIH-figshare (https://doi.org/10.6084/m9.figshare.c.5037854) or from the authors.

FIGURE 3.

Example of application of histogram equalization to a lateral cephalogram

2.3 ∣. Cephalometric tracing and averaging process

Cephalometric landmarks were digitized as Cartesian coordinates using the eDigit software program developed by the Craniofacial Research Instrumentation Lab (CRIL; Arthur A. Dugoni School of Dentistry, University of Pacific) (Baumrind & Miller, 1980). Needle-point perforations in the corner of each film serve as registration points, referred to as “fiducials.” Landmark coordinate values for multiple tracings of each film are then mathematically superimposed, registered on the fiducial points. The coordinate values for all points on each tracing are expressed according to an orthogonal system that is based arbitrarily on two of the corner fiducials. Each tracing was digitized separately by three individuals. The coordinates for sella from the three assessments were averaged, and the mean value redefined as the origin of the new coordinate system. Similarly, the coordinates for nasion were averaged, and the mean redefined as a point on the new x-axis (i.e., a point with a y-value of zero). All other landmarks on each tracing were then mathematically reexpressed in terms of this new “sella-nasion coordinate system.” The eDigit software proceeds to check the reliability of the three coordinates of the various landmarks for the multiple tracings of each film. The final coordinates will be the mean value of the triplicate tracings. If two or more coordinates for a landmark are grossly different, as determined from the established envelope of error for that landmark, the program will report no coordinates and the assessors have the option to retrace and rerun the averaging process. This process is built into the CRIL-Avepic software program. The error deletion and averaging processes are crucial to controlling systematic and random errors in landmark identification.

2.4 ∣. Computation of lengths

Based on the coordinates identified, 11 craniofacial traits were computed from ten landmarks (Tables 2 and 3, Figure 4). Measures were chosen based on common usage by clinicians. The set of measures includes aspects of posterior and anterior cranial base length, facial height, mandibular height and length, and palatal length. All linear distances were corrected for radiographic enlargement using the specific correction factors established for each study (Table 4). These correction factors are based on the formula $\frac{X(\text{TH}-D)}{\text{TH}}$ where X is the radiographic measurement, TH is the tube height or the tube-to-film distance, and D is the distance from object (in this case the cranial midline) to the film (Sherwood, Meindl, Robinson, & May, 2000).

TABLE 2.

Cephalometric points used in the current study (see Figure 4)

Number	Name	Abbreviation	Definition
1	Sella	S	Mid-point of the pituitary fossa
2	Basion	Ba	Inferior-most point on the anterior margin of the foramen magnum in the midsagittal plane
3	Gonion	Go	Lowest point of the curvature of the angle of the mandible where the inferior surface of the body of the mandible meets the ramus. When two mandibular images are seen, the average of the points for each image is used.
4	Menton	Me	Inferior-most point on the mandible at the symphysis
5	Pogonion	Pog	Anterior-most point of the bony chin at the midline
6	PNS	PNS	Intersection between the posterior extension of the superior surface of the palate and the downward extension of the pterygomaxillary fissure
7	ANS	ANS	Anterior-most point of the anatomical anterior nasal spine
8	Nasion	N	Most antero-inferior point on the frontal bone at the naso-frontal suture
9	Articulare	Ar	The intersection of the image of the posterior border of the ramus with the external surface of the basicranium
10	Condylion	Co	Point on the posterior-superior contour of the condyle that is the longest distance from pogonion

TABLE 3.

Sample size by trait

Trait	Females	Males
ANS-PNS	871	880
Ar-Pog	871	881
Co-Go	871	881
Co-Pog	871	881
Go-Pog	871	881
Me-ANS	871	879
N-ANS	871	880
N-Ba	870	879
N-Me	871	880
S-Ba	870	879
S-Go	871	881
S-N	871	881

FIGURE 4.

Cephalometric points and linear measures used in the current study

TABLE 4.

List of enlargement factors for each study^a

Study	Magnification
Michigan	87.1%
Bolton Brush	92%
Denver	96%
Oregon	92.2%
Fels	Variable 89.6–96.8% (see below)
Iowa	94% prior to March 16, 1956
	91% between March 16, 1956 and September 19, 1957
	87% for films on or after September 19, 1957 and before 1970
	87.75% for films in 1970 and later

Fels Longitudinal Study (enlargement factor based on age of child and visit date
Age (years)	1930–1940	1941–1947	1948–1952	1953–1964	1965–
0.1	93.2%	96.0%	96.1%	96.8%	91.7%
0.2–0.4	92.3%	95.1%	95.2%	96.0%	91.7%
0.5–0.7	91.9%	94.7%	94.8%	95.7%	91.7%
0.8	91.7%	94.5%	94.6%	95.5%	91.7%
0.9–1.5	91.6%	94.4%	94.5%	95.4%	91.7%
1.6–3.0	91.4%	94.2%	94.4%	95.3%	91.7%
3.1–5.0	91.1%	93.9%	94.1%	95.1%	91.7%
5.1–7.0	90.9%	93.7%	93.9%	94.9%	91.7%
7.1–9.0	90.7%	93.5%	93.7%	94.7%	91.7%
9.1–11.0	90.5%	93.3%	93.4%	94.5%	91.7%
11.1–13.0	90.3%	93.1%	93.3%	94.4%	91.7%
13.1–15.0	90.1%	92.9%	93.1%	94.2%	91.7%
15.1–17.0	89.9%	92.7%	92.9%	94.1%	91.7%
17.1–19	89.7%	92.5%	92.7%	93.9%	91.7%
19.1 +	89.6%	92.3%	92.5%	93.8%	91.7%

Note: Dates from January 1–March 31, 1941, have been recorded as 1940. Dates from January 1–August 15, 1948, have been recorded as 1947. Dates from January 1–May 31, 1953, have been recorded as 1952.

^aIn our work we discovered several errors regarding enlargement factors reported in prior publications, or originally provided by the AAOF Legacy Collection. The Legacy Collection scaling document has been updated to match the values reported here.

2.5 ∣. Statistical methods

2.5.1 ∣. Multilevel double logistic growth model

To estimate growth trajectories, we implemented a multilevel double logistic equation as described by Bock et al. (1973), and fit each trait separately to this model. The general form of this model for a single individual with a series of trait measurements (y) at different ages is:

$$y(age)=\frac{a_1}{1+e^{-b_1(age-c_1)}}+\frac{f-a_1}{1+e^{-b_2(age-c_2)}}$$

where the first term represents the prepubertal stage of growth, to which is added an adolescent growth spurt, the second term. In this double logistic model, y(age) is the trait measurement at a given age. Constants, f, a₁, b₁, c₁, b₂, and c₂, define the pattern of growth as a function of age. f is the asymptotic measurement; a₁ is the trait value contribution from the prepubertal component, and f − a₁ from the adolescent stage. b₁ and b₂ are the slopes and c₁ and c₂ are ages at maximal velocity for the prepubertal and adolescent stages, respectively. Although Bock et al. (1973) assumed f to be fixed and known, only estimating five model parameters, here we also estimate f from the data, given the overall goal of establishing population-level growth pattern and rate estimates. Originally proposed for estimating patterns of human stature (but see el Lozy (1978) for criticism in the context of height), this model performs well for any trait that exhibits two periods of growth.

Nonlinear growth models, such as the double logistic model above, have some advantages over more easily fit linear models (e.g., polynomials). Nonlinear models are often based on a natural process such as growth, which have predictable patterns of monotonicity (only increasing or decreasing over their range) and asymptotes (the trait's maximum predicted value), both of which are present in human growth. Second, although polynomials may fit as well as nonlinear models across the observed range of data, polynomials frequently deviate significantly outside of that range. Finally, the parameters of nonlinear models often have directly interpretable values (Pinheiro & Bates, 2000).

In the sample for the current study, each individual had a set of measurements at two or more ages, resulting in the need for a multilevel model to account for within-individual variation. Similarly, the six growth studies were assumed to be drawn from a random set of similar studies. Levels for data set and repeated measures per individual within set were modeled as multilevel or “random” effects (Gelman & Hill, 2007; Laird & Ware, 1982; Pinheiro & Bates, 2000). These estimates serve as intercept offsets, an adjustment to f per participant within each growth study.

2.6 ∣. Bayesian inference

Parameters in nonlinear models, such as the double logistic model used here, can be difficult to estimate (Bates & Watts, 1988), a challenge which is greatly increased for multilevel models, which can have many hundreds of estimated parameters (>875 in this study). Bayesian inference using Monte Carlo methods can effectively address this challenge because they do not rely on methods such as maximum likelihood or expectation maximization (EM), which can fail with even moderately complex models (e.g., those with uneven sampling or small numbers of within-group observations). Parameter estimation via Markov Chain Monte Carlo or Hamiltonian Monte Carlo estimation (Gelman et al., 2013) involves the probabilistic exploration of parameter space and results in the gradual accrual of sets of simultaneous parameter estimates that are most consistent with the data, given the model and priors for each parameter (see below). With sufficient samples, a distribution of the most plausible parameter estimates is produced (McElreath, 2015). Additionally, because distributions of parameter estimates are produced, credible intervals such as highest density posterior intervals (HDPI) result naturally.

The six parameters governing the shape of the double logistic function as well as intercepts for growth study and individual were estimated separately by sex for each of the 12 traits (Table 5). For each trait, we fit three types of models: (1) with both growth study set (each of the six growth studies, set) and participant ID as multilevel predictors, (2) participant ID as multilevel predictor and all growth studies pooled, and (3) with ID only and models fit separately for each growth study. The former analysis yields a population-level prediction, and the latter were necessary to compare results from among the six growth studies.

TABLE 5.

Example of double logistic analysis for the trait Nasion to Basion in each sex. Full parameter set is shown

Parameter	Mean	SD	Lower 89%	Upper 89%
Male
f	106.58	0.32	106.05	107.07
a1	101.70	0.37	101.11	102.28
b1	0.18	0.01	0.17	0.19
c1	−5.35	0.15	−5.58	−5.11
b2	0.97	0.07	0.86	1.08
c2	13.60	0.06	13.51	13.69
Female
f	99.27	0.31	98.76	99.75
a1	95.78	0.38	95.17	96.37
b1	0.22	0.01	0.21	0.23
c1	−4.65	0.16	−4.90	−4.40
b2	0.90	0.08	0.77	1.04
c2	10.80	0.10	10.64	10.96

2.6.1 ∣. Starting values and priors

The priors in Bayesian inference represent the plausible distribution of values that the parameter estimates can take before the data has been incorporated (Gelman et al., 2013; McElreath, 2015). One of the challenges to nonlinear model fitting is the choice of starting values, which Pinheiro and Bates (2000, p. 276) describe as “somewhat of an art.” In a Bayesian context, starting values for parameters in each chain are drawn from the prior distributions for those parameters. Thus, determining the priors for each parameter provides starting values for sampling. We used approximate Bayesian computation (ABC) to search the plausible parameter space for each of the six parameters.

Approximate Bayesian computation (Beaumont, Zhang, & Balding, 2002; Pritchard, Seielstad, Perez-Lezaun, & Feldman, 1999; Tavaré, Balding, Griffiths, & Donnelly, 1997) is a general purpose approach to optimization which has proven useful for estimating parameters when derivative-based methods often fail (Beaumont, 2010; Csilléry, Blum, Gaggiotti, & Francois, 2010). We briefly describe the algorithm here. At each iteration, sets of parameters are drawn from prior distributions and used to predict the observed outcome values, given a model. The difference between the observed and predicted values is summarized (here by the mean squared difference [MSD] across all observed values). In ABC, a posterior sample size is typically desired (e.g., 1,000 samples less than some target MSD). Based on knowledge of the underlying process, we loosely constrained the search space for parameters. f was drawn from a normal distribution centered on the mean value for observations over age 20 (standard deviation equal to 5% of the mean value), a₁ from a uniform distribution between 70% of f and f − 1 (mm), b₁ and b₂ from a uniform distribution between 0 and 2 (flat or monotonically increasing trait measurement; mm/yr), and c₁ and c₂ from a uniform distribution between −20 and 20 (yr). The model assumed that all observations were independent (i.e., a nonmultilevel model), which provided adequate starting values and means for prior distributions for Bayesian inference. Because we required only point estimates, rather than a posterior sample, we generated 10⁸ samples, which were sorted by MSD and filtered for values of c₂ greater than 8 (effectively constraining the second growth peak to occur after age 8). Although time consuming to generate samples, ABC proved effective for determining starting values for Bayesian multilevel modeling.

For final Bayesian inference, we used normal distributions in priors for the outcome trait value (y), as well as the parameters f, a₁, b₁, c₁, b₂, and c₂, with the latter centered on their estimates from ABC. Multilevel effects of growth set and ID were centered on zero (i.e., no mean difference from the population value). All standard deviations were assigned an exponential prior with rate λ = 0.25 (mean = 4; median ≈ 2.8).

Thus, the full model specification, with prior distributions (Pr) for parameters was

$$\begin{gathered} y_i\sim \text{Normal}({\mu }_i,\sigma ) \\ {\mu }_i=\frac{a_1}{1+e^{-b_1(ag{e}_i-c_1)}}+\frac{f-a_1}{1+e^{-b_2(ag{e}_i-c_2)}}+a_{ID}[ID]+a_{set}[set] \\ f\sim \text{Normal}(P{r}_f,2) \\ {a}_1\sim \text{Normal}(P{r}_{a_1},2) \\ {b}_1\sim \text{Normal}(P{r}_{b_1},0.2) \\ {c}_1\sim \text{Normal}(P{r}_{c_1},1) \\ {b}_2\sim \text{Normal}(P{r}_{b_2},0.2) \\ {c}_2\sim Normal(P{r}_{c_2},1) \\ \sigma \sim \text{Exponential}(0.25) \\ {a}_{ID,j}\sim \text{Normal}(0,{\sigma }_{ID}),\text{for}j=1\dots n_{ID} \\ {\sigma }_{ID}\sim \text{Exponential}(0.25) \\ {a}_{set,k}\sim \text{Normal}(0,{\sigma }_{set}),\text{for}k=1\dots 6 \\ {\sigma }_{set}\sim \text{Exponential (0.25)}. \end{gathered}$$

The second set of six models, one for each growth study, was identical, except that these models did not include separate estimates for a_set[set] or the associated priors.

2.6.2 ∣. Sampling details

Models were fit using the Stan programming language (Carpenter et al., 2017; Gelman, Lee, & Guo, 2015) via the rethinking package (McElreath, 2015; https://github.com/rmcelreath/rethinking) in R (ver. 3.6.2; (R Development Core Team, 2013). Each model was run with four chains in parallel for 10,000 iterations, yielding at least 4,000 effective samples for the six parameters of interest. Adequate sampling was assessed visually via rank histograms and $\widehat{R}$ values ≤1.01 (Vehtari, Gelman, Simpson, Carpenter, & Burkner, 2019).

3 ∣. RESULTS

Results for each trait are summarized below. Full results including all analysis-specific graphs and tables are provided on NIH Figshare (https://doi.org10.6084/m9.figshare.c.5037854). To detail the analyses conducted, we begin by focusing on the trait Nasion–Basion as an example. Table 5 provides the population parameter set from the double logistic analysis for Nasion-Basion in both sexes. Table 6 shows the milestone estimates for the population models. Despite the traits representing different regions of the craniofacial complex, basicranium, upper face, mandible, the age at peak growth velocity and age at cessation for the 12 traits is consistent. For example, age at peak growth velocity cluster within a period of about 2 years centered at ~11 years and ~14 for females and males, respectively. Values for age at cessation also tend to cluster around a 2–3 year period centered at 15 and 18 years for females and males respectively. There are clear extreme estimates for both parameters, when models are estimated separately for each growth study.

TABLE 6.

Population models by trait

Trait	aPGV	PGV	sPGV	aCess	sCess	% sCessPGV
Male
ANS-PNS	13.63	1.02	49.67	16.90	51.84	95.8
Ar-Pog	14.00	2.78	99.89	18.63	107.73	92.7
Co-Go	14.74	2.30	61.01	24.58	69.01	88.4
Co-Pog	14.05	3.21	106.84	NA	NA	90.7
Go-Pog	12.58	1.47	61.87	15.79	65.89	93.9
Me-ANS	13.84	1.70	62.79	16.53	66.44	94.5
N-ANS	13.26	1.39	49.40	19.42	52.85	93.5
N-Ba	13.42	1.77	100.50	16.74	104.45	96.2
N-Me	13.74	3.08	111.10	17.58	118.34	93.9
S-Ba	NA	NA	NA	16.84	45.38	NA
S-Go	14.26	2.72	77.89	19.90	86.27	90.3
S-N	14.37	1.10	68.15	22.26	71.18	95.8
Female
ANS-PNS	11.63	0.79	46.91	13.26	48.08	97.6
Ar-Pog	11.42	2.25	91.96	15.63	98.22	93.6
Co-Go	12.00	1.51	54.40	23.37	60.20	90.4
Co-Pog	11.53	2.39	98.57	15.95	105.10	93.8
Go-Pog	10.63	1.45	58.13	13.90	61.95	93.8
Me-ANS	12.53	1.25	58.91	15.26	61.23	96.2
N-ANS	NA	NA	NA	14.42	48.67	NA
N-Ba	10.32	1.47	93.60	13.79	97.29	96.2
N-Me	11.47	2.24	102.93	14.68	108.41	94.9
S-Ba	8.05	0.87	38.20	12.26	41.18	92.8
S-Go	11.79	2.05	70.34	19.42	76.77	91.6
S-N	11.47	0.79	63.81	14.16	65.36	97.6

Abbreviations: aCess, age (years) at cessation of growth; sCess, size at cessation of growth; PGV, peak growth velocity; aPGV, age (years) at peak growth velocity; sPGV, size at peak growth velocity; %Growth_PGV, percent of total growth at aPGV.

Figure 5a provides a spaghetti plot showing raw data for each individual in the total sample for each study for the trait Nasion-Basion. From this image, one can clearly see that measurements of Nasion-Basion from specific studies do not separate into unique clusters. That is, the overall distribution of Nasion-Basion measures for each set spans the range of the entire sample (gray lines). These plots provide clear information on the differences between the studies in terms of overall density of records as well as duration of observation. Because they are drawn from such a large number of individuals and observations the six studies likely approximate population-level variation. This observation is particularly striking given the geographic, temporal, and experimental variation among the individual studies.

FIGURE 5.

(a) Plots of ontogenetic trajectories (raw data) of individuals by study of origin for the trait Nasion–Basion. Plot of each study is superimposed over plots of the total sample in grey. (b) Double logistic model separately estimated for each study in both females and males. Dotted line = age (years) at peak growth velocity; solid line = age (years) at cessation. (c) Plot of first derivative for the longitudinal models. Dashed line = age (years) at peak growth velocity for the population, solid line = age (years) at cessation for the population

Figure 5b provides plots of ontogenetic trajectories of individuals by study of origin for the trait Nasion-Basion, with the double logistic model separately estimated for each study in both females and males. These plots identify age at peak growth velocity (aPGV; dashed vertical line) and age of cessation of the adolescent growth spurt (defined as age at 98% of asymptotic value; solid line). The separate growth trajectories per study are superimposed along with the trajectory estimated for the population model in Figure 5b. In general, the overall shape for each study is similar. Figure 5c provides a plot of the first derivatives over age for the growth model of Basion to Nasion for each study and for the population model as in Figure 5b. The first derivative shows the change in growth velocity over the course of the adolescent period. The middle peak in the velocity curves identifies the age of peak growth velocity.

Table 6 provides milestone estimates from the population models for all traits. For the trait Basion—Nasion in males, the population estimate of age at peak velocity was 13.42 years, with a peak velocity of 1.77 mm/yr. The estimate for growth cessation is 16.74 years with the average length of Basion to Nasion at cessation equal to 104.45 mm. Comparison of ontogenetic milestones between studies (Table 7) shows that, for estimates of maximum velocity (range 1.51 mm/yr (Michigan) to 2.05 mm/yr (Oregon)), age at peak growth (range 12.84 years (Michigan) to 13.74 years (Denver)), and size at peak growth (range 99.70 mm (Oregon) to 101.19 mm (Fels)), show only minor variation between studies. The estimates for age at cessation are similarly variable between studies with the minimal age estimate from the Iowa Growth Study at 15.84 years, and the maximal age estimate of 17.00 years for the Oregon Growth Study. Size at cessation was notably consistent among studies with the total difference between extremes only 1.1 mm (minimum 104.12 mm (Oregon); maximum 105.13 mm (Fels)). The percent of final growth achieved at aPGV was also calculated and was consistent among studies with a range from 95.7% (Michigan) to 96.5% (Iowa).

TABLE 7.

Milestone estimates for each study by sex for the trait Nasion to Basion

Set	aPGV	PGV	sPGV	aCess	sCess	% GrowthPGV
Male
Bolton Brush	13.42	1.81	100.48	16.79	104.63	96.0
Denver	13.74	1.94	100.53	16.68	104.37	96.3
Fels	13.26	1.66	101.19	16.47	105.13	96.3
Iowa	13.32	2.00	101.05	15.84	104.67	96.5
Michigan	12.84	1.51	100.46	16.84	104.98	95.7
Oregon	13.26	2.05	99.70	17.00	104.12	95.8
Female
Bolton Brush	10.11	1.53	93.33	13.58	97.24	96.0
Denver	10.32	1.41	93.34	13.63	96.87	96.4
Fels	9.53	1.38	93.49	13.79	98.12	95.3
Iowa	10.00	1.43	93.65	13.37	97.39	96.2
Michigan	9.79	1.56	94.37	13.00	98.37	95.9
Oregon	10.37	1.46	93.36	13.74	97.00	96.2

Abbreviations: aCess, age (years) at cessation of growth; sCess, size at cessation of growth; PGV, peak growth velocity; aPGV, age (years) at peak growth velocity; sPGV, size at peak growth velocity; %Growth_PGV, percent of total growth at aPGV.

In females, the pattern is similar to that described for males with age of milestone attainment earlier, and size smaller, as would be expected. The population estimate of age at peak velocity was 10.32 years, with a peak velocity of 1.47 mm/yr (Table 6). The estimate for growth cessation was 13.79 years with the average length of Basion–Nasion at cessation equal to 97.29 mm. Examination of ontogenetic milestones between studies (Table 7) shows that estimates of peak growth velocity (range 1.38 mm/yr (Fels) to 1.56 mm/yr (Oregon)), age at peak growth (range 9.53 years (Fels) to 10.37 years (Oregon), and size at peak growth (range 93.33 mm (Bolton Brush) to 94.37 mm (Michigan)) are, again, very consistent between studies. The estimates for age at cessation vary similarly between studies with the minimal age estimate from the Michigan Growth Study at 13.00 years, and the maximal age estimate of 13.79 years for the Fels Longitudinal Study. Size at cessation showed slightly more variation among studies all traits by study for each sex. With few exceptions, it is clear there is considerable consistency in milestones among the individual studies. It is important to note that there is no clear pattern in position of the studies among the milestones. That is, no one study consistently has the earliest or latest aPGV or age at cessation or the smallest or greatest PGV (Figures 6 and 7).

FIGURE 6.

Plots, as described in Figure 5, of all traits examined. (a) ANS-PNS; (b) Ar-Pog; (c) Co-Go; (d) Co-Pog; (e) Go-Pog; (f) Me-ANS; (g) N-Me; (h) S-Ba; (i) S-Go; (j) S-N; (k) N-ANS

FIGURE 7.

Graphical summary of the milestone estimates for all traits by study for each sex. (a) age (years) at peak growth velocity; (b) age (years) at cessation; (c) peak growth velocity

4 ∣. DISCUSSION

4.1 ∣. What is normal?

The motivation for a long-term growth study can be highly variable ranging from biological to clinical to environmental inquiries. The six studies described here were all established to investigate growth of normal individuals. Normal in this context means without large-scale clinical disruptions that would affect growth such as endocrine disorders or congenital defects. Historical growth studies were often not directed by physicians and did not include medical examinations or tests. Common procedures for growth studies included anthropometric assessments such as height, and weight and radiographic assessments of the skeleton such as hand/wrist, knee, and head. Health assessments and health histories were largely self-reported. A common goal amongst studies was the development of growth standards. In this context, a standard of normal growth provides a means of comparison for a patient or participant providing a simple assessment of a child relative to their peers. Deviation from the standard may indicate a growth discrepancy warranting further investigation.

For the studies used here, “normal” may be difficult to define. Approximately one-third of the current U.S. population undergoes some form of orthodontic treatment during their lifetime. Sharma, Narkhede, Sonawane, and Gangurde (2013) noted that, in orthodontic patients, approximately 50% of patients self-diagnosed a need for treatment based on a desire to improve their appearance. According to the American Association of Orthodontists, the number of people seeking orthodontic treatment in the USA and Canada is growing rapidly (43.75% increase in the decade prior to 2013).

As noted, the growth studies comprising the CGCS excluded individuals with congenital deformities or overt health problems. Considering that these historical studies extend back to the 1930s, it is clear that a good percentage of people are included in the CGCS who, by today's standards, would be defined as “needing treatment” but did not receive it, or who would, today, seek treatment to improve esthetic issues more than was previously done. Treatment regimens have clearly changed over time as well with extractions giving way to root canals, crowns, caps, and so on. The goal of the CGCS was to include only untreated individuals but it is expected that a small percentage of individuals may have undergone treatments that are unknown to us. We believe the percentage of treated individuals is sufficiently small as to not affect our results. We do not have a record of general oral health for the participants other than what can be assessed via radiographs. What this all means is that we are using a broad brush to define “normal” in the CGCS. The result of such a broad definition is assembly of a dataset with a magnitude of individual variation more representative of human populations than more selective sampling would produce.

4.2 ∣. Sources of error

As with any study, it is important to acknowledge the sources of error and the possible effects those may have on accuracy and bias within the study. A retrospective study using radiographs taken over the course of 50+ years at different localities by different technicians presents certain challenges. We report those here with a discussion on our efforts to minimize bias while maximizing accuracy.

4.2.1 ∣. Radiographic enlargement

Radiographic protocols varied between the studies examined in the current work and, for some studies, varied during the course of the studies (Table 4). As the primary goal of the CGCS is to combine the data from the studies described into a single study for the creation of predictive models, it is critical that the effects of enlargement related to the specific radiographic protocol for each cephalogram are applied to linear measurements as described in the methods. The Bolton-Brush and Oregon studies both used the Bolton Cephalostat, a technique which created unique enlargement factors for each radiograph. The information for each radiograph was not available for us and an average correction factor of 92% for Bolton-Brush and 92.2% for Oregon is used here. Two studies had invariant protocols with constant enlargement factors, Michigan with 87.1%, and Denver with 96%. The Iowa study used several protocols resulting in different enlargement factors ranging from 87 to 94% depending on visit date. The Fels Longitudinal Study changed protocols more than other studies. Fortunately, detailed notes were available on protocol changes noting the specific day of the change. Table 4 provides details of enlargement factors for all studies. Once measurements are adjusted for radiographic enlargement, the resultant metrics allow for straightforward comparison of differences in age-specific size between the studies as well as ontogenetic trajectories and parameters. It is important to note that once corrected, midline measures taken from these cephalograms should also be equivalent to measures derived from other imaging modalities including medical CT, Cone Beam CT, MRI, photogrammetry (e.g., 3DMD), and others assuming that proper calibration has been maintained.

4.2.2 ∣. Cephalostat usage

When capturing a complex, three-dimensional object, such as a head, onto a two-dimensional surface, a radiographic plate, positioning is critical. It is not surprising that, early in the history of radiographic imagery, a device was developed to position the head of patients relative to the plate such that the head was fully captured, and the midline plane was parallel to the plane of the film. Based on the craniostat, previously developed for radiography of the skull, Broadbent designed and implemented one of the first such cephalostats using ear posts and a nose clamp (Broadbent, 1931). The Broadbent cephalostat was designed to allow imaging in both lateral and posterior–anterior views thus allowing reconstruction of 3D coordinates if desired. The Bolton cephalostat was also used in the Oregon Growth study. Over time, multiple cephalostats were developed and utilized by different studies. All had the same goal of positioning the midline plane of the head parallel to the plane of the film. The Fels Longitudinal Study was unique amongst the studies used here in not adopting a cephalostat for the first 35 years or so. Even with use of a positioner such as a cephalostat, it is possible that significant rotation of the head can occur during imaging. Rotation in any plane is relatively easy to identify by comparison of, for example, the images of the right and left mandibular rami, or teeth or anterior cranial base. Prior to use in the present study, all images were assessed for factors that could negatively impact accuracy. Any images with excessive rotation were excluded from the study. Examples of each of the cephalostats used in the studies are given in Figure 8.

FIGURE 8.

Examples of cephalostats used in the studies comprising the CGCS

4.2.3 ∣. Landmark placement

Accurate placement of landmarks on radiographs is a skill, and we have taken great care to train all assessors. The nature of cephalometric points means that some points, such as those with clear boundaries (e.g., cephalometric point Nasion-the intersection of nasal and frontal sutures) are relatively easy to identify with minimal error. Other points falling along curves, such as the point Gonion, marking the change from the vertical mandibular ramus to the horizontal mandibular corpus, are more prone to error. The eDigit-Calibrator program used in the CGCS allows visualization of landmark placement from multiple assessors relative to each other on a cephalogram. This allows both the trainer and trainee to identify systematic error in landmark placement. If that occurs, additional instruction is provided, and the trainee is retested until performance is maximized. All assessors received training and approval from one investigator (H. S. Oh, Craniofacial Research Instrumentation Lab at the University of the Pacific).

It is clear that images dating back 50 years or more cannot be expected to have the clarity of those taken with modern digital technology. Image quality is another factor that can directly affect landmark placement. Faced with varying image quality, and multiple assessors, we chose to undertake the laborious task of triple determination of all points from all cephalograms. This allowed us to, first, identify potential errors in landmark identification, and second, minimize stochastic error by averaging the three observations.

4.2.4 ∣. Differences in age range

As noted, individual studies had varying protocols regarding how often observations were taken, or changes to density of observation based on age of the participant, or whether attempts to recall participants occurred after the study had ended. For instance, the collection of cephalograms for the Iowa growth study occurred over a 15-year period with a shift from biannual to annual radiographs at 12 years of age. Two attempts at recalling participants were made with some success. Compare this to the Fels Longitudinal Study which collected cephalograms for over 50 years with a schedule of biannual participant visits up to the age of 18 with variable periods after that. Obviously, truncation of craniofacial data at an age prior to attainment of peak growth velocity or the cessation of growth will have an impact on the ability to produce accurate statistical models of growth from a sample despite the value of those data. The CGCS benefits from the inclusion of even those studies with incomplete or sparse data during the adolescent growth spurt because they provide valuable information. At the same time, because of the sizable dataset available in the CGCS, analyses of growth trajectory and ontogenetic trajectories are not penalized due to the omission of data in these studies. The multilevel models that we use for estimating growth trajectories benefit from including all the available data, even individuals with as few as two observations. Truncated data from the Iowa and Michigan studies do not harm estimation, because data from the other studies are available to inform estimations where these studies lack observations.

It is clear that any attempt to create accurate descriptive or predictive models aimed towards further understanding of craniofacial growth with a potential clinical application must draw from a sample that maximally approaches population-level variation (Figure 4a). The studies comprising the CGCS are all drawn from geographically localized populations with some constraint on available variation. Combining these studies into the CGCS provides, for the first time, a dense longitudinal study of human craniofacial growth with population-level variation. The degree of variation represented by the CGCS is necessary to accurately predict growth at the individual level. We have shown here that we are able to estimate multilevel, nonlinear, double logistic growth models successfully for a wide range of craniofacial traits. Growth modeled separately by study generally mirrors the pattern seen for the whole population (Figures 4 and 5), with high overall consistency among derived parameters of aPGV, age at cessation, and peak growth velocity. Thus, prediction for a single individual, given even a limited number of observations, should be possible by placing them into the context of population-level variation and informing predictions using patterns estimated for the combined sample.

5 ∣. LIMITATIONS

Many of the growth models generated for our cranial traits appear as a sigmoid curve characteristic of adolescent growth spurts. These curves begin with a period of low growth transitioning to rapid growth as the spurt begins. A period of maximal growth (peak growth velocity) is followed by deceleration followed by a plateau indicating zero growth. The rapid changes in the beginning and end of the curve are identified by inflection points and can be considered as indicators of the onset and cessation of growth respectively. The double-logistic approach used here is successful in providing biologically meaningful estimates of peak growth, the age at peak growth, and cessation of growth periods in the vast majority of analyses.

In some situations, however, the modeled curve exhibits modest incremental growth without rapid changes in velocity. In other words, there are no discernible inflection points. In these instances, estimation of growth milestones is problematic using a double logistic growth model (e.g., aPGV for Sella-Basion in males and Nasion-ANS in females, age at cessation for Condylion-Pogonion in males). Several identifiable factors may influence this outcome. For instance, female craniofacial growth, more often than that of males, is characterized by a pattern of slow, steady growth. Traits of an overall small dimension, such as Menton-ANS, with a growth of ~10 mm over the entirety of the adolescent spurt, may also be less prone to present a growth curve with identifiable inflections, in part due to greater relative variation introduced by sources of error. As noted, two of our samples have age ranges that are truncated during the adolescent growth spurt. The Michigan growth study is truncated at ~18 years of age and the Iowa growth study truncated at ~15 years. The Iowa study recalled participants approximately 10 years later leaving a large gap between datapoints. In general, the age range of the observations being used to build the models may have an effect. If the age range is constrained to a portion of growth with a relatively constant rate, it will be difficult to estimate parameters. We have chosen an age range we believed would accommodate the beginning and end of the growth spurt. However, in some instances, our youngest age may be after the growth spurt has begun.

5.1 ∣. Future directions

We are fully aware of the irony of describing “population-level” variation when the growth studies included in the CGCS are not comprised of significant racial diversity, but, instead, primarily consist of white individuals based on what was available in the historic growth studies used. The dense longitudinal nature of these studies makes them invaluable for characterizing craniofacial growth, and, while there are studies inclusive of a more diverse population, they are rare, and we were unable to include them in the initial work of the CGCS. It is critical to expand upon the work presented by adding to the diversity so that critical questions of similarity or difference can be answered.

6 ∣. CONCLUSION

As noted, this article developed growth models based on a set of clinically relevant measurements in an effort to evaluate the individual studies making up the CGCS and to evaluate the validity, for future analyses, of combining craniofacial data derived from these geographically disperse studies. Due to the complexities of working with images from historical archives, we have detailed the potential sources of error and the efforts to minimize the effects on statistical analyses.

Results of the derived models of the individual studies show considerable consistency in milestone estimates and overall growth curves. Results of the study-specific analyses demonstrates a lack of bias imposed by any given study for population-level analyses. We believe the analyses presented provide no argument against combining the studies into the CGCS and conducting future analyses on the collective dataset. The CGCS, therefore, provides an unparalleled opportunity to examine craniofacial growth from childhood into adulthood. Future analyses will focus on the development of evidence-based models of craniofacial growth important to basic researchers and clinicians alike.