## INTRODUCTION

There is considerable individual variation in the timing, duration, and intensity of growth that occurs in the craniofacial complex during adolescence. The purpose of this article is to describe the extent of this variation between traits and between individuals within the Fels Longitudinal Study (FLS). While comprising a functionally integrated unit, the craniofacial components each have unique growth characteristics and these may be differentially affected in various growth disorders. Accurate characterization of the parameters governing ontogenetic trajectories such as onset and offset of the pubertal growth spurt, as well as the timing and magnitude of peak growth velocities, in normal individuals will lead to a better understanding of the alterations of growth and resultant phenotypes in a variety of pathological conditions (Sherwood et al., 1997). In a broader context, such characterization allows for examination of heterochrony in comparative morphology (Alberch et al., 1979).

It has long been recognized that there is a large amount of individual variation in the pattern of facial growth and that this heterogeneity must be taken into consideration when determining the optimal timing of orthodontic and prosthodontic treatment (Björk, 1969; Walker, 1972). Relative to an individual's adolescent growth spurt, the optimal timing varies across treatment objectives. For example, attempts to modify mandibular growth may be best suited to the period of highest growth velocity (Casutt et al., 2007); maxillary protraction therapy is more effective at younger ages (Kapust et al., 1998); and dental implants are most effective if placed after growth has ceased (Mishra et al., 2013). Growth and/or rotation of the mandible and maxilla after implant placement can result in poor alveolar crest positions relative to the adjacent teeth. Thus, numerous studies have estimated growth velocity using changes in facial traits (increments) between regularly spaced measurements (Tracy and Savara, 1966; Savara and Tracy, 1967; Baughan et al., 1979; Ekström, 1982; Lewis et al., 1982, 1985; Krieg, 1987; Nanda, 1988; Hunter et al., 2007), and some have additionally estimated the age of onset of the adolescent growth spurt (Björk, 1963; Roche and Lewis, 1974; Bishara et al., 1981; Jamison et al., 1982; Zionic Alexander et al., 2009; Ball et al., 2011).

For the purpose of creating clinically applicable reference values for changes in facial dimensions, increments are required in order to capture the actual between-individual variation in change over a specified period of time. However, if the goal is to estimate the age at which peak growth velocity occurs, then increments are not ideal as they are constrained by the coarseness of the measurement times (typically semiannual or annual). A more appropriate method for this purpose is multilevel modeling (MLM) (Goldstein, 1986; Goldstein, 2011). MLM has been used extensively to model growth of anatomical structures, providing a means to estimate not only population-average growth curves, but also the amount of individual variation in the pattern of growth.

Various mathematical models for craniofacial growth have been applied to data from individuals including the logistic (Roche et al., 1977), double logistic (Roche and Lewis, 1976), Gompertz (Maunz and German, 1996), and polynomial (Buschang et al., 1986). These can be fit to each individual, with the results then summarized over individuals. However, since the advent of MLM, these models can be fit to all individuals simultaneously. MLM was first applied to facial growth by Buschang et al. (1988a, b). Since then, polynomial MLM has been used in numerous studies to model facial growth.

For example, Buschang et al. (1989) used a 5th degree polynomial to model growth in sella-gnathion for six to 15-year-old French-Canadian girls. They presented estimates of the individual-level variation in the polynomial coefficients, as well as percentiles for size and yearly growth increments derived from the model. On the basis of this model, they estimated the average ages of two growth spurts—a childhood spurt at 7.6 years and an adolescent spurt at 12.7 years (where a “spurt” is defined as an age at which velocity peaks). A number of similar analyses, motivated by the desire to understand normal and abnormal growth patterns, in particular during the adolescent years, have been carried out for various other facial measurements, for boys and for girls, and for various age ranges and populations (Buschang et al., 1989, 1990; van der Beek et al., 1991; Henneberke and Prahl-Andersen, 1994; van der Beek et al., 1996; Buschang et al., 1999; Smith and Buschang, 2002; Chvatal et al., 2005; Bills et al., 2008; van Diepenbeek et al., 2009; Arboleda et al., 2011; Wolfe et al., 2011).

These analyses generally report average values of polynomial coefficients, as well as the amount of variation in the coefficients between individuals. However, polynomial coefficients themselves are difficult to interpret. In this article, we use data from the FLS and polynomial MLM to estimate a set of meaningful growth parameters (that are functions of the polynomial coefficients) in a representative set of measures from the maxilla, mandible, and basicranium. The measures examined are mandibular length (Ar-Me), maxillary length (PNS-PtA), cranial base length (Ba-N), gonial angle (Ar-Go-Me), and saddle angle (N-S-Ba) (Fig. 1). We begin by estimating the ages of onset of adolescent growth, peak growth velocity, and cessation of adolescent growth for each measure, both at the group level (average) and at the individual level (variation between individuals). Subsequently, the duration of time between each of these ages, the amount of growth between these ages, and the magnitude peak growth velocity is calculated. Finally, we report the amount of growth relative to adult size, and relative to adult length of the cranial base (Ba-N).