 Research article
 Open Access
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Insight into human pubertal growth by applying the QEPS growth model
BMC Pediatricsvolume 17, Article number: 107 (2017)
Abstract
Background
Computerized mathematical models describing absolute and relative individual growth during puberty in both cm and standard deviation (SD)scores are lacking. The present study aimed to fill this gap, by applying the QEPSmodel that delineates mathematically the specific pubertal functions of the total growth curve.
Methods
Study population used was the individual growth curves of the longitudinally followed cohort GrowUp1974 Gothenburg (n = 2280). The QEPSmodel describes total height as (T)otalfunction: a combination of four shapeinvariant growth functions, modified by timescale and heightscale parameters: a (Q)uadraticfunction for the continuous growth from fetal life to adulthood; a negative (E)xponentialfunction adds the rapid, declining fetal/infancy growth; a (P)ubertalfunction the specific pubertal growth spurt; a (S)topfunction the declining growth until adult height. A constructed variable, MathSelect, was developed for assessing dataquality. CIs and SDscores for growth estimates were calculated for each individual.
QEPSmodel estimates used for pubertal growth; from the Tfunction: onset of puberty as minimal height velocity (AgeT _{ ONSET }); midpuberty as peak height velocity (AgeT _{ PHV }); end of puberty as height velocity decreased to 1 cm/year (AgeT _{ END }); duration of different intervals and gain (AgeT _{ ONSET–END } and Tpubgain); from the Pfunction: onset of puberty, estimated as growth at 1% or 5% (AgeP1 _{ , } AgeP5); midpuberty as 50% (AgeP50) and PHV (AgeP _{ PHV }); end of pubertal growth at 95 or 99% (AgeP95, AgeP99); duration of different intervals and pubertal gain (Ppubgain; P _{ max }); from the QESfunction: gain (QESpubgain) _{ . }
Results
Application of these mathematical estimates for onset, middle and end of puberty of Pfunction, QESfunction, and Tfunction during puberty showed: the later the onset of puberty, the greater the adult height; pubertal gain due to the Pfunction growth was independent of age at onset of puberty; boys had higher total gain during puberty due to Pfunction growth than to QESfunction growth; for girls it was reversed.
Conclusions
QEPS is the first growth model to provide individualized estimates of both the specific pubertal growth function and the total growth during puberty, with accompanying SDscores and Cis for each individual. These QEPSderived estimates enable more indepth analysis of different aspects of pubertal growth than previously possible.
Background
Pubertal growth is unique to humans [1]. For the individual, puberty constitutes a dramatic change in both the magnitude and tempo of growth. In a healthy population, there is wide variation in when children enter puberty, both within and between genders [2, 3]. Thus, accurately describing this period of growth is challenging due to the complexity of the changes that occur and the differences observed between individuals. At present, methods for modelling pubertal growth are limited, and no existing growth references allow appropriate adjustments for the onset of puberty. Furthermore, variations between individuals add to the challenges of modeling growth, particularly when they are considered to be related to maturation (biological age) rather than to chronological age. The pattern of pubertal growth has also changed over time, and varies between different populations [4, 5]. The large variations in both the timing of puberty and amount of growth which are apparent among individuals and between populations highlight the need for individualized equations and estimates describing pubertal growth.
The years preceding puberty are characterized by a period of slowly declining height velocity [3, 6]. The onset of pubertal growth can be identified based on the smallest height velocity that precedes what has been referred to as the takeoff, onset, nadir or insertion point [2, 3, 7]. In previous studies it has also been described as the point at the beginning of the pubertal growth spurt where height increased by 0.3 standard deviation (SD) scores, or as the point 2 years before peak height velocity (PHV) [6, 8, 9]. Thereafter, height velocity rapidly increases, and the SD of both observed height and height velocity for any population increases due to the broad variation in the timing of puberty [3]. PHV – the midpoint in puberty where growth is most pronounced – has often been used as the only estimate of pubertal growth in previous research. The easiest, and probably most unreliable, way of defining age at PHV is by estimating the age at which height increases most from the growth curve, either by visual inspection or using a specific puberty ruler [10]. Age at PHV in contemporary research may also be defined by visual inspection of the change in growth velocity on a computergenerated height velocity chart [2, 11]. Another way of defining age at PHV is to take the age at the midpoint in the interval between the two height measurements with the greatest calculated yearly height increment [6, 12]. The latter is reliable when height measurements are available every 3 months, but less precise when measurements are taken at longer intervals; there is a risk of underestimating the age at PHV when measurements are at 6 or 12month intervals [9, 13]. The end of pubertal growth has typically not been specifically identified, and measurements have instead been based on when adult height was attained. Therefore, the total pubertal height gain has been defined as the amount of growth observed from the onset of pubertal growth until adult height, with the duration of pubertal growth defined as the time period from the onset of puberty to the attainment of adult height [2, 12, 14].
Few studies have attempted to describe the whole pattern of pubertal growth in a detailed manner, including separate estimations of growth for the onset, middle and end of the pubertal period. In 1980, Taranger & Hägg described a way to estimate the duration and gain of pubertal growth based on visual inspection of individual growth charts [15]. Mathematical models have also been used to describe growth from birth to adult height [16–19]. The ICPmodel (InfancyChildhoodPuberty), developed in Gothenburg by Karlberg et al. was the first model to use three different mathematical functions related to the periods of biological growth [12, 20, 21]. Thus, during the pubertal years, total growth can be separated into the childhood component and the pubertal component. However, the pubertal component of the ICPmodel has a fixed form, such that only the timing of pubertal growth, not the magnitude of the specific pubertal growth function, can be individualized. This means that the model assumes that all variations in pubertal growth in individuals of the same gender are related to differences in the childhood growth component that is still ongoing during the pubertal period [21]. The first published growth model that allowed for individualization of the pubertal growth was the SITARmodel by Cole et al. [22]. The model generates a growth curve and three subjectspecific parameters (size, tempo and velocity) that can be adjusted to describe individual growth patterns. However, this model cannot separate growth during puberty into different components, instead providing only one mean shapeinvariant growth function.
The first model to describe individual longitudinal human growth and its different phases from fetal life until adult height is the QEPSmodel by Nierop et al. [23]. The model was constructed with a combination of four distinct shapeinvariant growth functions: Quadratic (Q), Exponential (E), Pubertal (P) and Stop (S), (Fig. 1). All four functions have an individual heightscale parameter, and the E and Pfunctions also have individual timescale parameters; giving six modifying parameters in total to describe individual growth. Basic features of the Q and E functions have been used in previous prediction models [24–26].
In the present study we implement novel estimates of pubertal growth from the QEPS mathematical growth model in cm and SDscores both at the individual and the group level. The model calculates both the specific pubertal Pfunction during puberty and the from prepubertal period ongoing QESfunctions, as well as the combined total growth. Moreover, the model provides confidence intervals (CIs) of the different growth estimates that can be used to assess the quality of growth data at the individual level. Basic features of the QEPSmodel have been presented at meetings [27–29].
Methods
Ethical approval
Ethical approval was obtained from the ethics committee of the University of Gothenburg (91–92/131–93), and individual approval was given by the participants of the 1974 cohort study if they were 18 years or older, or by their legal guardian if they were not old enough to give consent (16 to 18 years of age).
Subjects – A healthy cohort born in 1974
The data used for analysis was from a communitybased, observational growth study the GrowUp1974 Gothenburg study that was conducted in all high schools in Gothenburg, Sweden in 1992 [3]. Longitudinal growth data from healthy individuals born at term (gestational age 37–42 weeks) within this study, together with data from the Swedish Medical Birth Registry, were used to create the Swedish national Growth References used from 2000 [3, 30]. A study group of individuals with longitudinal growth data was selected from the GrowUp1974 population for the present study using the following steps.

1.
Computerized selection of individuals with height measurements registered for each of the following ages were selected; at birth; as an infant 0 to 9 months (two or more measurements); as a toddler between 9 months and 3.5 years; as a child 3.5 to <6.0 years; as a schoolchild; 6.0 to <9.0 years; as a juvenile 9.0 to <12.0 years; in adolescence 12.0 to <16.0 years; and in adulthood > = 16 years.

2.
Visual growth curve analysis for confirmation of the growth characteristics of the individuals in the selected study group; see Growth curve analysis section below. This selection reduced the study group with 696 individuals from 2976 to 2280 individuals. The main characteristics of the study group are shown in Table 1.
Mathematical selection criterion (MathSelect)
To assess the quality of the fitted individual total height function, T(age), a mathematical selection criterion, MathSelect, was used that we developed for the QEPSmodel. The MathSelect criterion combines information from nine individual variables. Details on how MathSelect was constructed can be found in the Additional file 1: Section A2. Two different MathSelect values, 0.975 and 0.68 were used for computerized data quality check of the study group. For all figures MathSelect 0.975 was used.
Processing of the data
To construct a longitudinal growth curve for each individual in the present study group, data files were analysed with Matlab software (version 7.13.0 R2012b, The Mathworks). The Matlab Curve Fitting Toolbox was used for regular curve fitting and was customized to perform penalized curve fitting. Individual curves were estimated with 95% CIs for the fitted parameters.
Growth curve analysis
The quality of the height data were evaluated by visual inspection using QEPSmodelfitted growth charts. The quality of data, and the presence of potential errors that needed further assessment, were evaluated by stepwise observations:

1.
Assessment of outliers; assessment of individual height data that deviated from the individual growth curve, giving rise to suspicion of input or measurement errors.

2.
Assessment of the adult height; visual analysis of whether adult height was reached at the last measurement or not.

3.
Comparison between the new midpuberty parameter AgeP50 and visually evaluated age at PHV (AgePHV).
If there was a difference of more than 0.66 years between AgeP50 and AgePHV, or if observations 1 and 2 above gave rise to uncertainty regarding any data points, the original growth data were reevaluated; if uncertainty remained, the individuals were excluded from the study.
QEPS variables describing pubertal growth
Estimates from the QEPS total curve have the prefix T (Total), using the basic additive QEPS model in which T(age) = Q(age) + E(age) + P(age) − S(age) [23]. From the Total growth curve, onset of pubertal growth, AgeT _{ ONSET }, was calculated as the age at minimum height velocity (HV) of the total height function. Midpuberty was calculated as the age at PHV from the total growth curve (AgeT _{ PHV }) and end of puberty as the age at which HV had decreased to 1 cm/year (AgeT _{ END }). The total gain in height (cm) during puberty (TpubTgain) was based on growth during the time period AgeT _{ ONSET–END } . The total gain in adult height due to the specific Pfunction is estimated by the QEPSmodel as the maximum height of the Pfunction, P _{ max } in cm (P _{ max } = T _{ max }–QES _{ max }). Due to the specific form of the Pfunction, which is a quadratic, logistic function, P _{ max } can be calculated without defining a specific duration of puberty. The Pfunction starts before AgeT _{ ONSET } since the velocity of T will not increase until the velocity of the Pfunction increases more than the decreasing velocity of the QESfunction. If the relative influence of P and QESfunctions during the pubertal time period needs to be calculated, an age point is needed for both QES and Pfunctions. Thus, onset of pubertal growth can be estimated from the Pfunction as the age when 1% (AgeP1) or 5% (AgeP5) of the total Pfunctionestimated gain was reached. For midpuberty, we calculated the age when 50% of the Pfunction gain was achieved, AgeP50, and the age at PHV from the Pfunction growth curve, AgeP _{ PHV. } In order to identify the end of pubertal growth, we calculated the age when 95% (AgeP95) or 99% (AgeP99) of the specific pubertal gain was achieved, Fig. 2. The time from AgeP5 to AgeP95, as well as AgeP1 to AgeP99 and AgeT _{ ONSET } to AgeT _{ END } gives estimates for the duration of pubertal growth.
In general, pubertal gain can be described as the increase in height of the T, P, and QESfunction from AgeT _{ ONSET } until AgeT _{ END }, and corresponding decomposition of the total puberty gain in a P and a QES part: Tgain = Pgain + QESgain. The value of onset and end of pubertal growth can be any selected combination of onset of puberty age, expressed as AgeT _{ ONSET }, AgeP1 or AgeP5, with any selected end of puberty age, expressed as AgeT _{ END }, AgeP95 or AgeP99. During the selected pubertal period the model can separate the influence of the specific Pfunction from the ongoing Qfunction, and also their relationship as a ratio with the locations of AgeT _{ ONSET } and AgeT _{ PHV }; see Additional file 1: Section A1:3 for more information. Examples of pubertal growth for four individuals are shown adjusted for age at onset of puberty in Fig. 3, and their entire growth according to chronological age in Additional file 1: Figure S1, where the total pubertal growth is expressed in both cm and SDscores, and is also divided into the P and QESfunctions. The details of the equations describing pubertal growth can be found in Nierop et al. [23], with complementary information explaining the pubertal period in more detail in the Additional files 1: Section A1.
Statistical considerations
The measured and calculated variables in the tables are presented as mean, median, standard deviation, maximum and minimum. Lower and upper 95% CIs, skewness and kurtosis computations conducted in order to estimate any departure from the normal distribution are given in the Additional file 1: Tables. These computations were performed using SAS Software 9.3 (SAS Institute Inc., Cary, NC, USA).
Results
Pubertal growth estimates
The different pubertal growth estimates are shown in Tables 2 and 3, and in more detail in the Additional file 1: Tables S1 and S2. Differences in the mean between various pubertal estimates are shown below; grouped by the type of measurement; for pubertal duration and pubertal height results are given with the ±1 SD interval of the population in brackets.
Onset of pubertal growth
Estimates of timing for onset of pubertal growth vary depending on the variable used. For girls, the mean age at onset of puberty as AgeT _{ ONSET } from the total growth curve was 9.24 years, 0.53 years after AgeP1 and 0.62 years before AgeP5 from the Pfunction. For boys, there was no difference between AgeT _{ ONSET } (10.74) and AgeP1 (10.73), whereas AgeP5 occurred 1.0 years later, Table 2.
The median percentage of the Pfunction reached at the AgeT _{ ONSET } was 2.4% for girls and 1% for boys, Additional file 1: Figure S2.
Midpubertal growth estimates
The visually estimated age at PHV (AgePHV) was compared with the QEPScalculated AgeT _{ PHV } from the Tfunction and with AgeP _{ PHV } /AgeP50 from the Pfunction; the mean values of these four estimates of midpubertal growth showed minor differences from each other (maximal 3 months), Table 2. The difference in years between AgePHV and AgeP50 was −0.171 (±0.46 SD) for girls and 0.037 (±0.36 SD) for boys, Table 2. The median percentage of the Pfunction reached at mid puberty as AgeT _{ PHV } was 43% for girls and 45% for boys, Additional file 1: Figure S2, middle panel.
End of pubertal growth
For girls, the mean difference in years between AgeT _{ END } from the total curve and AgeP95 from the Pfunction was 0.35, and the difference between AgeT _{ END } and AgeP99 was −1.32. For boys, the corresponding values were 0.58 and −0.88 years, respectively, Table 2. For both genders, taller adult heights were found in individuals with later pubertal growth (later AgeP50); however, there was broad individual variation and apparent differences in the distribution of pubertal timing between genders, as seen in Fig. 4. When relating adult height to age at onset of puberty, the pattern was similar for both genders, see Additional file 1: Figure S4; a 1year delay in the onset of puberty, expressed as AgeT _{ ONSET, } will give an adult height that is taller by 1.2 cm in girls and 0.8 cm in boys. The percentage of Pfunction growth reached at S _{ 0 } was for girls 74% and for boys 89%, whereas at AgeT _{ END } it was 97% for both genders, Additional file 1: Figure S2 right panel.
Duration of pubertal growth
For girls, the mean duration in years for pubertal growth from AgeP5 to AgeP95 was 4.80 (4.59–5.01), the duration from AgeP1 to Age99 was 7.61 (7.28–7.94) and the duration from the total growth curves defined as AgeT _{ ONSET–END } was 5.77 (5.27–6.27).
The corresponding durations of pubertal growth in years for boys were 4.32 (4.10–4.54), 6.83 (6.49–7.17), and 5.94 (5.56–6.32), respectively, Table 2. A clear gender difference was seen in both timing and duration of pubertal growth when estimates were based on the Pfunction, Fig. 5, left panel, with not only a later, but also a shorter pubertal growth spurt seen in boys. This is in contrast to the less pronounced gender difference observed when the duration of pubertal growth was based on the total growth curve for the same agepoints, Fig. 5, right panel.
Gain of pubertal growth
From the total growth curve, the mean pubertal gain for girls from AgeP5 to Age95 was 26.34 cm (18.74–33.94), and from AgeP1 to Age99 it was 33.64 cm (24.52–42.76). For boys, the corresponding pubertal gains were 29.00 (21.72–36.28) and 35.62 cm (27.10–44.14), respectively, Table 3.
The pubertal gain can also be described as what the specific Pfunction adds to the ongoing QESfunction. The mean pubertal gain, from the Pfunction, P _{ max }, was 12.73 cm for girls and 17.34 cm for boys, and was not influenced by the timing of puberty, as seen in Fig. 6, upper left panel (with Ppubgain, 95% of P _{ max }). However, for both genders, the increase in total height during the pubertal years, Tpubgain, appeared to be higher for individuals with earlier puberty compared with those with later puberty, due to differences in the growth from the QESfunction during these years, Fig. 6, upper middleright panels. The Ppubgain was clearly negatively related to Q _{ max }, the higher the Q _{ max }, the lesser the Ppubgain, also the Tpubgain was negatively correlated to Q _{ max }, but to a lesser extent as seen in Fig. 6, lower panels.
Using the QEPSmodel, pubertal gain can also be shown for each individual both as total gain and divided into the individual components of the Pfunction and the ongoing QESfunction, Fig. 2, left. For the whole study population during the pubertal years, defined as the time period AgeP5–100 in Fig. 7, growth from the QESfunction dominated in girls, whereas growth from the Pfunction dominated in boys, but with large interindividual variations for both genders.
Tempoadjusted SDscores for pubertal age and height
The QEPSmodel calculates the age of the individual for all pubertal estimates, which enables these estimates to be compared with the mean of the background population as relative age in deviations from the mean (i.e. standardized age in SDscores). Thus, instead of showing the age of a child in chronological age, the age at onset of puberty can be visualized according to the mean age (zero) of the internal reference for onset of puberty, i.e. adjusted to pubertal age [23]. With this tempocorrection for the onset of puberty, the QEPSmodel enables an individualized reference of pubertal growth in which height_{SDS} can be expressed according to a pubertal tempoadjusted reference curve as shown in Fig. 3.
Moreover, in the examples of individuals presented in Fig. 3 and in the Additional file 1: Figure S1, (also presenting entire growth vs chronological age), the individual estimates of the different growth functions are presented not only in cm but also in individualized SDscores.
In Fig. 8, we show the relationship between the midpuberty variable, AgeP50, and its corresponding CI for standardized midpuberty (AgeP50 _{ SDS }). The scatterplot illustrates that the CI increases in those with later pubertal growth, especially in girls. Thus, there is a greater uncertainty in the estimate of midpuberty for individuals with later pubertal growth.
Individual CIs for precision and MathSelect for quality assurance
From the whole study group, only 49 individuals were removed from the study population/analysis when using MathSelect < 0.975, and the absolute differences in pubertal population estimates were small. Kurtosis and skewness decreased only slightly in the MathSelect < 0.975 group (excluding CI estimates). In contrast, using MathSelect < 0.68, the study group was reduced by 731 individuals, and mostly by affecting skewness estimates, Additional file 1: Tables S3A–C.
There was a clear gender difference with higher CIs for girls for all QEPS variables for the onset, middle and end of pubertal growth. Moreover, for both genders, the CIs for variables of onset and end of pubertal growth were broader, than for the estimates of midpubertal growth, Additional file 1: Tables S3A–C. These tables also show the resulting lower CIs when reducing the group by using the MathSelect function. The relationship between the CIs for AgeP50 and the MathSelect values is shown in Fig. 9. As expected from the modelling procedure, a lower MathSelect value corresponds to a lower maximum CI, whereas a higher maximum CI corresponds to a higher MathSelect value.
The relationship between the CI for AgeP50 and the Pfunction height gain (P _{ max }) showed a nonlinear correlation; higher CIs were associated with lower P _{ max }, Fig. 10. Independent of gender, a pubertal gain below 8 cm, gave a CI of more than 9 months, whereas a gain of at least 14 cm, gave a CI of less than 6 months as shown in Fig. 10. The apparent gender difference was related to the fact that a low P _{ max } was more common in girls.
Additional file 1: Figure S4 shows a QEPScalculated height velocity graph of an individual with low pubertal height gain, which further illustrates the problems in defining AgeT _{ PHV } and
AgeT _{ ONSET } when the Pfunction is low. To be distinguishable, P _{ max } must be greater than 50% of the CI, which for boys corresponds to a P _{ max } of 2.74 cm and for girls to a P _{ max } of 3.14 cm as seen in Additional file 1: Figure S5.
Discussion
Principal findings: QEPS variables for pubertal growth enable new information
The present study, as the first implementation of the QEPSmodel to describe pubertal growth, describes the pubertal growth variables generated by the model and their accompanying SDscores for the population and the individual. Furthermore, the study demonstrates the potential to use these variables to explore human pubertal growth in greater detail than has previously been possible. The variables were calculated for the total growth curve during the pubertal years, and were also separated to provide information on growth specific to puberty, the Pfunction (Ppubgain), and growth related to the still ongoing QESfunction (QESpubgain). The Ppubgain was found to be independent of age at onset of puberty, whereas the total height gain during puberty, also depending on the QESfunction, was greater in those with earlier puberty. Moreover, a gender difference was identified, with more QESfunction growth in girls and more Pfunction growth in boys.
As well as providing robust variables, the QEPSmodel is the first growth model to provide individual CIs. Moreover, it allows height SDscore estimations during puberty to be expressed in relation to an individualized tempoadjusted reference. This is a major achievement as it allows relative growth during the pubertal years to be expressed at any timepoint; previous models have only been able to present total pubertal gain from prepuberty to adult height which has limited in depth analysis regarding pubertal growth [8]. By applying the QEPSmodel to longitudinal growth data, we have identified new mathematical variables that are linked to specific timepoints and which can be used to describe pubertal growth in detail, thus enabling comparison of growth patterns between individuals and populations. A practical advantage of using the QEPSmodel compared to other growth models is that it automatically describes a wide variety of growthrelated variables without relying on visual inspection of growth data; thus, the model is not subject to the estimation errors that can occur when relying on visual assessments.
Onset of puberty
The QEPSmodel gives different timepoints that differ from each other for onset of puberty; from the total growth curve as well as from the specific pubertal growth curve. Based on the specific Pfunction, (AgeP1), the onset of puberty was estimated to be 1.4 years earlier than in previous studies of pubertal growth in Scandinavian populations. Similarly, the onset of puberty was 0.9 years earlier when estimated based on the total growth curve, AgeT _{ ONSET } [2, 31]. Our findings are consistent with other studies using mathematical models [17, 19, 32], which typically result in earlier estimates of pubertal onset compared with studies using visual estimates of the onset of puberty [13]. Future studies may show how the AgeP1 and AgeT _{ ONSET } estimates correlate in the individual with the time when gonadal steroids start to increase during the nighttime [33, 34] which is another way of identifying onset of puberty. In fact, AgeP5 (9.9 years) at onset of puberty in girls is approximately equal to the onset of puberty in other Scandinavian studies; our results were only 0.24 years earlier than in the Finnish study [31] and 0.34 years earlier than in the Danish study [2], both of which used a visually defined onset of puberty. For boys, a consistency between AgeP5 (11.8) and onset of puberty in other Scandinavian studies was even greater than for girls, with differences varying from −0.18 to +0.24 years [2, 31, 35].
Midpubertal growth estimates
Midpuberty, expressed as visual PHV, has so far been the main estimate of pubertal timing used in the literature [2, 6, 7, 17–19, 21, 31]. Here we compared three new estimates of midpubertal growth generated by the QEPSmodel; from the total growth curve AgeT _{ PHV } , and from the Pfunction growth curve AgeP _{ PHV } and AgeP50, and found their mean values to be close to each other; both of the QEPS mathematically calculated variables of age at PHV were similar to visual age at PHV. For both boys and girls, we found the strongest correlation to be between AgeP50 and visual age at PHV [3]; at a population level, the mean difference between AgeP50 and visual PHV for boys was only 13 days and for girls 62 days. This suggests AgeP50 to be a variable that could be considered for use to identify age at midpuberty in future studies of pubertal growth.
End of puberty and duration of pubertal growth
The QEPSmodel enables us to estimate the end of pubertal growth. In fact, there is no other growth model today that can precisely estimate the end of growth [12, 17–19, 21, 22]; therefore, little attention has been paid to the end of pubertal growth. In this work, we introduced 95 and 99% of the Pfunction curve (AgeP95 and AgeP99, respectively), as well as the end of the total curve, AgeT _{ END,} as possible variables for defining the end of pubertal growth. Due to the lack of variables with which to estimate the end of pubertal growth, the duration of puberty has rarely been included in studies of pubertal growth; the study by Taranger & Hägg [15] is one of few exceptions; however, they employed only visual inspection to identify an point corresponding to AgeT _{ END }. Using the new variables presented here, the duration of pubertal growth can be expressed for individuals and study populations in future research.
Total pubertal gain
The shapeinvariant QEPSmodel is the first growth model that can calculate and describe the specific pubertal height gain together with the total height gain during puberty at an individual level. The specific pubertal height gain was found to be independent of the age at onset of puberty. This was in contrast to the total height gain during the pubertal years which was greatest in those with an early onset of puberty, as reflected in the model by more growth associated with the QESfunction than the Pfunction. It has been debated whether or not adult height is dependent on the timing of puberty [13, 36, 37]. The results of the present study confirm that there is an impact of a delay in onset of puberty, with a taller adult height in both boys and girls who experienced a later onset of puberty and a later AgeP50; in fact a 1year delay gave approximately a 1 cm greater adult height. For some individuals, mainly girls, the estimated pubertal gain was so low that it was not possible to calculate either AgeT _{ PHV } or AgeT _{ ONSET } from the total growth curve. We can now also define the specific component of the pubertal growth spurt, and using CIs we are also able to assess the accuracy of the estimated measurements. This represents an advance on what was possible using the previous ICP and SITARmodels [12, 21, 22]. The relation between Ppubgain and QESpubgain varies between genders, but also between individuals, with more QESpubgain in those with earlier puberty.
Tempoadjusted individualized reference gives SDscores for pubertal growth
The relative age at onset of puberty is of major interest to both researchers and clinicians because of the great variation between individuals in biological maturity during the pubertal years [6]. So far, only changes in total pubertal height gain have been described with SDscores. For the analysis of individual growth patterns during puberty, Tanner et al. constructed “tempoconditional” heightvelocity curves [6], for individuals with early, average or late puberty, which were applied and further modified in recently updated growth charts for the UK [38], whereas Karlberg superimposed pubertal growth curves adjusted for the timing of the mean age of PHV [12]. In the present study, we describe the individual pubertal growth curve in relation to a pubertal reference adjusted to both time and to age. We calculate and present numerically the relative pubertal age for each individual in comparison to the mean for the population /reference curve, presented as SDscores. Up to now, only the shapeinvariant SITARmodel can adjust for individual tempo, amplitude, and size of total pubertal growth [22]. It is important to note that in contrast to the SITARmodel which only describes growth during the pubertal years, the QEPSmodel can describe growth from birth until adult height, where growth during the pubertal years is based on two different additive functions that separate growth from the continuously ongoing QESfunction from growth by the specific Pfunction. These different growthfunctions are probably regulated by different factors/hormones, and will therefore be of considerable use when searching for/identifying regulatory factors for growth. Thus, these QEPSvariables will enable us to make a more precise description of individual growth during puberty, related to the individual timing of puberty, as well as to the balance between the different growth functions of the model. However, not only pubertal, but also good prepubertal data is required for calculating the QESfunction as well as the Pfunction with good accuracy. Height expressed in SDscores versus a tempoadjusted height reference will serve as an individualized reference that is unique for this model.
Quality markers of individual and population growth data
Using CIs as a quality marker for growth in an individual has to our knowledge not been done before, despite the almost universal use of CIs to show the quality of data. Information on CIs makes it possible to visualize the quality of data for each individual; thereby providing information on the number of measurements that are required during the different periods of growth for the construction of a reliable growth curve at the individual level.
On the population level, data quality estimation by MathSelect enables the quality of growth data to be graded, and selection with the MathSelect function is easy and reproducible. We found it to be a useful instrument for identifying individuals with missing or unreliable height values; findings that were confirmed by visual inspection of the computerized growth charts of these individuals. Thus, using the MathSelect function can be a method for checking the quality of pubertal growth data in future studies, especially when it comes to the assessment of outliers and individuals for whom there may be measurement/input errors.
Limitations of the study
The current study presents results on pubertal growth that are specific for the population studied. Thus, the exact numerical values of the different pubertal variables cannot be generalized to pubertal growth in children born in other countries or during other times, with different tempo of secular changes. Instead, it can be used as a baseline for comparisons with studies in the future using either old or new data.
The implementation of the QEPSmodel in this study was based on the same study group as the development of the model [3, 30], which may also be regarded as a limitation. However, the model was developed based on mean values, whereas in this present study, the implementation and analyses were done at an individual level, for the 2280 individuals included.
As a model for puberty, it is also important to note that the QEPSmodel relies only on information about growth, without any information on the hormonal changes and/or other manifestations that characterise this period of development. Future studies in individuals should be undertaken in order to correlate the pubertal growth variables from the QEPSmodel with both hormonal changes [33, 34, 39, 40] and secondary sexual characteristics [41–43] in order to link the four growth functions with underlying biological processes.
Conclusion
During puberty, the QEPSmodel can mathematically delineate the total growth curve as well as identify growth resulting from both the specific pubertal growth Pfunction and the continuation of the prepubertal growth QESfunction, using four shapeinvariant growth functions, with four heightscale and two timescale parameters. Different variables estimating the onset, middle and end of pubertal growth will enable us to collect measures of both the duration of, and height gain associated with, the Pfunction in relation to total growth during puberty. The QEPSmodel is the first growthmodel that expresses the timing and amount of pubertal growth in individual SDscores, thereby indicating both the tempo and the amount of growth at any timepoint for the individual in relation to a reference population. Moreover, all pubertal variables are described with individual CIs for the first time, allowing both the population and individual measurements to be more precisely evaluated.
New insights have been achieved for genderspecific pubertal growth; the specific pubertal height gain was found to be independent of age at onset of puberty, whereas the total height gain during puberty, also depending on the QESfunction, was greater in those with earlier puberty. Moreover, a gender difference was identified, with more QESfunction growth in girls than boys and more Pfunction growth in boys than girls. The pubertal growth variables from the QEPSmodel implemented in this study, will enable us to standardize methods to assess, describe and compare pubertal growth in different populations and patient subgroups, and will also serve as a tool for gaining new insights into pubertal growth.
Abbreviations
 AgeP1 :

age at which 1% of the Pfunction growth is reached
 AgeP5 :

age at which 5% of the Pfunction growth is reached
 AgeP50 :

age at which 50% of the Pfunction growth is reached
 AgeP95 :

age at which 95% of the Pfunction growth is reached
 AgeP99 :

age at which 99% of the Pfunction growth is reached
 AgePHV:

visually estimated age at peak height velocity
 AgeT _{ END } :

age at the end of puberty where the HV has decreased to 1 cm/y for function T’(age)
 AgeT _{ ONSET } :

age at minimum height velocity of the Tfunction at start of the pubertal growth
 AgeT _{ PHV } :

age at Peak Height Velocity of the Tfunction
 CDF :

cumulative distribution function
 CI:

confidence interval
 E :

negative exponential growth function of age E(age) in cm
 E _{ heightscale } :

individual height scale ratio, modifying the height scale of the Efunction growth, with E _{ heightscale } = E _{ max } / mE _{ max }
 E _{ max } :

gain in adult height in cm due to Efunction growth
 E _{ timescale } :

individual time scale ratio; modifying the time scale of the Efunction growth, and therefore inversely related to the tempo of E. The origin is at t _{0}, the age when length is theoretically zero, E(t _{0} ) = 0, Q(t _{0} ) = 0
 HA:

height acceleration
 Height_{SDS} :

Height position related to the reference standard deviation score
 HV:

height velocity
 MaxCDF :

individual maximum cumulative probability out of nine MathSelect step one cumulative distribution functions: max(F _{ TSDerror }(T _{ SDerror }), F _{ AgeP50CI }(AgeP50 _{ CI }), F _{ PheightscaleCI }(P _{ heightscalePCI }), F _{ PtimescaleCI }(P _{ timescaleCI }), F _{ SPheightinterceptCI }(SP _{ heightinterceptCI }), F _{ SPheightscaleCI }(SP _{ heightscaleCI }), F _{ EtimescaleCI }(E _{ timescaleCI }), 2*abs(F _{ ΔTmaxAH }(ΔTmaxAH) − 0.5), F _{ Penalty }(Penalty))
 MathSelect :

criterion for assessing the quality of the fitted total individual height function T(age) by combining nine parameters: T _{ SDerror }, AgeP50 _{ CI } , P _{ heightscalePCI } , P _{ timescaleCI } , SP _{ heightinterceptCI } , SP _{ heightscaleCI } , E _{ timescaleCI } , ΔTmaxAH and Penalty.
 P :

quadratic logistic function describing the pubertal growth spurt P(age) in cm
 P _{ AUC } :

pubertal area under the curve of pubertal height velocity P′(age), equals maximum P _{ max } of the pubertal height function P(age)
 Penalty :

penalty ratio bias / (error + bias) from fitting T(age)
 Pgain _{ Px%y% } :

gain in total height in cm due to the pubertal growth of the Pfunction from x% till y% of the Pfunction, so Pgain _{ P5–95 } is the Pgain from AgeP5 to AgeP95.
 P _{ heightscale } :

individual height scale ratio, modifying the height scale of the Pfunction, with P _{ heightscale } = P _{ max } / mP _{ max }
 PHV:

Peak height velocity
 P _{ max } :

pubertal gain in adult height in cm due to the Pfunction growth, equal to P _{ AUC }
 Ppubgain :

Pgain _{ P5–100 } = P _{ max } – P(AgeP5) = 0.95*P _{ max }
 P _{ timescale } :

individual time scale ratio, modifying the time scale of the Pfunction and is therefore inversely related to the tempo of P. The origin is at AgeP50, the age at which 50% of the individual Pfunction is reached
 Q :

quadratic growth function of age Q(age) in cm
 QES(AgeP5) T(AgeP5) :

0.05 * P _{ max } .
 QESgain _{ Px%y% } :

gain in total height in cm due to the pubertal growth of the QESfunction from x% till y% of the Pfunction, so QESgain _{ P5–95 } is the QESgain from AgeP5 to AgeP95
 QES _{ max } :

T _{ max } – P _{ max }
 QESpubgain :

QESgain _{ P5–100 } = QES _{ max } – QES(AgeP5)
 Q _{ max } :

gain in adult height in cm due to Qfunction growth
 S :

stop function S(age) in cm, stopping the Qfunction growth at the end of growth
 SD:

standard deviation
 T :

total height function in cm; T(age) = Q(age) + E(age) + P(age) – S(age)
 Tgain _{ Px%y% } :

gain in total height in cm due to the pubertal growth of the Tfunction from x% till y% of the Pfunction, so Tgain _{ P5–95 } is the Tgain from AgeP5 to AgeP95.
 T _{ max } :

modelled total adult height in cm, T _{ max } = E _{ max } + Q _{ max } + P _{ max } − S _{ max }
 Tpubgain :

Tgain _{ P5–100 } = T _{ max } – T(AgeP5)
 TpubTgain :

T(AgeT _{ END } ) – T(AgeT _{ ONSET } )
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Acknowledgements
The authors are grateful for the contributions of the school nurses in the participating schools and the study team of I Larsson, A Olsson, B Samuelsson, and L Wirén for the collection of the original data. We would also like to thank the staff and all the students of the 11th grade 1992 of the Gothenburg schools. Deep thanks to A Ericson and B Svensson who computerized all the original data and made the original visual inspection for PHV and adult height. Thanks for knowledgeable editing and language revision by Harriet Crofts.
Funding
The authors acknowledge financial support from the Swedish Research Council (VR no 7509), EpiLifeTEENS research program (FORTE), Pfizer AB, the Governmental Grants for University Hospital Research (ALF), from RegionVästra Götaland, PhDgrants from the Southern Swedish healthcare region, the R&D department, County of Halland, and the Foundation Växthuset for children. The funding bodies were not involved in the design of the study, data collection, analysis or interpretation of data or in the writing of the manuscript.
Availability of data and materials
The present dataset for the analyses was made as described in the method section. This data used for the present dataset was after administrative permissions obtained from the ‘GrowUp Gothenburg study Database’, and is now a part of this database, which is stored in a server at the Gothenburg University. Swedish Data protection Act (1998:204) does not permit sensitive data on humans (like the GrowUp Questionnaires) to be openly shared. However, the authors are positive to collaborate with researchers worldwide. The data are available upon request from the principle Investigator Kerstin AlbertssonWikland (Kerstin.albertsson.wikland@gu.se); depending on the research question, ethical approval might be required.
Authors’ contributions
KAW is the principal investigator of the study population used. AFMN performed the modeling work for the QEPSmodel described here with contributions from AH, AN, LG, SA, and KAW on the specific pubertal growth estimates. AN performed the statistical analysis in SAS. AH performed the visual growth curve analysis of all individual growth charts, with second opinions from AN, SA, and KAW in unclear cases. AH, AN, LG, SA, AFMN, and KAW have all given substantial contribution to the conception, design, analysis and interpretation of these data, where all involved in writing the manuscript and also revised it critically for intellectual content, as well as giving approval for the final version of the manuscript to be submitted for publication.
Competing interests
AH has received an independent research grant from Pfizer AB. AFMN works for Muvara, Multivariate Analysis of Research Data, Statistical Consultation, The Netherlands. AN, LG, SA, and KAW declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Ethical approval was obtained from the ethics committee of the University of Gothenburg (91–92/131–93), and individual written consent was given by the participants of the 1974 cohort study if they were 18 years or older, or by their legal guardian if they were not old enough to give consent (16 to 18 years of age).
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Additional file
Additional file 1:
The first two sections explain pubertal variables of the QEPSmodel in more detail in texts, figures and tables for the general pubertal growth, section A.1:1 and the individual variation in pubertal growth, section A.1:2, and the PQ _{ ratio } in A1:3. The construction of the mathematical selection criterion, MathSelect, is described in section A.2:1, in texts, figures and tables, and extreme possible values of the nine input variables corresponding with MathSelect values are computed in section A.2:2. (ZIP 4423 kb)
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Keywords
 Puberty
 Growth model
 Onset of puberty
 Peak height velocity
 End of puberty
 Duration of puberty
 Data quality
 Cumulative distribution
 Confidence interval