Insight into human pubertal growth by applying the QEPS growth model

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).

The data used for analysis was from a community-based, 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.

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 QEPS-model. 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.

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.

The quality of the height data were evaluated by visual inspection using QEPS-model-fitted growth charts. The quality of data, and the presence of potential errors that needed further assessment, were evaluated by stepwise observations:

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. Mid-puberty 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 P-function is estimated by the QEPS-model as the maximum height of the P-function, P _max in cm (P _max  = T _max –QES _max ). Due to the specific form of the P-function, which is a quadratic, logistic function, P _max can be calculated without defining a specific duration of puberty. The P-function starts before AgeT _ONSET since the velocity of T will not increase until the velocity of the P-function increases more than the decreasing velocity of the QES-function. If the relative influence of P- and QES-functions during the pubertal time period needs to be calculated, an age point is needed for both QES- and P-functions. Thus, onset of pubertal growth can be estimated from the P-function as the age when 1% (AgeP1) or 5% (AgeP5) of the total P-function-estimated gain was reached. For mid-puberty, we calculated the age when 50% of the P-function gain was achieved, AgeP50, and the age at PHV from the P-function 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 QES-function 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 P-function from the ongoing Q-function, 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 SD-scores, and is also divided into the P- and QES-functions. 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.

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).

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.

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 P-function adds to the ongoing QES-function. The mean pubertal gain, from the P-function, 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 QES-function during these years, Fig. 6, upper middle-right 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 QEPS-model, pubertal gain can also be shown for each individual both as total gain and divided into the individual components of the P-function and the ongoing QES-function, 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 QES-function dominated in girls, whereas growth from the P-function dominated in boys, but with large inter-individual variations for both genders.

The QEPS-model 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 SD-scores). 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 tempo-correction for the onset of puberty, the QEPS-model enables an individualized reference of pubertal growth in which height_SDS can be expressed according to a pubertal tempo-adjusted 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 SD-scores.

In Fig. 8, we show the relationship between the mid-puberty variable, AgeP50, and its corresponding CI for standardized mid-puberty (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 mid-puberty for individuals with later pubertal growth.

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 mid-pubertal 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 P-function 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 QEPS-calculated 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 P-function 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.

The present study, as the first implementation of the QEPS-model to describe pubertal growth, describes the pubertal growth variables generated by the model and their accompanying SD-scores 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 P-function (Ppubgain), and growth related to the still ongoing QES-function (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 QES-function, was greater in those with earlier puberty. Moreover, a gender difference was identified, with more QES-function growth in girls and more P-function growth in boys.

As well as providing robust variables, the QEPS-model is the first growth model to provide individual CIs. Moreover, it allows height SD-score estimations during puberty to be expressed in relation to an individualized tempo-adjusted reference. This is a major achievement as it allows relative growth during the pubertal years to be expressed at any time-point; previous models have only been able to present total pubertal gain from pre-puberty to adult height which has limited in depth analysis regarding pubertal growth [8]. By applying the QEPS-model to longitudinal growth data, we have identified new mathematical variables that are linked to specific time-points 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 QEPS-model compared to other growth models is that it automatically describes a wide variety of growth-related 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.

The QEPS-model gives different time-points 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 P-function, (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].

Mid-puberty, 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 mid-pubertal growth generated by the QEPS-model; from the total growth curve AgeT _PHV , and from the P-function 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 mid-puberty in future studies of pubertal growth.

The QEPS-model 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 P-function 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.

The shape-invariant QEPS-model 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 QES-function than the P-function. 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 1-year 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 SITAR-models [12, 21, 22]. The relation between Ppubgain and QESpubgain varies between genders, but also between individuals, with more QESpubgain in those with earlier puberty.

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 SD-scores. For the analysis of individual growth patterns during puberty, Tanner et al. constructed “tempo-conditional” height-velocity 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 SD-scores. Up to now, only the shape-invariant SITAR-model can adjust for individual tempo, amplitude, and size of total pubertal growth [22]. It is important to note that in contrast to the SITAR-model which only describes growth during the pubertal years, the QEPS-model 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 QES-function from growth by the specific P-function. These different growth-functions are probably regulated by different factors/hormones, and will therefore be of considerable use when searching for/identifying regulatory factors for growth. Thus, these QEPS-variables 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 QES-function as well as the P-function with good accuracy. Height expressed in SD-scores versus a tempo-adjusted height reference will serve as an individualized reference that is unique for this model.

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.

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 QEPS-model 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 QEPS-model 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 QEPS-model 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.