## 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 QEPS-model 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 QEPS-model describes total height as *(T)otal-*function*:* a combination of four shape-invariant growth functions, modified by time-scale and height-scale parameters: a (*Q)uadratic-*function for the continuous growth from fetal life to adulthood; a negative *(E)xponential-*function adds the rapid, declining fetal/infancy growth; a *(P)ubertal-*function the specific pubertal growth spurt; a *(S)top-*function the declining growth until adult height. A constructed variable, *MathSelect*, was developed for assessing data-quality. CIs and SD-scores for growth estimates were calculated for each individual.

QEPS-model estimates used for pubertal growth; from the *T-*function: onset of puberty as minimal height velocity (*AgeT*
_{
ONSET
}); mid-puberty 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 *P-*function: onset of puberty, estimated as growth at 1% or 5% (*AgeP1*
_{
,
}
*AgeP5)*; mid-puberty 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 *QES-*function: gain (*QESpubgain)*
_{
.
}

### Results

Application of these mathematical estimates for onset, middle and end of puberty of *P-*function*, QES-*function, and *T-*function during puberty showed: the later the onset of puberty, the greater the adult height; pubertal gain due to the *P*-function growth was independent of age at onset of puberty; boys had higher total gain during puberty due to *P-*function growth than to *QES*-function 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 SD-scores and Cis for each individual. These QEPS-derived estimates enable more in-depth analysis of different aspects of pubertal growth than previously possible.