11.63 yr

Residual variance

0.144 cm2

Standard error of estimate

0.380 cm2

Age at takeoff

8.33 yr

Height at takeoff

121.2 cm2

Velocity at takeoff

5.1 cm2/yr

Age at peak velocity

11.18 yr

Height at peak velocity

139.8 cm2

Velocity at peak velocity

8.9 cm2/yr

Adolescent gain

33.2 cm2

• Inappropriate age ranges of the growth data for the chosen model (each model is designed to fit growth data in a particular age range, including data beyond that age range results in bad curve fittings).

• Inappropriate models for the type of variable (models designed for postcra-nial skeletal dimensions, such as the PB1 function, for instance, are inappropriate for fitting head dimensions or weight).

• The presence of particular features in the growth data that cannot be described by the growth model (prepubertal growth spurt(s), unusual variations of growth rate in pathological growth).

Growth variables that do not necessarily have a monotonously increasing pattern (such as weight, body mass index, skinfolds) cannot be successfully described by structural models such as nonlinear growth functions. Nonstructural approaches, such as polynomials, smoothing splines, and kernel estimations are more appropriate for these kinds of traits.24,25

In addition to producing a smooth continuous curve for growth and growth velocity and summarizing the growth data into a limited number of constants, the main goals of mathematical modeling of human growth data are to estimate

• Growth between measurement occasions (interpolation).

• Milestones of the growth process (the so-called biological parameters), such as age, size, and velocity at takeoff and peak velocity, for instance.

• The "typical average" curve in the population by means of the mean-constant curve (see later).

Table 3-1 also shows a number of biological parameters that were derived from the fitted curve shown in Figure 3-4. Note that adult height estimated by the PB1

fit is equal to h1 = 154.4 cm. Biological parameters, obtained by fitting a growth model, characterize the shape of the human growth curve and form a basis for studies of genetic and environmental factors that control the dynamics of human growth.

One should be suspicious about estimations of final size by structural growth models in cases where the growth data give no clear indication that the end of the growth phase is nearby. Estimated final height, for instance, is likely to be fairly unreliable if the last yearly increment in the growth data exceeds 2 cm/yr. Least-squares techniques are hopelessly weak in fitting parameters beyond the observation range and thus inapt to extrapolate. Analogous problems may arise when the lower bound of the age range does not include the takeoff of the adolescent growth spurt. In such a situation, the estimation of the age at takeoff and all derived biological parameters by a PB1 fit are not under control of the data and likely to be erroneous. A possible solution to the problem of extrapolation, like the prediction of mature stature (and also to the problem of incomplete data), is by using Bayesian estimations instead of least-squares techniques for the parameter estimation.18,26

tempo of growth

The famous American anthropologist Franz Boas, in the beginning of the twentieth century, already described that "some children are throughout their childhood further along the road to maturity than others."27,28 Indeed, individuals not only vary considerably in size but also in tempo of growth; that is, the speed at which they reach mature size. Tempo of growth, or maturation rate, is correlated with other markers of maturation, such as secondary sexual characteristics and bone age.

Figure 3-5 shows a theoretical example of the main effects of variation in tempo on the shape of the human growth curve. The figure shows the distance and velocity curves for the stature of typical early, average, and late-maturing children having the same size at birth and adulthood. These three theoretical subjects have, so to speak, the same potential for reaching a certain mature size, but they differ considerably in height at all ages along their growth period and in the shape of their growth pattern. We can see that the early maturer reaches final size earlier and is taller than the average maturer throughout childhood and adolescence. In turn, the average maturer reaches adult size earlier and is taller than the late maturer. The effects of differences in tempo of growth on attained height increase with age and are most apparent in periods where the slope of the growth curve is steepest. Therefore, variation in maturation rate affects attained height mostly during the adolescent period.

The relationship between the shape of the growth curve and the tempo of growth, as depicted in the preceding theoretical example, is also reflected in real population data. Longitudinal studies have repeatedly shown that little or no correlation exists between the timing of the pubertal spurt and adult stature; that is, early, average, and late-maturing children reach, on average, the same adult height.25,29-33 This

Age, years figure 3-5 Effect of tempo on the pattern of growth: Atheoretical example.

Age, years figure 3-5 Effect of tempo on the pattern of growth: Atheoretical example.

is also true for other postcranial body dimensions29 but not for weight. Early maturing children are, on average, heavier than late-maturing children.27 The shorter growth cycle in early maturers is compensated by a slightly but consistently greater growth velocity during childhood and, particularly, by a more intense puber-tal growth spurt. The opposite is seen in late-maturing children. This relationship is reflected in the negative correlation between peak velocity and age at peak velocity in height and several other traits.25'34'35

Studies on longitudinal growth of twin and family data have shown that tempo of growth is to a great extent genetically determined.13,36-39 In a more recent longitudinal growth study of monozygotic and dizygotic male twins, Hauspie et al.40 found a strong genetic component in the variance of various biological parameters characterizing the shape of the human growth curve; in particular, for age at peak velocity, reflecting tempo of growth. Similar findings were reported by Byard, Guo, and Roche41 on the basis of an analysis of familial resemblance in growth curve parameters in the Fels Longitudinal Growth Study. Tanner42 suggested that both the growth status and the tempo of growth are under genetic control but that the genetic factors might be quite different. Despite the strong genetic control over tempo of growth, there is also evidence that the human body can adapt to adverse environmental conditions by slowing down the developmental growth rate, probably allowing a child to better cope with the physiological and metabolic requirements for a balanced development in suboptimal situations. If the adverse conditions are reversed, then a child usually restores its growth deficit by a period of rapid growth to regain its original "growth channel," the so-called catch-up growth.43-45 If, however, the environmental stresses hold on for a long period or throughout the whole growth cycle, the resulting effect on growth may be a pattern that is typical for late-maturing children. Typical examples of this were found for children exposed to chronic mild undernutrition,46 chronic diseases such as asthma,47 psychosocial stress,48-50 socioeconomic deprivation,51 and living at high altitude.52 Those children tend to be slightly delayed in reaching the adolescent growth spurt, in achieving sexual maturity, and in attaining their final size. Final stature is usually not affected (i.e., is compatible with the population average) unless the long-lasting adverse conditions are too severe.45

individual versus average growth

Much of our knowledge on children's growth comes from longitudinal studies, that is, data comprising series of growth measurements of the same subjects over time, allowing to establish part or the whole of the individual growth pattern. However, the great majority of growth studies are cross-sectional; that is, based on single growth measurements taken from individuals who differ in age. Cross-sectional growth data allow to estimate the central tendency and variation of growth variables at each age in the population and to construct smooth centile lines showing the "average" growth and the limits of "normal" variation in that population. These centile lines form the basis of growth standards and reference curves (see Chapter 18). Despite the immense merits of cross-sectional growth surveys in constructing growth standards and in epidemiological studies of the genetic and environmental factors involved in growth, they can give only a static picture of the population variation in growth variables and are hopelessly weak in providing information on the dynamics of individual growth patterns over time.

The variation in the tempo of growth means that a cross-sectional mean curve, to some extent, smoothes out the phenomenon of the adolescent growth spurt. Figure 3-6 illustrates this effect very clearly on the basis of the longitudinal growth curves of two boys (taken from Chrzastek-Spruch's unpublished data on the Lublin Longitudinal Growth Study53), compared to the cross-sectional mean of their growth

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