Multiphasic analysis of growth

W.J. Koops

Research output: Thesisinternal PhD, WU


<p>The central theme of this thesis is the mathematical analysis of growth in animals, based on the theory of multiphasic growth. Growth in biological terms is related to increase in size and shape. This increase is determined by internal (genetical) and external (environmental) factors. Well known mathematical functions, used in studies to quantify growth in size from birth to maturity, assume growth to be a result of one growth phase. Over the course of time, body weight or other body measures first show an exponential increase, followed by a decreasing increase. For size-age relationships, this results in a S-shaped growth curve.<p>Multiphasic growth theory states that the total growth curve is a result of a summation of many smaller S-curves. Each cell or group of cells has its own genetically determined growth potential, with different ages where gain is maximum. The total growth curve reflects an average growth pattern. The number of detectable phases depends on frequency and variability of the measurements.<p>Multiphasic, or multicyclic, growth theory was a subject of discussion in growth literature from 1900 to 1945. In the Introduction of this thesis, a review of this literature is given. Multiphasic growth functions have been widely accepted to describe human growth. In the description of animal growth, however, single-phase (or monophasic) functions have been used, although in the extensive literature of animal growth studies there is sufficient evidence for the existence of more than one growth phase.<p>The main objective of this study was to investigate the application of a multiphasic growth function to quantify animal growth. Investigation includes: application of the multiphasic function for different growth data (mean and individual curves) under different circumstances (internal and external influences), comparison of a multiple-phase function to single-phase functions and consequences on morphometric growth studies of assuming multiphasic growth.<p>Application of a multiphasic growth function is demonstrated in Chapter 1, with four data sets taken from literature. The multiphasic growth function used was a summation of n logistic growth functions. Human height growth curves of this type are known as "double logistic" (n=2) or "triple logistic" (n=3) growth curves. When applied to the human height curve and to pika, mice and rabbit weight curves, the fit of the multiphasic growth function was superior to the monophasic model in terms of smaller residual variances and absence of autocorrelation of residuals. For pika weights, two phases could be distinguished and for the other data sets, three phases.<p>Application of a multiphasic function to individual weights is tested in Chapter 2. Growth curves of mean body weights were compared to those of individual weights, when fitted to data of male and female mice using monophasic and triphasic growth (logistic) functions. Because of the large variability in individual weights, it was necessary to set bounds on some parameters. Goodness-of-fit criteria suggested that the triphasic function, with smaller and less correlated residuals, described the data better than the monophasic function. For the triphasic function, residual variances were larger when fitting curves for individual weights than for mean weights. Means of parameters for the triphasic function were higher for individual weights than for mean weights. Differences in parameter estimates between curves within sex were small. Parameters were similar for males and females in the first phase of growth. For the second and the third phase, however, asymptotic weight was higher for males than for females. It could be concluded that the triphasic function was able to describe accurately individual weights of male and female mice.<p>In Chapter 3, a multiphasic growth function is applied to problems of growth in different circumstances. Seasonal influence on growth in length of Northsea herring is an example of an external factor causing phases of growth. By modifying the multiphasic function slightly, length growth was described. The most important internal factor causing phases of growth is the difference in growth patterns of body components. The multiphasic function was modified and applied to growth of body components in pigs. Growth of total dry matter was analyzed with a diphasic function, and growth of fat and fat-free components were each analyzed with a monophasic function. Results for total dry matter showed clearly that parameter estimates of the diphasic function for the two distinguishable phases were related closely to parameter estimates of a monophasic function for each of the two Components. In a second illustration on growth in pigs, also a relation between growth phases of the function and growth of different body components could be shown. By restricting parameters of the general multiphasic function, or treating some parameters as constants, growth functions can be constructed that have parameters that are easy to interpret.<p>The relation of the multiple-phase to the single-phase approach for describing growth curves is studied in Chapter 4. A multiple-phase growth function is compared with four single-phase growth functions. From a general five-parameter function, four functions were selected to achieve maximal differences in shape: the JohnsonSchumacher, Michaelis-Menten, Gompertz and Logistic function. The multiphasic function was fitted to the simulated data from each of the four functions. Body weights of a cow and a boar were analyzed with four single-phase functions and with the multiphasic function.<p>Results of simulation showed that a triphasic function satisfactorily described simulated data from the four functions, in terms of smaller residual standard deviation and absence of systematic deviations in residuals. it could be concluded, therefore, that single-phase functions, with early inflection points, show relatively high fractions of the asymptotic value in the first phase. Using a single-phase function, cow and boar data could be described best with the Johnson-Schumacher function. Using a diphasic function for each data set, systematic deviations were eliminated and residual standard deviation was no larger than when using the JohnsonSchumacher function for the two data sets. This comparison showed that a multiple-phase function is a reasonable alternative to a single-phase function. An important advantage of a multiple-phase over a single-phase function is not having to select the 'best' single-phase function. Problems of having to estimate a larger number of parameters for a multiple-phase function than for a sing lephase function can be overcome because parameters for a multiple-phase function are less correlated than those for a single-phase function.<p>Application of the multiphasic growth function to body weights and tail lengths of mice is studied in Chapter 5, in cases where large, genetically determined differences in size exist between littermates. Mice were progeny of one male that carried the human growth hormone gene (somatotropin) and random bred NMRI females. At week 12, ten litters, with at least four females or four males, were chosen. Within each litter, four females or four males were selected, two with highest and two with lowest body weight. Mice with highest body weight were considered to be transgenic. Although this was not tested biologically, differences in body weight were considerable and the assumption probably was correct. Body weight and tail length of these 40 mice were measured about weekly from week 3 to 26. Female transgenic mice reached<br/>26-week body weight that averaged 1.6 times that of their non- transgenic littermates; for males, this ratio was 1.9. A diphasic growth function was used either for body weight and for tail length with marked results, especially for tail length. In the first phase, transgenic females had .64 cm shorter tails and transgenic males had .92 cm shorter tails than nontransgenic littermates. In second phase, transgenic females grew 1.4 cm and males 1.58 cm more than non-transgenic littermates. Body weight differences in each phase were in favor of transgenic mice. Multiphasic growth functions fitted data for body weight and tail length satisfactorily and provided clearer insight into differences in growth patterns of transgenic and nontransgenic mice.<p>Body weights and tail lengths of these same mice were used in Chapter 6 to study consequences on morphometric studies of assuming multiphasic growth. These types of studies, frequently indicated as 'allometric growth studies', will have an extra dimension when phases are taken into consideration. Multiphasic growth functions are based on assigning weight or other body measures to different phases. The well-known allometric function is used most often to study relations in growth of different body dimensions of an individual. Complex allometry exists when age at maximum gain is shifted on the age scale. Growth functions can used to estimate these ages. In this chapter, the literature on this subject is reviewed. By using multiphasic functions to describe growth of different body measures, it is possible to relate the growth in different phases. In mice data, it could be shown that the second phase of tail length was related the strongest to the first phase of body weight. Gain in second phase of body weight seemed to be unrelated to other body measures. Multiphasic growth analysis provides a suitable extension to study relations of growth in different body dimensions.<p>Findings in these investigations can be summarized in the following conclusions:<br/>1. The existence of more than one phase in growth curves of humans and animals is supported by biological explanations in the literature.<br/>2. Application of the multiphasic growth function provides detailed insight into growth patterns of body weight or other body measures, for individuals or groups of individuals.<br/>3. Use of the multiphasic function requires frequent measurements during a relatively long period of life; this improves discrimination of the phases.<br/>4. The multiphasic function is applicable in circumstances where phases are caused by systematic external influences and in cases where phases are the result of internal factors.<br/>5. In each phase of the multiphasic function, body components that have maximum gain at a similar age will be grouped together, if external influences are negligible.<br/>6. In cases where detection of multiple phases in the growth curve is not the main objective, use of a multiphasic function is an attractive alternative to single-phasic functions. If a multiphasic function is modified according to circumstances in which growth took place, then parameters are less correlated then those of single-phase functions.<br/>7. Use of a multiphasic function in "allometric growth studies" leads to an extra dimension for comparison. In addition to comparing growth of different body measures, it is possible to compare growth of different phases. In light of conclusion 5, phases within one body measure will show "complex allometric" relations.<br/>8. Application of the multiphasic growth function can make an important contribution in determining stage of physiological maturity, which is especially of interest when using the "genetic scaling rules" defined by Taylor.
Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Politiek, R.D., Promotor, External person
  • Grossman, M., Promotor
Award date16 Jun 1989
Place of PublicationS.l.
Publication statusPublished - 1989


  • growth
  • development
  • models
  • research
  • statistics
  • mathematical models


Dive into the research topics of 'Multiphasic analysis of growth'. Together they form a unique fingerprint.

Cite this