Anat Embryol (1992) 186 : 537 541 Anatomy and

Embryology �9 Springer-Verlag 1992

Human mandibular prenatal growth: bivariate and multivariate growth allometry comparing different mandibular dimensions Carlos Alberto Mandarim-de-Lacerda, and Maria Urania Alves

Department of Anatomy, Biomedical Center, State University of Rio de Janeiro (UERJ), Av. Prof. Manuel de Abreu s/n, BR-20550, Rio de Janeiro (RJ), Brazil

Accepted August 3, 1992

Summary. Mandibular growth was studied in 36 human fetuses (both sexes) ranging from 13 to 37 weeks of gesta- tion by bivariate and multivariate analyses (bivariate al- lometry and principal components analysis, PCA). Sev- eral mandibular dimensions were measured and correlat- ed with fetal weight. Considering the different mandibu- lar dimensions in sequence of increasing component weights, PCA agreed with bivariate analysis. No mandi- bular dimension was considered to increase in isometric relationship. PCA showed the following distances with negative allometry: caput mandibulze-gnathion (both sides), gonion-processus coronoideus (both sides), caput mandibulze-processus coronoideus (both sides) and gon- ion-gnathion (right side). On the other hand, the follow- ing dimensions grew with positive allometry: gonion- gnathion (left side) and symphyseal height (both sides). PCA and bivariate analysis showed higher growth rates for the gonion-processus coronoideus distance and sym- physeal height on the right side than on the left. All other mandibular dimensions presented more elevated growth rates on the left than on the right side. During the second and third trimesters of prenatal life the man- dibular growth was allometrical; the mandibular body grew with more intensity than the ramus in both length and height. The greatest growth rate was found for the height at the symphysis. The angulus mandibula~ pre- sented a negative and slight correlation with the other linear dimensions of the mandible during prenatal life.

Key words: Mandible Growth - Fetus - Allometry - Anatomy

Abbreviations: CM-Gn, Caput mandibul~e-gnathion; Go-PC, Gon- ion-processus coronoideus; Go-Gn, Gonion-gnathion; CM-PC, Ca- put mandibulze-processus coronoideus; SH, Symphyseal height; AM, Angulus mandibul~e

Correspondence to: C.A. Mandarim-de-Lacerda

Introduction

Facial skeletal growth occurs by displacement and re- modeling in all directions. In mandibular growth, active remodeling (surface resorption and deposition of bone) has been considered as an important process (Bj6rk 1969; Enlow 1985, 1986). On the other hand, functional matrix theory of Moss and Salentijn (1969) states that as soft tissues grow they carry with them individual bones. Thus, a space is created at the articular junctions into which the bone can grow, either intramembranously or endochondrally, depending on the location. The dis- placement is primary, and bony growth occurs secondar- ily (Beals 1986).

The lower face is dominated by the mandible, of which the primary components are the mandibulary pro- cesses. Although the mandible appears in the adult as a single bone, during development it is divisible into several skeletal subunits: the mandibular body (to which are attached the alveolar portion), the coronoid, angular and condylar processes and the chin (Sperber 1989).

Growth is considered a process of self-multiplication of living substance (Huxley 1932) which does not occur synchronously in all components of the body. The facial skeleton increases in size in all three planes: height, width and depth. However, it grows in these three di- mensions of space differentially in time and rate (Sarnat 1971).

Multivariate allometries are rates of growth estimated with respect to overall body size. These rates are an estimate of biological age independent of chronological time, but concerned with the growth process (Strauss 1987).

In principal components analysis, morphometric data are interpreted as patterns of covariation in size and shape, the first principal component being a multivariate approximation to general size, with near-equal contribu- tions made by each character (Jolicoeur 1963a). Gener- ally, researchers assume that the first component repre- sents overall size, since all characters are positively corre-

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la ted with this c o m p o n e n t (Jol icoeur 1963 a, b; H o p k i n s 1966; Shea 1985; M a n l y 1986; Somers 1986).

This s tudy a ims to c o m p a r e the g rowth o f the differ- ent d imens ions o f the m a n d i b l e in the second and th i rd t r imesters o f ges ta t ion.

M a t e r i a l and m e t h o d s

Fetuses. Thirty-six human fetuses of single pregnancy were studied. On the basis of their greatest foot length the ages were estimated (Streeter 1920; Mandarim-de-Lacerda 1990) and the sample was divided into six age groups: 13-17 weeks (three fetuses); 17-21 weeks (seven fetuses); 21-25 weeks (eight fetuses); 25 29 weeks (ten fetuses); 29-33 weeks (four fetuses); 33-37 weeks (four fe- tuses).

Only well-preserved fetuses without external pathologies or ede- ma were analyzed (collection of the Department of Anatomy, State University of Rio de Janeiro). All material was obtained from spon- taneous abortions (death due to prematurity or perinatal asphyxia) and studied until 24 h after expulsion. Parity, sex or maternal age were not considered in the selection of the specimens.

Mandibular biometry. Mandibles were meticulously prepared after natural running water maceration of the skulls during a period of 3 to 4 weeks. They were blanched in hydrogen peroxide and then dried and identified. The bones for this study were selected taking into consideration their anatomical integrity after prepara- tion.

All measurements were made by one author (MU Alves). The right and left mandibular halves were analyzed; distances were measured to the nearest 0.01 mm in the following regions (Fig. 1): 1. Caput mandibul~e-gnathion; distance between the most posterior point in condyloid process and the gnathion 2. Gonion-processus coronoideus; distance between gonion and most superior point in coronoid process 3. Gonion-gnathion; length between gonion and gnathion 4. Caput mandibul0e-processus coronoideus; distance between the most posterior point in condyloid process and the most anterior point in coronoid process 5. Symphyseal height measured between the central incisors

The angulus mandibula~ was also measured, using a goniometer, between the posterior margin of the ramus mandibul0e and the inferior margin of the corpus mandibnlee.

Statistical analyses. Bivariate and multivariate analyses were per- formed in order to study the mandibular growth rates.

The bivariate study used log-transformed data and the allomet- tic model:

Ln y = Ln a + (b) Ln x

G~ 1 AM Go-Gn

Fig. 1. Schematic drawing with the points of reference where the mandibular lengths were measured. Abbreviations in the text

The dimensions of the mandible (dependent variable y) were correlated with fetal weight (independent variable x). Logs allowed us to work with the multiplicative properties of the data and facili- tated slope comparisons between variables. Likewise, this over- comes a latent heterocedasticity of the standardized residuals when the variabiiity of y increase with increasing values of x (Zar 1984).

Because the problem of biased estimates of slopes of y on x when both variables are subject to measurement error, the slope of the principal axis of the standardized variables, i.e., reduced major axis (RMA), was computed (Sokal and Rohlf 1981 ; Jolicoeur 1990). R 2 and F-statistics were used to determine the significance of each regression. A t-test was used to test for significance depar- ture from a predicted slope examined with residual analysis (Wit- tink 1988). Each regression slope was checked for departure from isometry. A slope of 0.333 indicates isometry when linear measure- ments are correlated with weight as in the present study (Gould 1966).

A covariance matrix was calculated from natural logarithms of the dimensions of the mandible. The relative growth rates of these dimensions were computed by multivariate analysis using a principal components analysis (PCA) (Jolicoeur 1963a; Maniy 1986; Somers 1986).

PCA is a statistical method of data reduction. A set of correlat- ed variables is transformed into a smaller set of uncorrelated vari- ables (named principal components) that account for most of the variability among subjects. The principal components are linear combinations (weighted averages) of the original variables. The first principal component (cpl) has the largest variance of any such linear combination. Successive principal components are meaningful and if they include most of the variability of the original variables, these new variables can efficiently substitute the original, larger set of variables in subsequent analyses (Nelson et al. 1991). In the present study, PCA was based on the covariance matrix. Multivariate isometry exist when all p dimensions increase at the same rate; this implies that all p loadings are equal :

cpl = (p-~ , . . . p -+ .... p-~)

A loading greater than p �89 indicates positive allometry, whereas one less than p-} signifies negative allometry (Jolicoeur 1963a; Jungers and German 1981).

Bivariate analysis, RMA, eigenvalues and eigenvectors of the matrix were computed by using the Lotus and Statgraphics soft- ware packages on an IBM PC computer.

R e s u l t s

Resul ts are set ou t in Tables 1-3 and Fig. 2. The s t a n d a r d g rowth coefficients in this s tudy were based on the s lopes o f the l inear m o d e l f i t ted to the logged da ta . S t a n d a r d stat ist ics (R 2 and F-s ta t i s t ics ) i nd ica t ed tha t all regres- sions were s ignif icant and l inear mode l s a p p r o p r i a t e .

Bivar ia te analysis i nd ica t ed posi t ive a l lome t ry only for G o - G n length (left side) and SH (bo th sides). Al l o ther m a n d i b u l a r d imens ions p resen ted negat ive al lo- met r ica l coeff icients (Table 1, Fig. 2). By tes t ing the sig- nif icance o f these coefficients wi th t-test all coeff icients were s ta t i s t ica l ly s ignif icant (P < 0.001).

D u r i n g the last two t r imesters o f ges ta t ion the angu- lus mandibul~e showed litt le change and no r e m a r k a b l e difference be tween r ight and left sides ( r ight s ide: mean _+ s t a n d a r d d e v i a t i o n = 1 3 9 _ 1 degree). I t was sl ightly negat ive ly co r re l a t ed wi th the l inear d imens ions o f the mand ib l e , while the o ther me a su re me n t s were h ighly cor- re la ted (Table 2). Since the P C A d id no t inc lude the an- gulus mandibular , it was h ighly effective in reduc ing the n u m b e r o f var iab les for analysis . The first p r inc ipa l corn-

Table 1. Mandibular dimensions (as the dependent variable y) and fetal weight (independent variable x) in bivariate analysis using the formula of allometry Ln y = Ln a + (b) Ln x. The analysis was fitted by Reduced Major Axis regression method (the probabil- ity that the coefficient of correlation is different from zero is lower than 0.0001 for all coefficients)

Ln y Slope a 95% CI (b) b Intercept r (mm) (b) (Ln a)

Go-PC right 0 . 2 9 2 0.243/0.340 0.627 0.879 left 0.290 0.243/0.336 0.632 0.885

CM-Gn right 0 . 2 9 8 0.268/0.328 t.502 0.958 left 0.301 0.271/0.331 1.487 0.957

Go-Gn right 0 . 3 2 2 0.273/0.371 1.004 0.899 left 0.347 0.289/0.405 0.820 0.877

CM-PC right 0 .311 0.260/0.363 0.300 0.880 left 0.320 0.268/0.373 0.241 0.882

SH right 0 . 4 0 0 0.322/0.477 -0.692 0.830 left 0.393 0.320/0.466 -0.640 0.847

Isometry = 0.333 b 95% confidence interval for the slope b

Table 2. Correlations among mandibular dimensions (right side, n = 36, P < 0.001 except for angulus mandibul~e)

Go-PC Go-Gn CM-PC SH CM-Gn

Go-Gn 0.96 CM-PC 0.96 0.98 SH 0.94 0.94 0.94 CM-Gn 0.90 0.93 0.92 0.85 AM -0.50 -0.44 -0.44 -0.34

(0.002) * (0.007) (0.007) (0.04) -0.43

(O.Ol)

* Probability

Table 3. Component weights (growth rates) for the first principal component of the Principal Components Analysis studying the in- crease in length of the mandible during the second and third trimes- ter of gestation. The condition of the isometry for this analysis is indicated

Mandibular Side Component dimensions weight

Caput mandibul~e-gnathion (R) 0.2750 (L) 0.2784

Gonion-processus coronoideus (L) 0.2796 (R) 0.2814

Caput mandibul~e-processus coronoideus (R) 0.3025 (L) 0.3119

Gonion-gnathion (R) 0.3136 Isometry 0.3162

Gonion-gnathion (L) 0.3369 Symphyseal height (L) 0.3785

(R) 0.3813

ponent accounted for 94.72% of the variation in the sample showing a greater significance than the other components. All characters were positively correlated with this component .

539

Mandibular growth (ram}

50 o ~k

Go-PC

Oo-On 3 0

-E~ CM-PC

~-- CM-Gn

1

200 900 1600 2300 3000

Fe ta l weigh t (g)

Fig. 2. Growth curves of the right side mandibular dimensions relative to the fetal weight. Equations are presented in Table i. Abbreviations in the text

The isometry hypothesis of the growth vector was checked with chi-square test on the assumption of multi- variate normal distribution of the variables. The test showed that the coefficients of the growth vector differed from the unit value (p -~ = 0.316, where p is the number of variables) with the chi-square value 30.12 (the value is significant at 0.0004 level under the degree of freedom p - 1 = 9) (Anderson 1963).

No mandibular dimension was considered to increase in isometric relationship: PCA showed that the follow- ing dimensions grew with negative allometry: C M - G n (both sides), Go-PC (both sides), CM-PC (both sides) and Go-Gn (right side). On the other hand, the following dimensions grew with positive allometry: G o - G n (left side) and SH (both sides) (Table 3).

PCA and bivariate allometry analyzed the growth of the mandible similarly: for the Go-PC and SH higher growth rates were found at the right side than at the left. All other mandibular dimensions presented more elevated growth rates at the left than at the right side.

Discussion

One of the first decisions in studies of ontogenetic allo- merry is the choice of a model to fit to the data. Linear and nonlinear models have been proposed, debated and used or rejected for both theoretical and practical rea- sons (German and Meyers 1989a). Some models (e.g., high order polynomial equations) can fit the data ade- quately, but have coefficients nearly impossible to inter- pret in biological terms. So, comparisons among these coefficients for different dependent variables are mean- ingless (German and Meyers 1989 b).

In this study the standard equation used for examin- ing the relationship between variables x and y is the log t ransformed Huxley's (1932) bivariate formula. Lin- ear model, whether the independent variable is weight, assumes that the rate of growth of the y variable relative to the x measured by the coefficient b, is constant (Ger- man and Meyers 1989a). Although a firm theoretical basis for the empirical success of Huxley's power func-

540

tion of relative growth has proved elusive, there are re- cent arguments to support Huxley's notion of multiplica- tive growth. The use of logarithmic t ransformation in studies of growth allometry was directly related to the analysis of specific, or multiplicative, growth (Katz 1980; Shea 1985; Jolicoeur 1989, 1990).

In studies of facial growth it is usual to analyze facial diameters and angular measurements: cephalometric measurements as made by orthodontists (Moss and Sa- lentijn 1970; Bhatia et al. 1979) or anthropometr ic mea- surements as made by physical anthropologists (Siebert 1986).

The slight negative correlation of the angulus mandi- bula~ with linear dimensions of the mandible and the lack of quantitative variation of this angle in prenatal life are not unexpected. The growth and action of the tongue and masticatory muscles are functional forces corrected with the mandibular growth that will be de- cisive in postnatal life (Sperber 1989).

On the other hand, we found allometric growth rates for the mandibular dimensions in agreement with pre- vious workers (Green 1933; Moss and Baer 1956; McKeown 1975) who determined that the cranium fol- lows the principle of allometric growth, and the basic form of the head was established at the earliest stages of fetal development. The logarithmic nature of the growth of the mandible has been varified previously (Moss and Salentijn 1970; Salentijn and Moss 1971).

Houpt (1970) studied the size of various structures within the craniofacial complex by radiographic cepha- lometry in the human fetus between the ages of 12 and 19 weeks. This author 's study is not comparable with the present one, since only four fetuses of our sample ranged f rom 12 to 19 weeks of age. On the other hand, the slopes indicated by Houpt are related to some linear craniofacial measurements, unlike those in the present study. Our findings therefore differ f rom those of Houpt, who reported constant growth rates of components of the craniofacial complex.

The present allometrical study confirmed the findings of Lavelle and Moore (1970) who observed the growth of the human jaws on radiographs obtained f rom 65 fetuses between the 4th and 7th months. They found different rates of growth between the ramus and the body of the mandible, the body growing more rapidly than the ramus in both length and height. Both studies found the greatest growth rate for the height of the man- dibular body at the symphysis level. This is probably related to the growth of the alveolar process of the man- dible as noted by Ford (1956) in the upper jaw.

Jolicoeur (1963 a) demonstrated that the eigenvector in PCA (first principal component) , extracted f rom the covariance matrix of logarithmic values, describes rela- tive changes in the measured characters during growth. In an additional paper (Jolicoeur 1963b) this author demonstrated that the ratio of the variable loadings on the first principal component is proport ional to the slope value b, given in bivariate allometry formula. In the pres- ent study no significant difference exists comparing the bivariate with the multivariate coefficients.

In this work the growth of some mandibular dimen-

sions was analyzed on dehydrated material. Although it may be considered a reliable approach for the study of size changes in human fetal bones, it does not neces- sarily, if at all, represent the rates of normal prenatal growth in living individuals. Nevertheless, the analysis performed in this study provides an approximat ion to the biological phenomenon of growth which may be use- ful for the interpretation of facial growth.

Acknowledgments. The authors would like to thank Mr. Jo~o Mendes for his technical assistance. Valuable comments were re- ceived from the referees of Anatomy and Embryology. This re- search was supported by the Brazilian agency CNPq (grant 50.00.427/91-7).

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