An allometric network model of craniofacial growth

An allometric network model of craniofacial growth

Melvin L. Moss, Richard Skalak, Himanshu Patel, Masanobu Shinozuka, Letty Moss-Salentijn, and Henning Vilmann New York, N. Y., and Copenhagen, Denmark

This study of cranial skeletal growth kinematics details the conceptual principles underlying the development of an allometric network model of such growth. This model is tested by the analysis of longitudinal rat and cross-sectional human growth data and by comparison of this model with a previously described allometric centered model. It is shown that the network model is superior to the centered model in three ways: (1) The allometric network model permits growth prediction when allometric constants are known; (2) the network model has significantly smaller errors than the centered model; and (3) the network model is capable of displaying growth kinematics of both the neural and facial skulls while in time there are marked transformations, such as relative rotations of two sets of cranial anatomic points.

Key words: Allometric network, craniofacial , growth, modeling.

I t is generally agreed that analysis of craniofacial skeletal growth would be materially assisted by development of appropriate models. Mathematically, such models should closely describe and graphically illustrate the kinematic behavior of the growing skele- ton. There is, however, little agreement as to which analytical model best serves this purpose. Recently it was shown that all analytical methods related to customary roentgenographic cephalometry are incapable of correctly modeling craniofacial skeletal growth kinematics, criticisms accompanied by suggestions of alternative models .I-’ Recently we suggested three additional modeling methods,‘j one of which was an allometric centered model. Using inspectional tech- niques alone, many workers independently suggested the potential utility of this visually attractive model .7-s However, our tests demonstrated that, although an allometric center model may be valuable for empirical research, no unique point can be readily located as the center for which errors are clearly reduced. Because of the inability to demonstrate a unique center, it would seem that validity of results based on any procedure selecting a unique center may be dubious. If a given case departs from centered allometry, there is no way to tell what center should be used to judge the error.9

A second suggestion is that of an allometric net-

From the Department of Anatomy and Cell Biology and the Department of Civil Engineering and Engineering Mechanics, Bioengineering Institute, CO- lumbia University, and the Department of Anatomy, Royal Dental College, Copenhagen. This study was aided in part by Grants DE-0514503 and HD-14371-03.

316

work model. This is of greater theoretical interest than a centered model for three reasons, as is to be shown below: (1) The allometric network model permits the display of the kinematic behavior occurring when there are marked transformations of particular cranial skeletal regions during growth (as in orthocephalization of the rodent skuWO-“); (2) the allometric network model permits growth prediction; and (3) the growth network model has significantly smaller error than a centered model when modeled and actual biologic data are compared. The conceptual and methodologic bases of an allometric network model are presented below and illustrated by analyses of rat and human craniofacial growth data.

ALLOMETRIC NETWORKS

In growing biologic systems allometry expresses the concept that a constant ratio exists between the specific growth rates of a pair of quantifiable variables during some time interval. This relationship, as origi- nally expressed by Huxley,12 is

y = bx’ (1)

where k represents the constant ratio between the specific growth rates of x and y that grow allometrically if

1 dx = k, 12) -- x dt

and

then

;3=kk, Y dt k = k,/k2.

(3)

(4)

Volume 85 Number 4

Allometric network model of craniofacial growth 317

Fig. 1. The three line elements (I,, I,, IS) of this figure can all grow allometrically with different growth rates only if it is not fixed and the line elements are allowed to rotate. If all growth rates are equal, the angles are constant, but not otherwise.

The allometric centered model was one way by which some degree of allometry could exist in a growing craniofacial skeletal system. Another way to achieve some degree of allometry is to assume that there exists a network of points in a plane (two- dimensional) or in space (three-dimensional), so that the length of each line element between points has its own specific growth rate. For example, in Fig. 1, 1,) l,, and 1, are each assumed to have differing specific growth rates. These three line elements cannot all grow allometrically if 0 is fixed; however, if the line elements are allowed to rotate, then each line element may grow allometrically. The future geometry of the triangle at any time is determined (“predictable”) if the initial size and shape are given and the growth rates of the three sides are known.

When networks more complex than a triangle are considered, the main problem is to determine the con- ditions under which allometric growth is possible, and the main answer is that allometric growth is possible, with arbitrary specific growth rates for each line element, if the network forms what is known in structural analysis as a statically determinate truss. For this purpose, the definition of simple networks is similar to the definition of simple trusses.13 A truss is a network in which the line segments are solid bars, as in a bridge truss. In the allometric networks considered here, the line segments are not considered to be. material bars and may, in fact, consist of different material points at different times. (See article by Moss and associates3 for definition of material points.)

A plane network will be called simple if it can be drawn in the following way: Start with a triangle made up of three segments and three nodes. Then add one node at a time, connecting it to two previous nodes by two new segments. If the entire network can be generated in this way, it is called a simple plane network.

m = 13 2j - 3 = ;3= 8 m=2j-3 Determinate

m = 14 2j - 3 = i3= I3 m>2j -3 Indeterminate

m = 12 2j - 3=13 m<2j-3 Unstable

C.

Fig. 2. Three plane networks of points and lines are shown. The relationship between the number of modes (i, and line segments (m) is such that A permits allometric growth of all line segments, and its future geometry is determined by the allometric constants. If it were a truss it would be stable. B is overdetermined, and it it were a truss it would be indeterminate. Growth rates cannot be independently assigned to each segment. C would be unstable.

A space network will be called simple if it can be drawn in the following way: Start with a tetrahedron made up of four nodes and six segments that form the edges of the four triangular faces of the tetrahedron. Then add one node at a time, connecting it to three previous nodes by three new segments. If the entire network can be generated in this way, it is called a simple space network.

Fig. 2, A shows a plane network of points and lines which divide the plane into triangles. If each element is assumed to grow allometrically , then the future geometry is determined. If it were a truss, it would be stable. It can be shown that the criterion for allowing arbitrary growth rates of every line segment of a plane network is

m=2j-3 (5)

where m = number of line segments and j = number of nodes (joints). Equation 5 is the same criterion as re- quired for a framework or truss to be statistically determinate and can be derived by similar methods.‘“-l5 If m > 2j - 3 (Fig. 2, B), the network is overdetermined. If it were a truss, it would be indeterminate. Growth rates of each member cannot be arbitrary, and they must be related to some extent. If m < 2j - 3, the network may be called undetermined. If it were a truss, it would be unstable (Fig. 2, C); as a growth network, it would not permit a unique prediction.

318 Moss et al. Am. J. Orthod. April 1984

&ji?J@@ @ C C

m = 15 j = 7 m = 16 ,=7 3j - 6 = 15 3, - 6 = 15 m = 3, = 6 m > 3j - 6 Determinate Indeterminate

m = 11 *j - 3 = :I= '

m = 12 ,=7 2, - 3 = II

"= 2, - 3 m > 2, - 3 Determinate Indeterminate

A. B

Fig. 4. These classic figures were studied by Cauchy in 1813 and are related to those shown in Figs. 2 and 3.

A. B.

Fig. 3. Two space (three-dimensional) networks are illustrated-one determinate and stable, the other indeterminate.

In Fig. 3, A and B, space networks are illustrated. It can be shown that the growth rates of each line segment can be arbitrary if

m=3j-6 (6)

Rule 6 is the same as for the static determinancy of space trusses and was given by MaxwelP in 1864. Cauchy17 studied the stability of networks of triangles such as those shown in Fig. 4, A and B. EuleP also derived some interesting results on geometry of poly- hedra. Caspar and Klugls consider a virus coat problem of similar nature. (See article by Skalak and col- leagues .6)

Fig. 5 shows some simple two-dimensional networks which might be used for the description of craniofacial growth. Fig. 5, A and B is sketched to resemble human proportions. Fig. 5, C and D is drawn as if for a bird or rodent. Topologically, however, all of the networks in Fig. 5 could be used for the skull of any animal. Some possible simple three-dimensional networks are shown in Fig. 6. The three-dimensional networks of Fig. 7 obey equation 6. Fig. 7, A andB shows types that might be used in structural design. Fig. 7, C might be used for craniofacial growth description. (See Badoux”, ‘l for closely similar representations of skull structure).

Glossary of Corresponding Terms Terminology for Terminology of growth networks structural mechanics

Network Truss Nodes Joints Segments (or line segments) Bars (or members) Unstable Unstable Determinate Statistically determinate Indeterminate Statistically indeterminate

Summary of criteria 2D 3D

Unstable (non-unique) m<2j-3 m<3j-6 Determinate (permits allo- m=2j-3 m=3j-6

metric growth)

Indeterminate m>2j-3

(Independent growth rates not possible) m = Number of line segments

j = Number of nodes

MATERIALS AND METHODS

m>3j-6

It is useful to restate the theory of allometric growth networks in a more concise manner. A network is defined as a finite collection of points that are connected by straight line segments such that at least two line segments ate incident at each point. The points of a network are referred to as nodes. A plane network is one in which all the nodes lie in a plane. Line segments of a network are said to grow allometrically if they increase or decrease according to the formula

li = loieklt (7)

where &i and ki are constants and i ranges from 1 to n, where n is the number of segments. Equation 7 leads to equations of the form of equation 1 between any two lengths li and lj, that is,

Ii = CijljMij (8)

where Cij and Mij are constants. From equation 8, it :r. follows that growth of a network is allometnc if the logarithms of lengths of any two line segments are lin- early related. An allometric growth of a plane network is possible with arbitrary constants kr if the network satisfies the relationship

n=2j-3 (9)

where n is the number of segments and j is the number of nodes.

Rat data

The allometric network theory is used to analyze rat craniofacial growth. Longitudinal data were available for the cranial growth of twenty-one male rats and

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@

m = 25 j = 14 2j - 3 m = 2j Determ

= 25 -3

inate Determ

= 27 -3

.inate

m = 21 j = 12 2j - 3 = 21 m=2j-3 Determinate

m = 37 j = 20 2j - 3 = 37

D. m=2j-3 Determinate

Fig. 5. Four determinate plane networks that might be used for the description of craniofacial growth are shown.

twenty-five female rats. lo, l1 Fifteen cranial anatomic points on the midsagittal plane of each rat skull were used as the network nodes and are connected by twenty-seven suitable line segments so as to satisfy equation 9. The anatomic points used were (1) basion; (2) opisthion; (3) posterior lambdoidal suture; (4) lambda; (5) bregma; (6) jugum olfactorium; (7) superior fronto-ethmoidal suture; (8) nasion; (9) anterior nasal point; (10) anterior palatal point; (11) posterior nasal spine; (12) intersphenoidal synchondrosis; (13) spheno-occipital synchondrosis; (14) presphenoidale; and (15) inferior fionto-ethmoidal suture. (See Vil- mann and Moss” for detailed descriptions.)

The growth period of each rat is from 7 to 150 days. Data points are available for the ages of 7, 14, 21, 30, 40, 60, 90, and 150 days. Since the rats are derived from a close-bred strain, it is observed that shape changes due to growth in one rat are similar to those in another rat. Therefore, rather than analyzing the growth of each rat skull, growth of an average rat skull cross- section, formed for each age by averaging corresponding line segments over all rats, is analyzed. Put another

way, the rat data are longitudinal but derived from a close-bred strain. The phrase close-bred indicates a high degree of inbreeding, not as close as in pure in- bred strains but, as our data showed, very closely resembling it. in our preliminary analyses of the kinematic behavior of each rat independently and following appropriate statistical analysis (of each sex independently), note that we observed that the shape changes due to growth of all rats was similar. The word similar is not used here in its customary, linguistic sense of ‘ ‘closely resembling’ ’ but, rather, in the more rigorous, mathematical (geometric) definition of identify in shape of several structures, that may differ only in size and position.

More precisely, our statistical analyses of these data showed that the shape change of ah rats, at all ages, did not differ significantly from the group mean at any age; that is, all such deviations were random in nature, and each rat constantly regulated its growth about the group mean. In a still more technical sense, these rats are closely ergodic.

For these reasons, we do not present the analysis of

320 Moss et al.

m = 10 j=6 3j - 6 = 12 m< 3j-6 Unstable

m = 12 j=6 3j - 6 = 12 m=3j-6 Stable

m = 13 j=6 3j - 6 = 12 m>3j -6 Indeterminate

Fig. 6.Three three-dimensional networks are shown. These are unstable, stable, and indeterminate, respectively.

any individual growth record. The use of mean values provided an additional advantage. They permitted statistical analysis and comparison with the previously published allometric centered model, such a comparison being one of the purposes of the present article. Clearly, in this instance, analysis of the growth behavior of an individual rat would be only anecdotal and would not permit statistical comparison. Our present purpose is to demonstrate a method. Any worker can use this method and apply it to any individual data at hand.

Fig. 8 shows the network used in the analysis. Line segment lambda-bregma (segment 8) is chosen as pseudotime; that is, constants Crj and Mij are evaluated with lambda-bregma as one of the line segment variables in equation 8 (Fig. 9). Constants Cij and Mij (the b and k, respectively, of equation 1) are evaluated by a linear regression analysis in which the sum of the squares of the distances between points and a regression line, in direction, perpendicular to the axis repre- senting the pseudotime variable is minimized. Two sets of constants Cij and Mu are obtained; one set characterizes growth for the time period of 7 to 2 1 days, while the second set characterizes growth for the period of 30 to 150 days. This is necessary to obtain a good fit of the data. This indicates that there are different, identifiable

Am. J. Orthod. April 1984

m = 18 j=8 3j - 6 = 18 m=3j-6 Determinate

m = 30 j = 12 3j -6=30 m=3j-6 Determinate

B.

Fig. 7. Three determinate space networks are figured.

stages of growth that are not describable by a single allometric law.

To test the results of the model, line segment lengths of the network are calculated using the derived Cij and Mij in equation 8. The networks are then recon- structed, using these calculated line segments by solv- ing simultaneous equations to determine coordinates of all nodes, with one node (basion) and one line segment (basion-posterior lambdoid suture) kept fixed for each age. The computed networks are then translated and rotated as rigid bodies, so that the sum of squares of distances between nodes and corresponding nodes of networks constructed from data are minimized. The root mean square (RMS) of the sum of squares of these distances gives a measure of accuracy with which the allometric network model describes actual growth data. Mean percent error is calculated by dividing RMS by the average length of all line segments of networks constructed from data, giving a nondimensional mea-

Volume 85 Number 4


m = 39 j = 15 C. 3j - 6 = 39

m = 3j - 6 Determinate

Fig. 7 (Cont’d). C is an arbitrary figure that might be used in the description of craniofacial growth.

Fig. 8. The determinate plane network used in the description of rat skull growth is shown. The fifteen anatomic points (nodes) are described in the text. An additional, but unanalyzed, line segment (10-l 1, interrupted line) is added to permit a closer visual approximation of network geometry to facial skull structure depicted. Segment 4-5, lambda-bregma serves as a pseudotime in this study.

sure of accuracy of fit between actual and predicted networks of the rat skull cross section. All calculations and graphic presentations of results were performed by computer programs written for this purpose.

The difference in representation of cranial skeletal growth behavior between the allometric network model and the previously reported allometric center model was analyzed statistically for male and female rats independently. The eight midsagittal anatomic points used (Nos. 1 to 5, 12, 13, and 14, as defined above) are those used previously with the centered model .s

The comparison of the models was performed by a t test for differences between means of distances from model to data nodes in each case as follows:

log Seg n n = 1,2-- 27

l4 8 I/+ 7 ’

Slope = tan 0

Itierceptf )

log Seg 8

Fig. 9. An arbitrary graphic representation of the relation between a pair of line segments, as expressed in equation 8. Segment 8 is a pseudotime. In this double-logarithmic plot the approximate linearity of two data sets is shown and is typical for the rat data. The actual mathematical structure of the interven- ing region (21 to 30 days for the rat) is unknown. Human neurocranial data also show two such linear regions when similarly plotted.

Fig. 10. The determinate plane network used in the description of average human skull growth is shown. The anatomic points (nodes) are described in the text. As in Fig. 8, an additional, unanalyzed line segment (GL-Na, interrupted line) is added to approximate more closely the customary depiction of the human neurocrania.

Let Ut be the distance from model to data node i for the centered model, i = 1 . . . n, let Vi be the distance from model to data node i for the network model, i = 1 . . . n, ui, vi are the distances between model and data nodes for each skull. Let ii be the mean of the distances ui between centered model nodes and data nodes.


0 1 I , I I I , I I I, 1, I, 0 20 40 60 00 100 120 140 160 180

TIME, DAYS

Fig. 11. The specific growth rate of segment 8, of the average male rat, is not constant but is a function of time, as depicted here. However, when we consider pseudotime (segment 8) as the variable, the ratio of the k’s of the two lines may be nearly a constant, as in Fig. 12.

log seg 8 vs. log sag 4

slope = 1.842

intercept = -2.159

slope = .5193

intercept = 2.094

Fig. 12. The double logarithmic plot of segment 4 and segment 8 for the average female rat in the period 7 to 150 days. The two sets of data points are shown, with their distinctive values of slope (M(i) and intercept (C$ are shown. Note that strict allometry is not present.

Let 7 be the mean of the distances vi between network model nodes and data nodes.

Define S:,. = Ji (u,-~)‘+~,(“~-v)’ ’ = ’ 2n - 2 (10)

where n is the number of nodes and

ii = f i$, ui andV = A ,$, vi (11)

then u-v t = __ degrees of freedom = 2n - 2. &I, ,lz

(12)

If t > t,, then there is a significant difference between the means of the variables ui and vi where ff = level of significance.

An alternate statistical method for testing the differences between the allometric network and centered models is provided by the Wilcoxen two-sample rank- sum test. Here the distance between the raw data and the model is calculated independently for each of the eight anatomic points (nodes). The values from both models are arranged together in ascending order, and the rank values are added separately for each model and compared. The differences between the rank sums of the two models (Nr) is compared with established tabu- lar values (Wcr) of significant differences between models (where Q = level of confidence). This test was performed separately for average male and female rats for each of the eight ages studied.

Human data

Similar procedures and tests were used to evaluate the differences between the allometric network and the allometric centered models in the analysis of human neurocranial anatomic point growth behavior. The data consisted of age-grouped mean location coordinates of eight anatomic points (nasion, glabella, midfrontal, bregma, lambda, sella, basion, and opisthion). These cross-sectional data were derived from 112 lateral skull films ordered in six age groups: Group B , 7.5 months (n = 5); Group C, 3 months postnatal (n = 15); Group D, 22 months (n = 17); Group E, 49 months (n = 16);

Volume 85 Number 4


Table I. Allometric constants, for average female and average male rat skull growth, in the periods 7 to 21 days and 30 to 150 days (Slope = k, Mu; Intercept = b, Ci,)

Segment No. Slope 1 Intercept Segment No. Slope Intercept

Period 7 to 21 days-Female I 2.321235 2 I .450866 3 1.334597 4 1.841626 5 2.257627 6 I .449342 7 I. I27235 8 I 000000 9 I .919268

10 I .789482 II 0.9596064 12 1.891778 13 I .733378 14 I .326762 15 I .374805 16 1 .017460 17 I .301948 18 2.012575 19 1.075496 20 0.9607571 21 1.406861 22 0.4496151 23 0.9569434 24 0.8571227 25 0.7967462 26 0.5793790 27 0.8087967

Period 30 to 150 days-Female I 0.3345824

2 0.6877849 3 -4.904709 4 0.5192878 5 0.9129800 6 4. I58745 7 0.3353775 8 1.000000 9 0.6692672

I 0 5.381304 II 0.2382249 12 1.121507 13 I .260444 14 4.562044 15 0.5192262 16 0.672 1463 17 0.9061723 18 0.2 150089 19 0.3755687 20 0.3200205 21 0.1250107 22 0.1258703 23 0.3103277 24 0.2812058 25 0.2850906 26 0.2400680 27 0.3046021

-3.411043 - 1.729625 -0.6268220 -2.159010 -5.321379 -2.032534

0.503824M - I o.ooooooo

-4.113944 -3.224173

0.5950802 -2.692414 -2.961972 -0.6138876 -0.89727709

0.5637001 -0.6680786 -0.3104178

0.2846939 0.7194309 1.242679

2.745972 0.30936697 0.4068062 0.4099032 1.488634 1.088799

2.737251 1.162292

19.95230 2.094272 0.9480482Ccf - 1

- 12.93869 2.644802 0.0000000 1.017242

- 17.25050 2.966693

-0.353865OCl -I -1.151459

- 11.47677 1.993560 1.614645 0.6274447 3.329623 2.523021 2.730039 3.533822 3.462477 2.621933 2.632973 2.544010 2.794108 2.741060

Period 7 to 21 days-Male 1 1.252148 2 1.189981 3 1.408024 4 1.385104 5 1.570478 6 1.190323 7 0.7779304 8 1.000000 9 1.614717

10 1.517231 I1 0.7853105 12 1.514880 13 1.389654 14 0.9556826 15 1.22493 1 16 0.8159074 17 1.082527 18 1.440811 19 0.9400022 20 0.9030383 21 0.8565738 22 0.4022379 23 0.8260326 24 0.7705330 25 0.6832470 26 0.4965956 27 0.7070473

Period 30 to 150 days-Male I 0.4785004

2 0.8686408 3 - 36.22267 4 0.4669868 5 0.9638048 6 2.878700 7 0.3807234 8 1 .oooooo 9 0.7363 180

10 2.669153 11 0.2755636 12 1.041318 13 1.151628 14 14.93656 15 0.5077753 16 0.4626642 17 0.8923798 18 0.2206194 19 0.4551096 20 0.4135047 21 0.1540435 22 0.1762470 23 0.3737572 24 0.3069507 25 0.3063621 26 0.2635036 27 0.3121119

-0.9462687(tc - 1 ~0.7338792 -0.8566521 -0.7214800 -2.565686 -1.004189

1.201245 O.OOOOOOO

-2.873361 -2.163913

11156776 - 1.403756 - 1.623344

0.6203690 ~0.3780516

1.200589 0.7358161((1 - 1 0.8490395 0.7159595 0.9245322 2.164449 2.846860 0.7711513 0.7641637 0.9064366 1.827751 1.427596

2.311139 0.4846996

124.2163 2.312718

-0.88152411(1 - 1 -7.832763

2.521554 0.0000000 0.7648082

-6.673488 2.871692 0.2617329

-0.6978225 -46.22083

2.066280 2.331681 0.7110298 3.343905 2.278400 2.461784 3.494188 3.367538 2.421667 2.564643 2.493244 2.736052 2.749726


Rms - .8012 Mean % errar = 1.999 % - DATLI

---- MODEL

- DATA ---- MODEL

Rms = 1.051 Mean % error = 2.749 %

Figs. 13 and 14. For each of the eight ages of rat growth studied, the average configuration of the network predicted by the model (interrupted line) and of the biologic data (solid line), with minimal deviation from the model, are illustrated. In each figure the rms and mean % error are shown for female (Fig. 13) and male (Fig. 14) rats.

Table II. Analysis of significance (t test) of differences of nodal point location in actual data between allometric centered and growth network models for average male and average female rats (t.,, = 1.761 at 13 degrees of freedom)

t (days) x Y t Significance

Average male I 7

14 21 30 40 60 90

150

*at 1.374192 0.07931363 3.85985 1 Significant difference

0.8556015 0.2863149 5.968698 Significant difference 0.7064557 0.2104376 4.960651 Significant difference 1.123337 0.6771443 I .882505 Significant difference 0.747776 0.4305335 1.675199 No significant difference 1.279843 0.9835330 1.090090 No significant difference

0.9054461 0.3644784 3.753273 Significant difference 1.323984 0.3322279 6.463045 Significant difference

Average female rat

7 14 21 30 40 60 90

150

1.608377 0.1265871 4.155054 Significant difference 0.8372000 0.4379171 3.103917 Significant difference 0.74763 13 0.3102697 2.285602 Significant difference 1.389899 0.8693065 1.852044 Significant difference

0.9839546 0.6405582 1.688762 No significant difference 0.7359416 0.5086502 1.619100 No significant difference 0.9413089 0.5197280 2.757096 Significant difference 1.24045 1 0.6126223 2.713571 Significant difference

x and y are mean values summed over all nodes used. x is the mean distance between the centered model nodes and the data nodes in position of best fit. y is the mean distance between network model node and data node in position of best fit. These are analyses of significance of difference between centered and network models and actual data (nodal point location).

Volume 85 Number 4


Rms residual : .9804 Mean%errar - 2.87%

Fig. 15. An allometric centered model of the average female rat skull. The center, from which the points along the several radial lines show minimal deviation, is situated 14 mm below the midpoint of a segment extending between the spheno-occipital and the intersphenoidal synchondrosis. Note that both the rms and mean % errors are larger than shown in Fig. 13.

Group F, 102 months (n = 17); Group G, 165 months (n = 52) .22. 23 (See Moss and associates.g)

Having only cross-sectional human data, the time history of any individual cannot be presented, and the analysis of these data, given below, demonstrates only the general applicability of the present method to human cranial growth data.

The network constructed for the analysis of the human data is shown in Fig. 10. Each of the seven peripheral network segments was used sequentially as a pseudotime, and the allometric constants of each segment pair were determined. The networks were recon- structed with glabella-midfrontal as a fixed point and the segment glabella-bregma fixed in direction. The seven computed networks were compared graphically with the actual data, by freely rotating and translating the figures to a position of optimum fit.

RESULTS Rat data

Allometry. With segment 8, bregma-lambda, used as a pseudotime, it can be shown in a graph of log (segment 8) versus time (in days), for the average male rat, for example, that k is a function of time in the relationship

1 = I,ekt (In,1 = In&, + kt))

as the graph is not a straight line (Fig. 11). However,

when we consider pseudotime as the variable, the ratio of k’s (equation 4) of the two segments may be very nearly a constant. This is observed from graphs of log (segment x) versus log (segment 8); that is, the relationship of log (segment x) and log (segment 8) typi- cally has two nearly linear segments, with two distinct slopes (Fig. 12). The values of the constants (k, Mu and b, Cij) for each phase of the rat skull growth are given in Table I.

Comparison of raw data and network model. By the above methods, the data in Table I were used to con- struct male and female rat growth network models. The male and female rat networks were separately translated and rotated as rigid bodies to minimize distances between the raw data and the corresponding nodes of the constructed network (Figs. 13 and 14).

Comparison of allometric centered and growth network models. An allometric centered model of average female rat skull growth is shown in Fig. 15 and may be compared with Fig. 14. Table II shows that, using the t test with the average male and female rat data separately, the allometric network model was significantly more capable of closely modeling rat neurocranial growth than the allometric centered model at six of the eight ages studied (p < 0.5) and that the network model gave a closer fit, although not significantly so, at the two remaining ages. These differences are shown graphically in Figs. 16 and 17.


1DO

VARIANCE

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-----. NETWORK MODEL .8(

1.50 I

MEAN % ERROR

3 \

I \ 2 N fi , ‘.OO: v U I I : ,‘, : 0.50 ’ : : : \“I : i

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)-

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GROWTH STAGE

Figs. 16 and 17. The abilities of the growth network (interrupted line) and allometric centered (solid line) models to approximate model predicted and biologic data are compared in these figures for male (Fig. 16) and female (Fig. 17) rats separately. At each of the eight growth stages studied (7 to 150 days), the values of the mean % error and of the variance are illustrated.

The Wilcoxen two-sample rank-sum test gave closely similar results for the male rat data; with the females, in only four of the eight ages studied was the network model significantly better than the centered model but again the rank sums of the network model were consistently lower than those of the centered

model at all ages, whether significant or not (Tables III and IV).

Human data

Allometly. The allometric constants Mrj and C& were determined independently for each of the seven

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Table III. Wilcoxen two-sample rank-sum test used for Table IV. Wilcoxen two-sample rank-sum test used for comparison of the allometric centered and comparison of the allometric centered and growth network models growth network models

A. Anatomic point Centered

model Nenvork

model

I 1.309 mm 0.448 6 1.313 0.279

25 0.143 0.243 5 1.342 0.553 4 0.598 0.375

28 0.142 0.328 9 0.364 0.175 8 0.770 0.0798

B. Rank Value Centered Network

1 0.798 1 2 0.142 2 3 0.143 3 4 0.175 4 5 0.243 5 6 0.279 6 7 0.328 I 8 0.364 8 9 0.375 9

10 0.448 10 11 0.553 11 12 0.597 12 13 0.710 13 14 1.309 14 15 1.313 15 16 1.342 16

Rank = 83 Rank = 53 = Nr

In this table the values of the average female rat 21 days of age are displayed. In A, for eight anatomic points, the distance between the raw data and the model is calculated. In B the values from both models are arranged together in ascending order (as rank); then the ranks from the centered model are added, and is 83, similarly for the network model the sum of ranks is 53; this equals the term Nr.

pseudotime segments (Table V). In each instance two sets of constants were observed for the periods of (1) 7.5 months prenatal to 22 months and (2) 22 to 165 months. Graphs of log (segment pseudotime) versus log (segment n) showed close approximations to linearity .

Comparison of raw data and the network model. The best-fitting network model of the human neurocranial growth data, with point S included, is shown in Fig. 18.

Comparison of allometric network and allometric centered models. An allometric centered model for the same eight neurocranial points is illustrated in Fig. 19 and may be compared with Fig. 18. Tables VI and VII show that the network model was consistently able more closely to model the growth than was the centered

Average male rat 7 100

14 98 21 100 30 86 40 82 60 82 90 93

150 100

Average female rat 7 97

14 91 21 83 30 84 40 82 60 80 90 91

150 89

36 Significant difference 38 Significant difference 36 Significant difference 50 Significant difference 54 No significant difference 54 No significant difference 43 Significant difference 36 Significant difference

39 Significant difference 45 Significant difference 53 No significant difference 52 No significant difference 54 No significant difference 56 No significant difference 45 Significance difference 47 Significant difference

Following the determination of Nr values, as shown in Table III, the statistical significance between the two models is tested as shown in the present table. Here the value of W is determined from published tables (W,,Os = 51), and if Nr < W there is a significant difference between the models, where cr = the level of confidence. The table shows that the network model was significantly more able to model biologic data at ten of the sixteen ages studied, and more so for average male than for average female rats (six of eight and four of eight ages, respectively).

model; with the t test, these differences were significant (p < 0.5) at five of the six ages studied and, with the Wilcoxen two-sample rank-sum test, significantly different at four of six ages (Fig. 20).

DISCUSSION Terminology

It is helpful to introduce here certain terms of the engineering and mathematical sciences and to indicate their equivalence with alternate terms customarily seen in the craniofacial growth literature. A particular two- or three-dimensional graphic model, such as an allometric network, represents a roentgenographic-cepha- lometric figure constructed by connecting a series of landmark points by straight line segments. In engineering practice, if this network undergoes a change in time, it is said to be deforming; that is, a growth change in time is equivalent to a deformation. There are two types of growth deformation: (1) a shear growth deformation and (2) a normal growth deformation. The shear deformation is related to an angular change in time. A


Table V. Allometric constants for average human neural skull growth in the periods 7.5 months prenatal to 22 months postnatal and 22 months to 165 months @lone = k. Mu; Intercept = b, Crj)

Segment No. Slope Intercept &pen/ No. Slope Intercept

Period 7.5 months prenatal fo 22 months (CL-MF) Period 7.5 months to 22 months (MF-Br) 1 0.7209577 1.007290 1 I 000000 o.ooooooo 2 0.8883447 -0.1219021 2 I .232176 ~ I .566250 3 1 .oooooa 0.0000000 3 1.386978 - I. 396925 4 0.6125518 0.9782275 4 0.8496188 -0.4023954(~( - 1 5 0.9233939 0.7847297(~ ~ I 5 I .280795 - I .288337 6 0.8109000 -0.11398581((-I 6 1.124707 -1.412765 7 1.218003 - 1.775048 7 I .689492 -3.859520 8 0.7301831 0.5013461 8 I .012750 -0.7015767 9 I .042419 -0.3667128 9 I .445815 - I .905556

10 0.7920154 I .064145 I 0 I .09853 I 0.7895079icr - 1 II I .630565 - I .630354 II 2.261826 -3.659031 I2 I .753481 -3.661 I44 I2 2.432045 -6.474870 I3 1.527419 -2.107320 I3 2. I 18765 -4.320718

Period 22 to 165 months (CL-MF) Period 22 to 165 months (MF-Br) I I .777375 -2.950686 I I 000000 0.0000000 2 1.107662 ~ I .052939 2 0.7107497 0.6707789 3 I .000000 o.oooomo 3 0.6683658 1.240439 4 1.650188 ~ 3.446344 4 0.9836406 -0.5269247 5 0.9734612 0.2014312(1/ -I 5 0.6568290 1.228747 6 3.260590 -11.15609 6 2.024368 -5.497356 7 1.913075 -4.968827 7 I .305542 -2.206272 8 2.001729 -4.997432 8 I (504008 -2.843801 9 0.8824552 0.3054088 9 0.5889707 1.448019

10 0.9177713 0.7487492 IO 0.5718565 I .886678 II 0.7112908 I .453415 II 0.5074843 2.098380 12 0.7877246 0.468961 I I2 0.5531666 I .435331 I3 0.7979762 0.7507168 I3 0.5580740 I .642433

The values are calculated after the use of either (GL-MF) or of (MF- Br) as pseudotime as indicated in parentheses.

normal growth deformation is related to an extension in time. The term extension is equivalent to what the cephalometrician and physical anthropologist mean when they speak of an increase in time of a distance or of length of a measurement (for example, nasion-sella). A shortening is called a compression (opposite of extension). Where the orthodontist speaks of dimensional and angular growth changes in time, the engineer indicates these changes as extensions, compressions, and shear deformations.

Allometry

The existence of allometric growth relationships is an underlying assumption of the allometric network, as well as of the allometric centered model. In the network model this relationship is presumed to exist for the extension of any arbitrary pair of line segments, each connecting two anatomic points in the network; in the centered model, allometry is presumed to exist for the growth of any arbitrary pair of anatomic points along two radial lines having a common origin at the presumed allometric center.

The methods for testing allometry in the network and centered models differ in detail but are similar in principle. The methods applicable to the network model are given above; those for the centered model are published elsewhere.g

It is possible to test quantitatively the extent to which strict allometry exists in real growth data, using both the network and centered models. This is possible without regard to the inability of the centered model to permit prediction or to display growth transformations in a nonarbitrary manner. More precisely, both models can display normal growth deformations (that is, extensions in time), but only the network model can also display shear growth deformations (that is, angular changes in time) between network segments which originate at the center.

It was not possible to demonstrate strict allometry in either the rat or human cranial growth data; only a close approximation was possible. The data suggest two factors related to this phenomenon: (1) that the specific growth rates of each cranial network segment seem to change constantly in time, and (2) that this fact

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Rms = .4033 Mean % error = .6663 %

Fig. 18. For each of six ages of human neurocranial growth studied, the configurations of the network predicted by the model (solid line) and of the biologic data (interrupted line) with minimal deviation from the model are illustrated. The pseudotime us@ (GL-MF) was the segment for which the growth of the other segments studied was most nearly allometric. The rms and mean % error are shown.

is then reflected in the relative growth rate of any arbitrary pair of such segments. Briefly, cranial skeletal structure considered as an allometric growth network closely approximates, but does not fully exhibit, the attributes of the actual growth; there is some degree of stochastic (random) behavior exhibited in the growth of the mammalian cranial skeletal framework.

The presence of two distinct phases of allometric growth is indicated by the two sets of allometric constants, each of which is closely approximated by sepa- rate straight lines. To derive a continuous model, these lines are extended until they intersect. The precise mathematical and biologic properties near such inter- sections (as in Fig. 9) are not yet fully described. The question of changes in allometric constants is an interesting one, a problem as old as the beginning of the study of relative growth by Huxely.12 In 1947 Huxley and Reeve25 considered this point again. They state that although “simple allometry ” (that is, an allometric relationship indicated by a single line segment on a double log graph) is common, deviations are often observed. While Gould25 studied only “simple allometry,” Huxley and Reeve noted many instances in which it was difficult to determine, precisely, if there existed a region of definite curvilinear trend on such graphs or whether there were, indeed, certain “critical points ” where abrupt alterations of allometric constants occurred and where shape changed suddenly. The relative growth literature is filled with many examples of such critical points. 26 We have previously referred to

Rms residual q .6350 mean % ermr = 1.557%

Fig. 19. The allometric centered model of the same human neural skull growth data shown in Fig. 18, with point S included. (This point is not located at the allometric center depicted.) In this figure with minimal deviation of anatomic point locations away from the radial lines having a common origin at the indicated location of the center, both the rms and mean % error values are greater than those shown in Fig. 18.

Table VI. A Wilcoxen two-sample rank-sum test comparing the abilities of the allometric centered and growth network models to depict human neurocranial growth

French data Rank Wilcoxon two-sample rank-sum test

Centered Network Age (months) model model wo.05 = 51

7.5 prenatal 87 49 Significant difference 3 17 59 No significant difference

22 98 38 Significant difference 49 87 49 Significant difference

102 83 53 No significant difference 165 95 41 Significant difference

See tables III and IV for further description

such regions as “interphases ” and suggested that, in the case of long-bone growth, an interphase was associated with the formation of the definitive epiphyseal growth plates and, in the case of neurocranial growth, that the interphase was associated with the differential timing of expansive growth of different portions of the brain .27* 28

The shape of the curve (Fig. 9) near such an intersection is unclear. It is possible that the two relatively straight line segments (Fig. 9) are connected by a smooth, continuous curved line, indicating a continuous change of growth rate, or a singularity could exist, with the sharp intersection indicating a sudden change.

330 Moss et al

MEAN % ERROR -

Am. .I. Orthod. April 1984

- CENTERED MODEL

------ NETWORK MODEL

L

1 2 3 4 5 6 AGE STAGE AGE STAGE

Fig. 20. As in Figs. 16 and 17, the comparison between the centered model (solid line) and the network model (interrupted line) in terms of mean % error and variance is shown for the six ages studied with the cross-sectional human neuronal data. The ability of the network model to model the actual growth more closely than does the allometric centered model generally is significant.

Table VII. A t test of the significance of the differences between an allometric centered and growth network model to depict human neurocranial growth

Age (months)

7.5 mpn 3m

22 m 49 m

102 m 165 m

Mean Mean cenlered network

model model

I .343215 0.703273 I .425398 0.95023 19 1.120086 0.2517929

0.7363097 0.4156401 0.9817138 0.6732088 1.189326 0.2538322

1 1 o,05 = I. 761

2.244976 Significant difference 1.233157 No significant difference 3.788953 Significant difference 2.307989 Significant difference I .857267 Significant difference

2.938548 Significant difference

At each age studied, for each model separately, the mean distances between the data nodes and location of the model nodes, with eight points, in- cluding sella point S, were tested at positions of best fit.

Analysis of our present data strongly suggests that the growth behavior of the cranial anatomic points studied is closely allometric as shown, for example, in our previous article in this JOURNAL on the allometric centered mode. Given this close allometry, the present manuscript goes on to show that, using one line segment as a pseudotime, for the reasons stated in the present manuscript, all other line segments grow in a closely allometric manner relative to the former.

Given such close approximations to allometry, it is expected that a double log plot of any such pairs will be best fitted by a straight line segment and not a curvilinear one. Given this, it is reasonable for us to have created a computer-assisted program that would test for such linearity and quantitatively determine the values

of the allometric constants Cij and Mij which, of course, is precisely what we did and in all cases we found two sets of constants.30

The observation that data were consistently fitted by fwo line segments, and, significantly, that this intersection occurred at identical “ages” for each pair of line segments studied seems to reflect some sort of reality. This finding, however, does not give information about the structure at the graphic intersection of two such line segments.

While no available statistical method can unam- biguously determine whether a sharp “break” or curvilinear segment exists at such intersection regions, consideration of the topics discussed above suggests the empirical propriety of our present methodology and

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lends strength to the suggestion that such changes in allometry do occur and that this is so without regard to the precise structure of the interphase region.

Modeling of craniofacial growth

The allometric network model has been shown to be superior to the allometric centered model in the analysis of craniofacial skeletal kinematic growth behavior in three ways: (1) The allometric network model permits growth prediction, when allometric constants are known; (2) the network model has significantly smaller errors than the centered model; and (3) the network model is capable of displaying growth kinematics of both the neural and facial skulls while marked transformations such as relative rotation of two sets of cranial anatomic points occur in time.

It is important to note that the allometric network model is not an alternative method of roentgenographic cephalometry; for example, the segments of maximal superimposition in representations of best fit between actual and theoretical rat and human cranial growtrr data (Figs. 12, 13, and 17) indicate only those regions with respect to which growth is most nearly allometric. Note also that the predictability of the allometric network model, while real, is related to these same super- impositions of best data fit and should not be under- stood as referring to any possibly underlying biolog- ically dynamic process(es). Segment bregma-lambda was used here as the pseudotime since it had been shown previously that, with this pseudotime, minimal root mean squares and mean percent errors were found.” In the present analysis of human cranial data, the segment glabella-lambda similarly gave minimal error values. It is noted that in both the present figures of the centered and network models the point S (sella) is included; these had been omitted previously.”

Despite the evident advantages of the allometric network model relative to an allometric centered model, the network model does not exhibit all possible attributes of cranial skeletal growth kinematics. For example, the network model displays differing shapes, in time, but not shape change, per se, in the sense of Bookstein.‘” The third of our suggested methods,6 that of a finite element model, is better able to provide a detailed description of cranial skeletal growth change. This model currently is being studied by us.

REFERENCES 1. Bookstein FL: The measurement of biological shape and shape-

change. Lecture Notes in Biomathematics, No. 24, Berlin, 1978, Springer-Verlag.

2. Moyers RE, Bookstein FL: The inappropriateness of con- ventional cephalometrics. AM J ORTHOD 75: 599-617, 1979.

3. Moss ML, Vilmann H, Dasgupta G, Skalak R: Space, time, and

space-time in craniofacial growth. AM J ORTHOD 77: 591-612, 1980.

4. Vilmann H, Moss ML: Studies on orthocephalization. 7. Behav- ior of the rat cranial frame in the period between 1 day before birth and 14 days after birth. Acta Anat 109: 157-160, 1981.

5. Vilmann H, Moss ML, Skalak R, Vilmann V: Space-time presentations of the skull of the bivalved mollusk Car&m edule. AM J ORTHOD 80: 417-428, 1981.

6. Skalak R, Dasgupta G, Moss M, Otten F, Dullemeijer P, Vil- mann H: Analytical description of growth. J Theor Biol 94: 555.577, 1982.

7. Ricketts R: New perspectives on orientation and their benefits to clinical orthodontics. Part 2. Angle Orthod 46: 26-36, 1976.

8. Todd JT, Mark LS: Issues related to the prediction of craniofacial growth. AM J ORTHOD 79: 63-80, 1981.

9. Moss ML, Skalak R, Shinozuka M, Pate1 H, Moss-Salentijn L, Vilmann H, Mehta P: Statistical testing of an allometric centered model of craniofacial growth. AM J ORTHOD 83: 5-18, 1983.

10. Moss ML, Vilmann H: Studies on orthocephalization of the rat head. 1. A model system for the study of adjustive cranial growth processes. Gegenbaurs Morph01 Jabrb 124: 559-579, 1978.

11. Vilmann H, Moss ML: Studies in orthocephalization. 2. Flexion of the rat head in the period between 14 and 60 days. Gegen- baurs Morph01 Jabrb 125: 572-582, 1979.

12. Huxley JS: Problems of relative growth. London, 1932. Met- huen & Co.

13. Timoshenko SP, Young D: Theory of structures, ed. 2, New York, 1965, McGraw-Hill Book Company.

14. DiMaggio FL: Statistical indeterminacy and stability of structures. Proc Am Sot Civil Engineers 89-ST3: 63-75, 1963.

15. Calladine CR: Buckminster Fullers “Tensegrity” structures and Clark Maxwell’s rules for the construction of stiff frames. Int J Solids Structures 14: 161-172, 1978.

16. Maxwell JC: On the calculation of the equilibrium and stiffness of frames. Phil Mag 27: 294-299, 1864. (Paper XXVI in Col- lected Papers, London, 1890, Cambridge University Press.)

17. Cauchy AL: Sur Les polygones et les polyedres. J. ecole polytech 14 Cahier 9: 87-99, 1813. Reprinted in: Oeuvres completes

d’Augustin Cauchy (Ser 2) 1: 26-38, 1905. 18. Euler L: Elementa doctrinae solidorium. Novi commentari.

Acad Sci Petropolitanae 4: 109.140, 1752-3. Reprinted in Opera Omnia Leonard Euler (Ser 1) 26: 71-93, 1953. Venditioni Ex- ponunt, Ore11 Fussli, Turici [A. Speisli, Ed.]).

19. Casper DL, Klug A: Physical processes in the construction of regular viruses. Cold Spring Harbor Symp Quant Bio127: l-24, 1962.

20. Badoux DM: Framed structures in the mammalian skull. Acta Morph01 Neerl Stand 6: 239-250, 1966.

21. Badoux DM: Cremona diagrams of framed structures in the skull of Canis familiaris and Sus scrofa scrofa. Koninkl Akad Wetensch Ser C 71: 229-244, 1968.

22. Cousin RP: Etude en projection sagittale de cranes d’enfants orientes dans les axes vestibulaires, Thesis, Fat. Med. Paris, 1969.

23. Obled J: Considerationes su la notion de vitesse dans le de- placement ontegenetique des points crania-faciaux en orientation vestibulaire, Thesis, University De Droit el la Santa le Lille, 1979.

24. Reeve ECR, Huxley JS: Some problems in the study of relative growth In LeGros Clark WE, Medewar P (editors): Essays on growth and form, Oxford, 1947, Clarendon Press, pp. 121-156.


25. Gould S: Allometry and size in ontongeny and phylogeny. Biol Rev 41: 587640, 1966.

26. Brody S: Bioengineering and growth, New York, 1945, Reinhold Publishing Corp.

27. Moss ML, Noback CR, Robertson GG: Critical developmental horizons in human fetal long bones: correlated quantitative and histological criteria. Am J Anat m 155-175, 1955.

28. Moss ML, Noback CR: Differential growth of the human brain. J Comp Neural 105: 539-552, 1956.

29. Bookstein FL: On the cephalometrics of skeletal change. AM J

ORTHOD 82: 177-198, 1982.

30. Goedbloed JF: The embryonic and postnatal growth of rat and mouse. I. The embryonic and early postnatal growth of the whole embryo. A model exponential growth and sudden changes in growth rate. Acta Anat 82: 305336, 1972.

Reprint requests to: Prof. Melvin L. Moss Department of Anatomy Columbia University 630 West 168th St. New York, NY 10032

An allometric network model of craniofacial growth

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