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The application of the concepts of continuum mechanics and of the numerical techniques of the finite element method permits the development of a new and potentially clinically useful method of describing craniofacial skeletal growth. This new method differs from those associated with customary roentgenographic cephalometry in that its descriptions and analyses are invariant; that is, they are independent of any method of registration and superimposition. Such invariance avoids the principal geometric constraint explicit in all analytical methods associated with conventional roentgenographic cephalometry. The conceptual and mathematical bases of the finite element method (FEM) are presented and illustrated by the numerical and graphic descriptions of the two-dimensional growth of the rat skull, for which two sets of longitudinal growth data are used. In practice, the FEM permits analysis of the skull at a scale significantly finer than previously possible, by considering cranial structure as consisting of a relatively large number of contiguous finite elements. For each such element, independently, it is then possible to describe and depict both the magnitude and the direction of temporal size and shape changes occurring in that element relative to itself at some initial time. It is emphasized that such descriptions are completely independent of any local reference frame.

An allometric centered model of craniofacial growth was tested by several computer-assisted statistical methods on the pure longitudinal growth data of twenty-four close-bred female rats and on cross-sectional human cranial growth data. The study demonstrated that such a model was heuristic and, being incapable of exact definition, was deemed inappropriate for further use in modeling of craniofacial skeletal growth. The necessity for vigorous testing of any hypothesis concerning the modeling of craniofacial growth is stressed.

Summary The growth of the human mandible in a series of orthodontically normal children was examined by methods designed to demonstrate both the vectors of active and passive growth and the logarithmic attributes of such growth. Both methods are described and illustrated. The individual differences in the vectorial properties of active and passive growth during relatively short periods of time suggest that is is hazardous to predict such growth increments within clinically useful tolerances. Further, all mandibular growth is shown to be logarithmic. As such, mandibular growth shares three logarithmic attributes: Its growth is allometric (expressed by the relative growth equation); growth of all human mandibles is capable of graphic presentation as a unitary logarithmic spiral; and its growth is gnomonic, that is, it changes its size while retaining the same shape. A synthesis of mandibular growth is presented in terms of the interrelationship between active and passive growth processes and their logarithmic attributes.

A brief review of the fundamental postulates of the method of functional cranial analysis is given, with particular emphasis on the definition of the functional matrix. Two basic types of such matrices—periosteal and capsular—are described. Periosteal matrices include muscles and teeth, while the capsular matrices are conceived of as volumes enclosed and protected by both the neurocranial and the orofacial capsules. In the neural skull the capsular matrix is the neural mass. In the facial skull this matrix is the functioning space of the oronasopharyngeal cavity. We note the following differences between the activity of periosteal and capsular functional matrices. Periosteal matrices act upon skeletal units in a direct fashion by the processes of osseous deposition and resorption (or of cartilaginous or fibrous tissue multiplication. Their net effect is to alter the form (size and shape) of their respective skeletal units. Capsular matrices act upon functional cranial components as a whole in a secondary and indirect manner. They do so by altering the volume of the capsules within which the functional cranial components are embedded. The effect of such growth changes is to cause a passive translation of these cranial components in space. Cranial growth is a combination of the morphogenetically primary activity of both types of matrix. Growth is accomplished by both spatial translation and charges in form.

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.

The position of the three neural foramina describing the course of the inferior alveolar nerves was studied, with the aid of a previously determined logarithmic spiral, in cases of anterior open-bite and deep overbite. In anterior open-bite, the position of the foramen ovale was shown to be characteristically lower on this spiral than it is in any other condition studied; indeed, it is a diagnostic characteristic. The significance of this finding in the discussion of the problems of mandibular size and shape in these conditions was discussed. The etiologic role of periosteal and capsular functional matrices in the development of anterior open-bite was analyzed, and capsular matrices, with their related processes of passive translative growth, were held to be developmentally primary. Finally, an associated difference in the spatial location of the oral space in anterior open-bite was discussed. In essence, anterior open-bite was found to be a condition associated with specific spatial and developmental abnormalities of the oral functioning space, acting as a capsular matrix. All observed skeletal tissue changes are but secondary, compensatory, and mechanically obligatory responses to the morphogenetically and temporally prior alterations of this capsular matrix. Clinicians should carefully differentiate between the causes of an observed deformation of the orofacial skeleton and its etiologic factors. In anterior open-bite, they are completely different.

The morphogenetic role of two types of functional matrix-periosteal and capsular—in craniofacial growth is examined. The term growth is defined inclusively to include the changes in size and shape as well as changes in spatial position in time. The periosteal matrices are responsible for transformative growth, the changes in size and shape. The capsular matrices are responsible for translative growth, the changes in position. The role of oronasopharyngeal functioning spaces as competent, primary, morphogenetic agencies is detailed. The volumetric expansion of these capsular functional matrices is capable of accurate quantitative description.