An allometric network model of craniofacial growth

A New Method for Landmark-Based Studies of the Dynamic Stability of Growth, with Implications for Evolutionary Analyses

A matrix manipulation new to the quantitative study of develomental stability reveals unexpected morphometric patterns in a classic data set of landmark-based calvarial growth. There are implications for evolutionary studies. Among organismal biology’s fundamental postulates is the assumption that most aspects of any higher animal’s growth trajectories are dynamically stable, resilient against the types of small but functionally pertinent transient perturbations that may have originated in genotype, morphogenesis, or ecophenotypy. We need an operationalization of this axiom for landmark data sets arising from longitudinal data designs. The present paper introduces a multivariate approach toward that goal: a method for identification and interpretation of patterns of dynamical stability in longitudinally collected landmark data. The new method is based in an application of eigenanalysis unfamiliar to most organismal biologists: analysis of a covariance matrix of Boas coordinates (Procrustes coordinates without the size standardization) against their changes over time. These eigenanalyses may yield complex eigenvalues and eigenvectors (terms involving $$i=\sqrt{-1}$$ i = - 1 ); the paper carefully explains how these are to be scattered, gridded, and interpreted by their real and imaginary canonical vectors. For the Vilmann neurocranial octagons, the classic morphometric data set used as the running example here, there result new empirical findings that offer a pattern analysis of the ways perturbations of growth are attenuated or otherwise modified over the course of developmental time. The main finding, dominance of a generalized version of dynamical stability (negative autoregressions, as announced by the negative real parts of their eigenvalues, often combined with shearing and rotation in a helpful canonical plane), is surprising in its strength and consistency. A closing discussion explores some implications of this novel pattern analysis of growth regulation. It differs in many respects from the usual way covariance matrices are wielded in geometric morphometrics, differences relevant to a variety of study designs for comparisons of development across species.

Centric Allometry: Studying Growth Using Landmark Data

The geometric morphometric (GMM) construction of Procrustes shape coordinates from a data set of homologous landmark configurations puts exact algebraic constraints on position, orientation, and geometric scale. While position as digitized is not ordinarily a biologically meaningful quantity, and orientation is relevant mainly when some organismal function interacts with a Cartesian positional gradient such as horizontality, size per se is a crucially important biometric concept, especially in contexts like growth, biomechanics, or bioenergetics. “Normalizing” or “standardizing” size (usually by dividing the square root of the summed squared distances from the centroid out of all the Cartesian coordinates specimen by specimen), while associated with the elegant symmetries of the Mardia–Dryden distribution in shape space, nevertheless can substantially impeach the validity of any organismal inferences that ensue. This paper adapts two variants of standard morphometric least-squares, principal components and uniform strains, to circumvent size standardization while still accommodating an analytic toolkit for studies of differential growth that supports landmark-by-landmark graphics and thin-plate splines. Standardization of position and orientation but not size yields the coordinates Franz Boas first discussed in 1905. In studies of growth, a first principal component of these coordinates often appears to involve most landmarks shifting almost directly away from their centroid, hence the proposed model’s name, “centric allometry.” There is also a joint standardization of shear and dilation resulting in a variant of standard GMM’s “nonaffine shape coordinates” where scale information is subsumed in the affine term. Studies of growth allometry should go better in the Boas system than in the Procrustes shape space that is the current conventional workbench for GMM analyses. I demonstrate two examples of this revised approach (one developmental, one phylogenetic) that retrieve all the findings of a conventional shape-space-based approach while focusing much more closely on the phenomenon of allometric growth per se. A three-part Appendix provides an overview of the algebra, highlighting both similarities to the Procrustes approach and contrasts with it.

Automatic computerized radiographic identification of cephalometric landmarks

Computerized cephalometric analysis currently requires manual identification of landmark locations. This process is time-consuming and limited in accuracy. The purpose of this study was to develop and test a novel method for automatic computer identification of cephalometric landmarks. Spatial spectroscopy (SS) is a computerized method that identifies image structure on the basis of a convolution of the image with a set of filters followed by a decision method using statistical pattern recognition techniques. By this method, characteristic features are used to recognize anatomic structures. This study compared manual identification on a computer monitor and the SS automatic method for landmark identification on minimum resolution images (0.16 cm2 per pixel). Minimum resolution (defined as the lowest resolution at which a cephalometric structure could be identified) was used to reduce computational time and memory requirements during this development stage of the SS method. Fifteen landmarks were selected on a set of 14 test images. The results showed no statistical difference (p > 0.05) in mean landmark identification errors between manual identification on the computer display and automatic identification using SS. We conclude that SS shows potential for the automatic detection of landmarks, which is an important step in the development of a completely automatic cephalometric analysis.

Reconstruction of three-dimensional anatomical landmark coordinates using video-based stereophotogrammetry

A method is described for reconstructing the 3-dimensional coordinates of anatomical landmarks from multiple video images. This method consists of 2 stages: (1) separate reconstructions of landmark coordinates visible in dorsal and ventral orientations using Direct Linear Transformation (Abdel-Aziz & Karara, 1972), and (2) least-squares alignment of the partial reconstructions using landmarks present in both. The repeatability of the overall reconstructions and of individual landmarks were assessed by performing multiple independent reconstructions of landmark coordinates for 2 camd specimens. Results indicate that the repeatability of interlandmark distances based on stereophotogrammetric reconstructions from standard-resolution video images is comparable to that of caliper measurement, and slightly superior to that of electronic 3-dimensional digitisation. At least 4 images should be available per landmark for optimal repeatability, but additional images allow only a slight increase in repeatability.

Studies on orthocephalization: growth behavior of the rat skull in the period 13-49 days as described by the finite element method

Rat cranial skeletal growth was studied, using a cross-sectional data set, for the period 13-49 days by the application of the concepts of continuum mechanics and the numerical techniques of the finite element method (FEM). In contrast to the methods of conventional craniometry (CM) and roentgenographic cephalometry (RCM) the FEM permits fine scale, reference frame invariant descriptions and analysis of growth behavior. This advantage was demonstrated by a numerical example of the use of FEM. The skull was discretized into a number of two-dimensional, triangular elements, whose enclosed areas corresponded closely to both specific skeletal structures and to related functional matrices. Since it was assumed presently that the growth behavior of all of the points enclosed within a given element was similar, the application of the functional matrix hypothesis permitted an integrated description of the growth of the skeletal structure and functional matrix related to each element. The principal locus of rotation of the facial skull, relative to the cranial base, is the inferior frontoethmoidal articulation, a motion that includes a rigid body rotation. Other active and passive skeletal and visceral growth events associated with orthocephalization were located and described. Finally it was shown that the morphogenetically important growth behavior of other portions of the rat head were not directly involved in orthocephalization.

Finite element method modeling of craniofacial growth

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.