### Introduction

A computational scheme has been developed to describe and analyze the human face. The study of an orthomorphic sampling has revealed an architectural balance of the anatomical components. The maxillo-facial model by Treil has now become a standard reference for scientists and medical practitioners.

The c2000 cepha software mathematically deduces the location of anatomical landmarks from CT scan data. Subsequently, three-dimensional images of the maxillo-facial anatomical structures are generated for human faces. The principal coordinate directions are determined on the basis of selected anatomical locations and teeth positions. All of these landmarks are situated on the trigeminal axes described by Moss [1], and they form an invariant frame of reference in the study of the maxillo-facial growth. Furthermore, via the calculation of these principal axes, the model is organized into a hierarchy of anatomical elements--for example, the teeth, half-arches, maxillar and mandibular arches separately, both arches together, all elements as a group, and the maxillo-facial frame. Each element's position and axes of inertia define a three-dimensional reference landmark. This allows for the calculation of relative spatial orientations of anatomical elements. Biologically significant components of the growth tensor with respect to invariant frames of reference (determined on the basis of principal moments of inertia) are calculated by Mathematica packages.

Individual research pursued by the authors motivated innovative computing in medical and engineering sciences, with symbolic computations aided by graphics and sound. In this presentation, the following items will be developed using Mathematica:

• Consistency of the geometrical model for lower-order interpolants
• Morphometric considerations in bioengineering computations
• Formulation of finite/boundary elements with specific emphasis on radiology and three-dimensional reconstruction

In the oral presentation, collaborating efforts in imaging and mathematical representation of anatomical elements were emphasized. Mathematica constructs for handling CT data were elaborated; especially the deeper biological implications were derived from the formulation. Calculation of true (not projected) parameters and application of symbolic computer mathematics in radiology were illustrated using the Mathematica graphics environment. Anatomical concepts via tensorial invariants are related to biological objects. Consequently, the axes of a single tooth, the coronoradicular axis, the bucalingual axis and the mesio-distal axis were identified. Statistical calculations of the principal axes of inertia were presented using color graphics. Mathematica definitions of geometrical location and local and global axes of symmetry were employed to model ontogenic and philogenic growth.

Representation of the growth tensor according to the computer mathematics formulation confirms the Function Theory of Biology proposed by Moss [1]. In the future, Maeder's parallel computing strategy [2] will be implemented to transform symbolic objects of biological significance, and the resulting (very large volume of) data (120GB) will be handled in external C programs (such as MathCode [3]).

NOTE: Some inputs in this article depend on an extensive set of programs contained in several large notebooks. In the interest of brevity, we have omitted these programs, and the affected inputs appear in comments. Interested readers are encouraged to contact Gautam Dasgupta (gd18@columbia.edu).
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