The proposed project will perform a differential geometric analysis of the nasal surface of a human being with a view to doing pre-operative analysis to assist the plastic surgeon in formulating a procedural strategy. We take rhinoplasty (plastic surgery of the nose) to be a microcosm of many typical plastic surgery problems. For the nose has an interesting and complex three-dimensional anatomy, and it also plays a critical role in the aesthetics of the face. The nose is of interest as a differential geometric object. Its exterior dermis is a hypersurface in space, and the nature of the nose is determined (i) by its curvatures and (ii) by its deviations as measured by quasiconformal mappings from certain standard geometric forms. Clinical experience indicates that reforming a convex surface into another convex surface is one of the easiest procedures for a plastic surgeon to perform. The most difficult surfaces to handle are surfaces of negative curvature; of course the typical nose has such surfaces on either side, near the nostrils. Preliminary discussions indicate that quasiconformal mappings may be one way to measure deviation from convexity, or distance from a negatively curved dermal surface to a positively curved one. The final goal of the project create software that will assist the plastic surgeon in developing medical procedures. The software that we propose to create will take a respected but largely non-scientific part of medicine and endeavor to make it analytical. The physician will input either numerical data, taken from measuring the patient's organ, or will scan in data with a suitable digitizing device. The software will solve an optimization problem and make a procedural recommendation. Of course the final decisions, and the interpretation, will be made by the plastic surgeon.
Plastic surgery in general, and aesthetic rhinoplasty in particular, is a rapidly developing area of medical technique and treatment. Laser surgery, microstitching, tissue transplants, man-made implants, and many other devices are now important (and quite recently developed) tools in the field. Definitive studies of aesthetic rhinoplasty concentrate on techniques that center around "aesthetics", "perspective", and "balance". In short, technique in aesthetic rhinoplasty, even at a very sophisticated level, is a sort of intuition informed by experience and by an artistic sense. To the extent that technology and mathematics are used in the process, they are used only in the pre-diagnosis stages, and only for descriptive purposes. Our goal in this project is to use a mathematical analysis of the nasal surface (and, ultimately, of other organs as well) to inform the plastic surgeon as he plans a surgical procedure. The specific short-term goal is to create a mathematical model of the nose, formulated in the language of the differential geometry of surfaces. The broader, long-term goal is to formulate questions of plastic surgery procedure as optimization problems for surfaces imbedded in space. In the final analysis, the questions that the plastic surgeon considers are surface design problems. And surface design has, in the last ten years, entered a new era of sophistication. Our goal is to examine the hitherto unexplored subject of local surface design problems in plastic surgery using techniques of analysis and differential geometry. We will create software for use by the plastic surgeon in assessing analytic/geometric data of the nasal and facial surfaces and formulating an operative procedure.
