Differential Geometric Analysis, Geometric Visualization, and Aesthetic Rhinoplasty

拟议的计划将对人的鼻表面进行微分几何分析，以便在手术前进行分析，以协助整形外科医生制定手术策略。我们把鼻整形术（鼻子整形手术）看作是许多典型整形手术问题的一个缩影。因为鼻子有一个有趣而复杂的三维解剖结构，它在面部美学中也起着至关重要的作用。鼻子是一个有趣的微分几何对象。它的外部真皮是空间中的超表面，鼻子的性质是由(i)它的曲率和（ii）它的偏差（由某些标准几何形式的拟共形映射测量）决定的。临床经验表明，对整形外科医生来说，将一个凸面改造成另一个凸面是最容易的手术之一。最难处理的曲面是负曲率曲面；当然，典型的鼻子在靠近鼻孔的两侧都有这样的表面。初步的讨论表明，拟共形映射可能是一种测量偏离凸性的方法，或者从负弯曲的皮肤表面到正弯曲的皮肤表面的距离。该项目的最终目标是开发一种软件，帮助整形外科医生制定医疗程序。我们打算创建的软件将会采用医学中受人尊敬但很大程度上非科学的部分，并努力使其具有分析性。医生将输入从测量患者器官中获得的数字数据，或者用合适的数字化设备扫描数据。该软件将解决一个优化问题，并进行程序推荐。当然，最终的决定和解释将由整形外科医生做出。一般来说，整形外科，特别是鼻整形术，是一个快速发展的医疗技术和治疗领域。激光手术、显微缝合、组织移植、人造植入物和许多其他设备现在都是该领域重要的（也是最近才开发出来的）工具。美学鼻整形术的权威研究集中在围绕“美学”、“透视”和“平衡”的技术上。简而言之，美学隆鼻术的技术，即使是在非常复杂的水平上，也是一种由经验和艺术感决定的直觉。在某种程度上，技术和数学在这个过程中被使用，它们只在诊断前阶段被使用，并且只用于描述目的。我们在这个项目中的目标是使用鼻表面的数学分析（最终，也包括其他器官）来告知整形外科医生他计划的手术过程。具体的短期目标是创建一个鼻子的数学模型，用曲面微分几何的语言来表述。更广泛的、长期的目标是将整形手术过程的问题表述为嵌入空间表面的优化问题。归根结底，整形外科医生考虑的问题是表面设计问题。在过去的十年里，表面设计进入了一个复杂的新时代。我们的目标是利用分析和微分几何技术来研究迄今为止尚未探索的整形外科局部表面设计问题。我们将开发软件，供整形外科医生用于评估鼻腔和面部表面的分析/几何数据，并制定手术程序。

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.