Care to compute your facial surgery results?
Yes, there are mathematical tools for predicting facial surgery results.
A plastic surgery procedure called Cranio-Maxillofacial Surgery is a medical specialty focusing on facial and skull reconstruction -- specifically patients with such disorders as cleft palate, malformations of the upper or lower jaw, and problems with the facial skeleton due to injury.
To ensure that the medical purposes of the surgery are achieved intensive pre-operative planning is needed. This is where the ‘math’ comes in – giving the patients a sense of what their faces will look like after the surgery is performed.
The article "Mathematics in Facial Surgery," by Peter Deuflhard, Martin Weiser, and Stefan Zachow (of the Konrad Zuse Zentrum (ZIB), Berlin) devised the mathematical techniques to assist cranio-maxillofacial surgeons in predicting the outcomes of surgery. Quite successfully, these techniques have produced predictions that ended up matching well the post-operative outcomes.
Important phases in the planning paradigm for such surgery are: (1) use of medical imaging data of the patient to construct a 3-dimensional computer model, called the "virtual patient", and (2) use of data to create a "virtual lab" in which various operative strategies can be tested – required here are modeling and solving partial differential equations (PDEs), which are equations that represent changing physical systems. The last step is to play back to the patient the outcomes of the various strategies.
The authors admitted that "biomechanical tissue modeling turns out to be a tough problem," apparently however, they cannot get any more precise than this: a mean prediction error of between 1 and 1.5mm for the soft tissue, which the authors write, "seems to be a fully acceptable result."
Surgeons be glad – the purveyors suggest that their computer assisted planning permits an improved preparation before the actual operation.
Only, there is a whole lot of computing to do, or so.

