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Mathematics And Beauty? You Bet!

Cranio-maxillofacial surgery is a division that deals with the reconstruction of the face and related skull areas after severe injuries// or the occurrence of birth defects like cleft palate. Since the procedure itself is rigorous, extensive pre-operative preparation is required. This will not only ensure a smooth operation, but will also give patients an idea of how their face will look like in future.

In their article "Mathematics in Facial Surgery," Peter Deuflhard, Martin Weiser, and Stefan Zachow (of the Konrad Zuse Zentrum (ZIB), Berlin) describe the mathematical techniques they have used to assist cranio-maxillofacial surgeons to predict the outcomes of surgery. These techniques have proven to be quite successful in producing predictions that end up matching well the post-operative outcomes.

The first step in the planning paradigm for such surgery is to use medical imaging data of the patient to construct a 3-dimensional computer model, called the "virtual patient". The second step, which is the one the article focuses on, uses the data to create a "virtual lab" in which various operative strategies can be tested. The last step is to play back to the patient the outcomes of the various strategies.

The second step in the paradigm requires modeling and solving partial differential equations (PDEs), which are equations that represent changing physical systems. One must identify which PDEs are appropriate for biomechanical modeling of soft facial tissue and bone. Standard methods for handling the equations need to be adapted for this particular application. One must also formulate ways to represent the interface between tissue and bone, as well as their interactions. Generally such PDEs cannot be solved exactly in closed form, so mathematics enters the picture once again to provide numerical techniques for producing approximate solutions.

With the "virtual patient" data as input, one can use the approximate solutions to
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