Providence, RI---Cranio-maxillofacial surgery is a medical specialty focusing on facial and skull reconstruction. This surgery can help patients with such disorders as cleft palate, malformations of the upper or lower jaw, and problems with the facial skeleton due to injury. Intensive pre-operative planning is needed not only to ensure that the medical purposes of the surgery are achieved, but also to give patients a sense of what their faces will look like after the surgery is performed.
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
Contact: Dr. Peter Deuflhard
American Mathematical Society