The first step in the planning paradigm for such surgery is to use medical imaging data of the patient to construct a three-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 modelling and solving partial differential equations (PDEs), which are equations that represent changing physical systems. One must identify which PDEs are appropriate for biomechanical modelling 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 generate an individualized model for that particular patient. The surgeons can then use the model as a "virtual lab" to predict the effects of surgical procedures and options, and patients can get a picture of approximately how they will look after the surgery.

The article by Peter Deuflhard et al. states that qualitative comparisons between the outcomes predicted by the model, and the actual surgical outcomes, have been surprisingly good. The authors have also made quantitative comparisons, by creating a post-operative model of the patient and comparing it quantitatively to the predicted outcome. They found a mean prediction error of between 1 and 1,5mm for the soft tissue, which they write "seems to be a fully acceptable result".

"Even though biomechanical tissue modelling turns out to be a tough problem, we are already rather successful in predicting post-operative appearance from pre-operative patient data", the authors write. "For the surgeon, our computer-assisted planning permits an improved preparation before the actual operation."

The article "Mathematics in Facial Surgery" appears in the October 2006 issue of the *Notices of the American Mathematical Society* (AMS). It is available at AMS web site.

Founded in 1888 to further mathematical research and scholarship, the more than 30.000-member American Mathematical Society fulfils its mission through programmes and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.