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Mathematical Tools For Predicting Facial Surgery Results
(press release from American Mathematical Society)
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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
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
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
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 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 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
"Even though biomechanical tissue modeling turns out to be a tough problem, we
are already rather successful in predicting postoperative appearance from
preoperative 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 AMS. It is available on the web at
Founded in 1888 to further mathematical research and scholarship, the more than
30,000-member American Mathematical Society fulfills its mission through
programs 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.
Contact: Dr. Peter Deuflhard
American Mathematical Society
Knowledge is Power.
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