1. Department of Orthodontics, University of Maryland Dental School, 650 W. Baltimore Street, Room 6408, Baltimore, MD 21201, USA
2. Department of Mathematical Sciences, the University of Bath, UK
3. Department of Orthodontics, the University of North Carolina at Chapel Hill, NC, USA
4. Department of Plastic and Reconstructive Surgery at the University of North Carolina School of Medicine, Chapel Hill, NC, USA