Mathematical Modeling of Vessel Geometry and Circumference in Microvascular Surgery.

Introduction: Microvascular anastomosis has traditionally been executed with a perpendicular transection through the vessel at the widest diameter to increase circumference and thus increase blood flow while decreasing resistance. In Chen's 2015 article, it was suggested that an "open Y" would improve vessel size match, and Wei and Mardini discuss angled transections of the vessels. This project aims to explore the geometric configurations feasible at the anastomotic transection and mathematically model the resulting hypothetical increases in circumference.

Materials and methods: The mathematical models were theoretically developed by our team. The formulas model increases in circumference of the transection at different distances in relation to the bifurcation of a blood vessel, as well as changes in circumference at different transection angulations. An in vitro exploration as to the anastomotic feasibility of each geometric cut was completed on ten poultry tissue specimens.

Results: The mathematical models demonstrated the change in vessel circumference, with multiple geometric designs calculated, best shown through diagrams. For example, if the vessel width is 1 mm, the distance from the increasing vessel diameter to the final bifurcation is 1 mm, and the bifurcation angle is 45°, the circumference of the transected vessel increases by 82.8%. Models of transections at different angulations, for instance 30°, 45°, and 60°, yield an increase in elliptical circumference of 8.0%, 22.5%, and 58.1%, respectively. Additional derivations calculate the elliptical circumference at any angle in a single vessel, and at any angle in a bifurcating vessel.

Conclusion: The theoretical and clinical aim of this project is to increase awareness of the anastomotic creativity and mathematically demonstrate the optimal anastomotic geometry, which has not been objectively explored to our knowledge. An in vivo study would further support clinical improvements, with the aim to map postoperative fluid dynamics through the geometric anastomoses.