Relationship between mN and mg in vessel contraction studies?
The difference between above two units is a matter of metric systems used:
The newton (symbol: N) is the International System of Units (SI) derived unit of force. One newton is the force needed to accelerate one kilogram of mass at the rate of one metre per second squared.

Another system:  kilogram (gram, milligram)-force
The kilogram-force (kgf or kgF) is a gravitational metric unit of force. It is equal to the magnitude of the force exerted by one kilogram of mass in a 9.80665 m/s2 gravitational field (standard gravity, a conventional value approximating the average magnitude of gravity on Earth). Therefore one kilogram-force is by definition equal to 9.80665 N. Similarly, a gram-force is 9.80665 mN, and a milligram-force is 9.80665 µN. 
Kilogram-force is a non-standard unit and does not comply with the SI Metric System.
See:
http://www.sengpielaudio.com/ConvForce.htm