Very very many variables... For example, burnt tissue, crushed tissue, injection trauma, infected tissue plus variable anatomy.

Yes, you can produce a model incorporating nonlinear elasticity, viscosity, plasticity, frictional contact, subfailure damage, and fracture / cutting. Repeat for each tissue involved. Some of those methods are well-developed, others are not. Damage and failure in particular is poorly developed, and simulating deformable body contacts can be tricky. Then you have to estimate parameters for all of that, validate that the model produces realistic outcomes, and get it to run in real time. You can ignore most of the complexity and make approximations, like treating everything as an isotropic linear elastic material, setting cutting resistance to a fixed value per tissue, treating the scalpel like a lightsaber, and accepting that contact will be imperfectly enforced (surfaces may interpenetrate). A crude simulation can still be good enough to be useful as a learning tool though, but it tends to work more like a video game than a true physics-based simulation. We'll still eventually get the kind of accurate simulation you're asking for, probably.