Kirchhoff's law-based velocity-controlled motion models to predict real-time cutting forces in minimally invasive surgeries

Abstract

A theoretical framework, united by a "system effect" is formulated to model the cutting/haptic force evolution at the cutting edge of a surgical cutting instrument during its penetration into soft biological tissue in minimally invasive surgery. Other cutting process responses, including tissue fracture force, friction force, and damping, are predicted by the model as well. The model is based on a velocity-controlled formulation of the corresponding equations of motion, derived for a surgical cutting instrument and tissue based on Kirchhoff's fundamental energy conservation law. It provides nearly zero residues (absolute errors) in the equations of motion balances. In addition, concurrent closing relationships for the fracture force, friction coefficient, friction force, process damping, strain rate function (a constitutive tissue model), and their implementation within the proposed theoretical framework are established. The advantage of the method is its ability to make precise real-time predictions of the aperiodic fluctuating evolutions of the cutting forces and the other process responses. It allows for the robust modeling of the interactions between a medical instrument and a nonlinear viscoelastic tissue under any physically feasible working conditions. The cutting process model was partially qualitatively verified through numerical simulations and by comparing the computed cutting forces with experimentally measured values during robotic uniaxial biopsy needle constant velocity insertion into artificial gel tissue, obtained from previous experimental research. The comparison has shown a qualitatively similar adequate trend in the evolution of the experimentally measured and numerically predicted cutting forces during insertion of the needle.

Keywords: Cutting force; Equations of motion; Force balance; Fracture force; Friction model; Kirchhoff's law; Minimally invasive surgery; Process damping; Soft biological tissue response; System effect.