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

Kostyantyn Malukhin*, Timon Rabczuk, Kornel Ehmann, Michael J. Verta

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Article number106523
JournalJournal of the Mechanical Behavior of Biomedical Materials
Volume154
DOIs
StatePublished - Jun 2024

Funding

We are very thankful to the National Science Foundation (DFG) of Germany (Bohn) for consideration of our research proposal and the approval of the DFG research Grant #: MA 8422/3–1 to support the current phase of this study during the 2020–2022 years at the Bauhaus University (Germany). Many thanks to the former President of Bauhaus University (Germany) Prof. Dr. Winfried Speitkamp, the university administration, and the university library for the comprehensive and timely supply of the related contemporary and rare literature on the subject via the extensive number of interlibrary book loans. Separate gratitude must be expressed to Prof. Dr. Thomas R. Kurfess (Georgia Institute of Technology, Chair in Fluid Power and Motion Control, School of Mechanical Engineering, Atlanta, GA, USA) for his continued support of our research efforts. We also would like to express our deep gratitude to the USA National Science Foundation (NSF) for the NSF Research Grant #: CMMI 0825722 that helped to support the starting phase of this research at Northwestern University (Mechanical Engineering Department, McCormick School of Engineering , Evanston, IL) for 2011–2012 years.

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

ASJC Scopus subject areas

  • Biomaterials
  • Biomedical Engineering
  • Mechanics of Materials

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