An Analytical Model of the Deflection of Hollow Needle-Like Structures Moving in Soft Materials

Significant research efforts have been focused on modeling the interaction between surgical needles and soft tissues which can be used to measure the deflection of a bevel-tip needle inside soft tissue. The development of such a model, which predicts the steering behavior of the needle during needle-tissue interactions, could improve the performance of many percutaneous needle-based procedures. In this work, we use the dynamic Euler-Bernoulli beam theory to model the needle as a cantilever beam that is subjected to forces inflicted by the tissue to predict the deflection of the needle. The model to estimate the deflection from the needle-tissue interaction is based on Euler-Bernoulli beam theory. In the Euler–Bernoulli thin beam theory (beams in which the length is much larger than the depth with aspect ratio of, at least, 10:1), the rotation of cross-sections of the beam is neglected compared to the translation and the angular distortion due to shear is considered negligible compared to the bending deformation. The needle is assumed to be a cantilever beam and the equation of motion for the transverse vibration of the beam is in the form of fourth-order partial differential equations with two boundary conditions at each end. The needle-tissue interactions can be modeled by a distributed load perpendicular to the needle shaft acting along the inserted needle portion and a point load acting at the needle tip representing reaction forces caused by the cutting of tissue by the beveled needle tip. The previous assumptions can be modeled as a single Euler-Bernoulli beam rests on an elastic foundation under the effect of a distributed moving load. This dynamic Euler-Bernoulli beam theory is used to acquire a governing equation for the needle tissue system. The resulting equation is a partial differential equation (PDE), which will be solved numerically and an analytical equation for the needle deflection will be derived. The needle can be divided into two sub-beams, the first sub-beam is the part of the needle that lays outside of tissue. The rest of the needle from the entry point to the tip point behaves like the second sub-beam. Thus, the needle is consisting of two uniform beams connected at the entry point. Each sub-beam is itself a Euler-Bernoulli beam which satisfies the Euler-Bernoulli beam PDE governing equation. The boundary conditions for each part can be determined knowing that the first segment is clamped at the needle holder whereas the second one has a free end, and it is subject to distributed interaction forces. Also, because of the continuity condition, both segments share the same deflection and slope, and they also experience the same amount of shear force and bending moment at the connection point. Our ongoing work is to improve the model for studying the mechanics of bioinspired surgical needles and to predict the deflection of these needles during insertion into multi-layered tissues. Finite element model and experimental data will be used to verify the analytical model.