Modeling and simulation of flexible needles
Introduction
Percutaneous insertion of long and flexible needles into soft tissue is involved in many clinical and therapeutic procedures such as biopsy and prostate brachytherapy. As a result of needle–tissue interaction, the needle base movements and, for needles with a beveled tip, the asymmetric cutting force on the tip, the tissue deforms and the needle bends (see Fig. 1). The targeting procedure is complicated by the bending of the needle shaft, target displacement due to tissue deformation, and insufficient visual feedback from medical imaging modalities. Accurate needle insertion requires significant skill and training of the performing physician. Modeling, simulation, and path planning of needle insertion are emerging fields of research aimed at providing the physicians with training devices and accurate pre-surgery plans.
Needle insertion simulators usually include a soft tissue model, a flexible needle model and a needle–tissue interaction model. The simulation of needle bending and tissue deformation together in one combined model is generally not feasible for two reasons: (i) a fast solution for a large tissue mesh requires exploiting simplifications in deformation equations, which are not suitable for estimating the needle behavior and (ii) the interaction surface of the needle and tissue changes during insertion, which many techniques like the finite element method (FEM) cannot inherently accommodate. Therefore, generally two separate models for the tissue and the needle are employed [1], [2], [3], [4] with a third needle–tissue contact model governing their interaction.
Deformable tissue models have been studied extensively in simulating tissue deformation during surgery and needle insertion [1], [2], [3], [4], [5]. Models based on the FEM are the most common techniques employed for tissue deformation simulation during needle insertion [1], [2], [4], [6]. Mass-spring models [7] have also been used for this purpose. The interaction of needles with such deformable tissue models has been studied widely [2], [8], [9], [5].
The aim of this paper, the preliminary results of which were presented in [10], is to compare different models for simulating the bending of flexible needles. In general, medical needles can be categorized into three major groups: rigid needles, highly flexible needles, and (moderately) flexible needles. Rigid needles keep a straight posture regardless of the forces applied on them during insertion. Due to their simplicity, rigid needle models have been used when the needle physical properties and the insertion procedure lead to negligible bending [2], [1], [6]. On the opposite end of the flexibility range are the highly flexible needles. These needles are assumed to bend in the direction of their tip bevel with a constant curvature, without applying considerable force to the tissue in the lateral direction. These needles were modeled as a non-holonomic system [3] and were used for needle insertion simulations and planning in [11], [12], [13].
Some needles, such as brachytherapy needles, cannot be categorized as either rigid or highly flexible. They are not rigid, since their deflection during procedures is significant. They are not highly flexible either, since a considerable lateral force is necessary to bend them. Several groups have modeled this type of needles. DiMaio and Salcudean simulated the needle as an elastic material using FEM with geometric nonlinearity and 3-node triangular elements and validated this method in phantom studies [14]. This method was extended to 3D using 4-node tetrahedral elements by Goksel et al. [4]. Glozman and Shoham used linear 2D beam elements to simulate the needle bending for needle steering [7]. Linear beam theory was also used to introduce a needle steering model with online parameter estimator [15], to estimate the needle tip deflection during insertion due to tip bevel [16], and to identify the shaft force profile due to the bevel [17]. Models based on linear beam theory are relatively simple and fast. However, they are not rotationally invariant and cannot preserve the needle length during large deformations/deflections.
Many needles, such as brachytherapy needles, consist of a stylet sliding inside a hollow cannula. Physical modeling of this combination without any simplifications is indeed very complicated as it requires separate models for the stylet and the cannula and an interaction model to simulate their interface. In this paper, flexible needles are approximated as solid bars and, accordingly, models applicable to solid bars are examined to simulate their flexion.
Three different models were used to simulate needle bending. The first two models are based on the FEM and were chosen due to their frequent use in the literature, while the third model is an angular springs model. The first model uses 4-node tetrahedral elements, where nonlinear geometry is accommodated to simulate large deformations. The second model also accommodates nonlinear geometry and uses Euler–Bernoulli nonlinear beam elements. In this work, nonlinear beam elements were preferred over more common linear ones in the literature due to their superiority in modeling large deformations. The third model is novel and utilizes angular springs for the quasi-static simulation of needle bending. In the literature, angular springs have been used to model cantilever-like structures [18] such as beams in mechanical engineering [19] and hair deformation in computer graphics [20]. They have been also incorporated in 3D mass-spring models to simulate large volume deformations [21]. In this paper, this type of a spring model is implemented using finite-differences for medical needles, for which its performance is compared with two other common types of physically based models using FEM.
The Young’s modulus is the parameter that describes needle bending in the first two models. Similarly, the third model is identified by its spring constant. In contrast to [10], in this paper all models are devised for needle bending in 3D. The parameter of each model is identified for a brachytherapy needle through experiments, where several lateral forces were applied to the needle tip and the shaft deflection was recorded. The parameters defining each model were identified and the models were studied for their accuracy in simulating the actual needle deflection observed during experiments.
In this paper, the needle bending models are derived in 3D, since their formulations are applicable in 3D settings as well as in 2D. Note that, for the single force applied at the needle tip during the experiments, the needle deformation is entirely planar. Therefore, the parameter identification and the model validations were performed in 2D. Nevertheless, the same identified parameters also describe a needle in 3D, since the shafts of most medical needles are built with axial symmetry resulting in symmetric deflection for the same force rotated around their long axis.
The following section derives the models in 3D. Next, the experimental method to validate the models is described in Section 3. The results and a discussion follow in Sections 4 Results, 5 Discussion, respectively. Finally, conclusions are presented in the last section.
Section snippets
Finite element method using tetrahedral elements
The finite element method is a powerful tool for approximating a solution to the continuum mechanics equations. In this method, an entire body is divided into several discrete elements . Then, the constitutive equations are approximated over each element and combined to give an approximation to the global solution. Various types of elements can be chosen depending on the nature of the problem. 4-Node tetrahedral (TET4) or 3-node triangular (TRI3) elements are the simplest elements to use in
Experiment
In order to show the feasibility of the models and compare their accuracy, the following experiments were conducted using an 18 gauge 20 cm Bard BrachyStar needle (C.R. Bard, Inc., Covington, GA), that is used in prostate brachytherapy seed-implant procedures. In these experiments, the needle was bent under several known forces and its bent shaft form is recorded for evaluating our model simulations.
During the experiments, the needle was clamped at its base while its shaft lay horizontal to
Results
In the experiments, the needle deformations were all planar. Therefore, the model equations were reduced to their 2D equivalents. The equations for a 2D nonlinear beam element are obtained by simply removing the rows and columns of the stiffness matrix in (16) that correspond to , and in the vector of nodal variables (reducing to its parts , , , and in Appendix A). The angular spring model can be used in 2D simply by neglecting the torsion. The FEM with tetrahedral elements
Discussion
Thanks to the fast computation of the presented models, and in particular, the angular springs model, they can easily be integrated into real-time medical training simulation systems. Due to their high accuracy, they can also be used in simulations for needle steering and path planning.
Computationally, the beam element model is more efficient than the triangular/tetrahedral element model. Note that a beam element and a triangular element both have stiffness matrices in 2D and, similarly in
Conclusions
Three different models were presented to simulate the deformations of a needle. The first two models use tetrahedral and beam elements, while the third model uses rigid bars connected through angular springs. All the models can preserve the needle length during moderately large deformations. The efficacy of the models in simulations of needle bending was evaluated through experiments during which several lateral forces were applied to a brachytherapy needle and the resulting deformations were
Conflict of interest statement
The authors declare that they have no competing interests.
Acknowledgement
This work was supported by the Natural Sciences and Engineering Research Council of Canada.
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