## Abstract

The insertion of the needle is difficult because the deformation and displacement of the organs are the key elements in the surgical act. Liver and tumor modeling are essential in the development of the needle insertion model. The role of the needle is to deliver into the tumor an active chemotherapeutic agent. We describe in this chapter the deformation of the needle during its insertion into the human liver in the context of surgery simulation of the high- robotic-assisted intraoperative treatment of liver tumors based on the integrated imaging-molecular diagnosis. The needle is a bee barbed type modeled as a flexible thread within the framework of the Cosserat (micropolar) elasticity theory.

### Keywords

- bee needle
- human liver
- Cosserat elasticity theory

## 1. Introduction

The flexible bee needles are useful tools to transport drugs into the liver tumors [1, 2]. The insertion trajectory of the needle must avoid the ribs, blood vessels, and other organs to protect the liver [3, 4, 5, 6] (Figure 1a). The bee needle assures reduced insertion forces and small tissue deformations because of the tip deflections. The furthermost current publications on the surgical needle navigation into the liver can be demonstrated in [7, 8, 9]. The bee needle is shown in Figure 1b. The front angle has 157 deg., the back angle, 110 deg., the height is 0.5 mm, and the tip thickness 0.15 mm.

A number of scientific researches have been carried out on the collision free trajectory of the needle to the target. The surgical event requires experience in imaging the tumor location based on the liver structure and the microstructural interaction between the needle and the liver. Several studies have revealed that the needle flexibility is essential to achieve a good precision in the handling.

The strain and stress fields and the topological changes of the liver are not to be neglected during the needle navigation towards the tumor [10, 11, 12, 13]. Details of the forces during needle insertion into the liver are find in [14], the real time collision detection for virtual surgery in [15] and the minimal hierarchical collision detection in [16]. Optimization is required to modify the needle trajectory in order to protect the liver [17, 18], to manage the tumor risk [19], and to change the robot architecture [20, 21, 22]. The inverse sonification problem for capturing hardly detectable details in a medical image is treated in [23], and the control in [24, 25, 26, 27]. Microscopic investigation of the human liver offers details on its microanatomy with emphases to the granular, fibrillar components and irregular solid–fluid interfaces [28, 29, 30]. The basic unit of the liver is the hepatic lobule which is a hexagonal element with comprised the portal triad -portal vein, hepatic artery and the bile duct [31, 32]. Lobuli form two layers membranes with internal space of 100A and the cellular elements with twisted, spiraling fibers braided into the helical and screw-shaped gaps (pores) of 40–100

In this chapter we try to answer a few questions such as how is the deformation of the needle and how the free-collision trajectories are determined.

## 2. Deformation of the needle

The needle is a bee barbed needle and it is modeled as a flexible thread within the framework of the Cosserat (micropolar) elasticity theory [37, 38, 39, 40, 41, 42]. The Cosserat elasticity is applied to describe the interaction between the needle and the human liver. Let us consider a serial surgical robot composed of a revolute joint and a flexible needle. A Lagrange frame

The Euler axes are oriented with respect to the Lagrange axes by Euler angles

The functions

where * s*which is the coordinate along the central line of the needle. The functions

The function

So, the needle is rigid along the tangential direction and the total length of the needle

The link between the position vector

We introduce the inertia of the needle characterized by

where _{0} the area of the cross section,

The equations which describe the deformation are

where * A*and

*are the bending stiffness and respectively the torsional stiffness of the needle, related to the Lame constants*C

*is the radius of the cross section of the needle, and*a

The Eqs. (7–11) are solved by using the cnoidal method [43].

In short, this method is reducible to a generalization of the Fourier series with the cnoidal functions as the fundamental basis function. This is because the cnoidal functions are much richer than the trigonometric or hyperbolic functions, that is, the modulus

To understand the cnoidal method, consider now a nonlinear system of equations that govern the motion of a dynamical system

with

where

This system of equations can be reduced to Weierstrass equations of the type

We introduce the function transformation

where the theta function

and

Further, we write the solution (15) under the form

for

In (20) we recognize the expression [43].

with

The second term

If

As a result, the cnoidal method yields to solutions consisting of a linear superposition and a nonlinear superposition of cnoidal waves.

Therefore, by applying the cnoidal method, the closed form solutions of the Euler angles

where

with

Our objective is to determine the functions which measure the bending of the needle (

Figures 5 and 6 show that the deformation of the needle for both routes. We see that the deformation is small with no tendency to chaos. The strains are described by localized solitons which propagate for a long time without changes. The soliton is a localized wave with an infinite number of degrees of freedom. This wave conserves its properties even after interaction with another wave. In short, this wave acts somewhat like particles [43]. The system of Eqs. (7–11) has unique properties. These properties are locally preserved such as an infinite number of exact solutions expressed in terms of the Jacobi elliptic functions or the hyperbolic functions, and the simple formulae for nonlinear superposition of explicit solutions.

## 3. Determination of the free-collision trajectories

Let us present a model for determining the collision-free trajectories by using the Fibonacci sequence. The trajectories are determined from the restrictions of avoiding the collisions with blood vessels, ribs and surrounding tissues, and also the interference of needles with each other. We consider that more needles are planned to be inserted into the liver [45].

The trajectory of each needle

The sequence terms in the Fibonacci problem are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. It is clear that each term is a sum of two preceding sequence term. In fact, the sequence can be recursively defined in the form

The Fibonacci sequence was highlighted in nature for example in the arrangement of the flower petals, in seeds and in the spiral arrangement of pine cones and pineapple. The keys on a piano are divided into Fibonacci numbers, and numerous classical compositions implement the golden section. Such an example is found in the Alleluia Choir in Handel’s Messiah and in many of Chopin’s preludes [50].

Development of a target drug delivery technique usually consisted of three steps. In step 1, a 2D ultrasound image of the tumor is obtained. The size and position of the tumor are analyzed by the surgeon who decides the number of required needles. In step 2, the insertion points on the skin of the needles and the positions of the targets in the tumor are established. In step 3, once the needles are placed at the target points on the skin, they are guided according to a precise surgical planning based on the optimization of the collision-free trajectories to avoid the ribs, blood vessels and tissues in the abdominal area.

The base coordinate frame is established on the first needle with the vertical axis

The first control parameter is the length of the

and

where

The choice of the scaling number

The possible collision point between the needle and the tissue is analyzed by an identifier to check the minimum distance between needle and the surrounding tissue [45]. The minimum distance is expressed as

with

where

As application, the case of a tumor with a difficult location in the vicinity of the portal tree of the vascular territory in the liver, is considered (Figure 8a). The tumor image seen on the microscope is shown in Figure 8b. White and gray denote forbidden areas while the shade of purple are safe regions. The tumor is drawn in red. The Fibonacci algorithm is applied to three-needles with restrictions to avoid the collision with the tissues, blood vessels, ribs and previously inserted needles.

The task of our simulation is to determine the boundaries of each needle as a collision-free surface which represents the feasible insertion area based on given constraints. Then, the optimal trajectory of each needle can be chosen in this surface automatically.

Once all needles are placed at the predetermined epidermis and the ordering of entry is chosen to be 1, 2 and 3, following the first needle’s insertion, the operation is repeated for the needles 2 and 3.

Simulation of the collision-free trajectories for three needles is presented in Figure 9. The insertion scheme is determined by the Fibonacci spirals. A set of free-collision trajectories (red) in the immediate vicinity of the epidermis, is suggested.

From these possible collision-free trajectories (red) the green paths corresponding to the Fibonacci spirals (black) are chosen. These trajectories avoid the blood vessels (purple) and the coasts (brown) in all directions until the tumor. Optimal solution with 3 collision-free trajectories to the target is displayed in Figure 10. The Fibonacci spirals with the centers in ribs (brown) are displayed for needles 1 and 3. For the needle 2, the Fibonacci.

Three locally optimal collision-free trajectories for the surgical needle corresponding to three different entry points into the skin A, B and C are displayed in Figure 11.

## 4. Conclusions

The study investigates the navigation of a flexible needle into the human liver. The role of the needle is to deliver into the tumor an active chemotherapeutic agent. The deformation of the needle during its insertion into the human liver is describe in this chapter in the context of intraoperative treatment of liver tumors based on the integrated imaging-molecular diagnosis. The needle is a bee barbed type modeled as a flexible thread within the framework of the Cosserat (micropolar) elasticity theory. The Cosserat elasticity describes the interaction between the needle and the human liver by incorporating the local rotation of points and the couple stress as well as the force stress representing the chiral properties of the human liver.

## Acknowledgments

This work was supported by a grant of the Romanian Ministry of Research and Innovation project PN-IIIP2-2.1-PED-2019-0085 CONTRACT 447PED/2020 (Acronim POSEIDON).

## Conflict of interest

The authors of this paper certify that they have no affiliations with or involvement in any organization or entity with any financial or nonfinancial interest in the subject matter or materials discussed in this manuscript.