A Magnetically Controlled Guidewire Robot System with Steering and Propulsion Capabilities for Vascular Interventional Surgery

Magnetically manipulated interventional robotic systems offer outstanding advantages for improving vascular interventions, including minimizing radiation exposure to physicians and increasing the controllability of magnetic interventional devices in hard‐to‐reach vessels. However, automatic control of magnetic guidewires (MGs) is still challenging in terms of modeling of guidewires and trajectory planning. Herein, a magnetically controlled guidewire robotic system (MCGRS) with steering and propulsion capabilities is proposed based on adequate modeling and trajectory planning methods. The steering kinematics of MG is first modeled by constant curvature theory. Then, a continuum mechanics model is built to predict the deformation of the magnetic tip by combining the dipole model and the Cosserat‐rod model. Moreover, a trajectory planning algorithm is developed to navigate the MG through vessels. Furthermore, trajectory following experiments within three vascular phantoms confirm that the proposed model and algorithm are reliable and capable of navigating the MG through the desired trajectory. Finally, two extra navigation experiments are implemented in 3D vascular phantom, which show that the MCGRS can be remotely controlled to manipulate the MG to actively steer and reach the target site. The system and methods will build the foundation for automatic control of MG and help to improve the autonomy.

Magnetically manipulated interventional robotic systems offer outstanding advantages for improving vascular interventions, including minimizing radiation exposure to physicians and increasing the controllability of magnetic interventional devices in hard-to-reach vessels.However, automatic control of magnetic guidewires (MGs) is still challenging in terms of modeling of guidewires and trajectory planning.Herein, a magnetically controlled guidewire robotic system (MCGRS) with steering and propulsion capabilities is proposed based on adequate modeling and trajectory planning methods.The steering kinematics of MG is first modeled by constant curvature theory.Then, a continuum mechanics model is built to predict the deformation of the magnetic tip by combining the dipole model and the Cosserat-rod model.Moreover, a trajectory planning algorithm is developed to navigate the MG through vessels.Furthermore, trajectory following experiments within three vascular phantoms confirm that the proposed model and algorithm are reliable and capable of navigating the MG through the desired trajectory.Finally, two extra navigation experiments are implemented in 3D vascular phantom, which show that the MCGRS can be remotely controlled to manipulate the MG to actively steer and reach the target site.The system and methods will build the foundation for automatic control of MG and help to improve the autonomy.interventionalists need to manually advance, retract, and rotate the guidewire during the VIS. [5,6]Therefore, manipulating the passive guidewire in the tortuous and complex vessels with large-angle bifurcations (such as the cerebral vessels) takes longer surgery time than the simple vessels. [7]0] Finally, the VIS require a high level of experience, and the number of interventionalists who can skillfully operate the procedure is far less than the amount of patient demand for interventional procedures.
In recent years, several commercial vascular interventional robotic systems for clinical practice have been developed to address the challenges in existing vascular interventional procedures. [11,12]For example, the CorPath GRX from Siemens Healthcare Corindus, USA, is one of the most successful vascular interventional robots for clinical application in percutaneous coronary intervention procedures, and has received Food and Drug Administration (FDA) and Conformité Européenne (CE) approval; the robot R-One developed by Robocath, France, has received CE approval; and Johnson & Johnson Medical, USA, has introduced the first peripheral vascular intervention for the robotic system Magellan, which has been clinically used in more than 1000 cases.However, existing vascular interventional robots generally adopt master-slave control, and the robots can only mimic the simple mechanical operations by the surgeon's hand, including advancing, retracting, and rotating the guidewire/catheter.Additionally, existing vascular interventional robots do not overcome the drawbacks of conventional pre-shaped guidewires, which means manipulating guidewires at complex vascular bifurcations remains challenging in terms of flexibility, efficiency, and autonomy.
][28] Furthermore, several internationally renowned academic teams have started to investigate magnetically manipulated interventional robotic systems.For example, Zhao's team [12] from MIT proposed a magnetically manipulated robotic remote neurointervention system capable of steering and propulsion functions, which mainly consists of a magnetically controllable guidewire, a movable permanent magnet drive system, and a pair of guidewire/catheter advancement units.The navigation capability of the system is verified by in vitro and in vivo experiments on pigs.In addition, Choi's team [29] from DGIST in Korea has developed an electromagnetically controllable microrobotic intervention system, capable of assisting interventionalists with teleoperated and real-time control of micro-guidewires.This system consists mainly of magnetic guidewires (MGs), a human-level electromagnetic actuation system, and guidewire/catheter advancement units.The capabilities of the system in terms of manipulation and navigation are also experimentally validated through in vitro and in vivo on pigs.[32] Zhang's team has designed a variety of magnetic steerable guidewire robotic systems based on ultrasound guidance and moving electromagnetic systems. [33,34]Xu's team proposed a magnetically controlled guidewire with an integrated large workspace magnetic manipulation system and guidewire propulsion device. [35,36]These aforementioned magnetically manipulated interventional robotic systems are more flexible and controllable than conventional passive guidewires, which are expected to reduce radiation exposure to interventionalist and patients by remote control.However, the autonomy of these systems still needs to be improved.In this case, the detailed modeling of MGs and the trajectory planning algorithm should be developed to achieve automatic control of MGs.
In this work, we propose a magnetically controlled guidewire robotic system (MCGRS) with steering and propulsion capabilities based on adequate modeling and trajectory planning methods.The steering kinematics of MG is first modeled by constant curvature theory.Then, a continuum mechanics model is built to predict the deformation of the magnetic tip by combining the dipole model and the Cosserat-rod model.Moreover, a trajectory planning algorithm is developed to navigate the MG through vessels.Furthermore, the proposed models and algorithm are verified by trajectory following experiments in three vascular phantoms, which show that the proposed methods are reliable and the trajectory planning algorithm can navigate the MG through the desired magnetic tip trajectory.Finally, two extra navigation experiments are implemented in 3D vascular phantom, which show that the MCGRS can be remotely controlled to manipulate the MG to actively steer and reach the target site.As a result, our system and methods will build the foundation for automatic control of MG, which will help to improve the autonomy of the magnetically manipulated interventional robotic systems.In addition, compared to our previous research, [35,36] we have upgraded the magnetic manipulation system and completed more detailed modeling for MGs and developed trajectory planning algorithms to further improve the system and methodology.

Design Overview of the MCGRS
Conventional passive guidewires lack path selection and propulsion capabilities because they use pre-shaped curved tips with fixed angles and rely on the interventionalist to manually push, pull, and rotate the guidewire for intervention manipulation.In this study, an MCGRS for VIS was proposed.The MCGRS comprised the following modules (Figure 1b): an MG, a magnetic actuation system (MAS), a guidewire/catheter advancer (GCA), a teleoperation console (TC), and X-ray imaging system, etc.This system could break through the limitations of traditional passive guidewires and provide active steering and automatic propulsion capabilities, allowing that interventionalists could manipulate the MG to actively steer and rapidly select the correct path at complex vascular bifurcations and reach the target site by TC in a flexible and controlled manner.(Figure1a) Therefore, our method could help the interventionalists to rapidly establish therapeutic access so that shorten the procedure time.

Design of the MAS
Compared with electromagnets, permanent magnets had a simple structure and did not need to pass current through the coil to generate a magnetic field like electromagnets, so there was no need to consider heat dissipation, and it could generate a stronger magnetic field in the same volume.The strength of the magnetic field was affected by its material, location, and other factors, and the strength and direction of the magnetic field could be adjusted by changing the pose of the permanent magnet to control the deflection direction of the MG.
The general 6-degree of freedom (6-DOF) serial robotic arm could achieve arbitrary motion of the end-effector in space with flexible control, and the working space could be expanded by increasing the size of the robotic arm to cover the operating space of clinical interventions.Therefore, we combined a mobile cart, a robotic arm, a permanent magnet, and a stepper motor module to form an MAS.In this article, an axially magnetized N52-grade cylindrical NdFeB permanent magnet with a dimension of D50 Â L25 mm was selected to generate the external actuating magnetic field, and its detailed parameters were described in Table S1, Supporting Information.The permanent magnet was mounted to the UR10 robotic arm as the end-effector through 3D-printed connectors, flexible coupling, and stepper motor (Figure 1b).We controlled the end motion of the arm to change the position of the permanent magnet and change orientation of the permanent magnet by controlling the rotation of the stepper motor, so that MG could be deflected to the desired position and orientation under the actuation of an external magnetic field.In particular, the control of the magnet was divided into attraction mode and repulsion mode, and the two operation modes were achieved by controlling the rotation of the stepper motor to switch the N-S pole (left and middle in Figure 1b).For the attraction mode, the magnetic field B1 was generated when the S pole of the magnet was close to the MG, which would be actuated to steer and approach the magnet; while for the repulsion mode, the magnetic field B2 was generated when the magnet was switched to the N pole and close to the MG, which would move away from the magnet.By expanding small magnetic field area of the magnet into the large working space of the arm and generating magnetic field at any position in space, the MAS could realize the magnetic actuate steering control of the MG in the large working space, which made the whole system flexible and controllable and closer to the clinical requirements.

Design of the GCA
Currently, the majority of vascular interventional robotic systems consisted of a positioning device and an interventional device pushing device, where the pushing device specifically performed the advancing, retracting, and rotating action of the interventionalist's hand on interventional devices such as guidewires, catheters, etc., to deliver the devices to the target vessel, thus completing the robot-assisted interventional treatment.In this article, we designed and manufactured a GCA based on the continuous propulsion principle, which was divided into guidewire and catheter advancers with a total of 4-DOF motion to achieve advancing and retracting and rotating operations of the guidewire/catheter respectively (Figure 1b).For linear advancing and retracting operation, we used stepper motor with a pair of friction wheel clamping principle to achieve linear propulsion motion of the guidewire/catheter (see Figure S1a, Supporting Information, for details).The function of the clamping adjustment mechanism in Figure S1a, Supporting Information, was to adjust the gap between the two friction wheels and the clamping force on the guidewire/catheter.The detailed compositional structure of this mechanism is shown in Figure S1b, Supporting Information, comprising an adjusting spring, an adjusting base, an adjusting screw, a floating screw, and a fixing screw.Specifically, the clamping force and two-wheel gap were able to be adjusted by rotating the adjusting base and the adjusting screw (in Figure S1c, Supporting Information).In addition, the instrument installation process and the state comparison of the before/after clamping are shown in Figure S1c, Supporting Information.In the process of instrument installation, according to the different diameter of the instrument, the adjusting base was first manually pressed down and rotated clockwise, at which time the spring was compressed so that a gap larger than the diameter of the instrument was left between the two wheels, and the guidewire/catheter passed through all cavities in turn and arrived at the appropriate position, and then the adjusting base was loosened so that the guidewire/catheter was clamped and fixed.As shown in Figure S1c, Supporting Information, the input/output cavities of the linear propulsion module were designed with a diameter of 5 mm, which meant that the cavities could accommodate guidewires/catheters with diameters of no more than 5 mm to pass through.As the diameter of commercial micro-guidewires was generally 0.014 inch (0.36 mm), we set the minimum diameter that could be accommodated by the GCA to 0.3 mm.As a result, our GCA could accommodate different guidewires/catheters with diameters ranging from 0.3 to 5 mm, which could expand the scope of application of the device.
For rotating operation, we used a stepper motor with a pair of gears to achieve the rotary propulsion of the guidewire/catheter.Among two gears, the driving gear was actuated by the motor to provide input power, and the slave gear followed the rotation by meshing with its gear teeth in the opposite direction, and provided output power for the linear propulsion module (see Figure S2, Supporting Information).As shown in Figure S2, Supporting Information, when two stepper motors were controlled to work synchronously, a pair of friction wheels were actuated by one of the motors to undergo synchronous rotation, enabling the guidewire/catheter to be clamped by the two wheels and thus obtaining advancement or retraction power; at the same time, a pair of cylindrical gears that meshed with each other were actuated by another motor to undergo synchronous rotation, and then the entire linear propulsion module was also driven to follow the rotation through the rotary output connector so that the guidewire/catheter gained rotational power, ultimately achieving simultaneous linear and rotational motion of the guidewire/ catheter.During the intervention, the catheter advancer was in the front and the guidewire advancer was in the rear, which could realize the simultaneous propulsion of the guidewire/ catheter.The intervention velocity of GCA ranged from 0 to 15 mm s À1 .

Design of the MG
In this article, an MG was proposed.The MG consisted of a commercial guidewire and a magnetic tip with flexible steering capability.The fabrication process of the MG was shown in Figure 1c.An empty silicone tube with an inner and outer diameter of 0.5 Â 1 mm and a length of not less than 12 mm was first prepared and then a certain amount of polymer consisting of polydimethylsiloxane soft silicone was injected into the empty silicone tube and cured naturally at 25 °C for 2 h.Two small axially magnetized N52-grade NdFeB cylindrical magnets with a diameter and length of 0.5 Â 2.5 mm were fully inserted from front end of the silicone tube in order, then one 0.035 inch (0.89 mm) commercial guidewire was inserted 2 mm from the rear end of the silicone tube.Next, the magnetic tip and commercial guidewire were bonded by medical grade silicone adhesive to ensure a reliable connection.As a result, an MG with a magnetic tip was fabricated, and it was able to exhibit a relatively strong magnetic response, a deflection capability of more than 90°, and good flexibility by applying an external actuating field (see Table S2, Supporting Information, for the detailed parameters).

Magnetic Actuation Model
Among the methods for describing the magnetic field of permanent magnets, the dipole model was often chosen because of its computational simplicity. [16]In our MAS, the permanent magnet could be regarded as a point source approximately, and the force model was simplified to a magnetic dipole model, where the field was nonuniform.The magnetic field B generated by a dipole source at a target point is denoted as where μ 0 = 4π Â 10 À7 T m A À1 is permeability of vacuum, p is the vector of distance from the center of the external magnet to the magnetic tip (p = p t À p m ), I is an identity matrix, and M is the magnetic dipole moment of external magnets.
The MG was actuated by the external dipole field produced by external magnet of the MAS (Figure 2a).The magnetic field could apply a magnetic force F m and a magnetic torque T m to the magnetic tip m, where the F m existed in a nonuniform magnetic field and the T m could deflect the magnetic tip from the direction of magnetization aligned to the direction of the magnetic field.Combined with the simplification requirement of Maxwell's equations, they can be expressed by the following equation (3) where B is the magnetic field vector, ∇ is a gradient operator, and m is the magnetic moment vector of the magnetic tip magnet, which is related to the pitch angle θ (first around Y-axis) and the yaw angle ϕ backward around X-axis).It can be expressed as  where m is the amplitude of the magnetic moment m; s and c denote sin and cos, respectively.

Steering Kinematic Model
According to the literature, [29,37] the magnetic tip could be modeled by Eulerian Bernoulli beam theory under the assumption of constant curvature, which could predict the steering angle of magnetic tip.Therefore, the curvature k can be expressed by the bending moment T b as [33] dκ dl where EI is mechanical stiffness of the magnetic tip, as shown in the following equation where t and m denote the silicone tube and the tip magnet, respectively; and E t and E m are Young's modulus; I t and I m are area moments of inertia; and d t and d m are outer diameter.Under constant curvature assumption, Equation ( 5) can be further derived as where θ and L are steering angle and total length of the magnetic tip, respectively.Reference to the right schematic in Figure 2a, based on the norm of magnetic field B and the norm of magnetic moment m, the magnetic torque in Equation ( 2) can be rewritten as where γ is direction angle of the external magnetic field.When the magnetic moment was aligned with the direction of the magnetic field, the magnetic torque would be balanced with the bending moment, and then steering angle θ of the magnetic tip would be derived as the following equation By using the Newton-Raphson equation, the steering angle θ of the magnetic tip in Equation ( 10) could be solved by numerical iteration for Equation (11). [29]The solution principle was to initialize an estimated values θ 0 (at n = 0) for the desired γ and B, and then iterate cyclically until it was within the range of the target solution (jθ nþ1 À θ n j < 0.001°)

Continuum Mechanics Model
The Cosserat elastic rod theory was well suited for effective numerical modeling and control of MCRs with heterogeneous structures and large deformations in nonuniform magnetic fields with minimal approximation. [38,39]Our MG fitted the characteristics of this continuum robot, so we combined the Cosseratrod model and the dipole field model previously to predict the deformation of MG.The model derivation mainly includes constitutive relations, equilibrium equations, and boundary conditions.The schematic diagram of the arbitrary rod mechanics model is shown in Figure 2b.
Constitutive Relations: The shape state of Cosserat rod along arc length s can be described by a pose matrix T(s) ∈ SE(3) as where p(s) ∈ R 3 and R(s) ∈ SO(3) are the position vector and the orientation matrix of the rod, respectively, and they can be represented as where v(s) and u(s) are the spatial linear and angular rate of change in T(s), respectively, and [] x denotes the operation of solving skew-symmetric matrices, as follows And the v(s) and u(s) obey the constitutive laws τðsÞ ¼ RðsÞK bt ðuðsÞ À u Ã ðsÞÞ (17)   where K se is the shear/extension stiffness matrix, and K bt is the bending/torsion stiffness matrix.v*(s) and u*(s) are the original value of linear and angular rate of change.The commercial guidewire was stiffer and the magnetic tip was shorter, here we assumed that magnetic tip rod could bend and twist, while shear and extension strains were ignored, which satisfied the Kirchhoff rod model (in which v = v* = [0 0 1] T , K se = 0). [38]For elastic axisymmetric materials (v* = [0 0 1] T ), the expression of K bt is as follows where I xx , I yy , and I zz are the second area moments of the cross section of magnetic tip rod about the principal axis of inertia; E is Young's elastic modulus, and G is the shear modulus.
Equilibrium Equations: In the static equilibrium state, the internal force and moment vectors of Cosserat rod could be described by a set of nonlinear ordinary differential equations (ODE).As shown in Figure 2b, the equilibrium differential equation is as follows where f(s) and τðsÞ are the rod's own internal force and moment vectors at s, respectively, and f e (s) and τ e ðsÞ are external distributed force and moment vectors, respectively.Boundary Conditions: According to the Cosserat rod theory, the deflection deformation of the magnetic tip rod could be described as a two-point boundary value problem (BVP), so we could solve the BVP over the arc length interval of the rod s ε [s a , L] (in Figure 2c).
In the middle of Figure 2a, the loads on both sides of a constraint point C 2 (s = σ), w(σ þ ), and w(σ À ) were dissimilar, the boundary condition relationship of the loads was as follows where w c is the constraint load.Suppose a constraint C 2 existed at the base of a magnetically magnetic tipped rod that limited the position and orientation of the rod centerline at p(s a ), but the base could be rotated about the z-axis to produce the torsional deformation caused by τ z ðs a Þ, which was the z-component of the internal moment at the base.In the coordinate system {b} with s a = 0 (in Figure 2c), the boundary conditions are as follows where R z (α) is the orientation matrix about the static z-axis, and α is determined by the parameter τ z ðs a Þ.
As shown in Figure 2a,c, in the coordinate system {b} with s = L, the boundary conditions are as follows where R x (ϕ) and R y (θ) is the two orientation matrix of magnetic tip about fixed x-axis and y-axis, respectively (the angle ϕ and θ corresponds to Equation (4)), l tm is the length of the magnetic tip magnet, and G is the gravitational force of the magnetic tip magnet.And F m and T m were magnetic force and magnetic torque generated by the dipole field in Equation (1), respectively.Finally, the deformation shape of the magnetic tip could be obtained by solving ODEs ( 13), ( 14), (19), and ( 20), together with the boundary conditions given by ( 22)-(31).

Propulsion Kinematic Model
For linear advancing and retracting operation of the GCA, we used stepper motor with a pair of friction wheel clamping principle to achieve linear propulsion motion of the guidewire/ catheter.Since linear propulsion motion was the most important function of GCA and directly affected the propulsion displacement and velocity of MG, it was necessary to analyze the propulsion kinematics of GCA The analysis schematic is shown in the left side of Figure 2a, and the propulsion velocity at any moment can be expressed as where r w is the radius of the friction wheel, and variable N p (t) is the real pulse number of motor at any moment, and constant N pr is the pulse number corresponding to one revolution of the motor (N pr = 360/δ), here δ is the step angle of stepper motor.
The propulsion displacement at a given time period [0,t] is as follows

Trajectory Planning
In the practical control of the proposed system in this work, the inverse kinematic was more useful, i.e., to plan the trajectory of the magnet and the velocity of the advancer based on the given magnetic tip trajectory.Thus, we needed to analyze and derive the trajectory planning method of the MG.As shown in Figure 2c, here we set the robot arm base frame {r} as the global frame.We set that the pose of the guidewire advancer frame {a} with respect to the robot frame {r} was definite and noted as T r a , in which the orientations of the two frames remained parallel and only had a fixed relative position.The transformation of the magnetic tip distal frame {t} with respect to the magnetic tip base frame {b} was T b t , and the transformation of the magnetic tip base frame {b} with respect to the advancer frame {a} was T a b .So the pose transformation matrix T r t of the magnetic tip in the robot frame {r} is expressed as where P t = (x t , y t , z t ) and R t are the central position value and orientation matrix of the magnetic tip in the robot frame {r}, respectively.Then, the magnetic tip position p(L) and magnetic tip orientation R(L) were obtained by calculating the BVP over the fixed interval [s a , L] in magnetic tip base frame {b}, and lðtÞ was the propulsion displacement at a given time period.
And the pose of the external magnet frame {m} relative to the robot frame {r} was T r m ðθÞ, where θ is the joint positions of the robot arm and determines the external magnet's pose T r m .Thus, we have the following expression where P m = (x m , y m , z m ) and R m are the central position value and orientation matrix of the external magnet in the robot frame {r}, respectively.The MAS and the advancer formed a cooperative system, in which the controlled trajectory q(t) was input including as θ(t) and lðtÞ, while the desired magnetic tip trajectory r(t) was the output of the cooperative system in the frame {r}.The q(t) and r(t) can be expressed as qðtÞ ¼ lðtÞ θðtÞ (38) where the dimension of r(t), which we denote as d, is generally six or less.Since the orientation of the magnetic tip was determined by the orientation angle and the pitch angle, d was equivalent to five after considering the magnetic tip position dimension.We denoted p as the DOF of the advancer, which was one in this problem.Thus, if the robot had j DOF, the dimension of q was p þ j.
The mapping relationship of the entire cooperative system from the configuration space to the task space can be expressed as Derivation of the aforementioned formula could get the velocity relationship by the Jacobian matrix j(q) ∈ R d Â (p þ j) r : ¼ JðqÞq : (41)   where r : is the spatial velocity vector of magnetic tip, and q : is the control velocity vector of the coordination system.We wished to obtain a control trajectory q(t) relative to a desired trajectory of magnetic tip r(t) by multiplying the pseudo-inverse of the Jacobian J † ðqÞ.

External Magnetic Field Characterization
In our design, the nonuniform external magnetic field is produced by a cylindrical magnet with a dimension of D50 Â L25 mm.The magnetic field distribution in the same radial section is isotropic, so we characterize its 2D magnetic field analytically, and here we choose the Y-Z plane.The external magnetic field is simulated by COMSOL Multiphysics v.5.6 (COMSOL Inc., Stockholm, Sweden).In the Y-Z axial section, we build the finite-element model (FEM) to estimate the magnitude of the magnetic flux density of the external magnet along its mid-center axis and the three nonzero magnetic field gradients (see Figure S3a-d, Supporting Information).Also, the variation relationship between the magnetic field strength in the axial direction Bz and the normal working distance d from the magnetic field center-of-mass point to the magnetic tip action point is depicted in Figure S3e, Supporting Information, where the magnetic field decays rapidly and tends to zero as the normal working distance increases.

Steering Characterization of the MG
We used the ODE solver module bvp5c [40] in MATLAB 2021a (MathWorks, Natick, MA, USA) to solve the magnetic tip kinematic ODEs described by BVP.To predict the complex nonlinear deformation behavior of the magnetic tip at each target position point, we need to reconstruct its shape, i.e., solve the pose matrix for each state point along the arc length of the magnetic tip rod.And by solving the BVP equation of the Cosserat-rod model in this paper, we are able to obtain the complete shape of the magnetic tip rod.Referring to Figure 2a, we place the magnetic tip alone in the X-Z plane with the initial state of the magnetic tip positive along the Z-axis and its base constrained to remain stationary and set as the origin.the external magnet center of the MAS is aligned directly above the magnetic tip apex with an orientation angle of 90°and moved uniformly downward along the X-axis at a distance range of [180 mm, 40 mm], and the magnetic tip is deflected counterclockwise to the target state from its initial state under the magnetic actuation, and the position curves of magnet and magnetic tip are shown in Figure 3a.By solving the aforementioned BVP, the shape of the magnetic tip at each target position is reconstructed (Figure 3b).
The quantitative analysis between steering angle, magnetic field strength, and working distance can verify the validity of steering model and magnetic actuation model of the magnetic tip.In the simulation calculations, the mechanical stiffness EI is calculated using Equation (6), where E contains the Young's modulus of the silicone tube and the NdFeB magnet at the magnetic tip.Since the two materials are relatively common, based on the parameters provided by the purchasing manufacturer, the E of the silicone tube is 1.2 Â 10 9 Pa, and referring to the literature, [9] the E of the N52 grade NdFeB is 1.6 Â 10 11 Pa.The equivalent stiffness EI of the magnetic tip is calculated as 5.46 Â 10 À4 Nm 2 referring to Equation (6).For the torsional stiffness GI of the magnetic tip, the value is set as 6.28 Â 10 À4 Nm 2 with reference to reference. [41]To verify the validity of the aforementioned parameters, we first measured the values of the steering angle of the magnet at different distances from the tip.Then, according to the established kinematic model of magnetic tip, we calculate the magnetic field strength corresponding to different distances and import it into the model to calculate the steering angle values of the simulation, and the results of the comparison between the two are shown in Figure 3c.From the figure, we can see that the simulation is closer to the experimental values, which confirms that the parameters EI and G are feasible.
While for magnetic field, we compare the magnitude of the magnetic field strength B obtained by simulation and experiment at different distances from the magnet and the tip, respectively, and give the error curves for both in Figure 3d.From the figure, it can be seen that the trend of B calculated by modeling is consistent with the experimental value, and the error is within the acceptable range, which can satisfy the use for the time being.Moreover, the steering angle of the tip in Figure 3c is relatively close in the experiment and under simulation, again indicating the validity of the model.If the model accuracy needs to be improved in the subsequent closed-loop control, the computational model can be corrected by obtaining calibration parameters through nonlinear fitting for the experimental values.
In addition, as the working distance between the external magnet and the magnetic tip is gradually shortened, the magnetic field at the point of action of the magnetic tip magnet becomes stronger (Figure 3d).As a result, the magnetic force and magnetic torque will increase as the magnetic field, thus making the steering angle also gradually increases, and the closer distance, the faster growth (Figure 3c).Finally, the magnetic tip deflects from the initial 0°to 90°, and the experimental state of the four angles is shown in Figure 3c, which show that the trend of experimental and simulated values is basically the same.In Figure 3e, we marked the four magnetic field strengths as B1, B2, B3, and B4 corresponding to the aforementioned four steering angles of the experiment.In COMSOL, a 2D FEM of the magnetic tip is built and the four magnetic field values are applied to the FEM, and the shape of the magnetic tip obtained by simulation is shown in Figure 3f.In short, the previous experiments confirm that the proposed models are reliable, and can predict the steering angle of magnetic tip and reconstruct its shape so that predict the complex nonlinear deformation of the magnetic tip.

Trajectory Following in Three Different Vascular Phantoms with Multiple Angles
In this section, we carry out experimental validation of trajectory following in three different vascular phantoms with multiple turning angles, including a 2D Phantom 1 and 3D Phantom 2 with 90°corners and a complex tree-structured circulatory Phantom 3 (Figure 4b).The whole trajectory following experiment was executed based on Robot Operating System (ROS Neotic), which contains four groups of control nodes, namely, motion command generator node, robot arm controller node, propulsion motor controller node, and camera measurement node.Among them, there are two input modes for motion commands, i.e., manual input and automatically generated preplanned trajectories.For the remote operation mode, we publish the commands via joystick, which are then subscribed by the robotic arm controller node and the propulsion motor controller node to execute manual commands.For the automatic operation mode, the discrete points of the preplanned trajectory will be published by the motion command generator node and subscribed by both controller nodes to automatically execute trajectory following.Here, the scripts of all nodes are implemented based on a mixture of python and Cþþ programming, while the camera measurement node also makes use of the OpenCV library to process the measured images and finally calculate the magnetic tip's poses in real time.The trajectory following program is executed in Linux Ubuntu 20.04 environment.The experimental setup is shown in Figure 4a.Three continuous curved vascular phantom with multiple angles were fabricated based on 3D-printing technology using transparent resin material, with inner diameters ranging from 10 to 13 mm and total length ranging from 160 to 180 mm.As shown in Figure 4b, we designed three desired trajectories marked as red lines, and set S1-S3 as starting points and G1-G3 as target points, then discretized the trajectories to obtain a set of position and orientation angles of the magnetic tip.Next, the trajectory planning method proposed in Section 2.2.5 was applied to plan the trajectory pose of the external magnet and the propulsion velocity of the guidewire advancer based on the desired trajectory of magnetic tip in MATLAB.
In the experiment, the MAS controls the end magnet to follow the preplanned trajectory, and the guidewire advancer provides forward power for the MG according to the given velocity.With the cooperation of steering and propulsion, the MG is enabled to follow the desired magnetic tip trajectory from the starting point S1-S3 through the straight and turning sections, and it finally reaches the target point G1-G3 (Figure 4b), and the detailed process is shown in Video S1, Supporting Information.The respective desired and actual trajectories of the magnetic tip and magnet in three following experiments are compared in Figure 4c-e, and it can be seen that the magnetic tip is basically able to follow the desired trajectory.Then, the respective desired and actual orientation angles of the magnetic tip and magnet with time are shown in Figure 4f-h, among which all the angles are absolute angles of the magnetic tip with respect to the initial position of guidewire.In the latter two experiments, the maximum deflection angle of the MG was able to approach 100°, as shown in the first peak of the tip angle curve in Figure 4g.Moreover, the respective position errors of the trajectory following are also shown in Figure 4i, and the mean position errors of the magnetic tip and the external magnet are 1.68 AE 0.31 and 1.81 AE 0.52 mm for first experiment, 1.88 AE 0.96 and 1.99 AE 0.77 mm for second experiment, and 1.52 AE 0.45 and 1.55 AE 0.46 mm for third experiment.And the respective angle errors of trajectory following are also shown in Figure 4j, and the mean angle errors of the magnetic tip and the external magnet are 2.36 AE 1.36°and 2.3 AE 1.47°f or first experiment, 2.44 AE 1.79°and 2.29 AE 1.44°for second experiment, and 2.57 AE 1.62°and 2.07 AE 1.36°for last experiment.
In addition, we have also plotted the curves of magnet and tip positions, relative distances, and actual velocities of tip over time in the three trajectory following experiments, as shown in Figure S4, Supporting Information.As shown in Figure S4d-f, Supporting Information, it can be seen that the propulsion velocity can be fast when passing through a straight section, while a slow velocity is required when passing through a turning section, so that the magnetic tip can respond to the external magnetic field and deflect to the desired angle in time.And the mean propulsion velocities in three experiments are 2.93, 4.63, and 6.06 mm s À1 , respectively.In summary, the previous experiments confirm that after the MAS and the advancer automatically execute the planned commands, the MG can be navigated to follow the desired trajectory through the phantom.

In Vitro Navigation validation of the MCGRS in 3D Vascular Phantom
To verify the potential of the proposed MCGRS for clinical applications in interventional procedures, we conducted an experimental validation in a complex 3D Vascular Phantom.The experimental setup is shown in Figure 5a.In addition to the modules mentioned in the previous system overview, we also customized a 3D vascular phantom and a flow pulsation pump (FPP) by commissioning a medical company (TRANDOMED, Ningbo, China).The phantom is based on the real vessel size of the human body and is manufactured by 3D printing technology using transparent silicone (overall length of 959 mm).The whole phantom includes starting from the femoral artery, going all the way through the aortic arch above the heart, the carotid artery, and finally reaching the intracranial artery.For the intracranial artery part of the phantom, which is difficult to access and has complex vascularity, we selected four vascular pathways labeled as 1, 2, 3, and 4 (Figure 5b), and the anatomy and nomenclature of the intracranial artery of the adopted phantom is shown in Figure S5, Supporting Information.While the FPP acts as a heart, the flow velocity and pulsation frequency of the simulating fluid are provided to simulate the blood circulation in human arteries.
Since the 3D vascular phantom is made of soft silicone (different from real blood vessels), the guidewire will be subjected to a lot of resistance when it touches the vessel wall for a long distance, when which the main body of the guidewire is twisted, and a great deal of tension is created between it and the vessel wall.To overcome the resistance and tension, the guidewire is simultaneously propelled and rotated, which may result in a sudden and rapid ejection when the propulsive force is too great.To reduce the occurrence of this phenomenon, we mixed the surface lubricant with water at a ratio of 1:100 to obtain a blood simulation fluid, which was then injected into the phantom through the FPP to make the advancement of guidewire smoother and more continuous and prolong the phantom's service life.
Next, we conducted navigation experiments for each of the four pathways in turn (see Video S2, Supporting Information, for more details).In each experiment, we controlled the end magnet position of the MAS and the propulsion velocity of the guidewire advancer synchronously in real time by operating the Xbox joystick, thus achieving the coordination of steering and propulsion for the MG.The detailed navigation processes are shown in Figure 5c (video frames after compression fast-forward 5x), which showed that the MG can be manipulated to actively steer and navigate through the four pathways to the target point G1, G2, G3, and G4 by remotely controlling MAS and guidewire advancer to work together.Herein, the 3D vascular phantom is more complex than the vascular phantoms in Section 3.3 due to its progressively decreasing vessel diameters, many turns, short tortuous paths, and the presence of multiple aneurysms.Therefore, in order for the magnetic tip to respond to the external magnetic field and deflect to the desired angle in time, we perform relatively slow propulsion velocities in the experiments.The mean propulsion velocity was between 1.5 and 2.5 mm s À1 for all four pathways, and navigation time from the starting point to the target point was within 1-2 min (actual time before video compression).
For interventionalists, the real intervention velocity is related to the operator's level of experience, the size of the vessel diameter, and the complexity of the tortuosity, etc.There is no standardized velocity range.However, for intervention time, a skilled surgeon may generally take 1-2 min for a conventional pathway, and 3-5 min for a particularly tortuous and complex pathway.Therefore, the proposed system MCGRS can basically match the intervention time of the conventional pathway.

In Vitro Verification of the MCGRS under X-ray Fluorescence Imaging
Real interventional procedures are performed in a catheterized operating room equipped with medical imaging equipment (e.g., digital subtraction angiography, DSA), during which the interventional surgeon needs to observe the position of the interventional instrument in the patient's vasculature under real-time X-ray fluorescence imaging to judge and perform subsequent operations.Therefore, further validation for the fluoroscopic effect of the proposed MG under DSA and integration effect of the complete system with DSA is needed so that the feasibility of the system for clinical application is validated.
To conduct the integration verification in the interventional catheterization room of the partner hospital, we customized a mobile cart for mounting and moving the UR robotic arm, as well as facilitating the deployment of the host and hardware modules.After the entire system was transferred to the catheterization room, we performed installation tests and adjusted the position of the system relative to the DSA and the surgical bed, and further adjusted the tilt angle of the DSA.Finally, the experimental scenario was arranged as shown in Figure 6a.The DSA used in the experiment was an ultrahighend large flat-panel angiography system (Azurion 7M20, Philips, Netherlands).We selected a path in the 3D vascular phantom mentioned earlier for testing the fluoroscopic and navigational effects as shown in Figure 6b.With the help of fluorescence imaging of X-rays, by operating the Xbox joystick outside the catheterization room, we were able to remotely control the proposed MCGRS to manipulate the MG to navigate through the selected path in real time.The navigation process of the MG under DSA is shown in Figure 6c (see Video S3, Supporting Information, for more details), where it can be seen that the magnetic tip can be fully visualized (red rectangle box) and the proposed system can be compatible with DSA and surgical bed.The mean propulsion velocity of the MG in this experiment is 1.2 mm s À1 .
In the experiment, although the end-effector of the robotic arm was exposed to X-rays and partially obscured the vascular model, the tip of the guidewire was not obscured during the whole process, and the interventionalist was still able to observe the state of the tip in real time with the help of X-rays.In addition, the interventionalist could still operate the MG safely by rotating the DSA.In fact, by coordinating the relative positions of the DSA and MAS, we are able to keep the tip of the MG visible so that this interference does not obstruct too much with its real-time tracking.In the future work, we will further improve the system and propose control strategies to solve the aforementioned problems.

Discussion
At this stage, our proposed system and methods have been preliminarily verified by conducting a series of experiments in 2D and 3D vascular phantoms.The proposed MCGRS can provide active steering and automatic propulsion capabilities for the MG by remote control, which can be manipulated to rapidly select the correct path in the tortuous and complex vessels.As a result, our work will build the foundation for automatic control of MG, which will help to improve the autonomy of the magnetically manipulated interventional robotic systems.
In terms of modeling, since the MG is a slender and soft structure with inconsistent stiffness between the main body and the magnetic tip, there are many difficulties in navigating through complex narrow vessels, such as strong nonlinear perturbations, incomplete kinematic constraints, and unfixed soft continuum morphology, which pose great challenges for its accurate modeling and control.
In future works, we will consider that several modeling factors such as frictional resistance, blood flow perturbations, and vascular elasticity, further improve our system and experimental setup, realize closed-loop tracking control, and also conduct some animal experiments such as arterial intervention in pigs, which will better validate the feasibility, safety, and application potential of the proposed system for clinical applications.

Conclusion
In this work, we propose an MCGRS with steering and propulsion capabilities based on adequate modeling and trajectory planning methods.The steering kinematics of MG is first modeled by constant curvature theory.Then, a continuum mechanics model is built to predict the deformation of the magnetic tip by combining the dipole model and the Cosserat-rod model.Next, the magnetic field characterization and steering characterization experiment are accomplished, and confirm that the proposed models are reliable, and can predict the steering angle of magnetic tip and reconstruct its shape so that predict the complex nonlinear deformation of the magnetic tip.Moreover, we develop a trajectory planning algorithm to obtain the trajectory of the external magnet and the velocity of the advancer for the desired magnetic tip trajectory, and also carry out experimental validation of trajectory following in three vascular phantoms.The experiments show that after the MAS and the advancer automatically execute the planned commands, the MG can be navigated to follow the desired trajectory through the phantom.As a result, our models and methods will build the foundation for automatic control of MG, which will help to improve the autonomy of the magnetically manipulated interventional robotic systems.Finally, to further validate the application potential of the proposed MCGRS in VIS, two in vitro navigation experiments are implemented in a 3D vascular phantom, which show that the MCGRS can be remotely controlled to manipulate the MG to actively steer and reach the target site.Therefore, our method can help the interventionalists to rapidly establish therapeutic access so that shorten the procedure time.In the future, we hope that the proposed system MCGRS can assist interventionalists in more controllable manipulation of interventional devices such as MG and minimize radiation exposure.

Figure 1 .
Figure 1.Schematic diagram of interventional surgery and the proposed magnetically controlled guidewire robot system (MCGRS).a) Schematic diagram of magnetic guidewire (MG) intervention.After entering from the puncture point of the femoral artery, the MG passes through the aorta to reach the aortic arch under an external magnetic field, and then is magnetically actuated to deflect precisely into the target vessel to complete superselection, thus establishing therapeutic access for subsequent interventions.b) Conceptual diagram of the future clinical application of the MCGRS.With the assistance of the X-ray imaging system, the magnetic actuation system (MAS) provides active steering capability for the MG, and the guidewire/catheter advancer (GCA) is able to advance, retract, and rotate the guidewire/catheter, and the teleoperation console (TC) is used to control the MCGRS.c) Diagram of the fabrication process of the MG.

Figure 2 .
Figure 2. Schematic diagram of the modeling of the MG.a) Schematic diagram of the overall magnetically actuation kinematic modeling of MG and the analysis schematic of propulsion kinematics of guidewire advancer.b) Schematic diagram of the Cosserat-rod mechanical model of the magnetic tip.c) Schematic diagram of the spatial coordinate transformation relationship between MG and other parts of the system.

Figure 3 .
Figure 3. Steering characterization.a) The position curves of magnet and magnetic tip.b) The shape reconstruction of the magnetic tip rod at each target position.c) The relationship between the magnetic tip steering angle and the working distance, and the magnetic tip state at four angles.d) Comparison of the magnetic field strength of experiment and simulation along the central axis under different working distance, also their error.e) Relationship between the steering angle and the magnetic field strength along the central axis.f ) The magnetic tip shape obtained by simulation at four magnetic field strength.

Figure 4 .
Figure 4. Analysis of experimental results of trajectory following.a) Diagram of trajectory following experimental platform.b) Schematic diagram of three vascular phantoms of different sizes and shapes, with the red lines showing the respective desired trajectories, S1-S3 as starting points and G1-G3 as target points.c-e) Comparison of the desired and actual trajectory of the magnetic tip and external magnet for the three trajectory following experiments, respectively.f-h) The desired and actual orientation angles of the magnetic tip and magnet with time for the three trajectory following experiments, respectively.i) The position error of the magnetic tip and external magnet respectively for the three trajectory following experiments, respectively.j) The angle error of the magnetic tip and external magnet for the three trajectory following experiments, respectively.

Figure 5 .
Figure 5.In vitro navigation validation of the MCGRS in 3D vascular phantom.a) The experimental setup of the MCGRS in vitro validation.b) The four vascular pathways labeled as 1, 2, 3, and 4 in 3D vascular phantom are selected to validate the navigation capability of the MCGRS (along the red line).c) Results of in vitro navigation experiments of the MCGRS through the four vascular pathways to the target point G1, G2, G3, and G4, respectively.The state of the magnetic tip at each of the four target points is shown in the red rectangular box.

Figure 6 .
Figure 6.In vitro verification of the MCGRS in 3D vascular phantom under real-time X-ray fluorescence imaging.a) System integration scenario of the MCGRS in the interventional catheterization room.b) Vascular paths for system integration testing, and G is the target point.c) Intravascular navigation process of MCGRS under real-time X-ray fluorescence imaging.The perspective effect of the magnetic tip at key points is shown separately in the red rectangular box.