A Realistic Elastic Rod Model for Real-time Simulation of ... - here

A Realistic Elastic Rod Model for Real-time Simulation of Minimally Invasive Vascular Interventions Wen Tang1, Pierre Lagadec1, Derek Gould2, Tao Ruan Wan3, Jianhua Zhai2, and Thien How 2 1

School of Computing, Teesside University, UK; 2 Radiology Department, Royal Liverpool University Hospital, UK; 3School of Computing, Informatics and Media, Bradford University, UK [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]

Abstract: Simulating intrinsic deformation behaviors of guidewire and catheters for interventional radiology (IR) procedures, such as minimally invasive vascular interventions is a challenging task. Especially real-time simulations for interactive training systems require not only the accuracy of guidewire manipulations, but also the efficiency of computations. The insertion of guidewires and catheters is an essential task for IR procedures and the success of these procedures depends on the accurate navigation of guidewires in complex 3D blood vessel structures to a clinical target, whilst avoiding complications or mistakes of damaging vital tissues and blood vessel walls. In this paper, a novel elastic model for modelling guidewires is presented and evaluated. Our interactive guidewire simulator models the medical instrument as thin flexible elastic rods with arbitrary cross sections, treating the centerline as dynamic and the deformation as quasi-static. Constraints are used to enforce inextensibility of guidewires, providing an efficient computation for bending and twisting modes of the physically based simulation model. We demonstrate the effectiveness of the new model with a number of simulation examples. Keywords: surgical simulation and training, interventional radiology, physically-based computer animation, discrete elastic rod model, medical education

1 Introduction Interventional radiology is a treatment paradigm, using medical imaging techniques such as x-ray, computed tomography (CT), ultrasound and touch to guide needles, guidewires (specialized wires) and catheters (tubes) to treat a range of conditions. Benefits of such procedures are reduced postoperative pain, hospital stay, cost and complications as compared with open surgery [1]. Yet high levels of expertise

are required to attain these benefits, presenting challenges to the safety and cost effectiveness of training in patients. Interactive virtual reality systems can be used o train skills and experience complications in safety remote from patients. Studies show that surgical skills learnt through computational simulators can significantly decrease the time of carrying out procedures and reduce the frequency of medical errors by up to six fold compared to traditional training methods [2]. One of the key technical aspects central to virtual reality based training systems for interventional radiology is realistic and real-time simulations of medical instrumental behaviours of guidewires and catheters within complex 3D vascular structures, since the success of these procedures depends on how guidewires are accurately maneuvered to a clinical target while avoiding complications or mistakes of damaging vital tissues and blood vessel walls. By twisting the proximal end of guidewires as it is inserted, a doctor can steer the tip of the wires to reach a clinical target while traveling through blood vessels. It is not easy to learn how to control steerable thin specialist wires, thus realistic training simulations if developed appropriately will accelerate this learning process [3]. We present a new simulator that models deformations of guidewires, and frictions and collisions between blood vessel walls and the guidewires. It realistically simulates bending and twisting of the instruments with minimal stretching. Bending occurs as thin flexible guidewires travelling through 3D complex vascular structures. Difficulties in simulating interactions between guidewires or catheters and blood vessels are due to several impediments: a) the fixed geometrical 3D mesh model of vascular structures produced by the rotational X-ray imaging technology do not accurately support the precise computation of contact forces and guidewire steering; b) there is notably lacking of physically based models for curve-like long elastic bodies to realistically simulate deformations characterized by bending and twisting with minimal stretching; c) the simulation must run at interactive rate. To address these challenges, we introduce a novel algorithm for modeling guidewires and catheters. This efficient algorithm simulates guidewire deformations in terms of bending and twisting at interactive frame rate. In order to facilitates the guidewire manipulation, several applicable techniques are also developed to reduce computation time for physically based simulations as well as taking into account of complex 3D vascular structures. We introduce a centerline algorithm that efficiently relocates and creates hierarchy navigation nodes along a curvilinear path in 3D vascular structures, enabling the simulation to apply multicontact collision forces and friction forces along the guidewire shaft while still maintaining low level of surface triangle count for efficient computation. A simple to implement and efficient to execute elastic rod model which is build on a recent model

introduced in [4] allows to incorporate dexterity interactions of pushing and twisting with greater accuracy and realism as compared with particle based models [5]. The finite element based model [6] had low accuracy in deformation computation due to the use of a linear finite element method. A previous elastic energy based model [7] was inherently of high cost in computations since a global optimization scheme was employed in order to find the solution of energy based deformation equations, thus it was a non-realtime solution for guidewire simulation. In contrast, our proposed new guidewire model is a real-time solution with greater accuracy by using a discrete geometry formulation. Speeding up computation scheme used in our simulator also includes accelerating the computation of stretching and bending of discrete centerline deformations with adaptive sampling, subjecting to curvatures of the curvilinear path of the blood vessel structures. Together, our proposed algorithms enable the simulator to model guidewire and catheter insertions realistically at interactive frame-rate in 3D vascular structures. The simulator uses material properties for guidewires and catheters measured from real guidewires, making it more challenge than more compliant materials for which numerical stability is usually reported. We present simulations for interventional radiology procedures with guidewires of different stiffness and simulations of guidewires reaching targets located at different vascular branches with varying geometry structural complexity.

2 Background and Related Work 2.1 Guidewire Simulations Main characteristics of guidewire dynamics include high tensile strength and low resistance to bending. Doctors steer a guidewire to reach a clinical target by pushing, pulling and twisting at the proximal end of the guidewires. For 3D simulations of guidewire insertions, modeling the wire like deformable object is a key difficulty. To model guidewire dynamics, Cotin et al proposed a multi-rigid-body linkage connected by spherical joints [8]. Guidewires were also modelled as 1D dynamic splines controlled by different constraints such as fixed contact points at predefined locations of vascular structures and only bending energy term was incorporated into the model [9]. A set of finite beam elements was used to simulate catheters based on linear elasticity [10]. Despite the ability of taking into account of various crosssectional geometries and mechanical properties of the medical devices, the linear elastic strain method of the beam model was incapability of simulating large non-linear geometric nonlinearity of deformations, a physical property that guidewires as long flexible curve like object naturally possesses and such non-linear deformations occur during insertions.

Figure 1: Guidewire model: the discrete setting of elastic rod is used to describe a guidewire as a set of vertices x0 , x1,…xn+1, which are connected by edges, e0, e1,….en (left); a guidewire insertion starts from the tip of the guidewire by turning and pushing at the end of the wire (right). An incremental finite beam element model was introduced with an optimization scheme operating on substructure decomposition to speed up the computation [6]. This beam model was further improved by dynamically changing its material properties to locally describe the combination of guidewires and catheters. Assuming no elongation and perfect torque control, a model that was similar to [11] but integrated more complex bending energies with respect to frictions and springs was described in [7]. This model only computed bending deformations, moreover, the equations of motion were solved with an iterative optimization convergence. Therefore, the speed of simulation is an issue for interactive applications. Nonetheless, this model was capable of demonstrating intrinsic curvatures of catheters with good accuracy.

2.2 Elastic Rod Models The study of elastic rod model is an active field in mechanics [12], numerical analysis [13], and geometry [14]. In computer graphics community, Pai [15] applied Cosserat theory to simulate elastic rod as a boundary value problem, while Bertails et al [16] used a piecewise helical discretization to the elastic rod model in order to improve the running time execution with few elements per strand. A recursive algorithm was proposed to further increase the speed of the computation to this model [17]. Maintaining efficient computation of complex contact and collisions is also a key aspect of 3D simulation of guidewire insertions. Elastic models presented in these papers described the centerline by a sequence of edges connected by linear and torsional springs, facilitating the simulation of complex contacts. Our simulation builds on the method of Bergou et al [4], who devised a discrete differential geometry to physical modelling of elastic rods.

3 Methods and Algorithms Guidewires and catheters are visualized as 1D elastic rods represented as thin and fixed centerlines that tend to bend or twist rather than stretch. We consider naturally straight rods with isotropic bending responses and denote curvilinear guidewire paths in 3D vascular structures as cubic spline

where Δt is the time step, 0≤β≤1 and 0≤γ≤1 are constants for approximately calculating the integration. In the simulation, we set β = 0.25 and γ =0.5. Solving (3) to find accelerations

curves C and denote the guidewire by a sequence of edges E, including additional edges that represent the portion of the guidewire outside the vessel, as shown in Figure 1. Multicontact frication forces are checked between the vessel wall and the guidewire shaft. Each node that is in contact with the vessel wall is in a dynamic frication state, sliding against each other. The guidewire tip, when it is moving through the interior of the blood vessel and met with obstacles such as blood clots, encounters much greater resistance than accounted for by dynamic friction alone. We know of no prior simulation that handles interactive simulation of blood clot resistance coupled with guidewire deformation as we do here. Our main contributions are the application of the new elastic rod model for medical simulations, the multi-contact friction algorithm that is independent of how the guidewire forces are computed. Algorithm 1 summaries the execution of force computations of one simulation step. During the simulation, we maintain for each node of the guidewire a material coordinate (recording the geometry of the un-deformed mesh) and a world coordinate (recording the deformed mesh). Let uik , xik , vik , aik ∈ ℜ 3 denote vectors of

are the Jacobian matrices of where ∂F ∂x ,∂F ∂v ∈ ℜ the force with respect to positions and velocities, evaluated at ( x k , v k ) . Substituting (1) and (2) into (4) yields a sparse linear system for the accelerations of guidewire nodes.

the material coordinates, the world coordinates, the velocity and the acceleration of the ith node at the time step k, omitting the node index for all the nodes in the guidewire.

Aa k +1 = b ,

a k +1 ∈ ℜ 3 n .

F ( x k +1 ,v k +1 ) = Ma k +1

(3)

where M ∈ ℜ 3 n×3 n is the mass matrix and F (.) ∈ ℜ 3 n is the sum of all internal forces such as stiffness and damping forces and external forces such as gravity and frictions. Equation (3) is linearized with one Newton-Raphson iteration by solving

F( x k ,v k ) +

∂F k +1 k ∂F k +1 ( x − xk ) + ( v − v ) ≈ Mak +1 (4) ∂x ∂v 3 n×3 n

(5) k +1

3.1 Implicit Time Integration Let n be the number of nodes in the guidewire mesh. Newmark’s method of direct integration is used for finding the node positions x ∈ ℜ 3 n and the velocities v ∈ ℜ 3 n over time.

⎛⎛ 1 ⎞ ⎞ x k + 1 = x k + Δ tv k + Δ t 2 ⎜⎜ ⎜ − β ⎟ a k + β a k + 1 ⎟⎟ ⎠ ⎝⎝ 2 ⎠

v k +1 = v k + Δt ((1 − γ )a k + γa k +1 )

(1)

(2)

Algorithm 1 Guidewire simulation (one time step) 1: Compute guidewire forces F, and Jacobians ∂F/∂x and ∂F / ∂v for

the Guidewire ( §3.2)

Having solved for a , we obtain x k +1 , v k +1 from equations (1) and (2). The matrix A is a structured sparse matrix, because each node has nonzero entries for the two nodes before and after it on the guidewire.

3.2 Force and Jacobian Computations We compute bending and twisting forces with discrete elastic rod model formulations [4]. Assuming that the dynamic deformation model of guidewires and catheters is a constrained system, in our application, the inextensible of the guidewires is enforced by the penalty method [18]. A constraint is formulated as an energy function E ( x ) : ℜ n → ℜ with E ( x ) = 0 if constraint is satisfied. Differentiating this energy function with respect to position x yields constrained forces Fc that are compensatory stretching forces added on the system. The total of internal forces of the system is the sum of bending forces, twisting forces, and stretching forces.

2: Solve dynamic system , yielding the friction configurations and the nodal accelerations in guidewire ( §3.3 and §3.4) 3: Update the positions and velocities of the guidewire (§3.1) 4: if the centerline curvature is greater than threshold then 5: Re-mesh the guidewire (§3.4.3) 6: Re-parameterize the guidewire (§3.4.4)

3.3 Contacts and Frictions between the Guidewire and the Blood Vessel Wall Another important difficulty in simulating guidewires or catheters navigation is solving the contacts between the guidewire device and the blood vessel wall, because sliding occurs at the point of contact. In our simulation, each node i of the device that is inside the vessel will have either a free friction state when it is not collide with the vessel wall, or a

dynamic friction state, while sliding against the vessel wall. All the nodes outside the vessel have free friction states. After detailed discussions with our research collaborators who are interventional radiologist, it is suggested that in diseased vessels, deformations of the blood vessel wall poked by the medical devices is a less prominent feature because diseased vessels are normally hard coated with hard surfaces. In contrast, global deformations of blood vessels, such as bending of the vessel as a whole induced by deformations of surrounding tissues, are important factors that should be taken into account in our future development. Therefore, we treat the blood vessel wall as an un-deformable mesh same as in [6-11]. Further improvements will be to simulate the surrounding soft tissues as deformable surface using tetrahedrons [19, 20, 21]. Friction plays an important role in guidewire and catheter manipulations, since reducing frictions helps the detection of curvatures and shape turns in blood vessels [22]. A considerable amount of work in medicine was conducted in order to reduce friction contacts by coating the devices with different materials [22]. The friction coefficients between various different types of guidewires/catheters and the surface of blood vessel walls are of interests in our simulation applications. We use the results presented in [23] as our guidance in choosing friction coefficients. Signorini’s law is used to resolve the contact between the guidewire and the blood vessel inner surface. To find contact forces induced by the penetration distance d between the contact points of two colliding objects, the contact collision force is evaluated as f N = −κdN , where κ is a chosen constant, N is the unit vector of the contact normal of the blood vessel surface at the contact point. For each contact node, friction force is along the tangential direction T. Friction force on each contact node is evaluated as fT = μRiT , where µ is the friction coefficient with reference to [23] and R i ∈ ℜ 3×3 is a rotation matrix transforming from the local coordinates to world coordinates to the guidewire node i. The total contact force is the sum of the collision force and the friction force.

f = f N + fT ,

(6)

Coulomb’s friction law describes behaviour in tangent contact space.

fT < μ f N

T T

fT = − μ f N

the

macroscopic

( stick ) T T

( slip )

We treat all contact nodes as in the dynamic friction state with slip frictions. The coefficient of friction depends on the surface characteristics and on the medium between the two contact objects as well as the stiffness of the guidewire.

Although there were studies shown the coefficient of friction of different types of guidewires with various coating materials between catheters and guidewires, we are dealing with the frictions between the surface of guidewires and the inner surface of the blood vessel. We choose 0.1≤ µ ≤ 1.0 as in [23]. If node i is the guidewire tip and it is colliding with a blood clot. We add to a resistance force f R that varies according to the type of blood clot (soft or hard) and the type of the guidewire tips. We also apply an implicit viscous frictional force induced by the relative velocity between guidewires and the vessel wall as f R = −λv to the guidewire node.

3.4 Collision Detection and Guidewire Manipulations 3.4.1 Collision detections At each simulation step, the contact state of each node has to be determined via an efficient collision detection process. To detect the contacts between the guidewire and the blood vessel wall, we use AABB (axis-aligned bounding boxes) tree data structures to facilitate real-time collision detections. The AABB tree component offers a static data structure and algorithms to perform efficient intersection and distance queries against sets of finite 3D geometric objects. We attach a spherical implicit geometry as a bounding sphere that has the same radius as that of the guidewire to each of the elastic rod nodes. Bounding spheres simplify signed distance functions as straightforward analytic expressions without the need to approximate signed distance with a numerical method. This method has the efficient of O(N log N) with the first-order accurate, where N = n3 is the number of nodes in the three dimensional guidewire mesh. As geometry primitives, geometrical surface dataset of the guidewire is pre-computed as a hierarchy AABB tree structure, consisting of all the AABB nodes. A heuristic scheme that defines the maximum depth of the tree and the minimum number of vertices that each node should contain is used to determine the best collision detection performance. Instead of using the ‘longest’ axis to construct the hierarchy AABB nodes, the ‘best’ axis creates a well-balanced tree structure, eliminating the problem of having empty leaf nodes or nodes that contains disproportional large number of geometry primitives.

3.4.2 Collision responses Our algorithm detects any guidewire that penetrates the blood vessel wall and applies impulses to push it to the surface. Assuming that sample nodes do adequately resolve the guidewire or catheter objects, hence the resolution of the nodes is adequate for checking the guidewire object inside the blood vessel object.

Therefore a friction force, in the direction of the pre-friction relative tangential velocity but opposite to it, has at the most magnitude of µfN.

3.4.3 Guidewire manipulations

Figure 2: Guidewire navigation: The main vessel consists of three sub vessel branches (left), the guidewire arrives at the junction node and selects one of the branch vessel as highlighted in green (right). Considering a collision between a node on guidewire node p with velocity vp and the blood vessel collision object. At the spatial location of p, the collision detection pipeline detects a set of polygon primitives on the blood vessel wall. We compute the outward normal N using the average normal of these primitives, given that the motion of the blood vessel is static. The penetration distance d along N is computed to resolve the collision of the point p with the blood vessel object. The anticipated collision/impulse distance during a time step ∆t is thus computed as d = -∆t vpdot(N), and we predict that the node will move to a new position inside the blood vessel wall as:

p new = p − Δt v p ⋅ N The second term in this equation accounts for the relative motion between the node and the blood vessel. We update the normal and tangential components to obtain:

vp

new

new = v new p ,N N + v p ,T

Following the collision detection, the new normal velocity is calculated as

v

new p ,N

= v p ,N + p

new

/ Δt

which predicts the node to lie exactly on the inner surface of the vessel wall after time ∆t. The collision force of magnitude new

| v p ,N | is applied on the opposite direction of the normal. The friction is incorporated into the relative tangential velocity update v new p ,F = max( 0 ,1 − μ

v new p ,N − v p ,N v p ,T − ( v p − v p ,N N

)( v p ,T − ( v p − v p ,N N )

where µ is the friction coefficient of the contact. The final post- collision velocity of the node is : new v new p ,T = v p − v p ,N N + v p ,F

Interventional radiology devices such as guidewires and catheters have characteristics of being thin flexible but extremely long objects. By twisting the proximal end of the devices as they are inserted, the tip of the guidewires or catheters needs to be steered to reach a clinical target while traveling through complex vascular structures. In order to reach targeted locations that are sufficiently far from the device’s insertion point, effective planning algorithms are required to direct or give guidance to the guidewire steering along a path or along a selected vascular branch among other branches that sprout out from a main vessel. Previous simulations have rarely dealt with the issue of guidewire navigations and path planning. To our knowledge, in this paper we are the first to present a solution to address path planning of guidewire navigations. We construct centerline paths for each vascular branch of a vascular dataset by utilizing a hierarchy tree structure, consisting of a set of node as control points for cubic splines, as shown in Figure 2. Each cubic spline represents the centerline of an intended vascular branch C that couples with the guidewire model during the simulation. Thus the relative location of a guidewire node to the spatial position of the blood vessel structure can be tracked by traversing through spline branches and also by identifying a point on the centerline at a given time step. The centerline structure offers a way of steering the guidewire towards the intended branch where the target is located. When the guidewire tip reaches a location where a number of arteries or veins are branched out, the simulator has the ability to select an intended branch by applying a steering force f s = δCT / CT to the tip of the guidewire along the tangential direction of the branch’s centerline curve where δ is the elastic coefficient of the steering force and CT is a vector computed from the first and the second nodes of the guidewire. Motions of the guidewire are controlled by two types of actions: insertion and twisting at the guidewire base. As it is inserted, the guidewire travels along a curved path due to the contact force inserted by the vessel wall or a pre-bent kink of the wire. Twisting at the base controls the arc direction, which allows the guidewire travels in circular arcs [24] by holding constant twist and helical trajectories while simultaneously twisting and inserting the base at constant velocity [25]. When it is in contact with an obstacle e.g. a blood clot in diseased vessels, the contact exerts a force that bends the guidewire, as shown in Figure 3. We approximate the effect by adding a resistance force opposite to the tangential direction of the

Figure 3: Contact with a blood clot: When in contact with an obstacle i.e. a blood clot (green sphere on the left image), a resistance force along the tangential direction of the guidewire bends the device (image to the right). guidewire at the tip. The magnitude of the force is computed as β T / T where β is a constant. Blood clot is also simulated firstly in this paper, no previous work in guidewire simulations has presented the feature for modeling the resistance of blood clot to the bending behaviours of guidewires, which is also an very important aspect of the simulation to be used for medical training purposes. Motions of guidewire are controlled by two parameters: vinsert , the speed at which the guidewire advances inside the blood vessel and φ, the angular velocity of the guidewire base. A time step advances the last segment of guidewire just inside the vessel by a length Δtvinsert . The rotation angle at the base equals φ∆t +Ɵ. The simulation propagates this rotation to the tip of the guidewire.

3.4.4 Guidewire re-parameterization As stated in [4], the smooth setting takes into account of certain quantities such as the twist and the curvature at each point of the rod, and the energy can be defined by integrating all of the points along the length of the rod. With the discrete formulation in the simulation, representing the rod as a set of vertices and edges as shown in Figure 2, the quantities of bending and twist energies often emerge naturally in an integrated rather than a point wise way. Converting the integrated quantities into the discrete formulations is carried out by dividing the length | | of the domain of integration ⁄2 where , However, ei is the with | | edge of the discrete setting of the guidewires. However, using the discrete formulation to model the guidewire traveling through complex 3D vascular structures gives rise to a challenge of achieving a balance between realism and realtime efficiency of the simulation. Especially we are computing the length of guidewires that are equivalent to 100 ~ 170 cm, which is a large length in comparison with rods [4].

Figure 4: Guidewire re-parameterization: The edge of the guidewire model is divided into two when the curvature greater than a threshold (top); some vertices of the guidewire are merged to form large edges with edge length = 2L when the edge of the guidewire is realistically representing the curvature of the blood vessel (bottom). During the simulation, it is often that the sample points of the guidewire do not adequately resolve the actual guidewire object in terms of the simulated length of the guidewires. Especially in the situation where there is a large curvature along the blood vessel sturctures. In order to maintain the realtime simulation speed at the same time enable the guidewire’s sampling resolution to fit as closely as possible to the blood vessel structure, we developed an algorithm that adaptively modifies the discrete setting of the guidewire according to curvatures of the blood vessel wall. The simulation system is optimized to perform the dynamic adaptive sampling of the physics model of the guidewire in order to achieve both simulation efficiency and accuracy. The adaptive algorithm dynamically splits the edges of the guidewire model into smaller edges when the blood vessel curvature is large and merges the vertices of the guidewire by connecting theses points with larger edges when they can sufficiently represent the guidewire on the straight portions of the blood vessel network. The algorithm is illustrated in Figure 4. To compute the acceleration at a node after a time step, we interpolate the parameters used in the guidewire force computations and use piecewise linear interpolation for velocity and acceleration.

4

Experimental

Results

and

Discussion We used 3D rotational X-ray images in DICOM dataset captured in a real patient and reconstructed the dataset into a 3D polygon mesh to generate an anatomically accurate model of the vasculature. The triangular faces of the vascular model confirm to the boundary of the blood vessel. A video accompanying this paper demonstrates the guidewire being inserted into the vasculature both slowly and vigorously to demonstrate the simulation stability.

Figure 5: Different stiffness of guidewire behaviors: the soft guidewire with flexural modulus equivalent to 10 GPa has more contact points on the vessel wall (left ) than the stiffer guidewire of 67 GPa (right). We measured the material properties of the guidewires with physical guidewires of different stiffness. The material’s 3 flexural modulus Eb was measured with E b = 2 Pl , where P 2 πd h is the normal force, l is the guidewire length, d is the guidewire diameter, and h is the deflection at a local point. Eight types of guidewires were taken into account with flexural modulus measured ranging from 5 GPa for very flexible guidewires to 67 GPa for very stiff guidewires. Figure 6 demonstrates the computation time for flexible and stiff guidewires.

Figure 6: Guidewire computation time with different stiffness: The more flexible guidewire (red) requires slightly more computational time than that of stiffer guidewire (blue). For guidewire of 1.7 meters length, the system takes about 36 ms to compute one step simulation, including collision detections, force computations, numerical integrations and rendering. Guidewires with different flexibility behave differently when advanced inside the vasculature. Flexible guidewires generally have more contact points with the vessel wall, whereas stiff guidewires have fewer contact points. We also constructed a number of artificial examples, shown in Figure 5: an artificial circular blood vessel structure, with which we tested the ability of the guidewire with different stiffness to be steered by rotating its base to advance it inside 3D structures

Figure 7: Overview of the procedure: A guidewire insertion procedure starting from the proximity end of the vasculature and moving towards a target allocated at the sub-arterial shown on the top left image, in which the guidewire is able to maneuver at very sharp bend; image to the bottom right shows the guidewire’s physical behavior completely inside the vessel structure. of different complexity settings. Figure 5 shows the bending behaviours of a flexible guidewire and a stiff guidewire. The more flexible the guidewire is, the more contacts with the surface of the vessel at the top of the insertion and it bends more easily near the end of the guidewire, where inserting and turning forces are induced to the device. We are able to simulation very long rod structures for about 1.7 meters, which is the standard length for guidewires. The computation time of stiff guidewires is compared with the soft guidewires that have the same diameters and lengths, shown in Figure 6. In general, soft guidewires are more intrinsic to be manipulated and takes slightly longer time to arrive a target location, thus more computation time for its configurations than that of stiff guidewires, yet both types of guidewires are simulated in real time on single core graphics processor. A number of experiments were performed to illustrate the result of the simulation in different circumstances. We run the simulation of advancing guidewires to travel through a blood vessel structure with intrinsic curvatures by rotating the guidewire at the base. Figure 7 demonstrates the simulation results of a guidewire bending and twisting behaviors while targeting at various clinical locations inside blood vessel model. We achieve the frame rate of 15 Hz on an Intel® Core™2 Duo CPU with T9600 PC. In Figure 7 a guidewire insertion procedure is performed, starting from the main arteries and reaching different subarteries.

5 Conclusion and Future Work Our elastic rod model for modeling guidewire and catheters is capable of interactive and accurate simulations for a wide range of guidewires with complex vasculatures, from very flexible to high stiff guideiwres. We achieve this in conjunction with an adaptive sampling algorithm and efficient collision detections. We aim at developing a surgical planning system by extending functionaries of the current simulator to model gudiewires with different tips in order to extend the versatility of our simulator, for example in medical operations, doctors also using guidewires with pre-bend tips, such that these types of wires require much smaller turning radius, facilitating the turning at sharp angles. We are also exploring ways to implement real-time planning and feedback control algorithms. These extended features would enable the ability of the simulator to maneuver guidewires to targets with paths that are truthfully replicating the paths in the clinic cases. Acknowledgment We would like to thank Dr Yi Song at Leeds University for constructing the vasculature model. This work is partially supported by the Regional Development Agency, One North East, England, UK. References 1

2

3 4 5

6

A. Lunderquist, K. Ivancev, S. Wallace, I. Enge, F. Laerum, and A. N. Kolbenstvedt, The acquisition of skills in interventional radiology by supervised training on animal models: A three year multicenter experience, Cardiovascular Interventional Radiology, vol. 18, no. 4, pp. 209–211. N. E. Seymour, A. G. Gallagher, S. A. Roman, M. K. O’Brien, V. K. Bamsal, D. K. Andersen, and R. M. Satava, Virtual reality training improves operating room performance: Results of a randomized, double-blinded study. Annals of Surgery 236, 2002, 4 (Oct.), 458–463. M. Cavusoglu, F. Tendick, M. Cohn, and S. Sastry. A laparoscopic telesurgical workstation. IEEE Transactions on Robotics and Automation, 15(4):728–739, Aug 1999. M. Bergou, M. Wardetzky, S. Robinson, B. Audoly., and E. Grinspun, Discrete Elastic Rods, SIGGRAPH, ACM Transactions on Graphics 27, 3 (Aug.), 63:1–63:12 2008. V. Luboz, L. Lai, J. Blazewski, D. Gould, and F. Bello, A virtual Environment for Core Skills Training in Vascualr Interventional Radiology. Proceedings of the 4th International Symposium on Biomedical Simulation. 5104:215-220, 2008 S. Cotin, C. Duriez, J. Lenoir, P. Neumann, S. Dawson, New Approaches to Catheter Navigation for Interventional Radiology Simulation. Medical Image Computing and Computer-assisted intervention, Vol. 8, pp 534-542 2005.

7

8

9 10 11 12

13 14 15 16

17 18

19 20 21

22

23

T. Alderliesten, M. K. Konings, W, J. Niessen, Modeling Friction, Intrinsic of Curvature, and Rotation Guidewires for Simulation of Minimally Invasive Vascular Interventions. IEEE Transactions on Biomedical Engineering, Vol. 54, no. 1, January 2007. S. Cotin, S. Dawson, D. Meglan, D. Shaffer, M. Ferrell, and P. Sherman. Icts, an interventional cardiology training system. In J.D. Westwood et al., editor, Proceedings of Medicine Meets Virtual Reality, pages 59–65. IOS Press, 2000. J. Lenoir, P. Meseure, L. Grisoni, and C. Chaillou. Surgical thread simulation. In MS4CMS (Proc. of ESAIM), volume 12, pages 102–107, 2002. W. L. Nowinski and C. K. Chui. Simulation of interventional neuroradiology procedures. In MIAR, pages 87 – 94, 2001. T. Alderliesten. Simulation of Minimally-Invasive Vascular Interventions for Training Purposes. PhD dissertation, Utrecht University, December 2004. G. H. M. Van Der Heijden, S. Neukirch, V. G. A. Goss, and J. M. T. Thompson, Instability and self-contact phenomena in the writhing of clamped rods. Int. J. Mech. Sci. 45, 161–196. 2003. R.S. Falk, and J. -M. Xu, Convergence of a secondorder scheme for the nonlinear dynamical equations of elastic rods. SJNA 32, 4, 1185–1209. 1995 C. -C. Lin and H. R. Schwetlick, On the geometric flow of Kirchhoff elastic rods. SJAM 65, 2, 720–736. 2004 D. K. Pai, STRANDS: interactive simulation of thin solids using Cosserat models. Computer Graphics Forum 21, 3. 2004 F. Bertails, B. Audoly, M. –P. Cani, B. Querleux, F. Leroy, and J. –L. Le’ve’que, Super-helices for predicting the dynamics of natural hair. ACM Transaction on Graphics 25, 3 (July), 1180–1187. 2006 F., Bertails, Linear time super-helices. Computer Graphics Forum 28, 2 (Apr.), 417–426. 2009. J. Spillmann, and M. Teschner, CORDE: Cosserat rod elements for the dynamic simulation of one-dimensional elastic objects. In Proceedings of the 2007 Symposium on Computer Animation, Eurographics Association, 63–72. 2007 M. Botsch, and O. Sorkine, On linear variational surface deformation methods. IEEE Transactions on Visualization and Computer Graphics (TVCG) 14, 1, 213–230, 2008. The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Dover Publications. W. Tang, C. Niquin, and A. Schildknecht , and T. R. Wan, Fast and Accurate Finite Element Method for Deformation Animations, in Proceedings of Theory and Practice of Computer Graphics, 2009, June Cardiff University, UK N. Ogata, K. Goto, and K. Uda, “An evaluation of the physical properties of current microcatheters and guidewires,” Interventional Neuroradiol., vol. 3, pp. 65– 80, 1997. J. Schroder, Technology Assessment: The Mechanical Properties of Guidewires Part III: Sliding Friction,

CardioVascular Interventional Radiology (1993) 16: 9397. 24 R. J. Webster III, N. J. Cowan, G. Chirikjian, and A. M. Okamura, “Nonholonomic modeling of needle steering,” in 9th International Symposium on Experimental Robotics, June 2004. 25 V. Duindam, R. Alterovitz, S. Sastry, and K. Goldberg, “Screw-based motion planning for bevel-tip flexible needles in 3d environments with obstacles,” in IEEE Intl. Conf. on Robot. and Autom., May 2008, pp. 2483–2488.

Dr Derek Gould is a Consultant clinical interventional radiologist (IR) for 27 years. He has been Chair (currently a co-opted member) of the British Society of Interventional Radiologists’ Education Committee, during which time he completed (2006) an interventional radiology training Syllabus. His current research interests are in development of novel interventional techniques and interventional skills training. In 2002 he formed a multidisciplinary group, ‘Collaborators in Radiological Interventional Virtual Environments’ to develop simulator models with appropriate fidelity, to remove the risks of skills training from patients.

Dr Wen Tang is a Reader in Computer Graphics at School of Computing, Teesside University, UK. Her research interests are in the areas of physically-based computer animations and applications, including motion synthesis, real-time finite element based deformations, and simulation & modelling for medical training and surgical planning.

Dr. Tao Ruan Wan is a senior lecturer in the School of Computing, Informatics and Media at the Bradford University, UK. His research interests are mainly focused on computational models for 3D visualisation and simulation, and real-time physically-based modelling and simulations. His research interest also includes imaging processing and artificial intelligence.

Mr. Pierre Lagadec is currently a PhD student in Sports Science at Teesside University, UK. Pierre's research interests mainly are concerned with computer-based solutions for human body analysis for medical and training purposes. Topics include real-time physically-based simulations, image processing, motion capture systems and biomechanics.

Dr Jianhua Zhai received his PhD in Engineering at University of Liverpool, UK. He is currently working as a postdoctoral researcher in the University of Liverpool. His current interest is to develop miniaturised sensors for medical applications.

Dr. Thien How is a Senior Lecturer in the Department of Clinical Engineering, University of Liverpool. His interests include vascular haemodynamics, design and evaluation of vascular prostheses for the bypass of peripheral arteries and clinical aspects of arterial reconstruction etc.