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1 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE 1 A Novel Approach for Modeling Separation Forces Between Deformable Bodies Mohsen Mahvash, Member, IEEE, Abstract Many minimally invasive surgeries (MIS) involve removing whole organs or tumors which are connected to other organs. Development of haptic simulators that reproduce separation forces between organs can help surgeons learn MIS procedures. Powerful computational approaches such as Finite Element Methods generally cannot simulate separation in realtime. This paper presents a novel approach for real-time computation of separation forces between deformable bodies. Separation occurs either due to fracture when a tool applies extensive forces to the bodies or due to evaporation when a laser beam burns the connection between the bodies. The separation forces are generated online from pre-calculated force-displacement functions that depend on the local adhesion/separation states between bodies. The pre-calculated functions are accurately synthesized from a large number of force responses obtained through either offline simulation, measurement or analytical approximation during the pre-processing step. The approach does not require online computation of force-global deformation to obtain separation forces. Only online interpolation of pre-calculated responses is required. The state of adhesion/separation during fracture and evaporation are updated by computationally simple models which are derived based on the law of conservation of energy. An implementation of the approach for the haptic simulation of the removal of a diseased organ is presented, showing the fidelity of the simulation. Index Terms haptic rendering, cutting simulation, surgery training, physically based modeling. I. INTRODUCTION MINIMALLY invasive surgery (MIS) is an efficient way to perform many surgical procedures since it causes less trauma than open surgery and consequently requires shorter hospitalization. However, MIS has a long learning curve and requires extensive practice on patients. Haptic simulators can help surgeons to learn minimally invasive procedures in a shorter period of time, outside the operating room. Certain MIS procedures involve removal of a body such as an organ or tumor via a long slender tool which is moved to the diseased organ through an incision or orifice. The diseased organ is separated from the other organs when it is pushed away by the tool or is vaporized by an optical fiber which transfers energy emitted from a laser. Haptic simulation of these procedures requires calculation of separation forces between bodies. Real-time calculation of separation forces is challenging. Calculation of forces requires real-time computation of contact Manuscript received January 10, This work was supported by Technology Gap Grant of the Institute for Robotics and Intelligent Systems (IRIS) (Canada s Network of Centers of Excellence). M. Mahvash was with RealContact Inc., Montreal, H2X 2B1, Canada. He is now with the Department of Mechanical Engineering, the Johns Hopkins University, Baltimore, MD USA ( mahvash@jhu.edu). between the tool and the bodies and the prediction of the separation between the bodies. Powerful computational methods such as finite element methods are capable of simulating contact and separation given sufficient computational power, but it is impractical to use these methods online since the main effects that need to be simulated result from nonlinear large deformation at microscopic scales. This potentially requires enormous amounts of discrete elements for adequate approximation and there are no simple computational shortcuts such as condensation methods or equivalent meshes [1], [2]. Mahvash and Hayward [3] introduced a nonlinear precalculation approach that allows accurate real-time simulation of contact between a tool and a deformable body. The approach was to pre-calculate a large number of responses in off-line processing and to look them up in real-time to generate interaction responses. However, this approach is not directly applicable to calculation of separation forces between deformable bodies. Separation changes the boundary condition of the bodies online, and as a result, the pre-calculated force responses obtained for certain boundary conditions are not enough to calculate all possible situations that may occur during online simulation. In this paper, we introduce a new pre-calculation method to allow real-time calculation of separation forces. The method presents a shortcut for simulating the separation of connected deformable bodies without depending on the details of the underlying deformation models. The method consists of a preprocessing step and online simulation. During pre-processing, a large number of calculations are performed to obtain force functions at a large number of initial contact points between a tool and bodies. The force functions depend on the local separation state around each initial contact point. The precalculated force functions are obtained either by accurate offline simulation, analytical approximations, or measurement. During online simulation, the force functions of contact everywhere are calculated by interpolation of pre-calculated force functions. The interpolation factors at each contact point are calculated online based on the local state of separation around the contact point. We simulate the separation which occurs due to either fracture or evaporation of connection between bodies. The fracture is caused by a blunt tool that deforms the bodies. Evaporation is caused by a laser beam that transmits energy to the bodies. The law of conversation of energy is applied to calculate the state of separation during fracture and evaporation. Fracture occurs over the surfaces that connect bodies if the tool can supply enough energy to overcome the work required to separate the bodies [4]. Evaporation occurs over a small volume of body if the material stays inside the laser

2 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE 2 beam focal spot for a sufficient period of time. The separation method of this paper is based on three theoretical approaches that were found in the literature. Each of the theoretical approaches had been tested against different materials to verify its results. First, an interpolation approach calculates the tool contact forces. It is shown in [3] and [5] that the force of contact between a tool and a deformable body can be represented by a continuous force-displacement field that is dependent on the shape of the tool. The continuous field can be approximated by the interpolation of a set of local fields. Second, a fracture mechanics approach defines the dynamics of separation due to fracture caused by a tool. Mahvash and Hayward in [4] verified the fracture mechanics approach against several materials including liver and potato. Third, a valid energy approach calculates the evaporation rate of the bodies during laser interaction. Jacques verified the results of the energy-based approach against several materials [6]. Combining these individual approaches presents a simulation of the removal of a prostate interior that demonstrates high fidelity [3]. The rest of this paper is organized as follows: Section II reviews related work. Section III gives an overview of the simulation framework applied in this paper. Section IV defines the body structure used for computation of contact and separation of bodies. Section V presents a model for synthesizing contact forces. Section VI discusses an implementation of the presented approach and Section VII concludes the paper. II. RELATED WORK Separation of bodies can be considered as a special case of cutting in which the route of cutting is pre-defined. Simulating cutting of a deformable body requires the simulation of contact of a cutting tool with the body as well as real-time computation of structural changes in the body and modification of the contact model that calculates forces. Delingette et al. considered a hybrid model for the simulation of the cutting that included a tensor spring model related to mass-spring models and a pre-calculated finite element linear elastic model [7]. The tensor-spring model simulated the local structural changes of the body made by cutting. They simulated cutting by eliminating tetrahedral elements from the model. Bro-Nielsen also simulated cutting by the removal of tetrahedral elements combined with a technique for updating the pre-computed inverse of the stiffness matrix of finite element when cutting takes place [1]. Force responses of these approaches become discontinuous since the extension of a crack inside the deformable body was processed in a discrete manner by removing the pieces of the body structure. A very small scale discritization or a huge number of discrete elements was needed to generate fairly continuous force responses. Basdogan et al. simulated the cutting force between a tool and a body surface as a spring force proportional to indentation depth along the surface normal and a damping force proportional to tangential velocity [8]. Mahvash and Hayward introduced a fracture mechanics approach for haptic rendering of cutting with a sharp tool [4]. Cutting was viewed as an energy interchange between the potential elastic energy stored in the deformed body, the work done by the tool, and the irreversible work spent in creating a fracture. The approach was applied to haptic rendering of cutting with a sharp blade when the crack was developed across a line in the direction of the tool movement. We apply the fracture mechanics approach here to predict the extension of a crack developed between bodies during separation caused by a blunt tool that can be moved in several directions. Mendoza et al. computed the cutting force by an exponential function of time with a damping factor depending on the energy lost during the cut [9]. The exponential function switched to a contact model when there was no cutting. Greenish et al. used force data recordings to generate haptic simulation of cutting with a pair of scissors [10]. Only the information obtained during data recording could be displayed online. Okamura et al. developed a force model for needle insertion based on experimental data [11]. Different approaches have been proposed in computer graphics to compute the structural changes of bodies during cutting and fracture [12] [15]. In [12], cutting was simulated when a virtual scalpel split tetrahedra elements of a deformable body into substructures whose connectivity followed the trajectory of the cut. Nienhuys et al. presented a method to keep the mesh well shaped with a small size during cutting of triangulated surfaces with a scalpel [13]. In [14], cutting was simulated by removing links as well as elements that were encountered along the path of the cutting tool as it passed through the body. Molino et al. presented a virtual node method that allows bodies to separate along arbitrary piecewise paths [15]. None of these approaches addressed calculation of forces for haptic rendering. Several techniques have been proposed to simulate deformation and forces of contact with a deformable body in realtime. Astley and Hayward proposed using a multi-layer spring damper model for modelling viscoelastic deformation [16]. Zhuang and Canny considered the effects of large deformations [17]. In [18], deformation was calculated by a massspring-damper model. LeDuc et. al. presented a mass-spring surface model with home springs to simulate interaction with a deformable body [19]. The home springs were connected to the nodes of the body to maintain its original shape. Wu et. al. applied explicit integration and adaptive meshing for simulating large deformations of non-hookean materials [2]. Debunne et. al. used adaptive multi-rate integration with multispatial resolution for simulating tool contact with viscoelastic materials [20]. Cotin et al. [21], Bro-Nielson et al. [1] and James and Pai [22] introduced the idea of pre-calculation for computation of deformation in linear-elastic material. Mahvash and Hayward developed a pre-calculation technique for haptic synthesis of contact with nonlinear bodies when there was no damage or structural change in the bodies [3], [23]. Several techniques have been proposed to make discretetime implementation of haptic simulation stable. Colgate et al. developed a virtual mass spring to make haptic simulation with a virtual wall stable [24]. Hannaford and Ryu introduced a time-varying damper to provide stability [25]. Mahvash and Hayward applied high force update and multi-rate simulation to provide stable simulation [26].

3 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE 3 III. OVERVIEW OF THE SIMULATION FRAMEWORK We construct the force responses of contact between a tool and a virtual scene from force responses obtained in the pre-processing step. While this framework was already introduced in [3], we extend it to the case where the bodies of the scene are separated during interaction. The simulation should provide forces for all possible interactions that could occur during online interaction. This includes force responses of contact for all possible points of contact and the ways the bodies of the scene can be separated. There are an infinite numbers of possibilities, however data interpolation and appropriate approximations allow us to construct online responses from a limited number of pre-calculated responses. When a tool comes in contact with elastic deformable bodies, the force of contact depends on the location of contact and the way the bodies of the scene are supported at their boundaries. The separation changes the boundary conditions of the bodies, so it also changes the force responses of the tool contact. This is the main problem that makes the simulation of separation more challenging. When the location of the separation is far from a contact point, the separation has small effects on force responses at that point. However, when the location of the separation is very close to the contact point or overlaps, the change in the force is significant at that point during separation. In order to model the effects of the location of the separation on tool contact forces, we subdivide the common surfaces that connect the bodies into small sub-surfaces. The state of the separation of a sub-surface is encoded by the separated area of the sub-surface. The change of force responses at a contact point due to separation on a sub-surface is linearly related to the change of the separated area size of the sub-surface by an influencing factor that depends on whether that region is close to or far from the contact point. The influencing factors of the separation of sub-surfaces close to a contact point are determined by looking at force responses at that contact point obtained at two different separation states. The influencing factors of all sub-surfaces far from a contact point are small and therefore are considered to be equal. In this way, the change of the area size of the entire common surface that contains all sub-surfaces can be related to the force responses at a contact point. In the next section, we define common surfaces and their sub-surfaces and formalize the calculation of forces during separation. IV. PROBLEM FORMULATION Consider two deformable bodies labelled by B 1 and B 2 (Fig. 1a). B 1 and B 2 represent the surfaces of the deformable bodies before they are deformed. B 1 and B 2 share a common surface, C = B 1 B2. At time t of the separation process, common surface C can be divided into two regions: S(t), the connected part of the common surface, and S(t), the disconnected part (Fig. 1b). S(t) and S(t) do not change during deformation of the bodies. The edge of the closed surface S(t) is defined as e(t). B 1 and B 2 consist of triangular meshes and share the same triangles at the common surface. A triangle of the common surface C is called a common triangle and it is labelled by C m, where m is the index of the triangle. At time t, C m can be divided into two parts: S m (t), the connected part of the common triangle that belongs to S(t), and S m (t), the disconnected part of the triangle that belongs to S(t) (Fig. 1c). e(t) consists of line segments that connect the edge of the common triangles. D n represents a region around node n of the common surface that consists of all the triangles that share that node (Fig. 1d). At time t, D n can be divided into two regions: R n (t), the connected part, and R n (t), the disconnected part of the region. S m (t) (c) B 2 C S (t) S m(t) B 1 B 2 B 1 C m e(t) R n (t) n (d) S (t ) C (t) R n Fig. 1. Bodies B 1 and B 2 at time t of separation. a) B 1 and B 2 share triangular mesh surface C at their intersection. b) Common surface C is divided into a connected and a disconnected section. c) Common triangle C m, its connected region S m (t), and its disconnected region S m (t). d) D n, its connected region R n (t), and its disconnected region R n (t). The mesh structure described in this section is applied to computation of contact forces and can be different from the one used for graphical representation of the bodies of the virtual scene. This mesh structure does not change during deformation and separation of the bodies. V. TOOL INTERACTION MODEL A model for calculating forces of contact during separation is presented by considering some assumptions. A. Assumptions It is assumed that the bodies of the virtual scene are nonlinear elastic and do not have rigid movement. It is assumed that fracture only occurs over the common surfaces that connect the bodies. It is a reasonable assumption for bodies that the connections at their boundaries are weaker than connections among their interior structure. It is assumed that the cracks only develop from the edges of the cracks previously created over the common surfaces. Magnified stresses and appropriate contact conditions at the crack edges make this assumption reasonable [4]. Heat conduction of the bodies of the virtual scene during laser evaporation is neglected [27]. This is an approximation to make online calculation of laser effects possible. D n

4 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE 4 B. Forces of Contact The forces of contacts between the tool and bodies B 1 and B 2 are separately calculated and added to the tool force. Here we explain how the force of contact with B 1 is calculated. The force of contact at a point on the surface of B 1 is calculated by a nonlinear function that depends on the deflection of the body at that point and the state of adhesion/seperation between the body and the rest of the virtual scene. We refer to [23] for a more detailed description of a force function. A point of the tool surface initiates the contact with the surface of the body (Fig. 2). Point c defines the initial contact point on the body before being deformed and x defines the initial contact point on the surface of the tool that moves when the tool deforms the body. Contact starts at the initial contact points c and x and extends to an area or areas over the surface of deformed body during deformation. δ = x c defines the deflection at point c. The force of contact at point c is calculated by a local force field F c (δ, t), which is considered to be time-dependent due to its dependency on the state of separation. F c (δ, t) can be expressed by its components in local cartesian coordinates at point c: F c (δ, t) = f x c (δ, t)u x c + f y c (δ, t)u y c + f z c (δ, t)u z c, (1) where u x c, u y c, and u z c are unit vectors of the local coordinates and f x c (δ, t), f y c (δ, t), and f z c (δ, t) are the components of the force field. Fig. 2. Contact between a tool and a deformable body. C. Pre-computation of Nodal Force Fields c x For a contact point located at node i of the body surface and node j of the tool surface, the force field is denoted by F i,j (δ, t), its coordinates by u x i,j, uy i,j, and uz i,j, and its force components by fi,j x (δ, t), f y i,j (δ, t), and f i,j z (δ, t). We define three pre-calculated force-deflection fields determined at different states of the body connection for the cases that node j of the tool contacts with the node i of B 1 : F i,j (δ): the field at the state in which the entire common surface between two bodies are connected. Fi,j (δ): the field at the state in which only all common triangles around node i are separated. Fi,j e (δ): the field when two bodies are completely separated. The above force-deflection fields are obtained in the preprocessing step by experimentally creating the three scenarios just enumerated or by off-line finite element methods (FEM). It should be mentioned that we prevent separation when force responses of a pre-calculated field at a state of connectivity Algorithm V.1: PREPROCESSING(tool, B 1 ) comment: preprocessing step for body B 1 comment: M: number of body points comment: N: number of tool points comment: obtain force fields before any separation for i 0 to M 1 for j 0 to N 1 do do { comment: obtain the force field Fi,j by applying a variety of deflections F i,j COMPUTEFIELD(B 1, tool, i, j) comment: obtain force fields after local separations for i 0 to M 1 comment: separate the region around point i SEPARATELOCALREGION(B 1, i) for j do 0 to N 1 comment: obtain the force field F i,j do by applying a variety of deflections F i,j COMPUTEFIELD(B 1, tool, i, j) comment: force fields after the bodies are separated comment: separate B 1 and B 2 SEPARATE(B 1, B 2 ) for i 0 to M 1 for j 0 to N 1 comment: obtain the force field F do i,j e do by applying a variety of deflections Fi,j e COMPUTEFIELD(B 1, tool, i, j) comment: Save force fields SAVE(F, F, F e ) are recorded. Algorithm V.1 shows the pseudo code of the pre-processing step for obtaining force fields. To obtain a force field at a point, local cartesian coordinates are defined at that point. For a range of deflection distances in the direction of each axis of the local coordinates, several force-deflection data are obtained. Then an appropriate force-deflection function fits to the data. Assume that a five-degree polynomial represents the force-deflection function in each direction. Six force-deflection points determine the six coefficients of each polynomial. This way, each force field is defined by 27 numbers: 18 for representing the coefficients and 9 for the frame coordinates. The computational time of obtaining a force field could be from a few seconds to several minutes depending on the required accuracy. At run time the force-deflection field at node i is obtained by interpolation of the above force-deflection fields at the same node: F i,j (δ, t) = [ A n (R i (t)) F i,j (δ) + A n ( R i (t)) F i,j (δ) ] A n (S(t)) + A n ( S(t)) F e i,j(δ). (2) The interpolation factors A n ( ) are used to calculate the normalized area sizes of connected and disconnected regions that are involved in the separation process. The interpolation factors are such that: A n (S(t)) + A n ( S(t)) = 1 A n (R i (t)) + A n ( R i (t)) = 1 (3) The first term in the brackets in (2) calculates the local effects of separation of the region close to node i on the forcedeflection field at that node, and the second interpolation term represents the effects of separation everywhere on the forcedeflection field at that node. The sizes of the separated area around node i and the total separated area of the common surface together do not define a unique connection state,

5 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE 5 however, the force-deflection fields of the most-likely states for defined area sizes should be almost equal. D. Online Computation of Feedback Forces The force of contact at points c and x are calculated by area interpolation of nodal force fields over triangles that carry point c and x. Functions α i (c) and β j (x) calculates the interpolation coefficients at nodes i and j of the body and the tool. Readers should see [3] for a more detailed description of interpolation approach used to calculate forces of contact in arbitrary points. Algorithm V.2 shows the pseudo code of online computation of separation forces. When the tool is sliding over the deformable body, either contact point c or x move over the surface of the body or the tool to simulate sliding friction [3], [28]. The contact point c stops sliding if it reaches a crack edge. E. Multi-Body Separation When a virtual scene contains more than two bodies, an independent contact model is assigned to each body that computes the force of contact with that body. Summation of contact forces of all contact models calculates the total force applied to the tool. Coupling of contact models can be taken into account during the pre-processing to prevent increase in tool forces at the places where contact points of several contact models approach each other, such as crack edges [29]. The separation process uses the union of all common surfaces that connect a body to the rest of the scene. This way, the same influencing factor is assumed for all common surfaces that connect a body to the scene on a point far from the common surfaces. An alternative approach is to consider different influencing factors for different common surfaces. However, this exponentially increases the number of required pre-calculated responses when the number of bodies is increased. F. Fracture Bodies B 1 and B 2 can be separated through fracture caused by the tool. We obtained the size of the separated area and level of force that causes fracture for two types of fracture that generally occur over the common surface C that connects two bodies: 1) Fracture Due to Excessive Tangent Force.: Fig. 3 shows the tool interaction and the state of the tool and connection between the bodies at times t and t + dt of the separation. The tool moves for dx over the common surface that connects the bodies. If fracture occurs during time dt, the crack edge is moved for dx and a small area of the connection between the bodies is separated. l t defines the length of the extended crack and can be evaluated by the length of contact between the tool and the body. The separated area is obtained by: = l t dx. (4) = dx When fracture occurs over the common triangle m: ( S m (t)) = = l t dx (5)!b B 2 Algorithm V.2: THREAD(tool, B 1 ) comment: online simulation for body B 1 comment: initialization LOAD(F, F, F e ) comment: real-time loop if COLLISION(B 1, tool) c, x INITIALCOLLISIONPOINTS(B 1, tool) δ x c comment: k 1, k 2, k 3 are nodes of the body triangle k 1, k 2, k 3 CONTACTTRIANGLENODES(B 1, c) comment: l 1, l 2, l 3 are nodes of the tool triangle l 1, l 2, l 3 CONTACTTRIANGLENODES(tool, x) for i k 1, k 2, k 3 for j l 1, l 2, l 3 do do { F t i,j A n (S)[A n (R i )F i,j (δ)+ A n ( R i )F i,j (δ)] + A n ( S) F e i,j (δ) comment: obtain force by area interpolation comment: α i (c) and β j (x) are interpolation factors at nodes i,j f t i=k 1,k 2,k 3 α i (c) j=l 1,l 2,l 3 β j (x)f t i,j (δ) comment: sliding friction if sliding then DISPLACE(c, x) comment: update separated areas during fracture if fracture then UPDATE(R i, S) comment: update separated areas during evaporation if evaporation then UPDATE(R i, S) B 1 t t dt Fig. 3. Fracture due to excessive tangent force applied to the edge of the crack. a) A general description of the interaction, b) the state of the tool and the state of connection between the bodies at times t and t + dt of the separation. The separation can be visualized as opening of an envelope of size and length l t via a pen that moves for dx through the surfaces of the envelope. We apply the energy-based approach of [4] to calculate the level of force that causes fracture. Based on the law of conservation of energy, the fracture can occur, if the energy created by the tool is enough to deform and to separate the connection between bodies: F t F t C d x l t dw t = du + dw A, (6) where W t is the tool work, du is the change in elastic energy stored in the bodies and dw A is the work of fracture. The tool work is calculated by: dw t = F dx (7) The work of fracture for separating area is obtained by dw A = J c, (8) where J c is the fracture toughness of the connection between two bodies. Fracture toughness is a material property and does not depend on the way the bodies are separated [30]. We ignore

6 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE 6 B 2 B 1 t t dt Fig. 4. Fracture due to excessive normal force applied to the surface of one of the bodies. a) A general description of the interaction, b) the state of the tool and connection between the bodies at times t and t+dt of the separation. The separation can be visualized as opening of an envelope of size and length l n via a pen that is pushed against one of the surfaces of the envelope. the change of elastic energy during fracture since deformation pattern around the crack edges translates, but does not change shape: du = 0. (9) By substituting (7), (8) and (9) into (6) the energy condition for fracture is calculated by: F n F n dz C l t F dx = J c. (10) Considering the (10), the tangent force F t fracture is calculated by: that causes F t = J c dx = J c l t (11) 2) Fracture Due to Excessive Normal Force.: Fig. 4 shows the tool interaction and the state of the tool and connection between the bodies at times t and t + dt of the separation. The tool moves dz in the direction normal to the common surface C. If fracture occurs during time dt, the crack edge is moved and a small area is separated. l n = dz defines the effective length of the extended crack which depends on the tool shape and can be set in the pre-processing step. The separated area is then calculated by: = l n dz = l n dz. (12) When fracture occurs over the common triangle m: ( S m (t)) = = l n dz. (13) In this case, energy condition 10 concludes: F n = J c dz = J c l n (14) where F n is the normal component of the tool force and J c is the toughness as defined in 8. G. Vaporization. Fig. 5 shows dissection with a laser fiber and the state of the fiber and the state of connection between bodies B 1 and B 2 at times t and t + dt. We consider a steady state mode for the process of laser dissection when a groove between the bodies is extended with the same speed as the laser fiber is moved. dx defines the displacement of the laser tip and dv is the volume of material vaporized during time dt. dv can be calculated by: dv = l 2 f dx, (15) where l f is the diameter of the focal spot of the gaussian laser beam formed at the tip of the laser fiber. defines the area of the projection of volume dv over the common surface C and is calculated by: = l f dx, (16) When the laser focal spot is located on the region m: B 2 ( S m (t)) = = l f dx. (17) B 1 t t dt Fig. 5. The laser fiber vaporizes the connection between two bodies: a) A general description of the laser interaction, b) The state of laser beam and the groove between bodies at times t and t + dt of laser dissection. The gaussian laser beam vaporizes volume dv of the groove. The dimensions of dv is defined by the size and displacement of the gaussian laser beam. Area defines the projection of the volume dv over the common surface C v s, the steady state speed of the laser dissection can be calculated by the law of conservation of energy [27], [31]. The absorbed energy by volume dv during time dt must be equal to the energy required to evaporate the volume: P P l f dx α P dt = q dv (18) where α is the ratio of the absorbed energy, P is the laser power, and q is energy required to evaporate the unit of volume of the body, The required energy q is related to the temperature and the material properties of the bodies: q = ρ (c p T + H) (19) where ρ is density of the body material, c p specific heat capacity of the material, H latent heat of vaporization and T is the difference between the body temperature and the boiling point of the material. 1 Speed v s is then calculated by v s = dx dt = dv αp = dt q lf 2 l 2 f (20) If a surgeon moves the laser faster than v s, no part of the bodies is vaporized. During simulation, if the laser beam targets the groove between the bodies, and if the fiber is moved toward the groove no faster than v s, the groove will be extended with the same speed as the speed of the laser fiber. 1 It is shown in [6] that the thermal properties of a variety of biological bodies that contain a large percentage of water can be approximated by the thermal properties of the water during laser dissection.

7 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE 7 VI. IMPLEMENTATION AND RESULTS The method described in this paper was applied to create a high-fidelity surgical simulator prototype for removal of the interior of the prostate (Fig. 6). The procedure involves the use of a long cylindrical instrument inserted through an orifice to perform the separation of the interior of the prostate from its shell (Fig. 7). During the procedure the interior is first divided into two or three lobes and then each lobe is separated from the shell [32]. Pivot Fig. 8. The surface representation of prostate lobes used for simulation. Configuration space obstacle for the left lobe in respect to the cylindrical tool whose center line always passes a fixed pivot point. The wire mesh shows the configuration space obstacle. Fig. 6. Haptic simulation of the process of separating the interior of the prostate from its shell by using a cylindrical tool that carries a laser fiber on its tip. a) Graphical user interface which includes two graphical panels. The left panel shows an endoscopic view of the surgical scene. The right panel shows a view set by the user. b) A user interacts with the simulator through a haptic device. Shell Lobes Pivot Tool Fig. 7. The procedure of separating prostate lobes from prostate shell by applying excessive normal force. A. Components of the Simulator The bodies of the surgical scene include three prostate lobes and a prostate shell. They are represented by triangular surface meshes (Fig. 8a). The convex hull of each lobe is used as a body for the haptic rendering. This smoothes rendering of sliding contact and simplifies the problem of interference detection. The simulation of contact of the cylindrical tool with each convex hull is performed separately by computing the contact of a point of the cylinder with the configuration space obstacle for that convex hull with respect to the cylindrical tool whose center line always passes a fixed pivot point [33] (see Fig. 8b). This way, the problem of collision detection between the cylindrical tool and a piece of the surgical scene was converted into contact between a point and an obstacle. Meanwhile, it should be mentioned that the separation model of this paper does not require the specific interference detection used in our implementation. The software is written in Java and C for a Real time Linux platform. It is organized around several threads: A hard real-time thread that updates device forces at high rate (1 khz) A thread in Java that performs the interference detection and separation process. This thread send its information to the high-rate process through two FIFO queues implemented in the Java native interface. This thread provides nodal force fields of active elements and calculates the connected areas of the common surfaces. Several graphics and user interaction threads written in Java 3D and for the Java virtual machine. The force component of a force field at each node in each cartesian direction was synthesized by a third-degree polynomial of the component of deflection in the same direction. If more accuracy is required, higher-order polynomials can be used to express the fields. A useful feature of the simulator allowed us to modify the force field information online, that is, while interacting with the virtual scene. This provided flexibility to define regional behaviors for the process of removal of the prostate to synthesize more realistic force responses at certain states of the surgical procedure which required more accurate responses. For graphical representation, the deformation at a point of a lobe surface is calculated by the deflection of the lobe at the contact point divided by the square root of its distance to the contact point. B. High-fidelity Force Response We performed an experiment to demonstrate the force responses of the simulator. During the experiment, a local region between a prostate lobe and prostate shell is separated by applying an excessive normal force to the lobe (Fig. 7). The z axis of the local frames of force fields for all nodes are defined by normals to the nodes of the body surfaces. The components of force fields in their local planes (x, y) are assumed to be zero. The z components of the force fields for all nodes at three states of separation are obtained by thirddegree polynomials shown in Fig. 9. Fig. 10 shows force responses of the simulator during separation. The force-deflection curve of Fig. 10c shows the force-deflection response. The response consists of three phases: phase AB, phase BC and phase CA. No separation occurs during phases AB and CA. During phase AB, the force response is equal to the pre-calculated force response of

8 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE 8 Fig. 9. (c) Pre-calculated force fields f z (δ z ), fz (δ z ) and f e z (δ z ) = 0. Fig. 10. Force responses of separation a) Displacement of the tool in z direction. b) Normalized area size of the separated region. c) Force-deflection response. The force response shows a deformation phase from point A to B, a fracture phase from point B to C and another deformation phase from point C to A. d) Force-time response. Fig. 9a. During phase CA, the force response is almost equal to the pre-calculated force response of Fig. 9b. In phase BC, fracture occurs and the level of force is limited to a threshold. The force-deflection response of the deformation of phase BC is obtained by (2). During phase BC, the curve shows nonsmooth behavior. This is due to the non-smooth movement of the the tool causing switching between deformation and fracture. Such non-smooth motion will occur for any human user when he or she moves an haptic device. However, the simulator generates continuous force response during separation. Also, it generates a passive force-displacement response that maintains the stability of haptic interaction [26]. Further, the simulator can generate separation forces of an actual interaction if accurate pre-calculated force responses are applied to enough number of nodes. C. Potential Training Capabilities In this section, we listed some potential training capabilities of the simulator that were expressed by three surgeons who tried the simulator. However, it should be mentioned that we have not conducted a thorough human study to determine the effectiveness of the haptic simulator. For this experiment, the pre-calculated force responses of the lobes and the shell were defined based on feedback that we obtained from surgeons who perform prostate surgery. One problem that frequently occurs during prostate surgery is that the surgeon loses the location of the tool inside the surgical scene. This usually occurs when the surgeon pushes against a interior piece via the tool to separate it from the shell and suddenly the tool slides over the piece and goes to another place. The surgeons that tried the simulator expressed that the simulator also generates a similar situation. The simulator could be capable of training novice surgeons how to push an interior piece such that the tool does not slide over it. Use of correct means of dissection at different places of the surgical scene is an important decision that surgeons are expected to make. When there is no initial crack between bodies, the tool cannot cause fracture in actual surgery. Here, only the laser could make a crack between lobes. The surgeons expressed that this behavior of the simulator was similar to what occurred in actual surgery. The simulator should have the capability to train novice surgeons about what kind of energy they should use at different places of surgery. During separation with excessive normal force, the visual feedback in actual surgery does not provide much information about the separating crack [32]. Surgeons rely on force to control the process of separation and then evaluate the depth of the crack they made. The surgeons expressed that the simulator provided similar force responses as those felt during actual surgery. The effects of separation on forces provided this capability for the simulator. The simulator presents several drawbacks mostly due to visual rendering of the surgical scene. The change of the graphical texture is not representative of the change of texture during laser dissection in actual surgery. The mesh structure of bodies is not dense enough to represent local graphical changes during separation. Interference detection of the endoscopic camera with the virtual scene and local deformation of the bodies are not accurate enough to prevent the camera to pass through the body surfaces. Bleeding and the circulation of water inside the prostate are some other graphical aspects which are not simulated [8], [34]. VII. CONCLUSION A novel approach for synthesizing the force of contact of a tool separating deformable bodies was described. Computation of the tool force during separation was performed through online interpolation of a set of pre-calculated force-deflections. The pre-calculated force-deflections were obtained from either off-line calculations or from measurement of the contact responses of the same tool with a set of bodies that shared the same surface properties. The law of conservation of energy was applied to predict the separation caused due to either fracture or evaporation. An implementation of the approach for simulation of the removal of the interior of prostate was explained. The method allowed realtime simulation of separation of nonlinear elastic deformable bodies. It permitted the simulation to be arbitrarily accurate as far as separation of a crack had no significant effects on contact responses at places far from the crack. More accurate responses could be obtained in exchange of having a larger number of pre-calculated responses. The method allows the simulation of separation of bodies that have inhomogeneous connection toughness. Each

9 IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE 9 common sub-surface that connects two bodies could have its own toughness. However the method may not be accurate for simulation of separation of the bodies that have significant global coupling on each other. A large number of possibilities should be considered in order to model the global coupling. Future work will be the establishment of an experimental setup that can provide appropriate experimental force data for tuning the available simulators and will also be the development of a graphics representation of the scene. Acknowledgments The author would like to thank the reviewers for helpful comments. The author also thanks the contributions of the Royal Victoria Hospital, Montreal, Canada, for allowing access to their facilities, and Immersion Corporation, USA, for providing the haptic device. REFERENCES [1] M. 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Schwarzkopf, Computational geometry, 2nd ed. Berlin, Germany: Springer, [34] C. Basdogan and M. Srinivasand, Haptic rendering in virtual environments, in Virtual Environments HandBook, 2001, pp Mohsen Mahvash graduated from K. N. Toosi University of Technology, Tehran, Iran and received a Master s degree in electrical engineering from Isfahan University of Technology, Isfahan, Iran. He received the Ph.D degree in electrical engineering from McGill University, Montreal, QC, Canada, in He is currently a postdoctoral fellow with the Department of Mechanical Engineering at The Johns Hopkins University, Baltimore, MD, USA. He is Co-Founder of Real Contact Inc, Montreal, QC, Canada. He is interested in surgical-assistant telerobotic systems, surgical simulation and control systems.