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1 Modélisation de linteraction avec objets déformables en temps-réel pour des simulateurs médicaux Diego dAulignac GRAVIR/INRIA Rhone-Alpes France

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2 Medical Simulators zMotivations ydanger to patients ycost ycertification zObjectives yGeometric Models yPhysical Models xdeformation xinteraction

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3 Problems zSimulation MUST be real-time! ydeformation yresolution zSimulation MUST be realistic! ymodel yidentification of parameters zSimulation MUST be interactive! ycollision detection yhaptic interaction

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4 Plan zDeformation Models yMass-Spring vs. FEM zReal-time Resolution Techniques yStatic yDynamic zEchographic Simulator yparameter identification zLiver Model yinteractive deformation

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5 Deformable Object zGeometry zElements ySprings [TW90] yTetrahedra FEM [OH99] zComparison yRealism ySpeed

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6 Geometrical Model z56 surface points z108 triangles z57 total points z120 tetrahedra z230 edges

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7 Mass-Spring Model Initial length Deformed length

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8 Finite Element Method (FEM) displacements Small strain Greens strain Cauchy Strain: Deformation tensor: Initial configuration Deformed configuration a x

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9 Strain-Stress Lamé coefficients force per unit area Deformation Energy

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10 Mass-Spring Model Springs are placed along the edges (230) Not very realistic: modeling a volume with springs! The force of each spring relatively cheap to evaluate globally fast

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11 Finite Element Method (FEM) 120 tetrahedra using Greens strain tensor Continuum is modeled with volumetric element. Dilatation may be controlled Approximately four times slower than mass-spring network

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12 Deformable Models (conclusions) zMass-Spring yOne dimentional elements yUnrealistic to model volume zTetrahedral FEM yGood realism for 3D continuum yControl of dilatation yApproximately 4 times slower to evaluate forces

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13 Contributions zQuantitative and qualitative comparison of mass-springs and tetrahedral elements zInteractive non-linear static resolution zFormal analysis of the real-time stability of integration methods ybased on parameters zIdentification of the parameters of a model from experimental data zRelevant medical applications

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14 Plan zDeformation Models yMass-Spring vs. FEM zReal-time Resolution Techniques yStatic yDynamic zEchographic Simulator yparameter identification zLiver Model yinteractive deformation

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15 Real-time Resolution zStatic Resolution ylinear resolution [Cotin97] xsmall displacements yOur approach: non-linear resolution xlarge displacements zDynamic resolution yexplicit [Picinbono01] yimplicit [BW98]

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16 Linear Static Resolution Principle of virtual work: internal and external forces are balanced Linear case: Pre-inversion (if enough space) No large strain No rotation No material non-linearity

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17 Nonlinear Static Resolution Non-linear case: Stiffness matrix changes with displacement: geometric material

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18 Newton Iteration Full Newton-Rapson method: Reevaluation of Jacobian Faster convergence Modified Newton-Rapson method: Constant Jacobian Slower Convergence

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19 Dynamic Analysis 2nd order non-linear differential equation Convert to 1st order system

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20 Explicit Integration Runge-Kutta method with s stages Order of consistency (accuracy) vs. stages s precision

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21 Explicit Integration Stability Im Re Timestep is limited by the the physical parameters! linearizing

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22 Implicit Integation linearisation Semi-implicit euler B-stable implicit euler: Stable for linear case (A-stable) any timestep any physical parameters If you know your history, then you would know where you are coming from. Bob Marley Over-damped case

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23 Resolution (conclusions) zStatic analysis ynon-linear resolution for large displacements zDynamic yexplicit xstrict stability criteria yimplicit xno limit on timestep, but resolution of non-linear system

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24 Contributions zQuantitative and qualitative comparison of mass-springs and tetrahedral elements zInteractive non-linear static resolution zFormal analysis of the real-time stability of integration methods ybased on parameters zIdentification of the parameters of a model from experimental data zRelevant medical applications

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25 Plan zDeformation Models yMass-Spring vs. FEM zReal-time Resolution Techniques yStatic yDynamic zEchographic Simulator yparameter identification zLiver Model yinteractive deformation

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26 Thigh Echography

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27 Echographic Simulator zData Acquisition zModel of the thigh yMass-Spring yNeural zInteraction ycollision yhaptics zGeneration of echographic image

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28 Data Acquisition 64 sample points are marked on the thigh. For each, the forces for some given penetrations are measured Two different probes (a) Indentor shaped probe for punctual force-penetration data (b) Probe with surface equal to that of a typical echographic probe Two different probes (a) Indentor shaped probe for punctual force-penetration data (b) Probe with surface equal to that of a typical echographic probe 1- The end effector advances in small steps (2mm) in the direction normal to the surface of the thigh. 2- The force depending on the penetration distance is measured (at LIRMM, Montpellier)

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29 Data Acquisition: Experimental Results zThe two probes do not offer the same resistance ydifference in surface area zDifferent curves for different points ydifferent depth of soft tissue zHighly non-linear behaviour Indentor probe Surface probe displacement Force displacement Force [dAulignac et al. MICCAI 99]

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30 Echographic Simulator zData Acquisition zModel of the thigh yMass-Spring yNeural zInteraction ycollision yhaptics zGeneration of echographic image

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Dynamic Model of the thigh Incompressibility of the tissue Elasticity of the epidermis Why mass-spring model? computationally efficient interior NOT discretized into tetrahedra

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32 Identification of the Parameters of a Dynamic Model New parameters (elasticity, plasticity, collision stiffness...) Desired behaviour Behaviour Error Optimization Algorithm Model Resolution - Measurements For each sample point, 10-12 deformation/force values with each probe => Total of ~1200 measurements.

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Parameter Estimation Least-squares minimisation: 1. find (a,b) for each non-linear spring 1. find (a,b) for each non-linear spring 2. find (a,b) for each non-linear spring, and (a) for all linear springs 2. find (a,b) for each non-linear spring, and (a) for all linear springs zError of the model with respect to the experimental data => Overall error less than 5% Distribution of Nonzero Error Values (in collaboration with UC Berkeley) [dAulignac et al., IROS 99] Error (N) => Avoid local minima

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34 Dynamic Analysis zExplicit integration yEuler stability xtoo small timesteps no real-time x...or large mass slow movement no gravity zImplicit integration ySemi-Implicit Euler xconstant Jacobian x100 steps per second h=1/100 (i.e. real time)

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35 Dynamic Resolution 100 Hz using semi-implicit integration

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36 Neural Networks Forces acting on particles: f Displacement of particles: u Static Analysis Multi-layer perceptron is a general approximizer Network is trained directly on experimental data back-propagation 64 inputs and outputs

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37 Neural Networks Displacement (mm) Force (N) Experimental data Neural Model

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38 Mass-Spring vs. Neural Model zMass-spring ytopology chosen xbased on measurements ydynamic resolution xsemi-implicit (100 Hz) zNeural model yno assuption on topology ystatic resolution xvery fast xno change of topology

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39 Echographic Simulator zData Acquisition zModel of the thigh yMass-Spring yNeural zInteraction ycollision yhaptics zGeneration of echographic image

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40 Interaction zCollision Detection zCollision Response zForce Feedback

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41 Collision Detection zFinds polygons in the OpenGL viewing frustrum zDetects collision between simple rigid body and any other object quickly

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42 Collision Response zInter-penetration distance must be computed zGenerates large forces (bad for haptics) Penalty forces [Hunt and Crossley 1975]

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43 Haptics zHaptic devices require high update frequency ytypically around 1kHz z….which the simulation normally cant meet y100 Hz (dynamic model)

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44 Haptic Interaction zLocal approximation of the contact ysimple local model running in a separate thread xfast collision detection xfast force computation [Balaniuk 99] Haptic loop (1kHz): collision detection and response with local model Simulation Loop (100Hz): deformation global collision detection and response position Local model update

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45 Haptic Feedback time With local model Without local model [dAulignac et al., ICRA, 2000] force

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46 Echographic Simulator zData Acquisition zModel of the thigh yMass-Spring yNeural zInteraction ycollision yhaptics zGeneration of echographic image

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47 Echographic Image Generation z64 images aquired yon each sample point zVoxel Map y120 Mb zInterpolation yfill in the blanks zProvide image yany rotation yany position [Vieira01] (in collaboration with TIMC- IMAG, France)

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48 Echographic Image Deformation zProblem ystructures deform differently xvein xbone, etc. ysegmentation zLinear deformation yPossible extension: precalculated deformation maps [Troccaz et al, 2000]

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49 A first Prototype

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50 Echographic Simulator (conclusions) zData Acquisition zModel of the thigh yMass-Spring yNeural zInteraction ylocal model zGeneration of echographic image ylinear deformation

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51 Contributions zQuantitative and qualitative comparison of mass-springs and tetrahedral elements zInteractive non-linear static resolution zFormal analysis of the real-time stability of integration methods ybased on parameters zIdentification of the parameters of a model from experimental data zRelevant medical applications

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52 Plan zDeformation Models yMass-Spring vs. FEM zReal-time Resolution Techniques yStatic yDynamic zEchographic Simulator yparameter identification zLiver Model yinteractive deformation

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53 Keyhole Surgery Surgery involves soft tissues Need to model deformation …in real-time! simulation

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54 Human Liver Interior composed of parenchyma Surounded by elastic skin or Glissons capsule Venous network Approximate weight: 1.5 kg

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55 Liver Model zGeometry zPhysical Model zDynamic Analysis yexplicit integration stability zStatic Analysis ynon-linear resolution

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56 Geometrical Model 187 Vertices 370 Triangles 299 Particles 1151 Tetrahedra 1634 Edges GHS3D

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57 Physical Model [Boux et al., ISER, 2000] HeterogenousNon-linear: skinParenchyma Weight distributed equaly on all particles (i.e. approximately 5g each) Strain Stress

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58 Explicit Integration 280 steps per second mass 5 grams

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59 Stability Analysis Im Re

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60 Simulation Achitecture SGI Onyx2 Compexity 370 facets 1151 tetrahedra 3399 springs Frequency 150Hz Explicit not stable!...large mass

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61 Static Resolution The large deformations of the organ during operation require non-linear resolution techniques.

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62 Calculate forces on nodes Evaluate stiffness matrix K? (analytically) Iteratively solve linear system for displacements u Ku = f by successive over- relaxation (SOR) until residual forces < epsilon through Newton-Rapson iteration Iterative Solution

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63 Modified Newton-Raphson Accurate solution (many SOR iterations) does not allow faster solution Inexact Jacobian limits convergence speed Of special importance for strong nonlinearities residual iterations

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64 Newton-Raphson Less iteration to converge then modified NR Exact Jacobian allows faster convergence Global time gain when solving linear system accurately iterations residual

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65 Pseudo-Dynamic Interactive resolution of the non-linear system.

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66 Result 1157 tetrahedra Iterative non-linear resolution Rotational invarience (N.B. Real-time animation) 1157 tetrahedra Iterative non-linear resolution Rotational invarience (N.B. Real-time animation) 60 NR iterations/sec on SGI Octane 175Mhz Pseudo-dynamic

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67 Liver Model (conclusions) zPhysical Model ymass-springs zDynamic Analysis yexplicit integration unstable zStatic Analysis yinteractive non-linear resolution

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68 Summary zPhysical Models yMass-Spring or FEM? zResolution yStatic xlinear or non-linear? yDynamic xexplicit or implicit? zMedical Simulators yThe choice of numerical methods must be guided by the application!

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69 Contributions zQuantitative and qualitative comparison of mass-springs and tetrahedral elements zInteractive non-linear static resolution zFormal analysis of the real-time stability of integration methods ybased on parameters zIdentification of the parameters of a model from experimental data zRelevant medical applications

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70 Local Model

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71 Explicit Integration Dynamic equations solved by Eulers method Linearizing by assuming constant matrices we can calculate derivative analytically The absolute value of (1+z) must be smaller than 1

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72 Backwards engineering Geometrical description Physical Model Results Physical Model Geometrical description forces displacements elasticity forces displacement

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