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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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Presentation on theme: "1 Modélisation de linteraction avec objets déformables en temps-réel pour des simulateurs médicaux Diego dAulignac GRAVIR/INRIA Rhone-Alpes France."— Presentation transcript:

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

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

3 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

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

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

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

7 7 Mass-Spring Model Initial length Deformed length

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

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

10 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

11 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

12 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

13 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

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

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

16 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

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

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

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

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

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

22 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

23 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

24 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

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

26 26 Thigh Echography

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

28 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)

29 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]

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

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

32 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.

33 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

34 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)

35 35 Dynamic Resolution 100 Hz using semi-implicit integration

36 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

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

38 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

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

40 40 Interaction zCollision Detection zCollision Response zForce Feedback

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

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

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

44 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

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

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

47 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)

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

49 49 A first Prototype

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

51 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

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

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

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

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

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

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

58 58 Explicit Integration 280 steps per second mass 5 grams

59 59 Stability Analysis Im Re

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

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

62 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

63 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

64 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

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

66 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

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

68 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!

69 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

70 70 Local Model

71 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

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


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