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**Diego d’Aulignac GRAVIR/INRIA Rhone-Alpes France**

Modélisation de l’interaction avec objets déformables en temps-réel pour des simulateurs médicaux Diego d’Aulignac GRAVIR/INRIA Rhone-Alpes France

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**Medical Simulators Motivations Objectives danger to patients cost**

certification Objectives Geometric Models Physical Models deformation interaction

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**Problems Simulation MUST be real-time! Simulation MUST be realistic!**

deformation resolution Simulation MUST be realistic! model identification of parameters Simulation MUST be interactive! collision detection haptic interaction

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**Plan Deformation Models Real-time Resolution Techniques**

Mass-Spring vs. FEM Real-time Resolution Techniques Static Dynamic Echographic Simulator parameter identification Liver Model interactive deformation

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**Deformable Object Geometry Elements Comparison Springs [TW90]**

Tetrahedra FEM [OH99] Comparison Realism Speed

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**Geometrical Model 56 surface points 108 triangles 57 total points**

120 tetrahedra 230 edges

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Mass-Spring Model Initial length Deformed length

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**Finite Element Method (FEM)**

Deformed configuration Deformation tensor: Initial configuration x a Green’s strain displacements Small strain Cauchy Strain:

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Strain-Stress Deformation Energy Lamé coefficients force per unit area

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**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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**Finite Element Method (FEM)**

120 tetrahedra using Green’s strain tensor Continuum is modeled with volumetric element. Dilatation may be controlled Approximately four times slower than mass-spring network

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**Deformable Models (conclusions)**

Mass-Spring One dimentional elements Unrealistic to model volume Tetrahedral FEM Good realism for 3D continuum Control of dilatation Approximately 4 times slower to evaluate forces

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Contributions Quantitative and qualitative comparison of mass-springs and tetrahedral elements Interactive non-linear static resolution Formal analysis of the real-time stability of integration methods based on parameters Identification of the parameters of a model from experimental data Relevant medical applications

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**Plan Deformation Models Real-time Resolution Techniques**

Mass-Spring vs. FEM Real-time Resolution Techniques Static Dynamic Echographic Simulator parameter identification Liver Model interactive deformation

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**Real-time Resolution Static Resolution Dynamic resolution**

linear resolution [Cotin97] small displacements Our approach: non-linear resolution large displacements Dynamic resolution explicit [Picinbono01] implicit [BW98]

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**Linear Static Resolution**

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

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**Nonlinear Static Resolution**

Non-linear case: Stiffness matrix changes with displacement: geometric material

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**Newton Iteration Full Newton-Rapson method: Reevaluation of Jacobian**

Faster convergence Modified Newton-Rapson method: Constant Jacobian Slower Convergence

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**Dynamic Analysis 2nd order non-linear differential equation Convert to**

1st order system

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**Explicit Integration Runge-Kutta method with s stages s**

Order of consistency (accuracy) vs. stages precision

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**Explicit Integration Stability**

linearizing Im Timestep is limited by the the physical parameters! Re

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**Implicit Integation B-stable implicit euler: linearisation**

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

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**Resolution (conclusions)**

Static analysis non-linear resolution for large displacements Dynamic explicit strict stability criteria implicit no limit on timestep, but resolution of non-linear system

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Contributions Quantitative and qualitative comparison of mass-springs and tetrahedral elements Interactive non-linear static resolution Formal analysis of the real-time stability of integration methods based on parameters Identification of the parameters of a model from experimental data Relevant medical applications

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**Plan Deformation Models Real-time Resolution Techniques**

Mass-Spring vs. FEM Real-time Resolution Techniques Static Dynamic Echographic Simulator parameter identification Liver Model interactive deformation

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Thigh Echography

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**Echographic Simulator**

Data Acquisition Model of the thigh Mass-Spring Neural Interaction collision haptics Generation of echographic image

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Data Acquisition (at LIRMM, Montpellier) 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 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

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**Data Acquisition: Experimental Results**

[d’Aulignac et al. MICCAI 99] Data Acquisition: Experimental Results displacement Force displacement Force Indentor probe Surface probe The two probes do not offer the same resistance difference in surface area Different curves for different points different depth of soft tissue Highly non-linear behaviour

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**Echographic Simulator**

Data Acquisition Model of the thigh Mass-Spring Neural Interaction collision haptics Generation 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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**Identification of the Parameters of a Dynamic Model**

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

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**Distribution of Nonzero Error Values**

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

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**Dynamic Analysis Explicit integration Implicit integration**

Euler stability too small timesteps no real-time ...or large mass slow movement no gravity Implicit integration Semi-Implicit Euler constant Jacobian 100 steps per second h=1/100 (i.e. real time)

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Dynamic Resolution 100 Hz using semi-implicit integration

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**Neural Networks Displacement of particles: u Static Analysis**

Multi-layer perceptron is a general approximizer Network is trained directly on experimental data back-propagation Forces acting on particles: f 64 inputs and outputs

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**Neural Networks Neural Model Experimental data Displacement (mm)**

Force (N) Neural Model Experimental data

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**Mass-Spring vs. Neural Model**

topology chosen based on measurements dynamic resolution semi-implicit (100 Hz) Neural model no assuption on topology static resolution very fast no change of topology

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**Echographic Simulator**

Data Acquisition Model of the thigh Mass-Spring Neural Interaction collision haptics Generation of echographic image

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Interaction Collision Detection Collision Response Force Feedback

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**Collision Detection Finds polygons in the OpenGL viewing frustrum**

Detects collision between simple rigid body and any other object quickly

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**Collision Response Inter-penetration distance must be computed**

Penalty forces [Hunt and Crossley 1975] Inter-penetration distance must be computed Generates large forces (bad for haptics)

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**Haptics Haptic devices require high update frequency**

typically around 1kHz ….which the simulation normally can’t meet 100 Hz (dynamic model)

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**Haptic Interaction Local approximation of the contact**

simple local model running in a separate thread fast collision detection fast force computation Haptic loop (1kHz): collision detection and response with local model [Balaniuk 99] Local model update position Simulation Loop (100Hz): deformation global collision detection and response

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**Haptic Feedback With local model force time Without local model**

[d’Aulignac et al. , ICRA, 2000] Haptic Feedback With local model force time Without local model

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**Echographic Simulator**

Data Acquisition Model of the thigh Mass-Spring Neural Interaction collision haptics Generation of echographic image

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**Echographic Image Generation**

[Vieira01] (in collaboration with TIMC-IMAG, France) 64 images aquired on each sample point Voxel Map 120 Mb Interpolation fill in the blanks Provide image any rotation any position

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**Echographic Image Deformation**

Problem structures deform differently vein bone, etc. segmentation Linear deformation Possible extension: precalculated deformation maps [Troccaz et al, 2000]

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A first Prototype

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**Echographic Simulator (conclusions)**

Data Acquisition Model of the thigh Mass-Spring Neural Interaction local model Generation of echographic image linear deformation

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Contributions Quantitative and qualitative comparison of mass-springs and tetrahedral elements Interactive non-linear static resolution Formal analysis of the real-time stability of integration methods based on parameters Identification of the parameters of a model from experimental data Relevant medical applications

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**Plan Deformation Models Real-time Resolution Techniques**

Mass-Spring vs. FEM Real-time Resolution Techniques Static Dynamic Echographic Simulator parameter identification Liver Model interactive deformation

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**Keyhole Surgery Surgery involves soft tissues simulation**

Need to model deformation …in real-time!

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**Human Liver Interior composed of parenchyma**

Surounded by elastic skin or Glisson’s capsule Venous network Approximate weight: 1.5 kg

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**Liver Model Geometry Physical Model Dynamic Analysis Static Analysis**

explicit integration stability Static Analysis non-linear resolution

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**Geometrical Model 187 Vertices 370 Triangles 299 Particles GHS3D**

1151 Tetrahedra 1634 Edges GHS3D

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**Physical Model Heterogenous Non-linear: Stress Strain**

[Boux et al., ISER, 2000] Heterogenous Non-linear: Stress Strain skin Parenchyma Weight distributed equaly on all particles (i.e. approximately 5g each)

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Explicit Integration 280 steps per second mass 5 grams

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Stability Analysis Im Re

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**Simulation Achitecture SGI Onyx2 Compexity 370 facets 1151 tetrahedra**

3399 springs Frequency 150Hz Explicit not stable! ...large mass

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

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**Iterative Solution 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

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**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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**Newton-Raphson Less iteration to converge then modified NR**

Exact Jacobian allows faster convergence Global time gain when solving linear system accurately residual iterations

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

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**Result Pseudo-dynamic 1157 tetrahedra Iterative non-linear resolution**

Rotational invarience (N.B. Real-time animation) 60 NR iterations/sec on SGI Octane 175Mhz

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**Liver Model (conclusions)**

Physical Model mass-springs Dynamic Analysis explicit integration unstable Static Analysis interactive non-linear resolution

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**Summary Physical Models Resolution Medical Simulators**

Mass-Spring or FEM? Resolution Static linear or non-linear? Dynamic explicit or implicit? Medical Simulators The choice of numerical methods must be guided by the application!

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Contributions Quantitative and qualitative comparison of mass-springs and tetrahedral elements Interactive non-linear static resolution Formal analysis of the real-time stability of integration methods based on parameters Identification of the parameters of a model from experimental data Relevant medical applications

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Local Model

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**Explicit Integration Dynamic equations solved by Euler’s 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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**Backwards engineering**

Geometrical description Geometrical description elasticity elasticity Physical Model Physical Model forces displacement forces displacements Results Results

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