Hervé DELINGETTE. INRIA Sophia-Antipolis

Transcription

1 Hervé DELINGETTE INRIA Sophia-Antipolis French National Research Institute in Computer Science and Automatic Control Asclepios : 3D Segmentation, Simulation Platform, Soft Tissue Modeling CardioSense3D : Cardiac Simulation, SOFA : Open Software Plateform for Medical Simulation

2 Towards the virtual physiological human Physically-based simulation of biological tissues

3 Soft Tissue Characterization Biomechanical behavior of biological tissue is very complex Most biological tissue is composed of several components : Fluids : water or blood Fibrous materials : muscle fiber, neuronal fibers, Membranes : interstitial tissue, Glisson capsule Parenchyma : liver or brain

4 Soft Tissue Characterization To characterize a tissue, its stressstrain relationship is studied Height h Radius r Stress σ = Strain ε = F πr h*-h h Height h* Force F Radius r* Rest Position Cylindrical Piece of Tissue Deformed Position

5 Linear Elastic Material Simplest Material behaviour Only valid for small deformations (less than 5%) 800 Stress Strain

6 Biological Tissue Many complex phenomena arises σ σ V 0 V 1 σ loading V V3 Hysteresis Unloading ε Visco-elasticity ε Linear Domain Non-Linearity Slope =Young Modulus ε Anisotropy

7 Continuum Mechanics Continuum Mechanics Fluid Mechanics Structural Mechanics Linear Elasticity (Anisotropic, heterogeneous) Hyperelasticity Elasticity Plasticity

8 Basics of Continuum Mechanics Deformation Function X Ωa ( X ) R Displacement Function U φ ( X ) = φ( X ) X 3 Rest Position Ω Φ Deformed Position X U(X)

9 Basics of Continuum Mechanics The local deformation is captured by the deformation gradient : = = X X X X X X X X X X F j i ij φ φ φ φ φ φ φ φ φ φ X F = φ Ω X φ(x) Rest Position Deformed Position F(X) is the local affine transformation that maps the neighborhood of X into the neighborhood of φ(x)

10 Basics of Continuum Mechanics Distance between point may not be preserved Rest Position X+dX X Ω Deformed Position φ(x) φ(x+dx) Distance between deformed points ( ) ( ) ( ) T ds X dx X dx ( T = φ + φ φ φ)dx Right Cauchy-Green Deformation tensor C T = φ φ Measures the change of metric in the deformed body

11 Basics of Continuum Mechanics ( ) T Example : Rigid Body φ X motion = RX entails + no deformation T F X ) = φ ( X ) = R ( C = R R = Id Strain tensor captures the amount of deformation It is defined as the distance between C and the Identity matrix E = 1 1 ( T φ φ Id ) = ( C Id )

12 Strain Tensor Diagonal Terms : ε i Capture the length variation along the 3 axis Off-Diagonal Terms :γ i Capture the shear effect along the 3 axis E = 1 1 ε γ γ x xy xz 1 1 γ ε γ y xy yz 1 1 γ γ ε z xz yz

13 Linearized Strain Tensor Use displacement φ ( X ) = rather Id + than U ( Xdeformation ) 1 T T E = U + U + U U ( ) Assume small displacements 1 T E = U + U Lin ( )

14 Hyperelastic Energy The energy required to deform a body is a function of the invariants of strain tensor E : Trace E = I 1 Rest Position Ω Deformed Position Trace E*E= I Determinant of E = I 3 ( φ ) (,, ) 1 W 3 = w I I I dx Ω Total Elastic Energy

15 Linear Elasticity Isotropic Energy ( λ, µ ) : Lamé coefficients λ w X ) = Lin µ ( ) tr E + tr E ( Lin Hooke s Law w (X ) : density of elastic energy Advantage : Quadratic function of displacement λ µ w = ( div U ) + µ U rot U Drawback : Not invariant with respect to global rotation Extension for anisotropic materials

16 Shortcomings of linear elasticity Non valid for «large rotations and displacements»

17 St-Venant Kirchoff Elasticity Isotropic Energy ( λ, µ ) : Lamé coefficients Advantage : Generalize linear elasticity Invariant to global rotations Drawback : Poor behavior in compression Quartic function of displacement Extension for anisotropic materials λ w ( X ) = µ E ( ) tr E + tr

18 St Venant Kirchoff vs Linear Elasticity Linear St Venant Kirchoff Rest Position Linear St Venant Kirchoff

19 Other Hyperelastic Material w ( X ) = µ tre + f I Neo-Hookean Model w( X ) = µ e tre + f ( I 3 ) Fung Isotropic Model µ tre k k w X ) = e + e k Fung Anisotropic Model Veronda-Westman Mooney-Rivlin : ( ) ( ( I 1) ( ) + f ( I ) 1 1 ( γ tre ) e + c tre + f ( ) w( X ) = c I w + 3 ( ) ( X ) = c10tre + c01tre f I3 3

20 Estimating material parameters Complex for biological tissue : Heterogeneous and anisotropic materials Tissue behavior changes between in-vivo and in-vitro Ethics clearance for performing experimental studies Effect of preconditioning Potential large variability across population

21 Soft Tissue Characterization Different possible methods In vitro rheology In vivo rheology Elastometry Solving Inverse problems

22 Soft Tissue Characterization In vitro rheology can be performed in a laboratory. Technique is mature Not realistic for soft tissue (perfusion, )

23 Soft Tissue Characterization In vivo rheology can provide stress/strain relationships at several locations Influence of boundary conditions not well understood Source : Cimit, Boston USA

24 Soft Tissue Characterization Elastometry (MR, Ultrasound) mesure property inside any organ non invasively validation? Only for linear elastic materials Source Echosens, Paris

25 Soft Tissue Characterization Inverse Problems well-suited for surgery simulation (computational approach) require the geometry before and after deformation

26 Soft Tissue Characterization Still difficult to find reliable soft tissue material parameters Example : Liver soft tissue characterization

27 Discretisation techniques Four main approaches : Volumetric Mesh Based Surface Mesh Based Meshless Particles

28 Different types of meshes Surface Elements : Triangle 3, 1 nodes and more Quad 4, 8 nodes and more Volume Elements Tetrahedron 4, 10 nodes Prismatic 6, 15 nodes and more Hexahedron 8, 0 nodes and more

29 Structured vs Unstructured meshes Example 1 : Liver meshed with hexahedra 3 months work (courtesy of ESI) Example : Liver meshed with tetrahedra Automatically generated (1s)

30 Volumetric Mesh Discretization Classical Approaches : Finite Element Method (weak form) Rayleigh Ritz Method (variational form) Finite Volume Method (conservation eq.) Finite Differences Method (strong form) FEM, RRM, FVM are equivalent when using linear elements

31 Rayleigh-Ritz Method Step1 : Choose Finite Element (e.g. linear tetrahedron) Mesh discrediting the domain of computation Hyperelastic Material with its parameters Boundary Conditions ( tr E ) + tr λ w ( X ) = µ E Tetrahedron 4 nodes Fixed nodes Young Modulus Poisson Coefficient

32 Rayleigh-Ritz Method Step Write the elastic energy required to deform a single P 1 element Q 1 u ( P ) = Q P = U i i i i P 4 K P 3 P = Q 4 Q 3 [ ] T K jk U k t i WT i U j jk 1 = µ Q T T ( λ M M + µ M M + ( M M )[ Id ]) Ti [ jk ] i k j i j k i j k 3x3 36.V( Ti ) u( X ) = tre 4 i= 1 λ ( X ) u( i P i M i λi ( X ) = 6V ( T ) = M i Ui 6V ( T ) i )

33 W Rayleigh-Ritz Method Step3 Sum to get the total elastic energy ( ) ( ) T U w I, I I dx = W = U K U = 1, 3 Ω h Write the conservation of energy W ( U ) F T U + ρ ( X ) ( X g )dx Internal Energy = Ω Nodal Forces T i T Gravity Potential Energy i

34 Rayleigh-Ritz Method Step3 Linear Elasticity Write first variation of the energy : KU = R Static case M U&& + C U& + KU = R (t ) Dynamic case HyperElasticity=NonLinear Elasticity K ( U ) = R M U&& + C U& + K = ( U ) R (t ) Static case Dynamic case

35 Surface-Based Methods Only consider the mesh surface under some hypothesis : Linear Elastic Material (sometimes homogeneous) Only interact with organ surface Pros : No need to produce volumetric meshes Much faster than volumetric computation Cons : Only linear material No cutting

36 Surface-Based Methods Possible approaches : Boundary Element Models (BEM) Based on the Green Function of the linear elastic operator Requires homogeneous material Matrix Condensation Full Matrix inversion Iterative Precomputed Generation Solve 3*Ns equations F=KU

37 Other Methods Meshless Methods Use only points inside and specific shape functions Can better optimize location of DOFs Can cope with large deformations Deformation accuracy unknown Particles Smooth Particles Hydrodynamics that interact based on a state equation

38 Dynamic evolution and numerical integration Dynamic evolution Discrete models = lumped mass particles submitted to forces Newtonian evolution (1 st order differential system): { δp= V.dt δv= M -1 F(P,V).dt Explicit schemes: { Euler: δp= V t.dt δv= M -1 F(P t, V t ).dt Runge-Kutta: several evaluations to better extrapolate the new state [press9] Semi-Implicit schemes: Euler: δp= V t+dt.dt Verlet [teschner04] Unstable for large time-step!! { { δv= M -1 F(P t, V t ).dt P t+dt = P t P t-dt.+m -1 F(P t, V t ).dt V t+dt = (P t+dt P t.)dt-1

39 Evolution Implicit schemes [terzopoulos87], [baraff98], [desbrun99], [volino01], [hauth01] First-order expansion of the force: F(P t+dt, V t+dt ) F(P t, V t ) + F/ PδP + F/ VδV Euler implicit { δp= V t+dt.dt H = I - M -1 F/ V dt - M -1 F/ P dt with δv= H -1 Y Y = M -1 F(P t, V t ) + M -1 F/ P V t dt Backward differential formulas (BDF) : Use of previous states Unconditionally stable for any time-step But requires the inversion of a large sparse system Choleski decomposition + relaxation Conjugate gradient Speed and accuracy can be improve through preconditioning (alteration of H)

40 Towards Realistic Interactive Simulation Surgery Simulation must cope with several difficult technical issues : Soft Tissue Deformation Collision Detection Collision Response Haptics Rendering Real-time Constraints : 5Hz for visual rendering Hz for haptic rendering

41 Example of Soft Tissue Models Pre-computed Elastic Model Tensor-Mass and Relaxation-based Model Non-Linear Tensor-Mass Model Computational Efficiency Cutting Simulation Large Displacements - - +

42 Precomputed linear elastic model 9517 Tetrahedra

43 Tensor-Mass Models (low resolution) N = 1394 (634 Tétraèdres)

44 Non-linear Tensor-Mass Models

45 Simulation of surgical gestures Gliding Gripping Cutting (pliers) Cutting (US)

46 Hepatic Surgery Simulation

47 Cardiac Simulation Volumetric Conditions: 4 Cardiac Phases: Filling Isovolumetric Contraction Ejection Isovolumetric Relaxation Pressure Field in the endocardium Isovolumetric Constraint of myocardium Slowed 6 times Color: Action potential 47

48 More Information in Asclepios : 3D Segmentation, Simulation Platform, Soft Tissue Modeling CardioSense3D : Cardiac Simulation, SOFA : Open Software Plateform for Medical Simulation