Soft Tissue Biomechanics

Transcription

1 Soft Tissue Biomechanics Prof. Maarja Kruusmaa Tallinna Tehnikaülikool

2 Where is it used? For modelling stresses and deformations on tissue

3 Where is it used? Implant simulation to develop better prosthesis

4 Where is it used? Surgery simulations

5 Where is it used? Entertainment and game industry

6 Stress σ = F A τ 11, τ 22,τ 33 Normal stresses τ 12,τ 13,τ 21,τ 23,τ 31,τ 23 Shearing stresses τ ij Stress tensor 1 st component of stress 2 nd component of stress Surface normal to X 1 τ 11 τ 12 τ 13 Surface normal to X 2 τ 21 τ 22 τ 23 Surface normal to X 3 τ 31 τ 32 τ 33 3 rd component of stress Stress tensor is symmetric τ 12 = τ 21,τ 23 = τ 32,τ 31 = τ 13

7 Strain ε = L L 0 e = L2 2 L 0 L 0 2L 2 Hooke s Law σ = Ee E Young s modulus

8 Deformation of a body x i = x i (a 1,a 2,a 3 ) a i = a i (x 1, x 2, x 3 ), i =1,2,3 ds 0 2 = da da da 3 2 ds 2 = dx dx dx 3 2 dx i = x i a j da j da i = a i x j dx j

9 δ ij ds 0 2 = δ ij da i da j = δ ij a i x l a j x m dx l dx m ds 2 = δ ij dx i dx j = δ x i a l x j a m da l da m, Kronecker delta, 1 if i=j, 0 if i j Green s strain tensor, Lagrangian strain tensor E ij = 1 2 δ αβ x α a i x β a j δ ij Almansi s strain tensor, Eulerian strain tensor e ij = 1 2 (δ ij δ αβ a α x i a β x j ) ds 2 ds 0 2 = 2E ij da i da j ds 2 ds 0 2 = 2e ij dx i dy j For an infinitesimal case, the distinction between the Lagrangian and Eulerian strain tensor disappears

10 The Hookean Elastic Solid Hooke s Law σ ij = C ijkl e kl Hooke s Law for isotropic materials σ ij = λe xx δ ij + 2µe ij λ, μ Lamé constants, μ=g- shear modulus σ xx = λ(e xx + e yy + e zz ) + 2Ge xx σ yy = λ(e xx + e yy + e zz ) + 2Ge yy σ zz = λ(e xx + e yy + e zz ) + 2Ge zz σ xy = 2Ge xy σ yz = 2Ge yz σ zx = 2Ge zx The inverted form e ij = 1+ν E σ ν E σ kkδ ij ν- Poisson s ratio

11 e ij = 1+ν E σ ν E σ kkδ ij e xx = 1 E [σ xx ν(σ yy +σ zz )], e yy = 1 E [σ yy ν(σ zz +σ xx )], e zz = 1 E [σ zz ν(σ xx +σ yy )], e xy = 1+ν E σ xy = 1 2G σ xy e yz = 1+ν E σ yz = 1 2G σ yz e zx = 1+ν E σ zx = 1 2G σ zx

12 λ = 2Gν G(E 2G) = 1 2ν 3G E = Eν (1+ν)(1 2ν) G = ν = E = λ(1 2ν) 2ν = E 2(1+ ν) λ 2(λ + G) = λ 3K λ = E 2G 1 G(3λ + 2G) λ + G = λ(1+ν)(1 2ν) ν = 2G(1+ν)

13 Solids resist shear deformation shear stress is proportional to shear strain. τ = Gtan(θ) Fluids resist rate of shear F S = µθ t For Newtonian fluids (dynamic) viscosity τ = µ du dy F θ µ = const.

14 Strain Rate Velocity vector field v i = (x 1, x 2.x 3 ) Difference between velocities at x i and x i +dx i dv i = v i x j x j v i = 1 v i + v j x j 2 x j x i 1 v j 2 x i V ij 1 v i + v j 2 x j x i Ω ij 1 v j 2 x i v i x j dv i = V ij dx j Ω ij dx j v i x j Strain rate tensor Vorticity tensor

15 Non-viscous fluids Stress tensor σ ij = pδ ij p pressure in ideal gas in related to density and temperature p ρ = RT For incompressible fluid ρ = const.

16 Newtonian Viscous Fluids Shear stress is linearly proportional to the strain rate σ ij = pδ ij + L ijkl V kl For an isotropic tensor L ijkl -tensor of viscosity coefficients L ijkl = λδ ij δ kl + µ ( δ ij δ kl +δ il δ ) jk σ ij = pδ ij + λv kk δ ij + 2µV ij σ kk = 3p + (3λ + 2µ)V kk If the mean normal stress 1/3σ kk is independent of the rate of dilation V kk Stokes fluid σ ij = pδ ij + 2µV ij 2 3 µv kkδ ij For incompressible fluid V kk =0 3λ + 2µ = 0 σ ij = pδ ij + 2µV ij

17 Non-Newtonian Fluids

18 Models of viscoelasticity Spring F = µu Dashpot F = η u

19 Maxwell model F u = µ + F η u(0) = F(0) µ c(t) = 1 µ + 1 η t 1(t) k(t) = µe µ t η 1(t)

20 Voigt model F = µu +η u u(0) = 0 c(t) = 1 µ (1 e µ t η )1(t) k(t) = ηδ(t) + µ1(t)

21 Standard linear solid (Kelvin model) u =u + u 1 1 ʹ F = F 0 + F 1 F 0 = µ 0 u F = η 1 u = µ u 1 ʹ F = µ 0 u + µ 1 u 1ʹ F + η 1 µ 1 F = µ 0 u +η 1 (1+ µ 0 µ 1 ) u

22 Elasticity, viscosity and deformation

23 Dynamic modulus Strain ε = ε 0 sin(tω) Stress σ = σ 0 sin(tω +δ) δ - Phase lag Elastic materials δ = 0 Viscous materials δ = π 2 Elasticity : Storage modulus energy conserved Viscosity: Loss modulus energy dissipated E ' = σ 0 ε 0 cosδ E '' = δ 0 ε 0 sinδ

24 Springs and chock absorbers Solids are springs Fluids are absorbers

25 Physial Laws of Soft Tissues Nonlinear stress-strain relationship - Hysteresis loop in cyclic loading and unloading - - Preconditioning in repeated cycles - When held at constant strain, soft tissue shows stress relaxation - When held at constant stress, soft tissue shows creep

26 Actin Present in muscles, leokocytes, red blood cells, endothelial cells etc. mechanical support to cells - in muscle cells, to be the scaffold on which myosin proteins generate force to support muscle contraction

27 Resilin Resilin is a protein found in arthropods Elastic joints of insect wings Base of the hind legs of fleas and locusts

28 Abductin An elastic protein found in scallops hinges

29 Elastin * Present in: - skin and areolar tissue (as thin strands) - walls of arteries and veins prominent component of lung tissue

30 Collagen Basic structural element for soft and hard tissues in animals * Provide our bodies mechanical integrity and strength * Collagen, in the form of elongated fibrils, is mostly found in fibrous tissues such as tendon, ligament and skin * Also abundant in cornea, cartilage, bone, blood vessels

31 Stress-strain curve of a tendon By Shantanu Sinha and Ryuta Kinugasa, 2012

32 3. PHYSICAL LAWS OF SOFT TISSUES * Strain 2D Elasticity Example * video 4 w w w. b i o r o b o t i c s. t t u. e e

33 4. DEFORMATION MECHANICAL MODELS * Hyperelasticity St. Venant Kirchhoff model is an extension of the linear elastic material to the nonlinear regime w w w. b i o r o b o t i c s. t t u. e e

34 4. DEFORMATION MECHANICAL MODELS * Deformable splines (electromagnetic behaviour) - Non-Linear Strain w w w. b i o r o b o t i c s. t t u. e e

35 4. DEFORMATION MECHANICAL MODELS * Deformable splines (mechanical behaviour) - e.g. FEniCS Project FEM computational meshes w w w. b i o r o b o t i c s. t t u. e e

36 4. DEFORMATION MECHANICAL MODELS * Choise of the model depends on: - type of the tissue - speed of the needed simulation - representation quality of the simulation * Heuristic Model: - Geometry based Model * Deformable splines * Mass springer damper model * Linked Volumes * Mass-tensor Model still in use mostly for surgical simulation w w w. b i o r o b o t i c s. t t u. e e

37 4. DEFORMATION MECHANICAL MODELS * Heuristic Model:. positive: computational simplicity. negative: low realism w w w. b i o r o b o t i c s. t t u. e e

38 4. DEFORMATION MECHANICAL MODELS * Continuum Mechanical Model:. solving analytically proves impossible. uses numerical solution scheme, such as: - FEM/FEA Finite Element Analysis w w w. b i o r o b o t i c s. t t u. e e

39 4. DEFORMATION MECHANICAL MODELS * Continuum Mechanical Model:. solving analytically proves impossible. uses numerical solution scheme, such as: - Boundary Element Methods w w w. b i o r o b o t i c s. t t u. e e

40 4. DEFORMATION MECHANICAL MODELS * Hybrid Models: - Is a mix of * Heuristic Model * Continuum Mechanical Model - This approach subdivided the organ into * Operational region (requires more accuracy) - use nonlinear continuum-mechanics law provides more accurary * Non-operational region (less accuracy) - use fast surface heuristic model w w w. b i o r o b o t i c s. t t u. e e

41 4. DEFORMATION MECHANICAL MODELS * Tissue Damage w w w. b i o r o b o t i c s. t t u. e e

42 4. DEFORMATION MECHANICAL MODELS * Tissue Damage w w w. b i o r o b o t i c s. t t u. e e

43 5. IMPLEMENTATION APPROACHES * Data Acquisition: - Clinical data is the best and more realistic approach - 3D model generation from CT data using segmentation techniques w w w. b i o r o b o t i c s. t t u. e e * video 5

44 5. IMPLEMENTATION APPROACHES * Data Acquisition: - Real Time 3D Visualization - CT (Computed Tomography) Scan Aortic Dissection Endoscope Navigated w w w. b i o r o b o t i c s. t t u. e e * video 6

45 5. IMPLEMENTATION APPROACHES * Physical Interaction: w w w. b i o r o b o t i c s. t t u. e e

46 5. IMPLEMENTATION APPROACHES * Physical Interaction: - Touch/Cut and Deformation Rendering Feedback control of Tissue Damage: - Define thresholds for tissue stress - avoid irreparable tissue damage * video 7 w w w. b i o r o b o t i c s. t t u. e e

47 5. IMPLEMENTATION APPROACHES * OpenFOAM: * video 8 w w w. b i o r o b o t i c s. t t u. e e

48 5. IMPLEMENTATION APPROACHES * COMSOL: * video 9 w w w. b i o r o b o t i c s. t t u. e e