3D constitutive modeling of the passive heart wall

Transcription

1 3D constitutive modeling of the passive heart wall Thomas S.E. Eriksson Institute of Biomechanics - Graz University of Technology Department of Biophysics - Medical University of Graz SFB status seminar, Schloß Röthelstein November 5-7, 2008

2 Outline Mechanical Behavior of Heart Tissue Fiber Example Constitutive Models Shortcomings Future Model Conclusions

3 Mechanical Behavior of Heart Tissue Nonlinear behavior Large deformation Fiber oriented layered structure Different behavior in each orthogonal direction Different behavior when sheared in each direction Stress or Force Strain or Displacement

4 Mechanical Behavior of Heart Tissue Nonlinear behavior Large deformation Fiber oriented layered structure Different behavior in each orthogonal direction Different behavior when sheared in each direction Stress or Force Strain or Displacement

5 Mechanical Behavior of Heart Tissue Nonlinear behavior Large deformation Fiber oriented layered structure Different behavior in each orthogonal direction Different behavior when sheared in each direction Nash, M. (1998), "Mechanics and Material Properties of the Heart using an Anatomically Accurate Mathematical Model", Thesis, University of Auckland, New Zealand. Modified.

6 Mechanical Behavior of Heart Tissue Nonlinear behavior Large deformation Fiber oriented layered structure Different behavior in each orthogonal direction Different behavior when sheared in each direction Sheet Sheet-normal Fiber

7 Mechanical Behavior of Heart Tissue Nonlinear behavior Large deformation Fiber oriented layered structure Different behavior in each orthogonal direction Different behavior when sheared in each direction Stress Fiber axis Sheet axis Sheet-normal axis Strain Hunter et. al. (1997), "A mathematical model of cardiac anatomy". In "Computational Biology of the Heart." Panfilov and Holden (eds.) John Whiley & Sons; Chichester, Modified.

8 Mechanical Behavior of Heart Tissue Nonlinear behavior Large deformation Fiber oriented layered structure Different behavior in each orthogonal direction Different behavior when sheared in each direction Shear Stress [kpa] FN FS SF SN NF, NS Shear Displacement [%] Dokos et. al. (2002) "Shear properties of passive ventricular myocardium" Am. J Physiol, 283:H2650-H2659. Modified.

9 Fiber Example Simple shear, plain stress, 3D incompressible material 2 The deformation gradient and left Cauchy-Green tensor is 2 1 F = 1 k , b = 1 + k 2 k 0 k k = tan(γ) 1 and m = FM = F cos(α) sin(α) 0

10 Fiber Example Simple shear, plain stress, 3D incompressible material 2 Invariant formulation gives I 1 = I 2 = 3 + k 2, I 3 = 1, I 5 = (I 1 1)I Ψ(I 1, I 4 ) in its most general form σ = p1 + 2ψ 1 b + 2ψ 4 m m in its most general form k = tan(γ) 1 where ψ 1 = Ψ I 1 and ψ 4 = Ψ I 4

11 Fiber Example Simple shear, plain stress, 3D incompressible material 2 σ 11 = p + 2(1 + k 2 )ψ 1 + 2ψ 4 (cos α + k sin α) 2 σ 22 = p + 2ψ 1 + 2ψ 4 sin 2 α 2 1 σ 33 = p + 2ψ 1 σ 12 = σ 21 = 2kψ 1 + 2ψ 4 sin α(cos α + k sin α) k = tan(γ) 1 Plain stress σ 33 = 0 p

12 Fiber Example Simple shear, plain stress, 3D incompressible material 2 2 k = tan(γ) 1 1 Without fibers σ 11 = 2k 2 ψ 1 σ 22 = 0 σ 12 = σ 21 = 2kψ 1 With fibers σ 11 = 2k 2 ψ 1 + 2ψ 4 (cos α + k sin α) 2 σ 22 = 2ψ 4 sin 2 α σ 12 = σ 21 = 2kψ ψ 4 sin α(cos α + k sin α)

13 Fiber Example Simple shear, plain stress, 3D incompressible material 2 2 k = tan(γ) 1 1 Without fibers σ 11 = 2k 2 ψ 1 σ 22 = 0 σ 12 = σ 21 = 2kψ 1 With fibers σ 11 = 2k 2 ψ 1 + 2ψ 4 (cos α + k sin α) 2 σ 22 = 2ψ 4 sin 2 α σ 12 = σ 21 = 2kψ ψ 4 sin α(cos α + k sin α)

14 Fiber Example Using the Holzapfel model Ψ GAH = 1 2 µ(i 1 3) + m 1 2m 2 {exp[m 2 (I 4 1) 2 ] 1} ψ 1 = 1 2 µ ψ 4 = m 1 (I 4 1) exp[m 2 (I 4 1) 2 ] Holzapfel et. al. (2000), A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models, J. Elasticity, 61:1-48.

15 Fiber Example Simple shear, plain stress, 3D incompressible material α = 30 o

16 Constitutive Models Several models tries to capture the behavior of heart tissue Don t use transversely isotropic models Pole-Zero Law Costa Law Separate Fung-type Law Tangent Law Langevin Eight-chain Law

17 Constitutive Models Several models tries to capture the behavior of heart tissue Don t use transversely isotropic models Pole-Zero Law Costa Law Separate Fung-type Law Tangent Law Langevin Eight-chain Law

18 Contstitutive Models Pole-Zero Law E 2 ff E 2 ss E 2 nn Ψ PZL =k ff a ff E ff b + k ss ff a ss E ss + k bss nn a nn E nn bnn E 2 fn E 2 sn Efs 2 +k fs a fs E fs b + k fn fs a fn E fn b + k sn fn a sn E sn bsn where E αβ is the strain in each direction k αβ is the strenght a αβ is the limiting strain or pole b αβ accounts for the non-linearity (Nash and Hunter (2000), Computational Mechanics of the Heart, J. Elasticity, 61: )

19 Contstitutive Models Costa Law Ψ CL = 1 a(exp[q] 1) 2 and Q =b ff E 2 ff + b sse 2 ss + b nn E 2 nn +2b fs [(E fs + E sf )/2] 2 + b fn [(E fn + E nf )/2] 2 + b sn [(E sn + E ns )/2] 2 where E αβ is the strain in each direction a is a copling term of b αβ b αβ represents the stiffness in direction αβ (Costa et. al. (2001), Modeling cardiac mechanical properties in three dimensions, Philos. T. Roy. Soc. A, 359: ).

20 Shortcomings of Constitutive Models Many material parameters Non-convexity, could be several solutions Pole-Zero law - all parameters uncoupled Costa law - all parameters coupled

21 Future Model Have only a few material parameters Show convexity Show strong ellipticity Invariant based

22 Future Model Shear tests Cubes of myocardial tissue (3mm) Where muscle layers were uniformly oriented Less than 1h after surgery Sheared in 6 directions F N N S S F NS NF Shear Stress [kpa] S F F 6 N N S FN FS SF SN NF, NS Shear Displacement [%] FN FS Dokos et. al. (2002) "Shear properties of passive ventricular myocardium" Am. J Physiol, 283:H2650-H2659. Modified. N S S F F N SF SN

23 Future Model Our proposed model Ψ = 1 2 µ(i 1 3) + m 1 2m 2 {exp[m 2 (I 4(F) 1) 2 ] 1} + n 1 2n 2 {exp[n 2 (I 4(N) 1) 2 ] 1} + b 12 2a 12 {exp[a 12 I 2 8(S,F) ] 1} K(J 1)2 G.A. Holzapfel, R.W. Ogden S N F

24 Future Model Our proposed model Ψ = 1 2 µ(i 1 3) + m 1 2m 2 {exp[m 2 (I 4(F) 1) 2 ] 1} + n 1 2n 2 {exp[n 2 (I 4(N) 1) 2 ] 1} + b 12 2a 12 {exp[a 12 I 2 8(S,F) ] 1} K(J 1)2 G.A. Holzapfel, R.W. Ogden Fits the characteristics of Dokos data Shows convexity Shows strong ellipticity Has only 7 material parameters

25 Future Model Our proposed model Ψ = 1 2 µ(i 1 3) + m 1 2m 2 {exp[m 2 (I 4(F) 1) 2 ] 1} + n 1 2n 2 {exp[n 2 (I 4(N) 1) 2 ] 1} + b 12 2a 12 {exp[a 12 I 2 8(S,F) ] 1} K(J 1)2 G.A. Holzapfel, R.W. Ogden Next step Add the active response Couple with electrophysiological model

26 Conclusions Heart tissue Nonlinear Fiber reinforced Layered structure Orthotropic Pole-Zero Law Components of strain with physical meaning (+) Many material paramaters (-) Does not show convexity (-) Costa Law Strains with physical meaning (+) Have shown better fitting capabilities then PZ (+) No separation between material directions (-) Does not show convexity (-)

27 Conclusions Heart tissue Nonlinear Fiber reinforced Layered structure Orthotropic Pole-Zero Law Components of strain with physical meaning (+) Many material paramaters (-) Does not show convexity (-) Costa Law Strains with physical meaning (+) Have shown better fitting capabilities then PZ (+) No separation between material directions (-) Does not show convexity (-)

28 Conclusions Heart tissue Nonlinear Fiber reinforced Layered structure Orthotropic Pole-Zero Law Components of strain with physical meaning (+) Many material paramaters (-) Does not show convexity (-) Costa Law Strains with physical meaning (+) Have shown better fitting capabilities then PZ (+) No separation between material directions (-) Does not show convexity (-)

29 Conclusions New model Has few material parameters Shows convexity Shows strong ellipticity Captures characteristic shear deformation

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