BioreactorMechanical models
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A microscopic model for cell-seeded material
J. Yi1, M. Stoffel1, D. Weichert1, K. Gavenis2, R. Muller-Rath2
1Institut fur Allgemeine Mechanik, RWTH Aachen2Klinik fur Orthopadie und Unfallchirurgie, RWTH Aachen
MSB-Net in Marburg, 5. February 2010
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
ProblemsRefereces
Contents
BioreactorWhat is the Bioreactor?Phenomenon in the Bioreactor
Mechanical modelsMacroscopic constitutive equationsMicroscopic constitutive equations
Problems
Refereces
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
ProblemsRefereces
What is the Bioreactor?Phenomenon in the Bioreactor
Sketch of the Bioreactor
=⇒ change of material properties & change of mass
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
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What is the Bioreactor?Phenomenon in the Bioreactor
Fotos in Bioreactor
(a) without stimulating (b) with stimulating
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
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Macroscopic constitutive equationsMicroscopic constitutive equations
Macroscopic Model
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
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Macroscopic constitutive equationsMicroscopic constitutive equations
Theory for the macroscopic model
Constitutive equation:
σ = σscaf + σunit
= C : ε+ C(ε) : ε− Dσ + Cunit: ε+ Cunit : ε
≈ C : ε+ C(ε) : ε− Dσ + Cunit : ε
Evolution equation:
Cunit11 (Ψ) = k
√Ψ(Cunit
11,crit − Cunit11
), 0 < Cunit
11 ≤ Cunit11,crit
Ψ =12λ ln2(J) +
12µ(IC
1 − 3)− µ ln(J)
where λ, µ are the Lame constants, IC1 = C : I = FtF : I, J = det F
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
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Macroscopic constitutive equationsMicroscopic constitutive equations
Evolution of Young’s modulus in macroscopic model
(a) t= 0sec (b) t= 6sec
(c) t= 12sec (d) t= 24sec (e) t= 30sec
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
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Macroscopic constitutive equationsMicroscopic constitutive equations
Microscopic Model
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
ProblemsRefereces
Macroscopic constitutive equationsMicroscopic constitutive equations
Theory for the microscopic model
General constitutive approach for transversely isotropicmaterials:
W (C,M(C)) = W [I1(C), I2(C), I3(C), I4 (C,M) , I5(C,M)]
where M is a structure tensor: M(C) = nM ⊗ nM ,nM is a unit vector in the growth direction of fiber andI4(C,M (C)) = C : M = nMCnM = η2 (η: stretch ratio of fibers)
I5(C,M(C)) = C2 : M = nMC2nM
Assumptions for the phenomen of the bioreactor:
I nM = nM(C, ~F
) nM⊥~F−→ , nM = nM(C) =???
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
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Macroscopic constitutive equationsMicroscopic constitutive equations
Suggested model:
W = ρtΨ −→ W (C,M(C)) = Ψ (I1, I2, I3) ρt (I4, I5)
Further simplified assumptions
I Ψ = Ψ (I1, I3) = Ψ (I1, J) =12λ ln2 J +
12µ(I1 − 3)− µ ln J
I ρt = ρt(I4; t) = ρt(η) = ρ0 + ρc(1− ce−η)
where ρ0 is the the initial mass density, ρc is the critical value ofthe density growth parameter and c is the growth parameter.
12S =
∂W∂C =
∂ (ρtΨ)
∂C =∂ρt∂C Ψ + ρt
∂Ψ
∂C :=12Srem +
12Smech
−→ S := Srem + Smech
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
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Macroscopic constitutive equationsMicroscopic constitutive equations
where
I∂ρt∂C =
∂ρt∂η
∂η
∂I4∂I4∂C = ρcce−η 1
2η (M + A) =ρcce−η
2η (M + A)
I∂Ψ
∂C =∂Ψ
∂I1∂I1∂C +
∂Ψ
∂J∂J∂C =
µ
2 I +1J (λ ln J − µ)
12JC−1
A =?
∂I4∂C =
∂ (C : M)
∂C = M : C,C + C : M,C︸︷︷︸:=P
= M : I + C : P︸ ︷︷ ︸:=A
= M + A
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
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Macroscopic constitutive equationsMicroscopic constitutive equations
The 2nd Piola-Kirchhoff stress tensor S is
S = 2∂W∂C = Srem + Smech
= 2 ρcce−η2η
[12λ ln2 J +
12µ(I1 − 3)− µ ln J
](M + A)
+ρc(1− ce−η
) [µ2 I +
12 (λ ln J − µ) C−1
]
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
ProblemsRefereces
Macroscopic constitutive equationsMicroscopic constitutive equations
The material tensor C is
C = 2∂S∂C
= 2 ρcce−η2η
[12λ ln2 J +
12µ(I1 − 3)− µ ln J
](M,C + A,C︸︷︷︸
:=Q
)
+ρc(1− ce−η
) [µ2 I,C +
12 (λ ln J − µ) C−1
,C
]= 2 ρcce−η
2η
[12λ ln2 J +
12µ(I1 − 3)− µ ln J
](P + Q)
+ρc(1− ce−η
) 12 (λ ln J − µ)
(−C−1 ⊗ C−1
)
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
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Macroscopic constitutive equationsMicroscopic constitutive equations
The anisotropy of the material due to the new added mass can beexplained with the two tensors:{
P = M,CQ = A,C = (C : P),C = (C : M,C),C
Summary: M,C plays a key role!!!
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
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Problems and future work
I n = n(C)? −→ M = M(C)?
I The evolution equation?
I The roll of fiber: Only against pull?
I The factors of the fiber growth?
I Micro level and macro level in tissue mechanics
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material
BioreactorMechanical models
ProblemsRefereces
References
E. Kuhl, P. Steinmann, 2003. Mass- and volume specific viewson thermodynamics for open system. Proc. R. Soc 459,2547-2568.V. A. Lubarda, A. Hoger, 2002. On the mechanics of solidswith a growing mass. International Journal of Solids andStructures 39, 4627-4664.
J. Yi, M. Stoffel, D. Weichert, K. Gavenis, R. Muller-Rath A microscopic model for cell-seeded material