A microscopic model for cell-seeded material · Bioreactor Mechanical models Problems Refereces A...
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BioreactorMechanical models
ProblemsRefereces
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
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BioreactorMechanical models
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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
![Page 3: A microscopic model for cell-seeded material · Bioreactor Mechanical models Problems Refereces A microscopic model for cell-seeded material J. Yi1, M. Stoffel1, D. Weichert1, K.](https://reader033.fdocuments.in/reader033/viewer/2022042306/5ed27ca7a3d9aa6a22567aad/html5/thumbnails/3.jpg)
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
![Page 4: A microscopic model for cell-seeded material · Bioreactor Mechanical models Problems Refereces A microscopic model for cell-seeded material J. Yi1, M. Stoffel1, D. Weichert1, K.](https://reader033.fdocuments.in/reader033/viewer/2022042306/5ed27ca7a3d9aa6a22567aad/html5/thumbnails/4.jpg)
BioreactorMechanical models
ProblemsRefereces
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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BioreactorMechanical models
ProblemsRefereces
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
![Page 6: A microscopic model for cell-seeded material · Bioreactor Mechanical models Problems Refereces A microscopic model for cell-seeded material J. Yi1, M. Stoffel1, D. Weichert1, K.](https://reader033.fdocuments.in/reader033/viewer/2022042306/5ed27ca7a3d9aa6a22567aad/html5/thumbnails/6.jpg)
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
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BioreactorMechanical models
ProblemsRefereces
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
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BioreactorMechanical models
ProblemsRefereces
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
![Page 9: A microscopic model for cell-seeded material · Bioreactor Mechanical models Problems Refereces A microscopic model for cell-seeded material J. Yi1, M. Stoffel1, D. Weichert1, K.](https://reader033.fdocuments.in/reader033/viewer/2022042306/5ed27ca7a3d9aa6a22567aad/html5/thumbnails/9.jpg)
BioreactorMechanical models
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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
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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
![Page 11: A microscopic model for cell-seeded material · Bioreactor Mechanical models Problems Refereces A microscopic model for cell-seeded material J. Yi1, M. Stoffel1, D. Weichert1, K.](https://reader033.fdocuments.in/reader033/viewer/2022042306/5ed27ca7a3d9aa6a22567aad/html5/thumbnails/11.jpg)
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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
![Page 12: A microscopic model for cell-seeded material · Bioreactor Mechanical models Problems Refereces A microscopic model for cell-seeded material J. Yi1, M. Stoffel1, D. Weichert1, K.](https://reader033.fdocuments.in/reader033/viewer/2022042306/5ed27ca7a3d9aa6a22567aad/html5/thumbnails/12.jpg)
BioreactorMechanical models
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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
![Page 13: A microscopic model for cell-seeded material · Bioreactor Mechanical models Problems Refereces A microscopic model for cell-seeded material J. Yi1, M. Stoffel1, D. Weichert1, K.](https://reader033.fdocuments.in/reader033/viewer/2022042306/5ed27ca7a3d9aa6a22567aad/html5/thumbnails/13.jpg)
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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
![Page 14: A microscopic model for cell-seeded material · Bioreactor Mechanical models Problems Refereces A microscopic model for cell-seeded material J. Yi1, M. Stoffel1, D. Weichert1, K.](https://reader033.fdocuments.in/reader033/viewer/2022042306/5ed27ca7a3d9aa6a22567aad/html5/thumbnails/14.jpg)
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
![Page 15: A microscopic model for cell-seeded material · Bioreactor Mechanical models Problems Refereces A microscopic model for cell-seeded material J. Yi1, M. Stoffel1, D. Weichert1, K.](https://reader033.fdocuments.in/reader033/viewer/2022042306/5ed27ca7a3d9aa6a22567aad/html5/thumbnails/15.jpg)
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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
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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