a microscopic model for cell-seeded material · bioreactor mechanical models problems refereces a...

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

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Contents

BioreactorWhat is the Bioreactor?Phenomenon in the Bioreactor

Mechanical modelsMacroscopic constitutive equationsMicroscopic constitutive equations

Problems

Refereces

ProblemsRefereces

What is the Bioreactor?Phenomenon in the Bioreactor

Sketch of the Bioreactor

=⇒ change of material properties & change of mass

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What is the Bioreactor?Phenomenon in the Bioreactor

Fotos in Bioreactor

(a) without stimulating (b) with stimulating

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Macroscopic constitutive equationsMicroscopic constitutive equations

Macroscopic Model

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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

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Evolution of Young’s modulus in macroscopic model

(a) t= 0sec (b) t= 6sec

(c) t= 12sec (d) t= 24sec (e) t= 30sec

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Microscopic Model

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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) =???

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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.

∂W∂C =

∂ (ρtΨ)

∂C =∂ρt∂C Ψ + ρt

∂C :=12Srem +

12Smech

−→ S := Srem + Smech

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I∂ρt∂C =

∂ρt∂η

∂I4∂I4∂C = ρcce−η 1

2η (M + A) =ρcce−η

2η (M + A)

I∂Ψ

∂C =∂Ψ

∂I1∂I1∂C +

∂J∂J∂C =

2 I +1J (λ ln J − µ)

12JC−1

∂I4∂C =

∂ (C : M)

∂C = M : C,C + C : M,C︸︷︷︸:=P

= M : I + C : P︸︷︷︸:=A

= M + A

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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

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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︸︷︷︸

+ρc(1− ce−η

) [µ2 I,C +

12 (λ ln J − µ) C−1

]= 2 ρcce−η

[12λ ln2 J +

12µ(I1 − 3)− µ ln J

](P + Q)

+ρc(1− ce−η

) 12 (λ ln J − µ)

(−C−1 ⊗ C−1

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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!!!

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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

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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.

a microscopic model for cell-seeded material · bioreactor mechanical models problems refereces a...

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