Chemotaxis in 3D scaffolds - a finite element approach Christoph - - PowerPoint PPT Presentation

Department of Mathematics Institute of Scientific Computing

Chemotaxis in 3D scaffolds - a finite element approach

Christoph Landsberg

Dresden

Introduction Modeling and simulation Results Conclusion

People

◮ Michael Gelinsky

Institute of Materials Science, TU Dresden

◮ Angela R¨

• sen-Wolff

University Clinic Carl Gustav Carus, Dresden

◮ Florian Stenger

Institute of Scientific Computing, TU Dresden

◮ Axel Voigt

Institute of Scientific Computing, TU Dresden

Chemotaxis in 3d scaffolds - a finite element approach 2

Introduction Modeling and simulation Results Conclusion

in situ tissue engineering for bone defect healing

◮ migration of stem cells into injured areas ◮ differentiation in specific cell types to set up new tissue

mesenchymal stem cells - subset of bone marrow stromal cells (BMSCs) can migrate and differentiate into osteoblasts idea: in large bone defects BMSCs could be attracted toward bone substitute material serving as a scaffold that could be colonized by BMSCs and consequently remodeled to new bone tissue additional effect: BMSCs can induce angiogenesis (e.g. VEGF) to ensure supply of nutrients

Chemotaxis in 3d scaffolds - a finite element approach 3

Introduction Modeling and simulation Results Conclusion

in situ tissue engineering for bone defect healing

◮ three-dimensional scaffolds

made from biomimetically mineralized collagen I interconnected pore structure (mean pore diameter 100µm) elastic properties

• M. Gelinsky et al. Chem. Eng. J 137 (2008), H. Domascke et al. Tissue Engineering 12 (2006)

Chemotaxis in 3d scaffolds - a finite element approach 4

Introduction Modeling and simulation Results Conclusion

in situ tissue engineering for bone defect healing

◮ chemoattractant for BMSCs

stromal cell-derived factor-1α (SDF-1α) binds to CXCR4 subpopulation of BMSCs express transmembrane receptor CXCR4 and migrate towards SDF-1α concentration gradient

• S. Thieme et al. Tissue Engineering 15 (2009)

goal: complete invasion of BMSCs into internal compartments

• f large 3D substitute scaffolds without in vitro preseeding

Chemotaxis in 3d scaffolds - a finite element approach 5

Introduction Modeling and simulation Results Conclusion

Modified Keller-Segel model for chemotaxis

BMSC population u on curved surface of pores Γ SDF-1α density v in 3D pore structure Ω1 f source of SDF-1α in center of scaffold ut

=

Du∆Γu − λ∇Γ · (u∇Γv)

• n Γ

vt

=

Dv∆v + f in Ω1

∆Γ − Laplace-Beltrami-operator

coupled bulk/surface model with boundary conditions ∂nv = 0 on Γ and u = g on ∂Γ

Chemotaxis in 3d scaffolds - a finite element approach 6

Introduction Modeling and simulation Results Conclusion

Implicit description of computational domain - surface

◮ convection - diffusion equation on Γ

ut = Du∆Γu − λ∇Γ · (u∇Γv)

• n Γ

◮ formulation in fixed domain

(uδΓ)t = Du∇ · (δΓ∇u) − λ∇ · (δΓu∇v)

in Ω

δΓ surface delta-function (

• Γ u dΓ =
• Ω uδΓdΩ)

◮ approximation of δΓ by phase-field function

δΓ ≈ |∇c|,

c = 1 2 tanh(1 − 3r

ǫ )

• A. R¨

atz, A. Voigt, Comm. Math. Sci. 4 (2006) Chemotaxis in 3d scaffolds - a finite element approach 7

Introduction Modeling and simulation Results Conclusion

Implicit description of computational domain - bulk

◮ diffusion equation in Ω1

vt = Dv∆v + f in Ω1 boundary condition e.g. Dv∇v · n = −j

◮ formulation in fixed domain

(vH)t = Dv∇ · (H∇v) + Hf − jδΓ

H Heaviside function, δΓ surface delta-function

◮ approximation of H and δΓ by phase-field function

H ≈ c,

δΓ ≈ |∇c|,

c = 1 2 tanh(1 − 3r

ǫ )

• S. Li, J. Lowengrub, A. R¨

atz, A. Voigt, Comm. Math. Sci. 7 (2009) Chemotaxis in 3d scaffolds - a finite element approach 8

Introduction Modeling and simulation Results Conclusion

Implicit description of computational domain

◮ requires a signed-distance representation of domain ◮ adaptive refinement at internal boundary

input: surface mesh, µCT-data, CAD-file, . . .

Chemotaxis in 3d scaffolds - a finite element approach 9

Introduction Modeling and simulation Results Conclusion

Model to be solved

◮ chemotaxis model in Ω

(u|∇c|)t =

Du∇ · (|∇c|∇u) − λ∇ · (|∇c|u∇v) in Ω

(vc)t =

Dv∇ · (c∇v) + cf in Ω with boundary condition u = g on ∂Ω and ∂nv = 0 on ∂Ω phase-field function c = 1

2 tanh(1 − 3r ǫ )

convergence for ǫ → 0 to original problem (asymptotic analysis)

Chemotaxis in 3d scaffolds - a finite element approach 10

Introduction Modeling and simulation Results Conclusion

BMSC population u and SDF-1α density v

Chemotaxis in 3d scaffolds - a finite element approach 11

Introduction Modeling and simulation Results Conclusion

SDF-1α density v

Chemotaxis in 3d scaffolds - a finite element approach 12

Introduction Modeling and simulation Results Conclusion

BMSC population u

Chemotaxis in 3d scaffolds - a finite element approach 13

Introduction Modeling and simulation Results Conclusion

Conclusion

◮ in situ tissue engineering

colonize bone substitute material by BMSCs leading to remodeling of the scaffold into new bone tissue and induce angiogenesis

◮ BMSCs have to invade into internal compartments of scaffold ◮ modified bulk/surface chemotaxis model on µCT data of

scaffold

◮ diffuse interface / diffuse domain approach to enable

efficient simulation

• A. R¨

atz, A. Voigt, Comm. Math. Sci. (2006); X. Li, J. Lowengrub, A. R¨ atz, A. Voigt, Comm. Math. Sci. (2009)

◮ adaptive finite element simulation toolbox AMDiS

Chemotaxis in 3d scaffolds - a finite element approach 14

Thanks for your attention!