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¨ osen-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 100 µ m) 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-1 α (SDF-1 α ) binds to CXCR4 subpopulation of BMSCs express transmembrane receptor CXCR4 and migrate towards SDF-1 α concentration gradient S. Thieme et al. Tissue Engineering 15 (2009) goal: complete invasion of BMSCs into internal compartments of 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-1 α density v in 3D pore structure Ω 1 f source of SDF-1 α in center of scaffold on Γ u t = D u ∆ Γ u − λ ∇ Γ · ( u ∇ Γ v ) v t = D v ∆ v + f in Ω 1 ∆ Γ − Laplace-Beltrami-operator coupled bulk/surface model with boundary conditions ∂ n v = 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 u t = D u ∆ Γ u − λ ∇ Γ · ( u ∇ Γ v ) on Γ ◮ formulation in fixed domain ( u δ Γ ) t = D u ∇ · ( δ Γ ∇ u ) − λ ∇ · ( δ Γ u ∇ v ) in Ω � � δ Γ surface delta-function ( Γ u d Γ = Ω u δ Γ d Ω ) ◮ approximation of δ Γ by phase-field function c = 1 2 tanh ( 1 − 3 r δ Γ ≈ |∇ c | , 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 Ω 1 v t = D v ∆ v + f in Ω 1 boundary condition e.g. D v ∇ v · n = − j ◮ formulation in fixed domain ( vH ) t = D v ∇ · ( H ∇ v ) + Hf − j δ Γ H Heaviside function, δ Γ surface delta-function ◮ approximation of H and δ Γ by phase-field function c = 1 2 tanh ( 1 − 3 r H ≈ c , δ Γ ≈ |∇ c | , 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, µ CT-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 = D u ∇ · ( |∇ c |∇ u ) − λ ∇ · ( |∇ c | u ∇ v ) in Ω ( vc ) t D v ∇ · ( c ∇ v ) + cf in Ω = with boundary condition u = g on ∂ Ω and ∂ n v = 0 on ∂ Ω phase-field function c = 1 2 tanh ( 1 − 3 r 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-1 α density v Chemotaxis in 3d scaffolds - a finite element approach 11
Introduction Modeling and simulation Results Conclusion SDF-1 α 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 µ CT 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
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