Modelling and simulation of mechano-chemical fluid-structure interaction with application to atherosclerotic plaque growth Stefan Frei University College London Thomas Richter Otto von Guericke University Magdeburg Thomas Wick Leibniz University Hannover Modeling, Simulation and Optimization of the Cardiovascular System Magdeburg, October 24, 2018 Gefördert durch
Plaque growth in blood vessels Mechano-chemical fluid-structure interaction Solid growth depending on concentration of monocytes/foam cells Migration of monocytes depends on wall shear stress Ω s ( t ) Γ i ( t ) Transport of Monocytes Ω f ( t ) Transendothelial migration and differentiation Plaque Γ i ( t ) Growth Ω s ( t ) Formation of foam cells Difficulties Strongly-coupled FSI problem: ρ f ≈ ρ s Large deformations up to full clogging Different time scales of fluid dynamics (milliseconds-seconds) and plaque growth (days-months) S. Frei | Mechano-chemical FSI 2
Overview Model 1 Temporal two-scale approach 2 Fluid-structure interaction 3 Discretisation 4 Numerical results 5 S. Frei | Mechano-chemical FSI 3
Overview Model 1 Temporal two-scale approach 2 Fluid-structure interaction 3 Discretisation 4 Numerical results 5 S. Frei | Mechano-chemical FSI 4
Equations Mechano-chemical FSI system based on Yang et al. 2016 (simplified) � ρ f ( ∂ t v f + v f · ∇ v f ) − div σ f = 0 in F ( t ) div v f = 0 � ρ s ( ∂ t v s + v s · ∇ v s ) − div σ s ( c s )= 0 in S ( t ) ∂ t u s + v s · ∇ u s − v s = 0 � σ f n f + σ s ( c s ) n s = 0 on Γ i ( t ) v f = v s ∂ t c s = γ ( σ WS ) f Material laws (simplifications) Newtonian fluid: σ f = ρ f ν f ( ∇ v + ∇ T v ) − p f I Solid material law based on St.Venant-Kirchhoff material σ s ( c s ) Dependence of the foam cell concentration c s on the wall shear stress σ WS f � − 1 � 1 + σ WS � γ ( σ WS f σ WS ) = γ 0 , = | σ f n f · � e 1 | d o f f σ Γ i S. Frei | Mechano-chemical FSI 5
Coupling FSI-Chemistry σ WS f Concentration of foam cells c s depends on the wall shear stress σ WS f c s FSI Feedback: Multiplicative decomposition of the (ODE) deformation gradient F s = F g ( c s ) F e ( Rodriguez et al. 1994 ) F s = F g ( c s ) F e Idea F s = I + ˆ ˆ ∇ ˆ u s V ˆ ˆ V V g F g = F − 1 ˆ F e = F − 1 ˆ g e F s = I − ∇ u s S. Frei | Mechano-chemical FSI 6
Solid material law with growth in Lagrangian coordinates Multiplicative decomposition of the deformation gradient F e = ˆ ˆ F s ˆ F − 1 ( c s ) = ( I + ˆ u s )ˆ F − 1 ∇ ˆ ( c s ) g g St. Venant-Kirchhoff in Lagrangian coordinates E e = 1 ˆ 2 (ˆ F T e ˆ Σ e = 2 µ s tr(ˆ ˆ E e ) + λ s ˆ F e − I ) , E e . Solid system of equations ρ 0 v s − div � ˆ F e ( c s )ˆ Σ e ( c s ) � ˆ s ∂ t ˆ = 0 in ˆ S ∂ t ˆ u s − ˆ v s = 0 Yang et al (2016), S.F., T.Richter, T.Wick, JCP (2016) S. Frei | Mechano-chemical FSI 7
Solid material law with growth in Eulerian coordinates Multiplicative decomposition of the deformation gradient F e = F − 1 ( c s ) F s = F − 1 ( c s )( I − ∇ u s ) g g St. Venant-Kirchhoff in Eulerian coordinates E e = 1 2 ( F − T F − 1 σ s = J s F − 1 Σ e F − T − I ) , Σ e = 2 µ s tr( E e ) + λ s E e , , e e e s where J s = det F s . Density: ρ = J − 1 ( c s ) J s ρ 0 s , where J g = det F g . g Solid system of equations J − 1 ( c s ) J s ρ 0 s ( ∂ t v s + v s · ∇ v s ) − div � J s F − 1 ( c s )Σ e ( c s ) F − T � = 0 g e s in S ( t ) ∂ t u s + v s · ∇ u s − v s = 0 S.F., T.Richter, T.Wick, JCP (2016) S. Frei | Mechano-chemical FSI 8
Overview Model 1 Temporal two-scale approach 2 Fluid-structure interaction 3 Discretisation 4 Numerical results 5 S. Frei | Mechano-chemical FSI 9
Time scales Different time scales for fluid dynamics ( t short = 1 s , period of inflow data) and plaque growth ( T long = 1 month) To resolve the short scale ( k ≤ 1 / 20 s ) over T long would require > 10 7 time-steps Assumption : c s is approximately constant on the short scale c s = � t Consider the averaged quantity ¯ t − 1 s c s ( r ) dr There holds � t � t γ ( σ WS γ ( σ WS ∂ t ¯ c s = ∂ t c s ( r ) dr = ( v , p )) dr =: ¯ ( v , p )) . f f t − 1 s t − 1 s Problem : No initial values available for u , v , as we can not resolve the short scale We assume the existence of a periodic solution v c s ( t − 1 s ) = v c s ( t ) for fixed c s For Stokes flow α = t short � � v ( t ) − v c s ( t ) � H 1 (Ω) + � � p ( t ) − p c s ( t ) � L 2 (Ω) = O ( α ) , � � T long ( v c s , p c s )) γ ( σ WS ∂ t ¯ c s ≈ ¯ f S. Frei | Mechano-chemical FSI 10
Time scales Different time scales for fluid dynamics ( t short = 1 s , period of inflow data) and plaque growth ( T long = 1 month) To resolve the short scale ( k ≤ 1 / 20 s ) over T long would require > 10 7 time-steps Assumption : c s is approximately constant on the short scale c s = � t Consider the averaged quantity ¯ t − 1 s c s ( r ) dr There holds � t � t γ ( σ WS γ ( σ WS ∂ t ¯ c s = ∂ t c s ( r ) dr = ( v , p )) dr =: ¯ ( v , p )) . f f t − 1 s t − 1 s Problem : No initial values available for u , v , as we can not resolve the short scale We assume the existence of a periodic solution v c s ( t − 1 s ) = v c s ( t ) for fixed c s For Stokes flow α = t short � � v ( t ) − v c s ( t ) � H 1 (Ω) + � � p ( t ) − p c s ( t ) � L 2 (Ω) = O ( α ) , � � T long ( v c s , p c s )) γ ( σ WS ∂ t ¯ c s ≈ ¯ f S. Frei | Mechano-chemical FSI 10
Time scales Different time scales for fluid dynamics ( t short = 1 s , period of inflow data) and plaque growth ( T long = 1 month) To resolve the short scale ( k ≤ 1 / 20 s ) over T long would require > 10 7 time-steps Assumption : c s is approximately constant on the short scale c s = � t Consider the averaged quantity ¯ t − 1 s c s ( r ) dr There holds � t � t γ ( σ WS γ ( σ WS ∂ t ¯ c s = ∂ t c s ( r ) dr = ( v , p )) dr =: ¯ ( v , p )) . f f t − 1 s t − 1 s Problem : No initial values available for u , v , as we can not resolve the short scale We assume the existence of a periodic solution v c s ( t − 1 s ) = v c s ( t ) for fixed c s For Stokes flow α = t short � � v ( t ) − v c s ( t ) � H 1 (Ω) + � � p ( t ) − p c s ( t ) � L 2 (Ω) = O ( α ) , � � T long ( v c s , p c s )) γ ( σ WS ∂ t ¯ c s ≈ ¯ f S. Frei | Mechano-chemical FSI 10
Averaging of FSI Averaging for linear FSI under periodicity � t ∂ t v c s ( r ) + A ( u c s ( r ) , v c s ( r ) , p c s ( r )) dr 0 = t − 1 s = v c s ( t ) − v c s ( t − 1 s ) + A (¯ u c s , ¯ v c s , ¯ p c s ) u c s , ¯ v c s , ¯ p c s ) . = A (¯ For non-linear FSI , additional terms would arise on the right, that will be neglected in this work Averaged system of equations (T) u c s , ¯ v c s , ¯ p c s ) = 0 A (¯ γ ( σ WS ( v c s , p c s )) ∂ t ¯ c s = ¯ f Short-scale influence enters the right-hand side of the ODE S. Frei | Mechano-chemical FSI 11
Two-scale algorithm Strategy to design a two-scale algorithm Discretise the total time interval in sub-intervals I m = [ T m , T m +1 ] , K = T m − T m − 1 Discretise for each m a short scale interval [ T m , T m + 1 s ] into sub-intervals i n = [ t n , t n +1 ] , k = t n +1 − t n For m = 1 , ..., M : Solve a short-scale problem on [ T m , T m + 1 s ] (non-stationary FSI) and γ ( σ WS compute ¯ ) f Advance c s by solving the ODE and ( v , p , u ) by solving the averaged stationary FSI problem In practice: Iterate on short-scale until v ( t m + 1 s ) ≈ v ( t m ) for periodicity Theoretically O ( k 2 + α K 2 + α ) for a Stokes system with growing boundary and second-order time-stepping (S.F., T. Richter, in preparation) , open for nonlinear FSI S. Frei | Mechano-chemical FSI 12
Two-scale and pure long-scale approach We will compare our approach to a pure long-scale approach (a) Pure long-scale algorithm: (b) Temporal two-scale-algorithm: A (¯ u , ¯ v , ¯ p ) = 0 A (¯ u , ¯ v , ¯ p ) = 0 ( v c s , p c s )) c s = γ ( σ WS γ ( σ WS ∂ t ¯ c s = ¯ ∂ t ¯ (¯ v , ¯ p )) f f u n − 1 c n ¯ s , ¯ s σ WS ¯ Short f Nonstat. FSI Long c s ¯ γ ( σ WS ) FSI f (ODE) Stat. FSI c n ¯ s ODE F = F g (¯ c s ) F e c n F s = F g (¯ s ) F e S. Frei | Mechano-chemical FSI 13
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