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Modelling coupled tissue mechanics-blood flow within the heart D. Kay, D. Nordsletten, M. McCormick and N. Smith web.comlab.ox.ac.uk/people/David.Kay/ email:[email protected] Outline 1. Motivation 2. Fluid Flow Within Moving Domains: ALE Formulation 3. Finite Element Approximation 4. Coupled Problem 5. Finite Element Approximation 6. Numerical Results 7. Scan, Mesh, Model. 8. Ongoing and Future Work Motivation Over the last decade computational modelling of biological processes has grown rapidly: • Data collection: MRI, Ultrasound • Computational Power: Vastly different length scales What do we hope to achieve? Motivation Computational Modelling Cardiac activation and contraction Regional work through the cardiac cycle Stresses during contraction Coronary Blood flow Blood flow Electrical Mapping Motivation HPC Enabled Applications Imaging • Image processing: extraction of patient specific data • Personalisation: Customising model to data • Virtual Imaging: visualisation of new information • Clinical Interfaces: Clinical interaction with models • Improved patient diagnosis, selection and treatment Personalisation Virtual Imaging Clinical Interfaces Motivation Clinical Application: LVAD Personalisation Personalisation of LVAD flow to improve recovery rate in Heart Failure Virtual Imaging Comprehensive MR LVAD and implantation Motivation Haematoma Fluid Solid Fluid Solid Figure 1: Haematoma under deformation Motivation Heart streamlines: Outer Heart streamlines: Inner Fluid Flow Within Moving Domains: ALE Formulation Mathematical Model b, pb ) (Velocity,Pressure) such that Find ( v ∂b v bv b − ν Dx v b + pb I) + ∇x · ( v ∂t b −∇x · v b v b − pb I) · n b (ν Dx v b( · , 0) v Gresho and Sani 1998 = 0 in Ωt, = = = = in Ωt, on ΓD t on ΓN t in Ω0, 0 btD g btN g b0 v Fluid Flow Within Moving Domains: ALE Formulation Arbitrary Lagrangian Eulerian (ALE) Formulation The ALE form of (3.3a-3.3a) – a hybrid of the Eulerian (viewing particles from a fixed spatial position) and Lagrangian (viewing particles from a given particles reference) frames. Reference Domain #$Frame Pt ! Malvern 1969 "t x-Frame Fluid Flow Within Moving Domains: ALE Formulation The conservative ALE form The spatial conservation law ∂t (J ) = −J ∇x · w. This leads to the problem: find (v, p) such that, ∂t (vJ ) + J Gη,t · ( (v − w)v − ν Dη v + pI ) −J Gη,t · v v (ν Dη v − pI) · n v( · , 0) = = = = = 0 in Λ, 0 in Λ, gtD on ΓD Λ, gtN on ΓN Λ, v0 in Λ.I Fluid Flow Within Moving Domains: ALE Formulation The Weak Form Multiplying (1.2a) by u ∈ H10(Λ) and (1.2a) by q ∈ W, using the incompressibility constraint, and then integrating initially over space and then over time (a, b) we obtain the weak formulation: find ( v, p ) ∈ VD × W, such that for any (a, b) ∈ I Z b ( v, u )Ω − ( v, u )Ω + cξ (v − w; v, u) dξ a b a Z b Z b + aξ (v, u) dξ + bξ (p, u) dξ = 0, ∀ u ∈ H10(Λ), a Za b bξ (q, v) dξ = 0, ∀ q ∈ W, a where, for ease, we have assume g N (η, t) := 0. The ALE Finite Element Method Spatial Mapping We want to preserve the reference mesh quality. Define t − tn−1 n d (ξ) Ph(ξ, t) := Ph(ξ, tn−1 ) + tn − tn−1 h wnh dn = . tn − tn−1 where atn−1 (dnh, uh) = 0 ∀ uh ∈ W0h(Λ). Bottasso, Detomi and Serra 2005, Farhat, Lesoinne and LeTallec 1998 The ALE Finite Element Method The Discrete Weak Form Given v0( · , 0) = π h (v0) find ( vh, p ) ∈ VDh × W h, such that for n = 0, 1, 2, . . . , N − 1 Z tn+1 n ( vn+1 cξ (vn+1 − w; vn+1 h , u )Ωn+1 − ( vh , u )Ωn + h h , u) dξ n t Z tn+1 Z tn+1 + aξ (vn+1 bξ (ph,n+1, u) dξ = 0, ∀ u ∈ V0h, h , u) dξ + tn tn Z tn+1 bξ (q, vn+1 ∀ q ∈ W h, h ) dξ = 0, tn Numerical Solver ALE: Large Sparse Linear System n+1,k+1 F Bn+1 n+1 T B 0 ! n+1,k+1 α β n+1,k+1 Fx = b Nordsletten et al 2010 ! = f1n+1,k+1 f2n+1,k+1 ! Coupled Problem t Γc s Γc Traction forces between Solid and Fluid Coupled Problem Fully Coupled Problem Fluid-Solid Coupled Mesh Coupled Problem Fluid, Ωf : Navier-Stokes ρ ∂v + ∇x · ( %v v − ν Dxv + p I) ∂t ∇x · v v ( − ν Dx v + p I) · n v( · , 0) = 0 = = = = in Ωf , 0 in Ωf , on ΓD gD f, f gN on ΓN f f , v0 in Ωf (0) Coupled Problem Solid, Ωs: Quasi-Static Finite Elasticity −∇x · ( σ(u) + p I ) ∂tJ(u) u ( − σ(u) + p I) · n u( · , 0) = = = = = 0 in Ωs, 0 in Ωs, gD on ΓD s s , gN on ΓN s s , u0 in Ωs(0) Coupled Problem Boundary Coupling ΓC Interface Forces: tf = ( − ν Dx v + pf I) · n ts = ( − σ(u) + ps I) · n Ωs and Ωf are coupled by the following conditions, tf + ts = 0 on ΓC , and v1 − ∂tu2 = 0 on ΓC . Coupled Problem Lagrange Multiplier On the coupled boundary Γc we introduce a Lagrange Multiplier 1 Space Mh ⊂ L2((0, T ); H− 2 (Γc)). This is used to satisfy the constraints. As before multiplying by test functions and integrating firstly in space and then in time. We derive the fully discrete coupled problem: Coupled Problem Find: n+1 n+1 {vn+1, un+1, pn+1 } ∈ VDh × UDh × Wfh × Wsh × Mh, f , ps , λ such that for n = 0, 1, 2, . . . , N − 1 Z tn+1 n ( vn+1 cξ (vn+1 − w; vn+1 h h , y) dξ h , y )Ωn+1 − ( vh , y )Ωn + n t Z tn+1 Z tn+1 + aξ (vn+1 bξ (ph,n+1, y) dξ h , y) dξ + tn tn +∆nt(λn+1, y)Γc = 0, ∀ y ∈ V0h, Z tn+1 tn bξ (q, vn+1 h ) dξ, = 0, ∀ q ∈ W h, Coupled Problem Z tn+1 Z tn+1 (σ(uhi ), G(atn ) y )Λ dt − bt(phi , y)Λi dt i tn tn Z tn+1 (f ni , y )Λ dt = 0, + (∆nt) (λh,n , y)Γc − ∀ y ∈ U0h i tn Z tn+1 bt(q, uh,n+1 − uh,n ∀ q ∈ Wsh i i )Λi dt = 0, tn , , Coupled Problem The previous two systems give (weakly) the first constraint on the common boundary, tns = λn = tnf . Hence, we require the final constraint to be satisfied. This is achieved by the equation n+1 n [uh − u ] z, vn+1 − = 0, ∀ z ∈ M h. h n ∆t Γc Nordsletten et al J. Comp. Phys. Coupled Problem Channel flow with pressure gradient Heart: Vector flow Heart: Streamlines Heart: Displacement Heart streamlines: Outer Heart streamlines: Inner Scan, Mesh, Model Scan, Mesh, Model Scan, Mesh, Model 1 Scan, Mesh, Model Heart wall motion under filling Heart wall motion under filling: transparent Passive filling Nordsletten et al J. Num. Bio. Eng. Ongoing and Future Work 1. Full analysis of the scheme (error estimate). 2. Efficient numerical solvers, Implementation of AMG. 3. Efficient numerical solver for fully coupled method. 4. Mesh refinement strategies. 5. Applications and Testing: LVADS, Efficiency of the heart, ...