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:david.kay@comlab.ox.ac.uk
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, ...