Numerical Simulation of Bloch Equations for
Dynamic Magnetic Resonance Imaging
A. Hazra, G. Lube, H.-G. Raumer
Institute for Numerical and Applied Mathematics,
Georg-August-University Göttingen
12th International Workshop on Variational Multiscale Methods
and Stabilized Finite Elements
University of Sevilla,
April 26-28, 2017
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
1 / 29
Motivation: Real-time MRI
Figure: Pouring Water from Top
Figure: Cerebrospinal Fluid Flow
Flip Angle 4◦ , 1.5 × 1.5 × 8 mm3 ,
Acquisition Time 30.0 ms, 33fps
Flip Angle 8◦ , 0.75 × 0.75 × 5 mm3 ,
Acquisition Time 50 ms,20fps
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
2 / 29
Dynamic Magnetic Resonance Imaging (MRI)
Real-time MRI
MRI – world-wide applied non-invasive imaging technique in medicine (108 MRT/ year)
FLASH (fast-low-angle-shot)-technology
by J. F RAHM, Biomed NMR Göttingen
Patent revenues ∼ 150.000.000 EUR (since
1985) for Max-Planck Society
http://www.biomednmr.mpg.de/
Real-time MRI of heart with
measurement time of 33 ms/image
Goals of the talk:
Well-posedness of Bloch model for dynamic MRI
Numerical simulation of multiscale problem
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
3 / 29
Bloch model for Magnetic Resonance Imaging
Outline
1
Bloch model for Magnetic Resonance Imaging
2
Well-posedness of Bloch model for MRI
3
Spatial semidiscretization with upwind dG-FEM
4
Temporal discretization
5
Numerical simulations
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
4 / 29
Bloch model for Magnetic Resonance Imaging
Bloch Model for Magnetic Resonance Imaging (MRI) - I
Schematic of a pulsed MR experiment
Equilibrium
Excitation
Decay
Precession
+
Recovery
Bloch model for MRI: Find magnetic field M(t, r) = (Mx , My , Mz )T (t, r)
My
dM
M0 − Mz
Mx
= γM × B +
êz −
êx −
êy
dt
T1
T2
T2
Excitation: B(t, r) = (Bx (t), By (t), Bz (t, r))T ,
Bz (t, r) := B0 + G(t) · r + ∆B
radio frequency (RF) pulse B1 (t) = (Bx (t), By (t), 0)T
Relaxation:
spin-lattice relaxation T1 , spin-spin time relaxation T2
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
5 / 29
Bloch model for Magnetic Resonance Imaging
Bloch Model for Magnetic Resonance Imaging (MRI) - II
Bloch model in rotating frame: Find M0 (t, r) = (Mx0 , My0 , Mz0 )T (t, r0 )
My0
dM0
M0 − Mz
Mx 0
= γM0 × Beff +
êz −
êx0 −
êy0
dt
T1
T2
T2
Elimination of (strong) static field B0 and simplification of expressions
Beff = B = (Bx0 , By0 , Bz )T ,
Include transport of magnetization via incompressible flow field u
My0
∂M0
M0 − Mz
M0
+ (u · ∇r )M0 = γM0 × Beff +
êz − x êx0 −
êy0
∂t
T1
T2
T2
Replace from now on M0 by M etc.
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
6 / 29
Bloch model for Magnetic Resonance Imaging
Variational formulation - I
Notation:
Flow domain Ω ⊂ R3 with outer normal n
S
S
Lipschitz boundary Γ = Γ− Γ+ Γ0 with
Γ± : sgn(u · n) = ±1, Γ0 : u · n = 0
Assume:
dist(Γ− , Γ+ ) := min(P,Q)∈Γ− ×Γ+ |P − Q| > 0
B = Beff = (Bx , By , Bz )T , D = diag
f = (0, 0,
1
, 1, 1
T2 T2 T1
,
M0 T
)
T1
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
7 / 29
Bloch model for Magnetic Resonance Imaging
Variational formulation - I
Notation:
Flow domain Ω ⊂ R3 with outer normal n
S
S
Lipschitz boundary Γ = Γ− Γ+ Γ0 with
Γ± : sgn(u · n) = ±1, Γ0 : u · n = 0
Assume:
dist(Γ− , Γ+ ) := min(P,Q)∈Γ− ×Γ+ |P − Q| > 0
B = Beff = (Bx , By , Bz )T , D = diag
f = (0, 0,
1
, 1, 1
T2 T2 T1
,
M0 T
)
T1
∂M
+ (u · ∇r )M + γB × M + DM = f,
∂t
M = MΓ ,
0
M=M ,
G. Lube (University of Göttingen)
(t, r) ∈ [0, T] × Ω,
(t, r) ∈ [0, T] × Γ−
(t, r) ∈ {0} × Ω
Numerical Simulation of Bloch Equations
April 26-28, 2017
7 / 29
Bloch model for Magnetic Resonance Imaging
Variational formulation - II
H := [L2 (Ω)]3 with (M, N)H :=
R
Ω
M · N dr and kNkH :=
p
(N, N)H
1
X = {N ∈ H : (u · ∇r )N ∈ H} with kNkX := k(u · ∇r )Nk2H + kNk2H 2
Multiply by arbitrary N ∈ X, integrate over Ω and impose boundary conditions weakly:
(∂t M, N)H + ((u · ∇r )M, N)H + γ(B × M, N)H
Z
Z
+ (DM, N)H + (u · n) M · N ds = (f, N)H + (u · n) MΓ · N ds
Γ
Γ
with w (r) := 21 (|w(r)| − w(r) and M × B = −B × M.
Variational formulation: Find M : (0, T] → X s.t.
(∂t M, N)H + a(t; M, N) = l(N),
∀ N ∈ X,
0
M|t=0 = M .
Z
a(t; M, N) := (u · ∇r )M, N)H + γ(B × M), N)H + (DM, N)H +
Z
l(N) := (f, N)H + (u · n) MΓ · N ds
(u · n) M · N ds,
Γ
Γ
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
8 / 29
Well-posedness of Bloch model for MRI
Outline
1
Bloch model for Magnetic Resonance Imaging
2
Well-posedness of Bloch model for MRI
3
Spatial semidiscretization with upwind dG-FEM
4
Temporal discretization
5
Numerical simulations
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
9 / 29
Well-posedness of Bloch model for MRI
Elliptic regularization
Variational form of Bloch problem: Find M : (0, T] → X s.t.
(∂t M, N)H + a(t; M, N) = l(N),
∀ N ∈ X,
0
M|t=0 = M .
Problem: Theory of Friedrichs systems not directly applicable to time-dependent coefficients.
Remedy: Elliptic regularization for 0 < 1
Find M : (0, T] → X := [W 1,2 (Ω)]d s.t.
(∂t M , N)H + a (t; M , N) = l(N),
∀ N ∈ X ,
0
M |t=0 = M .
with
a (t; M, N) := a(t; M, N) + (∇M, ∇N)H
Note: Do-nothing boundary conditions
G. Lube (University of Göttingen)
∇M · n = 0 on Γ0 ∪ Γ+
Numerical Simulation of Bloch Equations
April 26-28, 2017
10 / 29
Well-posedness of Bloch model for MRI
Application of main theorem on evolution problems - I
Spaces X ⊆ H and dual space X ∗ form evolution tripel (X , H, X ∗ ).
For p ≥ 1 and Banach space Y, denote by Lp (0, T; Y) Bochner space of vector-valued
functions v : (0, T) → Y.
Look for solution M ∈ L∞ (0, T; H) ∩ L2 (0, T; X ) of regularized problem.
Theorem (Well-posedness)
For all > 0, for u ∈ [L∞ (0, T; W 1,∞ (Ω)]3 with div u = 0 and B ∈ [L∞ (0, T; H)]3 , there
exists a unique solution M ∈ L∞ (0, T; H) ∩ L2 (0, T; X ) to regularized Bloch problem.
For t ∈ (0, T] and with σ :=
1
,
T1
there holds a-priori estimate:
t
Z
1
eσ(τ −t) k∇M (τ )k2H +
|(u · n)|(M · M )(s, τ ) ds dτ
2 Γ
0
Z t
1
1
≤ kM (0)k2H e−σt +
kf(τ )k2H eσ(τ −t) dτ
2
2σ 0
1
kM (t)k2H +
2
Z
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
11 / 29
Well-posedness of Bloch model for MRI
Application of main theorem on evolution problems - II
Sketch of proof:
1
2
kNkX := k(u · ∇r )Nk2H + kNk2H ,
h
i1
2
kMkX := k∇Mk2H + kMk2X .
Form t 7→ a (t; M, N) measurable ∀M, N ∈ X as u and B are sufficiently smooth
Form a (t; ·, ·) bounded for t ∈ [0, T] and ∀M, N ∈ X :
|a (t; M, N)| ≤ (1 + + γkBkL∞ + kDkL∞ + ks )kMkX kNkX
with kBk∞ := kBkL∞ (0,T;[L∞ (Ω)]3 ) , kDkL∞ := max{ T11 , T12 } and trace inequality
Z
(u · n| M · N ds ≤ ks kMkX kNkX
Γ
Form a is coercive: a (t; N, N) ≥ k∇Nk2H + σkNk2H + 21
with σ := min T11 , T12 as (B × N, N)H = 0 and div u = 0
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
R
Γ
|u · n|N · N ds
April 26-28, 2017
12 / 29
Well-posedness of Bloch model for MRI
Passage to the limit ε → +0
Finally, we can pass to the limit → +0, i.e. to the Bloch model.
Theorem
The Bloch model admits a unique solution M ∈ L∞ (0, T; H) ∩ L2 (0, T; X). The kinetic energy
of the magnetic field is bounded by:
Z
Z
1 t σ(τ −t)
1
e
kM(t)k2H +
|u · n|(M · M)(s, τ ) ds dτ
2
2 0
Γ
Z t
1
1
2 −σt
kf(τ )k2H eσ(τ −t) dτ.
≤ kM(0)kH e
+
2
2σ 0
Proof.
Inspection of previous proof shows that existence/uniqueness result together with
a-priori estimate remain valid for → +0 .
Regularized Bloch problem with do-nothing conditions ∇M · n = 0 on Γ0 ∪ Γ+
For passage → +0 at Γ0 ∪ Γ+ , can proceed as in L IONS, Lect. Not. Math. 323 [1979],
Chap. V.1. Note that inflow and outflow are separated.
G. Lube (University
Göttingen)
Numerical
Simulation
of Bloch
Equations
April 26-28,
2017
13 / 29
Remark:
Resultof and
a-priori estimate
remain
valid
form
case u = 0, i.e. Bloch
equations
for
Spatial semidiscretization with upwind dG-FEM
Outline
1
Bloch model for Magnetic Resonance Imaging
2
Well-posedness of Bloch model for MRI
3
Spatial semidiscretization with upwind dG-FEM
4
Temporal discretization
5
Numerical simulations
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
14 / 29
Spatial semidiscretization with upwind dG-FEM
Discontinuous Galerkin Formulation - I
Notation:
Admissible mesh Th := {Ωi }Ii=1 into convex, simplicial subdomains Ωi
Discont. FE space [Pk (Th )]d := {Nh ∈ H; Nh |Ωi ∈ [Pk (Ωi )]d
T
Xh = [Pk (Th )]d X
Left: 2d-simplicial mesh.
∀ Ωi , i = 1, 2, · · · , I}
Right: 1D-example of average and jump operators
T
For adjacent subdomains Ωi , Ωj with interface E = Γij = Ωi Ωj and unit normal vector
~nij , define average and jump of Nh ∈ X across Γij by
hNh iΓij (r) :=
G. Lube (University of Göttingen)
1
(Nh |Ωi (r) + Nh |Ωj (r)),
2
[Nh ]Γij (r) := Nh |Ωi (r) − Nh |Ωj (r)
Numerical Simulation of Bloch Equations
April 26-28, 2017
15 / 29
Spatial semidiscretization with upwind dG-FEM
Discontinuous Galerkin Formulation - II
Fhi – set of interior interfaces E ⊆ Ω
Upwind form:
Sh (t; M, N) :=
XZ
E∈Fhi
1
(u · nE )[M] · [N] + |u · nE |[M] · [N]
2
E
Penalization of gradient jumps over interior faces via
X 2Z
p (M, N) := ˜
hE |u · nE |[∇M]E : [∇N]E ds,
E∈Fhi
ds
˜ ≥ 0
E
Set
aupw
(t; M, N) := p (M, N) + a(t; M, N) + Sh (t; M, N)
Upwind dG-FEM:
Find Mh : (0, T] 7→ Xh s.t. ∀ Nh ∈ Xh
(∂t Mh , Nh )H + aupw
(t; Mh , Nh ) = l(Nh )
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
16 / 29
Spatial semidiscretization with upwind dG-FEM
Well-posedness of semidiscrete problem
Define norm:
k|Nh k|2U := ˜
X
E∈Fhi
h2E
Z
|u · nE | k[∇Nh ]E k2L2 (E) +
E
1
2
Z
|u · n|N2h ds +
Γ
Z
1 X
|u · nE |[Nh ]2E ds
2
E
i
E∈Fh
Theorem
Semidiscrete problem is well-posed and admits a-priori estimate
Z t
Z t
1
1
1
kMh (t)k2H +
eσ(τ −t) k|Mh (τ )k|2U dτ ≤ kMh (0)k2H e−σt +
eσ(τ −t) kf(τ )k2H dτ.
2
2
2σ 0
0
Proof:
Existence/ uniqueness follow lines of proof of previous Theorem.
Symmetric testing Nh = Mh yields
2
2
aupw
((t; Mh , Mh ) ≥ σkMh kH + k|Mh k|U
Young inequality shows
1 d
σ
1
kMh (t)k2H + kMh k2H + k|Mh k|2U ≤
kfk2H .
2 dt
2
2σ
Gronwall lemma yields a-priori estimate.
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
17 / 29
Spatial semidiscretization with upwind dG-FEM
Semidiscrete error estimate - I
Theorem
Spatial discretization error Mh − πh M with L2 -orthogonal projection M → πh M ∈ Xh :
Z
1
1
1 t σ(τ −t)
e
k|(M − Mh )(τ )k|2U dτ ≤ k(M − Mh )(0)k2H e−σt
k(M − Mh )(t))k2H +
2
2 0
2
Z t
σ(τ −t)
2
e
k|(M − πh M)(τ )k|U,∗ + δk(M − πh M)(τ )kH + p (M, M)(τ ) dτ
+
0
with δ := γkBkL∞ + kDkL∞ and
k|Ih k|2U,∗ := max{1; kukL∞ (0,T;W 1,∞ (Ω)) }k|Ih k|2U +
X
kukL∞ (∂T) kIh k2L2 (∂T)
T∈Th
For smooth solution M ∈ L∞ (0, T; [W k+1,2 (Ω)]3 ), the last term is of order O(h2k+1 ).
Sketch of proof:
Error equation: (∂t (M − Mh ), Nh )H + aupw
(t; M − Mh , Nh ) = p (M, Nh ) ∀ Nh ∈ Xh
Split error as M − Mh = (M − πh M) + (πh M − Mh ) ≡ Ih + Eh
Nh = Eh
G. Lube (University of Göttingen)
upw
(∂t Eh , Eh )H + aupw
(t; Eh , Eh ) = −a (t; Ih , Eh ) + p (M, Eh )
Numerical Simulation of Bloch Equations
April 26-28, 2017
18 / 29
Spatial semidiscretization with upwind dG-FEM
Semidiscrete error estimate - II
L.H.S. bounded from below as
(∂t Eh , Eh )H + aupw
(t; Eh , Eh ) ≥
1 d
kEh k2H + σkEh k2H + k|Eh k|2U
2 dt
R.H.S. terms from above as
p (M, Eh ) ≤ p (M, M)k|Eh k|U
≤ k|Ih k|U,[ + δkIh k2H k|Eh k|U
−aupw
(t; Ih , Eh )
(+)
with δ := γkBkL∞ + kDkL∞ and
k|Ih k|2U,[ := max{1; kukL∞ (0,T;W 1,∞ (Ω)) }k|Ih k|2U +
X
kukL∞ (∂T) kIh k2L2 (∂T)
T∈Th
Estimate (+) extends Lemma 2.30 in DI P IETRO /E RN [2012] from scalar- to
vector-valued case, exploiting L2 -orthogonality of subscales.
L.H.S. and R.H.S. estimates and Young inequality lead to
1 d
1
1
kEh k2H + σkEh k2H + k|Eh k|2U ≤ k|Ih k|2U,[ + δkIh k2H + p (M, M)
2 dt
2
2
Gronwall lemma and triangle inequality imply error estimate. G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
19 / 29
Temporal discretization
Outline
1
Bloch model for Magnetic Resonance Imaging
2
Well-posedness of Bloch model for MRI
3
Spatial semidiscretization with upwind dG-FEM
4
Temporal discretization
5
Numerical simulations
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
20 / 29
Temporal discretization
Temporal discretization - I
Starting point: find Mh : (0, T] 7→ Xh s.t. ∀Nh ∈ Xh
(∂t Mh (t), Nh )H + aupw
(t; Mh (t), Nh ) = l(Nh ),
Mh (0) = Mh0 .
Multiscale character: magnetization much faster than advection
Restricted temporal smoothness of data, in particular G = (Gx , Gy , Gz )T
Remedy: explicit time stepping (fully coupled or operator splitting approach)
Efficiency of simulation strongly improved via GPU computing
Fully coupled approach
Define
upw
Aupw
: X × Xh 7→ Xh via (Aupw
(t)M, Nw)H := a (t; M, N).
Mnh = Mh (tn ) etc.
Low-order explicit Runge-Kutta scheme on 0 = t0 < t1 < t2 < · · · < tN = T with time
steps τn := tn+1 − tn , n = 0, 1, · · · , N − 1, here of second-order, as compromise
between temporal accuracy and restricted data smoothness:
n
upw n
Mn,1
h = Mh − τn A Mh + τn L
1
1
1
upw n,1
Mn+1
= (Mnh + Mn,1
h
h ) − τn A Mh + τn L.
2
2
2
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
21 / 29
Temporal discretization
Temporal discretization - II
Statements on stability/ convergence for smooth data: see DI P IETRO /E RN [2012], Sec.
3.1.6, e.g., CFL-time step restriction for advective term.
Here: Relevant time step restriction from magnetization
CFL condition always valid
Do not repeat details, e.g. of stability/convergence RK2-analysis in DI P IETRO /E RN.
1
For smooth data in time: error of order O(τn2 + hk+ 2 ) with degree k of dG-FEM
Here: Such error estimate in time not valid, since data B(t, r) only in [C0,1 [0, T]]3 (for
FLASH-sequence studied in experiments) or even in [L∞ (0, T)]3 for next example
Alternatively: embedded RK-scheme of type RK3(2) or even of higher order like
RK5(4) for time step selection
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
22 / 29
Numerical simulations
Outline
1
Bloch model for Magnetic Resonance Imaging
2
Well-posedness of Bloch model for MRI
3
Spatial semidiscretization with upwind dG-FEM
4
Temporal discretization
5
Numerical simulations
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
23 / 29
Numerical simulations
Proof of concept for flowing spins - I
A basically one-dimensional test case for flowing spins
Effect of RF pulse (Blackman-windowed sinc pulse) on
magnetization for different through-plane velocities uz
Flip angle α = 90o , duration 1.5 × 2.6794 ms
Reciprocal of step size
slice selection gradient G ∈ [L∞ (0, T)]3
10
8
10
7
10
6
10
5
10
4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time [ms]
Left: Effect of penalty stabilization,
G. Lube (University of Göttingen)
Right: Time adaptivity
Numerical Simulation of Bloch Equations
April 26-28, 2017
24 / 29
Numerical simulations
Proof of concept for flowing spins - II
0.8
80
120
0.4
Left: Spatial mesh for flow
200
Mx
0
0
-0.4
-0.8
cell
160
-1.2
1
Slice Thickness
120
0.5
0
My
Y
X
0
160
Z
-0.5
Right: Comparison to Y UAN ET
Medic. Phys. 14 (6) (1987):
80
200
-1
1.2
AL .,
1
0
0.8
Magnetization Mx , My , , Mz for uz
80
M z0.6
in range 0 − 200 cm/s
120
160
0.4
200
0.2
0
-0.2
-10
-5
0
5
10
Z [mm]
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
25 / 29
Numerical simulations
Comparison with Experiments for Through-plane Flow - I Laminar unidirectional flow in a circular pipe
Echo
RF
Flow pump operates at different voltages to
generate flow field u, see table (below)
FLASH pulse sequence with flip angle = 8o ,
TR/TE = 1.96/1.22 ms,
G ∈ [C0,1 [0, T]]3
d
c
d
c
a
b
TE
TR
cell
Slice Thickness
Y
X
Voltage
6 [volt]
5 [volt]
4 [volt]
3 [volt]
Mean Velocity
49.19 [mm/s]
38.71 [mm/s]
28.84 [mm/s]
18.52 [mm/s]
Standard Deviation
2.26 [mm/s]
1.97 [mm/s]
1.47 [mm/s]
1.04 [mm/s]
Re
2217
1744
1300
834
Z
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
26 / 29
Numerical simulations
Comparison with Experiments for Through-plane Flow - II
Experiment with different velocities vs. numerical simulation
1.3
1.2
u z [mm s -1 ]
|M xy | [a.u]
1.1
49.19
38.71
28.84
18.52
49.19
38.71
28.84
18.52
1
0.9
0.8
2.26
1.97
1.47
1.04
0.7
0.6
5 10 15 20 25 30 35 40 45 50 55 60
Frame Number
Good agreement of numerical simulations with experiments
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
27 / 29
Numerical simulations
Comparison with Experiments for Pulsatile Flow
Laminar unidirectional pulsatile flow in a circular pipe
Motivated by: Cerebrospinal (unidirectional, pulsatile)
fluid flow
Identical domain, spatial, and temporal discretization as
in previous experiment
Good agreement for periodicity, but: amplitude deviation
10
2.5
u z estimated by pcMRI
9
fitted u
z
2
7
|M xy | [a.u]
u z [mm s -1 ]
8
6
5
1.5
4
Experiment pulsatile flow
Simulation Pulsatile Flow
3
2
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G. Lube (University of Göttingen)
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Numerical Simulation of Bloch Equations
April 26-28, 2017
28 / 29
Numerical simulations
Summary. Outlook
Bloch model for magnetization M
Friedrichs system
Well-posedness of Bloch model for dynamic MRI (via elliptic regularization)
Spatial semidiscretization with dG-FEM
Well-posedness and quasi-optimal error estimates for semidiscrete problem
Explicit time integration with embedded Runge-Kutta methods
Verification for basic applications via comparison with experiments
Further acceleration with multiple GPU’s to reduce the gap to real-time MRI
Application to more realistic experiments
Numerical simulation, e.g. for quantitative understanding of MRI signal alterations
Thanks you for your attention !
G. Lube (University of Göttingen)
Numerical Simulation of Bloch Equations
April 26-28, 2017
29 / 29