Design of optimal RF pulses for NMR as a discrete-valued control problem

Transcription

1 Design of optimal RF pulses for NMR as a discrete-valued control problem Christian Clason Faculty of Mathematics, Universität Duisburg-Essen joint work with Carla Tameling (Göttingen) and Benedikt Wirth (Münster) 9th Vienna International Conference on Mathematical Modelling Wien, February 2, 28 / 9

2 Motivation: MRT Magnetic resonance tomography (MRT): based on measurement of resonance frequency of hydrogen nuclei hydrogen nucleus (proton) acts like rotating magnet H N angular momentum : nuclear spin, axis randomly aligned S 2 / 9

3 Motivation: MRT nuclear spin in magnetic field: axis aligns with external field precesses around field direction 3 / 9

4 Motivation: MRT nuclear spin in rotating magnetic field: axis aligns with external field precesses around field direction field rotating at resonance frequency: energy is absorbed resonance frequency proportional to field strength 3 / 9

5 Motivation: MRT magnetic field off: spin relaxes......induces current in coil measurement 4 / 9

6 Mathematical model Bloch equation d dt M(t) = M(t) B(t), M() = M M(t) R 3 describes temporal evolution of spin ensemble B(t) = (u (t), u 2 (t), ω) T controlled time-dependent magnetic field ω resonance frequency control-to-state mapping S (ω) : u M(T) 5 / 9

7 Optimal control problem Goal: compute control u(t) = (u (t), u 2 (t)) such that M(T) M d M d desired magnetization state M d = M (ω) d selective to resonance frequency ( spectroscopy, slice selection) in addition: control with minimal specific absorption rate (SAR) min u L 2 2 ω S (ω) (u) M (ω) d α T u(t) 2 2 dt 2 6 / 9

8 Discrete control Technical limitation: device can only realize control from discrete set U = { u L 2 (, T; R 2 ) : u(t) {u,..., u d } a.e. } u,..., u d R 2 given (fixed amplitude, phases) non-convex discrete-valued control problem min u U 2 ω S (ω) (u) M (ω) d α T u(t) 2 2 dt 2 7 / 9

9 Multi-bang penalty convex relaxation: replace U by convex hull works only for d = 2, cf. bang-bang control (α = ) promote u(x) {u,..., u d } by convex pointwise penalty G(u) = g(u(x)) dx Ω generalize L norm: polyhedral epigraph with vertices u,..., u d not exact relaxation/penalization (in general)! 8 / 9

10 Multi-bang penalty generalize L norm: polyhedral epigraph with vertices u,..., u d 3 2 u u 2 u 3 v motivation: convex envelope of 2 u 2 + δ U multi-bang (generalized bang-bang) control non-smooth optimization in function spaces 8 / 9

11 Motivation 2 Approach Convex analysis Moreau Yosida regularization Semismooth Newton method 3 Vector-valued multi-bang 4 Numerical examples 9 / 9

12 Fenchel duality f : V R := R {+ } convex, V Banach space, V dual space subdifferential f( v) = { v V : v, v v V,V f(v) f( v) for all v V } Fenchel conjugate (always convex) f : V R, f (v ) = sup v, v V,V f(v) v V convex inverse function theorem : v f(v) v f (v ) / 9

13 Fenchel duality: application F(ū) + G(ū) = min F(u) + G(u) u Fermat principle: (F(ū) + G(ū)) 2 sum rule: F(ū) + G(ū), i.e., there is p V with { p F(ū) 3 Fenchel duality: p G(ū) { p F(ū) ū G ( p) / 9

14 Regularization G non-smooth subdifferential G set-valued regularize u, p L 2 (Ω) Hilbert space consider for γ > Proximal mapping prox γg (p) = arg min G (w) + w p 2 w 2γ single-valued, Lipschitz continuous coincides with resolvent (Id +γ G ) (p) (also required for primal-dual first-order methods) 2 / 9

15 Regularization Proximal mapping prox γg (p) = arg min G (w) + w p 2 w 2γ Complementarity formulation of u G (p) u = γ ( (p + γu) proxγg (p + γu) ) equivalent for every γ > single-valued, Lipschitz continuous, implicit 2 / 9

16 Regularization Proximal mapping prox γg (p) = arg min G (w) + w p 2 w 2γ Moreau Yosida regularization of u G (p) u = γ ( p proxγg (p) ) =: G γ (p) G γ = ( G + γ 2 2) G as γ single-valued, Lipschitz continuous, explicit nonsmooth operator equation, Newton method 2 / 9

17 Semismooth Newton method f locally Lipschitz, piecewise C : f(v) =, f : R n R Newton derivative D N f(v)δv C f(v)δv Clarke generalized gradient: convex hull of piecewise derivatives semismooth Newton method D N f(v k )δv = f(v k ), v k+ = v k + δv converges locally superlinearly 3 / 9

18 Semismooth Newton method f locally Lipschitz, piecewise C : F(u) =, F : L p (Ω) L q (Ω), [F(u)](x) = f(u(x)) Newton derivative [D N F(u)δu](x) C f(δu(x))δu(x) any measurable selection of Clarke generalized gradient semismooth Newton method D N F(u k )δu = F(u k ), u k+ = u k + δu converges locally superlinearly if p > q 3 / 9

19 Vector-valued multi-bang: penalty Here: admissible control set U of d radially distributed states, origin { ( U = ) ( ) ( )}, ω cos θ ω sin θ,..., ω cos θ d ω sin θ d fixed amplitude ω > phases θ <... < θ d < 2π multi-bang penalty g = ( δ U) convex envelope ( ( g (q) = 2 ) ) ( δ U (q) = 2 ) δ U (q) { q, u i = 2 ω2 for all i d q, u i 2 ω2 θ i +θ i 2 q θ i+θ i+ 2, q, u i 2 ω2 4 / 9

20 Vector-valued multi-bang: subdifferential Fenchel conjugate g (q) = { =: u q Q q, u i 2 ω2 q Q i Subdifferential { g {u i } q Q i i d (q) = co {u i,..., u ik } q Q i...i k i,..., i k d 5 / 9

21 Vector-valued multi-bang: subdifferential Subdifferential { g {u i } q Q i i d (q) = co {u i,..., u ik } q Q i...i k i,..., i k d Moreau Yosida regularization ( g ) γ (q) = u i ( q,ui α γω 2 2γ u i +u i+ 2 + q,u i u i+ (u i u i+ ) γ u i u i+ 2 2 q γ α γ q Q γ i ) u i q Q γ,i q Q γ i,i+ ( ω u i +u i+ 2 ) 2 (ui + u i+ ) q Q γ,i,i+ 5 / 9

22 Vector-valued multi-bang: subdifferential q 2 q 2 Q 3 Q 2 Q 2 Q 23 Q 2 Q 2 Q 23 Q Q 6 Q 3 Q Q 6 Q 4 Q 45 Q 5 Q Q 34 Q 34 Q 4 Q 56 Q 5 Q 45 Q 56 Q 6 Q 6 q Q Q γ γ 2 2 Q γ Q γ Q γ 23 2 Q γ 2 Q γ Q γ 23 Q γ Q γ 6 Q γ 3 Q γ 3 Q γ 6 Q γ q Q γ 6 Q γ Q γ Q γ 4 Q γ Q γ Q γ Q γ Q γ 56 4 Q γ Q γ 45 5 (a) subdomains for g (b) subdomains for ( g ) γ 5 / 9

23 Vector-valued multi-bang: Newton method Newton derivative q Q γ i u i u T i q Q γ D N ( g γω γ )(q) = 2,i (u i u i+ )(u i u i+ ) T q Q γ γ u i u i+ 2 i,i+ 2 γ Id q Qγ,i,i+ Superposition operator: [ DN H γ (p) ] (t) := D N ( g γ ) (p(t)) a.e. t [, T] 6 / 9

24 Vector-valued multi-bang: Newton method Newton system ( ) Id D N H γ (F (u k ))F (u k ) δu = u k + G γ (F (u k )) matrix-free Krylov method for semismooth Newton step F, F via linearized, adjoint Bloch equation discretization, adjoint from [Aigner/Clason/Rund/Stollberger 6] 6 / 9

25 Numerical examples goal: shift magnetization from M = (,, ) T at t = to M d = (,, ) T at t = T d = 3, 6 radially distributed admissible control states n =, 4 isochromats with different resonance frequencies shift all isochromats 2 shift only one isochromat α =, ω = example motivated by [Dridi/Lapert/Salomon/Glaser/Sugny 5] 7 / 9

26 Numerical examples M().5 u2 Mz.5.5 u t (a) control u(t) Mx.5 M(T) M d.5 My (b) state M(t).5 Figure: n = isochromat, d = 3 control states 8 / 9

27 Numerical examples M().5 u2 Mz.5.5 u t (a) control u(t) Mx.5 M(T) M d.5 My (b) state M(t).5 Figure: n = isochromat, d = 6 control states 8 / 9

28 Numerical examples M().5 u2 Mz.5.5 u t (a) control u(t) Mx.5 M(T) M d.5 My (b) state M(t).5 Figure: n = 4 isochromats, same target 8 / 9

29 Numerical examples M() M(T).5 u2 Mz.5.5 u t (a) control u(t) Mx.5 M(T) M d.5 My (b) state M(t).5 Figure: J = 4 isochromats, different targets 8 / 9

30 Conclusion Discrete controls in NMR: can be promoted by convex penalties linear complexity in number of parameter values efficient numerical solution (superlinear convergence) applicable to nonlinear, vector-valued problems Outlook: robust optimization, e.g., with respect to static field include signal acquisition other discrete continuous problems: switching, networks Preprint, MATLAB/Python codes: 9 / 9