Principles of MRI Lecure 0 EE225E / BIO265 Insrucor: Miki Lusig UC Berkeley, EECS
Bloch Eq. For Recepion No B() : 2 4 Ṁ x Ṁ y Ṁ z 3 5 = 2 6 4 T 2 ~ G ~r 0 ~G ~r T 2 0 0 0 T 3 2 7 5 4 M x M y M z 3 5 + 2 4 0 0 T 3 5 M 0 2
Bloch Eq. For Recepion No B() : 2 4 Ṁ x Ṁ y Ṁ z 3 5 = 2 6 4 ~ T 2 G() ~r 0 ~G() ~r T 2 0 0 0 T 3 2 Decouples ino wo independen eqns. 7 5 4 M x M y M z 3 5 + 2 4 0 0 T 3 5 M 0 3
Soluion for Recepion Mz: (2) is already solved! 4
Soluion for Recepion M xy (~r, )=M xy (~r, 0)e T 2 e i~r R 0 ~G( )d Define a Spaial frequency vecor: ~ k() =[kx (),k y (),k z ()] T ~ k() =2 Z ~G( )d 0 5
Soluion for Recepion Transverse Magneizaion Mxy is hen: M xy (~r, )=M xy (~r, 0)e T 2 e i2 ~ k() ~r Received Signal is proporional o Mxy inegraed over volume: Fourier kernel s() = Z ~R M xy (~r, 0)e T 2 e i2 ~ k() ~r d~r Also a funcion of r! 6
Signal Equaion Assume ha T2 is LARGE Z So, s() ~R M xy (~r, 0)e i2 ~ k() ~r d~r 7
k-space 8
Slice Selecive 2D Example Slice Selec in Z Resolve in x,y Need Gx(), Gy() Acquire kx(),ky() 9
k-space Trajecory k ymax k xmax k xmax Exen Resoluion Densiy FOV (nex ime) k ymax Ofen Square coverage Someimes circle Wha are some opions? 0
Radial Scanning (Projecion Reconsrucion) Image opional k-space wha is k-space rajecory? wha is s() A/D oupu?
Radial Scanning (Projecion Reconsrucion) Image 2 3 opional k-space wha is s() A/D oupu? 2
Radial Scanning (Projecion Reconsrucion) Image 2 3 opional k-space 3
Radial Scanning (Projecion Reconsrucion) k-space Repea for each diameer Peak a k()=0 4
2DFT (Spin Warp) k-space k x () k y () 5
2DFT (Spin Warp) k-space 2 pre-winder phase-encode readou/freq.-encode k x () 2 echo-ime (TE) k y () 6
2DFT (Spin Warp) k-space Repea for each line Obain 2D Caresian Grid By far he mos common rajecory! 7
2DFT (Spin Warp) k-space Repea for each line Obain 2D Caresian Grid Overlap prewinders wih slice refocus By far he mos common rajecory! 8
2DFT (Spin Warp) Gy gradien is Phase-Encode gradien Esablishes fixed linear-phase before readou Gx gradien is Frequency-Encode gradien, or Readou = 9
Wha Does This Sequence Do? k x () k y () 20
Muli-Echo 2DFT Echo-Time is when kx()=0 k x () TE TE 2 TE 3 k y () TE TE 2 TE 3 2
Echo Planar Imaging (EPI) More phase encode for each exciaion k x () TE TE 2 TE 3 k y () TE TE 2 TE 3 22
Echo Planar Imaging (EPI) More phase encode for each exciaion phase-encode k x () TE TE 2 TE 3 k y () TE TE 2 TE 3 23
Spiral k-space k x () k y () 24
EPI and Spiral EPI: Mos common high-speed acq. fmri, diffusion, real-ime Can use inerleaved muliple shos Easy DFT reconsrucion Spiral Very hardware efficien Useful for flow/ hears, also used for fmri Ofen muliple inerleaved shos Uses gridding for non-uniform FT reconsrucion 25
Sampling, Resoluion and FOV Signal is sample of FT of objec sampled a discree imes! small m s(n ) =M xy ( ~ k(n )) 26
2DFT Case k x k-space 2 k y 2kymax = Wky Nr - # readou samples Np - # phase encodes k ymax = G y 2 2k xmax = W kx W kx = G x 2 2 k x = W kx N r k y = W ky N p 27
Sampled Fourier Transform In k-space In Image space 28
FOV and Aliasing Assume infinie resoluion (for now...) hen, has an impulse when: 29
FOV and Aliasing k y k x k x no aliasing as long as objec is smaller han FOV k y 30
FOV and Aliasing k y k x k x no aliasing as long as objec is smaller han FOV k y 3
FOV and Aliasing k y k x k x no aliasing as long as objec is smaller han FOV k y 32
From HW Face 4 20 40 60 80 00 20 40 60 80 50 00 50 33
Aliasing in 2DFT k-space Readou is sampling a coninuous ime signal Phase encode is inherenly discree! Aliasing does no occur in frequency encode direcion 34
In pracice: No Aliasing Along Readou s() LPF x s(nδ) Ш(/Δ) ani-aliasing - -/2 FOV x /2 /2 FOV x /2 f[hz],x[cm] no aliasing -/2 /2 f[hz],x[cm] 35
FOV In readou/frequency-encode direcion Increase FOV by increasing sampling rae No change in acquisiion gradiens Jus more daa FOVx is free (almos) no aliasing In phase encode, aliasing can occur Phase-encodes are fundamenally discree FOV y * ) k y = W ky ) N p * N p More acquisiions more scan ime FOVy coss scan ime! 36
Aliasing FOVx pe FOVy fe Q: Wha would he reconsrucion image look like? 37
Aliasing Q: Wha is he difference in acquisiion beween he wo images? phase encode A/P frequency encode L/R frequency encode L/R phase encode A/P 38
r rouine use, now wih 6.3 fold acceleraion. ored delineaion of growh plae (arrowhead) and Aliasing II ying fibroma wih SPIR-iT. small FOV large FOV aliased IV ube resoluion wih Poisson-disc sampling and 5-fold 39
Aliasing in Radial Trajecory Nyquis crieria sill deermined by FOV Cos π imes more han Caresian However: Aliasing is incoheren Degrades gracefully wih reducion of spokes k< FOV 40
Aliasing in Radial 20cm objec, mm resoluion 256 spokes 28 spokes 64 spokes Poin Spread funcion: Reconsucion 4
Differen Samplings rajecory PSF x-secion 42