Biomedical Imaging Nuclear agneic Resonance Parícia Figueiredo IST, 213-214
The wide specrum of medical imaging echniques (F. Deconinck, Vrije Universi, Belgium).
Overview Nuclear magneism Precession: ineracion wih a saic magneic field Eciaion: ineracion wih a ime-varing magneic field Relaaion: reurn o equilibrium Pulse-acquire eperimens
Nuclear magneic resonance (NR) Nuclei wih a non-ero spin number spins (paricularl 1 H in H 2 ) are polaried b ineracion wih a srong saic magneic field B Precession around B a he Larmor frequenc L = γb Polaried nuclear spins are ecied b varing magneic fields B 1 applied a he nuclei s Larmor frequenc Eciaion Reurn of he ecied nuclear spins o equilibrium: Relaaion Inducion of a curren on a receive coil according o Farada s law: Free Inducion Deca (FID) This signal depends on he densi of nuclear spins, as well as on he properies of he medium refleced on he relaaion ime consans.
Nuclear magneism Nuclear spins A proon is a elecric charge roaing => magneic momenum Isoope Spin I γ [H/T] Naural abundance [%] 1 H 1/2 42.575 99.985 2 H 1 6.53.15 + 19 F 1/2 4.8 1 31 P 1/2 17.25 1 23 Na 3/2 11.27 1 14 N 1 3.78 99.63 13 C 1/2 1.7 1.1 Spin number I Spin operaor I Spin angular momenum S Dipolar magneic momen µ Gromagneic raio γ [H/T] S = " I µ = γs 17 O 5/2 4.1.48 Roaing veloci
Precession Ineracion wih a polariing magneic field B Zeeman ineracion (energ): Quanied energ levels (ineracion energ wih B o ) he ineracion onl occur along Nuclear magneic quanum number, m: The value of I depends on he number of proons and neurons in he nucleus Proons: 2 energ levels 2 orienaions I = 1/ 2 m = ±1/ 2 µ = ±γ 3 2 µ = ±γ 2 E ±1/2 = γb 2 ΔE = γb $ θ ±1 2 = ±cos 1 µ ' $ & ) = ±cos 1 γ 2 ' & ) % µ ( % γ 3 2( θ ±1 2 = ±54.7 # B o is aligned wih he ais E = µ B E = µ B B = B ẑ µ = γm m = I I 1 " I I = spin number
Precession Ineracion wih a polariing magneic field B Proons: I=1/2 E ±1/2 = γb 2 " θ ±1 2 = ±cos 1 µ % $ ' # µ & " ΔE = γb = ±cos 1 γ 2 % $ ' # γ 3 2& θ ΔE B ±1 2 = ±54.7 " m = ½ µ = γh 3 2 The proons are pariall aligned wih Bo because µ = γs = γhm I B θ B E=+½γħB m =+½ ΔE E= ½ γħb
Precession Ineracion wih a polariing magneic field B wih a populaion of proons Bolmann disribuion: N 1 2 N+ 1 2 = e T=37º, B =1.5T N -1/2 /N +1/2 =.999998 ΔE kt The populaion difference is abou: 1-4 % ΔE = γb = L Larmor (resonance) frequenc: L = γb B ~ 1 1 T L ~ 4 5 H [RF] N -1/2 =.4999951 ΔE, L, ΔN B B N +1/2 =.549
Precession Ineracion wih a polariing magneic field B agneiaion: (for an ensemble of nuclei) = µ ( N ) γ = + 1 2 N 2 1 2 B 2 2 γ B N 4kT N s = oal number of proons proon densi s B B
Precession Classical descripion Free Precession: d S = µ d B d µ = γµ d B µ roaes abou he ais, wih frequenc: Larmor (resonance) frequenc = γb = L µ,s θ B φ Longiudinal ais: Transverse place:
Precession Classical descripion B d d γ = L B B B ˆ = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = 1 cos sin sin cos = B B d d d d d d γ γ Equaions of moion: roaion abou he ais, clockwise, wih frequenc L B γ = = = ˆ Vecor model: L Bloch equaion:
Precession Classical descripion Laboraor reference frame: Roaing reference frame: B L L " = Lˆ = γb ˆ = saionar in he roaing reference frame.
Precession Classical descripion Effecive field (for a general applied field B and associaed magneiaion ): d d = = γb d d = γ B = γ B + = γ B eff γ Effecive field: B eff = B + γ In general, he magneiaion precesses around he effecive field B eff in he roaing frame, jus as i precesses around he applied magneic field B in he laboraor frame.
Eciaion Ineracion wih a radiofrequenc field B 1 : Perurbaion awa from equilibrium, hrough resonance: Energ equal o he energ difference beween spin populaions will induce ransiions beween saes, reducing he magneiaion. Δ E = γ cb = L Classical descripion: Roaion of he magneiaion awa from equilibrium, owards he plane Applicaion of B 1 in he plane, wih oscillaing frequenc equal o = L γb B Circularl polaried magneic field (roaing clockwise): ( ) = B ( ) ˆ B sin( )ˆ 1 1 cos RF 1 RF = = γ RF L B T=37º, B =1.5T N -1/2 /N +1/2 =.999998 B ~ 1 1 T L ~ 4 5 H [RF]
Eciaion Ineracion wih a radiofrequenc field B 1 : Nuaion in he laboraor reference frame: B eff d d d B + B = 1 d d = d + γ ( γb ) ( γb ) γb 1 γb 1 B eff B 1 B -/γ B B B L RF B 1 B 1 B 1
Eciaion Ineracion wih a radiofrequenc field B 1 : Precession in he roaing reference frame: roaion abou wih frequenc ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = cos sin sin cos 1 1 1 1 1 1 1 = γb 1 1 1 ˆ B B B B eff " = = = γ B 1 L Δ Δ = B B d d d d d d 1 1 γ γ If off-resonance occurs: ( ) B B B eff ˆ ˆ 1 γ γ Δ + = + Δ = "
Eciaion Ineracion wih a radiofrequenc field B 1 : RF pulse: Flip angle / ip angle: B 1 = B 1 (), [,τ] θ = γ τ ( ) B1 d 1 = γb 1 9º pulse: eciaion/sauraion 18 º pulse: inversion/refocusing θ = 9 θ = 18 B 1 B 1 = = -
Eciaion Eciaion/ Sauraion 9º RF pulses: B 1 = = The effec of an RF pulse is o ransfer energ from he ransmiing coil o proons. This ecess energ resuls in a non- Bolman disribuion (non equilibrium) of he populaions of he parallel and ani-parallel energ saes. is reduced and and are no longer. θ = 9 = = B 1 θ = 9 Inversion/ Refocusing 18º RF pulses: θ = 18 θ = 18 - B 1 - B 1
Relaaion Reurn o hermal equilibrium: relaaion Following eciaion: Energ sae hermal equilibrium = = = = Longiudinal relaaionl 1- ep{-/t 1 } Loss of coherence Transverse relaaion ep{-/t 2 }
Relaaion Bloch equaions for relaaion: d d = T Following eciaion: 1 d d = ( ) = + ( ( ) ) ep T1 ( ) ( ) = ep T2 T 2 ( ) = ( ) = 1 ep T1 ( ) = ( ) = ep T2 Longiudinal relaaion 1- ep{-/t 1 } Transverse relaaion ep{-/t 2 }
Relaaion Relaaion mechanims: Relaaion resuls from energ echange hrough he flucuaing magneic fields eperienced b he nuclei as a consequence of heir hermal molecular moion. The dominan mechanisms of flucuaing magneic fields for nuclei of spin ½ are dipolar ineracions wih oher nuclei. Longiudinal relaaion or spin-laice relaaion: requires flucuaions a he Larmor frequenc o produce ransiions beween energ saes and hus resore polariaion and. Transverse relaaion or spin-spin relaaion: is also promoed b flucuaions a ero frequenc, which produce random dephasing of spins and hus loss of coherence and cancellaion of.
Relaaion Relaaion mechanims: Transverse relaaion or spin-spin relaaion: is also promoed b flucuaions a ero frequenc, which produce random dephasing of spins and hus loss of coherence and cancellaion of. Phase dispersion (roaing frame): ΔB = B B Δ = γδb ΔB Δφ = γ Δ
Relaaion Relaaion mechanims: olecular moion (roaions, ranslaions, vibraions) flucuaions in nuclear magneic fields J() Fas Low Viscosi High emperaure Specral densi funcion = frequenc specrum of flucuaing magneic fields of nuclei J τ c ( ) 2 2 1+ τc Slow High Viscosi Low emperaure L (low B ) τ c -1 L (high B ) τ c is he correlaion ime: characerisic ime scale of hermal molecular moion: τ c T -1
Relaaion Relaaion ime consans T 1 and T 2 : Relaaion ime consans for dipolar ineracions (spin 1/2): J() Longiudinal relaaion (spin-laice): slow fas 1 T γ r 4 c [ J ( L ) + J ( 2L )], J ( L ) 2 2 6 1 1 + τ τ c L Transverse relaaion (spin-spin): τ -1 c L 1/T 2 1/T 1 1 T 2 4 γ 6 r [ J( ) + J( L ) + J( 2L )], J( ) τc T 1 - T 2 is alwas shorer han T 1 - In soluion: T 1 ~T 2 (fas moion, shor τ c ) T 2 slow L fas τ c -1 - In vivo, T 1 up o ~1 T 2 (slow moion, long τ c ) - T 1 increases wih field srengh while T 2 is roughl independen.
Relaaion Off-resonance conribuions o relaaion mechanims - Chemical shifs, J-couplings, ec. specral peaks - Saic field inhomogeneiies T 2 * deca - Imperfec B uniformi - Inrinsic sample suscepibili differences - Eernall applied gradien fields
Relaaion Off-resonance conribuions o relaaion mechanims Chemical shif: - shif in he precessional frequenc of a nucleus due o he magneic field associaed wih he elecronic momen (elecron spin), of opposie polari o B : eff = γ B o (1 - σ), σ = shielding consan. - in order o allow direc comparison a differen field srenghs, he chemical shif is defined as he frequenc shif scaled o a reference peak: δ = 1 6 ( - ref )/ ref [ppm] (e.g., σ{η 2 Ο} = 1.3 ppm, σ{-cη 2 -} = 4.5 ppm - differen chemical compounds produce signals a differen precessional frequencies and a specrum is obained (RS). E: 1H - NR specrum of human brain 4.4 4. 3.6 3.2 2.8 2.4 2. 1.6 1.2.8.4 (ppm)
Relaaion Off-resonance conribuions o relaaion mechanims: T 2 * deca T 2 * deca: Addiionall o spin-spin relaaion mechanisms, loss of coherence of he ransverse magneiaion also occurs as a resul of bulk magneic field effecs: saic field inhomogeneiies due o applied and/or inrinsic gradiens. ΔB Δ = γδb Δφ = γδb Δ Field inhomogenei Phase dispersion (roaing frame)
Inducion Signal deecion: Farada s Law of Inducion: For a solenoid wih N urns and surface area A: ε = Φ ( L) L B ε = NA sin L
Inducion Signal deecion: Farada s Law of Inducion: For a solenoid wih N urns and surface area A: Φ ε = = NA sin ε L ( L ) L B Quadraure deecion
Inducion Free Inducion Deca (FID): 1 1 1 T 2 * deca: = + + T 2 * < T T * T T 2 2 2 2 T * ( ) = ( ) e 2 signal ~1/T 2 ~1/T 2 * ~ep{-/t 2 } FT ime real L frequenc ~ep{-/t 2 *} T 2 * processes speed up signal deca and broaden specral linewidh.
Pulse-acquisiion eperimens Spin echo Eciaion Spin dephasing
Pulse-acquisiion eperimens Spin echo 18º RF pulse Spin refocusing
Pulse-acquisiion eperimens Spin echo Eciaion Spin dephasing
Pulse-acquisiion eperimens Spin echo 18º RF pulse Spin refocusing
Pulse-acquisiion eperimens Spin echo 18º RF pulse Spin refocusing
Pulse-acquisiion eperimens Spin echo pulse sequence (SE) RF eciaion 9 o refocusing 18 o TE = echo ime acquire Signal TE / 2 TE T 2 * T 2 FID echo
Pulse-acquisiion eperimens Spin echo pulse sequence (SE) A spin-echo can refocus spins ha are siing in a ime invarian (saic) B field, i.e., phase dispersion due o saic field inhomogenei (T 2 * processes). A spin-echo canno refocus spins ha have eperienced a ime varing B field, i.e., phase dispersion due o diffusion and T 2 processes.
Pulse-acquisiion eperimens Spin echo pulse sequence (SE) easuremen of T 2 b muliple spin-echo: T2 ( nte) = e nte T 2 * T 2 9 18 18 18 TE/2 TE 2 TE 3 TE
Pulse-acquisiion eperimens Inversion recover pulse sequence (IR) easuremen of T 1 b inversion recover: ( TI ) n 1 2e TI = T 1 n 18 9 9 9 T 1 TI 1 TI 2 TI 3
Pulse-acquisiion eperimens Sead-sae magneiaion ( ss ) and repeion ime (TR) Erns angle α Erns : maimies he ransverse magneiaion: α α α α cos α Erns α TR ep T = 1 TR ss
References Webb, Inroducion o Biomedical Imaging, Wile 23. Cho, Foundaions of edic E. ark Haacke, Rober W. Brown, ichael R. Thompson, Ramesh Venkaesan, agneic Resonance Imaging: Phsical Principles and Sequence Design, Wile 1993.