Principles of Magnetic Resonance Imaging

,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Principles of Magnetic Resonance Imaging,J. Peter Mustonen,(from David J. Michalak),Presentation for Physics 250,05/01/2008,Motivation,Principles of NMR,Interactions of spins in B,0,field,Principles of 1D-MRI,Principles of 2D-MRI,Summary,Outline,Magnetic Resonance Imaging provides a non-invasive imaging technique.,Pros:,-No injection of potentially dangerous elements (radioactive dyes),-Only magnetic fields are used for imaging no x-rays,Cons:,-Current geometries are expensive, and large/heavy,Motivation,B,0,Principles of NMR,Application of prepolarizing magnetic field, B,0, aligns the spins in a sample to give a net magnetization,M,.,M,rotates about B,0,at a Larmor precession frequency,w,0,=,g,B,0,M,=,S,M,i,RF,Pulse,B,0,Principles of NMR,Application of prepolarizing magnetic field, B,0, aligns the spins in a sample to give a net magnetization,M,.,M,rotates about B,0,at a Larmor precession frequency,w,0,=,g,B,0,y,x,z,B,0,Application of a rf pulse,w,0,=2,p,f,0,along the,x,-axis will provide a torque that displaces,M,from the,z,axis towards,y,axis. A certain pulse length will put M right on,xy,plane,M,=,S,M,i,M,RF,Pulse,B,0,Principles of NMR,Application of prepolarizing magnetic field, B,0, aligns the spins in a sample to give a net magnetization,M,.,M,rotates about B,0,at a Larmor precession frequency,w,0,=,g,B,0,y,x,z,B,0,y,x,z,B,0,M,=,S,M,i,M,precesses in the transverse plane. In the absence of any disturbances,M,continues to rotate indefinitely in,xy,plane.,M,M,Time,Application of a rf pulse,w,0,=2,p,f,0,along the,x,-axis will provide a torque that displaces,M,from the,z,axis towards,y,axis. A certain pulse length will put M right on,xy,plane,exp-,i,w,0,t,RF,Pulse,B,0,Principles of NMR,Application of prepolarizing magnetic field, B,0, aligns the spins in a sample to give a net magnetization,M,.,M,rotates about B,0,at a Larmor precession frequency,w,0,=,g,B,0,y,x,z,B,0,y,x,z,B,0,M,=,S,M,i,M,precesses in the transverse plane. In the absence of any disturbances,M,continues to rotate indefinitely in,xy,plane.,Detector,M,M,Time,Application of a rf pulse,w,0,=2,p,f,0,along the,x,-axis will provide a torque that displaces,M,from the,z,axis towards,y,axis. A certain pulse length will put M right on,xy,plane,exp-,i,w,0,t,RF,Pulse,B,0,Principles of NMR,Application of prepolarizing magnetic field, B,0, aligns the spins in a sample to give a net magnetization,M,.,M,rotates about B,0,at a Larmor precession frequency,w,0,=,g,B,0,y,x,z,B,0,y,x,z,B,0,M,=,S,M,i,M,precesses in the transverse plane. In the absence of any disturbances,M,continues to rotate indefinitely in,xy,plane.,Detector,M,M,Time,Assume:,All spins feel same B,0,.,No other forces on,M,i,(including detection).,Application of a rf pulse,w,0,=2,p,f,0,along the,x,-axis will provide a torque that displaces,M,from the,z,axis towards,y,axis. A certain pulse length will put M right on,xy,plane,exp-,i,w,0,t,RF,Pulse,B,0,Principles of NMR,Application of prepolarizing magnetic field, B,0, aligns the spins in a sample to give a net magnetization,M,.,M,rotates about B,0,at a Larmor precession frequency,w,0,=,g,B,0,y,x,z,B,0,y,x,z,B,0,M,=,S,M,i,Detector,M,M,Time,time,t,signal, s,r,(,t,),Application of a rf pulse,w,0,=2,p,f,0,along the,x,-axis will provide a torque that displaces,M,from the,z,axis towards,y,axis. A certain pulse length will put M right on,xy,plane,(,w,0,/2,p,),-1,exp-,i,w,0,t,RF,Pulse,B,0,Principles of NMR,Application of prepolarizing magnetic field, B,0, aligns the spins in a sample to give a net magnetization,M,.,M,rotates about B,0,at a Larmor precession frequency,w,0,=,g,B,0,y,x,z,B,0,y,x,z,B,0,M,=,S,M,i,Detector,FT,M,M,Time,s,r,(,t,),s,r,(,w,),t,Application of a rf pulse,w,0,=2,p,f,0,along the,x,-axis will provide a torque that displaces,M,from the,z,axis towards,y,axis. A certain pulse length will put M right on,xy,plane,w,0,= 2,p,f,0,w,exp-,i,w,0,t,(,w,0,/2,p,),-1,RF,Pulse,B,0,Principles of NMR,Application of prepolarizing magnetic field, B,0, aligns the spins in a sample to give a net magnetization,M,.,M,rotates about B,0,at a Larmor precession frequency,w,0,=,g,B,0,y,x,z,B,0,y,x,z,B,0,M,=,S,M,i,Detector,(,w,0,/2,p,),-1,FT,M,M,Time,Boring Spectrum!,w,0,= 2,p,f,0,s,r,(,t,),s,r,(,w,),t,w,Application of a rf pulse,w,0,=2,p,f,0,along the,x,-axis will provide a torque that displaces,M,from the,z,axis towards,y,axis. A certain pulse length will put M right on,xy,plane,exp-,i,w,0,t,Principles of NMR,y,x,z,B,0,In Reality:,Relaxation (Inherent even if B,0,is homogeneous),T,1,: Spins move away from,xy,plane towards,z,.,T,2,: Spins dephase from each other.,B,0,inhomogeneity.,Chemical Shift.,Complexity Makes Things Interesting,Principles of NMR,y,x,z,B,0,T,1,Spin Relaxation,: return of the magnetization vector back to,z,-axis.,Spin-Lattice Time Constant:,Energy exchange between spins and surrounding lattice.,Fluctuations of B field (surrounding dipoles,receivers) at,w,0,are important. Larger E exchange necessary for larger B,0,longer,T,1,.,Math: d,M,/d,t,= -(,M,z,-,M,0,)/T,1,Solution:,M,z,=,M,0,+ (,M,z,(0)-,M,0,)exp(-,t/T,1,),After 90 pulse:,M,z,=,M,0,1-exp(-,t/T,1,),M,0,= net magnetization based on B,0,.,M,z,= component of M,0,along the,z,-axis.,t,= time,T,1,Spin Relaxation,Principles of NMR,y,x,z,B,0,T,2,Spin Relaxation,: Decay of transverse magnetization,M,xy,.,T,1,plays a role, since as,M,xy,M,z,M,xy,0,But dephasing also decreases,M,xy,:,T,2,T,1,.,T,2,: Spin-Spin Time Constant,Variations in,B,z,with time and position.,Pertinent fluctuations in,B,z,are those near dc frequencies (independent of,B,0,) so that,w,0,is changed.,Molecular motion around the spin of interest.,Liquids: High Temp more motion, less,D,B, high,T,2,Solids: slow fluctuations in B,z, extreme,T,2,.,Bio Tissues: spins bound to large molecules vs. those free in solution.,y,x,z,B,0,+,D,B(,r,t,),T,2,Spin Relaxation,M,xy,Principles of NMR,y,x,z,B,0,Comparison of,T,1,and,T,2,Spin Relaxation,:,y,x,z,B,0,+,D,B(,r,t,),Tissue,T,1,(ms),T,2,(ms),Gray Matter,950,100,White Matter,600,80,Muscle,900,50,Fat,250,60,Blood,1200,100-200*,*200 for arterial blood, 100 for venous blood.,B,0,= 1.5 T, 37 degC (Body Temp),Magnetic Resonance Imaging: Physical Principles and Sequence Design, Haacke E.M.,et al., Wiley: New York, 1999.,T,1,/,T,2,Spin Relaxation,Math: d,M,/d,t,= -,M,xy,/T,2,After 90 pulse:,M,xy,=,M,0,exp(-,t/T,2,),Principles of NMR,y,x,z,B,0,Comparison of,T,1,and,T,2,Spin Relaxation,:,y,x,z,B,0,+,D,B(,r,t,),Tissue,T,1,(ms),T,2,(ms),Gray Matter,950,100,White Matter,600,80,Muscle,900,50,Fat,250,60,Blood,1200,100-200*,*200 for arterial blood, 100 for venous blood.,Magnetic Resonance Imaging: Physical Principles and Sequence Design, Haacke E.M.,et al., Wiley: New York, 1999.,FID,w,0,Spectrum,T,2,T,1,M,xy,decays,exp(-t/T,2,),Detector,FT,2/,T,2,Because,T,2,is independent of B,0, higher B,0,gives better resolution,T,1,/,T,2,Spin Relaxation,s,r,(,t,),t,Principles of NMR,y,x,z,B,0,y,x,z,B,0,+,D,B(,r,t,),Inclusion of,T,1,and,T,2,Spin Relaxation,:,Inclusion of mathematical expression:,Bloch Equation,g,= gyromagnetic ratio,T,1,= Spin-Lattice (longitudinal-,z,) relaxation time constant,T,2,= Spin-Spin (transverse-x/y) relaxation time constant,M,0,= Equilibrium Magnetization due to B,0,field.,i,j,k,= Unit vectors in,x,y,z,directions respectively.,T,1,/,T,2,Spin Relaxation,Principles of NMR,y,x,z,B,0,y,x,z,B,0,+,D,B(,r,t,),Inclusion of,T,1,and,T,2,Spin Relaxation,:,Inclusion of mathematical expression:,Bloch Equation,g,= gyromagnetic ratio,T,1,= Spin-Lattice (longitudinal-,z,) relaxation time constant,T,2,= Spin-Spin (transverse-x/y) relaxation time constant,M,0,= Equilibrium Magnetization due to B,0,field.,i,j,k,= Unit vectors in,x,y,z,directions respectively.,Precession,Transverse,Decay,Longitudinal,Growth,Net magnetization is not necessarily constant:,e.g., very short,T,2, long,T,1,.,T,1,/,T,2,Spin Relaxation,Principles of NMR,y,x,z,B,0,Chemical Shift,: Nuclei are shielded (slightly) from B,0,by the presence of their electron clouds.,Effective field felt by a nuclear spin is B,0,(1-,s,).,Larmor precession freq,w,=,g,B,0,(1-,s,).,Shift is often in the ppm range.,500,000 precessions before,M,xy,= 0,Chemical environment determines amount of,s,.,H,2,O vs. Fat (fat about 3.5 ppm lower,w,0,),y,x,z,B,0,(1-,s,),Discrete Shift,H,H,O,d,+,d,+,2d,-,H,H,C,Detector,Less Shielding,More Shielding,Chemical Shift,Principles of NMR,y,x,z,B,0,Chemical Shift,: Nuclei are shielded (slightly) from B,0,by the presence of their electron clouds.,y,x,z,B,0,(1-,s,),Discrete Shift,w,0,2/,T,2,Because,T,2,is independent of B,0, higher B,0,gives better resolution,Detector,w,0,(1-,s,),Ability to resolve nuclei in different chemical environments is key to NMR,Chemical Shift,Principles of NMR,y,x,z,B,0,T,2,*: B,0,Inhomogeneity,: Additional decay of,M,xy,.,In addition to,T,2, which leads to,M,xy,decay even in a constant B,0, application of,d,B,0,(,x,y,z,t,) will cause increased dephasing: 1/,T,2,* = 1/,T,2,+ 1/,T, where,T, is the dephasing due only to,d,B,0,(,x,y,z,t,),.,T,2,*,T,2, and depends on,d,B,0,(,x,y,z,t,),.,Additional loss of resolution between peaks.,y,x,z,B,0,+,d,B(,r,t,),time,t,Field Inhomogeneity,Principles of NMR,y,x,z,B,0,T,2,*: B,0,Inhomogeneity,: Additional decay of,M,xy,.,In addition to,T,2, which leads to,M,xy,decay even in a constant B,0, application of,d,B,0,(,x,y,z,t,) will cause increased dephasing: 1/,T,2,* = 1/,T,2,+ 1/,T, where,T, is the dephasing due only to,d,B,0,(,x,y,z,t,),.,T,2,*,T,2, and depends on,d,B,0,(,x,y,z,t,),.,Additional loss of resolution between peaks.,If,d,B,0,(,x,y,z,),is not time dependent, then it can be corrected by an echo pulse.,y,x,z,B,0,+,d,B(,r,t,),time,t,Field Inhomogeneity,Principles of NMR,y,x,z,B,0,T,2,*: B,0,Inhomogeneity,: Additional decay of,M,xy,.,In addition to,T,2, which leads to,M,xy,decay even in a constant B,0, application of,d,B,0,(,x,y,z,t,) will cause increased dephasing: 1/,T,2,* = 1/,T,2,+ 1/,T, where,T, is the dephasing due only to,d,B,0,(,x,y,z,t,),.,T,2,*,T,2, and depends on,d,B,0,(,x,y,z,t,),.,Additional loss of resolution between peaks.,If,d,B,0,(,x,y,z,),is not time dependent, then it can be corrected by an echo pulse.,y,x,z,B,0,+,d,B(,r,t,),y,x,z,B,0,+,d,B(,r,t,),y,x,z,B,0,+,d,B(,r,t,),180,x,pulse,(,x x,y,y,),time,t,time,t,Field Inhomogeneity,Echo!,Principles of NMR,y,x,z,B,0,T,2,T,2,*,Field Inhomogeneity,T,2,*: B,0,Inhomogeneity,: Additional decay of,M,xy,.,If echo pulse applied at time,t, then echo appears at 2,t,.,Only,T, can be reversed by echo pulsing,T,2,cannot be echoed as the field inhomogeneities that lead to,T,2,are not constant in time or space.,4)Signal after various echo pulsed displayed below.,t,180,pulse,applied,2t,Echo,t,= 0,90,pulse,t,180,pulse,applied,2,(t,-t),Echo,s,r,(,t,),t,Principles of 1DMRI,Single B,0, No Spatial Information,Measured response is from,all,spins in the sample volume,. Detector coil probes all space with equal intensity,90,pulse,B,0,B,0,FID,w,0,Spectrum,FT,2/,T,2,If only B,0,is present (and homogeneous) all spins remain in phase during precession (as drawn).,- B(,x,y,z,t,) = B,0,; thus,w,(,x,y,z,) =,w,0,=,g,B,0,time,B,0,No Spatial Information,(Volume integral),Detector,coil,s,r,(,t,),t,Principles of 1DMRI,Slice Selection:,z,-Gradient,Slice selection along,z,-axis,. Gradient in,z,and selective excitation allows detection of a single slice.,B(z) = B,0,+ G,z,z,Field strength indicated by line thickness,G,z,G,z,= dB,z,/dz,integrate,B,z,=G,z,z,It follows that:,B(z=0)=B,0,Principles of 1DMRI,Slice Selection:,z,-Gradient,Slice selection along,z,-axis,. Gradient in,z,and selective excitation allows detection of a single slice.,Selective,90 pulse,w,rf,=,w,0,+,g,G,z,z,B(z) = B,0,+ G,z,z,Field strength indicated by line thickness,G,z,G,z,= dB,z,/dz,integrate,B,z,=G,z,z,It follows that:,B(z=0)=B,0,Principles of 1DMRI,Slice Selection:,z,-Gradient,Slice selection along,z,-axis,. Gradient in,z,and selective excitation allows detection of a single slice.,B(z) = B,0,+ G,z,z,Field strength indicated by line thickness,Larmor Precession frequency is,z,-dependent:,w,(,z,) =,g,B(,z,),w,(,z,) =,g,(B,0,+ G,z,z),w,(,z,) =,w,0,+,g,G,z,z,G,z,Selective,90 pulse,w,rf,=,w,0,+,g,G,z,z,Excite only one plane of,z,D,z,by using only one excitation frequency for the 90 pulse. For example, using B,0,for excitation: only spins at,z,=0 get excited. All other spins are off resonance and are not tipped into the transverse plane.,G,z,= dB,z,/dz,integrate,B,z,=G,z,z,It follows that:,B(z=0)=B,0,FT,Principles of 1DMRI,Slice Selection:,z,-Gradient,Slice selection along,z,-axis,. Gradient in,z,and selective excitation allows detection of a single slice.,B(z) = B,0,+ G,z,z,Field strength indicated by line thickness,G,z,In practice, you must bandwidth match the frequency of the 90 pulse with the desired thickness (,D,z,) of the z-slice. (i.e., with a linear gradient, the Larmor precession of spins within z = 0 ,D,z oscillate with frequency,w,0,g,G,z,D,z. Thus, BW = 2,g,G,z,D,z.),4)To apply a “boxcar” of frequencies,w,g,G,z,D,z, we need the 90 deg excitation profile to be a sinc function in time.,FT(sinc) = rect,Selective,90 pulse,w,rf,=,w,0,+,g,G,z,z,t,w,90,z,D,z,G,z,= dB,z,/dz,integrate,B,z,=G,z,z,It follows that:,B(z=0)=B,0,sinc = (sin,x,)/,x,w,0,-,g,G,z,D,z,Principles of 1DMRI,Slice Selection:,z,-Gradient,Gradient Echo Pulse,. Gradient Echo pulse restores all spins to have the same phase within the slice,D,z,.,Selective,90,pulse,B(z) = B,0,+ G,z,z,G,z,Pulse Sequence,RF,G,z,Gradient Echo,w,0,w,0,+,g,G,z,D,z,Before Gradient Echo,t,=,t,0,t,3,t,/2,time,Spins out of phase on,xy,plane,z,w,0,-,g,G,z,D,z,Principles of 1DMRI,Slice Selection:,z,-Gradient,Gradient Echo Pulse,. Gradient Echo pulse restores all spins to have the same phase within the slice,D,z,.,Selective,90,pulse,B(z) = B,0,+ G,z,z,G,z,Pulse Sequence,RF,G,z,Gradient Echo,z,w,0,w,0,+,g,G,z,D,z,Before Gradient Echo,t,=,t,0,t,time,Spins out of phase on,xy,plane,Top View of xy plane,t,=,t,w,0,w,0,+,g,G,z,D,z,w,0,-,g,G,z,D,z,3,t,/2,Principles of 1DMRI,Slice Selection:,z,-Gradient,Gradient Echo Pulse,. Gradient Echo pulse restores all spins to have the same phase within the slice,D,z,.,Selective,90,pulse,B(z) = B,0,+ G,z,z,G,z,Pulse Sequence,RF,G,z,z,z,Before Gradient Echo,t,=,t,After Gradient Echo,t,= 3,t,/2,0,t,time,Spins out of phase on,xy,plane,Spins all IN phase,Gradient Echo,t,=,t,t,=,3,t,/,2,w,0,w,0,+,g,G,z,D,z,w,0,-,g,G,z,D,z,Top View of xy plane,w,0,-,g,G,z,D,z,w,0,w,0,+,g,G,z,D,z,3,t,/2,Principles of 1DMRI,Slice Selection:,z,-Gradient,Slice selection along,z,-axis,. Gradient in,z,and selective excitation allows detection of a single slice.,Selective,90,pulse,B(z) = B,0,+ G,z,z,G,z,Principles of 1DMRI,Slice Selection:,z,-Gradient,Slice selection along,z,-axis,. Gradient in,z,and selective excitation allows detection of a single slice.,Selective,90,pulse,B(z) = B,0,+ G,z,z,G,z,Detector,coil,time,Principles of 1DMRI,Slice Selection:,z,-Gradient,Slice selection along,z,-axis,. Gradient in,z,and selective excitation allows detection of a single slice.,Selective,90,pulse,B(z) = B,0,+ G,z,z,G,z,Detector,coil,time,FID,w,0,Spectrum,FT,2/,T,2,No,x,y,Information, but only spins from the,z,D,z,slice contribute to the signal.,exp(-,t,/T,2,),If we can encode along,x,and,y,dimensions, we can iterate for each,z,slice.,s,r,(,t,),t,Principles of 1DMRI,Frequency Encoding,Perform z-slice,. Now only look at 2D plane from now on. Use Gradient along,x,to generate different Larmor frequencies vs.,x,-position.,Selective,90,pulse,in,z,D,z,time,z,y,x,2D,z,Apply,x,-Gradient,G,x,= dB,z,/d,x,Precession Frequency varies with x,z,y,x,z,x,B,z,(,x,) - B,0,Principles of 1DMRI,Frequency Encoding,Perform z-slice,. Now only look at 2D plane from now on. Use Gradient along,x,to generate different Larmor frequencies vs.,x,-position.,Selective,90,pulse,in,z,D,z,time,z,y,x,2D,z,Apply,x,-Gradient,G,x,= dB,z,/d,x,Precession Frequency varies with x,z,y,x,z,x,w,0,w,0,+,g,G,x,x,w,0,-,g,G,x,x,w,(,x,),Frequency Encoding along,x,B,z,(,x,) - B,0,Principles of 1DMRI,Frequency Encoding,Perform z-slice,. Now only look at 2D plane from now on. Use Gradient along,x,to generate different Larmor frequencies vs.,x,-position.,z,x,Detector,coil,Pulse Sequence,RF,G,z,0,time,G,x,Detect Signal,“readout”,G,x,on while detecting,B,z,(,x,) - B,0,w,0,w,0,+,g,G,x,x,w,0,-,g,G,x,x,w,(,x,),Principles of 1DMRI,Frequency Encoding,Perform z-slice,. Now only look at 2D plane from now on. Use Gradient along,x,to generate different Larmor frequencies vs.,x,-position.,Apply,x,-Gradient DURING acquisition.,Precession Frequency varies with x.,z,x,Detector,coil,FID,FT,exp(-,t,/T,2,*),T,2,* is based on the intentionally applied gradient.,w,0,2/,T,2,*,w,0,-,g,G,x,x,w,0,+,g,G,x,x,s,r,(,t,),t,B,z,(,x,) - B,0,w,0,w,0,+,g,G,x,x,w,0,-,g,G,x,x,w,(,x,),w,0,+,g,G,x,x,Principles of 1DMRI,Frequency Encoding,Perform z-slice,. Now only look at 2D plane from now on. Use Gradient along,x,to generate different Larmor frequencies vs.,x,-position.,Apply,x,-Gradient DURING acquisition.,Precession Frequency varies with x.,Spins at various x positions in space are,encoded to a different precession frequency,z,x,Detector,coil,FID,FT,exp(-,t,/T,2,*),T,2,* is based on the intentionally applied gradient.,w,0,2/,T,2,*,w,0,-,g,G,x,x,w,0,+,g,G,x,x,s,r,(,t,),t,B,z,(,x,) - B,0,w,0,w,0,-,g,G,x,x,w,(,x,),Principles of 1DMRI,90,pulse,time,z,x,B,z,(,x,),Imaging Example,Two Microfluidic Channels,. Water only exists in two microfluic channels as shown.,z,y,x,D,z,z,y,x,Application of G,x,Principles of 1DMRI,90,pulse,time,z,x,B,z,(,x,),No spins exist at,x,=0 where G,x,=0 (,w,0,): FT of signal has no intensity at,w,0,.,Signal is the line integral along,y,. (Still no info about,y,distribution of spins.),Imaging Example,Two Microfluidic Channels,. Water only exists in two microfluic channels as shown.,z,y,x,D,z,w,0,w,0,-,g,G,x,x,w,0,+,g,G,x,x,Image,z,y,x,m,(,x,y,) = spin density(,x,y,),Application of G,x,Principles of 1DMRI,1DFT Math,Signal is the 1DFT of the line integral along,y,.,Homodyne the signal,(from,w,0,to 0).,Principles of 1DMRI,1DFT Math,Signal is the 1DFT of the line integral along,y,.,Homodyne the signal,(from,w,0,to 0).,Let,g,(,x,) = Line integral along,y,for a given,x,position,.,Principles of 1DMRI,1DFT Math,Signal is the 1DFT of the