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UCSD BENG 280A - MRI Lecture 2

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1!TT Liu, BE280A, UCSD Fall 2010!Bioengineering 280A"Principles of Biomedical Imaging""Fall Quarter 2010"MRI Lecture 2"!TT Liu, BE280A, UCSD Fall 2010!Relaxation!An excitation pulse rotates the magnetization vector away from !its equilibrium state (purely longitudinal). The resulting vector !has both longitudinal Mz and tranverse Mxy components.!!Due to thermal interactions, the magnetization will return to its !equilibrium state with characteristic time constants. !T1 spin-lattice time constant, return to equilibrium of Mz! !T2 spin-spin time constant, return to equilibrium of Mxy !TT Liu, BE280A, UCSD Fall 2010!Longitudinal Relaxation!! dMzdt= "Mz" M0T1Due to exchange of energy between nuclei and the lattice (thermal!vibrations). Process continues until thermal equilibrium as !determined by Boltzmann statistics is obtained. !!The energy ΔE required for transitions between down to up spins, !increases with field strength, so that T1 increases with B. !! Mz(t) = M0(1" e"t /T1)After a 90 degree pulse!TT Liu, BE280A, UCSD Fall 2010!T1 Values!Gray Matter!muscle!White matter!kidney!liver!fat!Image, caption: Nishimura, Fig. 4.22!TT Liu, BE280A, UCSD Fall 2010!Transverse Relaxation!! dMxydt= "MxyT2Each spinʼs local field is affected by the z-component of the field!due to other spins. Thus, the Larmor frequency of each spin will be!slightly different. This leads to a dephasing of the transverse !magnetization, which is characterized by an exponential decay.!!T2 is largely independent of field. T2 is short for low frequency !fluctuations, such as those associated with slowly tumbling!macromolecules. !z!x! y!z!x! y!z!x! y!TT Liu, BE280A, UCSD Fall 2010!T2 Relaxation!! Mxy(t) = M0e"t /T2After a 90 degree excitation!TT Liu, BE280A, UCSD Fall 2010!T2 Relaxation!x!x!x!x!x!x!x!x!x!x!x!x!x!x!x!x!x!x!x!x!x!Runners!Net signal!Credit: Larry Frank!x!x!x!x!x!x!x!TT Liu, BE280A, UCSD Fall 2010!T2 Values!Table: adapted from Nishimura, Table 4.2 Tissue T2 (ms) gray matter 100 white matter 92 muscle 47 fat 85 kidney 58 liver 43 Solids exhibit very short T2 relaxation times because there are many low frequency interactions between the immobile spins. !!On the other hand, liquids show relatively long T2 values, because the spins are highly mobile and net fields average out. !CSF 4000!3!TT Liu, BE280A, UCSD Fall 2010!Example!T1-weighted! T2-weighted!Density-weighted!Questions: How can one achieve T2 weighting? What are the relative T2ʼs of the various tissues? !TT Liu, BE280A, UCSD Fall 2010!Example!Hanson 2009!TT Liu, BE280A, UCSD Fall 2010!Bloch Equation!! dMdt= M "#B $Mxi + MyjT2$Mz$ M0( )kT1i, j, k are unit vectors in the x,y,z directions. !Precession!!Transverse!Relaxation!Longitudinal!Relaxation!TT Liu, BE280A, UCSD Fall 2010!! dMdt= M "#B=#ˆ i ˆ j ˆ k MxMyMzBxByBz=#ˆ i BzMy$ ByMz( )$ˆ j BzMx$ BxMz( )ˆ k ByMx$ BxMy( )% & ' ' ' ( ) * * * Free precession about static field!B!Μ#dΜ#4!TT Liu, BE280A, UCSD Fall 2010!! dMxdtdMydtdMzdt" # $ $ $ % & ' ' ' =(BzMy) ByMzBxMz) BzMxByMx) BxMy" # $ $ $ % & ' ' ' =(0 Bz)By)Bz0 BxBy)Bx0" # $ $ $ % & ' ' ' MxMyMz" # $ $ $ % & ' ' ' Free precession about static field!TT Liu, BE280A, UCSD Fall 2010!Precession !! dMxdtdMydtdMzdt" # $ $ $ % & ' ' ' =(0 B00)B00 00 0 0" # $ $ $ % & ' ' ' MxMyMz" # $ $ $ % & ' ' ' ! M " Mx+ jMy! dM dt = d dt Mx+ iMy( )= " j#B0MUseful to define!Mx!jMy!! M(t) = M(0)e" j#B0t= M(0)e" j$0tSolution is a time-varying phasor !Question: which way does this rotate with time?!TT Liu, BE280A, UCSD Fall 2010!Matrix Form with B=B0 !! dMxdtdMydtdMzdt" # $ $ $ % & ' ' ' =(1/T2)B00()B01/T200 0 (1/T1" # $ $ $ % & ' ' ' MxMyMz" # $ $ $ % & ' ' ' +00M0/T1" # $ $ $ % & ' ' ' TT Liu, BE280A, UCSD Fall 2010!Z-component solution!! Mz(t) = M0+ (Mz(0) " M0)e"t /T1! If Mz(0) = 0 then Mz(t) = M0(1" e"t /T1)Saturation Recovery!Inversion Recovery!! If Mz(0) = "M0 then Mz(t) = M0(1" 2e"t /T1)5!TT Liu, BE280A, UCSD Fall 2010!Transverse Component!! M " Mx+ jMy! dM dt = d dt Mx+ iMy( )= " j#0+ 1/T2( )M! M(t) = M(0)e" j#0te"t /T2TT Liu, BE280A, UCSD Fall 2010!Summary!1) Longitudinal component recovers exponentially.!2) Transverse component precesses and decays exponentially.!Source: http://mrsrl.stanford.edu/~brian/mri-movies/ !TT Liu, BE280A, UCSD Fall 2010!Summary!1) Longitudinal component recovers exponentially.!2) Transverse component precesses and decays exponentially.!Fact: Can show that T2< T1 in order for |M(t)| ≤ M0 Physically, the mechanisms that give rise to T1 relaxation also contribute to transverse T2 relaxation. !TT Liu, BE280A, UCSD Fall 2010!Gradients!Spins precess at the Larmor frequency, which is proportional to the local magnetic field. In a constant magnetic field Bz=B0, all the spins precess at the same frequency (ignoring chemical shift). !!Gradient coils are used to add a spatial variation to Bz such that Bz(x,y,z) = B0+Δ Bz(x,y,z) . Thus, spins at different physical locations will precess at different frequencies. !!6!TT Liu, BE280A, UCSD Fall 2010!Simplified Drawing of Basic Instrumentation. Body lies on table encompassed by coils for static field Bo, gradient fields (two of three shown), and radiofrequency field B1. MRI System!Image, caption: copyright Nishimura, Fig. 3.15 TT Liu, BE280A, UCSD Fall 2010!Imaging: localizing the NMR signal!Resonant Frequency: !#ν(x) = γB0+γΔB(x) !!!RF and Gradient Coils!The local precession frequency can be changed in a position-dependent way by applying linear field gradients!ΔB(x)!x!Credit: R. Buxton!TT Liu, BE280A, UCSD Fall 2010!Gradient Fields!! Bz(x, y,z) = B0+"Bz"xx +"Bz"yy +"Bz"zz= B0+ Gxx + Gyy + Gzzz!! Gz="Bz"z> 0! Gy="Bz"y> 0yTT Liu, BE280A, UCSD Fall 2010!Interpretation!∆Bz(x)=Gxx!Spins Precess at!at γB0+ γGxx!(faster)!!!Spins Precess !at γB0- γGxx!(slower)!!x!Spins Precess at γB0!7!TT Liu, BE280A, UCSD Fall 2010!Rotating Frame of Reference!Reference everything to the magnetic field at isocenter. !TT Liu, BE280A, UCSD Fall 2010!Spins!There is nothing that nuclear spins will not do for you, as long as you treat them as human beings. !Erwin Hahn!TT Liu, BE280A, UCSD Fall 2010!Phasors!! "= 0! "= #$/2! "=#! "=#/2TT Liu, BE280A, UCSD Fall 2010!Phasor Diagram!Real!Imaginary!! "= #2$kxx! G(kx) = g(x)exp " j2#kxx( )"$$%dx! "= #2$kxx! "= 0! kx= 1; x = 02"kxx = 0! "= #$/2! x = 1/ 42"kxx ="/2! "=#! x = 1/22"kxx ="! "=#/2! x = 3 /22"kxx = 3"/ 48!TT Liu, BE280A, UCSD Fall 2010!Interpretation!∆x! 2∆x!-∆x!-2∆x!


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UCSD BENG 280A - MRI Lecture 2

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