1TT. Liu, BE280A, UCSD Fall 2007Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2007MRI Lecture 1TT. Liu, BE280A, UCSD Fall 2007Topics• The concept of spin• Precession of magnetic spin• Relaxation• Bloch EquationTT. Liu, BE280A, UCSD Fall 2007Spin• Intrinsic angular momentum of elementaryparticles -- electrons, protons, neutrons.• Spin is quantized. Key concept in QuantumMechanics.TT. Liu, BE280A, UCSD Fall 2007The History of Spin• 1921 Stern and Gerlach observed quantization ofmagnetic moments of silver atoms• 1925 Uhlenbeck and Goudsmit introduce theconcept of spin for electrons.• 1933 Stern and Gerlach measure the effect ofnuclear spin.• 1937 Rabi predicts and observes nuclear magneticresonance.2TT. Liu, BE280A, UCSD Fall 2007Classical Magnetic MomentIA ! r µ = IAˆ n TT. Liu, BE280A, UCSD Fall 2007Energy in a Magnetic FieldLorentz ForceB ! E = "r µ • B= "µzBMinimum Energy StateMaximum Energy StateTT. Liu, BE280A, UCSD Fall 2007Stern-Gerlach ExperimentImage from http://library.thinkquest.org/19662/high/eng/exp-stern-gerlach.html?tqskip=1TT. Liu, BE280A, UCSD Fall 2007Force in a Field Gradient! F = "#E =µz$Bz$zDeflected upDeflected downIncreasingvertical B-field.3TT. Liu, BE280A, UCSD Fall 2007Stern-Gerlach ExperimentImage from http://library.thinkquest.org/19662/high/eng/exp-stern-gerlach.html?tqskip=1TT. Liu, BE280A, UCSD Fall 2007Quantization of Magnetic MomentThe key finding of the Stern-Gerlach experiment is thatthe magnetic moment isquantized. That is, it can onlytake on discrete values.In the experiment, the findingwas that the component ofmagnetization along thedirection of the applied field wasquantized: µz = + µ0 OR - µ0http://www.le.ac.uk/biochem/mp84/teachingTT. Liu, BE280A, UCSD Fall 2007Magnetic Moment and Angular MomentumA charged sphere spinning about its axishas angular momentum and a magnetic moment.This is a classical analogy that is useful forunderstanding quantum spin, but remember thatit is only an analogy!Relation: µ = γ S where γ is the gyromagnetic ratio andS is the spin angular momentum.TT. Liu, BE280A, UCSD Fall 2007Quantization of Angular MomentumBecause the magnetic moment is quantized, so is theangular momentum.In particular, the z-component of the angular momentumIs quantized as follows: ! Sz= mshms" #s,#(s #1),....s{ }s is an integer or half integer4TT. Liu, BE280A, UCSD Fall 2007Nuclear Spin Rules2H1H, 23Na, 31P17O12C, 16OExamples j/2EvenOdd jOddOdd j/2OddEven 0EvenEven SpinNumber ofNeutronsNumber ofProtonsTT. Liu, BE280A, UCSD Fall 2007Hydrogen Proton ! Spin 1/2Sz=+h /2"h /2# $ % & µz=+'h /2"'h /2# $ % & TT. Liu, BE280A, UCSD Fall 2007Magnetic Field Units! 1 Tesla = 10,000 GaussEarth's field is about 0.5 Gauss 0.5 Gauss = 0.5x10-4 T = 50 µTTT. Liu, BE280A, UCSD Fall 2007Boltzmann DistributionB0ΔΕ = γhΒ0Ε = −µzΒ0Ε = µzΒ0Number Spins DownNumber Spins Up= exp(-ΔE/kT)Ratio = 0.999990 at 1.5T !!!Corresponds to an excess of about 10 up spins per million5TT. Liu, BE280A, UCSD Fall 2007Equilibrium Magnetization ! M0= Nµz= Nnupµz( )" ndownµz( )N# $ % & ' ( = NµeµzB / kT( )" e"µzB / kT( )eµzB / kT( )+ e"µzB / kT( )) Nµz2B /(kT)= N*2h2B /(4kT)N = number of nuclear spins per unit volumeMagnetization is proportional to applied field. TT. Liu, BE280A, UCSD Fall 20077T Human imager at U. Minn.3T Human imager at UCSD.7T Rodent Imager at UCSD9.4T Human imager at UICBigger is betterTT. Liu, BE280A, UCSD Fall 2007Gyromagnetic Ratios 75 mM17.25 1.1311/2 31P 80 mM11.27 2.2163/2 23Na 88 M42.58 2.7931/2 1HAbundance γ/(2π)(MHz/Tesla)MagneticMomentSpinNucleusSource: Haacke et al., p. 27TT. Liu, BE280A, UCSD Fall 2007TorqueBµNN = µ x BTorqueFor a non-spinning magnetic moment, the torque will try to align the moment with magnetic field (e.g. compass needle)6TT. Liu, BE280A, UCSD Fall 2007PrecessionN = µ x B= NdSdtµ = γ S= µ x B= µ x γBdµdtTorqueChange in Angular momentumRelation betweenmagnetic moment andangular momentumdSdtTT. Liu, BE280A, UCSD Fall 2007PrecessionBµ= µ x γBdµdtdµAnalogous to motion of a gyroscopePrecesses at an angular frequency ofω = γ ΒThis is known as the Larmor frequency.http://www.astrophysik.uni-kiel.de/~h haertel/mpg_e/gyros_free.mpgTT. Liu, BE280A, UCSD Fall 2007Larmor Frequencyω = γ Βf = γ Β / (2 π)Angular frequency in rad/secFrequency in cycles/sec or Hertz, Abbreviated HzFor a 1.5 T system, the Larmor frequency is 63.86 MHzwhich is 63.86 million cycles per second. For comparison,KPBS-FM transmits at 89.5 MHz. Note that the earth’s magnetic field is about 50 µΤ, so thata 1.5T system is about 30,000 times stronger. TT. Liu, BE280A, UCSD Fall 2007Notation and Units! 1 Tesla = 10,000 GaussEarth's field is about 0.5 Gauss 0.5 Gauss = 0.5x10-4 T = 50 µT"= 26752 radians/second/Gauss"="/2#= 4258 Hz/Gauss = 42.58 MHz/Tesla7TT. Liu, BE280A, UCSD Fall 2007Recap• Spins: angular momentum and magnetic momentare quantized.• Spins precess about a static field at the Larmorfrequency.• In MRI we work with the net magnetic moment.• In the presence of a static field and non-zerotemperature, the equilibirum net magnetic momentis aligned with the field (longitudinal), sincetransverse components cancel out.• We will use an radiofrequency pulse to tip thislongitudinal component into the transverse plane.TT. Liu, BE280A, UCSD Fall 2007Magnetization Vector! M =1Vµiprotons in V"! dMdt="M # BVector sum of the magneticmoments over a volume.For a sample at equilibrium in amagnetic field, the transversecomponents of the moments cancelout, so that there is only alongitudinal component.Equation of motion is the sameform as for individual moments.http://www.easymeasure.co.uk/principlesmri.aspxTT. Liu, BE280A, UCSD Fall 2007Simplified Drawing of Basic Instrumentation.Body lies on table encompassed bycoils for static field Bo, gradient fields (two of three shown), and radiofrequency field B1.Image, caption: copyright Nishimura, Fig. 3.15TT. Liu, BE280A, UCSD Fall 2007RF ExcitationFrom Levitt, Spin Dynamics, 20018TT. Liu, BE280A, UCSD Fall 2007RF ExcitationImage & caption: Nishimura, Fig. 3.2B1 radiofrequency field tuned toLarmor frequency and applied intransverse (xy) plane inducesnutation (at Larmor frequency) ofmagnetization vector as it tipsaway from the z-axis. - lab frame of referenceAt equilibrium, net magnetizaionis parallel to the main magneticfield. How do we tip themagnetization away
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