Spin • Intrinsic angular momentum of elementary Bioengineering 280A" Principles of Biomedical Imaging" particles -- electrons, protons, neutrons. " • Spin is quantized. Key concept in Quantum Fall Quarter 2010" Mechanics. MRI Lecture 1" TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 Magnetic Moment and " Angular Momentum Nuclear Spin Rules A charged sphere spinning about its axis has angular momentum and a magnetic moment. Number of Number of Spin Examples Protons Neutrons This is a classical analogy that is useful for understanding quantum spin, but remember that Even Even 0 12C, 16O it is only an analogy! Even Odd j/2 17O Odd Even j/2 1H, 23Na, 31P Relation: µ = γ S where γ is the gyromagnetic ratio and S is the spin angular momentum. Odd Odd j 2H TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 1 Classical Magnetic Moment Energy in a Magnetic Field Maximum Energy State r ! E = "µ • B A µ = IAnˆ B I = "µzB Lorentz Force Minimum Energy State TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 ! Energy in a Magnetic Field Magnetic Field Units 1 Tesla = 10,000 Gauss Earth's field is about 0.5 Gauss 0.5 Gauss = 0.5x10-4 T = 50 µT ! TT. Liu, BE280A, UCSD Fallwww.qi-whiz.com/images/ 2010 Earth-magnetic-field.jpg TT. Liu, BE280A, UCSD Fall 2010 2 ! Boltzmann Distribution Equilibrium Magnetization M N 0 = µ z 2 " Nµz B /(kT) 2 2 = N# ! B /(4kT) B N = number of nuclear 0 ! spins per unit volume Magnetization is proportional to applied field. TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 Hansen 2009 Bigger is better Torque B µ For a non-spinning magnetic moment, the torque will try to align the moment with 3T Human imager at UCSD. 7T Human imager at U. Minn. magnetic field (e.g. compass needle) N N = µ x B Torque 7T Rodent Imager at UCSD 9.4T Human imager at UIC TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 3 Precession Precession Analogous to motion of a gyroscope dµ Torque = µ x γB dt Precesses at an angular frequency of N = µ x B B dS ω = γ Β = µ x B dS dt = N dµ This is known as the Larmor frequency. dt = µ x γB dt dµ µ Change in µ = γ S Angular momentum Relation between magnetic moment and angular momentum http://www.astrophysik.uni-kiel.de/~hhaertełmpg_e/gyros_free.mpg TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 Magnetization Vector Gyromagnetic Ratios Vector sum of the magnetic moments over a volume. For a sample at equilibrium in a magnetic field, the transverse Nucleus Spin Magnetic γ/(2π) Abundance components of the moments cancel Moment (MHz/Tesla) out, so that there is only a longitudinal component. 1H 1/2 2.793 42.58 88 M Equation of motion is the same form as for individual moments. 1 23Na 3/2 2.216 11.27 80 mM M = "µi V protons in V 31P 1/2 1.131 17.25 75 mM dM = "M # B dt Hansen 2009 Source: Haacke et al., p. 27 ! TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 ! 4 Larmor Frequency Notation and Units ω = γ Β Angular frequency in rad/sec 1 Tesla = 10,000 Gauss Earth's field is about 0.5 Gauss f = γ Β / (2 π) Frequency in cycles/sec or Hertz, Abbreviated Hz 0.5 Gauss = 0.5x10-4 T = 50 µT For a 1.5 T system, the Larmor frequency is 63.86 MHz which is 63.86 million cycles per second. For comparison, " = 26752 radians/second/Gauss KPBS-FM transmits at 89.5 MHz. " = " /2# = 4258 Hz/Gauss Note that the earth’s magnetic field is about 50 µΤ, so that a 1.5T system is about 30,000 times stronger. = 42.58 MHz/Tesla TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 ! RF Excitation 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. Image, caption: copyright Nishimura, Fig. 3.15 From Levitt, Spin Dynamics, 2001 TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 5 RF Excitation RF Excitation At equilibrium, net magnetizaion is parallel to the main magnetic field. How do we tip the magnetization away from equilibrium? B radiofrequency field tuned to Image & caption: Nishimura, Fig. 3.2 1 Larmor frequency and applied in transverse (xy) plane induces nutation (at Larmor frequency) of magnetization vector as it tips away from the z-axis. - lab frame of reference http://www.drcmr.dk/main/content/view/213/74/ http://www.eecs.umich.edu/%7EdnolłBME516/ TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 On-Resonance Excitation RF Excitation Hanson 2009 http://www.eecs.umich.edu/%7EdnolłBME516/ TT. Liu, BE280A, UCSD Fall 2010 http://www.drcmr.dk/JavaCompass/ TT. Liu, BE280A, UCSD Fall 2010 6 Rotating Frame of Reference Reference everything to the magnetic field at isocenter. a) Laboratory frame behavior of M b) Rotating frame behavior of M Images & caption: Nishimura, Fig. 3.3 TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 http://www.eecs.umich.edu/%7EdnolłBME516/ RF Excitation Free Induction Decay (FID) From Buxton 2002 http://www.easymeasure.co.uk/principlesmri.aspx TT. Liu, BE280A, UCSD Fall 2010 TT. Liu, BE280A, UCSD Fall 2010 7 .