• 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 = "µ B I z
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
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TT. Liu, BE280A, UCSD Fallwww.qi-whiz.com/images/ 2010 Earth-magnetic-field.jpg TT. Liu, BE280A, UCSD Fall 2010
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! 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
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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
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