Biomedical Engineering Reference
In-Depth Information
important since continuum mechanics constitute the mathematical framework of
the finite element method, extensively used in Chaps. 5 , 6 and 7 for numerical
simulation.
3.1.2 Fundamental Equations
The terms T 1 and T 2 , as weighting terms used in MRI, are discussed by introducing
basic mathematical relations.
In a static magnetic field, nuclear magnetic resonance permits non-invasive
body tissue imaging. To avoid a misleading association with radioactivity the
technique is commonly termed magnetic resonance or MRI. The diagnostic
principle relies on the influence of an external magnetic radiofrequency field on
the magnetic alignment of the atomic nuclei of the 1 H-isotope, which occurs in
living tissue. Three-dimensional images can be obtained. Atomic nuclei, like those
of the 1 H-atom, with an odd atomic number (odd number of protons, neutrons or
both) possess a fundamental intrinsic physical property, the intrinsic angular
momentum or spin angular momentum or simply spin for short. The spin s of the
nucleus is a result of the spin of its nucleons, i.e. protons and neutrons, and their
constituent parts, quarks and gluons, respectively. Due to s, i.e. the intrinsic
spinning motion of a charged particle, which is independent of its motion in space
(cf. e.g. ''electron orbital motion''), an atomic current is produced (in a closed
circuit of atomic dimensions). Elementary particles and nuclei behave as if they
possess a magnetic dipole moment, commonly denoted l, oriented in the direction
of the spin axis. The magnetic dipole moment is proportional to the intrinsic
angular momentum spin, l s where the proportionality constant is given through
the gyromagnetic ratio c:
s ¼ 1
l ¼ c s
and
c l :
ð 3 : 1 Þ
The direct relationship ( 3.1 ) is empirically deduced since the value of c is a
constant for any particular type of elementary particle or nucleus. The differences
in c-factors make electrons unusable for MR-imaging. Since the c-factor also
depends on the particle mass and the proton mass is more than three orders of
magnitude larger than the mass of the electron, a much higher frequency, and thus
higher energy, would be needed to change the initial alignment of electron mag-
netic dipoles.
In ( 3.1 ), as an experimental fact (e.g. S TERN -G ERLACH -experiment), the angular
momentum spin is not spread continuously, since in quantum mechanics, it is
quantized, i.e. it cannot take any indiscriminate value but only discrete multiples
of h ; where h is the reduced Planck's constant. Equation ( 3.1 ) thus yields
with k s q h
p
l ¼ c s q
s ð s þ 1 Þ
ð 3 : 2 Þ
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