Magnetic Resonance Imaging
295
the magnetic field vector H depends on the product of the magnetic field intensity
and the time frame of interaction (the pulse duration). Most commonly, the angles are
perpendicular or antagonistic: α = 90° or α = 180°, which are appropriately referred
to as 90° pulse and 180° pulse, respectively. Note that the radiofrequency coils in
the NMR devices are designed to produce magnetization only in the x–y plane and
only detect magnetization in the x–y plane as well. The key to MR imaging is the
design of pulse sequences, which are applied in order to obtain images with desired
contrast.
If a 90° pulse followed by an 180° pulse produces an entirely different spectral
profile. The detected spectrum will be observed at a characteristic echo time delay,
TE, which is twice the time interval between the two respective pulses. The sequence
of RF pulse delivery and FID acquisition is illustrated in Figure 15.12.
While MRI is often used to create images identifying the anatomical features
of the biological tissues, it has the potential to image metabolic activities. Such a
specialized MR imaging is called functional MRI ( f MRI). f MRI is typically used to
generate maps of brain function and localize regions with increase metabolic activity such as tumors. We will discuss f MRI and its applications after reviewing the
reconstruction methods used to form MR images.
15.4 FORMULATION OF MRI RECONSTRUCTION
An MR image is generally reconstructed using the Fourier slice theorem
described in the previous chapters. During the tomography process, the signals
produced by the inhomogeneity of the tissues are processed using signal processing methods. This process creates an image of the tissue as function of space. As
described in the previous chapters, the computational processes involved in the
Fourier slice theorem are primary performed using the discrete Fourier transform or DFT.
Following the 90° shift in magnetic field gradient, the characteristic FID pulse
can be acquired. At time t = 0, the nuclear spins will be aligned with the magnetization M along the z-direction. This condition is referred to as thermal equilibrium.
During the pulse, the magnetization vector will be tipped into the x–y plane by an
oscillating field: H = H 0 cos(ωt). When the RF pulse is switched off, the magnetization will precess and the measured value in the x–y plane will exhibit a damped
oscillation type signal. The x–y projection of the magnetization as a function of
time is the FID signal. The FID signal is the result of many nuclear spins decaying
simultaneously while observing their projection in the x–y plane, which converges
to zero. This net x–y projection is inherent to the fact that all nuclei will experience
a slightly different magnetic field and will thus experience a different precession
frequency. Additionally, the magnetization will attempt to return to its prior equilibrium that was at zero net magnetization. Using the rotating frame of reference, where
the reference frame rotates with the RF frequency, the magnetization spins will have
a frequency that is slightly off from this RF frequency. By measuring the difference
in frequency, the detection process is simplified dramatically. The magnetization
can now be written for both the x–y and z-projections using the time constant of the
295
the magnetic field vector H depends on the product of the magnetic field intensity
and the time frame of interaction (the pulse duration). Most commonly, the angles are
perpendicular or antagonistic: α = 90° or α = 180°, which are appropriately referred
to as 90° pulse and 180° pulse, respectively. Note that the radiofrequency coils in
the NMR devices are designed to produce magnetization only in the x–y plane and
only detect magnetization in the x–y plane as well. The key to MR imaging is the
design of pulse sequences, which are applied in order to obtain images with desired
contrast.
If a 90° pulse followed by an 180° pulse produces an entirely different spectral
profile. The detected spectrum will be observed at a characteristic echo time delay,
TE, which is twice the time interval between the two respective pulses. The sequence
of RF pulse delivery and FID acquisition is illustrated in Figure 15.12.
While MRI is often used to create images identifying the anatomical features
of the biological tissues, it has the potential to image metabolic activities. Such a
specialized MR imaging is called functional MRI ( f MRI). f MRI is typically used to
generate maps of brain function and localize regions with increase metabolic activity such as tumors. We will discuss f MRI and its applications after reviewing the
reconstruction methods used to form MR images.
15.4 FORMULATION OF MRI RECONSTRUCTION
An MR image is generally reconstructed using the Fourier slice theorem
described in the previous chapters. During the tomography process, the signals
produced by the inhomogeneity of the tissues are processed using signal processing methods. This process creates an image of the tissue as function of space. As
described in the previous chapters, the computational processes involved in the
Fourier slice theorem are primary performed using the discrete Fourier transform or DFT.
Following the 90° shift in magnetic field gradient, the characteristic FID pulse
can be acquired. At time t = 0, the nuclear spins will be aligned with the magnetization M along the z-direction. This condition is referred to as thermal equilibrium.
During the pulse, the magnetization vector will be tipped into the x–y plane by an
oscillating field: H = H 0 cos(ωt). When the RF pulse is switched off, the magnetization will precess and the measured value in the x–y plane will exhibit a damped
oscillation type signal. The x–y projection of the magnetization as a function of
time is the FID signal. The FID signal is the result of many nuclear spins decaying
simultaneously while observing their projection in the x–y plane, which converges
to zero. This net x–y projection is inherent to the fact that all nuclei will experience
a slightly different magnetic field and will thus experience a different precession
frequency. Additionally, the magnetization will attempt to return to its prior equilibrium that was at zero net magnetization. Using the rotating frame of reference, where
the reference frame rotates with the RF frequency, the magnetization spins will have
a frequency that is slightly off from this RF frequency. By measuring the difference
in frequency, the detection process is simplified dramatically. The magnetization
can now be written for both the x–y and z-projections using the time constant of the