## And Techniques Of Cardiac

2.1. Magnetic Resonance

Felix Bloch and Edward Purcell discovered independently, in 1946 (1,2), that nuclei with a magnetic dipole moment gave rise to a resonance phenomenon when immersed in a magnetic field and subjected to a pulsed or continuous radiofrequency excitation. MR is based on the interaction of nuclear magnetic moments with externally applied magnetic fields, both static fields and time-varying fields. For the sake of simplicity, we can think of atomic nuclei having an intrinsic angular momentum called spin because these are microscopic magnetic dipoles. Immersion of atoms with a nuclear magnetic dipole moment in a magnetic field, in the absence of other interactions or disturbances, would cause alignment of the magnetic dipoles along the direction of the applied magnetic field, similar to what happens to a compass needle placed in a magnetic field. The interaction of a nuclear magnetic dipole with an external magnetic field can be considered quite weak (e.g., in comparison to the

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Fig. 1. A single magnetic dipole moment in a static magnetic field of strength B0. It is customary to align the z-axis of a rectangular coordinate system with the direction of the externally applied static magnetic field B0. In this example, the magnetic moment, initially aligned with the applied magnetic field, was tipped away from the z-direction by an angle a through application of an oscillating magnetic field (not shown). The oscillating magnetic field is kept on only for the time necessary to tip the magnetic dipole moment by a certain angle, a in this example. After turning the oscillating magnetic field off, the magnetic dipole moment precesses about the B0 direction at a frequency vL = y B0, where y is a constant, the gyromagnetic ratio, and represents a property of the nucleus. For 1H nuclei, y equals 42.6 MHz/ T. The angle <p denotes the phase angle of the magnetization component in the x-y plane, orthogonal to the direction of B0.

Fig. 1. A single magnetic dipole moment in a static magnetic field of strength B0. It is customary to align the z-axis of a rectangular coordinate system with the direction of the externally applied static magnetic field B0. In this example, the magnetic moment, initially aligned with the applied magnetic field, was tipped away from the z-direction by an angle a through application of an oscillating magnetic field (not shown). The oscillating magnetic field is kept on only for the time necessary to tip the magnetic dipole moment by a certain angle, a in this example. After turning the oscillating magnetic field off, the magnetic dipole moment precesses about the B0 direction at a frequency vL = y B0, where y is a constant, the gyromagnetic ratio, and represents a property of the nucleus. For 1H nuclei, y equals 42.6 MHz/ T. The angle <p denotes the phase angle of the magnetization component in the x-y plane, orthogonal to the direction of B0.

Fig. 2. The transverse magnetization component of a nuclear dipole precesses at the Larmor frequency and produces an oscillating magnetic flux density that can be detected with a wire loop that is part of a resonant circuit. The induced voltage is amplified and mixed with the signal of an oscillator. The low-frequency component from the mixer is a free induction decay with frequency vL - v0. Often, two coils, oriented perpendicular to each other, are used to detect the signal from the Mx and My components of the transverse magnetization, which are in quadrature; that is, they have a relative phase difference of 90°. By detection of the quadrature components, it is possible to determine the sign of the difference vL - v0, and by combination of the two signals, after phase shifting one by 90°, one improves the signal-to-noise ratio by a factor of J 2. FID, free induction decay; MR, magnetic resonance.

Fig. 2. The transverse magnetization component of a nuclear dipole precesses at the Larmor frequency and produces an oscillating magnetic flux density that can be detected with a wire loop that is part of a resonant circuit. The induced voltage is amplified and mixed with the signal of an oscillator. The low-frequency component from the mixer is a free induction decay with frequency vL - v0. Often, two coils, oriented perpendicular to each other, are used to detect the signal from the Mx and My components of the transverse magnetization, which are in quadrature; that is, they have a relative phase difference of 90°. By detection of the quadrature components, it is possible to determine the sign of the difference vL - v0, and by combination of the two signals, after phase shifting one by 90°, one improves the signal-to-noise ratio by a factor of J 2. FID, free induction decay; MR, magnetic resonance.

thermal energy at room temperature of atoms or molecules that carry some nuclear magnetic moment). Therefore, the detection of the magnetic resonance signal benefits from the application of strong magnetic fields. For current MRI systems, magnetic field strengths ranging from 1 to 3 tesla (T) are typical.

### 2.2. Larmor Precession

A magnetic dipole subjected to a static magnetic field, if tipped away from the direction of the magnetic field, will pre-cess about the direction of the static magnetic field (Fig. 1). This precession has a rotation frequency vL that is directly proportional to the magnetic field strength B0. For hydrogen nuclei, the precession frequency varies with field strength as v L = 42.6 [MHz/Tesla ]• B0 [Tesla ]

The precession frequency is also known as the Larmor frequency.

Tipping a nuclear magnetic moment away from the direction of the z-axis (B0 direction) can be accomplished by applying an oscillating magnetic field, denoted by B1, in a direction perpendicular to B0. The radiofrequency transmitter should be tuned to a frequency close to the Larmor frequency to elicit a resonant excitation. On MRI scanners, the oscillating magnetic fields can be switched on and off for very precise fixed durations, therefore controlling the angle by which the magnetic moment is tipped away from the B0 direction.

It is customary to refer to the magnetic fields oscillating at radiofrequencies and turned on for brief durations as radio-frequency pulses. A pulse that tips the magnetic moment from the z-axis into the x-y plane is referred to as a 90° radiofrequency pulse. A pulse that inverts the orientation of the magnetic moment is called a 180° or inversion pulse.

### 2.3. Free Induction Decays

After a radiofrequency excitation pulse, the static magnetic field B0 causes precession of the transverse magnetization component, which can be detected with an external coil as shown in Fig. 2. Immediately after a radiofrequency excitation, individual magnetic moments that were tipped into the transverse plane are in phase; that is, they have the same phase angle. If all magnetic moments were to precess at exactly the same Larmor frequency, this phase coherence would persist.

Residual magnetic field inhomogeneities, magnetic dipole interactions between neighboring nuclei, molecule-specific shifts of the precession frequency, and other factors produce a distribution of Larmor frequencies. The frequency shifts relative to a reference frequency can be tissue specific, as in the case of 1H nuclei in fat. The spread of Larmor frequencies results in a slow loss of phase coherence of the transverse magnetization; that is, the sum of all transverse magnetization components decays with time.

The decay following a radiofrequency excitation is called free induction decay and often has the shape of an exponential function with an exponential time constant denoted as T2, roughly on the order of approx 0.1-102 ms for 1H nuclei in biological systems. In the laboratory frame of reference, the transverse magnetization can be expressed as a complex quantity Mxy:

Mxy = Mx + i ■ My =[M 0 cos fflLt + iM0 cos ®Lt]' exp(_t /T2) = M0 exp(imLt) • exp(-1 /T2)

where mL = 2kvl, and Mx and My denote the x and y components of the transverse magnetization.

In magnetic resonance research, the useful information extracted by spectroscopic or imaging studies is derived from analysis of the frequency modulation of the MR signal. To detect the frequency-modulated signal, heterodyne detection is used; that is, the received signal is mixed with an oscillating waveform of frequency m0 = 2ft v0, close to the Larmor frequency, to obtain a low-frequency signal that can be digitized. If the Ara denotes the difference between mL = 2kvl and m0, then the mixed-down signal component can be described as

90° pulse

180° pulse

90° pulse

180° pulse

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