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EDUCATION Revista Mexicana de F´ısica E 63 (2017) 48–55 JANUARY–JULY 2017

Quantum-mechanical aspects of magnetic imaging

J.A. Soto, T. Cordova,´ and M. Sosa Division´ de Ciencias e Ingenier´ıas Campus Leon,´ Universidad de Guanajuato Loma del bosque no. 103, Col. Lomas del Campestre, Leon,´ Guanajuato, Mexico,´ phone:(+52)477-7885100 email: jasa@fisica.ugto.mx; theo@fisica.ugto.mx; modesto@fisica.ugto.mx S. Jerez Centro de Investigacion´ en Matematicas´ A.C. Jalisco S/N, Col. Valenciana, Guanajuato, Guanajuato, Mexico,´ phone: (+52)473-7327155 email: [email protected] Received 7 March 2016; accepted 23 September 2016

The Magnetic Resonance Imaging (MRI) is a non-invasive technique which uses the physical phenomenon of nuclear magnetic resonance to obtain structural and compositional information about human body regions. In this imaging study we use the - and a powerful static magnetic field, which aligns the magnetization of hydrogen nuclei. Nowadays there are many types of clinical equipment that conduct MRI studies, which have intensities of magnetic fields from 0.2T to 7.0T. Moreover, liquid is required for the superconducting coil. This paper presents an analysis of the magnetic resonance phenomenon; by doing a review of the quantum-mechanical aspects as the spin and Zeeman effect.

Keywords: Magnetic resonance; magnetic field; spin; magnetization; voxel.

PACS: 01.40.Fk; 03.65.Ca; 41.20.Gz

1. Introduction are fermions and are subject to the Pauli exclusion principle, being in a crystal lattice, the energy between them Nowadays, there are many techniques to observe and analyze becomes negative (attractive) so that pairs are created to min- the inside of the human body in order to obtain a better diag- imize the energy and behave as bosons. nosis. One non-invasive and high-resolution technique is the In this paper we discuss the basic physics involving mag- Magnetic Resonance Imaging (MRI), which takes advantage netic resonance, providing an analysis from the viewpoint of of hydrogen nuclei, a powerful static magnetic field, and a quantum . computer system to process and get images. The Nuclear Magnetic Resonance is a physical phe- 2. The Hydrogen nomenon in which certain atomic nuclei (with an odd num- ber of protons or ) are placed under a high-intensity MRI uses the properties of hydrogen nuclei when they are magnetic field and can selectively absorb energy from elec- exposed to a high magnetic field and a radio-frequency field, tromagnetic in the radio-frequency range. Once the so it is important to analyze the physical characteristics of the nuclei have absorbed the energy, the excess energy returns to hydrogen atom [11]. the surroundings through a process called relaxation which is The hydrogen atom is the simplest atom since, there is a accompanied by a local magnetic variation, which induces a proton in its core and an orbiting it experimenting an signal to a receiving for digital processing and thus attractive Coulomb potential. obtain an image or to perform a spectrometric analysis [6]. e2 V (r) = − . (1) MRI equipment consists of a magnet (usually supercon- 4π²0r ductor), radio frequency coils, magnetic field gradients, a To perform the analysis, the Schrodinger¨ equation indepen- bore or tunnel, and a computer for signal processing. MRI dent of time is used: requires the use of a high-intensity magnetic field. Clinical µ ¶ p2 equipment use field strengths ranging from 0.5-3.0 T, which HΨ(r) = + V (r) Ψ(r) = EΨ(r), (2) have been achieved by replacing the permanent magnets by 2µ superconducting electromagnets resulting in a very wide line where µ is the reduced of the system and p, the momen- research. BCS theory [12–14] is the dominant physical the- tum operator ory of superconductivity and was proposed by John Bardeen, m · m µ = p e , (3) Leon Cooper, and Robert Schrieffer. The theory is based on mp + me the fact that the charge carriers are not free electrons but, rather, pairs of electrons known as Cooper pairs. Although p = −i~ 5 . (4) QUANTUM-MECHANICAL ASPECTS OF MAGNETIC RESONANCE IMAGING 49

Because there is a central potential V (r), natural coordinates multiplying the above equation by sin2θ: are spherical coordinates, and Eq. (2) becomes: sen2θ d2Θ senθcosθ dΘ · µ ¶ + ~2 ∂ ∂ Θ(θ) dθ2 Θ(θ) dθ − sen θ r2 2µr2sen θ ∂r ∂r 1 d2Φ µ ¶ ¸ + = −l(l + 1)sen2θ, (12) ∂ ∂ 1 ∂2 Φ(φ) dφ2 + sen θ + Ψ(r) ∂θ ∂θ sen θ ∂φ2 separating the azimuth part, e2 − Ψ(r) = EΨ(r), (5) 1 d2Φ 4π²0r = −m. (13) Φ(φ) dφ2 re-arranging the equation above: The solution to Eq. (13) is given as: 2 µ 2 ¶ 2 ∂ Ψ ∂Ψ 2µ re 2 r 2 + 2r + 2 + Er Ψ −imφ imφ ∂r ∂r ~ 4π²0 Φ(φ) = Ae + Be , (14)

∂2Ψ cosθ ∂Ψ 1 ∂2Ψ where A and B are constants. Equation (12) is as follows: + 2 + + 2 2 = 0. (6) ∂θ senθ ∂θ sin θ ∂φ sen2θ d2Θ senθcosθ dΘ + − m2 = −l(l + 1)sen2θ, (15) To solve Eq. (6) the separation of variables is: Θ(θ) dθ2 Θ(θ) dθ

Ψ(r, θ, φ) = R(r)Y (θ, φ), (7) multiplying the equation by Θ(θ) we get:

d2Θ dΘ replacing the solution proposed in Eq. (6) and dividing by sen2θ + senθcosθ R(r)Y (θ, φ) we obtain: dθ2 dθ £ 2 2¤ 2 2 µ 2 ¶ + l(l + 1)sen θ − m Θ(θ) = 0. (16) r ∂ R(r) 2r ∂R(r) 2µ re 2 2 + + 2 + Er R(r) ∂r R(r) ∂r ~ 4π²0 Making the following change in the Eq. (16) cos θ → x and 1 ∂2Y (θ, φ) cosθ ∂Y (θ, φ) Θ → y, to get: = − − Y (θ, φ) ∂θ2 Y (θ, φ)senθ ∂θ dΘ dΘ = −senθ ; 1 ∂2Y (θ, φ) dθ dx − . (8) Y (θ, φ)sen2θ ∂φ2 d2Θ d2Θ dθ = sen2θ − cosθ ; dθ2 dx2 dx We can separate the Eq. (8) into two with a separation con- 2 2 stant. For reasons that will eventually become clear, we will sen θ = 1 − x . write this separation constant as l(l + 1), i.e. Performing the above changes, Eq. (16) is written as: 1 ∂2Y (θ, φ) cosθ ∂Y (θ, φ) · ¸ + d2y dx m2 Y (θ, φ) ∂θ2 Y (θ, θ) θ ∂θ (1 − x2) − 2x + l(l + 1) − y = 0. (17) sen dx2 dy 1 − x2 1 ∂2Y (θ, φ) + = −l(l + 1), (9) Equation (17) is known as Associated Legendre Differential Y (θ, φ)sen2θ ∂φ2 Equation, and also reveals why we chose the constant above r2 ∂2R(r) 2r ∂R(r) to be l(l+1). That is, its solutions are given by the Associated + R(r) ∂r2 R(r) ∂r Legendre Polynomials: µ ¶ 2µ re2 m l+m + + Er2 = l(l + 1). (10) m (−1) 2 m/2 d 2 l 2 Pl (x) = (1 − x ) (x − 1) . (18) ~ 4π²0 2ll! dxl+m For the solution of Eq. (9) the separation of variables method The solution Y (θ, φ) = Θ(θ) · Φ(φ), comprising the angular is used again, this time making Y (θ, φ) = Θ(θ) · Φ(φ), to part of the Eq. (8) is of the form: get: m imφ Yl,m(θ, φ) = NPl (cosθ)e ; (19) 1 d2Θ cosθ dΘ + Θ(θ) dθ2 Θ(θ)senθ dθ where N is a normalization constant defined as: s 2 1 d Φ (2l + 1)!(l − 1)! + = −l(l + 1); (11) N = , (20) Φ(φ)sen2θ dφ2 4π(l + 1)!

Rev. Mex. Fis. E 63 (2017) 48–55 50 J.A. SOTO, T. CORDOVA,´ M. SOSA, AND S. JEREZ and thus it ensures that the functions Yl,m(θ, φ) are orthonor- 3. Nuclear magnetic resonance mal. Equation (19) is known as Spherical . Re- turning to the radial part of Schrodinger¨ Eq. (10), rearranging Nuclear magnetic resonance is a physical phenomenon asso- it to be: ciated with the intrinsic of the spin and the magnetic properties of atomic nuclei. When a nucleus d2R(r) 2 dR is placed in a magnetic field an interaction occurs between + dr2 r dr the of the nucleus and the field, resulting · µ ¶ ¸ 2µ e2 l(l + 1) in an energy splitting. By the absorption and emission of + + E − R(r) = 0, (21) photons with the right frequency, transitions between these ~2 4π² r r2 0 energy states may occur. whole solution is given by the Associated Laguerre Polyno- Local magnetic changes produced by the absorption and mials, which meet the following normalization condition: emission of photons are detected by an antenna that sends the signal to a computer for decoding and image generating. Z∞ 2n [(n + l)!]3 e−ρρ2l[L2l+1(ρ)]2ρ2dρ = ; (22) 3.1. Nuclear Spin n+l (n − l − 1)! 0 The particles that make up the atomic nuclei (protons and neutrons) have the intrinsic quantum mechanical property of so the solution of Eq. (8) for the radial part is expressed as: spin. In nuclear physics the total angular momentum that s the nucleus has is called nuclear spin [18], though the term (n − l − 1)! R(r) = should not be confused with spin of each or total 2n[(n + l)!]3 spin as the sum of all , because this is just one of the

µ ¶3/2 two contributions to the nuclear spin, the other is the angular 2 −ρ/2 l 2l+1 momentum of the nucleons. So each nucleon will have a net × e ρ Ln−l−1(ρ), (23) na0 angular momentum such that:

2 where ρ = 2r/na0 y a0 = ~/me . ji = li + si, (26) Since we have the solutions of each of the equations de- then the nuclear spin will be: pending only on one variable, we proceed to build the exact X X solution of the Schrodinger¨ equation independent of time for J = ji = (li + si), (27) the hydrogen atom, i i s or just: µ ¶3/2 (n − l − 1)! 2 = + . Ψ(r, θ, φ) = J L S (28) 2n[(n + l)!]3 na 0 Each of these vectors have a similar quantum number. The −ρ/2 l 2l+1 magnitude of orbital angular momentum satisfies: × e ρ Ln−l−1(ρ)Yl,m(θ, φ); (24) L2 = l(l + 1)~2 con l = 0, 1, 2,... (29) explicitly writing in terms of the spherical harmonics: The direction of the angular momentum vector L is associ- s s µ ¶ (2l + 1)!(l − 1)! (n − l − 1)! 2 3/2 ated with the quantum number m. For a given quantum or- Ψ(r, θ, φ) = bital momentum number l, there are 2l + 1 integral magnetic 4π(l + 1)! 2n[(n + l)!]3 na 0 quantum numbers m ranging from −l to l. Along the z axis −ρ/2 l 2l+1 m imφ the component of angular momentum is given as: × e ρ Ln−l−1(ρ)Pl (cosθ)e , (25)

Lz = ml~. (30) where the principal, azimuthal, and magnetic (n, l, m) quan- tum numbers take the following values: For the vector of spin angular momentum S, its quantum number s can take integer or half-integer values: n = 1, 2, 3,...

l = 0, 1, 2, . . . , n − 1 S2 = s(s + 1)~2 con s = 0, 1/2, 1, 3/2,... (31)

m = −l, −l + 1,..., 0, . . . , l − 1, l and for the z component, we find that sz = ms~, there are 2s + 1 values of ms. To exhibit the property of MRI, the The function Ψ(r, θ, φ) by itself has no physical nucleus must have a non-zero value of s. In medical applica- 2 meaning, but the expression |Ψ| calculated for a place and tions, the proton (1H) is the nucleus of most interest, due to a given time is proportional to the probability of finding an its high natural abundance, however other nuclei have been electron in that place, at that instant. studied.

Rev. Mex. Fis. E 63 (2017) 48–55 QUANTUM-MECHANICAL ASPECTS OF MAGNETIC RESONANCE IMAGING 51

The proton is a fermion whose value of s is 1/2, there- fore there will be two possible values for Sz, i.e. ±~/2. The eigenfunctions describing the nucleus can be written as | + 1/2i and | − 1/2i, and since in every physical observable has an associated operator, we can write an eigenvalue equation to describe the observation of the spin state as: Sz|msi = ms|msi, (32) where Sz is the operator describing the measurement of an- gular momentum along the z axis.

3.2. Voxel magnetization

To measure the energy of the spin system it is necessary to construct a Hamiltonian operator. The shape of the Hamilto- nian can be derived from classical electromagnetism, for the energy of a magnetic moment placed in a magnetic field [15]. The nuclei have a magnetic moment ~µ which is proportional to the spin angular momentum [4]: FIGURE 1. Energy level diagram for a Zeeman interaction.

~µ = γS, (33) Thus obtaining the Larmor equation which underpins the whole phenomenon of Nuclear Magnetic Resonance. where γ is a constant of proportionality called gyro- magnetic ratio, which for the proton it has a value of ω0 = γB0. (41) 2.675×108 rad/s·T. When this magnetic moment is placed in a magnetic field B its state is degenerated and the The characteristic frequency ω0 is called Larmor frequency energy of a particular level will be: and B0 = Bz. In a real system there is not only an isolated nucleus, but many nuclei which may occupy a particular spin E = −~µ · B; (34) state. This means that our development must be extended combining Eqs. (6), (7) the Hamiltonian is obtained: to consider an ensemble of spins. To do this we define a Ψ eigenstate, which is a linear combination of possible spin H = −~γS · B, (35) states for a single nucleus as: X and since the field B is oriented in the z direction, the Hamil- |Ψi = ams |msi. (42) tonian becomes: ms H = −~γS B ; (36) z z When performing a measurement on the system, the expected now using Schrodinger’s¨ equation, the energy eigenstates are value of the operation on this superposition of states is: found X 2 hΨ|Sz|Ψi = |am | ms, (43) H|m i = −~γB S |m i = −~γB m |m i, (37) s s z z s z s s ms and they will be: 2 where the value |ams | represents the probability of finding a single nucleus in the m state. For the case of a proton, E = −~γB m (38) s z s which is the particle of interest, with two spin states we have: ∆E = E (m = −1/2) − E (m = 1/2) = ~γB . (39) ¯ E ¯ E s s z ¯ 1 ¯ 1 |Ψi = a 1 ¯ + + a 1 ¯ − . (44) + 2 − 2 For the proton ms = ±1/2, a transition between the two 2 2 states represents a change in energy. This is called Zeeman The ratio of the population in the two energy states follows splitting [17] and is shown in Fig. 1, where it shows that as the Boltzmann statistics [4], the intensity of the magnetic field increases, the energy differ- 2 µ ¶ µ ¶ ence between the states is bigger. The two possible states are |a−1/2| −∆E −~γB0 2 = exp = exp , (45) more commonly known as “spin − up” and “spin − down”, |a+1/2| kBT kBT where the latter has a higher energy state than the “spin−up” state. Transitions between the two states can be induced by and since kBT À ~γB0, expanding the first order exponen- tial we have absorption or emission of a photon such that: 2 |a−1/2| ~γB0 2 ≈ 1 − , (46) ∆E = ~γBz = ~ω. (40) |a+1/2| kBT

Rev. Mex. Fis. E 63 (2017) 48–55 52 J.A. SOTO, T. CORDOVA,´ M. SOSA, AND S. JEREZ

The process of the radio-frequency emission is of the or- der of microseconds called radio-frequency pulse and quan- tified by the value α.A 90◦ pulse moves the magnetization vector about the xy plane, and a 180◦ pulse inverts the mag- netization respect to its equilibrium position. After a 90◦ pulse, the longitudinal component of the mag- netization vector is zero, in this position the number of nuclei in the lower energy state equals the number of nuclei in the higher energy state, it is called a state of saturation.

3.4. MRI selectivity

If we have multiple voxels placed under different magnetic fields, we can selectively excite either by simply changing the transmission frequency of the antenna. A magnetic gradient enables atomic nuclei to perceive a FIGURE 2. Magnetization vector of the volume element [6]. distinct magnetic field. In addition to changes in biochemical environment, one can selectively resonate all nuclei within and the difference in the number of spins between the spin-up positions that are being excited by the frequency bandwidth state and the spin-down state will be: used in the pulse emitter. Therefore all voxels contained in a plane perpendicular to the direction of the gradient plane 2 2 2 ~γB0 ~γB0 and whose thickness depends, once defined the value of the |a+1/2| − |a−1/2| ≈ |a+1/2| · ≈ . (47) kBT 2kBT gradient, on the bandwidth used on the pulse emitter, will be The magnitude of the magnetization of a voxel containing a excited. nucleus density ρ is obtained by multiplying the density by Magnetic resonance images are obtained by sending pulses the difference between the number of spins in both states and to different spaced time intervals with suitable values, which is a pulse sequence. the magnetic moment of the nuclei µz = ±(γ~/2)

2 2 ργ ~ B0 3.5. Evolution of magnetization under temporal varia- M = . (48) 4kBT tion of the magnetic field The magnetization of the volume element also has a direction which is equal to the orientation of the main magnetic field Taking a volume element (Voxel) that has a magnetization for that reason it should be considered as a vector quantity, as vector M, the set of equations describing the temporal evo- it shown in Fig. 2. lution of said vector are called Bloch equations [1] and are defined as follows: 3.3. Excitation of the Magnetization by radio-frequency pulse ∂Mx Mx = γ (M × B)x − (49) A volume element of the sample is defined as a voxel. Hav- ∂t T2 ing placed a transmitting antenna in the direction of maxi- ∂My My = γ (M × B)y − (50) mum emission to the voxel in the vertical plane and changing ∂t T2 the transmission frequency, when we emit the frequency of ∂My Mz − M0 precession (Larmor frequency), the nuclei are able to absorb = γ (M × B) − (51) ∂t z T energy, i.e. enter resonance. 1 When the nuclei of the voxel are entering into resonance, where γ is the gyromagnetic ratio of the proton and M0 = M the magnetization vector M travels performing a spiral rota- is the magnitude of the magnetization in equilibrium, T1 and tional movement relative to the direction of the field B0. Each T2 are time constants that will be discussed later. The above nucleus entering into resonance at a specific frequency spec- equations can be solved with the appropriate conditions; for ified by the Larmor Law, depending on B0 perceived and example, immediately after when the radio-frequency pulse biochemical environment in which it is located. Therefore, is turned off: the radio-frequency emission contains an approximate band- −t/T2 width of 100 kHz. Mx(t)= [Mx(0)cos(ω0t)+My(0)sen(ω0t)] · e , (52) Separation of M with respect to its equilibrium posi- −t/T2 tion is determined by the angle of inclination or FLIP angle. My(t)= [My(0)cos(ω0t)−Mx(0)sen(ω0t)] · e , (53) h i Its value depends on the power and time of radio-frequency M (t)=M (0)e−t/T1 +M 1 − e−t/T1 . (54) emission, among other factors. z z 0

Rev. Mex. Fis. E 63 (2017) 48–55 QUANTUM-MECHANICAL ASPECTS OF MAGNETIC RESONANCE IMAGING 53

3.6. Nuclear relaxation The magnetic flux through the coil can be written as: Z Once the radio emission is completed, the magnetization Φ = Br(r0) · M(r0, t)dr0. (63) vector, M, returns to its initial position by releasing energy M through a process called relaxation. Relaxation occurs be- muestra cause nuclei emit the excess energy absorbed when entering Finally, the electromotive is expressed as: resonance. Relaxation ends when the proportion of nuclei Z voxel between higher energy states and lower energy states d fem = − Br(r0) · M(r0, t)dr0. (64) match the Boltzmann equilibrium. dt The return to the equilibrium position of M produces muestra magnetic field modifications that may be collected by a re- Two voxels placed under different magnetic fields in ceiving antenna, since the magnetic field variations induce the moment of relaxation, will have different relaxation fre- an electrical signal called Free Induction Decay (FID) [16], quencies and, therefore, their signals may be differentiated which is a damped sinusoidal signal. The frequency of the through a frequency analysis such as . By sinusoid is the precession frequency imposed by the value of studying signals of relaxation, one can obtain information on the magnetic field during relaxation. the density (D) of hydrogen nuclei in the voxel and infor- Given the definition for the electromotive force (fem) mation related with the medium by the parameters T1, T2, and the magnetic flux through a coil: and T2 *. Z dΦ fem = − Φ = B · ds. (55) dt 3.7. Spin-lattice relaxation (T1 relaxation) S On the other hand, for the magnetization vector M of a sam- During the relaxation, the hydrogen nuclei release their ex- ple, there is an associated magnetic field from the current cess energy. Once reaching complete relaxation, the magne- density tization vector M recovers its initial value aligned with the ~ direction of the main magnetic field B0. An analysis after a JM (r, t) = 5 × M(r, t), (56) pulse of radio-frequency of the variations in time of the pro- the vector potential derived from a source is defined as: jection of the magnetization vector on the longitudinal axis Z (Mz), called longitudinal relaxation, once the value of the µ0 J(r) A(r) = dr,0 (57) projection is identical to the initial value of M, the relaxation 4π |r − r0| will be over. Thus the study of the longitudinal relaxation with this, the magnetic field can be calculated as B = 5~ × A, gives an idea of the rapidity with which it again reaches its and we can write the flux through the coil as: initial state. Z Z ³ ´ I As we can see in the Fig. 3, the longitudinal relaxation has Φ = B · ds = 5~ × A · ds = A · dl. (58) the form of a growing exponential regulated by a time con- s stant expressed in milliseconds called T1, which is also called Substituting the vector potential in the term for the current Longitudinal Relaxation Time. density and later in the line integral, we have: The values of T1 increases with the value of the magnetic field. I " Z 0 # µ 5~ × M(r0) Φ = dl · 0 dr0 , (59) M 4π |r − r0| simplifying the above equation and performing integration by parts: Z · µI ¶¸ µ dl Φ = 0 dr0M(r0) · 5~ × . (60) M 4π |r − r0| The expression in the parentheses of Eq. (60) can be com- pared with the vector potential A, obtaining an expression for circuits: I µ Idl A(r0) = 0 , (61) 4π |r − r0| thus the curl of the line integral is indeed the magnetic field per unit current that would be produced by the coil at point r0 µ I ¶ FIGURE 3. The recovery of Mz to M0 is controlled by an exponen- B(r0) 0 µ dl tial. T1 is the necessary time to recover the 63% of the equilibrium Br(r0) = = 5~ × 0 . (62) I 4π |r − r0| value of the magnetization [4].

Rev. Mex. Fis. E 63 (2017) 48–55 54 J.A. SOTO, T. CORDOVA,´ M. SOSA, AND S. JEREZ

or T2 if neither are influenced by the external magnetic field’s inhomogeneities or local magnetic variations acting perma- nently on the nuclei. Therefore T2 indicates asynchronism of nuclei of the voxel during relaxation due to spin-spin random influences which depend on the composition and structure of the tissue. T2 is the time that must elapse before the trans- verse magnetization loses 63% of its value. The time constant T2 is called the Transverse Relaxation Time. Returning to the , it is lost in two ways:

• Energy transfer between spins as a result of a local change in the magnetic field. FIGURE 4. T1 and T2 simultaneously happen, but T2 is much faster than T1 [4]. • Inhomogeneities independent of time of the external magnetic field B. 3.8. Spin-spin relaxation (T2 Relaxation)

We can get information related to the biochemical structure 4. Conclusions of the medium, by studying the variations over time of the MRI is a technique for visualizing tissues that takes use of the component for the magnetization vector in the vertical plane physical phenomenon of nuclear magnetic resonance, which (x, y) during the transverse relaxation. is the union of quantum mechanics with classical electrody- When M is zero it implies that the magnetization vec- x,y namics, that uses the quantum-mechanical properties of the tor is aligned on the z-axis with the main magnetic field. hydrogen atom to produce high resolution images that help Immediately after excitation, part of the spins precess syn- with medical diagnosis. Nowadays, there are many MRI ma- chronously, these spins have a 0◦ and are said to be in chines used for both medicine and research, where, for the phase, this state is called coherent phase. second case it is essential to have knowledge of the facets The coherent phase is gradually lost as the spins advance at a quantum level of the physical process involved. Like- and others are delayed on their way precession. In other wise there is a wide variety of research topics such as digital words the transverse relaxation is the decay of transverse processing of images acquired during a study, magnetic res- magnetization due to loss of coherence in the spins. onance , designing new radio frequency coils, The transverse relaxation differs from longitudinal ralax- pulse frequency design, etc., thus the broad field for the de- ation in that the spins do not dissipate energy to the surround- velopment and monitoring of research. ings but, rather, exchange energy with each other. The evolution of the transverse magnetization over time, until it vanishes, corresponds to a sinusoid with a dampened Acknowledgement relaxation frequency generating an exponential decay (see Fig. 4). This exponential decay of the surroundings is regu- Authors thanks Investigacion-DAIP,´ AL Proof-Reading Ser- lated by a parameter T2* [2]. When we take into considera- vice for the english reviewing and Conacyt by the PhD fel- tion all factors that influence the asynchronism of the nuclei, lowship.

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