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**“MAGNETIC RESONANCE IMAGING PRINCIPLES”**

Alejandro Jaramillo Robles Soluciones Globales en Imagenología MRI-DT Julio 22, 2005

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Antecedentes MRI is based on the principles of nuclear magnetic resonance (NMR), a spectroscopic technique used to obtain microscopic chemical and physical information about molecules Julio 22, 2005 Imagenología por Resonancia Magnética

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Phisics … Spin …is a fundamental property of nature. Spin comes in multiples of 1/2 and can be + or -. … two or more particles with spins having opposite signs can pair up to eliminate the observable manifestations of spin Julio 22, 2005 Imagenología por Resonancia Magnética

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**Phisics… Max Plank E= υ h**

a particle can undergo a transition between the two energy states by the absorption of a photon The energy E of a photon is related to its frequency υ called resoanance frequency, by Plank´s constant h. This particle can undergo a transition between the two energy states by the absorption of a photon E= υ h

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**Properties of Spin γH= 42.58 MHz/T υ = γB**

A particle with a net spin in a magnetic field B, can absorb a photon of a frequency υ. The frequency υ depends on the gyromagnetic ratio γ of the particle υ = γB Properties of Spin When placed in a magnetic field of strength B, a particle with a net spin can absorb a photon, of frequency . The frequency depends on the gyromagnetic ratio, of the particle. = B For hydrogen, = MHz / T. γH= MHz/T Julio 22, 2005 Imagenología por Resonancia Magnética

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**Nuclei of interest inMRI**

Unpaired Protons Unpaired Neutrons Net Spin Biologic Abundance (MHz/T) 1H 1 0 1/2 0.63 42.58 31P 0.0024 17.25 23Na 1 2 3/2 11.27 14N 0.015 3.08 13C 0.094 10.71 The human body is primarily fat and water. Fat and water have many hydrogen atoms which make the human body approximately 63% hydrogen atoms. Hydrogen nuclei have an NMR signal. For these reasons magnetic resonance imaging primarily images the NMR signal from the hydrogen nuclei. Each voxel of an image of the human body contains one or more tissues. If we zoom into one of the hydrogens past the electron cloud we see a nucleus comprised of a single proton. The proton possesses a property called spin which: can be thought of as a small magnetic field, and will cause the nucleus to produce an NMR signal. Properties of Spin Because nucleons have spin, just like electrons do, their spin can pair up when the orbitals are being filled and cancel out. Almost every element in the periodic table has an isotope with a non zero nuclear spin The principle behind all magnetic resonance imaging is the resonance equation, which shows that the resonance frequency n of a spin is proportional to the magnetic field, Bo, it is experiencing. n = g Bo Recall from the spin physics chapter that g is the gyromagnetic ratio.

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**Electromagnetic Spectrum**

Magnetic resonance imaging is based on the absorption and emission of energy in the radio frequency range of the electromagnetic spectrum. It is clear from the attenuation spectrum of the human body why x-rays are used, but why did it take so long to develop imaging with radio waves, especially with health concerns associated with ionizing radiation such as x-rays? Many scientists were taught that you can not image objects smaller than the wavelength of the energy being used to image. MRI gets around this limitation by producing images based on spatial variations in the phase and frequency of the radio frequency energy being absorbed and emitted by the imaged object. Julio 22, 2005 Imagenología por Resonancia Magnética

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**Energy Level Diagrams E = hγB**

The energy of the two spin states can be represented by an energy level diagram. the energy of the photon needed to cause a transition between the two spin states is E = h B When the energy of the photon matches the energy difference between the two spin states an absorption of energy occurs. In the NMR experiment, the frequency of the photon is in the radio frequency (RF) range. In NMR spectroscopy, is between 60 and 800 MHz for hydrogen nuclei. In clinical MRI, is typically between 15 and 80 MHz for hydrogen imaging. E = hγB When the energy of the photon matches the energy difference between the two spin states an absorption of energy occurs Julio 22, 2005 Imagenología por Resonancia Magnética

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**Energy Level Diagrams E = hγB Variable Magnetic Field**

The energy of the two spin states can be represented by an energy level diagram. the energy of the photon needed to cause a transition between the two spin states is E = h B When the energy of the photon matches the energy difference between the two spin states an absorption of energy occurs. In the NMR experiment, the frequency of the photon is in the radio frequency (RF) range. In NMR spectroscopy, is between 60 and 800 MHz for hydrogen nuclei. In clinical MRI, is typically between 15 and 80 MHz for hydrogen imaging. Variable Magnetic Field Constant Frequency Constant Magnetic Field Variable Frecuency E = hγB Julio 22, 2005 Imagenología por Resonancia Magnética

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Spin Packets A spin packet is a group of spins experiencing the same magnetic field strength. In this example, the spins within each grid section represent a spin packet.

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Magnetization Vector At any instant in time, the magnetic field due to the spins in each spin packet can be represented by a magnetization vector. The size of each vector is proportional to (N+ - N-).

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Net Magnetization The vector sum of the magnetization vectors from all of the spin packets is the net magnetization. In order to describe pulsed NMR it is necessary from here on to talk in terms of the net magnetization. Adapting the conventional NMR coordinate system, the external magnetic field and the net magnetization vector at equilibrium are both along the Z axis.

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**At Equilibrium Mz = Mo Mxy = 0**

At equilibrium, the net magnetization vector lies along the direction of the applied magnetic field Bo and is called the equilibrium magnetization Mo. In this configuration, the Z component of magnetization MZ equals Mo. MZ is referred to as the longitudinal magnetization. There is no transverse (MX or MY) magnetization here. Mz = Mo Mxy = 0

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Relaxation Time T1 Processes It is possible to change the net magnetization by exposing the nuclear spin system to energy of a frequency equal to the energy difference between the spin states. If enough energy is put into the system, it is possible to saturate the spin system and make MZ=0. The time constant which describes how MZ returns to its equilibrium value is called the spin lattice relaxation time (T1). The equation governing this behavior as a function of the time t after its displacement is: Mz = Mo ( 1 - e-t/T1 ) T1 is the time to reduce the difference between the longitudinal magnetization (MZ) and its equilibrium value by a factor of e. If the net magnetization is placed along the -Z axis, it will gradually return to its equilibrium position along the +Z axis at a rate governed by T1. The equation governing this behavior as a function of the time t after its displacement is: Mz = Mo ( 1 - 2e-t/T1 ) Again, the spin-lattice relaxation time (T1) is the time to reduce the difference between the longitudinal magnetization (MZ) and its equilibrium value by a factor of e. If the net magnetization is placed in the XY plane it will rotate about the Z axis at a frequency equal to the frequency of the photon which would cause a transition between the two energy levels of the spin. This frequency is called the Larmor frequency. If enough energy is put into the system with a frequency equal to the resonance frequency…

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**Relaxation Time Mz = 0 Mxy ≠ 0 T1 Processes**

It is possible to change the net magnetization by exposing the nuclear spin system to energy of a frequency equal to the energy difference between the spin states. If enough energy is put into the system, it is possible to saturate the spin system and make MZ=0. The time constant which describes how MZ returns to its equilibrium value is called the spin lattice relaxation time (T1). The equation governing this behavior as a function of the time t after its displacement is: Mz = Mo ( 1 - e-t/T1 ) T1 is the time to reduce the difference between the longitudinal magnetization (MZ) and its equilibrium value by a factor of e. If the net magnetization is placed along the -Z axis, it will gradually return to its equilibrium position along the +Z axis at a rate governed by T1. The equation governing this behavior as a function of the time t after its displacement is: Mz = Mo ( 1 - 2e-t/T1 ) Again, the spin-lattice relaxation time (T1) is the time to reduce the difference between the longitudinal magnetization (MZ) and its equilibrium value by a factor of e. If the net magnetization is placed in the XY plane it will rotate about the Z axis at a frequency equal to the frequency of the photon which would cause a transition between the two energy levels of the spin. This frequency is called the Larmor frequency. Mz = 0 Mxy ≠ 0

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**Relaxation Time T2 Mxy ≠ 0 Mz = 0 T2 Processes**

In addition to the rotation, the net magnetization starts to dephase because each of the spin packets making it up is experiencing a slightly different magnetic field and rotates at its own Larmor frequency. The longer the elapsed time, the greater the phase difference. Here the net magnetization vector is initially along +Y. For this and all dephasing examples think of this vector as the overlap of several thinner vectors from the individual spin packets. The time constant which describes the return to equilibrium of the transverse magnetization, MXY, is called the spin-spin relaxation time, T2. MXY =MXYo e-t/T2 T2 is always less than or equal to T1. The net magnetization in the XY plane goes to zero and then the longitudinal magnetization grows in until we have Mo along Z. Any transverse magnetization behaves the same way. The transverse component rotates about the direction of applied magnetization and dephases. T1 governs the rate of recovery of the longitudinal magnetization. In summary, the spin-spin relaxation time, T2, is the time to reduce the transverse magnetization by a factor of e. In the previous sequence, T2 and T1 processes are shown separately for clarity. That is, the magnetization vectors are shown filling the XY plane completely before growing back up along the Z axis. Actually, both processes occur simultaneously with the only restriction being that T2 is less than or equal to T1. Two factors contribute to the decay of transverse magnetization. 1) molecular interactions (said to lead to a pure T2 molecular effect) 2) variations in Bo (said to lead to an inhomogeneous T2 effect The combination of these two factors is what actually results in the decay of transverse magnetization. The combined time constant is called T2 star and is given the symbol T2*. The relationship between the T2 from molecular processes and that from inhomogeneities in the magnetic field is as follows. Mxy ≠ 0 Mz = 0

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Relaxation Time T2 T2 Processes In addition to the rotation, the net magnetization starts to dephase because each of the spin packets making it up is experiencing a slightly different magnetic field and rotates at its own Larmor frequency. The longer the elapsed time, the greater the phase difference. Here the net magnetization vector is initially along +Y. For this and all dephasing examples think of this vector as the overlap of several thinner vectors from the individual spin packets. The time constant which describes the return to equilibrium of the transverse magnetization, MXY, is called the spin-spin relaxation time, T2. MXY =MXYo e-t/T2 T2 is always less than or equal to T1. The net magnetization in the XY plane goes to zero and then the longitudinal magnetization grows in until we have Mo along Z. Any transverse magnetization behaves the same way. The transverse component rotates about the direction of applied magnetization and dephases. T1 governs the rate of recovery of the longitudinal magnetization. In summary, the spin-spin relaxation time, T2, is the time to reduce the transverse magnetization by a factor of e. In the previous sequence, T2 and T1 processes are shown separately for clarity. That is, the magnetization vectors are shown filling the XY plane completely before growing back up along the Z axis. Actually, both processes occur simultaneously with the only restriction being that T2 is less than or equal to T1. Two factors contribute to the decay of transverse magnetization. 1) molecular interactions (said to lead to a pure T2 molecular effect) 2) variations in Bo (said to lead to an inhomogeneous T2 effect The combination of these two factors is what actually results in the decay of transverse magnetization. The combined time constant is called T2 star and is given the symbol T2*. The relationship between the T2 from molecular processes and that from inhomogeneities in the magnetic field is as follows. The time constant which describes the return to equilibrium of the transverse magnetization Mxy = 0 Mz = 0

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Chemical Shift The magnetic field that each nuclei experiment varies according to the type of nuclei and bonds in the molecule δ 3.5ppm Agua Grasa Chemical Shift When an atom is placed in a magnetic field, its electrons circulate about the direction of the applied magnetic field. This circulation causes a small magnetic field at the nucleus which opposes the externally applied field. The magnetic field at the nucleus (the effective field) is therefore generally less than the applied field by a fraction s. B = Bo (1-s) The electron density around each nucleus in a molecule varies according to the types of nuclei and bonds in the molecule. The opposing field and therefore the effective field at each nucleus will vary. This is called the chemical shift phenomenon. Consider the methanol molecule. The resonance frequency of two types of nuclei in this example differ. This difference will depend on the strength of the magnetic field, Bo, used to perform the NMR spectroscopy. The greater the value of Bo, the greater the frequency difference. This relationship could make it difficult to compare NMR spectra taken on spectrometers operating at different field strengths. The term chemical shift was developed to avoid this problem. The chemical shift of a nucleus is the difference between the resonance frequency of the nucleus and a standard, relative to the standard. This quantity is reported in ppm and given the symbol delta, d. d = (n - nREF) x106 / nREF In NMR spectroscopy, this standard is often tetramethylsilane, abbreviated TMS. In the body there is no TMS, but there are are two primary hydrogen containing substances, water and fat. The chemical shift difference between these two types of hydrogens is approximately 3.5 ppm. El cero del espectro se toma arbitrariamente como la señal singlete que da el tetrametilsilano TMS (Me4Si). Julio 22, 2005 Imagenología por Resonancia Magnética

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Relaxation Time T1 T1 Processes It is possible to change the net magnetization by exposing the nuclear spin system to energy of a frequency equal to the energy difference between the spin states. If enough energy is put into the system, it is possible to saturate the spin system and make MZ=0. The time constant which describes how MZ returns to its equilibrium value is called the spin lattice relaxation time (T1). The equation governing this behavior as a function of the time t after its displacement is: Mz = Mo ( 1 - e-t/T1 ) T1 is the time to reduce the difference between the longitudinal magnetization (MZ) and its equilibrium value by a factor of e. If the net magnetization is placed along the -Z axis, it will gradually return to its equilibrium position along the +Z axis at a rate governed by T1. The equation governing this behavior as a function of the time t after its displacement is: Mz = Mo ( 1 - 2e-t/T1 ) Again, the spin-lattice relaxation time (T1) is the time to reduce the difference between the longitudinal magnetization (MZ) and its equilibrium value by a factor of e. If the net magnetization is placed in the XY plane it will rotate about the Z axis at a frequency equal to the frequency of the photon which would cause a transition between the two energy levels of the spin. This frequency is called the Larmor frequency. The time constant which describes how MZ returns to its equilibrium value Mz = Mo

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**Relaxation Time T1, T2 T2 ≤ T1 T2 Processes**

In addition to the rotation, the net magnetization starts to dephase because each of the spin packets making it up is experiencing a slightly different magnetic field and rotates at its own Larmor frequency. The longer the elapsed time, the greater the phase difference. Here the net magnetization vector is initially along +Y. For this and all dephasing examples think of this vector as the overlap of several thinner vectors from the individual spin packets. The time constant which describes the return to equilibrium of the transverse magnetization, MXY, is called the spin-spin relaxation time, T2. MXY =MXYo e-t/T2 T2 is always less than or equal to T1. The net magnetization in the XY plane goes to zero and then the longitudinal magnetization grows in until we have Mo along Z. Any transverse magnetization behaves the same way. The transverse component rotates about the direction of applied magnetization and dephases. T1 governs the rate of recovery of the longitudinal magnetization. In summary, the spin-spin relaxation time, T2, is the time to reduce the transverse magnetization by a factor of e. In the previous sequence, T2 and T1 processes are shown separately for clarity. That is, the magnetization vectors are shown filling the XY plane completely before growing back up along the Z axis. Actually, both processes occur simultaneously with the only restriction being that T2 is less than or equal to T1. Two factors contribute to the decay of transverse magnetization. 1) molecular interactions (said to lead to a pure T2 molecular effect) 2) variations in Bo (said to lead to an inhomogeneous T2 effect The combination of these two factors is what actually results in the decay of transverse magnetization. The combined time constant is called T2 star and is given the symbol T2*. The relationship between the T2 from molecular processes and that from inhomogeneities in the magnetic field is as follows. T2 ≤ T1

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**Free Induction Decay - FID**

The Time Domain NMR Signal As transverse magnetization rotates about the Z axis, it will induce a current in a coil of wire located around the X axis. Plotting current as a function of time gives a sine wave. This wave will of course decay with time constant T2* due to dephasing of the spin packets. This signal is called a free induction decay (FID). We will see in Chapter 5 how the FID is converted into a frequency domain spectrum. Julio 22, 2005 Imagenología por Resonancia Magnética

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**Free Induction Decay - FID**

The Time Domain NMR Signal As transverse magnetization rotates about the Z axis, it will induce a current in a coil of wire located around the X axis. Plotting current as a function of time gives a sine wave. This wave will of course decay with time constant T2* due to dephasing of the spin packets. This signal is called a free induction decay (FID). We will see in Chapter 5 how the FID is converted into a frequency domain spectrum. Time Domain Frequency Domain Julio 22, 2005 Imagenología por Resonancia Magnética

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**FID 90o Sequence The 90-FID Sequence**

A set of RF pulses applied to a sample to produce a specific form of NMR signal is called a pulse sequence. In the 90-FID pulse sequence, net magnetization is rotated down into the X'Y' plane with a 90o pulse. The net magnetization vector begins to precess about the +Z axis. The magnitude of the vector also decays with time. A timing diagram is a multiple axis plot of some aspect of a pulse sequence versus time. A timing diagram for a 90-FID pulse sequence has a plot of RF energy versus time and another for signal versus time. When this sequence is repeated, for example when signal-to-noise improvement is needed, the amplitude of the signal after being Fourier transformed (S) will depend on T1 and the time between repetitions, called the repetition time (TR), of the sequence. In the signal equation below, k is a proportionality constant and is the density of spins in the sample. S = k ( 1 - e-TR/T1 ) Julio 22, 2005 Imagenología por Resonancia Magnética

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**FID 90o Sequence The 90-FID Sequence**

A set of RF pulses applied to a sample to produce a specific form of NMR signal is called a pulse sequence. In the 90-FID pulse sequence, net magnetization is rotated down into the X'Y' plane with a 90o pulse. The net magnetization vector begins to precess about the +Z axis. The magnitude of the vector also decays with time. A timing diagram is a multiple axis plot of some aspect of a pulse sequence versus time. A timing diagram for a 90-FID pulse sequence has a plot of RF energy versus time and another for signal versus time. When this sequence is repeated, for example when signal-to-noise improvement is needed, the amplitude of the signal after being Fourier transformed (S) will depend on T1 and the time between repetitions, called the repetition time (TR), of the sequence. In the signal equation below, k is a proportionality constant and is the density of spins in the sample. S = k ( 1 - e-TR/T1 ) Julio 22, 2005 Imagenología por Resonancia Magnética

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**Spin Echo Sequence The Spin-Echo Sequence**

Another commonly used pulse sequence is the spin-echo pulse sequence. Here a 90o pulse is first applied to the spin system. The 90o degree pulse rotates the magnetization down into the X'Y' plane. The transverse magnetization begins to dephase. At some point in time after the 90o pulse, a 180o pulse is applied. This pulse rotates the magnetization by 180o about the X' axis. The 180o pulse causes the magnetization to at least partially rephase and to produce a signal called an echo. A timing diagram shows the relative positions of the two radio frequency pulses and signal. The signal equation for a repeated spin echo sequence as a function of the repetition time, TR, and the echo time (TE) defined as the time between the 90o pulse and the maximum amplitude in the echo is S = k ( 1 - e-TR/T1 ) e-TE/T2 Julio 22, 2005 Imagenología por Resonancia Magnética

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**Magnetic Resonance Imaging**

Information generated by a proton in a magnetic field, returning to its equilibrium, after being exposed to a RF signal. Todos los puntos experimentan el mismo campo magnético, por lo tanto la señal de retorno solamente nos dirá que elementos químicos tenemos y su concentración, pero no su posición. The intensity of a pixel is proportional to the NMR signal intensity of the contents of the corresponding volume element or voxel of the object being imaged. Julio 22, 2005 Imagenología por Resonancia Magnética

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Gradient …is a variation in the magnetic field with respect to position υ = γB Each region experience a unique magnetic field Magnetic Field Gradient If each of the regions of spin was to experience a unique magnetic field we would be able to image their positions. A gradient in the magnetic field is what will allow us to accomplish this. A magnetic field gradient is a variation in the magnetic field with respect to position. A one-dimensional magnetic field gradient is a variation with respect to one direction, while a two-dimensional gradient is a variation with respect to two. The most useful type of gradient in magnetic resonance imaging is a one- dimensional linear magnetic field gradient. A one-dimensional magnetic field gradient along the x axis in a magnetic field, Bo, indicates that the magnetic field is increasing in the x direction. Here the length of the vectors represent the magnitude of the magnetic field. The symbols for a magnetic field gradient in the x, y, and z directions are Gx, Gy, and Gz. Frequency Encoding The point in the center of the magnet where (x,y,z) =0,0,0 is called the isocenter of the magnet. The magnetic field at the isocenter is Bo and the resonant frequency is no. If a linear magnetic field gradient is applied to our hypothetical head with three spin containing regions, the three regions experience different magnetic fields. The result is an NMR spectrum with more than one signal. The amplitude of the signal is proportional to the number of spins in a plane perpendicular to the gradient. This procedure is called frequency encoding and causes the resonance frequency to be proportional to the position of the spin. n = g ( Bo + x Gx ) = no + g x Gx x = ( n - no ) / ( g Gx ) This principle forms the basis behind all magnetic resonance imaging. To demonstrate how an image might be generated from the nmr spectra, the backprojection method of imaging is presented in the next section. Julio 22, 2005 Imagenología por Resonancia Magnética

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Pulse Diagram Now we will examine the sequence from a macroscopic perspective of the spin vectors. Imagine a cube of spins placed in a magnetic field. The cube is composed of serveral volume elements each with its own net magnetization vector. Suppose we wish to image a slice in the XY plane. The Bo magnetic field is along the Z axis. The Slice selection gradient is applied along the Z axis. The RF pulse rotates only those spins packets within the cube which satisfy the resonance condition. These spin packets are located within an XY plane in this example. The location of the plane along the Z axis with respect to the isocenter is given by Z = n / g Gs where n is the frequency offset from no ( i.e. n - no ), Gs the magnitude of the slice selection gradient, and g the gyromagnetic ratio. Spins located above and below this plane are not affected by the RF pulse. They will therefore be neglected for purposes of this presentation. t Julio 22, 2005 Imagenología por Resonancia Magnética

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Pulse Diagram Suppose we wish to image a slice in the XY plane. The Bo magnetic field is along the Z axis. The Slice selection gradient is applied along the Z axis. The RF pulse rotates only those spins packets within the cube which satisfy the resonance condition. These spin packets are located within an XY plane in this example. The location of the plane along the Z axis with respect to the isocenter is given by Z = n / g Gs where n is the frequency offset from no ( i.e. n - no ), Gs the magnitude of the slice selection gradient, and g the gyromagnetic ratio. Spins located above and below this plane are not affected by the RF pulse. They will therefore be neglected for purposes of this presentation. t Julio 22, 2005 Imagenología por Resonancia Magnética

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Pulse Diagram Now we will examine the sequence from a macroscopic perspective of the spin vectors. Imagine a cube of spins placed in a magnetic field. The cube is composed of serveral volume elements each with its own net magnetization vector. Suppose we wish to image a slice in the XY plane. The Bo magnetic field is along the Z axis. The Slice selection gradient is applied along the Z axis. The RF pulse rotates only those spins packets within the cube which satisfy the resonance condition. These spin packets are located within an XY plane in this example. The location of the plane along the Z axis with respect to the isocenter is given by Z = n / g Gs where n is the frequency offset from no ( i.e. n - no ), Gs the magnitude of the slice selection gradient, and g the gyromagnetic ratio. Spins located above and below this plane are not affected by the RF pulse. They will therefore be neglected for purposes of this presentation. t Julio 22, 2005 Imagenología por Resonancia Magnética

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Pulse Diagram Once rotated into the XY plane these vectors would precess at the Larmor frequency given by the magnetic field each was experiencing. If the magnetic field was uniform, each of the nine precessional rates would be equal. In the imaging sequence a phase encoding gradient is applied after the slice selection gradient. Assuming this is applied along the X axis, the spins at different locations along the X axis begin to precess at different Larmor frequencies. When the phase encoding gradient is turned off the net magnetization vectors precess at the same rate, but possess different phases. The phase being determined by the duration and magnitude of the phase encoding gradient pulse. Once the phase encoding gradient pulse is turned off a frequency encoding gradient pulse is turned on. In this example the frequency encoding gradient is in the -Y direction. The frequency encoding gradient causes spin packets to precess at rates dependent on their Y location. Please note that now each of the nine net magnetization vectors is characterized by a unique phase angle and precessional frequency. If we had a means of determining the phase and frequency of the signal from a net magnetization vector we could position it within one of the nine elements. A simple Fourier transform is capable of this task for a single net magnetization vector located somewhere within the 3x3 space. For example, if a single vector was located at (X,Y) = 2,2, its FID would contain a sine wave of frequency 2 and phase 2. A Fourier transform of this signal would yield one peak at frequency 2 and phase 2. Unfortunately a one dimensional Fourier transform is incapable of this task when more than one vector is located within the 3x3 matrix at a different phase encoding direction location. There needs to be one phase encoding gradient step for each location in the phase encoding gradient direction. The point is you need one equation for each unknown you are trying to solve for. Therefore if there are three phase encoding direction locations we will need three unique phase encoding gradient amplitudes and have three unique free induction decays. If we wish to resolve 256 locations in the phase encoding direction we will need 256 different magnitudes of the phase encoding gradient and will record 256 different free induction decays. t Julio 22, 2005 Imagenología por Resonancia Magnética

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Pulse Diagram To simplify the remainder of the presentation, we shall concentrate on a 3x3 subset of the net magnetization vectors. The picture of these spins in this plane looks like this. Once rotated into the XY plane these vectors would precess at the Larmor frequency given by the magnetic field each was experiencing. If the magnetic field was uniform, each of the nine precessional rates would be equal. In the imaging sequence a phase encoding gradient is applied after the slice selection gradient. Assuming this is applied along the X axis, the spins at different locations along the X axis begin to precess at different Larmor frequencies. When the phase encoding gradient is turned off the net magnetization vectors precess at the same rate, but possess different phases. The phase being determined by the duration and magnitude of the phase encoding gradient pulse. Once the phase encoding gradient pulse is turned off a frequency encoding gradient pulse is turned on. In this example the frequency encoding gradient is in the -Y direction. The frequency encoding gradient causes spin packets to precess at rates dependent on their Y location. Please note that now each of the nine net magnetization vectors is characterized by a unique phase angle and precessional frequency. If we had a means of determining the phase and frequency of the signal from a net magnetization vector we could position it within one of the nine elements. A simple Fourier transform is capable of this task for a single net magnetization vector located somewhere within the 3x3 space. For example, if a single vector was located at (X,Y) = 2,2, its FID would contain a sine wave of frequency 2 and phase 2. A Fourier transform of this signal would yield one peak at frequency 2 and phase 2. Unfortunately a one dimensional Fourier transform is incapable of this task when more than one vector is located within the 3x3 matrix at a different phase encoding direction location. There needs to be one phase encoding gradient step for each location in the phase encoding gradient direction. The point is you need one equation for each unknown you are trying to solve for. Therefore if there are three phase encoding direction locations we will need three unique phase encoding gradient amplitudes and have three unique free induction decays. If we wish to resolve 256 locations in the phase encoding direction we will need 256 different magnitudes of the phase encoding gradient and will record 256 different free induction decays. t Julio 22, 2005 Imagenología por Resonancia Magnética

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Pulse Diagram Once the phase encoding gradient pulse is turned off a frequency encoding gradient pulse is turned on. In this example the frequency encoding gradient is in the -Y direction. The frequency encoding gradient causes spin packets to precess at rates dependent on their Y location. Please note that now each of the nine net magnetization vectors is characterized by a unique phase angle and precessional frequency. If we had a means of determining the phase and frequency of the signal from a net magnetization vector we could position it within one of the nine elements. A simple Fourier transform is capable of this task for a single net magnetization vector located somewhere within the 3x3 space. For example, if a single vector was located at (X,Y) = 2,2, its FID would contain a sine wave of frequency 2 and phase 2. A Fourier transform of this signal would yield one peak at frequency 2 and phase 2. Unfortunately a one dimensional Fourier transform is incapable of this task when more than one vector is located within the 3x3 matrix at a different phase encoding direction location. There needs to be one phase encoding gradient step for each location in the phase encoding gradient direction. The point is you need one equation for each unknown you are trying to solve for. Therefore if there are three phase encoding direction locations we will need three unique phase encoding gradient amplitudes and have three unique free induction decays. If we wish to resolve 256 locations in the phase encoding direction we will need 256 different magnitudes of the phase encoding gradient and will record 256 different free induction decays. t Julio 22, 2005 Imagenología por Resonancia Magnética

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**Image Processing MR – Back Projection**

one-dimensional field gradient is applied at several angles, and the NMR spectrum is recorded for each gradient Julio 22, 2005 Imagenología por Resonancia Magnética

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Image Processing Julio 22, 2005 Imagenología por Resonancia Magnética

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