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When a magnetic field is applied to an atom, the nuclear spin will be orientated with the field. The spin allows absorption of a photon of frequency Î½L, which is dependent on the magnetic field applied.
Î½L = Î³ * B
In the equation above, B is the magnetic field, Î³ is the gyromagnetic ratio and Î½L is the larmor frequency. When atoms are placed in a non-uniform magnetic field, the nuclei of the atom have varying larmor frequencies due to the equation.
(b) Why does the RF field have to be applied exactly at the Larmor frequency to tip the net nuclear magnetisation?
Some nuclei are said to be parallel to the magnetic field with some nuclei orientated anitparallel which are at a higher energy state. The energy difference between the two states is give by Î”E. When this energy difference is applied there is a transition from the lower energy level to the higher energy level. There is now more nuclei in an anitparallel orientation. When using the RF pulse it has to be resonance with the Larmor frequency. As the RF energy is applied the nuclei will orient themselves in the anitparallel position, this results in more nuclei being in a higher energy state.
(c)Classically the spin population states are divided into "parallel" and "anitparallel": what is the approximate population difference between these two states at 1.5T?
In order to calculate the population difference use the larmor equation to find the frequency at 1.5 T
f = Î³B
Î³ = 42.6 MHz/T
B= 1.5 T
f = (42.6 x 10 ^6) Î‡ (1.5)
f= 63.9 x 10^6 secâ»Â¹
Now use the following equation Î”E = hf to find the difference in population where
h is plancks constant which is 4.14 x 10^- 15 eVÎ‡sec
f is 63.9 x 10^6 secâ»Â¹
Î”E = hf
Î”E =4.14 x 10^-15 )(63.9 x 10^-6 secâ»Â¹)
Î”E = 2.65 x 10^-7 eV
(d) Describe what is meant by the Free Induction Decay (FID) of the MR signal
"In the rotating frame, the net magnetisation vector decays as spins lose phase coherence and begin to cancel each other out. They do this because they experience slightly different magnetic field strengths due to interaction between spins (via their own oscillating magnetic fields). Slightly different magnetic fields means slightly different precession (Larmor) frequencies. This causes some spins to "lag behind" the average, and some "get ahead" of the average. Eventually the spins point in arbitrary directions and the Mxy component of the net magnetisation vector is lost. This decaying, oscillating signal is the free induction decay." 
(e) Describe what is meant by spin-lattice relaxation. How is it described mathematically?
"One relaxation process involves a return of the protons to their original alignment with the static magnetic field. This process, called longitudinal or spin-lattice relaxation, is characterized by a time constant T1. The term spin-lattice refers to the interaction of the protons (spins) with their surroundings (the "lattice" or network of other spins). This interaction causes a net release of energy to the surroundings as the protons return to the lower energy state of alignment." 
The spin- lattice relaxation is given by M_z=M_0 [1-exp (-t/T_1)] where M_z and M_0 components of the magnetisation vector and t is the time.
(f) Describe what is meant by spin-spin relaxation. How is it described mathematically?
"Before a radio wave is applied, the precessional orientation of the protons is
Random. The application of a radio wave brings the protons into synchronous precession, or "in phase." When the radio wave is switched off, the protons begin to interact with their neighbours and give up energy in random collisions. In so doing, they revert to a state of random phase. As the protons revert to random orientation, the bulk signal decreases because the magnetic moments tend to cancel each other. This process is called transverse or spin-spin relaxation and is characterized by a time constant T2." 
The spin-spin relaxation is given by M_x, y=M_0 [exp (-t/T2)].
(g) Using the spin echo sequence, describe how the timing parameters can be adjusted to reflect T1, T2 and proton density in the image.
Figure 2.1 : Image given in assignment
(i)What are the two user variable parameters marked as 'a' and 'b'? Explain briefly in general terms how 'a' and 'b' might be chosen to achieve different image contrast weightings.
The user variable 'a' is the echo time and 'b' is the repetition time. When there is a short repetition time and short echo time the result will be a T1-weighted image. A long repetition time and short echo time will give a proton density image, and long repetition time and long echo time will give a T2-weighted image.
(ii) Identify the gradients marked'd'. Explain what is illustrated by the 'dotted' horizontal lines in this part of the diagram. What user variable parameter directly influences what is happening in'd'? Briefly relate this to total scan time.
The gradients marked'd' represents the phase encoding gradient. The user variable parameter that directly influences the phase encoding gradient is the frequency encoding gradient. "The total scan time for a standard spin echo or gradient echo sequence is number of repetitions x the scan time per repetition (means the product of repetition time (TR), number of phase encoding steps, and NSA)." 
(iii)Identify the gradient 'e'. If all other variables are considered fixed what user variable parameter would change the strength of this applied gradient?
The gradient is 'e' is the slice selection gradient. The user variable parameter that would change the strength of this applied gradient is the slice thickness parameter. "The slice thickness is governed by the following equation:
thk = BWtrans / (Ï‰0Â·GS)
Where thk is the slice thickness, BWtrans is the transmitted RF bandwidth (the range of frequencies it covers), 0 is the gyromagnetic ratio and GS is the magnitude of the slice selection magnetic field gradient. So, slice thickness is inversely proportional to GS; increasing GS will yield a thinner imaging slice." 
(iv) What is the waveform show in 'g'? Give a brief description of how the signal from 'g' is captured by the MR computer system and subsequently processed to form an image.
The waveform 'g' represents the echo signal received.
(A) Describe briefly the physical basis of the following MRI artefacts:
Truncation or Gibbs
Gibbs or truncation artefacts arise from going from a bright are to a dark area resulting in bright or dark lines which are present at parallel and adjacent borders where there is a sudden change in intensity. This type of artefact is associated with the number of steps used in the fourier transform to reconstruct the image . To minimize the effect of Gibbs or truncation artefacts more encoding steps are performed in order to lessen the intensity and narrow the artefacts.
Figure 3.1 : Diagram shows the Gibbs effect resulting from a Fourier transformation of a sharp change in image intensity
"There are various causes for zipper artefacts in images. Most of them are related to hardware or software problems beyond the radiologist immediate control. The zipper artefacts that can be controlled easily are those due to RF entering the scanning room when the door is open during acquisition of images. RF from some radio transmitters will cause zipper artefacts that are oriented perpendicular to the frequency axis of your image. Frequently there is more than one artefact line on an image from this cause. Other equipment and software problems can cause zippers in either axis." 
Figure 3.2 : When this image was taken the scanner room door was left open during the acquisition causing the zipper artefacts shown.
Chemical shift arises from a variation in the resonance frequency due to the nuclear spin of protons in different environments like fat or water. Due to the magnetic shielding of different protons , will result in different resonance frequency and hence lead to miss registration of protons in the same slice during the fourier transform. The chemical shift artefact will appear as bright or dark band at the edge of the anatomy. High field strength increases the miss registration of the protons.
Figure 3.3 : In this image the chemical shift artefact is seen as a small bright line in front of the femoral bone.
(b) In a magnetic field, fat precesses at a lower frequency that water giving rise to a "Chemical shift". At 1.5T, the frequency difference is 210Hz. For an image matrix of 256 x 256 and a receive coil bandwidth of 16 kHz, calculate the size of the "shift" between water and fat in pixels.
To find the chemical shift use the following equations
d = (n - nref) x 106 / nref
But know that
nref = gBo
The equation now becomes
d = (n - nref) x 106 / gBo
nref is resonance frequency
g is gymagnetic ratio where g is 42.58 MHz/T
Bo is magnetic field strength where Bo is 1.5 T
n is resonance frequency of second component
d is chemical shift difference
Can now put these values into the above equations. Given that the frequency difference is 220 Hz
d = (n - nref) x 106 / gBo
d= (220) x 106 / (42.58 x 106) (1.5)
d= 3.44 ppm
Using diagrams where appropriate, briefly describe k-space under the following headings
What does k-space represent
K space corresponds to a matrix of the MR data and represents the image before processing like fourier transforms are performed. Within k-space each line represents a measurement, with a separate line for varying phase gradients. A line of height 0 represents a line with no phase gradient.
Figure 4.1: In this diagram, Kx represents frequency, Ky represents phase directions. Each measurement is positioned at a different Ky coordinate ("height")
How is k-space normally filled
The polarity and amplitude of the frequency and phase encoding gradients directly affect how k-space is filled. The amplitude of the frequency encoding gradients establish how far the k-space goes to the left or right and therefore gives the field of view of the image in the frequency direction. Positive values go from left to right while negative values go from right to left.
The amplitude of phase encoding gradient estimates how far up and down k-space is filled up and down in the phase direction and hence determines the field of view in this direction. Positive values fill the top half while the negative values fill the bottom half.
What determines k-space co-ordinates
"The k-space location (kx and ky coordinates) of data is governed by the accumulated effect of gradient events and excitation pulses." 
How does an absence of data in k-space affect the image
If there is some date missing in k-space this will result in a loss of resolution of the image.
(a) Describe four important safety hazards in MRI.
Static magnetic fields
Depolarization can be caused by voltages induced in flowing blood and the movement of muscles in the heart which can be detected by an electrocardiogram. In order to minimize this effect the magnetic field strength kept withing the following limits. 2.5 T for body of patients; 0.2 T for arms and hands of staff and 0.02 T for whole body of staff.
Switching of the gradient magnetic fields
Involuntary muscular contraction, breathing difficulties and ventricular fibrillation arise from eddy currents induced in the body. MRI should not be performed on patients with implants, as it can cause harm to the patient. In the case of strong fields, taste sensations can be experienced by the patient while flashes of light can be present on the patient's retina. Magnetic field build up is usually 1-5 T's to avoid symptoms.
In strong static fields, at high frequencies, heating can occur. The temperature should not rise above 1 degree Celsius on the patient's skin. In order to minimize the heating affects the specific absorption ratio should not surpass 0.4 W/Kg and pulsed RF field should not go beyond 70 W
Mechanical attraction of ferromagnetic objects
This changes with the square of the magnetic field and inverse of the distance. Metal objects are made into projectiles if they come into the fringe field. For this reason non-magnetic material should be used.
What is meant by SAR? Define the units used to measure this parameter.
"The Specific Absorption Rate is defined as the RF power absorbed per unit of mass of an object, and is measured in watts per kilogram (W/kg). The SAR describes the potential for heating of the patient's tissue due to the application of the RF energy necessary to produce the MR signal. Inhomogeneity of the RF field leads to a local exposure where most of the absorbed energy is applied to one body region rather than the entire person, leading to the concept of a local SAR." 
What is the whole body limit?
"4 W/kg averaged over the whole body for any 15-minute period." 
Which sequences are a particular concern and may give rise to radio-frequency effects?
Radio frequency effects occur when a patient is exposed to static magnetic fields in MRI. "The Radio frequency pulses mainly produce heat, which is absorbed by the body tissue. If the power of the RF radiation is very high, the patient may be heated too much. To avoid this heating, the limit of RF exposure in MRI is up to the maximum specific absorption rate (SAR) of 4 W/kg whole body weight (can be different from country to country). For MRI safety reasons, the MRI machine starts no sequence, if the SAR limit is exceeded."