Please answer the following questions, and do well!

A cylindrical test tube of 1cm in diameter, filled with a mixture of equal volumes of water (H20) and acetone (CH3-O-CH3), is put in an MR scanner. Water protons give one resonance peak at 4.7 ppm, and the hydrogen protons in acetone resonate a the single frequency of 2.1 ppm. For simplicity, assume an equal density for acetone and water (same number of molecules per unit volume). The gyromagnetic ratio of protons is 26,752.2 g^-1s^-1.

Assume:  T₁(water) = 3s, T₁(acetone) = 6s, T₂(water) = T₂(acetone) = 1.5s.

a. If the experiment is performed at 4.7T, what is the frequency difference between the resonances of the water and the acetone protons? Which protons (water or acetone) resonate at a higher frequency?

b. Draw a schematic NMR spectrum of the mixture described above, obtained at 4.7T. Use a ppm scale in a proper way, and briefly explain on what would the properties of the spectral lines you drew (intensity, area, line width) depend, and if possible - provide actual numbers.

c. Based on what we learned throughout the course, describe at least two methods to obtain a spectrum in which only the water peak is visible. Provide enough information in your method description so that someone interested in performing this experiment could actually do so, i.e. don't just write e.g. "use the Hahn Spin Echo sequence", but provide the necessary parameters of this sequence to obtain the goal of obtaining only the water peak visible.

For the following sections we will consider the position of the test tube in the scanner as described below. The scanner has gradients that can act along the X and the Z axes. Assume that the gradients can reach instantaneously any desired strength (rectangular gradients, no slopes).

d. We are interested in selecting a slice along the Z-axis (doesn't matter exactly where). The bandwidth of the RF pulse we are using is of 2KHz (2000 Hz), and its total duration is 5ms. The pulse shape is symmetric. Draw the pulse sequence (RF pulse + gradients) we need to use for selecting the slice, and calculate what should be the value of gz to obtain a 4mm slice thickness. Make sure all the elements for a proper slice selection are present in the "sequence".

e. What would we need to change in the parameters of the pulse sequence, had we wanted to change the slice thickness to 2mm? Provide two options.

f. What parameter in the pulse sequence would we need to change, had we wanted to shift the slice upwards by lcm? By how much? Provide a number based on the gradient strength you found in (d).

g. Is the position of the slice selected for the water the same as the position of the slice selected for the acetone? Explain your answer, and if you think that the answer is "no", calculate the distance between these positions.

h. Following the slice-selective excitation described in (d), the FID is frequency-encoded with a gradient along the X-axis. The gradient is switched on immediately after the slice selection module is finished. Assuming we want a field-of-view (FOV) of 5 cm along the X-axis, what should be the gradient amplitude?

i. Suppose we want to have a 0.1mm resolution along the X-axis. How many k_x data points we need to acquire to reach this resolution?

j. How long would it take to acquire these data points? Consider Nyquist rule for answering this section.

k. Draw schematically the one-dimensional image (along the X-axis) that results from the Fourier transformation of the k_x data acquired in this experiment.

I. Will the acetone and the water "images" be at the same location? If not - explain, and calculate by how many pixels will the two images be separated.

m. Based on the information on the water and the acetone, what kind of contrast-based imaging method would you use to enhance the signal of either the water or the acetone in an MRI image? Provide two possible ways to do so and briefly discuss.