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MIT HST 583 - Problem Set 3

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HST.583: Functional MRI Data Acquisition and AnalysisProblem Set 3MRI PhysicsProblem 1: Advantages of using larger magnetic fieldsa) In class it was stated that you could reduce the number of spins that move from the lower energy state to the upper state due to thermal excitation by either reducing the temperature of the sample or increasing the field strength. Assume that you could reduce your subject’s body temperature by 10 C (note that normal human temperature varies in the range of 36.5-37.5 C). What is the change in the ratio of the number of spins in the upper energy level to those in the lower energy level due tothis temperature decrease assuming that you were scanning on a 1.5 T machine? On the other hand, you might be able to move across the hall from the 1.5 Tesla scanner and use a 3.0 Tesla machine to image the patient. What is the change in the ratio of the number of spins in the upper energy level to the lower energy level due to this change in B0 (assuming a 37 C person)?b) How does this answer change if you were interested in measuring the signal from carbon 13?Problem 2: Playing with T1 and T2 contrastUsing the table of tissue parameters below, we will investigate how the choice of pulse sequence parameters can change the contrast in the image. We will assume that the pulse sequence is the following:  - TI - /2 – TE/2 -  - TE/2 – acq and make use the relationship: S = 1 – 2exp -TI/T1 ) exp -TE/T2, where S is the MR signal strength, is the proton spin density, TR the repetition time and TE the echo time.Tissue T1 T2Gray matter 1.2 s 70 ms .98White matter 800 ms 45 ms .80a) Plot the signal equation versus TI for both white matter and gray matter assuming TE = 50 ms. At what inversion time can you completely null the signal from gray matter?What is the phase of the white matter signal at this value of TI (positive or negative)?b) In another figure, plot the signal difference versus TI between white matter and gray matter (given the constant TE = 50 ms). At what TI is this difference maximized? How does this relate the tissue T1 s? c) Now set TI = 400 ms. In another figure, plot the signal difference between white matter and gray matter as TE is changed. At what TE is this difference maximized?Problem 3: Investigation of Diffusion Weighting and Signal-to-Noise RatioIn the lecture we saw that the signal intensity in a diffusion-weighted experiment was given by Si = exp(-biADC), where S is the signal, ADC is the apparent diffusion coefficient, and b is the b-value. Let’s define Li = ln Si and L0 – ln S0, where ‘0’ denotes no diffusion weighting and ‘i’ defines which b-value we are talking about. If we are using only two b-values (b = 0 and some other b  0) then we can write bLLADC01The variance of the ADC can then be written as )1()(122022222210bADCLLADCeIbbwhere  is the sample standard deviation for a reasonable SNR (>2). Given this, we can write the SNR for the diffusion experiment as 02/12)1(IebADCADCSNRbADCADCADC.Given this: a) find the optimum b-value to use for an ADC of 0.45  10-5 cm2/s (this is thevalue of the ADC in ischemic brain tissue). Now find the optimum b-value for normal brain tissue (say, 0.9  10-5 cm2/s). b) At the optimum bADC value, what is the SNR of the diffusion-weighted image in terms of that for the b=0


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