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HST.583 Functional Magnetic Resonance Imaging: Data Acquisition and Analysis
Fall 2006
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HST.583: Functional Magnetic Resonance Imaging: Data Acquisition and Analysis, Fall 2006
Harvard-MIT Division of Health Sciences and Technology
Course Director: Dr. Randy Gollub.
HST.583: Functional MRI Data Acquisition and Analysis
Problem Set 3
MRI Physics
Problem 1: Advantages of using larger magnetic fields
a) 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 to
this 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 contrast
Using 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: π - TI - π/2 – TE/2 - π - TE/2 – acq and make use the
relationship: S = ρ(1 – 2exp -TI/T1 ) exp -TE/T2, where S is the MR signal strength, ρ is the
proton spin density, TR the repetition time and TE the echo time.
T1
T2
ρ
Tissue
Gray matter
1.2 s
70 ms
.98
White matter
800 ms
45 ms
.80
a) 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?
Cite as: Dr. Randy Gollub, HST.583 Functional Magnetic Resonance Imaging: Data Acquisition and Analysis, Fall 2006.
(Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed MM DD, YYYY). License: Creative
Commons BY-NC-SA.
Problem 3: Investigation of Diffusion Weighting and Signal-to-Noise Ratio
In 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
ADC =
L1 − L0
b
The variance of the ADC can then be written as
2
≈
σ ADC
1 2
σ2
2
(
+
)
=
(1 + e 2bADC )
σ
σ
L0
L1
2
2 2
b
b I0
where σ is the sample standard deviation for a reasonable SNR (>2). Given this, we can
write the SNR for the diffusion experiment as
SNR ADC =
ADC
σ ADC
=
I0
bADC
.
2 bADC 1 / 2
(1 + e
) σ
Given this: a) find the optimum b-value to use for an ADC of 0.45 × 10-5 cm2/s (this is
the value 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 image?
Cite as: Dr. Randy Gollub, HST.583 Functional Magnetic Resonance Imaging: Data Acquisition and Analysis, Fall 2006.
(Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed MM DD, YYYY). License: Creative
Commons BY-NC-SA.