
ELG 7173  Assignment #2
 Nuclear Medical Imaging
Figure Q1: Diagram for question 1.
Figure Q1 shows the nuclear medical imaging
projection data acquired from a simple
object with nine regions R_{1} to R_{9},
and twenty four data projections,
P_{1} to P_{24}.
The projection calculations are indicated by the arrows;
for example, P_{1} is calculated as
the photons received along the axis through
R_{1}, R_{4} and R_{7}.
Assume each region is 2 cm wide, no matter whether it
is traversed from the edge or sides.
Consider a SPECT Imaging model, where
region R_{5} contains a radioisotope, and is
emitting 1000 γ particles along each of the eight projection
lines
per unit time. All other regions have no activity.
Each projection is the number of γ's received
per unit time.
 Initially, consider all regions to have
zero attenuation.
Calculate the value of all projections
P_{1} to P_{24}, for this condition.

Using simple (ie. unfiltered) backprojection, calculate
the reconstructed activity in each region,
for these projection data.
 Now, consider regions
R_{1}, R_{2}, and R_{3} to
have attenuation of 0.2 cm^{1},
regions
R_{4}, R_{6},
R_{7}, R_{8}, and R_{9} to
have attenuation of 0.1 cm^{1}.
Calculate the value of all projections
P_{1} to P_{24}, for this condition.

Using simple (ie. unfiltered) backprojection, calculate
the reconstructed activity in each region,
for these projection data.

Comment briefly on the impact of attenuation
on Nuclear Medical Images
 Nuclear Medical Imaging

In class we have discussed three kinds of collimation:
1) Pinhole collimation
2) Electronic collimation
3) Parallel hole collimation.
Discuss each collimation technology, describing
one medical imaging application.

Discuss two issues (of the three presented in class)
to be considered in the choice of isotopes
for nuclear medical imaging (max 100 words/answer).

A new detector technology is invented for PET which allows
the difference in timing of coincidence events to be measured
to an accuracy of 1 ns.
Describe how such information can be used to improve PET image
reconstruction.
Specifically, describe in general terms how backprojection
reconstruction could be modified to use this information.

Why is attenuation correction important for Nuclear Medical
image reconstruction?
In class we discussed three methods to estimate the data
necessary for attenuation correction.
Describe one method to estimate attenuation.
 MRI Fourier transform imaging
Figure Q2: Diagram for question 2.
Figure Q2 shows two contrasting regions, A and B, in an otherwise
empty space. The MRI z axis (frequency encoded)
and
x axis (phase encoded)
are shown. The timing diagram for the
pulse sequence (90°FID) is also shown.
The magnetic field B_{0} is 1.5T. All gradient
field strengths are 1 Gauss/cm. The RF field strength
is 0.05 Gauss, and the shape of the RF pulse is
rectangular. Hydrogen is being imaged (γ=42.58 MHz/T)

What is the bandwidth of the RF receiver required
to image the field of view including A and B.

It is desired that the phase difference between A
and B be 180°.
Which timing parameter does
this contrain, and what is its value?

The RF pulse length is set to achieve the 90° rotation in
Nuclear Magnetic Field.
Which timing parameter does
this contrain, and what is its value?
 MRI Imaging Protocols

Consider the following MRI pulse sequences:
90°FID and Spinecho.
Explain, using graphs as necessary, how SpinEcho
allows distinguishing tissues which differ in
T_{2} but not T_{1}, while
90°FID does not.
Two tissue regions (A and B) are being imaged with a SpinEcho
pulse sequence, and have parameters
T_{1}, T_{2}, and ρ^{*}
as follows:
Tissue
 T_{1} (s)
 T_{2} (ms)
 ρ^{*}


A
 1.0
 80
 50

B
 0.8
 120
 40

The MRI signal for SpinEcho imaging is proporional to
ρ^{*}
×
( 1 − exp(−T_{R} / T_{1} ) )
×
exp (−T_{E} / T_{2} )

Derive an expression for the conditions
(T_{R} and T_{E}) which maximizes
the contrast between regions A and B.
Note: you should be able to solve for one parameter.
The solution for the other parameter can be left as
a nonlinear differential equation.

A particular MRI machine has settings of
T_{R} of 1.0s, 1.5s and 2.0s
and settings of
T_{E} of 100ms and 150ms.
Which settings should be chosen for maximum contrast
between tissues A and B?
 MRI Simulation (Bonus question)
The following Matlab function mri_signal
simulates an MRI RF signal output, for a given
image matrix img,
scale [pixels/cm],
xGradient [Gauss/cm],
yGradient [Gauss/cm],
main Magnetic field B0 [Tesla],
T2^{*} T2star,
Simulation time Tlim [µs], and
Sample time Ts [µs].
function rf=mri_signal( img, scale , Gx, Gy, Ty, B0, T2star, Tlim, Ts)
% simulate rf signal for MRI pulse sequence
% rf=mri_signal( sig_params, scale , Gx, Gy, B0, T2star, Tlim, Ts)
% img [szY x szX] image of MRI signal (ie rho*(1exp(TR/T1)*exp(TE/TR) for spin echo)
% scale image scale (pixels/ cm)
% Gx is x gradient (freq encoding) in Gauss/cm
% Gy is y gradient (phase encoding) in Gauss/cm
% Ty is phase encoding time in microseconds
% B0 is magnetic field in Telsa
% T2star T2* spin relaxation in microseconds
% Tlim the recording time (in microseconds)
% Ts RF signal sampling time resolution (microseconds)
Gx= Gx*1e4; Gy= Gy*1e4; %convert to Tesla
Gyromagenticratio = 42.6; %Mhz/Tesla
% Note this assumes a point phase source for each voxel,
% this introduces artefacts into the images
[szX, szY] = size( img );
phase= 2*pi*Ty*Gy*Gyromagenticratio*( (szY1)/2:(szY1)/2 )*scale ;
imgX= exp( 1j*phase ) *img;
f_min= Gyromagenticratio*(B0  Gx*(szX1)/2*scale );
f_max= Gyromagenticratio*(B0 + Gx*(szX1)/2*scale );
ft_img= interp1( linspace(f_min, f_max, szX ), imgX, ...
0:1/Tlim:1/Ts, ... % size of FFT domain,
'linear',0); % freq is zero outsize domain
ft= [ ft_img, fliplr( conj(ft_img(2:end1)) ) ];
rf= ifft(ft);
if std(imag(rf))>1e10; error('something wrong with code'); end
time1= 1:Tlim/Ts;
rf=real(rf(time1)).*exp( Ts*time1/T2star );
The following sample code allows reconstruction
of an MRI image based on multiple simulations
using this software.
spc=.05 ; rlim=1;
[x,y]= meshgrid(rlim:spc:rlim,rlim:spc:rlim);
plen= size(x,1);
img = .0*((x.^2 + y.^2) > rlim);
img( x >.45 & x<.65 & y>.05 & y<.45) =1;
img( x >.55 & x<.25 & y>.45 & y<.65) =1;
scale=1;
T2star= 50;
B0= 0.5; % To increase B, decrease Ts;
G= 10;
Gyromagenticratio = 42.6; %Mhz/t
Ts= 0.01;
Tlim= 100;
szX= size(img,2);
max_f= Gyromagenticratio*(B0 + G*1e4*(szX1)/2*scale ) *Tlim/2;
min_f= Gyromagenticratio*(B0  G*1e4*(szX1)/2*scale ) *Tlim/2;
freq_range= floor(min_f):ceil(max_f);
mri= [];
for Ty= you need to figure this out here
rf = mri_signal(img, scale , G, G, Ty, B0, T2star, Tlim, Ts);
ft= fft(rf);
mri= [mri;ft(freq_range)];
end
fm= ifftshift(ifft(mri));imagesc(abs(fm)');
Question: Find an appropriate selection of time steps Ty
to reconstruct the image. Show your reconstructed image, and briefly
describe any defects in it, in terms of the simulation or reconstruction
software.
Last Updated:
$Date: 20070202 14:12:04 0500 (Fri, 02 Feb 2007) $
