MR image formation
Static magnetic field and Larmor precession
• situating a sample (such as human body) in a strong static magnetic field will cause water hydrogen protons (i.e. spins) to precess at the Larmor frequency: w0 = gB0. (where B0 is by convention along the z-axis)
• In order to ascertain that the spin sample is behaving in a predictable manner, it is essential that the static magnetic field B0 is as homogeneous (uniform) as possible.
•The volume where the static field is spatially uniform to typical +/- 5ppm (parts- per-million) is called the ‘diameter spherical volume’ (DSV) region – (i.e. for a 1.5T scanner, the field in the DSV region is very close to 1.5T, while the field may be higher or lower outside the DSV).
•A strong static magnetic field (1.5T-3T clinical and 4T-11.7T research) is generated by a superconducting magnet/ coil (‘MR hardware’ lecture).
iso-centre
Static magnetic field and Larmor precession
Central Slice through DSV showing |B0|
Calling for a localization method
• With the B0 field applied, all the spins precess at (virtually) the same Larmor frequency and are thus in phase.
• The B1 field pulse excites a volume of spins irrespective of their position in Cartesian coordinate system.
• Then the B1 field is removed. During the relaxation of net magnetization (i.e. excited spins return to the low energy state
– energy is emitted), we receive MR signals that are (virtually) of identical frequency and phase (we see only one peak in the frequency spectrum).
• We can not tell these signals apart – i.e. we do not know from which spatial location they are coming from and what their respective magnitudes are.
• In order to obtain an MRI image, we require a localization method.
Influence of magnetic field gradients on spins
• In MRI, magnetic field gradients are used to encode the spatial position of the spins in the patient by varying the local magnetic field in a controlled manner. This is the basis of MR image formation.
• The gradient is imposed in addition to the static field. This causes the Larmor frequency of the nuclear spins to vary as a function of their position.
• The result is an MR spectrum with more than one signal.
• Normally the position ‘encoding’ is achieved by varying the magnetic field in
a linear manner with axial position. Any axis can be used (i.e. x, y or z).
• The magnetic field gradients are generated by magnetic field gradient coils, (‘MR hardware’).
Influence of magnetic field gradients on spins
• Consider a gradient imposed along the z-axis of the magnet.
20 mT/m
10 mT/m
0 mT/m
Influence of magnetic field gradients on spins
-10 mT/m -20 mT/m
• Such a magnetic field, when applied to a sample of homogeneous material like water, causes the spins on one side of the sample with respect to the z-direction to have a different frequency from spins on the other side of the sample.
• A distribution of frequencies will be obtained along the sample. In the absence of a field gradient, the frequency of the spins in a homogeneous sample is the same.
Magnetic field gradients
• In the presence of a field gradient, spins along a line perpendicular to the field gradient, all have the same frequency. Thus, the resultant signal intensity at a single frequency is the sum of the signals along that line.
• It is very important to realise that the gradient that is being added to the main static field is a gradient of Bz – (i.e. Bz and B0 are along the z-axis)
z-gradient x-gradient
Examples of typical spatial magnetic field Bz variation of real gradient coils. The magnetic field at the iso-center is B0 and the resonant frequency is w0.
Magnetic field gradients
• In the DSV, the magnetic field that we work with always
points along the z-axis (i.e. Bz) with a spatial gradient along
x, y or z-direction. We define the gradient variable G:
G=Gx+Gy+Gz xˆyˆzˆ
!
G =¶Bz ; G =¶Bz ; G =¶Bz x ¶x y ¶y z ¶z
• From the Larmor equation we know that the frequency at which the spin precesses is proportional to the magnetic field strength it sees. Since we are able to make gradients
that change the magnetic field so that the field strength varies linearly in space, then the precession frequency of the magnetization in the rotating frame will be determined by its location in space:
where
æ ¶B ¶B ¶Bö w(x,y,z)=g(B0 +xGx +yGy +zGz)=gçB0 +x z +y z +z z ÷
è ¶x ¶y ¶zø
where the Gn is the slope of the change in magnetic field in the direction of n.
Example
• A spin is located 10cm away from the iso-centre of a 1.0T system in the x-direction, 15cm in the y-direction and 0cm in the z-direction. If the following gradients are
applied simultaneously Gx = 10mT/m, Gy = 20mT/m and Gz = 35mT/m. What is the precessional frequency in MHz at that point? (remember g/2p = 42.58 MHz/T)
w=g(B0 +Gxx+Gyy+Gzz)
= 42.58×(1.0+0.01×0.1+0.02×0.15+0)
= 42.7503MHz Or the offset from resonance is 170.3 kHz.
•Now reverse the direction (sign) of the applied gradients and recalculate the frequency – what do you notice?
The offset is now -170.3 kHz and the frequency is decreased to 42.4097MHz.
The gradients are typically switched on and off.
Pulsing of field gradients
• As we have seen, the effect of the magnetic field gradients is to alter the precessional frequency with spatial position. The gradients are normally pulsed and so the complete effect on a spin at position r! is:
!!!
w(r)=g(B0 +G(t)×rdt)
Where G!(t) is the time-varying gradient vector (field) and r! is the position vector:
!
r = r xˆ + r yˆ + r zˆ xyz
MR image orientation
•In MRI we differentiate between axial, sagittal, coronal and oblique image orientation amongst possible others.
z
Axial
(x – y plane)
Sagittal (y – z plane)
Coronal (x – z plane)
y
x
www..nns.com
Slice selection
• Slice selection in MRI is achieved by applying a linear magnetic field gradient (slice select gradient), which results in frequency encoding of spatial position. A B1 frequency selective pulse can then be used to select a slice of magnetization from the sample. The width and definition of the slice is dependent on the excitation bandwidth of the RF pulse.
• Thus, only magnetization with frequencies within the bandwidth of the RF pulse are excited (tipped into the x – y plane) and thus create a voltage in the receive coil.
ideal
in reality
Slice selection
(a) Time-domain representation of a sine-wave carrier that is amplitude modulated with a low- frequency sinc-pulse; (b) frequency spectrum of the B1 field pulse; (c) B1 excitation pulse (d) selection of the axial slice zs; the slice thickness Dz = |z1-z2| is determined by the combination of the gradient field strength Gz and the frequency bandwidth of the B1 excitation pulse.
Slice selection
• Classically, the slice selection is along the z-axis, in which case axial MR images are obtained. Other slice/axis selections are also regularly used (coronal and sagittal).
• For reasons of simplicity, in the subsequent slides we shall assume that the slice
selection is along the z-axis (i.e. axial). Later on we may discuss the other orientations.
• The frequency profile of the RF pulse, in combination with the magnetic field gradient specifies the thickness of the slice.
Dz = DwRF gGs
• The position of the axial slice is given by the frequency of the RF pulse.
zs = wRF gGs
Is slice selection enough to get an MR image?
• With the slice selection gradient alone, it is impossible to distinguish signals uniquely in reference to their spatial position in the slice, as all signal frequencies are equal to the Larmor precessional frequency. It is therefore
necessary to somehow further differentiate the RF signals.
Phase encoding Frequency encoding
Phase encoding
• Once the axial slice is selected, the ‘phase encoding gradient’ is turned on.
• For axial imaging, by convention, the phase encoding gradient is the y-gradient (i.e.
Gp = Gy).
• The frequency of spin precession as a function of space (that is x – y plane) is
therefore:
w(x,y)=g(B0 +yGy)=w0(x,y)+gyGy
• The phase encoding gradient is used to impart a specific phase angle to a transverse
magnetization vector.
• The specific phase angle depends on the location of the transverse magnetization vector.
Phase encoding
w0(x, y) = g (B0) Slice selected
No gradients
B0
x
y
z
Same frequency and in phase
Phase encoding
w(x,y)=g(B0 +yGy)=w0(x,y)+gyGy Phase encoding
gradient ON
B0
x
y
z
Gp
Different frequency
phase accumulation along y
Phase encoding
w0(x, y) = g (B0) Phase encoding
gradient OFF
B0
x
y
z
Same frequency but out of phase along y
Phase encoding
• So we can cause a vector to accumulate change in phase based on its position by using the phase encoding gradient.
• The amount of phase that it acquires is dependent on the strength of the gradient and the length of time that gradient is applied. Thus, the phase f acquired is based on the following equation:
f = gGy yTy
gradient pulse Gy instead of the length of time the gradient pulse Ty is applied.
p=y
• The same change in phase could be accomplished if we changed the strength of the
• With the phase encoding gradient we can obtain more than one signal peak in the frequency spectrum. Unfortunately, we are still unable to uniquely differentiate MR signals along the spatial x-axis, as the spins along this axis have same frequency and are all in phase.
• For unique identification of MR signals in the x – y plane (2 dimensions), we need one more gradient, i.e. one along the x-axis!
Frequency encoding
•After slice selection and pulsed application of the phase encoding gradient, the frequency encoding gradient is turned on.
• We assume here that the frequency encoding gradient is the x-gradient (i.e. Gf = Gx).
• The frequency of spin precession as a function of space (that is x – y plane) is
therefore:
w(x,y)=g(B0 +xGx)=w0(x,y)+gxGx • The signal is in the form of a free induction decay (FID).
Frequency encoding
• Now, we are in position to uniquely localize the spin pockets in space as function of their unique frequency and phase.
w(x,y)=g(B +xG )=w (x,y)+gxG 0x0 x
Frequency encoding gradient ON
Gf
B0 Frequency and phase in the axial 2D slice (x
– y plane)
Normally, the MR signal is acquired during the application of the frequency encoding gradient (more in upcoming lectures)
Signal processing
• In MRI, the time-domain FIDs are received (by a receiver unit) and Fourier transformed into frequency domain.
• Using the frequency and phase information, with some straightforward maths one can deduce the magnitudes and 2-dimensional spatial positions of the MR signals. This ultimately leads to the formation of an MR image.
• In reality, a 1-dimensional Fourier transform (1DFT) is capable of processing a single net magnetization vector located somewhere within an example 3×3 space.
Signal processing
• Unfortunately a 1DFT is incapable of this task when more than one vector is located within the 3×3 matrix at a different phase encoding direction location.
• There needs to be one phase encoding gradient step for each location in the phase encoding gradient direction.
• The point is you need one equation for each unknown you are trying to solve for. Therefore if there are three phase encoding direction locations we will need three unique phase encoding gradient amplitudes and have three unique free induction decays. For example: If we wish to resolve 256 locations in the phase encoding direction we will need 256 different magnitudes of the phase encoding gradient and will record 256 different free induction decays – (i.e. 256 measurements).
Maths of image formation
• The phase shift of the magnetization at a point along that phase encoding gradient is given by:
f(y)=gnTgp y 1
• where the gradient is being applied in the y direction, n stands for the nth measurement being made, T is the length of time the phase encoding gradient is applied and gp is the size of the step of the amplitude of the gradient.
• n usually varies from -(N-1)/2 to N/2 where N is the number of resolution steps to be acquired in the phase encoding direction.
Maths of image formation
• The signal without a gradient applied in the rotating frame will be the following:
S(t) = r
• where r is the component of the magnetization vector in the x – y plane and is
the sum of all the signal in our imaging plane.
• If we apply a phase encoding gradient along the y direction prior to the period we are acquiring the signal, then the spin at a given point along y, yo, has the following signal on the nth measurement:
Sn (t) = r(y0 )eignTg p y0
3
2
• This equation makes use of the fact that we can describe a 2D vector rotation as an exponential.
Maths of image formation
•If we add the frequency encoding gradient during the readout (data acquisition), then the signal precesses in time if it is not exactly at the centre of the frequency encoding gradient.
• We will assume that the magnetization vector we are observing is at the spatial location (xo, yo). During the data acquisition following the phase encoding gradient that was applied in equation 3, we get the following signal for the magnetization:
Sn (t) = r(x0, y0 )eignTg p y0 eigGx x0t 4
=r(x y )eig(nTgpy0+Gxx0t) 0, 0
Maths of image formation
•The actual signal received is the sum of all signals from the individual magnetization vectors at each location in space. Mathematically this is described as a 2D integral over x and y.
¥¥
S (t)= òòr(x,y)eig(nTgpy+Gxxt)dxdy 5
-¥-¥
• The imaging problem is to recover the spatial distribution of r(x,y) from a set of measured signal Sn(t).
n
k – space
• We can turn equation 5 into a Fourier transform. To do this we need to define two
new variables:
kx =gGxt 6 2p
ky =gngyT 2p
• Sn(t) is a function of two parameters, n and t, which are scaled versions of ky and kx. Thus, by suitably stretching and/or compressing the coordinates of Sn(t), the 2D function S(kx,ky) can be obtained.
• Using equation 6, equation 5 becomes:
Fourier Transformation
¥¥
S(k ,k )= òòr(x,y)ei2p(kxx+kyy)dxdy
-¥-¥
7
xy
• and now Eq.7 looks like a 2D Fourier transform with conjugate variables kx, ky, x and y. kx and ky are spatial frequencies.
• To obtain the image of the distribution of the magnetization vector, r(x,y), we need to perform the inverse Fourier transform.
r(x, y) = òòS(kx ,ky )e-i2p (kx x+ky y)dkxdky
8
MR image reconstruction
There are two important take home messages from this discussion:
1. Reconstruction of MR data into an image happens by using a Fourier
transform.
2. Magnetic resonance image data is acquired in the spatial frequency domain called k-space. This has a lot of important implications that we will discuss as we continue describing MR imaging. One important point that follows now from equation 5 is that k-space is acquired one row at a time. A line of kx points, for a given ky value, is acquired with each individual measurement.
Definitions in MRI
• The selected object slice is considered with thickness d and the square area of the slice that is imaged is the field of view (FOV).
• The position within the slice is determined by the horizontal frequency encode (x) and vertical phase encode (y) direction.
• The slice is thought to be equally partitioned into N x N voxels where the volume
of any voxel is given by:
• The magnitude of the net magnetization associated with a voxel is displayed as
brightness of a point, known as the pixel (surface element).
• The final image constitutes of an array of N x N pixels and is displayed on a computer monitor. Typically, MRI images are presented in digital form with 256 x 256 pixels (or 128 x 128 pixels) and therefore require 256 (or 128) phase encode measurements.
V =d(FOV/N)2 vox
• •
MRI k-space
The distance between points in image space, or image resolution, is measured in units of length while in k-space it is spatial frequency or length-1.
Thus, spatial resolution in image space is determined by how far out in k- space we sample. A small region of k-space being acquired will translate into a low resolution in image space whereas covering a large area in k-space will lead to high resolution in image space.
The field of view (FOV) in image space is determined by the distance between
samples in k-space. Large sampling distances in k-space mean a small field of view in image space whereas a small sampling distance in k-space will provide a large field of view.
Most MRI systems only allow a small number of points to be acquired (512 or lower). If we keep the number of points we acquire for our image constant, then we can discuss the density of our sampling of k-space. For a large FOV image, we will sample around the centre of k-space densely and the rest of k-space coarsely.
•
•
MRI k-space
MRI k-space
• What does this k-space sampling mean in terms of the imaging sequence?
• To travel out further from the centre of k-space, we need to use stronger gradients. Thus, to get high resolution images we need to use strong gradients traversing a larger region of k-space. This implies that the path we take through k-space will be dependent on the gradients.
• Conventional MR imaging acquires a line of k-space at a time. Each data acquisition with a frequency encoding gradient applied will provide such a line. The data points acquired are points in k-space.
• Which line is acquired will be determined by the phase encoding gradient. The centre of k-space in the phase encoding direction is acquired when the phase encoding gradient is zero.
MRI k-space
• Above: k-space conventions for MRI. In k-space, the frequency encoding is taken to be along the x direction, kx while the phase encoding direction is in the y, ky. The centre of k-space is the (0,0) spatial frequency. Phase encoding gradients move you up and down in the y direction with positive gradients moving up and negative gradients moving down. Frequency encoding gradients or gradients along the x direction move you across the space, with positive being toward the right and negative toward the left.