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Basic Mathematics of image reconstruction in MRI

Magnetic resonance imaging (MRI) is able to resolve the response of a system to a magnetic field in space, thereby creating images. This blog introduces a simplified description of magnetic resonance, the vector model using the Bloch equations. Although there are aspects of magnetic resonance that cannot be explained by this phenomenological description (mainly relevant in spectroscopy), it is good enough to see how images are formed, and to design pulse sequences to manipulate systems to produce images with different contrast types and information content.


The Bloch Equations

The Bloch equations in the laboratory frame can be stated as a vector cross product without relaxation as


is the magnetisation from the sample,

is the external magnetic field and γ is the gyromagnetic ratio. Relaxation can be included by re-writing this as

where R_1 and R_2 are longitudinal and transverse relaxation rate constants respectively. M_{Eq} is the equilibrium magnetisation, whose only non-zero entry is the longitudinal term. These coupled differential equations can also be written in matrix form as

or, more concisely,

In the laboratory frame, we can make the identification

where ω0 is the Larmor frequency given by ω0 = γB0 with B0 the static applied field,

ωcs is the chemical shift. G(t) is the B0 field gradient and X is the spatial coordinate.

It is this term which is responsible for spatial encoding in MRI. We can then identify the transverse components of the external field B(t) as being due to radiofrequency (RF) irradiation. It is useful to deal with complex-valued transverse magnetisation as a single quantity (representing −1 quantum coherence which motivates a change of basis:

This change of basis can be brought about by the matrix A, defined as

so that in the new basis the Bloch equations are

By evaluating the inverse of A we obtain

with a new basis for the RF field defined as

Whilst we will deal with more general solutions later, it is insightful to consider the solution to the Bloch equations in the absence of relaxation or RF irradiation. We have then only a single equation

In the simplest case that the gradient field is constant this has the solution

This solution has 3 terms; Larmor precession, chemical shift evolution and precession due to the field gradient. It is dominated by Larmor precession, which is at a much higher frequency than the other terms. It is therefore more convenient to work in a rotating reference frame to remove the large contribution to M}(t) from Larmor precession. We therefore define a new variable

where Rz is a rotation operator about the z axis. In Cartesian space it is given by

Equivalently, the laboratory frame magnetisation is

In the Cartesian basis the Bloch equations are then

The solution to the (transverse) Bloch equations in the absence of RF irradiation or relaxation in the rotating frame is therefore simply

If there were no gradient field applied, this solution would not have a spatial dependence and would instead (inclusive of relaxation) read

If a Fourier transform were taken, then a spectrum of -1 quantum coherence would be obtained, which is the NMR spectrum

We are more interested, however, in how we can make use of the spatial information encoded by the field gradients.


Image Formation

We will henceforth omit the subscript r (for rotating frame) and assume the use of a rotating frame with frequency ω0. Let us assume that we have a sample in a magnet and will detect protons, and that the sample has a distribution of proton density ρ(X). We will also ignore chemical shift, which is equivalent to incorporating it into the rotating frame frequency. After an excitation pulse of 90 degrees the transverse magnetisation is

If a constant field gradient is then applied this evolves according to

Therefore the frequency of resonance for spins is determined by their position in space by the use of the magnetic field gradient. The field gradient has encoded position.

The receiver detects the total signal from the entire sample, such that we observe

We now introduce a new variable k defined (for a constant gradient amplitude) as

such that in the space of k

The observed signal M+(k) is therefore the Fourier transform of the proton density in the sample. An image of the proton density in the sample can therefore be obtained by taking the inverse Fourier transform of the observed signal

Although this derivation has ignored relaxation and chemical shift, it serves to establish a fundamental relationship in MRI, that a field gradient allows an image to be extracted from the observed FID by inverse Fourier transformation of the signal.

As the simplest example, suppose there is a region in the centre of the scanner with constant spin density up to some boundary then nothing detectable beyond, such as a ball full of water. In one dimension the spin density being imaged is then a top-hat function. Its Fourier transform is well known as the sinc function. So the observed k-space signal would be a sinc function, maximum at k=0 and decaying to zero once k gets large enough. Rationally, this makes good sense: at the centre of k-space (k=0, implying no no field gradient), all resonance frequencies are the same so the signal adds constructively. As we go to greater k, the frequency distribution gets broader and so too does the effect of destructive interference.

In a real imaging experiment, ρ(X) can be manipulated by RF pulses so that different pulse sequences generate differently contrasted images with different chemical or biological information content. The "measurement" coordinate k depends on X by gradient amplitude and time and represents the phase accumulated due to field gradients. Both gradient amplitude and time for which the field gradient is applied are controllable. If the amplitude is constant and a signal recorded in time (i.e. simply turn on the ADC), the line of k-space recorded is frequency-encoded. If the time for which the gradient pulse is fixed and we carry out an array of experiments with different gradient amplitudes, recording a single k-space point per experiment, the signal is phase encoded. The two methods can be mixed to obtain images in 2 or 3 dimensions.


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