You learned in Chapter 1 that magnetic resonance imaging is an imaging modality which is primarily used to construct pictures of the NMR signal from the hydrogen atoms in an object. In medical MRI, radiologists are most interested in looking at the NMR signal from water and fat, the major hydrogen containing components of the human body.

The principle behind all magnetic resonance imaging is the resonance equation, which shows that the resonance frequency ν of a spin is proportional to the magnetic field, B_{o}, it is experiencing.

Recall from the spin physics chapter that γ is the gyromagnetic ratio.

For example, assume that a human head contains only three small distinct regions where there is hydrogen spin density. In reality the entire head would contain signal. When these regions of spin are experiencing the same general magnetic field strength, there is only one peak in the NMR spectrum.

If each of the regions of spin was to experience a unique magnetic field we would be able to image their positions. A gradient in the magnetic field is what will allow us to accomplish this. A magnetic field gradient is a variation in the magnetic field with respect to position. A one-dimensional magnetic field gradient is a variation with respect to one direction, while a two-dimensional gradient is a variation with respect to two. The most useful type of gradient in magnetic resonance imaging is a one- dimensional linear magnetic field gradient. A one-dimensional magnetic field gradient along the x axis in a magnetic field, B_{o}, indicates that the magnetic field is increasing in the x direction. Here the length of the vectors represent the magnitude of the magnetic field.
The symbols for a magnetic field gradient in the x, y, and z directions are G_{x}, G_{y}, and G_{z}.

The point in the center of the magnet where (x,y,z) =0,0,0 is called the isocenter of the magnet. The magnetic field at the isocenter is B_{o} and the resonant frequency is
ν_{o}.
If a linear magnetic field gradient is applied to our hypothetical head with three spin containing regions, the three regions experience different magnetic fields.
The result is an NMR spectrum with more than one signal. The amplitude of the signal is proportional to the number of spins in a plane perpendicular to the gradient. This procedure is called frequency encoding and causes the resonance frequency to be proportional to the position of the spin.

This principle forms the basis behind all magnetic resonance imaging. To demonstrate how an image might be generated from the nmr spectra, the backprojection method of imaging is presented in the next section.

Backprojection imaging is a form of magnetic resonance imaging. It was one of the first forms of magnetic resonance imaging to be demonstrated. Backprojection is an extension of the frequency encoding procedure just described. In the backprojection technique, the object is first placed in a magnetic field. A one-dimensional field gradient is applied at several angles, and the NMR spectrum is recorded for each gradient. For example, say you wished to produce an YZ plane image of an object. A magnetic field gradient in the +Y direction is applied to the object and an NMR spectrum is recorded.

A second spectrum is recorded with the gradient now at a one degree angle to the +Y axis. The process is repeated for the 360° between 0° and 359°. Once this data has been recorded the data can be backprojected through space in computer memory.

Once the background intensity is suppressed an image can be seen. The actual backprojection scheme is called the inverse Radon transform.

In a conventional 90-FID imaging sequence this procedure might be applied with the aid of the following pulse sequence.
Varying the angle γ of the gradient is accomplished by the application of linear combinations of two gradients.
Here the Y and X gradients are applied in the following proportions to achieve the required frequency encoding gradient G_{f}.

For the backprojection technique to be a viable tomographic imaging technique we need to have the ability to image the spins in a thin slice.
The G_{z} gradient in the last graphic accomplishes this.
The following section will describe how slice selection is accomplished.

Slice selection in MRI is the selection of spins in a plane through the object. The principle behind slice selection is explained by the resonance equation. Slice selection is achieved by applying a one-dimensional, linear magnetic field gradient during the period that the RF pulse is applied. A 90° pulse applied in conjunction with a magnetic field gradient will rotate spins which are located in a slice or plane through the object. Picture what this would look like if we had a cube of small net magnetization vectors. To understand this we need to examine the frequency content of a 90° pulse. A 90° pulse contains a band of frequencies. This can be seen by employing the convolution theorem. The frequency content of a square 90° pulse is shaped as a sinc pulse. The animation window displays the real components of this pulse. The amplitude of the sinc function is largest at the frequency of the RF which was turned on and off. This frequency will be rotated by 90° while other smaller and greater frequencies will be rotated by lesser angles.

The application of this 90° pulse with a magnetic field gradient in the x direction will rotate some of the spins in a plane perpendicular to the x axis by
90°. The word some was used because some of the frequencies have a B_{1} less than that required for a 90° rotation.
As a consequence the selected spins do not actually constitute a slice.

A solution to the poor slice profile is to shape the 90° pulse in the shape of a sinc pulse. The sinc pulse, as first seen in Chapter 5, has a square frequency distribution. The animation window displays the real components of this function.

A backprojection tomographic image can be achieved by the application of the following pulses.
An apodized sinc pulse shaped 90° pulse is applied in conjunction with a slice selection gradient.
A frequency encoding gradient is turned on once the slice selection pulse is turned off.
The frequency encoding gradient is composed of a G_{x} and G_{y} gradient in this example.
The FIDs are Fourier transformed to produce the frequency domain spectrum, which is then backprojected to produce the image.

The backprojection imaging technique is highly educational but never used in state of the art imagers. Instead, Fourier transform imaging techniques are used. These techniques are described in the next chapter.

- A sample contains water at two locations, x = 0 cm and x = 2.0 cm. A one-dimensional magnetic field gradient of 1 G/cm is applied along the x-axis during the acquisition of an FID. What frequencies (relative to the isocenter frequency) are contained in the Fourier transformed spectrum?
- An NMR spectrum is recorded from a sample containing two water locations. The frequency encoding gradient is 1 G/cm along the y-axis. The spectrum contains frequencies of +1000 Hz and -500 Hz relative to the isocenter frequency. What are the locations of the water?
- You want to excite spins in an xy-plane located at z = -5.0 cm. The resonance frequency at the isocenter is 63.85 MHz and your slice selection gradient is 1 G/cm. Describe in detail the RF pulse which should be used.
- A sample contains water at two locations, y = 1.0 cm and y = -2.0 cm. A one-dimensional magnetic field gradient is applied along the y-axis during the acquisition of an FID. The frequency encoding gradient is 1 G/cm. What frequencies (relative to the isocenter frequency) are contained in the Fourier transformed spectrum?
- An NMR spectrum is recorded from a sample containing two water locations. The frequency encoding gradient is 0.5 G/cm along the z-axis. The spectrum contains frequencies of -1000 Hz and +500 Hz relative to the isocenter frequency. What are the locations of the water?
- You want to excite spins in an xy-plane located at z = -2.0 cm. The resonance frequency at the isocenter is 63.85 MHz and your slice selection gradient is 2 G/cm. Describe in detail the RF pulse which should be used.

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