Do You Know about the Larmor Frequency

Anyone who has sat on a swing moving legs and trunk in slow rhythm to swing ever higher, or who was the "swing pusher on duty" for a little sister or brother, daughter, or son, realizes that objects have a certain inherent frequency at which they swing (or resonate): their resonance frequency. If you do not know or feel this frequency or are not able to move your body accordingly (like a small child), you will never be able to swing on your own. If you are, however, able to apply the frequency appropriately, you will go a long way with very little force. The same holds true for atoms and molecules, of course. The nuclei of atoms spin about their axes with high frequency and some nuclei (such the hydrogen nucleus— the proton) have resultant magnetic moments. We are actually looking at small rapidly spinning "magnets." As the atoms move randomly, these "magnets" tumble about chaotically and thus neutralize each other's magnetic fields. A call to order is necessary before anything good can come out of this.

I Magnetic Resonance Tomography

Y-axis

Z-axis

Y-axis

Z-axis

Small Magnets Gradient

Magnetic field increasing in strength along the " Z-axis active during B1 pulse (Z-gradient)

Salami slices with increasing Larmor frequency

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RF pulse B1 of the LarmorfrequencyE

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RF pulse B1 of the LarmorfrequencyE

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wwwvvvwwwwwwwwvvwvvwvvwwwwvvwv c Temporary magnetic field of varying strength along the Y-axis active after B1 pulse (Y-gradient)

Y-axis

Magnetic field increasing in strength along the " Z-axis active during B1 pulse (Z-gradient)

Salami slices with increasing Larmor frequency

-----------Z-axis c Temporary magnetic field of varying strength along the Y-axis active after B1 pulse (Y-gradient)

Y-axis

Y-axis

Salami Mri

Magnetic field increasing in strength along the X-axis active during read-out (X-graaient)

Selected salami slice

Cubicles (voxels) of same phase but different Larmor frequency A-F

Receiving antenna coil

Magnetic field increasing in strength along the X-axis active during read-out (X-graaient)

Y-axis

Selected salami slice

Cubicles (voxels) of same phase but different Larmor frequency A-F

Receiving antenna coil

You probably remember this physics experiment from back in school: iron dust arranges itself along the lines of a magnetic field. In MR a constant external magnetic field (called B0 by the MR physicists) calls the little nuclear "magnets" to order. The protons align themselves along the axis of the magnetic field and, in addition to their spin, begin to rotate around the axis of the B0 magnetic field much like gyroscopes wobble in the Earth's gravitational field.

f This rotational frequency is identical to the resonance ! frequency, which is also named the Larmor frequency. This frequency varies with the strength of the magnetic field.

The Irishman Whose Frequency We Can-^^ not Do Without tL A Sir Joseph Larmor was an Irish physicist who p j^,_^ taught in Cambridge, England, around the beginning of the last century. One of his special fields was the mathematical theory of electromagnetism. The Larmor frequency is just one of several physical phenomena

a The coordinate system with three axes Z, X, and Y inside the MR machine is shown. Inside of the gantry you can see the salami and the antenna alongside it in which the MR signal is induced. b If a gradient is superimposed on the static field along the Z-axis (Z-gradient), every slice of the salami gets its own "Larmor frequency address." An excitation pulse B1 of a frequency E will now only excite slice E. c Right after the B1 excitation of slice E, a temporarygradient is superimposed along the Y-axis (Y-gradient). As protons within the slice now rotate with different Larmor frequencies, the signals dephase except in the rod that keeps the original frequency. The phase shift persists until read-out. d During read-out, a third gradient is superimposed along the X-axis (X-gradient). Each cube in the rod now has its own "Larmor frequency address." The measured signal of that specific frequency can now be assigned to a specific voxel in the image.

e This is a modern whole-body MRI scanner (by Siemens Medical Systems). ▼

Larmor Fregence Mri

that carry his name. He was a conservative man, at times opposing most of Einstein's ideas and the introduction of baths in his college in Cambridge: "We have done without them for 400 years, why begin now?" As it turns out, he became an avid bather right after the public baths were installed.

What Is So Special about the "External" and "Internal" Magnetic Fields?

The magnets for the external applied magnetic field (B0) are large and incredibly strong (0.5,1.0, or 1.5 tesla, the last corresponding to 30 000 times the force of the natural terrestrial magnetic field). Why do we need such a strong field? Our protons do align themselves along the field axis and wait patiently for coming sensations—they may, however, choose a parallel and an antiparallel orientation. This is where the simple magnet story comes to an end. The parallel orientation is the least energy-consuming, which is why more than half of the protons choose it. The other protons assume the antiparallel orientation. As the external magnetic field increases in power, the antiparallel orientation requires ever more energy and thus becomes less and less popular. The dominance of the parallel protons increases and thereby the magnetization of the examined body. This "internal" magnetic field initially has the same orientation as the external field B0. Its axis corresponds to the longitudinal axis of the MR gantry, also called the Z-axis (Fig. 3.6a). Now the stage is set: Enter a biological sample to examine—how about a nice salami?

How Do We Generate an MR Signal in a Salami?

It so happens that protons (i.e., nuclei of hydrogen atoms)—which can be beautifully studied by MR—are abundant in salamis and other organic material: in excess of 90% of organic material consists of hydrogen. After having been moved into the B0 external magnetic field of the MR system the majority of protons inside the sausage have aligned themselves parallel to B0 and have generated an "internal" magnetic field. If we now want them to tell all, we'd better get them excited. This is done by a radio-frequency pulse (RF pulse), a temporary outer RF magnetic field that oscillates with the Larmor frequency of hydrogen (also called B1 by MR physicists). Remember: Hydrogen protons could not care less about RF pulses of higher or lower frequencies. The longer the B1 RF pulse is active and the stronger it is, the more the axis of the protons is tilted away from the Z-axis into the X-Y-plane. For simplicity's sake, let us consider a pulse that has the power and duration to tilt the proton axis by 90°. As this happens not only to one proton but synchronously to many protons in the salami, the "internal" magnetic field also tilts 90° and rotates with the Larmor frequency of hydrogen, much like a propeller—or the magnet inside the bicycle dynamo (in the X-Y plane; Fig. 3.6a). If you now position a wire coil along the sausage (corresponding to the receive coil or antenna of the MR machine), a measurable alternating current is induced—much like in the coils of a bicycle dynamo.

This current is the MR signal we can start our work with. Remember for later that the field signal is strongest if all protons are in phase ("listen to the same beat") which is always the case right after the B1 pulse. After the RF pulse B1 and the resulting 90° tilt of the "internal" magnetic field, the current measured by the antenna—our signal—decreases again. The reasons are twofold: For one thing, the axis of the "internal" magnetic field moves back to the Z-axis—remember that the "external" magnetic field B0 is always present and is very strong. For another thing, the protons lose the phase synchronization they have been forced into by the RF pulse Bl. As they dephase, the "internal" magnetic field power also shrinks. You will learn more about these processes later.

We now have proof that there are protons inside that salami; of course we had a hunch there would be. To look at slices of the sausage we have to assign the signals to locations in a three-dimensional coordinate system.

Spatial Allocation of the MR signal

The frequency with which I swing or push my swinging child depends, besides other things, on the terrestrial gravity. The Larmor frequency with which I can excite a proton depends on the strength of the magnetic field surrounding it. Magnetic fields can be built asymmetrically so that their strength increases along an axis. These types of fields are called gradients.

Z-gradient: If such a gradient is positioned along the longitudinal or Z-axis of the system (Z-gradient) (Fig. 3.6a) the magnetic field increases along the length of the salami, giving every slice of the sausage a different Larmor frequency address. If we now give the Bl pulse, it excites not the whole salami but only one slice—the one with the Larmor frequency of the Bl pulse (Fig. 3.6b). The bandwidth and form of the Bl pulse determine the thickness of the selected slice.

Y-gradient: After the excitation Bl pulse is over, a second gradient is positioned along the Y-axis of the system (Y-gradient). During the duration of this gradient the protons thus have different Larmor frequencies depending on their position along the Y-axis; that is, they rotate with different speeds. The subsequent phase shift persists after the Y-gradient is turned off again. The sausage slice now consists of rods of different phase (Fig. 3.6c). Here is the analogy to illustrate the phenomenon: If three different cars drive on a three-lane highway and adhere to a speed limit, they stay side by side. Once the speed limit is lifted, they drive with different speeds and the gap between them grows. As the speed limit is enforced again, they drive at the same speed (same Larmor frequency for the protons) and the gap between them (the phase shift) persists. This applies to law-abiding drivers only, of course. The gradient can be designed to leave the Larmor frequency unchanged in one rod that subsequently does not undergo the phase shift. Frequency and phase are thus identical to the original Bl pulse. We will now dice this rod into volume elements (voxels).

X-gradient: The last gradient is switched on during the read-out phase and is positioned along the X-axis (X-gra-dient). It divides the rod into cubes, assigning a Larmor frequency address to each (Fig. 3.6d).

Now we have the single cubes (or voxels) that we need for a two-dimensional image: a selectively excited slice of a defined thickness, and a rod in correct phase that is subdivided into cubes of different Larmor frequencies assignable to locations in a coordinate system. To calculate the image, a separate measurement must be performed for every rod (voxel or pixel row) of the image matrix; that is, for a matrix of 256 x 256 voxels, we need to repeat the process 256 times. The rest is complex electrical engineering.

Analysis of the MR Signal

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