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### QUESTIONS» Basic Physics

Where does the MRI signal come from? This section explores the basic physics of magnetic resonance imaging.

# Time Constants T1 and T2

## What are the time constants T1 and T2?

T1 is the spin-lattice or longitudinal relaxation time, and T2 is the spin-spin or transverse relaxation time.

T1 and T2 are magnetic timing parameters which differ from one tissue to the next. They can be used as a source of contrast in MRI images.

The T1 and T2 time constants dictate the shape of the exponential recovery and decay curves of the longitudinal and transverse magnetisation, respectively.

M at thermal equilibrium, followed by a 90° RF excitation pulse.

Consider the net magnetisation vector of a group of spins. At thermal equilibrium (M = M0), it is aligned with the external magnetic field, B0. Now consider the rotation of the net magnetisation vector by 90° (π/2 radians). The net magnetisation will precess around the external magnetic field direction (diagrams are drawn in the rotating frame of precession). Eventually, the net magnetisation vector will return to its thermal equilibrium position. The T1 and T2 time constants (measured in milliseconds) describe how this happens; the T1 time constant describes the recovery of the Mz component of the net magnetisation vector, and the T2 time constant describes the decay of the Mxy component of the net magnetisation vector.

(M does not simply "rotate" back to M0, because the T1 and T2 processes are separate, and the T1 and T2 times for a tissue type are not the same. Mxy decays away faster than the regrowth of Mz; T2<T1. Typically T1 ≈ 5T2.)

Decay of Mxy (time constant T2), and recovery of Mz (time constant T1).

The T1 time is related to the transfer of energy from a nuclear spin system to its environment (historically called the lattice, because this process was first observed in crystals). The transfer of energy to the "lattice" is not spontaneous—T1 "relaxation" can only occur when a proton encounters an oscillating magnetic field at (or near) the Larmor frequency. The frequency of the oscillating magnetic field a molecule produces is dependent on how fast it moves (rotation, translation, vibration). The source of such a field is other protons in the same molecule or in different molecules. The energy transferred to the lattice causes a very slight rise in temperature. This loss of energy to the "lattice" is caused by the same process which forms the net magnetisation vector in the first place: Thermal energy of molecules in within which nuclear spins reside.

The T2 time is related to the effect of nuclear spins on each other. This may sound alot like the T1 process described above, but it is slightly different. The spin-spin interation purely refers to the loss of phase coherence of spins as they interact with each other via their own oscillating magnetic fields. (Phase coherence means spins are all precessing together.) The slight changes in magnetic field which a proton experiences causes its Larmor frequency to change. As a result, the precession of spins moves out of phase and the overall net magnetisation is reduced. This T2 "relaxation" can occur without loss of energy to the spin system (spins going out of phase only), and it can occur with loss of energy to the spin system at the same time (which is T1 relaxation, see above).

T1 relaxation occurs when a proton encounters an oscillating magnetic field at (or near) the Larmor frequency. This field depends on the molecular environment, which differs from one tissue to the next. As a result, different tissues have different T1 relaxation times. This can be used as a source of contrast between tissues in MRI images.

A similar argument can be made for the T2 relaxation time. Different tissues have different T2 relaxation times, which can also be used to generate contrast between different tissues in an MRI image.

Example recovery curve of Mz (time constant T1).

The decay of Mxy and the recovery of Mz (see response to the first answer on this page) can be described mathematically.

After a 90° rotation of M0, the movement of these components of M are given by

Mz(t) = M0(1-exp(-t/T1)) and

Example decay curve of Mxy (time constant T2).

Mxy(t) = Mxy(0)exp(-t/T2).

The Mz equation says the following: the changing magnitude of the longitudinal component of M is an exponential process. However, it is an increasing exponential (produced by the one minus exponential decay part), towards an asymptote equal to M0 (thermal equilibrium). The time constant T1 (simply the denominator of the fraction by which the exponential is powered) describes the speed of the recovery; it describes the shape of the recovery curve.

The Mxy equation says the following: the changing magnitude of the transverse component of M is an exponential process. Mxy starts at a value at time = 0 (equal to M0 in this case), and decays exponentially away to zero. The time constant T2 (again simply the denominator of the fraction by which the exponential is powered) describes the speed of the decay; it describes the shape of the decay curve.