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### TOOLS» Time Constants

An understanding of time constants allows easier understanding of MRI signal equations. This tool allows you to build up the equations describing the change of magnetisation in basic pulse sequences.

# Time Constant Tool: y = 1-exp(-x/400)

An exponential decay may be turned into an exponential recovery by subtracting the decay equation from 1.

The construction of this recovery curve is simply 1 (representing full recovery,) minus exponential decay. Hey presto: exponential recovery! This represents recovery after a 90° pulse. The 90° pulse saturates the spin system (sets all longitudinal magnetisation to zero). The equation of this curve is, therefore, the equation of saturation recovery:

S = 1 - exp(-TR/T1)

where S is signal intensity (normalised to M0), TR is the repetition time and T1 is the longitudinal relaxation time.

## How do we define the time constants for a particular tissue?

Or: what value does τ take? (See the introduction. In the example above, τ = T1 = 400.) This depends on the sample / voxel under investigation. For longitudinal recovery (of Mz) after a 90° pulse, the magnetisation in this direction is described by this equation:

Mz(t) = M0(1-e-t/τ).

We define τ (which in this case is T1) to be equal to the time for the magnetisation to recover to (1-e-1) of M0, or 63.2% (i.e. when t = τ). On the following graph, the recovery is normalised to M0.

For transverse decay (Mxy) after a 90° pulse, the equation is

Mxy(t) = M0(e-t/τ)

and so τ (which in this case is T2) is equal to the time for the magnetisation to decay to e-1 of M0, or 36.8% (i.e. when t = τ).

Navigate below to the next equation in the list: 1 - 2·exp (-x/400).

Navigate through the tool...

Other sections of this tool:

Remember, T1, T2 and T2* are time constants.