The Bloch equations that describe the signal of a given tissue for a spin echo sequence are $S\left(\mathrm{TR},\mathrm{TE}\right)=K{\rho }_m\left(1-e^{\frac{\mathrm{-TR}}{\mathrm{T1}}}\right)e^{\frac{\mathrm{-TE}}{\mathrm{T2}}}$ Where K is a scaling function, TR is the time between RF pulses, TE is the time between the RF pulse and signal detection, and ${\rho }_m$ T1 and T2 are the tissue properties of the tissue being imaged.

The equation has three segments. The first is simply the proton density. The second segment, which is in parentheses, describes the recovery of longitudinal magnetization. The third segment describes the decay of transverse magnetization.

These equations describe how the signal of a specific tissue (with proton density ${\rho }_m$ and relaxation times T1 and T2) changes with the TR and TE of a spin echo pulse sequence. This useful for predicting the signal of a specific tissue on a specific sequence but not for predicting the contrast between different tissues or for predicting how tissue signal will change with disease. This is because the dependent variables are the time parameters of the sequence which remain constant for a given sequence. In real life the dependent variables are the tissue properties which vary between tissues and change with disease.

If the above equation is separated into three separate segments and then modified so that the tissue properties become the dependent variables (i.e. T1 instead of TR and T2 instead of TE) then it becomes easy compare the signal from different tissues.

The segment describing signal due to proton density becomes $S\left({\rho }_m\right)={\rho }_m$.

The segment describing signal due to the T1 tissue property becomes $S\left(\mathrm{T1}\right)=1-e^{\frac{\mathrm{-TR}}{\mathrm{T1}}}$.

The segment describing signal due to the T2 tissue property becomes $S\left(\mathrm{T2}\right)={\mathrm{TE}}^{\frac{\mathrm{-TE}}{\mathrm{T2}}}$.

These are the filters for each tissue property and describe how signal intensity changes as a function of tissue ${\rho }_m$, T1, and T2. Because each filter can be graphed separately it is easy to demonstrate how signal changes as a function of each tissue property.

The ${\rho }_m$ tissue filter describes how signal changes with ${\rho }_m$.
The T1 tissue filter describes how signal changes with tissue T1.
The T2 tissue filter describes how signal changes with tissue T2.

Weighting, or how signal changes as a function of PD, T1, and T2 is the slope of each filter. Since the slope of these curves is not constant weighting also changes as a function of tissue property and is therefore different for different tissues. The overall signal of a given tissue on a pulse sequence is the product of the tissue filters.

Subsequent sections discuss each tissue filter independently. Each filter is graphed to demonstrate the relative signal of different soft tissues. You can interactively modify the tissue properties to see how this affects the signal of different tissues. Weighting is introduced as the slope of the tissue filter; how signal changes in response to small changes in tissue properties. Differential calculus is be used to determine sequence parameters to produce optimal contrast between different tissues and produce maximal sensitivity to changes from normal. The filters are combined to demonstrate how contrast is really generated with the spin echo sequence. Finally, the central Contrast Theorem is introduced and is used to explain the relative contributions of each tissue property to image contrast.

The discussion starts with the spin echo tissue filters discussed above but continues to include more complicated filters including inversion recovery and gradient echo. The final sections explain how these tissue filters can be manipulated to create sequences highly sensitive to small changes in tissue properties and also use tissue properties synergistically.