There are two adjacent tissues in an image with a T2 of 30 and 50 ms. You wish to produce a spin-echo image of the tissues such that the contrast between the tissues is maximized. What TE should be used?

C = S1 - S2

where:

    C = contrast
    S1 = signal from tissue 1
    S1 = signal from tissue 2

We need to find the maximum C, or dC/dTE = 0.
    dC/dTE = d(S1 - S2)= 0
    d(e-TE/T2(2))/dTE - d(e-TE/T2(1))/dTE = 0
    (e-TE/T2(2))/T2(2) = (e-TE/T2(1))/T2(1)
    T2(2)/T2(1) = (e-TE/T2(2))/(e-TE/T2(1))
    T2(2)/T2(1) = exp(-TE/T2(2)+ TE/T2(1))
    ln(T2(2)/T2(1)) = TE/T2(1) + TE/T2(2)
    ln(T2(2)/T2(1)) = TE (T2(2) - T2(1))/ (T2(2) T2(1))
    TE = T2(2) T2(1) ln[T2(2)/T2(1)]/(T2(2) - T2(1))
    TE = 38.3 ms