When using a 90-FID pulse sequence and a sample containing all the tissues in question number two, what TR value would guarantee at least 98% of the signal from all the tissues?

S(TR) = kr(1-e-TR/T1)

where:

    TR = repetition time
    S(TR) = signal at time TR
    r = spin density
    k = proportionality constant
    T1 = spin-spin relaxation time
Taking the ratio of the observed signal (S(TR) from above) to the signal at infinitely long TR, gives:

S(TR) / S(TR=inf) = kr(1-e-TR/T1) / kr(1-e-inf/T1)

S(TR) / S(TR=inf) = kr(1-e-TR/T1) / kr(1)

Setting S(TR) / S(TR=inf.) = 0.98 we have

0.98 = 1-e-TR/T1
ln( 1 - 0.98) = -TR/T1
TR = -T1 * ln( 0.02 )

Using the longest T1 (20 s) to determine TR would ensure 98% signal or better for all of the listed tissues.

TR = -20 s * ln( 0.02 ) = 78.24 s