Numerical T1 computation from NMR intensity ratios

Two-point measurement of tissue T1 from NMR intensity ratios consists of forming an a priori ratio function describing a T1 dependence of the ratio R(T1) and computing T1 from an observed ratio Q by numerically solving R(T1)-Q = 0 or an equivalent equation. Impact of R(T1) designs on the numerical computation and dependence of relative speeds of three numerical methods on desired computational precisions q and on other factors are examined. All three methods begin with computing a table of R(T1) entries in uniform T1 steps (delta T1). In two iterative methods, a step containing the T1 root is looked up, and the precise T1 location within the step is pinpointed to within a q value by either linear-interpolative (LI) or Newton-Raphson (NR) iteration. The third method simply consists of computing a large table of delta T1 = q for a mere "look-up" with no iterative search. All three methods require a monotonous R(T1) for uniformly effective computation over wide T1 ranges. Speeds of either iterative method for computing T1 images are expected to vary with delta T1 and q with unsharp speed maxima at delta T1 near 20, 6, and 2 ms for q = 10(-1), 10(-2), and 10(-3) ms, respectively. Either iterative method is suitable for both low- and high-precision computations, the LI method being generally faster. The simple look-up is the fastest of the three for T1 image computation to low precisions of q greater than or equal to 1 ms, is likely the slowest for that to q = 0.1 ms, and is impractical for that to q less than or equal to 0.01 ms.