Data Fitting
At first glance, one could be tempted to fit VFA data using a non-linear least squares fitting algorithm such as Levenberg-Marquardt with Eq. 1, which typically only has two free fitting variables (T1 and M0). Although this is a valid way of estimating T1 from VFA data, it is rarely done in practice because a simple refactoring of Equation 1 allows T1 values to be estimated with a linear least square fitting algorithm, which substantially reduces the processing time. Without any approximations, Equation 1 can be rearranged into the form y = mx+b (Gupta 1977):

As the third term does not change between measurements (it is constant for each θn), it can be grouped into the constant for a simpler representation:

With this rearranged form of Equation 1, T1 can be simply estimated from the slope of a linear regression calculated from Sn/sin(θn) and Sn/tan(θn) values:

If data were acquired using only two flip angles – a very common VFA acquisition protocol – then the slope can be calculated using the elementary slope equation. Figure 5 displays both Equation 1 and 4 plotted for a noisy dataset.
Figure 5. Mean and standard deviation of the VFA signal plotted using the nonlinear form (Equation 1 – blue) and linear form (Equation 4 – red). Monte Carlo simulation details: SNR = 25, N = 1000. VFA simulation details: TR = 25 ms, T1 = 900 ms.
(View simulation code)
(View plot code)