# Analytical Dual Flip Angle R1 Calculation Outside the Small‐Angle Regime

## TL;DR

This paper introduces a new method for calculating R1 and proton density from MRI data that works well even with large flip angles.

## Contribution

A novel analytical estimator for R1 and apparent proton density that avoids the small flip angle approximation.

## Key findings

- The novel method avoids flip angle-dependent bias seen in traditional small-angle methods.
- The new method preserves precision with negligible differences in coefficients of variance.
- The method is more accurate and broadly applicable for a wider range of MRI data.

## Abstract

To evaluate a new analytical estimator for R1 and apparent proton density (A) from short‐TR dual flip angle data which does not rely on the small flip angle approximation and can thus be applied to a broader range of data, especially where relatively large flip angles are needed to achieve sufficient T1‐weighting for R1 estimation.

A rational approximation of the Ernst equation was derived for small R1·TR and rearranged to give analytical estimators of R1 and A from dual flip angle data. Unlike previously used analytical estimators, this method relies neither on the flip angles being small nor the two TRs being equal or integer multiples of each other. R1 and A estimated using the novel method were compared to estimates using the conventional small‐angle approximation approach in simulations and data measured at 7T from six in vivo human participants and a postmortem chimpanzee brain. Test–retest scans of the six participants were used to evaluate within‐participant coefficients of variance (WCV) of R1 and A using the two methods.

The small‐angle approximation gave rise to a flip angle‐dependent bias in all cases. This bias was not observed using the novel method, demonstrating its higher accuracy. There were only negligible differences in WCV between the methods, demonstrating that precision is preserved.

The increase in accuracy and preservation of precision suggest that the novel method should be used instead of the current small flip angle method.