Modified Look-Locker Inversion recovery (MOLLI) sequence is increasingly performed for myocardial T1 mapping but is known to underestimate T1 values. The aim of the study was to quantitatively analyze several sources of errors when T1 maps are derived using standard post-processing of the sequence and to propose a reconstruction approach that takes into account inversion efficacy (η), T2 relaxation during balanced steady-state free-preces-sion readouts and B1+ inhomogeneities. Contributions of the different sources of error were analyzed using Bloch equations simulations of MOLLI sequence. Bloch simulations were then combined with the acquisition of fast B1+ and T2 maps to derive more accurate T1 maps. This novel approach was evaluated on phantoms and on five healthy volunteers. Simulations show that T2 variations, B1+ heterogeneities and inversion efficiency represent major confounders for T1 mapping when MOLLI is processed with standard 3-parameters fitting. In vitro data indicate that T1 values are accurately derived with the simulation approach and in vivo data suggest that myocardium T1 are 15% underestimated when processed with the standard 3-parameters fitting. At the cost of additional acquisitions, this method might be suitable in clinical research protocols for precise tissue characterization as it decorrelates T1 and T2 effects on parametric maps provided by MOLLI sequence and avoids inaccuracies when B1+ is not homogenous throughout the myocardium.