['To develop and validate closed form mathematical expressions that predict the optimal contrast agent concentration for the maximum T1-weighted spoiled gradient echo (SGRE) signal.', 'Gadolinium and iron-based contrast agents can have significant transverse relaxivity that leads to signal dropout with increasing contrast agent concentration. A mathematical expression for the "optimal" contrast agent concentration where recovery of longitudinal magnetization is offset by increasing transverse signal decay was derived. Expressions for the maximum possible SGRE signal were also derived. Three phantoms were constructed, each with varying concentrations of one of the following three agents: gadoteridol, gadobenate dimeglumine, and ferumoxytol. After measuring the longitudinal and transverse relaxivity of the three agents, the SGRE signal was measured in the phantoms over a wide range of flip angles and echo times.', 'Excellent qualitative agreement between the SGRE signal behavior, optimal concentration, and optimal flip angle were observed between experimental measurements and theoretical predictions.'] Do mathematical optimization of contrast concentration for T1-weighted spoiled gradient echo imaging?