Fast Field Cycling (FFC) MRI ¹ offers a new degree of freedom in contrast manipulation by switching rapidly the magnetic field B₀, enabling to access the NMRD profile for tissue. State-of-the-art systems generate, within milliseconds, offsets going up to ±500 mT for preclinical applications ^2,3, and up to 200 mT for clinical ones ⁴. When used as an insert to a static B₀, delta relaxation enhanced Magnetic Resonance (dreMR) can be performed providing new information further characterizing MR properties: the slope of the dispersion profile. ⁵ DreMR consists in a time evolution during which relaxation occurs at a different field prior to imaging. While inversion-recovery-prepared spin-echo-based sequences are usually performed, as T₁-mapping approaches at static B₀, it requires a long acquisition time and SNR per time unit could be limited. ⁶ Similarly to static field applications, fast sequence involving repeated low flip angles could be used. Here, a theoretical analysis is performed to take into account relaxation dispersion in such sequences modified to include FFC pulses. The derivation generalizes the Ernst angle with relaxation dispersion and provides insight into the potential of FFC approaches for contrast manipulation.

The practical case of an FFC pulse included in the sequence is depicted in Fig.1. The sequence is repeated every TR: after a low flip angle α followed by an acquisition window at B₀ field, a final ΔB pulse is applied during T_Δ. The magnetization behavior can be calculated from the Bloch equation. However, polarization at thermal equilibrium and R₁ relaxation rates need to be modified as a function of ΔB. If M₀ is the thermal equilibrium at B₀, M₀.(1+ΔB/B₀) should be considered when ΔB is applied. Additionally, as a first approximation, longitudinal relaxation rate can be assumed to evolve linearly as a function of ΔB as R₁ + β.ΔB, with β the slope of the dispersion profile at B₀. Each RF pulse flips the longitudinal magnetization by an angle α and the transversal magnetization is considered to be spoiled before the next excitation. After few excitations, it can be shown that a steady-state is reached with a transverse component provided by the following expression:

$${{\text{M}}_{\text{xy}}}\left( +\Delta B \right)\text{=}{{\text{M}}_{\text{0}}}\sin (\alpha )\frac{1-\exp (-{{\text{R}}_{\text{1}}}TR-\beta \Delta B{{T}_{\Delta }})+\frac{\Delta B}{{{\text{B}}_{\text{0}}}}(1-\exp (-{{\text{R}}_{\text{1}}}{{T}_{\Delta }}-\beta \Delta B{{T}_{\Delta }}))}{1-\cos (\alpha )\exp (-{{\text{R}}_{\text{1}}}TR-\beta \Delta B{{T}_{\Delta }})}$$ (Eq.1)

Assuming that R₁ is known (obtained easily using mapping technique at fixed B₀), this expression can be normalized to isolate the effect of the polarizing field and non-varying relaxation rate R₁ as:

$${{\text{M}}_{\text{xy}\text{,norm}}}\text{=}{{\text{M}}_{\text{xy}}}\frac{1-\exp (-{{\text{R}}_{\text{1}}}TR)}{1-\exp (-{{\text{R}}_{\text{1}}}TR)+\frac{\Delta B}{{{\text{B}}_{\text{0}}}}\left( 1-\exp (-{{\text{R}}_{\text{1}}}{{T}_{\Delta }}) \right)}$$ (Eq.2)

As can be seen in Eq.2, without dispersion (β=0), one recovers the usual Ernst equilibrium at fixed field which maximizes the signal for cos(α)=exp(-R₁.TR). The dreMR contrast is obtained by combining several acquisitions with different FFC pulses, such as +ΔB and –ΔB. Without dispersion, the normalized transverse magnetization is not changing and the difference will not provide any contrast. However, when a dispersive contrast occurs (β≠0), the normalized signal is changing with the term exp(-β.ΔB.T_Δ) providing the dreMR contrast.

According to Eq.2, the normalized signal intensity is maximal for a given angle that can be determined after differentiation and which expression is similar to the standard Ernst angle: cos(α)=exp(-R₁.TR-β.ΔB.T_Δ), generalizing the expression to the case of FFC. Fig.2 displays simulated signals with realistic values that can be obtained using state-of-the art FFC-MRI systems. Different maxima are obtained as a function of flip angle for the case of dispersion with +ΔB and –ΔB pulses. The difference can be defined as the dreMR contrast in fast low angle sequences. The dreMR contrast is then maximized for an angle larger than the generalized Ernst angle. For small products β.ΔB.T_Δ, the angle maximizing the dreMR contrast corresponds to Buxton angle that maximizes contrast in T₁-weigthed sequences: cos(α) = (1-2.exp(-R₁.TR))/(exp(-R₁.TR)-2), thus independent of the dispersion parameters. ⁷DreMR experiments could benefit from fast low angle sequences to enhance contrast-to-noise ratio and sensitivity to NMR dispersion profile variations. The theoretical analysis of the steady-state signal dependence in such sequence was performed. DreMR contrast generation requires the calibration of background relaxation rate at the static B₀ field. When normalized, the signal is maximized at a generalized Ernst angle and the contrast is maximized at the Buxton angle. Simulations were performed with realistic hardware parameters such as feasible ΔB amplitudes. By using dispersive contrast agents with favorable NMRD profiles, it should be possible to estimate the detection limit in β using the proposed derivations and define optimized imaging protocols (TR, α, ΔB, T_Δ) for dreMR. Future work will focus on experimental validations.