Accurate measurement of tissue-specific relaxation parameters is an ultimate goal of quantitative MRI. The cross-relaxation effects between “free” and “bound” water are known to affect MRI signal and can bias quantitative measurements of tissue relaxation parameters in gradient echo^1-3 and inversion recovery studies⁴. The objective of this study is to theoretically account for the cross-relaxation effects in a gradient recalled echo (GRE) experiment and to demonstrate that the new theoretical approach can be used as a basis for a novel technique - Simultaneous Multi-Angular Relaxometry of Tissue with Magnetic Resonance Imaging (SMART MRI). The technique provides naturally co-registered quantitative maps of spin density, longitudinal and transverse relaxation rates along with parameters characterizing magnetization transfer (MT) effects, thus allowing measurements of some MT parameters without exploring off-resonance MT pulses.

A system comprising a pool of free water molecules (f-pool) and a pool of water bounded to macromolecules (b-pool) is analyzed in the framework of Bloch-McConnell equation⁵. Since T2 relaxation time of bound water is very short (~ 10 μsec ^6,7), the b-pool does not directly contribute to the measured MR signal. However, the cross-relaxation among the pools between RF excitation pulses substantially affects the free-water MR signal. In the present study, we derive a theoretical expression describing the GRE signal affected by the cross-relaxation effects between the f- and b-pools.

Experimental studies were approved by the local IRB. Experiments were performed using a 3T Trio MRI scanner (Siemens, Erlanger, Germany) equipped with a 32-channel phased-array head coil. High resolution SMART data with a voxel size of 1×1×1 mm³ were acquired using a 3D multi-gradient-echo sequence with TR=18ms, TE1=2.3ms, ΔTE=3.9ms, five flip angles of 5°, 10°, 20°, 40° and 60° with a rectangular RF pulse of 400 μs duration. A navigator-based method was used to correct the physiological artifacts⁸. GRAPPA⁹ acquisition with an acceleration factor of two and 24 auto calibrating lines in each phase encoding direction, resulted in the total acquisition time of 17 min 30 sec.

The following analytical expression describing the GRE signal S in the presence of MT effects, as a function of gradient-echo time TE, repetition time TR and flip angle α, has been derived:

$$S(\alpha, TE, TR)=S_{0}\cdot\psi(\alpha, TR)\cdot\exp(-R_2^*\cdot TE)$$

$$\psi(\alpha, TR)=\frac{1-E(TR)-G(\alpha, TR)}{1-E(TR)}\cdot sin\alpha$$

where the amplitude S₀ is proportional to the magnetization of the f-pool; R₂* describes the transverse relaxation of the f-pool affected by the cross-relaxation effect; the functions E and G (not shown due to space restrictions) depend on the relaxation and cross-relaxation parameters, the volume fraction of the b-pool, and a structure and duration of RF pulses. The above equations can be considered as a generalized Ernst-Anderson expression¹⁰, accounting for the cross-relaxation effects with the b-pool. The function ψ can be shown to depend on 3 independent model parameters: R₁, k_f', and λ:

$$R_{1}=R_1^f+k_f^\prime$$

$$k_f^\prime=k_{f}\cdot(1+k_{b}/R_1^b)^{-1}$$

$$\lambda=R_2^b\cdot(R_1^b+k_{b})$$

where R₁^f and R₁^b are the R₁-relaxation rate constants in the f- and b-pool, respectively; R₂^b is the R₂-relaxation rate constant in the b-pool; k_f and k_b are the (f→b) and (b→f) exchange constants, correspondingly. Importantly, the parameter k_f' is proportional to the volume fraction of the f-pool.

Given the pulse sequence parameters, the five model parameters (S₀, R₂*, R₁, k_f', and λ) can be found by fitting the model to the measured multi-flip angle, multi-echo GRE signal. The intrinsic R₁^f - relaxation rate constant in the f-pool can be readily obtained as R₁^f = R₁ – k_f’. Examples of the parameters’ maps obtained from one healthy subject (24 y.o.) and one MS subject (41 y.o.) are shown in Fig.1 and Fig.2, respectively. In Fig.1, all the maps show good white matter (WM) / gray matter (GM) contrast. WM shows higher k_f' values, indicating more exchange activity in WM, which usually contains highly myelinated fibers. As only short TEs were used, the R₂* map is highly weighted by the fast decaying signal from myelin water. In Fig.2, different maps exhibit different contrasts: lesions are hyperintense on the S₀ map and hypointense on R₁, R₁^f, k_f' and R₂* maps, consistent with a myelin loss. MS lesions exhibit different details on different maps, thus providing a potential for discrimination of MS lesion severity and structure.

Our approach establishes a theoretical basis for a SMART MRI technique that is based on derived theoretical model and multi-flip angle, multi-gradient-echo time MRI experiment. The technique provides quantitative measurements of longitudinal, transverse, and some essential MT parameters without exploring off-resonance MT pulses.