Cross-relaxation
Fig. 1 shows a cross-relaxation model between two water pools. A cross-relaxation model between two components is the basis of our method. Here, we assumed only proton exchange between both pools without chemical exchange [6]. Although normal longitudinal relaxation (i.e., from β to α) occurs between the same molecules, the relaxation times are different between different molecules. Thus, it is necessary to consider each as having a different spin, i.e., cross-relaxation. In our study, the spin-lattice relaxation rate and abundance ratio of each material were defined to be equal [7].
Two-component analysis with QPM
General approach for T1 relaxometry for spoiled gradient-echo (SPGR) is described as follows. A single-component model of SPGR is described as:
Fig. 2: Equation 1
where M is the MR signal, T1 is the longitudinal magnetization, M0 is the proton density, E1 = exp(-TR/T1), and α is flip angle. This model only can be estimate a single T1 value, it does not capture the structural and compositional complexity of tissue.
Here, the single-component model of SPGR signal function is expanded to the multi-component model (8). The multi-component model is described by the following equation:
Fig. 3: Equation 2
where
Fig. 4: Equation 3
where fs and fl are volume fractions of short and long species, respectively, I represents the 2×2 identity matrix, and ks,l and kl,s represent the exchange rates between the two components. Chemical equilibrium is assumed to fsks,l = flkl,s. Fig. 5 shows a multi-component exchange relaxation scheme.
If the above function in Eq. [2] is given as an expression of nonlinear function, then the calculation cost can be controlled. We propose a method using single-component relaxation time undergoing the assumption of cross-relaxation and applying linear algebra. When α«π, the exponential function is replaced using linear algebra, and trigonometric functions such as sine and cosine are transformed:
Fig. 6: Equation 4
Then, Eq.[1] is described as follows:
Fig. 7: Equation 5
When solving for T1, Eq. [5] is described as follows:
Fig. 8: Equation 6
Next, we expand two components from a single component using Eq. [3]. Each T1 value is derived from a two-component SPGR signal and is given by
Fig. 9: Equation 7
where T1,i represents short and long T1 relaxation times (T1,s,T1,l). In this study, TR was used as the setting value for an imaging parameter of QPM. Thus, after expanding about TR, the TR is inserted into Eq. [5] as follows:
Fig. 10: Equation 8
Fig. 11: Equation 9
MR imaging
Fig. 12 shows the procedure for producing a two-component T1 map with QPM. On a 3 Tesla MR scanner, a QPM dataset of the brain of a healthy volunteer (22-year-old male) was acquired using 3D-RSSG method. Table 1 shows the settings for QPM imaging parameters. Basic QPM estimates T1, T2, B1, and PD was fitted by the intensity function which is pre-calculated using Bloch simulation.
Table 1 Setting parameters of QPM.
Imaging parameters |
Parameter values |
Pulse sequence |
3D-RSSG |
TE |
4.6 - 32.1 ms (ΔTE = 4.6 or 6.9 ms) |
TR |
10,20 and 40 ms |
FA |
10, 25, 35, and 40 degrees |
Voxel size |
100 × 192 × 192 |
Slice thickness |
2 mm |
Data analysis
In the analysis, ks,l was defined as an abundance ratio of myelin and kl,s which was defined as water; then, the exchange coefficient ratio was set at 3:7. The obtained quantitative values for PD and T1 were substituted into Eq.[6], and M and α was selected from part of the QPM datasets. The datasets was taken by 10 ms TR, 4.6 ms TE, and 40 degrees FA. In addition, we compared the T1 value obtained by QPM with T1 values (T1,s, T1,l) divided into two components calculated by this method.
All data analysis was conducted using MATLAB (MathWorks, Natick, MA, USA) using an in-house program. The region of interest (ROI) settings are shown in Fig. 13 . After ROIs were set as the posterior thalamic radiation (PTR; green area) and gray matter (red area), mean T1 values in each ROI were measured. The mean value was used as the measured value.
Results
Table 2 shows each measurement T1 value derived from two-component analysis with QPM at each region. From the T1,s and T1,l values, we can be recognize that the PTR and gray matter are divided according the applied exchange coefficient ratio.
Table 2 Each measured T1 value of each tissue.
Tissues |
Each T1 value derived from QPM [ms] |
T1 |
T1,s |
T1,l |
Gray matter |
2203 ± 440 |
749 ± 268 |
2044 ± 931 |
Posterior thalamic radiation |
1354 ± 64.0 |
343 ± 18.0 |
792 ± 65.0 |
Notes: Data are presented as mean ± standard deviation.
Fig. 14 shows general T1 map with QPM and T1 maps derived from two-component analysis with QPM. From these images we could recognized as having a different T1 range according to each calculation method.
Fig. 15 shows the whole brain T1 maps derived from QPM. These images were able to be generated while maintaining high signal QPM quality.