2.1 Phantom characteristics
The methodology employs a phantom consisting of two sets of samples. The samples are contained preferably in cylindrical vials, to best allow the selection of homogeneous sections. The reference set, contains samples with well-known T1 and T2 values and the second set CA samples whose relaxivities need to be determined. The CA sample concentration should not exceed the maximum dose allowed for the corresponding organ or tissue in which they will be employed. Both sets will be prepared to obtain the same relaxation time range.
In all cases, sample volume should be enough to guarantee an adequate SNR and to permit the acquisition of at least two slices in a homogeneous region. Consequently, the estimates of the relaxation times can be done for each slice, thus allowing for comparison and ensuring that the sample is stable and homogeneous.
Another factor to be considered is the sample's relative position within the phantom. To prevent the propagation of magnetic field disturbances, a distance of at least one diameter between any two samples is necessary. The samples must be fixed with their vertical axes parallel to each other so that the excited slice plane is orthogonal to all samples.
Usually in MRI, coil volume is larger than those used in NMR relaxometers, which facilitates the characterization of several samples simultaneously, reducing the required time.
2.2 Pulse sequences
Relaxation times can be determined using spin-echo sequences (SE) for T1 and T2 and inversion-recovery spin-echo sequences (IR-SE) for T1.
For SE, the equation describing pixel intensity can be expressed as:
$$S(x,y)={S_0}(x,y)(1-A(x,y){e^{-\frac{{TR}}{{{T_1}(x,y)}}}}){e^{-\frac{{TE}}{{{T_2}(x,y)}}}}{e^{-\vec {b}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftrightarrow}$}} {D} }}$$
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Since no diffusion sensitizing gradients are applied, the last exponential factor in Eq. (2) can be discarded. The parameter\(A(x,y)=1 - \cos \theta (x,y)\)accounts for RF inhomogeneity, which is discussed in more detail below. Repetition time TR and echo time TE can then be adjusted to separate, as much as possible, the contribution from each relaxation time.
In SE T1 estimation, TR is varied while keeping TE minimum, to minimize T2 weighting. Eq. (2) is simplified to:
$$S(x,y)={S_0}(x,y)(1-A(x,y){e^{-\frac{{TR}}{{{T_1}(x,y)}}}})$$
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The T1 measurement error is affected by the minimum TR attainable by the MRI equipment. The error is higher in cases of short T1 values.
For SE T2 measurement, TE is varied and TR is chosen to fulfill TR ≈ 5T1, minimizing T1 weighting. Eq. (2) now becomes:
$$S(x,y)={S_0}(x,y){e^{-\frac{{TE}}{{{T_2}(x,y)}}}}$$
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The T2 measurement error is affected by the minimum TE attainable. The error is higher for short T2 values.
In the case of IR-SE sequence the signal equation can be expressed as:
$$S(x,y)={S_0}[(1-A(x,y){e^{-\frac{{TI}}{{{T_1}(x,y)}}}})+{e^{-\frac{{TR}}{{{T_1}(x,y)}}}}]{e^{-\frac{{TE}}{{{T_2}(x,y)}}}}$$
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Where TI is the inversion time. By choosing TR ≥ 5T1 and setting minimum TE, Eq. (5) is simplified to:
$$S(x,y)={S_0}(x,y)[1-A(x,y){e^{-\frac{{TI}}{{{T_1}(x,y)}}}}]$$
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When using IR-SE, the T1 estimate standard deviation is proportional to [13]:
$${\sigma _{{T_1}}} \propto \frac{\sigma }{{DR}}$$
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Here σ refers to image noise standard deviation and DR to signal dynamic range. Hence the IR-SE has a two-fold advantage over the SE because its dynamic range is twice that of the SE. However, the IR-SE method is slower than SE.
2.3 RF field inhomogeneity
The aforementioned spatial inhomogeneity of the B1 excitation field results in flip angle deviations depending on the spatial position according to:
$$\theta (x,y)=\gamma \int\limits_{0}^{{{t_p}}} {{B_1}(x,y)d\tau }$$
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Several factors contribute to B1 inhomogeneities, like RF coil configuration, RF pulse shapes, and RF penetration depth. These inhomogeneities reduce the magnetization vector dynamic range, thus reducing the estimated T1.
This spatial distribution of flip angles due to B1 inhomogeneity can be mapped with the dual-angle method using a homogeneous phantom. This method acquires two images, S1 and S2, with flip angles related by θ2 = 2θ1, and the flip angle distribution is obtained by[14]:
$$\theta (x,y)=\arccos \left( {\frac{{{S_2}(x,y)}}{{2{S_1}(x,y)}}} \right)$$
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Another point related to the B1 excitation to be taken into account in MRI is that the duration of the selective pulse is several times longer than the non-selective used in the relaxometry. Then, during excitation, flip angle is more affected for those spin systems having very short T1, which is equivalent to the B1 inhomogeneity.
In addition, to minimize the influence of B1 inhomogeneity, the phantom must be always placed in the more homogeneous region of the coils.
2.4 Signal to Noise Ratio
The image SNR is another factor influencing the relaxation times estimation. We estimated SNR for each sample by region of interest (ROI) measurements. Let Ssample be the ROI image intensity in a sample and Sbackground be the ROI image intensity on the image background (air surrounding the samples), then SNR can be calculated as [15]:
$$SNR=0.665\frac{{mean({S_{sample}})}}{{std({S_{background}})}}$$
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The factor 0.665 accounts for the Rayleigh distribution of the noise in the magnitude image. If a real image is evaluated no correction factor is needed.