Optimizing variable flip angles in magnetization-prepared gradient-echo sequences for efficient 3D-T1ρ mapping

TABLE S1. Comparison of different magnetization-prepared $$$ {T}_{1\rho } $$$ pulse sequences with their weighting parameters ($$$ {\lambda}_A $$$, $$$ {\lambda}_F $$$, $$$ {\lambda}_S $$$, $$$ {\lambda}_R $$$), and $$$ {\boldsymbol{m}}_{\mathrm{ref}} $$$ and $$$ {\upalpha}_{\mathrm{ref}} $$$ values, used for optimization.

TABLE S2. Tested sequences with human volunteers comparing mean of normalized absolute difference results at the entire knee joint and the cartilage region. Cartilage was manually segmented in all slices of volunteers, including patellar, medial, and lateral cartilage.

TABLE S3. Summary of results from repeated model agarose gel phantom scans. Model phantoms were repeated on four different days to evaluate the repeatability and stability of measurements.

TABLE S4. Summary of results for knee joint for 5 different healthy subjects.

TABLE S5. Comparing different postprocessing corrections for magnetization-prepared gradient-echo with magnetization reset (MPGRE-WR) on agarose gel phantoms. MPGRE-WR C1 accounts for $$$ {e}_{\tau } $$$ and ($$$ 1-{e}_{\tau } $$$) in Eqs. (3) and (4). MPGRE-WR C2 accounts for $$$ {T}_1 $$$ contamination and imperfect Mz reset, using Eq. (12). The corrections follow the same postprocessing as proposed in Johnson et al.²⁶

TABLE S6. Comparison of different postprocessing corrections for magnetization-prepared gradient-echo with magnetization reset (MPGRE-WR) on 5 healthy volunteers.

TABLE S7. SNR versus SNR efficiency (SNR/sqrt(min)) of each tested method.

TABLE S8. Mean $$$ {T}_{1\rho } $$$ values of all 5 healthy volunteers in the manually segmented cartilage regions and its difference concerning the reference method. The last column shows the minimum and maximum differences.

FIGURE S1. Illustration of the composition of $$$ {\boldsymbol{m}}_k\left(\boldsymbol{\alpha} \right) $$$ for a signal evolution (SE) with the triplet ($$$ {T}_1\left[k\right]=3000\ \mathrm{ms} $$$, $$$ {T_2}^{\ast}\left[k\right]=50\ \mathrm{ms} $$$, $$$ {T}_{1\rho}\left[k\right]=58\ \mathrm{ms} $$$). This example has $$$ \mathrm{VPS}=128 $$$, $$$ D=2 $$$ (dummy shots), $$$ S=4 $$$ (acquired shots), $$$ S+D=6 $$$ (total shots), and $$$ {T}_{\mathrm{TSLs}}=2 $$$. In this work, $$$ k $$$ is the index of the relaxation triplet, and $$$ m $$$ is the index of the vector position, which depends on $$$ q $$$, $$$ s $$$, and $$$ n $$$, whereas $$$ q $$$ is the index of the spin-lock time (TSL); $$$ s $$$ is the index of the shot; and $$$ n $$$ is the index of the flip angle (FA).

FIGURE S2. Illustration of the composition of the matrix $$$ \boldsymbol{A} $$$ and how it performs on $$$ {\boldsymbol{m}}_k\left(\boldsymbol{\alpha} \right) $$$. Example of a signal evolution (SE) with the triplet ($$$ {T}_1\left[k\right]=3000\ \mathrm{ms} $$$, $$$ {T_2}^{\ast}\left[k\right]=50\ \mathrm{ms} $$$, $$$ {T}_{1\rho}\left[k\right]=58\ \mathrm{ms} $$$). This example has $$$ \mathrm{VPS}=128 $$$, $$$ D=2 $$$ (dummy shots), $$$ S=4 $$$ (acquired shots), $$$ S+D=6 $$$ (total shots), and $$$ {T}_{\mathrm{TSLs}}=2 $$$. The matrix $$$ \boldsymbol{A} $$$ computes the finite difference between all pairs of $$$ {M}_{xy}\left(k,p,s,1\right)/{e}^{-\frac{t_p}{T1\rho (k)}} $$$ and $$$ {M}_{xy}\left(k,q,s,1\right)/{e}^{-\frac{t_q}{T1\rho (k)}} $$$, where $$$ {t}_p $$$ and $$$ {t}_q $$$ are two different spin-lock times (TSLs). This is done for the first elements for all $$$ S $$$ acquired shots (not for the dummy shots). This leads to several columns that are $$$ S $$$ times the number of combinations of $$$ {T}_{\mathrm{TSLs}} $$$ items (2 at a time).

FIGURE S3. Illustration of the composition of the matrix $$$ \boldsymbol{F} $$$ and how it performs on $$$ {\boldsymbol{m}}_k\left(\boldsymbol{\alpha} \right) $$$. Example of a signal evolution (SE) with the triplet ($$$ {T}_1\left[k\right]=3000\ \mathrm{ms} $$$, $$$ {T_2}^{\ast}\left[k\right]=50\ \mathrm{ms} $$$, $$$ {T}_{1\rho}\left[k\right]=58\ \mathrm{ms} $$$). This example has $$$ \mathrm{VPS}=128 $$$, $$$ D=2 $$$ (dummy shots), $$$ S=4 $$$ (acquired shots), $$$ S+D=6 $$$ (total shots), $$$ {T}_{\mathrm{TSLs}}=2 $$$. The matrix $$$ \boldsymbol{F} $$$ computes the finite difference on the SE inside the shot, for $$$ 1\le ns\le VPS $$$, and it is repeated for all TSLs and all $$$ S $$$ acquired shots (except dummy shots). This leads to a block matrix, consisting of finite differences blocks.

FIGURE S4. Illustration of the composition of the matrix $$$ \boldsymbol{S} $$$ and how it performs on $$$ {\boldsymbol{m}}_k\left(\boldsymbol{\alpha} \right) $$$. Example of a signal evolution (SE) with the triplet ($$$ {T}_1\left[k\right]=3000\ \mathrm{ms} $$$, $$$ {T_2}^{\ast}\left[k\right]=50\ \mathrm{ms} $$$, $$$ {T}_{1\rho}\left[k\right]=58\ \mathrm{ms} $$$). This example has $$$ \mathrm{VPS}=128 $$$, $$$ D=2 $$$ (dummy shots), $$$ S=4 $$$ (acquired shots), $$$ S+D=6 $$$ (total shots), and $$$ {T}_{\mathrm{TSLs}}=2 $$$. In this work, $$$ \boldsymbol{S} $$$ has ones on the positions related to $$$ {M}_{xy}\left(k,1,s,1\right)/{e}^{-\frac{t_1}{T1\rho (k)}} $$$, the first element of the first spin-lock time (TSL), for all $$$ \boldsymbol{S} $$$ acquired segments, except the dummy segments.

FIGURE S5. Some of the flip angles (FAs) used in the experiments reported in Table 1, comparing optimization of the variable flip angles (OVFA) and standard choices for MAPSS (magnetization-prepared angle-modulated partitioned k-space spoiled gradient-echo snapshots),²⁸ MPGRE-WR (magnetization-prepared gradient-echo with magnetization reset),²⁶ or the constant FA for magnetization-prepared gradient echo (MPGRE). The FAs for the MPGRE-WR OVFA with the perfect Mz reset model (Eq. 2) and with the imperfect Mz reset model (Eq. 12, with $$$ c=0.05 $$$) can be compared in this figure. The FAs for the MPGRE OVFA with zero recovery time (ZRT) sequences are also shown, comparing the choice for more SNR (MPGRE OVFA ZRT SNR) against the choice of less filtering effects (MPGRE OVFA ZRT FLAT).

FIGURE S6. Individualized histograms of the agarose gel phantoms (3%, 4%, 5%, 6%, and 8%) for the MAPSS methods reported in Table 1.

FIGURE S7. Individualized histograms of the agarose gel phantoms (3%, 4%, 5%, 6%, and 8%) for the MPGRE-WR methods reported in Table 1.

FIGURE S8 Individualized histograms of the agarose gel phantoms (3%, 4%, 5%, 6%, and 8%) for the MPGRE methods reported in Table 1.

FIGURE S9. Individualized histograms of the agarose gel phantoms (3%, 4%, 5%, 6%, and 8%) for the MPGRE ZRT methods reported in Table 1.

FIGURE S10. Individualized histograms of the agarose gel phantoms (3%, 4%, 5%, 6%, and 8%) for the MPGRE-WR with different corrections. MPGRE-WR C1 accounts for $$$ {e}_{\tau } $$$ and ($$$ 1-{e}_{\tau } $$$) in Eqs. (3) and (4). MPGRE-WR C2 accounts for $$$ {T}_1 $$$ contamination and imperfect Mz reset, using Eq. (12). The corrections follow the same postprocessing as proposed in Johnson et al.,²⁶ using an expected $$$ {T}_1 $$$ value of $$$ 1200\ \mathrm{ms} $$$, the same used in the FA optimization proposed by Johnson et al.²⁶