compares the proposed scalable solution with the conventional simulation of multi-pool CEST contrast. The conventional simulation () utilizes multiple sets of Bloch equations coupled by the exchange terms, with each set representing a single labile proton group [28
]. The Bloch matrix size, for a general N-saturation transfer sites, is 3N by 3N (O(N2
)). In comparison, the proposed simplified solution utilizes multiple 2-pool models, with a correction term that takes into account of the indirect coupling effect (). As such, it includes (N−1) sets of 2-pool exchangeable models (O(N)), significantly reduced from that of the conventional approach when N becomes large.
Fig. 1 a) The conventional modeling of multi-pool CEST contrast was based on the Bloch-McConnell equations, with each saturation transfer site represented by a set of Bloch equations. b) The proposed simplified solution describes the multi-pool CEST contrast (more ...)
We evaluated the proposed simplified solution using a representative 3-pool CEST system, i.e., two 2-pool solution vs. a conventional 3-pool solution. The two labile protons were assumed to be at 4 and 5 ppm, and their concentrations were assumed to be 1:250 and 1:500, with exchange rates of 100 s−1
and 300 s−1
, respectively (). Moreover, T1
for bulk water were assumed to be 3 s and 100 ms, being 1 s and 15 ms for both labile protons. In addition, we set the magnetic field strength to be 4.7T, and solved the CEST contrast for a long (7.5 s) continuous wave (CW) RF irradiation field of 1 μT. Z-spectra for the two labile protons were simulated independently using the 2-pool model (). shows the CEST contrast for 4 and 5 ppm, in green and blue line, respectively, obtained by subtracting the direct RF saturation contribution from the two Z-spectra. In addition, the linear superposition of two CEST spectra was shown in gray line, while the proposed simplified solution is shown in a red dotted line, slightly less than the simplistic linear superposition. compared the Z-spectra obtained by the simplistic linear superposition (gray, Eq. 5
), the proposed simplified solution (red dotted, Eq. 6
) and the conventional 3-pool simulation solution (black). While the simplistic linear superposition solution differed from conventional simulation by 7%, the maximal difference between the proposed solution and the conventional simulation is less than 0.2%. As such, we showed that the proposed solution, by including the coupling term, agreed very well with conventional Bloch-McConnell solution.
Fig. 2 Comparison of the proposed simplified solution with the conventional simulation using a representative 3-pool mode: bulk water and two dilute labile groups at 4 and 5 ppm. a) Simulated spectrum showing the dilute labile proton (magnified by 100) and bulk (more ...)
We also evaluated the precision of the proposed simplified numerical solution by comparing it with the conventional 3-pool model for a typical range of labile proton concentration and exchange rate (). When the labile proton concentration for both exchangeable groups was increased from 1:2000 to 1:500, the CESTR, calculated as the MTR asymmetry at 4 and 5 ppm, increased almost linearly with the concentration (). In fact, the CESTR obtained from both solutions nearly overlapped, and their maximal difference is less than 0.2% (). In addition, CESTR also increased with exchange rate (). It is noticeable that the slope of CESTR increment with respect to exchange rate decreased at high exchange rates, likely because the saturation efficiency is reduced when the CEST contrast competes with the T1 relaxation (). The two solutions agreed very well, with their maximal difference being less than 0.1% ().
Fig. 3 Evaluation of the proposed solution as a function of labile proton concentration and exchange rate. a) The labile proton concentration for both groups was varied from 1:2000 to 1:500, with their exchange rates assumed to be 100 and 300 s−1 for (more ...)
Our study also evaluated the proposed solution while varying the RF amplitude and chemical shift (). shows that the CESTR initially increased with RF amplitude, peaking at approximately 1 μT for labile protons at 4 ppm, and 2 μT for labile protons at 5 ppm (). In fact, the two solutions agreed reasonably well, with their maximal difference less than 1% (). Moreover, when the chemical shift of one labile proton was varied from 4 to 6 ppm, while the other group remained at 5 ppm, the proposed solution also agreed well with the conventional simulation (). Because the CEST contrast was derived as the MTR asymmetry at their individual chemical shift, the calculated CESTR is susceptible to the coupling term of two CEST groups. This was particularly the case when the chemical shift of the second pool was 5 ppm, overlapping with that of the first pool (). Nevertheless, our proposed method agreed very well with the conventional 3-pool simulation, within 0.3% (). Hence, our results showed that the proposed solution provides simplified yet reasonably accurate modeling of multi-pool CEST contrast.
Fig. 4 Evaluation of the proposed solution as a function of RF amplitude and offset. a) The CEST contrast initially increased with RF amplitude, but peaked and then subsequently decreased with higher RF amplitude. b) Simulation from both methods agreed reasonably (more ...)
To demonstrate the scalability of the proposed numerical solution, we simulated CEST contrast for an illustrative 5 saturation transfer sites, equivalent to the conventional 6-pool CEST model (). The hypothetical CEST system includes an exchangeable amide proton pools (3.65 ppm) and two hydroxyl protons (2.3 and 1.2 ppm) [9
]. In addition, two aliphatic groups (−3 and −3.5 ppm) were added to model saturation transfer via NOE [22
]. We also simulated two typical RF amplitudes of 1 and 2 μT. shows that at low RF amplitude, the individual CEST contrast can be reasonably distinguished, while the frequency resolution of the Z-spectrum degraded at stronger RF amplitude due to the direct RF saturation effect and the coupling term among multiple saturation transfer sites [15
]. The direct RF saturation effect was also simulated, shown in dotted and dash dotted lines in . It is important to point out that very little modification was required when we extended the proposed solution from 3-pool to 6-pool, simply providing the inputs of the labile proton concentration, chemical shift, exchange and relaxation rates, without modification of the subroutines. shows the CEST-specific contrast, as determined by subtracting each Z-spectrum from the corresponding direct RF saturation effect. At 1 μT, the CEST contrast of the -NH and -OH groups can be reasonably resolved, whereas two NOE sites showed significant overlap due to their relatively short T2
and small chemical shift difference (). Moreover, the MTR asymmetry (MTRasym
) was calculated by taking the difference between downfield and highfield MTR symmetrically around the bulk water signal (). It showed that the CEST contrast may be underestimated unless the negative shift induced by saturation transfer effect from highfield offset (NOE effects) can be properly taken into account.
Fig. 5 Simulation of CEST contrast of five saturation transfer sites, being at 3.65, 2.3, 1.2, −3 and −3.5 ppm. The concentration of each site was assumed to be 1:1000, 1:1000, 1:500, 1:500 and 1:500, with their exchange rates being 50, 50, 50, (more ...)