In this appendix, the implementation of model spectra simulation by the density matrix formalism within MATLAB is described. When no RF pulse and magnetic field gradient is applied, the Hamiltonian for a metabolite is described in the rotating frame at the RF pulse frequency by
where and stand for the spin operator of the ith and jth observable (i.e., nonexchangeable) protons, respectively, Ki is the difference of its resonance frequency and the RF pulse frequency due to chemical shift, Jij is the J-coupling constants between spins i and j. The values of Ki and Jij are taken from Ref. . During the experiment, field gradients are applied for voxel localization and for dephasing coherences other than those involved in the stimulated echo generation. When a gradient is applied, the following term should be added to the Hamiltonian in Eq. (A1):
where Λ is the spatial dependent frequency shift due to the gradient field. Furthermore, when an RF pulse is applied, another term should be added to the Hamiltonian:
where θ is the angle of the B1 field in the rotating frame, and γ is the gyromagnetic ratio of protons. The shaped RF pulses each consist of 256 segments, with constant Hrf during each segment. The evolution of the density matrix (σ) under a constant Hamiltonian is
where Hi is the Hamiltonian from time ti−1 to ti and . The evolution of the density matrix from the initial thermal equilibrium value can thus be calculated for the entire pulse sequence using Eq. (A1), (A2), (A3), (A4) without numerical integration. In our program, all spin operators, the Hamiltonians, and σ are represented by 2n×2n matrices in the Hilbert space formed by the eigenvectors of Ii,z, where n is the number of observable protons in the metabolite.
The spatial-encoding gradients were not simulated. Their effect is equivalent to shifting the frequencies of the RF pulses which does not affect the shape of the simulated spectrum as long as all resonance frequencies are still within the selective band of the RF pulses. In contrast, the crusher gradients cannot be ignored in order to preserve only the experimentally relevant coherence transfer pathway. Their effects can be simulated by calculating σ evolution with a large set of values, where τ is the duration of Hg, and then taking the average over all results . Such an approach is time-consuming and complete removal of unwanted coherences is not guaranteed. However, since all unwanted coherences in the resulting σ depend on as for crushing gradients applied during the first and third TE/2 periods, it is sufficient to run the simulation twice at two values which differ by 1/4 and sum over the results to remove those coherences. We note that a similar approach was taken in Ref. ; however, four consecutive values were used with steps of 1/4, resulting in doubling of computing time. The crusher gradient during the mixing time (TM) period can be simulated in a similar way. Now the dependence becomes , as both single- and double-quantum coherences are nonzero during the TM period. Therefore, the simulation needs to be run at four values with steps of 1/4 to cancel all the oscillating terms. Alternatively, the effect of the crusher gradients during the TM period can be simulated by simply setting the matrix elements in σ that corresponds to single or higher-order quantum coherences to zero . Both approaches were tested and found to give identical results. In the final implementation, the second approach was taken to reduce computation time by a factor of 4.
The final NMR signal was calculated as
where l=1, 2, 3,…, sw (= 4000 Hz) is the spectral width in the experiment, and the exponential term with R2*=25 s−1 simulates line broadening (full width at half maximum=8 Hz) due to apparent transverse relaxation. Additional broadening needed to match the real experimental condition was determined during LCModel fitting. The T1 and T2 relaxation effects were not simulated because T1 >> TE+TM and T2 >> TE in our measurements. For all simulated metabolite and water signals, the signal amplitude |S| is proportional to the number of protons n.