# define constants
te <- 0.028 # MRS sequence echo-time
tr <- 2.0 # MRS sequence repetition-time
t1_gm <- 1.331 # T1 of grey matter water
t2_gm <- 0.110 # T2 of grey matter water
t1_metab <- 1.317 # T1 of metabolites
t2_metab <- 0.207 # T2 of metabolites
w_conc <- 55510 # molal concentration of pure water (moles/kg)
# calculate water and metabolite relaxation attenuation
R_h2o <- exp(-te / t2_gm) * (1.0 - exp(-tr / t1_gm))
R_met <- exp(-te / t2_metab) * (1.0 - exp(-tr / t1_metab))
# read the .csv of unscaled values
vals_unsc <- read.csv("fit_res_unscaled.csv")
# calculate tNAA and tCr molal concentrations
tNAA_molal_conc <- (vals_unsc$tNAA / R_met) / (vals_unsc$w_amp / R_h2o) * w_conc
tCr_molal_conc <- (vals_unsc$tCr / R_met) / (vals_unsc$w_amp / R_h2o) * w_conc3 Water scaling details
3.1 Default assumptions
Assuming all metabolite and water signals are measured from a single tissue compartment, and neglecting \(\mathsf{T_1}\) and \(\mathsf{T_2}\) relaxation effects, we calculate the molal (moles/kg) metabolite concentration as follows:
\[ \mathsf{[M]_{molal} = \frac{S_M}{S_{H2O}}}\mathsf{[H_2O]_{molal}} \tag{3.1}\]
where \(\mathsf{S_M}\) and \(\mathsf{S_{H2O}}\) are the metabolite and water signal strengths respectively, and \(\mathsf{[H_2O]_{molal}}\) is the molal concentration of pure water, 55.51 moles/kg. We also assume \(\mathsf{S_M}\) and \(\mathsf{S_{H2O}}\) signals are correctly scaled according to the number of contributing protons.
To correct for relaxation effects we define the attenuation factor, \(\mathsf{R_x}\), of a signal \(\mathsf{x}\) undergoing relaxation with \(\mathsf{T_{1x}}\) and \(\mathsf{T_{2x}}\) time constants:
\[ \mathsf{R_x = \exp(-TE/T_{2x})[1-\exp(-TR/T_{1x})]} \]
where \(\mathsf{TE}\) and \(\mathsf{TR}\) represent the MRS sequence echo time and repetition time. We modify Equation 3.1 to include relaxation effects as follows:
\[ \mathsf{[M]_{molal} = \frac{S_M / R_M}{S_{H2O} / R_{H2O}}}\mathsf{[H_2O]_{molal}} \]
Assumed relaxation values, based on published studies of healthy adults, are given in Table 3.1.
| Relaxation value | 1.5 T | 2.9, 3.0 T | 7.0 T |
|---|---|---|---|
| GM water T1 (s) | 1.304 [1] | 1.331 [2] | 2.132 [3] |
| GM water T2 (s) | 0.093 [1] | 0.110 [2] | 0.050 [4] |
| WM water T1 (s) | 0.660 [1] | 0.832 [2] | 1.220 [3] |
| WM water T2 (s) | 0.073 [1] | 0.0796 [2] | 0.055 [4] |
| CSF water T1 (s) | 2.39 [5] | 3.817 [6] | 4.425 [3] |
| CSF water T2 (s) | 0.23 [5] | 0.503 [7] | 1.050 [8] |
| Metabolite T1 (s) | 1.153 [1] | 1.317 [9] | 1.583 [10] |
| Metabolite T2 (s) | 0.347 [1] | 0.207 [9] | 0.141 [10] |
When the voxel contains a mix of compartments, eg 80% white matter and 20% CSF, it is necessary to correct for water densities in each compartment. We assume the following water densities for gray matter, white matter and CSF: \(d_{\mathsf{GM}} = 0.78\); \(d_{\mathsf{WM}} = 0.65\); \(d_{\mathsf{CSF}} = 0.97\); based on values from Ernst et al [11].
For further details on MRS water concentration scaling, including partial volume effects and tissue specific relaxation correction, please see the following papers: [1], [12], [13].
3.2 Manual scaling
Whilst the defaults described above provide reasonable values for healthy brain tissue, they are unlikely to be appropriate for all cases, and alternate assumptions and correction schemes may be required. For example, the default approach assumes all metabolites have the same relaxation properties, which could result in strong bias for certain acquisition protocols (e.g. longer echo-times) or pathology. Unscaled metabolite and water signal amplitudes are therefore provided in the fit_res_molal_conc.csv file in the results directory to allow custom corrections. Note that these values have already been scaled for the number of contributing protons. Example R code is listed below to calculate the molal concentrations of tNAA and tCr for a dataset acquired at 3 Tesla with an echo time of 28 ms and repetition time of 2 seconds, assuming the voxel contains 100% grey matter tissue:
3.3 Legacy scaling
Metabolite concentration scaling using the default method for the popular LCModel MRS analysis software [14] may be added to the analysis output by setting the legacy_ws argument for the fit_svs function to TRUE. Concentrations are calculated as a ratio between the metabolite and water signal intensities (as in Equation 3.1) but with the additional scaling factors representing the NMR-visible water concentration (mM) in the voxel: \(\mathtt{WCONC}\); and the attenuation of the water signal due to relaxation: \(\mathtt{ATTH2O}\).
\[ \mathsf{[M]_{lcm} = \frac{S_M}{S_{H2O}}} \times \mathtt{WCONC} \times \mathtt{ATTH2O} \]
The default values for \(\mathtt{WCONC}\) and \(\mathtt{ATTH2O}\) are 35880 and 0.7 respectively. The LCModel manual [15] states that concentrations derived from this method:
“should be labelled ‘mmol per Kg wet weight’. We use the shorter (incorrect) abbreviation mM. The actual mM is the mmol per Kg wet weight multiplied by the specific gravity of the tissue, typically 1.04 in brain.”
Note that we don’t recommend this method of water scaling, but have implemented it to simplify comparison with legacy analysis packages such as LCModel [14] and TARQUIN [16].
3.4 Legacy relaxation assumptions
The following relaxation values were assumed for spant versions older than 3.1.0.
| Relaxation value | 1.5 T | 2.9, 3.0 T |
|---|---|---|
| GM water T1 (s) | 1.304 | 1.331 |
| GM water T2 (s) | 0.093 | 0.110 |
| WM water T1 (s) | 0.660 | 0.832 |
| WM water T2 (s) | 0.073 | 0.0792 |
| CSF water T1 (s) | 2.93 | 3.817 |
| CSF water T2 (s) | 0.23 | 0.503 |
| Metabolite T1 (s) | 1.15 | 1.15 |
| Metabolite T2 (s) | 0.3 | 0.3 |
References
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