Absolute metabolite concentrations of a metabolite can be calculated by using an internal water reference. This internal water reference is the unsuppressed water signal collected at the same time as the water-suppressed spectra. Instead of using a ratio of each metabolite signal over the signal of a reference metabolite (e.g. creatine), a ratio over the signal of water is used. This is advantageous because the water signal from brain tissue is consistent between individuals, whereas a reference in vivo metabolite signal can vary between individuals because its concentration varies.

Using an internal reference to determine metabolite concentration can be written as follows:

Equation 1

where

• concentration of the metabolite
• concentration of pure water (55.14 M)
• corrected water signal
• corrected metabolite signal

As Equation 1 shows, the measured water signal is mapped to the known concentration of pure water, This mapping is then used to determine what concentration the measured metabolite signal represents. The measured water and metabolite signals are not used directly; they are first corrected for a variety of factors. (It is important to note that spectroscopic voxel volume is not considered in the present discussion, as it is assumed the unsuppressed water signal and the water-suppressed spectra are measured using the same spectroscopic voxel. If this is not the case, voxel volume would have to be considered.)

The most important of these corrections is the correction for and relaxation. The objective of the and relaxation correction is to determine what the measured signal would be if there was no relaxation. Relaxation of the transverse magnetization is described by the following equation:

Equation 2

where is the measured signal, and is the signal before and relaxation begins.

Equation 2 describes relaxation in a single tissue compartment. However, in the ontext of in-vivo brain metabolite quantification, the GM, WM, and CSF compartments must be considered:

Equation 3

, , and may be expressed in terms of . Considering only the GM compartment for the moment,

where is the volume fraction of GM in the spectroscopic voxel, and is the relative proton density in GM.

To explain why , it is important to remember that is proportional to the total number of protons in the spectroscopic voxel. Since proton density is the number of protons in a volume, multiplying the volume fraction by the relative proton density allows us to determine the fraction of the total number of protons contributed by the GM. This fraction can then be used to properly attribute the proportion of to GM. Using the same reasoning,

where is the volume fraction of WM in the spectroscopic voxel, is the volume fraction of CSF in the spectroscopic voxel, is the relative proton density in WM, and is the relative proton density in CSF.

Thus, Equation 3 may be rewritten as:

Equation 4

Since the objective of the and relaxation correction is to determine the signal without or relaxation, applying a correction for and is the same as solving for . Solving Equation 4 for results in:

Equation 5

where

Equation 4 may be applied directly to correct the measured water signal for and relaxation:

Equation 6

where

• = 0.82 = the relative proton desnity of water in GM as compared to that of pure water
• = 0.73 = the relative proton density of water in WM as compared to that of pure water
• = 1.00 = the relative proton density of water in CSF as compared to that of pure water

and , , and are the corresponding relaxation terms of Equation 4 using the tissue specific and relaxation rates of water. , the amplitude of the fitted water peak.

Unlike the measured water signal, Equation 4 cannot be directly applied to correct the measured metabolite signal because of a couple of reasons. First, it is assumed that the CSF contribution to the metabolite signal is negligible because there are little to no metabolites in the CSF. No part of the metabolite signal should be attributed to the CSF compartment and any metabolite signal should be equally attributed to the WM and GM compartment. Secondly, the relative proton density in GM, WM, and CSF for metabolites are not known. Because of this, , , and are assumed to be 1.00 for metabolites. The equation to correct the measured metabolite signal for and is then:

Equation 7

where and are the corresponding relaxation terms of Equation 4 using the tissue specific and relaxation rates of metabolite m. , the sum of the amplitudes of the metabolite as determined by the fitted prior-knowledge model.

In addition to corrections for and relaxation, the following corrections are applied to the measured water and metabolite signals:

1. A correction for the number of averages used to acquire the water signal
2. A correction for the number of MRS-visible ¹H nuclei in the water molecule
3. Corrections for any gain and scaling factors applied by the scanner and during post-processing

These corrections are applied as follows:

Equation 8

Equation 9

Combining Equations 6 and 8, the final equation for the corrected water signal is:

Equation 10

Combining Equations 7 and 9, the final equation for the corrected metabolite signal is:

Equation 11

The equation to calculate the concentration of a metabolite with respect to the entire of the spectroscopic voxel volume is obtained by substituting Equations 10 and 11 into Equation 1:

Equation 12

Equation 12 may be simplified into:

Equation 13

With Equation 13, a metabolite concentration with respect to the entire volume of the spectroscopic voxel was found. To determine the concentration of the metabolite in brain tissue alone, the dilution equation can be applied:

Thus, the metabolite concentration in brain tissue is:

Equation 14

Equation 14 can be simplified by multiplying the dilution volume correction through the relaxation correction term for the metabolite signal:

Equation 15