quantification-tissue-correction.Rmd

---
title: "Quantification & tissue correction"
date: "Last updated: `r format(Sys.time(), '%B %d, %Y')`"
bibliography: bibliography.bib
csl: american-medical-association.csl
link-citations: yes
output:
  html_document:
    toc: TRUE
    toc_depth: 2
    toc_float:
      collapsed: FALSE
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

```{r, child = "js/back-to-top.js"}
```

```{css, echo = FALSE}
.info {
  margin-bottom: 20px;
}

table {
  margin: auto;
}

table thead th {
  border-bottom: 1px solid #ddd;
}

th, td {
  padding: 5px;
}

tfoot, tr:nth-child(even) {
  background: #eee;
}
```

<br>

::: info
<i class="fa fa-info-circle" style="color: white"></i>&nbsp; The formulas below apply to releases 3.1+ and may not be applicable to older releases.
:::

Gannet quantifies metabolite levels in different ways at various points in the pipeline depending on what reference signals are present and if structural image data are available. It is important to understand what the underlying quantification formulas are for the interpretation and reporting of results.

<u>A note on units of concentration</u>

There is a perennial issue surrounding how quantified in vivo MRS measurements should be expressed; that is, in what biochemical units (if any) they should be reported. In vivo MRS measurements can be, and have been, reported in molar, molal, and institutional units (i.u.) or as simple signal area ratios. However, this makes interpreting and comparing such values, especially across published studies, highly challenging. While Gannet does apply a number of signal scaling factors to water-referenced metabolite measurements (that are required when reporting "absolute" concentrations), it is our philosophy that these measurements should be considered pseudo-absolute because accounting for all empirical signal scaling factors is impractical (perhaps even impossible) to permit truly absolute quantification. Therefore, Gannet reports water-referenced metabolite measurements in i.u.

## GannetFit

`MRS_struct.out.vox1.(metab).ConcCr`, `MRS_struct.out.vox1.(metab).ConcCho`, and `MRS_struct.out.vox1.(metab).ConcNAA` are simple signal integral ratios of the metabolite of interest and the metabolite reference signals Cr, Cho, and NAA, respectively:

$$C = \frac{I_{M}}{I_{ref}}$$

If a water reference is provided, `MRS_struct.out.vox1.(metab).ConcIU` is also calculated. It is defined as the signal integral ratio of the metabolite of interest and water reference scaled by a number of global signal scaling factors:

$$
C_{corr} =
\frac{I_{M}}{I_{W}}\cdot
\frac{H_{W}}{H_{M}}\cdot
\frac{MM}{\kappa}\cdot
C_{W}\cdot
W_{vis}\cdot
\frac{\exp\left(-\frac{TE_{W}}{T_{2W}}\right)\left[1-\exp\left(-\frac{TR_{W}}{T_{1W}}\right)\right]}
{\exp\left(-\frac{TE_{M}}{T_{2M}}\right)\left[1-\exp\left(-\frac{TR_{M}}{T_{1M}}\right)\right]}
$$

where:

| <u>Parameter</u> | <u>Description</u> | <u>Default value</u> |
| :- | :-------- | :------ |
| $C_{corr}$ | Estimated metabolite concentration in i.u. | |
| $I_{M}$ | Metabolite signal integral | |
| $I_{W}$ | Water signal integral | |
| $H_{M}$ | Number of ^1^H protons that give rise to the metabolite signal | Metabolite dependent; see `GannetFit.m` for default values |
| $H_{W}$ | Number of ^1^H protons that give rise to the water signal | 2 |
| $MM$ | Correction factor for the contribution of the co-edited macromolecule signal in the metabolite signal | 0.45 for GABA editing and 1 for all other edited metabolites |
| $\kappa$ | Editing efficiency | Acquisition dependent; 0.5 for GABA editing |
| $C_{W}$ | Molal concentration of pure water | 55.51 mol/kg |
| $W_{vis}$ | Approximate relative MR visibility of water in brain tissue | 0.65 [@Ernst1993] |
| $TE_{W}$ | Echo time of the water reference acquisition | Acquisition dependent |
| $TR_{W}$ | Repetition time of the water reference acquisition | Acquisition dependent |
| $TE_{M}$ | Echo time of the metabolite acquisition | Acquisition dependent |
| $TR_{M}$ | Repetition time of the metabolite acquisition | Acquisition dependent |
| $T_{2W}$ | Average transverse relaxation time of water in GM and WM | 1.100 s [@Wansapura1999] |
| $T_{1W}$ | Average longitudinal relaxation time of water in GM and WM | 0.095 s [@Wansapura1999] |
| $T_{2M}$ | Transverse relaxation time of metabolite | Metabolite dependent; see `GannetFit.m` for default values |
| $T_{1M}$ | Longitudinal relaxation time of metabolite | Metabolite dependent; see `GannetFit.m` for default values |

## GannetSegment

When segmenting structural images to obtain voxel volume fractions of GM, WM, and CSF, and if a water reference is available, a CSF-only correction is applied to the `ConcIU` measurement.

`MRS_struct.out.vox1.(metab).ConcIU_CSFcorr`:

$$
C_{CSFcorr} =
\frac{I_{M}}{I_{W}}\cdot
\frac{H_{W}}{H_{M}}\cdot
\frac{MM}{\kappa}\cdot
C_{W}\cdot
W_{vis}\cdot
\frac{\exp\left(-\frac{TE_{W}}{T_{2W}}\right)\left[1-\exp\left(-\frac{TR_{W}}{T_{1W}}\right)\right]}
{\exp\left(-\frac{TE_{M}}{T_{2M}}\right)\left[1-\exp\left(-\frac{TR_{M}}{T_{1M}}\right)\right]}\cdot
\frac{1}{1-f_{CSF}}
$$

where $C_{CSFcorr}$ is the estimated metabolite concentration in i.u. corrected for CSF and $f_{CSF}$ is the voxel volume fraction of CSF.

## GannetQuantify

`GannetQuantify.m` goes a step further and corrects for partial volume effects that attenuate the observed water and metabolite signals. There are two approaches that are employed. The first is termed the Gasparovic et al. [@Gasparovic2006] method and the second is termed the Harris et al. [@Harris2015] method. Although similar, the difference between these approaches is that the Harris et al. method additionally accounts for intrinsic differences in metabolite concentrations in GM and WM. 

### <u>The Gasparovic et al. method</u>

`MRS_struct.out.vox1.(metab).ConcIU_TissCorr`:

$$
C_{TissCorr} =
\frac{I_{M}}{I_{W}}\cdot
\frac{H_{W}}{H_{M}}\cdot
\frac{MM}{\kappa}\cdot
C_{W}\cdot
\frac{\sum_{i}^{GM,WM,CSF}f_{i}\beta_{i}\exp\left(-\frac{TE_{W}}{T_{2W,i}}\right)\left[1-\exp\left(-\frac{TR_{W}}{T_{1W,i}}\right)\right]}
{(1-f_{CSF})\exp\left(-\frac{TE_{M}}{T_{2M}}\right)\left[1-\exp\left(-\frac{TR_{M}}{T_{1M}}\right)\right]}
$$

where:

| <u>Parameter</u> | <u>Description</u> | <u>Default value</u> |
| :- | :-------- | :------ |
| $C_{TissCorr}$ | Estimated metabolite concentration in i.u. corrected for partial volume effects of water | |
| $f_{i}$ | Voxel volume fraction of GM, WM, or CSF | |
| $\beta_i$ | Relative MR visibility of water in GM, WM, or CSF | 0.78, 0.65, and 0.97 [@Ernst1993] |
| $T_{2W,i}$ | Transverse relaxation time of water in GM, WM, or CSF | 0.110, 0.0792, and 0.503 s [@Wansapura1999; @Piechnik2009] |
| $T_{1W,i}$ | Longitudinal relaxation time of water in GM, WM, or CSF | 1.331, 0.832, and 3.817 s [@Wansapura1999; @Lu2005] |

### <u>The Harris et al. method</u>

`MRS_struct.out.vox1.(metab).ConcIU_AlphaTissCorr`:

$$
C_{AlphaCorr} =
\frac{I_{M}}{I_{W}}\cdot
\frac{H_{W}}{H_{M}}\cdot
\frac{MM}{\kappa}\cdot
\frac{\sum_{i}^{GM,WM,CSF}f_{i}C_{W,i}\exp\left(-\frac{TE_{W}}{T_{2W,i}}\right)\left[1-\exp\left(-\frac{TR_{W}}{T_{1W,i}}\right)\right]}
{\exp\left(-\frac{TE_{M}}{T_{2M}}\right)\left[1-\exp\left(-\frac{TR_{M}}{T_{1M}}\right)\right]}\cdot
\frac{1}{f_{GM}+\alpha{f_{WM}}}
$$

where:

| <u>Parameter</u> | <u>Description</u> | <u>Default value</u> |
| :- | :-------- | :------ |
| $C_{AlphaCorr}$ | Estimated metabolite concentration in i.u. corrected for partial volume effects of water and metabolite | |
| $C_{W,i}$ | Molal concentration of water in GM, WM, or CSF | 43.30, 36.08, and 53.84 mol/kg [@Ernst1993; @Gasparovic2006] |
| $\alpha$ | Ratio of intrinsic WM:GM metabolite concentrations | Metabolite dependent; see `GannetQuantify.m` for default values |

A modification of the Harris et al. method is also calculated where `ConcIU_AlphaTissCorr` is further scaled by the average voxel GM and WM composition of all datasets in a group.

`MRS_struct.out.vox1.(metab).ConcIU_AlphaTissCorr_GrpNorm`:

$$
C_{AlphaCorrNorm} =
\frac{I_{M}}{I_{W}}\cdot
\frac{H_{W}}{H_{M}}\cdot
\frac{MM}{\kappa}\cdot
\frac{\sum_{i}^{GM,WM,CSF}f_{i}C_{W,i}\exp\left(-\frac{TE_{W}}{T_{2W,i}}\right)\left[1-\exp\left(-\frac{TR_{W}}{T_{1W,i}}\right)\right]}
{\exp\left(-\frac{TE_{M}}{T_{2M}}\right)\left[1-\exp\left(-\frac{TR_{M}}{T_{1M}}\right)\right]}\cdot
\frac{\mu_{GM}+\alpha{\mu_{WM}}}{(f_{GM}+\alpha{f_{WM}})(\mu_{GM}+{\mu_{WM}})}
$$

where:

| <u>Parameter</u> | <u>Description</u> |
| :- | :-------- |
| $C_{AlphaCorrNorm}$ | Estimated metabolite concentration in i.u. corrected for partial volume effects of water and metabolite and adjusted to the average voxel GM and WM composition across a group |
| ${\mu_{GM}}$ | Group-averaged voxel volume fraction of GM |
| ${\mu_{WM}}$ | Group-averaged voxel volume fraction of WM |

<br>

### References