\documentclass[a4paper]{article}

\usepackage{filecontents}

\begin{filecontents}{qmriIR.bib}

@Article{MRMLilaj21b,
  author        = {Lilaj, L. and Fischer, T. and Guo, J. and Braun, J. and Sack, I. and Hirsch, S.},
  title         = {Separation of fluid and solid shear wave fields and quantification of coupling density by magnetic resonance poroelastography},
  journal       = {Magnetic Resonance in Medicine},
  year          = {2021},
  volume        = {85},
  number        = {3},
  pages         = {1655-1668},
  note          = {cited By 7},
  document_type = {Article},
  doi           = {10.1002/mrm.28507},
  source        = {Scopus}
  }

@Article{MRMLilaj21a,
  author        = {Lilaj, L. and Herthum, H. and Meyer, T. and Shahryari, M. and Bertalan, G. and Caiazzo, A. and Braun, J. and Fischer, T. and Hirsch, S. and Sack, I.},
  title         = {Inversion-recovery MR elastography of the human brain for improved stiffness quantification near fluid-solid boundaries},
  journal       = {Magnetic Resonance in Medicine},
  year          = {2021},
  volume        = {86},
  number        = {5},
  pages         = {2552-2561},
  note          = {cited By 3},
  document_type = {Article},
  doi           = {10.1002/mrm.28898}
}

@Book{MRBIbook,
  title =     {Magnetic Resonance Brain Imaging: Modeling and Data Analysis Using R, 2nd Ed.},
  publisher = {Springer},
  year =      {2023},
  author =    {J\"org Polzehl and Karsten Tabelow},
  series =    {Use R!},
  doi =       {10.1007/978-3-030-29184-6}
}
\end{filecontents}

\usepackage[style=authoryear,backend=bibtex,url=false]{biblatex} %backend tells biblatex what you will be using to process the bibliography file
\addbibresource{qmriIR}

\newcommand{\pkg}[1]{{\normalfont\fontseries{b}\selectfont #1}\index{Packages!#1}}
\let\proglang=\textsf
\let\code=\texttt
%\VignetteIndexEntry{An example session for analyzing Inversion Recovery MRI and MR Elastography data}

\title{An example session for analyzing Inversion Recovery MRI and MR Elastography data}
\author{J\"org Polzehl and Karsten Tabelow}

\begin{document}
\SweaveOpts{concordance=TRUE}

\maketitle

\setkeys{Gin}{width=\textwidth}
This document illustrates the workflow of analyzing Inversion Recovery Magnetic Resonance Imaging (IRMRI)
data. 
The example uses noisy IR data created from a small sub cube of an artificial IR image (Infinity Inversion Time), a corresponding segmentation image and MR Elastography data. For neuroimaging bacckground we refer to~\parencite{MNRLilai21b} (IRMRI) and~\parencite{MNRLilai21a} (MRE).



For an more extended introduction we refer to \cite{MRBIbook2} Chapter 6.

\section{Generating the IR MRI data}
<<0,echo=FALSE>>=
old <- options(digits=3)
on.exit(options(old))
@
First, we specify the directory where the data are stored within the package
<<0b>>=
dataDir0 <- system.file("extdataIR", package = "qMRI")
dataDir <- tempdir()
library(oro.nifti)
@

We now generate IRMRI data following a model that assumes voxel to contain a mixture of a solid tissue (either DM or WM) and fluid. 
<<1>>=
library(qMRI)
segm <- readNIfTI(file.path(dataDir0,"Brainweb_segm"))
Sf <- 900
Rf <- 0.000285
Sgm <- 400
Rgm <- 0.00075
fgm <- .15
Swm <- 370
Rwm <- 0.0011
fwm <- .05
InvTimes <- c(100, 200, 400, 600, 800, 1200, 1600, 2000, 2500, 3000, 
              3500, 4000, 4500, 5000, 6000, Inf)
InvTimes0 <- c(100, 200, 400, 600, 800, 1200, 1600, 2000, 2500, 3000, 
              3500, 4000, 4500, 5000, 6000, 15000)
@

Typical intensities as functions of inversion times an tissue type (black for CSF, red for GM and green for WM) are illustrated in Figure~\ref{Fig:curves}

<<2, fig=TRUE, width=12, height=6>>=
x <- seq(100,12000,10)
fintCSF <- qMRI:::IRhomogen(c(Sf,Rf),InvTimes0)
fintGM <- qMRI:::IRmix2(c(fgm,Rgm,Sgm),InvTimes0,Sf,Rf)
fintWM <- qMRI:::IRmix2(c(fwm,Rwm,Swm),InvTimes0,Sf,Rf)
plot(InvTimes0,fintCSF,xlab="InvTime",ylab="Intensity")
points(InvTimes0,fintGM,col=2)
points(InvTimes0,fintWM,col=3)
lines(x,qMRI:::IRhomogen(c(Sf,Rf),x))
lines(x,qMRI:::IRmix2(c(fgm,Rgm,Sgm),x,Sf,Rf),col=2)
lines(x,qMRI:::IRmix2(c(fwm,Rwm,Swm),x,Sf,Rf),col=3)
@

We generate artificial Rician distributed data with standard deviation $\sigma=40$
<<3>>=
sigma <- 40
nTimes <- length(InvTimes0)
nCSF <- sum(segm==1)
nGM <- sum(segm==2)
nWM <- sum(segm==3)
IRdata <- array(0,c(nTimes,prod(dim(segm))))
IRdata[,segm==1] <- sqrt(rnorm(nTimes*nCSF,fintCSF,sigma)^2+
                         rnorm(nTimes*nCSF,0,sigma)^2)
IRdata[,segm==2] <- sqrt(rnorm(nTimes*nGM,fintGM,sigma)^2+
                         rnorm(nTimes*nGM,0,sigma)^2)
IRdata[,segm==3] <- sqrt(rnorm(nTimes*nWM,fintWM,sigma)^2+
                         rnorm(nTimes*nWM,0,sigma)^2)
dim(IRdata) <- c(nTimes,dim(segm))
for(i in 1:9) writeNIfTI(as.nifti(IRdata[i,,,]), 
                         file.path(dataDir,paste0("IR0",i)))
for(i in 10:nTimes) writeNIfTI(as.nifti(IRdata[i,,,]), 
                         file.path(dataDir,paste0("IR",i)))
@
\section{Analysis of IR MRI data}

We now illustrate the analysis pipeline for IRMRI data. First we generate an IRdata object

<<4>>=
library(qMRI)
t1Files <- list.files(dataDir,"*.nii.gz",full.names=TRUE)
segmFile <- file.path(dataDir0,"Brainweb_segm")
IRdata <- readIRData(t1Files, InvTimes0, segmFile, sigma=sigma,
                     L=1, segmCodes=c("CSF","GM","WM"))
@

In a first analysis step parameters $S_f$ and $R_f$ characterizing fluid are obtained from voxel that are classified as CSF using the model
\begin{equation}
   \label{I-mono} 
   \xi(TI; S^f, R_1^f) = |S^f \left( 1 - 2{\mathrm e}^{-TI \cdot R_1^f} \right)|
\end{equation}
with data for inversion time $TI$ distributed as $Rician(\xi(TI; S_f, R_1^f), \sigma)$.

The parameters $S_f$ and $R_1^f$ are assumed not to vary within CSF.

<<5>>=
setCores(2) # parallel mode using 2 threads
IRfluid <- estimateIRfluid(IRdata, method="NLR", verbose=FALSE)
cat("Estimated parameters Sf:", IRfluid$Sf, 
                        " Rf:", IRfluid$Rf, "\n")
@

We here use nonlinear regression instead of the more adequate quasi-likelihood method (\code{method="QL"})
In the next step we evaluate a mixture model 

\begin{equation}\label{I-mixture}
     \xi(TI; f, S^f, R_1^f, S^s, R_1^s) = |(1-f) S^f \left( 1 - 2{\mathrm e}^{-TI \cdot R_1^f} \right)+ f S^s \left( 1 - 2{\mathrm e}^{-TI \cdot R_1^s}\right)|,
 \end{equation}

for voxel classified as GM or WM with parameters $S_f$ and $R_1^f$ plugged in. 

<<6>>=
IRmix <- estimateIRsolid(IRfluid, verbose=FALSE)
@

Parameters $S^s$ and $R_1^s$ characterizing solid material in GM and WM can be assumed to be spatially smooth within the respective tissue types. Parameter $f$ characterizes the proportion of fluid within a voxel. This parameter is difficult to estimate in model \ref{I-mixture}. 
We therefor apply an adaptive smoothing procedure within segments characterizing GM and WM to reduce the variance of the estimates  of $S^s$ and $R_1^s$

<<7>>=
sIRmix <- smoothIRSolid(IRmix, alpha=1e-4, verbose=FALSE,partial=FALSE)
@

and then re-estimate the fluid proportion $f$

<<8>>=
sIRmix <- estimateIRsolidfixed(sIRmix, verbose=FALSE)
@

We shortly illustrate the estimated maps (central slice) that we gain

<<9, fig = TRUE, width=16,height=3>>=
oldpar <- par(mfrow=c(1,4),mar=c(3,3,3,.5),mgp=c(2,1,0))
on.exit(par(oldpar))
library(adimpro)
rimage(segm[,,2])
title("Segmentation")
rimage(sIRmix$Sx[,,2],zlim=c(250,500))
title("solid intensity map")
rimage(sIRmix$Rx[,,2],zlim=c(0,.0015))
title("solid relaxation rate map")
rimage(sIRmix$fx[,,2],zlim=c(0,.4))
title("fluid proportion map")
@

All analysis steps can be combined, in this case using quasi-likelihood, simply calling

<<10, eval=FALSE>>=
sIRmix <- estimateIR(IRdata, method="QL")
@

\printbibliography

\end{document}
