# Introduction and Background

## Introduction

Diffusion weighted magnetic resonance imaging (MRI) is an imaging technique that has allowed unique insights into both the microstructural properties and the organization of cerebral white matter tissues. This technique measures the relative displacement of water molecules in the biological tissues and is highly sensitive to any microstructure which restricts this diffusion. Diffusion MRI is particularly useful in tissues with a high degree of organization, such as the white matter tissues of the brain. This organization means that the restricted diffusion is coherent and measuring the diffusion signal allows us to not only infer the presence or absence of diffusion restricting elements, but also how they’re organized and other tissue properties. For example, in the white matter of the brain, the diffusion characteristics of the tissue can give us insight into the organization of axons, the level of myelination of those axons and possible pathology in these tissues. The level of information contained in diffusion imaging makes this imaging technique both powerful and challenging to work with.

Diffusion images cannot generally be read by human specialists, but instead must be modeled using computational techniques. These modeling techniques can produce either composite images or 3d renderings, which can then be used in clinic or for research, or quantitative measurements which can then be used as biomarkers of brain heath or disease progression. However, in order to maximize the utility of diffusion imaging one needs to pick the right imaging protocol and modeling approach for a given problem. The modeling of the diffusion signal can be approached both as a local problem, modeling the characteristics of a given brain region or voxel using the diffusion signal specific to that structure. The model can also be thought of as a whole brain model, trying not only the estimate local tissue properties, but also the organization of connections and pathways that contribute to the architecture of the whole brain. Fiber tracking, or tractography, describes the process of using local directional information from brain tissues to build reconstruct these tracts and pathways. These techniques are fairly new and actively being developed and have had some success in describing the organizational properties of the human brain, or the human connectome.

In this work, I present some common diffusion modeling techniques that have been applied to diffusion MRI, focused specifically on modeling techniques that estimate directional information which can be used for fiber tacking. I present and compare several methods for estimating diffusion MRI noise and in the process discuss how noise estimates for diffusion MRI acquisitions can help identify model failures inform the choice of model for an acquisition type. I further present a framework for thinking about and implement fiber tract reconstruction from diffusion MRI data. Finally I present an application of these methods to a large, public data set for the purpose of understanding the impact of high body mass index on brain health.

## Background

### Diffusion MRI Acquisition

Diffusion weighting in modern neuroimaging applications is most often done using the echo planer imaging (EPI) technique(Poustchi-Amin 2001, Ordidge 1984). Because diffusion MRI is attempting to measure the tiny, random motion of water molecules, it is highly sensitive to gross motion of the subject. EPI helps mitigate issues related to subject motion by reducing the time required to acquire each image to less than 100ms. This short acquisition time is also important in diffusion MRI because a complete diffusion MRI data set often includes dozens, or even hundreds, of brain volumes. The time efficiency of the EPI sequence makes it practical to collect complete diffusion MRI data sets in MRI scan sessions that are acceptable for research subjects and patients.

The EPI sequence uses a series of gradient echos, called an echo train, to acquire a large portion of each image slice in a single radio frequency (RF) excitation. Multi-shot EPI sequences have been proposed in order to improve SNR and reduce distortions in diffusion MRI. However, these sequences require longer acquisition times and can produce phase errors in the the data(Feinberg 1994). The result of these limitations has been that single-shot EPI, where the entire image slice is acquired in a single excitation, has become dominant in diffusion MRI acquisitions. The diffusion weighting in a diffusion MRI sequence is introduced by using a pair of diffusion gradients on either side of a 180 degree RF pulse. The first of these diffusion gradients serves to dephase the MRI signal. The following 180 degree pulse inverts the signal so that the second diffusion gradient can now rephase the signal. In the absence of motion, these two gradients have no net effect on the diffusion signal. However, when magnetic particles move during this diffusion weighting procedure, their phases can only partially align, resulting in a single fallout. Equation \ref{eqn:diffSignal} gives the equation for the diffusion signal, $$S$$, defined in terms of $$S_{0}$$, the expected signal when no diffusion signal is applied (Magnetic Resonance Im...). In this equation, $$\gamma$$ is a constant, called the gyromagnetic ratio, $$G$$ is the gradient strength defined in units of Tesla per meter, $$\delta$$ is the length, in seconds, of each of two gradient pulses applied, and $$\Delta$$ is the time, in seconds, between the first and second diffusion gradient. Finally, $$D(\vec{u})$$ is the net diffusivity along the direction of the diffusion gradient, $$\vec{u}$$.

$$\label{eqn:diffSignal}S(\vec{u})=S_{0}e^{-\gamma^{2}G^{2}\delta^{2}(\Delta-\frac{1}{3}\delta)D(\vec{u})}\\$$

In practice, all the terms that contribute to the diffusion weighting are grouped into one factor called the b value, where $$b=\gamma^{2}G^{2}\delta^{2}(\Delta-\frac{1}{3}\delta)$$. This b value has the inverse units of diffusivity, $$\frac{s}{m^{2}}$$, and represents the net diffusion weighting of the sequence. In some applications, like storke imaging, diffusivity is so altered in the brain tissues that imaging with a few gradient directions is enough to observe and measure the effect (Mukherjee 2000). In stroke imaging a metric called apparent diffusion coefficient (ADC) is used. The ADC is the average diffusivity measured along three orthogonal directions, usually the x, y, and z gradient directions. The ADC can be estimated by acquiring three diffusion weighted scans and one scan with no diffusion weighting. However, to do any more complex modeling of the diffusion signal requires acquiring diffusion weighted volumes. The next section discusses different diffusion models, and the types of data need to support the modeling.