## Introduction to reconstruction methods

DSI Studio supports several reconstruction methods, but what are their difference and similarity? To better understand the pros and cons of each reconstruction method, we can categorize them into either model-based methods or model-free methods. This categorization is similar to the classification of the parametric and non-parametric methods in statistics. The parametric approaches assume a known distribution/model (e.g. Gaussian) to obtain inference, whereas non-parametric approaches assume no underlying distribution/model and obtain inference using empirical distribution. The model-based method is comparable to parametric approaches, and the model-free methods are comparable to non-parametric approaches.

Model-based methods include DTI, ball-and-sticks model, NODDI as well as more complicated model like CHARM and AxCaliber. Model based methods assume a particular diffusion distribution pattern/function, and the parameters are calculated by fitting diffusion signals with the model. For example, DTI assumes that velocity of water diffusion follows a 3D Gaussian distribution, and the tensor calculated is exactly the covariance matrix of the Gaussian. Ball-and-sticks models is a kind of multiple tensor model, whereas the ball is the isotropic Gaussian, and the sticks is a purely anisotropic Gaussian. The strength of model-based methods, similar to the parametric methods in statistics, is that they only requires few samples to get the whole distribution. However, the results of model-based methods are limited by the model, and it is common that the diffusion pattern does not follows the assumption. A complicated model may also have overfitting problem. For example, past studies have used bi-exponential model to fit intra-cellular and extra-cellular diffusion, but it turned out that they failed to reveal such biophysics.

Model-free methods estimate the empirical distribution of the water diffusion, and there is no assumption on the distribution itself. The methods include diffusion spectrum imaging (DSI), q-ball imaging (QBI), and generalized q-sampling imaging (GQI). DSI use Fourier transformation and numerical integration to calculate the orientation distribution function (ODF, which is the empirical distribution of water diffusion at different orientations) of water diffusion. The Fourier transform requires a specific grid structure for the diffusion sampling scheme (multiple b-values multiple directions). QBI uses either Funk-Randon transform or spherical harmonics to calculate the ODF. The calculation also requires a shell like diffusion sampling scheme (i.e. HARDI acquisition, single b-value, multiple directions). GQI is based on an analytical relation between diffusion signals and the spin distribution function (SDF), which is the density of diffusing water at different direction. While DSI and QBI are numerical estimation of the ODF, GQI offers a direct calculation of SDF and thus free from the error in the numerical estimation. In the original GQI paper, it was shown that GQI is a general form for both QBI and DSI, and thus GQI can use any types of diffusion sampling schemes, including the one used in DSI or QBI. The SDF calculated from GQI also provides density-based measurements such as QA, ISO, and RDI. It is different from ODF, which is a probability density function on a unit sphere. The advantage of model-free methods is that they are not limited by a model. They do not assume a particular diffusion structure, and there is no risk for violation of the model. It does not have the overfitting problem in the model-based method. The calculation does not require complicated optimization or fitting and thus is less affected by outliers in comparison with model-based method. The down side of model-free methods is that they often need more diffusion samplings, at least 60 to get a more robust estimation (In comparison,DTI only needs 6 sampling in addition to b0).

There are still other methods which are both model-based and model-free. The spherical deconvolution methods (including CSD and the derived approaches) can be viewed as model-based because they assume a underlying distribution for a fiber population (i.e. response function) and used it for deconvolution. It can be also viewed as model-free because they also get an ODF for the fiber distribution (also termed fiber orientation distribution, FOD). The advantage of these methods is that they have both the benefit of model-based and model-free method (i.e. less sampling to get a sharp ODF). The disadvantage is that they also have flaws from both (model violated, sensitive to outliers, prone to create false fibers).

The choice of the method really depends on the application. If fiber orientation and the tractography itself is the only thing needed, model-based methods such as ball-and-stick model and spherical deconvolution or even DTI usually give good results since the violating the model assumption does not really affect the resolved fiber orientation. Model-free methods are also good for tractography if there is enough diffusion sampling. A recent comparison study have shown that the methods that offered the highest validation connections are GQI and DTI methods (see http://biorxiv.org/content/early/2016/11/07/084137 ). If the study needs to investigate the diffusion pattern and property beyond DTI, model-free methods are better because they can reveal the underlying pattern without assuming any distribution. GQI is a better choice over DSI and QBI because it provides a more accurate estimation of the diffusion property without the error from numerical estimation. GQI further provides its further derived form called q-space diffeomorphic reconstruction (QSDR), a method that reconstructs GQI diffusion pattern directly in the MNI space. This makes group comparison and regression studies much easier. QSDR enables template construction, connectome fingerprinting, and connectometry analysis.

## Introduction to diffusion Indices

A common question about diffusion indices is that how do they differ in term of their biophysical meaning? Here is a brief explanation. First of all, all tensor derived measurements, including FA, AD, RD, are based on "diffusivity", which by definition measures how fast water diffuses. By contrast, measurements derived from GQI or QSDR, such as QA, iso, RDI, and local connectome fingerprint are based on "density", which by definition measure how much water diffuses in a particular direction. It is like two different approaches to measure the "traffic". One way is to quantify traffic by measuring how fast vehicles travels on the highway (diffusivity) or count how many vehicles are traveling (density). These measurement may be related, but entirely different.

In application, diffusivity measurements are more sensitive to pathological conditions, whereas density measurements are more sensitive to individual/physiological difference (see Yeh et al. PLoS Comput Biol 12(11): e1005203, where local connectome fingerprint is a type of density measurement). To better understand this difference, we can compare axons to water pipes. If the pipes are in good condition, they will have same water transfusion rate (diffusivity), even though the amount of water (density) being transfused can vary a lot. This indicates that diffusivity is good for detecting whether the structural is still intact, whereas the density measurement is good for quantifying the "connectivity" because it quantifies the total quantity of the diffusing water. This is shown in Yeh et al. PLoS Comput Biol 12(11): e1005203, where the individuality can be only revealed by density-based measurement, not diffusivity measurement.

The following is a brief explanation for the diffusion indices provided by DSI Studio:

FA: fractional anisotropy AD: axial diffusivity RD: radial diffusivity (the average of two radial eigenvalues) MD: mean diffusivity (the average of all eigen values). The diffusivity (either RD, MD, or AD) calculated in DSI Studio has a unit of 10-3 mm2/s

GFA: generalized fractional anisotropy. GFA is calculated from an ODF function. The definition is documented in the q-ball imaging paper [3]. GFA has a high correlation with FA (Fritzsche, K. H., et al. (2010). Neuroimage 51(1): 242-251.), and it also suffers from partial volume effect. The value deceases in fiber crossing or voxels with CSF partial volume. GFA values may not be comparable across different reconstruction methods since the sharpness of ODF may differ due to the reconstruction method used. It may not be comparable if different b-value and diffusion sampling scheme are used. Studies using GFA should be conducted in a careful control manner to make sure that the results are not due to other confounding factors.

QA: quantitative anisotropy (QA). QA is calculated from the peak orientations on a spin distribution function (SDF). Each peak orientation defines a QA value. The definition for QA is defined in the generalized q-sampling imaging paper [9]. One should note that QA is defined for each fiber orientation, whereas FA and GFA are defined for each voxel. This forms a big difference in fiber tracking, since QA can be used to filter out false fibers in crossing fiber scenario. In DSI Studio, qa0 represents the QA value for the most prominent fiber orientations, and qa1 the second, ...etc.

Another difference between QA and FA, GFA is that QA scales with spin density. These features make it less susceptible to partial volume effect (see a comparison in Yeh F-C et al. PLoS ONE 8(11): e80713.2013). However, since QA scales with spin density, it is affected by T2-shine through, receiver gain, and B1 inhomogeneity. The QA value may not be comparable across subject if different TE and diffusion sampling scheme are used.

QA scales with diffusion signals and thus can be affected by the gain of the receiver coil. To ensure a better consistency/reproducibility, DSI Studio uses a voxel with free water diffusion to calibrate QA. The "CSF calibration" option use spatial normalization to search for voxels in the 3rd ventricle and use it to calibrate QA and achieve better reproducibility.

NQA is the normalized QA. Normalizing QA is another solution to ensure QA consistency across subjects. DSI Studio has adopts strategies to calibrate QA (see the above description under the QA subtitle), but sometimes the calibration may not be accurate enough, causing a large variability in QA. A solution to this problem is to scales the maximum QA value of a subject to 1 so that QA may be more comparable across the subject. This assumes that all subjects share the same compactness of the white matter bundle.

ISO: the isotropic value of the ODF. The definition is documented in [9]. ISO is the minimum distribution value of an ODF, and thus it represent background isotropic diffusion.

RDI: an index quantifying the density of restricted diffusion given a displacement distance (L). nRDI quantifies non-restricted diffusion. See the "restricted diffusion imaging" reconstruction above for detail.

## Introducion to diffusion MRI analysis methods

### Voxel-based analysis

SOURCE: Zhang, Yu, Norbert Schuff, An-Tao Du, Howard J. Rosen, Joel H. Kramer, Maria Luisa Gorno-Tempini, Bruce L. Miller, and Michael W. Weiner. "White matter damage in frontotemporal dementia and Alzheimer's disease measured by diffusion MRI." Brain 132, no. 9 (2009): 2579-2592.

Spatially transform the mapping of diffusion indices to the standard space and analysis them voxel-by-voxel.

Pros:
1. Simple and straightforward.

Cons:
1. After correction for family wise error, the significance is usually very low due to low SNR of diffusion MRI
2. Most spatial transformation method is designed for gray matter regions.

### ROI-based analysis

source: Zhang, Junying, Yunxia Wang, Jun Wang, Xiaoqing Zhou, Ni Shu, Yongyan Wang, and Zhanjun Zhang. "White matter integrity disruptions associated with cognitive impairments in type 2 diabetic patients." Diabetes 63, no. 11 (2014): 3596-3605.

Manually or using an atlas to define a study region and analysis the average of diffusion indices
Pros:
1. Simple and straightforward.
2. There is no need to correct for multiple comparison.

Cons:
1. White matter region is often ill-defined. Multiple fiber pathways may share the same region, making result interpretation difficult.
2. Manually defining the region can be time consuming.

### Track-specific analysis

source: Advanced NTUH MRI Lab, http://abmri.mc.ntu.edu.tw/en/technique.php

Use diffusion MRI fiber tracking to obtain the track trajectories and average diffusion indices along the pathways.

Pros:
1. More accurate way to define white matter region than ROI-based analysis
2. There is no need to correct for multiple comparison.

Cons:
1.The analysis is often time-consuming.
2.The accuracy depends on fiber tracking and subject to human error.
3.The finding be limited to a local segment of a track. Averaging the indices along the pathway may decrease the significance of the findings.

### Tract-Based Spatial Statistics

source: Haller, Sven, Pascal Missonnier, F. R. Herrmann, Cristelle Rodriguez, M-P. Deiber, Duy Nguyen, Gabriel Gold, K-O. Lovblad, and Panteleimon Giannakopoulos. "Individual classification of mild cognitive impairment subtypes by support vector machine analysis of white matter DTI." American Journal of Neuroradiology 34, no. 2 (2013): 283-291.

TBSS projects voxel-based indices to a "skeleton" and analyze them. It handles the
Pros:
1. Fully automatic

Cons:
1. TBSS is only available in FSL
2. Interpreting the result on the skeleton can be challenging since there can be multiple pathways passing around the skeleton.

Reference:
Smith, Stephen M., Mark Jenkinson, Heidi Johansen-Berg, Daniel Rueckert, Thomas E. Nichols, Clare E. Mackay, Kate E. Watkins et al. "Tract-based spatial statistics: voxelwise analysis of multi-subject diffusion data." Neuroimage 31, no. 4 (2006): 1487-1505.

### Graph theoretical analysis

source:
Bernhardt, Boris C., SeokJun Hong, Andrea Bernasconi, and Neda Bernasconi. "Imaging structural and functional brain networks in temporal lobe epilepsy." (2013).

1.Construct a connectivity matrix by first partitioning the brain into regions and then running fiber tracking to determine which regions are connected.
2.Apply graph and network analysis to extract network measures such as clustering coefficient, centrality, robustness, modularity.

Pros:
1. Study the brain structure at the network level.
2. There are several readily available graph analysis approaches that can be used.

Cons:
1. Connectivity matrix is sensitive to tracking parameter and limited by the accuracy of the tracking algorithm.
2. How to partition the brain regions is a challenge

Reference:
Bullmore, Ed, and Olaf Sporns. "Complex brain networks: graph theoretical analysis of structural and functional systems." Nature Reviews Neuroscience10.3 (2009): 186-198.

### Connectometry

1. Mapping the difference of diffusion properties at the voxel level.
2. Conduct comparison at each fiber orientation.
3. Tracking the finding along the fiber pathways.
4. Obtain statistical inference.

Pros:
1. Fully automatic
2. Connectometry identifies only the affected segment of the fiber tracks
3. Allows for multiple regression, group comparison, paired group comparison.

Cons:
1. Cannot work on distorted brain structure (e.g. brain tumor)
2. Relatively poor at identifying short-ranged focal difference.

Reference:

Yeh, Fang-Cheng, David Badre, and Timothy Verstynen. "Connectometry: A statistical approach harnessing the analytical potential of the local connectome." NeuroImage 125 (2016): 162-171.

## Questions

1.  Which analysis methods can be applied to 30-direction, b-value=1500 DTI data?