## Q-space imaging

`Q-space imaging maps the spatial distribution of diffusion displacement.`

source: Sosnovik, D. E., Wang, R., Dai, G., Reese, T. G., & Wedeen, V. J. (2009). Diffusion MR tractography of the heart. Journal of Cardiovascular Magnetic Resonance, 11(1), 1-15. [link]

Callaghan proposed the idea of "q-space imaging" [1], which can be compared to the k-space relation. The k-space encoding (coordinates) is defined by the readout gradient, whereas q-space encoding is defined by the diffusion gradient. The k-space signals have a Fourier relation with the spatial distribution of spins, whereas the q-space signals also have a Fourier relation with the "ensemble average propagator" (EAP).

$S(q) = \int P(R) e^{2\pi iqR} dR$

Q-space imaging estimates the "ensemble average propagator" (EAP). EAP, in short, is a 3-dimensional probability density function of diffusion displacement. The terms "ensemble" and "average" indicate that EAP provides an averaged estimation of the diffusion environment (regardless of its heterogeneity). The term propagator indicates the function considers only the displacement of the diffusing spins. It does not discriminate on the absolute location of the spins.

Q-space imaging is a nonparametric approach to study the diffusion distribution. It is thus categorized under model-free methods because it does not assume a particular diffusion model (e.g. tensor model, multiple-tensor model...).

The q value is defined as follows:

$b = (2 \pi q) ^{2} \Delta$

q = gGd/2p and thus

The bandwidth resolution relation in Fourier transform indicates that higher q band width results in higher resolution in R. For example, a DSI 101 sequence has b-max = 4,000 and min b-value=307 s/mm2, diffusion time = 80 ms (diffusion distance=38 micron). The q interval = 12 mm-1. Thus Rmax = 1/2(q interval) = 41 microns.

Q-space imaging compared with DTI and multi-shell.

### Significance

QSI provides rich information for characterizing the diffusion environment. It has been used to study the microscopic geometry of the tissues such as the axonal diameters, pore diameters.

Pros
1. No model violation problem in DTI
2. Provide rich diffusion information for resolving crossing fibers and accessing their integrity.

### Cons

1. The number of sampling in q-space imaging is limited by the scanning time. Simultaneous acquisition of k-space and q-space imaging results in 6D data, making q-space imaging more intractable.
2. Due to the low SNR of diffusion signal, QSI data are often acquired with limited diffusion gradient strength. This suffers substantial truncation artifact.
3. The q-space imaging assumes that the diffusion gradient encoding duration is sufficiently smaller than diffusion time. This often results in long TE. A feasible approach is using stimulated echo.
4. EAP often has to be transformed into another form to make it more feasible for analysis.

### Recital

The Fourier transform of diffusion signals provides the ensemble average propagator, which is the density distribution of the diffusion displacement.

Questions:
1. A study proposes to use EAP to study a microscopic tissue with a tiny tube structure. How to set up the q-space imaging parameter? (e.g. maximum b-value, diffusion time, ..etc.)
2. A study proposes to use EAP to study non-restricted diffusion. How to set up the q-space imaging parameter?

## Diffusion orientation distribution function (dODF)

`Diffusion orientation distribution function (dODF) quantifies ``probability distribution of the diffusion spins on a sphere. `

fiber geometry     Signals            EAP (PDF)          dODF

One way to analyze EAP (3D PDF) is by projecting the values on a unit sphere to calculate the diffusion orientation distribution function (dODF), which is defined as the probability distribution of the diffusion spins on a sphere.

One should note that there are several ways to project the EAP to a sphere, and thus the calculation methods of dODF can be different. For example,

none-weighted infinite-upper-bound dODF:

$\Psi (\hat{u}) = z\int_{0}^{\infty }P(r\hat{u})dr$

The psi function is the dODF. u is a unit orientation vector. P is the EAP. r is the displacement distance.

r-squared weighted infinite-upper-bound dODF:

$\Psi (\hat{u}) = z\int_{0}^{\infty }P(r\hat{u})r^2dr$

The upper bound of the integral can be finite, e.g., none-weighted finite-upper-bound dODF:

$\Psi (\hat{u}) = z\int_{0}^{L}P(r\hat{u})dr$

DSI [3] and GQI [4] have adopted different definitions to compute the dODF.

DSI used r-squared infinite-upper-bound weighted dODF
GQI uses a finite version of none-weighted or r-squared weighted ODF.

DSI first used inverted Fourier transform to calculate EAP, and then the EAP was then integrally radially to estimate the calculates dODF by integrating EAP. However, the EAP is often too noisy, and DSI reconstruction has to add a hanning filter to smooth it (see Reconstruction(DTI, QBI, DSI, GQI, QSDR) for detail).

### Significance

dODF provides a model-free approach to study diffusion characteristics and resolve crossing fibers.

Pros
1. dODF is a model-free approach to resolve crossing fiber.
2. dODF can provide anisotropy measures.

### Cons

1. DSI can only be applied to grid scheme (if re-griding is not used). It is limited by specific diffusion sampling schemes.
2. dODF has a limited angular resolution to resolve cross fibers.
3. dODF does not have "radial information" to further explore its biophysical meanings.

## Spin Distribution Function (SDF)

`Spin distribution function is an ODF that quantifies the density of the diffusing spins`

The dODF was formulated as a probability distribution. It can be further scaled by spin density to quantify the amount of diffusing spin at a different orientation. This idea gives rise to GQI, which allows for computing the SDF directly from q-space signals by a simple analytical relation between them.

$\Psi (\hat{u})= z_0 \sum_{ \forall S} S(\hat{g},b)H( \sigma \sqrt{6Db} <\hat{g},\hat{u}>)$
z0: a constant that unifies spin density with respect to the amount of water.
S: diffusion signals
b: b-value
sigma: the upper limit of the diffusion distance.
H(.): is a basis function. Using a sinc function as the H(.) makes SDF a none-weighted version of the ODF (also known as GQI0, the non-weighted version of GQI). It is also possible to calculate GQI1 (the r-weighted version) and GQI2 ( the r-squared weighted version, see [4] for definition) using different basis function.

It is noteworthy that SDF scaled with spin density, and thus SDF can be viewed as a resampling of spin density along the radial direction. This turns it into quantity measurement, not a probabilistic measurement. By definition, SDF quantifies the density of the diffusing spins with displacement less than L (finite upper bound) and oriented at u. This is different from the dODF from DSI because dODF was originally defined as a probability density function and the sum of its support is equal to one.  The quantity of diffusing water at axonal direction is defined as the quantitative anisotropy.

The parameter sigma allows for exploring the radial information and calculating the restricted diffusion given any length variables.

QA is less susceptible to the partial volume of crossing fibers

The diffusion phantom examines three types of partial volume effect: (a) fibers/free water, (b) fibers crossing, and (c) fibers/non-diffusive materials.

Studies have shown that QA is less susceptible to the partial volume effect:

A: FA map
B: GFA map
C: QA at horizontal
D: QA at vertical
source: Yeh, Fang-Cheng, et al. "Deterministic diffusion fiber tracking improved by quantitative anisotropy." (2013): e80713.

QA has a specific pattern in each individual

source: http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1005203
B: QA from repeat scans shows an identical pattern for each individuals C: FA does not have much difference between normal subjects.

### Significance

SDF provides a density-based measurement that can be correlated with biological characteristics. It can be calculated from any diffusion sampling schemes (e.g. DTI, DSI, HARDI, multishell) and also allows for quantifying restricted diffusion.

Pros
SDF provides a absolute measurement for diffusion quantity.

### Cons

SDF has a limited angular resolution to resolve cross fibers.
Spin density has an arbitrary unit. How to normalize SDF to allows inter-subject comparability is challenging.

### Question

What is the biological basis for QA and FA?

Exercise
1. Reconstruct diffusion data with GQI (see instructions in Reconstruction(DTI, QBI, DSI, GQI, QSDR)). Evaluate the result by inspecting the dODF and axonal direction map.
2. Change the GQI reconstruction parameters and observe how this affect the smoothness/sharpness of the dODF.
3. Remove one or more DWI and reconstruct DTI or GQI. Does it affect the result?

## Reference

[1] Callaghan, P.T., Principles of Nuclear Magnetic Resonance Microscopy. 1994: Oxford University Press.
[2] Tuch, David S. "Q‐ball imaging." Magnetic Resonance in Medicine 52.6 (2004): 1358-1372.
[3] Wedeen, Van J., et al. "Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging." Magnetic Resonance in Medicine 54.6 (2005): 1377-1386.
[4] Yeh, Fang-Cheng, Van Jay Wedeen, and Wen-Yih Isaac Tseng. "Generalized-sampling imaging." Medical Imaging, IEEE Transactions on 29.9 (2010): 1626-1635. [pdf]
[5] Barnett, Alan. "Theory of Q‐ball imaging redux: Implications for fiber tracking."Magnetic Resonance in Medicine 62.4 (2009): 910-923.