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" , 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).
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:
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.
QSI provides a 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.
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 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 to another form to make it more feasible for analysis.
The Fourier transform of diffusion signals provides the ensemble average propagator, which is the density distribution of the diffusion displacement.
fiber geometry Signals EAP (PDF) dODF
One way to analyze EAP (3D PDF) is by projecting the values on an 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 way 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:
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:
The upper bound of the integral can be finite, e.g., none-weighted finite-upper-bound dODF:
QBI , DSI , and GQI  have adopted different definitions to compute the dODF.
QBI used the none-weighted infinite-upper-bound ODF (only an approximation, see  for discussion).
DSI used r-squared infinite-upper-bound weighted dODF.
GQI uses a finite version of none-weighted or r-squared weighted ODF.
To calculate the dODF, QBI used Funk-Radon transform to approximate the dODF.
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).
dODF provide a model-free approach to study diffusion characteristics and resolve crossing fibers.
1. QBI can only be applied to HARDI (uniform b-value), DSI can only be applied to grid scheme (if re-griding is not used). They are limited by specific diffusion sampling schemes.
2. dODF has limited angular resolution to resolve cross fibers.
3. dODF does not have "radial information" to further explore its biophysical meanings.
The dODF was formulated as a probability distribution. It can be further scaled by spin density to quantify the amount of diffusing spin at 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.
z0: a constant that unifies spin density with respect to the amount of water.
S: diffusion signals
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  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 QBI and 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.
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.
SDF has limited angular resolution to resolve cross fibers.
Spin density has arbitrary unit. How to normalize SDF to allows inter-subject comparability is challenging.
The diffusion weighted images can be acquired with a variety of combinations of gradient sensitization settings (schemes). The classification is often based on the spatial distribution of gradient encoding vectors (Gx,Gy,Gz).
(Gx,Gy,Gz) distributes equally on a sphere, where |G| is constant. It is noteworthy that there is no perfect tessellation of a unit sphere (except icosahedron), and thus it is impossible to keep sampling vectors perfectly equal distributed. Certain vectors may inevitably more close to each other, causing bias.
A combination of more than two single shells is termed multi-shell. Multi-shell scheme shares the same distribution problem as the single shell. In addition, the distance between shell points can be highly nonuniform and add more bias into the scheme.
The most popular non-shell scheme is the grid scheme, also known as the diffusion spectrum imaging (DSI) scheme. (Gx,Gy,Gz) distributes on the Cartesian points within a sphere, where |G| is less than a maximum magnitude value. The major advantage of a grid scheme is its equal distance between sampling points. The limitation is that not every modeling method can work on grid sampling data. Other non-shell scheme includes body-center-cubic and face-center-cubic.
Single shell maximizes the sensitive to orientation, whereas the grid scheme is a balance between orientation and radial information. The choice is thus depending on the application. If the study concerns only axonal direction, then the single shell is the most efficient approach. If the study involves pathological condition, then grid scheme gives the best coverage on both orientation and radial information.
2. Can we reconstruct multishell and DSI data using QBI? Why or why not?
3. A study proposes to use EAP to study a microscopic tissue with a tiny tube structure. How to setup the q-space imaging parameter? (e.g. maximum b-value, diffusion time, ..etc.)
4. A study proposes to use EAP to study non-restricted diffusion. How to setup the q-space imaging parameter?
1. Reconstruct diffusion data with QBI, DSI, GQI (see instructions in Reconstruction(DTI, QBI, DSI, GQI, QSDR)). Evaluate the result by inspecting the dODF and axonal direction map.
2. Change the QBI, DSI, and GQI reconstruction parameters and observe how this affect the smoothness/sharpness of the dODF.
3. Remove one or more DWI and reconstruct DSI, QBI, or GQI? Does it affect the result?
4. Load the sample images provided from Sample Images. Could you tell what diffusion scheme was used by the b-table?
 Callaghan, P.T., Principles of Nuclear Magnetic Resonance Microscopy. 1994: Oxford University Press.
 Tuch, David S. "Q‐ball imaging." Magnetic Resonance in Medicine 52.6 (2004): 1358-1372.
 Wedeen, Van J., et al. "Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging." Magnetic Resonance in Medicine 54.6 (2005): 1377-1386.
 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]
 Barnett, Alan. "Theory of Q‐ball imaging redux: Implications for fiber tracking."Magnetic Resonance in Medicine 62.4 (2009): 910-923.