### Synopsis

**Diffusion MRI can be modeled as sampling the Fourier transform of the
Ensemble Average Propagator (EAP). This is potentially advantageous because of
extensive theory that has been developed to characterize sampling requirements,
accuracy, and stability for Fourier reconstruction. However, previous work has not taken advantage
of this characterization. This work
presents a novel theoretical framework that precisely describes the
relationship between the estimated EAP and the true original EAP. The framework is applicable to arbitrary
linear EAP estimation methods, and for example, provides new insights into the
design of q-space sampling patterns
and the selection of EAP estimation methods.**### Purpose

Under the short-pulse approximation, the data measured in diffusion
MRI can be modeled as the Fourier transform of the Ensemble Average Propagator
(EAP) [1], a probability distribution that characterizes the molecular diffusion
of the spins within each voxel. This is potentially advantageous because of extensive
theory that has been developed to characterize sampling requirements, accuracy,
and stability of Fourier reconstruction. However, existing diffusion MRI
sampling and EAP estimation approaches have largely been developed and tuned
without the benefit of such theory, instead relying on intuition, approximations,
modeling assumptions, and extensive empirical evaluation.
This
work introduces a novel theory that can be used to characterize the performance
of arbitrary linear EAP estimation methods with arbitrary q-space sampling
schemes, building off of previous theoretical characterizations of orientation
estimation [2-4]. For the first time,
this provides a precise theoretical relationship between the true EAP and the
estimated EAP. This relationship is similar to the point-spread function (PSF) relationship
that is widely used to characterize the spatial resolution of MR images. In the context of EAP estimation, this
theoretical framework provides direct insight into issues such as resolution,
aliasing, and noise amplification. This
valuable information can be directly used when choosing between different q-space sampling schemes and EAP
estimation methods. Due to space constraints, we only describe the noiseless
case in this abstract, though we also have a full noise-based theory.

### Theory

Noiseless diffusion data $$$E(\textbf{q}_m)$$$ measured at q-space location $$$(\textbf{q}_m)$$$ can be modeled as the Fourier transform of the EAP, and a variety of linear and nonlinear EAP estimation methods
have been designed for such data. This
work focuses on linear methods like DSI [5], GQI [6], 3D-SHORE [7], SPF [8],
SPFdual [9], SoH [10], and directional radial basis function estimation [11]. While each of these methods is based on
different assumptions, they are all linear and can be characterized by the
common reconstruction formula:
$$EAP_{estimated}(\textbf{x})=∑_{m=1}^ME(\textbf{q}_m)G(\textbf{x},\textbf{q}_m)$$ where $$$G(\textbf{x},\textbf{q}_m)$$$ are the method-dependent linear estimation coefficients. Due to the symmetry Fourier transform relationships, it is
possible to expand this expression as
$$EAP_{estimated}(\textbf{x})=∫EAP_{true}(\textbf{ξ})g(\textbf{x},\textbf {ξ})d\textbf{ξ}$$
where $$$g(\textbf{x},\textbf{ξ})=∑_{m=1}^MG(\textbf{x},\textbf{q}_m)cos(2π\textbf{q}_m^T\textbf{ξ})$$$
is the “EAP response function”. The EAP response function is
similar to the standard PSF used in imaging.
Similar to a PSF, the EAP response function should be similar to a delta
function in order to achieve high fidelity EAP reconstruction, though some
amount of blurring and aliasing is inevitable due to the finite sampling of q-space.
Importantly, the EAP response function is easy to compute,
and provides direct insight into the quality of a given q-space sampling-scheme and linear EAP estimation method.

### Illustration

An illustration of the usefulness of our proposed theoretical
framework is shown in Fig. 1, where we compare EAP responses for different q-space sampling schemes (multi-shell
and Cartesian, each using 204 and 515 samples and maximum b-values of 4000 and 10000s/mm2) for GQI EAP
estimation. As can be seen, the Cartesian
sampling method suffers from coherent aliasing (seen as multiple peaks in the
EAP response), while, multi-shell sampling scheme suffers from incoherent
aliasing (side lobe interference). This
expectation is confirmed with simulated data as shown in Fig. 2. Note that there is a clear trade-off between
the number of measured samples, the amount of aliasing interference (controlled
by the sample spacing in q-space),
and the resolution of the EAP (controlled by the size of the largest b-value). These effects are expected from traditional
Fourier sampling theory, though have not usually been considered when designing
diffusion experiments.
As
further illustration, Figs. 3 and 4 show that the EAP response can be used to
theoretically predict the impact of user-selected reconstruction parameters. Specifically, we show EAP responses and
estimated EAPs for 3D-SHORE EAP estimation using different values of the 3D-SHORE
regularization parameter [7]. The EAP response behavior in Fig. 3 suggests that
increasing regularization would initially improve accuracy by reducing sidelobe
interference, though will eventually lead to a loss in EAP resolution if the
regularization is increased too far.
This theoretical prediction is confirmed by the estimation results shown
in Fig. 4. This kind of characterization
enables a new mechanism for optimizing reconstruction parameters.

### Conclusion

We have proposed novel theoretical tools that can be
used to characterize the performance linear diffusion MRI estimation methods
without requiring extensive empirical testing. Notably, these tools do not
require any assumptions about the EAP, and can be used even when traditional
modeling assumptions are inaccurate. We
expect these tools to be useful when designing sampling and estimation methods
for a wide range of diffusion MRI experiments.

### Acknowledgements

This work was supported in part by NSF CAREER award CCF-1350563 and NIH grant R01-NS089212.### References

[1] Callaghan, P. T. Principles of Nuclear Magnetic Resonance
Microscopy. Clarendon Press, 1991.
[2] Tuch, D.
S. Q-Ball imaging. Magn. Reson. Med., 2004, 52, 1358-1372.
[3 Haldar, J. P. & Leahy, R. M. Linear transforms for
Fourier data on the sphere: Application to high angular resolution diffusion
MRI of the brain. NeuroImage, 2013, 71, 233-247.
[4]
Varadarajan, D. & Haldar, J. MS-FRACT: Optimized linear transform methods
for ODF estimation in multi-shell diffusion MRI. IEEE ISBI, 2015, 1172-1175.
[5] Wedeen, V. J.; Hagmann, P.; Tseng, W.-Y. I.; Reese, T. G.
& Weisskoff, R. M. Mapping complex tissue architecture with diffusion
spectrum magnetic resonance imaging. Magn. Reson. Med., 2005, 54, 1377-1386.
[6] Yeh,
F.-C.; Wedeen, V. J. & Tseng, W.-Y. I. Generalized q-sampling imaging. IEEE Trans. Med. Imag., 2010, 29, 1626-1635.
[7] Ozarslan,
E.; Koay, C.; Shepherd, T. M.; Blackb, S. J. & Basser, P. J. Simple
harmonic oscillator based reconstruction and estimation for three-dimensional
q-space MRI. ISMRM 17th Annual Meeting and Exhibition, Honolulu, 2009, 1396.
[8] Assemlal,
H.-E.; Tschumperlé, D. & Brun, L. Efficient and robust computation of PDF
features from diffusion MR signal. Med. Image Anal., 2009, 13, 715-729.
[9] Merlet, S.; Cheng, J.; Ghosh, A. &
Deriche, R. Spherical Polar Fourier EAP and odf reconstruction via compressed
sensing in diffusion MRI. IEEE ISBI, 2011, 365-371.
[10]
Descoteaux, M.; Deriche, R.; LeBihan, D.; Mangin, J.-F. & Poupon, C.
Multiple q-shell diffusion propagator imaging. Medical Image Analysis,
Elsevier, 2011, 15, 603-621.
[11] Ning,
L.; Westin, C.-F. & Rathi, Y. Estimating diffusion propagator and its
moments using directional radial basis functions. IEEE Transactions on Medical
Image, 2015, 34, 1-21.