Generalized mean apparent propagator (GMAP) MRI to measure and image advective and dispersive flows in medicine and biology

Abstract

Water transport in biological systems spans different regimes with distinct physical behaviors: diffusion, advection, and dispersion. Identifying these regimes is of paramount importance in many in vivo applications, among them, measuring microcirculation of blood in capillary networks and cerebrospinal fluid (CSF) transport in the glymphatic system. Diffusion magnetic resonance imaging (dMRI) can be used to encode water displacements, and a Fourier transform of the acquired signal furnishes a displacement probability density function, known as the propagator. This transformation normally requires the use of a fast Fourier transform (FFT), which presents major feasibility challenges when scanning in vivo, mainly because of dense signal sampling, resulting in long acquisition times. A second approach to reconstruct the propagator is by using analytical representation of the signal, overcoming many of the FFT’s limitations. In all analytical implementations of dMRI to date, the translational motion of water has been assumed to be exclusively diffusive, which is the case only in the absence of flow. However, retaining the phase information from the diffusion signal provides the ability to measure both mean coherent velocity and random diffusion from a single experiment. We implement and extend an analytical framework, mean apparent propagator (MAP), which can account for non-zero flow conditions. We call this method generalized MAP, or GMAP. We describe a numerical optimization scheme and implement it on data from an MRI flow phantom constructed from a pack of 10 µm beads. The advantages of GMAP over the FFT-based method in the context of sampling density and low-flow detection were demonstrated, and analytically derived propagator moments were shown to agree with theoretical values even after data subsampling. GMAP would enables the detection of microflow in vivo that could help elucidate many important biological processes.

I. INTRODUCTION

WATER translational motion provides valuable information on the viability, structure, and function of biological tissue. Diffusion magnetic resonance imaging (dMRI) is the most commonly used method to noninvasively measure the microscopic displacements of diffusing water molecules interacting with their environment [1]. In conjunction with tissue models, dMRI experiments can be used to infer macroscopic (diffusion tensor imaging, DTI) [2], [3], [4] and microscopic structural features [5], [6], [7], [8], [9], [10], [11], [12], on the basis of the diffusive transport of water molecules in heterogeneous systems.

Diffusion-based approaches that do not assume a specific tissue model generally fall into two categories: Laplace and Fourier transform-based. In the former case, the underlying microstructure is captured by a distribution of effective diffusivities that is related to the diffusion signal data via a Laplace transform [13], [14]. This approach can be extended by combining it with other magnetic resonance (MR) observables, e.g., relaxation, to deliver multidimensional information with high sensitivity and specificity to different microenvironments within biological tissue [15].

The second model-free approach exploits the Fourier relation between the diffusion-weighted signal and an important quantity, referred to as the propagator [16]

$$P(\text{r})=\int e^{i2\pi \text{q}\cdot \text{r}}E\left(\text{q}\right)\hspace{0.17em}\hspace{0.17em}\text{dq},$$ (1)

where P (r) denotes the propagator, indicating the likelihood for particles to undergo net displacement, r. The spin echo signal, E(q), depends on the reciprocal space vector $q={\text{(2π)}}^{-1}\gamma G\delta ,$ which is determined by the acquisition [17], [18], composed of a pair of magnetic pulsed gradients [19] of duration δ, separation ∆, and amplitude and direction G, with γ being the gyromagnetic ratio. In general, the coherent phase shift between the real and imaginary channels and the attenuation of the signal intensity provide the ability to measure both mean coherent velocity and random diffusion from a single experiment. In this case, the signal describes the sum of the phase shifts associated with molecular displacements, which is shown in Eq. 1.

Practically, the propagator can be reconstructed using two approaches: fast Fourier transform (FFT) and analytical. The FFT-based method is usually referred to as q-space imaging (QSI) or diffusion spectrum imaging (DSI) [20], [21], [22], and it involves the direct inversion of the acquired signal, E(q), using the discrete Fourier transform. Although Eq. 1 presents a simple relationship between the diffusion signal profile and the propagator, with a well-posed inversion procedure, a major disadvantage of DSI is that it requires dense sampling of the diffusion signal on a Cartesian q-space lattice, resulting in long acquisition times compared with clinically acceptable methodologies [22], [23]. Conversely, the second approach involves analytical representations of the signal. Those methods are convenient because they are inherently less susceptible to noise, and they provide compact representations of the signal and the estimated quantities [24], [25], [26], [27].

To date, all of the analytical representations have assumed that the odd moments of the estimated propagator are zero (e.g., symmetric propagator with zero mean), which is the case only in the absence of flow. Nevertheless, FFT-based propagator estimation has been used to accurately measure flow in a range of plants physiology and industrial applications for over two decades [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38]. While these systems do not present crucial experimental limitations, biological systems, and specifically in vivo applications, suffer from low signal-to-noise ratio (SNR) and limited scan time. Likely for these reasons, as well as other confounds, velocity imaging (i.e., velocimetry) in in vivo systems has been limited to measuring fast flows [39], [40], [41], [42], [43], [44].

Water transport in biological systems may have different properties: diffusive, advective (i.e., bulk motion), dispersive (i.e., shear flow that increases the effective diffusivity), and the mixture of them. Despite growing interest in the medical and biological communities, microscopic flow and low-flow in in vivo systems were not the focus of these previous studies. Examples of important biological mechanisms associated with low-flow include the glymphatic system and blood microcirculation. The former is a recently discovered [45] cerebral spinal fluid (CSF)–mediated transport system that clears metabolic and cellular waste products in the brain, which currently cannot be detected noninvasively. Although MRI methods were developed to measure microcirculation of blood in the capillary network (e.g., intravoxel incoherent motion, IVIM [46], [47]), separation of advection from diffusion is prone to artifacts from other bulk flow phenomena [48], [49].

A clinically feasible propagator estimation framework that accounts for flow would characterize diffusive, advective, and dispersive transport processes in each voxel of brain parenchyma. Furthermore, with recently developed feasible MRI methods that measure water molecular exchange [50], [51], an array of complementary approaches would provide a promising step toward in vivo microflow detection.

We present here an implementation of the mean apparent propagator (MAP) MRI method [27] that accounts for flow, both macroscopic and microscopic, which we call generalized MAP, or GMAP. We describe a numerical optimization framework and demonstrate it by using data from an MRI flow phantom constructed from a pack of 10 µm beads. The advantages of GMAP over the FFT-based method are compared on the basis of their q-space sampling density, maximal q-value, and ability to detect low-flow.

II. THEORY

For the purpose of the current experimental design a 1D version of MAP, termed simple harmonic oscillator-based reconstruction and estimation (SHORE) [52] was used here. In this case, q has a fixed direction and therefore reduces to a scalar, q.

For completeness, we have briefly summarized the essential derivation of the SHORE framework. The q-space MR signal is expressed in terms of the eigenfunctions of the quantummechanical simple harmonic oscillator Hamiltonian (i.e., the Hermite functions), which form a complete orthogonal basis for the space of square integrable functions:

$$\hspace{0.17em}S(q)=\sum _{n=0}^{N-1}{a}_n^{\prime }{\varphi }_n(u,q),$$ (2)

With

$${\varphi }_n\left(u,q\right)=\frac{i^{-n}}{\sqrt{2^{n}n!}}{e}^{-2{\pi }^2{q}^2{u}^2}{H}_n\left(2\pi uq\right).$$ (3)

H_n(x) is the nth order Hermite polynomial, and u is a characteristic length to be estimated. The parameter u is a tissue/data driven parameter that depends on tissue microstructure, orientation, anisotropy, and flow patterns, and should be interpreted similarly to the way in which the apparent diffusion coefficient is interpreted. The signal, E(q) = S(q)/S(q = 0), can be expressed in the same basis as

$$E(q)=\sum _{n=0}^{N-1}{a}_n{\varphi }_n\left(u,q\right),$$ (4)

where

$$a_n=\frac{a_n^{\prime }}{S\left(q=0\right)},$$ (5)

and

$$S\left(q=0\right)=\sum _{n=0}^{N-1}{a}_n^{\prime }{\varphi }_n\left(u,0\right)=\sum _{n=0,2,4,\mathrm{...}}^{N-1}\frac{\left(n-1\right)!!}{\sqrt{n!}}{a}_n^{\prime }.$$ (6)

Because Fourier transforms of the Hermite functions are Hermite functions themselves, the propagator can be expressed in the same set of basis functions as the signal,

$$P(x)=\sum _{n=0}^{N-1}{a}_n{\psi }_n\left(u,x\right),$$ (7)

With

$${\psi }_n\left(u,x\right)=\frac{1}{\sqrt{2^{n+1}\pi n!u}}{e}^{x^2}/(2u^2)H_n\left(x/u\right).$$ (8)

Apart from estimation of the displacement probability density function, the MAP framework can be used to analytically and directly compute all moments of the propagator using the Hermite function representation of the E(q) profile, regardless of the choice of x grid discretization. The explicit relationship between the mth moment of the propagator, µ_m, and the GMAP coefficients is given by [52]

$${\mu }_m=u^m\sum _{k=0,2,\dots }^{N-1}\frac{(k+m-1)!!}{k!}\times \sum _{l=0,2,\dots }^{N-k-1}{(-1)}^{l/2}\frac{\sqrt{s^{k-l}(k+l)!}}{(l/2)!}{a}_{k+1},$$ (9)

when m is even. Odd-ordered moments can be computed by using Eq. 9 where the index k takes odd values, i.e., k = 1,3,5,….

The above expressions can capture and describe any type of water translation. This approach was demonstrated by analyzing a simulated flowing fluid with the assumption that the molecules undergo a net coherent displacement [53]. The purpose of this work is to generalize the method to fit experimental data from porous media flow, and to use it on a biomimetic MRI phantom.

III. METHODS

A. Optimization procedure

The implementation here follows the ideas outlined in the original SHORE publications [52], [53], with some modifications and additional steps to account for flow conditions. GMAP can be extended from 1D to 3D in a similar manner as SHORE [52] was generalized to MAP [53].

• 1)The first step involves finding the characteristic length u. The magnitude of the signal attenuation at the low-q regime was fitted to $E\left(q\right)=e^{-2{\pi }^2{q}^2{u}^2}$ with a trust-region-reflective nonlinear optimization procedure by using MATLAB’s lsqcurvefit function (The Mathworks, Natick, MA).
• 2)
  Once u was determined, the search for the optimal a_n coefficient vectors, a, begins. According to Eq. 4 the acquired signal E(q) is a linear combination of the function ϕ_n(u,q) weighted by the appropriate a_n coefficients. This function can be written in a matrix form Φ, where one dimension accounts for different orders of Hermite functions (up to N ) and the other dimension for the number of experimental q-values, N_q. Eq. 4 can then be written as

  $${\text{E}}_N={\text{Φ}}_N{\text{a}}_N,$$ (10)

  where the subscript N indicates the chosen maximal order (and subsequent number of estimated coefficients). The procedure of choosing the optimal N is detailed in section III-B. Similarly, Eq. 7 can be turned into a matrix equation, with the propagator P defined over a grid with N_x samples in the displacement space, and spacing between adjacent points, δx.
• 3)
  Finding a_N is performed by minimization,

  $${\text{a}}_N=\arg \hspace{0.17em}\min \left(\parallel {\mathbf{\Phi }}_N{\text{a}}_N-\text{E}\right)\left.{\parallel }^2\right),$$ (11)

  subject to two sets of constraints:

  $${\mathbf{\Psi }}_N{\text{a}}_N\hspace{0.33em}\ge 0,$$ (12)

  and

  $$\delta {\text{x}}^{\text{T}}\left({\mathbf{\Psi }}_N{\text{a}}_N\right)=1,$$ (13)

  where δx^T is a 1×N_x vector containing δx. Eq. 12 enforces the nonnegativity condition that P (x) ≥ 0 for any x. Eq. 13 ensures that the propagator is a proper probability density function, i.e., that $\int P(x)\hspace{0.33em}\hspace{0.17em}\text{d}x\hspace{0.17em}=1.$ To solve Eq. 11 subject to the above constraints we used CVX, a package for specifying and solving convex programs [54], [55].
• 4)
  For addressing cases with non-zero flow, the phase shift information from the signal must be retained and E(q) is complex. Complex functions were optimized by separating their real and imaginary parts. The matrix Φ_N, which is N_q ×N, was split into two matrices: the even columns (n = 0, 2, 4,…), which contain exclusively real values, and the odd columns (n = 1, 3, 5, … ), which contain exclusively imaginary values, were stored as ${\mathbf{\Phi }}_N^e$ and ${\mathbf{\Phi }}_N^{\circ }$, respectively. We then defined ${\mathbf{\Phi }}_N^{ℝ}=ℝ\text{e}\left({\mathbf{\Phi }}_N^e\right)$ and ${\mathbf{\Phi }}_N^{\mathbb{I}}=\mathbb{I}\text{m}\left({\mathbf{\Phi }}_N^o\right)$ and solved

  $${\text{a}}_N=\arg \hspace{0.17em}\min \left(\parallel \left[{\mathbf{\Phi }}_N^{ℝ}{\text{a}}_N^{ℝ}-ℝ\text{e(}\mathbf{E}\text{)}\right]\right.\oplus \left[{\mathbf{\Phi }}_N^{\mathbb{I}}{\text{a}}_N^{\mathbb{I}}-\mathbb{I}\text{m(}\mathbf{E}\text{)}\right]\left.{\parallel }^2\right),$$ (14)

  where ${\text{a}}_N^{ℝ}$ and ${\text{a}}_N^{\mathbb{I}}$ are the even and odd coefficients vectors, respectively, and the ⊕ operation indicates vector concatenation. The constraints in Eqs. 12 and 13 were separated into real and imaginary components in a similar fashion.

B. Parsimonious order selection

The maximal order of the Hermite function directly affects the fitting quality. Higher order N would allow sharper oscillations in E(q) to be captured, with the downside of additional coefficients and the risk of overfitting or fitting the noise. We therefore present a scheme to select the optimal N : the optimization in section III-A was repeated iteratively with an increasing maximal value of N in each iteration, Ñ. In each iteration the estimated real and imaginary parts of the signal, $E_{\tilde{N}}^{ℝ}={\mathbf{\Phi }}_{\tilde{N}}^{ℝ}{a}_{\tilde{N}}^{ℝ}$ and $E_{\tilde{N}}^{\mathbb{I}}={\mathbf{\Phi }}_{\tilde{N}}^{\mathbb{I}}{a}_{\tilde{N}}^{\mathbb{I}},$ were computed, and a distance measure from the complex signal is defined as

$$\xi =\frac{{\xi }^{ℝ}+{\xi }^{\mathbb{I}}}{2},$$ (15)

where

$${\xi }^{ℝ}=\sum _{i=0}^{N_q}{\text{D}}_M{\left[{\mathbf{E}}_{\tilde{N}}^{ℝ},ℝ\text{e(}\mathbf{E}\text{)}\right]}_i$$ (16)

and

$$\hspace{0.17em}{\xi }^{\mathbb{I}}=\sum _{i=0}^{N_q}{\text{D}}_M{\left[{\mathbf{E}}_{\tilde{N}}^{\mathbb{I}},\mathbb{I}\text{m(}\mathbf{E}\text{)}\right]}_i$$ (17)

The D_M operator indicates the Mahalanobis distance operation. The average distances sum in Eq. 15 was then combined with a Bayesian information criterion (BIC)[56], [57] to yield a parsimonious order selection

$${\in }_{\tilde{N}}=\log (\frac{\xi }{N_q})+\tilde{N}\frac{\log (N_q)}{N_q}.$$ (18)

The resulting computed index,ϵ_Ñ, balances the choice of N by imposing penalties for models with a larger number of free parameters and a larger mean residual error. The index,ϵ_Ñ, was stored and compared with the one in the previous iteration. The selection process for N stops when ϵ_Ñ stops decreasing.

IV. EXPERIMENTS

A. Phantom preparation

The MRI flow phantom was constructed from 10 µm monodisperse polystyrene beads (Duke standard) loosely packed under flow conditions in a 5 mm Tricorn column (GE Healthcare) into which water was pumped. This phantom had two regions: beads and bulk water. The bulk water compartment modeled the advective motion in the CSF-filled ventricles, while the packed beads represented dispersive flow in brain interstitial space. We chose this bead size since it roughly corresponds to the diameter of neurons in gray matter. A peristaltic water pump (Pharmacia Biotech) pumped water at two constant flow rates of Q = 0.876 ml/min, and Q = 0.041 ml/min. The flow in the bulk water region of the column (with radius R = 2.50 mm) exhibits Poiseuille flow, in which the velocity profile is

$$\upsilon (r)=\frac{2Q}{\text{π}\hspace{0.17em}{R}^2}\left[1-{\left(\frac{r}{R}\right)}^2\right].$$ (19)

Theoretical values can then be computed, resulting in average velocities of ${\overline{\upsilon }}_{bulk}=743,34.8\hspace{0.33em}\hspace{0.17em}\mu m/s$ and peak velocities of $\max ({\upsilon }_{bulk})=1490,69\hspace{0.33em}\mu \text{m/s,}$ for the high- and low-flow rates, respectively.

B. MRI data acquisition

MRI data were collected on a 7 T Bruker wide-bore vertical magnet with an AVANCE III MRI spectrometer equipped with a Micro2.5 microimaging probe and three GREAT60 gradient amplifiers, which have a nominal peak current of 60 A per channel. This configuration can produce a maximum nominal gradient strength of 24.65 mT/m/A along each of the three orthogonal directions.

Diffusion-weighted (DW) data were acquired with the diffusion-encoding direction set to the direction of flow, such that |G|=G, with 61 linear steps from 0 to 300 mT/m for high-flow, and with 41 linear steps from 0 to 400 mT/m for low-flow. In both cases δ = 3.15 ms, resulting in a maximal q-value of 402 cm⁻¹ and 536 cm⁻¹ for the high- and low-flow cases, respectively. The high-flow acquisition was repeated with two diffusion periods, ∆ = 25, 75 ms, and the low-flow data were acquired with a single ∆ = 50.

The high-flow rate data were acquired using a spin-echo DW echo planar imaging (DWI–EPI) sequence with TR = 7 s, TE=90 ms. Two 2 mm axial slices imaging CSF (bulk water) and interstitial (beads water) model phantoms, with a matrix size of 64×64 and an in-plane resolution of 187×187 µm², were acquired with two averages and two segments. The low- flow rate data were acquired using the same sequence with TR = 3 s, TE = 59 ms. Two 2 mm axial slices imaging CSF (bulk water) and interstitial (beads water) model phantoms, with a matrix size of 64 64 and an in-plane resolution of 125×125 µm², were acquired with four averages and eight segments.

In the case of the highest flow rate the spatial image in-plane resolution was 187×187 µm² while the third dimension of the voxel (i.e., direction of the flow) was much larger (2000 µm). These space-time scales ensured that the fluid elements moved several pore spacings during the measurement, but did not leave the voxel.

V. RESULTS AND DISCUSSION

A. Effect of increased observation time

We first discuss the high-flow rate case, in which the effect of the observation time, ∆, is most prominent. The propagators were obtained by finding the coefficients in Eq. 4 by using the procedure described in the Methods section. Once the coefficients have been estimated, E(q) can be evaluated analytically at any q-value on the lattice. An example of such a fit to the data acquired with a high-flow rate through the packed beads with ∆ = 25 ms is shown in Fig. 1A. Both the real and imaginary components of the evaluated signal profile (solid lines) and the corresponding data (symbols) are presented, and the good agreement between them is evident. The residuals from the fit are presented in Fig. 1B, and mostly appear randomly scattered around zero (within two standard deviations) indicating that the model describes the data well.

Fig. 1.

Complex signal from water pumped through the packed beads with ∆ = 25. (A) The data (symbols) are shown along with the analytical form of E(q) estimated by using the GMAP framework (solid lines). Error bars indicate standard error from the voxels within the ROI. (B) The GMAP standardized residuals relative to the fit for the real and imaginary parts.

Next we examine the GMAP-based propagators in Fig. 2, reconstructed at ∆ = 25 ms and ∆ = 75 ms, both in the bead pack and the bulk water regions. When considering bulk water, a characteristic Gaussian displacement distribution due to pure diffusion, is expected in the absence of flow (Fig. 2A, dashed lines). As ∆ increases, the Gaussian curve gets broader, proportional to $\sqrt{2D\Delta }$ (D is the diffusion coefficient). Bulk water flow under the current experimental conditions exhibits a laminar flow profile, in which the entire Gaussian distribution simply shifts with some gradual broadening. The bulk water flow propagators from a small region of interest (ROI) at the center of the tube are shown in Fig. 2A (solid lines). For reference, black lines are Gaussian or shifted Gaussian distributions, plotted using the apparent diffusivity found from fitting the low-q data and the theoretical mean velocity. All of the theoretical Gaussian curves fit nicely to the estimated propagators, with the exception of bulk flow at ∆ = 75 ms where some unexpected asymmetry is evident, conceivably due to noise in the data.

Fig. 2.

Propagators reconstructed at different observation times and different flow regimes. (A) Bulk and (B) bead pack water propagators at Q=0, 0.876 ml/min (dashed and solid lines, respectively). Solid black lines are Gaussian or shifted Gaussian propagators.

When the GMAP framework is used, there are two possible ways to obtain moments of the propagator: (1) numerically integrating over the propagator with respect to the displacement axis, x, and (2) analytically compute the moment by using Eq. 9. The first moments, which are the average water displacements, ${\overline{x}}^{num}$ and ${\overline{x}}^{ana}$ were obtained by using numerical integration and analytical methods, respectively. Dividing those by the observation time, ∆, resulted in the average fluid velocity. The maximal bulk velocity (center of the tube) averaged from the ∆ = 25 ms and ∆ = 75 ms propagators can be computed and compared to the known Poiseuille flow theoretical velocity from Eq. 19. The GMAP-derived values were ${\overline{\upsilon }}^{num}=1430\hspace{0.33em}\pm \hspace{0.33em}114$ and ${\overline{\upsilon }}^{ana}=1470\hspace{0.33em}\pm \hspace{0.33em}122\hspace{0.33em}\hspace{0.17em}\mu \text{m/s,}$ compared with the theoretical value of 1490 µm/s, indicating high accuracy.

In the case of the bead pack (Fig. 2B), the reduced fluid volume leads to higher average and peak velocities compared with the bulk water flow. The velocity distribution within the bead pack puts stationary fluid and fast-moving fluid in close proximity, such that fluctuations and diffusion across streamlines are expected, causing the propagator to broaden. The propagators in Fig. 2B have distinct non-Gaussian characteristics, e.g., long tails and asymmetric shapes, which clearly indicate both diffusive and dispersive effects through the column.

Repeating the GMAP estimation in a voxelwise manner results in a water displacement distribution in each of the image voxels. Integrating with respect to the displacement variable results in quantitative images of average water displacement. Such images are shown in Fig. 3 for the bulk and beads regions, with different observation times (all at Q = 0.876 ml/min). The expected parabolic flow profile (Poiseuille) is evident in the bulk water images (top row). The relatively uniform plug flow profile in the beads water images (bottom row) was also expected. Note the higher overall average displacement of the bead pack region compared with the bulk water.

Fig. 3.

Average displacement images of the different flow regimes under different experimental conditions. Note the signature Poiseuille flow profile of the bulk water images and the overall higher displacement of water transported through the bead pack compared with bulk water.

B. q-space subsampling

One of the limiting factors of the FFT (and implementations such as DSI) is the dense sampling of the diffusion signal on a Cartesian q-space lattice. In biological applications, where the sample has three-dimensional heterogeneity, the q-space lattice is three-dimensional, leading to a time-intensive acquisition protocol. These data would be difficult to acquire when strong gradient coils, high SNR, and/or long acquisition times are unavailable, which is invariably the case in clinical settings.

A good indicator of the clinical feasibility of this approach is whether it is possible to reduce the amount of data while maintaining the reliability and stability of the reconstructed propagators. The q-space data sampling scheme has a direct impact on the FFT-based propagator. The reciprocal displacement vector, x, is computed directly from the q vector; thus, reducing the number of q samples leads to the same reduction in x. The resolution in x, δx, is determined by the maximal q-value; therefore, reduction in the q-space sampling density reduces the range of the reciprocal displacement space. In theory, optimal q-space sampling may not be uniform. Although the GMAP framework can evaluate the propagator regardless of the q-space sampling scheme, FFT-based reconstruction requires uniform sampling. Thus, we limited our analysis here to uniform subsampling to allow for comparison between the methods.

Even though the current work uses a 1D flow phantom with 1D acquisition, these results can be generalized to the 3D case. Focusing on the more challenging case of low-flow, the Q = 0.041 ml/min full data set contained 41 q-values, which in itself is not a large number of acquisitions. However, when considering a 3D generalization, sampling of a 3D grid at such density results in 41³ images, which is infeasible. To test the robustness to data reduction, we tested a range of data subspaces, spanning from N_q = 41 (full) to N_q = 6 data points. Adhering to the uniform subsampling requirement (for FFT) resulted in 7 subspaces.

The full data and one instance of subsampled q-space (N_q = 11) propagators are shown, respectively, in Figs. 4A and B. No flow and Q = 0.041 ml/min flow at ∆ = 50 ms were compared, along with FFT- and GMAP-based reconstruction (symbols and solid lines, respectively). Qualitatively, GMAP- based analysis is almost not affected by partial data reconstruction, despite significant reductions in the number of images. Aside from the inherent reduction in the displacement space range, the FFT-based analysis resulted in evident wiggles and nonphysical negative values of the displacement probability.

Fig. 4.

Bead pack propagators at Q = 0, 0.041 ml/min, reconstructed by using GMAP- and FFT-based methods, and the effect of limited data. (A) Full data set (Nq = 41). (B) Partial data set (Nq = 11). Error bars indicate standard error from the voxels within the ROI.

The odd moments of the propagator contain most of the coherent water motion information and, therefore, should be very useful for flow characterization. As shown above, average velocities can be computed from the first moment. In addition, the asymmetry of the propagator is another important feature that is captured by the third moment, µ₃. It can be used to derive a dimensionless quantity, the skewness of the propagator, which is defined as γ₁ = µ₃/µ^3/2. For imaging purposes, $\overline{\upsilon }$ and γ₁ can be used to generate quantitative maps that reflect information about water dynamics. It is therefore important that estimations of these quantities are robust to data reduction.

Computing the average velocity and skewness of the different propagators (i.e., fully sampled and subsampled) would result in a quantitative assessment of the impact of subsampling. Using the three methods that were described above, numerical integration on FFT- and GMAP-based propagators, and analytically using the GMAP coefficients, the “baseline” quantities were first computed (i.e., using full data set):${\overline{\upsilon }}^{num}\hspace{0.33em}55.6\hspace{0.17em}\pm 8.1\hspace{0.33em}\mu \text{m}/s$, and ${\Upsilon }_1^{num}=0.635\pm 0.36$ from the FFT-based propagator, and ${\overline{\upsilon }}^{num}/{\overline{\upsilon }}^{ana}=61.0\pm 4.9/71.1\pm 5.9\hspace{0.33em}\hspace{0.17em}\mu \text{m}/s,$ and ${\Upsilon }_1^{num}/{\Upsilon }_1^{ana}=0.498\pm 0.10/0.620\pm 0.090$ from the GMAP-based propagator.

Figures 5 A and B show the percent error from the baseline values of the velocity and skewness estimations as a function of the number of q data points, respectively. The velocity estimates, which are based on the first moment of the propagator, all generate a relatively large error using the FFT-based method, with percent errors mostly larger than 20%, and in two instances larger than 40%. Conversely, both approaches to calculate the velocity using the GMAP framework resulted in quite robust velocity estimates, with all subspaces but one yielding percent errors lower or about 10%. Because the third moment is more susceptible to effects of noise than the first moment of the propagator, skewness estimation accuracy should be expected to be lower than that of velocity. And indeed, Fig. 5B shows a similar trend in the skewness parameter estimation compared to the velocity estimation, with the distinction of much higher percent error values in the case of the FFT-based method. While the percent error values of the FFT-based skewness estimation reached 600%, these values did not cross 40% and 30% using the numerical and analytical GMAP methods, respectively. These results point to the analytically derived moments having increased stability and robustness to reduced data, which is a unique feature of GMAP.

Fig. 5.

The effect of reduced data on the (A) velocity and (B) skewness estimations, using GMAP- and FFT-based methods. Error bars indicate standard error from the voxels within the ROI.

The fully sampled and subsampled q-value ranges in this experiment resulted in an FFT x lattice that ranges from x ± 377 µm for N_q = 41 to x = 59.2 µm for N_q = 6, with δx = 9.44 µm. Integrating over the entire x range of the FFT-based propagators resulted in very poor estimates of the moments, and subsequent $\overline{\upsilon }$ and γ₁. For example, when FFT analysis was used in the no-flow case, the estimates of the average velocities were 62.2 and −9.39 µm/s for full and the N_q = 11 data, respectively, when they both should have been 0. To permit a fair comparison between the FFT- and GMAP- based results, we computed the moments by integrating over the same displacement range for all propagators at a given N_q, which was usually smaller than the full FFT reciprocal displacement range: x = ±85.2 µm for N_q > 9, and the full FFT reciprocal space for N_q≤9. This partial-range integration, in fact, uses prior knowledge about the flow conditions and may not be possible to generalize, emphasizing another limitation to the FFT approach.

C. Low-flow sensitivity and detection

Sensitivity to low-flow conditions and the ability to mea- sure and differentiate between microscopic and macroscopic flows are crucial in detecting features of glymphatic transport. Because GMAP provides an analytical closed-form expression for the displacement probability density function, it is continuous and does not suffer from the resolution limit the FFT- based propagators have. The discrete nature of the FFT-based distributions is directly determined by the maximal q-value, which is inversely proportional to the spatial resolution, δx. Strong diffusion gradients, and subsequently high q-values, are unavailable on conventional clinical scanners, leading to a (relatively low) upper boundary on the maximal q-value. In some cases, this limitation can be circumvented by zero filling the signal profile and artificially increasing the maximal q- value and, with it, the spatial resolution [58]. This procedure requires that the acquired signal decays sufficiently into the noise before zero filling, which is rarely the case for biological tissue, in which the experimentally applied q-values usually do not attenuate the signal close to zero [59], [60], [15]. Because the context of the present work is biological, we avoid the use of the zero-filling technique. For the FFT, not sampling the data to a sufficiently high q-value is effectively the same as multiplying the true E(q) decay by a hat filter, which results in the measured propagator being convolved with a sinc function. In that case, the propagator exhibits wiggles that are difficult to discern from true flow. GMAP is well behaved for subsampled data, thus eliminating this limitation.

The lowest flow rate in our experimental setup was that of bulk water pumped at Q = 0.041 ml/min, within close proximity to the tube wall. These data were analyzed by selecting an ROI at a distance of r = 2.20 mm from the center of the tube; then the theoretical velocity was calculated by using Eq. 19 and resulted in $\overline{\upsilon }=15.7\hspace{0.33em}\hspace{0.17em}\mu \text{m/s}.$ Estimating the average velocities from the GMAP- and FFT-based propagators resulted in ${\overline{\upsilon }}^{num}=17.9\hspace{0.17em}\pm \hspace{0.17em}0.13$ and ${\overline{\upsilon }}^{ana}=16.47\pm 0.11\hspace{0.33em}\hspace{0.17em}\mu \text{m/s,}$ and ${\overline{\upsilon }}^{ana}=16.47\pm 0.11\hspace{0.33em}\hspace{0.17em}\mu \text{m/s,}$ respectively (again here the integration range was set at ± 85.2 µm; the FFT-based velocity estimation over the entire FFT x axis range was - 127 µm/s). The velocity estimated by using GMAP was quite close to the theoretical value, with overestimations of 14% and 4.9% using the numerical integration and analytical methods to compute the moments, respectively. In contrast, the FFT- based velocity overestimated the theoretical value by 119.0%.

Both the current and the previous subsections demonstrated the unreliability in estimating the average displacement and velocity from the FFT-based propagators when integrating over the entire range of x. A frequently used alternative method for the determination of the mean molecular velocity is locating the peak of the displacement distribution and dividing it by the observation time [58]. This alternative method also presents a challenge for the FFT-based estimation because its discrete nature prevents it from being able to detect very small water displacements. To illustrate, a magnified range of ‒5≤x≤15 µm of both GMAP- and FFT-based low-flow propagators are shown in Fig. 6. The densely dashed vertical line indicates zero displacement, and the second vertical line indicates the theoretical displacement, 0.78 µm. While the peak of the GMAP-based propagator (0.89 µm) is closely aligned with the theoretical displacement value, the peak of the FFT-based propagator is at 0, which would lead one to conclude zero flow.

Fig. 6.

Bulk water propagators at Q = 0, 0.041 ml/min at 2.2 mm from the center of the tube. Vertical dashed lines indicate zero and theoretical displacements. GMAP- and FFT-based propagators are compared.

VI. CONCLUSION

We presented a generalization of a previously suggested analytical propagator representation, MAP-MRI. In our implementation we account for fluid flow and adjust the numerical optimization accordingly. We tested the GMAP framework on a biomimetic MRI flow phantom with different flow rates, different flow regimes, and under different experimental conditions. The presented framework provides a stable and robust means to analytically derive all moments of the water displacement probability density function, P (r), and propagator-derived features such as the mean displacement and skewness. GMAP MRI resulted in accurate estimates of the expected theoretical flow rates and exhibited high sensitivity in detecting very low-flow rates. The advantages over the traditional FFT- based method in the context of limited data and low-flow detection were experimentally demonstrated. GMAP is expected to be especially useful with nonuniform q-space sampling, limited data and when the expected signal does not fully decay. The current imaging framework was developed to enable the characterization of in vivo diffusive, advective, and dispersive transport processes such as glymphatic transport–induced CSF flow and blood microcirculation. GMAP can be extended from 1D to 3D in a straightforward manner (as SHORE [52] was generalized to MAP [53]). Three-dimensional GMAP would provide a full description of the water displacement probability density function in the voxel, thus it would not require that the direction of the flow be known a priori.