Estimation of pore size distribution using concentric double pulsed-field gradient NMR

Double pulsed field gradient diffusion MRI to assess skeletal muscle microstructure

PURPOSE

Preliminary study to determine whether double pulsed field gradient (PFG) diffusion MRI is sensitive to key features of muscle microstructure related to function.

METHODS

The restricted diffusion profile of molecules in models of muscle microstructure derived from histology were systematically simulated using a numerical simulation approach. Diffusion tensor subspace imaging analysis of the diffusion signal was performed, and spherical anisotropy (SA) was calculated for each model. Linear regression was used to determine the predictive capacity of SA on the fiber area, fiber diameter, and surface area to volume ratio of the models. Additionally, a rat model of muscle hypertrophy was scanned using a single PFG and a double PFG pulse sequence, and the restricted diffusion measurements were compared with histological measurements of microstructure.

RESULTS

Excellent agreement between SA and muscle fiber area (r2  = 0.71; p < 0.0001), fiber diameter (r2  = 0.83; p < 0.0001), and surface area to volume ratio (r2  = 0.97; p < 0.0001) in simulated models was found. In a scanned rat leg, the distribution of these microstructural features measured from histology was broad and demonstrated that there is a wide variance in the microstructural features observed, similar to the SA distributions. However, the distribution of fractional anisotropy measurements in the same tissue was narrow.

CONCLUSIONS

This study demonstrates that SA-a scalar value from diffusion tensor subspace imaging analysis-is highly sensitive to muscle microstructural features predictive of function. Furthermore, these techniques and analysis tools can be translated to real experiments in skeletal muscle. The increased dynamic range of SA compared with fractional anisotropy in the same tissue suggests increased sensitivity to detecting changes in tissue microstructure.

Double diffusion encoding and applications for biomedical imaging

Diffusion Magnetic Resonance Imaging (dMRI) is one of the most important contemporary non-invasive modalities for probing tissue structure at the microscopic scale. The majority of dMRI techniques employ standard single diffusion encoding (SDE) measurements, covering different sequence parameter ranges depending on the complexity of the method. Although many signal representations and biophysical models have been proposed for SDE data, they are intrinsically limited by a lack of specificity. Advanced dMRI methods have been proposed to provide additional microstructural information beyond what can be inferred from SDE. These enhanced contrasts can play important roles in characterizing biological tissues, for instance upon diseases (e.g. neurodegenerative, cancer, stroke), aging, learning, and development. In this review we focus on double diffusion encoding (DDE), which stands out among other advanced acquisitions for its versatility, ability to probe more specific diffusion correlations, and feasibility for preclinical and clinical applications. Various DDE methodologies have been employed to probe compartment sizes (Section 3), decouple the effects of microscopic diffusion anisotropy from orientation dispersion (Section 4), probe displacement correlations, study exchange, or suppress fast diffusing compartments (Section 6). DDE measurements can also be used to improve the robustness of biophysical models (Section 5) and study intra-cellular diffusion via magnetic resonance spectroscopy of metabolites (Section 7). This review discusses all these topics as well as important practical aspects related to the implementation and contrast in preclinical and clinical settings (Section 9) and aims to provide the readers a guide for deciding on the right DDE acquisition for their specific application.

JEDI: Joint Estimation Diffusion Imaging of macroscopic and microscopic tissue properties

PURPOSE

A new method for enhancing the sensitivity of diffusion MRI (dMRI) by combining the data from single (sPFG) and double (dPFG) pulsed field gradient experiments is presented.

METHODS

This method uses our JESTER framework to combine microscopic anisotropy information from dFPG experiments using a new method called diffusion tensor subspace imaging (DiTSI) to augment the macroscopic anisotropy information from sPFG data analyzed using our guided by entropy spectrum pathways method. This new method, called joint estimation diffusion imaging (JEDI), combines the sensitivity to macroscopic diffusion anisotropy of sPFG with the sensitivity to microscopic diffusion anisotropy of dPFG methods.

RESULTS

Its ability to produce significantly more detailed anisotropy maps and more complete fiber tracts than existing methods within both brain white matter (WM) and gray matter (GM) is demonstrated on normal human subjects on data collected using a novel fast, robust, and clinically feasible sPFG/dPFG acquisition.

CONCLUSIONS

The potential utility of this method is suggested by an initial demonstration of its ability to mitigate the problem of gyral bias. The capability of more completely characterizing the tissue structure and connectivity throughout the entire brain has broad implications for the utility and scope of dMRI in a wide range of research and clinical applications.

ActiveAxADD : Toward non-parametric and orientationally invariant axon diameter distribution mapping using PGSE

PURPOSE

Non-invasive axon diameter distribution (ADD) mapping using diffusion MRI is an ill-posed problem. Current ADD mapping methods require knowledge of axon orientation before performing the acquisition. Instead, ActiveAx uses a 3D sampling scheme to estimate the orientation from the signal, providing orientationally invariant estimates. The mean diameter is estimated instead of the distribution for the solution to be tractable. Here, we propose an extension (ActiveAx_ADD ) that provides non-parametric and orientationally invariant estimates of the whole distribution.

THEORY

The accelerated microstructure imaging with convex optimization (AMICO) framework accelerates mean diameter estimation using a linear formulation combined with Tikhonov regularization to stabilize the solution. Here, we implement a new formulation (ActiveAx_ADD ) that uses Laplacian regularization to provide robust estimates of the whole ADD.

METHODS

The performance of ActiveAx_ADD was evaluated using Monte Carlo simulations on synthetic white matter samples mimicking axon distributions reported in histological studies.

RESULTS

ActiveAx_ADD provided robust ADD reconstructions when considering the isolated intra-axonal signal. However, our formulation inherited some common microstructure imaging limitations. When accounting for the extra axonal compartment, estimated ADDs showed spurious peaks and increased variability because of the difficulty of disentangling intra and extra axonal contributions.

CONCLUSION

Laplacian regularization solves the ill-posedness regarding the intra axonal compartment. ActiveAx_ADD can potentially provide non-parametric and orientationally invariant ADDs from isolated intra-axonal signals. However, further work is required before ActiveAx_ADD can be applied to real data containing extra-axonal contributions, as disentangling the 2 compartment appears to be an overlooked challenge that affects microstructure imaging methods in general.

Anisotropic diffusion phantoms based on microcapillaries

Diffusion MRI is an efficient and widely used technique for the investigation of tissue structure and organisation in vivo. Multiple phenomenological and biophysical diffusion models are intensively exploited for the analysis of the diffusion experiments. However, the verification of the applied diffusion models remains challenging. In order to provide a "gold standard" and to assess the accuracy of the derived parameters and the limitations of the diffusion models, anisotropic diffusion phantoms with well known architecture are demanded. In the present work we built four anisotropic diffusion phantoms consisting of hollow microcapillaries with very small inner diameters of 5, 10 and 20μm and outer diameters of 90 and 150μm. For testing the suitability of these phantoms, we performed diffusion measurements on all of them and evaluated the resulting data with a set of popular diffusion models, such as diffusion tensor and diffusion kurtosis imaging, a two compartment model with intra- and extra-capillary water spaces using bi-exponential fitting, and time-dependent diffusion coefficients in Mitra's limit. The perspectives and limitations of these diffusion phantoms are presented and discussed.

Anisotropic phantom to calibrate high-q diffusion MRI methods

A silicon oil-filled glass capillary array is proposed as an anisotropic diffusion MRI phantom. Together with a computational/theoretical pipeline these provide a gold standard for calibrating and validating high-q diffusion MRI experiments. The phantom was used to test high angular resolution diffusion imaging (HARDI) and double pulsed-field gradient (d-PFG) MRI acquisition schemes. MRI-based predictions of microcapillary diameter using both acquisition schemes were compared with results from optical microscopy. This phantom design can be used for quality control and quality assurance purposes and for testing and validating proposed microstructure imaging experiments and the processing pipelines used to analyze them.

Use of marginal distributions constrained optimization (MADCO) for accelerated 2D MRI relaxometry and diffusometry

Measuring multidimensional (e.g., 2D) relaxation spectra in NMR and MRI clinical applications is a holy grail of the porous media and biomedical MR communities. The main bottleneck is the inversion of Fredholm integrals of the first kind, an ill-conditioned problem requiring large amounts of data to stabilize a solution. We suggest a novel experimental design and processing framework to accelerate and improve the reconstruction of such 2D spectra that uses a priori information from the 1D projections of spectra, or marginal distributions. These 1D marginal distributions provide powerful constraints when 2D spectra are reconstructed, and their estimation requires an order of magnitude less data than a conventional 2D approach. This marginal distributions constrained optimization (MADCO) methodology is demonstrated here with a polyvinylpyrrolidone-water phantom that has 3 distinct peaks in the 2D D-T1 space. The stability, sensitivity to experimental parameters, and accuracy of this new approach are compared with conventional methods by serially subsampling the full data set. While the conventional, unconstrained approach performed poorly, the new method had proven to be highly accurate and robust, only requiring a fraction of the data. Additionally, synthetic T1-T2 data are presented to explore the effects of noise on the estimations, and the performance of the proposed method with a smooth and realistic 2D spectrum. The proposed framework is quite general and can also be used with a variety of 2D MRI experiments (D-T2,T1-T2,D-D, etc.), making these potentially feasible for preclinical and even clinical applications for the first time.

White matter microstructure from nonparametric axon diameter distribution mapping

We report the development of a double diffusion encoding (DDE) MRI method to estimate and map the axon diameter distribution (ADD) within an imaging volume. A variety of biological processes, ranging from development to disease and trauma, may lead to changes in the ADD in the central and peripheral nervous systems. Unlike previously proposed methods, this ADD experimental design and estimation framework employs a more general, nonparametric approach, without a priori assumptions about the underlying form of the ADD, making it suitable to analyze abnormal tissue. In the current study, this framework was used on an ex vivo ferret spinal cord, while emphasizing the way in which the ADD can be weighted by either the number or the volume of the axons. The different weightings, which result in different spatial contrasts, were considered throughout this work. DDE data were analyzed to derive spatially resolved maps of average axon diameter, ADD variance, and extra-axonal volume fraction, along with a novel sub-micron restricted structures map. The morphological information contained in these maps was then used to segment white matter into distinct domains by using a proposed k-means clustering algorithm with spatial contiguity and left-right symmetry constraints, resulting in identifiable white matter tracks. The method was validated by comparing histological measures to the estimated ADDs using a quantitative similarity metric, resulting in good agreement. With further acquisition acceleration and experimental parameters adjustments, this ADD estimation framework could be first used preclinically, and eventually clinically, enabling a wide range of neuroimaging applications for improved understanding of neurodegenerative pathologies and assessing microstructural changes resulting from trauma.

Joint radius-length distribution as a measure of anisotropic pore eccentricity: an experimental and analytical framework

In this work, we present an experimental design and analytical framework to measure the nonparametric joint radius-length (R-L) distribution of an ensemble of parallel, finite cylindrical pores, and more generally, the eccentricity distribution of anisotropic pores. Employing a novel 3D double pulsed-field gradient acquisition scheme, we first obtain both the marginal radius and length distributions of a population of cylindrical pores and then use these to constrain and stabilize the estimate of the joint radius-length distribution. Using the marginal distributions as constraints allows the joint R-L distribution to be reconstructed from an underdetermined system (i.e., more variables than equations), which requires a relatively small and feasible number of MR acquisitions. Three simulated representative joint R-L distribution phantoms corrupted by different noise levels were reconstructed to demonstrate the process, using this new framework. As expected, the broader the peaks in the joint distribution, the less stable and more sensitive to noise the estimation of the marginal distributions. Nevertheless, the reconstruction of the joint distribution is remarkably robust to increases in noise level; we attribute this characteristic to the use of the marginal distributions as constraints. Axons are known to exhibit local compartment eccentricity variations upon injury; the extent of the variations depends on the severity of the injury. Nonparametric estimation of the eccentricity distribution of injured axonal tissue is of particular interest since generally one cannot assume a parametric distribution a priori. Reconstructing the eccentricity distribution may provide vital information about changes resulting from injury or that occurred during development.

Magnetic-resonance pore imaging of nonsymmetric microscopic pore shapes

Imaging of the microstructure of porous media such as biological tissue or porous solids is of high interest in health science and technology, engineering and material science. Magnetic resonance pore imaging (MRPI) is a recent technique based on nuclear magnetic resonance (NMR) which allows us to acquire images of the average pore shape in a given sample. Here we provide details on the experimental design, challenges, and requirements of MRPI, including its calibration procedures. Utilizing a laser-machined phantom sample, we present images of microscopic pores with a hemiequilateral triangular shape even in the presence of NMR relaxation effects at the pore walls. We therefore show that MRPI is applicable to porous samples without a priori knowledge about their pore shape and symmetry. Furthermore, we introduce "MRPI mapping," which combines MRPI with conventional magnetic resonance imaging (MRI). This enables one to resolve microscopic pore sizes and shapes spatially, thus expanding the application of MRPI to samples with heterogeneous distributions of pores.

Nonparametric pore size distribution using d-PFG: comparison to s-PFG and migration to MRI

Here we present the successful translation of a pore size distribution (PSD) estimation method from NMR to MRI. This approach is validated using a well-characterized MRI phantom consisting of stacked glass capillary arrays (GCA) having different diameters. By employing a double pulsed-field gradient (d-PFG) MRI sequence, this method overcomes several important theoretical and experimental limitations of previous single-PFG (s-PFG) based MRI methods by allowing the relative diffusion gradients' direction to vary. This feature adds an essential second dimension in the parameters space, which can potentially improve the reliability and stability of the PSD estimation. To infer PSDs from the MRI data in each voxel an inverse linear problem is solved in conjunction with the multiple correlation function (MCF) framework, which can account for arbitrary experimental parameters (e.g., long diffusion pulses). This scheme makes no a priori assumptions about the functional form of the underlying PSD. Creative use of region of interest (ROI) analysis allows us to create different underlying PSDs using the same GCA MRI phantom. We show that an s-PFG experiment on the GCA phantom fails to accurately reconstruct the size distribution, thus demonstrating the superiority of the d-PFG experiment. In addition, signal simulations corrupted by different noise levels were used to generate continuous and complex PSDs, which were then successfully reconstructed. Finally, owing to the reduced q- or b- values required to measure microscopic PSDs via d-PFG MRI, this method will be better suited to biomedical and clinical applications, in which gradient strength of scanners is limited.

Pore size distribution of bioresorbable films using a 3-D diffusion NMR method

Pore size distribution (PSD) within porous biomaterials is an important microstructural feature for assessing their biocompatibility, longevity and drug release kinetics. Scanning electron microscopy (SEM) is the most common method used to obtain the PSD of soft biomaterials. The method is highly invasive and user dependent, since it requires fracturing of the sample and then considers only the small portion that the user had acquired in the image. In the current study we present a novel nuclear magnetic resonance (NMR) method as an alternative method for estimation of PSD in soft porous materials. This noninvasive 3-D diffusion NMR method considers the entire volume of the specimen and eliminates the user's need to choose a specific field of view. Moreover, NMR does not involve exposure to ionizing radiation and can potentially have preclinical and clinical uses. The method was applied on four porous 50/50 poly(dl-lactic-co-glycolic acid) bioresorbable films with different porosities, which were created using the freeze-drying of inverted emulsions technique. We show that the proposed NMR method is able to address the main limitations associated with SEM-based PSD estimations by being non-destructive, depicting the full volume of the specimens and not being dependent on the magnification factor. Upon comparison, both methods yielded a similar PSD in the smaller pore size range (1-25μm), while the NMR-based method provided additional information on the larger pores (25-50μm).

Modeling of the diffusion MR signal in calibrated model systems and nerves

Diffusion NMR is a powerful tool for gleaning microstructural information on opaque systems. In this work, the signal decay in single-pulsed-field gradient diffusion NMR experiments performed on a series of phantoms of increasing complexity, where the ground truth is known a priori, was modeled and used to identify microstructural features of these complex phantoms. We were able to demonstrate that, without assuming the number of components or compartments, the modeling can identify the number of restricted components, detect their sizes with an accuracy of a fraction of a micrometer, determine their relative populations, and identify and characterize free diffusion when present in addition to the components exhibiting restricted diffusion. After the accuracy of the modeling had been demonstrated, this same approach was used to study fixed nerves under different experimental conditions. It seems that this approach is able to characterize both the averaged axon diameter and the relative population of the different diffusing components in the neuronal tissues examined.