Inferring Microstructural Information of White Matter from Diffusion MRI

C H A P T E R

9 Inferring Microstructural Information of White Matter from Diffusion MRI Yaniv Assaf*, Yoram Coheny *

Department of Neurobiology, George S. Wise Faculty of Life Sciences, Tel Aviv University, Tel Aviv, Israel School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

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O U T L I N E 9.1 The Morphological Features of White Matter

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9.6 Q-Space Analysis

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9.2 Diffusion MRI and Tissue Microstructure

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9.7 Models of Diffusion in White Matter

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9.3 Diffusion Tensor ImagingdA Tool for White Matter Microstructural Mapping 187

9.8 Towards Virtual Biopsy of White Matter With Diffusion MRI

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9.4 Diffusion Tensor ImagingdA Tool for White Matter Microstructural Mapping? 188

9.9 Summary

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References

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9.5 Types of Diffusion Processes in the Tissue

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9.1 THE MORPHOLOGICAL FEATURES OF WHITE MATTER The following section will briefly review the morphological features of white matter essential for this chapter. More in-depth description of the white matter is given in Chapter 7. The white matter is a hierarchically ordered tissue: from aligned microtubular fibers within the axons to large fascicles of neuronal fibers connecting different brain regions (Kandel et al., 2000; Filley, 2001; Nicholls et al., 2001; Bear et al., 2007). The white matter is composed of several cellular components, some of them unique to this type of tissue. The axonal projections leaving and entering the cortical layers are the main feature of the white matter, whereas the other cellular components support this complex array of neuronal fiber network. In general, we can divide the cellular components of white matter into four Diffusion MRI http://dx.doi.org/10.1016/B978-0-12-396460-1.00009-3

categories: the vascular capillary bed, glial cells, axons, and extracellular/axonal space. One of the glial cells, the oligodendrocyte, a unique cell of the white matter, wraps itself around the axons to form the myelin multilamella membrane. While the size of the glial cell is in the order of 5–10 mm, the axon size and myelin thickness vary. Some axons have a large diameter (>2 mm) with a thick myelin membrane, while others have a smaller diameter (<1 mm) with a thin myelin membrane or without a myelin membrane at all (non-myelinated axons). The axon is a long fiber whose membrane is enwrapped by the myelin lamellas in a segmented manner: each segment is encapsulated by the nearby oligodendrocyte where the myelin lamellas are spaced by a myelin-free node called the node of Ranvier, and it is here where different ion channels interact with the extra-axonal milieu to progress the electric signal. Within the axon, microtubular fibers strengthen the

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Copyright Ó 2014 Elsevier Inc. All rights reserved.

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cylindrical shape and orientation of the fiber; the microtubular fibers are aligned in the same manner as the axon. The diameter of the axons and the myelin membrane thickness are proportionally related to the conduction velocity along the axon, and are usually characterized by an axon diameter distribution function (Ritchie, 1982). Figure 9.1 shows an electron micrograph image of a section through the corpus callosum of a rat. Although the corpus callosum is considered to be one of the most ordered white matter structures, the microstructure heterogeneity is evident: axons of different sizes, non-evenly spaced extracellular matrix, and glial cells of different shapes and sizes. From a biophysical point of view the aforementioned components of white matter should have different physical properties. Physical parameters or properties such as viscosity, elasticity, permeability, density, and diffusivity may depend on tissue microstructure and morphology. For instance, the amount of myelin that encapsulates an axon is known to directly influence tissue viscosity; loss of myelin results in reduction in tissue viscosity, elasticity, and permeability (Kuroiwa et al., 2006). Another example is the change in tissue density and viscosity with the increase in water content of a tissue (Mellergard et al., 1989). MRI is a macroscopic imaging tool as it is able to measure different physical properties of the tissue at a resolution of millimeters. However, as the biophysical properties and tissue microstructure are linked, there is a possibility that such information can also be retrieved from MRI. The following sections will describe how diffusion MRI can be used to probe

FIGURE 9.1 The microstructure of white matter. An electron microscopy image of a cross-section of the rat’s corpus callosum showing the diversity of structures in the tissue: (1) large-diameter axons, (2) small-diameter axons, (3, 4) glial cells.

tissue microstructure, and the methodology limits and advantages with respect to tissue anatomy and morphology.

9.2 DIFFUSION MRI AND TISSUE MICROSTRUCTURE Diffusion MRI is considered a microstructural probe because it measures the micrometer-scale displacement of water molecules (Stejskal and Tanner, 1965; Wesbey et al., 1984; Le Bihan et al., 1986; Le Bihan, 1995)da displacement that is in the order of magnitude of the microstructure of the tissue. Intuitively, the ability of diffusion MRI to detect microstructural parameters depends on the effect of tissue permeability and viscosity of the measured diffusion properties. In the early 1990s, three experimental observations pointed out that tissue geometry does influence the diffusion properties of water molecules: (i) water diffusion in ischemic tissue (Moseley et al., 1990b; Warach et al., 1995); (ii) water diffusion following cortical depolarization (Latour et al., 1994), and (iii) anisotropic diffusion of water in white matter (Moseley et al., 1990a; Basser et al., 1994; Basser, 1995). The apparent diffusion coefficient (ADC) of water molecules is decreased immediately following ischemia (Moseley et al., 1990b; Warach et al., 1995). This is the basis for the early detection of ischemic stroke. For many years the exact biophysical mechanism leading to this observation remained unclear. Several experiments pointed to a decrease in intracellular diffusion, while others gave much higher importance to diffusion changes in the extracellular matrix as well as other physiological features (Benveniste et al., 1992; Busza et al., 1992; Dardzinski et al., 1993; Hasegawa et al., 1994; Hoehn-Berlage, 1995; van der Toorn et al., 1996). There is no clear agreement on the the weighted contributions of each of the suggested processes. However, it is generally agreed that the process of cell swelling is an important contributor, suggesting that changes in tissue microstructure may be related to changes in the observed tissue diffusivity. The indication that cell swelling causes changes in the diffusion properties of water was corroborated following experiments on spreading waves of cortical depolarization (Latour et al., 1994). It is known from neurophysiology that a cell swells following depolarization due to changes in energetic cycles (Andrew and MacVicar, 1994). Using diffusion MRI, the time course of spreading cortical depolarization waves could be detected (Latour et al., 1994). As in ischemic stroke, following cortical depolarization, the water ADC decreased. Much of this observation was attributed to the increase of water molecules that experienced highly

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9.3. DIFFUSION TENSOR IMAGINGdA TOOL FOR WHITE MATTER MICROSTRUCTURAL MAPPING

hindered or restricted diffusion in the extracellular matrix (Latour et al., 1994). The third indication that diffusion MRI could provide information on tissue microstructure was from diffusion MRI of healthy white matter tissue. It was found that the diffusivity in white matter structures is much slower when the diffusion is measured perpendicular to the neuronal fiber orientation, than in parallel (Chenevert et al., 1990; Moseley et al., 1990a, 1991; Pierpaoli and Basser, 1996; Pierpaoli et al., 1996). This anisotropy of diffusion is most probably a reflection of the alignment of neuronal fibers. The high packing of the fibers and their thick myelin membrane most probably hinders or restricts water diffusion perpendicular to the fibers; while parallel to the fibers, the diffusivity is almost undisturbed. These three observations lead to the hypothesis that diffusion MRI can be considered as a microstructural probe. In addition, it appears that tissue contents (membranes, organelles) reduce water mobility according to their size and shape.

9.3 DIFFUSION TENSOR IMAGINGdA TOOL FOR WHITE MATTER MICROSTRUCTURAL MAPPING The following section briefly reviews a basic description of diffusion tensor imaging (DTI) and its relevance to tissue microstructure. More in-depth description of DTI is given in Chapter 1. Macroscopic diffusion anisotropy in the CNS seems to be unique to white matter. Low degrees of macroscopic anisotropy (if any) are found in gray matter and CSF, and thus an anisotropy-weighted image should enhance the white matter signal (Pierpaoli and Basser, 1996; Pierpaoli et al., 1996). DTI is the framework that provides such information (Basser et al., 1994; Basser, 1995; Pierpaoli and Basser, 1996; Pierpaoli et al., 1996; Basser and Pierpaoli, 1998; Basser and Jones, 2002). By sampling the diffusivity along multiple directions evenly spaced on a sphere, a 3D representation of the diffusion process can be computed. Such analysis assumes Gaussian distribution of the diffusion (i.e. a diffusion that is characterized by an ellipsoid) (Basser et al., 1994; Basser, 1995). While in gray matter and CSF the 3D diffusion is characterized by a sphere, in white matter it is represented by an ellipsoid having one axis much bigger than the two others. The axes of the ellipsoid are commonly referred to as the principal diffusivities, and represent the diffusion parallel (the biggest axis) and the diffusion perpendicular (the two smaller axes) to the fibers passing through the sampled region. Besides the quantitative principal diffusivities, two other quantitative parameters can be extracted from conventional

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DTI analysis (Pierpaoli and Basser, 1996): (i) the mean of the principal diffusivities, often called the mean diffusivity or 1/3 trace of the ADC; (ii) The normalized variance of the three diffusivities, often called the fractional anisotropy (FA). The FA has low values in areas of the CSF and gray matter and high values in areas of white matter. In addition to these quantitative measures, DTI offers an additional two qualitative measurements that stem from the 3D orientation of the ellipsoid (Conturo et al., 1999; Mori et al., 1999; Pajevic and Pierpaoli, 1999; Basser et al., 2000). If we assume that the orientation of the ellipsoid represents the underlying arrangement of neuronal fibers then this orientation can be used for the visualization of tracts. This can be done in 2D by the directionally encoded colored FA maps where each ellipsoid is colored based on its orientation (in a formal RGB color scale) (Pajevic and Pierpaoli, 1999) or in 3D by a line that connects the ellipsoids that may represent a fiber pathway (tractography) (Conturo et al., 1999; Mori et al., 1999; Basser et al., 2000). DTI has a few advantages: it is fast (it can be performed in as little as 3–4 min although research scans often take up to 20 min), it is quantitative, it is rotationally invariant, it enhances significantly the white matter signal, and it allows 3D mapping of white matter trajectories. These advantages have turned DTI into a robust technique for white matter mapping. The methodology has been applied to many neurological diseases and disorders, indicating the white matter involvement. The most common observation in these studies is that a neurodegenerative process will most likely end up with reduction in FA, elevation in the principal diffusivities (especially the smallest two) and an increase in the mean diffusivity (Horsfield and Jones, 2002; Filley, 2005; Kim et al., 2006; Wozniak and Lim, 2006; Budde et al., 2007; Assaf and Pasternak, 2008). In general, those changes are referred to as loss of tissue order without the ability to distinguish between the two main neurodegenerative processes of white matter: axonal loss and demyelination. Several studies have tried to investigate the relation between microstructure and DTI parameters. For example, it was found that tissue cellularity is well correlated with the mean diffusivity in a brain tumor patient (Gauvain et al., 2001). A study of demyelination in the mouse corpus callosum found a strong correlation between the radial diffusivity and the amount of demyelination (Sun et al., 2006). The FA index was found to be in correlation with the process of Wallerian degeneration that occurs following a stroke, indicating sensitivity to axonal loss (Pierpaoli et al., 2001). However, none of the DTI indices are a direct measurement of specific white matter compartments (Wozniak and Lim, 2006). The abovementioned studies, and many others, suggest that changes in these parameters are correlated with

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morphological changes that occur in the affected compartment (but not a direct measurement of the compartment). For example, diffusion anisotropy is also found in non-myelinated white matter tissue, suggesting that myelin is not the sole contributor to the observed anisotropy (Beaulieu and Allen, 1994; Beaulieu, 2002). Thus, loss of the anisotropy cannot be unequivocally related to myelin loss. Decrease in the mean diffusivity occurs when tissue cellularity index increases (Gauvain et al., 2001) but it also happens when the cell swells (Benveniste et al., 1992). What limits the capacity of DTI to uncover tissue microstructure? Can we increase DTI’s specificity to different compartments? To answer these questions we need a better understanding of the DTI model and diffusion processes in neuronal tissue.

9.4 DIFFUSION TENSOR IMAGINGdA TOOL FOR WHITE MATTER MICROSTRUCTURAL MAPPING? Despite the great advantages of DTI, it has become apparent in recent years that DTI suffers from inherent limitations embedded in its model. The DTI model assumes a single diffusing process that propagates in time and space in a Gaussian manner (Chenevert et al., 1990; Basser et al., 1994; Basser, 1995). Combining the fact that white matter tissue has few compartments and that each image pixel contains roughly half a million cellular elements (cells or axons), the following issues with DTI become significant: (i) partial volume effectdpixels contain several types of tissues (gray matter and white matter or CSF and white matter) (Shimony et al., 1999; Alexander et al., 2001; Papadakis et al., 2002); (ii) averaging effectdpixels contain more than one predominant fiber pathway (Pierpaoli et al., 1996; Wiegell et al., 2000; Tuch et al., 2002); and (iii) non-Gaussian diffusion (Assaf and Cohen, 1998; Cohen and Assaf, 2002; Assaf and Basser, 2005). The DTI model cannot handle any of these inherent issues, which may result in measurement artifacts. For example, it is well known that certain areas within the white matter are characterized by significantly reduced anisotropy. These areas are considered to be regions where two or more fiber systems cross, leading to an averaging effect imposed by the diffusion model as it assumes only one diffusing component (Pierpaoli et al., 1996; Wiegell et al., 2000; Tuch et al., 2002). CSF contamination is another known artifact, leading to reduced FA in pixels that are on the borderline between CSF and white matter (Shimony et al., 1999; Alexander et al., 2001; Papadakis et al., 2002). Lastly, it is unwise to assume that, in such a complicated tissue where water

molecules diffuse across membranes, with various thicknesses and with different viscosities encountering a variety of obstacles, the diffusion will be represented by a Gaussian model. Taking into account the aforementioned limitations of DTI, can we refer to this methodology as a microstructural probe? Although several frameworks have been suggested to cope with the effects of partial volume and crossing fibers (q-ball imaging) (Tuch et al., 2003), persistent angular structure (PAS) MRI (Jansons and Alexander, 2003), and high angular resolution diffusion imaging (HARDI) (Frank, 2001), none of these frameworks deal with the Gaussian assumption of the model. Most of these frameworks are used to correct artifacts in the tractography process caused by the above problems. One repeated observation that is overlooked in conventional DTI analysis is the deviation of the signal decay (with respect to the b-value) from single exponential decay (Figure 9.2). In the early years of diffusion MR, it was calculated that for a single, free diffusion component, an exponential relation is present between the signal decay and diffusion weighting factordb. In the mid 1990s it was evident that in neuronal tissue there is a deviation from this relation when performing the experiment at high enough b-values. It became evident that sampling a wide range of b-values necessitates fitting the data to a high order of exponential function. This led to the suggestion that the measurement of the diffusion signal at high b-values can be used to probe different diffusion processes occurring in a tissue.

FIGURE 9.2 The deviation for the signal decay at high b-values: Diffusion MR signal decay of water molecules in bovine optic nerve (measured perpendicular to the nerve’s long axis) as a function of the b-value and diffusion time. Note the significant deviation from mono-exponential behavior noticed around b-value of 2  106 s/cm2 (2000 s/mm2).

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9.5. TYPES OF DIFFUSION PROCESSES IN THE TISSUE

9.5 TYPES OF DIFFUSION PROCESSES IN THE TISSUE Diffusion MRI can become a microstrucutral probe if a certain distinguishable diffusion process can be assigned to a certain tissue compartment. There are three major modes of diffusion: free diffusion, hindered diffusion, and restricted diffusion (Le Bihan, 1995). These can be further divided into isotropic and anisotropic (Basser, 1995). Free and hindered diffusion follow the Gaussian displacement distribution (Basser, 1995; Assaf et al., 2004). One way to distinguish between diffusion processes is to look at the ratio between the mean displacement and diffusion time (Le Bihan, 1995). For free diffusion there is a linear relation between the square of the mean displacement and the diffusion time with a constant coefficient also known as the diffusion coefficient. This relation is formally known as the Einstein’s equation (equation (9.1)). qffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi (9.1) hxi2 ¼ nDt where Ohxi2 is the mean displacement, n is the measurement dimensions (2 for one dimension, 4 in two dimensions, and 6 in three dimensions), D is the diffusion coefficient, and t is the diffusion time. In contrast to free/hindered diffusion, restricted diffusion will be characterized by a constant mean displacement as a function of the diffusion time. It should be noted that in brain tissue, hindered diffusion has a linear relation between the diffusion time and square of the mean displacement. In that case, the coefficient between these two parameters is referred to as the ADC (Le Bihan, 1995). The relations between mean displacement and diffusion time for the three types of diffusion processes are presented schematically in Figure 9.3.

FIGURE 9.3 Diffusion processes. The three possible diffusion processes that can occur in biological tissues: free, hindered, and restricted. While free and hindered diffusion will show linear relation between the mean displacement and square root of the diffusion time, partially restricted (or highly hindered) and restricted diffusion will show nonlinear dependency.

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The motivation to search for different diffusion processes in the CNS stems from the fact that diffusion MRI measures the displacement of water molecules rather than the actual diffusion coefficient. Therefore, if processes such as restricted or hindered diffusion occur we should be able to identify them in theory. For several years it was believed that one could not identify restricted diffusion in the CNS. As will be described in the following paragraphs, the diffusion time and the diffusion weighting are the key factors that enable us to probe water displacements that will be indicative of restricted and hindered diffusion. In order for it to be possible to probe microstructural properties with diffusion MRI it is essential that the diffusion in the “restricted” compartment is truly restricted at the timescales used in diffusion MRI. One of the structures that can cause hindered or restricted diffusion in brain tissue is cellular membranes. Assuming that membranes are partially permeable to water molecules, at a certain range of diffusion time, their effect on the displacement should be significant. The exchange rate of water molecules between the different compartments should reflect the membrane permeability and determine the range of diffusion time that should be used. In fact, the exchange rates for the brain’s cellular compartments are not known and can only be estimated. The exchange rate for water in red blood cells was reported to be around 15–30 ms (Kuchel et al., 1997); one can speculate that a similar exchange rate or a higher rate will be found for neurons and glial cells. More importantly, it seems reasonable to assume that the exchange rate for water across the myelin membrane of axons will be much lower. Unknown factors that can influence the exchange rate are the number of functioning water and ion channels and the axonal activity state. Indeed, the exchange rate for water molecules across axons was estimated to be around 500 ms (Meier et al., 2003). Hence, in the timescale of diffusion times that are generally used in diffusion MRI (50–100 ms) the cellular (hindered) and axonal (restricted) compartments should be distinguishable. While the cellular compartment should exhibit hindered diffusion, the axonal compartment should appear restricted. In the mid 1990s, various studies showed that water signal decay in diffusion MR experiments of neuronal tissue (brain and nerve) was not mono-exponential, a fact that could result from restricted diffusion or from the existence of few diffusing components (or both factors) (King et al., 1994; Niendorf et al., 1996; Assaf and Cohen, 1998). Two diffusing components were described as fast- and slow-diffusing components. Additional experimental evidences pointed out that the fast-diffusing component had the characteristics of free or slightly hindered diffusion, while the slow-diffusing component appeared to

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represent mainly restricted diffusion (Assaf and Cohen, 2000). Among these experimental evidences were: (i) the slow diffusion component, which was larger in areas of white matter than in gray matter (Assaf and Cohen, 2000; Yoshiura et al., 2001); (ii) the magnitude of the slow-diffusing component, which increased tremendously when measured perpendicular to the neuronal fibers (Beaulieu and Allen, 1994; Stanisz et al., 1997; Stanisz and Henkelman, 1998; Assaf and Cohen, 2000); (iii) the measured mean displacement of the slowdiffusing component, which did not change when the diffusion time was increased (Assaf and Cohen, 1998, 2000). Several studies suggested that the main compartment contributing to the slow-diffusing component is

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the axonal compartment (Assaf and Cohen, 2000). Figure 9.4 summarizes the experimental evidence connecting the slow-diffusing component to restricted diffusion within axons. In principle, it should be possible to identify populations of water molecules that experience hindered or restricted diffusion since their displacement would be smaller or even significantly smaller than that of free diffusing water. This is achieved by increasing the diffusion weighting (i.e. the b-value) of the diffusion experiment (usually by varying the diffusion time). While typical b-values of DTI experiments are in the range 700–1500 mm2/s (probing minimum diffusion distance of about 5–10 mm at diffusion time range of 50–100 ms),

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FIGURE 9.4 Evidence that the slow-diffusing component reflects restricted water diffusion in white matter: (a) MR images of in situ rat brain at the hippocampal level taken with a b-value of (1) 0, (2–4) 1.1  106 s/cm2, diffusion gradient applied in the y, x, and z directions, respectively. (5) Histological diagram of horizontal rat brain section showing areas of macroscopically ordered white mater fiber tracts. st, striatum; cc, corpus callosum; ce, cerebelum; sc, superior colliculus; hc, hippocampus. (b) Normalized attenuation of water signal (I/I0) as a function of the diffusion time for excised optic nerve samples. Full and open symbols represent nerve data in which the diffusion gradient direction was parallel and perpendicular to the long axis of the nerve, respectively. Error bars were omitted for clarity. Note that the signal decays much more rapidly in the parallel orientation that in the perpendicular orientation. (c) The “restriction test” for different diffusing components of brain water as obtained by a bi-exponential fit. The plot of the diffusion distance as a function of the square root of the diffusion time should give a straight line that passes through the origin if no restriction occurs. It is apparent that the slow-diffusing component deviates significantly from free or hindered diffusion behavior and matches that of restricted diffusion. Reproduced with permission from (a, b) Assaf and Cohen (2000) and (c) Assaf and Cohen (1998).

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9.6. Q-SPACE ANALYSIS

experiments aimed at probing restricted diffusion should be much more sensitive to smaller displacements and thus much larger b-values are required (w8000– 15 000 mm2/s probing for minimum diffusion distance of about 1–2 mm and less at diffusion time range of 50–100 ms) (Assaf and Cohen, 2000). Therefore, MRI diffusion experiments aimed at revealing the slow diffusion component are also termed high b-value diffusion imaging. In that sense diffusion MRI can be used as a filter for certain diffusing components; at a high b-value, mostly water molecules that experience restricted diffusion will be probed. At low b-value, hindered compartments, whose molecules experience greater displacements, will dominate Without going into the theory of diffusion MRI, the signal in DWI decays exponentially with b when the diffusion is free and Gaussian, and non-monoexponentially in any other situation (Callaghan, 1991; Assaf and Cohen, 1998). Although experimental data showed that nonmonoexponential signal decay exists in brain tissue (when measured in a high b-value range) (Beaulieu and Allen, 1994; Niendorf et al., 1996; Assaf and Cohen, 1998, 2000; Stanisz and Henkelman, 1998; Mulkern et al., 1999) the analysis framework that can relate this decay to morphological/physiological components or compartments of the tissue is not straightforward. In recent years, several approaches have been suggested to analyze the non-monoexponential signal decay. These have been divided into two categories: parametric and nonparametric approaches. Non-parametric approaches do not enforce a mathematical model of the signal and try to extract indices that can be related indirectly to a compartment or a diffusion process. On the other hand, the parametric approach uses a mathematical model to describe the signal decay and thus finds the weighted contribution of each of the model component; yet such a model might not be comprehensive enough to describe the entire complexity of the tissue. The nonparametric approaches are based on the q-space theory (Callaghan, 1991) and include q-space imaging (Assaf et al., 2000; Cohen and Assaf, 2002), diffusion spectrum imaging (DSI) (Wedeen et al., 2005), and diffusion kurtosis imaging (DKI) (Jensen et al., 2005). The parametric approaches construct a model of diffusion in brain tissue either based on the morphological components or based on the diffusion components (Niendorf et al., 1996; Stanisz et al., 1997; Assaf and Basser, 2005; Assaf et al., 2006).

9.6 Q-SPACE ANALYSIS The q-space theory relates the diffusion signal decay to the displacement distribution function of the molecules (Callaghan, 1991). In the early works of Stejskal

and Tanner and later on in the first diffusion imaging applications and DTI, the displacement distribution function had a Gaussian distribution. In the q-space analysis we start with the same description of the relation between the signal decay and the displacement distribution function but assume that this function is non-Gaussian. Developed at the end of the 1980s to study the structure of porous materials (Callaghan, 1991), the q-space approach was applied to extract the dimensions of yeast cells in the early 1990s (Cory and Garroway, 1990) and to neuronal tissue and red blood cells shortly afterwards (King et al., 1994; Kuchel et al., 1997; Assaf and Cohen, 1999; Assaf et al., 2000; Cohen and Assaf, 2002). The theory is based on the Fourier relation between the signal decay and the displacement distribution profile (equation (9.2)) (i.e. a probabilistic function that describes the probability of molecules to diffuse a certain distance at a given diffusion time): Z ED ðqÞ ¼

Ps ðR; DÞexpði2pq$RÞdR:

(9.2)

In this equation, E is the signal decay, Ps ðR; DÞ is the displacement propagator, D is the diffusion time, and q is defined as gdG/2p (g is the gyromagentic ratio, d is the diffusion gradient duration, and G is the diffusion gradient amplitude and relates to the b-value with the expression: b ¼ 4p2q2D). The first demonstration that the q-space analysis could provide structural information on biological specimens was shown on yeast cells (Cory and Garroway, 1990). Diffusion spectroscopy measured over a wide range of b-values (or q-values) and diffusion times combined with q-space analysis (i.e. simple Fourier transform of the diffusion signal decay) revealed the dimensions of yeast cells by exploring the displacement distribution function (Cory and Garroway, 1990). Experiments performed in vivo on mouse brains and ex vivo on nerve samples revealed a more complicated displacement distribution function than in the yeast cells experiment (King et al., 1994; Assaf and Cohen, 1999, 2000; Assaf et al., 2000; Cohen and Assaf, 2002). It appeared that the displacement distribution function is a superposition of at least two diffusing components: one with a broad displacement distribution and one with a very narrow displacement distribution (Figure 9.5) (Assaf and Cohen, 2000). The diffusion time-dependent experiment revealed that the mean displacement of the broad component follows the expectations from free diffusion, while the narrow component’s mean displacement does not change much with the diffusion time (Figure 9.5) (Assaf and Cohen, 2000). The broad and narrow components were assigned to the extra- and intra-axonal compartments based on the observation that the narrow component was more prominent in white matter-rich

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FIGURE 9.5 Displacement distribution profiles for water in neuronal tissue. (a) A free diffusing reference sample of t-butanol depicting the broadening of the displacement profile with the increase of the diffusion time. The displacement profiles for excised rat brain (b), an excised bovine optic nerve measured parallel (c) and perpendicular (d) to the long axis of the nerve show mixed behavior of two components: a broad component that follows the t-butanol behavior as a function of the diffusion time and a narrow component that does not change its width with the increase in the diffusion time (i.e. restricted diffusion). Reproduced and edited with permission from Cohen and Assaf (2002).

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regions and showed higher restricted diffusion and anisotropy as compared to the broad component. The advantages of using the q-space approach are evident: (i) it is a model-free approach that provides information on the measured quantitydthe displacement; (ii) it is a simple mathematical relation allowing fast and robust computation; (iii) it is compatible with non-Gaussian diffusion as it does not enforce a Gaussian model on the system. The disadvantages of using the q-space approach are: (i) the acquisition times are much longer than those for conventional DTI since it requires sampling of the entire range of signal decay (roughly 5–10 times longer); (ii) the output of the analysis is a function that should be further analyzed to provide quantitative information; (iii) the q-space theory requires an extreme acquisition parameter (short gradient pulse) that is difficult to achieve with conventional clinical scanners (Mitra and Halperin, 1995). The q-space framework can be applied in one dimension to average the displacement distribution function over a certain direction (Assaf and Cohen, 2000; Assaf

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et al., 2000; Cohen and Assaf, 2002) or in three dimensions to extract the displacement distribution function itself (Wedeen et al., 2005). The 1D approach is formally termed q-space imaging (QSI) and two parameters are extracted from the displacement profile: the full width at half maximum (FWHM, termed apparent displacement), and the peak intensity of the displacement profile (termed probability for zero displacement, P0) (Assaf et al., 2000). The 3D reconstruction of the displacement distribution function, formally termed diffusion spectrum imaging (DSI), from which the contours of the displacement function can provide a more accurate orientation distribution function of the diffusion components in the tissue, is mainly used to resolve orientationally complex white matter structures (Wedeen et al., 2005). Another model-free approach that uses the q-space concept is diffusion kurtosis imaging, which measures the deviation of the displacement profile from Gaussian distribution (Jensen et al., 2005). The benefits of using the q-space approach over DTI are not evident at first glance. Like DTI, q-space does not provide information on specific compartments.

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FIGURE 9.6 Origins of restricted diffusion in optic nerve: Root mean square (rsm) displacement of the slow-diffusing components as the function of square root of the diffusion time (td) for optic nerve along with the theoretical rms displacement expected from the Einstein equation.

However, as the q-space analysis incorporates the contribution of non-Gaussian diffusion it should be more sensitive to restricted diffusion processes (Assaf and Cohen, 2000; Assaf et al., 2002a). Thus it is assumed that compartments that exhibit restricted diffusion will be more

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apparent in the q-space analysis than in conventional DTI. Experiments performed on nerve samples (Assaf and Cohen, 2000; Bar-Shir and Cohen, 2008b), neuronal development (Assaf et al., 2000), degeneration (Assaf et al., 2002a; Assaf et al., 2003; Biton et al., 2007), and connectivity (Wedeen et al., 2005) support this hypothesis. Bar-Shir and Cohen (2008b) have shown that measuring the narrow displacement component at very short diffusion times and incrementing it in small steps (of 1 ms) allows identification of the time point at which the diffusion becomes restricted (Figure 9.6). The q-space imaging performed on maturing rat spinal cords revealed significant changes in the displacement distribution function as age increases (Assaf et al., 2000). It is well known that myelin forms in the first few weeks following birth in rats (and first few years in humans) and thus it is expected that the magnitude of restricted diffusion in the white matter will be significantly increased with age. Indeed at age of 3 days the displacement profiles of gray and white matter tissues were similar, while later on, the displacement profile in the white matter became much narrower, indicating a higher degree of restricted diffusion (Assaf et al., 2000). Those results are presented in Figure 9.7. In that figure the excised spinal cords were analyzed according

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to the q-space approach on a pixel-by-pixel basis, from which the peak intensity of the displacement profile (termed probability index) and the FWHM (termed the displacement index) were extracted. The probability and displacement maps are shown on a color scale. During development of white matter, several processes occur in parallel with the formation of myelin and packing of the axons; the tissue density increases, extracellular water fraction decreases, and the viscosity of the tissue increasesda factor that may also influence the diffusivity of water (Neil et al., 2002; Miller et al., 2003; Mukherjee and McKinstry, 2006). Demyelination is a white matter-specific pathology in which the myelin membrane surrounding the axon disintegrates due a neurodegenerative process (as in multiple sclerosis). Experiments performed on demyelinated white matter tissues showed that the magnitude of the slow-diffusing component reduces dramatically, leading to a reduced weighting and a broadening of the narrow displacement component (Assaf et al., 2002a, 2002b, 2003; Biton et al., 2005). Figure 9.8 shows the q-space images of a demyelinated rat spinal cord due to chronic hypertension, along with the electron micrographs (EM) of the same tissue (Assaf et al., 2003). From the EM images, clear demyelination and other morphological myelin changes are apparent (formation of vacuoles). These changes were reflected by an increase in the apparent displacement index in the white matter when diffusion was measured perpendicular to the fibers in the spinal cord. Figure 9.9 shows q-space images as well histology of porcine spinal cords at a severe phase of an experimental autoimmune encephalomyelitis (EAE) model of multiple sclerosis

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(Biton et al., 2005). The q-space images provide very similar information to the histological staining identifying the demyelinating lesion. For example, regions that are identified as a lesion are also characterized by increased displacement. On the other hand, regions identified as normal white matter also appear normal in the histological sections. This implies that, at least in this example, q-space imaging can be regarded as “virtual histology.” In addition, the same work has shown that the sensitivity of conventional diffusion anisotropy to the lesion was much lower (Biton et al., 2005). Figure 9.10 shows q-space images of human brains from healthy and diseased (multiple sclerosis) subjects, revealing an increase in the displacement in areas of multiple sclerosis lesions and surrounding them (Assaf et al., 2002a) in a similar manner to the animal model of the disease (Biton et al., 2005). With the help of histology (Figures 9.8 and 9.9) we can explain the observations in the human brain: since the number and thickness of myelin lamellas reduces in demyelination, the exchange with extracellular water becomes more rapid, leading to the reduction in restricted diffusion. In addition, the formation of water-filled vacuoles between the myelin lamellas also contributes to an increase in the free diffusion water component. Also, it should be noted that DTI was able to reveal these changes but with less sensitivity (i.e. the differences between the healthy and diseased groups were smaller) (Assaf et al., 2002a, 2002b, 2003). In addition, the q-space approach enables the detection of significant changes in the “normal appearing white matter” (NAWM) of people with multiple sclerosis, suggesting that this methodology

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might be sensitive even to fine tissue changes. This provides further evidence that the high b-value diffusion imaging combined with q-space analysis emphasizes the contribution of restricted diffusion and thus increases the sensitivity to cellular processes that end up with changes in this property (such as demyelination or remyelination). Figure 9.4 showed that the contrast between gray and white matter is significantly increased when measured in the high b-value range (Assaf and Cohen, 2000; Yoshiura et al., 2001). Assuming that at high b-values

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FIGURE 9.10 Q-space imaging in multiple sclerosis: conventional MRI (FLAIR) images (a, c) and displacement q-space imaging (b, d) of a healthy (a, b) and multiple sclerosis diseased (c, d) subjects. Note the reduction of displacement in the lesions and surrounding areas in the normal-appearing white matter. Reproduced and edited with permission from Assaf et al. (2002a).

the contribution to the diffusion signal mostly originates for water molecules experiencing restricted diffusion, and if we assume that the axons are the main compartment where restricted diffusion occurs, then a fiber orientation analysis will be much more accurate when measured with high b-values. Diffusion spectrum imaging is an expansion to the 1D q-space analysis to 3D (Wedeen et al., 2005). It allows visualization of the displacement distribution function in 3D and thus reveals complicated white matter structure (i.e. areas where multiple white matter fiber bundles cross). Figure 9.11 shows the advantage of using the displacement profile for estimation of the orientation density function over conventional DTI (Tuch et al., 2003).

9.7 MODELS OF DIFFUSION IN WHITE MATTER The model-free approach does not provide compartment-specific information, but rather quantitative measures that increase the sensitivity to nonGaussian diffusion. Imposing a model on the signal decay will enable extraction, assuming the model is valid, of specific information on various compartments or components.

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FIGURE 9.11 Cerebral diffusion spectrum imaging (DSI) of normal human subjects. (a) The complete coronal brain slice under study. Diffusion is represented by a tensor fit from the DSI data. Tensors are represented as boxes shaped by the eigenvalues and eigenvectors, and color-coded based on leading eigenvector orientation (blue: axial; red: transverse; green: anteroposterior orientation). (b) Zoom image of the brainstem. Diffusion spectra are represented as polar plots of the orientation distribution functions (ODFs), which are color-coded depending on diffusion orientation. Here, the corticospinal tract contributes spectral maxima of axial orientation (blue lobes along the vertical axis) and the pontine decussation of the middle cerebellar peduncle contributes horizontally (green lobes crossing at center). Many local spectra show contributions of both structures. (c) This image shows the centrum semiovale and contains elements of the corticospinal tract (blue), corpus callosum (green), and superior longitudinal fasciculus (red), including voxels with two- and three-way intersections of these components. Orientational correspondence between tensor and spectral data is best at locations with simple unimodal spectra, while locations with multimodal diffusion spectra correspond to relatively isotropic DTs. Reproduced with permission from Wedeen et al. (2005)).

The most basic model that was used was the biexponential model (Niendorf et al., 1996; Assaf and Cohen, 1998; Mulkern et al., 1999). This model assumes that the diffusion in white matter tissue is composed of two diffusing components exhibiting Gaussian diffusion with no exchange between them. In early studies attempts were made to assign the two components extracted from the bi-exponential fit to the intra- and extracellular compartments. It is approximated that intracellular compartments in brain tissue occupy 80% of the volume of the tissue while the extracellular compartments occupy only 20%. Assuming that water in the intracellular space should exhibit slower diffusion than water in the extracellular space, the slowdiffusing component can be assigned to the intracellular space. However, the bi-exponential fit estimated that only 20% of the signal is that of the slow-diffusing

component and the rest is the fast-diffusing component (Niendorf et al., 1996; Assaf and Cohen, 1998). Thus, the population fraction of the two components does not fit their physiological fraction and although the biexponential model can fit the experimental signal decay, it cannot be related to tissue compartments. In addition, it was found that when using a very high diffusing weighting (b-values of more than 15 000 mm2/s) the biexponential model does not fit the data, requiring more exponents to fit the data (Assaf and Cohen, 1998). The mono-, bi-, and tri-exponential fits of the diffusion-weighted signal in extracted rat brain tissue are presented in Figure 9.12. It is evident that in the very high b-values regime even the tri-exponential function does not fit the data with high precision. Stanisz et al. (1997) suggested using a morphological model to describe the diffusion in ordered white matter

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tissue. The model was composed of three diffusing components: diffusion in spheres (representing glial cells), diffusion in ellipsoids (representing axons), and the extracellular matrix. The signal decays for each of the compartments were derived from an analytical model developed on the basis of Monte-Carlo simulations of the diffusion signal decay. In addition, exchange between compartments was also modeled. The model was evaluated on a diffusion MR spectroscopy data of bovine optic nerve that was measured parallel and perpendicular to the fibers. The model fit the data very

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well, and provided the diffusivities and the mean size of each of the compartments (Stanisz et al., 1997). The Stanisz model was one of the first models to incorporate and account for restricted diffusion in white matter tissue. However, the model was very specific to the optic nerve and therefore did not take into account the variability in sizes of the compartments which are relatively small in the optic nerve. In other white matter tissues (i.e. fascicles in the brain, peripheral nervous system nerves), a much wider distribution of fiber size and type exists, which the model needs to account for. In addition, the Stanisz model required knowledge of the fiber alignment in order to perform the acquisition in the perpendicular and parallel directions, and thus could not be applied to analysis of diffusion imaging in the brain. Peled et al. (1999) suggested a model for diffusion in the sciatic nerve. Instead of modeling the diffusivity in axons as an ellipsoid, here the axonal diffusion is modeled as impermeable cylinders, taking into account a diameter distribution function based on histological analysis. In the same work it was shown that the diffusion–diffraction phenomenon (Callaghan, 1991; Callaghan et al., 1991) that occurs in restricted geometries could be used to explain the non-monoexponential signal decay (as shown in Figure 9.13). This supported the hypothesis that the slow-diffusing component reflects restricted diffusion. While the model introduced tissue heterogeneity, it was not implemented in 3D space and did not take into account other diffusion processes. The composite hindered and restricted model of diffusion (CHARMED) (Assaf et al., 2004; Assaf and Basser, 2005) tries to overcome the limitations of the Stanisz and Peled models. Instead of describing the tissue by the morphology, the CHARMED model decomposes the measured signal to diffusion processes. The model further assumes the assignment of each of

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FIGURE 9.14 The composite and hindered model of diffusion (CHARMED). (a) The CHARMED model, a composition of two diffusion processes (hindered and restricted). (b) The CHARMED acquisition framework, samples of the raw-data images. Data were collected at ten shells of b-value (only six are shown) starting from 0 to 10 000 s/mm2. The angular resolution of gradient directions increased with the b-value starting from 6 at the lowest (714 s/mm2) up to 30 at the highest (10 000 s/mm2). Image quality was reasonable even at the highest b-value shells (see darkest blue ring, b ¼ 8571 s/mm2, SNR >5) where signal originates only from white matter fibers placed perpendicular to the gradient direction. Reproduced with permission from Assaf and Basser (2005).

the diffusion processes to specific compartments. The CHARMED model assumes two diffusing processes: one restricted and one hindered (Figure 9.14) (Assaf et al., 2004). Each process can be composed of several diffusing components having the same physical parameters but different alignment in space. It is assumed that the cells and extracellular compartments, in the time diffusion scale of the experiment, are in full exchange and thus can be considered as one compartment. On the other hand, it is assumed that the intra-axonal compartment completely restricts water motion (perpendicular to the fibers) and that the exchange of this compartment with extra-axonal media is limited. Under these assumptions, the signal decay can be modeled in 3D and even accounts for axon size variability. The hindered part can be modeled by a full diffusion tensor, while the diffusion within the axon can be treated as diffusion within impermeable cylinders to which several solutions are present, depending on experimental conditions (Neuman, 1974; van Gelderen et al., 1994; Codd and Callaghan, 1999). In order to model the diffusion in 3D, CHARMED also incorporates an experimental framework that measures the diffusion in shells of multiple b-values (Figure 9.14). This way the space is covered homogeneously in the low b-value range (where the hindered part is more significant) as well as in the high b-value range (where the restricted part is more significant). On its basic level, CHARMED can be applied to the multi-shell acquisition data with only two independent parameters: the extra-axonal diffusivity and the extraaxonal volume fraction (with the intra-axonal volume fraction being the completion to one of the extraaxonal volume fraction) (Assaf et al., 2004; Assaf and

Basser, 2005). The assumptions of this framework are: (i) the intra-axonal diffusion does not affect the signal decay at the typical diffusion times (i.e. at long enough diffusion time, the only factor affecting the signal decay will be the cell dimensions); and (ii) the axon diameter distribution is fixed and modeled by a known histological distribution. One of the most straightforward uses of CHARMED is to homogeneously resample q-space and by 3D Fourier transformation of the resampled space (in a similar manner to DSI; Wedeen et al., 2005) to reproduce the 3D displacement density function. Using this procedure, it was shown that the restricted component can delineate the orientation of the fiber system with much greater accuracy than the hindered component (as in DTI) (Assaf et al., 2004). Avram et al. (2004) showed on phantom data, and later on Bar-Shir and Cohen (2008a) on nerve samples, that the angular sensitivity of the restricted diffusion component is extremely high (Figure 9.15). This means that signal from the slowdiffusing component changes dramatically when the angle between the nerve fibers and diffusion gradient approaches 90 . Concomitantly, the changes in radial diffusivity extracted from conventional diffusion anisotropy analysis (at low b-values) revealed a much weaker dependency. When the ADC obtained from low b-value data and the fast component of the q-space were analyzed as a function of the rotational angle, the maximal changes were observed for a ¼ 45 , implying that these parameters are less reliable in depicting the correct orientation of the fibers. As a consequence, the separation of complex white matter systems become more feasible and the uncertainty in fiber orientations is reduced dramatically (Figure 9.16) (Assaf and Basser,

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2005). In other words, by modeling tissue microstructure, one can use the information to refine fiber tractography data. The abovementioned models allow estimation of diffusivities of the different compartments or components and their relative weightings. Assuming the models do represent tissue diffusion, the extracted parameters turn diffusion MRI into a microstructural probe. The Stanisz model was the first to use the geometric features of the sample in order to decompose the complicated signal decay. By doing so, the mean size of the geometric components (ellipsoids and spheres representing axons and cells) could be estimated with their volume fractions. The Peled model incorporated the axon size distribution in the fitting procedure, thus accounting for geometrical variability. The CHARMED model uses a framework similar to the one suggested in the Stanisz and Peled models, but defines the diffusion process in 3D, which shifts the nonparametric approaches to the imaging playground rather than the spectroscopic one. The advantages of this are enormous; for example, images of axonal density can be produced (reflected by the volume fraction of the restricted component) or the axon diameter distribution estimated (Assaf et al., 2006), features that bring diffusion MRI to the verge of virtual biopsy or histology.

9.8 TOWARDS VIRTUAL BIOPSY OF WHITE MATTER WITH DIFFUSION MRI The use of parametric approaches enabled extraction of quantitative, physically and physiologically meaningful parameters. The axonal density is such a parameter

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derived from the volume fraction of the restricted diffusion component of the CHARMED model (Figure 9.14) (Assaf and Basser, 2005). The CHARMED model can be further developed to extract another morphological parameterdthe axon diameter distribution (ADD) (Assaf et al., 2006). The ADD is a quantity of white matter that is linearly correlated with the functionality of the fibers (Ritchie, 1982). The ADD, measured conservatively with histological procedures, is known to differ between brain regions, change during maturation, as well as being a consequence of neurodegenerative processes. The AxCaliber framework expands CHARMED in the sense that it also incorporates the ADD as an independent parameter. To be able to fulfil such a function, a multi-diffusion time, multi b-value CHARMED

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acquisition is required (Assaf et al., 2006). Assuming that different axon populations will be selectively weighted at different diffusion times, the model simultaneously fits the multi-diffusion time signal decay to extract the ADD. At first, the methodology was implemented on excised nerve samples (Figure 9.17) and a comparison between AxCaliber ADD and histological ADD was

performed, showing a high correlation between the two methods (Assaf et al., 2006). In addition, this methodology can be combined with MRI sequences and clustering algorithms to segment white matter tissue to regions that differ in their ADD. Very recent works indicate that AxCaliber is also feasible in in vivo segmenting of the corpus callosum of the rat to distinct regions of

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In vivo AxCaliber of the rat corpus callosum. (a) AxCaliber cluster analysis revealing five clusters of distinct axon diameter distribution shown in (b). The colors of the graphs in (b) match the cluster colors in (a). Note that in the genu and splenium of the CC (green, red, and cyan cluster) the ADD is narrow and sharp compared with the body (red and orange). (c, d) Conventional histological analysis of the same specimen as in (a), revealing similar axon diameter distribution as in (b) but shifted to smaller axonal diameter values. For more information please refer to Barazany et al. (2009).

different ADD (Barazany et al., 2009) (Figure 9.18). Combined with cluster algorithm, AxCaliber allows the segmentation of the corpus callosum to distinct regions that differ in their ADD (Figure 9.18). From the in vivo experiment it became evident that AxCaliber provides estimates on the ADD that are 2–3 times larger than those from histology. This discrepancy was attributed to the shrinkage effect of the histological process that may lead to underestimation of the ADD. The advantages of measuring the ADD with AxCaliber include the ability to: (i) estimate the ADD on large sections of the sample; (ii) monitor this function in vivo without referring to invasive histological procedures that also introduce fixation artifacts (shrinkage effect), and (iii) study these parameters in the living brain because no in vivo markers of axonal microstructure and physiology exist. AxCaliber’s original framework has one main limitation: the orientation of the axon should be known a priori in order to acquire the diffusion MRI signal exactly perpendicular to the neuronal fiber (Assaf et al., 2008;

Barazany et al., 2009). Two advanced methods tried to overcome this limitation. The first, ActiveAx, developed by Alexander et al. (2010), uses the four-compartment minimal model of white matter diffusion (MMWMD) which is an extension of the CHARMED model. The four compartments in ActiveAx are the intra-axonal water, adjacent extra-axonal water, CSF water, and glial cell water (Alexander et al., 2010). While each of these compartment has different diffusion properties, the first one, intra-axonal water, is influenced, as described above, by the axonal diameter. Within ActiveAx, only the mean axon diameter is computed but in a rotational invariant matterdmeaning that the mean axon diameter can be computed to any orientation of the fiber system without any a priori information. The experimental setup of ActiveAx requires the acquisition of the HARDI (90 directions) in multiple shells of b-values (from b of 500 s/mm2 to w3000 s/mm2) where each shell has different diffusion time and diffusion gradient duration (Alexander et al., 2010; Drobnjak et al., 2010; Dyrby et al.,

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FIGURE 9.19 ActiveAx mean axon diameter analysis in the vervet and human corpus callosum. ActiveAx analysis of the vervet (a and b) and human (c and d), revealing the mean axon diameter (a and c) and axonal density (b and d) at the cross-section of the corpus callosum. Reproduced with permission from Alexander et al. (2010).

2010; Zhang et al., 2011b). First implementation of this framework was on the excised vervet brain and in vivo human brain (Figure 9.19). As in AxCaliber, the extracted mean axon diameter in ActiveAx is larger than the expected histological one, however, there is a correlation between the two measures (Zhang et al., 2011a, 2012a; Dyrby et al., 2012). More recent studies with ActiveAx increased the precision of the measured mean axon diameter by including factors for dispersion of white matter fibers within the voxel (Zhang et al., 2011b). In addition, the ActiveAx model was recently simplified for feasibility in the clinical environment in a framework called NODDI (neurite orientation dispersion and density imaging). NODDI separates conventional diffusion tensor indices (FA, MD, etc.), into more specific microstructural information (such as the axon density and orientation dispersion). The technique supports measures of connectivity (via tractography) combined with microstructure from the same acquisition (Zhang et al., 2012b). Both ActiveAx and NODDI provide a powerful and robust framework to study both micro- and macrostructure of white matter for the whole brain. Another approach for extracting axonal diameter information for any oriented fiber system is the extension of the original AxCaliber framework from 1D to 3D. In AxCaliber 3D (Barazany et al., 2011) simultaneous

analysis of multiple CHARMED data sets at different diffusion times (Ds) provides, for each voxel, the conventional CHARMED parameters (i.e. the volume fraction of hindered and restricted diffusion, the fibers orientations (two in each voxel), the hindered and restricted diffusivities and the noise floor). From the multi-CHARMED fitting routine, a full AxCaliber data set can be resampled exactly perpendicular to the fitted fiber systems (and if two fiber systems reside in the same voxel the AxCaliber data set can be resampled to each independently). In this manner the AxCaliber3D model can be used to calculate the ADD of each voxel. AxCaliber 3D was demonstrated on a fixed excised porcine spinal cord phantom. The phantom contained two fiber systems: one sectioned out from the fasciculus gracilis and the second from the anterior corticospinal. The two sections were placed in a plate perpendicular one above the other (Figure 9.20). The multiCHARMED analysis recognized the two fiber systems at the crossing region (green frame in Figure 9.20). Following resampling of the data to create a full AxCaliber data set it was possible to calculate, for each image voxel, and for each fiber within it, the ADD as shown in Figure 9.20. One of the main challenges of adapting axon diameter estimation frameworks to the human brain is the limitations of clinical scanners. Specifically, the

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FIGURE 9.20 AxCaliber 3D. Implementation of the AxCaliber 3D framework. (a) The multi-diffusion time CHARMED analysis revealing a two-fiber population at the crossing fiber area (green) and a single-fiber population elsewhere (yellow and cyan). (b) Conventional spherical deconvolution analysis of DTI scans on the same sample that was used for tractography of the two-fiber systems. (c) Following resampling of the CHARMED data into 1D AxCaliber data set, for each point on the reconstructed fiber tracts we could estimate the ADD which is different for the two segments of the spinal cord (see yellow and orange ADDs). Note that the fiber tracts are colored according to their mean ADD. For more information please see Barazany et al. (2011).

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FIGURE 9.21 In vivo AxCaliber of the human corpus callosum. AxCaliber analysis of 1D AxCaliber acquisition on the corpus callosum revealing four clusters of distinct axon diameter distributions. As expected, in the genu and splenium the ADD is much narrower and sharper than in the body of the corpus callosum. For more information please refer to Horowitz et al. (2012).

low-gradient amplitude is a conceptual limitation as the axon diameter estimation frameworks requires, in theory, a strong diffusion gradient amplitude with relatively short exposure time (d) (see next section). While the low-gradient amplitude may lead to inaccuracy in the calculation of very small diameter axons and to general overestimation of the axonal diameter, both methods were recently implemented in clinical scanners, underscoring the great potential of these methods in studying the microstructure of the human brain. As was shown in Figure 9.19, ActiveAx was able to nicely demonstrate the mean axon diameter variation of the corpus callosum (Alexander et al., 2010; Dyrby et al., 2010). In the same way, Horowitz et al. (2012) modified the AxCaliber framework and replaced the formula for modeling diffusion within axons (Callaghan’s formula; Codd and Callaghan, 1999) in the original framework with another model that suits better the clinical conditions (van Gelderen’s formula; van Gelderen et al., 1994). Such experiment and analysis was able to extract the ADD for different parts of the corpus callosum (CC) (Figure 9.21) that resemble the known morphology and fiber composition of the CC.

9.9 SUMMARY The parametric and nonparametric approaches to the analysis of high b-value diffusion imaging allow the extraction of compartment-specific information. The methodologies presented so far emphasize the contribution of diffusion within the intra-axonal space and thus enhance the sensitivity towards this compartment. It should be noted that high b-value acquisition and the use of each of these analysis approaches have a few limitations. At the acquisition level, high b-value diffusion imaging attenuates the diffusion signal, which may reach noise level before enabling full characterization of the decay curve (Cohen and Assaf, 2002). This may lead to an increase in number of acquisitions per b-value, which, in conjunction with the large number of b-values needed for adequate analysis, produces an extremely long acquisition time (20–45 min with echo planar imaging). In addition, clinical MRI scanners use long gradient pulses to compensate for the limited diffusion gradient strength they can produce (Callaghan, 1991). This leads to the lengthening of the total echo

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time of the experiment and further reduction in the signal-to-noise ratio. More importantly, this leads to experimental conditions under which the abovementioned analysis routines are not valid, thus producing inaccurate structural information (as most of them are developed under the short gradient pulse approximation) (Mitra and Halperin, 1995). The length of the gradient pulse may influence the accuracy of the extracted parameters but not their relative proportions. An additional limitation of the parametric approaches is the fitting procedure. Since few of the independent parameters cause similar effects on the signal decay, local minima and initial conditions become a critical issue of the fitting procedure. However, in recent years, additional experimental and analysis frameworks have been developed to cope with these issues. For example, the use of the general waveform approach to deal with “fat” diffusion gradient pulses allows correction of their effect on the model (Codd and Callaghan, 1999). Methods for dealing with the violation of the short gradient pulse approximation have also been presented (Codd and Callaghan, 1999), allowing accurate structural information to be obtained from even when long d are used to acquire the data. This problem becomes even smaller with the introduction of new gradient technology that can reach higher and higher amplitudes. Future applications of diffusion MRI should be targeted at other tissue compartments, such as neurons, glial cells, and dendrites. Very recently though, the double pulsed gradient spin echo sequence (Komlosh et al., 2007) was extended to MR imaging. With this sequence the diffusion gradient can be applied along different directions within the same acquisition, enabling local anisotropy to be extracted. Such an acquisition has revealed that microscopic anisotropy can be detected in areas that are macroscopically isotropic (e.g. gray matter). Another direction that should be considered is measuring the diffusion signal at very short diffusion times. By using spherical functions instead of (or in parallel with) cylindrical ones, such data can be used to fit the cellular (glial/neuronal) components. Finally, these approaches should be implemented in various neurological diseases and disorders in human and animal models. Comparison of the histological appearance of tissue with the quantitative microstructural parameters extracted from diffusion MRI could indicate the sensitivity, specificity, and applicability of the high b-value approach and its various analysis procedures. It is evident that high b-value diffusion imaging includes additional information complementary to that obtained through DTI, and with the use of advanced analysis routines it has the potential to turn diffusion MRI into a microstructural probe which will allow us an in vivo glance into tissue micromorphology.

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