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I am studying tractography technique which aims to reconstruct bundles of axons in brain by following the diffusion direction of water. It is very interesting because it is non-invasive. It exploits the difference between grey matter and white matter: in white matter (axons) we find an anisotropic environment since there is a preferential direction of diffusivity of water molecules that is along the axons direction, while in grey matter the environment is isotropic since molecules do not move with a preferential direction. The diffusion of molecules is generally given by a difference of concentration of molecules between two points but they should move even though this gradient is not present because of the temperature so because of the fact that they have a certain thermal energy. Therefore my question is: why do molecules move along the axons direction ? Is there a biological aspect that causes like a difference in concentration between the starting point and the ending point ? Maybe the answer is trivial but since I am not a biologist I do not know it :) and I am not able to find something on web . It seems that I find only explanations for diffusion of water molecules from outside the cell to inside… I would like to know the reason why they move in the same direction of the axon.
The movement is just the thermal energy motion you describe, not due to any concentration difference.
The anisotropy is due to the geometry. White matter tracts are effectively a dense bundle of thick, fatty hoses. Water can freely move in the tubes but cannot move as freely through the dense walls, so you see more movement along the axon length rather than perpendicular to them.
You can read more at https://en.wikipedia.org/wiki/Diffusion_MRI
The spontaneous redistribution of a substance is due to the random motion of the molecules (or atoms or ions) of the substance. Because of the random nature of the motion of molecules, the rate of diffusion of molecules out of any region in a substance is proportional to the concentration of molecules in that region, and the rate of diffusion into the region is proportional to the concentration of molecules in the surrounding regions. Thus, while molecules continuously flow both into and out of all regions, the net flow is from regions of higher concentration to regions of lower concentration. Generally, the greater the difference in concentration, the faster the diffusion.
Since an increase in temperature represents an increase in the average molecular speed, diffusion occurs faster at higher temperatures. At any given temperature, small, light molecules (such as H2, hydrogen gas) diffuse faster than larger, more massive molecules (such as N2, nitrogen gas) because they are traveling faster, on the average (see heat heat,
nonmechanical energy in transit, associated with differences in temperature between a system and its surroundings or between parts of the same system. Measures of Heat
. Click the link for more information. kinetic-molecular theory of gases kinetic-molecular theory of gases,
physical theory that explains the behavior of gases on the basis of the following assumptions: (1) Any gas is composed of a very large number of very tiny particles called molecules (2) The molecules are very far apart compared to their sizes,
. Click the link for more information. ). According to Graham's law (for Thomas Graham), the rate at which a gas diffuses is inversely proportional to the square root of the density of the gas.
Diffusion often masks gravitational effects. For example, if a relatively dense gas (such as CO2, carbon dioxide) is introduced at the bottom of a vessel containing a less dense gas (such as H2, hydrogen gas), the dense gas will diffuse upward and the less dense gas will diffuse downward. It is true, however, that at equilibrium the two gases will not be uniformly mixed. There will be some variation in the density and composition of the gas mixture at the top of the vessel the gas mixture will be slightly less concentrated, and there will be a slight preponderance of molecules of the less dense gas. These differences, which are due to gravity, are almost impossible to measure in the laboratory, although they interact with other factors in determining the distribution of gases in planetary atmosphere.
Diffusion is not confined to gases it can take place with matter in any state. For example, salt diffuses (dissolves) into water water diffuses (evaporates) into the air. It is even possible for a solid to diffuse into another solid e.g., gold will diffuse into lead, although at room temperature this diffusion is very slow. Generally, gases diffuse much faster than liquids, and liquids much faster than solids. Diffusion may take place through a semipermeable membrane, which allows some, but not all, substances to pass. In solutions, when the liquid solvent passes through the membrane but the solute (dissolved solid) is retained, the process is called osmosis osmosis
, transfer of a liquid solvent through a semipermeable membrane that does not allow dissolved solids (solutes) to pass. Osmosis refers only to transfer of solvent transfer of solute is called dialysis.
. Click the link for more information. . Diffusion of a solute across a membrane is called dialysis dialysis
, in chemistry, transfer of solute (dissolved solids) across a semipermeable membrane. Strictly speaking, dialysis refers only to the transfer of the solute transfer of the solvent is called osmosis.
. Click the link for more information. , especially when some solutes pass and others are retained.
A great deal of confusion exists in the way the clinicians and radiologists refer to diffusion restriction, with both groups often appearing to not actually understand what they are referring to.
The first problem is that the term "diffusion-weighted imaging" is used to denote a number of different things:
- isotropic diffusion map (what most radiologists and clinicians will refer to as DWI)
- the pulse sequence that results in the generation of the various images (e.g. isotropic map, b=0, ADC)
- a more general term to encompass all diffusion techniques including diffusion tensor imaging
Additionally, confusion also exists in how to refer to abnormal restricted diffusion. This largely stems from the initial popularisation of DWI in stroke, which presented infarcted tissue as high signal on isotropic maps and described it merely as "restricted diffusion", implying that the rest of the brain did not demonstrate restricted diffusion, which is clearly not true. Unfortunately, this shorthand is appealing and is more widespread than using the more accurate but clumsier "diffusion demonstrates greater restriction than one would expect for this tissue."
To make matters worse, many are not aware of the concept of T2 shine-through, a cause of artifactual high signal on isotropic maps, or interpret it as a binary feature with T2 contribution to signal either present or absent when in reality there is always a T2 component even to regions with true T2 diffusion restriction.
A much safer and more accurate way of referring to diffusion restriction is to remember that we are referring to actual apparent diffusion coefficient (ADC) values, and to use wording such as "the region demonstrates abnormally low ADC values (abnormal diffusion restriction)" or even "high signal on isotropic images (DWI) is confirmed by ADC maps to represent abnormal restricted diffusion".
As opposed to essentially free diffusion of water kept inside a container, diffusion of water inside brain tissue, for example, is hindered primarily by cell membrane boundaries. The overall diffusion characteristics of a single volume represent the combined water diffusion in a number of compartments:
- diffusion within the intracellular fluid
- within the cytoplasm generally
- within organelles
- interstitial fluid
- various biological cavities, e.g. ventricles of the brain
The contribution of each one of these will depend on the tissue and pathology. For example, in acute cerebral infarction it is believed that the decrease in ADC values is the result of a combination of water moving into the intracellular compartment (where its diffusion is more impeded by organelles than it is in the extracellular space) and the resulting cellular swelling narrowing the extracellular space 6 . Similar mechanisms result in low ADC values in highly cellular tumors (e.g. small round blue cell tumors (e.g. lymphoma/PNET) and high grade gliomas (GBM)).
The further an individual water molecule diffuses during the sequence the more it will be exposed to varying gradient strength and the more it will be dephased reducing the amount of signal returned. This occurs at a much smaller scale than a single voxel. The strength of this effect (in other words how much the signal will be attenuated by diffusion) is determined by the b value.
Diffusion-weighted imaging has a major role in the following clinical situations 3-5 :
- early identification of ischemic stroke
- differentiation of acute from chronic stroke
- differentiation of acute stroke from other stroke mimics
- differentiation of epidermoid cyst from an arachnoid cyst
- differentiation of abscess from necrotic tumors
- assessment of cortical lesions in Creutzfeldt-Jakob disease (CJD)
- differentiation of herpes encephalitis from diffuse temporal gliomas
- assessment of the extent of diffuse axonal injury
- grading of diffuse gliomas and meningiomas
- assessment of active demyelination
- grading of prostate lesions (see PIRADS)
- differentiation between cholesteatoma and otitis media 9
A variety of techniques for generating diffusion maps have been developed. By far the most commonly used technique relies on a spin-echo echo-planar sequence (SE-EPI) although non-EPI techniques (e.g. turbo spin-echo) are also available and are of use particularly where tissue is adjacent or within bone where T2* effects cause artifact, distortion and signal loss on EPI sequences 7,8 .
General principle of diffusion-weighted imaging
The fundamental idea behind diffusion-weighted imaging is the attenuation of T2* signal based on how easily water molecules are able to diffuse in that region. The more easily water can diffuse (i.e. the further a water molecule can move around during the sequence) the less initial T2* signal will remain. For example, water within cerebrospinal fluid (CSF) can diffuse very easily, so very little signal remains and the ventricles appear black. In contrast, water within brain parenchyma cannot move as easily due to cell membranes getting in the way and therefore the initial T2* signal of the brain is only somewhat attenuated. An important consequence of this is that if a region of the brain has zero T2* signal it cannot, regardless of the diffusion characteristics of that tissue, show signal on isotropic diffusion-weighted images.
The way in which diffusion information is extracted from the tissue is to first obtain a T2* weighted image with no diffusion attenuation. This is known as the b=0 image.
Next, the ease with which water can diffuse is assessed in various directions the minimum is 3 orthogonal directions (X, Y and Z) and we will use this for the rest of this explanation.
This is done by applying a strong gradient symmetrically on either side of the 180-degree pulse. The degree of diffusion weighting is dependent primarily on the area under the diffusion gradients (which is in turn related to the amplitude and duration of the gradient) and on the interval between the gradients. The combination of these factors generates the b value. The higher the number the more pronounced the diffusion-related signal attenuation.
Stationary water molecules acquire phase information by the application of the first gradient. After the 180-degree pulse, however, they are exposed to the exact same gradient (because they have not changed location) which undoes all the effects of the first (since they have flipped 180-degrees). Hence at the time the echo is generated they have retained their signal.
Moving water molecules, on the other hand, acquire phase information by the first gradient but as they are moving when they are exposed to the second gradient they are not in the same location and thus are not exposed to precisely the same gradient after the 180-degree pulse. Hence they are not rephased and they lose some of their signal. The further they are able to move the less successfully they will be rephased and the less signal will remain.
Generating isotropic DWI and ADC maps
The aforementioned process generates four sets of images: a T2* b=0 image and three diffusion-weighted images (one for each X, Y and Z direction) with the T2* signal attenuated according to how easily water can diffuse in that direction.
These images can then be combined arithmetically to generate maps that are devoid of directional information (isotropic): isotropic diffusion-weighted images (what we usually refer to as DWI) and ADC maps.
To generate the isotropic DWI maps, the geometric mean of the direction-specific images is calculated.
The ADC map, in contrast, is related to the natural logarithm (ln) of the isotropic DWI divided by the initial T2* signal (b=0). These can either be calculated directly from the isotropic DWI images or by finding the arithmetic mean of ADC values generated from each directional diffusion map.
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The time-dependent dMRI signal measured in vivo in brain WM provides a signature for along-axon caliber variation. The specificity to this microstructural feature is determined here from a characteristic power-law decay of the diffusivity and validated by performing realistic MC simulations of diffusion inside axons from 3d EM images of mouse brain. In particular, our simulation results are consistent with in vivo measurements and the corresponding theoretical prediction that diffusion along axons is characterized by short-range disorder in 1d, with the dynamical exponent ϑ = 1/2 for Eq. (1). This short-range disorder was confirmed by the power spectrum analysis of the actual shape of segmented myelinated axons in the 3d EM sample of mouse brain in this study, and it was also observed in a preliminary study 32 performing MC simulations within realistic axons segmented from a large 3d EM sample of human subcortical WM.
Furthermore, simulations in different microgeometries based on this EM sample allow us to disentangle the contributions of different microstructural features to the overall 1d structural disorder and reveal that the diffusivity and kurtosis time dependence along axons is dominated by caliber variations rather than axonal undulations. For example, in Supplementary Information, simulations of diffusion in fiber bundles composed of fibers without caliber variations, such as undulation-only fibers (geometry IV in Fig. 2a) or perfectly straight cylinders, demonstrate very small axial diffusivity time dependence along the main direction, even for highly dispersed case (Supplementary Fig. 3). Similarly, mitochondria have negligible impact on the time dependence, due to their low volume fraction (Supplementary Fig. 1). Yet, mitochondria are shown to correlate with axon caliber (Fig. 6) and hence could indirectly impact the time dependence, as discussed below for the MS pilot study.
a Approximately 1300 mitochondria (red) in the 227 axons (gray) of Fig. 1a, b were manually segmented. b The inner axonal diameter 2r approximately correlates with each axon’s mitochondrial volume per unit length (Vmito/L) via a quadratic function. c Axonal diameters in cross-sections where mitochondria are present (red) are significantly larger, compared to cross-sections where mitochondria are absent (blue) (P-value < 0.001).
Using PGSE dMRI in vivo in human brain, the power-law scaling (Eq. (1)) was found in all WM ROIs (Fig. 3a, b). In particular, in genu, the fit parameters of PGSE measurements (Table 3) are of the same order as those in the MC simulations for dispersion angles θ = 15°–30° (Table 2), which is consistent with the fiber dispersion ≃ 20° observed in histology 23,33 .
The fitted power-law parameters show similar patterns over different WM regions between subjects (Fig. 3a, b), and especially in CC composed of highly aligned axons, demonstrating the potential of clinical applications in the future, as discussed later. Admittedly, the regional variations in the bulk diffusivity D∞ and strength c of restrictions are noisy for individual subjects however, we are still able to observe the general trends across the CC for the average over all subjects: the trends relate remarkably well to the pattern of axonal density in CC observed in histology 34 and of axonal volume fraction in CC estimated via dMRI 17,35 , as well as the higher spectrum of large axon diameters in the midbody according to ref. 34 . On the one hand, the high–low–high trend in D∞ in CC could be related to the pattern of axonal density in CC observed in histology, with the assumption that the axial diffusivity in IAS is larger than that in extra-axonal space 28 . On the other hand, the low–high–low trend in c in CC could be related with the bead width and/or distance between local caliber maxima along individual axons. This observation cannot be supported or rejected by 2d histology and remains incompletely explained. For example, fiber bundles composed of (1) caliber-varying axons or (2) perfectly straight cylinders can have exactly the same 2d cross-sectional diameter distribution (Supplementary Fig. 3). Three-dimensional histology and analysis in different regions of CC are needed in the future to better understand our empirical observation of the trends across the CC.
In addition to in vivo PGSE measurements in human brain WM reported here, the power-law dependence has also been reported using STE measurements in vivo in human brain WM 21 and ex vivo in the spinal cord WM 36 , where the diffusion time is varied by changing the mixing time. Both studies reported somewhat stronger time dependence, as manifested by larger amplitude c for the time dependence (cf. Table 2 of ref. 21 and Table 1 of ref. 36 as compared to current study Tables 2 and 3), a potential overestimation caused by water exchange between intra-/extra-axonal water (fast diffusion, long T1, T2 values) and myelin water (slow diffusion, short T1, T2 values) during the mixing time of the STE sequence 37 . Furthermore, in gray matter, the power-law dependence has been observed for the mean diffusivity using oscillating gradients in human brain 38 and rat brain 15,39 , suggesting that the characteristics of short-range disordered restrictions to diffusion along axons and dendrites are a universal feature of neuronal tissue.
Conventionally, diffusion in WM has been modeled using the featureless stick model (reviewed by ref. 4 ), thereby assuming Gaussian diffusion, corresponding to a negligible axial intra-axonal kurtosis. Here, however, based on realistic simulations, combined with theory and experimental verification, we conclude that the intra-axonal axial kurtosis is non-negligible at clinical diffusion times. Indeed, for t = 20–100 ms, the intra-axonal kurtosis along axons is
0.7 for θ = 30° based on simulations (Fig. 2f), and
0.8 for monopolar PGSE measurements in the human brain WM (Fig. 3c, d). The measured kurtosis in experiment is slightly larger than the intra-axonal kurtosis in simulations, likely due to the additional contribution of the extra-axonal space to the overall K(t) in the measurement (Eq. (10b) in “Methods”).
Simulations also demonstrated that the intra-axonal K(t) increases with dispersion angle, especially for θ ≳ 30° (Fig. 2f), which can be understood by the corresponding increasing range of intra-axonal diffusivity values when projected to the fiber bundle’s main direction, resulting in a larger contribution to the overall K(t), i.e., the first right-hand-side term in Eq. (10b). Hence, the higher-order cumulants of the intra-axonal signal, including K, are very sensitive to the fiber dispersion (i.e., the functional form and the degree of orientation distribution) and should be incorporated in future biophysical models of dMRI in WM.
Besides the nominal (nonzero) value of the axial kurtosis, the observed time dependence of both D(t) and K(t) are nontrivial and should be considered in WM biophysical modeling. For D(t) time dependence along axons, proportional to (Delta (1/sqrt
) sim Delta tcdot ^<-3/2>) , it is negligible only when the time range Δt is small (e.g., Δt < 5 ms), or the diffusion time is long (e.g., t > 200 ms). For K(t), our simulations in IAS show 7% changes over the clinical time range t = 20–100 ms.
The observation of axon caliber variation and beading with non-invasive time-dependent dMRI calls for evaluating the role of this microstructural feature in pathology. In this work, we demonstrated altered diffusion time dependence along axons in WM lesions vs. NAWM of five MS patients (Fig. 5), with corresponding changes in the fit parameters that are potentially related to specific pathological changes. In particular, the increase in the bulk diffusivity D∞ along axons in MS lesions vs. NAWM (Fig. 5b) may suggest ongoing demyelination and axonal loss 40 . Although this observation alone has been reported before with dMRI 41,42,43 , our results reveal, for the first time to our knowledge, that the diffusivity time dependence along WM axons, i.e., c in Eq. (1), is smaller in MS lesions than that in NAWM, with c ∝ the correlation length lc. 21 This observation is potentially indicative of an increase in mitochondria density, a feature of chronic demyelination documented from histology in axons and astrocytes of WM lesions 44 . As mitochondria and axon caliber are shown to correlate (Fig. 6c) 45 , an increase in mitochondria would shorten the correlation length lc that characterizes the distance between local maxima of the axon. Hence, the parameter c potentially targets the specific pathology of mitochondria increase in MS. However, the exact relation of c and restriction properties (e.g., their width and distance between them) is non-trivial and requires further explorations, for example, by solving the Fick-Jacobs equation for a caliber-varing fiber with randomly distributed beads along the fiber.
Conventional MRI methods (e.g., T2-FLAIR) are well-known to distinguish MS lesions from NAWM and the distinction is typically attributed to demyelination 44 . Here, however, we aim to use MS lesion data to in vivo validate the strength c of restrictions in Eq. (1) as a specific measure for changes in mitochondria: we demonstrate significant difference in c between MS lesions and NAWM (Fig. 5c), and attribute it to an increase in mitochondria as a response to demyelination in MS lesions 44 . This observation may contribute to understanding the underlying pathological mechanisms taking place in MS lesion formation. In addition, our finding also suggests diffusion time dependence measurements as potential biomarker suitable for monitoring other pathologies presenting increased neurite beadings due to other mechanisms (rather than mitochondrial increase).
In addition to MS 46 , axonal beading in WM has been observed in several other pathologies, such as traumatic brain injury (TBI) 47,48 and ischemic stroke 49 . Axonal varicosities, or axonal beading along axons, can be a pathological change caused by accumulation of transported materials in axonal swellings after TBI 47,48 it has been observed that varicosities arise during dynamic stretch injury, caused by microtubule breakdown and partial transport interruption along axons. Furthermore, varicosities due to ischemic injury to WM axons can be caused by Na + loading of the axoplasm, which leads to a lethal Ca + overload through reversed Na + -Ca + exchange 49 . Hence, the average distance between varicosities is potentially a biomarker for axonal injury in TBI and ischemia, facilitating evaluation of the effectiveness of treatment and rehabilitation services. As the average distance between varicosities along axons is of the order of 10 μm 47,48,49 , much smaller than the resolution of most of the clinical imaging techniques, dMRI is the method of choice to estimate in vivo the pathological change of TBI 50,51 and of ischemic stroke 52 . In particular, time-dependent diffusion tensor imaging may enable the estimation of the correlation length of varicosities along axons, related to the average distance between varicosities, a potential biomarker for monitoring TBI and ischemic stroke patients.
Besides beading in WM, the ubiquitous (1/sqrt
) time dependence along neurites in gray matter 15,39,53 suggests possible applications in other neurodegenerative diseases. For instance, reduced density of axonal varicosities was observed in the human superior frontal cortex of mild to moderate Alzheimer disease 54 decreased dendritic spine density was observed in the human prefrontal cortex of Schizophrenia 55 and an increased density of axonal varicosities was observed in injured dopaminergic neurons in the rat substantia nigra, an animal model of Parkinson’s disease 56 . The ability to evaluate restriction changes along neurites opens a door to monitoring the progression and therapy response of these diseases.
Thanks to recent advances in 3d EM 57 , our work, for the first time to our knowledge, demonstrate the feasibility to employ EM-derived microstructure as numerical phantoms for realistic 3d simulations. By fully controlling the microgeometry of numerical phantoms, MC simulations provide complete flexibility to evaluate the influence of different microstructural features. Here, they were employed to elucidate that the time-dependent diffusion signal along axons mainly originates from the caliber variations, with the contributions from mitochondria and axonal undulations having relatively small effects.
While we demonstrate here the value of realistic simulations as a validation tool, one can think of extending this approach further to study the sensitivity of MRI to microstructure. Larger EM samples 58 would be needed to enable diffusion simulations at longer diffusion times. For the EM sample used in the current study, the maximal axon length L
18 μm corresponds to a length-related correlation time τL = L 2 /(2D0) ≃ 80 ms for D0
2 μm 2 /ms, which sets the maximum feasible diffusion time for the simulation. This maximum time
τL also prevents us from validating the relation of power-law scaling in D(t) and K(t), i.e., ΔD(t)/D∞ = ΔK(t)/2 68 , since the power-law scaling in K(t) happens at longer diffusion times. Furthermore, we only focused on the intra-axonal geometry of myelinated axons in WM. Although the contribution of extra-axonal space is non-negligible, extra-axonal signals are relatively smaller than the intra-axonal ones because of the shorter T2 in extra-axonal space and long echo time applied in experiments 28 . For the diffusivity time dependence along the fiber bundle, we expect that the diffusivity time dependence in extra-axonal space is similar to that in IAS, as water molecules experience similar beading arrangement in either intra- or extra-axonal spaces. Faithfully segmenting and simulating the diffusion in the extra-axonal space is needed to understand how robust the observed power-law is with increasing dispersion. In addition, other structures, such as unmyelinated axons, glia cell, and blood vessels, may have nontrivial contributions to the (time-dependent) diffusion signal and can be added to the numerical microgeometry. Ultimately, a large human EM sample 32 (comparable to MRI voxel size), prepared with extra-cellular space preserving technique if possible, would provide the most representative numerical phantom to the human tissue microstructure after fully segmenting all the cells inside the sample.
Finally, although the proposed framework here focuses on performing MC to model diffusion in realistic WM microstructure, it can also be applied to gray matter, or tissue samples with pathology. In addition, the framework can be extended to include other MR contrast mechanisms, e.g., magnetization transfer, mesoscopic susceptibility 59 , T1 and T2 relaxation 60 , and water exchange 61 , thereby facilitating the exciting ability to validate non-invasive MR-based tissue microstructural mapping.
Diffusion MRI: what water tells us about the brain
Diffusion MRI has been used worldwide to produce images of brain tissue structure and connectivity, in the normal and diseased brain. Diffusion MRI has revolutionized the management of acute brain ischemia (stroke), saving life of many patients and sparing them significant disabilities. In addition to stroke, diffusion MRI is now widely used for the detection of cancers and metastases (breast, prostate, liver). Another major field of application of diffusion MRI regards the wiring of the brain. Diffusion MRI is now used to map the circuitry of the human brain with incredible accuracy, opening up new lines of inquiry for human neuroscience and for the understanding of brain illnesses or mental disorders. Here, as a pioneer of the field, I provide a personal account on the historical development of these concepts over the last 30 years.
Among the sensational 1905 Albert Einstein papers, there is one that unexpectedly gave birth to a powerful method to explore the brain. Einstein explained molecular diffusion on the basis of the random translational motion of molecules, which results from their thermal energy (Einstein, 1905 ). Moving fast forward, in the mid 1980s, I was able to show that water diffusion could be imaged in the human brain through magnetic resonance imaging (MRI). This move was triggered by the idea that water diffusion could provide unique information on the functional architecture of tissues, since during their random displacements water molecules probe tissue structure at a microscopic scale (Fig 1). What I did not expect was that this pioneering work would end up some day interesting the readers of a major molecular medicine journal. Sure, this story is about an apparently simple molecule, water. However, although water is an essential molecule for life, its importance in biology has perhaps been often overlooked, if not forgotten. Water diffusion MRI has proven to be extraordinarily popular. Its main clinical domain of application (Fig 2) has been neurological disease, especially in the management of patients with acute brain ischemia. With its unmatched sensitivity, water diffusion MRI provides patients with the opportunity to receive suitable thrombolytic treatment at a stage when brain tissue might still be salvageable, thus avoiding them terrible consequences. On the other hand, water diffusion turned out to be anisotropic in brain white matter, because axon membranes limit molecular movement perpendicularly to the axonal fibers. This feature can be exploited to produce stunning maps of the organization in space of white matter bundles and brain connections in just a few minutes, as well as to provide information on white matter integrity. Diffusion MRI has been also used in a full-body setting for the detection and treatment monitoring of cancer lesions and metastases (liver, breast, prostate), because water diffusion slows down in malignant tissues in relation to the cell proliferation. The versatility and the potential of diffusion MRI, both for research and clinical applications, have been reviewed elsewhere (Le Bihan, 2003 Le Bihan & Johansen-Berg, 2012 ). Here, I will provide a more personal account on the historical development of these concepts and how they inspired my research in the last 30 years.
Figure 1. Principles of diffusion MRI
Figure 2. Main applications of water diffusion MRI
The birth of water diffusion MRI
Back in 1984 (when I was a radiology resident), a colleague came to me with a challenge: How could one differentiate liver tumors from angiomas? I had some fuzzy intuition that, perhaps, water molecular diffusion measurements would result in lower values in solid tumors because of steric hindrance to water molecular movement compared to flowing blood. Based on the pioneering work of physicists such as Stejskal and Tanner in the 1960s, I knew that specific encoding of the diffusion process could be achieved using magnetic field gradients, but the problem lied in the integration of such gradients with those used in the MRI scanner to generate images. The problem was not trivial and indeed many colleagues at that time thought that diffusion MRI was not feasible. The idea was to localize the diffusion measurements, that is, to map water diffusion coefficients in tissues this had never been done before, especially in vivo. As a junior student in physics and medicine, I was very excited about this potential, and in a matter of weeks, diffusion MRI as we know it was conceived and implemented. My first diffusion images were obtained on a 0.5-tesla MRI scanner from the late CGR (Companie Générale de Radiologie, Buc, France). The first trials on liver were very disappointing, as the images were hampered by huge motion artifacts from respiration. This was due to the fact that the hardware to generate the magnetic gradients was sub-powered and also that the imaging methods at the time were slow and consequently very sensitive to patient motion. I thus decided to switch to the brain, as this was my original background and I must confess that I started scanning my own before actually moving to patients! It worked beautifully and that move established the great neurological potential for diffusion MRI the rest, as the saying goes, is history.
The world's first diffusion images of the brain were made public in August 1985 at the Society of Magnetic Resonance in Medicine (SMRM) meeting in London. My first diffusion MRI paper appeared in 1985 in the journal of the French Academy of Sciences (Le Bihan & Breton, 1985 ). This paper described all the necessary ingredients to successfully perform diffusion MRI. The paper did not receive much attention, possibly because it was written in French. My next paper in Radiology (Le Bihan et al, 1986 ) was much better received, with more than 2000 citations up to now (3 rd most cited paper of all times for this journal). However, because of the sensitivity of diffusion MRI to motion artifacts, many colleagues remained sceptical at the time. It was discouraging, but I kept pressing on and it paid off, as diffusion MRI progressively gained momentum. Ironically, some of my early detractors started working full time on diffusion MRI. At the same time, it was becoming clear that the results of diffusion measurements with MRI in tissues would largely differ from those obtained for water in a glass where water diffuses freely, and I therefore introduced the apparent diffusion coefficient (ADC) concept to describe diffusion MRI findings (Le Bihan et al, 1986 ).
Acute brain ischemia
Shortly thereafter, Michael Moseley at UCSF made an unexpected but crucial discovery in an acute cat brain ischemia model (Moseley et al, 1990 ): Water diffusion dropped significantly (30–50%) during the very early phase of acute brain ischemia. This finding tremendously boosted diffusion MRI, then still essentially a pure research tool, by attracting clinicians and convincing manufacturers to improve their systems. This move to the clinical field became possible after my encounter with Robert Turner at the National Institutes of Health (NIH) in Bethesda, where I had just moved. Dr Turner was an expert on EPI (Echo-Planar Imaging) MRI. With this technique, MRI images could be acquired as a snapshot in just a fraction of a second, virtually freezing patient motion: We were thus able to obtain the first “clean” diffusion images using EPI (Turner et al, 1990 ). This endeavor was not that simple, however, as we had to obtain gradient coils and power supplies with exceptional performances for the time, but we finally managed to obtain diffusion images in a matter of seconds (instead of minutes) and motion artifacts became history. The setup was also installed at Harvard Medical School in Boston where the first patients suffering from acute stroke (within 3–6 h of onset) were scanned with whole brain diffusion MRI in just a few seconds. The results immediately and directly confirmed Dr. Moseley's observations on cat brains: Water diffusion was found to be decreased in the infarcted areas, where dying brain cells undergo swelling through cytotoxic edema, clearly highlighting those areas as bright spots, while most of the time, standard MRI images (as well as CT scans) would not show any clear sign of abnormality (Warach et al, 1992 ). Around that time a drug company was developing an intravenous recombinant tissue plasminogen activator (rt-PA) drug aimed at thrombolytic therapy for acute stroke patients. Clearly, diffusion MRI could not have arrived at a better time, and this coincidental match became a milestone in the history of the management of acute stroke patients.
The wirings of the brain
Michael Moseley's group had made another great discovery: In the diffusion images, contrast seemed to change according to the spatial direction of the diffusion measurement in white matter (spinal cord and brain). Water diffusion in white matter fibers was anisotropic, faster in the direction of the fibers and slower perpendicularly to them (and no one appeared aware of it until that time). With Philippe Douek, a French student working with me at the NIH, we suggested that this feature could be used to determine and map the orientation of white matter fibers in the brain, assuming the direction of the fibers was parallel to the direction with the fastest diffusion. The first attempt was very crude, with diffusion being measured along two directions only, but the concept of white matter fiber orientation color mapping was born and a proof of principle provided (Douek et al, 1991 ). Progress from those basic images to the gorgeous fiber tract 3D displays that now make up the covers of journals and anatomy textbooks, implied a big step, which was made possible by my encounter with Peter Basser in 1990. Peter was working at NIH on ionic fluxes in tissues. Peter quickly came up with the view that a better way to deal with anisotropic diffusion was to switch to a tensor formalism to properly determine the true direction of the highest diffusivity. The problem was to determine each of the terms of the diffusion tensor with diffusion MRI. After some brainstorming, Peter and I devised a solution in 1992, which we published and patented under the name of diffusion tensor imaging (DTI Basser et al, 1994 ). Experiments were first conducted on vegetables with fibers, and soon after white matter fiber orientation could be obtained on a pixel-by-pixel basis within the whole brain in vivo, completely non-invasively and in just a few minutes. Algorithms were developed at the end of the 1990s to connect those pixels together, resulting in the world's first 3D representations of the fiber bundles (with very colorful representations of the “white” matter) within the human brain.
Water, the forgotten biological molecule
Once back in France a few years later, my priority was to understand the basic mechanisms of water diffusion in biological tissues, which underlie what we visualize with diffusion MRI. This is a huge and certainly not a simple endeavor. We still cannot clearly explain why diffusion decreases so much in brain acute ischemia, how cell swelling leads to decreased diffusion, or precisely why diffusion anisotropy occurs in white matter. Some relevant and often sophisticated models have been proposed, but we always manage to find some piece of experimental evidence that questions those models. I have recently reviewed these inconsistencies and suggested that, beside mechanical or geometric constraints (such as cell membranes) hindering water diffusion, the physical structure of water networks in tissues, especially close to membranes, might play a role (Le Bihan & Johansen-Berg, 2012 ). The presence (or the amount) of structured water in cells is in itself a subject of great controversy among physicists and biologists, and we should be prepared for yet more exciting years of brainstorming and great workshops. I believe we have largely underestimated the importance of water in biology, from protein and membrane dynamics to cell physiology. This emerging research area should also greatly benefit from diffusion MRI.
Imaging brain function
Another important possible application of diffusion MRI is the measurement of brain activity. Functional neuroimaging has become an essential means to study the brain and the mind. Thus far, positron emission tomography (PET) and current functional MRI (based on the Blood Oxygen Level Dependant or BOLD contrast mechanism) have relied on the principle that neuronal activation and blood flow are coupled through metabolism, and brain activation can be indirectly imaged through variations in local blood flow. With diffusion MRI, a new paradigm has emerged whereby we can look at brain activity through the observation of water molecular diffusion. In collaboration with my colleagues from Kyoto University, we have shown that water diffusion is indeed modulated by brain activity (Le Bihan et al, 2006 ). The diffusion signal response is characterized by a sharp peak, faster than the indirect hemodynamic response (increased in blood flow) observed with BOLD functional MRI imaging (Aso et al, 2009 ). The diffusion response persists after inhibition of neurovascular coupling (which suppresses the BOLD fMRI response) and shares the features of the underlying neuronal response (Tsurugizawa et al, 2013 ). This “diffusion fMRI” approach thus appears to be a paradigm shift in the way we visualize brain activity, and more directly linked to neuronal function, pointing out to changes in the neural tissue microstructure to which diffusion MRI is exquisitely sensitive. Such activation-driven structural events, for example cell swelling, have been reported in many instances in the literature. As cell swelling in tissues has been shown to result in a water diffusion decrease detectable with MRI, we have hypothesized that the slowdown in diffusion observed during neuronal activation could reflect cell swelling (probably at the dendrite and spine levels) occurring within the activated cortical ribbon. Based on this “electromechanical coupling,” the thought occurred to me that neural cells could perhaps be seen as piezoelectric sensors: Variations in cell shape should, in return, induce cell depolarization, potentially allowing a very fast, non-synaptic transmission mechanism within neural clusters of the cortical ribbon. Indeed, dynamic changes in neuronal structure (especially the dendritic spines) are now thought to play an important role in the functioning of such cell clusters, as envisioned by Crick (Crick, 1982 ) and even Ramon y Cajal (Ramon y Cajal, 1899 ): “The state of activity would correspond to the swelling and elongation of the [dendritic] spines, and the resting state (sleep or inactivity) to their retraction.” Diffusion MRI has the potential to address such questions and might allow us to further our understanding of the biophysical mechanisms associated with neuronal activation.
Perspectives and conclusions
In summary, diffusion MRI has the potential to provide, non-invasively and in vivo, information on the cellular organization of the brain cortex, the connections between regions and the underlying activity. Still, the exact mechanisms governing water diffusion processes in tissues, notably in the brain, remain unclear. Future research should aim at gathering diffusion data at the individual neuron and neuron cluster scale to better understand water diffusion behavior in neural tissues. Indeed, a more profound understanding of such processes is necessary to develop the application of diffusion MRI further and to directly obtain information on tissue microstructure. Access to such micro- and mesoscopic scales will benefit from the unique ultra-high magnetic field MRI systems we have assembled in our institute dedicated to ultra-high field MRI (NeuroSpin, Saclay, France), in particular its preclinical 17.2-T MRI system and an experimental clinical 11.7-T MRI system (available from 2015), both unique in the world. Using magnetic resonance microscopy (MRM), we have shown that water diffusion measured inside isolated neuronal soma and in the region of cell bodies of the Aplysia buccal ganglia under exposure to ouabain, resulted in a water diffusion increase inside isolated neurons, but a decrease at the tissue level (Jelescu, 2014 ). Such opposite findings cannot be explained with current “mechanistic” tissue diffusion models. The scenario involving a layer of water molecules bound to the inflating cell membrane surface could conciliate this apparent discrepancy.
The potential of diffusion MRI to probe human brain connectivity has attracted worldwide interest and is now widely used in clinical practice. Recent results from the European FP7 CONNECT project (Assaf et al, 2013 ) and the Human Connectome Project (HCP, USA) have clearly underlined the enormous potential of this approach, yielding insight into how brain connections underlie function and opening up new lines of inquiry for human neuroscience and brain dysfunction in aging, mental health disorders, addiction and neurological disease (Le Bihan & Johansen-Berg, 2012 ). The increased spatial resolution expected with the NeuroSpin 11.7-T MRI scanner could allow for the detection of smaller white matter bundles. Of special interest are the short connections between adjacent cortical regions, or even within cortical regions. Similarly, there are also hints that diffusion imaging could become a very important tool to probe the functional architecture of the brain cortex. It has been long established that cortical cells are not organized along the brain cortex in a random, homogeneous way. Cells are rather well characterized (in terms of size, geometry, receptor type and density) and arranged in specific patterns, identified by Brodmann in 1908. The Brodmann areas are deemed to be associated with specific brain functions, and most functional neuroimaging studies have relied on the seminal classification of Brodmann to report the location of activated regions. With diffusion MRI, one may envision that cytoarchitectonic areas could be determined on an individual basis. Such specific cellular arrangements in space along the brain cortex might establish a “neural code,” a set of basic functions from which, once connected together on a timely manner, higher order functions could emerge.
While MRI is merely a means to visualize diffusion, molecular diffusion itself (of water or other molecules, metabolites, neurotransmitters) has a life of its own and remains a powerful, genuine multidisciplinary concept at our disposal to understand cell physiology and life. After all, all biological processes require molecules to interact, whether for DNA replication, RNA transcription, protein translation, protein and enzyme activity, cross-membrane transport, and so on. For molecules to interact, however, they must first meet: Diffusion appears to be the universal process through which this occurs. In a sense, diffusion rates set the speed limit for life, just as the speed of light sets the limit in the physical world. Indeed, diffusion MRI is just emerging from adolescence and has a great future.
Growing axons have a highly motile structure at the growing tip called the growth cone, which "sniffs out" the extracellular activities in the environment for signals that instruct the axon which direction to grow. These signals, called guidance cues, can be fixed in place or diffusible they can attract or repel axons. Growth cones contain receptors that recognize these guidance cues and interpret the signal into a chemotropic response. The general theoretical framework is that when a growth cone "senses" a guidance cue, the receptors activate various signaling molecules in the growth cone that eventually affect the cytoskeleton. If the growth cone senses a gradient of guidance cue, the intracellular signaling in the growth cone happens asymmetrically, so that cytoskeletal changes happen asymmetrically and the growth cone turns toward or away from the guidance cue. 
A combination of genetic and biochemical methods (see below) has led to the discovery of several important classes of axon guidance molecules and their receptors: 
- : Netrins are secreted molecules that can act to attract or repel axons by binding to their receptors, DCC and UNC5. : Secreted proteins that normally repel growth cones by engaging Robo (Roundabout) class receptors. : Ephrins are cell surface molecules that activate Eph receptors on the surface of other cells. This interaction can be attractive or repulsive. In some cases, Ephrins can also act as receptors by transducing a signal into the expressing cell, while Ephs act as the ligands. Signaling into both the Ephrin- and Eph-bearing cells is called "bi-directional signaling." : The many types of Semaphorins are primarily axonal repellents, and activate complexes of cell-surface receptors called Plexins and Neuropilins. : Integral membrane proteins mediating adhesion between growing axons and eliciting intracellular signalling within the growth cone. CAMs are the major class of proteins mediating correct axonal navigation of axons growing on axons (fasciculation). There are two CAM subgroups: IgSF-CAMs (belonging to the immunoglobulin superfamily) and Cadherins (Ca-dependent CAMs).
In addition, many other classes of extracellular molecules are used by growth cones to navigate properly:
- Developmental morphogens, such as BMPs, Wnts, Hedgehog, and FGFs
- Extracellular matrix and adhesion molecules such as laminin, tenascins, proteoglycans, N-CAM, and L1
- Growth factors like NGF
- Neurotransmitters and modulators like GABA
Integration of information in axon guidance Edit
Growing axons rely on a variety of guidance cues in deciding upon a growth pathway. The growth cones of extending axons process these cues in an intricate system of signal interpretation and integration, in order to ensure appropriate guidance.  These cues can be functionally subdivided into:
- Adhesive cues, that provide physical interaction with the substrate necessary for axon protrusion. These cues can be expressed on glial and neuronal cells the growing axon contacts or be part of the extracellular matrix. Examples are laminin or fibronectin, in the extracellular matrix, and cadherins or Ig-family cell-adhesion molecules, found on cell surfaces.
- Tropic cues, that can act as attractants or repellents and cause changes in growth cone motility by acting on the cytoskeleton through intracellular signaling. For example, Netrin plays a role in guiding axons through the midline, acting as both an attractant and a repellent, while Semaphorin3A helps axons grow from the olfactory epithelium to map different locations in the olfactory bulb.
- Modulatory cues, that influence the sensitivity of growth cones to certain guidance cues. For instance, neurotrophins can make axons less sensitive to the repellent action of Semaphorin3A.
Given the abundance of these different guidance cues it was previously believed that growth cones integrate various information by simply summing the gradient of cues, in different valences, at a given point in time, to making a decision on the direction of growth. However, studies in vertebrate nervous systems of ventral midline crossing axons, has shown that modulatory cues play a crucial part in tuning axon responses to other cues, suggesting that the process of axon guidance is nonlinear. For example, commissural axons are attracted by Netrin and repelled by Slit. However, as axons approach the midline, the repellent action of Slit is suppressed by Robo-3/Rig-1 receptor.  Once the axons cross the midline, activation of Robo by Slit silences Netrin-mediated attraction, and the axons are repelled by Slit.
Cellular strategies of nerve tract formation Edit
Pioneer axons Edit
The formation of a nerve tract follows several basic rules. In both invertebrate and vertebrate nervous systems initial nerve tracts are formed by the pioneer axons of pioneer neurons.  These axons follow a reproducible pathway, stop at intermediate targets, and branch axons at certain choice points, in the process of targeting their final destination. This principle is illustrated by CNS extending axons of sensory neurons in insects.
During the process of limb development, proximal neurons are the first to form axonal bundles while growing towards the CNS. In later stages of limb growth, axons from more distal neurons fasciculate with these pioneer axons. Deletion of pioneer neurons disrupts the extension of later axons, destined to innervate the CNS.  At the same time, it is worth noting that in most cases pioneer neurons do not contain unique characteristics and their role in axon guidance can be substituted by other neurons. For instance, in Xenopus retinotectal connection systems, the pioneer axons of retinal ganglion cells originate from the dorsal part of the eye. However, if the dorsal half of the eye is replaced by less mature dorsal part, ventral neurons can replace the pioneer pathway of the dorsal cells, after some delay.  Studies in zebrafish retina showed that inhibiting neural differentiation of early retinal progenitors prevents axons from exiting the eye. The same study demonstrated aberrant growth trajectories in secondary neurons, following the growth of pioneer neurons missing a guidance receptor.  Thus, while the extent of guidance provided by pioneer axons is under debate and may vary from system to system, the pioneer pathways clearly provide the follower projections with guidance cues and enhance their ability to navigate to target.
Role of glia Edit
The first extending axons in a pathway interact closely with immature glia cells. In the forming corpus callosum of vertebrates, primitive glia cells first migrate to the ependymal zones of hemispheres and the dorsal septum wall to form a transient structure that the pioneer axons of the callosal fibers use to extend.  The signaling between glia and neurons in the developing nervous system is reciprocal. For instance, in the fly visual system, axons of photoreceptors require glia to exit the eye stalk whereas glia cells rely on signals from neurons to migrate back along axons. 
The growing axons also rely on transient neuronal structures such as guidepost cells, during pathfinding. In the mouse visual system, proper optic chiasm formation depends on a V-shaped structure of transient neurons that intersect with specialized radial glia at the midline of the chiasm. The chiasm axons grow along and around this structure but do not invade it.  Another example is the subplate in the developing cerebral cortex that consists of transient neuronal layer under the subventricular zone and serves as a guidepost for axons entering permanent cortical layers. The subplate is similar to the chiasmatic neurons in that these cell groups disappear (or transit into other cell types) as the brain matures.  These findings indicate that transitory cell populations can serve an important guidance role even though they have no function in the mature nervous system.
The earliest descriptions of the axonal growth cone were made by the Spanish neurobiologist Santiago Ramón y Cajal in the late 19th century.  However, understanding the molecular and cellular biology of axon guidance would not begin until decades later. In the last thirty years or so, scientists have used various methods to work out how axons find their way. Much of the early work in axon guidance was done in the grasshopper, where individual motor neurons were identified and their pathways characterized. In genetic model organisms like mice, zebrafish, nematodes, and fruit flies, scientists can generate mutations and see whether and how they cause axons to make errors in navigation. In vitro experiments can be useful for direct manipulation of growing axons. A popular method is to grow neurons in culture and expose growth cones to purified guidance cues to see whether these cause the growing axons to turn. These types of experiments have often been done using traditional embryological non-genetic model organisms, such as the chicken and African clawed frog. Embryos of these species are easy to obtain and, unlike mammals, develop externally and are easily accessible to experimental manipulation.
Axon guidance model systems Edit
Several types of axon pathways have been extensively studied in model systems to further understand the mechanisms of axon guidance. Perhaps the two most prominent of these are commissures and topographic maps. Commissures are sites where axons cross the midline from one side of the nervous system to the other. Topographic maps are systems in which groups of neurons in one tissue project their axons to another tissue in an organized arrangement such that spatial relationships are maintained i.e. adjacent neurons will innervate adjacent regions of the target tissue.
Commissure formation: attraction and repulsion Edit
As described above, axonal guidance cues are often categorized as "attractive" or "repulsive." This is a simplification, as different axons will respond to a given cue differently. Furthermore, the same axonal growth cone can alter its responses to a given cue based on timing, previous experience with the same or other cues, and the context in which the cue is found. These issues are exemplified during the development of commissures. The bilateral symmetry of the nervous system means that axons will encounter the same cues on either side of the midline. Before crossing (ipsilaterally), the growth cone must navigate toward and be attracted to the midline. However, after crossing (contralaterally), the same growth cone must become repelled or lose attraction to the midline and reinterpret the environment to locate the correct target tissue.
Two experimental systems have had particularly strong impacts on understanding how midline axon guidance is regulated:
The ventral nerve cord of Drosophila Edit
The use of powerful genetic tools in Drosophila led to the identification of a key class of axon guidance cues, the Slits, and their receptors, the Robos (short for Roundabout). The ventral nerve cord looks like a ladder, with three longitudinal axon bundles (fascicles) connected by the commissures, the "rungs" of the ladder. There are two commissures, anterior and posterior, within each segment of the embryo.
The currently accepted model is that Slit, produced by midline cells, repels axons from the midline via Robo receptors. Ipsilaterally projecting (non-crossing) axons always have Robo receptors on their surface, while commissural axons have very little or no Robo on their surface, allowing them to be attracted to the midline by Netrins and, probably, other as-yet unidentified cues. After crossing, however, Robo receptors are strongly upregulated on the axon, which allows Robo-mediated repulsion to overcome attraction to the midline. This dynamic regulation of Robo is at least in part accomplished by a molecule called Comm (short for Commissureless), which prevents Robo from reaching the cell surface and targeting it for destruction. 
The spinal cord of mice and chickens Edit
In the spinal cord of vertebrates, commissural neurons from the dorsal regions project downward toward the ventral floor plate. Ipsilateral axons turn before reaching the floor plate to grow longitudinally, while commissural axons cross the midline and make their longitudinal turn on the contralateral side. Strikingly, Netrins, Slits, and Robos all play similar functional roles in this system as well. One outstanding mystery was the apparent lack of any comm gene in vertebrates. It now seems that at least some of Comm's functions are performed by a modified form of Robo called Robo3 (or Rig1).
The spinal cord system was the first to demonstrate explicitly the altered responsiveness of growth cones to cues after exposure to the midline. Explanted neurons grown in culture would respond to exogenously supplied Slit according to whether or not they had contacted floor plate tissue. 
Topographic maps: gradients for guidance Edit
As described above, topographic maps occur when spatial relationships are maintained between neuronal populations and their target fields in another tissue. This is a major feature of nervous system organization, particular in sensory systems. The neurobiologist Roger Sperry proposed a prescient model for topographic mapping mediated by what he called molecular "tags." The relative amounts of these tags would vary in gradients across both tissues. We now think of these tags as ligands (cues) and their axonal receptors. Perhaps the best understood class of tags are the Ephrin ligands and their receptors, the Ephs.
In the simplest type of mapping model, we could imagine a gradient of Eph receptor expression level in a field of neurons, such as the retina, with the anterior cells expressing very low levels and cells in the posterior expressing the highest levels of the receptor. Meanwhile, in the target of the retinal cells (the optic tectum), Ephrin ligands are organized in a similar gradient: high posterior to low anterior. Retinal axons enter the anterior tectum and proceed posteriorly. Because, in general, Eph-bearing axons are repelled by Ephrins, axons will become more and more reluctant to proceed the further they advance toward the posterior tectum. However, the degree to which they are repelled is set by their own particular level of Eph expression, which is set by the position of the neuronal cell body in the retina. Thus, axons from the anterior retina, expressing the lowest level of Ephs, can project to the posterior tectum, even though this is where Ephrins are highly expressed. Posterior retinal cells express high Eph level, and their axons will stop more anteriorly in the tectum.
The retinotectal projection of chickens, frogs and fish Edit
The large size and accessibility of the chicken embryo has made it a favorite model organism for embryologists. Researchers used the chick to biochemically purify components from the tectum that showed specific activity against retinal axons in culture. This led to the identification of Ephs and Ephrins as Sperry's hypothesized "tags."
The retinotectal projection has also been studied in Xenopus and zebrafish. Zebrafish is a potentially powerful system because genetic screens like those performed in invertebrates can be done relatively simply and cheaply. In 1996, large scale screens were conducted in zebrafish, including screens for retinal axon guidance and mapping. Many of the mutants have yet to be characterized.
Cell biology Edit
Genetics and biochemistry have identified a large set of molecules that affect axon guidance. How all of these pieces fit together is less understood. Most axon guidance receptors activate signal transduction cascades that ultimately lead to reorganization of the cytoskeleton and adhesive properties of the growth cone, which together underlie the motility of all cells. This has been well documented in mammalian cortical neurons.  However, this raises the question of how the same cues can result in a spectrum of response from different growth cones. It may be that different receptors activate attraction or repulsion in response to a single cue. Another possibility is the receptor complexes act as "coincidence detectors" to modify responses to one cue in the presence of another. Similar signaling "cross-talk" could occur intracellularly, downstream of receptors on the cell surface.
In fact, commissural axon growth responses have been shown to be attracted, repressed, or silenced in the presence of Netrin activated DCC receptor.  This variable activity is dependent on Robo or UNC-5 receptor expression at growth cones. Such that Slit activated Robo receptor, causes a silencing of Netrin’s attractive potential through the DCC receptor. While growth cones expressing UNC-5 receptor, respond in a repulsive manner to Netrin-DCC activation. These events occur as consequence of cytoplasmic interactions between the Netrin activated DCC receptor and Robo or UNC-5 receptor, which ultimately alters DCC’s cytoplasmic signaling. Thus, the picture that emerges is that growth cone advancement is highly complex and subject to plasticity from guidance cues, receptor expression, receptor interactions, and the subsequent signaling mechanisms that influence cytoskeleton remodeling.
Growth cone translation in guided axons Edit
The ability for axons to navigate and adjust responses to various extracellular cues, at long distances from the cell body, has prompted investigators to look at the intrinsic properties of growth cones. Recent studies reveal that guidance cues can influence spatiotemporal changes in axons by modulating the local translation and degradation of proteins in growth cones.  Furthermore, this activity seems to occur independent of distal nuclear gene expression. In fact, in retinal ganglion cells (RGCs) with soma severed axons, growth cones continue to track and innervate the tectum of Xenopus embryos. 
To accommodate this activity, growth cones are believed to pool mRNAs that code for receptors and intracellular signaling proteins involved in cytoskeleton remodeling.  In Xenopus retinotectal projection systems, the expression of these proteins has been shown to be influenced by guidance cues and the subsequent activation of local translation machinery. The attractive cue Netrin-1, stimulates mRNA transport and influence synthesis of β-Actin in filopodia of growth cones, to restructure and steer RGC growth cones in the direction of Netrin secretion.  While the repulsive cue, Slit, is suggested to stimulate the translation of Cofilin (an actin depolymerizing factor) in growth cones, leading to axon repulsion.  In addition, severed commissural axons in chicks, display the capability of translating and expressing Eph-A2 receptor during midline crossing.  As a result, studies suggest that local protein expression is a convenient mechanism to explain the rapid, dynamic, and autonomous nature of growth cone advancement in response to guidance molecules.
The axon growth hypothesis and the consensus connectome dynamics Edit
Contemporary diffusion-weighted MRI techniques may also uncover the macroscopical process of axonal development. The connectome, or the braingraph, can be constructed from diffusion MRI data: the vertices of the graph correspond to anatomically labelled brain areas, and two such vertices, say u and v, are connected by an edge if the tractography phase of the data processing finds an axonal fiber that connects the two areas, corresponding to u and v. Numerous braingraphs, computed from the Human Connectome Project can be downloaded from the http://braingraph.org site. The Consensus Connectome Dynamics (CCD) is a remarkable phenomenon that was discovered by continuously decreasing the minimum confidence-parameter at the graphical interface of the Budapest Reference Connectome Server.   The Budapest Reference Connectome Server depicts the cerebral connections of n=418 subjects with a frequency-parameter k: For any k=1,2. n one can view the graph of the edges that are present in at least k connectomes. If parameter k is decreased one-by-one from k=n through k=1 then more and more edges appear in the graph, since the inclusion condition is relaxed. The surprising observation is that the appearance of the edges is far from random: it resembles a growing, complex structure, like a tree or a shrub (visualized on this animation on YouTube. It is hypothesized in  that the growing structure copies the axonal development of the human brain: the earliest developing connections (axonal fibers) are common in most of the subjects, and the subsequently developing connections have larger and larger variance, because their variances are accumulated in the process of axonal development.
Axon guidance is genetically associated with other characteristics or features. For example, enrichment analyses of different signaling pathways led to the discovery of a genetic association with intracranial volume. 
In diffusion weighted imaging (DWI), the intensity of each image element (voxel) reflects the best estimate of the rate of water diffusion at that location. Because the mobility of water is driven by thermal agitation and highly dependent on its cellular environment, the hypothesis behind DWI is that findings may indicate (early) pathologic change. For instance, DWI is more sensitive to early changes after a stroke than more traditional MRI measurements such as T1 or T2 relaxation rates. A variant of diffusion weighted imaging, diffusion spectrum imaging (DSI),  was used in deriving the Connectome data sets DSI is a variant of diffusion-weighted imaging that is sensitive to intra-voxel heterogeneities in diffusion directions caused by crossing fiber tracts and thus allows more accurate mapping of axonal trajectories than other diffusion imaging approaches. 
Diffusion-weighted images are very useful to diagnose vascular strokes in the brain. It is also used more and more in the staging of non-small-cell lung cancer, where it is a serious candidate to replace positron emission tomography as the 'gold standard' for this type of disease. Diffusion tensor imaging is being developed for studying the diseases of the white matter of the brain as well as for studies of other body tissues (see below). DWI is most applicable when the tissue of interest is dominated by isotropic water movement e.g. grey matter in the cerebral cortex and major brain nuclei, or in the body—where the diffusion rate appears to be the same when measured along any axis. However, DWI also remains sensitive to T1 and T2 relaxation. To entangle diffusion and relaxation effects on image contrast, one may obtain quantitative images of the diffusion coefficient, or more exactly the apparent diffusion coefficient (ADC). The ADC concept was introduced to take into account the fact that the diffusion process is complex in biological tissues and reflects several different mechanisms. 
Diffusion tensor imaging (DTI) is important when a tissue—such as the neural axons of white matter in the brain or muscle fibers in the heart—has an internal fibrous structure analogous to the anisotropy of some crystals. Water will then diffuse more rapidly in the direction aligned with the internal structure, and more slowly as it moves perpendicular to the preferred direction. This also means that the measured rate of diffusion will differ depending on the direction from which an observer is looking.
Traditionally, in diffusion-weighted imaging (DWI), three gradient-directions are applied, sufficient to estimate the trace of the diffusion tensor or 'average diffusivity', a putative measure of edema. Clinically, trace-weighted images have proven to be very useful to diagnose vascular strokes in the brain, by early detection (within a couple of minutes) of the hypoxic edema. 
More extended DTI scans derive neural tract directional information from the data using 3D or multidimensional vector algorithms based on six or more gradient directions, sufficient to compute the diffusion tensor. The diffusion tensor model is a rather simple model of the diffusion process, assuming homogeneity and linearity of the diffusion within each image voxel.  From the diffusion tensor, diffusion anisotropy measures such as the fractional anisotropy (FA), can be computed. Moreover, the principal direction of the diffusion tensor can be used to infer the white-matter connectivity of the brain (i.e. tractography trying to see which part of the brain is connected to which other part).
Recently, more advanced models of the diffusion process have been proposed that aim to overcome the weaknesses of the diffusion tensor model. Amongst others, these include q-space imaging  and generalized diffusion tensor imaging.
Diffusion imaging is a MRI method that produces in vivo magnetic resonance images of biological tissues sensitized with the local characteristics of molecular diffusion, generally water (but other moieties can also be investigated using MR spectroscopic approaches).  MRI can be made sensitive to the motion of molecules. Regular MRI acquisition utilizes the behavior of protons in water to generate contrast between clinically relevant features of a particular subject. The versatile nature of MRI is due to this capability of producing contrast related to the structure of tissues at the microscopic level. In a typical T 1
> -weighted image, water molecules in a sample are excited with the imposition of a strong magnetic field. This causes many of the protons in water molecules to precess simultaneously, producing signals in MRI. In T 2 > -weighted images, contrast is produced by measuring the loss of coherence or synchrony between the water protons. When water is in an environment where it can freely tumble, relaxation tends to take longer. In certain clinical situations, this can generate contrast between an area of pathology and the surrounding healthy tissue.
To sensitize MRI images to diffusion, the magnetic field strength (B1) is varied linearly by a pulsed field gradient. Since precession is proportional to the magnet strength, the protons begin to precess at different rates, resulting in dispersion of the phase and signal loss. Another gradient pulse is applied in the same magnitude but with opposite direction to refocus or rephase the spins. The refocusing will not be perfect for protons that have moved during the time interval between the pulses, and the signal measured by the MRI machine is reduced. This "field gradient pulse" method was initially devised for NMR by Stejskal and Tanner  who derived the reduction in signal due to the application of the pulse gradient related to the amount of diffusion that is occurring through the following equation:
In order to localize this signal attenuation to get images of diffusion one has to combine the pulsed magnetic field gradient pulses used for MRI (aimed at localization of the signal, but those gradient pulses are too weak to produce a diffusion related attenuation) with additional "motion-probing" gradient pulses, according to the Stejskal and Tanner method. This combination is not trivial, as cross-terms arise between all gradient pulses. The equation set by Stejskal and Tanner then becomes inaccurate and the signal attenuation must be calculated, either analytically or numerically, integrating all gradient pulses present in the MRI sequence and their interactions. The result quickly becomes very complex given the many pulses present in the MRI sequence, and as a simplification, Le Bihan suggested gathering all the gradient terms in a "b factor" (which depends only on the acquisition parameters) so that the signal attenuation simply becomes: 
Although this ADC concept has been extremely successful, especially for clinical applications, it has been challenged recently, as new, more comprehensive models of diffusion in biological tissues have been introduced. Those models have been made necessary, as diffusion in tissues is not free. In this condition, the ADC seems to depend on the choice of b values (the ADC seems to decrease when using larger b values), as the plot of ln(S/So) is not linear with the b factor, as expected from the above equations. This deviation from a free diffusion behavior is what makes diffusion MRI so successful, as the ADC is very sensitive to changes in tissue microstructure. On the other hand, modeling diffusion in tissues is becoming very complex. Among most popular models are the biexponential model, which assumes the presence of 2 water pools in slow or intermediate exchange   and the cumulant-expansion (also called Kurtosis) model,    which does not necessarily require the presence of 2 pools.
Diffusion model Edit
Given the concentration ρ
and flux J , Fick's first law gives a relationship between the flux and the concentration gradient:
where D is the diffusion coefficient. Then, given conservation of mass, the continuity equation relates the time derivative of the concentration with the divergence of the flux:
Putting the two together, we get the diffusion equation:
Magnetization dynamics Edit
With no diffusion present, the change in nuclear magnetization over time is given by the classical Bloch equation
which has terms for precession, T2 relaxation, and T1 relaxation.
In 1956, H.C. Torrey mathematically showed how the Bloch equations for magnetization would change with the addition of diffusion.  Torrey modified Bloch's original description of transverse magnetization to include diffusion terms and the application of a spatially varying gradient. Since the magnetization M
is a vector, there are 3 diffusion equations, one for each dimension. The Bloch-Torrey equation is:
For the simplest case where the diffusion is isotropic the diffusion tensor is a multiple of the identity:
then the Bloch-Torrey equation will have the solution
The standard grayscale of DWI images is to represent increased diffusion restriction as brighter. 
An apparent diffusion coefficient (ADC) image, or an ADC map, is an MRI image that more specifically shows diffusion than conventional DWI, by eliminating the T2 weighting that is otherwise inherent to conventional DWI.   ADC imaging does so by acquiring multiple conventional DWI images with different amounts of DWI weighting, and the change in signal is proportional to the rate of diffusion. Contrary to DWI images, the standard grayscale of ADC images is to represent a smaller magnitude of diffusion as darker. 
Cerebral infarction leads to diffusion restriction, and the difference between images with various DWI weighting will therefore be minor, leading to an ADC image with low signal in the infarcted area.  A decreased ADC may be detected minutes after a cerebral infarction.  The high signal of infarcted tissue on conventional DWI is a result of its partial T2 weighting. 
Diffusion tensor imaging (DTI) is a magnetic resonance imaging technique that enables the measurement of the restricted diffusion of water in tissue in order to produce neural tract images instead of using this data solely for the purpose of assigning contrast or colors to pixels in a cross-sectional image. It also provides useful structural information about muscle—including heart muscle—as well as other tissues such as the prostate. 
In DTI, each voxel has one or more pairs of parameters: a rate of diffusion and a preferred direction of diffusion—described in terms of three-dimensional space—for which that parameter is valid. The properties of each voxel of a single DTI image is usually calculated by vector or tensor math from six or more different diffusion weighted acquisitions, each obtained with a different orientation of the diffusion sensitizing gradients. In some methods, hundreds of measurements—each making up a complete image—are made to generate a single resulting calculated image data set. The higher information content of a DTI voxel makes it extremely sensitive to subtle pathology in the brain. In addition the directional information can be exploited at a higher level of structure to select and follow neural tracts through the brain—a process called tractography. 
A more precise statement of the image acquisition process is that the image-intensities at each position are attenuated, depending on the strength (b-value) and direction of the so-called magnetic diffusion gradient, as well as on the local microstructure in which the water molecules diffuse. The more attenuated the image is at a given position, the greater diffusion there is in the direction of the diffusion gradient. In order to measure the tissue's complete diffusion profile, one needs to repeat the MR scans, applying different directions (and possibly strengths) of the diffusion gradient for each scan.
Mathematical foundation—tensors Edit
Diffusion MRI relies on the mathematics and physical interpretations of the geometric quantities known as tensors. Only a special case of the general mathematical notion is relevant to imaging, which is based on the concept of a symmetric matrix. [notes 1] Diffusion itself is tensorial, but in many cases the objective is not really about trying to study brain diffusion per se, but rather just trying to take advantage of diffusion anisotropy in white matter for the purpose of finding the orientation of the axons and the magnitude or degree of anisotropy. Tensors have a real, physical existence in a material or tissue so that they don't move when the coordinate system used to describe them is rotated. There are numerous different possible representations of a tensor (of rank 2), but among these, this discussion focuses on the ellipsoid because of its physical relevance to diffusion and because of its historical significance in the development of diffusion anisotropy imaging in MRI.
The following matrix displays the components of the diffusion tensor:
The same matrix of numbers can have a simultaneous second use to describe the shape and orientation of an ellipse and the same matrix of numbers can be used simultaneously in a third way for matrix mathematics to sort out eigenvectors and eigenvalues as explained below.
Physical tensors Edit
The idea of a tensor in physical science evolved from attempts to describe the quantity of physical properties. The first properties they were applied to were those that can be described by a single number, such as temperature. Properties that can be described this way are called scalars these can be considered tensors of rank 0, or 0th-order tensors. Tensors can also be used to describe quantities that have directionality, such as mechanical force. These quantities require specification of both magnitude and direction, and are often represented with a vector. A three-dimensional vector can be described with three components: its projection on the x, y, and z axes. Vectors of this sort can be considered tensors of rank 1, or 1st-order tensors.
A tensor is often a physical or biophysical property that determines the relationship between two vectors. When a force is applied to an object, movement can result. If the movement is in a single direction, the transformation can be described using a vector—a tensor of rank 1. However, in a tissue, diffusion leads to movement of water molecules along trajectories that proceed along multiple directions over time, leading to a complex projection onto the Cartesian axes. This pattern is reproducible if the same conditions and forces are applied to the same tissue in the same way. If there is an internal anisotropic organization of the tissue that constrains diffusion, then this fact will be reflected in the pattern of diffusion. The relationship between the properties of driving force that generate diffusion of the water molecules and the resulting pattern of their movement in the tissue can be described by a tensor. The collection of molecular displacements of this physical property can be described with nine components—each one associated with a pair of axes xx, yy, zz, xy, yx, xz, zx, yz, zy.  These can be written as a matrix similar to the one at the start of this section.
Diffusion from a point source in the anisotropic medium of white matter behaves in a similar fashion. The first pulse of the Stejskal Tanner diffusion gradient effectively labels some water molecules and the second pulse effectively shows their displacement due to diffusion. Each gradient direction applied measures the movement along the direction of that gradient. Six or more gradients are summed to get all the measurements needed to fill in the matrix, assuming it is symmetric above and below the diagonal (red subscripts).
In 1848, Henri Hureau de Sénarmont  applied a heated point to a polished crystal surface that had been coated with wax. In some materials that had "isotropic" structure, a ring of melt would spread across the surface in a circle. In anisotropic crystals the spread took the form of an ellipse. In three dimensions this spread is an ellipsoid. As Adolf Fick showed in the 1850s, diffusion exhibits many of the same patterns as those seen in the transfer of heat.
Mathematics of ellipsoids Edit
At this point, it is helpful to consider the mathematics of ellipsoids. An ellipsoid can be described by the formula: ax 2 + by 2 + cz 2 = 1. This equation describes a quadric surface. The relative values of a, b, and c determine if the quadric describes an ellipsoid or a hyperboloid.
As it turns out, three more components can be added as follows: ax 2 + by 2 + cz 2 + dyz + ezx + fxy = 1. Many combinations of a, b, c, d, e, and f still describe ellipsoids, but the additional components (d, e, f) describe the rotation of the ellipsoid relative to the orthogonal axes of the Cartesian coordinate system. These six variables can be represented by a matrix similar to the tensor matrix defined at the start of this section (since diffusion is symmetric, then we only need six instead of nine components—the components below the diagonal elements of the matrix are the same as the components above the diagonal). This is what is meant when it is stated that the components of a matrix of a second order tensor can be represented by an ellipsoid—if the diffusion values of the six terms of the quadric ellipsoid are placed into the matrix, this generates an ellipsoid angled off the orthogonal grid. Its shape will be more elongated if the relative anisotropy is high.
When the ellipsoid/tensor is represented by a matrix, we can apply a useful technique from standard matrix mathematics and linear algebra—that is to "diagonalize" the matrix. This has two important meanings in imaging. The idea is that there are two equivalent ellipsoids—of identical shape but with different size and orientation. The first one is the measured diffusion ellipsoid sitting at an angle determined by the axons, and the second one is perfectly aligned with the three Cartesian axes. The term "diagonalize" refers to the three components of the matrix along a diagonal from upper left to lower right (the components with red subscripts in the matrix at the start of this section). The variables ax 2 , by 2 , and cz 2 are along the diagonal (red subscripts), but the variables d, e and f are "off diagonal". It then becomes possible to do a vector processing step in which we rewrite our matrix and replace it with a new matrix multiplied by three different vectors of unit length (length=1.0). The matrix is diagonalized because the off-diagonal components are all now zero. The rotation angles required to get to this equivalent position now appear in the three vectors and can be read out as the x, y, and z components of each of them. Those three vectors are called "eigenvectors" or characteristic vectors. They contain the orientation information of the original ellipsoid. The three axes of the ellipsoid are now directly along the main orthogonal axes of the coordinate system so we can easily infer their lengths. These lengths are the eigenvalues or characteristic values.
Diagonalization of a matrix is done by finding a second matrix that it can be multiplied with followed by multiplication by the inverse of the second matrix—wherein the result is a new matrix in which three diagonal (xx, yy, zz) components have numbers in them but the off-diagonal components (xy, yz, zx) are 0. The second matrix provides eigenvector information.
Measures of anisotropy and diffusivity Edit
In present-day clinical neurology, various brain pathologies may be best detected by looking at particular measures of anisotropy and diffusivity. The underlying physical process of diffusion causes a group of water molecules to move out from a central point, and gradually reach the surface of an ellipsoid if the medium is anisotropic (it would be the surface of a sphere for an isotropic medium). The ellipsoid formalism functions also as a mathematical method of organizing tensor data. Measurement of an ellipsoid tensor further permits a retrospective analysis, to gather information about the process of diffusion in each voxel of the tissue. 
In an isotropic medium such as cerebrospinal fluid, water molecules are moving due to diffusion and they move at equal rates in all directions. By knowing the detailed effects of diffusion gradients we can generate a formula that allows us to convert the signal attenuation of an MRI voxel into a numerical measure of diffusion—the diffusion coefficient D. When various barriers and restricting factors such as cell membranes and microtubules interfere with the free diffusion, we are measuring an "apparent diffusion coefficient", or ADC, because the measurement misses all the local effects and treats the attenuation as if all the movement rates were solely due to Brownian motion. The ADC in anisotropic tissue varies depending on the direction in which it is measured. Diffusion is fast along the length of (parallel to) an axon, and slower perpendicularly across it.
Once we have measured the voxel from six or more directions and corrected for attenuations due to T2 and T1 effects, we can use information from our calculated ellipsoid tensor to describe what is happening in the voxel. If you consider an ellipsoid sitting at an angle in a Cartesian grid then you can consider the projection of that ellipse onto the three axes. The three projections can give you the ADC along each of the three axes ADCx, ADCy, ADCz. This leads to the idea of describing the average diffusivity in the voxel which will simply be
We use the i subscript to signify that this is what the isotropic diffusion coefficient would be with the effects of anisotropy averaged out.
The ellipsoid itself has a principal long axis and then two more small axes that describe its width and depth. All three of these are perpendicular to each other and cross at the center point of the ellipsoid. We call the axes in this setting eigenvectors and the measures of their lengths eigenvalues. The lengths are symbolized by the Greek letter λ. The long one pointing along the axon direction will be λ1 and the two small axes will have lengths λ2 and λ3. In the setting of the DTI tensor ellipsoid, we can consider each of these as a measure of the diffusivity along each of the three primary axes of the ellipsoid. This is a little different from the ADC since that was a projection on the axis, while λ is an actual measurement of the ellipsoid we have calculated.
The diffusivity along the principal axis, λ1 is also called the longitudinal diffusivity or the axial diffusivity or even the parallel diffusivity λ∥. Historically, this is closest to what Richards originally measured with the vector length in 1991.  The diffusivities in the two minor axes are often averaged to produce a measure of radial diffusivity
This quantity is an assessment of the degree of restriction due to membranes and other effects and proves to be a sensitive measure of degenerative pathology in some neurological conditions.  It can also be called the perpendicular diffusivity ( λ ⊥
Another commonly used measure that summarizes the total diffusivity is the Trace—which is the sum of the three eigenvalues,
If we divide this sum by three we have the mean diffusivity,
which equals ADCi since
is the matrix of eigenvectors and D is the diffusion tensor. Aside from describing the amount of diffusion, it is often important to describe the relative degree of anisotropy in a voxel. At one extreme would be the sphere of isotropic diffusion and at the other extreme would be a cigar or pencil shaped very thin prolate spheroid. The simplest measure is obtained by dividing the longest axis of the ellipsoid by the shortest = (λ1/λ3). However, this proves to be very susceptible to measurement noise, so increasingly complex measures were developed to capture the measure while minimizing the noise. An important element of these calculations is the sum of squares of the diffusivity differences = (λ1 − λ2) 2 + (λ1 − λ3) 2 + (λ2 − λ3) 2 . We use the square root of the sum of squares to obtain a sort of weighted average—dominated by the largest component. One objective is to keep the number near 0 if the voxel is spherical but near 1 if it is elongate. This leads to the fractional anisotropy or FA which is the square root of the sum of squares (SRSS) of the diffusivity differences, divided by the SRSS of the diffusivities. When the second and third axes are small relative to the principal axis, the number in the numerator is almost equal the number in the denominator. We also multiply by 1 / 2 >> so that FA has a maximum value of 1. The whole formula for FA looks like this:
The fractional anisotropy can also be separated into linear, planar, and spherical measures depending on the "shape" of the diffusion ellipsoid.   For example, a "cigar" shaped prolate ellipsoid indicates a strongly linear anisotropy, a "flying saucer" or oblate spheroid represents diffusion in a plane, and a sphere is indicative of isotropic diffusion, equal in all directions.  If the eigenvalues of the diffusion vector are sorted such that λ 1 ≥ λ 2 ≥ λ 3 ≥ 0
geq lambda _<2>geq lambda _<3>geq 0> , then the measures can be calculated as follows:
Each measure lies between 0 and 1 and they sum to unity. An additional anisotropy measure can used to describe the deviation from the spherical case:
There are other metrics of anisotropy used, including the relative anisotropy (RA):
and the volume ratio (VR):
The most common application of conventional DWI (without DTI) is in acute brain ischemia. DWI directly visualizes the ischemic necrosis in cerebral infarction in the form of a cytotoxic edema,  appearing as a high DWI signal within minutes of arterial occlusion.  With perfusion MRI detecting both the infarcted core and the salvageable penumbra, the latter can be quantified by DWI and perfusion MRI. 
DWI showing necrosis (shown as brighter) in a cerebral infarction
DWI showing restricted diffusion in the medial dorsal thalami consistent with Wernicke encephalopathy
DWI showing cortical ribbon-like high signal consistent with diffusion restriction in a patient with known MELAS syndrome
Another application area of DWI is in oncology. Tumors are in many instances highly cellular, giving restricted diffusion of water, and therefore appear with a relatively high signal intensity in DWI.  DWI is commonly used to detect and stage tumors, and also to monitor tumor response to treatment over time. DWI can also be collected to visualize the whole body using a technique called 'diffusion-weighted whole-body imaging with background body signal suppression' (DWIBS).  Some more specialized diffusion MRI techniques such as diffusion kurtosis imaging (DKI) have also been shown to predict the response of cancer patients to chemotherapy treatment. 
The principal application is in the imaging of white matter where the location, orientation, and anisotropy of the tracts can be measured. The architecture of the axons in parallel bundles, and their myelin sheaths, facilitate the diffusion of the water molecules preferentially along their main direction. Such preferentially oriented diffusion is called anisotropic diffusion.
The imaging of this property is an extension of diffusion MRI. If a series of diffusion gradients (i.e. magnetic field variations in the MRI magnet) are applied that can determine at least 3 directional vectors (use of 6 different gradients is the minimum and additional gradients improve the accuracy for "off-diagonal" information), it is possible to calculate, for each voxel, a tensor (i.e. a symmetric positive definite 3×3 matrix) that describes the 3-dimensional shape of diffusion. The fiber direction is indicated by the tensor's main eigenvector. This vector can be color-coded, yielding a cartography of the tracts' position and direction (red for left-right, blue for superior-inferior, and green for anterior-posterior).  The brightness is weighted by the fractional anisotropy which is a scalar measure of the degree of anisotropy in a given voxel. Mean diffusivity (MD) or trace is a scalar measure of the total diffusion within a voxel. These measures are commonly used clinically to localize white matter lesions that do not show up on other forms of clinical MRI. 
Applications in the brain:
- Tract-specific localization of white matter lesions such as trauma and in defining the severity of diffuse traumatic brain injury. The localization of tumors in relation to the white matter tracts (infiltration, deflection), has been one of the most important initial applications. In surgical planning for some types of brain tumors, surgery is aided by knowing the proximity and relative position of the corticospinal tract and a tumor.
- Diffusion tensor imaging data can be used to perform tractography within white matter. Fiber tracking algorithms can be used to track a fiber along its whole length (e.g. the corticospinal tract, through which the motor information transit from the motor cortex to the spinal cord and the peripheral nerves). Tractography is a useful tool for measuring deficits in white matter, such as in aging. Its estimation of fiber orientation and strength is increasingly accurate, and it has widespread potential implications in the fields of cognitive neuroscience and neurobiology.
- The use of DTI for the assessment of white matter in development, pathology and degeneration has been the focus of over 2,500 research publications since 2005. It promises to be very helpful in distinguishing Alzheimer's disease from other types of dementia. Applications in brain research include the investigation of neural networksin vivo, as well as in connectomics.
Applications for peripheral nerves:
- : DTI can differentiate normal nerves  (as shown in the tractogram of the spinal cord and brachial plexus and 3D 4k reconstruction here) from traumatically injured nerve roots.  : metrics derived from DTI (FA and RD) can differentiate asymptomatic adults from those with compression of the ulnar nerve at the elbow  : Metrics derived from DTI (lower FA and MD) differentiate healthy adults from those with carpal tunnel syndrome
Early in the development of DTI based tractography, a number of researchers pointed out a flaw in the diffusion tensor model. The tensor analysis assumes that there is a single ellipsoid in each imaging voxel— as if all of the axons traveling through a voxel traveled in exactly the same direction.  This is often true, but it can be estimated that in more than 30% of the voxels in a standard resolution brain image, there are at least two different neural tracts traveling in different directions that pass through each other. In the classic diffusion ellipsoid tensor model, the information from the crossing tract just appears as noise or unexplained decreased anisotropy in a given voxel. David Tuch was among the first to describe a solution to this problem.   The idea is best understood by conceptually placing a kind of geodesic dome around each image voxel. This icosahedron provides a mathematical basis for passing a large number of evenly spaced gradient trajectories through the voxel—each coinciding with one of the apices of the icosahedron. Basically, we are now going to look into the voxel from a large number of different directions (typically 40 or more). We use "n-tuple" tessellations to add more evenly spaced apices to the original icosahedron (20 faces)—an idea that also had its precedents in paleomagnetism research several decades earlier.  We just want to know which direction lines turn up the maximum anisotropic diffusion measures. If there is a single tract, there will be just two maxima pointing in opposite directions. If two tracts cross in the voxel, there will be two pairs of maxima, and so on. We can still use tensor math to use the maxima to select groups of gradients to package into several different tensor ellipsoids in the same voxel, or use more complex higher rank tensors analyses,  or we can do a true "model free" analysis that just picks the maxima and go on about doing the tractography.
The Q-Ball method of tractography is an implementation in which David Tuch provides a mathematical alternative to the tensor model.  Instead of forcing the diffusion anisotropy data into a group of tensors, the mathematics used deploys both probability distributions and a classic bit of geometric tomography and vector math developed nearly 100 years ago—the Funk Radon Transform. 
For DTI, it is generally possible to use linear algebra, matrix mathematics and vector mathematics to process the analysis of the tensor data.
In some cases, the full set of tensor properties is of interest, but for tractography it is usually necessary to know only the magnitude and orientation of the primary axis or vector. This primary axis—the one with the greatest length—is the largest eigenvalue and its orientation is encoded in its matched eigenvector. Only one axis is needed as it is assumed the largest eigenvalue is aligned with the main axon direction to accomplish tractography.
Apparent Diffusion Coefficient
As discussed above, DWI images are inherently T2-weighted. Therefore, lesions with long T2 relaxation will appear bright, even if they do not restrict diffusion. This effect will be particularly apparent on low b-value images, where the diffusion weighting is less (i.e. lesions with fast diffusion have not lost much signal and so will still be bright). Because of its extremely long T2, free water (e.g. CSF, cysts) will be bright even on relatively high b-value images. We would like to eliminate the T2 effects to obtain a more accurate idea of diffusion restriction and eliminate spurious bright spots. In order to do this, we can actually calculate the diffusion coefficient by using several DWI series with different b values.
Apparent Diffusion Coefficient. The diffusion coefficients we measure with MRI represent averages of the entire voxel and of each direction of diffusion (see discussion about anisotropy and DTI later). Therefore, we use the word apparent to describe the values we calculate. The signal of a particular tissue decreases exponentially with increasing b-value. Given an apparent diffusion coefficient D, the signal intensity I is
I = I0 * e -b * D , where I0 depends on T2 characteristics
If we acquire at least 2 DWI sequences with different b-values, we can plug them into the equation to solve for D. Typically, at least 3 different b-values are used to improve noise (e.g. 40, 400, and 800) we can take the log of the intensity to linearize the graph and then use linear regression to get a best-fit D. By plotting D for each pixel, we obtain the ADC image (sometimes called an ADC map).
Simulation of how differing b values affect the appearance of DWI images and how to calculate ADC. Left, simulated DWI images showing bright CSF on the low b-value images and increased visibility of the left frontal stroke on the high b-value images. Center, plot of the log of the signal intensity of different tissues (blue, CSF gray, brain brown, stroke) at varying b-values. The slope of the line connecting the point is the ADC. Right, simulated ADC image areas of diffusion restriction, which have the flattest slope on the center plot, have the darkest signal on the ADC image.
Areas of diffusion restriction will lose the least signal on high b-value images (because their protons are not moving). The slope of the line on the DWI plot (see above) will be flat, and thus the ADC value will be small (thus, dark pixels). On the other hand, areas of fast diffusion will lose the most signal as b values increase, giving a large slope - and bright pixels on the ADC image. Importantly, the T2 shine-through - i.e. the brightness on DWI images related to the underlying T2 signal in the tissue only affects the initial position of the points in the DWI plot it does not affect the slope, thus the ADC image is independent of T2 shine-through - it reflects diffusion alone.
Clinically, we typically still use DWI images because bright abnormalities are much easier to see than dark abnormalities (you may notice this in the simulation above). People have developed several strategies to transform the ADC image into a "bright = bad" map for example, the exponential ADC (EADC) map takes the exponential of the ADC values, leading to an inverted scale more similar to the DWI (but again, eliminating T2 shine-through effects). Another important reason to use DWI images is that because the ADC image relies on several DWI images, it is inherently more susceptible to artifacts than individual DWI images. Finally, for many abnormalities, they not only restrict diffusion but are bright on T2 thus, we can actually take advantage of the T2 shine-through effect to make lesions more conspicuous, then confirm true diffusion restriction on the ADC map.
Imaging of Oligodendrogliomas
William Ankenbrandt , Nina Paleologos , in Handbook of Neuro-Oncology NeuroImaging , 2008
Diffusion Tensor Imaging – Fractional Anisotropy
Diffusion tensor imaging is sometimes employed preoperatively to attempt to establish the location of important fiber tracts, such as the corticospinal tract, and to determine whether there is displacement, infiltration or destruction of such tracts. The diffusion-weighted images displayed as part of routine MRI are ‘isotropic’ – that is the differences in directional water diffusivity (greater ease of diffusion along axonal tracts, for example) are essentially nullified by averaging of directional anisotropy. The goal of diffusion tensor imaging is essentially to discover for each voxel which way water diffuses most readily and this has been shown, in an experimental cat model, to be along the orientation of major white matter fiber tracts [ 30 ]. A map of directional anisotropy (most commonly calculated as ‘fractional anisotropy’) can approximate white matter tract configurations by displaying diffusivity along three principal axes in three different colors [ 31 , 32 ].