Diffusion Tensor Imaging

Physics Underlying the Image Signal in Nuclear Magnetic Resonance

The physical basis of the DTI technique is that water tends to diffuse more rapidly along the long axis of an axon as opposed to diffusing transversely through the walls of the axon. At a theoretical level, water should be able to diffuse transversely through axon membranes and myelin sheaths, but the directionality of water diffusion in neural tissue is well proven. When water diffuses uniformly in all directions in a material or tissue, that pattern is described as isotropic, or equal in all directions. When there is a preferred direction of diffusion, then the behavior of water in that tissue is termed anisotropic.

Fundamentally, MRI scanning is based on measurement of nuclear spins of water protons. A proton in a given compound (such as the hydrogen proton in a water molecule) spins around its axis at a rate that is related to a surrounding magnetic field based on the Larmor equation. Stated simply, for example, if the applied magnetic field is at 4.7 T, the water protons will have an intrinsic spin rate of exactly 200 MHz (200 million times around per second). If radiofrequency (RF) energy is pumped into the volume of water in the 4.7-T field at a frequency of 200 MHz, then that will be introducing energy at exactly the resonant frequency of those protons, and they will commence absorbing the energy.

At the start, all of the protons can be thought of as standing vertically, each spinning on its axis and all aligned with the direction of the main magnetic field of the scanner. The absorbed energy cannot make the proton spin any faster; however, like a top, the proton’s magnetic axis may be pushed off of vertical so that the protons begin to precess, all coherently with one another, in response to the incoming wave of the applied RF signal. In MR, a pulse of RF is applied at the exact necessary frequency and then the RF is turned off and an antenna is brought into use to detect an RF signal coming back from the spinning protons as they coherently precess en masse, giving off their excess energy as the “spinning tops” return toward vertical. With elapsing time—a few microseconds—the excess energy will gradually be transmitted out of the coherently spinning protons, and the signal from the protons will decay away as their vertical axis of spin returns to alignment with the main magnetic field.

In T1 or T2 MR paradigms, the rate of decay provides the image contrast, with some tissues having faster or slower rates of decay. T1 decay relates to interactions between the spinning protons and magnetically active components in surrounding tissue components, whereas T2 contrast is related to decay interactions with other spinning protons.

Use of Location Gradients to Create a Magnetic Resonance Image

The difference between nuclear magnetic resonance (NMR)—a laboratory technique going back to the 1950s—and MRI is the application of “locational” magnetic gradients that establish a three-dimensional Cartesian grid in MRI. A gradient is an addition to the main magnetic field such that, for instance, for voxels on the left side of the imaging volume the field strength is at 4.72 T and for voxels on the right side of the volume the field strength is at 4.68 T. The spin frequency for protons in the voxels on the left will be higher (e.g., 200.1 MHz), and the resonant frequency for protons in the voxels on the right side will be lower (e.g., 199.8 MHz).

The gradient is turned on briefly by running an electrical current through a loop of wire that results in generation of a magnetic field. The field is strongest near the loop and lowest at a distance, so there is a gradient from weak to strong along one direction. A first gradient is applied from head to foot (axial or z -axis—“slice select”), a second gradient is applied from back to front (anteroposterior or y -axis—“phase encode”), and a third gradient is applied right to left (transverse or x -gradient— “frequency encode”). When all three gradients are used, each voxel in the three-dimensional volume will have a unique and different field strength, so each will have a unique and distinct resonant frequency and phase. The system can learn about signal strength in each voxel and know exactly where in the three-dimensional grid each of the voxel signal return sources is located.

With this—the invention of MRI (by Paul Lauterbur and by Raymond Damadian )—it became possible to individually and separately address different voxels in an image. If one wanted to test a voxel in the lower left part of the volume in the back, one would use a lower precise frequency to stimulate and the antenna to learn the decay rate in that voxel, and so on. In this fashion the decay rate could be individually measured in each separate voxel in the imaged volume, such as the brain. Operationally, all the different frequencies are pumped in at the same time (often slice by slice).The receiving antenna receives a complex cacophony of thousands of signals at the full array of frequency and uses the mathematics of Fourier transforms to sort the data into an ordered array of data measurements. The measurements are repeated a number of times over—for example, a second—and some voxels will begin to decay in strength faster. For a T2 image, voxels with cerebrospinal fluid (CSF) will decay slowly and will still have a bright strong signal at the end of the second, whereas voxels with denser brain tissue will have return resonance signals that decay more rapidly and so can be shown as a darker voxel (three dimensional) or a darker pixel (two dimensional) on an image on the screen.

Diffusion-Weighted Magnetic Resonance Imaging

Diffusion MRI is based on the addition of an additional magnetic field gradient during the image acquisition. In this case the gradient is used to make a measurement of the diffusion of water. When the diffusion gradient is applied there will be (as for the imaging gradients), for example, a higher field strength on the left side of the voxel and a lower strength on the right side of the voxel. If water molecules with their protons diffuse (physically move or spread owing to brownian forces of energy such as heat) equally in all directions, then the diffusion is isotropic or equal in all directions.

At the initial instant of the imaging experiment, all the water molecules in the voxel are set to precessing coherently, all spinning at their resonant frequency for the applied field strength (e.g., 4.7 T, so 200 MHz). However, as some of the water molecules wander, by diffusion, into an area of the voxel where the applied diffusion gradient is stronger, they will speed up. Others will diffuse or spread into areas of the voxel where the applied main magnetic field is weaker and will slow down by a small amount. Very soon the various water protons in the voxel will each have their own distinct spin rate frequencies. The coherence of the signal will have decayed owing to their diffusion or physical movement in the voxel.

In a voxel with isotropic diffusion (e.g., CSF), this decay of the coherent signal due to diffusion movements will happen at exactly the same rate whether the diffusion gradient is applied from right to left or from head to toe, and so on. However, in axons, diffusion is anisotropic. Water molecules diffuse preferentially up and down the length of an axon. If, for instance, a peripheral nerve is to be imaged as to diffusion, we can consider what happens when a gradient is applied perpendicular to the nerve direction. The entire length of the nerve will travel along at one level of the gradient strength. The water diffusion preferentially up and down the length of the axon will not experience decay from diffusion, because as the molecules diffuse up and down along the axon, they remain at the same gradient strength. However, if the gradient is then turned so that it is applied down the length of the nerve—higher strength proximally, lower strength distally—then as the water diffuses the molecules will move rapidly into higher or lower magnetic field strengths at a rapid rate and there will be accelerated decay from diffusion.

In isotropic tissue, the gradual decay rate caused by the diffusion gradient is the same no matter which way the gradient is oriented. However, for an anisotropic tissue such as a nerve axon, the rate of diffusion decay is highly dependent on the orientation of the applied diffusion gradient. Because of this difference, a given voxel will remain brightest when the gradient is applied perpendicular to the long axis of an axon passing through it; will be intermediate in decay rate if it contains isotropic tissue; and will decay rapidly, becoming dark in intensity, if it contains an axon and the gradient is applied parallel to the direction of the axon. Because of this, we can learn about the direction in which an axon is oriented by the way its image signal strength responds to the orientation of a particular applied diffusion gradient.

Diffusion MRI was first introduced into clinical use because of its capability to detect stroke in an MRI scan. There is a characteristic pattern of image intensities in a brain MRI image slice when diffusion gradients are applied. However, in the setting of stroke, axons start to lose their apparent diffusion when they suffer infarction. Therefore the image characteristics of a diffusion-weighted MR image will start to change in an area affected by stroke. The affected area of neural tissue will begin to act as if it were restricted or decreased rather than unrestricted. For diffusion-weighted MRI, there is no need to know the precise direction of diffusion.

One additional technical aspect of diffusion MRI is that the diffusion gradient is pulsed on twice during each acquisition to actually detect the effect. In essence, it is turned on briefly to “label” the protons by giving them a particular coherent spin at their location at the beginning of the acquisition and then it is pulsed on a second time a few milliseconds later with the two pulses symmetrically placed on either side of the 180-degree inversion pulse. The difference in physical position in the gradient between the two pulses will sharply and precisely delineate the effects of diffusion. This is called a Stejskal-Tanner pulse after its two developers. ^,

Diffusion Anisotropy Imaging: Tensor and Vector Techniques

As described earlier, when the orientation of a nerve is simple and linear and known in advance (such as the median nerve in the upper arm), diffusion-weighted gradients can be applied parallel or perpendicular to the nerve to show the anisotropy, or health, of the nerve. However, as in many locations for peripheral nerves and for most locations in the brain, neural tracts will curve and turn as they progress, so there is no simple method of planning a perpendicular or parallel diffusion gradient. In DTI or in vector versions (e.g., HARDI, Q-Ball imaging), a larger number of gradient directions are applied and mathematical methods are then used to calculate neural direction on a voxel-by-voxel basis.

Note, however, that the diffusion gradients have to be set to about 10 times the strength of the location gradients in order to work. Without diffusion gradients, there is an ordered array of magnetic field strengths voxel by voxel in a cube generated by the x, y, and z gradients. With the addition of a far more powerful gradient from some off angle, the ordered array of field strengths set up by the location gradients, particularly the “phase encode” or y gradient, turns to chaos. These interactions are called cross terms, and they made it impossible to actually get a useful image when off-angle gradients were applied to generate tensor data.

For standard diffusion-weighted MRI, a single diffusion gradient could be applied parallel to one of the location gradients. The cross terms could be managed in this situation by using a balanced opposing pair of gradients—for example, one pointing along the x -axis and a second one pointing in the same orientation but from the opposite direction. However, to calculate a tensor, it was necessary to send in diffusion gradients from multiple different angles. As Mori pointed out, the two balanced pulses Neeman had proposed have to arrive at substantially different times, so diffusion is progressing between the two arrival times and therefore full correction is not possible. With the added the complexity of a gradient that is not parallel to the location gradient, the geometry of correction becomes inordinately complex. The cross terms get far more complicated. There is then the additional issue that multiple images, at least seven, will be obtained with different gradient orientations, so failures of compensation will be different from one gradient direction to the next, even as the water molecules are diffusing in complex, anatomically determined pathways that do not follow a Cartesian grid.

The fundamental invention that created the field of DTI—which is wholly attributable to the Filler group in their US patent number 5,560,360—was a method for allowing a non-Cartesian gradient (a gradient not directed along the x-, y-, or z -axis) to be applied without destroying the locational gradients ( x, y, and z ). Unlike in other scientific fields (such as the use of tensor analysis to determine the direction of magnetization in anisotropic rocks by the Harvard group where Filler trained [Tauxe and colleagues]) the same physical process (precisely oriented magnetic field gradients) is used to create the image in MRI and to apply the method for disruption to assess diffusion direction and anisotropy.

Filler identified the problem in 1991 and tasked Franklyn Howe, an MRI physicist, with finding a refocusing method that would work intrinsically with each gradient orientation (no balanced pair). Howe developed two different methods of refocusing the location gradients immediately after each application of a diffusion gradient so that diffusion could be measured in any direction without disrupting the image being created simultaneously by the image gradients. This discovery was published in 1992 showing detailed high-resolution imaging despite application of an off-angle diffusion gradient. With this problem solved, there was no limit to how many diffusion gradient directions could be obtained without the image being significantly affected by cross terms.

In addition, based on work by co-inventor Todd Richards, the group showed that the image could be constructed in an entirely novel way. In radiography, CT, MRI, and DWI, the result of imaging is that each voxel in the image is ultimately assigned a single image intensity number (a scalar quantity). These various shades of black, gray, and white make up the image, whether as exposed silver grains on a film or as pixel intensities on a computer monitor screen.

In the diffusion anisotropy techniques (e.g., DTI, HARDI, Q-Ball), each voxel becomes the subject of a mathematical calculation, with the result that it contains one or more vectors and not a single scalar quantity. The direction(s) of the vector will show the orientation of axons in the voxel, and the length of the vector will show the health or relative degree of anisotropy in that voxel. For simple DTI methods, each voxel can contain only one vector. For the vector methods HARDI and Q-Ball, for instance, there can be multiple vectors in the voxel, illustrating the fact that there may be crossing, joining, or splitting brain tracts captured within a single voxel. Richards et al. published this methodology for mathematically combining the several different diffusion direction image acquisitions into a single image whose voxels contained the directional information in 1992 as well.

For all of these diffusion anisotropy methods, the fundamental paradigm is to collect the image by using multiple diffusion gradients (each as a Stejskal-Tanner paired pulse ^, ). The gradients for a tensor method must be applied in a minimum of six different directions; modern DTI/DVI (diffusion vector imaging) scanning is typically done with about 30 directions of diffusion, but there may be as many as 256 directions of diffusion gradients applied for super-high resolution analysis. The MRI unit may still have the original design of just three gradient coils used for the three imaging gradients ( x, y, and z directions). However, by turning two or all three of them on together in multiple varying strengths, the scanner can generate an infinite number of diffusion gradient orientations. The gradient refocusing is carried out in the course of each gradient acquisition.

Prior to this advance, in papers such as the Moseley work on anisotropy, the researcher would publish a grid of four images, one routine with no diffusion, one showing how the slice appeared with an x -direction gradient, one with the y -direction appearance, and one with the z -direction appearance. Using Todd Richards’s method, it was possible to do a calculation for each voxel that incorporated all of the image results for that voxel and produced an invariant result. The data in the voxel would show the calculated orientation in space of an effective vector (either truly a vector or as the summary primary axis of a tensor). In addition, it would encode in the length of the vector—just how relatively isotropic or anisotropic the tissue was within that particular voxel.

The word tensor is involved for two entirely separate reasons. In physics the phenomenon of diffusion is called a tensor quantity. In anisotropic diffusion in various materials (rocks, polymer liquid crystal solutions, and so on) there is no single pure direction of movement, but rather the result is a sort of oriented ovoid or ellipsoid. It cannot be accurately and fully described by a single vector, but rather forms a sort of ovoid cloud of movement, with a long axis and two short axes. This is described as the tensor quantity phenomenon. One therefore can speak of the diffusion tensor as the quantity to be measured, although in clinical brain MRI we are interested in the orientation of axons and have no clinical interest in describing the niceties of “the diffusion tensor” of axonal tissue as a general physical phenomenon.

Separately, there is a type of mathematical analysis developed by Einstein called tensor analysis, ^{, ,} which he applied to the study of space and time and also to the “Brownian motion” of diffusion. This mathematics is one of a number of mathematical techniques that can take an input of multiple vector measurements and convert them into a result that reveals the orientation and length of the principal axis of the ellipsoid and the orientation and length of the two short axes ( Fig. 14.5 ). This will show whether the ellipsoid is nearly spherical (as occurs in tissues with isotropic diffusion) or whether it is thin and elongated (as occurs in tissues with anisotropic diffusion), as well as determining the precise orientation in space of the principal or long axis of the ellipsoid. Typically in DTI of the brain, we are mostly interested in the long axis—its direction and how long is it (relative to the short axes). These are individually calculated for each voxel in the brain.

By contrast, in a vector mathematical analysis, multiple crossing, joining, or splitting tracts can be identified. A typical vector concept is based on treating each voxel as a multitessellated icosidodecahedron (polyhedron with 60 sides), like an elaborate buckminsterfullerene. We then simply ask if there are any faces of the polyhedron that are particularly bright on image acquisition. These will be potential multiple different axonal orientations within the single imaged voxel. There are also statistical spherical deconvolution methods ( Fig. 14.6 ) that can find multiple maxima for image directional intensity within the data space of each single vector. For certain regions of the brain, such as the optic radiation, the ability to follow multiple tracts and to follow tracts through sharp turns is critical for patient safety when tractograms are used to guide glioma surgery.

The Method of Tractography

As a fourth aspect of the invention, Jay Tsuruda, a neuroradiologist, introduced the use of the connected voxel algorithm. This was a method under development at the time to improve carotid artery MR angiography. Tsuruda reprocessed brain image data produced by Richards to show that the algorithm could progress from voxel to voxel in a stepwise fashion. He also applied whole-volume ray projection (maximum intensity projection), but that method is not well suited to sorting out individual tracts. This led to the discovery that it was actually possible to produce images of brain tracts based on postprocessing of the voxel by voxel diffusion direction anisotropy data. The first DTI tractogram of the pyramidal tract appeared as Figure 17 in US patent number 5,560,360 (see Fig. 14.4 ).