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/ Diffusion Tensor Imaging A New View of the Brain O n battlefields and playing fields, from Iraq to Cowboys Stadium, one of the signature injuries of the past decade has been concussion. More than 300,000 soldiers suffered suspected concussions between 2001 and 2007. Nevertheless, it remains a difficult condition to diagnose because the damage to the brain is hard to see with conventional imaging techniques. The brain may look completely normal on magnetic resonance imaging (MRI) or on a computed axial tomography (CAT) scan, yet patients report ongoing effects, such as memory loss, headaches, sensitivity to light or noise, and depression. A new imaging technique—a variant of MRI called diffusion tensor imaging—has revealed that the damage to concussed brains may lie not in the gray matter but in the white matter. For decades, neurologists considered the white matter (consisting of axons and glial cells) to be less important than the gray matter (which consists of neurons). They saw the white matter as a passive scaffolding for the brain’s architecture. However, this view has changed dramatically in the last decade. If the brain is like a computer, then the gray matter can be compared to the processors, while the white matter can be compared to the communications grid that links those processors. Even the most powerful processors cannot work correctly if the pathways are destroyed or disrupted. Besides concussion, a whole host of other brain functions and malfunctions are now linked to the white matter. Patients with schizophrenia, Alzheimer’s disease, or deterioration due to a stroke, autism, and attention deficit disorder all have detectable changes in diffusion tensor imaging images of their white matter. Even during normal development and learning, the diffusion tensor imaging changes in intriguing ways. FUELING innovation and discovery 24
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The revolution in our understanding of white matter—which has only just begun— would never have been possible without diffusion tensor imaging. And diffusion tensor imaging, in turn, would never have been possible without the mathematical sciences. The mathematics is hidden in plain sight: in that mysterious word “tensor” in diffusion tensor imaging. A tensor is a mathematical concept, developed in the 19th century, that generalizes the notion of vectors. Tensors have proved useful in a number of areas of physics. To explain what a tensor has to do with white matter in the brain, it helps to start with how MRI works. An MRI machine (see Figure 8) creates a strong magnetic field, Already this which causes the protons in the body to rotate and line up in a predictable way. Most of imaging capability these protons are actually hydrogen atoms in water molecules; thus MRI is especially sensitive to the water (or fluids) in your body. It is an excellent complement to traditional has led to x-rays, which see the dense, hard structures in your body but are relatively blind to the fundamental soft tissues. One of the most informative parts of the body to image with MRI is the insights about brain, because it is squishy and it uses a lot of blood. By modulating or pulsing the magnetic field in various ways, doctors can tune the normal and MRI scan to detect different kinds of tissue in the body. In particular, one technique abnormal brains. allows them to measure the displacement of water molecules over a short period of time—displacements that are due not to blood flow but to random jitters of the molecules, called Brownian motion. Because Brownian motion underlies the process of diffusion, this technique measures what is called the “apparent diffusion coefficient” in a tiny cubic region of the brain. 8 / Magnetic resonance imaging (MRI is an im- portant medical imaging technique that allows in- ternal structures to be vi- sualized. Image courtesy of the National Institutes of Health Clinical Center, Center for Interventional Oncology. / in the 21st Century The Mathematical Sciences 25
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9 / Diffusion tensor imaging used to reconstruct network connections in the brain (tractogra- phy). Similarly oriented fibers are shown in the same color. Reprinted from Moriah E. Thomason and Paul M. Thompson, 2011, Diffusion imaging, white matter, and psychopathology, Annual Review of Clinical Psychology 7:63-85, with permission from Annual Reviews, Inc. / Beginning in the early 1990s, researchers noticed a puzzling fact: In the white matter, the apparent diffusion coefficient of a sample seemed to depend on its orientation with respect to the magnetic field. Tilt the sample and you would get a different diffusion coefficient. In 1991, a biomedical engineer had a eureka! moment: The dependence of the apparent diffusion coefficient on orientation wasn’t a problem, it opened a path toward a solution. This engineer knew something that most doctors didn’t. In an anisotropic material—a material that is directionally dependent, such as a wood with a grain going in a particular direction or brain tissue that consists of layers or fibers—water doesn’t diffuse equally rapidly in all directions. Water molecules move faster along the fibers and more slowly perpendicular to them. Over time, a tiny blob of water molecules will diffuse into an ellipsoid (or football) shape, with the long axis of the ellipsoid pointing along the fibers. The diffusion tensor contains all the mathematical information needed to graph this ellipsoid. It is not just a single number (like the apparent diffusion coefficient) but a 3 × 3 array of numbers. Starting with pork loins and working up to living human tissue, the experimental and mathematical procedures were developed for measuring the diffusion tensor from point to point within a sample and putting it into a three- dimensional image. Diffusion tensor imaging was made to order for visualizing white matter, which consists mostly of axons, elongated cells that convey electrical impulses. Water diffuses rapidly along the length of an axon but slowly across the width. In addition, many but not all axons have a fatty sheath, called a myelin layer, which impedes the diffusion of water. (The myelin sheath is also what gives white matter its color.) Thus diffusion tensor imaging can both map out the direction of the brain’s electric fibers (this is called “tractography” and is illustrated in Figure 9) and also detect the extent of myelination in various parts of the brain. FUELING innovation and discovery 26
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Already this imaging capability has led to fundamental insights about normal and abnormal brains. For example, biologists have known for a long time that the human brain starts out with little myelin, and that the axons gradually myelinate over childhood and adolescence. The myelination process seems to be associated with learning. With diffusion tensor imaging, researchers can now see this process in living humans. For example, they can see which parts of the brain are associated with reading and language acquisition. People with higher IQs in general tend to have longer, skinnier diffusion ellipsoids, suggesting greater fiber integrity or a greater amount of myelination. The fiber integrity (or “fractional anisotropy”) seems to peak in the early 30s and gradually decreases thereafter; this may explain why memory and other cognitive processes decline gradually with age. Likewise, diffusion tensor imaging points out areas of the white matter that are compromised in particular diseases. In schizophrenic patients, the fiber integrity is reduced in the part of the brain called the cingulate (responsible for error detection), the corpus callosum (responsible for communication between the brain hemispheres), and the frontal lobe. In autism, the deficits in fractional anisotropy occur in regions that are associated with processing social cues. Attention deficit hyperactivity disorder seems to be an exceptional case where the fractional anisotropy is too high rather than too low. And in concussion injuries, the fiber integrity is reduced near the site of the injury. This finding could be useful as both an objective criterion for diagnosis and a way of predicting which patients will suffer more serious long-term symptoms. In the decade of the 2000s, research on diffusion tensor imaging took off, with the number of research papers doubling roughly every 2 years. Probably the most fundamental problem that remains is to distinguish when two fibers cross within a single cube (or “voxel,” the three-dimensional analogue of a pixel) of the image. It has been estimated that as many as 30 percent of the voxels in a diffusion tensor imaging scan have more than one fiber passing through them. Unfortunately, the standard diffusion tensor cannot detect this fact. An ellipsoid has only one longest axis, and it cannot have two separate “bumps.” If there are actually two fibers, diffusion tensor imaging will produce not two ellipsoids but a single, rounder ellipsoid. It will thus underestimate the fractional anisotropy in that voxel, and it may also draw the fiber pathways incorrectly. One way to address the problem of crossing fibers would be to improve the resolution of the scans, so that each voxel is smaller. This would require MRI scanners with stronger magnetic fields—a trend that has continued throughout the past decade. But a less expensive alternative is to develop mathematical methods that would replace ellipsoids with more complicated diffusion surfaces. For example, a method called high angular resolution diffusion imaging (as shown in Figure 10 on page 28) combines magnetic resonance data with the principles of tomography, and it produces in the 21st Century The Mathematical Sciences 27
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10 / High angular resolution diffusion imaging data used to show diffusion surfaces in three dimensions in the brain. This allows for a higher-resolution image than conventional diffusion tensor imaging but generates a great deal more data, making data mining and analysis more complex. Reprinted from Moriah E. Thoma- son and Paul M. Thomp- son, 2011, Diffusion imaging, white matter, and psychopathology, Annual Review of Clini- cal Psychology 7:63-85, with permission from Annual Reviews, Inc. / spectacular detailed images of crossing fibers that would confuse an ordinary diffusion tensor imaging scan. However, it generates a great deal more data, necessitating advances in data mining and analysis. It is safe to say that much work remains to be done, from both the experimental and analytical sides. FUELING innovation and discovery 28