Handelsmakler wikipedia english
With no diffusion present, the change in nuclear magnetization over time is given by the classical Bloch equation. Torrey mathematically showed how the Bloch equations for magnetization would change with the addition of diffusion.
The Bloch—Torrey equation is:. For the simplest case where the diffusion is isotropic the diffusion tensor is a multiple of the identity:. Anisotropic diffusion will have a similar solution for the diffusion tensor, except that what will be measured is the apparent diffusion coefficient ADC. In general, the attenuation is:. The standard grayscale of DWI images is to represent increased diffusion restriction as brighter.
Conventional DWI without DTI 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. DWI showing necrosis shown as brighter in a cerebral infarction.
DWI showing restricted diffusion in the mesial dorsal thalami consistent with Wernicke encephalopathy. 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 weighing will therefore be minor, leading to an ADC image with low signal in the infarcted area. 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: 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.
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 cerebro-spinal 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 it 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. 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. 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. Historically, this is closest to what Richards originally measured with the vector length in 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.
Another commonly used measure that summarizes the total diffusivity is the Trace —which is the sum of the three eigenvalues,. 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. However, this proves to be very susceptible to measurement noise, so increasingly complex measures were developed to capture the measure while minimizing the noise.
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. 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. 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.
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:. 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. 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.
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. 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. 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.
Some clinical applications of DTI are in the 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.
The use of DTI for the assessment of white matter in development, pathology and degeneration has been the focus of over 2, research publications since It promises to be very helpful in distinguishing Alzheimer's disease from other types of dementia.
Applications in brain research cover e. DTI also has applications in the characterization of skeletal and cardiac muscle. The sensitivity to fiber orientation also appears to be helpful in the area of sports medicine where it greatly aids imaging of structure and injury in muscles and tendons. 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. 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 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. 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: 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. 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 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 s, diffusion exhibits many of the same patterns as those seen in the transfer of heat.
At this point, it is helpful to consider the mathematics of ellipsoids. An ellipsoid can be described by the formula: 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: 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. In any event, Wednesday dawned with bright sunshine and a brilliant blue sky, and the cross country trail system described to us by a fellow NH tourist as the best in North America beckoned. So we set aside our downhill skis and took off into the woods, on beautifully groomed trails that seemed to travel into Narnia.
During our whole outing, we barely saw another person. This may have been because they heard us coming and hurriedly decided to try a different trail. The kids held up well, given that they had little to no experience on cross country skis. Within the first ten minutes most of us were down to our shirtsleeves.
Our 8km loop was just the right distance for us to get in before returning to the lodge for lunch. Zoe and I decided to persevere on the ski trails, and managed to get in another 10km before the end of the day. We even tried a blue intermediate trail, despite my reservations. I was more afraid of the downhill than the up, but apparently the laws of physics do not apply the same way in Canada, because I swear the loop was uphill both ways. The next day, of course, our legs were even more tired…but with several inches of snow having fallen the night before, the siren song of one more day on the mountain was too much to resist.
A return trip to the toboggan ride, which we discovered goes 44 mph! No wonder it was so terrifying. If we thought we were in for an easier time than the day before, we were greatly mistaken. We found that the mountain had groomed only a few trails, leaving the others with large mounds of powder scattered across them in uneven lumps. After one run, our thighs were screaming. Most of our crew called it a day by lunchtime.
Bob, Chris and I dragged ourselves back after a long break in the condo, and were glad we did. After what we assumed would be our last run, we saw the lift was.
Given how the weather has been in NH, this may be our last skiing of the year, so we wanted to make the most of it. We know the mountain a little bit so a lot of the surpises have been revealed. Our legs are very sore. Today, almost every one of us did something of championship calibre. Eliza skied her first black diamond — out of necessity, because the adjacent blue trail was closed — and immediately said she wanted to go back up and ski it again, on purpose this time.
Trisha skied the same green pea trail over and over again, then said she was ready for an easy blue square. She skied it like a pro, then decided she was happy going back to the green pea trail a few more times, si vou plait. She would end the day on one of the trickiest trails any of us has seen here, and she did it without complaint. She had her ski pants on and was working on her boots before most of us had put our dishes in the dishwasher.
It should be said that all the kids — and adults — were eager to get back to the mountain after lunch today. It was nice to see. Not only did Nadia go back out after lunch today, she was filled with energy all day long, encouraging several of us to try a new trail for our last run.
It turned out to be a horrible decision that had us skiing down a frozen waterfall several stories high, but she still showed good initiative. Zoe stuck with Eliza and talked her through the steepest part of her first black diamond, snatching a victory from the jaws of defeat. Emma, who is nursing a knee injury, combined self-control and true Olympic grit.
She seemed to know just when to turn in for lunch, so she could make it through the afternoon until last run. She also stuck around at the exit of the Enchanted Forest the last time until all her comrades made it out, which required quite a bit of waiting. The Enchanted Forest is lovely, but also very bumpy. Chris spoke clear, if urgent, French to some ski patrollers while trying to explain that some of his kids might still be out on a closed trail at the end of the day.
Although, he admits he might have said some of his children were under the trail or perhaps that some of his children lived on the trial, Chris eventually got his message across. More snow is coming. A lot more tomorrow night. This may have not been our last ski day after all. When we left you last, reader, it was snowing. It was lovely to watch from inside our condo and anticipate the impact the flakes would have on the slopes just beyond our vision. It was difficult to wait. Jen, Chris and I did some cross country skiing yesterday on a neighboring golf course, and after a particularly steep climb and a glide through a sled-dog kennel, we found ourselves on one the the Mt.
Anne downhill trails, and hour after the lifts stopped running. Today, though, everything fell into place. We had the right skis on the right hills. The right snow had fallen on the right trails. We finally got to ski. Anne gondolas until 9: Much praise for Trisha, who decided to straight to the top and not mess around with the magic carpet and training hill.
Trisha was ready to tackle them. The stars all lined up: We had reasonable temperatures — no one asked for hand or boot warmers.
There was a fair amount of sun and good visibiltiy, except for the brief blizzard that caused a little havoc right around lunchtime. Beaupre is steep for a blue square, though it might not look like it here. We had lots of fun. Most of us made it back out for afternoon runs after lunch and were rewarded with very forgiving snow and very little ice. The girls found several glades to explore and displayed enough competence that eventually the adults no longer felt obligated to follow them into the woods.
This crew, and the guy who took the picture, made it until the very end.. Our legs became sore, but we stuck it out to the very end. Tomorrow looks like another good day for skiing.
Rumor has it the deal extends to drinks at the bar, as well. This might make for an interesting blog post. Jen and Trish sat down yesterday and made a plan for the week.
This is what they drew up for day We went to the city. Then we got croissants. First we picked them out. Almost everyone got chocolate. We stopped into the tourist center to warm up some more. Eliza was the only one who came over to meet Bonhomme.
Even she was a little skeptical. But we made it back to the mountain and had a quiet evening, including cider by the fire. Last fall Jen and I returned for a weekend after a year absence. It made us want to get right back up here, preferably with a bunch of people. The timeshare gods provided us with a vacancy here at Mt. Anne for February break and our friends the Halls were on board to help fill the place. Then we hit Franconia Notch. But mahem as we passed through the notch.
Snow, sleet and wind made passage from Lincoln to Littleton a strenuous and slow slog. After some rest and Thai food in Littleton a nicer town than I had anticipated, and the Thai restaurant is great , we hit the road again. We secured two cabins in possibly the northernmost cabin outpost currently operating in Northern Vermont.
Thanks for your hospitality, Barton, VT, and congratulations to your two winter Olymians. Our ne- return to Quebec would have to wait another half day. Smack-dab in the heart of the Province du Quebec, Canada. The sun is out. There are beds for everyone. Some of us ventured out to check out the lodge and the trail maps and such. Others stayed in the condo and hatched a plan to bake cookies. Other things to consider:. It might be tough not to ski tomorrow.
There are notable exceptions: The cayes of Belize. We might not even make it back to the St. So, readers might infer that there is something special about Quebec City. Peak foliage and apple harvesting time make for a nice setting for cider tasting.
In some ways, it feels like a new city to us. Also good for a vineyard visit — note the Chute-Montmorency is that white smudge in the distance toward the top left of the picture. This was during a break in the fog. This island in the St. Lawrence River is 15 minutes from the walls of the old city, but calm, quiet and pastoral.
To find a setting like this outside their city, a Bostonian might have to drive two hours to get to Vermont or lakes region New Hampshire or far-western Massachusetts. Once across the bridge and onto the island, we had plus miles of vineyards, cideries, bakeries and farm stands for us to wander through, and some of them were in view of the city!
That is to say they would be in view of the city except that for much of the morning we were frustrated by drizzle and fog. Then the fog came back and rain. Then rain and wind. We kept on driving along. The attractions on the north side of the island were mostly art galleries, we were told, and we were ok passing them by while staying dry in the car.
Just before noon, we stopped at almost the halfway point along the route. It was the nicest weather for the whole trip so far, and it allowed for extensive views east and west along the river. The soup was nice, too. The tastings offered at the latter locales were small and Jen and I were sharing them; still, the day started to take on a bacchanal-like feeling. The last vineyard we went to was even named after Bacchus. All along the route were farms, some for hay and livestock, and others for the main produce crops on the island: Such is the climate here that all three of those crops were being harvested as we wandered past.
Not only could we see well down the St. With the this reconnoitering goal in mind, we turned ourselves loose on the Old City again this evening in search of the fondue restaurant we visited during our first visit here — or something similar. It seems like the kind of dinner the girls would appreciate.