A theoretical validation of the B-matrix spatial distribution approach to diffusion tensor imaging

Magnetic Resonance Imaging 36 (2017) 1–6

Contents lists available at ScienceDirect

Magnetic Resonance Imaging journal homepage: www.mrijournal.com

Original contribution

A theoretical validation of the B-matrix spatial distribution approach to diffusion tensor imaging Karol Borkowski a,⁎, Krzysztof Kłodowski a, Henryk Figiel a, Artur Tadeusz Krzyżak b a b

Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, 30-059, Cracow Faculty of Geology, Geophysics and Environmental Protection, AGH University of Science and Technology, al. Mickiewicza 30, 30-059, Cracow

a r t i c l e

i n f o

Article history: Received 20 June 2016 Revised 31 August 2016 Accepted 5 October 2016 Keywords: MRI DTI Phantoms Diffusion Anisotropy

a b s t r a c t The recently presented B-matrix Spatial Distribution (BSD) approach is a calibration technique which derives the actual distribution of the B-matrix in space. It is claimed that taking into account the spatial variability of the B-matrix improves the accuracy of diffusion tensor imaging (DTI). The purpose of this study is to verify this approach theoretically through computer simulations. Assuming three different spatial distributions of the B-matrix, diffusion weighted signals were calculated for the six orientations of a model anisotropic phantom. Subsequently two variants of the BSD calibration were performed for each of the three cases; one with the assumption of high uniformity of the model phantom (uBSD-DTI) and the other taking into account imperfections in phantom structure (BSD-DTI). Several cases of varying degrees of phantom uniformity were analyzed and the distributions of the B-matrix obtained were used for the calculation of the diffusion tensor of a model isotropic phantom. The results were compared with standard diffusion tensor calculation. The simulations confirmed the improvement of accuracy in the determination of the diffusion tensor after the calibration. BSD-DTI improves accuracy independent of both the degree of uniformity of the phantom and the inhomogeneity of the B-matrix. In cases of a relatively good uniformity of the phantom and minor distortions in the spatial distribution of the B-matrix, the uBSD-DTI approach is sufficient. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Diffusion tensor imaging is a powerful technique with multiple clinical applications. It can provide additional information about anatomical details which are not achievable by means of any other imaging techniques [1]. White matter fiber tracking and fractional anisotropy (FA) maps can be useful during pre-surgical planning as well as in intraoperative MRI, especially in the case of brain and spinal cord tumors [2–5]. DTI can also improve the diagnostic capabilities of MRI in cases of neurodegenerative disorders like epilepsy [6,7], multiple sclerosis [7,8], Alzheimer's disease [9–11], ischemic stroke [12,13] and other brain injuries which cause changes in the diffusion properties in a given brain region [1]. A novel potential application of DTI emerges in the field of brain functional MRI, which can be faster and more accurate than BOLD fMRI [14]. The DTI experiment requires at least six diffusion weighted imaging (DWI) measurements with non-collinear directions of the diffusion sensitizing gradients and one additional reference measurement, usually performed without any diffusion gradient. The

⁎ Corresponding author. E-mail address: Karol.Borkowski@fis.agh.edu.pl (K. Borkowski). http://dx.doi.org/10.1016/j.mri.2016.10.002 0730-725X/© 2016 Elsevier Inc. All rights reserved.

parameters of these gradients for a given sequence are incorporated in the 3 × 3 symmetric matrix for each gradient direction, so called B-matrix [15]. It should be pointed out that the B-matrix also depends weakly on other factors like imaging gradients, eddy current effects and other background interference [15–17]. Therefore, it is virtually impossible to calculate the B-matrix analytically taking into account all of those effects and thus commercial systems usually provide its approximated form. Such a practice can lead to systematic errors and a decrease in accuracy. There are several ways of indicating the B-matrix more accurately, e.g. refocusing each diffusion gradient before the imaging gradient is turned on and refocusing each imaging gradient before the next diffusion gradient [18] or acquiring data twice in each direction; once with the given diffusion gradient and once with the opposite polarity [19,20]. However, applying one of these methods entails prolonged acquisition time. The imaging gradient effect can also be reduced by establishing the optimal diffusion gradient scheme [21]. Another important fact is that, in general case, due to the inhomogeneity of the gradients which affect the B-matrix, it is not spatially constant [22–24]. For diffusion imaging, the spatial variability of the B-matrix results in serious inaccuracies of the quantitative characterization of the diffusion. Most of the effects caused by gradient uniformities can be removed by means of an

2

K. Borkowski et al. / Magnetic Resonance Imaging 36 (2017) 1–6

Table 1 Diffusion gradient components along readout (GR), phase (GP) and slice (GS) encoding directions for each of the six diffusion gradient directions. Index

GR

GP

GS

1 2 3 4 5 6

1.00 0.45 0.45 0.45 0.45 −0.45

0.00 −0.90 −0.28 0.72 0.72 0.28

0.00 0.00 0.85 −0.53 0.53 0.85

approximation of the magnetic field using spherical harmonic expansion [22,25]. The BSD-DTI (B-matrix spatial distribution in DTI) is an alternative method of determining the exact form of the B-matrix. The actual values of the matrix in a particular region of interest (ROI) are derived experimentally with the use of an anisotropic phantom with a well-defined structure. Unlike previous methods, in this technique, knowledge of almost all imaging sequence parameters is not required. In order to proceed, the BSD calibration phantom is situated inside an MRI scanner and the standard DTI measurement is performed. Subsequently the phantom is mechanically rotated using a certain set of Euler angles and the measurement is repeated to complete the data set [23,26]. Previous studies [23,27,28] were performed under the assumption of the high uniformity of the applied phantom. It means that the diffusion tensor was assumed to be the same across its entire volume. We recently called this approach uniform BSD-DTI or just uBSD-DTI. This method is simpler to implement and less time-consuming, but on the other hand, it requires using phantoms of very high quality, which are expensive and difficult to manufacture. Nevertheless, the BSD-DTI method is not limited only to such cases. If the spatial distribution of the diffusion tensor is well known in the coordinate system associated with the phantom, it can be derived in the laboratory coordinate system by a rotation transformation [29]. This approach allows us to use basically any anisotropic phantom with well-known diffusion properties. The only requirement is that the phantom parameters must be constant during the calibration. In this study we have examined the impact of the B-matrix spatial variability and the anisotropic phantom imperfections on the accuracy of the uBSD-DTI and BSD-DTI approaches and compare the results with standard DTI. 2. Material and methods The simulation process was divided into three stages. The first concerned establishing the simulated experiment conditions, the structure of the virtual anisotropic phantom, patterns of the

B-matrix spatial distribution and the outcome of the diffusion tensor imaging under such assumptions. The second pertained to the BSD calibration and the third to the threefold calculation of the diffusion tensor. The workflow presenting the whole simulation process was depicted in Fig. 8. 2.1. The preparation stage In order to simulate a realistic experiment, six diffusion gradient directions were taken from a clinical scanner (GE 3.0 T Discovery 750 MR) DTI sequence (Table 1). The standard B-matrix (constant in the entire imaging volume) for each diffusion gradient direction was derived accordingly to the formulas proposed by Mattiello [15] for the particular sequence parameters (Eqs. (1a) to (1f)). The simulated in-plane image size was 25 × 25 pixels with 25 slices without interleaves. The BDS calibration procedure does not depend on the applied imaging protocol, therefore the presented study remains valid for any other set of parameters. bRR ¼ 3:51 þ 136:89GR þ 2228:52GR

2

ð1aÞ

2

ð1bÞ

bPP ¼ 3:19 þ 130:63GP þ 2228:52GP 2

bSS ¼ 1:76−58:50GS þ 2228:52GS

ð1cÞ

bRP ¼ bPR ¼ 3:17 þ 68:45GP þ 65:31GR þ 2228:52GR GP

ð1dÞ

bRS ¼ bSR ¼ −1:84−29:25GP þ 65:31GS þ 2228:52GP GS

ð1eÞ

bPS ¼ bSP ¼ −1:93−29:25GR þ 68:45GS þ 2228:52GR GS

ð1fÞ

The set of six equations above forms a B-matrix: 2

bRR B ¼ 4 bPR bSR

bRP bPP bSP

3 bRS bPS 5 bSS

ð2Þ

The B-matrix Spatial Distribution calibration rests on the assumption that the B-matrix varies spatially, so for each gradient direction and each point in space, and thus for each voxel of the image, an individual B-matrix should be derived. In order to reproduce such conditions, the standard B-matrix was spatially altered with three different patterns. Each pattern was generated by the multiplication of the diffusion gradient along three orthogonal directions (R, P, S) with one of the pattern functions listed below: p1 ¼ σ∙ðR þ P þ S−36Þ=10

ð3aÞ

p2 ¼ σ∙ðjR−12j þ jP−12j þ jS−12j−13:83Þ=4:05

ð3bÞ

 p3 ¼

σ ∙ðR þ P þ S−36Þ=10; for : GR ; GS σ ∙ð36−R−P−SÞ=10; for : GP

Fig. 1. Visualization of the pattern p1 = σ ∙ (R + P + S − 36)/10 across three slices. From left to right: R = 1, R = 13, R = 25, respectively.

ð3cÞ

K. Borkowski et al. / Magnetic Resonance Imaging 36 (2017) 1–6

3

Fig. 2. Visualization of the pattern p2 = σ ∙ (|R − 12| + |P − 12| + |S − 12| − 13.83)/4.05 across three slices. From left to right: R = 1, R = 13, R = 25, respectively.

Where R, P, S are the coordinates of a given voxel and σ is the standard deviation across the chosen region of interest (ROI). For the purpose of this study a spherical ROI of 25 voxels diameter, located in the center of the slice was established and used in all further analysis. All the patterns ensure a mean value equal to zero. For better understanding how the patterns modify the B-matrix spatial distribution, let us consider the following example. The B-matrix associated with gradient G1 for the voxel of coordinates (1, 2, 3) in read-out, phase-encoding and slice-selection direction, respectively, is distorted with pattern p1 and σ equal to 2%. Then, the bRR element of the matrix becomes: 2

bRR ¼ 3:51 þ 136:89∙ðGR ∙p1 Þ þ 2228:52∙ðGR ∙p1 Þ ¼   1 þ 2 þ 3−36 ¼ 3:51 þ 136:89∙ 1:00∙0:02∙ 10   1 þ 2 þ 3−36 2 þ 2228:52∙ 1:00∙0:02∙ 10



2.2. The BSD calibration The BSD calibration requires conducting DTI measurements of the phantom in six various positions defined by the following Euler angles: (0°, 0°, 0°), (90°, 0°, 0°), (45°, 0°, 0°), (0°, 90°, 0°), (0°, 45°, 0°), (45°, 45°, 0°), where in the initial position the eigenvectors of the diffusion tensor describing the phantom were aligned with the laboratory coordinate frame. For each voxel and each diffusion gradient direction the logarithm of the ratio of the diffusion weighted signal to the reference signal was derived from the Stejskal–Tanner equation [30]: ln

ð4Þ

The calculated patterns are visualized in Figs. 1–3. An additional pattern p0 being simple Gaussian noise was also considered. In order to carry out the BSD-DTI calibration an anisotropic phantom is required. In this study the model phantom was assumed to be a cube of 25 voxel side length, characterized by the following eigenvalues of the diffusion tensor: 0.002 mm 2/s, 0.002 mm 2/s, and 0.001 mm 2/s, which is an idealized model of the cubic glass plate phantom consisting of an array of thin glass plates separated with thin layers of water [27]. Since in the real situation the phantom structure is not ideal and the eigenvalues should vary in some range, they were altered with patterns similar to those applied to the B-matrices. Another pattern applied to the phantom was Gaussian noise with mean value equal to 0. Such pattern was denoted as p0. The BSD calibration is likely to depend on the quality of the phantom, thus several cases with a standard deviation of noise varying from 0.2% to 2% were analyzed. From now on, the standard deviation concerning the B-matrix

Fig. 3. Visualization of the pattern p3 ¼

distribution is denoted by σB, while the standard deviation concerning the diffusion properties of the phantom, by σph.

  Sx ¼ −B∙D; S0

ð5Þ

where: B is the standard B-matrix, and D is diffusion tensor in a particular voxel, derived in the preparation stage. For simplicity, the left hand side of the above equation is further called just signals. All the signals were distorted with Gaussian noise with 1% relative standard deviation (RSD) being similar to the noise level in a real measurement. Based on the signals generated, two types of spatial distribution of the B-matrix were derived from Eq. (5); the former under the assumption of the high uniformity of the phantom, i. e. for each phantom position an average diffusion tensor across the whole phantom was calculated and introduced to the Eq. (5) (further this distribution is referred to as b-uniform), and the latter with individual tensors characterizing each particular voxel in the imaging space (referred to as b-complete). 2.3. Calculation of the diffusion tensors The validation of the calibration was done using a model isotropic phantom with diffusion coefficient equal to 0.002 mm 2/s in each direction. The phantom shape and size were adjusted to the

σ∙ðR þ P þ S−36Þ=10; for : GR ; GS across three slices. From left to right: R = 1, R = 13, R = 25, respectively. σ∙ð36−R−P−SÞ=10; for : GP

4

K. Borkowski et al. / Magnetic Resonance Imaging 36 (2017) 1–6

Table 2 The KSD values for the p1 pattern (σB = 0.5%) of the B-matrix spatial distribution. σph

0,2% 0,4% 0,6% 0,8% 1.0% 1,2% 1,4% 1,6% 1,8% 2.0%

Pattern of the phantom in uBSD-DTI

Pattern of the phantom in BSD-DTI

p0

p1

p2

p3

p0

p1

p2

p3

2,12 1,88 1,60 1,38 1,18 1,04 0,92 0,81 0,73 0,67

2,13 1,87 1,59 1,35 1,16 1,02 0,89 0,80 0,72 0,65

2,04 1,67 1,34 1,09 0,91 0,78 0,68 0,60 0,54 0,49

2,11 1,81 1,53 1,28 1,10 0,95 0,83 0,74 0,67 0,60

2,34 2,34 2,34 2,35 2,34 2,34 2,33 2,34 2,34 2,34

2,32 2,35 2,35 2,32 2,34 2,35 2,34 2,33 2,32 2,34

2,33 2,34 2,35 2,33 2,32 2,34 2,33 2,33 2,34 2,34

2,35 2,35 2,35 2,36 2,34 2,34 2,34 2,35 2,34 2,33

geometry of the analyzed ROI. Simulated signals of the isotropic phantom were prepared in a manner similar to the signals of the anisotropic phantom. Finally the diffusion tensors of the isotropic phantom were calculated threefold applying the following B-matrices: 1) standard, constant B-matrix (standard DTI approach) 2) b-uniform B-matrix distribution (uBSD-DTI approach) 3) b-complete B-matrix distribution (BSD-DTI approach) Each approach was analyzed for all possible combinations of the assumed B-matrix pattern and anisotropic phantom uniformity level. 3. Results The resultant diffusion tensors were diagonalized in order to obtain their eigenvalues. Since the model isotropic phantom was used for the validation of the analyzed approaches, all three eigenvalues are expected to be equal, and the standard deviation of their distribution to be as low as possible. An improvement factor KSD defined as a ratio of the standard deviation of the distribution of the eigenvalues derived through standard DTI approach to the standard deviation of the approach in question. The higher the KSD the better the uniformity of the eigenvalues distribution for the analyzed approach. The comparison of the results was twofold. The KSD factor was calculated for a given B-matrix pattern with various σB and each phantom pattern in the former (Tables 2 and 3 and Figs. 4 and 5); while a given phantom pattern with various σph and each B-matrix pattern was considered in the latter (Tables 4 and 5 and Figs. 6 and 7).

Fig. 4. The dependence of the KSD for the p1 pattern (σB = 0.5%) of the B-matrix spatial distribution on the relative standard deviation of the diffusion properties of the phantom disturbed with various patterns and calibrated with various approaches. Uncertainty of the KSD in order of 10−3 was not presented in the figure.

The results were very similar for each B-matrix pattern in the first representation, as well as for each phantom pattern in the second. Thus only one representative data set was presented in each case. 4. Discussion and conclusions The standard DTI approach serves as a reference for all the remaining simulations. Its KSD by definition is equal to one in every case. Depending on the noise superimposed on either the B-matrix patterns or the calibration phantom structure, the KSD of BSD-DTI approach ranges from 2.3 to 9.0. Interestingly it does not change with the RSD of the phantom for a B-matrix pattern with a given SD, as well as remaining constant for the opposite (the given SD of the phantom structure and changing RSD of the B-matrix pattern). Such behavior is a straightforward result of the definition of the KSD factor, which for S-DTI is always equal to one, even when the SD of the diffusion tensor eigenvalues distribution change. Worth noting is the

Table 3 The KSD values for the p1 pattern (σB = 2.0%) of the B-matrix spatial distribution. σph

0,2% 0,4% 0,6% 0,8% 1.0% 1,2% 1,4% 1,6% 1,8% 2.0%

Pattern of the phantom in uBSD-DTI

Pattern of the phantom in BSD-DTI

p0

p1

p2

p3

p0

p1

p2

p3

4,34 3,87 3,35 2,84 2,49 2,15 1,90 1,70 1,54 1,39

4,36 3,86 3,31 2,83 2,43 2,11 1,87 1,67 1,50 1,36

4,20 3,45 2,76 2,26 1,89 1,61 1,40 1,24 1,10 0,99

4,31 3,76 3,17 2,68 2,28 1,97 1,74 1,55 1,39 1,26

8,66 8,70 8,69 8,68 8,75 8,71 8,69 8,72 8,69 8,76

8,71 8,71 8,71 8,74 8,69 8,58 8,71 8,68 8,70 8,65

8,72 8,72 8,67 8,74 8,75 8,72 8,74 8,73 8,70 8,78

8,67 8,70 8,75 8,74 8,68 8,72 8,69 8,68 8,67 8,69

Fig. 5. Deviation of the diffusion properties of the phantom disturbed with various patterns and calibrated with various approaches. Uncertainty of the KSD in order of 10−3 was not presented in the figure.

K. Borkowski et al. / Magnetic Resonance Imaging 36 (2017) 1–6

5

Table 4 The KSD values for the p0 pattern (σph = 0.5%) of the phantom. σB

0,2% 0,4% 0,6% 0,8% 1.0% 1,2% 1,4% 1,6% 1,8% 2.0%

Pattern of the B-matrix in uBSD-DTI

Pattern of the B-matrix in BSD-DTI

p1

p2

p3

p1

p2

p3

2,12 1,88 1,60 1,38 1,18 1,04 0,92 0,81 0,73 0,67

2,10 1,86 1,59 1,37 1,18 1,03 0,90 0,80 0,71 0,66

2,15 1,91 1,63 1,38 1,20 1,04 0,92 0,82 0,75 0,68

2,34 2,34 2,34 2,35 2,34 2,34 2,33 2,34 2,34 2,34

2,33 2,30 2,32 2,31 2,30 2,33 2,30 2,31 2,33 2,30

2,37 2,39 2,39 2,37 2,38 2,37 2,37 2,38 2,37 2,37

fact that, for a given B-matrix pattern, the KSD is the same for all considered patterns superimposed on the phantom structure. The performance of the uBSD-DTI approach is much more sensitive to both the quality of the phantom and the B-matrix Spatial Distribution. In the case of a given B-matrix pattern with relatively small SD and small RSD of the phantom structure the uBSD calibration works almost as well as BSD. However with the increase of the phantom structure RSD its performance drops and reveals its dependence on the phantom structure pattern. This is awaited, since for a given B-matrix pattern a good phantom whose structure is defined by measurement should reflect the structure of the B-matrix. An increase in the B-matrix pattern SD increases the gap between the performance of the BSD and uBSD but also gives better results of uBSD in the whole range of the analyzed RSD of the phantom structure. For a given pattern of the phantom structure, the performance of uBSD does not depend on the actual pattern of the B-matrix, but decreases with its RSD. The BSD technique allows to identify and minimize the systematic errors resulting from the diffusion gradient inhomogeneities, therefore it should enhance the quality of virtually any kind of DTI measurement. Such an improvement can be clinically important in the case of DTI-based tractography. The spatial variations of the B-matrix result in differences between the real and measured orientations of fibers [23]. This can also lead to the erroneous indication of white matter tracts. Therefore, the application of BSD-DTI technique in this field may result in the improvement of the reliability of tractography. Moreover, the accuracy and precision of indicating the diffusion tensor is important while comparing multi-center measurements [31]. Enhancement of these parameters may contribute to reproducibility of the DTI measurements.

Fig. 6. The dependence of the KSD for the p0 pattern (σph = 5.0%) of the phantom on the relative standard deviation of the diffusion properties for various patterns of the B-matrix and various calibration approaches. Uncertainty of the KSD in order of 10−3 was not presented in the figure.

In conclusion, the simulation confirmed the very good performance of the BSD-DTI calibration technique, which neither depends on the inhomogeneity of the phantom nor on the SD of the B-matrix. The simulations rested on the assumption that the phantom structure can be exactly defined, which corresponds to the very well defined structure of a real phantom. Additionally, if the structure is also highly homogeneous, then the uBSD calibration approach might be sufficient.

Acknowledgements This work was supported by the National Centre of Research and Development (contract No. PBS2/A2/16/2013) and by the Polish Ministry of Science and Higher Education. K.B. and K.K. would like to thank the Marian Smoluchowski Cracow Scientific Consortium – KNOW for their support.

Table 5 The KSD values for the p0 pattern (σph = 2.0%) of the phantom. σB

0,2% 0,4% 0,6% 0,8% 1.0% 1,2% 1,4% 1,6% 1,8% 2.0%

Pattern of the B-matrix in uBSD-DTI

Pattern of the B-matrix in BSD-DTI

p1

p2

p3

p1

p2

p3

4,34 3,87 3,35 2,84 2,49 2,15 1,90 1,70 1,54 1,39

4,30 3,80 3,27 2,83 2,43 2,12 1,88 1,65 1,50 1,36

4,51 3,99 3,44 2,88 2,53 2,20 1,94 1,73 1,56 1,42

8,66 8,70 8,69 8,68 8,75 8,71 8,69 8,72 8,69 8,76

8,57 8,55 8,59 8,56 8,55 8,59 8,55 8,59 8,58 8,55

8,98 8,97 8,96 8,94 8,91 8,97 8,92 8,93 8,95 8,94

Fig. 7. The dependence of the KSD for the p0 pattern (σph = 2.0%) of the phantom on the relative standard deviation of the diffusion properties for various patterns of the B-matrix and various calibration approaches. Uncertainty of the KSD in order of 10−3 was not presented in the figure.

6

K. Borkowski et al. / Magnetic Resonance Imaging 36 (2017) 1–6

Fig. 8. The workflow of the simulated process. The preparation stage is presented on the top (green and yellow boxes), the calibration stage in the middle (blue and red) and the DTI measurements on the bottom (violet).

References [1] Lerner A, Mogensen MA, Kim PE, Shiroishi MS, Hwang DH, Law M. Clinical applications of diffusion tensor imaging. World Neurosurg 2014;82(1):96–109.

[2] Berman J. Diffusion MR tractography as a tool for surgical planning. Magn Reson Imaging Clin N Am 2009;17(2):205–14. [3] Coenen VA, Krings T, Mayfrank L, Polin RS, Reinges MH, Thron A, et al. Threedimensional visualization of the pyramidal tract in a neuronavigation system during brain tumor surgery: first experiences and technical note. Neurosurgery 2001;49(1):86–92-93. [4] Behrens TEJ, Johansen-Berg H, Woolrich MW, Smith SM, Wheeler-Kingshott CAM, Boulby PA, et al. Non-invasive mapping of connections between human thalamus and cortex using diffusion imaging. Nat Neurosci 2003;6(7):750–7. [5] Maesawa S, Fujii M, Nakahara N, Watanabe T, Wakabayashi T, Yoshida J. Intraoperative tractography and motor evoked potential (MEP) monitoring in surgery for Gliomas around the Corticospinal tract. World Neurosurg 2010; 74(1):153–61. [6] Kimiwada T, Juhász C, Makki M, Muzik O, Chugani DC, Asano E, et al. Hippocampal and thalamic diffusion abnormalities in children with temporal lobe epilepsy. Epilepsia 2006;47(1):167–75. [7] Bammer R, Augustin M, Strasser-Fuchs S, Seifert T, Kapeller P, Stollberger R, et al. Magnetic resonance diffusion tensor imaging for characterizing diffuse and focal white matter abnormalities in multiple sclerosis. Magn Reson Med 2000;44(4): 583–91. [8] Calabrese M, Rinaldi F, Seppi D, Favaretto A, Squarcina L, Mattisi I, et al. Cortical diffusion-tensor imaging abnormalities in multiple sclerosis: a 3-year longitudinal study. Radiology 2011;261(3):891–8. [9] Bozzali M, Falini A, Franceschi M, Cercignani M, Zuffi M, Scotti G, et al. White matter damage in Alzheimer's disease assessed in vivo using diffusion tensor magnetic resonance imaging. J Neurol Neurosurg Psychiatry 2002;72(6):742–6. [10] Huang J, Auchus AP. Diffusion tensor imaging of normal appearing white matter and its correlation with cognitive functioning in mild cognitive impairment and Alzheimer's disease. Ann N Y Acad Sci 2007;1097:259–64. [11] Medina DA, Gaviria M. Diffusion tensor imaging investigations in Alzheimer's disease: the resurgence of white matter compromise in the cortical dysfunction of the aging brain. Neuropsychiatr Dis Treat 2008;4(4):737–42. [12] Mukherjee P, Bahn MM, McKinstry RC, Shimony JS, Cull TS, Akbudak E, et al. Differences between gray matter and white matter water diffusion in stroke: diffusion-tensor MR imaging in 12 patients. Radiology 2000;215(1):211–20. [13] Mukherjee P. Diffusion tensor imaging and fiber tractography in acute stroke. Neuroimaging Clin 2005;15(3):655–65. [14] Mandl RCW, Schnack HG, Zwiers MP, van der Schaaf A, Kahn RS, Hulshoff Pol HE. Functional diffusion tensor imaging: measuring task-related fractional anisotropy changes in the human brain along white matter tracts. PloS One 2008; 3(11):e3631. [15] Mattiello J, Basser PJ, Le Bihan D. The b matrix in diffusion tensor echo-planar imaging. Magn Reson Med 1997;37(2):292–300. [16] Mattiello J, Basser PJ, Lebihan D. Analytical expressions for the b matrix in NMR diffusion imaging and spectroscopy. J Magn Reson A 1994;108(2):131–41. [17] Boujraf S, Luypaert R, Osteaux M. B matrix errors in echo planar diffusion tensor imaging. J Appl Clin Med Phys 2001;2(3):178–83. [18] Hsu EW, Mori S. Analytical expressions for the NMR apparent diffusion coefficients in an anisotropic system and a simplified method for determining fiber orientation. Magn Reson Med 1995;34(2):194–200. [19] Neeman M, Freyer JP, Sillerud LO. A simple method for obtaining cross-term-free images for diffusion anisotropy studies in NMR microimaging. Magn Reson Med 1991;21(1):138–43. [20] Güllmar D, Haueisen J, Reichenbach JR. Analysis of b-value calculations in diffusion weighted and diffusion tensor imaging. Concepts Magn Reson Part A 2005;25A(1):53–66. [21] Ozcan A. Minimization of imaging gradient effects in diffusion tensor imaging. IEEE Trans Med Imaging 2011;30(3):642–54. [22] Bammer R, Markl M, Barnett A, Acar B, Alley MT, Pelc NJ, et al. Analysis and generalized correction of the effect of spatial gradient field distortions in diffusion-weighted imaging. Magn Reson Med 2003;50(3):560–9. [23] Krzyżak AT, Olejniczak Z. Improving the accuracy of PGSE DTI experiments using the spatial distribution of b matrix. Magn Reson Imaging 2015;33(3):286–95. [24] Krzyzak AT, Klodowski K. The b matrix calculation using the anisotropic phantoms for DWI and DTI experiments. Conf Proc IEEE Eng Med Biol Soc 2015; 2015:418–21. [25] Janke A, Zhao H, Cowin GJ, Galloway GJ, Doddrell DM. Use of spherical harmonic deconvolution methods to compensate for nonlinear gradient effects on MRI images. Magn Reson Med 2004;52(1):115–22. [26] Krzyzak A. Anisotropic diffusion phantom for calibration of diffusion tensor imaging pulse sequences used in MRI; 2014[US8643369 B2]. [27] Kłodowski K, Krzyżak AT. Innovative anisotropic phantoms for calibration of diffusion tensor imaging sequences. Magn Reson Imaging 2016;34(4):404–9. [28] Krzyzak AT, Klodowski K. The b matrix calculation using the anisotropic phantoms for DWI and DTI experiments. 2015 37th annual international conference of the IEEE engineering in medicine and biology society (EMBC); 2015. p. 418–21. [29] Krzyzak A, Borkowski K. Theoretical analysis of phantom rotations in BSD-DTI. 2015 37th annual international conference of the IEEE engineering in medicine and biology society (EMBC); 2015. p. 410–3. [30] Stejskal EO, Tanner JE. Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient. J Chem Phys 1965;42(1):288–92. [31] Zhu T, Hu R, Qiu X, Taylor M, Tso Y, Yiannoutsos C, et al. Quantification of accuracy and precision of multi-center DTI measurements: a diffusion phantom and human brain study. Neuroimage 2011;56(3):1398–411.