梅沢 栄三

S. Naka, H. Toyoda, T. Takanashi, E. Umezawa. Prog Theor Exp Phys 2014;2014(4):043B03.

PURPOSE: q-ball imaging (QBI) reconstructs the orientation distribution function (ODF) that describes the probability for a spin to diffuse in a given direction, and it is capable of identifying intravoxel multiple fiber orientations. The local maxima of ODF are assumed to indicate fiber orientations, but there is a mismatch between the orientation of a fiber crossing and the local maxima. We propose a novel method, multi-shelled QBI (MS-QBI), that gives a new ODF based on the moment of the probability density function of diffusion displacement. We test the accuracy of the fiber orientation indicated by the new ODF and test fiber tracking using the new ODF. METHODS: We performed tests using numerical simulation. To test the accuracy of fiber orientation, we assumed that 2 fibers cross and evaluated the deviation of the measured crossing angle from the actual angle. To test the fiber tracking, we used a numerical phantom of the cerebral hemisphere containing the corpus callosum, projection fibers, and superior longitudinal fasciculus. In the tests, we compared the results between MS-QBI and conventional QBI under the condition of approximately equal total numbers of diffusion signal samplings between the 2 methods and chose the interpolation parameter such that the stabilities of the results of the angular deviation for the 2 methods were the same. RESULTS: The absolute value of the mean angular deviation was smaller in MS-QBI than in conventional QBI. Using the moment-based ODF improved the accuracy of fiber pathways in fiber tracking but maintained the stability of the results. CONCLUSION: MS-QBI can more accurately identify intravoxel multiple fiber orientations than can QBI, without increasing sampling number. The high accuracy of MS-QBI will contribute to the improved tractography.

q-Space diffusion analysis is a method to obtain the probability density function of the translational displacement of diffusing water molecules. Several quantities can be extracted from the function that indicate a characteristic of the water diffusion in tissue, e.g., the mean displacement of the diffusion, probability for zero displacement, and kurtosis of the function. These quantities are expected to give information about the microstructure of tissues in addition to that obtained from the apparent diffusion coefficient (ADC); however, this method requires high q (i.e., high b) values, which are undesirable in practical applications of the method using clinical magnetic resonance (MR) imaging equipment. We propose a method to obtain certain quantities that indicate a characteristic of the diffusion and that uses low q-value measurements. The quantities we can obtain are the moments of translational displacement, R; the n-th order moment is defined as the average of Rn (n: integer). Kurtosis can also be calculated from the second and fourth moments. We tried to map the moments and kurtosis using clinical MR imaging equipment. We also estimated the inherent errors of the moments obtained. Our method requires precision in measuring spin echo signals and setting q values rather than using high q-value measurements. Although our results show that further error reductions are desired, our method is workable using ordinary clinical MR imaging equipment.

Interslice sensitivity variation in magnetic resonance imaging (MRI) is said to be attributable to crosstalk. Crosstalk is a phenomenon in which deterioration of the rectangular profile of the RF pulse frequency characteristics leads to poor slice profiles, resulting in interference between slices. In the present study, the ideal characteristics of multislice profiles were analyzed through numerical simulation with the Bloch equation using sinc symmetrical RF pulses. The effects of the slice excitation order on the slice profile were also examined, comparing interleaved acquisition and sequential acquisition. The results of the simulation showed that the order of image acquisition affects the slice profile.

It is shown that the one-loop two-point amplitude in {\it Lorentz-invariant} non-commutative (NC) $\phi^3$ theory is finite after subtraction in the commutative limit and satisfies the usual cutting rule, thereby eliminating the unitarity problem in Lorentz-non-invariant NC field theory in the approximation considered.

It is argued that the familiar algebra of the non-commutative space-time with $c$-number $\theta^{\mu\nu}$ is inconsistent from a theoretical point of view. Consistent algebras are obtained by promoting $\theta^{\mu\nu}$ to an anti-symmetric tensor operator ${\hat\theta}^{\mu\nu}$. The simplest among them is Doplicher-Fredenhagen-Roberts (DFR) algebra in which the triple commutator among the coordinate operators is assumed to vanish. This allows us to define the Lorentz-covariant operator fields on the DFR algebra as operators diagonal in the 6-dimensional $\theta$-space of the hermitian operators, ${\hat\theta}^{\mu\nu}$. It is shown that we then recover Carlson-Carone-Zobin (CCZ) formulation of the Lorentz-invariant non-commutative gauge theory with no need of compactification of the extra 6 dimensions. It is also pointed out that a general argument concerning the normalizability of the weight function in the Lorentz metric leads to a division of the $\theta$-space into two disjoint spaces not connected by any Lorentz transformation so that the CCZ covariant moment formula holds true in each space, separately. A non-commutative generalization of Connes' two-sheeted Minkowski space-time is also proposed. Two simple models of quantum field theory are reformulated on $M_4\times Z_2$ obtained in the commutative limit.

The bi-local model of hadrons is studied from the viewpoint of non-commutative geometry formulated so that Higgs-like scalar fields play the role of a bridge, the bi-local fields, connecting different spacetime points. We show that the resultant action for Higgs-like scalar fields has a structure similar to that of the linear sigma model. According to this formalism, we can deduce the dual nature of meson fields as the Nambu-Goldstone bosons associated with chiral symmetry breaking and bound states of quarks.

We study constraints among coupling constants of the standard model obtained in the noncommutative geometry (NCG) method. First, we analyze the evolution of the Higgs boson mass under the renormalization group by adopting the idea of Álvarez et al. For this analysis we derive two certain constraints by modifying Connes's way of constructing the standard model. Next, we find renormalization group invariant (RGI) constraints in the NCG method. We also consider the relation between the condition that a constraint among coupling constants of a model becomes RGI and the condition that the model becomes multiplicative renormalizable by using a simple example.

We study Sogami's generalized covariant derivative method for a $SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L} \times SU(3)_{c}$ model that contains bi-doublet and triplet Higgs bosons. In particular, a detailed study is made on the minimization conditions of the Higgs potential. It is known that a minimization condition and certain phenomenology for a extra gauge boson mass derive a restriction on potential parameters, which requires fine-tuning. We show that the restriction can be reduced to a condition of Yukawa coupling constants giving a heavy mass of the right-handed tau neutrino in our model. We also discuss the consistency among parameter restrictions of our model by taking phenomenology of the Higgs boson masses into account.

S. Naka, E. Umezawa. Prog Theor Phys 1994;92(1):189-204.