1　Introduction

       The tortuosity parameter^{[1,2,3,4,5,6,7,8]} describes geometrical hindrance of an environment relative to an obstacle-free medium. If the obstacles exhibit some directional preference, the hindrance becomes anisotropic, that is, it depends on direction. For example, the molecules diffuse more readily along the white matter fibers than across them. In a macroscopically homogeneous and anisotropic environment, the tortuosity takes the shape of a symmetric tensor of the second order, which can be represented by a 3 × 3 matrix with six independent values. The tortuosity tensor combines with the scalar free diffusion coefficient into a tensor of apparent diffusion . However, despite all these complexities, the pulse field gradient method proposed by Stejskal and Tanner is still used to calculate signal attenuation due to diffusion along any single direction of the diffusion gradient. The only real difference from the homogeneous and isotropic case is that the experiment must be repeated using at least six non-collinear directions of the diffusion gradient to obtain six independent components of the apparent diffusion tensor. It is therefore fair to conclude that the pulse field gradient method proposed by Stejskal and Tanner is at the heart of most modern diffusion and DTI experiments.

       DTI has become a very popular MR imaging modality and is developing into an important tool for non-invasive study and characterization of the brain white matter. It has been applied to the study of many neurological brain disorders such as schizophrenia, cocaine addiction, HIV infection, alcoholism, geriatric depression and Alzheimer's disease. An overview of theoretical issues surrounding the DTI technique can be found in literature^[9,10,11]. A review of DTI applications in neuroscience is presented by Lim and Helpern^[12].

       It is our view that good understanding of the Stejskal-Tanner formulae is essential for logical design or interpretation of any MR diffusion experiment. Unfortunately, the derivation in a completely general case is nontrivial as it involves solving the Bloch-Torrey partial differential equations^[4,5]. We need to briefly review a simple connection between Gaussian diffusion and random walks. A comprehensive treatment can be found, for example, in works of Callagan, Haacke et al and Chandrasekhar^[13,14,15].

       If we begin with a qualitative description (within the rotating frame of reference) of diffusion on the NMR signal, an RF pulse turns all the equilibrium magnetization M0 into the transverse plane, perpen-dicular to the main static magnetic field B0. The magnetization vector will rotate around it at angular frequency given by the Larmor equation:ω= γB0.

       Where γ is the gyromagnetic ratio (for a hydrogen proton). After the excitation, a short and strong gradient G is applied along the x-axis, changing the constant main field B0 to a spatially variable field

       Larmor frequencies therefore become different at different places along the x-axis. When the gradient is switched off again, some phase differences will have accumulated between the spins at different positions. At this point, we just wait for a period equal to the diffusion time t=nΔt. Then an opposite but otherwise identical gradient, -G, is applied. If the atoms did not change their positions during the diffusion time, all the phase differences would be perfectly reversed and the magnetization would be fully restored (apart from the neglected relaxation effects). However, if the spins move, the second gradient finds them at different locations than the first one and the phases will be reversed "incorrectly" . The result is phase dispersion in the measured sample and loss of signal when all the spins are eventually summed up to form the magnetization vector. Faster diffusion (larger D) means that the spins have bigger chance to travel farther and therefore experience larger magnetic field changes due to diffusion gradients. This causes larger spread in the phases and therefore results in a smaller signal.

       Bloch NMR flow equations, describing the dynamics of the magnetization vector, have been solved analytically by imposing specific adiabatic and non adiabatic conditions^{[16,17,18,19,20,21,22]}:

       It may be very important to solve the flow equation for general cases without imposing these conditions.

       In this investigation, we solved the NMR flow equation for general cases using the Bessel properties and functions as mathematical tools to obtain the NMR transverse magnetization and the diffusion coefficient which depends on NMR relaxation parameters. The NMR system is considered in both real and complex domains which can easily provide additional information to understand the basic concept of NMR/MRI without the usual exponential functions.

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2　Mathematical formulation

       Specifically, we consider the fluid particle (on the atomic scale) which either initially or in some average sense is in steady rotation. We apply a mathematical algorithm to describe in detail the dynamical state of the flowing fluid particle starting from the Bloch NMR flow equations^[18,19,20]. We study the flow properties of the modified time independent Bloch NMR flow equations which describe the dynamics of fluid flow under the influence of RF magnetic field as derived in the earlier studies^{[16,17,18,19,20,21,22]} when resonance condition exists at Larmor frequency:

       Then the x, y, z components (in the rotating frame) of the magnetization of a fluid bolus moving with variable velocity v(x) is given by the Bloch equations which may be written as follows:

       Subject to the following conditions:

       i. Mo ≠ Mz a situation which holds well in general and in particular when the RFB1(x) field is strong say of the order of 1.0G or more.

       ii. Before entering signal detector coil, fluid particles has magnetization Mx = 0, My = 0.

       iii. If B1(x) is large, B1(x) >> 1G or more so that My of the fluid bolus changes appreciably from Mo.

       iv. In anisotropic diffusion, the diffusion coefficient, D=D(x),v(x) =u0x and where t is considered a constant.

       γ denotes the gyromagnetic ratio of fluid spins, ω/2π is the RF excitation frequency, f0/γ is the off-resonance field in the rot ating frame of reference.T1 and T2 are the spin-lattice and spin-spin relaxation times respectively, the reciprocals of T1 and T2 are defined as relaxation rates. RF B1 is the spatially varying magnetic field and v is the fluid flow velocity. Equations (1f and 1g) give a second order non-homogeneous differential equation which may be fundamental for the development of new magnetic resonance techniques. Hence, the Bloch NMR flow equation becomes:

       Subject to the following definition:

       In a typical MRI procedure, G is the pulsed gradient applied for the length of time δ and q=1. However, for spin magnetization which is independent of the length of time t (constant time), we shall substitute for equations (1a) and (3) in equation (2) :

       When the radiofrequency field turns all the equilibrium magnetization M0 into the transverse plane, perpendicular to the main static magnetic field B0, maximum signal is expected for maximum RF B1(x) field and M0 is minimum (say M0 ≈ 0). For a maximum value of My, we can write equation (4) as follows:

       Equation (5) is a general form of an equation transformable into Bessel’s equation of order β with parameter λ. A general solution of equation (5) is in the form:

       where k, m0, η, β are all constants defined as:

       Equation (6) therefore becomes:

       Since the transverse magnetization needs to have a finite limit as x tends to zero, c2 =0 and if we set the constant, c1=1, we have:

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Tab. 1  Values of the relaxation times^[23] of human tissues at 1.5 T as they relate to the Bessel function parameters

3　Complex radio frequency magnetic field

       In equation (3a), if the applied radio frequency field when k = 0 and q=1 is complex for a complex system, we may write:

       It should be noted that in the light of equations (6) and (7), we may write Therefore, we can write equations (11) as

       In this case, β is not restricted. If β is not an integer, we have the alternative form

       It may be useful especially in diffusion MRI to consider the applied radio frequency field in equation (3b) as complex function when k = 0 and q=1 given as follows:

       We can write the NMR transverse magnetization as

       In equation (17), for real values of x the equation

       Is a complex number with characteristic length or modulus Fm(x) and a characteristic angle or amplitude

       It may be more useful to replace the expression berm(x)+ibeim(x) by the exponential expression Fm (x)e^iAm^(x). Similarly, the equation

       has a characteristic length or modulus, Hm(x) and a characteristic angle of amplitude

       The expression kermx+i keimx can also be replaced by the exponential expression Hm (x)e^iBm^(x).

       We can finally write equation (17) as

       The Bessel functions and properties when k =0 in equation (9) are readily available for the analysis of MRI signals as derived in equations (12, 14, 15, 17,22) , These functions have been graphically displayed for certain important values of relaxation parameter m when λ is a unit parameter either real of complex^{[24,25,26,27]}. The measurement may be carried out by varying either the magnitude G, of the pulsed gradient or the length of time τ=δ, the time between pulses in the experiment. This sum is best tackled using the Euler theorem exp iA(x)=cosA(x) + i sinA(x).

       Abbreviating the expression, we have:

       Therefore, we derive^[13,14,15]

       The NMR signal undergoes various stages of amplification, filtering and other transformations. It is therefore not possible to measure M0 in absolute terms. We can remedy this unfortunate drawback of NMR by measuring the same sample twice: once without the diffusion gradients, to obtain un-attenuated signal S0, and once with them, to obtain signal S. The M0 term in the signal will stay the same but the attenuation term will disappear from S0. We then calculate the ratio of the signals with and without the diffusion gradients:

       The last step is to eliminate Δx and express the result in terms of experimentally accessible variables.

       where the so-called b -value is introduced^[13,14,15],

       S is the signal strength in a pulse sequence with a pair of balanced diffusion-sensitizing gradients of strength G, each of a duration δ and with a delay τ between them. S0 is the signal strength in an identical experiment but without the diffusion gradient pair. When it can be safely assumed that δ<< τ, the expression for b (usually called b-value) simplifies to equation (24). Equation (22) becomes

       where D1m and D2m may represent diffusion coefficients in different locations in the voxel of anisotropic tissue. These diffusion coefficients can be analyzed using the Bessel properties in equations (19, 21) as:

       If we set m=1/2 in equation (26a), we have the following expressions:

       In equations (26a, 27a and 27b),Γ (m+f+1) is the gamma function.

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Tab. 2  Values of the relaxation times of human tissues at 1.5 T^[23] as they relate to the Bessel parameters in the solutions (14), (15) and (22)

4　Conclusion

       We have solved the time independent NMR flow equation and obtained analytical solutions for the NMR transverse magnetizations in terms of Bessel properties and functions which can reveal wealth of MRI physics and make NMR theory, dynamics and applications more interesting, appealing, motivating and exciting. It is significant to note from equations (8, 11) that the duration of time when the pulse is applied depends completely on the relaxation parameters of the sample. This makes the technique presented in this report uniquely applicable to anisotropic tissues. For illustration, from figures 1,figures 2,figures 3, the changes in the MRI experimental parameters λ=γGδ and the signal are spatially represented at the microscopic level. It is observed that MRI signal decreases with increase in either T1 or T2 relaxation parameters. This can be very useful in planning for appropriate MRI experiments. Figure 3, is based on equations(10,12) which is completely described by the Bessel functions and properties in terms of MRI experimental parameters.The behaviour of the transverse magnetizations and the MRI parameters are completely controlled by Bessel functions. This can be extremely useful for MRI simulations of Biological and Physiological systems.

       Based on equations (3b, 5, 24, 25), quantity b depends on the NMR hardware especially the radiofrequency coil in use and the pulse program controlling it but does not depend on the diffusion constant. The computer program can be designed accordingly using the Bessel functions and properties. It is noteworthy that in the traditional NMR/MRI experiments, the expression for the b-value usually becomes more complicated if various NMR sequence intricacies are taken into account. Most importantly, it is assumed that the diffusion gradients are switched on for a negligible period of time in comparison with the diffusion time. However, Equation (24) does capture the most important features-square dependency on the gradient moment Gδ and linear dependency on the diffusion time t.

       The NMR signal from nuclei following the simple diffusion model is attenuated due to phase randomization as

       This is a "discrete" form of the Stejskal-Tanner equation; equation (28) offers the possibility to measure the diffusion constant. This can be done by acquiring signals with and without diffusion gradients and calculating D. Better still, one can obtain the signal many times with different b-values and obtain D by a fitting procedure from equation (28). However, equation (4) can generate functions as solutions to various differential equations applicable in MRI physics, typically describing wavelike oscillatory behavior or a combination of oscillation and exponential decay or growth. For example, if q=2 in equation (3b) gives the phase of the spins as

       where the b-value is,

       For the NMR transverse magnetization represented in figures 4,figures 5, we can easily determine the value of D, G or τ based on equations (8, 9, 29) for anisotropic tissue environment where the value of T1 and T2 relxation parameters may vary acording to directional preference. In this way the diffusion coefficient is proportional to the characteristic angle of the transverse magnetization. Figure 6 shows that the field gradient G, depends completely on T1 and T2 relaxation parameters. Figure 7 gives unique contrasts for different human tissues at 1.5 T, and the images show the possibility of using Equations (26a and 26b) for spatial diffusion mapping. This may prove to be very useful in the study of tissue diseases in which diffusion properties are progressively affected.

       It may be significant to note for example in edema with variable wall thickness, a solution for the rotation, moment and shear force on the cylinder requires successive differentiation of equation (17) starting from a knowledge of the deflection of the structure. The radial deflection of the cylinder can be shown to be:

       where ber’(λx), bei’(λx), ker’(λx) and kei’(λx) are the first derivatives of equation (17). The terms c1 and c2 are constants that are to bedetermined from the boundary conditions for the particular problem being investigated^[24]. The derivative dMy/dx gives the rotation of the cylinder. The bending moment is obtained from μx=-DdMy²/dx² and the radial shear force Sx= dμx/dx^[24]. These parameters, with some modifications, can be used to obtain solutions for a range of medical and biomedical problems. In our next investigation, a mathematical model and computational analysis of nano drugs based on Bloch NMR flow equation and Bessel functions will be in focus where the properties of equation (4) will be explored to monitor the effects of drugs on the functional activities of different tissues especially the brain.

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Fig. 1  The plots of transverse magnetization against x and λ [using equation (12)] for a time of 2 s, using the relaxation-time values at 1.5 T^[23] of (A) skeletal muscle (B) heart muscle (C) liver (D) kidney (E) spleen (F) fatty tissue (G) gray brain matter (H) white brain matter.
Fig. 2  The plots of transverse magnetization against x and λ [using equation (12)] for a time of 2 s and the relaxation-time values at 1.5 T^[23] of fatty tissue within the ranges (A) x: 0—5 m, (B) x: 0—0.05 m, (C) x: 0—0.005 m, (D) x: 0—0.000005 m.

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Fig. 3  Plots of D1m against λ and x, for τ=1.0 s and the relaxation-time values at 1.5 T^[23] of (A) skeletal muscle (B) heart muscle (C) liver (D) kidney (E) spleen (F) fatty tissue (G) gray brain matter (H) white brain matter.
Fig. 4  Plots of D1m against λ and τ, for x=3 µm and the relaxation-time values at 1.5 T^[23] of (A) skeletal muscle (B) heart muscle (C) liver (D) kidney (E) spleen (F) fatty tissue (G) gray brain matter (H) white brain matter.

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Fig. 5  Plots of D2m against λ and x, for τ =1.0 s and the relaxation-time values at 1.5 T^[23] of (A) skeletal muscle (B) heart muscle (C) liver (D) kidney (E) spleen (F) fatty tissue (G) gray brain matter (H) white brain matter.
Fig. 6  Plots of D2m against λ and τ, for x=3 µm and the relaxation-time values at 1.5 T^[23] of (A) skeletal muscle (B) heart muscle (C) liver (D) kidney (E) spleen (F) fatty tissue (G) gray brain matter (H) white brain matter.

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Fig. 7  Density plots of (A) D1m against λ and x, τ=1.0 s for fatty tissue at 1.5 T^[23] (B) D1m against λ and x, τ=1.0 s for gray brain matter at 1.5 T (C) D1m against λ and τ, x=3 µm for fatty tissue at 1.5 T (D) D1m against λ and τ, x=3 µm for gray brain matter at 1.5 T (E) D2m against λ and x, τ=1.0 s for fatty tissue at 1.5 T^[23] (F) D2m against λ and x, τ=1.0 s for gray brain matter at 1.5 T (G) D2m against λ and τ, x=3 µm for fatty tissue at 1.5 T (H) D2m against λ and τ, x=3 µm for gray brain matter at 1.5 T.

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