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Edited by: Federico Giove, Centro Fermi, Italy

Reviewed by: Silvia Capuani, National Research Council, Italy; Jonathan Doucette, University of British Columbia MRI Science Lab, Canada

*Correspondence: Felix T. Kurz

This article was submitted to Biomedical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In biological tissue, an accumulation of similarly shaped objects with a susceptibility difference to the surrounding tissue generates a local distortion of the external magnetic field in magnetic resonance imaging. It induces stochastic field fluctuations that characteristically influence proton spin dephasing in the vicinity of these magnetic perturbers. The magnetic field correlation that is associated with such local magnetic field inhomogeneities can be expressed in the form of a dynamic frequency autocorrelation function that is related to the time evolution of the measured magnetization. Here, an eigenfunction expansion for two simple magnetic perturber shapes, that of spheres and cylinders, is considered for restricted spin diffusion in a simple model geometry. Then, the concept of generalized moment analysis, an approximation technique that is applied in the study of (non-)reactive processes that involve Brownian motion, allows deriving analytical expressions of the correlation function for different exponential decay forms. Results for the biexponential decay for both spherical and cylindrical magnetized objects are derived and compared with the frequently used (less accurate) monoexponential decay forms. They are in asymptotic agreement with the numerically exact value of the correlation function for long and short times.

When exposed to an external magnetic field as in magnetic resonance imaging (MRI), spatial variations of magnetic susceptibility in heterogeneous systems such as biological tissue induce local inhomogeneities of the magnetic field that are usually visible for macroscopic structures. Susceptibility-weighted imaging and quantitative susceptibility mapping are MRI sequences that make use of this effect, e.g., to better visualize a cerebral thromboembolism or to quantify cerebral blood oxygen saturation [

Previously, local susceptibility gradients and their effect on MR-signal behavior have been examined through spin-echo relaxation rates _{2} = 1/_{2} and gradient-echo relaxation rates _{2}) or

Spin diffusion leads to a decrease of correlation over time that is mostly assumed to occur monoexponentially with a characteristic time constant or correlation time [

To account for the time dependence of microstructural quantities of the diffusion process, generalized moment expansion analysis can be employed to approximate their long time behavior. Generalized moment analysis is an extension of the first passage time approximation [

In this work, the generalized moment analysis will be introduced and utilized to determine Padé approximants in the biexponential approximation for correlation functions of spherical and cylindrical magnetic objects. This analysis extends and furthers a preceding analysis that only considered the monoexponential decay forms for cylindrical magnetic perturbers [

The following analysis is based on the consideration of compact and impermeable objects _{Voxel} that are exposed to an external static magnetic field with strength _{0} (see Figure _{Voxel} −

_{i} embedded in a surrounding medium _{e}; both volumes constitute the voxel volume _{Voxel} and the volume fraction is η = _{Voxel}. The spin is subjected to stochastic field fluctuations in _{ts}_{+t/2} that are shifted with time _{s} away from the standard spin echo value of

Generally, the distortion of the magnetic field is connected to the susceptibility difference Δχ = χ_{i} − χ_{e} between the magnetized object and its surrounding medium and leads to a variation in the local Larmor frequency
_{0}Δχ and gyromagnetic ratio γ = 2.675 × 10^{8}s^{−1}T^{−1} [_{0}, _{0} and time _{0}, _{0}, _{0}.

Then, to investigate stochastic field fluctuations around object _{0}) = 1/_{tp} is a spin echo that is measured after a 180° refocusing pulse given at a time _{p} ≠ _{t/2}. When |_{p} − _{s}, the following equality in our notation holds for low orders of _{s} according to Jensen et al. [

In this work, two kinds of magnetic perturbers are considered: spheres and cylinders (see Figures

_{0}. (A)_{i} (red), that is surrounded by a dephasing volume sphere with radius _{i} of the cross section circle perpendicular to the cylinder axis. The cylinder is surrounded by a cylindrical dephasing volume with radius

The magnetic field around a spherical magnetic perturber is that of a magnetic dipole, i.e., with the external magnetic field _{0} being parallel to the _{i}. Its dephasing volume

For a cylinder, the angle θ between its axis and the external magnetic field _{0} has to be acknowledged in the frequency shift to obtain the local frequency (see Figure

For ensembles of several (microscopic) magnetic objects, it is helpful to avail oneself of the principle of supply areas as in the Krogh capillary model [_{Voxel} where spin diffusion around the magnetic object does not reach the outer surface of

The diffusion equation (Equation 2) can be solved with the help of an eigenfunction expansion of the transition probability _{0}, _{n} are solutions of the eigenvalue equation
_{n}, and, they fulfill the orthogonality condition
_{i} is the radius of the inner sphere or cylinder, respectively. Thereby, the lowest eigenvalue κ_{0} = 0 corresponds to the lowest eigenfunction _{0}, _{0} = 0 does not contribute to the correlation function.

Explicit expressions can be given for both spherical and cylindrical geometries; in the case of spheres, _{n} obeying the eigenvalue equation
_{1} is provided in Appendix A.

For cylinders, we have with cylindrical Bessel functions _{2} and _{2} of the first and second kind, respectively,
_{n} are the solution of
_{1} only:
_{L}(_{0}, 0) = δ(_{0}):

The generalized moment expansion method or generalized moment approximation (GMA) has been used recently as an effective algorithm to approximate the time dependence of observables that are connected to reactive and non-reactive processes involving Brownian motion [

In using the Laplace transform _{n} hereby correspond to the short time behavior and the low-frequency moments μ_{−(n+1)} to the long time behavior. They are defined as
_{n} only depend on diffusion coefficient _{n} will be used below that refers to high-frequency moments for _{−n}(_{−n}(

The frequency autocorrelation function _{(Nh, Nl), n} and decay constants Γ_{(Nh, Nl), n} that are related to the moments μ_{n}. The sum of the number of high frequency moments _{h} and that of low frequency moments _{l} equals the 2_{h}+_{l} = 2_{h} high- and _{l} low-frequency moments. The resulting description of the (_{h}, _{l})-generalized-moment approximation, denoted as (_{h}, _{l})-GMA, is a two-sided Padé approximation around _{(Nh, Nl), n} and Γ_{(Nh, Nl), n} can be obtained through
_{n} are closely related to the shape of the local field inhomogeneity as shown above and parameters _{(Nh, Nl), n} and Γ_{(Nh, Nl), n} can be determined by the multi-exponential approximation of the correlation function in Equation (31). Therefore, an approximation of the correlation function is given through the solution of Equation (33), provided that the generalized moments μ_{n} are known.

In the following, the correlation function within the GMA is derived for the biexponential approximation (

For _{−2}, μ_{−1}, μ_{0}, and μ_{1} of the exact correlation function. To find the best approximation for the long time behavior, we choose the (1, 3)-GMA that reproduces the moments μ_{−3}, μ_{−2}, μ_{−1}, and μ_{0}, i.e., more low-frequency moments are being taken into account. Introducing the following abbreviations

The second case of the (1, 3)-GMA yields parameters

Introducing the above results for the biexponential approximation in the (2, 2)- and (1, 3)-GMA into Equation (31) determines the corresponding correlation function at

In order to solve differential equation (Equation 30) for spherical magnetic objects, it is helpful to start with the ansatz
_{−n}(_{n}(_{0}(

We have used the abbreviation
_{1}, …, _{6} that are given by
_{n}(η) are necessary to determine the moments μ_{n} in Equation (46) for negative indices from Equation (45). They are given by

It is convenient to define a ratio τ_{n} of successive moments as
_{n} converges to the lowest eigenvalue of Equation (11) and [_{1} is the solution to the geometry-dependent eigenvalue equation, i.e., Equation (17) for spheres and Equation (19) for cylinders. In Figure _{n} ranging from _{1} for spherical magnetic objects and Koenig's theorem is nicely exemplified for η → 1. The asymptotic value of the first eigenvalue for large volume fractions could be determined as _{1} is provided in Appendix A. Figure

_{−n}(η)/μ_{−(n+1)}(η) = _{−n}(η)/_{−(n+1)}(η) in dependence on volume fraction η is obtained from Equation (46) [functions _{−n}(η) are given in Equation (50)]. The numerically exact eigenvalue (solid line) can be obtained by solving Equation (17). An approximate solution for _{n}(η) determined from the cylindrical geometry.

Different cases of the monoexponential functions _{i}(_{n}. Evidently, the initial value of _{i}(_{i}(_{i} of 3–4 μm, a frequency shift of δω = 184 s^{−1}, a tissue density of η = 0.04 and ^{−9}m^{2}s^{−1}, leading to τ = 16.12 ms; see [_{(1, 3)}(_{(2, 2)}(_{i}(_{i} of the spherical perturbers affect the correlation time τ as well according to Equation (13). The correlation function in dependence on correlation time τ is therefore shown in Figure _{0}(τ) and _{1}(τ) coincide with

_{i}(_{L}(_{1} is determined by Equation (17) (η = 0.001 in _{(1, 3)}(_{(2, 2)}(_{(1, 3)}(_{(2, 2)}(

In analogy to spherical magnetic objects, the correlation functions for spins that diffuse between two concentric cylinders can be obtained within the GMA by employing a similar ansatz as the one for a spherical geometry and by using Equation (42) for _{−n}(

and the constants _{1}, …, _{6} are given by
_{n}(η) are obtained in analogy to the spherical case and are given as
_{−n} for cylinders from Equation (28) can be expressed as
_{n}(η) for cylinders (see Equation 56).

In analogy to Figure _{i}(_{L}(_{i}(_{L}(

_{i}(_{L}(_{1} is determined by Equation (19) (η = 0.01 in _{(1, 3)}(_{(2, 2)}(_{0}(τ), _{1}(τ), _{(1, 3)}(τ), and _{(2, 2)}(τ) coincide with

In comparison, the biexponential approximation is shown in Figure _{i} = 4 μm [^{−1} at 3 Tesla [^{−9}m^{2}s^{−1} [_{(1, 3)}(_{(2, 2)}(_{2}(_{3}(_{i} of the cylindrical perturbers affect the correlation time τ as well according to Equation (13). The correlation function in dependence on correlation time τ is therefore shown in Figure _{0}(τ) and _{1}(τ) coincide with

The incentive of this work was to gain a better understanding of the dynamic frequency correlation function of spins that diffuse around microscopic magnetized objects. The analysis is based on an eigenfunction expansion of the correlation function where eigenfunctions and eigenvalues can be determined in the case of restricted diffusion through reflective boundary conditions. A two-sided Padé approximation is then utilized to derive analytical expressions in the short and long time limit. The different regimes of the functional form of the correlation function as well as the case of unrestricted diffusion have been studied in a preceding analysis that considered a monoexponentially decaying correlation function in the generalized moment approximation (GMA) for cylindrical magnetic objects [

The model with its boundary conditions is based on a presumed regular arrangement of accumulations of similarly shaped objects that can then be considered in analogy to Krogh's model for supply volumes around capillaries [_{(1, 3)}(_{(2, 2)}(

_{0}(_{1}(_{(2, 2)}(_{(1, 3)}(

Recently, it could be shown that Carr-Purcell-Meiboom-Gill (CPMG) relaxation rates are connected to the correlation function [

This work was carried out by the seven authors, in collaboration. CZ designed research; CZ, FK, LB, and TK performed research; CZ and FK contributed numerical tools; and FK, TK, LB, HS, MB, SH, and CZ wrote the paper. Moreover, all authors have read and approved the final manuscript.

This work was supported by grants from the Deutsche Forschungsgemeinschaft (DFG ZI 1295/2-1 and DFG KU 3555/1-1). FK was also supported by a postdoctoral fellowship from the medical faculty of Heidelberg University and the Hoffmann-Klose Foundation (Heidelberg University).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

^{*}: a theoretical approach for the vasculature of myocardium

_{1}-based FAIR-ASL method: the influence of heart anatomy, cardiopulmonary blood flow and look-locker readout

To determine the eigenvalues in Equation (17), we must find the roots of the function

The derivatives of the spherical Bessel functions _{2} and _{2} can be expanded in a Taylor series for small arguments of κ_{n} and, thus, an expression for small arguments of _{n}) is given as

The first zero of Equation (A1) then approximates the first zero of Equation (17) as

see also Figure

In the monoexponential approximation (

The first ratio τμ_{0}/μ_{−1} = τ/τ_{−1} is determined in the (1, 1)-GMA that can be regarded as the “mean correlation time approximation,” since, for _{(1, 1)} = μ_{0}, Γ_{(1, 1)} = 1/τ_{−1} and with Equation (27)

where _{c}(_{t}_{−1} represents a “mean correlation time.” This is in analogy to the “mean relaxation time approximation” in nuclear spin dephasing [_{0} and μ_{1}, the corresponding (2, 0)-GMA with _{(2, 0)} = μ_{0} and Γ_{(2, 0)} = μ_{1}/μ_{0} = 1/τ_{0} describes the short time behavior of _{−2} and μ_{−1} and parameters _{(0, 2)} = μ_{−1}/μ_{−2} = 1/τ_{−2} is a measure for the long time behavior. Since the MCTA is an approximation of the dynamic monoexponentially decaying correlation function, the deviation in Figure _{−1} reflects a strongly decaying non-exponential correlation function.

It is useful to define monoexponential functions _{i}(

where the index _{i}(_{L} is given in Equation (20).

The relaxivity of particulate MR contrast media was examined in Muller et al. [_{i} = 50nm, δω = 34·10^{6}s^{−1}, η = 2·10^{−6}, ^{−9}m^{2}s^{−1}, and τ = 1.09·10^{−6}s for iron-oxide nanoparticle suspensions. This extreme example of a very short correlation time was chosen to demonstrate the coincidence of mono- and bi-exponential approximations of the correlation function, as shown in Figure _{(1, 3)}(_{(2, 2)}(_{1}(_{0}(

The influence of magnetic susceptibility properties of peripheral lung tissue with near-spherical air-containing alveoli on MR signal decay was examined recently, see e.g. [_{i} = 150 μm, δω = 1500 s^{−1}, η = 0.85, ^{−9}m^{2}s^{−1} (τ = 22.5 s) [_{0}(