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Magnitude and phase based mapping of particle concentrations - effects of diffusion

Lukas Buschle¹, Christian Ziener¹, Sabine Heiland², Martin Bendszus², Heinz-Peter Schlemmer¹, and Felix Kurz²

¹Radiology, German Cancer Research Center, Heidelberg, Germany, ²Neuroradiology, University Hospital Heidelberg, Heidelberg, Germany

Synopsis

The magnitude and phase of the gradient echo signal in biological tissue highly depend on its iron concentration. A quantitative evaluation of the iron concentration, however, is complicated due to the complex interplay between susceptibility and diffusion effects. In this work, we analyze the gradient echo signal as well as the spin echo signal of uniformly distributed particles, with inclusion of diffusion and susceptibility effects, and provide analytical relations that connect magnitude, phase and iron concentration. This allows a quantitative description of the iron concentration based on magnitude or phase images.

Introduction

The quantification of iron is clinically important for e.g. neurodegenerative diseases or hepatic diseases since excessive iron may damage the tissue [1]. Iron-containing protein-complexes like ferritin, hemosiderin or hemoglobin have paramagnetic properties and, thus, strong susceptibility differences to the surrounding tissue occur. The resulting local magnetic field inhomogeneities cause a fast $$$T_2^*$$$ decay and can also be observed in the phase images. Therefore, the gradient echo relaxation rate $$$R_2^*=1/T_2^*$$$ as well as the quantitative susceptibility value $$$\chi$$$, obtained from gradient echo phase images, were used as marker for iron deposition. Both methods could be verified in numerous publications that reveal a linear dependence of the relaxation rate or susceptibility on the iron concentration [2]. The slope of this linear dependence was so far only analyzed in the static dephasing regime, where diffusion effects are ignored. The experimentally determined slope, however, of the phase is smaller than expected from the static dephasing regime [3]. This effect can be described by the influence of diffusing water molecules. In this work, we provide an exact expression of the gradient echo relaxation rate $$$R_2^*$$$ and the phase $$$\Phi^*$$$ as well as the spin echo relaxation rate $$$R_2$$$ for arbitrary diffusion coefficient that allows a quantitative determination of the iron concentration.

Material and Methods

We assume a large number $$$N$$$ of randomly distributed magnetically labelled particles. The Larmor frequency $$$\omega(\mathbf{r})$$$ is a superposition of the magnetic field inhomogeneity around each sphere with radius $$$R$$$:

$$ \omega_\text{loc}(\mathbf{r}) = \delta\omega R^3 \frac{3\cos^2(\theta)-1}{r^3},$$

where $$$\delta\omega =\gamma B_0 \Delta\chi/3$$$ is determined by the susceptibility difference $$$\Delta\chi$$$ between sphere and surrounding tissue. In general, the local magnetization is not only influenced by the magnetic field inhomogeneities but also by diffusion of spin-bearing molecules. Both effects are treated within the Bloch-Torrey-equation $$$\partial_t m(\mathbf{r},t) = [D\Delta - \mathrm{i} \sum_{j=1}^N \omega_\text{loc}(\mathbf{r}-\mathbf{r}_j)]m(\mathbf{r},t)$$$, where $$$\mathbf{r}_j$$$ denotes the position of the magnetically labelled particles and $$$D$$$ is the diffusion coefficient.

In this work, we solve the Bloch-Torrey-equation for the gradient echo and spin echo signal around a single sphere with an eigenfunction expansion of the Bloch-Torrey-equation. Then, we generalize the approach of Yablonskiy and Haacke [4] to obtain the gradient echo and spin echo signal around randomly distributed particles in the physiological limit of small particle volume fraction $$$\eta$$$.

Results

The total magnetization around randomly distributed spheres is closely connected with the local magnetization around a single particle. The local magnetization around a single particle is shown in Fig. 1.

A monoexponential approximation of the gradient echo signal ($$$M(t)=M_0\mathrm{e}^{-R_2^* t}\mathrm{e}^{\mathrm{i}\Phi^*t}$$$) and spin echo signal ($$$M(t)=M_0\mathrm{e}^{-R_2t}$$$) around randomly distributed particles leads to the relaxation rates $$$R_2^*$$$ and $$$R_2$$$ and phase $$$\Phi^*$$$. Both relaxation rates are shown in Fig. 2 in comparison with random walk simulations, the weak field approximation [5] and the strong collision approximation [6]. Similarly, the linear phase $$$\Phi^*$$$ is shown in Fig. 3 in comparison with random walk simulations. Obviously, the presented methods allow a precise description of the signal decay for arbitrary diffusion effects. The results enable us to improve the empirical models presented by Yung [7] as shown in Fig. 4: the red lines represent empirical models found in this work in comparison to the green curves proposed by Yung and the exact solution shown in black. The applicability of the empirical model is also shown in Fig. 5, where the empirical models are compared with measurements of Weisskoff et al [8].

Discussion and Conclusion

In this work, we analyze the phase and magnitude of the gradient echo and spin echo signal around randomly distributed particles. Several publications showed a linear dependence of the relaxation rate and phase of the gradient echo signal on the concentration of particles (see Table 1 in [9]). However, a quantitative determination of the particle concentration is so far difficult since this linear relation is neither well described with the static-dephasing regime, where diffusion effects are neglected, nor with the motional-narrowing regime, where diffusion effects predominate. In this work, we provide exact expressions of the relaxation rates $$$R_2^*, R_2$$$ and the phase $$$\Phi^*$$$ for arbitrary particle size and diffusion strength. Adapting the empirical model of Yung, the relaxation rates and phases can be easily parameterized. The results provide an explanation for the measurement of Weisskoff et al. Finally, the model may be used to obtain quantitative values for particle concentrations from QSM-, R2- or R2*-maps for arbitrary particle sizes and diffusion strengths.

Acknowledgements

This work was supported by grants from the Deutsche Forschungsgemeinschaft (Contract Grant number: DFG ZI1295/2-1 and DFG KU 3555/1-1). L. R. Buschle was supported by Studienstiftung des deutschen Volkes.

References

[1] T. G. S. Pierre, P. R. Clark, W. Chua-anusorn, A. J. Fleming, G. P. Jeffrey, J. K. Olynyk, P. Pootrakul, E. Robins, and R. Lindeman. Noninvasive measurement and imaging of liver iron concentrations using proton magnetic resonance. Blood, 105(2):855–861, 2005.

[2] R. J. Ogg, J. W. Langston, E. M. Haacke, R. G. Steen, and J. S. Taylor. The correlation between phase shifts in gradient-echo MR images and regional brain iron concentration. Magn Reson Imaging, 17(8):1141–1148, 1999.

[3] C. Langkammer, F. Schweser, N. Krebs, A. Deistung, W. Goessler, E. Scheurer, K. Sommer,G. Reishofer, K. Yen, F. Fazekas, et al. Quantitative susceptibility mapping (QSM) as a means to measure brain iron? A post mortem validation study. Neuroimage, 62(3):1593–1599, 2012.

[4] D. A. Yablonskiy and E. M. Haacke. Theory of NMR signal behavior in magnetically inhomogeneous tissues: the static dephasing regime. Magn Reson Med, 32:749–763, 1994.

[5] A. L. Sukstanskii and D. A. Yablonskiy. Gaussian approximation in the theory of MR signal formation in the presence of structure-specific magnetic field inhomogeneities. Effects of impermeable susceptibility inclusions. J Magn Reson, 167:56–67, 2004.

[6] W. R. Bauer, W. Nadler, M. Bock, L. R. Schad, C. Wacker, A. Hartlep, and G. Ertl.The relationship between the BOLD-induced T2 and T2*: a theoretical approach for thevasculature of myocardium. Magn Reson Med, 42:1004–1010, 1999.

[7] K. T. Yung. Empirical models of transverse relaxation for spherical magnetic perturbers. Magn Reson Imaging, 21:451–463, 2003.

[8] R. M. Weisskoff, C. S. Zuo, J. L. Boxerman, and B. R. Rosen. Microscopic susceptibility variation and transverse relaxation: theory and experiment. Magn Reson Med, 31:601–610,1994.

[9] K. Ghassaban, S. Liu, C. Jiang, and E. M. Haacke. Quantifying iron content in magnetic resonance imaging. NeuroImage, 2018.

Figures

Fig. 1: Local magnetization m(x,y,z,t) around a magnetically labelled sphere under the influence of diffusion ($$$t=5R^2/D$$$, $$$\delta\omega R^2/D = 10$$$). The local magnetization is independent on the azimuthal angle and, thus, depends only on $$$\sqrt{x^2+y^2}$$$.

Fig. 2: Relaxation rates $$$R_2, R_2^*$$$ for randomly distributed particles in dependence on the product $$$\delta\omega R^2/D$$$. The exact solution presented in this work agrees very well with random walk simulations for both, spin echo and gradient echo. The relaxation rates are proportional to the particle volume fraction $$$\eta$$$. SD: static dephasing limit, WFA: weak field approximation, SCA: strong collision approximation, GE: gradient echo, SE: spin echo.

Fig. 3: Phase $$$\Phi^*$$$ of the gradient echo signal around randomly distributed particles. This work first describes the dependence of the phase on the particle size $$$R$$$, whereas all previous models either used the static dephasing phase or ignored the phase offset induced by the magnetically-labelled particles.

Fig. 4: Empirical models for (a) $$$R_2$$$ and $$$R_2^*$$$ as well as (b) the phase $$$\Phi^*$$$. The empirical model of Yung (green line) was improved in this work (red line) allowing an easy but precise description of the exact solution.

Fig. 5: Gradient echo (GE) and spin echo (SE) relaxation rates obtained from measurements of Weisskoff et al. compared with the adapted empirical model. The good agreement between the presented model and the measurements is remarkable, in particular since no parameters were fitted.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)

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