International Journal on Magnetic Particle Imaging

Vol 8 No 1 Suppl 1 (2022): Int J Mag Part Imag

https://doi.org/10.18416/IJMPI.2022.2203049

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Magnetoviscoelastic models in the context of magnetic particle imaging

### Main Article Content

This work is licensed under a Creative Commons Attribution 4.0 International License.

### Abstract

Some mathematical models of magnetic particle imaging include the Landau-Lifshitz-Gilbert equation that is known to model the dynamic behavior of the magnetization vector in the micromagnetic theory. Bearing in mind the fluid-structure interaction of the magnetic particles in a viscoelastic environment like blood or tissue, we discuss a modeling approach of the underlying physics that takes a magnetoviscoelastic coupling into account. In particular, we discuss applicability of models for the evolution of magnetoviscoelastic materials consisting of the incompressible Navier-Stokes equations, an evolution equation for the deformation gradient and the Landau-Lifshitz-Gilbert equation. We also consider potential implications of recent work by the authors about two-component magnetoviscoelastic materials for an advanced mathematical modeling of magnetic particles embedded into viscoelastic materials.

### Article Details

### References

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[14] M. Kalousek, S. Mitra, A. Schlömerkemper, Existence of weak solutions to a diffuse interface model involving magnetic fluids with unmatched densities, arXiv:2105.04291.

[2] T. Knopp, T.M. Buzug, Magnetic Particle Imaging: an Introduction to Imaging Principles and Scanner Instrumentation, Springer, 2012.

[3] B. Kaltenbacher, T. T. N. Nguyen, A. Wald, T. Schuster, Parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging, arXiv:1909.02912.

[4] T. Kluth, P. Szwargulski, T. Knopp, Towards Accurate Modeling of the Multidimensional Magnetic Particle Imaging Physics, New Journal of Physics, vol. 21, pp. 103032, 2019.

[5] T. Kluth, Mathematical models for magnetic particle imaging, Inverse Problems, vol. 34(8), pp. 083001, 2018.

[6] J. Weizenecker, The Fokker-Planck equation for coupled Brown-Néel-rotation, Phys. Med. & Biol., vol. 63, pp. 035004, 2018.

[7] M. Kružík, A. Prohl, Recent Developments in the Modeling, Analysis, and Numerics of Ferromagnetism, SIAM Review, vol. 48, pp. 439-483, 2006.

[8] B. Benešová, J. Forster, C. Liu, A. Schlömerkemper, Existence of weak solutions to an evolutionary model for magnetoelasticity, SIAM J. Math. Anal., vol. 50, pp. 1200-1236, 2018.

[9] M. Kalousek, J. Kortum, A. Schlömerkemper, Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity, arXiv:1904.07179.

[10] M. Kalousek, A. Schlömerkemper, Dissipative solutions to a system for the flow of magnetoviscoelastic materials, arXiv:1910.12751.

[11] A. Schlömerkemper, J. Žabenský, Uniqueness of solutions for a mathematical model for magneto-viscoelastic flows, Nonlinearity, vol. 31, pp. 2989-3012, 2018.

[12] H. Sun, C. Liu, The slip boundary condition in the dynamics of particles immersed in Stokesian flows, Solid State Comm.,vol. 150, pp. 990-1002, 2010.

[13] M. Kalousek, S. Mitra, A. Schlömerkemper, Existence of weak solutions to a diffuse interface model for magnetic fluids, Nonlinear Analysis: Real World Applications, Vol. 59, 2021, 103243.

[14] M. Kalousek, S. Mitra, A. Schlömerkemper, Existence of weak solutions to a diffuse interface model involving magnetic fluids with unmatched densities, arXiv:2105.04291.