^{1,a),b)}, Y. Gossuin

^{1}, P. Gillis

^{1}and S. Delangre

^{1,b)}

### Abstract

Superparamagneticnanoparticles are used as negative contrast agents in magnetic resonance imaging: owing to their large magnetic moment the water proton spins are dephased, which accelerates the nuclear magnetic relaxation of an aqueous sample containing these particles. Transverse and longitudinal relaxation times depend on several parameters of the nanoparticles such as radius and magnetization and on experimental parameters such as the static magnetic field or echo time. In this work, we introduce a new simulation methodology, using a classical formalism, allowing the simulation of the NMR signal during transverse and longitudinal relaxation induced by superparamagnetic particles in an aqueous solution, which, to our knowledge has never been done before. Nuclear magnetic relaxation dispersion profiles are obtained for a wide range of nanoparticle radii and magnetizations. The results can be classified in two regimes—the well-known motional averaging and static regimes. This generalizes previous studies focusing on transverse relaxation at high magnetic field (larger than 1 T). Simulation results correspond to analytical theories in their validity range and so far unknown dependences of the relaxation with magnetization and radii of the NMR dispersions profiles are observed, which could be used to characterize experimental samples containing large superparamagnetic particles.

The authors would like to thank Dr. Alain Roch for his help on the theory of MR relaxation induced by SPM, Professor Roberto Lazzaroni, Professor David Beljonne, and Dr. David Colignon for providing access to the computation resources. Dr. Quoc Lam Vuong and Delangre Sebastien, F.R.S.-FNRS Research Fellows, acknowledge the F.R.S.-FNRS for financial support. This research used resources of the *Interuniversity Scientific Computing Facility* located at the University of Namur, Belgium, which is supported by the F.R.S.-FNRS under Convention No. 2.461707 and resources of the *SEGI* located at the University of Liège, Belgium.

I. INTRODUCTION

II. THEORY

A. Superparamagnetic particles

B. Analytical theories

C. Simulation methodologies

D. Classical formalism

III. METHODOLOGY

A. Models of the SPM spin

B. Simulation steps

C. Simulation parameters

IV. RESULTS AND DISCUSSION

A. Isotropic model

1. NMRDs

2. Radius

3. Magnetization

B. Weak anisotropy model

1. NMRDs

2. Radius

3. Magnetization

C. Strong anisotropy model

1. NMRDs

2. Radius

3. Magnetization

D. Spin echoes

V. CONCLUSION

### Key Topics

- Protons
- 44.0
- Magnetic moments
- 43.0
- Magnetic anisotropy
- 19.0
- Superparamagnetism
- 15.0
- Nanoparticles
- 12.0

##### H01F13/00

## Figures

Proton diffusion in the dipolar magnetic field produced by a superparamagnetic particle. The proton magnetic moment rotates around a local magnetic field composed of the static field *B* _{0} and the dipolar field *B* _{1}.

Proton diffusion in the dipolar magnetic field produced by a superparamagnetic particle. The proton magnetic moment rotates around a local magnetic field composed of the static field *B* _{0} and the dipolar field *B* _{1}.

NMRDs obtained from the isotropic model for SPM radii of 5, 20, and 230 nm.

NMRDs obtained from the isotropic model for SPM radii of 5, 20, and 230 nm.

Positions of the second inflection point obtained from simulated NMRDs at different radii (isotropic model). Two regimes can be distinguished and can be expressed by Eqs. (31) and (32).

Positions of the second inflection point obtained from simulated NMRDs at different radii (isotropic model). Two regimes can be distinguished and can be expressed by Eqs. (31) and (32).

Influence of the SPM radius on the longitudinal and transverse relaxation rates at different *B* _{0} fields (10^{−1}, 10^{−3}, and 10^{−6} T) for the isotropic model. Theoretical lines are traced with Eqs. (16) and (17), points corresponding to large radii are fitted with a law.

Influence of the SPM radius on the longitudinal and transverse relaxation rates at different *B* _{0} fields (10^{−1}, 10^{−3}, and 10^{−6} T) for the isotropic model. Theoretical lines are traced with Eqs. (16) and (17), points corresponding to large radii are fitted with a law.

Influence of SPM magnetization on relaxation rates at different *B* _{0} fields and for a SPM radius of 20 nm. Longitudinal and transverse relaxation rates are equal at a *B* _{0} = 10^{−6} T. The points corresponding to low magnetization are fitted by a *ax* ^{ b } law (solid lines): *b* is equal to 1.71 for full squares, 1.75 for empty squares, and 1.95 for full circles. Dashed lines are linear fits of higher magnetization points.

Influence of SPM magnetization on relaxation rates at different *B* _{0} fields and for a SPM radius of 20 nm. Longitudinal and transverse relaxation rates are equal at a *B* _{0} = 10^{−6} T. The points corresponding to low magnetization are fitted by a *ax* ^{ b } law (solid lines): *b* is equal to 1.71 for full squares, 1.75 for empty squares, and 1.95 for full circles. Dashed lines are linear fits of higher magnetization points.

NMRDs obtained from the weak anisotropy model for SPM radii of 5 nm and 230 nm. Theoretical curves (for *R* _{ S } of 5 nm) are obtained with Eqs. (21) and (22). For comparison, data (dashed lines) are included from simulations of the isotropic model, Eqs. (16) and (17), for particles of 5 nm and 230 nm, respectively. *R* _{1} and *R* _{2} ^{*} curves of a same model can be distinguished knowing that *R* _{2} ^{*} always saturate at high fields while *R* _{1} tends to zero.

NMRDs obtained from the weak anisotropy model for SPM radii of 5 nm and 230 nm. Theoretical curves (for *R* _{ S } of 5 nm) are obtained with Eqs. (21) and (22). For comparison, data (dashed lines) are included from simulations of the isotropic model, Eqs. (16) and (17), for particles of 5 nm and 230 nm, respectively. *R* _{1} and *R* _{2} ^{*} curves of a same model can be distinguished knowing that *R* _{2} ^{*} always saturate at high fields while *R* _{1} tends to zero.

Influence of SPM magnetization on relaxation rates for different *B* _{0} fields (10^{−6}, 10^{−3}, 1 T) and for an SPM radius of 20 nm in the WA model. Longitudinal and transverse relaxation rates are equal at a *B* _{0} = 10^{−6} T. The points corresponding to low magnetization are fitted by an *ax* ^{ 2 } law (dashed lines). Solid line corresponds to the static model (7).

Influence of SPM magnetization on relaxation rates for different *B* _{0} fields (10^{−6}, 10^{−3}, 1 T) and for an SPM radius of 20 nm in the WA model. Longitudinal and transverse relaxation rates are equal at a *B* _{0} = 10^{−6} T. The points corresponding to low magnetization are fitted by an *ax* ^{ 2 } law (dashed lines). Solid line corresponds to the static model (7).

NMRDs obtained from the SA model for SPM radii of 5 nm and 230 nm. SA theoretical curves are obtained with Eqs. (23) and (24) (solid lines) and WA theoretical curves from (21) and (22) for *R* _{ S } = 5 nm. For SPM radius of 230 nm, points from simulations of the WA models are also shown for comparison. *R* _{1} and *R* _{2} ^{*} curves of a same model can be distinguished knowing that *R* _{2} ^{*} always saturate at high fields while *R* _{1} tends to zero.

NMRDs obtained from the SA model for SPM radii of 5 nm and 230 nm. SA theoretical curves are obtained with Eqs. (23) and (24) (solid lines) and WA theoretical curves from (21) and (22) for *R* _{ S } = 5 nm. For SPM radius of 230 nm, points from simulations of the WA models are also shown for comparison. *R* _{1} and *R* _{2} ^{*} curves of a same model can be distinguished knowing that *R* _{2} ^{*} always saturate at high fields while *R* _{1} tends to zero.

Dependence of the longitudinal and transverse relaxation rates on the SPM radius in the SA model at different *B* _{0} fields (10^{−3}, 10^{−6}, and 1 T). Solid line is obtained with Eq. (23). Dashed lines were obtained by fitting an law to the data.

Dependence of the longitudinal and transverse relaxation rates on the SPM radius in the SA model at different *B* _{0} fields (10^{−3}, 10^{−6}, and 1 T). Solid line is obtained with Eq. (23). Dashed lines were obtained by fitting an law to the data.

Influence of the SPM magnetization on the relaxation rates for a particle with 20 nm-radius in the SA model. Dashed lines are square law fit of the low magnetization points. Solid line is obtained from equation (7).

Influence of the SPM magnetization on the relaxation rates for a particle with 20 nm-radius in the SA model. Dashed lines are square law fit of the low magnetization points. Solid line is obtained from equation (7).

## Tables

The different equations used in this work and their corresponding limitations.

The different equations used in this work and their corresponding limitations.

Short summary of the observations made on the three simulated models.

Short summary of the observations made on the three simulated models.

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