Magnetic nanoparticles are powerful tools in biomedical applications, where they are employed in both diagnosis and therapy. A prerequisite for the efficiency of these applications are precisely characterized particles. Several magnetic measurement techniques are established, which all have in common that they measure the magnetic response of the particles exposed to an externally applied magnetic field, which may change the magnetic state of the particles [1]. Therefore, Thermal Noise Magnetometry (TNM) has been developed, a magnetic nanoparticle characterization technique that does not rely on an external magnetic excitation [2]. It detects amplitude variations in the net magnetic moment of the sample caused by thermal fluctuations. TNM measurements have been proven to be feasible, and complementary to other characterization techniques due to their diminutive impact on the sample [2, 3]. Two mechanisms are responsible for the directional fluctuations of the magnetic moments. Brownian fluctuations occur if the particles are free to move in the sample, i.e. when they are suspended in a solution. The hydrodynamic size of the particles and the viscosity of the fluid influence the Brownian fluctuation rate. In parallel, the magnetization of each particle can also switch in the frame of the particles themselves, called Néel switching. The Néel switching rate is determined by the volume of the particles’ magnetic core and anisotropy constant. Additionally, the timescales of both mechanisms also depend on the temperature since the fluctuations are induced by the thermal energy in system. TNM experiments are typically performed at room temperature, but the particles are characterized for biomedical applications at body temperature. From preliminary results, a considerable effect of temperature variations in TNM is visible, influencing the particles’ fluctuation rate distributions and noise spectrum. As expected from the theory, the fluctuation rates shift towards higher values for higher temperatures, which leads to a decrease in the noise power in the considered frequency range.

![](https://assets.underline.io/uploads/markdown_image/1/image/cb671696c9bc121f01d6d3494d47ebc4.jpg)

Power spectrum densities for Perimag® particles at different temperatures.

The Effect of Temperature Variations in Thermal Noise Magnetometry

Achieving homogeneous tumour heating is one of the most long-standing challenges in magnetic hyperthermia for cancer therapy. Clinical studies must meet strict safety regulations that require knowledge and control over the physical properties of the chosen magnetic nanoparticles [1]. Whereas it is well-established that the heat released by magnetic nanostructures is influenced by interparticle interactions, as of today there is no method to analyze heat exchange at single-particle level in interacting systems at nonzero temperature. This gap needs to be bridged so to harness nanoparticle magnetization dynamics for the sake of biomedical applications. Building on previous theoretical work [2], we present an equation to estimate the heat dissipation of individual, interacting particles at nonzero temperature that perform both field-induced and thermal switching. After validating this equation, we use the macrospin simulation tool Vinamax [3], to investigate a system of interacting particles with different anisotropies, as we showed in this contribution [4]. Our preliminary results indicate (see figure) that the generated heat becomes more homogeneously distributed at larger fields. We believe that this homogenization of the particle heating will help to achieve a more homogenized heating of tumors during hyperthermia treatments. The use of the proposed equation would simplify the selection process of optimum nanoparticle distributions leading to optimal tumour heating. Its simplicity and flexibility also allow it to be integrated into multi-scale, multi-physics simulations to accurately assess the magnetic hyperthermia therapy without impacting the overall computational time.

![](https://assets.underline.io/uploads/markdown_image/1/image/cd66b63a757f78697365ef1ca41b5bd1.jpg)

Figure: Heat generated in a system consisting of two interacting nanoparticles with different anisotropies. Both particles show switching behavior as soon as the excitation field amplitude overcomes their anisotropy barrier. The heat generation tends to become more uniform with increasing excitation fields.

Homogenization of Heating in Magnetic Hyperthermia Through Exploitation of Magnetisation Dynamics of Interacting Particles

The end of Moore’s Law for complementary metal-oxide semiconductor (CMOS) technology has prompted the search for low-power computing, resulting in several promising proposals based on magnetic logic. One approach aims at tailoring arrays of nanomagnetic islands in which the magnetostatic interactions constrain the equilibrium orientation of the magnetization to embed logical functionalities [1]. Despite the realization of several proofs of concepts of such nanomagnetic logic, it is still unclear what the advantages are compared to the widespread CMOS designs, due to their need for clocking and/or thermal annealing for which fast convergence to the ground state is not guaranteed. In fact, it seems increasingly evident that “beyond CMOS” technology will require a fundamental rethinking of our computing paradigm. In this respect, digital memcomputing machines have been proposed [2]. These machines consist of self-organizing logic gates (SOLGs) which are able to dynamically “self-organize” into their logically correct states irrespective of whether the signal is applied to the traditional input terminals, or the traditional output terminals. This type of terminal-agnostic logic thus allows for reversed computing. Here [3], we introduce a novel concept of SOLGs that employs stray-field coupled nanomagnetic islands to perform terminal-agnostic logic: nanomagnetic self-organizing logic gates. We thereby focus on the design of a functionally complete NAND gate with which any Boolean circuit can be constructed in a bottom-up approach. Our strategy to build such a nanomagnetic SO-NAND gate relies on two main properties: balancedness and dynamic error suppression (DES) – see Figure 1. For reversible logic to work, the logically correct states should be occupied equally in thermodynamical equilibrium (and not only after a relaxation process into the ground states, as balancedness was originally defined [4]). This contrasts the balanced NAND gate proposed in Ref. [5]. To allow for larger circuits, the probability of being in a logically incorrect state should suppressed. To this end, our DES scheme periodically checks the logical correctness of the gate/coupling and applies additional fields on each island of the gate/coupling if necessary. Our DES scheme is thus based on local information, i.e. the magnetization state of a single gate or coupling. This locality is in sharp contrast to the biasing used in p-bit logic inversion, for which the logical behavior of a collection of super-paramagnetic tunnel junctions (sMTJ) is fully imposed by globally-determined time-varying spin currents [6]. Owing to its locality, our approach does not suffer from scaling issues. To show the usefulness of our SO-NAND gate, we use it as a building block to construct a self-organizing two-bit multiplier – see Figure 2. Our SO-2BM is capable of decomposing any output number into its (prime)factors, finding each possible decomposition without clearly favoring nor penalizing any solution. This fact indicates that the balancedness remains largely conserved throughout the whole system, which allows the extension to larger systems in order to solve more complex problems.

**References:**

[1] A. Imre, G. Csaba, L. Ji et al., Science, Vol. 311, p. 205-208 (2006). [2] F. Traversa and M. Di Ventra, Chaos, Vol. 27, p. 023107 (2017). [3] P. Gypens, J. Leliaert, M. Di Ventra et al., arXiv preprint arXiv:2012.12721 (2020). [4] J. Lust and B. Dixon, Inf. Sci., Vol. 113, p. 193-204 (1999). [5] P. Gypens, J. Leliaert and B. Van Waeyenberge, Phys. Rev. Appl., Vol. 9, p. 034004 (2018). [6] W. Borders, A. Pervaiz, S. Fukami et al., Nature, Vol. 573, p. 390-393 (2019). [7] J. Leliaert, A. Vansteenkiste, A. Coene et al., Med. Biol. Eng. Comput., Vol. 53, p. 309-317 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7d77a99461c449a155c9bf16943ac1b1.jpg)
Fig. 1: Schematic representation of a self-organizing NAND gate consisting of four nanomagnetic islands with perpendicular anisotropy. The probability distribution is obtained with the macrospin simulation tool Vinamax [7] which numerically solves the Landau-Lifshitz-Gilbert equation taking thermal fluctuations into account. Logically incorrect states are suppressed by a dynamic error suppression (DES) scheme, while still preserving the balancedness of the gate. Prel is the probability relative to the most likely logical state.

![](https://assets.underline.io/uploads/markdown_image/1/image/bdd9a7d3218e170d709dc62ca31af92c.jpg)
Fig. 2: Schematic representation of a self-organizing two-bit multiplier and the relative probability distribution obtained with Vinamax for an output fixed to 0. Prel is the probability relative to the most likely logical state. The four output islands of the SO-2BM, labeled from a to d, are fixed such that they represent a certain number (0, 1, 2, 3, 4, 6, or 9) and the system is allowed to explore its state space. The magnetization of the input islands, corresponding to the solution of the computation, are labeled from A to D. All states of which one of the inputs is logically inconsistent are aggregated into the "??"-state.

 Nanomagnetic Self-Organizing Logic Gates

Magnetic nanoparticles are an important asset in many biomedical applications ranging from the local heating of tumours[1] to targeted drug delivery towards diseased sites[2]. Recently, magnetic nanoflowers showed a remarkable heating performance[3,4] in hyperthermia experiments thanks to their complex structure leading to a broad range of magnetic dynamics. To grasp their full potential and to better understand the origin of this unexpected heating performance, we propose the use of Kaczmarz’ algorithm[5] in interpreting magnetic characterisation measurements[6]. It has the advantage that no a priori assumptions need to be made on the particle size distribution, contrasting current magnetic interpretation methods that often assume a lognormal size distribution. Both approaches are compared on DC magnetometry, magnetorelaxometry and AC susceptibility characterisation measurements of the nanoflowers. We report that the lognormal distribution parameters vary significantly between data sets, whereas Kaczmarz’ approach achieves a consistent and accurate characterisation for all measurement sets. Additionally, we introduce a methodology to use Kaczmarz’ approach on distinct measurement data sets simultaneously. It has the advantage that the strengths of the individual characterisation techniques are combined and their weaknesses reduced, further improving characterisation accuracy. Our findings are important for biomedical applications as Kaczmarz’ algorithm allows to pinpoint multiple, smaller peaks in the nanostructure’s size distribution compared to the monomodal lognormal distribution. The smaller peaks permit to fine-tune biomedical applications with respect to these peaks to e.g. boost heating or to reduce blurring effects in images. Furthermore, the Kaczmarz algorithm allows for a standardised data analysis for a broad range of magnetic nanoparticle samples. Thus, our approach can improve the safety and efficiency of biomedical applications of magnetic nanoparticles, paving the way towards their clinical use.

**References:**

 [1] A. Espinosa, R. Di Corato, J. Kolosnjaj-Tabi, et al, ACS Nano, Vol. 10, p. 2436–2446 (2016) [2] K. T. Al-Jamal, J. Bai, J. T.-W. Wang, et al, Nano Letters, Vol. 16, p. 5652–5660 (2016) [3] P. Bender, J. Fock, C. Frandsen, et al, J. Phys. Chem. C, Vol.122, p. 3068–3077 (2018) [4] C. Blanco-Andujar, D. Ortega, P. Southern, et al, Nanoscale, Vol. 7, p. 1768–1775 (2015) [5] S. Kaczmarz, Bulletin International de l’Academie Polonaise des Sciences et des Lettres, Vol. 35, p. 355–357 (1937) [6] J. Leliaert, D. Schmidt, O. Posth, et al, Journal of Physics D: Applied Physics, Vol. 50, p.195002 (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d71e9f523e4a62d6a20fc7c82d90f1d3.jpg)
Best combined fit to a) the magnetorelaxometry measurement data and b) imaginary part of the AC-susceptibility measurement data of suspended nanoflowers using a lognormal distribution and using Kaczmarz’ method. c) The obtained relaxation time distributions corresponding to the fits in panel a) and b).

 Advanced analysis of magnetic nanoflower measurements to leverage their use in biomedicine

**Session chair:** Koji Sekiguchi, Yokohama National University


IOG - Microscopy, Imaging and Characterization II

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The Effect of Temperature Variations in Thermal Noise Magnetometry

Homogenization of Heating in Magnetic Hyperthermia Through Exploitation of Magnetisation Dynamics of Interacting Particles

Nanomagnetic Self-Organizing Logic Gates

Advanced analysis of magnetic nanoflower measurements to leverage their use in biomedicine

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