An efficient computational model of the in-flow capturing of magnetic nanoparticles by a cylindrical magnet for cancer nanomedicine

TL;DR

This work addresses the challenge of predicting in-flow capture of magnetic nanoparticles by external magnets in cancer nanomedicine by introducing a continuum framework based on the Smoluchowski advection-diffusion equation for the nanoparticle concentration φ^{\textsf{NP}}. The model integrates diffusion, advection, and magnetophoresis via a mobility tensor M(x) and employs an impenetrable boundary to simulate wall capture, solved with FEM and SUPG stabilization in a BACI-based platform. A central contribution is the analytical derivation of the magnetic field and force for a finite-length cylindrical magnet, with a magnetisation model for superparamagnetic nanoparticles that includes saturation and alignment with the field, enabling efficient 3D force calculations without solving Maxwell’s equations. Through 2D and 3D numerical experiments, the study demonstrates how magnet orientation, boundary conditions, and inter-particle forces influence nanoparticle distribution and capture, providing insights to guide magnet configurations and prototype design for in vivo/in silico investigations. Overall, the framework offers a computationally efficient, physics-based tool to probe magnetic nanoparticle transport and capture, serving as a foundation for more complex biomechanical models and experimental optimization in cancer nanomedicine.

Abstract

Magnetic nanoparticles have emerged as a promising approach to improving cancer treatment. However, many novel nanoparticle designs fail in clinical trials due to a lack of understanding of how to overcome the in vivo transport barriers. To address this shortcoming, we develop a novel computational model aimed at the study of magnetic nanoparticles in vitro and in vivo. In this paper, we present an important building block for this overall goal, namely an efficient computational model of the in-flow capture of magnetic nanoparticles by a cylindrical permanent magnet in an idealised test setup. We use a continuum approach based on the Smoluchowski advection-diffusion equation, combined with a simple approach to consider the capture at an impenetrable boundary, and derive an analytical expression for the magnetic force of a cylindrical magnet of finite length on the nanoparticles. This provides a simple and numerically efficient way to study different magnet configurations and their influence on the nanoparticle distribution in three dimensions. Such an in silico model can increase insight into the underlying physics, help to design novel prototypes and serve as a precursor to more complex systems in vivo and in silico.

Figures (7)

• Figure 1: Idealised test setup. The magnetic nanoparticles are dispersed in the fluid flowing through the channel. A cylindrical permanent magnet is placed below the channel and exerts a magnetic force on the magnetic nanoparticles to capture them at the bottom wall, which is impenetrable.
• Figure 2: Magnetisation curve for a superparamagnetic nanoparticle with linear magnetisation and saturation above an applied magnetic field of $H_{\textsf{sat}}$, assuming a saturation magnetisation of $M_{\textsf{sp}} = 478kA\per m$Furlani2006 and a magnetic susceptibility of $\chi^{\textsf{NP}} \gg 1$Furlani2006Sun2008McNamara2017.
• Figure 3: Magnetic field $\boldsymbol{H}$ (A) and magnetic force $\boldsymbol{F}_{\textsf{mag}}$ (B) on the nanoparticles of a cylindrical magnet with radius $R_{\textsf{mag}} = 2mm$ and length $L_{\textsf{mag}} = 7mm$.
• Figure 4: Investigation of the influence of the mobility tensor field on the nanoparticle distribution A) Computational setup. B) Functions for the $z$-component $\mathcal{M}_{zz}(z)$ of the mobility tensor field. C) Results for the nanoparticle distributions. The colourbar applies to all plots.
• Figure 5: Results for the nanoparticle capture with a cylindrical magnet of finite length positioned below the domain. The colourbar applies to all plots.

Theorems & Definitions (2)

• Remark : Stabilisation
• Remark : Parameter and sign conventions in the elliptic integrals