An in silico model of the capturing of magnetic nanoparticles in tumour spheroids in the presence of flow

Abstract

One of the main challenges in improving the efficacy of conventional chemotherapeutic drugs is that they do not reach the cancer cells at sufficiently high doses while at the same time affecting healthy tissue and causing significant side effects and suffering in cancer patients. To overcome this deficiency, magnetic nanoparticles as transporter systems have emerged as a promising approach to achieve more specific tumour targeting. Drug-loaded magnetic nanoparticles can be directed to the target tissue by applying an external magnetic field. However, the magnetic forces exerted on the nanoparticles fall off rapidly with distance, making the tumour targeting challenging, even more so in the presence of flowing blood or interstitial fluid. We therefore present a computational model of the capturing of magnetic nanoparticles in a test setup: our model includes the flow around the tumour, the magnetic forces that guide the nanoparticles, and the transport within the tumour. We show how a model for the transport of magnetic nanoparticles in an external magnetic field can be integrated with a multiphase tumour model based on the theory of porous media. Our approach based on the underlying physical mechanisms can provide crucial insights into mechanisms that cannot be studied conclusively in experimental research alone. Such a computational model enables an efficient and systematic exploration of the nanoparticle design space, first in a controlled test setup and then in more complex in vivo scenarios. As an effective tool for minimising costly trial-and-error design methods, it expedites translation into clinical practice to improve therapeutic outcomes and limit adverse effects for cancer patients.

Keywords: Cylindrical permanent magnet; Magnetic drug targeting; Magnetic nanoparticles; Multiphase porous media; Tumour-growth model.

Fig. 1

Experimental and computational setup a Experimental setup for magnetic accumulations of superparamagnetic iron oxide nanoparticles (SPIONs) in a flow system. The tumour spheroid is placed in a MIVO® chamber connected to a peristaltic pump. Permanent magnets guide the SPIONs to the tumour spheroid. The pictures are adapted from Behr et al. (2022), licensed under CC BY 4.0. b Computational setup combining the flow of the free fluid in ${\mathrm{\Omega }}^{\ell }$ with a multiphase porous medium for the tumour spheroid in ${\mathrm{\Omega }}^t$. A cylindrical permanent magnet is positioned below the flow chamber. The nanoparticles (NPs) are transported with the fluid and guided by the magnetic field

Fig. 2

Porous medium with the pore space of the extracellular matrix (ECM) occupied by the tumour cells and the culture medium. The brown arrows indicate the flow of the culture medium, which is transporting the nanoparticles (NP). At the microscale, the different phases can be distinguished (left), while at the macroscale, the phases are described by their volume fractions ${\epsilon }^{\alpha }$ (right). Up-scaling based on the thermodynamically constrained averaging theory (TCAT) bridges the gap between the two scales

Fig. 3

Velocities in the flow chamber for different tumour spheroid sizes and positions a Small tumour spheroid centred in the flow chamber. b Large tumour spheroid centred in the flow chamber. c Small tumour spheroid lying at the bottom of the flow chamber. d Large tumour spheroid lying at the bottom of the flow chamber. e Velocity magnitude for the large tumour spheroid lying at the bottom of the flow chamber (case d): velocity magnitude in the free fluid (left) in mm/s and in the tumour spheroid (right) in nm/s

Fig. 4

Magnetic flux density and force for the cylindrical magnet vertically positioned below the flow chamber at a distance of $0.25\text{mm}$ from the bottom of the domain. The magnet has a radius of 0.5 mm and a length of 1 mm. a Magnetic flux density $\mathit{B}$. b Magnetic force ${\mathit{F}}_{\mathsf{mag}}$

Fig. 5

Results for the nanoparticle mass fractions ${\omega }^{\mathsf{NP}\ell }$ at $t=20\text{s}$ for different tumour spheroid sizes and positions