A plethora of magnetic nanoparticles has been developed and investigated under different alternating magnetic fields (AMF) for the hyperthermic treatment of malignant tissues. Yet, clinical applications of magnetic hyperthermia are sporadic, mostly due to the low energy conversion efficiency of the metallic nanoparticles and the high tissue concentrations required. Here, we study the hyperthermic performance of commercially available formulations of superparamagnetic iron oxide nanoparticles (SPIOs), with core diameter of 5, 7 and 14 nm, in terms of absolute temperature increase ΔT and specific absorption rate (SAR). These nanoparticles are operated under a broad range of AMF conditions, with frequency f varying between 0.2 and 30 MHz; field strength H ranging from 4 to 10 kA m−1; and concentration cMNP varying from 0.02 to 3.5 mg ml−1. At high frequency field (∼30 MHz), non specific heating dominates and ΔT correlates with the electrical conductivity of the medium. At low frequency field (<1 MHz), non specific heating is negligible and the relaxation of the SPIO within the AMF is the sole energy source. We show that the ΔT of the medium grows linearly with cMNP, whereas the SARMNP of the magnetic nanoparticles is independent of cMNP and varies linearly with f and H2. Using a computational model for heat transport in a biological tissue, the minimum requirements for local hyperthermia (Ttissue >42°C) and thermal ablation (Ttissue >50°C) are derived in terms of cMNP, operating AMF conditions and blood perfusion. The resulting maps can be used to rationally design hyperthermic treatments and identifying the proper route of administration – systemic versus intratumor injection – depending on the magnetic and biodistribution properties of the nanoparticles.

In this work, we develop a computational tool to predict the patient-specific evolution of a highly malignant brain tumour, the glioblastoma multiforme (GBM), and its response to therapy. A diffuse-interface mathematical model based on mixture theory is fed by clinical neuroimaging data that provide the anatomical and microstructural characteristics of the patient brain. The model is numerically solved using the finite element method, on the basis of suitable numerical techniques to deal with the resulting Cahn-Hilliard type equation with degenerate mobility and single-well potential.The results of simulations performed on the real geometry of a patient brain quantitatively show how the tumour expansion dependens on the local tissue structure. We also report the results of a sensitivity analysis concerning the effects of the different therapeutic strategies employed in the clinical Stupp protocol. The simulated results are in qualitative agreement with the observed evolution of GBM during growth, recurrence and response to treatment. Taken as a proof-of-concept, these results open the way to a novel personalized approach of mathematical tools in clinical oncology. K E Y W O R D S diffuse-interface model, finite element, glioblastoma multiforme, mixture theory, personalized medicine M S C ( 2 0 1 0 ) 35K57, 35K65, 35Q92, 65M60, 92C50

Glioblastoma multiforme (GBM) is the most aggressive and malignant among brain tumors. In addition to uncontrolled proliferation and genetic instability, GBM is characterized by a diffuse infiltration, developing long protrusions that penetrate deeply along the fibers of the white matter. These features, combined with the underestimation of the invading GBM area by available imaging techniques, make a definitive treatment of GBM particularly difficult. A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of GBM evolution in every single patient throughout his/her oncological history, in order to target therapeutic weapons in a patient-specific manner. In this work, we propose a continuous mechanical model and we perform numerical simulations of GBM invasion combining the main mechano-biological characteristics of GBM with the micro-structural information extracted from radiological images, i.e. by elaborating patient-specific Diffusion Tensor Imaging (DTI) data. The numerical simulations highlight the influence of the different biological parameters on tumor progression and they demonstrate the fundamental importance of including anisotropic and heterogeneous patient-specific DTI data in order to obtain a more accurate prediction of GBM evolution. The results of the proposed mathematical model have the potential to provide a relevant benefit for clinicians involved in the treatment of this particularly aggressive disease and, more importantly, they might drive progress towards improving tumor control and patient’s prognosis.

Self-organization in developing living organisms relies on the capability of cells to duplicate and perform a collective motion inside the surrounding environment. Chemical and mechanical interactions coordinate such a cooperative behaviour, driving the dynamical evolution of the macroscopic system. In this work, we perform an analytical and computational analysis to study pattern formation during the spreading of an initially circular bacterial colony on a Petri dish. The continuous mathematical model addresses the growth and the chemotactic migration of the living monolayer, together with the diffusion and consumption of nutrients in the agar. The governing equations contain four dimensionless parameters, accounting for the interplay among the chemotactic response, the bacteria-substrate interaction and the experimental geometry. The spreading colony is found to be always linearly unstable to perturbations of the interface, whereas branching instability arises in finite-element numerical simulations. The typical length scales of such fingers, which align in the radial direction and later undergo further branching, are controlled by the size parameters of the problem, whereas the emergence of branching is favoured if the diffusion is dominant on the chemotaxis. The model is able to predict the experimental morphologies, confirming that compact (resp. branched) patterns arise for fast (resp. slow) expanding colonies. Such results, while providing new insights into pattern selection in bacterial colonies, may finally have important applications for designing controlled patterns.

In this paper, we study the mechanical behaviour of multicellular aggregates using the notion of multiple natural configurations. In particular, we extend the elasto-visco-plastic model proposed in Preziosi et al. (2010, An elasto-visco-plastic model of cell aggregates. J. Theor. Biol., 262, 35-47) taking into account the liquid constituent present in cellular spheroids. Aggregates are treated as porous materials, composed of cells and filled with water. The cellular constituent is responsible for the elastic and the plastic behaviour of the material. The plastic component is due to the rearrangement of adhesion bonds between cells and it is translated into the existence of a yield stress in the macroscopic constitutive equation. On the other hand, the liquid constituent is responsible for the viscous-like response during deformation. The general framework is then applied to describe uniaxial homogeneous compression both when a constant load is applied and when a fixed deformation is imposed and subsequently released. We compare the results of the model with the dynamics observed during the experiments in Forgacs et al. (1998, Viscoelastic properties of living embryonic tissues: a quantitative study. Biophys. J., 74, 2227-2234).

The mechanical properties of cell nuclei have been demonstrated to play a fundamental role in cell movement across extracellular networks and micro-channels. In this work, we focus on a mathematical description of a cell entering a cylindrical channel composed of extracellular matrix. An energetic approach is derived in order to obtain a necessary condition for which cells enter cylindrical structures. The nucleus of the cell is treated either (i) as an elastic membrane surrounding a liquid droplet or (ii) as an incompressible elastic material with Neo-Hookean constitutive equation. The results obtained highlight the importance of the interplay between mechanical deformability of the nucleus and the capability of the cell to establish adhesive bonds and generate active forces in the cytoskeleton due to myosin action.

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