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Optimization methods are presented to design Halbach arrays to maximize the forces applied on magnetic nanoparticles at deep tissue locations. In magnetic drug targeting, where magnets are used to focus therapeutic nanoparticles to disease locations, the sharp fall off of magnetic fields and forces with distances from magnets has limited the depth of targeting. Creating stronger forces at depth by optimally designed Halbach arrays would allow treatment of a wider class of patients, e.g. patients with deeper tumors. The presented optimization methods are based on semi-definite quadratic programming, yield provably globally optimal Halbach designs in 2 and 3-dimensions, for maximal pull or push magnetic forces (stronger pull forces can collect nano-particles against blood forces in deeper vessels; push forces can be used to inject particles into precise locations, e.g. into the inner ear). These Halbach designs, here tested in simulations of Maxwell’s equations, significantly outperform benchmark magnets of the same size and strength. For example, a 3-dimensional 36 element 2000 cm3 volume optimal Halbach design yields a ×5 greater force at a 10 cm depth compared to a uniformly magnetized magnet of the same size and strength. The designed arrays should be feasible to construct, as they have a similar strength (≤ 1 Tesla), size (≤ 2000 cm3), and number of elements (≤ 36) as previously demonstrated arrays, and retain good performance for reasonable manufacturing errors (element magnetization direction errors ≤ 5°), thus yielding practical designs to improve magnetic drug targeting treatment depths.

Magnetic fields have the potential to noninvasively direct and focus therapy to disease targets. External magnets can apply forces on drug-coated magnetic nanoparticles, or on living cells that contain particles, and can be used to manipulate them in vivo. Significant progress has been made in developing and testing safe and therapeutic magnetic constructs that can be manipulated by magnetic fields. However, we do not yet have the magnet systems that can then direct those constructs to the right places, in vivo, over human patient distances. We do not yet know where to put the external magnets, how to shape them, or when to turn them on and off to direct particles or magnetized cells-in blood, through tissue, and across barriers-to disease locations. In this article, we consider ear and eye disease targets. Ear and eye targets are too deep and complex to be targeted by a single external magnet, but they are shallow enough that a combination of magnets may be able to direct therapy to them. We focus on how magnetic fields should be shaped (in space and time) to direct magnetic constructs to ear and eye targets.

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