  Magnetic drug delivery is a promising therapeutic because of magnetic fields'
ability to permeate unperturbed in human tissue. One of the long-standing
challenges in magnetic drug delivery is the inability to generate 3D
aggregation non-invasively within the interior of the body. Earnshaw's theorem,
which proves the impossibility of creating an energetic minimum in a curl-free
and divergence-free field such as a magnetic field. However, one of the
assumptions of Earnshaw's theorem is a static field. Here we show that it is
possible to utilize a dynamically changing field and a dissipative force such
as the drag, which is generally present, to create a stable aggregation point
for magnetic particles. We also introduce a theoretical framework for designing
the suitable magnetic fields for controlling a given magnetic particle in a
particular fluid. This framework enables accurate determination of the
necessary parameters for aggregation across a wide variety of magnetic
particles and across multiple biologically-relevant fluids. By coating magnetic
particles with desired therapeutic agents or attaching them to cells, a new
class of treatment methodologies can be created in therapies such as targeted
drug delivery and cell-based therapies. By dynamically changing the aggregation
point, agents can also be guided along a particular path in the body. This
technique of using dissipative forces to create a stable 3D aggregation point
for particles could possibly be extended to a broad range of applications such
as microscopic and macroscopic manipulation, robotics, guided self-assembly,
magnetic plasma confinement, tissue engineering, and ion traps for quantum
computers.
