  With the goal of determining strategies to maximise drug delivery to a
specific site in the body, we developed a mathematical model for the transport
of drug nanocarriers (nanoparticles) in the bloodstream under the influence of
an external magnetic field. Under the assumption of long (compared to the
radius) blood vessels the Navier-Stokes equations are reduced, to a simpler
model consistently with lubrication theory. Under these assumptions, analytical
results are compared for Newtonian, power-law, Carreau and Ellis fluids, and
these clearly demonstrate the importance of shear thinning effects when
modelling blood flow. Incorporating nanoparticles and a magnetic field to the
model we develop a numerical scheme and study the particle motion for different
field strengths. We demonstrate the importance of the non-Newtonian behaviour:
for the flow regimes investigated in this work, consistent with those in blood
micro vessels, we find that the field strength needed to absorb a certain
amount of particles in a non-Newtonian fluid has to be larger than the one
needed in a Newtonian fluid. Specifically, for one case examined, a two times
larger magnetic force had to be applied in the Ellis fluid than in the
Newtonian fluid for the same number of particles to be absorbed through the
vessel wall. Consequently, models based on a Newtonian fluid can drastically
overestimate the effect of a magnetic field. Finally, we evaluate the particle
concentration at the vessel wall and compute the evolution of the particle flux
through the wall for different permeability values, as that is important when
assessing the efficacy of drug delivery applications. The insights from our
work bring us a step closer to successfully transferring magnetic nanoparticle
drug delivery to the clinic.
