Comparison of a Single Optimized Coil and a Helmholtz Pair for Magnetic Nanoparticle Hyperthermia

Magnetic nanoparticles in a tumor can induce therapeutic heating when energized by an alternating magnetic field from a current-carrying coil outside the body. We analyzed a single-turn, air-core coil carrying a filamentary current to quantify the power absorbed by: a) magnetic nanoparticles at depth in tissue and b) superficial tissue in response to induced eddy currents; we defined this quotient as power ratio (PR). Given some limit on the eddy current heating tolerated by an alert patient, maximizing the PR maximizes the power absorbed in the tumor; all else being equal, this increases the thermal dose delivered to the tumor. The mean eddy current heating rate tolerated in four clinical studies we reviewed equaled 12.5 kW/m
³
. We differentiated our analytical expression for PR with respect to the radius of the coil to find the value of radius that maximizes PR. Under reasonable simplifying assumptions, the optimal value of coil radius equaled 1.187 times the depth of the nanoparticle target below the body surface. We also derived the PR of two coils surrounding the body configured as a Helmholtz pair. We computed PR for combinations of nanoparticle depths below the surface and axial locations with respect to the coils. At depths less than 4.6 cm, the optimized single coil had a higher PR than that of the Helmholtz pair and furthermore produced less total ohmic heating within the coil. These results were independent of driving frequency, nanoparticle concentration, tissue electrical conductivity, and magnetic nanoparticle heating rate, provided the latter is assumed to be proportional to the product of frequency and the square of the local magnetic field. This paper supports the clinical application of current-carrying coils to deliver efficacious hyperthermia therapy to tumors injected with magnetic nanoparticles.