Adapting source grid parameters to improve the condition of the magnetostatic linear inverse problem of estimating nanoparticle distributions.

The problem of estimating magnetic nanoparticle distributions from magnetorelaxometric measurements is addressed here. The objective of this work was to identify source grid parameters that provide a good condition for the related linear inverse problem. The parameters investigated here were the number of sources, the extension of the source grid, and the source direction. A new measure of the condition, the ratio between the largest and mean singular value of the lead field matrix, is proposed. Our results indicated that the source grids should be larger than the sensor area. The sources and, consequently, the magnetic excitation field, should be directed toward the Z-direction. For underdetermined linear inverse problems, such as in our application, the number of sources affects the condition to a relatively small degree. Overdetermined magnetostatic linear inverse problems, however, benefit from a reduction in the number of sources, which considerably improves the condition. The adapted source grids proposed here were used to estimate the magnetostatic dipole from simulated data; the L2-norm, residual, and distances between the estimated and simulated sources were significantly reduced.