IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 29, NO. 11, NOVEMBER 2010 1851 The X-Space Formulation of the Magnetic Particle Imaging Process: 1-D Signal, Resolution, Bandwidth, SNR, SAR, and Magnetostimulation Patrick W. Goodwill* and Steven M. Conolly Abstract—The magnetic particle imaging (MPI) imaging process is a new method of medical imaging with great promise. In this paper we derive the 1-D MPI signal, resolution, bandwidth require- ments, signal-to-noise ratio (SNR), specific absorption rate, and slew rate limitations. We conclude with experimental data mea- suring the point spread function for commercially available SPIO nanoparticles and a demonstration of the principles behind 1-D imaging using a static offset field. Despite arising from the nonlinear temporal response of a magnetic nanoparticle to a changing magnetic field, the imaging process is linear in the magnetization distribution and can be de- scribed as a convolution. Reconstruction in one dimension is exact and has a well-behaved quasi-Lorentzian point spread function. The spatial resolution improves cubically with increasing diameter of the SPIO domain, inverse to absolute temperature, linearly with saturation magnetization, and inversely with gradient. The bandwidth requirements approach a megahertz for reasonable imaging parameters and millimeter scale resolutions, and the SNR increases with the scanning rate. The limit to SNR as we scale MPI to human sizes will be patient heating. SAR and magnetostimula- tion limits give us surprising relations between optimal scanning speeds and scanning frequency for different types of scanners. Index Terms—Biomedical imaging, magnetic particle imaging, signal detection, x-space. I. BACKGROUND M AGNETIC particle imaging is a new medical imaging modality that holds significant promise for high-sensi- tivity, high-resolution imaging in small animals and humans. Recent papers show real time imaging of tracer at physiologic concentrations being passed between the chambers of a mouse heart [1], methods for single-sided magnetic particle imaging (MPI) that do not require surrounding the patient or animal with a gradient magnet [2], and novel methods for improving receiver coil matching with inter-modulation excitation [3]. Manuscript received March 23, 2010; revised May 29, 2010; accepted May 31, 2010. Date of publication June 07, 2010; date of current version November 03, 2010. This work was supported in part by the California Institute for Regen- erative Medicine (CIRM) graduate fellowship under Training Grant T1-00007, in part by CIRM Tools and Technology Grant RT1-01055-1, in part by a Univer- sity of California Berkeley Bioengineering graduate fellowship, and in part by the National Institutes of Health training grant. The contents of this publication are solely the responsibility of the authors and do not necessarily represent the official views of CIRM or any other agency of the State of California. Asterisk indicates corresponding author. *P. W. Goodwill is with the Department of Bioengineering, University of California, Berkeley, CA 94720 USA (e-mail: goodwill@berkeley.edu). S. M. Conolly is with the Department of Bioengineering, University of Cali- fornia, Berkeley, CA 94720 USA (e-mail: sconolly@berkeley.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMI.2010.2052284 Fig. 1. (Left) The magnetization of the system when the FFP is at location and SPIO nanoparticles positioned at the small circles. The magnetization is shown for one particle and with two particles. (Right) Signal produced by the magnetization when rapidly scanning the FFP back and forth with trajectory . The signal is shown graphed against the position of the FFP. The signal changes in sign when the FFP is scanning in the opposite direction. In tandem with progress in building new hardware, there have been efforts to develop a working theory on the MPI process and how to rapidly reconstruct images in a linear, repeatable manner. Excellent progress towards this goal is shown in Rahmer et al. [4], where it is shown that the frequency-space signal contribution of a magnetic particle when using a linear or Lissajous excitation pat- tern can be described using Chebychev polynomials of the second kind, which form an excellent basis set for reconstruction. This shows that MPI excitation gives a readily invertible signal that can be converted to an image through algebraic reconstruction tech- niques or a Chebychev basis set. Simulation based approaches are discussed in [2] and [5]. However, there is not yet a unified theory describing the MPI signal, resolution, bandwidth, signal-to-noise ratio (SNR), SAR, and magnetostimulation. In this paper, we will show that MPI can be understood as a x-domain scanning process and, as such, reconstructed quickly and simply. We begin by approaching MPI as a 1-D system and solve for the point spread function (PSF). The PSF gives us bandwidth requirements, which we use to derive the signal-to- noise ratio. We then separately discuss SAR and magnetostim- ulation, which give surprising results regarding excitation slew rates and excitation frequencies. We conclude with a brief test of reconstruction and system linearity with a real instrument. II. FUNDAMENTAL RELATIONS OF MPI IN ONE DIMENSION Let us consider a 1-D MPI system with a convenient linear gradient and an time-changing offset field such as what would be created by a Helmholtz coil (Fig. 2). If the gradient is zero at the origin, we can find the 0278-0062/$26.00 © 2010 IEEE