© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

4.8.3 Magnetic Resonance Cytotomography

Electrostatic scanning is largely ineffective because of Debye-Huckel shielding (Section 4.7.1). Magnetic stray field probes allow resolution of 10 nm physical features, but only for materials with substantial magnetic domains -- which most biological substances lack. But nuclear magnetic resonance (NMR) imaging²¹⁸¹ may allow cellular tomography by creating 3-D proton (hydrogen atom) density maps.^451,489 Atomic density maps of other biologically important elements with nonzero nuclear magnetic moments (including D, Li⁶, B¹⁰, B¹¹, C¹³, N¹⁴, N¹⁵, O¹⁷, F¹⁹, Na²³, Mg²⁵, P³¹, Cl³⁵, K³⁹, Fe⁵⁷, Cu⁶³ and Cu⁶⁵)^396,3257 may also be compiled. For example, sodium imaging is already used clinically to assess brain damage in patients with strokes, epilepsy, and tumors.

In a hypothetical NMR cytotomographic nanoinstrument, a large permanent magnet is positioned near the surface of the cell or organelle to be examined. This creates a large static background magnetic field that polarizes the protons. The spatial gradient of this field establishes a unique resonant frequency, called the Larmor frequency, within each isomagnetic surface throughout the test volume. A second time-varying magnetic driver field (e.g., vibrating permanent magnet or weak rf field) is then scanned through the full range of resonant frequencies, exciting into resonance the protons (nuclear spins) in each isomagnetic surface, in turn. Depending on the sensor implementation chosen, each resonance detected may cause absorption of driver field energy, an increase in measured impedance, or even a return echo of magnetic energy (if the driver field is operated in pulse echo mode) as the excited protons relax to equilibrium in ~1 sec.⁴⁹⁰ The large polarizing magnet is then rotated to a new orientation, moving the isomagnetic surfaces to new positions within the test volume, and the scan is repeated. A 3-D map of the spatial proton distribution may be computed after several scan cycles.

Only those regions within a linewidth of resonance DB will generate an appreciable signal, and the expected spatial resolution (say, along the z-axis) Dz = DB / (dB/dz), where dB/dz is the spatial flux gradient of B. For a polarizing field B = 1.4 tesla placed adjacent to a (20 micron)³ cell, the cross-cell gradient dB/dz ~ 7 x 10⁴ tesla/m; placed next to a (1 micron)³ organelle, dB/dz ~ 1.4 x 10⁶ tesla/m. Assuming an effective NMR linewidth ~ 0.002 tesla,⁴⁸⁹ minimum spatial resolutions are 29 nm and 1.4 nm, respectively.

Unfortunately, the smallest reliably detectable energy in the sensor element is ~ kT e^SNR. The energy required to flip a single proton is E_flip = 4 p n_L L_proton, where L_proton = 5.28 x 10³⁵ joule-sec (the quantized spin angular momentum of the proton) and the Larmor resonance frequency n_L = g_proton B; g_proton is the gyromagnetic ratio (4.26 x 10⁷ Hz/tesla for protons) and B is the polarizing magnetic flux. However, the population difference between spin-up and spin-down nuclei in NMR is very small. For small energy differences, the Boltzmann distribution only allows a fraction E_flip / 2 kT to be flipped before the upper and lower state populations are made equal and the absorption disappears. In order to flip N_flip ~ kT e^SNR / 4 p n_L L_proton protons, the sample volume must include at least:

{Eqn. 4.50}

For T = 310 K, SNR = 2 and B = 1.4 tesla, n_L = 60 MHz and N_min ~ 1.7 x 10¹¹ protons.

When scanning a cell or organelle, the vast majority of the protons present are in water molecules (Table 3.2). Water at 310 K has a proton density n_water = 6.7 x 10²⁸ protons/m³, compared to n_fat ~ 6.4 x 10²⁸ protons/m³ (palmitic acid). Distinguishing fat from water thus requires a minimum sample volume of N_min / (n_water - n_fat) ~55 micron³.

Proteins range from 4.4 x 10²⁸ protons/m³ (aspartic acid minus one water) to 7.6 x 10²⁸ protons/m³ (leucine minus one water), average n_protein ~ 5.6 x 10²⁸ protons/m³; n_carbohydrate ~ 5.8 x 10²⁸ protons/m³ (glucose minus one water), allowing NMR cellular tomography to resolve minimum feature sizes of 2-4 microns (560 micron³). This resolution may include intracellular structures such as the endoplasmic reticulum (rough ER ~1100 micron³, smooth ER ~400 micron³), the Golgi complex (~500 micron³), the nucleus (~270 micron³), possibly the nucleolus (~50 micron³) (Table 8.17), and even large localized ferritin granule concentrations.

In 1998, the minimum sample size for microcoil (~470 micron diameter) NMR scanning was ~10,000 micron³ or ~10¹⁶ protons for a 1-minute measurement cycle.¹⁰⁶¹ Individual T-cell vacuoles ~1 micron in diameter labeled with dextran-coated iron oxide particles have been imaged by MRI.²³¹⁷

Last updated on 17 February 2003