Microfabricated High-Moment Micrometer-sized MRI Contrast Agents

Transverse dephasing

For the particle types considered in this paper, calculations are simplified by working in the static dephasing regime. In the vicinity of high-moment, micrometer-sized particles the static dephasing assumption, Δω·τ_c >> 1, is easily satisfied. Here, as usual, the diffusion correlation time τ_c represents the time for water to diffuse a distance of order the particle size, while Δω characterizes the local precession frequency due to the particle’s field (taken, for concreteness, in the particle equatorial plane). Temporarily ignoring k-space shifting effects, the time-dependent modification to the signal S caused by the magnetic particle is therefore simply proportional to an integral over the precessing proton spins within the volume of interest:

Here t measures time following an initial π/2 excitation, r⃗ is the spin location relative to the magnetic particle, ρ is the spin density, and φ, working in the rotating frame, is the additional accrued transverse phase due to the particle field. Since the microparticle size is always far less than the voxel size, signal is dominated by spins from the particle’s far-field region. Regardless of particle shape, therefore, the field may be modeled as that of a dipole, and the particle assumed, for concreteness, to be a sphere of radius determined by its net dipole moment p_m and the magnetic saturation of its constituent material. For a B₀-field aligned in the z-direction, the z-component of the field produced by such a particle when magnetized by B₀ is: B_z = p_m·(μ₀/4π)(3cos^2θ−1)/|r⃗|³ for magnetic permeability of vacuum μ₀ = 4π·10⁻⁷ H/m and polar angle θ. For a ferro- or superparamagnetic particle of radius a and saturation magnetic polarization J_s, the dipole moment is p_m = (J_s/μ₀)·4πa³/3, yielding an equatorial precession frequency of Δω = γJ_s/3 for gyromagnetic ratio γ. Here, neither B₀ nor the magnetic susceptibility difference Δχ, in terms of which dipole moments are usually expressed, appears, since ferro- or superparamagnetic substances are magnetized to saturation by typical B₀ fields. The normalized signal decay from such a particle centered in a (spherical) voxel of radius R and of homogeneous spin density therefore becomes:

Similar integral forms have been described previously (16); focus shifts, however, when considering single-voxel-level imaging of high-moment particles. For a start, if the argument of the exponential were always small, that is, Δω· t << 1, then Taylor expansion would lead to the well-known quadratic time dependence for the initial signal decay. Although this characterizes dephasing across regions with low susceptibility differences, it is physically irrelevant for strongly magnetized particles: since J_s is of order one tesla for ferromagnetic materials, for such an expansion to be valid everywhere about the particle it could describe just the first few nanoseconds of signal evolution.

While high moments preclude immediate expansion of the above integrand, simplification is still possible if the ratio of voxel to particle radius is such that Δω·t·(a/R)³ is of order unity or less. For millisecond timescales and micrometer-sized ferromagnetic particles, this becomes true at resolutions of the order of a hundred micrometers or larger. Integrating first, and then simplifying resultant functions of Δω·t and of Δω·t·(a/R)³ through asymptotic and power series expansions, respectively, approximates the signal magnitude to second order in Δω·t·(a/R)³ as:

The leading order time coefficient c₁ is equivalent to one originally derived by Yablonskiy and Haacke (16); the c₂ coefficient, however, is not present in that work since, focusing on NMR, their spatial integration was unbounded. Here the quadratic term indicates the onset of saturation due to finite voxel size. Despite the limited expansion, Eq. [3] accurately approximates Eq. [2] for initial signal decay and saturation (see Fig. 1a). Comparing linear and quadratic terms in Eq. [3] estimates the dephasing period required to appreciably saturate out the voxel signal, which we label t_sat. The asymptotic nature makes definition somewhat arbitrary, but as a first measure we consider the point where S(t) becomes stationary, at least as approximated by Eq. [3], giving:

Defining the fractional hypointensity, i ≡ 1−|S(t)/S(0)|, at t = t_sat, the saturation hypointensity i_sat therefore amounts to c₁²/4c₂ ≈ 90 %, coincidentally matching conventional rise- or decay-time definitions for exponential-like processes. Alternatively, inverting Eq. [3] yields the period needed to reach any particular fractional hypointensity level, i < i_sat, as t_sat·[1−(1−i/i_sat)^1/2], which simplifies to (t_sat/2)·i/i_sat in the linear regime where i << i_sat.

The above assumes spherical voxels but can be adapted to approximate more realistic cubic voxels. Estimates are imperfect since different voxel shapes capture different precession frequency distributions, but a simple adaptation yielding fair approximation, as seen in Fig. 1b, is to replace R with an effective cubic “radius” R_c that incorporates at least the different voxel volumes. For a cubic voxel of half-width R, (4/3)π R_c³=8R³, implying signal decays about twice as slow for cubic voxels as they would be for spherical voxels of similar diameter. As a quantitative example then, according to Eq. [4] a hypothetical 1-µm diameter, spherical ferromagnetic particle with J_s = 1 T would substantially saturate out a surrounding 100-µm diameter spherical volume in about 15 ms, and a surrounding cubic 100-µm voxel in a little over twice that. Note, however, that Eqs. [2–4] represent a “best-case” scenario that assumes particles centered within the surrounding voxel. As such, they determine only upper- and lower-limits for signal decay and t_sat, respectively. Depending on actual particle location with respect to the imaging voxel grid, however, voxel dephasing rates may often be substantially slower (explained below).

With the above caveat in mind, t_sat gives a quick order-of-magnitude starting point for estimating whether an intended label is likely to be large enough to give decent detectability. For example, consider 100-µm resolution imaging limited by endogenous T₂ dephasing to a 10 ms echo time. Assuming cubic voxels, a 1-µm pure magnetite particle (J_s ≈ 0.5 T) would have a (best-case) t_sat of over 60 ms rendering it barely visible, if at all. Switching to a higher-moment material such as iron (J_s ≈ 2 T) would cut t_sat four-fold; staying with magnetite, on the other hand, would require a substantially larger particle. Being able to fabricate particles from pure, solid, high-moment materials is clearly advantageous, however, with chemically synthesized MPIOs generally only a fraction (often 10 % – 50 %) of the particle is comprised of relatively low-moment magnetite. Thus, microfabrication can increase sensitivity by increasing both the magnetic volume fraction of the particle as well as the moment of the material used.

Image distortion

A limitation of Eqs [2–4] is that they address signal loss due to transverse dephasing only. While such dephasing often dominates, for high-resolution imaging and short echo times, geometrical image distortion can also appreciably modify signal intensity. Effective k-space shifting due to superposition of the particle field onto the read gradient (and in the case of 2D imaging also onto the slice select gradient) translates into image distortions that yield local hypo- and hyperintense signal regions near the particle. As with transverse dephasing, numerous studies exist on such image distortion (23–27), but that work is not focused at the individual voxel level where the concern is no longer the exact distortion geometry, but simply its overall size. In particular, if the resulting hypo- and hyperintense regions fall within the same imaging voxel, then conservation of spin number ensures that spatial variations in apparent spin density cancel and image distortion is negligible. For high resolution imaging and high-moment particles, however, the apparent spatial translation of spins may yield hypo- and hyperintense signal regions separated over a distance greater than the voxel size, thereby appreciably changing voxel signal intensities.

In approximating distortion length scales, we ignore higher-order slice selection effects and assume 3D imaging with a read gradient of strength G in the x-direction. Ignoring the B₀ offset, the field during read-out is therefore Gx + B_z. Accordingly, spins located at position x, map to an apparent position x + B_z/G, and hyper- and hypointense signal maxima result when |∂B_z/∂x| takes on a minimum or maximum value, respectively. Simplifying to y = z = 0, setting ∂B_z/∂x = 0 gives x = −(J_sa³/G)^1/4 and an associated hypointense signal maximum mapped a distance d away from the particle given by:

With compensating hyperintense signal maxima similarly displaced, the ratio of d to the voxel size therefore predicts whether image distortion significantly modifies initial signal magnitude. Although distortion differs depending on whether B₀ is parallel or perpendicular to the image plane, the distance d (determined as it was for the case y = z = 0) is unchanged, and overall distortion sizes do not depend strongly on B₀ direction. Because d scales as the fourth root of dipole moment, even with high-moment particles image distortion is significant only for high-resolution imaging. However, for detecting single particles, such high-resolution imaging is common. For example, substituting again for a hypothetical 1-µm diameter, J_s = 1 T, ferromagnetic particle and typical high-resolution imaging gradients of a few tens of mT / m, Eq. [5] shows that image distortion starts contributing non-negligibly around voxel sizes of order one hundred micrometers.

For high particle moments and high-resolution imaging, therefore, distortion will often dominate in the first few milliseconds following the initial excitation before the signal transitions over to the dephasing-dominated regime described by Eq. [3], which remains valid until the voxel signal starts to appreciably saturate around the time t_sat defined by Eq. [4]. To capture, simultaneously, dephasing, distortion, and saturation effects, we numerically simulate gradient-echo imaging of individual magnetic particles of various moments and at various image resolutions and echo times. The model tracks the phases of a volume of spins precessing in the field surrounding a magnetic dipole, with intravoxel dephasing captured through a grid spacing many times smaller than any voxel size. Apparent image location of each spin is determined by the field at that spin’s real location, given by the sum of the perturbing dipole field and a simulated readout gradient. Figure 2 compares simulated results, which include image distortion and dephasing, to those derived from numerical solution of Eq. [2], which include dephasing only. As expected, deviations from Eq. [2] are largest for high resolution imaging, showing increased hypointensity, but slower signal decay, at short timescales.

Image simulation

To compare directly with imaging experiments described below, we model specific chemically synthesized MPIOs as well as the sample microfabricated disks described above. As a reference MPIO we take a commonly used commercial 1.63-µm mean diameter bead (Bangs Laboratories Inc., Fishers, IN (28)), which is composed of 42.5 % magnetite by weight and has a total magnetite content of 1.5 pg (6). These chemically synthesized beads are contrasted against the microfabricated disk particles that comprise 2-µm diameter, 300-nm thick disks of magnetic material surrounded by 50 to 100-nm thick gold shells. With these dimensions all particles have roughly comparable total volumes; however, since microfabrication allows for pure, solid cores of magnetic material, and since iron oxide, pure nickel, and pure iron have different J_s values of approximately 0.5 T, 0.6 T, and 2.2 T (29), respectively, the respective magnetic dipole moments of these particles are approximately 0.1·10⁻¹² A·m², 0.45·10⁻¹² A·m², and 1.65·10⁻¹² A·m² (that is, scaling roughly as 1:4:4²). Connection between the simulation and imaging experiments is also facilitated by the intermediate moment (that of the nickel disk) corresponding closely to that of the hypothetical J_s = 1 T, 1-µm sphere used in the above theory and in the example graphs of Figs. 1 and 2.

Figure 3 shows imaging simulations for the MPIO, nickel, and iron particles (based on the same simulation code described above for calculations in Fig. 2) that capture image darkening over several voxels rather than in only the central voxel as quantified by Eqs. [2–4] and Figs. 1 and 2. The images show theoretical (noise-free) pixelized gradient-echo signals for various echo times from individual particles at 50 and at 100 micrometer (cubic) isotropic resolution and for B₀ oriented in-plane and perpendicular to the imaging plane. As expected, image distortion modifies high-moment particle signals initially, before dephasing begins to dominate.

The above simulations assume particles centered within a voxel; signals differ for particles offset from the voxel center. Fractional voxel offsets have little impact at those resolutions where image darkening extends over many voxels, but for lower resolution imaging (larger voxel sizes), signal hypointensities drop substantially due to increased signal dilution arising from partial volume effects. Even for particles aligned in the middle of an imaging slice, therefore, identical particles can appear different depending on their lateral registration with regard to the imaging voxel. Figure 4 models such position-dependent dilution for the same particles modeled in Fig. 3, at 100-µm resolution and for a typical 10 ms echo time. Signal hypointensity decreases as dephasing effects are averaged over an increasing number of voxels, disadvantaging lower moment particles, which can be rendered invisible against background noise if unfortunately located. Although non-centered particles complicate analytic calculation, overall signal hypointensity ranges can still be approximated, at least in those cases where image distortion is negligible and where there is not yet much saturation of the surrounding voxel. For example, in moving from a central point, labeled “A”, to a corner point, labeled “D”, (see Fig. 4), signal hypointensity drops nearly four-fold, as can be understood by noting that the offset particle at “D” could be regarded as centered in a surrounding voxel whose linear dimensions were double that of the original voxel. Similarly, in three dimensions an offset from voxel center to corner reduces signal as much as eight-fold.

Sub-voxel localization

The several-fold variation in signal intensity over a displacement range of just one half of a voxel, has several consequences. On the one hand, it complicates quantification of signal hypointensities, demanding a more statistical description, and it can pull weaker MPIO signals below the noise level. On the other hand, it suggests the possibility of a form of sub-voxel level tracking where the large changes in neighboring voxel intensities may allow for more accurate determination of particle position than that normally afforded by the voxel size.

Such sub-voxel tracking or locating of magnetic particles is not the main focus of this paper but we outline the possibility here because it reflects on two differences between microfabricated and chemically synthesized particles, namely, magnetic moment and variability in that moment. An obvious requirement for sub-voxel tracking is that position-dependent variations in voxel hypointensities exceed random signal variations from one voxel to the next due to background noise. Since signal intensities scale with particle moment, so too do the position-dependent variations in those signals (provided that one is not too far into the saturation limit). The higher the particle moment, therefore, the easier it is for position-dependent variation to dominate over experimental noise. Another advantage to microfabricated particles stems from the low variability in magnetic moment from one particle to the next, allowing absolute intensities to be used in addition to ratios of neighboring voxel intensities when back-calculating particle location. Taken together, high moment, high monodispersity, microfabricated particles appear well suited for potential sub-voxel tracking.

Although not shown here, note that contrast variation due to a shift in particle location must be equivalent to that due to an opposite shift in the spatial image field of view (FOV). Since shifting the FOV is, by the Fourier Shift Theorem, equivalent to multiplying the underlying k-space data by a linear phase in the shift direction, sub-voxel level particle position could also be determined by observing voxel hypointensities as a function of the magnitude of an applied linear phase modulation. Either way, well-defined, high moments are advantageous.

Microfabrication

Apart from recently described examples of microengineered multi-spectral agents (12–14) and in-situ B₁-field modifiers (30), MRI contrast agents are typically chemically synthesized. Chemical synthesis has much to recommend it, being often simpler and more efficient than top-down microfabrication. Nevertheless, the enhanced geometrical and compositional control possible through common microfabrication techniques (31) can increase agent functionality, compensating for increased processing complexity. The previously described multi-spectral agents are one example of this trade-off between complexity and functionality. Here, as a second example, the goal was to demonstrate microfabrication’s ability to produce agents that generate greater, and more consistent, T₂^*-contrast than chemically synthesized counterparts.

While chemically synthesized MPIOs are often spherical, the microfabricated agents described here are disk-shaped objects, better suited to microfabrication technology. However, since particle sizes fall well below resolution limits, in terms of average T₂^*-contrast across a voxel, particle shapes play little role and only overall magnetic moments matter.

Disk-shaped particles can be easily fabricated using standard micropatterning techniques. For example, to create the particles described in this paper, a 10-nm thick titanium adhesion layer was first evaporated onto a supporting substrate followed by a 100-nm thick sacrificial copper layer and a 100-nm thick gold layer. A double-layer of resist was then spin-coated over this titanium-copper-gold trilayer with the top layer being photosensitive while the bottom layer was an isotropically developing, lift-off resist (LOR). Following exposure through a mask containing an array of 2 µm-diameter circular holes, patterned development and dissolution of the top resist layer then led to isotropic development and dissolution of those physically exposed portions of the underlying LOR. Appropriately timed, undercut profiles, such as those sketched in Fig. 5a, result. An approximately 300-nm thick layer of iron or of nickel, followed by a 200-nm thick layer of gold, were then evaporated (Fig. 5b). Because of local shadowing by the undercut resist profile, metal deposited on top of the photoresist is physically disconnected from metal deposited on the substrate. Subsequent removal of the resist bi-layer therefore removes the top metal layers while leaving the metal on the substrate untouched. A 100-nm deep argon ion-milling can then remove the exposed gold on the substrate and half of the top 200-nm gold layer. During this process some of the back-sputtered gold ion-milled from the substrate redeposits on the iron / nickel sidewalls (Fig. 5c) leaving magnetic disks completely encased in gold, with 100-nm thick top and bottom gold coatings, and approximately 50-nm thick gold coatings around their circumferential sidewalls (Fig. 5d). Finally, a selective wet-etch of the underlying copper releases the particles from the substrate and, using simple bulk magnetic separation, they can then be repeatedly washed to remove any remaining etchant solution. Note that the above sequence of steps and chosen particle dimensions demonstrate only one possible fabrication protocol; many other particle sizes and microprocessing strategies would be equally plausible.

Figure 5 shows only two representative particles but the parallel nature of photolithographic patterning allows arrays of many millions of particles to be simultaneously fabricated. While the magnetic moments of these nominally identical particles are far more monodisperse than those of MPIOs, it should be noted that, even with top-down microfabrication, inter-particle variation is never completely non-existent. Small position-dependent process variations will typically introduce variations in particle moments on the order of several percent across the full area of typical 3- to 6-inch processing wafers. However, because millions of microparticles can easily fit within just a few millimeter square area of the processing wafer, realized experimental inter-particle variations may be as low as fractions of a percent, rendering them effectively negligible.