2.1 Experimental Setup
The experimental setup consists of a central cylindrical core made of a carrier material that releases small molecules into surrounding agar contained in a plastic enclosure, creating a finite-source, finite-sink setup. A silicone-elastomer mold to form the core was created from a two-part negative mold designed in Solidworks (Dassault Systèmes, Vélizy-Villacoublay, France) (Fig. 1A) that was 3D-printed in PLA. The mold was designed to form a cylinder of 17 mm radius and 40 mm length, which constitutes the central core. The enclosure, which contains both the core and its surrounding agar, has a diameter of 70 mm and either a raised inset to place the core (free diffusion experiment) or a 42 mm high wall that prevents the release of the molecules (Fig. 1B). The “blocked” phantom was needed to separate the effect of the susceptibility induced field inhomogeneity generated by the core from the effect of the increased concentration of the diffused molecules. The enclosures were 3D-printed on a Dremel 3D45 (Dremel, Mt. Prospect, Illinois) using clear PETG (Polyethylene terephthalate glycol) for the diffusion experiment or white PLA (Polylactic Acid) for the blocked experiment.
2.2 Sample Preparation
Gadobutrol solution (50 mmol/L) was prepared by diluting Gadovist (Bayer Inc., Leverkusen, Germany) with distilled water at a 1:20 ratio. Calcium sulphate hemihydrate powder (Stimulan, Biocomposites Ltd., England) was mixed with 6 ml of the dilute Gadovist. The resulting paste was poured into the silicone mold and left to set, forming a cylindrical core. Agar gel was prepared by boiling 1 L of distilled water to remove dissolved gas, allowing it to cool to room temperature in a sealed container and then mixing it with 35 g of agar and 80 mL of glycerol. The agar was heated to approximately 90°C and skimmed to remove impurities and air bubbles. The core was set into the enclosure and the agar was poured when cooled to 60°C. The assembled samples were left to set for 3 hours at room temperature prior to scanning. The samples are kept at room temperature between scans.
2.3 Phantom Design
An “external” phantom was also built to hold the assembled samples; this phantom serves two purposes: holding a set of calibration vials and providing a set of fiducial markers needed to ascertain the phantom’s precise orientation and center. Eight 2 mL calibration vials (Fisher Scientific, Waltham, Massachusetts) were prepared at 0.125, 0.25, 0.5, 1, 1.5, 2 and 2.5% of the raw (0.5 mmol/mL Gd) contrast agent (Magnevist; Berlex Laboratories; Wayne, NJ) in distilled water, along with a vial of only distilled water.34 Three additional vials were filled with agar made with 1.0%, 0.5%, and without Magnevist to assess the effect of agar on the quantitative images. Two vials of peanut oil were included to evaluate the effect of fat on our acquisition. The vials were held within a 3D-printed PLA construct; the configuration of these vials is shown in Fig. 1C. The 3D-printed construct was also built with three 7-mm spherical markers used to determine the phantom’s orientation relative to the coronal imaging plane. These markers are placed along a planar circle with a 51.45 mm radius. The assembled sample (Fig. 1D; the enclosure holding the core and surrounding agar) snugly fits into an inner ring, keeping the core consistently in the center of the ring formed by the markers; this enabled consistent repositioning of the assembled sample prior to each scan for phantom co-registration and alignment (see section 2.7). The outer ring was designed to be placed within a 130 mm diameter plastic container. The PLA construct, vials, and sample enclosure were placed in a plastic container and embedded in agar (Fig. 1E).
2.4 Gyroid-Based Porous Metal Core
A metal scaffold was designed based on a sheet-based gyroid – a triply periodic minimal surface that has been shown to have favorable mechanical properties for orthopedic applications, such as stiffness similar to bone and an appropriate strength for load-bearing implants.27 The scaffold, whose effective susceptibility has been previously studied,35 was designed with a porosity of 90% using Blender (Version 2.79, blender.org, Amsterdam, Netherlands), repeating a 6 mm3 unit cell (Fig. 2A) with a 0.2 mm wall thickness arranged into a 3x3x8 array, which was then truncated into a cylinder matching the central core (Fig. 2B). The resulting model was exported as STL (stereolithography) files and sliced for manufacture using the QuantAM build preparation software (Renishaw plc, Wotton-under-Edge, United Kingdom). The structures were 3D printed in Ti6Al4V medical grade titanium alloy (Ti6Al4V ELI-0406, Renishaw plc, United Kingdom, particle size 15–45 µm) using laser powder-bed fusion (AM400, Renishaw plc, Wotton-under-Edge, United Kingdom) at ADEISS (London, Canada) with a laser spot diameter of 70 µm and layer thickness of 40 µm. The manufactured porous titanium scaffold was loaded by filling the silicone mold with the gadobutrol-loaded calcium sulphate and slowly inserting the metal core prior to setting, which allows the fluid paste to fill the void spaces of the gyroid structure (Fig. 2C).
2.5 Imaging
Imaging was performed using a 3T Prisma scanner (Siemens Healthineers, Erlangen, Germany) with a Siemens 32-channel head coil. Scans were acquired using a 3D multi-echo GRE sequence with echo times at 4.16, 5.52, 6.88, 8.26, 9.76, 11.67, 14.00, 16.34, 18.67, and 21.00 ms (echo train length = 10) at 1 mm3 resolution. The other parameters include TR = 24 ms, 15° flip angle, BW = 1010 Hz/pixel, matrix size = 160x160x60, 16 cm FOV and a total acquisition time of 3 minutes and 50 seconds. The phantoms were scanned in the coronal configuration, with the samples perpendicular to B0. Scans were performed 3 hours and 10 hours after pouring the agar, followed by scans at 32, 56, and 80 h, then at 1, 2 and 4 weeks. Phase and magnitude images were channel combined and reconstructed on the scanner then exported as DICOM files.
2.6 Quantitative Mapping
Multi-echo complex data were assembled and processed in Matlab (Mathworks, Natick, Massachusetts) from the scanner reconstructed magnitude and phase images. Complex images were processed using the B0-NICE algorithm34,36,37 to generate fat fraction, R2*, and B0 maps from the 10 echoes. The R2* maps were calculated based on data-fitting of the magnitude images from the 10 echoes with echo spacing shortened (relative to previous B0-NICE applications) to reduce the effects of field inhomogeneity in the late echoes. QSM maps were generated using the MEDI algorithm38 implemented on the 10-echo complex data and normalized to the values of the distilled-water vial. The QSM maps were used to measure the drop in core susceptibility as the contained gadobutrol is released into the surrounding agar.
2.7 Phantom Co-registration and Alignment
Because imaging experiments over a prolonged period require sample repositioning, alignment of the images was required. The three alignment markers were identified in each image by thresholding, morphologically eroding the supporting structures, and calculating the markers’ centroids. The center of the circle formed by the 3 markers indicates the precise coordinates of the center of the sample core. The markers also provide a measurement of both the in-plane rotation and through-plane tilt of the phantom within the magnet. Rotation matrices were calculated based on the marker centroids and used to co-align all images.
2.8 Data analysis
The aligned magnitude images, R2* and QSM maps were analyzed at each time point for both sample types (calcium sulphate only and calcium sulphate in metal) in both free diffusion and blocked enclosures. Calibration vial data were averaged within 8-mm circular regions of interest (ROI), 3-slices thick, centered on the plane defined by the markers. Similar regions were defined to calculate average core QSM values. Analysis of the diffusion samples was performed using radially averaged line profiles centered on the core, culminating in the quantification of gadobutrol release, as described below.
2.8.1 Radial Averaging
Radial averaging was employed to improve SNR, minimize non-uniformities within the agar (e.g., small air bubbles) and simplify quantification and analysis while examining the slow diffusion of contrast agent through the agar. Radial averaging takes advantage of the radial symmetry of the sample and the radial nature of the diffusion of gadobutrol from the cylindrical core to average sample points that are equidistant to the core. In our case, where the images are aligned with the axis of the phantom, this was achieved by first averaging five slices, then creating and averaging 120 line profiles (spaced 3° apart) radiating from the center of the phantom. All data analysis was performed using Matlab.
2.8.2 Gadobutrol Release
The diffusion of small molecules can be modelled generally using Fick’s laws of diffusion.39 Our system (Gadovist in calcium sulphate formed into a cylinder) imitates a cylindrical diffusion-controlled (predominantly controlled by diffusional mass transport) drug delivery system, which has a known model.39 The calcium sulphate core constitutes a ‘monolithic solution,’ as the molecule of interest is dispersed homogeneously throughout the calcium sulphate matrix, and thus the “drug” (Gadovist) release can be described by the approximation:39
\(\frac{{M}_{t}}{{M}_{\infty }}=1-\frac{4}{{2.405}^{2}}\text{exp}\left(-\frac{{2.405}^{2}Dt}{{R}^{2}}\right) ,\)
|
(Eq. 1)
|
where M(t) is the cumulative amount of drug released at time t, M(∞) is the cumulative amount of drug released at infinity, D is the diffusion coefficient of the drug within the system, and R is the radius of the inner core cylinder. If R2* and concentration are linearly correlated, we can use the mean R2* value (R2*avg(t)) within the agar to measure the release of gadobutrol from the core into the enclosed agar through the ratio: |
\(\frac{{M}_{t}}{{M}_{\infty }}=\frac{{R2}_{avg}^{*}\left(t\right)}{{R2}_{avg}^{*}\left(\infty \right)}\cong \frac{{R2}_{avg}^{*}\left(t\right)}{{R2}_{avg}^{*}\left(final\right)}\) | (Eq. 2) |
with the approximation that R2*avg(∞) is equivalent to R2*avg(final), taken from the 4-week time point. For each time point, R2*avg(t) were calculated by averaging the values with a 26-mm annulus encompassing the agar surrounding the 9-mm central core. A baseline R2*avg(0) value was derived from the blocked (non-diffusing) phantom using the same averaging approach and subtracted from each R2*avg(t). For the calcium sulphate-only core (non-metal), best-fit curves (exponential plateau) were calculated while constraining the constants (Eq. 1) to find the diffusion coefficient D of the system. The addition of an internal scaffold structure (metal core) invalidates the model of Eq. 1. Therefore, the R2* data derived from the metal core phantom was fit without constraints. To determine the relationship between core QSM and Gd release, the average QSM values from the calcium sulphate core were fit to an exponential decay model.
2.8.3 Statistical analysis
All line fitting was evaluated for quality of fit by coefficient of determination (R-squared value). All curve fitting and statistics were done in Prism 9 (version 9.0.0, Graphpad Software, San Diego, California).