Design and experimental validation of a unilateral magnet for MRI-guided small animal radiation experiments

Abstract

Small animal radiation experiments are of paramount importance for the advancement of human radiation therapy. These experiments use a dedicated radiation platform to deliver radiation to small animals, such as mice and rats, similar to how human radiation therapy is performed. By acquiring images immediately before radiation delivery to guide positioning of the animals, image guidance plays a critical role to ensure accuracy of the experiments. Recently MR-based image guidance has been enabled in human radiation therapy. This paper proposed a new concept using a unilateral magnet-based MRI scanner to realize image guidance for small animal radiation experiments. We reported our design, optimization, construction, and characterization of the magnet. The magnet was designed using eight 2-inch neodymium magnet cubes approximately in a Halbach ring configuration. The ring has an opening to allow for animal positioning. We considered a spherical region of interest (ROI) located outside of the ring’s plane to allow radiation delivery to the ROI without obstruction of the magnet. An optimization problem was formulated and solved to determine the positions and orientations of the magnet cubes to generate a magnetic field with desired properties in the ROI. The optimization improved the average magnetic flux density from 55 mT to 72 mT and reduced variation from 1.2 T/m to 1.0 T/m. We constructed the magnet using 3D-printed templates to hold the neodymium magnet cubes with the optimized positions and orientations. We measured the spatial distribution of the magnetic flux density. The measurement results and computed results agreed with an average difference of 0.35% through the ROI.

1. Introduction

1.1. Image-guided small animal radiation experiment

Small Animal Radiation Experiments (SARE) refer to the applications of ionizing radiation, typically x-rays, on small animals, e.g. mice or rats, for the purpose of investigating radiobiological mechanisms and for developing new radiotherapeutic strategies. Serving as the counterpart to human radiotherapy (RT), SARE is of paramount importance, allowing for comprehensive preclinical studies with a large number of subjects with highly controllable experimental conditions at low costs. SARE relies on dedicated platforms (irradiators) to deliver radiation to small animals in a similar way to human RT[1]. This is critically important to ensure validity and relevance of the radiobiological studies to human RT. Over the years, irradiators have been developed to incorporate technologies available in human RT to support SARE studies. For instance, Fig. 1(a) shows the inside of the SmART system (Precision X-ray irradiation Inc., CT, USA) that is equipped with a motion controlled animal bed and an x-ray tube mounted on a rotating gantry. With gantry rotation, the radiation generated by the x-ray tube can be delivered from multiple angles towards a region at the center of the gantry.

Figure 1:

(a) Geometry of an SARE irradiator from Precision X-ray irradiation Inc. (b) Proposed unilateral magnet MRI scanner integrated onto the irradiator.

Image guidance plays an important role for accurately delivering radiation in both clinical RT and preclinical SARE. This refers to the process of acquiring images with the patient/animal on the bed, based on which the bed position can be adjusted to precisely align the radiation beam with the target defined in the acquired image. Achieving this function requires the integration of an onboard imaging device with the radiation delivery system. At present, Cone beam CT is the most widely used image guidance tool in clinical RT[2]. In SARE, existing irradiators have employed this technique using the x-ray tube and a flat panel detector on the gantry, as shown in Fig. 1(a). By rotating the gantry to acquire x-ray projection images, a volumetric cone beam CT image of the animal can be reconstructed.

1.2. MR-guided SARE and technical challenges

The use of cone-beam CT for image guidance is impeded by poor soft-tissue contrast, image artifacts, radiation exposure, and lack of real-time imaging capabilities. Recently, MR-guided RT has become available in human RT based on technical advancements by combining an MRI scanner with a medical linear accelerator[3]. This has permitted the acquisition of MR images of the patient for positioning and adaptive treatment planning before treatment delivery. Similarly, it is desired to have the function of MR-based image guidance in SARE for precise radiation experiments.

Nonetheless, there is no SARE platform with MR image guidance capabilities. Current strategies for implementing MR-based image guidance into the SARE workflow require the animal to be moved between a separate MRI scanner and the irradiator, which inevitably leads to positioning uncertainties. Attempts to remediate these uncertainties (e.g., immobilization techniques to fix the animal position) do not account for changes in internal anatomy or small shifts in the animal’s location. In addition, many institutions may have their irradiator and MRI scanner in separate rooms, or even different buildings, making this scan-and-reposition approach logistically unfavorable or even infeasible.

The geometric and electromagnetic interference between the MRI scanner and the x-ray irradiation system pose substantial challenges to integrate the two systems. Firstly, conventional MRI scanners necessitate a highly homogeneous magnetic field, while an irradiator in close proximity to the magnet distorts the field and disrupts the homogeneity. Secondly, an x-ray tube in the presence of a strong magnetic field can become inoperable or even damaged due to the deflection of electron trajectories missing the tungsten target. Lastly, when integrating an MRI scanner with an irradiator, the MRI components cannot block the radiation beam for experiments, posing strong geometric constraints in the system design, especially under the compact size of SARR systems.

1.3. Our contributions

To address these challenges and develop a compact MRI scanner that can be integrated to a SARE platform for image guidance, we propose to employ a non-conventional MRI scanner design with a unilateral magnet. Fig. 1(b) illustrates this idea. Specifically, we propose to position the magnet close to the x-ray irradiation plane formed by the x-ray beams delivered at different gantry angles. The fringe field extending to the region of interest at the center of the gantry is used for MR imaging purposes. A unilateral magnet-based MRI scanner, in general, is not a new idea. Over the years, several systems with this setting have been developed for a variety of different applications[4, 5, 6, 7, 8, 9, 10]. Compared to traditional closed-bore MRI system design, the unilateral magnet design allows for imaging a region of interest outside the magnet, and hence provides a flexible geometry to accommodate different needs. In our study, we innovatively make use of this unique geometry to allow an easy integration of the scanner with the small animal irradiator. More importantly, the magnet is positioned outside of the radiation beam plane to avoid blocking the x-rays and to maintain a relatively low magnetic field, ultimately preserving normal functions for radiation experiments. Moreover, the monotonically decaying magnetic field along the axial direction can be naturally used for spatial encoding along that direction, eliminating one gradient coil, which is beneficial for the limited space of this configuration.

In this paper, we will report our recent progress on the design, optimization, construction, and characterization of the unilateral magnet for the proposed MRI scanner. The rest of the paper is organized as follows. Sec. 2 will present our approach to design the magnet by solving an optimization problem and Sec. 3 will present the construction process of the magnet and the characterization of its performance. After a discussion on relevant issues and future works in Sec. 4, Sec. 5 will conclude this paper.

2. Magnet design

2.1. Geometry configuration and initial design

The primary constraint guiding the design of the magnet is that it has to fit in the limited space in the irradiator while not obstructing the radiation beam or preventing other essential operations such as couch alignment or rotation of the imager and x-ray tube. Furthermore, additional space must also be allocated for gradient coils and an RF coil to complete the MRI system. The magnetic field must be as high as possible to improve signal-to-noise ratio (SNR). At the same time, the field strength at the x-ray target must be kept low to not interfere with x-ray tube operations [11].

All of these considerations led to the initial design of the magnet system shown in Fig. 2(a). The initial design, which will be further optimized in the next section, was comprised of a Halbach cylinder with eight permanent neodymium (Nd) magnets that are each sized at 5.8 cm and graded at N52. The center of the eight magnets are located on a circle with a radius of 10.56 cm. This sizing, grading, and number of magnets was chosen, as they are the strongest commercially available magnets that can be arranged in a Halbach cylinder to achieve a bore opening size of ~ 12 cm in diameter, permitting clearance of the irradiator’s animal bed through the bore and allowing for additional space of the gradient coils.

Figure 2:

A) Geometry of the proposed magnet design relative to the ROI (red sphere). Orientations and postitions of the eight magnet cubes are not drawn exactly as in the final results. B) Illustration of a Halbach cylinder that produces a magnetic field that is vertically oriented. The circle shows an approximate spherical region with a homogeneous magnetic field.

A Halbach cylinder consists of an arrangement of permanent magnetic cubes that augment the field in one direction. A continuous 4π rotation in magnetization around a cylinder can be used to create a homogeneous field that is aligned along a single direction at the interior region of the cylinder. Fig. 2(b) illustrates a Halbach cylinder with eight magnets and the resulting homogeneous vertical field orientation within a spherical region in the plane of the Halbach cylinder. Prior studies have determined feasibility of Halbach cylinders for low-field MRI of ~ 50 mT. However, these studies developed Halbach cylinders for imaging at a region of interest (ROI) located at the center of the cylinder.[12, 13, 14]. In our particular problem, this setting would cause the magnets to block the radiation beam during experiments. Hence, the ROI must reside outside of the magnet. Specifically, let us consider the coordinates as shown in Fig. 2(a) with the origin at the center of the Halbach cylinder. We considered an ROI with 4 cm in diameter with its center located at z = 5.2 cm on the z axis. The ROI is large enough to encompass a typical target in animal experiments, e.g. a tumor, and surrounding tissues. The center position ensures that radiation delivered to the entire ROI is not blocked by the magnets, even with applied rotations when optimizing the magnet design (see the next section.)

We first evaluated the magnetic flux density in the ROI under this initial design. As such, we performed computations using COMSOL Multiphysics (Burlington, MA) with finite element analysis. The magnets were meshed and the remanent flux density was set to 1.44 T and relative permeability to 1 for the N52-graded magnet cubes. A rectangular prism of size 50 cm³ was defined with an infinite element domain surrounding this region to serve as an air domain with no induced magnetization. Results of this calculation determined a field strength of ~ 55 mT at the center of the ROI and the gradient along the z axis from this configuration was ~1.2 T/m.

2.2. Numerical modeling of magnetic field and Optimization problem

To further optimize the field distribution for imaging purposes, we solved an optimization problem to determine the orientations and positions of the eight magnets to push the magnetic flux towards the ROI. Let us consider the geometry as shown in Fig. 2(a). Denote the position and orientation of the ith magnet as x_i and Ω_i, respectively. x_i is a vector and Ω_i is a unit vector in three-dimensional space. At a point of interest r, the magnetic flux density B(r) was considered as the linear superposition of that of all the eight magnets:

$$\mathbf{B}(\mathbf{r})=\sum _{j=1}^8\mathbf{B}({\mathbf{r};\mathbf{x}}_j,{\mathbf{\Omega }}_j),$$ (1)

where B(r; x_j, Ω_j) is the contribution of the magnetic flux density at r from the jth magnet. We considered the dipole approximation in this study:

$$\mathbf{B}(\mathbf{r};{\mathbf{x}}_j,{\mathbf{\Omega }}_j)=\frac{{\mu }_0}{4\pi }\left[\frac{3(\mathbf{r}-{\mathbf{x}}_{\mathbf{j}})[{\mathbf{m}}_j\cdot (\mathbf{r}-{\mathbf{x}}_{\mathbf{j}})]}{{|\mathbf{r}-{\mathbf{x}}_j|}^5}-\frac{{\mathbf{m}}_j}{{|{\mathbf{r}-\mathbf{x}}_j|}^3}\right],$$ (2)

where m_j is the magnetic moment of the magnet, located at x_j with an orientation defined by Ω_j. The validity of this calculation, as well as the superposition approximation in Eq. (1) was evaluated by comparing the computed magnetic flux density in this form with that computed by COMSOL Multiphysics.

The optimization problem we considered was

$$\{{\mathbf{x}}_i,{\mathbf{\Omega }}_i\}={\mathrm{argmin}}_{\{{\mathbf{x}}_i,{\mathbf{\Omega }}_i\}}[-\sum _{i=0}^6|\mathbf{B}({\mathbf{r}}_i)|+{\lambda }_1||\mathbf{B}({\mathbf{r}}_1)|-|\mathbf{B}({\mathbf{r}}_2)||+{\lambda }_2\sum _{i=1}^6{\cos }^{-1}\left(\frac{\mathbf{B}({\mathbf{r}}_i)\cdot \mathbf{B}({\mathbf{r}}_0)}{|\mathbf{B}({\mathbf{r}}_i)\cdot \mathbf{B}({\mathbf{r}}_0)|}\right)],\mathrm{s.t.}\mathbf{B}({\mathbf{r}}_i)=\sum _{j=1}^8\mathbf{B}({\mathbf{r}}_i;{\mathbf{x}}_j,{\mathbf{\Omega }}_j).$$ (3)

While not explicitly written in the optimization problem, it also included the constraint that the positions and orientations of the magnets cannot overlap, and that the configuration must allow for a bore opening of at least 12 cm in diameter. Only the x and y components of xj were considered, whereas the z component was set to zero to constrain the centers of the magnets on the z = 0 plane.

Seven points of interest were considered in this optimization problem, located at the center of the spherical ROI, r₀, and at six points where the sphere intersects with the x, y, and z axis, r_i, i = 1, …, 6, as shown in Fig. 2(a). There are three objective function terms considered in Eq. (3) to achieve the desired properties of the field for MR imaging purposes. The first term attempted to increase the overall field strength in the ROI. Note that a minus sign was used in this term to increase the field strength when minimizing the objective function. The second term tried to minimize the field decay along the z axis. As for the third term, it enforced that the variation in the field direction was minimized in the ROI and that the field vector was aligned along the y direction in order to minimize signal loss. λ₁ and λ₂ are weighting factors to balance the contributions of the three objectives. These were manually adjusted in this study to achieve satisfactory results.

As the optimization problem is highly non-convex, we solved the problem using a genetic algorithm[15]. We first randomly perturbed positions and orientations of the magnets in the configuration of the initial Halbach cylinder shown in Fig. 2(b) for a total of 3000 times to generate a number of 3000 different configurations, or genes, as the initial solution set of the genetic algorithm, namely the initial population. The genetic algorithm iteratively updated the gene population by repeatedly performing selection, crossover, and mutations. Each iteration was termed as one generation with the maximal generation number set to 500 in our study. In each generation, the selection process occurred first by picking 500 genes that achieved the lowest objective function values from the previous generation and placing them to the current generation pool. Then the crossover operation randomly selected a pair of genes from this pool and allowed them to exchange their gene with a random crossover point to create a new pair of genes that were added to the pool of the current generation. We employed 750 crossover operations in each generation, creating 1500 new genes. Lastly, a number of mutations occurred in which the genes were randomly perturbed to create new genes. The mutation operation was helpful to enhance the diversity in the gene population, preventing early convergence of the genetic algorithm. In our study, we allowed for 1000 mutations in each generation, generating 1000 new genes in the pool. The selection, crossover, and mutation processes maintained the total number of 3000 genes unchanged in each generation. This process was repeated until the algorithm converged, or the maximal generation number was reached. At the end, the gene with the lowest objective function value was selected as the final solution.

2.3. Optimization results

We first investigated the validity using Eq. (2) to compute the magnetic flux density. As such, we compared the magnetic flux density distribution around a single magnet computed by this equation and that computed by COMSOL. As for the validity of the linear superposition assumption in Eq. (1), we considered a test case with two magnet cubes separated by 5 cm. We computed the magnetic flux density using Eq. (1) and COMSOL. The results in these two cases are shown in Fig. 3. As expected, the dipole approximation and the linear superposition approximation are valid for points relatively far away from the magnet cubes, as indicated by the small difference between the analytical and the COMSOL calculation results at regions ~> 5 cm from the surface of the magnets.

Figure 3:

Computed magnetic flux density field of a magnet (a) by Eq. (2), (b) by COMSOL, and (c) the difference between them. Computed magnetic flux density field of two magnet cubes (d) by Eq. (1), (e) by COMSOL, and (f) the difference between them. Gray region illustrates where the magnet is located.

With the validity of our magnetic field flux density calculation verified, we solved the optimization problem in Eq. (3). The results are presented in Fig. 4. The optimization successfully increased the magnetic flux density strength from on average 0.55 T within the spherical ROI to 0.72 T. The variation across the ROI was also reduced from 1.2 T/m to 1 T/m. The angular range of the magnetic field direction over the ROI was reduced by up to 9 degrees.

Figure 4:

(A) Optimized Halbach cylinder with the ROI illustrated by the white sphere. (B) COMSOL calculations comparing the resulting magnetic flux density magnitude of the optimized Halbach cylinder (blue) and non-optimized (orange). The profile of this plot is illustrated in (A) as the line extending along the z-axis. The shaded gray region depicts the intended imaging ROI. (C)-(D) Angles of the magnetic flux vector directed away from the y-axis is shown in (C) for the optimized solution and in (D) for the non-optimized solution. (E) Difference between the optimized and non-optimized. The x and y plane is illustrated in (A) by the red plane in the ROI.

3. Magnet construction and characterization

3.1. Magnet construction

The magnet assembly was constructed using eight cube-shaped magnets with side lengths of 5.8 cm, and graded at N52 (Applied Magnets, Plano, TX). These were arranged based on our optimization results. The optimized magnet orientations are in an unstable configuration, thus requiring additional restraints to prevent movement. As such, we designed molds to isolate the magnets using computer-aided design (CAD) in Autodesk Inventor (San Rafael, CA) as shown in Fig. 5. Each mold was sliced along the center (z = 0 plane) to allow for the magnet to be inserted. Additionally, two aluminum plates were used to hold these molds. They were fixed with brass bars to serve as guide-rails for the magnet molds to be positioned and fixed to the plates.

Figure 5:

(A) Half of the eight designed molds showing the opening to house the magnets. (B) Final concept of 3D printed molds for fixing the magnets to the optimized orientations and locations. One-half of a mold is left open to illustrate the magnet sealed inside. (C) Molds mounted between two aluminum plates with holes to allow for screws to hold everything in place. (D) Completed magnet assembly.

We created the molds using 3D printing on a Makerbot Replicator+printer (New York City, NY). The molds were printed with a 20% infill density and layer height of 0.12 mm. Each magnet was individually seated inside of a mold according to the optimized configuration. The magnets were sealed well enough to not warrant additional adhesives such as glue. This was then fixed with brass screws and tightened with nuts on the ends outside the aluminum plates to prevent movement. Tolerances of ±1 mm were expected for the final design due to 3D printing errors and alignment with the screw holes in the aluminum plates. This design and the final magnet assembly are shown in Fig. 5.

3.2. Magnet characterization

The magnetic flux density field generated by the completed optimized Halbach cylinder was measured with a hall probe (FP-2X-250-ZS15M-6, Lake Shore Cryotronics) and a teslameter (F71, Lake Shore Cryotronics). We scanned the field using three positioning stages (Velmex BiSlides, Bloomfield, NY) with a custom C++ code for motor control. The probe was moved to measure line profiles along the three major axes with a spacing of 1 mm.

Fig. 6 shows the comparison of the three components of the magnetic flux density between the measured and calculated results along the z axis from the center of the cylinder through a distance of up to 10 cm. The shaded region is the designed imaging ROI. Fig. 7 is the comparison of the magnitude of the magnetic flux density along the x and y axes at a distance of 2 cm from the surface of the magnet. These lines extend through the center of the irradiator’s gantry rotation, representing the area of interest in irradiation experiments. Generally, the agreements between calculations and measurements were satisfactory with an average relative difference of 0.35%. The discrepancies in the x and y components in Fig. 6 can be attributed to the misalignment of the hall probe from the targeted positions and orientations. While the discrepancies are also obvious in Fig. 7, this is indeed caused by the small range of the vertical axes in these plots.

Figure 6:

Calculated vs measured results along z axis.

Figure 7:

Comparison of the measured and calculated magnetic flux density magnitude along the x and y axes.

We also measured the temporal stability of the magnetic flux density over time due to changes in ambient temperature inside of the irradiator. For this purpose, we moved the magnet assembly to the inside of the irradiator at the position illustrated in Fig. 1(b). We then positioned the Hall probe at the center of the magnet’s front surface and recorded the results every 30 minutes for 20 hours. The magnetic susceptibility of Nd magnets is inversely proportional to temperature. As the temperature inside the irradiator is higher than the room temperature, a decrease in field strength was observed as the magnet was warmed to a settling temperature of about 24.6 °C, as shown in Fig. 8. The change in magnitude over this time, ΔB, was approximately 0.07mT. In the planned MRI data acquisition, we will use the gradient along the z axis as slice selection. The drift of field strength due to temperature will result in a shift of slice location, which can be estimated as ΔB/G_ss with G_ss ~ 1T/m being the gradient strength. This gave a slice shift of 0.07 mm, which is acceptable given that the slice thickness of MRI images is usually on the order of millimeters.

Figure 8:

Magnetic flux density field strength and temperature measured inside of the irradiator over a 20 hour period with 30 min intervals.

We also remark that this drift of the magnetic field is likely the worst-case scenario, as the measurement was recorded after moving the magnet assembly from the room to the inside of the irradiator. A more realistic condition will consist of having the magnet assembly permanently fixed inside the irradiator, resulting in a temperature change that is mainly caused by temporarily opening and closing the irradiator door. The resulting change in magnetic field strength is expected to be smaller than what is shown in Fig. 8.

4. Discussion

Unilateral, or single-sided, MR systems experience a natural decay of the magnetic field as a function of distance from the surface of the magnets. Relatively large unilateral systems compared to a respectively small ROI can create the so-called ”sweet spots”, which utilizes a limited homogeneous magnetic field region for imaging [9, 16, 17, 18, 19]. These sweet spots are typically limited to <1cm in diameter with a relatively low magnetic field strength. For larger ROI’s that extend beyond the surface of the magnets, a homogeneous sweet spot is not practical to achieve on a system that could be fixed inside of a small animal irradiator while addressing the challenges outlined in Sec. 2. Hence, in our design, we chose to use an inhomogeneous magnetic field design for the MR system. While the decaying field can be naturally used as slice selection, this poses multiple challenges that must be addressed down the road.

The first is the relatively large gradient of ~1 T/m that necessitates a large radiofrequency (RF) bandwidth in order to excite spins across the ROI. A major consideration when designing a unilateral magnet for MRI is the RF bandwidth requirements due to the gradient strengths. Conventional gradient magnetic fields produced by gradient coils in relatively small MRI systems, are typically on the order of a few mT, necessitating a BW that is usually <0.5MHz. Our unilateral magnet design has a gradient of ~1T/m, spanning from 91.3mT to 53.3mT through the ROI. This translates to a total bandwidth of 1.7MHz. This large bandwidth makes it difficult to excite spins during the slice select and refocusing pulses, and increases concerns regarding SNR. We have ongoing work to develop a transmit and receive RF coil that has a relatively narrow bandwidth but with a central frequency that can be electronically tuned across the required frequency range. The second challenge is on image processing. The inhomogeneous magnetic field design necessitates a 2D slice selection approach for data acquisition, as the spins are excited along the isosurfaces of the magnetic field. These isosurfaces span a curvature in 3D space that varies depending on the specific magnetic field distribution. This requires the reconstructed 2D images to be rebinned to form a volumetric image defined on the conventional Cartesian grid to prevent artifacts and facilitate image guidance tasks.

Another consideration for using a permanent gradient is the limitation of pulse sequences to those of spin-echo based methods, since gradient echo sequences require control of the slice select gradient. While $T_2^{*}$ inhomogeneities can be corrected with the use of spin-echo sequences, slice-rephasing lobes cannot be implemented. Hence, some loss in signal is inevitable. We plan to implement image denoising algorithms, e.g. using recent advancements in deep learning[20, 21], to allow imaging data acquisition within a reasonable amount of time while still achieving sufficient image quality.

5. Conclusion

In this paper, we proposed a new concept using a unilateral magnet-based MRI scanner to realize image guidance for SARE. We reported our design, optimization, construction, and characterization of the magnet. The magnet was designed using eight 5.8 cm Nd magnet cubes arranged in a Halbach cylinder configuration. To avoid obstruction of radiation beam, the spherical ROI was located outside the magnet plane. We formulated an optimization problem to determine the positions and orientations of the magnets to generate the desired magnetic field properties in the ROI. The optimization improved the average magnetic flux density from 55 mT to 72 mT and reduced variation from 1.2 T/m to 1.0 T/m. The magnet was constructed using 3D-printed molds to precisely hold the magnets. We measured the spatial distribution of the magnetic flux density. The measurement and computed results agreed with an average difference of 0.35%.

Highlights.

• Proposed a new concept to realize MR-guided small animal irradiation experiment

• Designed and optimized a unilateral MR magnet to meet constraints

• Constructed the magnet using 3D printed templates with neodymium magnet cubes

• Measured field distribution agreed with calculations with an average of 0.35% difference