Development of an optimized dome-shaped magnet for rapid magnetic immunostaining

Immunostaining plays an important role in the identification of various cancers and their subtypes.^1,2 Dye-labeled antibody is introduced to a specimen to bind with a cancer-specific antigen in the tissue sample. Immunostaining also enables investigation of tumor characteristics, such as the type of cancer, grade of malignancy, and anti-cancer drug resistance, dependent upon the detected antigens. Despite these benefits, the immunostaining process takes several hours and is not a feasible approach for intraoperative examination. Thus, the tissue specimen for immunostaining is excised during surgery and if postoperative immunostaining indicates that cancer cells likely remain in the patient’s body, it is recommended that the patient undergo second surgery for re-excision of the remaining tumor tissue. This causes a significant increase in treatment invasiveness for the patient. Thus, there is a need for a rapid immunostaining technique to facilitate intraoperative examination.

Our group previously developed a technique for magnetic immunostaining using antibody-labeled fluorescent magnetic beads and a magnetic field.^3–5 The magnetic beads were fabricated by coating ferrite nanoparticles with a polymer which incorporated fluorescent dye excitable by ultraviolet light, and fixing antibodies on the surface of the polymer.^6,7 These beads were then dispersed in a solution and introduced to the tissue specimen on a glass slide. A magnetic force was applied by a permanent magnet placed below the glass slide and the beads were attracted toward the specimen by the magnetic force. The locally increased density of antibody promoted an accelerated antigen-antibody reaction, with a completion time of as short as one minute.^3,4 Finally, the presence and distribution of cancer cells on the specimen were visualized by fluorescent imaging. Similar magnetic immunoassay techniques and detection systems have been developed by various groups worldwide.^8–11 

The design of a permanent magnet to produce the magnetic force field is an important aspect of the effectiveness of this technique.⁵ In addition to the strength of the magnet, the spatial distribution of force is an important factor in achieving uniform specimen staining. Since non-uniformity of staining can hinder accurate diagnosis, the magnet must apply a uniform force to the beads.

In this study, we analyzed the dynamics of magnetic beads under the applied force field. An optimal permanent magnet geometry for the application of homogeneous force was explored. Finally, a prototype magnet was fabricated for proof-of-concept immunostaining.

We investigated the motion and trajectory of antibody-labeled magnetic beads in response to the application of a magnetic field generated by a permanent magnet. In particular, numerical methods were used to calculate the resulting magnetic field distribution and magnetic force acting on the ferrite beads in response to a given magnet shape. The motion of beads in an aqueous solution in response to the magnetic field was examined to evaluate the homogeneity of the bead distribution on the tissue specimen. These investigations were performed using various parameters to the magnet shape, with the aim of finding the optimal shape. Numerical analysis was performed with MATLAB.

As shown in Figure 1(a), a magnet was designed within the region z ≤ 0. The magnet shape was composed of cubic elements with a side length of 0.1 mm. The magnetic flux density B at position r was calculated as the sum of magnetic flux densities arising from these small element as follows.⁵ 

where m_e = (0, 0, m₀) is the magnetic moment of the elements and r_i is the position of i-th element. In order to estimate the value of m₀ for neodymium magnets, the magnetic field distribution of a ring-shaped sample magnet was measured and a numerical simulation was performed to find an m₀ value which provided the best agreement between the experiment and the simulation. The estimated value of m₀ was thus 1.6×10⁻⁵ A m².

When a bead with the magnetic moment m_b is subjected to a magnetic flux density B, the magnetic energy is

Thus, the magnetic force F_m acting on the bead is

The value of m_b was measured by a magnetic property measurement system. Magnetic fields ranging from −300 mT to +300 mT were applied to the beads, and the magnetic moments of the beads were measured using a superconducting quantum interference device (SQUID). As shown in Figure 1(b), the magnetic moment at 300 mT was 7.1×10⁻¹⁷ A m² per bead.

The regions beyond the permanent magnet had no current source, satisfying the relation $∇×B=0$⁠. Assuming the bead was magnetized homogeneously, $∇× m b =0$ and $B ⋅ ∇ m b =0$⁠. Thus, the magnetic force acting on the bead was given by⁵ such that

This equation indicates that magnetic force on the bead is given by the product of magnetic moment and the gradient of the magnetic field.

The dynamics of the bead under the attractive force of the magnet was analyzed by equations of motion. The hydrodynamic diameter of a bead was measured using the dynamic light scattering method and was d = 224.6 ± 2.8 nm in deionized water and d = 833.4 ± 166.9 nm in a phosphate buffered saline (PBS) solution. These results suggested that the beads were monodispersed in the deionized water but that they aggregated in the PBS solution. The following analyses were conducted with beads in PBS solution with d = 833.4 ± 166.9 nm. Given Reynolds Number < 2, we considered the force of gravity F_g, buoyancy F_b, and viscous drag F_d given by the following formulae, in addition to the magnetic force¹² such that

where η is the viscosity and r_b is the position of the bead. Table I summarizes the parameters for Eqs. (5–7). While the magnetic force F_m ranged from 1.0 to 3.0×10⁻¹³ N, the gravity F_g and buoyancy F_b were approximately 4.4×10⁻¹⁵ N and 3.0×10⁻¹⁵ N, respectively. Since the magnetic force is significantly stronger than the forces of gravity and buoyancy, the magnetic field predominantly facilitates the accumulation of beads on the specimen. In addition, Brownian motion and diffusion play a considerable role when the following condition is satisfied¹² 

where F is the total external force on the bead. Since this condition does not hold in this study, the bead motion was analyzed based on Newtonian dynamics.

Hence the following equation gives the motion of magnetic beads.¹² 

The solution to this equation indicated that the beads reached the terminal velocity of v_t = (F_m + F_g + F_b)/3πηd in 1 ms. Thus, the motion was estimated by v_t, which was dependent on the location of the bead.

The tested shapes of the permanent magnet were designed to be symmetrical around the z-axis. The resulting accumulation of beads on the sample was also expected to be axisymmetric. To reduce computation time, the motion of the beads was only analyzed for x ≥ 0 in the x-z plane. Figure 2 shows the geometry and structure of the magnetic immunostaining device. Since typical tissue specimens are smaller than 20 mm, the homogeneity was evaluated in the region of 0 ≤ x ≤ 10 mm (with the resulting diameter of 20 mm). In addition, considering the thickness of a glass slide, we assumed that the tissue specimen was located at z = 1 mm. The solution with magnetic beads was placed above the specimen with a thickness of 2 mm. Thus, the initial heights of the beads were 1 ≤ z ≤ 3 mm. The homogeneity of beads was evaluated in the region of 0 ≤ x ≤ 10 mm at z = 1 mm.

As the initial condition at t = 0 s, the beads were uniformly distributed in the region of 1 ≤ z ≤ 3 mm with a spacing of 0.01 mm. The line of 0 ≤ x ≤ 10 mm at z = 1 mm was divided into 1000 sections with a width of 0.01 mm. The number of beads arriving at each section was counted every second. The mean and standard deviation of the number of beads were calculated across these 1000 sections and the coefficient of variation (c_v = standard deviation/mean) was also calculated as an index of homogeneity. c_v = 0 in the case of a perfectly homogeneous distribution, and thus, smaller c_v values indicate greater homogeneity. The magnitude of force was evaluated based on the time required for all beads to land on the specimen. Notably, this time should be approximately 60 seconds or less to make the system appropriate for intraoperative use.

Simulation analysis was first carried out for the cylindrical magnet shown in figure 3(a) which was 24 mm in diameter and 30 mm in height. The top surface of the magnet was positioned at z = 0. The results are shown in figures 3(d, g and j). Figure 3(d) presents a map of the magnetic force on the bead, with white curves indicating the trajectories of bead migration in the aqueous solution. The applied force was stronger around the edge of the magnet compared to the center as the magnetic field exhibited a larger gradient on the periphery. The force field was generated such that the beads gathered toward the edge of the magnet. Along the line z = 2 mm, the magnetic force was 2.8×10⁻¹³ N at x = 0 mm and 5.0×10⁻¹³ N at x = 10 mm. Figure 3(g) shows the distribution of beads for z = 1 mm recorded at t = 20, 40, 60, and 80 s. The coefficient of variance for each timepoint was calculated and is shown in figure 3(j). The c_v value at the end of the simulation was 0.023, but c_v > 0.200 prior to t = 30 s because the beads around the edge of the magnet rapidly approached z = 1 mm. Thus, the 24 mm cylindrical magnet transiently led to an inhomogeneous distribution of beads.

In light of the transient heterogeneous distribution, another cylindrical magnet, which is shown in figure 3(b), was evaluated with a larger diameter of 30 mm and the same height of 30 mm. This magnet was expected to generate more uniform force due to the increased diameter. The results of the simulations are presented in figures 3(e, h and j). The resulting attractive force was more uniform compared to the 24 mm magnet, as expected. However, the c_v value was approximately 0.12 throughout the simulation which indicated that further optimization of the bead distribution was required.

The results indicated that the increased diameter of the cylindrical magnet led to improved force field homogeneity. However, the increased diameter simultaneously caused a decrease in the gradient of the magnetic field. As shown in Figure 3(d), the magnetic force for the 30 mm diameter magnet at z = 2 mm and x = 0 mm was 2.0×10⁻¹³ N, demonstrating a reduced magnetic force compared to the smaller diameter magnet and in turn, decreased immunostaining efficiency. Furthermore, since the motion of beads induced by the 30 mm magnet was not completed within 60 seconds, as shown in Figure 3(j), further increases in magnet diameter were not desirable as this would extend the movement time.

Given the limitations of the cylindrical magnets, we investigated the effectiveness of dome-shaped magnets for producing a strong and uniform force field. The dome-shaped magnets were formed by tapering the surface of a cylindrical magnet to lowering the edges. Due to this modification, beads did not gather toward the edge of the magnet, and the attractive force increased due to the steeper force gradient generated above the center of the magnet. The dome-shape was designed such that both the diameter and height of the magnet were 30 mm.

Optimization of the shape of the magnet was performed by investigating a variety of monotonically decreasing curves in the x-z plane and by evaluating magnetic force and the dynamics of magnetic beads. The monotonically decreasing curves were defined within the regions of −6 ≤ z ≤ 0 mm and 0 ≤ x ≤ 15 mm. Two types of functions, z = −q(x/15)^γ passing through the point (15, −q) and $z= R 2 − x 2 −R$ , with the magnet having a curve radius of R, were considered. The parameters q, γ, and R were arrayed as {0.00, 0.25, 0.50, 0.75, … 5.75, 6.00 mm}, {1.0, 1.1, 1.2, 1.3, … 2.9, 3.0}, and {15.0, 17.5, 20.0, … 132.5, 135.0 mm}, respectively. The gap d between the top surface of magnet and the tissue specimen was arrayed as {1.0, 1.5, 2.0, 2.5 mm}. A total of 4024 shapes were investigated. The optimal shape was defined as the shape with the minimal time-averaged c_v value from 0 to 60 s and for which all beads had landed on the specimen in less than 60 s.

The optimal magnet shape had a curve radius of R = 50 mm, a diameter of 30 mm, a height of 30 mm, and was placed d = 2.0 mm from the specimen. Figures 3(f, i and j) show the results of simulation with this dome-shaped magnet. The magnetic forces at z = 2 mm were 3.0×10⁻¹³ N at x = 0 mm and 3.3×10⁻¹³ N at x = 10 mm. This force distribution was more uniform than that of the cylindrical magnet of the same diameter and was stronger at the center of the magnet. The time-averaged c_v value from 0 to 60 s was 0.015. All of the beads landed on the specimen within 59 s, with c_v = 0.013 at the completion of the simulation. The dome-shaped magnet exhibited smaller c_v values and more homogeneous distribution of beads on the specimen compared to the cylindrical magnets. Furthermore, the dome shaped magnet had stronger magnetic forces due to the increased gradient above the center of the magnet. These characteristics were suitable for use in efficient immunostaining applications.

Magnetic beads (TAB8849N2140, Tamagawa Seiki Co. Ltd., Japan) were labeled with an antibody which recognized epidermal growth factor receptor (EGFR). The beads contained a europium-based fluorescent dye which was excited by ultraviolet light and emitted red fluorescence. The A431 cell line originating from human epidermoid carcinoma, which is known to exhibit high EGFR expression, was used for the evaluation of the various magnet geometries. We fabricated and used three neodymium magnets: a cylindrical magnet with a diameter of 24 mm and a height of 20 mm, a cylindrical magnet with a diameter of 36 mm and a height of 25 mm, and a dome-shaped magnet with a diameter of 30 mm, a height of 30 mm, and a dome radius of 50 mm.

Figures 4(a–c) show the respective results of staining using the 24 mm cylindrical magnet, the 36 mm cylindrical magnet, and the 30 mm dome-shaped magnet. The left photographs present top views of the specimens stained using the magnetic anti-EGFR-labeled beads. The time of antigen-antibody reaction under the force field was 60 s for all three samples, and the images were obtained after washing out the residual beads. The three samples exhibited red fluorescence, indicating successful immunostaining of the samples. However, the intensity and distribution of fluorescence differed between the magnets used. These results were compared by plotting the distribution of red, green, and blue (RGB) values within the region of interest covering 0 to 12 mm from the center of the magnet.

Figure 4(a) shows that the R values for the specimen stained with the 24 mm cylindrical magnet were 175 au at x = 0 mm but 200 au at x = 12 mm, indicating an inhomogeneous distribution of beads. The increase in R value was observed for x > 7 mm. As expected from the results of the simulations, facilitation of the antigen-antibody reaction differed between the center and edge of the magnet. The resulting staining homogeneity was thus relatively low.

In the case of the 36 mm cylindrical magnet, the R value at x = 0 mm was 120, as shown in figure 4(b), which was significantly lower than that of the 24 mm magnet. The B value generated by autofluorescence of the cells was greater than the R value. This result was attributed to the decreased magnetic force intensity.

Finally, for the 30-mm dome-shaped magnet, R values of 150 ± 5 were maintained from x = 0 mm to x = 10 mm, which was relatively high. Unlike the cylindrical magnets, no edge effects appeared to be generated by the dome-shaped magnet. The curve of RGB values for the dome-shaped magnet demonstrated a homogeneous distribution of fluorescence. Furthermore, the intensity of red fluorescent was sufficiently high, showing the effectiveness of this magnet for immunostaining applications. Unlike the results of numerical simulations, the R value of the dome-shaped magnet was lower than that of 24 mm cylindrical magnet. One of the limitations in our experimental evaluation is fluorescence photobleaching which affects the intensity of R values. However, homogeneity of the fluorescence can be evaluated regardless of the photobleaching.

We designed a dome-shaped permanent magnet for the application of an attractive force to magnetic beads for immunostaining. The shape was optimized based on the homogeneity of the magnetic force applied to the beads. Numerical simulations of the dynamics of the beads showed that the accumulation of beads on the tissue specimen was completed within one minute and with sufficient homogeneity (c_v = 0.013). Finally, experiments using the dome-shaped and cylindrical magnets showed that the dome-shaped magnet empirically produced a more homogeneous distribution of beads on the specimen compared to the cylindrical magnets.

This research was supported by the Project for Medical Device Development from the Japan Agency for Medical Research and Development, AMED (grant number 18he0902006h0004).