Proposed device physics
When carrying an alternating current (AC), solenoid coils generate time-varying magnetic fields which produce looping eddy currents within conductive materials according to Ohm’s Law. Non-destructive techniques based on eddy current damping (ECD) are widely used to test the presence, quantity, and integrity of various conductive materials, most commonly metals [9]. Eddy currents generated within the coil by a conductive material flow in the direction that decreases coil inductance and increases coil resistance, which may be used to quantify the degree of ECD produced. These ECD signals may be measured as a change in parallel resistance (Rp) through the use of an electrical resonant circuit or LC tank, with a magnetic coil paired with a capacitor. When presented with a single or multi-dimensional space of magnetic flux density, the flux generated by eddy currents weakens the flux density within the entire field [10]. As such, it has been demonstrated that a correlation exists between magnetic flux density and ECD signals [11].
Because blood (0.65 S/m) is more electrically conductive than brain tissue (0.2 S/m) [12], we hypothesized that the presence of a more conductive material (blood) within a defined space in the brain (due to either ICH or increased local brain activity) may result in generation of a specific ECD signal, which is proportional to the magnitude, distance, and size of the blood-induced signal change according to a set of analytical solutions developed by Dodd and Deeds (Fig. 1) [13]. As such, a magnetic coil sensor capable of producing and measuring magnetic flux on the microtesla scale could detect ECD signals generated by local level intracranial hemodynamic changes.
Finite element model
Using COMSOL Multiphysics version 5.5 (COMSOL Inc., Burlington, MA), we generated both three-dimensional (3D) and two-dimensional (2D) models of hemodynamic changes within a human head model, in addition to one-dimensional (1D) modeling of magnetic field decay as a function of distance. Text files of coordinate data from MRI head scanning were imported as an interpolation curve and organized in sectionwise format, and lofting of the closed curve allowed for the generation of a solid object roughly 20 cm in height and 10 cm in width [14, 15]. A novel material with a density of 2000 kg/m3, electrical conductivity of 0.4 S/m, relative permeability of 1, and relative permittivity of 568 was added to simulate the properties of a human head. An ellipsoid with dimensions of 0.035 m in the a-axis, 0.045 m in the b-axis, and 0.025 m in the c-axis was added within the head to simulate the brain. A novel material with a density of 1000 kg/m3, electrical conductivity of 0.2 S/m, relative permeability of 1, and relative permittivity of 827 was added to simulate the properties of the human brain. Lastly, a smaller ellipsoid with dimensions of 0.014 m in the a-axis, 0.018 m in the b-axis, and 0.010 m in the c-axis was added within the head to simulate an area of increased blood concentration. A novel material with a density of 1,060 kg/m3, electrical conductivity of 0.65 S/m, relative permeability of 1, and relative permittivity of 3,030 was added to simulate an area of increased blood concentration [16, 17]. Using the simplified formula for intracerebral hemorrhage volume, which estimates blood volume as half the product of all three ellipsoid semiaxes (ABC/2), we determined our small ellipsoid to represent a local increase of 1.26 mL of blood [18,19,20,21]. This volume was decided because previous studies have established that local brain activation may increase local blood concentration by approximately 1.5mL, and while adjusting the parameters of our COMSOL model, a volume of 1.26mL was the closest volume available to the 1.5mL target volume that we wanted to investigate [22].
Two scenarios were explored with reference to magnetic coil design: (1) a small handheld coil with 12.1 cm external diameter that could be moved across the head and (2) a larger coil with an external diameter of 27 cm that resembles a headband and is placed on the head circumferentially. Both scenarios shared the following parameters: peak current of 0.0027 amperes, 6 coil turns, coil width of 4.2 cm, and material properties including copper wire with a density of 8960 kg/m3, electrical conductivity of 5.998 × 107 S/m, relative permeability of 1, and relative permittivity of 1. An infinite element with the material properties of air and modeled as a sphere was added encapsulating the entire head and sensor model.
Mathematical models and computations
Ampere’s Law was implemented to calculate the electric current density as the curl of the magnetic field: \(\nabla xH=J\)(where \(\nabla xH\) represents the curl of the magnetic field and J represents the electric current density). Electrical conductivity was modeled through Ohm’s Law: \(J=\sigma E\)(where J represents the electric current density, σ represents the material’s conductivity, and E represents the electric field). Lastly, magnetization (\(B={\mu }_{0}{\mu }_{r}H\)) and dielectric (\(D={\epsilon }_{0}{\epsilon }_{r}E\)) models were included, which utilized the relative permeability and permittivity values respectively to solve their corresponding equations. Testing frequency was set at 1 MHz and a stationary solver was implemented and finite elements included in the model utilized fine meshing with a total of 286,184 degrees of freedom in the small coil and 314,478 degrees of freedom in the larger coil. Several factors were considered when choosing 1 MHz. First, the LDC1101 inductance-to-digital converter chip used for experimental testing only operates at frequencies between 500 kHz and 10 MHz. Within this frequency range, only a handful of studies have thoroughly explored and published the electrical conductivities of biological tissues [12]. In many of these studies, conductivities within the 1 MHz region were thoroughly explored and validated. Lastly, we know that higher frequencies have shorter penetration into tissues and lower frequencies have higher penetration into tissues. After much thought, we settled to build the coil to operate at a frequency of 1 MHz to balance the pros and cons of sensor circuit limitations, available conductivity measurements from the literature, and penetration depth into the head.
Cadaver model
A fresh, unfixed human body with no history of neurological or cerebrovascular disease was procured and laid in supine position. The quality of the brain was evaluated to ensure that it still retained properties of a live human brain. The head was lifted using pillows to about a 45-degree angle. We ensured that no metallic objects (i.e. metal table, head and neck implants) were in the range of detection of the coils. A head cap with 8 equidistant tubes was placed such that the most anterior tube crossed the forehead horizontally, and the most posterior tube crossed the nape of the neck horizontally. The neurosurgeon made an incision just above the right eyebrow to retract the skin and proceeded to drill a right supraorbital burr hole. The head was scanned by hand across all 8 rows with all three sensors to procure control brain data. Frozen uncoagulated porcine blood was left to thaw, and 60mL was withdrawn into a syringe and injected 7 cm deep into the brain parenchyma. The brain was scanned once more with all three sensors, this time with the bleed in place. Data was stored on a local computer for analysis. The cadavers were procured from the University of California, San Diego as part of the UC Anatomical Donation Program. Through this program, the deceased consented their bodies to research. Written consent was obtained.