A total of eight adapters were initially designed and fabricated by 3D-printing for full cranial cavity coverage. Mechanical accuracy and phantom testing were performed with all eight adapters. Mock-surgical testing was performed with three adapters using one cadaveric specimen. Figure 1a depicts an N-bar localizer mounted on a Skull Anchor Key. Figure 1b shows the frame mounted on a ground truth fixture to test mechanical accuracy. Figure 1c illustrates the frame mounted on a human cranium. Figure 1d shows the work envelope of the stereotactic frame with the proposed adapters. The stereotactic frame followed the same principles as other arc-centered frame-based stereotactic systems [10].

Fig. 1

Design of frame and localizer. a 3-Dimensional image of CT localizer box. b 3-Dimensional image of the frame on the image calibration fixture that was used to test the mechanical accuracy of the frame. c 3-Dimensional image of the frame that is currently being used in real world applications, especially in deep brain stimulation surgeries. d Work envelope of the stereotactic frame

2.1 Adapter design

The stereotactic system included the new targeting device (frame) and the computed tomography (CT) N-bar or magnetic resonance (MR) imaging localizers that were attached to a small skull-mounted device platform called the “Skull Anchor Key” [10]. To develop the work envelope expanding adapters for the “Skull Anchor Key,” several design criteria were considered. These criteria included the ability to have both rotational and translational change capability, fit to the existing key and be re-attachable. The design of the adapter was performed with the aid of Solidworks™ (Dassault Systèmes™, France). After design, 3D-printing was conducted with Ultimaker™ S5 3D Printers (Ultimaker™, Netherlands) using tough polylactic acid. Tough polylactic acid was chosen due to its stability as functional prototype that minimized delamination and warping. Each adapter was capable of varying translations and rotations about all six degrees of freedom as seen in Table 1.

Table 1 Changes made to the original Cartesian coordinate plane in each of the six degrees of freedom by all adapters2.2 Localizers

The CT localizer utilized N-Bar fiducials and has two side plates and one anterior plate. The side plates were positioned 190 mm laterally from each other while the midpoint of the anterior plate was exactly 110 mm from the focus of the four parallel rods. For visualization in CT, the rods were made of carbon fiber and with a diameter of 2 mm. The parallel bars of the ‘N’ in each plate were positioned 120 mm apart, and the diagonal rod connecting the parallel rods was at a 45° angle. This design allowed for existing surgical software to be used for target planning.

2.3 Mechanical accuracy testing

To assess the mechanical accuracy of the adapter system, an acrylic imaging phantom was built in-house that contained 25 points of known coordinates that are accurate to 1/1000 of an inch (25.4 μm). These points, in reference to the targeting device, lie at locations providing a significant expansion to the previous work envelope. The frame was mounted to the testing device along with each respective adapter, and a 150 mm targeting probe was secured to the targeting device delivery platform. The frame was adjusted to target multiple phantom points with the tip of the probe, and the frame X, Y, and Z coordinate readouts were compared against the true coordinates of the point to determine the 3D Euclidean error. 3D distances were calculated using the 3D Euclidean distance equation:

$$D= \sqrt_-_)}^+_-_)}^+_-_)}^}$$

For each variable in the equation above, subscripts 1 and 2 represent observed and expected coordinates, respectively. The process of targeting test points was repeated by two independent examiners to account for inter-user variance. All data are presented as mean ± standard error of the mean (SEM).

2.4 Human cadaver testing

A CT-guided mock-DBS surgical procedure using the 3D-printed adapters with the stereotactic system was developed and performed on a human cadaver and deemed exempt by the Mayo Clinic’s Institutional Review Board (Supplemental Information). The specimen group consisted of one male human cadaver head and the CT imaging was conducted in a Siemens™ Somatom Definition Flash scanner (Slice thickness 5 mm, Rotations time 1 s, 120 kV, CTDI 107 mGy, FOV 300 mm). The device platform was secured to the cadaveric specimen for a CT scan with the localizer box to establish the reference coordinate system for the brain. A total of 10 surgical plans were developed from the CT scan on the Medtronic Stealth™ Station (Medtronic, Inc. Minneapolis, MN, USA) to plan the target and trajectory. The error (3D Euclidean distance from the distal end of the test stylet to the intended target) between the actual and planned electrode position was calculated using the post-operative CT image.

2.5 System mathematics

Mathematical formulas were developed for the transformations achieved by the new adapters using both traditional transformation matrices and geometric algebra. As previously described [10], the primary targeting area of the stereotactic system can be defined as a cube (100 × 110 × 70 mm) where the work envelope is defined. The main surgical targets are within this work envelope, which can be thought of as a sphere whose center may be moved to any target within the work envelope. With this structure, the stereotactic system can be defined to possess 6 degrees of freedom, the X, Y, Z, collar angle, arc angle, and radial distance. The X is the frame’s lateral/medial movement, Y is the anterior/posterior movement, and Z is the superior/inferior movement. These three linear degrees of freedom and the platform placement are in a constant relationship with the work envelope, and the sphere’s center can be linearly moved to any coordinates within the work envelope. This makes it possible to target any point with a probe directed normal to the surface of the sphere. The stereotactic system uses the center of arc principle [10] where the arc and collar angles are oriented along the X–Z and Y–Z planes, respectively, to add two angular degrees of freedom about the center of the sphere. Finally, the last degree of freedom is the radial distance, which is measured by the length of the probe along the trajectory. This distance is set at 150 mm and is a fixed number in the present system.

2.6 System mathematics for adapters—transformation matrices

To determine how usage of each of the adapters would impact the frame’s work envelope coordinate configuration, the Skull Anchor Key was first applied without any adapters, and a CT scan was taken to establish a baseline coordinate system and initial plan values (X, Y, Z, Arc, Collar) using the right superior posterior coordinate as the origin (x = 0, y = 0, z = 0). The initial target and trajectory plan were converted to a target point and cranial entry point, and transformation functions were applied to each. The calculated entry point (Xe, Ye, Ze) was derived using basic trigonometric functions in a three-dimensional space. Starting from the initial target coordinates, the arc and collar angles provided the angles to be used in the equations, and an arbitrary distance of 100 mm was used as the span between target and entry. The resulting equations for the calculated entry point were as follows [23, 24]:

$$\begin x_ = x_ - \rho \cos \theta = x_ - 100 \cos \left( \right) \hfill \\ y_ = y_ + \rho \sin \emptyset \cos \theta = y_ + 100\sin \left( \right) \cos \left( \right) \hfill \\ z_ = z_ - \rho \sin \emptyset \sin \theta = z_ - 100\sin \left( \right)\sin \left( \right) \hfill \\ \end$$

In the above equations, \(\theta\) is the arc angle and \(\varnothing\) is the collar angle. To determine the effect of each adapter on the work envelope, 3-dimensional transformations were applied to both the target point (Xt, Yt, Zt) and the entry point (Xe, Ye, Ze) based on each individual adapter’s geometry. These physical transformations, such as rotation or translation performed by the adapter, establish in the new coordinate system the new position of each point in step operations. First,rotations about the Z- and X-Axes are applied to (Rx (q)) and (Rz (q)), respectively Subsequently, a translation to adjust to the correct superior posterior coordinate about the origin is applied. Each rotation uses a standard 3D rotation matrix multiplication, and an example of Z-Axis rotation is provided below. The values used for the rotational and translational adjustments are unique to each adapter and derived using measurements in Solidworks™.

Z-Axis Rotation Matrix Equations:

$$\left[ \begin x^ \hfill \\ y^ \hfill \\ z^ \hfill \\ \end \right] = \left[ c} & & 0 \\ & & 0 \\ 0 & 0 & 1 \\ \end } \right] \, \times \, \left[ \begin x \hfill \\ y \hfill \\ z \hfill \\ \end \right]$$

The Prime values are the new coordinates and θ is the amount that the x axis is rotated towards the y axis (counterclockwise). After transforming the target and entry points to the new coordinate system X¢t, Y¢t, Z¢t and X¢e, Y¢e, Z¢e, the inverse methods to those described above are used to convert these transformed target and entry points back to the (X¢, Y¢, Z¢, Arc¢, Collar¢) values for use with the positioner on the stereotactic frame.

2.7 System mathematics for adapters—geometric algebra

Rotation of the stereotactic system using geometric algebra was achieved using custom-written software. Rotation in geometric algebra is defined by rotors [21, 25]. If the rotor is given by R, then the equation for rotation of any vector v is given by v’ = R~vR, where R~ is the inverse of R. Because the adapters transform the coordinate system with each rotation, the code for representing the rotor must allow for flexible coordinate bases. In this case, this was handled by representing each rotation sequentially from the perspective of the frame (rather than the perspective of a fixed coordinate plane), and then multiplying the corresponding rotors together in the proper order. Each rotor is defined by the plane of rotation. For example, the rotor describing the rotation that is equivalent to rotating about the Z-axis in Cartesian space is given by:

where Rz is the rotation about the z-axis, \(\phi\) is twice the angle of rotation, and e12 is the bivector representing the XY Cartesian plane (Fig. 2). In geometric algebra, for each multiplication of \(\sqrt,\) a 90° rotation occurs, and after two rotations (180°), it would be multiplied by −1. In general, to perform multiple rotations sequentially, one can multiply the associated rotors in order, from left to right. The rotor equations were developed in Python, and rotations were achieved by multiplying these rotors together in the same order as the rotations described by the adapters.

Fig. 2

Illustration of geometric algebra. For each multiplication of \(\sqrt\), a 90° rotation occurs. After two rotations, the original figure has been rotated 180°, and therefore multiplied by − 1. In general, to perform multiple rotations, sequentially, one can multiply the associated rotors in order, from left to right