mathematical software wolfram mathematica 13.1  (Wolfram Research)

Journal: Scientific Reports

Article Title: New method for neuromodulation against pain using minimally invasive electrodes outside the epidural space

doi: 10.1038/s41598-025-07750-8

Figure Lengend Snippet: Schematic of the entire implant, consisting of two clips attached onto two non-adjacent vertebrae, a thin flexible coiling wire connecting the two clips (thin black line in the figure), and a few tiny electronic components (not shown in the figure, but explained in Appendix A). ( a ) Schematic overview of the position of a single clip attached onto the lamina, with a primary electrode Sp on the bottom (i.e., ventral) side and a secondary electrode Ss on the top (i.e., dorsal) side. ( b ) The primary current (red arrows) flowing from electrode Sp(2) to electrode Sp(1) passes partially through the dorsal side of the spinal cord (yellow), thus giving rise to the desired electric stimulation of the dorsal columns of the spinal cord. The indices (1) and (2) refer to the left and the right clip, respectively. The secondary electrodes Ss(1) and Ss(2) on the dorsal side serve a dual purpose : (i) to serve as an entry point for the high-frequency (non-stimulating) current field that originates from the non-invasive external skin pad electrodes (see Appendix A and Figs. and for details). This high-frequency current constitutes the energy source for the implant, and is converted into a quasi-DC current by the rectifier circuit that is part of the implant. (ii) to emit a secondary current field, which is weaker, and opposite in polarity with respect to the primary current field, and serves to locally annihilate the unwanted effects of the primary current field in the small dorsal muscle fibers located near the secondary electrodes. A flexible coiling wire connects the two clips in order to make the entire circuitry complete. The colors red and blue of the electrode surfaces indicate opposite phases (i.e., opposite polarity) of the injected current; this implies that if the color red of the electrodes in the picture indicates a positive phase, then the blue color indicates a negative phase, and vice versa. Images were created by authors using Wolfram Mathematica 13.3 software ( www.wolfram.com/mathematica ).

Article Snippet: The simulation was performed on the commercial software Wolfram Mathematica (Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL (2020)).

Techniques: Injection, Software

Journal: Scientific Reports

Article Title: New method for neuromodulation against pain using minimally invasive electrodes outside the epidural space

doi: 10.1038/s41598-025-07750-8

Figure Lengend Snippet: Illustration of wireless energy transfer method. ( a ) A couple of pad skin electrodes (light blue, indicated as “Skin electrode (1)” and “Skin electrode (2)”) are applied onto the patient’s skin to deliver the high frequency current. The implant itself, which consists of two clips including two electrodes each (i.e. four electrodes in total : Sp(1), Ss(1), Sp(2) and Ss(2) ), has been securely and rigidly attached onto two vertebrae. The electrodes Ss(1) and Ss(2) are connected by a flexible (slightly coiling), electrically conducting wire W. This wire W, however, was electrically insulated from the tissues surrounding it over its entire length, except for the two very endings of the wire. Furthermore, a small Schottky diode D has been incorporated in the implant, in order to serve as a rectifier. ( b ) Schematic of the electric aspects of the complete implant (black solid lines, including the diode D and the electrode surfaces Ss(1),Sp(1),Ss(2) and Sp(2)), together with the applied HF AC current field (blue), the effective resistance R (green) of the pathway of the rectified, quasi-DC current traveling through the tissues between Sp(1) and Sp(2). This resistance R also includes the contact impedances between each electrode surface and the tissue directly nearby that it is in contact with. Furthermore, in (b), the measurement equipment is rendered in red (with dashed lines); this measurement is essentially a voltage measurement as function of time (as will be shown in Fig. in more detail). In this measurement set-up, we have added a small capacitor C (also in red), having a capacitance of C = 1 µF, that allows for the assessment of the total resistance R between the stimulation electrodes Sp(1) and Sp(2) on the basis of the measurement of the RC-time on the oscilloscope in combination with the known capacitance C. Images were created by authors using Wolfram Mathematica 13.3 software ( www.wolfram.com/mathematica ).

Article Snippet: The simulation was performed on the commercial software Wolfram Mathematica (Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL (2020)).

Techniques: Software

Journal: Scientific Reports

Article Title: New method for neuromodulation against pain using minimally invasive electrodes outside the epidural space

doi: 10.1038/s41598-025-07750-8

Figure Lengend Snippet: Schematic of the essentials of multi-electrode local field annihilation. Notably, the local field in proximity of only one single clip is illustrated here, whereas both clips are required to 'close the circuit,' as demonstrated in Fig. . ( a ) The primary electrode Sp on the lower side of the clip administers a current towards the target of the stimulation, i.e. the dorsal columns in the white matter marked in yellow (label A). Meanwhile, the secondary electrode Ss placed on the upper side generates a weaker current of opposite phase, aimed at annihilating the stimulation of the (red) muscles in the back (marked with X). The color red of the electrodes in the picture indicates a positive phase of the output current, the blue one a negative phase. ( b ) Schematic representation of the “local quenching field” designed to prevent undesired muscle stimulation in the back. The generation of a main strong field by the primary electrode Sp close to the spinal cord, in combination with the opposite, counteracting, weaker field from the secondary electrode Ss in proximity to the muscle tissue, results in a “local quenching effect”. This effect ensures a predominant stimulating field in the red/purple/blue regions, where the intensity is such as to enable the fibers stimulation, while maintaining the field intensity below the fibers stimulation threshold in the light blue/yellow/green regions. Consequently, the target A region is readily stimulated while in the X region the stimulation is nearly absent. c ) Sketch of the “On/off function” called EnvelopeOfPulses(t)”, as described in Eqs. and , which describes the envelope of the current pulses administered during the spinal cord stimulation, consisting of a number of current pulses of duration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau$$\end{document} and inter-pulse interval T. Images were created by authors using Wolfram Mathematica 13.3 software ( www.wolfram.com/mathematica ).

Article Snippet: The simulation was performed on the commercial software Wolfram Mathematica (Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL (2020)).

Techniques: Muscles, Software

Journal: Scientific Reports

Article Title: New method for neuromodulation against pain using minimally invasive electrodes outside the epidural space

doi: 10.1038/s41598-025-07750-8

Figure Lengend Snippet: ( a ) FEM 3D model including the clips with the electrodes, three thoracic vertebrae (T2,T3,T4), spinal cord, cerebrospinal fluid and dura mater. The SNAP clips are located between the spinous process and the inferior articular facet. For simplicity, it is assumed that the region surrounding the above-cited structures is entirely composed of fat. ( b ) The two clips including the primary and secondary electrodes are positioned onto T2 and T4. The two clips rendered in ( b ) are both located at the right side of the spinal processes; hence they are named “Clip 1Right” and “Clip 2Right”. These two clips could, however, equally well be positioned at the left side if stimulation of the left part of the dorsal columns would be desired, in which case the two electrodes would be named “Clip 1Left” and “Clip 2Left”. c ) More generally, one may even choose to use a “four clip method”, in which “Clip 1Right” and “Clip 2Right” as well as “Clip 1Left” and “Clip 2Left” are clamped onto their respective positions on the laminar bone structures. This “four clip method”, shown in this figure in ( c ), allows for better fine-tuning in the left–right direction. The basic mechanism, however, remains the same as in the case of the two clamps shown in ( b ). Therefore, in the following, only “Clip 1Right” and “Clip 2Right” are used in the simulations. In Appendix B, however, an example of the “four clip method” is shown. Images were created by authors using Wolfram Mathematica 13.3 software ( www.wolfram.com/mathematica ).

Article Snippet: The simulation was performed on the commercial software Wolfram Mathematica (Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL (2020)).

Techniques: Software

Journal: Scientific Reports

Article Title: New method for neuromodulation against pain using minimally invasive electrodes outside the epidural space

doi: 10.1038/s41598-025-07750-8

Figure Lengend Snippet: ( a ) The primary electrodes (red and blue parallelepipeds) are placed outside the spinal cord, and outside the epidural space, in such a way to generate a stimulating electric field targeting the region of interest, which is marked in yellow with the letter “A”, i.e. the longitudinal Aβ fibers in the dorsal columns. ( b ) The various tissue types inside the vertebral canal that were included in the simulation, were dura mater, cerebrospinal fluid, white matter, gray matter. The fatty tissue, which covers the space between the dura mater and the vertebral bone, is not indicated in the picture. ( c ) Neurostimulation target region (marked in yellow) : longitudinal Aβ fibers in the white matter tracts in the vertical dorsal columns (parallel to the z direction). Images were created by authors using Wolfram Mathematica 13.3 software ( www.wolfram.com/mathematica ).

Article Snippet: The simulation was performed on the commercial software Wolfram Mathematica (Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL (2020)).

Techniques: Software

Journal: Scientific Reports

Article Title: New method for neuromodulation against pain using minimally invasive electrodes outside the epidural space

doi: 10.1038/s41598-025-07750-8

Figure Lengend Snippet: ( a ) Longitudinal section of the dorsal columns in the white matter (0.2 mm below the white matter, target region “A” highlighted in yellow), in which the electric field distribution is computed. ( b ) Results from simulation nr 1 in which only a primary current of 5.80 mA is delivered (inside the red rectangle). ( c ) Results from simulation nr 2 (inside the green rectangle) in which a primary current of 6.30 mA is released from electrodes Sp(1) and Sp(2), and, furthermore, an additional, weaker and opposite secondary current of 1.26 mA is released from the secondary electrodes Ss(1) and Ss(2). Images were created by authors using Wolfram Mathematica 13.3 software ( www.wolfram.com/mathematica ).

Article Snippet: The simulation was performed on the commercial software Wolfram Mathematica (Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL (2020)).

Techniques: Software

Journal: Scientific Reports

Article Title: New method for neuromodulation against pain using minimally invasive electrodes outside the epidural space

doi: 10.1038/s41598-025-07750-8

Figure Lengend Snippet: Results of the computer simulation. ( a ) Overview of the anatomical structures used in the simulation. ( b ) Results from simulation nr 1 (inside the red rectangle) in which only a primary current of 5.80 mA is delivered. As can be seen in the top graph inside the red rectangle in (b), large parts of the Multifidus and the Semispinalis Cervicis muscles are within or near the stimulation threshold of 11.5 V/m, giving rise to (possible) muscle spasms ins these muscles. ( c ) Results from simulation nr 2 (inside the green rectangle) in which a primary current of 6.30 mA is released from electrodes Sp(1) and Sp(2), and, furthermore, an additional, weaker and opposite secondary current of 1.26 mA is released from the secondary electrodes Ss(1) and Ss(2). These secondary electrodes are rendered in yellow in (c). In contrast to the situation in (b), the region (marked by the letter X ) that contains the Multifidus and the Semispinalis Cervicis muscles, is now well inside the “purple zone” of the top graph in (c), which indicates that the electric field hitting these muscles around region X is now well below 9.2 V/m, i.e. well below the stimulation threshold of these muscles. Images were created by authors using Wolfram Mathematica 13.3 software ( www.wolfram.com/mathematica ).

Article Snippet: The simulation was performed on the commercial software Wolfram Mathematica (Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL (2020)).

Techniques: Muscles, Software

Journal: Scientific Reports

Article Title: New method for neuromodulation against pain using minimally invasive electrodes outside the epidural space

doi: 10.1038/s41598-025-07750-8

Figure Lengend Snippet: Computer simulation of the electric field distribution in two planes : (i) A transverse XY-plane with z = -22 mm (i.e., the “front plane” in the figure), which cuts perpendicularly through the spinal cord and vertebra T2. In this plane, the Dura Mater (DM), two Dorsal Rootlets (DR), as well as the white matter and the grey matter are rendered. (ii) A coronal XZ-plane with y = -9 mm (i.e. the “ceiling plane” in the figure), which cuts through the top layer of the dorsal columns of the spinal cord. Inside this plane, the region near the centerline (− 2 < x < 2 ) of the plane, which is equal to the ”yellow section plane” in Fig. a, represents the region to be stimulated by the SNAP method. The central midline (black dotted line) in this coronal “ceiling plane” corresponds to the x = 0 line inside this coronal XZ-plane. In this simulation, the “four clip method” from Fig. c has been used ; these four clips are rendered on vertebrae T2 and T4 in the small schematic in the left bottom corner of this Figure. Images were created by authors using Wolfram Mathematica 13.3 software ( www.wolfram.com/mathematica ).

Article Snippet: The simulation was performed on the commercial software Wolfram Mathematica (Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL (2020)).

Techniques: Software

Journal: Scientific Reports

Article Title: New method for neuromodulation against pain using minimally invasive electrodes outside the epidural space

doi: 10.1038/s41598-025-07750-8

Figure Lengend Snippet: Results of in-vitro measurements of the Electric Field Vector at five different positions within the spinal canal inside a cross-sectional transverse XY plane cutting through vertebra T2. The electronic equipment equals that of the in-vitro setup depicted in Fig. , but this time without the components rendered in red in Fig. b, i.e. without the red dotted lines and the capacitor and voltage meter depicted in Fig. b. ( a ) Electric field probe consisting of 6 tiny point-shaped electrodes distributed on both sides of a thin insulating piece of circuit board, enabling the measurement of voltage differences in all of the three directions (X, Y, Z). This small field probe was mounted on a thin plastic rod and positioned at each of the five positions inside the spinal canal, as indicated by the five dots in ( b ). ( c ) Results of the E-field measurements using the probe depicted in ( a ), for each of the five positions within the cross-sectional plane. The five E-field vectors are rendered in red, whereas the grey structure indicating the spinal canal is a section of the inner surface of the bone structure (i.e., vertebra T2) rendered in ( b ). The strengths of the E-field (i.e. the length of the E-vectors depicted in ( c ), ( d ), and ( e )) are all expressed as a percentage of the strongest E-field measured, which is the E-field measured at point number 1 in ( c ). In ( c ), the resulting five E-vectors are rendered in full 3D. For clarity, however, we have also added a graphic that renders only the (X,Y)-components of each E-vector ( in ( d )), as well as a graphic rendering only the Z-component of each E-vector (in ( e )). In ( c ), ( d ), and ( e ), all numbers along the coordinate axes are in mm. Images were created by authors using Wolfram Mathematica 13.3 software ( www.wolfram.com/mathematica ).

Article Snippet: The simulation was performed on the commercial software Wolfram Mathematica (Wolfram Research, Inc., Mathematica, Version 12.1, Champaign, IL (2020)).

Techniques: In Vitro, Plasmid Preparation, Software