Predicted effects of pulse width programming in spinal cord stimulation: a mathematical modeling study

A new three dimensional finite element mathematical (FEM) model was created of the low-thoracic spinal cord and its surrounding environment. The FEM model consisted of spinal cord white and gray matter, cerebrospinal fluid, dura, epidural space tissue, vertebral bone, and a cylindrical multicontact lead. Spinal cord (white and gray matter boundary) was modeled on the shape/size of the cord at the low-thoracic vertebral level (using images from human cadaver samples) [21]. Dura cross section and thickness (270 μm) was approximated from a recent model of motor cortex stimulation [24]. The inner portion of the vertebral bone was round in shape, but its outer surface was square/flat for computational simplicity. The surrounding layer 2 (Fig. 1) was treated as ground, while surrounding layer 1 was used to adjust electrode impedance between contact and the ground layer. Once cross sections of necessary compartments (white and gray matter, dura etc.) were identified, each compartment was extruded into three blocks in the rostro-caudal direction. Each block has a rostrocaudal length of 70, 35, and 70 mm. Therefore, total rostrocaudal length of the model is 175 mm which is long enough to minimize the boundary effect on potential distribution. The model of the stimulating lead is placed in the middle block, which has a length of 35 mm and has significantly higher density (Table 1) of nodes compared to the first and last block.

The lead consisted of eight cylindrical platinum–iridium contacts (conducting domains were 3 mm length and 1.25 mm diameter), separated by 1 mm lengths of insulating polymer (non-conducting domains of 1 mm length). The lead was positioned dorsally, atop the dura and aligned with the midline of the spinal cord. The geometry of the model is illustrated in Fig. 1a and electrical resistivities and node density are given in Table 1, values coming predominantly from the literature [4, 25]. The volume was meshed with over 1 million nodes with tetrahedron shaped elements, with a high-density mesh in the region close to where electrodes are located as illustrated in Fig. 1b.

The spinal cord geometry (Fig. 2) was created using a combination of features from relevant literature sources. The cross-section of the cord was derived from Kameyama et al., and the DR trajectory of Struijk et al. was adopted [11, 21]. Each DR was modeled as a larger diameter ‘mother’ fiber connected to bifurcated ‘daughter’ fibers of smaller diameter (Fig. 2b).

The electric potential was computed by solving the model using the ANSYS (Canonsburg, PA) Emag analysis tool, employing the iterative equation solver with Jacobi Conjugate Gradient method. A unit current was injected at the center point of any active contact with current return to ground (surrounding layer 2). To compute the electric potential generated by multiple contacts, multiple simulations with current injection from each individual contact alone were performed. The solutions were then superimposed to generate the net electric potential in the spinal cord (white and gray matter). Finally, the solution was extracted and post-processed in Matlab (Natick, MA).

The double cable axon model was developed by McIntyre et al. based on mammalian motor nerve fibers with geometrically and electrically accurate parameters (http://​senselab.​med.​yale.​edu/​) implemented in NEURON [23, 26]. The model incorporated explicit representations of the nodes of Ranvier, paranodal and internodal sections of the axon as well as a finite impedance myelin sheath. This model matched experimental data by combining an accurate representation of the ion channel and the geometry of the paranode, internode, and myelin. The model has nonlinear membrane dynamics of ion channels at the nodes of Ranvier. The ion channel includes fast Na+, persistent Na+, and slow K+ conductances. The node of Ranvier contains these ion channels in addition to a linear leakage conductance, and the membrane capacitance.

For the various model investigations, the voltage data from the FEM model was ported to Matlab for interpolation on regular grid points in 3D space. DC fibers were placed on regular grid (200 μm for mediolateral direction and 100 μm for dorsoventral direction; see Fig. 2a) and projected in the rostrocaudal direction. The interpolated voltage along the corresponding fiber (Fig. 2c) was applied to non-linear axon models [23] with different fiber diameters for DC and DR fibers using the extracellular mechanism capability in NEURON. To apply a guarded cathode (anode–cathode–anode, where each anode provides half of the total anode current; a commonly programmed contact combination in SCS), the electric potential generated from the three contacts individually was summed based on superposition principle of current sources in space. Fiber activation was defined when an action potential was observed at a minimum of five nodes for a depolarized fiber. During simulation, anodic break did not occur because the anode intensity was relatively weaker than cathode, and the axon’s ion channel dynamic was not sensitive to anodic break.

The new FEM SCS model includes several myelinated fiber sizes (5.7–14 μm diameter) for both DC and DR fibers. In the human DC, the density and distribution of various fiber sizes changes from the midline to the DR entry zone, with medium and larger sized fibers trending toward greater density in the lateral DCs [20] (see Fig. 3a, b). Thus, the distribution of DC fibers in the model was designed to reflect these variations.

The number of stimulated fibers for specific diameter and location was estimated from the density (Fig. 3b) assuming that the characteristic of mediolateral distribution is preserved dorsoventrally.

To study the effect of PW on DC fibers, a longitudinal guarded cathode combination (anode–cathode–anode with separations of 8 mm center-to-center) parallel to DC fibers was applied to the model with different PWs (60, 210, 450, and 1000 μs) monophasic square pulse. The center-to-center contact separation of the modeled lead is 4 mm, which matches commercially available SCS percutaneous leads (Boston Scientific Linear ST). The 8 mm separation between anode and cathode is selected because it reflects commonly-programmed combinations. Although commercial current-controlled stimulators use square pulses, followed by an interphase interval (typical: 100 μs), and a low-voltage passive recharge interval (typical: 6 ms), our simulation ignored the effect of recharge phase because a 100 μs interval after the stimulation phase is long enough to eliminate any impact of recharge phase [27]. Selection of PWs was intended to match the breadth of programmable values in commercial devices.

The thickness of the CSF was set at 3.2 mm [15]. The upper limit of stimulation current was defined using both the threshold of the most excitable DR fiber (DRth) fiber (15 μm diameter), and 1.4 times the threshold of the first-activated 11.5 μm DC fiber (1.4 * Pth), which is believed to generate initial paresthesia during SCS [10]. The rationale for having both limits was the clinical observation that some patients report that their maximum comfortable level of stimulation does not activate fibers believed to be DR fibers (i.e., they do not report abdominal cramping); in these cases, the stimulation is perceived as ‘too intense’ in caudal regions and the patient becomes uncomfortable [28].

A companion clinical study into the technical outcomes of PW programming (paresthesia thresholds and body coverage) in SCS was also performed [5, 29]. Briefly, subjects utilizing fully-implanted Precision™ SCS systems for chronic intractable pain were enrolled. Clinical data was obtained from an IRB-approved post-market clinical investigation of PW programming in spinal cord stimulators (ClinicalTrials.gov # NCT00399516). All subjects had percutaneous leads positioned in the dorsal epidural space at the T7–T9 vertebral level to treat low back and lower extremity pain with the Boston Scientific Precision SCS device and were screened for inclusion and exclusion criteria [5]. Patients who met the necessary criteria underwent the informed written consent process per the Declaration of Helsinki. Table 2 summarizes the gender, age, diagnosis, location of pain, and time-since-implant of the subjects.

In the study, the favorite program of each subject was used, with only the PW parameter varied from 50 to 1000 μs, in a pre-randomized order. At each PW, the pulse amplitude was increased from zero, first to perception threshold, which was recorded, then to a strong-but-comfortable level, at which point the subject was asked to draw the extent of their paresthesia over a displayed human figure on a pen tablet PC. Analyses of the perception thresholds were performed to create strength-duration curves from which chronaxie and rheobase estimates were made. Also, the paresthesia drawings were analyzed to look for changes in total body coverage, as well as shifts in the location of the paresthesia on the body as they related to PW. A total of 19 subjects completed testing and the clinical results were published [29].

In SCS programming, a common clinical observation is that the paresthesia ‘spreads’ to multiple dermatomes as the pulse amplitude is increased [30]. In previous mathematical modeling studies of SCS, increased stimulation amplitude increased the calculated cross-sectional area of stimulated DC fibers [4]. The combination of these phenomena suggests that, as amplitude is increased, a greater depth and breadth of DC fiber activation is responsible for the increased spread of paresthesia. Based on this relationship, the cross-sectional area of stimulated DC fibers was used to estimate ‘relative’ paresthesia coverage change in the computational model to study the effect of PW.

Anatomic studies of the DC have suggested a spatiotopic mapping of dermatomes, where lateral DC fibers represent relatively rostral dermatomes, and medial DC fibers are projections of DR fibers from more caudal dermatomes [31]. Although the boundary of each dermatome in the DC is not clearly divided, we assumed that the gross characterization of rostral versus caudal dermatomes could be assessed in the model. Therefore, for all fibers recruited by stimulation, medial fibers (MedF) were defined as those located within 600 μm of the spinal cord midline, and lateral fibers (LatF) as those located greater than 600 μm from the midline [32]. Using these definitions, the location of the focus of paresthesia for each PW was estimated by computing the total number of stimulated fibers from the medial (MedF) and lateral (LatF) regions and their ratio (LatF/MedF) (Fig. 4).