Keywords

1 Introduction

Transcutaneous spinal direct current stimulation (tsDCS) is a non-invasive stimulation technique considered as a possible therapeutic resource for spinal cord dysfunctions such as chronic pain or motor system lesion [1]. tsDCS consists in the application of low intensity direct currents of 2–4 mA to the spinal cord (SC) using electrodes placed over the vertebral column near target regions. Exploratory experimental studies in humans pursued in the last decade demonstrated the neuromodulatory potential of tsDCS to change signal transmission along nociceptive ascending pathways and spinal motor and reflex circuits in cervical and thoracic spinal segments [2,3,4,5].

The effects of tsDCS rely mainly on the electric field (EF) induced in the nervous tissue, similarly to brain stimulation techniques. The EFs may contribute to inhibit or facilitate neuronal responses, by transiently changing the resting membrane potential. This effect will depend on how each spinal neuron or circuit is orientated relative to the EF. Just as in brain stimulation, the spatial distribution of the EF induced by tsDCS depends on electrode number, design (shape and structure), placement relative to target, current intensity and polarity (anodal/cathodal). Computational studies using numerical methods to predict the EF and current distribution in realistic human models may be powerful tools to fine-tune experimental tsDCS protocols targeting specific spinal regions of interest [6,7,8,9].

Modelling studies in tsDCS generally consider square or rectangular electrodes in bipolar montages, which originate EFs in the spinal cord that increase with distance between electrodes at the cost of a larger (less focal) target region. Cortical stimulation studies predict that montages using smaller electrodes in a multi-electrode setup can induce maximum EF in a smaller target region [10,11,12]. Thus, one question of interest is to determine if a multi-small-electrodes paradigm in tsDCS can also originate a more focal EF in the spinal cord, as in brain stimulation, which could be relevant for neuromodulation of specific spinal circuits.

The effects of different montage settings are also dependent of the electrical properties of tissues located in the current path to the stimulation target. Sensitivity analyses performed on transcranial direct current stimulation (tDCS) demonstrated changes of the EF magnitude when varying the conductivities of tissues. Scalp, CSF, gray matter (GM) and skull were the tissues that introduced larger variability on EF values in the cortex [13, 14]. The EF induced in tsDCS also show spatial characteristics, such as local hotspots, that seem to be related with anatomical features such as vertebrae edges and disks protrusions into the spinal canal. These local features may be related with an interplay between the conductivities of different tissues, such as vertebrae, intervertebral disks and CSF [7,8,9].

This chapter is dedicated to study the interplay between the conductivities of tissues and two different types of electrode montages on the spatial distribution of the EF induced by tsDCS. These types of montages will be modelled considering tsDCS over the lower thoraco-lumbar spinal cord: a montage with two large and square electrodes (50×50 mm2) and a grid with 4×4 small circular electrodes (∅ = 10 mm). For these two types of montages, EF changes will be investigated considering rotation of the grid or increases in distance between the two large electrodes. The impact of electrical conductivities of the tissues located between the electrodes and the SC will be addressed in each montage considering three different types of models: homogeneous, semi-homogeneous and heterogeneous. The clinical relevance of differences in the EF induced by the multi-electrode grid system and two-electrode montages in tsDCS application will be discussed. Furthermore, the influence of different conductivity properties of tissues will be studied since it can be determinant to optimize stimulation delivery for intended spinal targets.

2 Methods

2.1 Realistic Human Model and Electrode Placement

This study used the same realistic human model as in previous works (e.g [9]). This model was based on relevant tissue masks from the 34 years-old male Duke model of the Virtual Population Family [15], comprising 15 tissues (Fig. 1). The spinal GM was artificially designed considering general anatomical knowledge and measurements from the Visible Human Data Set (National Library of Medicine, NLM, Visible Human Project®, www.nlm.nih.gov/research/visible/visible_human.html). The full model was truncated at the level of the thighs and above the elbows, to reduce model size and computational time.

Fig. 1
Three human models at the thighs and elbow level; an abdominal model; spine anatomy; and 4 images of vertebrae from the electrical conductivity study.

Tissues included in the human model and corresponding isotropic values of electrical conductivity σ, as assessed in Fernandes et al. [9]. (av) indicates that conductivities were averaged over different tissues’ components or longitudinal/transversal values (CSF – cerebrospinal fluid; WM – white matter; GM – gray matter)

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Electrodes were represented as hydrogel layers with 1 mm thickness, considering two geometries available for standard electrodes used in transcutaneous electrical nerve stimulation (TENS): 50 × 50 mm2 square electrodes (Fig. 2a); circular electrodes with a diameter of 10 mm (Fig. 2b). TENS electrode design was considered in this study to adequately reproduce the electrodes that will compose the grid montages to be studied by one of our clinical teams. Two types of electrode settings were considered: bipolar montages with two square electrodes (2-electrode); multi-electrode montages with 16 circular electrodes placed in a 4×4 electrode array (Grid). Both montages were centered midway between T9 and T11 vertebrae spinous processes. The 2-electrode montage was varied in alignment relative to the vertebral column – transversal (2-T), diagonal (2-D), longitudinal (2-L) – and distance – 5 mm and 65 mm between electrode edges (Fig. 2c). The electrode Grid was rotated by 45° and 90° relative to the first position (rotation 0°, Fig. 2d). Surface meshes were optimized and assembled, and volume meshing was performed with the 3-MATIC module from MIMICS (MIMICS software, v16), resulting in 2×107 tetrahedral elements for the entire human model with electrodes. Average meshing time was 4 hours per electrode montage.

Fig. 2
Two electrode types: a) square b) circular, and their respective settings between T9 and T11, c) 2-T, 2D, 2-L d) rotation at 0, 45, and 90 degrees.

Geometry and dimensions of the square (a) and circular (b) electrodes represented in the model. Variations considered for 2-electrode montages (c) and Grid montages (d); anodes and cathodes are represented by red and blue electrodes, respectively

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2.2 Electrical Properties of Tissues and Electrodes

Tissues and hydrogel were assumed to be purely resistive with isotropic electrical conductivities. An electrical conductivity of 0.02 S/m was assigned to hydrogel in all simulations [16]. Three different levels of tissue heterogeneity were considered (Table 1):

  • Heterogeneous model – Tissues and hydrogel were assumed to be purely resistive with isotropic electrical conductivities, considering DC electrical tissue properties compiled in our previous work [9] (Fig. 1);

  • Homogeneous model – All tissues were assigned an isotropic electrical conductivity of 0.289 S/m, that corresponds to the volume-weighted average of the isotropic conductivities of each tissue included in the model;

  • Semi-homogeneous model – Considers the same conductivity as the homogeneous model for all tissues, except for one tissue in the vertebral column (spinal gray matter (GM), spinal white matter (WM), CSF, dura, vertebrae or disks) resulting in six different models.

Table 1 Isotropic electrical conductivity models
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2.3 Electric Field Calculations

Electric field (EF) spatial distribution was simulated with COMSOL Multiphysics using the finite element method (FEM). Total current intensity was considered as 4 mA (the maximum delivered by a typical tsDCS device): 4 mA/electrode in 2-electrode montages and 0.5 mA/electrode in Grid montages. Boundary conditions were applied according to [17], considering the hydrogel top surface as isopotential. A total of 72 simulations (9 montages variations × 8 conductivity models) were performed, with 2.2×107 degrees of freedom and solution time of about 30 minutes per simulation on a computer with 2 quad-core Intel® Xeon® processors clocked at 3.2 GHz and 48 GB of RAM.

Modelling studies in tDCS reported volume-average E-field values larger than 0.15 V/m over the hand knob, when reproducing clinical settings with observed neuromodulatory effects [17, 18]. tsDCS neuromodulation will be assumed if the average EF exceeds this value in the SC.

EF components were defined as 3 orthogonal vectors: Elong – caudal-rostral oriented and tangent to SC axis; Evd – ventral-dorsal oriented and perpendicular to SC axis; Erl – right-left oriented and perpendicular to SC axis. Spatial profiles of the magnitude of the total EF and of its components along the spinal WM and GM were determined by considering EF averages calculated at 1-mm thick axial slices along the SC length.

3 Results

3.1 Current Delivery for 2-Electrode and Grid Placements: Safety Considerations

The electrode montages considered in this study may present some safety concerns in terms of tissue damage that should be addressed: (1) in 2-electrode placements, an inter-electrode distance of 5 mm may be too small to prevent current local maxima at electrode edges; (2) small circular electrodes in Grid montage may deliver large current densities below each electrode.

Current density was predicted at skin and target tissues (spinal GM and WM) for the heterogeneous model considering all placements addressed and is summarized in Fig. 3. Current density magnitude shows hotspots in skin regions near electrode edges (Fig. 3a). For inter-electrode distances of 5 mm, these hotspots are higher between electrodes. In Grid montages, the current density is larger at electrodes’ edges and maximum in skin regions located below the grid where electrodes’ polarity changes from anodal to cathodal. Maximum values of the current density vary from 6.89 to 24.60 A/m2 in skin regions near and below the electrodes, with the highest values predicted in Grid montages and the lowest values in 2-electrode montages with larger inter-electrode distances (Fig. 3b). The opposite occurs at the target tissue, where current density maxima are larger for 2-electrode montages with more distance between anode and cathode.

Fig. 3
The current density results for 2-electrode and grid placement are a) depicted in hotspots, and b) table of predicted values at the skin and target tissues.

(a) Current density magnitude distribution in skin below and near electrodes for each montage. A color scale is presented at the top right: all distributions are normalized to the respective maximum value. (b) Maximum values predicted for the current density magnitude in skin and spinal WM and GM for each montage

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Current density threshold for cutaneous and nerve lesion was established to be 143 A/m2 for DC stimulation [19]. This value has not been reached in previous tsDCS and tDCS experimental studies with no report of tissue damage [1, 5, 20, 21]. These studies applied currents of 2–4 mA using larger electrodes at larger distances. Models of these tsDCS protocols predicted current densities maxima of 12.6–21.7 A/m2 and 0.11–0.15 A/m2 at skin and spinal-GM, respectively [9]. These values are of the same order of magnitude of the values predicted for the 2-electrode and Grid montages and variations presented in this study, thus the montages proposed here can be considered safe for future experimental studies.

3.2 EF Distribution for Grid and 2-Electrode Montages in the Heterogeneous Model

The heterogeneous model includes isotropic electrical conductivities for each tissue in the model (Fig. 1), thus providing a more realistic insight on the spatial characteristics of the EF at target (spinal cord) and surrounding tissues. Figure 4 shows the EF magnitude in a target volume of the spinal cord, comprising lower thoracic to lumbosacral spinal regions, from T7 to Co segments for all montages and corresponding variations in rotation, alignment and distance between electrodes. 2-electrode and Grid montages can be grouped regarding similar EF spatial profiles: transversal/rotation 0° (T0); diagonal/rotation 45° (D45); longitudinal/rotation 90° (L90). The EF magnitude distribution along the SC is wider and reaches higher EF maximum in 2-electrode montages, especially for a larger electrode distance, whereas the distribution due to 2-electrode at 5 mm is more similar to that of Grid montages (Fig. 4). The EF direction varies with montage groups, which can be seen by comparing the maximum values of the EF components, normalized to the maximum EF magnitude (Emax; Fig. 4, table at top right). In T0 montages, the EF is mostly transversal, with a larger right-left component of 0.96–0.98 Emax. In D45 and L90, the EF direction is longitudinal, (Elong (max) = 0.94–0.99 Emax). A maximum contribution of Elong is present in L90 and 2-D at 65 mm. Maximum values occur mainly at T12 and L1 spinal segments, which are located midway between the electrodes or grid upper and lower borders. Figure 5 shows contributions of different montage variations inside the SC, considering an axial slice of the spinal cord at T12 level. EF similarities also translate in L0, D45 and L90 grouping at local level. D45 and L90 present different spatial patterns in the WM and GM when compared to T0. EF orientation projected in the axial plane is also represented in Fig. 5 (black arrows) and changes from group to group. D45 presents a EF axial projection with a dorsal-ventral, right-left direction, whereas L90 projection is almost only dorsal-ventral. This can also be seen when comparing values of Evd and Erl normalized to maximum in Fig. 4b. This orientation is consistent with anode-cathode relative position in each case.

Fig. 4
An illustration of three 2-electrode at distance equals 5 and 65 millimeters and 3 grid montage a) setting b) table of E F values at spinal W M and G M.

(a) EF distribution in a selected volume of the SC, between T7 and Co spinal segments for all montages and variations; maximum value and name of montage are on top of each plot; location of the selected volume is marked by a red rectangle over a representation of the vertebral column and colour scale for the EF magnitude are at top and middle left, respectively. (b) Maximum values of the EF components normalized to Emax in the spinal WM (top) and GM (bottom). Values of the largest component are in blue

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Fig. 5
A heat map of EF distribution for three 2-electrode at distance equals 5 and 65 millimeters and 3 grid montage settings, in the axial plane at T12.

EF distribution in an axial slice at T12 spinal level in spinal WM and GM for all montages. Left: location of the slice is indicated by a red dashed line over a representation of the vertebral column. Right: colour scale, with the red colour corresponding to the maximum EF in the WM in each case and orientation of the slices at the top. The black arrows have size proportional to the EF magnitude and represent the orientation of the field projected in the axial plane

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3.3 Homogeneous Model: How the Relative Positions of Electrodes Influence the EF Direction

The homogeneous model provides insight on the effect of electrode positions on the EF direction and magnitude, reducing variability that arises due to the different properties of tissues that surround the SC. 2-electrode montages originate larger EF magnitudes in the spinal WM and GM when compared to the 4×4 electrode grid. Furthermore, larger distance between the electrode edges results in an EF increase, whereas grid rotation does not change the EF considerably in terms of maximum values (Table 2). Again, 2-electrode and grid montages can be grouped regarding similar spatial profiles of the magnitude of the total EF and of its components (Fig. 6). Grid montages are more similar to 2-electrode montages with inter-electrode distance of 5 mm, compared to the montages with 65 mm. The latter also originate EF magnitude larger than 0.50 Emax in a wider region, by approximately two spinal segments than the grid or the 2-electrode at 5 mm.

Table 2 Maximum values of the EF in the spinal WM and GM in the homogeneous model, in V/m
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Fig. 6
Nine graphs depict EF magnitude, longitudinal, ventral dorsal, and right-left curves at different spinal segments for 9 Electrode montages in spinal-VM.

Magnitude of the EF and components in spinal-WM averaged over 1-mm thick slices along the spinal cord length and normalized to EF maximum (Emax). The corresponding electrode montage and orientation and the legend for each element are indicated in each plot at the top left and right, respectively. The Erl component is superimposed on the total EF magnitude in the first top row

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The direction of the EF is consistent with the relative position of anodes and cathodes in each case. T0 montages originate an EF with a Erl component that almost matches the total magnitude (Fig. 6, top row). The diagonal and longitudinal alignments (D45 and L90) increase Elong and Evd contributions to the total EF (Fig. 6, middle and bottom rows). Erl contribution is similar to the other components in D45, decreasing more in 2-D at 65 mm; this contribution disappears almost entirely at L90 montages.

The Elong component peaks in the anode-cathode transition region; Evd presents two peaks of opposite sign, one below each anode and cathode regions. The larger distance between electrodes in 2-D and 2-L at 65 mm increases Evd below the electrodes, resulting in a total EF spatial profile with two peaks, below anode and cathode regions. The anode-cathode relative placement and distance have a strong influence on the direction of the EF, allowing to establish which component can be more significant to the total EF magnitude. This could be determinant to target specific spinal neurons according to their orientation inside the SC.

3.4 Semi-homogeneous Models: How Tissues’ Different Conductivities Influence the EF

The EF spatial profiles in the semi-homogeneous models can reveal the influence of the electrical conductivity of each tissue on EF components and total magnitude along the SC. The EF spatial profile is very similar to the profile for the Erl component in T0 montages in the spinal WM (Fig. 7). The same type of profiles occurs in the spinal GM (not shown). This profile presents peak-like features in the spinal segments below and near the Grid or the 2-electrode. Also, the total magnitude and Erl profiles mimic the semi-homogeneous vertebrae model profile, thus the conductivity of vertebrae seems to be the main factor determining the existence of the peak-like features of the EF spatial variation in the spinal WM (Fig. 7, light grey dashed line in each plot). The Erl component contributes to 97–99% of Emax in all T0 models, which highlights the importance of the anode-cathode placement in the total EF spatial distribution (Fig. 10).

Fig. 7
Eight graphs depict 8 curves GM, WM, CSF et cetera along the spine for 4 EF profiles at grid 0 degrees and 2 EF profiles for 2-T at 2 distances.

EF profiles of the total and components magnitude normalized to Emax in the spinal-WM for T0 montages. The first two rows present the spatial profiles for the total EF, Elong, Evd and Erl for Grid 0° montage. The middle and bottom rows present the spatial profiles for the total EF and Erl for Transversal 2-pads at d = 5 mm and d = 65 mm, respectively

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D45 montages show similar profiles for the total EF and components (Fig. 8). It presents the same type of global profiles as in the homogeneous model, where Elong represents 69–75% of Emax in the homogeneous models. This percentage is around 95–99% for the heterogeneous (isotropic) model, the larger value corresponding to 2-D at d = 65 mm (Fig. 10). This increase in Elong contribution should arise due to the influence of the CSF conductivity, since the total EF has a profile more similar with semi homogeneous CSF model (Fig. 8, dot line in dark grey). There are also some peak-like features in the spatial distribution at the same spinal level as in the semi-homogeneous vertebrae model, which supports the influence of the vertebrae conductivity in local hotspots. The Evd component is larger in the semi-homogeneous models, presenting the same peaks below the anode and cathode regions as in the homogeneous model, but almost disappears in the heterogeneous isotropic model. This indicates that the combined effect of the different conductivities cancels out the contribution of the electrode placement to the Evd component.

Fig. 8
Eight graphs depict 8 curves GM, WM, CSF et cetera along the spine for 4 EF profiles at grid 45 degrees and 2 EF profiles for 2-D at 2 distances.

EF profiles of the total and components magnitude normalized to Emax in the spinal-WM for D45 montages. The first two rows present the spatial profiles for the total EF, Elong, Evd and Erl for Grid 45° montage. The middle and bottom rows present the spatial profiles for the total EF and Elong for Diagonal 2-pads at d = 5 mm and d = 65 mm, respectively

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L90 montages have similar distributions to D45 montages, with the Elong component contributing to 99% of Emax value, and with an almost negligible Erl component (Figs. 9 and 10). The Evd is present in homogeneous and semi-homogeneous models, but the overall effect of different conductivities results in a less meaningful contribution in the heterogeneous model, just as seen in the D45 montages.

Fig. 9
Eight graphs depict 8 curves GM, WM, CSF et cetera along the spine for 4 EF profiles at grid 90 degrees, and 2 EF profiles for 2-L at 2 distances.

EF profiles of the total and components magnitude normalized to Emax in the spinal-WM for L90 montages. The first two rows present the spatial profiles for the total EF, Elong, Evd and Erl for Grid 90° montage. The middle and bottom rows present the spatial profiles for the total EF and Elong for longitudinal 2-pads at d = 5 mm and d = 65 mm, respectively

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Fig. 10
Two sets of 9-line graphs for mean and max E long, Evd, Erl in the spinal WM and GM respectively for the 3 grid and six 2-electrode montage settings.

Mean and maximum values of Elong, Evd, Erl normalized to Emax in the spinal WM and GM for all models

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What determines the “peak-like” profile seen in the Erl component? Figure 11 compares the normalized EF magnitude profile with the distribution of volume of vertebrae and disks for Grid 0° montage, where the Erl component largely influences the total EF magnitude. The positions of vertebral spaces and intervertebral disks (volume minima and volume maxima, respectively) are coincident with the “peak-like” features in the EF spatial distribution. Considering that the profile of semi-homogeneous vertebrae is the only one that reflects these “peak-like” features, vertebral conductivity must have the largest influence in the formation of local hotspots.

Fig. 11
A graph to compare the normalized EF magnitude profile with the distribution of disk volume, and vertebrae volume depicts "peak-like” features.

Distribution of vertebral volume, disk volume and EF magnitude in Grid 0° montage, averaged over 1-mm thick slices along the SC length and normalized to maximum values in each case (horizontal bottom and upper axes correspond to EF and vertebral/disk volume, respectively). To the left, the EF magnitude volume distribution is represented with the same length for comparison and uses the same colour scale as in Fig. 4. The red lines show the spatial correspondence between local hotspots, EF peaks, disk volume maxima and vertebral volume minima

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4 Discussion

This study addresses the interplay of different electrode placements and electrical conductivity modelling paradigms in setting the main characteristics of the EF induced by tsDCS over the spinal cord. Our main finding is that the relation between the two factors is complex: anode-cathode placement determines the EF orientation, however electrical properties of tissues can change this orientation and originate local hotspots. Conversely, local hotspots that could be originated by tissue conductivity heterogeneities can disappear when the electrodes are oriented along the SC, by increasing the contribution of the highly conductive CSF. This variability in the interplay between electrodes and electrical properties of tissues in the current path is determinant to optimize EF delivery using tsDCS for a specific spinal target.

4.1 Anode-Cathode Placement Interplays with Tissues’ Conductivities to Define the EF Direction

The homogeneous model is a useful tool to isolate the effect of electrode placement since it considers the human model as a completely uniform volume conductor. The main observations using this model are:

  • Different electrode geometries and number are not determinant in the EF direction if the anode-cathode spatial relation is preserved – a grid of 16 small circular electrodes or a two-electrode configuration resulted in similar EF spatial profiles, if presenting the same anode-cathode relative position (Fig. 6);

  • Larger distances between electrodes originate higher EF magnitudes in wider regions (Fig. 6, Table 2).

When turning to the isotropic heterogeneous model, the EF shows peak-like features and does not always preserve the spatial profiles of the EF and of its components (Figs. 4, 7, 8, and 9). For instance, Evd profile in D45 and L90 homogeneous models has two main peaks and with opposite orientations below anodes and cathodes, that almost disappear in the heterogeneous model. This indicates that electrical conductivities considered in the heterogeneous (isotropic) model interfere with the EF orientation previously established by anode-cathode relative position. This raises the question: which tissues have the most impact on the EF spatial distribution in the spinal cord? Realistic numerical models of DC cortical stimulation refer CSF and skull as the tissues that can induce large variability in EF magnitude [13, 14]. Modelling studies on tsDCS revealed a negative relation between CSF volume and EF magnitude over the thoracic region when two electrodes are placed over T10 spinous process and right arm [7]. In our previous modelling study on thoraco-lumbar tsDCS, locations of vertebrae bony edges and CSF narrowing were associated with local hotspots, regardless of electrode placement [9]. The semi-homogeneous models were considered above to isolate the contribution of each tissue’s conductivity. In all cases, vertebrae and CSF are the two tissues that have the most impact in local features and contribute to the global spatial profile. The EF profiles in vertebrae and CSF semi-homogeneous models closely resemble the EF in the heterogeneous model for T0 and D45/L90 montages, respectively. Vertebrae and CSF conductivity values differ by almost one order of magnitude relative to those of other tissues, which may explain a larger effect in current spreading to the surrounding tissues, thus originating the profiles observed: vertebrae has a low conductivity, which favors the passage of current through inter-vertebral space to the spinal cord, favoring the Erl component; CSF’s higher conductivity contributes to an increase in the longitudinal component of the EF and also contributes to current focusing effects in narrower regions originating, for instance, the sharp peak observed at T12 level in D45/L90 montages. The effect of CSF thinning was also observed in transcranial magnetic stimulation models, where the induced current flows tangential to the skull, leading to a local increase of the EF magnitude at the gyral crowns due to local CSF thinning [22].

The effect of distance was already addressed in previous tsDCS modelling works. Anode-cathode relative positions and tissues conductivities should be carefully considered when choosing a tsDCS montage. The interplay between these two factors will determine the EF orientation. The modulation of spinal neurons, just as in cortical neurons, will occur if these are aligned with the induced EF. Cellular models of spinal motor neurons identified axon terminals as the dominant cellular target of tsDCS and misplacements of 5 cm in electrodes’ positions lead to a change in EF direction and a consequent reversal of polarization at target, thus the influence of electrode distance in position should be carefully considered [8]. However, DCS does not change excitability of peripheral axons, which suggests that synapses are the main targets for excitability modulation induced by DCS [23, 24]. Spinal dysfunction can have many causes but are most frequently associated with aberrant triggering of spinal reflexes and muscle tone, due to vertebral lesions, tumors or neuron degeneration. Many different neurodegenerative, inflammatory or traumatic disorders can affect spinal cord networks, with variable modulation by the same tsDCS montage. This leads to some considerations when translating modelling findings to the clinical context:

  1. 1.

    The selection of the topography and polarity of electrodes in tsDCS is strongly dependent on the position and orientation of the targeted spinal neurons position and their nearby anatomical frame – it is important to consider the location of the surrounding vertebrae and nearby CSF narrowing due to protruded or herniated disks;

  2. 2.

    Electrode grid should position the target below the anode-cathode transition region; two-electrode montages should comprise the target between the positions of the two electrodes;

  3. 3.

    Longitudinal/diagonal placements will be more favorable to modulate longitudinally-oriented targets; transversal placements will preferentially modulate neural targets with a mediolateral orientation;

  4. 4.

    In longitudinal/diagonal placements, increasing the distance between the electrodes increases the EF magnitude at the cost of focality.

4.2 Multi-electrode Montages Can Be Relevant for Differential Targeting in tsDCS

Safety concerns are a major issue when considering transcutaneous application of direct currents. Most applications of tsDCS have been more conservative, with delivery of current intensities below 3 mA. However, intensities up to 4 mA are being considered to increase neuromodulation outcomes in brain stimulation, with promising results regarding patients’ tolerability [21]. The use of a multi-electrode grid may raise some issues because current density will be larger than 2-electrode montages at the skin located below the anode-cathode transition region. However, our calculations predict maximum values around one order of magnitude below the threshold for tissue damage [19]. Also, multi-electrode grid montages can reproduce the same type of EF spatial distribution than the traditional two-electrode montages, as demonstrated in this study, however with lower EF values. The maximum EF obtained in the grid rotations considered above are larger than 0.15 V/m in the heterogeneous (isotropic) model (Emax (grid 0°) = 0.166 V/m; Emax (grid 45°) = 0.320 V/m; Emax (grid 90°) = 0.438 V/m; Fig. 4), an assumed threshold value for effective neuromodulation (Sect. 2.3). What would be the advantage of multi-electrode grid montages, if the EF orientations reproduced with the rotations can be obtained with 2-electrode montages and with higher EF magnitude? In the clinical setting, a multi-electrode grid system may be an advantage if spinal targets with different orientations are considered, because the grid allows to obtain different anode-cathode alignments without changing the placement of the grid, for example, to have a stimulation paradigm alternating mediolateral and longitudinally oriented EFs. A grid setting using only 8 of the 16 electrodes can reproduce an EF with a spatial distribution similar to the D45 group, by considering the 4 top right electrodes as anodes and the 4 bottom left electrodes as cathodes and leave the other electrodes turned off (Fig. 12). Thus, the multi-electrode grid can be used for differential targeting, i.e. to stimulate different spinal networks according to their orientation at specific periods within the same session, which can be of great relevance for clinical applications of tsDCS for multi-factorial spinal dysfunctions.

Fig. 12
Two graphs depict EF magnitude, longitudinal, ventral dorsal, and right-left curves along the spine, for grid montage at E subscript max values 0.460 and 0.320.

Distribution of total EF magnitude and EF components in the diagonal grid (left) and grid rotation 45° (right) montages, averaged over 1-mm thick slices along the SC length and normalized to maximum values in each case. The EF magnitude volume distribution over T6-Co segments in WM is represented at the top left corner of each plot, with the value of Emax

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4.3 Limitations in Modelling tsDCS: What Lies Ahead

The present study only considers isotropic electrical conductivities. In a previous study, we included artificial anisotropy tensors of WM and muscle, considering the typical direction of fibers. Muscle anisotropy only contributed to change slightly the EF magnitude, however the anisotropy of WM increased the EF sensitivity to anatomical characteristics and spinal curvature, which can have an impact on the predicted neuromodulation on certain spinal regions. Thus, we recommend that the use of models to optimize tsDCS protocols should include, whenever possible, anisotropic considerations for the WM and GM.

The model presented here has an artificial design of the GM and does not represent the spinal rootlets with dorsal ganglia, where sensory neurons somas are located. Realistic representation of these components will be a difficult step, since it will require MRIs of very high resolution, with a segmentation procedure which will take many semi-automatic or even manual time-consuming tasks. Even so, this will be a necessary update of human realistic models to address the true neuromodulatory potential of tsDCS.

Connecting macroscale representations of the SC with neuronal and circuit models is also essential to determine what are the effects at neuronal and synaptic level. The long-term effects of tsDCS may be similar to the processes of long-term potentiation (LTP) observed in cortical stimulation, related with neuroplasticity [18, 25]. Neuronal and network models can unveil how tsDCS may be applied to repair spinal network communication and recovery of function at long-term, thus providing a solid support of tsDCS as an effective therapeutic resource for sensorimotor rehabilitation.

5 Conclusion

Non-invasive stimulation of the spinal cord in the form of tsDCS has a promising therapeutic potential to address sensorimotor dysfunctions of the spinal cord. Computational numerical studies can provide useful information to optimize the delivery of currents, indicating how to combine knowledge of the electrical properties of tissues and relative placement of electrodes to increase stimulation selectivity aiming at a specific spinal target. Even if not very different from traditional 2-electrode montages, multi-electrode grid stimulation paradigms can provide differential stimulation, by changing EF directions over different periods without changing the position of the electrode grid. This could be helpful to modulate different spinal circuits in a single session, when these vary in orientation inside the spinal cord. Furthermore, the grid system may be further optimized by adjusting electrode size and inter-electrode distance, since these characteristics can change EF magnitudes, as predicted in 2-electrode montages.