Spinal Cord Stimulator Design

Electric Field Propagation with Spinal Cord Stimulation O. Bernard1,2,∗R. Fontaine3, O. Eddaoui4, J. Harnie4, A. Frigon4, and C. Iorio-Morin2

Université De Sherbrooke, Québec, Canada, J1K 2R1

2Service de Neurochirurgie, Département de Chirurgie,

Résumé

Spinal cord stimulation is routinely used for the treatment of chronic pain, and is increasingly being investigated for the restoration of movement after paralysis. However, most of our current knowledge on epidural spinal cord stimulation relies on empirical approaches and semi-hybrid mo- dels. Hence, optimizing this therapy requires a better theoretical understanding of how stimulation affects the target neural structures. Using the physical properties of tissues combined with an elec- tromagnetic, dielectric, and electrolytic conductor description of them, we calculated the upper analytical limit of electric field energy transmission in target neural tissues for different frequencies in the spinal cord using a dual-frequency framework : a quasi-static description for low frequencies (< 1 MHz) and a multi-layer transfer-matrix model for high-frequency radiation regime (> 1 MHz).

Simulations suggest that the propagation through the tissues of the spinal cord of an epidurally applied electric field has a complex relationship with the field pulse width. In addition the electric field energy is strongly absorbed at the junction between dura mater and cerebrospinal fluid, and within the cerebrospinal fluid as induced current. These currents hit the different bodies floating within the cerebrospinal fluid, and/or dorsal horn in a very localized region of the spinal cord.

The proposed models allow theoretical calculation of the transmission factor for different numbers of layers of tissue within the spinal canal, and for different electric field pulse widths. This eases the prediction of the expected energy reaching each layer of the spinal cord for different tissues, parameters, and electrode configurations, simplifying the optimization of spinal cord stimulation treatment.

Introduction

Electric Field (EF) stimulation has long been used to treat various human diseases, and injuries [1, 2]. An alternating electric current is applied using one or more pairs of electrodes positioned on a region of interest. This generates an EF that propagates into underlying tissues. The response varies depending on the target (e.g., muscle, nerve, etc.), and often results in a physiological output that can be used for therapeutic benefit. Examples include pacemakers leading to heart muscle contractions, transcutaneous electrical nerve stimulation

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(TENS) used for pain relief, and deep brain stimulation (DBS) used in Parkinson’s disease, obsessive-compulsive disorder, or epilepsy . Spinal cord stimulation (SCS) is a specific application in which electrodes are typically positioned on the dorsal surface of the spinal cord, within the spinal canal— composed of dura mater (dura), cerebrospinal fluid (CSF), white matter (WM), and grey matter (GM)— constrained circumferentially by the bony vertebrae (figure 1) . SCS is used routinely for the treatment of chronic pain and is increasingly being investigated for the restoration of movement after paralysis. Therefore, optimizing this therapy requires an understanding of how stimulation affects the target neural structures and how the EF propagates in and around the spinal cord. Mathematical descriptions of the neuronal potential activation thre-

A

Figure 1 – Cut view of a reconstruction of a cat lumbar spinal cord from CT and MRI images done with paraviewer as to show the different tissues making a spinal cord and electrode placement after surgery. The dura is brown (view A), the CSF is blue (View A), WM is light pink while GM is dark pink. The green rectangle represents the dorsal electrodes pad and the turquoise belts are the ventral electrodes pads. Human geometry is used in calculation. The human spinal cord dimensions used are extracted from and

Presented Supplementary Material

shold with the application of an EF date at least as far back as 1986 and form the basis of contemporary models treating how an EF stimulates different tissues . These results describing spinal cord activation during SCS [4, 20] led to semi-empirical and semi- hybrid computational models that discuss fiber activation by a propagating applied field .

In turn, this inspired computational modeling , making the correlation between patient response and experimental stimuli possible.

3

Together, these modeling approaches were successfully used to predict spinal cord sti- mulation targets for movement generation in paralyzed individuals [17, 22–24]. Although these studies have considerably improved empirical tools for spinal cord stimulation mode- ling [17, 22, 25, 26], they remain largely based on experimental evidence and results. This limits our ability to probe the fundamental mechanisms of action of spinal cord stimulation, which remain debated .

The purpose of this paper is to use analytical continuous calculation of electric field trans- mission to explore the normal incident EF transmission of a stimulation performed in the vicinity of the spinal cord and to discuss its behavior and propagation. Using electroma- gnetic concepts, we first describe key physical characteristics of biological tissues, such as complex permittivity ˜ε, complex wave vector ˜k, and complex refractive index ˜n, then derive the transmission coefficient T. These quantities (˜ε, ˜k, and ˜n) are then used in the equation of transmission T of an applied field through the spinal canal. Our hypotheses are that 1) weak transmission of the EF will be obtained after passing through three layers of tissue ; 2) medium transmission of the EF will be observed after a passage through two layers ; and 3) the CSF will absorb most of the EF energy. In this context, we propose a theoretical approach to describe and explore the transmission coefficient T mechanism of an externally applied EF by a pair of electrodes.

Physical Properties

In this paper, the theoretical discussion will be based on the lumbar spinal cord of a cat implanted with dorsal and ventral epidural electrode arrays allowing circumferential stimulation and recording. The anatomy is reconstructed from high-resolution microCT images and MRI acquisitions (figure 1). In addition, we describe pulses as continuous waves as illustrated in figure 2. We are using the frequency f (in Hz), which is the number of times a pulse repeats itself per cycle within a second. Here, the pulse width (pwd) is half the period T of a cycle, which we can relate to the frequency f with the relation f = 1/(2 pwd). The wavelength λ is the length of the pulse in meters (m) and is λ = v/f, with v the velocity (m/s) of the pulse in the medium. Permittivity, although usually considered a real

T

. . . . . .

Pwd

Figure 2 – Schematization of an ideal square wave with f = 1.6 kHz. The dashed line is the oscillation axis. Here, pwd is the pulse width. number, can be expressed as a complex quantity (˜εr) with a real (ε′) and a complex (ε′′) part. The paper will use complex representation.

As neurons and axons can be described as chemical capacitors [2, 3, 6, 14, 28], the behavior of the bulk tissue can be approximated as polar dielectrics, which are described by electromagnetic theories . The spinal cord and surrounding regions can be described as layers of complex dielectric mediums responding to electromagnetic stimuli.

Tissue Permittivity

The first property to describe for such medium is the relative permittivity ˜εr which is complex due to the lossy nature of biological tissues. The work of S. Gabriel [34, 35] and C. Gabriel [36, 37] on dielectric properties of human tissue provides rich information on ˜εr and conductivity σ, in addition to being a good starting point. Their work relies on the modified

(1)

with i the complex number (i2 = −1), τn a time constant characterizing a polarization mechanism of a relaxation region of the dielectric spectrum. Then, ε∞is the permittivity at ωτ >> 1, εs is the permittivity at ωτ << 1, ∆εn = ε∞−εs , αn is the measure of the broadening of a dielectric regime region, σn is the static ionic conductivity and ε0 is the permittivity of free space . The calculated values of ˜εr real and complex parts for the studied tissues are presented in figure 3. The complex part of ˜εr is yet to be reported

5

exhaustively in the literature and little work is done on ˜εr in the frequency region of 10 Hz to 0.1 THz.

(D)

Figure 3 – Calculated real (blue circles- left y axes) and complex (red triangles- right y axes) part of permittivity of (a) Grey matter, (b) White matter, (c) CSF, and (d) Dura tissues. The vertical grey lines indicate where the real and complex parts are equal.

Electromagnetism theories state that the greater the complex part, the greater the loss (or absorption), hence the weakening of a propagating pulse, much like in a lossy medium such as a bad conductor or a bad dielectric. The observed results are in agreement with the findings of previous work[34–36, 39–48].

The parameters reported by Gabriel et al. (1996) were used and extended by fitting the calculations on data from preceding work, allowing us to expand the frequency range below

6

400 MHz, which is yet to be reported.

Quasi-Static Model

With the permittivity ˜εr of the tissues well defined, we can look at how an EF propagates within the spinal cord and surrounding tissues. At low frequencies, the wavelength λ is multiple order of magnitude the size of the medium, and ohmic boundaries are dominant.

In these conditions, traveling waves and refractive boundary reflections do not physically manifest. Hence, we must use the quasi-static description of the EF to describe its influence on– and propagation within– said medium. In this regime, one can use Laplace volume

That Has A Solution Of The Following Form :

φ = Ax + B.

(3)

Using this solution, potential conservation and current density conservation, we can write



.

(4)

Generalizing for three junctions (four tissues), we get :



.

(5)

Solving this matrix multiplication, using the fact that Einitial = V0/r and T = |Etarget/Einitial|,

(6)

with QS standing for quasi −static. Here, M11 stands for



.

(8)

The detailed derivation of these relations is presented in supplementary material . Since spinal cord tissues physical properties are frequency sensitive, we use their frequency de- pendent value of conductivity in the calculation of TQS [37, 52]. Finally, for the transmission between two layers, the quasi-static Fresnel equation is used :

(9)

with the kiz the ones described in the cited article.

Complex Wave Vector

With the quasi-static model defined, we can explore the higher frequencies range where the spinal cord size is not neglegible in front of λ. In this regime, we can use the wave description of the EF in matter. In a given medium, a propagating EF can be described

(10)

with the complex wave vector ˜k = k + iκ, which are the phase velocity and extinction coefficient, respectively. Both parts can be described as [29–33, 58] :

(12)

with ˜ni the complex refractive index for the ith tissue, n′

I Being The Real And Imaginary

part of the index, respectively, and c the speed of light in vacuum.

Complex Refractive Index

The refractive index is proportional to the permittivity and magnetic permeability as[30,

Ε0Μ0

.

8

One can write relative magnetic permeability (µr) proportional to magnetic susceptibility (χM) as µr = (1 + χM) . In addition, previous work on the χM of biological tissue [59, 60] shows that the susceptibility of human tissues is of the order of magnitude χM ∼10−8 m2 kg−1, which is small enough to write µ ≃µ0 even at low frequencies. This is due to the fact that the EF intensity of interest/used in clinical situations will not be sufficiently strong to generate a great enough magnetic field to see a significant magnetic response. This leads

P

˜εr(f).

(14)

Using (14) and the fact that permittivity (˜εr = ε′

R) As Well As The Refractive Index

(˜n = n′ + in′′) are complex, we can use n′, n′′, ε′, and ε′′ as in the Appendix to get the

(16)

with variables ζ′ and ζ′′ present to fit calculations to the scarcity of data from previous

Work ,

Table I – Refractive index variable values for the different tissues.

N/A

with the exception of the complex part of dura, which behaves as an exponential with an inflection point near 60 THz, which justifies the multiplication of its n′′(f) by exp[−f ·63.4× 10−12]. The variables ζ′ and ζ′′ are exponents, in a similar fashion to αn of the modified Cole-Cole equation [34–36, 38]. This variable is introduced for ˜n rather than ˜εr as it is already well defined by the modified Cole-Cole equation. The values of ζ′ and ζ′′ are presented in table I. These equations are valid for a propagating electromagnetic field in the high frequencies region, as the magnetic properties of matter do not have time to react. They would not hold at lower frequencies if χM was great, as the magnetic interaction would have enough time to

9

influence the system. However, as mentioned earlier, it is reported that human tissues χM are too weak for the used EF to generate a magnetic response. Hence, we assume that the previous equations are valid at lower frequencies.

The results of the calculations of n′ and n′′ of the different tissues of the spinal cord are shown in figure 4, which has yet to be extensively reported for frequencies below 0.1 THz. The calculated values at higher frequencies are in agreement with previous work .

More measurements at low f are required to confirm these results.

(D)

Figure 4 – Calculated real (blue circles- left y axes) and complex (red triangles- right y axes) part of refractive index of (a) Grey matter, (b) White matter, (c) CSF, and (d) dura tissues. Vertical grey lines indicate the frequency at which the real and complex parts are equal.

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Even so, values at higher frequencies are slightly higher than reported values. Still, the theoretical values presented are valid as they are for human tissues and not other animal tissues (such as bovine, pig, rat, or mouse as examples) like the ones reported in the literature, which often differ from humans. It is good to underline the need for a greater quantity of data at lower frequencies to confirm the calculated results, down to the low radio frequencies (lRF).

Impedance

Finally, even if at first we are interested in the transmission coefficient T, there is reflec- tion of the EF within the system due to characteristic impedance mismatch η between the different tissues. Such an impedance can be written as :

(17)

as discussed earlier, we can assume ˜µ →µ0 in the given tissues and frequency range explored.

1

˜n(f).

(18)

Hence, the higher the refractive index, the lower the characteristic impedance. If the im- pedance of the first medium is greater than that of the second (η1 > η2), the reflection of the EF is hard and is reflected with a phase difference of a factor π with the incident field, resulting in a lower proportion of the EF being transmitted. In the opposite case (η1 < η2), the reflection of the EF will be soft and the reflected field will be in phase with the incident field.

Wave Propagation Model

Since most SCS modeling studies reported in the known literature use the finite element method [6, 18, 27], we would like to complement existing tools with an updated theoretical model of normal incident EF propagation in the spinal cord and surrounding regions at higher frequencies (> 1 MHz). This model, applied exclusively as a radiation absorption model in the high-frequency regime, will address the lack of a more theoretical description of

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EF propagation, considering real and complex components of the system elements, through spinal cord tissues. This could help us better understand the spinal cord’s reaction to an applied EF and open avenues for different approaches to SCS.

The SCS paradigm most frequently used in clinical practice consists of a monophasic or biphasic square pulse current (at f ∼1.6 kHz) injected by a pair of electrodes. This generates an EF in the epidural space [4, 6, 19, 21, 24, 70] that will propagate through the tissues in the vicinity of the spinal cord. These include dura, CSF, WM, and GM. Because the physiological target of stimulation are the neurons of the spinal cord, it is important to know how much of the applied EF reaches the WM and GM. Using the full complex wave vector ˜k, the boundary conditions can be written as sinusoidal solutions

(20)

with subscripts R (Right) and L (Left) giving the propagation direction of the wave, and rm the position in space, as shown in figure 5. Figure 5 – Schematization of the electric field direction of propagation and position in a medium with multiple junctions. The vertical black lines represent the different junction.

The time-dependent argument of the exponentials cancels out at the boundaries as they are at the same t and f. Using those, considering continuity, and specifying that the incident pulse is normal to the plane of incidence, we can write the matrices for each interface of two

Which Can Be Written As :

MjEi = NjEi+1.

(22)

Repeating the exercise for each interface, we can get the scattering matrix F for four

(23)

Completing the matrix product, we get the detailed transfer matrix F. Then, we can use

(24)

where wp stands for wave propagation. We use this relation to get

(25)

which depends on ˜n, ˜k, f, and tissue thickness. The detailed expression of F00 is written in the supplemental material . Finally, one should describe the propagation of EF when there are only three layers of tissue as well. This will allow us to know how much of EF energy reaches the WM and how much reaches the GM when it touches the CSF at the dorsal horn of the spinal cord. The same method as for four layers is used, stopping the calculation at two junctions instead of three as previously. This is written as another transfer matrix named F′, and leads to :

(26)

which is of the same form as the previous equations. The procedure of calculation to get F ′

00

is written in the in the supplemental material . For Twp from dura to CSF, the Fresnel equation for two tissues is used, which we multiply by Beers law to take into account absorption by the dura as (here, D stands for dura) :

(27)

The models presented in section II C (quasi-static) and II G (wave propagation) are then run using a Python script that calculates the value of T, using the proper model (QS or wp) for a given frequency range, with f going from 10 Hz up to 0.1 THz.

Physiological Considerations

As water is a major element of biological tissues, it is relevant to discuss its physics here. With a very large dipolar moment, in addition to having hydrogen bonds to the ions in the medium, we know that the water content has a significant contribution to the dielectric behaviors and the dipolar moment of the molecular structures .

In the frequency range of 10 Hz to 4 GHz, we expect water to have little damping effect on tissue vibration modes. In fact, even if the water molecules can rotate freely, they are unified into molecular structures of tissues, moving as a unit with the structure instead of individuals. Such massive structures can only present bulk modes of vibration in the low band of microwaves . With an applied EF around 0.1 GHz, rotation of water molecules starts to be inhibited by their rotative inertia (the “dielectric friction”). In other words, the absorbed energy is lost via collisions or interactions with the next neighbors (we can also talk of molecular viscosity). Near f ∼4 GHz, the water starts to exhibit a larger damping of the molecular modes, and there is an increase in the viscosity of the medium. At 10 GHz, water becomes overdamped and inhibits the vibrational modes of molecules. From then on, the water damping effect will weaken with frequency, as can be seen by its dielectric constant at infinity (ε∞= 1.8) that tends toward a frequency-independent behavior [77, 78]. Then, in the hundreds of GHz ranges, from 300 GHz and higher, we observe molecular resonance and vibration modes that can absorb EF energy.

We expect all of this to be truer for tissues with a greater part of water constituting them, such as CSF. In tissues with a lower water content, we still expect a strong influence on their behavior by the water element, but with a greater proportion of absorption linked to the presence of proteins, lipids, and other constituents present.

Results

Using the results of figure 4 as well as of table I in (6), (9), (25), (26), and (27), we calculate the transmission coefficient of a propagating EF with normal incidence and for different paths in tissues (lines a and b are the paths through two layers targetting white and grey matter respectively whereas c and d are the paths through three layers both targetting grey matter) which is presented in figure 6. It is important to underline that these results are

14

theoretical in nature and its validation limited to existing experimental results[34–36, 61– 64, 68]. In addition, in an experimental setting, contemporary stimulation devices usable for SCS can generate a stimuli pulse with a frequency that goes up to f ∼30 kHz , which is far from the maximum explored frequency (0.1 THz) of this work. If a stimulation device capable of generating a pulse with a frequency in the GHz region is developed, results in figure 6 could be used as a point of reference to guide future research on SCS.

Short

Figure 6 – (i) Calculated transmission of the EF to the last tissue of different path according to the frequency. The vertical grey line corresponds to the clinically used biphasic pulse centered at 1.6 kHz (pulses up to 30 kHz can be used in stimulation) and the horizontal one is a guide for the eye for the origin of the y axis. (ii) and (iii) are slice views of a cat reconstructed spinal cord with the electrodes (green) installed on the dura (brown) tissue. The white lines a (para for parallel to the WM fibers and perp for perpendicular to the fibers) and b are the EF paths through two layers targetting white and grey matter respectively (both ∼2.78 mm), whereas c (∼4 mm) and d (∼7.3 mm) are the EF paths going through three layers both targetting grey matter. The dashed line in (ii) is there to separate two possible form of the GM. Human geometry is used for calculation. The human tissues measurements used are extracted from previous work and presented supplementary material .

15

In perfect dieletrics, an increase in f would lead to a linear decrease in impedance, hence a greater EF wave transmission. This behavior is not seen in the results, as the mediums that make up the spinal cord behave more like a conductor than a dielectrics for f lower than a given threshold for each tissue. This is seen by the weak T at low f. This gives place to complex interactions dependent on the f of the EF.

Transmission Between Dura Mater And Csf

As the applied EF reaches the junction of dura and CSF, ∼34% of its energy is transmit- ted to the CSF for f going from 10 Hz up to 0.1 MHz. The great loss in energy is attributed to induced ionic current in the CSF, as it still behaves as a conductor in this frequency range.

At 0.1 MHz, it is where dura and CSF σ start to diverge, leading to a rise of the shunt in the CSF due to its great conductivity. Then, T continues to lower up to f ∼2 MHz, where its slope rises again. In the following, the slope of T decreases at f ∼60 MHz and rises at f ∼0.3 GHz. The first inflection point (at 2 MHz) coincides where dura behavior goes from conductor to dielectric and the later two are where impedance differences between the two tissues varies, changing T of the EF. Lastly, T for the dura to CSF interface peaks at f ∼2.4 GHz with T ∼96 %, then decreases due to absorption by the dura and shunting effects at the interface [76, 78].

B.

Transmission through multiple layers, low frequency regime In the low-frequency region (0.1 kHz to 1 MHz), or the quasi-static regime, T for the path a presents two distinct curves since its target tissue, WM is highly anisotropic (σx = σz = σperp and σy = σpara). The first one is for when the energy is propagating perpendicular to the WM fibers (path a, perp– empty black squares) which present a T that goes from 2.65 % to 2.78 % as it presents a weaker conductivity. The second solution is when the induced current goes along the aligned myelinated axon tracts of WM (path a, para– full black squares), which presents a T that goes from 3.5 % to 3.7 %. Since the CSF behaves as a conductor at this frequency range, we see induced current within the CSF. Then after most of the current is lost within the CSF due to shunting effect. What little energy that

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reaches WM follows the path of least resistance, which is along the WM fibers at the surface of the junction between CSF and WM. Considering this, we assume that, for f < 1 MHz, the solution of path a- para is the one that properly describes the EF propagation within the spinal cord with WM as the target tissue. For this reason, it is this conductivity (σW M−para) that is used in the calculation of T for path b, c, and d in this frequency range. The T for b goes from 2.8 % to 3 %, T for c goes from 1.75 % to 2 %, and T for d goes from 0.7 % to 0.9 %.

The differences between the results of T for path a and b, although within 1 % from one another, are due to the fact that the target tissues are different for both path, with WM an anisotropic tissue that differs from the isotropic GM. As for the difference between c and d, it is due to the thickness of the WM that has a small but visible impact on T. These results underscore that the EF reaches GM or WM with a weak intensity, yet the transmission is greater from CSF to WM (parallel to fibers) than from CSF to GM due to its greater conductivity (σW M > σGM) along the fibers.

For the clinical frequency (f ∼1.6 kHz), the a and b paths exhibit T ≃3.6 % and T ≃ 2.9 % respectively, whereas c and d paths exhibit T ≃1.93 % and T ≃0.82 % respectively. With these results, we can hypothesize that the stimulation EF centered at 1.6 kHz is strongly absorbed in dura and CSF as induced current (shunt effect), which also explains the low values of T calculated. In addition, one must consider that some current is coming from the electrodes as well as it follows the path of the least impedance. The total current then interacts with bodies floating in the CSF at first rather than with the GM and/or WM, as more energy reaches them. These results are consistent with previous work observations [4, 6, 20, 82, 83].

Lastly, the QS description of EF propagation converges to the wp model for frequencies ranging from 0.5 MHz (path c and d) up to 12 MHz (T from dura to CSF). This frequency dependence of the model convergence is attributed partly to the frequency-dependent pro- perties of the different tissues composing the spinal cord. Additionally to this, the wp model presented assumes a normal incidence and perfectly perpendicular EF. This in turns implies a neglect of 3D geometrical effects on the propagating EF, which results in an overestimation of T for frequencies below 5 MHz. These limitations are further discussed at the end of this work.

C.

Transmission through multiple layers, intermediate frequency regime Starting from f ≥1.6 MHz (where λ ≲6.5 m in the matter) the wave propagation model can be used to some extent to describe T. At 1.6 MHz, the slope of T increases, which is where the dura behavior to an applied EF transitions from conductor to dielectric. This results in a conductor (electrode)-dielectric (dura)-conductor (CSF) scenario, or a capacitor. The slope of T continues to increase for every configuration until the WM behavior changes to dielectric at f ∼85 MHz. It is also where we see a drop in tissues ε′′

Ε′

r and where molecular viscosity starts to rise. As water begins to exhibit dielectric friction, it is appropriate to discuss the behavior of T for the case of four- and three-layer path of the propagating EF in this frequency range.

In the case of four layers, this creates a periodic dielectric-conductor configuration. This means that the system is akin to a capacitor with CSF and GM as conductors, and with dura and WM as dielectrics. However, this state holds up to f ∼0.2 GHz where GM behavior changes to the one of a dielectric, which makes the capacitor description of the situation insufficient to explain the increase of T and the observed maximum at higher f, which will be discussed in more detail in the text. Let us move on to the case where there are only three layers of tissue.

For the three-layer scenario, we either have a dielectric-conductor-dielectric (dura-CSF- WM) or a dielectric-conductor-conductor (dura-CSF-GM) interface– so no capacitor-like configuration. In both cases, a reflection is occuring and a T value that is determined with f. This can be attributed to interference with reflected wave and tissue absorption. Hence, we need to verify the characteristic impedance of the tissues to know if the reflections at the junction are hard or soft. Using (18) and the fact that ˜n of CSF is greater than those of WM and GM, one can conclude that the impedances of WM and GM will also be greater than the one of CSF (η2 > η1). This results in soft reflection, meaning that there is no phase shift for the reflected wave, in addition to generating standing waves in the CSF and weaker wave transmission of the EF to GM and/or WM. As the impedance of the CSF and other tissues gets closer to one another with rising f, there is a lowering the impedance mismatch between tissues and therefore an increase of T is observed. The same point can be made for the four-layer case.

In the same order of idea, the T of path c (four layers, thin WM) is close to that of path

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