Novel mechanistic model and computational approximation for electromagnetic safety evaluations of electrically short implants

$\vec {g}(x)$ can easily be computed by solving the quasi-static equation in homogeneous medium under homogeneous exposure conditions and normalizing the current density $\vec {j}(x)$ distribution by the total (integrated) current J that leaves the tip. However, the determination of $f(s)$ can be more complex. If both tip-shapes are identical, $f(s)$ can simply be determined using equation (1). If the two tip-shapes differ, an additional semi-homogeneous simulation must be performed, in which the conductivity of the region around one of the tips is changed. Application of equation (1) to the two simulations results in a system of two equations for f(s₁) and f(s₂) that can be solved analytically or numerically. This approach can be generalized to more than two 'tips', in which case as many (semi-)homogeneous simulations as tips are required. In these simulations the combinations of local tissue conductivities around the tips must be linearly independent to obtain a uniquely solvable set of equations.

To test and validate the proposed mechanistic model, a range of simulations were performed on simple, straight insulated wires of radius 0.75 mm with bare tips (ASTM 2182-11 a (2011), figure 3) exposed to a homogeneous field, while varying the:

• overall tissue conductivity: 0.05–0.77 S m⁻¹, relevant for muscle, fat, heart, and grey matter in the frequency range under study (table 1, based on Gabriel et al (1996) and Hasgall et al (2016)); default: muscle
• frequency: 3 kHz–128 MHz
• tip shape: see figure 4; default: 10 mm of straight, bared wire as in the IEC 60601-2-33 (2010) standard; while some of these tip shapes are realistic (helical, sharp), others have been selected because they are either standardized (IEC 60601-2-33 (2010)), or to generically cover a wide variety of tip shapes and potentially relevant features (sharp versus spherical, short versus long)
• implant length: 100–800 mm; longer for the analysis of phase gradient
• insulation thickness: 0–1 mm; default: 0.5 mm
• structure of the lead: straight versus helical
• local tissue conductivity: one or both tips embedded in cylindrical muscle tissue with twice the background conductivity
• phase gradient of the tangential E along the lead: 0–5 rad m⁻¹

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For frequencies  >10 MHz, the finite differences time domain (FDTD) method of Sim4Life v.3.0 (ZMT Zurich MedTech AG, Switzerland) was used to perform simulations (Taflove et al 2005). The partially insulated leads were exposed to uniform plane waves with the E-field tangential to the lead. A plane-wave source box was placed at a distance of 30 mm from the lead, and the perfectly matched layer (PML) boundary condition was used to truncate the computational domain after 20 mm. For the phase gradient simulations, the Huygens Box source functionality of Sim4Life was used to define the corresponding incident field conditions. The computational domain was discretized with a rectilinear grid of 1 mm resolution and refinement of 0.5 mm in the tip region. A convergence analysis was performed for the case of a 10 cm wire at 64 MHz by varying the tip region resolution from 0.2–0.9 mm in five steps and computing the variability of the averaged quantities-of-interest (see section 2.2.4), which is found to be  <3%. The deviation between the standard resolution and the highest resolution is also  <3%.

For frequencies  ⩽10 MHz, where the wavelength is much larger than the length of the implant, the finite element method (FEM) simulator of Sim4Life was used to solve the electro-quasistatic (EQS) approximation (suitable when: $\newcommand{\e}{{\rm e}} \omega^2\epsilon\mu d^2 \ll 1$ and $\omega\sigma\mu d^2 \ll 1$, where ω is the angular frequency, the electric permittivity, σ the electric conductivity, μ the magnetic permeability, and d the characteristic length of the computational domain). The suitability of the EQS approximation was verified against FDTD simulations at 10 MHz, the most critical (highest) frequency: for all quantities-of-interest (QoI, see below), the deviation remains  <0.13 dB (1.5% for $E_{{\rm av}}$,j_av, J and 3% for psSAR), even for the longest lead. Only the unaveraged $\vec {E}$-field, which is very sensitive due to steep gradients (figure 5), was found to deviate by up to 0.9 dB (11%).

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In all cases, the background exposure (field in the absence of the implant) was subtracted before computing the QoI, to assess the impact of the presence of the implant.

To investigate magnetic exposure, two wires of circular cross-section were embedded along trajectories forming circular arcs in a cylindrical muscle tissue and were exposed at frequencies  ⩽10 MHz to circularly symmetric fields generated by a current loop. The insulated wires (wire radius: 0.75 mm, insulation thickness: 1 mm) with bare tips (10 mm exposed) were inserted along trajectories with constant, tangential E-fields. One wire covered one-quarter and the other three-quarters of a circle of radius 17.5 cm. The mesh had an overall resolution of 1 mm with refinements of 0.3 mm near the wire and of 0.5 mm in the regions around the tips.

To confirm the validity of the model in realistic inhomogeneous setups, simulations involving detailed anatomical models and a realistic implant were performed. The male adult human computational anatomical model 'Fats' from the Virtual Population (ViP) (Gosselin et al 2014) was used in combination with the model of an abandoned neuro-stimulator lead shown in figure 1. It features an insulated helical wire structure embedded in additional insulating material around an air-filled core. One end consists of a large and rounded electrode and the other ('tail') is the cut lead with only a small cross-section of the wire exposed. The implant models were positioned on the left and right of the spine, based on medical images of a patient. The right one was completely embedded in muscle, while the left one was partly in muscle (tail) and partly in fat (electrode). In all cases, the tissue environment was inhomogeneous, involving tissue interfaces with high dielectric contrast (e.g. to bone) in implant proximity. Exposure to E and magnetic (B) fields was simulated (figure 6). The B-field originated from a 10-turn rectangular spirangle coil, while electric exposure was to a capacitively-applied, longitudinally-oriented E-field. Tissue dielectric properties were assigned according to published values for exposure at 100 kHz (Hasgall et al 2016). Simulations were performed with the magneto-quasistatic and the ohmic-current dominated EQS solvers on a mesh with body resolution of 2 mm, refinement of 0.5 mm in the tip regions, and refinement of 0.03 mm of the wire and insulation. To determine $\vec {g}(x)$ and $f(s)$ (see section 2.1), two simulations were performed: homogeneous implant exposure within a homogeneous medium (0.5 S m⁻¹) and semi-homogeneous exposure within two half-spaces (0.5 S m⁻¹ and 0.1 S m⁻¹). For the semi-homogeneous exposure the electrode was embedded in the 0.5 S m⁻¹ medium and the tail in the 0.1 S m⁻¹ medium. From these two simulations, in vivo J and psSAR at the electrodes and tails of the left and right implant were estimated, using the mechanistic model and computed incident field conditions from electric and magnetic exposure in the absence of implants. For additional validation, another semi-homogeneous exposure simulation with inverted implant orientation was performed and compared to the model predictions.

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The following quantities were extracted from the EM exposure simulations: (i) the maximum induced $\vec {E}$, SAR, and $\vec {j}$ at the lead tip; (ii) the peak of the E-field vector averaged on a cube of 2 mm side according to ICNIRP 2010 ($E_{{\rm ICNIRP}}$, (ICNIRP 2010)); (iii) the peak of the E-field vector averaged over a 5 mm long straight line ($E_{{\rm IEEE}}$, (IEEE C95.1 2005)); (iv) the peak of the induced current $\vec {j}$ averaged over a disc of 1 cm² surface ($j_{{\rm ICNIRP}}$, (ICNIRP 1998)); (v) the psSAR in cubes of 0.1, 0.5, 1, and 10 g tissue mass (psSAR$_{0.1{\rm g}}$, psSAR$_{0.5{\rm g}}$, psSAR$_{1{\rm g}}$ and psSAR$_{10{\rm g}}$); and (vi) the total current J. $E_{{\rm ICNIRP}}$, $E_{{\rm IEEE}}$, $j_{{\rm ICNIRP}}$, and psSAR are standardized quantities that are routinely used in safety assessment.

In addition, thermal simulations were performed, using the EM simulation results as sources, to assess the temperature increase after one hour of exposure. The temperature increase was estimated by solving the thermal diffusion equation with the transient thermal solver in Sim4Life:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \rho C \frac{\partial T}{\partial t}=\nabla k \nabla T + \rho {\rm SAR} \nonumber \end{align} \tag{ 5 }$$

where T is the temperature, t the time, k the thermal conductivity, and C the specific heat capacity. Table 2 summarizes the thermal tissue property values employed. As in Kyriakou et al (2012), thermal conduction along the implant and heat removal by perfusion were not considered to obtain conservative results. See Kyriakou et al (2012) for an analysis of their impact.

The QoI described above were extracted in all simulations of homogeneously exposed insulated straight wires with bare tips for frequencies  <10 MHz and normalized according to the expected dependencies (equations (1)–(4)) on the tissue- and frequency-dependent σ, lead length, and incident field strength, in line with the proposed mechanism. The variability of the normalized psSAR, E, $E_{{\rm ICNIRP}}$, $E_{{\rm IEEE}}$, and $j_{{\rm ICNIRP}}$ is small; the coefficients of variation (standard deviations normalized by means) of the dependence on frequency, σ, and length are  <2%, <5%, and  <5%, respectively, over an 8-fold range of length. The small variability supports the proposed mechanistic model of voltage-driven current through an implant short circuit. Accordingly, differences of  ⩽5% are found between the simulated quantities and those obtained by rescaling a single simulation (800 mm lead at 10 MHz) according to the expected dependence on material properties and implant length (see figure 7).

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The ratio of temperature to SAR is stable for all muscle cases (coef. of var.  <1%), but—as expected—it is affected by the different thermal and perfusion properties of other tissues. For example, at 10 MHz, the temperature to SAR ratio of fat is higher than that of muscle by 26%.

When σ is doubled locally at one wire tip, the local SAR is expected to change by a factor of (4/3)²/2 (equation (4)). The 1g and 10g averaged SAR show deviations from the expected change by 1% and 5%, respectively. Insulation of one end of the lead reduces the SAR by a factor  >100 at a frequency of 1 MHz. When the frequency is increased from the low frequency regime to MRI frequencies, this insulation efficacy is reduced as capacitive coupling across the insulation becomes possible (as discussed in section 4.2), e.g. at 128 MHz deposited power near the tip is reduced by a factor of four.

With increasing electrical length, the normalized temperature increase and averaged SAR quantities (e.g. psSAR$_{{\rm norm}}={\rm psSAR}/(\sigma l^2 E^2/\rho)$, according to equation (4)) are no longer independent of length and frequency, as resonance and phase phenomena become relevant. Figure 8 shows the length and frequency dependence of psSAR_0.1g in a plot that is directly comparable to figure 6 from Kyriakou et al (2012). While the focus of Kyriakou et al (2012) is on higher frequencies and that of this paper on lower ones, there is visual agreement in the results of the two data sets in the overlapping frequency range. When the normalized quantities are plotted as a function of the electrical length l_E (lead length normalized by the 'wavelength' $\lambda_K$ of thin, insulated antennas embedded in lossy media, according to equation 13(b) of King et al (1974)), the values change substantially for $l_E \geqslant 0.2$–0.3 (see figure 9(a)). A universal scaling behavior is apparent for insulated implants embedded in muscle tissue. Comparison of the rescaled behavior curves in muscle and in fat tissues reveal very similar values for non-thermal quantities below $l_E \leqslant 0.2$–0.3, but differ strongly for longer electrical lengths. A resonance is apparent in fat tissue at $l_E \approx 0.5$, but absent in muscle tissue—this is likely due to dampening of the resonant mode by electrical losses in the conductive medium along the implant ($\sigma_{\rm muscle}>\sigma_{\rm fat}$), when the fields surrounding the insulated implant become non-negligible at resonance. Figure 9(b) shows similarly rescaled curves for bare implants; the psSAR values at the tips of the bare wire decay rapidly at electrical implant lengths much shorter than the lengths at which the psSAR values of partially insulated wires drop. While universal behavior is still apparent in muscle tissue, it is not as consistent as that seen with insulated implants. In conclusion, the proposed mechanistic model is valid for insulated implants with electrical lengths $\leqslant 0.2-0.3$. Values for the electrical length $\lambda_K$ at varying frequencies can be found for different tissues in table 3, which also puts $\lambda_K$ in relationship to the tissue-specific wavelength $\lambda=2\pi/\Im(\gamma)$ (where $\newcommand{\e}{{\rm e}} \gamma=\sqrt{i \omega \mu (\sigma + i \omega \epsilon)}$ and $\Im$ denotes the imaginary part). $\lambda_K$ considers the impact of the metal wire and its dielectric insulation on the 'wavelength', while λ only accounts for the dielectric tissue properties.

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Table 3. $\lambda_K$ (computed for insulation thickness of 0.5 mm) in different tissues over frequencies ranging from 0.15–128 MHz; $\lambda_K$ is the effective wavelength of thin, insulated antennas embedded in lossy media, according to King et al (1974). The wavelength (λ) is also provided for comparison.

Freq. (MHz)	Muscle		Fat		Grey matter
	$\lambda_K$ (m)	λ (m)	$\lambda_K$ (m)	λ (m)	$\lambda_K$ (m)	λ (m)
0.15	290	12	280	39	280	20
1	47	4.0	44	15	46	6.8
10	5.2	2.1	3.7	4.8	5.1	1.4
64	0.9	0.4	0.8	1.1	0.9	0.4
128	0.5	0.3	0.4	0.6	0.5	0.2

The ratio of temperature increase to SAR is not affected by length or frequency, showing a variation  <5% for partially insulated wires and  <7% between partially insulated and bare wires for all analyzed lead lengths and frequencies.

At 10 MHz, the insulation thickness (varying between 0.1–1 mm) has a minimal impact on psSAR and induced E-field (<5%). However, as discussed before, the normalized quantities decrease with increasing frequency, and that decrease is steeper for bare wires. With decreasing insulation thickness, thinly insulated wires start to behave more like bare wires the higher the frequency (see figure 10), because, at higher frequencies, capacitive coupling through the insulation becomes increasingly important and thin insulation layers less relevant.

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The impact of the tip shape of electrically short implants on the normalized quantities was investigated at 10 MHz, and more important variability than that observed for the other parameters was observed: 14% (coefficient of variation) for psSAR_1g, 22% for psSAR_10g, 7% for temperature increase, 11% for E_ICNIRP. The variability in peak E is high (60%), with a clear trend towards higher fields for sharper tip shapes. The variability of temperature increase in muscle of only 3% is likely due to the smoothing effect of thermal diffusion. The variability in the averaged quantities for electrically short implants is sufficiently small as to void the necessity for tip-specific analysis. However, the variability increases substantially in implants that are not electrically short. The tip-shape-dependent correlation between peak E, E_ICNIRP, or averaged SAR and temperature has been determined for a 20 cm lead at 10 MHz. The highest correlation is found for averaged SAR quantities (slightly higher for 0.5 g than for other averaging masses). The main configuration that reduces correlation is the short (2 mm) tip, but this case is an outlier: the two field maxima, one at the end of the insulation and the other at the end of the tip, are distinct on the field level or for very small averaging volumes, but contribute jointly to 1 g- and 10 g-averaged SAR values (due to the averaging volume) and temperature (due to thermal diffusion).

At 10 MHz, replacing the straight lead with a helical wire has only a minor impact (2–13%) on the averaged SAR quantities, E_ICNIRP and temperature increase. However, a helical wire behaves very differently with higher frequencies: at 64 MHz, the decrease in 0.1 g averaged SAR for the 20 cm helical wire is a factor of four greater than that for the straight wire, and at 128 MHz, the difference is a factor of 20. This reflects the increased effective length of the helical wire—in comparison with the straight wire—which now becomes comparable to the resonance length.

With increasing length, the phase gradient also starts to be relevant. Figure 11 shows (for partially insulated lead wires of varying length) the normalized variation in induced E-field at the lead tip under application of an incident 10 MHz field of constant strength with a constant phase gradient along the implant, as a function of the phase gradient. For lead lengths on the order of $\lambda_k/2$ (2.6 m; shorter at higher frequencies or for coiled leads), phase-gradient effects are increasingly important. The worst-case phase gradient is actually non-zero, which is equivalent to a complex-valued transfer function.

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