### Synopsis

**We performed 3-D electromagnetic simulations of coaxial
leads and numerically obtained the lead models to evaluate power deposition
and the voltage induced at
the lead proximal end with the lead models. No correlation between peak volume loss density and deposited powers
at the tip and the ring was observed.
In some cases deposited power at the ring exceeded deposited power at the tip. However further extensive simulations of
induced heating behavior should be done before
final conclusions regarding coax lead design preferences are made.**### Introduction

In some bipolar leads
significant RF induced heating can appear not only at the wire tips, but also in
a comparable level at the ring electrodes [1]. Numerical simulations of leads are useful for understanding the complex interaction between the incident RF field and a lead, which is important for reliably designing MR Conditional leads. The ISO/TS 10974 Tier
3 procedure [2] separates analysis
of clinically relevant incident tangential electric
fields (

**E**_{tan}(z)) along the lead pathways from
calculations of the lead responses to these

**E**_{tan}(z) using a lead
(implant) model expressed as: $$$p = A \times\
|\int_{0}^{L}S(z)\cdot E_{tan}(z)\cdot dz|^2 $$$,
where

**p**
is power deposited at a hot spot,

**A** is the calibration factor,

**L**
is lead length, complex

**S**(z) is the transfer function (TF).
Similar expression is used to model voltage (

**V**_{p}) induced at the lead proximal end. TF
based numerical analysis of coax leads has not been published yet. It is still
a challenge to simulate some commercial bipolar leads that consist of helical
inner and outer conductors. Simulations of generic RF coaxial cable structures not only allow to gather experience in analysis of a rather complex coaxial geometry
but also provides results for design optimization of some medical surgery
instruments that can be represented as a coaxial structure from an RF point of
view. Our goals in this study were: 1) to perform 3-D electromagnetic
simulation of coaxial leads; 2) to numerically obtain the lead models; 3) to calculate
the largest p and Vp for uniform RF excitations using the lead
models.

### Method

The leads were coaxial
titanium alloy structures: straight inner wire of 1.5 mm in diameter with insulation thickness of 0.5 mm, straight outer tube of 3 mm in outer diameter
with outer insulation thickness of 0.25 mm.

**L** was
varied from 40 mm to 800 mm in steps of 10 mm. At the proximal end the lead was
capped and inner wire and outer tube were connected via a resistor

**R**_{proximal}.
At the distal end, wire tip and ring electrode lengths were 10 mm long (Fig.1a).
Electrical properties of both insulators were ε

_{r }= 2.7 and σ = 0.000024 S/m.

**R**_{proximal }= {10

^{-6}, 1.25, 2.5, 50, 10

^{6}} Ohm.

**S**(z) for

**p** of tip,
ring and

**V**_{p} were calculated using the reciprocity approach
described in [3]. The calibration factors for the tip and the ring were calculated
from

**p** (

**p**_

_{uf}), obtained
for the leads excited from one side by a uniform 64 MHz source. The
same simulations that provided

**p**_

_{uf} were used to calculate calibration factors for the

**V**_{p} model. Hot spot integration volumes
(Fig. 1b) enveloped an area where the volume loss density (

**VLD**) decayed more
than 30 dB. |

**E**_{tan}(z)| being constant, the largest

**p** (

**p**_

_{wc}) was
generated if φ(

**E**_{tan}(z)) = - φ(

**S**(z)).

### Results and discussion

**VLD** distribution in
proximity of the wire tip and ring depended significantly on length of the coaxial
structure (Fig. 1b – 1g). Despite significantly large

**VLD** peak value in
proximity of the wire tip,

**p** values at the tip and ring were of the same order of
magnitude. For

**R**_{proximal}=1 GOhm, ring

**p**
values were even larger than tip

**p**
values (Fig. 2a and 2b). Tip and ring

**p** values,

**V**_{p}
(Fig 2.c), and

**S**(z) for both the tip and the ring (Fig. 3 and 4)
significantly depended on

**R**_{proximal}. For

**R**_{proximal }= {10

^{-6}, 1.25, 2.5}, tip φ(

**S**(z))
varied significantly
depending on distance from the distal end. The large variation of φ(

**S**(z))
near the proximal end was due to a sharp drop to zero of |

**S**(z)|. Coaxial structures properties were

**Z****0 **= 18.6 Ohm, λ/4 = 712 mm. The λ/4 impedance transformation resulted in an effective
high impedance between the tip and the ring for

**L ≈ **700 mm and even significantly (close to zero)
decreased values of
tip

**p_**_{wc} (Fig. 5). However, ring

**p_**_{wc}
value decrease was noticeably smaller.

### Conclusion

Numerical simulation of coaxial structures can provide
important requirements for experimental validation of TF, for
example, maximum acceptable distance between measurement points for proper
determination of spatial TF variations and required precision of

**R**_{proximal}. Correlations
existed neither between peak

**VLD** and

**p** at the tip and the ring nor between

**R**_{proximal} and

**p** at the tip and the ring.
Larger

**p** at the ring was observed for

**R**_{proximal}
equal to 50 and 10

^{6} Ohm. However

**p** results cannot be simply extrapolated to induced heating
evaluation. Further extensive thermal simulations should be conducted to obtain
dependences of induced heating at the tip and rings on coaxial structure
geometry and material electrical and thermal properties.

### Acknowledgements

This work was supported by the German Federal Ministry of Education and Research (BMBF) and within the European Joint Undertakings ENIAC JU, grant # 16ES0028, DeNeCoR.### References

[1] P. Nordbeck, et al. MRM 68:1963–1972 (2012).

[2] Technical specification ISO/TS 10974 1st edition 2012.

[3] Shi Feng et al. MTT, Vol.63,No.1,305-313,2015.