# Finite volume analysis of temperature effects induced by active MRI implants with cylindrical symmetry: 1. Properly working devices

- Martin HJ Busch
^{1, 4}Email author, - Wolfgang Vollmann
^{2}, - Jörg Schnorr
^{3}and - Dietrich HW Grönemeyer
^{4}

**4**:25

https://doi.org/10.1186/1475-925X-4-25

© Busch et al; licensee BioMed Central Ltd. 2005

**Received: **16 November 2004

**Accepted: **08 April 2005

**Published: **08 April 2005

## Abstract

### Background

Active Magnetic Resonance Imaging implants are constructed as resonators tuned to the Larmor frequency of a magnetic resonance system with a specific field strength. The resonating circuit may be embedded into or added to the normal metallic implant structure. The resonators build inductively coupled wireless transmit and receive coils and can amplify the signal, normally decreased by eddy currents, inside metallic structures without affecting the rest of the spin ensemble. During magnetic resonance imaging the resonators generate heat, which is additional to the usual one described by the specific absorption rate. This induces temperature increases of the tissue around the circuit paths and inside the lumen of an active implant and may negatively influence patient safety.

### Methods

This investigation provides an overview of the supplementary power absorbed by active implants with a cylindrical geometry, corresponding to vessel implants such as stents, stent grafts or vena cava filters. The knowledge of the overall absorbed power is used in a finite volume analysis to estimate temperature maps around different implant structures inside homogeneous tissue under worst-case assumptions. The "worst-case scenario" assumes thermal heat conduction without blood perfusion inside the tissue around the implant and mostly without any cooling due to blood flow inside vessels.

### Results

The additional power loss of a resonator is proportional to the volume and the quality factor, as well as the field strength of the MRI system and the specific absorption rate of the applied sequence. For properly working devices the finite volume analysis showed only tolerable heating during MRI investigations in most cases. Only resonators transforming a few hundred mW into heat may reach temperature increases over 5 K. This requires resonators with volumes of several ten cubic centimeters, short inductor circuit paths with only a few 10 cm and a quality factor above ten. Using MR sequences, for which the MRI system manufacturer declares the highest specific absorption rate of 4 W/kg, vascular implants with a realistic construction, size and quality factor do not show temperature increases over a critical value of 5 K.

### Conclusion

The results show dangerous heating for the assumed "worst-case scenario" only for constructions not acceptable for vascular implants. Realistic devices are safe with respect to temperature increases. However, this investigation discusses only properly working devices. Ruptures or partial ruptures of the wires carrying the electric current of the resonance circuits or other defects can set up a power source inside an extremely small volume. The temperature maps around such possible "hot spots" should be analyzed in an additional investigation.

## Keywords

## Background

The amplification is adjustable by the quality factor *Q* of the resonance circuit and allows the imaging of the inner volume of such metallic implants as well as gathering functional information from this location, for example, velocity or flow. For these special investigations inside the implant volume the quality factor should be as high as possible to increase the signal to noise ratio for this region and to get as low signal as possible from spins outside the volume of interest. For an overall overview of the implant inside the anatomic surrounding it is only necessary to raise the decreased signal inside the Faraday cage to the normal level, which is possible with a lower quality factor. The active implant locally increases the absorbed power during MR investigations. The temperature increase induced by this extra power loss around vessel implants with a cylindrical geometry is calculated with a finite volume analysis for a "worst-case scenario". For most of the simulations only thermal heat conduction is assumed as "worst case" without cooling due to blood perfusion and without cooling due to blood flow inside the active implant. Partly also a blood flow is considered carrying energy out of the simulation volume. The calculations are carried out with respect to power loss, time of rf-exposure and size of the aMRIi. The aim of this investigation was to perform a detailed assessment of the safety for such implants under "worst case" assumptions.

## Methods

### Theory

#### Specific Absorption Rate (SAR)

The local specific absorption rate (SAR; [W/kg]) for electromagnetic radiation is defined as power absorbed per unit of mass of an object at location **r** [11–16],

where *σ* [S/m] is the electric conductivity, *ρ* [kg/m^{3}] is the mass density, *E* [V/m] is the amplitude of a sinusoidal time dependent electric field and **r** is the position vector. Assuming that the amplitude of the electric field only arises by induction from a uniform, linearly polarized magnetic field with amplitude *B*_{m} [T] and angular frequency *ω* [rad/s] in a homogeneous body of rotational symmetry, the following equation holds [11]

with *r* as component of cylindrical coordinates and *r* = 0 on the rotational symmetry axis aligned parallel to the magnetic field *B*_{m}. Combining Eq. (1,2) the spatially averaged SAR over a cylinder with radius *R*_{cyl} and using dV = 2 *π* r h dr yields

For pulsed MR sequences two correction factors are necessary,

The duty cycle factor *c*_{dc} equals *N** *τ* */TR*, where *TR* [s] is the MR-sequence repetition time, *N* is the number of identical excitations during *TR* and *τ* [s] is the duration of one excitation pulse. *c*_{dc} equals to the time ratio of "rf-on" to "rf-on + rf-off" during an MR scan. *c*_{pwm} corrects for rf-pulses which are not rectangular in shape and is the ratio between the energy of the MR excitation pulse and the energy of a rectangular shaped pulse with identical amplitude and identical duration.

#### Principles of B_{1}-field amplification inside a resonator

A circularly polarized magnetic field with magnitude *B*_{1} or a linearly polarized magnetic field with amplitude *B*_{m} for excitation is amplified inside an aMRIi with respect to the used resonance circuit. Because of the shape of vascular implants and to simplify calculations, a solenoid is chosen as inductor of the resonator and is investigated as a magnetic antenna. For the theoretical estimation, the axis of the solenoid resonator is assumed to be parallel to a sinusoidal linearly polarized magnetic field with amplitude *B*_{m}, or equivalently to be in the plane of a circularly polarized magnetic field with magnitude *B*_{1}. For the resonance case (*2πν*_{0} = *ω*_{0} = (*LC*)^{-1/2}), where the overall resistance of the resonator is just *R*, the following equation can be derived from the law of induction, Ohms law and the definition of the quality factor Q of a resonator,

where *V*_{ind} [V] and *V*_{self} [V] are the induced and self-induced voltage respectively, *Z* [Ω] is the impedance, *R* [Ω] is the resistance, *L* [H] is the inductance of the inductor and *α*_{res} and *α* are the flip angles inside the resonator inductance and outside the resonator respectively. The total magnetic field inside the resonator arises from both components *B*_{m} and *B*_{res}. For large quality factors (*Q*>>1) *B*_{res} dominates *B*_{m}. With Eqs(3, 4) it is possible to calculate the magnetically induced SAR and with the SAR the corresponding power loss inside a resonator. But this takes into account only power losses due to eddy currents and not the total power loss of a resonator.

#### Total power loss of a resonator

The overall power loss P [W] also includes the electric losses on and around the resonator. It is given by the following basic equation [16],

where *W*_{total} [J] is the energy stored inside the resonator.

The total energy *W*_{total} can be expressed by the energy stored in the magnetic field of maximum amplitude using the following basic relations of a solenoid inductance with n_{r} turns (
),

where V, A, ℓ with index imp are the volume, cross section and length of the implant inductor. For a pulsed MR sequence with excitation magnitude *B*_{1} the total power loss can be calculated by combining Eqs. (4, 5, 6) to

A comparison of Eqs. (3b, 7) shows that for a certain resonator tuned to the resonance frequency of a specific MR field strength, the supplementary absorbed power is proportional to the SAR of the MR sequence due to the identical dependence on *c*_{dc}, *c*_{pwm} and B_{1}^{2}. For a specific sequence on a MRI system with a definite resonance frequency, the extra power is proportional to the quality factor *Q* and the volume *V*_{imp} of the inductance (Eq. (7)). The proportionality to the quality factor *Q* may be surprising, because it states that for better quality factors the power loss is larger. This is due to the total energy of the resonator, which depends quadratically on the field strength inside the resonator. This field strength is proportional to the exciting field *B*_{1} (or *B*_{m}) as well as the quality factor *Q*. Combining Eq. (5) and Eq. (6) alters the inverse proportionality to *Q* (Eq. (5)) to a proportionality (Eq. (7)) with respect to the field established by the transmit coil. Examining the power loss with respect to the magnetic field inside the resonator confirms the inverse proportionality to *Q*.

### Finite volume analysis

#### Principles

All temperature increases are calculated as temperature difference maps by using a simulation volume divided into many small simulation cells. The energy exchange Δ*E* [J] between two simulation cells with a specific contact area *A* [m^{2}], temperature difference Δ*T* [K] and heat diffusion path length Δ*x* [m] for a time interval Δ*t* [s] is given by the equation for heat conduction as,

where *λ* [W/(m K)] is the thermal conductivity.

The total energy change ΔE_{tot} of one cell during a time interval Δt is the sum of exchanges of this cell with all adjacent cells with non zero contact area and the energy change due to the power loss p_{cell} inside the cell. From this total energy change the temperature difference ΔT* can be calculated,

where *c* [J/(kg K)] is the specific thermal capacity of the cell material and *V*_{cell} is the cell volume.

This investigation did not use any commercially available finite volume package. The simulation is self-coded for problems with cylindrical geometry in Kylix and Delphi, a software development environment, based on object oriented Pascal. The graphical outputs are mostly generated by an evaluation (registered *β*-test) version of Teechart 7 used within the Delphi and Kylix environment. The simulation describes the time developing temperature difference maps around an aMRIi with cylindrical geometry for a constant total power loss *P*.

*L*

_{sim}and radius

*R*

_{sim}. Adequate for such a volume are cylinder coordinates. Because of the rotational symmetry, the temperature distribution is only dependent on the axial position x and the distance r from the cylinder axis (Figures 2, 3). Because of the mirror symmetry to the plane orthogonal to the cylinder axis at center, the simulation is designed to calculate the temperature distribution only on one side of the center plane. (Figures 2, 3). Using both symmetries one can divide the entire cylindrical simulation volume with length

*L*

_{sim}and radius

*R*

_{sim}into the following cells, which are sufficient to obtain complete information of the entire simulation volume. The calculation volume consists of n cylindrical shaped pieces with length Δ

*x*and index

*x*= 1, 2, ...., n denoting the position on the cylinder axis. Each sub cylinder of length Δ

*x*is divided in one cylinder with radius Δ

*r*and index r = 1 and m-1 shells with outer radius r* Δr, thickness Δ

*r*and index

*r*= 2, 3, ...., m. The entire simulation volume is represented by a two dimensional array of cells

*C*[

*r*,

*x*] (

*r*= 1,2,3,....m;

*x*= 1,2,....n). Most of the cells have contact to 4 adjacent cells with contact areas different from zero (Eqs (10a, 10b)). These contact areas as well as the volumes of the cells (Eq. (10c)) only vary with the index

*r*.

*A*_{l}[r] [m^{2}] is the contact area in both cylinder axis directions (from index *x* to *x*-1 and to *x*+1) whereas *A*_{r}[r] [m^{2}] is the contact area in radial direction from index r to r+1. The contact area in radial direction from index *r* to *r*-1 is identical to the area *A*_{r}[r-1]. A last "shell" with cells as heat (energy) sink is placed as one boundary of the simulation volume at *r* = m + 1 and *x* = n + 1. These cells always keep a constant temperature (Δ*T* = 0), even when receiving energy during one simulation step. Partly the simulations use a second heat sink representing a blood flow through the inner volume of the implant. For this second heat sink the temperature differences of all cell elements below a radius r_{flow}*Δr are also kept zero independent of the energy transfer from the cell elements with higher index r. This approximation assumes that the energy applied to the inner cylinder volume with blood flow is completely transported away from the simulation volume during one time step Δt. At *r* = 1 and *x* = 1 as well as at r = m + 1 and x = n + 1 the simulated volume has its boundaries. The symmetry and the use of cylinder coordinates imply that cells with index r = 1 or x = 1 can not exchange energy with cells at a lower index. For *x* = 1 the symmetry plane defines an identical temperature at index *x* = -1, (Figure 3) with no energy exchange across the symmetry plane. For *r* = 1 no cells with lower index r exist and therefore also no energy exchange is possible.

_{cell}[r,x]>0 Eq. (9)) the physical parameters

*λ*,

*ρ*and

*c*can be set freely. For all other cell elements they are set as tissue. The presented simulations use different metal parameters for heat generating cells (Table 1). The different physical parameters of metal and tissue for different simulation cells results in discontinuities at the metal tissue interface. Assuming the heat conduction path is from the center of one finite volume to the center of a neighboring finite volume, it takes place over two different materials with only half the diffusion length for each of the materials. Because of the much lower thermal conductivity of tissue compared to that of metal it is a good approximation to use only the diffusion length through tissue.

Physical constants of tissue, titanium, iron and tantalum

material | density | specific heat c | thermal conduct. | electric conduct. |
---|---|---|---|---|

[kg/m | [J/(kg K] | [W/(m·K)] | [S/m] | |

tissue | 1000 [22] | 3650 [23] | 0.5 [24] | 0.8–8.0 [22] |

saline | 1003 | 4185 | 1.45* | |

titanium | 4510 | 523 | 21.9 | 2.56 10 |

iron | 7870 | 449 | 80.2 | 10.4 10 |

tantalum | 16680 | 140 | 57.5 | 7.63 10 |

The simulation starts with a temperature field *T* [*r*,*x*] = 0 for all *r* and *x*. For each iteration step with duration Δ*t* the total energy change for each cell element is calculated according to Eq. (9a). This change Δ*E*_{tot}[*r*,*x*] of one cell is the sum of five different energies. One is due to the power loss p_{cell} = *p*[*r*,*x*] defined for a cell element at index r and x. The four others are energy exchanges (Eq. (8), depending on the contact areas (Eq. (10a, b)) and the temperature differences between adjacent cells with respect to the equivalent heat diffusion length through the material of the examined region. From Δ*E*_{tot} [*r*,*x*] the temperature change Δ*T**[*r*,*x*] is calculated according to Eq. (9b). This value is added to the prior value (*T*_{new} [*r*,*x*] := *T*_{old}[*r*,*x*] + Δ*T**[*r*,*x*)) for each cell during the whole iteration process. The entire simulated time *t*_{sim} consists of q iterations with time interval Δ*t* (*t*_{sim} = *q* * Δ*t*).

### Analytical model for a heat generating linear wire as test of the simulation

One test of the principal correctness of the self-coded simulation is possible by a comparison of the simulation results with those of an analytical solution. An easy straightforward analytical solution is available for an endless linear wire with radius *r*_{wire}, which is heated by a constant power density *P** [W/m], defined as power per wire length [W/m]. After reaching the thermal equilibrium, the power associated with *P** penetrates through every cylinder surface surrounding the linear power source independent of the radius *r* (with *r*>*r*_{wire}). The temperature difference ΔT between the wire surface and a point at radial position R can be calculated in a homogeneous medium from the equation for heat conduction.

*P**and with energy exchange only in radial direction should approach, after sufficient time, similar temperature differences between the wire surface and a point at distance (R-r

_{wire}) (Figure 4).

### Control of simulation results

The implemented algorithm was controlled with different checks, beside the afore-mentioned comparison with an analytical model. First, the total energy uptake of the heat sink surface is added up during all iteration steps. This energy summed with the energy stored inside the simulation volume must equal the total applied energy. The energy inside the simulation volume is calculated independently from the finite volume algorithm by using the final temperature increase, the specific heat, the density and the volume of each cell.

Second, the algorithm was tested as to whether it provided similar results for identical geometries with different spatial resolution, different time resolution as well as a different size of the simulation volume surrounding the implant.

### Simulations

*ρ*,

*λ*, c) from Table 1 and length

*L*

_{sim}and radius

*R*

_{sim}. A cylindrical implant with sizes

*L*

_{imp}and

*R*

_{imp}is placed at the center of this volume. The simulation is calculated for three different models.

- 1.
Heat generation inside a cylinder of length

*L*_{imp}and radius*R*_{imp}.The heat generation is assumed to be uniform over the entire volume of

*V*_{imp}=*π**R*^{2}_{imp}*L*_{imp}; the entire simulation volume is assumed to be tissue. - 2.
Heat generation in n

_{r}metallic rings with rectangular cross section of*r*_{wire}times*x*_{wire}at radius*R*_{imp}.The rings are equally spaced over the length

*L*_{imp}. These power generating rings are used instead of a solenoid with*n*_{r}turns to retain the cylindrical symmetry and they stand for a model where the power loss is equally distributed on and nearby the circuit paths forming the inductor. For the rings freely selectable physical parameters*c*,*λ*and*ρ*can be used (Table 1). Partly these simulations assume a flow inside the implant volume not allowing a temperature increase for the region of flow. - 3.
Heat generation in a cylinder shell of length

*L*_{imp}and thickness*r*_{wire}at radius*R*_{imp}.

This is an approximation of a modified solenoid structure with a high density of circuit paths on the cylinder surface. In this case, despite the assumed dense wire, most of the shell volume is tissue. Therefore the physical properties of tissue are used for the entire simulation volume.

None of the three cases exactly match the real situation. However, consistent results for all cases, which also are checked to the order of magnitude with the linear wire model of Eq. (11), could certify a realistic estimation.

## Results

### Test of Simulations: Endless wire model

*P** with Eq. (7). Using this power density

*P** it is possible to calculate the temperature difference between the wire surface and the difference between the wire surface and a specific radial distance after reaching the thermal equilibrium according to the model of Eq.(11). The results are summarized in Figure 5. The simulation of a linear wire is compared with a theoretical curve with respect to the same wire radius. To simulate an endless wire the energy exchange is restricted to radial direction by setting the heat sink shell temperature at maximal index x = m + 1 after each iteration not to zero but to the same value as at index x = m for all r respectively. Power generating metallic cells are assumed within the radius

*r*

_{wire}. In radial direction, just a few cells are sufficient to describe the heat generating wire; the radial resolution Δr is chosen in such a way that n Δr - Δr/2 is equal to r

_{wire}. This choice takes into consideration that the center of the last cell is the reference radius r

_{wire}for the theoretical description according to Eq. (11). For the comparison with theory, the temperature increase in a simulation cell is set to a radial position just half between the walls of a simulation cell. During the temporal development the simulated radial temperature differences increasingly approach the analytical solution for the thermal equilibrium.

### Test of Simulations: Temporal and spatial resolution

For sufficiently short time steps Δt, the simulations show the expected behavior that the temperature increases monotonically in each simulation cell. For overly large time steps the energy flow out of a cell during a time step can become larger than the energy loss inside it. This leads to a physical incorrect situation with decreasing temperatures inside a cell. For this case the simulation produces huge deviations and the temperature differences oscillate in an unpredictable fashion during the temporal development.

### Finite volume analysis

*Q*normalized volumetric power loss density

*P*

_{V}= P/(Q V

_{Imp}) for resonators exposed to an MRI sequence with maximum SAR (Table 2) can be calculated to 4 mW/cm

^{3}. Realistic vascular implants are in the order of 0.1 cm

^{3}(e.g. coronary stents; r = 1.5 mm; l = 14 mm) to 50 cm

^{3}, (e.g. aortic stent grafts, vena cava filters; r = 18 mm; l = 50 mm). The achievable quality factor is low in ionic surroundings, such as tissue, but increases with the insulation of the solenoid wire (unpublished diploma thesis, Karsten Behrens, University of applied sciences TFH Berlin, 2001, Department Mathematik-Physik-Chemie, Prof. Vollmann). For thin insulations, which are necessary for the retention of the mechanical properties of the vascular implants, the achievable quality factor is below 5. However, the simulation as a worst case scenario partly uses quality factors far above this value to test the safety of the implants also for these conditions. As examples, the resonators described in Table 3 were used with the physical parameters of Table 1. As simulated time, an MR sequence (Table 2) exposure of ten minutes was assumed.

Data for calculation of power density *P*_{V} according to Eq. (7) for an MRI sequence with an SAR of 4 W/kg (manufacturer declaration)

permeability of vacuum |
| [V s / (A m)] | 12.6E-7 |

magnitude of magnetic excitation field | B | [ | 25 |

repetition time of MRI sequence | TR | [s] | 2.23 |

duration of one excitation |
| [ms] | 0.80 |

number of identical excitations during TR | N | 246 | |

duty cycle c | c | 0.09 | |

pulse waveform modulation factor c | c | 0.45 | |

Larmor frequency = resonance frequency of LC circuit |
| [MHz] | 63.8 |

power loss density P | P | [mW/(cm | 4.0 |

Geometrical data and power losses for example resonators

Resonator | No | 1 | 2 | 3 | |
---|---|---|---|---|---|

Radius of ring resonator | R | [mm] | 18 | 5.55 | 1.5 |

Length of ring resonator | L | [mm] | 50 | 9.9 | 14 |

Dimension of ring wire axial | x | [mm] | 0.25 | 0.15 | 0.10 |

Dimension of ring wire radial | r | [mm] | 0.25 | 0.15 | 0.10 |

Number of rings (turns) | n | 2 | 6 | 12 | |

Volume of implant | V | [cm | 50 | 1 | 0.1 |

Quality factor | Q | 5 | 12.5 | 80 | |

Total power loss of resonator | P | [mW] | 1000 | 50 | 32 |

SAR inside resonator (Eq. (3b)) | SAR | [W/kg] | 1.8 | 1.7 | 5.1 |

Power loss of resonator due to SAR | P | [mW] | 0.09 | 0.0016 | 0.0005 |

### Heat generation inside a cylinder

_{loss}was verified experimentally. Figures 7,8,9 show the results of these simulations. In no simulation the temperature increases reach a value of 5 K. Literature [17–21] on hyperthermia permits an estimation of the temperature resistance of normal animal cells and human cancer cells, depending on exposure time and temperature increase. The value chosen as physiologically critical, 5 K for 10 min to 15 min exposure time, is a worst case assumption, as various cells tolerate higher temperature increases as well as longer exposure times.

### Heat generation in n_{r} metallic rings with rectangular cross section

### Heat generation in a cylinder shell

## Discussion

### Test of simulations: endless wire model

The simulation of an endless wire uses the first of the three above mentioned models. Especially, the radius R_{imp} is taken as equal to r_{wire}. The condition that the wire is endless can be realized in two ways. The first way is to choose a very long L_{imp} and to evaluate the temperature increases only at positions on the center plane (index x = 1) and at radial distances r small compared to L_{imp}. The second way is to modify the simulation software in such a way that the energy transfer in axial direction is excluded. This can be done by considering only volume elements of the center plane (index x = 1) and setting the heat sink shell temperature (index x = 2) after each iteration not to zero, but to the same value as at index x = 1 for all r respectively.

The second way has been realized. The common and expected result was that during the temporal development the simulated radial temperature differences increasingly approach the analytical solution for the thermal equilibrium (Eq. (11)). One comparison between a simulated and the analytical result is shown in Figure 6. Although a test on agreement of both results is independent of the specific choice of the power loss density P* [W/m], we have chosen a realistic value of P* in order to obtain realistic temperature increases. The power loss density P* was calculated with Eq. (7) for the MRI sequence of Table 2 for the example resonators (Table 2) to be between 0.24 W/m and 4.4 W/m. The simulation uses value of 1 W/m.

### Test of simulations

A correct simulation of an endless wire should deliver results similar to the analytical curve of Eq. (11) using an implementation of the algorithm with no energy exchange in longitudinal (x) direction. The physical model of such a linear wire claims that the temperature difference between the wire and a cell at a certain distance increases monotonically with that distance. The theoretically calculated curve is valid after reaching the thermal equilibrium. With increasing simulated time the calculated curves show an increasingly close coincidence with the prediction of Eq.(11) (Figure 5), indicating the correctness of the algorithm.

The analytical linear wire model can also indicate which power per unit of wire length *P** is safe for active magnetic implants (Figure 4). The theoretical estimation according to Eq.(11) shows for a power density larger than 2 W/m a risk of reaching temperatures increases greater than 5 K. The results do not vary with the time resolution Δ*t* as long as the value is small enough to prevent the simulation from oscillating (Figure 6c,d). Also the spatial resolution (Δ*r*, Δ*x*) does not affect the results very much as long as the temperature maps appears smooth (Figure 6d,e,f). On the other hand, the calculation time explodes with a finer resolution. An increase of the spatial resolution always requires a better temporal resolution. Especially if a metallic cell completely surrounded by tissue is divided into a few metallic cells, a superior temporal resolution is necessary. For a metallic cell completely surrounded by tissue the shortest diffusion length equals roughly half of the spatial resolution with the diffusion path through tissue. For adjacent metallic cells the diffusion length equals the spatial resolution, but the diffusion path is completely through metal with a much larger thermal conductivity. The energy transfer for an identical temperature difference through this path is at least an order of magnitude larger than the one between a metal cell and a tissue cell. This higher energy transfer requires the drastically reduced time step Δ*t* for a stable temporal development of the simulation. The calculation time for an identical simulated time increases by several orders of magnitude with metallic rings and a high spatial resolution and can easily reach computing times of several days on a normal desk top computer. Some examples for the calculation time is given in the second Table below Figure 6.

### Power loss of resonator

Achievable quality factors were derived from the construction of experimental solenoid resonator prototypes (Table 3). The quality factor for a given geometry is strongly dependent on the environment and the thickness of the wire insulation. It is reduced substantially after changing the environment from air or distilled water to saline solution. Also *Q* decreases with a thinner insulation of the wire. Using this information it is possible to conclude that the main power losses are electric losses. Losses due to eddy currents induced by the magnetic field would not change with the thickness of the electric insulation. The dominance of the electric losses is also verified by comparing the power loss from eddy currents inside the active implant (Eq. (3)) with the total power loss (Eq. (7)). For all example prototypes in Table 3, the power loss calculated with the SAR inside the resonator (Eq. (3)) is lower by several orders of magnitude than the total power loss according to Eq. (7) (Table 3).

Realistic values for the quality factor of a resonator placed in ionic surroundings such as tissue are below 5 for all example resonators of Table 3. The calculated examples are worst case assumptions as they assume the maximum achievable *Q* of 5 for resonator no 1 and therefore the maximum power loss. For resonator no 2 (Q = 12.5) and no 3 (Q = 80) the chosen value is far beyond the reachable inside an ionic surrounding. These high values for resonator 2 and 3 are chosen to pronounce the safety statement for smaller resonators with quality factor values well above achievable values. The unrealistic high quality factors do not show physiologically critical temperature increases. For resonator number 1 (with a large volume) a realistic value was chosen, because in this case the temperature increase reaches physiologically dangerous values.

Part of the simulation results could be confirmed on stents implanted inside the aorta of rabbits [9, 10]. Small coated resonators with a volume below 0.5 cm^{3} and quality factors of 3 to 4 were used. After excision and histopathologic examination, the tissue did not show any indications of heating after several MRI investigations. This is in coincidence with the simulation results, because even for small resonators with a much higher quality factor no dangerous heating was calculated.

Ten minutes of simulated time is 5 minutes less than the critical time according to the FDA regulations for imaging of the trunk assuming a SAR of 4 W/kg. But all simulations show only small changes after 10 minutes. Therefore it is unnecessary to increase the simulation time further. Nevertheless, one example of 15 minutes simulated time is given in the movie (see additional file 1) and shows only small changes in the last five minutes.

The weak influence on the distance between power generating cells and heat sink (the outer most layer) is also shown in the movie (see additional file 1). The difference between two almost identical calculations is shown. Starting from a first calculation with simulation parameters L_{sim} and R_{sim}, the second calculation uses doubled values of L_{sim} and R_{sim} without changing the spatial resolution or the other simulation parameters. This approach shifts the heat sink to a larger distance from the heat generating cell elements. As expected the heat sink layer drags down the temperature increase in its vicinity, but the effect is small if the volume is adequately chosen. The differences for identical cells between both simulations are shown by alternating both views a few times at the end of the movie (see additional file 1).

### Finite volume analysis

For all three example resonators (Table 3) the first simulation model assumes a uniform heat generation over the whole volume of the inductor. Figures 7,8,9 show the results of the simulations. All temperature increases are below the critical value of 5 K. The maximum temperature is at the cylinder axis, corresponding to the center of the vessel for a vessel implant. The uniform heat generation inside homogeneous tissue is – as discussed above – not the best approximation. The ring model is a more appropriate match for realistic conditions.

Therefore resonator 1 is assumed as a solenoid with 2 turns (Figure 10). The simulation assumes these turns as metallic rings, which should be similar to a solenoid for the question of temperature increases. For resonator no 2, six rings are calculated (Figure 12) and resonator 3 is simulated with twelve rings (Figure 13). Only the simulation for resonator no 1 shows temperature increases over 5 K, which are located on and directly adjacent to the power generating rings. For resonator 2 the temperature increase with 6 rings is slightly above 1 K (Figure 12). The same is true for resonator no 3 with twelve rings (Figure 13).

For 6 or 12 rings and after 600 s calculated simulation time, the peak values near the location of the power generating rings inside the tissue become more and more negligible. Therefore a model using a power generating cylinder shell is appropriate for resonators with a reasonable number of turns.

For resonators no 2 and no 3, the cylinder shell simulation did not show any unsafe heating and is similar to the ring calculations. Neglecting the peaks of the ring simulation the maximum temperature increase is nearly identical.

Only the simulation for resonator no 1 with two rings reaches a critical temperature value above 5 K, all others stay below this value. For a large number of rings or, stated more precisely, a high wire density at the cylinder surface, it is possible to use the cylinder shell model (compare Figures 12, 14 and Figures 13, 15). As expected, the rings cause higher maximum temperature increases according to the higher local power density. The difference between the shell generating the entire power (Figures 14, 15) and a uniform power generating volume (Figures 8, 9) is mainly a different temperature distribution inside the inductor. The former simulation first shows a moderate increase in radial direction up to the power generating shell inside the volume, followed by a decrease. The latter has a larger peak value at the axis of the cylinder and falls off continuously with greater distance from the axis. The ring model is the most preferable because of the dominance of the electric losses on and near the wire of the inductor. Using different metals without altering the other simulation parameters changes the results only marginally (Figures 10b,c; 11 b-d; 12 a, b).

The simulation worked on a "worst case" basis, neglecting in most cases all cooling effects except the energy transport due to the thermal conductivity. Especially for vascular implants in the "normal case", some blood flow and blood perfusion, causing a much faster energy transport, will reduce the temperature increases. The effect of blood flow is simulated for the example of resonator 1 with the two rings. This was the only simulation reaching physiologically critical temperature increases. Blood flow is simulated by keeping the temperature difference at zero for a central part of the cylindrical implant. Changing the dimension of the central part can simulate an implant, which may be more or less infiltrated to the vessel wall or covered by tissue (thrombosis, calcification or intimal hyperplasia). The reduced temperature increase for resonator no 1 and for various sizes of the cooling blood flow is shown in Figure 11. For an implant placed directly inside the wall, with none or only a small amount of tissue between wire and blood flow the temperature increases drop to a physiologically tolerable value (Figure 11). Nevertheless for implants with a tissue coverage of a few mm, which can be plaque or a thrombosis, critical temperatures can be reached.

With respect to the overall power absorption greater than 100 Watts inside the human body during MRI investigations with a maximum SAR, one additional Watt inside the entire body is negligible.

## Conclusion

This investigation assumes a "worst-case scenario" in different ways. Firstly, the resonator is assumed being perfectly aligned within the plane of the excitation field B_{1}. Secondly, to some extent no blood flow inside the inductor (vessel) of the implant is included. Thirdly, no blood perfusion inside the tissue around the resonator is taken into account. Lastly, too large quality factors are used.

For most "normal" cases, intact active implants will therefore be less critical, but it is not possible to exclude the worst case conditions. Pathologies (thrombosis, intimal hyperplasia, plaque) may alter blood flow and perfusion in the area directly adjacent to the current paths of the resonator and certainly the resonator can be perfectly aligned to the plane of the exciting rf-field B_{1}.

Especially for resonators with a large volume (such as resonator no 1) and with a small number of rings it is possible to reach critical temperature increases above 5 K (Figures 10, 11). For peripheral and especially cardiac vessels, even with an unrealistically high quality factor, the power loss is too low for dangerous heating. On the other hand, stent grafts or vena cava filters built as active implants can reach volumes of a few ten cm^{3}, which may be dangerous, if the tissue around the wires of the implant is not exposed to blood flow or sufficient blood perfusion. To reduce the risk for such active implants, the quality factor has to be low. This reduces also the amplification of the MR signal the resonator is made for and can thus be dispensed with altogether.

As the above simulations assume properly working resonance circuits without any failures on the electric paths of the system, one worst case scenario was not presented. Defects such as ruptures or partial ruptures may generate a relatively high resistance over very short distances. The current flow through this resistance can produce a large power loss inside an extremely small volume. This can generate a very high power density which, even for small implants, may induce physiologically critical temperature increases for a small volume. In fact the analysis of these "hot spots" is important, because ruptures of stent struts are likely and a high power density can occur also for smaller implants. An additional investigation estimating the maximum possible power loss inside such "hot spots" and the resulting temperature maps around them is necessary to check the safety of active implants under these circumstances. This is being prepared for future publication.

The study protocols for the cited animal experiments were approved by the responsible authority (Landesamt für Arbeitsschutz, Gesundheitsschutz und technische Sicherheit, Berlin, G 0142/99).

## Declarations

### Acknowledgements

Many thanks to Peter van Leeuwen, a bilingual native English speaker who has reviewed and corrected the entire "German English" manuscript.

## Authors’ Affiliations

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