# Solving the Time- and Frequency-Multiplexed Problem of Constrained Radiofrequency Induced Hyperthermia

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{target}= 19 mm, frequency range: 500–2000 MHz) and for human brain models including brain tumors of various size (r

_{1}= 20 mm, r

_{2}= 30 mm, frequency range 100–1000 MHz) and locations (center, off-center, disjoint) demonstrate the applicability and capabilities of the proposed approach. Due to its high performance, the algorithm can solve typical clinical problems in a few seconds, making the presented approach ideally suited for interactive hyperthermia treatment planning, thermal dose and safety management, and the design, rapid evaluation, and comparison of RF applicator configurations.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Problem Statement

**Q**is a N × N positive-semidefinite (psd) power correlation matrix obtained by forming correlation integrals of the electric fields inside lossy tissues. The elements q

_{ij}of the matrix

**Q**are hence calculated via

**r**) the electrical conductivity; and

**E**(

_{i}**r**) represents the electrical field of the i-th source using a unit excitation. By choosing the volume of integration, matrices representing said volume can be formed such as a tumor volume or cubes containing a given amount of tissue mass (e.g., 1 g or 10 g) in order to calculate spatially averaged local SAR.

**Q**, constraint matrices

**S**

_{i}, and associated constraint limits c

_{i}, and neglecting the linear and constant terms that do not arise in the problem at hand, a QCQP in general can be stated as

**Q**and

**S**

_{i}are psd, the problem is convex and can be readily solved. While the matrices involved in RF heating applications are indeed psd, we did not seek to minimize RF power deposition but to maximize it. Accordingly,

**Q**in Equation (3) is replaced by the negative semidefinite matrix −

**Q**, which renders the problem non-convex and thus hard to solve, with the general QCQP being in the class of non-deterministic polynomial-time (NP)-hard problems [42].

#### 2.2. Semidefinite Relaxation

**X**is formed as the outer product of the excitation vector x with its Hermitian transpose x

^{H}. The optimization problem now reads

^{3}[46,47], SDPA [48,49] and CSDP [50]. Commercial solvers such as MOSEK [51] can offer significantly reduced computation times for large problems. Using these different solvers is conveniently facilitated by high-level modeling interfaces such as YALMIP [52] or CVX [53].

**X**is 1, the solution to the relaxed problem is also the solution to the original problem. x can then be retrieved from

**X**using an eigendecomposition that will only yield one non-zero eigenvalue λ and its associated eigenvector v. The desired solution vector is then given by $x=\sqrt{\lambda}v$.

#### 2.3. Time-Multiplexed Radiofrequency (RF) Heating

**Y**is the sum of the outer products of all excitation vectors with their Hermitian transpose:

**Y**can be made. First,

**Y**is psd by construction. Second, depending on the number of excitation vectors m and whether the individual x

_{k}are linearly dependent, the inequality 1 ≤ rank(

**Y**) ≤ N always holds true, as a Hermitian matrix of dimension N can be of rank N at most. This means, that for any number of time-multiplexed arbitrary excitation vectors, we can find at most N alternative excitation vectors u

_{k}, resulting in an identical power deposition. With u

_{k}being the orthonormal eigenvectors of

**Y**and λ

_{k}the associated eigenvalues,

**Y**can be written as a weighted sum of outer products from its eigenvectors:

- A rank >1 solution to the semidefinite relaxation corresponds to the time-multiplexed excitation scenario.
- The individual excitation vectors for the time-multiplexed application can be retrieved using the eigendecomposition of
**X**. - Any arbitrary number of excitation vectors can be effectively compressed to at most N vectors.

#### 2.4. Arbitrary Heating Patterns

**Q**

_{i}are the power correlation or SAR matrices of the target regions. Additionally, we defined a target power vector r, whose entries represent the desired local power deposition and a diagonal matrix

**W**, where the (i,i) element contains a weighting factor between 0 and 1, which ranks the importance of the ith target region. The optimization problem can now be stated as a constrained norm minimization:

#### 2.5. Frequency-Multiplexed RF Heating

_{F}is defined as

**X**

_{f}, and

**S**

_{i,f}representing the ith constraint matrix at frequency f.

#### 2.6. Iterative Solution

- EMF simulation of the RF applicator with an appropriate model of the object under investigation for the desired frequencies.
- Calculation of appropriately averaged SAR matrices for regions targeted for RF heating and for regions outside the target region for all frequencies.
- Solution of the optimization problem using the calculated SAR and target matrices.
- Retrieval of the individual excitation vectors for each frequency.

^{5}–10

^{7}distinct matrices for each individual frequency. Solving an optimization problem of this magnitude is out of reach for readily available computing workstations and available semidefinite programming solvers. Different approaches have been suggested to tackle this such as dedicated algorithms incorporating highly parallel SAR calculation algorithms [56,58,59], or compression algorithms to reduce the number of constraint matrices to a smaller set of virtual observation points (VOPs) [60,61,62]. Notwithstanding this progress, the former is incompatible with readily available semidefinite programming solvers, while the latter suffers from a significant one-time computational burden to calculate the compressed matrix set, which will be further exacerbated once multiple frequencies have to be considered. Additionally, the intrinsic overestimation of the VOPs will produce non-optimal results.

#### 2.7. Retrieval of Excitation Vectors

**X**

_{f}.

- Perform an eigendecomposition of each matrix
**X**_{f}, each yielding N eigenvectors**v**_{k}and their associated eigenvalues λ_{k}. Very often, the solutions will be strongly rank-deficient, having only a few large eigenvalues. Each individual excitation vector is given by ${\mathit{v}}_{k}\sqrt{{\lambda}_{k}}$. - Compute the local SAR distribution for each of the N · F excitations and evaluate their respective influence on the target region (e.g., by calculating their maximum and mean SAR inside the target region).
- Discard all excitations that do not significantly contribute to the solution (e.g., all excitations whose maximum SAR contribution to the target region falls below a certain threshold). In our examples, we chose to discard all vectors contributing less than 0.1% to the overall solution.
- Scale the remaining vectors for time-multiplexing. If M solutions belonging to the same frequency remain, this indicates that time-multiplexing is required (i.e., the excitations are played out in succession during the application and each solution vector needs to be scaled by $\sqrt{M}$. Excitations at different frequencies do not interact coherently and can in principle be played out concurrently (i.e., their SAR patterns are purely additive). If the different frequency solutions are also applied in a time-multiplexed fashion, a similar scaling needs to be performed.

#### 2.8. Implementation and Validation

_{1}= 3 cm, r

_{2}= 2 cm) were embedded at off-center locations. The simulated setup is shown in Figure 3.

^{3}spherical averaging kernel [68] as an approximation to spatially averaged 10 g SAR [19]. Voxels containing more than 10% air after the averaging procedure were discarded.

## 3. Results

#### 3.1. Phantom Setup Using a 32-Channel 2D Applicator

#### 3.2. Human Brain Model Setup Using a 40-Channel 3D Helmet Grid Applicator

#### 3.3. Runtimes of Demonstration Examples

#### 3.4. Iterative vs. Non-Iterative Approach

#### 3.5. Dependence on Frequency and Channel Number

^{3}− N

^{4}and shows an almost linear dependence on the number of frequencies F. Due to the heuristic nature of the iterative solver, the performance graph is not entirely monotonic. Reducing the number of target points by a factor of 8 decreased the runtimes by 30% on average.

#### 3.6. Comparison of Multiplexed Vector Field Shaping (MVFS) to Focused Constrained Power Optimization (FOCO)

_{∞}150”) achieves the same mean SAR as FOCO, but raised the minimum value inside the target by 55% and increased TC50 from 0.87 to 0.95. All MVFS results differed over the given metrics, which would allow for weighing different treatment aspects and choosing the optimum result for the given treatment scenario. As demonstrated in Figure 8, computation times for current clinical channel numbers were shown to be 2.5 s for 20 channels and 0.7 s for 12 channels. These short runtimes conveniently allow solving for multiple target patterns to iteratively approach a power deposition pattern that is expected to perform best.

## 4. Discussion

**X**(i.e., its 2-norm), which comprises a convex constraint and is thus valid within the proposed framework. While this does not limit individual RF channel power, it nevertheless constrains total forward power over all channels during a single excitation. Additionally, since local SAR itself acts as a dampening factor preventing a single channel to be overly dominant, we do not expect overly unrealistic solutions to arise in other setups [77,78].

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Lo, Y.; Lee, S.W. Antenna Handbook; Lo, Y.T., Lee, S.W., Eds.; Springer: Boston, MA, USA, 1988; ISBN 978-1-4615-6461-4. [Google Scholar]
- Lin, J.C. Electromagnetic Fields in Biological Systems; Lin, J.C., Ed.; CRC Press: Boca Raton, FL, USA, 2016; ISBN 9780429106958. [Google Scholar]
- Brace, C.L. Microwave Tissue Ablation: Biophysics, Technology, and Applications. Crit. Rev. Biomed. Eng.
**2010**, 38, 65–78. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Brace, C. Thermal Tumor Ablation in Clinical Use. IEEE Pulse
**2011**, 2, 28–38. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Besse, H.; Barten-van Rijbroek, A.; van der Wurff-Jacobs, K.; Bos, C.; Moonen, C.; Deckers, R. Tumor Drug Distribution after Local Drug Delivery by Hyperthermia, In Vivo. Cancers
**2019**, 11, 1512. [Google Scholar] [CrossRef] [Green Version] - Oldenborg, S.; van Os, R.; Oei, B.; Poortmans, P. Impact of Technique and Schedule of Reirradiation Plus Hyperthermia on Outcome after Surgery for Patients with Recurrent Breast Cancer. Cancers
**2019**, 11, 782. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mei, X.; ten Cate, R.; van Leeuwen, C.M.; Rodermond, H.M.; de Leeuw, L.; Dimitrakopoulou, D.; Stalpers, L.J.A.; Crezee, J.; Kok, H.P.; Franken, N.A.P.; et al. Radiosensitization by Hyperthermia: The Effects of Temperature, Sequence, and Time Interval in Cervical Cell Lines. Cancers
**2020**, 12, 582. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Moroz, P.; Jones, S.K.; Gray, B.N. Status of hyperthermia in the treatment of advanced liver cancer. J. Surg. Oncol.
**2001**, 77, 259–269. [Google Scholar] [CrossRef] - Wust, P.; Hildebrandt, B.; Sreenivasa, G.; Rau, B.; Gellermann, J.; Riess, H.; Felix, R.; Schlag, P. Hyperthermia in combined treatment of cancer. Lancet Oncol.
**2002**, 3, 487–497. [Google Scholar] [CrossRef] - Paulides, M.M.; Bakker, J.F.; Neufeld, E.; Zee, J.; van der Jansen, P.P.; Levendag, P.C.; van Rhoon, G.C. The HYPERcollar: A novel applicator for hyperthermia in the head and neck. Int. J. Hyperth.
**2007**, 23, 567–576. [Google Scholar] [CrossRef] - Crezee, J.; Van Haaren, P.M.A.; Westendorp, H.; De Greef, M.; Kok, H.P.; Wiersma, J.; Van Stam, G.; Sijbrands, J.; Zum Vörde Sive Vörding, P.; Van Dijk, J.D.P.; et al. Improving locoregional hyperthermia delivery using the 3-D controlled AMC-8 phased array hyperthermia system: A preclinical study. Int. J. Hyperth.
**2009**, 25, 581–592. [Google Scholar] [CrossRef] - Gellermann, J.; Wlodarczyk, W.; Feussner, A.; Fähling, H.; Nadobny, J.; Hildebrandt, B.; Felix, R.; Wust, P. Methods and potentials of magnetic resonance imaging for monitoring radiofrequency hyperthermia in a hybrid system. Int. J. Hyperth.
**2005**, 21, 497–513. [Google Scholar] [CrossRef] - Guérin, B.; Villena, J.F.; Polimeridis, A.G.; Adalsteinsson, E.; Daniel, L.; White, J.K.; Rosen, B.R.; Wald, L.L. Computation of ultimate SAR amplification factors for radiofrequency hyperthermia in non-uniform body models: Impact of frequency and tumour location. Int. J. Hyperth.
**2017**, 34, 1–14. [Google Scholar] [CrossRef] [PubMed] - Winter, L.; Niendorf, T. Electrodynamics and radiofrequency antenna concepts for human magnetic resonance at 23.5 T (1 GHz) and beyond. Magn. Reson. Mater. Phys. Biol. Med.
**2016**, 29, 641–656. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Winter, L.; Oezerdem, C.; Hoffmann, W.; van de Lindt, T.; Periquito, J.; Ji, Y.; Ghadjar, P.; Budach, V.; Wust, P.; Niendorf, T. Thermal magnetic resonance: Physics considerations and electromagnetic field simulations up to 23.5 Tesla (1GHz). Radiat. Oncol.
**2015**, 10, 201. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Oberacker, E.; Kuehne, A.; Nadobny, J.; Zschaeck, S.; Weihrauch, M.; Waiczies, H.; Ghadjar, P.; Wust, P.; Niendorf, T.; Winter, L. Radiofrequency applicator concepts for simultaneous MR imaging and hyperthermia treatment of glioblastoma multiforme. Curr. Dir. Biomed. Eng.
**2017**, 3, 473–477. [Google Scholar] [CrossRef] [Green Version] - Trefná, H.D.; Vrba, J.; Persson, M. Time-reversal focusing in microwave hyperthermia for deep-seated tumors. Phys. Med. Biol.
**2010**, 55, 2167–2185. [Google Scholar] [CrossRef] - Takook, P.; Trefna, H.D.; Zeng, X.; Fhager, A.; Persson, M. A computational study using time reversal focusing for hyperthermia treatment planning. Prog. Electromagn. Res. B
**2017**, 73, 117–130. [Google Scholar] [CrossRef] [Green Version] - Rijnen, Z.; Bakker, J.F.; Canters, R.A.M.; Togni, P.; Verduijn, G.M.; Levendag, P.C.; Van Rhoon, G.C.; Paulides, M.M. Clinical integration of software tool VEDO for adaptive and quantitative application of phased array hyperthermia in the head and neck. Int. J. Hyperth.
**2013**, 29, 181–193. [Google Scholar] [CrossRef] - Bardati, F.; Borrani, A.; Gerardino, A.; Lovisolo, G.A. SAR optimization in a phased array radiofrequency hyperthermia system. IEEE Trans. Biomed. Eng.
**1995**, 42, 1201–1207. [Google Scholar] [CrossRef] - Böhm, M.; Louis, A. Efficient algorithm for computing optimal control of antennas in hyperthermia. Surv. Math. Indust
**1993**, 3, 233–251. [Google Scholar] - Mestrom, R.M.C.; van Engelen, J.P.; van Beurden, M.C.; Paulides, M.M.; Numan, W.C.M.; Tijhuis, A.G. A Refined Eigenvalue-Based Optimization Technique for Hyperthermia Treatment Planning. In Proceedings of the The 8th European Conference on Antennas and Propagation (EuCAP 2014), The Hague, The Netherlands, 6–11 April 2014; IEEE: Piscataway, NJ, USA, 2014; pp. 2010–2013. [Google Scholar]
- Köhler, T.; Maass, P.; Wust, P.; Seebass, M. A fast algorithm to find optimal controls of multiantenna applicators in regional hyperthermia. Phys. Med. Biol.
**2001**, 46, 2503–2514. [Google Scholar] [CrossRef] - Cappiello, G.; Drizdal, T.; Mc Ginley, B.; O’Halloran, M.; Glavin, M.; Van Rhoon, G.C.; Jones, E.; Paulides, M.M. The potential of time-multiplexed steering in phased array microwave hyperthermia for head and neck cancer treatment. Phys. Med. Biol.
**2018**, 63, 135023. [Google Scholar] [CrossRef] [PubMed] - Zastrow, E.; Hagness, S.C.; Van Veen, B.D.; Medow, J.E. Time-Multiplexed Beamforming for Noninvasive Microwave Hyperthermia Treatment. IEEE Trans. Biomed. Eng.
**2011**, 58, 1574–1584. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kok, H.P.; Van Stam, G.; Bel, A.; Crezee, J. A Mixed Frequency Approach to Optimize Locoregional RF Hyperthermia. In Proceedings of the 2015 European Microwave Conference (EuMC), Paris, France, 7–10 September 2015; pp. 773–776. [Google Scholar]
- Converse, M.; Bond, E.J.; Van Veen, B.D.; Hagness, S.C. A computational study of ultra-wideband versus narrowband microwave hyperthermia for breast cancer treatment. IEEE Trans. Microw. Theory Tech.
**2006**, 54, 2169–2180. [Google Scholar] [CrossRef] [Green Version] - Bellizzi, G.G.; Crocco, L.; Battaglia, G.M.; Isernia, T. Multi-frequency constrained SAR focusing for patient specific hyperthermia treatment. IEEE J. Electromagn. RF Microwaves Med. Biol.
**2017**, 1, 74–80. [Google Scholar] [CrossRef] - Takook, P.; Persson, M.; Gellermann, J.; Trefná, H.D. Compact self-grounded Bow-Tie antenna design for an UWB phased-array hyperthermia applicator. Int. J. Hyperth.
**2017**, 33, 387–400. [Google Scholar] [CrossRef] - Jacobsen, S.; Melandsø, F. The Concept of Using Multifrequency Energy Transmission to Reduce Hot Spots during Deep-Body Hyperthermia. Ann. Biomed. Eng.
**2002**, 30, 34–43. [Google Scholar] [CrossRef] - Bucci, O.M.; D’elia, G.; Mazzarella, G.; Panariello, G. Antenna Pattern Synthesis: A New General Approach. Proc. IEEE
**1994**, 82, 358–371. [Google Scholar] [CrossRef] - Liontas, C.A.; Knott, P. An Alternating Projections Algorithm for Optimizing Electromagnetic Fields in Regional Hyperthermia. In Proceedings of the 2016 10th European Conference on Antennas and Propagation (EuCAP), Davos, Switzerland, 10–15 April 2016; IEEE: Piscataway, NJ, USA, 2016; pp. 1–5. [Google Scholar]
- Liontas, C.A. Alternating Projections of Auxiliary Vector Fields for Electric Field Optimization in Temperature-guided Hyperthermia. In Proceedings of the 2019 13th European Conference on Antennas and Propagation (EuCAP), Krakow, Poland, 31 March–5 April 2019; pp. 1–5. [Google Scholar]
- Bellizzi, G.G.; Drizdal, T.; Van Rhoon, G.C.; Crocco, L.; Isernia, T.; Paulides, M.M. The potential of constrained SAR focusing for hyperthermia treatment planning: Analysis for the head & neck region. Phys. Med. Biol.
**2019**, 64, 1. [Google Scholar] - Iero, D.; Isernia, T.; Morabito, A.F.; Catapano, I.; Crocco, L. Optimal constrained field focusing for hyperthermia cancer therapy: A feasibility assessment on realistic phantoms. Prog. Electromagn. Res.
**2010**, 102, 125–141. [Google Scholar] [CrossRef] [Green Version] - Iero, D.A.M.; Crocco, L.; Isernia, T. Thermal and microwave constrained focusing for patient-specific breast cancer hyperthermia: A robustness assessment. IEEE Trans. Antennas Propag.
**2014**, 62, 814–821. [Google Scholar] [CrossRef] - Bellizzi, G.G.; Battaglia, G.M.; Bevacqua, M.T.; Crocco, L.; Isernia, T. FOCO: A Novel Versatile Tool in Hyperthermia Treatment Planning. In Proceedings of the 2019 13th European Conference on Antennas and Propagation (EuCAP), Krakow, Poland, 31 March–5 April 2019; pp. 1–4. [Google Scholar]
- Kuehne, A.; Goluch, S.; Waxmann, P.; Seifert, F.; Ittermann, B.; Moser, E.; Laistler, E. Power balance and loss mechanism analysis in RF transmit coil arrays. Magn. Reson. Med.
**2015**, 74, 1165–1176. [Google Scholar] [CrossRef] [PubMed] - Pennes, H.H. Analysis of Tissue and Arterial Blood Temperatures in the Resting Human Forearm. J. Appl. Physiol.
**1948**, 1, 93–122. [Google Scholar] [CrossRef] [PubMed] - Boulant, N.; Wu, X.; Adriany, G.; Schmitter, S.; Uʇurbil, K.; Van De Moortele, P.F. Direct control of the temperature rise in parallel transmission by means of temperature virtual observation points: Simulations at 10.5 tesla. Magn. Reson. Med.
**2016**, 75, 249–256. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kowalski, M.; Behnia, B.; Webb, A.G.; Jin, J.M. Optimization of electromagnetic phased-arrays for hyperthermia via magnetic resonance temperature estimation. IEEE Trans. Biomed. Eng.
**2002**, 49, 1229–1241. [Google Scholar] [CrossRef] - Luo, Z.; Ma, W.; So, A.; Ye, Y.; Zhang, S. Semidefinite Relaxation of Quadratic Optimization Problems. IEEE Signal Process. Mag.
**2010**, 27, 20–34. [Google Scholar] [CrossRef] - Bellizzi, G.G.; Iero, D.A.M.; Crocco, L.; Isernia, T. Three-Dimensional Field Intensity Shaping: The Scalar Case. IEEE Antennas Wirel. Propag. Lett.
**2018**, 17, 360–363. [Google Scholar] [CrossRef] - Vandenberghe, L.; Boyd, S. Semidefinite Programming. SIAM Rev.
**1996**, 38, 49–95. [Google Scholar] [CrossRef] [Green Version] - Sturm, J.F. Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw.
**1999**, 11, 625–653. [Google Scholar] [CrossRef] - Toh, K.C.; Todd, M.J.; Tütüncü, R.H. SDPT3—A Matlab software package for semidefinite programming, Version 1.3. Optim. Methods Softw.
**1999**, 11, 545–581. [Google Scholar] [CrossRef] - Tütüncü, R.H.; Toh, K.C.; Todd, M.J. Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program.
**2003**, 95, 189–217. [Google Scholar] [CrossRef] - Yamashita, M.; Fujisawa, K.; Kojima, M. Implementation and evaluation of SDPA 6.0 (Semidefinite Programming Algorithm 6.0). Optim. Methods Softw.
**2003**, 18, 491–505. [Google Scholar] [CrossRef] - Yamashita, M.; Fujisawa, K.; Fukuda, M.; Kobayashi, K.; Nakata, K.; Nakata, M. Handbook on Semidefinite, Conic and Polynomial Optimization; Springer: Boston, MA, USA, 2012; pp. 687–713. [Google Scholar]
- Borchers, B. CSDP, A C library for semidefinite programming. Optim. Methods Softw.
**1999**, 11, 613–623. [Google Scholar] [CrossRef] - Mosep ApS, The MOSEK Optimization Toolbox for MATLAB Manual, Version 9.2.; Mosep ApS: Copenhagen, Denmark, 2019.
- Löfberg, J. YALMIP: A Toolbox for Modeling and Optimization in MATLAB. In Proceedings of the CACSD Conference, New Orleans, LA, USA, 2–4 September 2004. [Google Scholar]
- CVX Research, Inc. CVX: Matlab Software for Disciplined Convex Programming. version 2.0. 2012. [Google Scholar]
- Graesslin, I.; Homann, H.; Biederer, S.; Börnert, P.; Nehrke, K.; Vernickel, P.; Mens, G.; Harvey, P.; Katscher, U. A specific absorption rate prediction concept for parallel transmission MR. Magn. Reson. Med.
**2012**, 68, 1664–1674. [Google Scholar] [CrossRef] [PubMed] - Gebhardt, M. Method and High-Frequency Check Device for Checking a High-Frequency Transmit Device of a Magnetic Resonance Tomography System. U.S. Patent 9,229,072, 2012. [Google Scholar]
- Pendse, M.; Rutt, B.K. A Vectorized Formalism for Efficient SAR Computation in Parallel Transmission. In Proceedings of the ISMRM, Toronto, ON, Canada, 30 May–5 June 2015; p. 0379. [Google Scholar]
- Wolpert, D.H.; Macready, W.G. No free lunch theorems for optimization. IEEE Trans. Evol. Comput.
**1997**, 1, 67–82. [Google Scholar] [CrossRef] [Green Version] - Pendse, M.; Rutt, B.K. Method and Apparatus for SAR Focusing with an Array of RF Transmitters. U.S. Patent 10,088,537 B2, 2 October 2018. [Google Scholar]
- Pendse, M.; Stara, R.; Mehdi Khalighi, M.; Rutt, B. IMPULSE: A scalable algorithm for design of minimum specific absorption rate parallel transmit RF pulses. Magn. Reson. Med.
**2018**, 1–15. [Google Scholar] [CrossRef] - Eichfelder, G.; Gebhardt, M. Local specific absorption rate control for parallel transmission by virtual observation points. Magn. Reson. Med.
**2011**, 66, 1468–1476. [Google Scholar] [CrossRef] - Lee, J.; Gebhardt, M.; Wald, L.L.; Adalsteinsson, E. Local SAR in parallel transmission pulse design. Magn. Reson. Med.
**2012**, 67, 1566–1578. [Google Scholar] [CrossRef] [Green Version] - Kuehne, A.; Waiczies, H.; Niendorf, T. Massively accelerated VOP compression for population-scale RF safety models. In Proceedings of the ISMRM, Honolulu, HI, USA, 2017; p. 0478. [Google Scholar]
- Volken, W.; Frei, D.; Manser, P.; Mini, R.; Born, E.J.; Fix, M.K. An integral conservative gridding--algorithm using Hermitian curve interpolation. Phys. Med. Biol.
**2008**, 53, 6245–6263. [Google Scholar] [CrossRef] - Iacono, M.I.; Neufeld, E.; Akinnagbe, E.; Bower, K.; Wolf, J.; Oikonomidis, I.V.; Sharma, D.; Lloyd, B.; Wilm, B.J.; Wyss, M.; et al. MIDA: A multimodal imaging-based detailed anatomical model of the human head and neck. PLoS ONE
**2015**, 10, e0124126. [Google Scholar] [CrossRef] [PubMed] - Paska, J.; Raya, J.; Cloos, M.A. Fast and Intuitive RF-Coil Optimization Pipeline Using a Mesh-Structure. In Proceedings of the ISMRM, Montréal, QC, Canada, 11–16 May 2019; p. 0571. [Google Scholar]
- Hasgall, P.; Neufeld, E.; Gosselin, M.; Klingenböck, A.; Kuster, N. IT’IS Database for Thermal and Electromagnetic Parameters of Biological Tissues, Version 4.0. Available online: www.itis.ethz.ch/database (accessed on 4 March 2020).
- Lu, Y.; Li, B.; Xu, J.; Yu, J. Dielectric properties of human glioma and surrounding tissue. Int. J. Hyperth.
**1992**, 8, 755–760. [Google Scholar] [CrossRef] [PubMed] - Kuehne, A.; Seifert, F.; Ittermann, B. GPU-accelerated SAR computation with arbitrary averaging shapes. In Proceedings of the ISMRM, Melbourne, Australia, 5–11 May 2012; p. 4260. [Google Scholar]
- Lee, H.K.; Antell, A.G.; Perez, C.A.; Straube, W.L.; Ramachandran, G.; Myerson, R.J.; Emami, B.; Molmenti, E.P.; Buckner, A.; Lockett, M.A. Superficial hyperthermia and irradiation for recurrent breast carcinoma of the chest wall: Prognostic factors in 196 tumors. Int. J. Radiat. Oncol.
**1998**, 40, 365–375. [Google Scholar] [CrossRef] - Myerson, R.J.; Perez, C.A.; Emami, B.; Straube, W.; Kuske, R.R.; Leybovich, L.; Von Gerichten, D. Tumor control in long-term survivors following superficial hyperthermia. Int. J. Radiat. Oncol.
**1990**, 18, 1123–1129. [Google Scholar] [CrossRef] - Canters, R.A.M.; Wust, P.; Bakker, J.F.; Van Rhoon, G.C. A literature survey on indicators for characterisation and optimisation of SAR distributions in deep hyperthermia, a plea for standardisation. Int. J. Hyperth.
**2009**, 25, 593–608. [Google Scholar] [CrossRef] - Bellizzi, G.G.; Drizdal, T.; van Rhoon, G.C.; Crocco, L.; Isernia, T.; Paulides, M.M. Predictive value of SAR based quality indicators for head and neck hyperthermia treatment quality. Int. J. Hyperth.
**2019**, 36, 456–465. [Google Scholar] [CrossRef] [Green Version] - Grissom, W.A.; Xu, D.; Kerr, A.B.; Fessler, J.A.; Noll, D.C. Fast Large-Tip-Angle Multidimensional and Parallel RF Pulse Design in MRI. IEEE Trans. Med. Imaging
**2009**, 28, 1548–1559. [Google Scholar] [CrossRef] [Green Version] - Eryaman, Y.; Akin, B.; Atalar, E. Reduction of implant RF heating through modification of transmit coil electric field. Magn. Reson. Med.
**2011**, 65, 1305–1313. [Google Scholar] [CrossRef] - Eryaman, Y.; Turk, E.A.; Oto, C.; Algin, O.; Atalar, E. Reduction of the radiofrequency heating of metallic devices using a dual-drive birdcage coil. Magn. Reson. Med.
**2013**, 69, 845–852. [Google Scholar] [CrossRef] - Guerin, B.; Angelone, L.M.; Dougherty, D.; Wald, L.L. Parallel transmission to reduce absorbed power around deep brain stimulation devices in MRI: Impact of number and arrangement of transmit channels. Magn. Reson. Med.
**2020**, 83, 299–311. [Google Scholar] [CrossRef] - Oberacker, E.; Kuehne, A.; Oezerdem, C.; Millward, J.M.; Diesch, C.; Eigentler, T.W.; Zschaeck, S.; Ghadjar, P.; Wust, P.; Winter, L.; et al. Power Considerations for Radiofrequency Applicator Concepts for Thermal Magnetic Resonance Interventions in the Brain at 297 MHz. In Proceedings of the ISMRM, Montréal, QC, Canada, 11–16 May 2019; p. 3828. [Google Scholar]
- Oberacker, E.; Kuehne, A.; Waiczies, H.; Nadobny, J.; Zschaeck, S.; Ghadjar, P.; Wust, P.; Niendorf, T.; Winter, L. Radiofrequency Applicator Concepts for Simultaneous MR Imaging and Hyperthermia Treatment of Glioblastoma Multiforme: A 7.0 T (298 MHz) Study. In Proceedings of the ISMRM, Honolulu, HI, USA, 22–27 April 2017; p. 2604. [Google Scholar]

**Figure 1.**Flowchart of the iterative solution algorithm. The algorithm initially selects a small subset of all healthy (=constraint) voxels to be considered during the targeted heating calculation. After performing the optimization, the resultant specific absorption rate (SAR) in the unconsidered voxels is calculated to find regions where the found solution violates the constraints (i.e., leads to undesired heating in healthy regions). A small number of the healthy voxels experiencing the strongest heating are added to the constraint subset and the optimization is repeated. This process is iterated until no further constraints are violated by the solution. The last step in the dashed outline is optional and only required if the number of constraints has increased to a level that significantly impacts each iterative solution runtime.

**Figure 2.**Depiction of the simulated 2D setup using a pure water phantom. A plane wave with E

_{Z}-polarization incident on a 2 cm aperture in a perfectly conducting shield is used as an excitation source to mimic a localized broadband radiofrequency applicator. In consecutive simulations, the aperture was rotated around the sample in 32 steps to provide completely circumscribing field sources. An exemplary E-Field plot is shown on the right.

**Figure 3.**(

**a**) Synthetic 40-channel helmet grid applicator for 3D evaluation of the proposed algorithm. The conductive paths conform to the contours of the head and can generate electromagnetic fields with varying polarizations. Simulated power sources are marked as red arrows. The RF applicator is shielded by a continuous conformal perfectly conducting shield positioned 2 cm away from the conductors (grey shaded area). (

**b**) Sagittal and (

**c**) coronal view of a spherical tumor (radius 2 cm) that was incorporated into the brain of the human voxel model and positioned at the center of the applicator (light green shaded area). (

**d**) and (

**e**) show oblique slices through the center of the second tumor model using two differently sized tumors (radius 2 and 3 cm).

**Figure 4.**Demonstration of shaped RF power deposition in the 2D water sample. The target region is delineated with a cyan contour (r = 19 mm), and the region between the cyan and magenta contour indicates a “safety margin” where no constraints are enforced. (

**a**) Optimization result using only 500 MHz sources minimizing the 2-norm. The large plot on the left details the total achieved local SAR deposition, with statistical measures of SAR within the target region detailed in the header (Mean ± SD (Min–Max)). The smaller plots to the right show the contributions of multiple time-multiplexed modes scaled to their respective maximum, with their respective peak contribution inside the target region shown in the header. In (

**b**), all frequencies between 500 and 1500 MHz could be used, resulting in an improved power deposition pattern. The green arrows indicate a region where SAR in the constrained regions is spread out between the different frequencies, which demonstrates a key principle of the proposed algorithm. For (

**c**), the optimization was set to minimize the worst-case deviation from the target. This resulted in a much more homogeneous power deposition, albeit with lower mean and peak values. Again, all excitations contributed to the target region but occupied complementary regions outside it as indicated by arrows. The image succession in (

**d**) shows the cumulative effects of the first five excitations from (

**c**) to demonstrate how the different patterns build up the final superposition. Here, the resulting mean value is given in the header.

**Figure 5.**Excitation of a complex disjoint shape inside the water phantom (the logo of the first author’s affiliation). The top row shows the target pattern with 60 and 80 W/kg regions (

**a**), constraints of 40 W/kg (

**b**), and weighting distribution (

**c**) used for the optimization. The achieved heating pattern is shown in (

**d**), which requires time- and frequency-multiplexed excitation using 43 different modes between 500 and 2000 MHz. The first nine modes with the strongest peak impact are shown in (

**e**). Measures of SAR within the target region are detailed in the header (Mean ± SD (Min–Max)).

**Figure 6.**Results of the 3D heating example using the human head model. The target is again outlined in cyan and target SAR was set to 150 W/kg, minimizing the 2-norm of the deviation from this value. Measures of SAR within the target region are detailed in the header (Mean ± SD (Min–Max)), along with TC25, TC50, and TC80 values for the tumor. Images (

**a**) and (

**g**) show sagittal slices through the tumor center, whereas the other SAR plots represent axial maximum intensity projections over the whole volume. Images (

**a**) and (

**b**) show the achieved power deposition pattern when using only 600 MHz fields. This pattern is achieved by two time-multiplexed modes, whose patterns are shown scaled to their individual maximum in (

**c**) and (

**d**) analogous to the previous figures. These two modes correspond to two counterrotating circularly polarized electric fields inside the tumor. E-field vector snapshots and the rotation direction are shown in (

**e**) and (

**f**). This example demonstrates that the algorithm can arbitrarily mix polarizations within a single excitation (a circular polarization being comprised of two linear components) as well as yield differently polarized time-multiplexed complementary solutions. The second row of results in (

**g**–

**l**) shows the optimum result when allowing the use of all frequencies between 100 and 1000 MHz. From a target coverage standpoint, this solution performed only slightly better, with an identical mean but modestly lowered maximum, elevated minimum, and lower standard deviation.

**Figure 7.**Results for the differently weighted two-tumor optimization. The leftmost image column displays axial slices through the target center; all other images are axial maximum-intensity projections with the results for individual contributing frequencies and modes being scaled to their individual maximum that is stated above them along with the frequency. Measures of SAR within the target region are detailed in the header (Mean ± SD (Min–Max)) along with TC25, TC50, and TC80 values for the tumor. Targeting each tumor separately leads to SAR patterns shown in (

**a**) and (

**b**). From (

**c**) to (

**e**), the weight of the smaller tumor increased from an equal to threefold weighting. Due to their different sizes, the target voxels belonging to the smaller tumor require a higher relative weighting for an approximately equal SAR deposition.

**Figure 8.**Runtime dependence on the number of available RF channels and frequencies used during the optimization. Since performance varied from the sub-second range up to about three minutes for the most complex scenario, a logarithmic scale was used. All calculations were performed by the iterative algorithm using the single-tumor model with 257 target points and approximately 30,000 total constraint voxels. For typical clinical problem sizes (single frequency, maximum of 20 channels during the planning stage [34]), the optimization times stayed below 2.5 s. The largest problem with all 40 channels and 10 frequencies was completed in 3 min and 24 s.

Example Figure # | # of Channels | # of Target Points | # of Frequencies | # of Constraint Voxels | % of Used Constraint Voxels | # of Iterations | Computation Time [hh:mm:ss] |
---|---|---|---|---|---|---|---|

4 (a) | 32 | 149 | 1 | 4952 | 17.4 | 25 | 00:00:52 |

4 (b) | 32 | 149 | 10 | 4952 | 15.3 | 10 | 00:03:21 |

4 (c) | 32 | 149 | 10 | 4952 | 13.9 | 16 | 00:04:42 |

5 | 32 | 8693 | 16 | 22,289 | 1.7 | 19 | 02:36:18 |

6 (a–f) | 40 | 257 | 1 | 29,914 | 0.7 | 9 | 00:00:22 |

6 (g–l) | 40 | 257 | 10 | 29,914 | 0.8 | 10 | 00:03:24 |

7 (a) | 40 | 247 | 10 | 29,908 | 0.6 | 8 | 00:03:10 |

7 (b) | 40 | 67 | 10 | 29,908 | 0.4 | 9 | 00:02:52 |

7 (c) | 40 | 314 | 10 | 29,908 | 0.6 | 8 | 00:03:04 |

7 (d) | 40 | 314 | 10 | 29,908 | 0.5 | 9 | 00:03:16 |

7 (e) | 40 | 314 | 10 | 29,908 | 0.6 | 8 | 00:03:03 |

**Table 2.**Comparison of multiplexed vector field shaping (MVFS) to focused constrained power optimization (FOCO) for the single tumor model using 40 channels and single frequency fields at 600 MHz. Results are separated into SAR statistics, tumor coverage, and solution details. FOCO was performed using a rank-1 approximation of the central tumor SAR matrix and compared to six different MVFS application scenarios. “Center” uses the full-rank central SAR matrix, “Averaged” utilizes a single target matrix built from averaging over the whole tumor volume. The remaining four examples used all 257 tumor SAR matrices to derive a field shaping (“S”) result. Here, the subscript defines the target norm used (either 2 or ∞) and the following number stands for the target SAR in W/kg (either 75 or 150 W/kg). The “Rank” row describes how many time-interleaved solutions were identified for the respective solution. As expected, the FOCO solution was of rank 1 while MVFS provided multiple excitations to better cover the target volume. The results for S

_{2}150 are visualized in Figure 6a–f.

Performance | MVFS | |||||||
---|---|---|---|---|---|---|---|---|

Metrics | FOCO | Center | Averaged | S_{2} 150 | S_{2} 75 | S_{∞} 150 | S_{∞} 75 | |

Local 10 g-SAR [W/kg] | Mean | 69 | 75 | 76 | 76 | 68 | 69 | 68 |

Max | 107 | 120 | 123 | 122 | 95 | 103 | 99 | |

Min | 33 | 41 | 43 | 43 | 45 | 51 | 51 | |

SD | 15 | 17 | 17 | 17 | 11 | 12 | 12 | |

Coverage | TC25 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

TC50 | 0.87 | 0.80 | 0.77 | 0.79 | 0.99 | 0.95 | 1.00 | |

TC80 | 0.13 | 0.14 | 0.12 | 0.12 | 0.23 | 0.16 | 0.18 | |

Solution | Time [s] | 16.5 | 14 | 22.2 | 22.8 | 21.7 | 24 | 26 |

Rank | 1 | 2 | 2 | 2 | 3 | 3 | 3 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kuehne, A.; Oberacker, E.; Waiczies, H.; Niendorf, T.
Solving the Time- and Frequency-Multiplexed Problem of Constrained Radiofrequency Induced Hyperthermia. *Cancers* **2020**, *12*, 1072.
https://doi.org/10.3390/cancers12051072

**AMA Style**

Kuehne A, Oberacker E, Waiczies H, Niendorf T.
Solving the Time- and Frequency-Multiplexed Problem of Constrained Radiofrequency Induced Hyperthermia. *Cancers*. 2020; 12(5):1072.
https://doi.org/10.3390/cancers12051072

**Chicago/Turabian Style**

Kuehne, Andre, Eva Oberacker, Helmar Waiczies, and Thoralf Niendorf.
2020. "Solving the Time- and Frequency-Multiplexed Problem of Constrained Radiofrequency Induced Hyperthermia" *Cancers* 12, no. 5: 1072.
https://doi.org/10.3390/cancers12051072