Bastien Guerin^{1,2}, Jorge F. Villena^{3}, Athanasios G. Polimeridis^{4}, Elfar Adalsteinsson^{5,6,7}, Luca Daniel^{5}, Jacob K. White^{5}, Bruce R. Rosen^{1,2,6}, and Lawrence L. Wald^{1,2,6}

^{1}A. A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Charlestown, MA, United States, ^{2}Harvard Medical School, Boston, MA, United States, ^{3}Cadence Design Systems, Feldkirchen, Germany, ^{4}Skolkovo Institute of Science and Technology, Moscow, Russian Federation, ^{5}Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, United States, ^{6}Harvard-MIT Division of Health Sciences Technology, Cambridge, MA, United States, ^{7}Institute for Medical Engineering and Science, Massachusetts Institute of Technology, Cambridge, MA, United States

### Synopsis

**We
propose a framework for the computation of the ultimate hyperthermia, which is
the best possible hyperthermia treatment for a given frequency and non-uniform
body model achievable by any multi-channel hyperthermia coil. We compute the
ultimate hyperthermia treatment of two shallow (close to skull) and deep (close
to ventricle) brain tumors in the realistic “Duke” body model and for treatment
frequencies ranging from 64 MHz to 600 MHz. We characterize the convergence to
the ultimate SAR pattern as well as temperature increase associated with the
ultimate SAR distribution in the presence of non-uniform perfusion effects.**### Target audience

RF
engineers, oncologists, hyperthermia physicists.

### Purpose

To
compute the best achievable RF hyperthermia treatment performance using sources
outside the head (“ultimate hyperthermia”) in realistic body models with
non-uniform electrical and thermal properties and non-uniform tissue perfusion.

### Methods

Electromagnetic basis
set: Our first step in computing the optimal external RF hyperthermia treatment is to obtain a
basis set for electromagnetic (EM) fields in a realistic non-uniform body model. We
have shown previously how to compute such a numerical basis [1] and used this
approach to compute the ultimate signal-to-noise ratio in MRI [2]. First, we place
a large number of dipoles (>105) around the body model at a
distance no less than D=3 cm (Fig. 1). Second, we compute the incident E and H
fields (i.e., in free space without the body model) created by random
excitations of the dipole cloud. This is done by (i) exciting all the dipoles
simultaneously using random excitations and (ii) computing the resulting E and
H fields using the free-space Maxwell Green function. Finally, we compute the
scattered E and H fields in the body model using the fast volume integral equation EM solver “MARIE” (https://github.com/thanospol/MARIE)
[3]. Although bases of the solutions of Maxwell equations are strictly speaking
infinite, we truncate the infinite basis to a finite number of basis vectors which converges to the ultimate with accuracy better than 1%. We model the Virtual
Family “Duke” body model (head only, the body was cut off below the neck) [4]
with a resolution of 3 mm isotropic (63×80×85 pixels), to which we added two
tumors (one shallow, i.e. close to the skull, and one deep in the brain close
to a ventricle). We compute EM basis sets in Duke at 64, 128, 297 and 600 MHz. Treatment
optimization: We optimize the weights assigned to each basis vector (equivalent
to voltage levels in a multi-channel hyperthermia coil) by maximization of the
average specific absorption rate (SAR) inside the tumor while enforcing that
the average SAR outside the tumor is below 3 W/kg. This is a convex constrained optimization
problem that we solve using Matlab’s function “fmincon” using analytical
expressions of the first and second derivatives of the objective and constraint
functions for robust convergence. Temperature simulation: We solve
Pennes bio-heat equation using the non-uniform thermal conductivity, heat
capacity and perfusion maps of “Duke” using a Crank-Nicholson finite-difference
time domain scheme with a time step computed using the Von Neuman stability criterion
(code available at www.martinos.org/~guerin)
[5]. Thermal parameter maps were obtained from the Gabriel database [6].
Perfusion maps were obtained from the Virtual Family database, except for tumors
which were each decomposed in their necrotic core (perfusion 10% that of grey
matter) and well-perfused penumbra (perfusion 200% that of grey matter) as
shown in Fig. 3.

### Results/Discussion

Fig. 2 shows that convergence of the hyperthermia treatment efficacy (“SAR
amplification ratio”) is achieved using ~500 basis vectors for all frequencies
and both tumor depths. Roughly 230 and 100 basis vectors are needed to achieve
80% of the ultimate for the shallow and deep tumors, respectively. This suggests
that, in practice, multi-channel hyperthermia coils with as many channels are
needed to achieve results similar to the ultimate. The treatment frequency seems
to affect the specificity of the ultimate treatment for the deep tumor only.
This is likely because complex E fields interferences are required to “focus”
the energy deposited in the deep tumor, which is more easily achieved at high
frequency (=small wavelength). We point out that it is likely that the optimal treatment
efficacy drops above a certain frequency however because of the shallower skin
depth with increasing frequency. Fig. 3 shows SAR maps of the ultimate treatment
at 600 MHz. Although the treatment energy is correctly focused in the tumor, there
is considerable “spillover” to neighboring tissues. Note that we constrained
the SAR outside the tumor to be below 3 W/kg, however the peak local SAR can –
and obviously is – much greater in healthy tissues. For the deep tumor, the
horn of the ventricle was significantly heated because of its proximity to the
tumor and the high electrical conductivity of CSF. This suggests that an
explicit SAR constraint corresponding to the ventricle volume should be added
to the optimization algorithm. Temperature simulations in Fig. 4 reflect the SAR
results of Fig. 3 and show a temperature amplification factor of x18 and x1.7
in the shallow and deep tumors, respectively, compared to the average healthy tissues.

### Acknowledgements

R01EB006847, P41EB015896,
K99EB019482### References

[1] Villena JF (2014). ISMRM 22:623. [2] Guerin B (2014). ISMRM 22:5125. [3] Polimeridis A. (2013). JCP 269:280-296. [4] Christ A. (2010). PMB 55(2): N23. [5] Karaa S (2005). MCS 68(4):
375-388. [6] Gabriel C (1996). DTIC
Document AL/OE-TR-1996-0037.