Article Contents

2024, Volume 2, Issue 2: 159-179. Doi: 10.3934/cac.2024008

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Tumor Treating Fields: Modeling and a numerical algorithm

1.
School of Mathematics, Nanjing University of Aeronautics and Astronautics, 211106, Nanjing, China
2.
Department of Biomedical Engineering, Nanjing University of Aeronautics and Astronautics, 211106, Nanjing, China
3.
Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China
4.
Department of Radiology, GuiQian International General Hospital, 550018, Guiyang, China

^*Corresponding author: Catharine W. K. Lo, Chunxiao Chen and Ming Lu
^*Corresponding author: Catharine W. K. Lo, Chunxiao Chen and Ming Lu

Received: April 2024

Revised: May 2024

Early access: May 2024

Published: June 2024

This work is supported by National Natural Science Foundation of China (Grant No. 12071215)

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Abstract

This paper investigates the optimal control problem generated in Tumor Treating Fields (TTFields). TTFields is an emerging cancer treatment method with several advantages including the convenience of treatment, fewer side effects, and a better quality of life for patients. Therefore, it holds significant promise for applications in cancer treatments and other fields. In view of such an immense potential, in this work, we are motivated to determine the optimal arrangement of electrodes for TTFields. The paper begins by presenting a comprehensive modeling process for TTFields, followed by the establishment of an optimal control problem that involves finding the optimal configuration of the electrode array. To solve this problem, the particle swarm optimization (PSO) algorithm, which is a type of intelligent algorithm, is employed, and is further improved on to address various situations. Finally, the model is validated and the effectiveness of the PSO algorithm is verified through numerical examples.

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Mathematics Subject Classification: 35R30, 78-10, 78M10, 90C59.

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Figure 1. The NovoTTF-100A System components and assembly

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Figure 2. Electric field strength, (a) the resolution of model (5) with FEMs, (b) COMSOL simulation

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Figure 3. The relationship between $ \omega $ and iteration number, (a) one pair of electrodes, (b) two pairs of electrodes

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Figure 4. Disk

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Figure 5. Particle aggregation (left) and solutions (right) for PSO

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Figure 6. Particle aggregation (left) and solutions (right) for MPSO

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Figure 7. Particle aggregation of PSO with coordinate transformation for two pairs of electrodes (left) and three pairs of electrodes (right)

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Figure 8. Particle aggregation using coordinate transformation PSO algorithm with segmented weights for two pairs of electrodes (left) and three pairs of electrodes (right)

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Figure 9. Sphere

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Figure 10. IPSO in 3D, (a) Particle motion graph in two pairs of electrodes, (b) solution in two pairs of electrodes

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Figure 11. MPSO in 3D, (a) particle motion graph in two pairs of electrodes, (b) solutions in three pairs of electrodes

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Figure 12. CPSO in 3D, (a) (b) particle motion graph and solution in two pairs of electrodes, (c) (d) particle motion graph and solution in three pairs of electrodes

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Figure 13. Brain and internal tumor

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Figure 14. The process of the candidate solution to the relatively good solution, in the case of one pair of electrodes, (a) electrode distribution at $ k = 0 $, (b) electrode distribution at $ k = 200 $, (c) electrode distribution at $ k = 250 $, in the case of two pairs of electrodes, (d) electrode distribution at $ k = 0 $, (e) electrode distribution at $ k = 200 $, (f) electrode distribution at $ k = 250 $

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Figure 15. Electrode array layouts and the electric field intensity of the tumor, (a) AP array layout, (b) LR array layout, (c) array layout obtained from PSO, (d) (c) array layout obtained from CPSO

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Table 1. The physical parameters for different tissues [37]

Tissue	Conductivity, $ \sigma(S/m) $	Permittivity, $ \epsilon_r(F/m) $
Scalp	0.2500	5000
Skull	0.0211	204
Gray matter	0.1410	2010
White matter	0.0868	1290
Tumor	0.2500	110
Cerebrospinal fluid	2	109
Gel	0.1000	100
Electrodes	0	6000

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Table 2. The mean $ E $ and $ \phi $ over tumor area

	Numerical solutions with model (5)	Simulation solution
$ \phi(V) $	395.4736	400.0800
$ E(V/cm) $	1.8814	1.8595

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Table 3. The electrical conductivity and permittivity of each tissue

Tissue	Conductivity, $ \sigma(S/m) $	Permittivity, $ \epsilon_r(F/m) $
Scalp	0.2500	5000
Skull	0.0211	204
Tumor	0.2500	110
Others	0.1589	1638

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Table 4. The iterative process of particles $ x $ in the case of one and two pairs of electrodes

Iterations	One pair of electrodes	Two pairs of electrodes
$ k = 0 $	[3608; 5367]	[5812; 8195; 8545; 10549]
$ k = 200 $	[4525; 7662]	[4395; 5367; 10544; 10549]
$ k = 250 $	[7718; 7485]	[6653; 6736; 10432; 10785]

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Table 5. The mean and minimum electric field intensity in tumor area of four different strategies for electrode array layout

Strategy for electrode array layout	$ \overline{E} $(V/cm)	$ E_{min} $(V/cm)
AP	1.0565	0.8826
LR	1.0768	0.7843
PSO	1.4342	1.1210
CPSO	1.5010	1.1454

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