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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. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical method for an optimal control problem governed by an obstacle problem. An approximate optimization problem is proposed by regularizing the original non-differentiable constrained problem with a simple method. The connection between the two formulations is established through some convergence results. A sufficient condition is derived to decide whether a solution of the first-order optimality system is a global minimum. The method with a second-order in time dissipative system is developed to solve the optimality system numerically. Several numerical examples are reported to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

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In this paper, we study a parameter identification problem for elliptic partial differential equations. We reconstruct the coefficient with additional boundary measurements, including both Dirichlet and Neumann boundary conditions. To solve the problem, the coupled complex boundary method (CCBM), originally proposed in Cheng et al. [A novel coupled complex boundary method for solving inverse source problems, Inverse Probl. 30 (2014), p. 055002] is used. With CCBM, a complex boundary problem is introduced in such a way that the boundary conditions are coupled in a complex Robin boundary condition with a parameter τ. Using Tikhonov regularization, the coefficient is sought such that the imaginary part of the solution of the forward Robin boundary value problem vanishes in the problem domain, which brings advantages on robustness in reconstruction. Besides, the reconstruction is feasible even for very small regularization parameter through choosing the values of τ properly. Some theoretical analyses are given. Moreover, noise model is analysed and the finite element method is used for discretization. Numerical examples show the feasibility and stability of the proposed method. [ABSTRACT FROM AUTHOR]

Inspired by the paper (Gong, W., Li, R., Yan, N. N. & Zhao, W. B. (2008) An improved error analysis for finite element approximation of bioluminescence tomography. J. Comput. Math., 26, 1–13), we consider an optimal finite element error estimate for an inverse problem in multispectral bioluminescence tomography. Different from achromatic or monochromatic measurements, hyperspectral or multispectral data can reduce the ill-posedness of the inverse problem and yield improved depth reconstruction. Compared with Gong et al. (2008, An improved error analysis for finite element approximation of bioluminescence tomography. J. Comput. Math., 26, 1–13), which just improved the error order of piecewise constant light source function, error order in this paper is optimal and all error estimates here are valid for a general smooth domain rather than a polyhedral/polygonal one. Moreover, under a boundedness assumption for the admissible source set, the constants in our error estimates do not depend on the regularization parameter, and therefore are bounded. [ABSTRACT FROM AUTHOR]

Diffuse optical tomography (DOT) and bioluminescence tomography (BLT) are different but complementary modalities in optical molecular imaging. The goal of DOT is to reconstruct optical parameters of a medium from surface measurements induced by external sources, whereas that of BLT is to reconstruct running a bioluminescent source distribution in the medium from surface measurements induced by internal bioluminescent sources. A pre-requisite for BLT reconstruction is knowledge on the distribution of optical parameters within the medium, which is the output of DOT. In this paper, we propose a mathematical model integrating BLT and DOT at the fundamental level; that is, performing the two types of reconstructions simultaneously instead of doing them sequentially. The model is introduced through minimizing the difference between predicted quantities and boundary measurements, as well as incorporating regularization terms. In practice, the optical parameters are assumed to be piecewise constants or piecewise smooth functions. Hence, we seek the optical parameters in the space of functions with bounded variation (BV). For the BV-regularized data-fitting functional and its discretized approximations, we demonstrate the existence of minima and show convergence of the minima of the discretized functionals to that of the non-discretized one. We also present numerical results to illustrate the strength of our proposed model. [ABSTRACT FROM AUTHOR]

Tumor treating fields (TTFields) is a novel therapeutic approach for the treatment of glioblastoma. The electric field intensity is a critical factor in the therapeutic efficacy of TTFields, as stronger electric field can more effectively impede the proliferation and survival of tumor cells. In this study, we aimed to improve the therapeutic effectiveness of TTFields by optimizing the position of electrode arrays, resulting in an increased electric field intensity at the tumor. Three representative head models of real glioblastoma patients were used as the research subjects in this study. The improved subtraction‐average‐based optimization (ISABO) algorithm based on circle chaos mapping, opposition‐based learning and golden sine strategy, was employed to optimize the positions of the four sets of electrode arrays on the scalp. The electrode positions are dynamically adjusted through iterative search to maximize the electric field intensity at the tumor. The experimental results indicate that, in comparison to the conventional layout, the positions of the electrode arrays obtained by the ISABO algorithm can achieve average electric field intensity of 1.7887, 2.0058, and 1.3497 V/cm at the tumor of three glioblastoma patients, which are 23.6%, 29.4%, and 8.5% higher than the conventional layout, respectively. This study demonstrates that optimizing the location of the TTFields electrode array using the ISABO algorithm can effectively enhance the electric field intensity and treatment coverage in the tumor area, offering a more effective approach for personalized TTFields treatment. [ABSTRACT FROM AUTHOR]