No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Optimizing Breast Cancer Treatment Using Hyperthermia: a Single and Multi-Objective Optimal Control Approach
ResearchGate has not been able to resolve any citations for this publication.
This study aimed at developing an inverse heat transfer approach for predicting the time-varying freezing front and the temperature distribution of tumors during cryosurgery. Using a temperature probe pressed against the layer of tumor, the inverse approach is able to predict simultaneously the metabolic heat generation and the blood perfusion rate of the tumor. Once these parameters are predicted, the temperature-field and time-varying freezing fronts are determined with the direct model. The direct model rests on one-dimensional Pennes bioheat equation. The phase change problem is handled with the enthalpy method. The Levenberg-Marquardt Method (LMM) combined to the Broyden Method (BM) is used to solve the inverse model. The effect (a) of the thermal properties of the diseased tissues; (b) of the initial guesses for the unknown thermal properties; (c) of the data capture frequency; and (d) of the noise on the recorded temperatures is examined. It is shown that the proposed inverse approach remains accurate for all the cases investigated.
Simple Summary
Magnetic hyperthermia therapy (MHT) is a promising nanotechnology-based treatment for cancer. Its widespread adoption is hampered by the imprecise control of tumor temperature during treatment and by a lack of automatic shutdown to avoid adverse events. Here, we verify the functionality and validate performance of an automated temperature controller, and the functionality of automated safety shutdowns. Experiments were performed ex vivo in liver tissue and in vivo in the brain of a healthy live research subject to demonstrate the potential for superior energy control with temperature stability using user-defined inputs, and real-time temperature monitoring for feedback. Performance of this device was validated against design criteria using FDA guidelines. It will be used to treat canine glioblastoma tumors in a future study.
Abstract
We present in vivo validation of an automated magnetic hyperthermia therapy (MHT) device that uses real-time temperature input measured at the target to control tissue heating. MHT is a thermal therapy that uses heat generated by magnetic materials exposed to an alternating magnetic field. For temperature monitoring, we integrated a commercial fiber optic temperature probe containing four gallium arsenide (GaAs) temperature sensors. The controller device used temperature from the sensors as input to manage power to the magnetic field applicator. We developed a robust, multi-objective, proportional-integral-derivative (PID) algorithm to control the target thermal dose by modulating power delivered to the magnetic field applicator. The magnetic field applicator was a 20 cm diameter Maxwell-type induction coil powered by a 120 kW induction heating power supply operating at 160 kHz. Finite element (FE) simulations were performed to determine values of the PID gain factors prior to verification and validation trials. Ex vivo verification and validation were conducted in gel phantoms and sectioned bovine liver, respectively. In vivo validation of the controller was achieved in a canine research subject following infusion of magnetic nanoparticles (MNPs) into the brain. In all cases, performance matched controller design criteria, while also achieving a thermal dose measured as cumulative equivalent minutes at 43 °C (CEM43) 60 ± 5 min within 30 min.
The direct method has been widely used in various fields of trajectory optimization due to the need to optimize the performance of a controlled dynamic system. An essential issue in the development of the direct method is the selection of the finite element mesh. Although there have been many pieces of research on adaptive mesh refinement, few works focused on finite element layout optimization before the refinement. This paper proposes an enhanced moving finite element method (EMFE), taking the length of each finite element as a variable, based on the geometric estimation of non-collocation-point error (NCPE) and generalized finite element length (GFEL). The features of EMFE include flexible problem reformulation, stable numerical calculation and the ability to locate breakpoints for singular control problems (the control variables appear linearly in the problems) and finding the local minimum numerical error. The theoretical basis of the ability to locate the breakpoints for singular control problems and the feature of NCPE estimate has also been investigated. Three numerical examples, two of which are singular control problems for lunar landing and the other is a multi-phase nonlinear problem for launch vehicles, demonstrated the performance and features of the proposed EMFE method.
Magnetic fluid hyperthermia (MFH) is a novel reliable technique with excellent potential for thermal therapies and treating breast tumours. This method involves injecting a magnetic nanofluid into the tumour and applying an external AC magnetic field to induce heat in the magnetic nanoparticles (MNPs) and raise the tumour temperature to ablation temperature ranges. Because of the complexity of considering and coupling all different physics involves in this phenomenon and also due to the intricacy of a thorough FEM numerical study, few FEM-based studies address the entire MFH process as similar to reality as possible. The current study investigates a FEM-based three-dimensional numerical simulation of MFH of breast tumours as a multi-physics problem. An anatomically realistic breast phantom (ARBP) is considered, some magnetic nanofluid is injected inside the tumour, and the diffusion phenomenon is simulated. Then, the amount of heat generated in the MNP-saturated tumour area due to an external AC magnetic field is simulated. In the end, the fraction of tumour tissue necrotized by this temperature rise is evaluated. The study’s results demonstrate that by injecting nanofluid and utilizing seven circular copper windings with each coil carrying 400 A current with a frequency of 400 kHz for generating the external AC magnetic field, the temperature in tumour tissue can be raised to a maximum of about 51.4°C, which leads to necrosis of entire tumour tissue after 30 minutes of electromagnetic field (EMF) exposure. This numerical platform can depict all four various physics involved in the MFH of breast tumours by numerically solving all different equation sets coupled together with high precision. Thus, the proposed model can be utilized by clinicians as a reliable tool for predicting and identifying the approximate amount of temperature rise and the necrotic fraction of breast tumour, which can be very useful to opt for the best MFH therapeutic procedure and conditions based on various patients. In future works, this numerical platform’s results should be compared with experimental in-vivo results to improve and modify this platform in order to be ready for clinical applications.
Malignant tumor (cancer) is the leading cause of death globally and the annual cost
of managing cancer is trillions of dollars. Although, there are established therapies including
radiotherapy, chemotherapy and phototherapy for malignant tumors, the hypoxic environment of
tumors and poor perfusion act as barriers to these therapies. Hyperthermia takes advantage of
oxygen deficiency and irregular perfusion in the tumor environment to destroy malignant cells.
Despite successes recorded with hyperthermia, there are concerns with the post-treatment condition
of patients as well as the required thermal dose to prevent harm. The investigation of the dynamics
of tumor-induced immune suppression with hyperthermia treatment using mathematical analysis
and optimal control theory is potentially valuable in the development of hyperthermia treatment.
The role of novel tumor-derived cytokines in counterattacking immune cells is considered in this
study as a mechanism accounting for the aggressiveness of malignant tumors. Since biological
processes are not instantaneous, a discrete time delay is used to model biological processes involved
in tumor inhibitory mechanisms by secretion, the elaboration of suppressive cells, and effector cell
differentiation to produce suppressive cells. Analytical results obtained using Lyapunov’s function
indicate the conditions required for global stability of the tumor-present steady-state. A thermal
optimal control strategy is pursued based on optimal control theory, and the best strategy to avoid
adverse outcomes is obtained. We validate the analytical results numerically and demonstrate the
impact of both inadequate and excessive heat on the dynamics of interactive cell functioning.
A linear combination of spherically symmetric heat sources is shown to provide optimal stationary thermal distribution in magnetic hyperthermia. Furthermore, such spatial location of heat sources produces suitable temperature distribution in biological medium even for assemblies of magnetic nanoparticles with a moderate value of specific absorption rate (SAR), of the order of 100–150 W/g. We also demonstrate the advantage of using assemblies of spherical magnetic nanocapsules consisting of metallic iron nanoparticles covered with non magnetic shells of sufficient thickness in magnetic hyperthermia. Based on numerical simulation we optimize the size and geometric structure of biocompatible spherical capsules in order to minimize the influence of strong magneto-dipole interaction between closely spaced nanoparticles. It is shown that assembly of capsules can provide sufficiently high SAR values of the order of 250–400 W/g at moderate amplitudes H0 = 50–100 Oe and frequencies f = 100–200 kHz of alternating magnetic field, being appropriate for application in clinics.
In this contribution the solution of Fractional Optimal Control Problems (FOCP) by using the Orthogonal Collocation Method (OCM) and the Multi-objective Optimization Stochastic Fractal Search (MOSFS) algorithm is investigated. For this purpose, three classical case studies on engineering are considered. Initially, the concentration profiles of laccase enzyme production process are analyzed to evaluate the influence of fractional order. Then, two classical FOCP (Catalyst Mixing and Batch Reactor) are solved by using the association between OCM and MOSFS approachesthrough the formulation and solution of a multi-objective optimization problem. The results indicate that the variation of the fractional order impliesdifferent values for the original objective function. In addition, physicallyincoherent profiles can be obtained by considering the fluctuation of the fractional order. Finally, the proposed MOSFS is considered as apromising methodology to solve multi-objective optimization problems.
Improvement and optimization of hyperthermia as one of the most effective and conventional cancer treatments have been tackled by many researchers in recent years. The main benchmark for hyperthermia treatment is to elevate the cancerous tissue temperature to a specified level to initiate damage or ablation and simultaneously to prevent any irreversible thermal damage to the surrounding healthy cells. Therefore, the heat source pattern for hyperthermia treatment should be adjusted according to the distribution of cancerous cells. To do this, in the present study the fractional bioheat equation with unknown heat source term is implemented. Two different hyperthermia scenarios with single and dual tumors are considered and the appropriate damage level at both healthy and cancerous tissues is planned according to the Arrhenius model. Moreover, the optimal control criterion is set to simultaneously produce the goal temperature level and minimize the thermal power consumption. The mentioned task is accomplished by the definition of a priority coefficient as a weight function which determines whether the establishment of desired thermal field or the reduction of applied thermal power is more important. Based on the results, the optimal control approach can precisely determine the heat source distribution for single and dual tumor cases (with 1.03 and 0.13% deviation error, respectively). Whereas, by shifting the priority to the minimization of applied energy, up to 23.1 and 38.7% deviation is observed in formation of the desired temperature field (while the overall cost function is minimized). In order to investigate the numerical results, a computational method based on the Legendre cardinal functions is proposed. The main advantage of the proposed method is that it transforms solving such fractional models into solving systems of algebraic equations which greatly simplify the problem.
The presented paper focuses on a numerical analysis of temperature in the anatomical model of the female breast with a strictly defined level of power generated by the EMF source in pathological tissue saturated with ferrofluid. The aim of this study was to examine the effect of blood perfusion rate models on the resultant tumor temperature. The four tumor perfusion models were subjected to comparative analysis: constant, linear, nonlinear and completely free of blood flow. The authors have shown that taking into account the various temperature dependences of blood perfusion models within the treated tissue might play an important role in the complex process of female breast cancer treatment planning.
In oncology, hyperthermia is understood as a planned, controlled technique of heating cancerous changes in order to destroy their cells or stop their growth. In clinical practice, hyperthermia is used in combination with radiotherapy, chemotherapy, or immunological therapy. During the hyperthermia, the tissue is typically exposed to a temperature in the range of 40–45 °C, the exception is thermoablation, during which the temperatures reach much higher values. Thermoablation is characterized by the use of high temperatures up to 90 °C. The electrode using the radiofrequency is inserted into the central area of the tumor. Interstitial thermoablation is used to treat, among others, breast and brain cancer. The therapy consists of inducing coagulation necrosis in an area that is heated to very high temperatures. Mathematical modeling is based on the use of a coupled thermo-electric model, in which the electric field is described by means of the Laplace equation, while the temperature field is based on the Pennes equation. Coupling occurs at the level of the additional source function in the Pennes equation. The temperature field obtained in this way makes it possible to calculate the Arrhenius integral as a determinant of the destruction of biological tissue. As a result of numerical calculations regarding the temperature field and the Arrhenius integral, it can be concluded that, with the help of numerical tools and mathematical modeling, one can simulate the process of destroying cancerous tissue.
The article presents the mathematical modeling of the phenomenon of artificial hyperthermia which is caused by the interaction of an electric field. The electric field is induced by the applicator positioned within the biological tissue with cancer. In addition, in order to estimate the degree of tumor destruction under the influence of high temperature an Arrhenius integral has been used. The distribution of electric potential in the domain considered is described by the Laplace system of equations, while the temperature field is described by the Pennes system of equations. These problems are coupled by source function being the additional component in the Pennes equation and resulting from the electric field action. The boundary element method is applied to solve the coupled problem connected with the heating of biological tissues.
Over the past decades, cancer is the major cause of incidence of death increasing every day. Different forms of tumor therapy including radiotherapy and chemotherapy are used to treat cancer. However, hyperthermia is the technique that neglects the use of chemicals or harmful radiations. The elevated body temperature can damage the cancerous cells with minimum injury to the normal cells. Successful therapy method in combination with radiation therapy and/or chemotherapy is provided to the cancer patient which proved to be beneficial to the patients. In this review, different studies of the clinical trials are reported on the patients with tumor and the therapy associated with it.
Hyperthermia (HT) is a method used to treat tumors by increasing the temperature of the cells. The treatment can be applied in combination with other verified cancer treatments using several different procedures. We sought to present an overview of the different HT tumor treatment, recent advances in the field, and combinational treatment sequences and outcomes. We used a computer-aided search to identify articles that contained the keywords hyperthermia, cancer treatment, chemotherapy, radiotherapy, nanoparticle, and cisplatin. There are three types of HT treatment, which each need the use of applicators that are in contact with or in the proximity of the patient for the purpose of heating. Heating can be achieved using different types of energy (including microwaves, radio waves, and ultrasound). However, the source of energy will depend on the cancer type and location. The temperature used will also vary. HT is rarely used alone, and can be combined with other cancer treatments. When used in combination with other treatments, improved survival rates have been observed. However, despite in vitro and in vivo studies that support the use of concurrent hypothermia treatments, contradictory results suggest there is a need for more studies to identify other hidden effects of HT.
This paper serves as an introduction to the theory of optimal control applied to systems of ordinary differential equations with emphasis on disease models. We outline the steps in formulating an optimal control problem and derive necessary conditions. Several simple examples provide detailed methodology in characterizing the optimal control through use of Pontryagin's Maximum Principle. An SEIR (Susceptible, Exposed, Infected, Recovered) model with control acting as a rate of vaccination is presented and an optimal control problem is formulated to include an isoperimetric constraint on the vaccine supply. Numerical results illustrate how such a constraint alters the optimal vaccination schedule and its effect on the population.
A novel scheme to obtain the optimum tissue heating condition during hyperthermia treatment is proposed. To do this, the effect of the controllable overall heat transfer coefficient of the cooling system is investigated. An inverse problem by a conjugated gradient with adjoint equation is used in our model. We apply the finite difference time domain method to numerically solve the tissue temperature distribution using Pennes bioheat transfer equation. In order to provide a quantitative measurement of errors, convergence history of the method and root mean square of errors are also calculated. The effects of heat convection coefficient of water and thermal conductivity of casing layer on the control parameter are also discussed separately.
Dynamic optimization problems can be numerically solved by direct, indirect and Hamilton- Jacobi-Bellman methods. Direct methods can provide sub-optimum solutions and less accurate results when compared to indirect methods. Otherwise, in despite of attending the constraints of the problem, indirect methods present convergence difficulties. The literature shows algorithms where these methods are combined into a hybrid methodology where direct methods are applied to a more simplified problem and the results are used as estimates by the indirect methods through the solution refinement of homotopy continuation. In this work, the differential algebraic approach is incorporated into a hybrid method class, extending the concepts of structural and differential indexes, consistent initialization analysis, index reduction and dynamic degrees of freedom of the optimal control problem. The resultant Differential Algebraic Optimal Control Problem (DAOCP) and the hybrid method are called as the Mixed Method. In the proposed algorithm, the computational code Dircol is used to obtain the NLP equations and the Coldae code solves the general index-1 boundary value problems given by the Pontryagin Minimum Principle extended to differential- algebraic problems (augmented system). The DAOCP characterization is made with the use of structural algorithms. The Dircol algorithm initially solves the original problem of searching the best possible solution, which also provides the events estimates and costate variables profiles. The switching structures are built based upon the estimated events and the analysis of results. The augmented system relates to each phase of the problem and it uses the slack variable method. The index-1 boundary value problems concerning to each phase are sequentially solved by the Coldae code. In this paper, the Mixed Method is applied to a pressure- constrained batch reactor optimization problem with great success.
The aim of this study was to evaluate and compare temperature distributions for different tissues being treated at the time of interstitial microwave hyperthermia. A coaxial-slot antenna implemented into the tissue is the source of microwave radiation. The described model takes into account the wave equation for the TM mode and the Pennes equation determining the temperature distribution within the tissue in the stationary case. The simulation results for the three fundamental microwave frequencies of tissue heating devices are presented.
The new concept of keeping primary tumor under control in situ to suppress distant foci sheds light on the treatment of metastatic tumor. Hyperthermia is considered as one of the means for controlling tumor growth. To simulate the tumor growth, a continuum mathematical model has been introduced. The newest understanding of the Warburg effect on the cellular metabolism and diffusion of the nutrients in the tissue has been taken into consideration. The numerical results are compared with the in vivo experimental data by fitting the tumor cell doubling time/tumor cell growth rate under different thermal conditions. Both the tumor growth curve and corresponding average glucose concentration have been predicted. The numerical results have quantitatively illustrated the controlling effect on tumor growth under hyperthermia condition in the initial stage.
A new heuristic approach for minimizing possiblynonlinear and non-differentiable continuous spacefunctions is presented. By means of an extensivetestbed it is demonstrated that the new methodconverges faster and with more certainty than manyother acclaimed global optimization methods. The newmethod requires few control variables, is robust, easyto use, and lends itself very well to parallelcomputation.
This paper gives a brief list of commonly used direct and indirect efficient methods for the numerical solution of optimal control problems. To improve the low accuracy of the direct methods and to increase the convergence areas of the indirect methods we suggest a hybrid approach. For this a special direct collocation method is presented. In a hybrid approach this direct method can be used in combination with multiple shooting. Numerical examples illustrate the direct method and the hybrid approach.
Magnetic nanofluid hyperthermia (MNH) damages malignant cells by the heat generated by magnetic nanoparticles (MNPs) exposed to an alternating magnetic field during therapy. The key point for magnetic hyperthermia is to maintain the treatment temperature to a safe range for bio-tissue, in which malignant tissue will be damaged due to its higher heat sensitivity but not the healthy tissue. However, the therapeutic system for MNH should be a non-linear one since the treatment temperature is generally determined by many certain and uncertain factors, which result in the difficulty to modulate the temperature to a specific range in a practical application. This study proposes a control method for a MNH system by introducing a proportional-integral-derivative control (PID) algorithm, and dynamically optimizes the PID coefficients for the proposed system by considering simulated annealing algorithm during therapy. The treatment temperature distribution for bio-tissue is obtained by solving an improved Pennes bio-heat transfer equation using finite element method. The simulation results demonstrate that the proposed control system can effectively modulate the power dissipation for MNPs and further exactly regulate the transient treatment temperature to an expected value. In addition, this proposed system can also automatically adapt to different cases during therapy, in which the nanofluid concentration distribution inside tumor region changes with nanofluid injection strategy.
Theoretical investigation of heat transport and temperature responses within biological tissues is of significant importance for medical hyperthermia treatment of cancer. This paper develops a biothermomechanical model of skin for hyperthermia treatment based on the non-Fourier bioheat transfer. The skin is considered as a three-layer tissue including epidermis, dermis and hypodermis, and the properties of each layer is assumed to be homogeneous and linear thermoelasticity. Analytical solutions of temperature profile and associated stress distribution in skin tissue are derived accounting for the blood perfusion, sweating, metabolic heat generation, specific boundary condition at skin surface and heating strategy inside biological body. Numerical results show that both of blood perfusion and sweating have great influence on temperature, thermal damage and stress, and the maximum stress induced by external heating during hyperthermia treatment always exceeds pain threshold and contributes to thermal pain sensation. Comparing with the convective cooling boundary at skin surface, the temperature field inside target domain is more uniform and the peak stress is reduced significantly when the prescribed temperature boundary is used. This paper may be helpful to improve the fundamental understanding of skin biothermomechanics and in designing of various biomechanics and biomedical related devices for hyperthermia treatment.
Due to the number of cancer cases diagnosed each year and the fatality rate resulting from some more severe types, the improvement of less invasive and more efficient treatment techniques is of great importance. In this context, hyperthermia is a medical procedure in which the tumor region is heated by using an applicator for a certain period, aiming to destroy pathological cells. Computational models can be used to simulate the heating effect of tumors and adjacent cells. In general, the solution to an optimization problem considering factors such as heating temperature, applicator position, and the time in which the region will be subjected to heating can provide important information about the procedure. Traditionally, this type of problem has been addressed in a single objective context, focusing on minimizing the destruction of adjacent healthy tissue considering the area of the applicator constant. Our fundamental objective is to propose a multi-objective design problem considering the minimization of the area subject to the procedure and the time required for the process of hyperthermia in a breast cancer treatment. The problem is constrained by the degree of tissue destruction and by a partial differential equation that describes the phenomenon of heat transfer in both healthy and tumor tissues. The results obtained demonstrate that a point with a good compromise between the objectives can be chosen in such a way that a particular strategy can be defined for each patient.
Laser ablation is a rising technique used to induce a localized temperature increment for tumor ablation. The outcomes of the therapy depend on the tissue thermal history. Monitoring devices help to assess the tissue thermal response, and their combination with a control strategy can be used to promptly address unexpected temperature changes and thus reduce unwanted thermal effects. In this application, numerical simulations can drive the selection of the laser control settings (i.e., laser power and gain parameters) and allow evaluating the thermal effects of the control strategies. In this study, the influence of different control strategies (On-Off and PID-based controls) is quantified considering the treatment time and the thermal effect on the tissue. Finite element model-based simulations were implemented to model the laser-tissue interaction, the heat-transfer, and the consequent thermal damage in liver tissue with tumor. The laser power was modulated based on the temperature feedback provided within the tumor safety margin. Results show that the chosen control strategy does not have a major influence on the extent of thermal damage but on the treatment duration; the percentage of necrosis within the tumor domain is 100% with both strategies, while the treatment duration is 630 s and 786 s for On-Off and PID, respectively. The choice of the control strategy is a trade-off between treatment duration and unwanted temperature overshoot during closed-loop laser ablation. Clinical Relevance-This work establishes that different temperature-based control of the laser ablation procedure does not have a major influence on the extent of thermal damage but on the duration of treatment.
This paper studies the three-dimensional (3D) trajectory optimization problem for unmanned aerial vehicle (UAV) aided wireless communication. Existing works mainly rely on the kinematic equations for UAV’s mobility modeling, while its dynamic equations are usually missing. As a result, the planned UAV trajectories are piece-wise line segments in general, which may be difficult to implement in practice. By leveraging the concept of state-space model, a control-based UAV trajectory design is proposed in this paper, which takes into account both the UAV’s kinematic equations and the dynamic equations. Consequently, smooth trajectories that are amenable to practical implementation can be obtained. Moreover, the UAV’s controller design is achieved along with the trajectory optimization, where practical roll angle and pitch angle constraints are considered. Furthermore, a new energy consumption model is derived for quad-rotor UAVs, which is based on the voltage and current flows of the electric motors and thus captures both the consumed energy for motion and the energy conversion efficiency of the motors. Numerical results are provided to validate the derived energy consumption model and show the effectiveness of our proposed algorithms.
The aim of this book is to furnish the reader with a rigorous and detailed exposition of the concept of control parametrization and time scaling transformation. It presents computational solution techniques for a special class of constrained optimal control problems as well as applications to some practical examples. The book may be considered an extension of the 1991 monograph A Unified Computational Approach Optimal Control Problems, by K.L. Teo, C.J. Goh, and K.H. Wong. This publication discusses the development of new theory and computational methods for solving various optimal control problems numerically and in a unified fashion. To keep the book accessible and uniform, it includes those results developed by the authors, their students, and their past and present collaborators. A brief review of methods that are not covered in this exposition, is also included.
Knowledge gained from this book may inspire advancement of new techniques to solve complex problems that arise in the future. This book is intended as reference for researchers in mathematics, engineering, and other sciences, graduate students and practitioners who apply optimal control methods in their work. It may be appropriate reading material for a graduate level seminar or as a text for a course in optimal control.
This article addresses the problem of spacecraft reorientation subject to attitude pointing, angular velocity, and actuator constraints. In order to tackle the attitude pointing constraints, a potential function is proposed, which is parameterized to allow for a wider range of optimal repointing solutions with attitude pointing constraints. Based on the potential function, a feedback control law is then designed for ensuring the system stability. The controller gains are optimized using a gradient-based optimal parameter tuning method. Furthermore, by imposing the angular velocity constraints and the actuator constraints into the formulation, the optimal parameter tuning method also ensures the satisfaction of these constraints. Therefore, stability, optimality, and feasibility are achieved simultaneously. Numerical simulations demonstrate accurate repointing maneuvering under different types of constraints.
Magnetic nanoparticles (MNPs)-induced hyperthermia is capable of heating the tumor without side effects. In this technique, the tumor temperature is elevated to 41–43 °C from a normal temperature of 37 °C of the body. The Pennes' bio-heat transfer equation is widely used to transfer heat in living organs with blood perfusion rate. The elevated temperature destroys cancer cells and keeps normal cells harmless. This state-of-the-art review describes the basic physical mechanisms behind this treatment modality and recent advances in the mathematical modeling approach toward this therapy. Firstly, we throw light on different MNPs applicable in hyperthermia, heat generation mechanism, basic parameters affecting the efficiency of heating, thermophysical properties of MNPs, in vivo studies, and clinical studies. Secondly, we have discussed in detail the mathematical modeling of hyperthermia including analytical solutions, computational modeling, and prominent optimization techniques applied in thermotherapy. We shortly discuss hyperthermia integrated with chemotherapy, laser therapy, radiotherapy, and immunotherapy. In the end, we narrate some major challenges and opportunities for hyperthermia and discussed future directions.
Background and objective:
For decades, mathematical models have been used to predict the behavior of physical and biological systems, as well as to define strategies aiming at the minimization of the effects regarding different types of diseases. In the present days, the development of mathematical models to simulate the dynamic behavior of the novel coronavirus disease (COVID-19) is considered an important theme due to the quantity of infected people worldwide. In this work, the objective is to determine an optimal control strategy for vaccine administration in COVID-19 pandemic treatment considering real data from China. Two optimal control problems (mono- and multi-objective) to determine a strategy for vaccine administration in COVID-19 pandemic treatment are proposed. The first consists of minimizing the quantity of infected individuals during the treatment. The second considers minimizing together the quantity of infected individuals and the prescribed vaccine concentration during the treatment.
Methods:
An inverse problem is formulated and solved in order to determine the parameters of the compartmental Susceptible-Infectious-Removed model. The solutions for both optimal control problems proposed are obtained by using Differential Evolution and Multi-objective Optimization Differential Evolution algorithms.
Results:
A comparative analysis on the influence related to the inclusion of a control strategy in the population subject to the epidemic is carried out, in terms of the compartmental model and its control parameters. The results regarding the proposed optimal control problems provide information from which an optimal strategy for vaccine administration can be defined.
Conclusions:
The solution of the optimal control problem can provide information about the effect of vaccination of a population in the face of an epidemic, as well as essential elements for decision making in the economic and governmental spheres.
Microwave ablation is a promising minimally invasive treatment for cancer. However, due to the respiratory movement of the lungs, it is very difficult to accurately predict and control the microwave ablation zone. Therefore, the influence of the changes of the physical parameters of the respiratory process on the microwave ablation zone is studied. Firstly, based on the 4D-CT describing the respiratory process of the lungs, all the image data are from 100 non-small cell lung cancer radiotherapy patients (50 males and 50 females, average 58 years, range 55–61 years). According to the theory of porous media, the change of the effective thermal conductivity of the lung tissue during the breathing process is obtained. The effective thermal conductivity of the lung parenchyma during respiration varies from 0.16 to 0.20 W/m °C, with the lowest vale at the end of inspiration and the highest at the end of expiration. The transient problems during microwave ablation of pulmonary tissue are analyzed by finite element method. The changes of relative permittivity, conductivity and density changes during the breathing process are also considered. The results show that the microwave ablation zone is significantly larger under dynamic physical parameters. At the end of expiration, when the tissue parameter is set to constant, the ablation lesion area is more concentrated around the tip and slot of the antenna, and the backward heating effect is smaller, Ablation volume was superior in nonventilated lungs. Therefore, single-lung ventilation can be considered during pulmonary ablation to reduce the impact of breathing on the ablation area. These findings can be useful to further our understanding the MWA and hold promise towards achieving successful treatment objective as well as enhanced therapeutic output via improved treatment planning and strategy. This study provides the basis for clinical pulmonary ablation and can also be used as a preoperative plan to provide guidance to physicians.
In this paper, a novel scheme based on strongly continuous semigroup is proposed to find a pointwise optimal control function in a biological tissue. Here, mathematical model for hyperthermia therapy involves solution to the thermal wave equation as state while the control is given by the pointwise time dependent heat source. The target is the temperature at a given point within the tumor. Pointwise optimal control problem on and inside a tissue is solved subject to thermal wave model with Dirichlet and Rubin boundary conditions. The pointwise heating source induced by heating probe inserted at the tumor site as control at specific depth inside the biological body. Solutions for both thermal wave problem and its associated adjoint problem are proposed. Approximate controllability of the thermal wave problem is derived with the help of strongly continuous semigroups theory. We prove that the system is pseudo-port Hamiltonian. Pointwise time dependent optimal control problem is solved by using time discretization, conjugate gradient technique and strongly continuous semigroups theory. A set of numerical experiments concerning the design of optimal heating power strategy in cancer treatment by hyperthermia are presented.
This fully revised 3rd edition offers an introduction to optimal control theory and its diverse applications in management and economics. It brings to students the concept of the maximum principle in continuous and discrete time by using dynamic programming and Kuhn-Tucker theory. While some mathematical background is needed, the emphasis of the book is not on mathematical rigor, but on modeling realistic situations faced in business and management. The book exploits optimal control theory to the functional areas of management science including finance, production and marketing and to economics of growth and of natural resources. In addition, this new edition features materials on stochastic Nash and Stackelberg differential games and an adverse selection model in the principal-agent framework. The book provides exercises for each chapter and answers to selected exercises to help deepen the understanding of the material presented. Also included are appendices comprised of supplementary material on the solution of differential equations, the calculus of variations and its relationships to the maximum principle, and special topics including the Kalman filter, certainty equivalence, singular control, a global saddle point theorem, Sethi-Skiba points, and distributed parameter systems.
Optimal control methods are used to determine optimal ways to control a dynamic system. The theoretical work in this field serves as a foundation for the book, which the author has applied to business management problems developed from his research and classroom instruction. The new edition has been completely refined and brought up to date. Ultimately this should continue to be a valuable resource for graduate courses on applied optimal control theory, but also for financial and industrial engineers, economists, and operational researchers concerned with the application of dynamic optimization in their fields.
A sensitivity analysis has been conducted to quantify the relative influence of several critical parameters on the size of ablation volume generated during temperature-controlled radiofrequency ablation (RFA) of breast tumor. In order to minimize the number of experiments, Taguchi's L16 orthogonal array has been utilized to determine the effect of four parameters with 4-levels each. The parameters considered are breast density composition, target tip temperature, tumor blood perfusion rate and location of tumor from body core. A three-dimensional heterogeneous numerical model of breast with a spherical tumor of 2.2 cm has been developed for this purpose. Temperature-controlled RFA has been performed by incorporating the closed-loop feedback proportional-integral-derivative (PID) controller in the numerical model using monopolar multi-tine electrode. The size of the tumor ablation volume has been taken as the response variable that has been obtained from finite element analysis by incorporating the coupled electric field distribution, the Pennes bioheat equation and the first-order Arrhenius rate equation. A non-linear piecewise model of blood perfusion has been considered to achieve better correlation with the clinical RFA. Also, the effects of temperature-dependent changes in electrical and thermal conductivity have been incorporated. Further, analysis of variance (ANOVA) has been performed to quantify the ranking and contribution of each parameter on the size of ablation volume produced during RFA. The results obtained from the numerical study revealed that the target tip temperature and tumor blood perfusion, followed by breast density composition, have a maximum influence on the ablation volume generated during temperature-controlled RFA.
The mathematical modeling of physical and biologic systems represents an interesting alternative to study the behavior of these phenomena. In this context, the development of mathematical models to simulate the dynamic behavior of tumors is configured as an important theme in the current days. Among the advantages resulting from using these models is their application to optimization and inverse problem approaches. Traditionally, the formulated Optimal Control Problem (OCP) has the objective of minimizing the size of tumor cells by the end of the treatment. In this case an important aspect is not considered, namely, the optimal concentrations of drugs may affect the patients' health significantly. In this sense, the present work has the objective of obtaining an optimal protocol for drug administration to patients with cancer, through the minimization of both the cancerous cells concentration and the prescribed drug concentration. The resolution of this multi-objective problem is obtained through the Multi-objective Optimization Differential Evolution (MODE) algorithm. The Pareto's Curve obtained supplies a set of optimal protocols from which an optimal strategy for drug administration can be chosen, according to a given criterion.
A distributed optimal control problem on and inside a homogeneous skin tissue is solved subject to Pennes' equation with Dirichlet boundary condition at one end and Rubin condition at the other end. The point heating power induced by conducting heating probe inserted at the tumour site as an unknown control function at specific depth inside biological body is preassigned. Corresponding pseudo-port Hamiltonian system is proposed. Moreover, it is proved that bioheat transfer equation forms a contraction and dissipative system. Mild solution for bioheat transfer equation and its adjoint problem are proposed. Controllability and exponentially stability for the related system is proved. The optimal control problem is solved using strongly continuous semigroup solution and time discretization. Mathematical simulations for a thermal therapy in the presence of point heating power are presented to investigate efficiency of the proposed technique.
The application of supraphysiological temperatures (>40°C) to biological tissues causes changes at the molecular, cellular, and structural level, with corresponding changes in tissue function and in thermal, mechanical and dielectric tissue properties. This is particularly relevant for image-guided thermal treatments (e.g. hyperthermia and thermal ablation) delivering heat via focused ultrasound (FUS), radiofrequency (RF), microwave (MW), or laser energy; temperature induced changes in tissue properties are of relevance in relation to predicting tissue temperature profile, monitoring during treatment, and evaluation of treatment results. This paper presents a literature survey of temperature dependence of electrical (electrical conductivity, resistivity, permittivity) and thermal tissue properties (thermal conductivity, specific heat, diffusivity). Data of soft tissues (liver, prostate, muscle, kidney, uterus, collagen, myocardium and spleen) for temperatures between 5 to 90°C, and dielectric properties in the frequency range between 460 kHz and 3 GHz are reported. Furthermore, perfusion changes in tumors including carcinomas, sarcomas, rhabdomyosarcoma, adenocarcinoma and ependymoblastoma in response to hyperthmic temperatures up to 46°C are presented. Where appropriate, mathematical models to describe temperature dependence of properties are presented. The presented data is valuable for mathematical models that predict tissue temperature during thermal therapies (e.g. hyperthermia or thermal ablation), as well as for applications related to prediction and monitoring of temperature induced tissue changes.
Purpose: The aim of this study was to investigate the relationship between the target tissue necrosis volume and the target tissue size during the radiofrequency ablation (RFA) procedure.
Materials and methods: The target tissues with four different sizes (dxy = 20, 25, 30 and 35 mm) were modelled using a two-compartment radiofrequency ablation model. Different voltages were applied to seek the maximum target tissue necrosis volume for each target tissue size. The first roll-off occurrence or the standard ablation time (12 min) was taken as the sign for the termination of the RFA procedure.
Results: Four different maximum voltages without the roll-off occurrence were found for the four different sizes of target tissues (dxy = 20, 25, 30 and 35 mm), and they were 36.6, 35.4, 33.9 and 32.5 V, respectively. The target tissues with diameters of 20, 25 mm can be cleanly ablated at their own maximum voltages applied (MVA) but the same finding was not found for the 35-mm target tissue. For the target tissue with diameter of 30 mm, the 50 °C isothermal contour (IT50) result showed that the target tissue can be cleanly ablated, but the same result did not show in the Arrhenius damage model result. Furthermore, two optimal RFA protocols with a minimal thermal damage to the healthy tissues were found for the target tissues with diameters of 20 and 25 mm, respectively.
Conclusions: The study suggests that target tissues of different sizes should be treated with different RFA protocols. The maximum target tissue volume was achieved with the MVA without the roll-off occurrence for each target tissue size when a constant RF power supply was used.
The problem of Pennes' bioheat boundary control through a skin with its inner medium kept at a constant steady-state temperature where the outer surface subjected to conductive condition is solved by different methods. Analytical and mild solutions of Pennes' boundary control problem yield a discrete optimization problem for temperature profile at the specific depth point as well as temperature profile at all points of the skin in the final time. The pointwise optimal control problem has been solved by discretized form of the unconstrained optimization problem using analytical solution for Pennes' boundary control problem. The analytical solution has been used to solve the optimal control problems via an unconstrained/constrained optimization in Methods I and II. In Method III first, Pennes' boundary control problem is solved using mild solution by strongly continuous semigroup theory. Then, nonlinear optimization problem with linear constraints is proposed to calculate the corresponding piecewise optimal control function. All methods, have been applied to a homogeneous tissue. Mathematical simulation is done to show the optimal controls for the related states when minimum effort is used.
We investigate two well-known basic optimal control problems for chemotherapeutic cancer treatment modified by introducing a time-dependent “resistance factor”. This factor should be responsible for the effect of the drug resistance of tumor cells on the dynamical growth for the tumor. Both optimal control problems have common pointwise but different integral constraints on the control. We show that in both models the usually practised bang-bang control is optimal if the resistance is sufficiently strong. Further, we discuss different optimal stategies in both models for general resistance.
In a fixed total time of operation of the process, a distributed optimal control problem for a system described by one-dimensional bio-heat equation for a single layered homogeneous plane tissue is analytically investigated such that a desired temperature of the tissue at a particular point of location of tumour in hyperthermia can be attained during specific time by controlling point heating power induced by conducting heating probe inserted at the tumour site and also optimal surface cooling temperature[4]. Here the temperature of the tissue versus the length of the tissue at different times of operation of the process are considered for investigation of the desired temperature of the tumour.
Optimal control has guided numerous applications in chemical engineering, and exact determination of optimal profiles is essential for operation of separation and reactive processes, and operating strategies and recipe generation for batch processes. Here a simultaneous collocation formulation based on moving finite elements is developed for the solution of a class of optimal control problems. Novel features of the algorithm include the direct location of breakpoints for control profiles and a termination criterion based on a constant Hamiltonian profile. The algorithm is stabilized and performance is significantly improved by decomposing the overall NLP formulation into an inner problem, which solves a fixed element simultaneous collocation problem, and an outer problem, which adjusts the finite elements based on several error criteria. This bilevel formulation is aided by a nonlinear programming (NLP) solver (the Interior Point Optimizer (IPOPT)) for both problems as well as an NLP sensitivity component, which provides derivative information from the inner problem to the outer problem. This approach is demonstrated on 11 dynamic optimization problems drawn from the optimal control and chemical engineering literature. © 2013 American Institute of Chemical Engineers AIChE J, 2013
Many practical chemical engineering problems involve the determination of optimal trajectories given multiple and conflicting objectives. These conflicting objectives typically give rise to a set of Pareto optimal solutions. To enhance real-time decision making efficient approaches are required for determining the Pareto set in a fast and accurate way. Hereto, the current paper illustrates the use of the freely available toolkit ACADO Multi-Objective (www.acadotoolkit.org) on several chemical examples. The rationale behind ACADO Multi-Objective is the integration of direct optimal control methods with scalarisation-based multi-objective methods enabling the exploitation of fast deterministic gradient-based optimisation routines.
We describe an optimization process specially designed for regional hyperthermia of deep seated tumors in order to achieve desired steady-state tem perature distributions. A nonlinear three-dimensional heat transfer model based on temperature-d ependent blood perfusion is applied to predict the temperature. Using linearly implicit methods in time and adaptive multilevel finite elements in space, we are able to integrate efficiently the instationary nonlinear heat equation with high accuracy. Optimal heating is obtained by minimizing an integral ob ject function which measures the distance between desired and model predicted temperatures. A sequence of minima is calculated from successively improved constant-rate perfusion models employing a damped Newton method in an in ner iteration. We compare temperature distributions for two individual patients calculated on coarse and fine spatial grids and present numerical results of opti mizations for a Sigma 60 Applicator of the BSD 2000 Hyperthermia System.
In the treatment of cancerous tumors, the thermal dose is the time temperature history required to treat or destroy the undesirable tissue. The aim of this article is to calculate the optimum history of the heat source that, in the one-dimensional bioheat transfer model, results in the desired thermal dose. The time dependent strength of this source defines the accumulated energy at the end of a single heat treatment period. First the optimum control problem is formulated in infinite-dimensional form. The associated adjoint problem is obtained using the calculus of variations and an analytical formula is derived for the gradient of the functional of interest. Then a parametric representation of the control parameter is developed and the adjoint state approach is performed in conjugation with the conjugate gradient method for the solution of this control problem in finite-dimensional form. A one-dimensional numerical case is analyzed and discussed to demonstrate the performance and the robustness of the present method.
Problems demanding globally optimal solutions are ubiquitous, yet many are intractable when they involve constrained functions having many local optima and interacting, mixed-type variables.The differential evolution (DE) algorithm is a practical approach to global numerical optimization which is easy to understand, simple to implement, reliable, and fast. Packed with illustrations, computer code, new insights, and practical advice, this volume explores DE in both principle and practice. It is a valuable resource for professionals needing a proven optimizer and for students wanting an evolutionary perspective on global numerical optimization. A companion CD includes DE-based optimization software in several programming languages.
Abstract The standard method for assessing hyperthermia treatment has been calculation of cumulative equivalent minutes at 43 °C, CEM43 and its variations. This parameter normalises treatment thermal histories rather than predicts treatment results. Arrhenius models have been widely used in analysing higher temperature thermal treatments and successfully employed to predict irreversible thermal alterations in structural proteins. Unfortunately, in many, but not all cases they fail to represent thermally induced damage or cell death at hyperthermic temperatures, 43-50 °C, exhibiting significant over-prediction of the initial 'shoulder' region. The failure arises from the simplifying assumptions used to derive the irreversible reaction format that has been used in thermal damage studies. Several successful multi-parameter fit methods have been employed to model cell survival data. The two-state statistical thermodynamic model was derived from basic thermodynamic principles. The three-state model results from relaxing the assumptions under the Arrhenius formulation that result in an irreversible reaction. In other cell processes studied in vitro the irreversible Arrhenius model holds, and is sufficient to provide an accurate and useful estimate of thermal damage and cell death. It is essential in numerical model work to include multiple thermal damage processes operating in parallel to obtain a clear image of the likely outcome in tissues. Arrhenius and other C(t) models have that capability, while a single value for CEM43, does not.