Design of HIFU Treatment Plans Using Thermodynamic Equilibrium Algorithm

Abstract

High-intensity focused ultrasound (HIFU) is a non-invasive medical procedure, which is mainly used to ablate tumors externally by focusing on them with high-frequency ultrasound. Because a single ablation can process only a small volume of tissue, a succession of ablations is required to treat a large volume of cancerous tissue. In order to maximize the therapeutic effect and reduce side effects such as skin burns, careful preoperative treatment planning must be performed to determine the focal location and sonication time for each ablation. This paper proposes a novel optimization algorithm, called the thermodynamic equilibrium algorithm (TEA), inspired by the behavior of thermodynamic systems reaching their equilibrium states. Like other evolutionary algorithms, TEA starts with an initial population. Gas chambers at various thermodynamic states are employed as representatives of the population individuals, and the equilibrium state is regarded as the global minimum. The movement of thermodynamic parameters in the direction of reducing the temperature gradient forms the basis of the proposed evolutionary algorithm. During this movement, the second law of thermodynamics is checked to ensure that entropy will increase in each process. This movement leads to the state where most of the systems are at equilibrium. In this state, the systems are localized at the same position and have the same cost as the global minimum. The TEA was applied to several well-known unconstrained and constrained benchmark cost functions, and the performance was compared with other well-known optimization algorithms. The results showed that the TEA has high potential to handle various types of optimization problems with a good convergence rate and high precision. Finally, the suggested evolutionary approach is applied to HIFU treatment regimens adopting a map of patient-specific material properties and an accurate thermal model. High-quality treatment plans could be created using the suggested method, and the average amount of tissue that is over- or under-treated was less than 0.08 percent.

1. Introduction

A number of solid malignant tumors, such as pancreatic, liver, prostate, breast, fibroids, and soft-tissue sarcomas, have recently been treated with High-Intensity Focused Ultrasound (HIFU). HIFU is a non-invasive and non-ionizing treatment modality with far fewer post-treatment problems than traditional tumor/cancer treatment techniques such as open surgery, radiotherapy, and chemotherapy. Over 100,000 patients have been successfully treated throughout the globe over the last decade [1].

The basic principles of HIFU ablation include coagulative thermal necrosis caused by ultrasonic energy absorption and ultrasound-induced cavitation damage. The thermal mechanism for tissue ablation is often employed when treatments are conducted under MR guidance, since it is better understood and simpler to manage. The fundamental idea behind hyperthermic HIFU treatment is to use enough ultrasonic energy to raise the temperature by several tens of degrees, triggering coagulative necrosis, which would destroy the cancerous tissue. Targeted HIFU reduces the danger of thermal injury to tissue beyond the focused zone by producing cytotoxic temperatures only at a specific location within a small volume (e.g., 1 mm in diameter and 10 mm in length). Normally, the margin between damaged cells and healthy tissue is less than 50 micrometers [2]. By creating a continuous lesion lattice that surrounds the tumor and has appropriate tissue margins, large tumors can be completely ablated.

Despite its advantages, HIFU has disadvantages compared to other existing treatment methods, including a longer extended treatment time, lack of haptic feedback, and monitoring methods. Furthermore, treatment becomes much more difficult when essential blood vessels pass near the tumor. Blood perfusion may remove a large quantity of energy from the body, compromising the therapy effect, and even small damage to blood vessels can have serious consequences [3]. With recent improvements throughout numerical methods and IT, it is now possible to run detailed simulations that accurately predict how focused ultrasound waves behave and how temperature is distributed in heterogeneous tissue [4]. This can be effectively used for a variety of purposes, such as identifying a patient’s suitability for a certain therapy based on their unique anatomy, or verifying the effectiveness of a treatment. Model-based treatment planning, which determines the best transducer placement and sonication settings to deliver ultrasonic energy to the intended target volume, is currently carried out in a crude way, depending on heuristics rather than actual physical models of the therapy.

Evolutionary algorithms are a subset of evolutionary computations inspired by Darwinian natural evolution, which tackle complex optimization problems. It is a metaheuristic optimization algorithm based on the concept of population [5]. The major part of these algorithms is a search procedure for an optimized solution, which begins with multiple points in the solution space. Compared to traditional optimization methods or search algorithms, evolutionary algorithms do not search from a single point, but study a population of potential solutions simultaneously in parallel. These algorithms require neither implicit information nor complementary knowledge; only objective function and related merits affect the search process [6,7]. Use of evolutionary algorithms is generally very straightforward, as there are no limitations in objective functions [8]. The outputs of these algorithms usually include a number of acceptable solutions. Therefore, the final choice rests on the user, especially when the problem does not have a singular solution (for example, a family of Pareto efficiency problems). Naturally, evolutionary algorithms are suitable for providing such multiple responses simultaneously [9].

Evolutionary algorithms can be further categorized on the basis of their implementation and the method of application. Evolutionary strategy (ES) is an optimization method which is based on the idea of compatibility and evolution. This mimics the biological evolution strategy, including mutation, recombination, selection, annidation, dominance, recession, etc. The genetic algorithm (GA) is a special kind of evolutionary algorithm which is inspired by heredity and mutation actions in biology. This algorithm is a search technique used in computer science to find an approximate solution for the optimization and search problems [10]. GA is also a good option for regression-based predictive techniques. In artificial intelligence, GA is a programming technique that employs genetic evolution as a problem-solving model. The desired solution of a problem can be achieved by the genetic evolution process. These solutions are evaluated by the fitness function, and if the solution is provided with the desired accuracy, the algorithm is terminated. Particle swarm optimization (PSO) is a global mathematical optimization method which can treat the problems with the solution of one point or a surface in multi-dimensional space [11,12,13]. In such a space, initially, a number of starting points would be selected, and an initial speed of movement would be assigned. Communication channels between the particles are also considered [14]. Then, the particles are moved in the solution space, and results are calculated based on the eligibility criteria after each time interval. Over time, this algorithm allows all particles to communicate with all the other particles. Thus, the whole swarm moves toward the optimal position found. The PSO method has shown great performance in solving continuous mathematical optimization problems [15,16]. The artificial bee colony (ABC) algorithm is another mathematical optimization algorithm based on swarm intelligence, i.e., intelligent behavior of honeybees [17]. This method offers use of swarm intelligence in the development of artificial systems to solve complex problems in traffic and transportation. Based on the foraging behavior of bee colonies, metaheuristic bee colony optimization (BCO) is able to solve combinatorial problems, as well as combined problems with uncertain characteristics [18]. Ant colony optimization (ACO) is inspired by the pheromone trail laying and following behavior of some ant species. ACO is known to be one of the best methods to tackle shortest path problem, which sometimes is very difficult and time consuming to solve. In ACO, artificial ants are moving around and leaving traces on the chart graph of the problem, like real ants leaving signs on their way, to help the next artificial ants find better solutions to the problem [19,20]. Simulated annealing (SA) is a simple and effective metaheuristic algorithm used to solve mathematical optimization problems. In order to solve difficult optimization problems, Kirkpatrick et al. proposed a SA optimization method based on the analogy with gradual annealing in solids [21]. The gradual annealing technique is used by metallurgists to achieve a state in which the solid material is well groomed while minimizing energy consumption. This technique involves placing the material at a high temperature and then gradually lowering the temperature [22]. Recently, the hunter algorithm (HA) was proposed as a novel optimization algorithm which mimics the behavior of a group of hunting dogs [23]. Hunting dogs allow humans to pursue and kill prey by following the smell of the prey, which is the key concept of this algorithm. In the HA, the hunters perform the action of hunting for trapping the prey. High performance can be achieved by setting few parameters, which is the most important benefit of this algorithm.

The aforementioned optimization algorithms are extensively used to solve diverse kinds of optimization problems in applied sciences, engineering, industry, biology, and computer science. Furthermore, optimization techniques are widely adopted in various fields such as artificial intelligence, hardware synthesis and testing, design and optimization of digital and analog filters, multi-processor systems, robots control, and logic cells layout [24,25,26,27,28,29,30,31,32,33,34,35,36]. However, each of these algorithms suffers from its own deficiencies. For example, a major problem of the proposed formula for PSO is that the speed is constantly increasing, while the particle should slow down as it gets closer to the global optimum so that it does not go past the global optimum. There are several solutions to this problem, but they are effective only for specific cases [37,38]. The genetic algorithm has the inherent disadvantages of producing unpredictable and unripe results [39,40]. Moreover, another major drawback of these algorithms is the need to set many parameters for each problem to achieve high performance [41]. Therefore, finding high-efficiency algorithms has been one of the most important tasks for optimization in engineering and AI applications [42,43].

In the present paper, a new novel algorithm for global optimization, called the thermodynamic equilibrium algorithm (TEA), is proposed. TEA is inspired by the equilibrium process of thermodynamic systems that move toward equilibrium according to the second law of thermodynamics, while satisfying the first law of thermodynamics. During the process, the equilibrium state is checked by using the first and second law of thermodynamics. This allows the systems to converge to the global minimum of the cost function. This paper describes how the thermodynamic equilibrium procedure can be modeled with physically relevant systems and implemented for optimization purposes. Moreover, this work tackles the HIFU treatment planning by utilizing an optimization-based method based on TEA. The inputs to the optimization are the anticipated target volume and the organs that have to be ablated, and the TEA produces a workable treatment strategy. This scheme has the potential to improve targeting accuracy of HIFU and shorten treatment durations, ultimately improving clinical outcomes for HIFU patients.

2. Materials and Methods

2.1. Thermodynamics

Thermodynamics is a branch of the natural sciences that discusses the relationship between heat, energy, and work. Macroscopic variables (such as temperature, internal energy, entropy, and pressure) are used in thermodynamics to characterize the state of the system and to derive the physical laws that govern the change in the state. Thermodynamics describes the average behavior of the system composed of a large number of microscopic particles. The rules governing thermodynamics can also be inferred through statistical mechanics.

2.2. Thermodynamic Equilibrium Algorithm

TEA is inspired by thermodynamic phenomena, and can be categorized as an evolutionary algorithm. In this method, the coordinates of each system are converted to temperature or volume, both of which are thermodynamic parameters. Figure 1 shows the flowchart of TEA’s working sequence. Like other evolutionary algorithms, TEA begins with an initial population (thermodynamic systems with different states). Given that each system is in its own condition, it is coupled with another system in the next step. The two systems are placed side by side so that they can exchange heat freely (Figure 2), and thermodynamic equilibrium processes of the two interactive systems are initiated. The algorithm intends to develop a mathematical expression mimicking thermodynamic phenomena occurring in the coupled systems to find the best result.

Then, according to the first law of thermodynamics, the equilibrium temperature and volume are calculated at each stage. Knowing the equilibrium state, both systems move toward equilibrium and update their recent states.

During the processes for equilibrium, specific conditions must be met and the new state must comply with the second law of thermodynamics. The process will be completed when the coupled systems reach equilibrium. Some systems may be trapped in the domain of the function (local extrema), but a large number of systems will reach equilibrium, and their final thermodynamic state will be declared as the optimum parameter.

2.3. Working Sequence of TEA

2.3.1. Initialize Thermodynamic Systems

The goal of optimization is to find the optimal solution according to the problem variables. In genetic algorithm (GA) terminology, an array of variables that need to be optimized is frequently called the chromosome, but here, the term is replaced with the thermodynamic system. In a $N_{var}$-dimension optimization problem, each system is a dimensional array defined by Equation (1).

$$x=[T,V_1,V_2,\dots ,V_{N_{var}-1}]$$

(1)

where T is the temperature and V is the volume, which may have more than one parameter. Therefore, for convenience, the problem can be turned into a two-dimensional problem by defining a new variable ∀ called “overall volume”:

$$\forall =\frac{{\sum }_{i=1}^{N_{var}-1}{V}_i}{N_{var}-1}$$

(2)

Now, the thermodynamic state of each system can be simply defined as:

$$x=[T,\forall ]$$

(3)

The cost of each system is determined by the cost function f, as shown in Equation (4).

$$cost=f\left(x\right)=f(T,\forall )$$

(4)

To start the algorithm, the initial optimization population is created by the number of $N_{sys}$. Systems are randomly scattered in the domain of the function based on the upper and lower limits.

2.3.2. Coupling of the Thermodynamic Systems

Initially, each system is coupled with the closest system in the function domain according to its coordinates. Then, the coupled systems exchange heat or perform work together to gradually reach equilibrium. Over time, the temperature gradient decreases and, as a result, the scope of the optimization function is well studied and the optimal value is found.

2.3.3. Compute Equilibrium Temperature and Volume

After the systems are coupled, heat exchange and work are performed, resulting in the changes in the temperature and the volume of the systems. If we consider the two systems as two cylinders side by side, the temperature and volume are different, but the pressure is the same in both systems (Figure 2). Additionally, the pistons in both systems are free to move, so each system can perform either positive or negative work. In each thermodynamic process, the pressures in both systems are equal.

According to the first law of thermodynamics, the internal energy change of a system can be calculated as follows:

$$\Delta E=Q-W$$

(5)

where Q is the amount of heat transfer into the system and W represents the work performed by each system. Due to the law of conservation of energy, the heat absorbed from one system is equal to the heat lost by the other ($Q_1=-Q_2$). Thus, the following relationship is established between these two systems:

$$\Delta E_1+W_1=-(\Delta E_2+W_2)$$

(6)

The energy change for ideal gases can be calculated by $\Delta E=m{c}_v\Delta T$, and the thermodynamic work by $W=P\Delta \forall$. By substituting these terms into Equation (6) and dividing with P, Equation (7) is obtained:

$$\frac{m_1{c}_v}{P}(T_{eq}-T_1)+({\forall }_{eq}-{\forall }_1)=-\frac{m_2{c}_v}{P}(T_{eq}-T_2)-({\forall }_{eq}-{\forall }_2)$$

(7)

where the subscript $eq$ represents the thermodynamic state at equilibrium. By assuming that $m_1=m_2$ and for simplicity $\alpha =\left(m_1{c}_v\right)/P=\left(m_2{c}_v\right)/P=1$, Equation (7) can be simplified as:

$$(T_{eq}-T_1)+({\forall }_{eq}-{\forall }_1)=(T_2-T_{eq})+({\forall }_2-{\forall }_{eq})$$

(8)

Note that Equation (8) is only dependent on temperature and overall volume at equilibrium state, which are still unknown at present.

On the other hand, the ideal gas law can be implemented as follows:

$$P\forall =nRT$$

(9)

Based on the conservation law of mass, the total molar mass at equilibrium state is equal to the sum of the molar mass of each system:

$$n_1+n_2=2n_{eq}$$

(10)

By substituting Equation (9) into Equation (10),

$$\frac{{\forall }_1}{T_1}+\frac{{\forall }_2}{T_2}=2\frac{{\forall }_{eq}}{T_{eq}}$$

(11)

By combining Equations (8) and (11), $T_{eq}$ is obtained as follows:

$$T_{eq}=\frac{T_1^2{T}_2+T_1{T}_2^2+T_1{T}_2{\forall }_1+T_1{T}_2{\forall }_2}{T_1{T}_2+T_2{\forall }_1+T_1{\forall }_2}$$

(12)

Additionally, ${\forall }_{eq}$ can be derived as:

$${\forall }_{eq}=\frac{T_1^2{\forall }_2+T_1{T}_2{\forall }_1+T_1{T}_2{\forall }_2+T_1{\forall }_1{\forall }_2+T_1{\forall }_2^2+T_2^2{\forall }_1+T_2{\forall }_1^2+T_2{\forall }_1{\forall }_2}{2(T_1{T}_2+T_2{\forall }_1+T_1{\forall }_2)}$$

(13)

At this stage, with the help of the first law of thermodynamics and the ideal gas law, the equilibrium state of the hypothetical systems can be calculated (Figure 3).

2.3.4. Update System Thermodynamic State

After calculating the equilibrium temperature ($T_{eq}$) and overall volume (${\forall }_{eq}$), the systems will move in the direction of equilibrium. In fact, the new thermodynamic state of the system is the result of calculation according to the thermodynamic laws. If each system reaches equilibrium immediately, the states of the two systems will be equal and they will not exchange energy. In this case, the domain of the function will not be searched further, and the optimal answer cannot be obtained. To prevent this, the following relationship is recommended for new temperature and overall volume:

$$T_{i,new}=\frac{T_i+T_{eq}}{2}$$

(14)

$${\forall }_{i,new}=\frac{{\forall }_i+{\forall }_{eq}}{2}$$

(15)

where subscript i represents system number.

If ∀ value is composed of more than one parameter, such as the cases in a three or higher dimensional problem, $V_i$ can be obtained as follows:

$$V_{i,new}=V_i+{\forall }_{i,new}$$

(16)

Using the proposed relationships above, the two systems grow closer to equilibrium at each stage, which causes the entire domain to be gradually searched and the optimal point to be found.

2.3.5. Check for Entropy Increase

After updating the thermodynamic state, we need to ensure that the systems are moving toward the optimal point of the function. Given that the second law of thermodynamics requires processes to move in the direction of increasing entropy, each system in the proposed algorithm must proceed in the direction of decreasing the cost function. For this purpose, at this stage, it is necessary to examine whether the cost function has decreased in the new state. The relationship between cost function and entropy can be defined as follows:

$$cost=\frac{1}{entropy}\equiv \frac{temperature}{heat}$$

(17)

Therefore, the attempt of the algorithm to minimize the cost function results in maximizing the system’s entropy (S). The desired condition is set by the following relation:

$$S_{new}-S_{eq}\le 0$$

(18)

By converting entropy into cost function, the condition can be translated as follows:

$$f(T_{new},{\forall }_{new})-f(T_{eq},{\forall }_{eq})\ge 0$$

(19)

The above condition causes the algorithm to move in the direction to optimize the functions. If the condition is not met, a counter is defined to count the number of unsuccessful attempts and thermodynamic conditions are updated as follows:

$$T_{i,new}=\frac{T_i+T_{eq}}{2^{j_{new}}}$$

(20)

$${\forall }_{i,new}=\frac{{\forall }_i+{\forall }_{eq}}{2^{j_{new}}}$$

(21)

where $j_{new}=j+1$. Due to the above relationships, in such a case, the system moves towards the balance state at a slower speed compared to Equations (14) and (15) to satisfy the aforementioned condition (Equation (19)). This part of the algorithm is similar to examining the second law of thermodynamics and prevents blind search of the function domain.

2.3.6. Check for Thermodynamic Equilibrium

Finally, a large number of coupled systems reach to equilibrium and accumulate at one point, except for a few that may have been trapped. This method has the advantage that not all systems have the same state, so the cost of each system varies with the thermodynamic equilibrium process. Therefore, the system with the lowest cost and its state will be reported.

There are two criteria for stopping the proposed algorithm. The algorithm will be stopped after a certain number of iterations, or when the systems reach an equilibrium state. For the latter, if the difference of the cost from last two steps remains lower than a certain value, stop condition is met and the iterative process of the algorithm is completed, as expressed by the following equation:

$$T_{new}-T<ϵ$$

(22)

where $ϵ$ is a threshold value which can be defined based on the required accuracy.

2.4. HIFU Treatment Plan

To properly cover a vast target region with HIFU ablation, a number of sonications are required. The potential solution I during the therapy, which is the HIFU transducer location, follows a certain course in the tissue. The HIFU therapy is not a continuous process, but the ultrasound is intermittently focused on a specific location in the tissue during each sonication. The total number of sonications for a specific tumor is restricted to N, which is often in the low tens. Sonication duration $t_{on}$ and cooling interval $t_{off}$ determine the amount of energy supplied in a single sonication. One sonication is defined as a quintuple $S_i$, which is a function of spatial coordinates of beam focus, sonication and cooling intervals $t_{on}$ and $t_{off}$:

$$I=(S_1,S_2,\dots ,S_N),where{S}_i=(x_i,y_i,t_{on,i},t_{off,i})$$

(23)

Determining the best location and sonication times for the specified number of ablations in order to thoroughly treat the intended target volume while protecting the surrounding organs is how the treatment planning problem is described in this case. The four parameters which formed $S_i$ in Equation (23) are the optimization parameters, and they are converted to T and ${\forall }_i$ based on Equation (1).

2.5. Fitness Function

There are various phases to evaluate the quality of prospective treatment programs. First, the expected form and location of the ultrasonic focal region, as well as the sonication duration, are used to calculate the heat deposition for each sonication. Second, the temperature distribution in the domain is calculated during the treatment using a numerical thermal model. The treatment plan simulates the whole sonication process using a heat model. Third, the treatment domain is then examined, and the locations where the thermal dose threshold has been exceeded are discovered. Finally, the areas that have not been adequately treated are highlighted, and the quality of the treatment plan is evaluated.

2.5.1. Calculation of the Heat Deposition

The first step in analyzing a treatment plan is to compute the heat deposition. The physical characteristics and the placement of the focal point are determined by considering the nonlinear behavior of the ultrasonic wave as it travels through heterogeneous tissue. The amount of thermal energy that is deposited at the focal point is proportional to the medium’s absorption of ultrasonic energy. It is possible to get an exact forecast of the heat deposition by using various models such as basic ray-tracing models [44], simplified forward wave propagation models [45], or precise full-wave models [46]. Unfortunately, none of these models can be used for TEA optimization because of their lengthy execution times (on the order of minutes).

As a result, some assumptions are required to simplify the problem. The initial assumption is that the focus’s center may be located at the coordinates determined by sonication $S_i=[x_i,y_i]$. This is accomplished via the HIFU transducer’s precise electrical steering of the ultrasound beam [47]. Second, the focus is assumed to be an elliptical shape in form, with the length of the axis proportional to the distance from the HIFU transducer (see Figure 4). Based on the power of the transducer, elliptical axes can be determined by the variance and used to estimate the volume rate of heat deposition using the Gaussian distribution of energy in the focus [48,49,50], i.e., the acoustic model is not utilized to compute the heat source term. Finally, in order to keep things simple, only two-dimensional issues are considered, and it is assumed that the transducer axes are always parallel to the domain axes. At this stage, the total energy deposited in the tissue is proportional to the sonication time $t_{on}$.

2.5.2. Thermal Model Execution

The numerical thermal model is then used to determine heat diffusion in the tissue in the second stage. Heat diffusion is predicted using the so-called Pennes’ bioheat equation [51], which includes a matching source term for ultrasonic energy absorption (see Equation (24)).

$$\rho C\frac{\partial T}{\partial t}=\nabla .(k\nabla T)+W_b{C}_b(T-T_a)+Q$$

(24)

The tissue-specific heat C and blood $C_b$ [J/kg°C] perfusion-related parameter $W_b$ [kg/m³s], arterial temperature $T_a$, and ultrasonic transducer power Q [W/m³] are all taken into account in this equation. The arterial temperature is considered to be 37 degrees Celsius.

Blood perfusion, which can cool the tissue and hinder the thermal ablation of cancerous cells close to blood vessels, should be carefully considered, especially when determining the energy balance in heat transmission. This effect becomes more important as the heating period grows longer; for example, for volumetric ablation of vast tissue regions.

Figure 5 depicts the temperature at the focus center during the duration of a 3-s sonication and a 7-s cooling interval. The temperature rises as soon as the transducer is turned on. Due to heat diffusion and blood circulation, the temperature gradually drops back to ambient levels after the transducer is turned off (during the cooling interval).

Following sonication, the primary focus axis was subjected to temperature measurements every 2 s (see Figure 6). The temperature distribution in space is Gaussian when a major blood artery is absent. The temperature drops from a peak of around 65 °C at the end of the sonication phase to about 38 °C at the end of the cooling phase. On the other hand, the surrounding region with temperatures above 37 °C is progressively expanding.

Thermal damage is often calculated using the Sapareto–Dewey iso-effect thermal dose equation [52], which gives the equivalent time in seconds required to induce the same biological effects at 43 °C. This statistic is known as cumulative equivalent minutes (CEM₄₃). After calculating CEM₄₃ at each location in the tissue (see Equation (25)), $S_i$ averages the results from all sonications.

$$CE{M}_{43}={\int }_0^{t_{on}+t_{off}}{R}^{(43-T)}dt,whereR=\left\{\begin{array}{cc}0\hfill & forT\le 39 °\mathrm{C}\hfill \\ 0.25\hfill & for39 °\mathrm{C}<T\le 43 °\mathrm{C}\hfill \\ 0.5\hfill & forT>43 °\mathrm{C}\hfill \end{array}\right.,$$

(25)

Thermal doses of 240 min at 43° Celsius irreversibly destroy and coagulate vital cellular protein, tissue structural components, and the vasculature, leading to rapid tissue death [1]. In Figure 6, the region where the dose exceeds 240 CEM₄₃ is marked by yellow bar. Note that the threshold may vary depending on the kind of tissue.

Patient-specific models of tissue anatomy are used to generate exact tissue parameter settings for implementation in Matlab using the FDA HIFU toolbox [53]. The Simulator predicts many important characteristics of continuous wave, high-intensity therapeutic ultrasound beams and their heating effects. The formulation is accurate and unconditionally stable for a homogeneous medium. Compared to standard pseudospectral time-domain approaches, the time scheme allows for greater time steps with the same degree of accuracy in a heterogeneous medium. As a result, the convergence testing determines the resolutions in space and time for the grid that divides the simulation domain.

2.5.3. Evaluation of the Treated Area

A cumulative spatial map of CEM₄₃ for treatment solution I is produced by the thermal model. To create a binary mask of damaged tissue, this map is thresholded at 240. The effectiveness of HIFU treatment is evaluated using the criterion of destroying all tissues of the target site without damaging the prohibited site (the organs at risk). Do not care zones may also be set to offer the optimization algorithm more flexibility.

For a two-dimensional instance, the fitness function can be represented as:

$$cost=f={\int }_0^Y{\int }_0^X((D\ast \overline{C})+(\overline{D}\ast C))dxdy,R=\left\{\begin{array}{cc}0\hfill & forCE{M}_{43}\le 240\hfill \\ 1\hfill & forCE{M}_{43}>240\hfill \end{array}\right.,D\in {\mathbb{R}}^{+},\overline{D}\in {\mathbb{R}}^{+},$$

(26)

where X, Y are the x and y dimensions of the domain, C is the bi-mask corresponding to the treated region, $\overline{C}$ is the complementary mask corresponding to the untreated area, D is the target map designating the treated area, and $\overline{D}$ is the banned area. Since D and $\overline{D}$ are described as functions over a two-dimensional space, users may choose the level of urgency for each point in the space. This enables the inclusion of two crucial concepts to the fitness function: (1) a fine-tuning of the target and prohibited regions’ shapes; and (2) the addition of a gradient to the fitness function space that directs evolution away from the prohibited areas and toward the desired areas.

3. Results & Discussions

3.1. Optimization Performance

The proposed TEA was tested with 25 optimization test functions. All of them were minimization problems. Details of the functions are listed in

Table 1. Details of the functions [54].

Function Name	3D Plot	Function Formula	Minimum Value	Search Domain	Optimum Value of the Cost Function
					GA	TEA
Ackley’s function		$f_1(x,y)=$ $-20exp(-0.2\sqrt{(0.5(x^2+y^2)}))$ $-exp\left(0.5\right(cos\left(2\pi x\right)+$ $cos\left(2\pi y\right)\left)\right)+e+20$	$f_1(0,0)=0$	$-5<x,y<5$	$3.9096$$\times {10}^{-3}$	0
Sphere function		$f_2(x,y)=x^2+y^2$	$f_2(0,0)=0$	$-\infty <x,y<\infty$	$2.5002$$\times {10}^{-3}$	0
Rosenbrock function		$$\begin{array}{l}{f}_3(x,y)=100{(y-x^2)}^2\\ +{(x-1)}^2\end{array}$$	$f_3(1,1)=0$	$-\infty <x,y<\infty$	$2.8738$	0
Beale’s function		$\begin{array}{l}{f}_4(x,y)={(1.5-x+xy)}^2\\ +{(2.25-x+x{y}^2)}^2\\ +{(2.625-x+x{y}^3)}^2\end{array}$	$f_4(3,0.5)=0$	$-4.5<x,y<4.5$	$5.2628$$\times {10}^{-1}$	0
Goldstein-Price function		$\begin{array}{l}{f}_5{(x,y)=(1+(x+y+1)}^2\\ (19-14x+3x^2\\ -14y+6xy+3y^2\left)\right)\\ (30+{(2x-3y)}^2\\ (18-32x+12x^2\\ +48y-36xy+27y^2\left)\right)\end{array}$	$f_5(0,-1)=108$	$-2<x,y<2$	$111.0029$	$108.1436$
Booth’s function		$\begin{array}{l}{f}_6(x,y)={(x+2y-7)}^2\\ +{(2x+y-5)}^2\end{array}$	$f_6(1,3)=0$	$-10<x,y<10$	$7.6235$$\times {10}^{-4}$	0
Matyas’s function		$\begin{array}{l}{f}_7(x,y)=\\ 0.26(x^2+y^2)-0.48xy\end{array}$	$f_7(0,0)=0$	$-10<x,y<10$	$2.3056$$\times {10}^{-5}$	0
Lévi function		$\begin{array}{l}{f}_8(x,y)=si{n}^2\left(3\pi x\right)\\ +{(x-1)}^2(1+si{n}^2\left(3\pi y\right))\\ +{(y-1)}^2(1+si{n}^2\left(2\pi y\right))\end{array}$	$f_8(1,1)=0$	$-10<x,y<10$	$3.3600$$\times {10}^{-4}$	$1.3498$$\times {10}^{-31}$
Three-hump camel function		$\begin{array}{l}{f}_9(x,y)=2x^2-1.05x^4\\ +\frac{x^6}{6}+xy+y^2\end{array}$	$f_9(0,0)=0$	$-5<x,y<5$	$8.1648$$\times {10}^{-6}$	0
Cross-in-tray function		$\begin{array}{l}{f}_{10}(x,y)=\\ -0.0001\left(\right|sin\left(x\right)sin\left(y\right)\\ exp\left(\right|100-\frac{\sqrt{x^2+y^2}}{\pi }{\left|\right)|+1)}^{0.1}\end{array}$	$\begin{array}{l}{f}_{10}(1.6494,\\ -1.3494)=\\ -2.0626\end{array}$	$-10<x,y<10$	$-2.0626$	$-2.0626$
Eggholder function		$\begin{array}{l}{f}_{11}(x,y)=\\ -(y+47)sin\left(\sqrt{|y+\frac{x}{2}+47|}\right)\\ -xsin\left(\sqrt{|x-(y+47\left)\right|}\right)\end{array}$	$\begin{array}{l}{f}_{11}(512,\\ 404.2319)=\\ -959.6407\end{array}$	$-512<x,y<512$	$-8.0351$$\times {10}^2$	$-9.51$$\times {10}^2$
Rastrigin function		$\begin{array}{l}{f}_{12}(x,y)=\\ 20+x^2-10cos\left(2\pi x\right)+y^2\\ -10cos\left(2\pi y\right))\end{array}$	$f_{12}(0,0)=0$	$-5.12<x,y<5.12$	$7.17$$\times {10}^{-8}$	0
Bukin function		$\begin{array}{l}{f}_{13}(x,y)=\\ 100\sqrt{|y-0.01x^2|}+0.01|x+10|\end{array}$	$f_{13}(-10,1)=0$	$-15<x<-5$ $-3<y<3$	$0.1165$	$0.01119$
Himmelblau’s function		$\begin{array}{l}{f}_{14}(x,y)=\\ {(x^2+y-11)}^2+{(x+y^2-7)}^2\end{array}$	$f_{14}(3,2)=0$	$-5<x,y<5$	$8.55$$\times {10}^{-9}$	0
Easom function		$\begin{array}{l}{f}_{15}(x,y)=\\ {-cosxcosyexp(-((x-\pi )}^2\\ +{(y-\pi )}^2\left)\right)\end{array}$	$f_{15}(\pi ,\pi )=-1$	$-100<x,y<100$	$-8.11$$\times {10}^{-5}$	$-1$
Holder table function		$\begin{array}{l}{f}_{16}(x,y)=\\ -\left|sinxcosyexp\right(|1\\ -\frac{\sqrt{x^2+y^2}}{\pi }\left|\right)|\end{array}$	$\begin{array}{l}{f}_{16}(8.05502,\\ 9.66459)=\\ -19.2085\end{array}$	$-10<x,y<10$	$-19.2084$	$-19.2085$
McCormick function		$\begin{array}{l}{f}_{17}(x,y)=\\ sin(x+y)+{(x-y)}^2\\ -1.5x+2.5y+1\end{array}$	$f_{17}(-0.54719$ $,-1.54719)=$ $-1.9133$	$\begin{array}{l}-1.5<x<4\\ -3<y<4\end{array}$	$-1.9132$	$-1.9132$
Schaffer function N. 2		$\begin{array}{l}{f}_{18}(x,y)=\\ 0.5+\frac{si{n}^2(x^2-y^2)-0.5}{{[1+0.001(x^2+y^2)]}^2}\end{array}$	$f_{18}(0,0)=0$	$-100<x,y<100$	$0.0045$	$0.004139$
Schaffer function N. 4		$\begin{array}{l}{f}_{19}(x,y)=\\ 0.5+\frac{co{s}^2\left[sin\right(|x^2-y^2\left|\right)]-0.5}{{[1+0.001(x^2+y^2)]}^2}\end{array}$	$\begin{array}{l}{f}_{19}(0,1.25313)=\\ 0.292579\end{array}$	$-100<x,y<100$	$0.3615$	$0.2535$
Styblinski-Tang function		$\begin{array}{l}{f}_{20}(x,y)=\\ \frac{x^4-16x^2+5x+y^4-16y^2+5y}{2}\end{array}$	$\begin{array}{l}{f}_{20}(-2.903534,\\ -2.903534)=\\ -78.33233\end{array}$	$-5<x,y<5$	$-78.0297$	$-78.2461$

. The first problem $f_1$ is Ackley’s function, which was frequently used as an optimization benchmark function.

For $f_1$, a 3D plot of the function is shown in Figure 7. The global minimum of the function in interval $-5<x,y<5$ is located at $f_1(0,0)$ and the cost of 0.

In the optimization process using TEA, the initial population of 100 systems was chosen in Figure 8a. Every coupled system individually explored the minimum point. The systems are indicated by × marks. Figure 8b–d show the movements of systems at iterations 17, 34, and 50. It can be seen that at iteration 17, while most of the systems were located at local minima of the function, a couple of them had reached the global minimum point, at which the best cost is $3.591\times {10}^{-6}$. At iteration 34, although a couple of systems were trapped, the majority of them reached the equilibrium and the best cost was $8.141\times {10}^{-11}$. At iteration 50, almost all systems had reached the optimal location. The final answer was $9.251\times {10}^{-15}$, which is very close to zero.

As examples of other optimization algorithms, continuous GA and PSO were also applied to this problem. To make a fair comparison of the performance among these methods, the initial population of 100 was chosen for all of them. In continuous GA, mutation and selection rates were set to 0.3 and 0.5, respectively, and for PSO, both $p\_increment$ and $g\_increment$ were set to 2. The minimum cost of all population versus generation is shown in Figure 9a. At the fifth iteration, TEA found the minimum to near one-thousandths and is attempting to increase the accuracy of the answer in the following iterations. GA and PSO converged to the global minimum at iteration 31 and 36, respectively, by the accuracy of the one-thousandth.

Problems $f_2$ and $f_3$ are Sphere function and Rosenbrock function, respectively. The minimum cost against the iteration for $f_2$ and $f_3$ achieved by TEA are compared with those by GA and PSO, as shown in Figure 9b,c, respectively. Both of the plots indicate that TEA performed best, followed by GA and PSO.

Table 1 shows the results of optimizations performed by GA and TEA for 20 different test functions to compare the performances when dealing with various types of problems.

Table 2. Details of the functions [54].

Function Name	3D Plot	Function Formula	Minimum Value	Search Domain	Optimum Value of the Cost Function
					GA	TEA
Rosenbrock function constrained with a cubic and a line		$\begin{array}{l}{f}_{21}(x,y)=100{(y-x^2)}^2\\ +{(x-1)}^2\\ \mathrm{subjected} \mathrm{to}: {(x-1)}^3-y+1\le 0\\ \mathrm{and} x+y-2\le 0\end{array}$	$f_{21}(1,1)=0$	$-1.5<x<1.5$ $-0.5<y<2.5$	$3.4817$	0
Rosenbrock function constrained to a disk		$\begin{array}{l}{f}_{22}(x,y)=100{(y-x^2)}^2\\ +{(x-1)}^2\\ \mathrm{subjected} \mathrm{to}: x^2+y^2\le 2\end{array}$	$f_{22}(1,1)=0$	$-1.5<x,y<1.5$	$2.8727$	0
Mishra’s Bird function- constrained		$\begin{array}{l}{f}_{23}(x,y)=sinyexp\left[{(1\ast cosx)}^2\right]+\\ cosxexp\left[{(1-siny)}^2\right]+{(x-y)}^2\\ \mathrm{subjected} \mathrm{to}: {(x+5)}^2+{(y+5)}^2<25\end{array}$	$\begin{array}{l}{f}_{23}(-3.1302468,\\ -1.5821422)=\\ -106.7645367\end{array}$	$-10<x<0$ $-6.5<y<0$	$-106.7645$	$-106.7645$
Townsend function		$\begin{array}{l}{f}_{24}(x,y)=-co{s}^2\left((x-0.1)y\right)-xsin(3x+y)\\ \mathrm{subjected} \mathrm{to}: x^2+y^2<[2cost-\frac{1}{2}cos2t-\\ \frac{1}{4}cos3t-\frac{1}{8}{cos4t]}^2+{\left[2sint\right]}^2\\ \mathrm{where}: r=Atan2(x,y)\end{array}$	$\begin{array}{l}{f}_{24}(2.0052938,\\ 1.1944509)=\\ -2.0239884\end{array}$	$-2.25<x<2.5$ $-2.5<y<1.75$	$-1.1486$	$-2.0195$
Simionescu function		$\begin{array}{l}{f}_{25}(x,y)=0.1xy\\ \mathrm{subjected} \mathrm{to}: x^2+y^2\le [1+\\ 0.2cos\left(8ta{n}^{-1}\frac{x}{y}\right){]}^2\end{array}$	$\begin{array}{l}{f}_{25}(0.84852813,\\ -0.84852813)=\\ -0.072\end{array}$	$-1.25<x,y<1.25$	$-0.01562$	$-0.0717$

presents the optimization results for five different constrained test functions, also by GA and TEA to investigate the precision of the proposed algorithm in constrained optimization. The numbers of initial systems and iteration used in problem $f_1$ to $f_{25}$ were 100 and 50, respectively. Table 1 and Table 2 show the minimum cost of all systems in problems $f_1$ to $f_{20}$ and $f_{21}$ to $f_{25}$, respectively.

In all test functions, TEA yields better results than GA. In smooth functions, such as $f_2$, $f_6$, and $f_9$, both algorithms have comparable performance. GA could find the answers within the given number of iterations and tried to improve the accuracy in further iterations. However, for complex functions which include several local optima, such as $f_{11}$ and $f_{13}$, GA was trapped in the local minima and required a much larger number of iterations to achieve an acceptable performance. Additionally, in complex constrained function such as $f_{24}$, GA showed higher tendency to converge towards local optima instead of the global minimum of the function. In contrast, by checking the second law of thermodynamics, TEA ensured that most of the systems were able to obtain the optimal answer and avoided local minima convergency.

3.2. Application to HIFU

The experimental study given in this research has three primary goals: (1) testing the idea that the TEA can develop appropriate HIFU treatment plans, (2) discovering appropriate TEA parameters, and (3) assessing the TEA’s computing complexity.

3.2.1. HIFU Treatment Setup

For the application of TEA to HIFU treatment, realistic parameters and situations were employed. The material properties were established using data from real patients, and the transducer parameters were created using data from a true-to-life HIFU transducer. Simulation parameters were selected using numerical analysis.

Maps of the material characteristics of the human body were needed to illustrate the suggested algorithm’s capacity to develop appropriate treatment regimens. AustinWoman, an open-source voxel model, was used in this case [55]. This is a segmentation of a digital image dataset obtained as a result of the Visible Human Project run by the US National Library of Medicine. The original dataset consists of digital cryosection images with a 0.33 mm pixel spacing of a 59-year-old female corpse taken at 0.33 mm intervals along the axial direction. There are also radiological imaging (CT and MRI) accessible. The AustinWoman segmentation divides the digital image data into 58 material labels, each of which stands for one of the main tissue types in the human body. The distribution of tissue density, specific heat, and thermal conductivity was obtained using the IT’IS tissue property database V3.0 [56]. Capillary blood perfusion was solely considered throughout this examination for the purpose of simplicity.

One thorax target inside the left lobule was utilized for the case study of HIFU therapy planning. The divided section of the body used for treatment planning is shown in Figure 10. The target location is near the center of the breast. D (target map) at this area is equal to 1 ($D_{x,y}=1$) and elsewhere equals to zero ($D_{x,y}=0$).

The HAIFU JC-200’s nominal characteristics and a single-element transducer were used to size the heat source [57]. The focusing beam has an oval shape with a diameter of around 19 × 2.5 mm, a radius of curvature of 145 mm, an aperture diameter of 200 mm, and a frequency of 0.95 MHz. This dimension was used to calculate the half-maximum full-width ellipsoidal heat source term. The peak rate of heat deposition in space was set to 100 W/cm², which is close to the values used in clinical treatments.

Following the results of numerical convergence testing, the following parameters were included into the numerical thermal model:

• 495 × 495 grid points, discretized simulation domain; periodic boundary condition.
• 0.2 mm of spatial resolution. Linear interpolation was used to upsample the original AustinWoman data.
• 0.1 s temporal resolution.
• The total length of the simulation ${\sum }_{i=0}^N(t_{on,i}+t_{off,i})$.
• Positions of the ultrasound focus center are limited to the bounding box at grid positions [270, 230] × [345, 295].
• Maximum sonication and cooling periods $t_{on}$ = [0, 5 s], $t_{off}$ = [0, 20 s].
• Number of sonications considered $N\in {\mathbb{Z}}^{+}$.

The numerical model and the TEA were run in Matlab 2021b. The Matlab Parallel Computing Toolbox was used to parallelize the execution over 16 processor cores.

3.2.2. Evaluation of the Equilibrium Process

This section assesses the TEA’s ability to determine a viable treatment plan for a specific patient. For statistical analysis, 15 separate runs were conducted with a maximum execution duration of 48 h per run. To evaluate the performance, three key metrics were looked at, including: (1) the best treatment plan’s quality, (2) the amount of fitness assessments that must converge, and (3) the best system’s progress over time.

The capacity of the TEA to identify a suitable treatment plan is the first and most significant criterion. This was assessed with the fitness value defined in Equation (26). With the fitness value equal to zero, the ideal solution would cover the entire treated region while leaving the rest completely undamaged. Figure 11 displays the histogram of fitness values compiled from the top solutions produced in 15 distinct runs. Here, the effects of numbers of systems and sonications on the fitness value are presented.

The overall trend in Figure 11 suggests that the average quality of the treatment plan improves as the number of sonications permitted by the TEA increases. Naturally, the finer the coverage control, the more (possibly shorter or smaller) sonications are performed. However, more sonications lead to longer treatment times, patient discomfort, and heavier computational load (more thermal model runs). The best compromise between the quality of the treatment plan and the number of sonications seems to be eight sonications with 20 to 30 systems. Finally, with ten sonications and 30 systems, the best fitness value approaches zero, but most of the systems still cannot achieve the best quality. As the numbers of sonication and system increase, the evaluation time also increases and the computation time quickly reaches 48 h before convergence is reached.

Since Equation (26) demonstrates that the fitness value is the product of two metrics (non-treated and mistreated), these metrics are also provided for the treatment plan’s quality.

Table 3. Median percentage of non-treated/mistreated area from 15 independent runs as a function of the number of systems N and the number of sonications.

Number of Systems	Number of Sonications
N	4	5	6	8	10
10	0.027/0.09	0.018/0.054	0.018/0.036	0/0	0/0
20	0/0.099	0/0.081	0/0.009	0/0	0/0
30	0/0.081	0/0	0/0	0/0	0/0

and

Table 4. Average percentage of non-treated/mistreated area from 15 independent runs as a function of the number of systems N and the number of sonications.

Number of Systems	Number of Sonications
N	4	5	6	8	10
10	0.09/0.09	0.027/0.045	0.045/0.045	0/0.018	0/0.009
20	0.036/0.09	0.009/0.072	0/0.036	0/0.009	0/0.009
30	0.036/0.063	0/0.027	0/0.027	0/0.009	0.018/0

, respectively, illustrate the median and average percentages of untreated and improperly treated locations. The non-treated region denotes areas in which the treatment was unsuccessful (due to insufficient energy deposition), while the mistreated region denotes areas where the normal tissue was harmed. According to the data from both tables, the non-treated and mistreated regions make up less than 0.1 percent of the highlighted areas. This is a remarkable achievement, and is considered clinically applicable.

In Table 3, given that the medians of both measures are zero for 8 and 10 sonications, regardless of the number of systems, it can be inferred that more than fifty percent of runs resulted in flawless treatment plans. The comparison between Table 3 and Table 4 indicates that in most circumstances, it was easier to completely cover the treated area than to completely protect the normal tissue. Energy seeping from the focus point (heat diffusion) to the protected zone was the cause of mistreatment.

Convergence speed is another crucial aspect that determines the quality of the suggested evolutionary method. Figure 12 demonstrates the number of iterations required to obtain an ideal solution, whereas

Table 5. Median of the execution time of the Matlab implementation running on a 16-core node computer measured over 15 runs as a function of the number of systems N and the number of sonications.

	Number of Sonications
N	4	5	6	8	10
10	10 h	14.42 h	18.08 h	21.4 h	28.46 h
20	13.68 h	17.42 h	25.26 h	25.43 h	29 h
30	23.22 h	28.6 h	38.8 h	30.48 h	42.26 h

shows the wall clock time required to get a suboptimal solution to quantify this metric.

As shown in Figure 12, the number of systems obviously influences the number of iterations. To achieve convergence in a population of 30 systems as opposed to a population of 20 systems, approximately 30% more TEA iterations are required. Unfortunately, the increase in evolution time due to the increase of the number of systems cannot be justified by the gain in solution quality.

Increases in the number of sonications, on the other hand, have large influence on the number of iterations. This is due to the TEA’s quality, which is supposed to scale linearly with the issue size.

Finally, the development of the TEA is looked at by tracking the evolution of the finest system. On a logarithmic scale, Figure 13 depicts the statistics of the evolution of a best system over the course of 15 separate runs, each of which had 8 sonications and 20 systems. It can be observed that a normal run converges very quickly, with 200 iterations generally sufficing.

3.2.3. Visualization of Treatment Plan

The illustration of the ideal treatment strategy with four sonication cycles is shown in Figure 14. In the order they appear in the treatment plan, each row in Figure 14 corresponds to each sonication out of four sonication cycles. The left column depicts the lesion map (yellow region), namely the places where $CE{M}_{43}$ has surpassed 240 min. In this column, the red line shows the boundary of the area which must be treated properly. After sonication, the actual temperature distribution of the domain is shown in the right column (including the cooling phase). It can be observed that some of the energy from prior sonications had not yet dissipated, but was utilized to increase the temperature by subsequent sonications. In Figure 14, the whole target area is ultimately covered.

3.2.4. Validation Results

Next, the numerical HIFU treatment was evaluated three times with the actual HIFU experiments on chicken breast. The results of the three trials are shown in Figure 15. The left column shows the shape of lesion patterns based on the numerical simulation. The right column demonstrates the actual treated region after the HIFU irradiation. Each experiment consisted of five sonications and a one-second cooling cycle. The distances between sonications were set to 3 mm, 5 mm, and 3 mm, and the heating time was set to 3 s, 3 s, and 2 s, respectively. Even though there are a few tiny patches of tissue that have not received the intended dosage of 240 $CE{M}_{43}$, comparisons of the planned and actual treated regions indicate excellent agreements.

4. Conclusions

High-Intensity Focused Ultrasound (HIFU) is a state-of-the-art, minimally invasive method of cancer treatment. However, significant preoperative treatment planning is necessary to accurately aim the ultrasonic beam and ablate only the tumor without damaging healthy tissue. Manual and semi-automated planning are time consuming and computationally complex tasks. As a result, new techniques for fully automated treatment planning are of great interest.

In this paper, an optimization algorithm based on the behavior of the thermodynamic systems, called the thermodynamic equilibrium algorithm (TEA), was proposed. In TEA, each individual of the population was modeled as a gas container. The global minimum of the function was defined as the thermodynamic equilibrium state. Changes in thermodynamic states of the systems toward the equilibrium direction, in which the temperature difference is reduced, formed the core of this algorithm. The proposed approach showed good performance in the convergence of systems to the global minimum of the problem. While the thermodynamic states were changing, most of the systems could reach the global minimum through thermodynamic equilibrium process. The TEA was tested with 25 benchmark functions, and the results confirmed that the algorithm was able to find the global minimum of these functions successfully. Additionally, the algorithm was evaluated by comparing its performance with PSO and GA for three of the benchmark functions, which showed that the convergence rate was faster and the global minimum was reached sooner with the TEA than with PSO or GA.

The TEA was applied to HIFU planning and produced excellent outputs with mistreated and undertreated areas on the order of 0.1 percent. These results shows that TEA can be utilized to create treatment plans that are close to ideal. The major benefit of the proposed method is that it just needs a map of the patient’s anatomy and the corresponding target and prohibited areas. Evolution will take care of the rest.

The proposed method, on the other hand, has a few drawbacks. First, even when no acoustic model was used and just 2D thermal simulations were performed, it took anywhere between 36 and 48 h to construct a good treatment plan. This will be addressed in the next phase of the study, which entails rewriting the whole algorithm in high-performance languages such as parallel C/C++ with the aim of achieving at least a five-fold reduction in computation time. Second, owing to computational complexity, it is only able to operate in two dimensions. Third, it requires a numerical model that accounts for acoustic heterogeneities in treatment plans by linking the ultrasonic and temperature modes.