Optimization of dynamic multi-leaf collimator based on multi-objective particle swarm optimization algorithm

Abstract

Background

The dynamic multi-leaf collimator (DMLC) plays a crucial role in shaping X-rays, significantly enhancing the precision, efficiency, and quality of tumor radiotherapy.

Objective

To improve the shaping effect of X-rays by optimizing the end structure of the DMLC leaf, which significantly impacts the collimator's performance.

Methods

This study introduces the innovative application of the multi-objective particle swarm optimization (MOPSO) algorithm to optimize DMLC parameters, including leaf end radius, source-to-leaf distance, leaf height, and tangent angle between the leaf end and the central axis. The main optimization objectives are to minimize the width and variance of the penumbra, defined as the distance between the 80% and 20% dose of X-rays on the isocenter plane, which directly impacts treatment accuracy.

Results

Structural optimization across various scenarios showed significant improvements in the size and uniformity of the penumbra, ensuring a more precise radiation dose. Based on the optimized structure, a three-dimensional model of the MLC was designed and an experimental prototype was fabricated for performance testing. The results indicate that the optimized MLC exhibits a smaller penumbra.

Conclusion

The proposed optimization method significantly enhances the precision of radiotherapy while minimizing radiation exposure to healthy tissue, representing a notable advancement in radiotherapy technology.

Keywords

multi-leaf collimator radiotherapy penumbra particle swarm optimization algorithm multi-objective optimization

Introduction

The most recent global cancer statistics, as reported by the International Agency for Research on Cancer (IARC), underscore the profound threat posed by cancer to human health.¹ Radiation therapy, applicable to tumor lesions, has become increasingly prominent in cancer treatment due to its non-invasive nature, minimal side effects, and high precision. It has emerged as one of the mainstream methods for cancer therapy.

The DMLC is a core device developed with the advancement of radiation therapy technology.^2,3 This sophisticated apparatus orchestrates the movement of its individual leaves to shape a radiation field that precisely conforms to the projected shape of the target area. This dynamic adaptation ensures the accurate delivery of radiation doses to the tumor, thereby safeguarding adjacent healthy tissues and enhancing the therapeutic impact on the tumor.⁴

The DMLC is characterized by its high speed of motion, superior field efficiency, wide irradiation field range, and compact structure. It has replaced traditional, intricately manufactured shaping alloy blocks, and has become an essential component of radiotherapy.

This advancement holds significant importance and value in enhancing the precision of radiotherapy. The penumbra of the DMLC is influenced by various factors, such as the energy of the radiation source, the shape of the leaf end, the height of the leaves, and the position of the leaves.⁵ The width and uniformity of the penumbra significantly impact the accuracy of the dose in the target area, with a smaller penumbra allowing for a larger dose gradient. Therefore, researchers have consistently focused on improving the geometric parameters of the leaves to achieve an ideal minimum penumbra width and consistent variance. The physical properties of the DMLC are crucial for precise dose delivery. Numerous research groups have studied and evaluated various DMLC designs to further enhance the precision of radiation therapy.^6–11

The particle swarm optimization (PSO) algorithm, proposed by Kennedy and Eberhart,¹² draws inspiration from the social behavior observed in animal groups. This algorithm seeks optimal solutions through a collaborative and competitive approach among individuals. In contrast to genetic algorithms, the particle swarm algorithm employs a population-based global search strategy. Notably, it distinguishes itself by avoiding complex coding and genetic operations, featuring a structurally simple design with fewer parameters, and proving easier to implement.¹³ This simplicity has led to its widespread application in various domains, including structural optimization problems.^14,15 In recent years, researchers have conducted extensive studies on the MOPSO algorithm.¹⁶ Aiming to address multi-objective optimization problems, MOPSO differs from PSO in that it can simultaneously optimize multiple conflicting objectives and seek a balance among them.

This paper develops a multi-objective algorithm for DMLC based on MOPSO, characterized by its simplicity, fast convergence, ease of programming implementation, and effectiveness in addressing multi-objective problems while avoiding local optima. Considering the impact of both the width and uniformity of the penumbra on dose delivery, the optimization objectives are set to minimize the penumbra width and achieve consistent variance. This study aims to further enhance the precision of radiation therapy by optimizing the structural design of the DMLC.

Material and methods

Particle swarm optimization algorithm

The PSO algorithm originates from the study of foraging behavior in bird swarms. Birds in flight often exhibit sudden changes in direction, dispersion, and aggregation. While their behavior is unpredictable, the overall swarm maintains coherence. When the entire group is searching for a target, an individual typically adjusts its next search step by referencing both the individual currently in the optimal position within the group and its own previously achieved optimal position. Kennedy and Eberhart modified and formulated models representing the interactions within these simulated populations into a general method for solving optimization problems, known as the PSO algorithm.^17,18 The basic PSO algorithm is outlined as follows:

v_{i d} (t + 1) = w v_{i d} (t) + c_{1} r_{1} (p_{i d} (t) - x_{i d} (t)) + c_{2} r_{2} (p_{g d} (t) - x_{i d} (t))

(1)

x_{i d} (t + 1) = x_{i d} (t) + v_{i d} (t + 1)

(2)

In the given equations, v and x represent the velocity and position of each particle, respectively. The particle's best position encountered during its flight is the solution that the particle itself has found. The best position experienced by the entire swarm represents the current optimal solution found by the entire group. Subscripts i and d represent the index of the particle and the index of the dimension, respectively, and t denotes the iteration step in the computation.

c_{1}

and

c_{2}

are system control parameters, often referred to as learning factors or acceleration constants. w is the inertia weight, and

r_{1}

and

r_{2}

are random numbers within the interval (0, 1) used to prevent excessive particle velocity. An upper limit for velocity, Vmax, can be set.

$P_{i d}$ represents the position of particle i, which is the best optimization result obtained by the particle up to the current moment and is referred to as the individual best (P_best). $P_{g d}$ represents the position of the particle corresponding to the best optimization result obtained by the entire particle swarm up to the current iteration, known as the global best (G_best).

Algorithmic processes

Problem Description: Formulate a mathematical model for the optimization design problem and determine the dimensionality of particles based on the optimization parameter variables.

Initialization: Specify the number of particles and randomly generate initial values for the particles to form an initial population. Set acceleration constants $c_{1}$ , $c_{2}$ , maximum velocity V_max, and maximum iteration T_max.

Evaluate Population $x (t)$ : Calculate the extreme value of the optimization function, identify the best individual particle P_best, and determine the global best particle G_best.

Generate a New Population: Generate a new population based on equations (1) and (2).

Check Termination Conditions: Evaluate whether termination conditions are satisfied, specifically whether the maximum number of iterations T_max has been reached or a threshold condition with a given precision ε (where ε>0). If these conditions are met, terminate the optimization process. If not, increment t by 1, proceed to step (3), and continue with the next iteration until termination conditions are satisfied.

Optimal Parameter Values and Results: The optimal values of particles that satisfy the termination conditions represent the design parameter values. The corresponding objective function values indicate the optimal value of the objective function. Typically, the average of multiple optimization runs may be considered to enhance reliability.

Penumbra analysis model

The end of the DMLC consists of a circular arc surface with a radius R and a flat surface with a height of 2Rsinθ. A numerical geometric analysis model is developed to establish functional relationships between various design parameters and the penumbra, as illustrated in Figure 1.

Figure 1.

Penumbra analysis model (a) Geometric penumbra model (b) Transmission penumbra model.

The total penumbra width $P_{a l l}$ in the DMLC is defined as $P_{a l l} = \sqrt{{P^{2}}_{g} + {P^{2}}_{t}}$ , where $P_{g}$ represents the geometric penumbra and $P_{t}$ represents the transmission penumbra.

The geometric penumbra be represented as:¹⁹

\begin{aligned} P_{g} = \frac{S \cdot (F - (h_{1} + h_{2} + R \cdot \sin θ))}{h_{1} + h_{2} + R \cdot \sin θ} (θ \geq β) \\ P_{g} = \frac{S \cdot (F - (h_{1} + h_{2} + R \cdot \sin (\arctan (\frac{N M}{F}))))}{h_{1} + h_{2} + R \cdot \sin (\arctan (\frac{N M}{F}))} (θ < β) \end{aligned}

(3)

When calculating the transmission penumbra on the isocenter plane, consider the shielding properties of leaf material for a certain energy radiation based on the Berger formula,²⁰ with a tungsten alloy leaf attenuation coefficient of µ=0.843cm⁻¹, the distances

d_{20}

and

d_{80}

, where radiation beams

{SD}_{20}

and

{SD}_{80}

penetrate through the leaf, can be computed.¹⁹ At a specific leaf location, determine the positions of the 20% dose point

P_{t 20}

and the 80% dose point

P_{t 80}

on the isocenter plane based on the shape of the leaf's end curve. Ensure that the distance between the two intersection points of line

{SD}_{20}

and the leaf's end curve is

d_{20}

, and similarly, for line

{SD}_{80}

, the distance is

d_{80}

. The difference between

P_{t 20}

and

P_{t 80}

represents the transmission penumbra value

P_{t}

. The transmission penumbra be represented as:

\begin{aligned} P_{t (80 - 20 %)} = & F \tan (\arcsin (\frac{\sqrt{R^{2} - {\frac{d_{80}}{4}}^{2}}}{{(h_{1} + h_{2})}^{2} + (h_{1} + h_{2}) \frac{N M}{F} + R \frac{N M}{F} \sin (β) + R \cos (β)}) \\ - F \tan (\arcsin (\frac{\sqrt{R^{2} - {\frac{d_{20}}{4}}^{2}}}{{(h_{1} + h_{2})}^{2} + (h_{1} + h_{2}) \frac{N M}{F} + R \frac{N M}{F} \sin (β) + R \cos (β)})) \end{aligned}

(4)

The design parameters included in the penumbra analysis model are as follows:

Distance from the radiation source to the isocenter (F)

Position of the leaf end in the isocenter projection (NM)

Distance from the radiation source to the surface of the DMLC leaf ( $h_{1}$ )

Half-height of the leaf ( $h_{2}$ )

Angle between the end and the central axis (β)

Source size (S)

Leaf end radius (R)

Angle between the end plane and the DMLC central line (θ)

Distance between the tangent point and the leaf centerline (y)

Path lengths $d_{20}$ and $d_{80}$

Here,

β

= arctan (NM/F) and y = R·sin (

β

). The output variable of the model is the total penumbra width

P_{a l l}

In linear accelerator radiation therapy, the parameters F and S are relatively fixed, with F set to 1000 mm and S to 2 mm. Parameters NM, β, and y are associated with the leaf position, while parameters $h_{1}$ , $h_{2}$ , θ, and R significantly impact the penumbra and serve as optimization parameters in this study.

Optimizing mathematical models

The width and uniformity of the penumbra in a DMLC significantly impact the precision of dose delivery to the target area. Achieving a consistently narrow penumbra allows for a larger dose gradient. In this study, distance from the radiation source to the surface of the DMLC leaf ( $h_{1}$ ), half-height of the leaf ( $h_{2}$ ), angle between the end plane and the DMLC central line (θ), and leaf end radius (R) are selected as optimization variables, with the optimization objectives being the penumbra mean width and the uniformity of the penumbra. Based on the penumbra analysis model, the following multi-objective optimization mathematical model for the DMLC can be established:

Find (h_{1}, h_{2}, θ, R)

Min F (x) {\begin{matrix} P 1 (h_{1}, h_{2}, θ, R) \\ P 2 (h_{1}, h_{2}, θ, R) \end{matrix}

P 1

represents the penumbra

mean

width, while

P 2

denotes the penumbra uniformity. At various projection angles

β_{i}

(where

i

=1, 2, … , N) within the maximum field, the penumbra is denoted as

P_{i}

. N = 226 is chosen based on a balance between computational efficiency and the need for adequate sampling resolution to ensure accurate calculation of the penumbra characteristics. The penumbra's mean width and uniformity are expressed mathematically as follows:

P 1 = \frac{1}{N} \sum_{i}^{N} P_{a l l i} (h_{1}, h_{2}, θ, R)

(5)

P 2 = \sqrt{\frac{1}{N - 1} \sum_{i = 1}^{N} {(P_{a l l i} (h_{1}, h_{2}, θ, R) - P 1)}^{2}}

(6)

Constraint condition

In the optimization design of the DMLC structure, apart from satisfying the designated objective functions, certain constraints must be taken into account. These constraints encompass both performance and boundary restrictions to ensure that, while optimizing the primary objectives, other performance indicators meet fundamental requirements. Essentially, the optimization design of the DMLC is conducted within reasonable parameter ranges, guaranteeing the engineering practicality and reliability of the optimization results. The limitations of each parameter are detailed in Table 1.

Table 1.

Limitation of parameter.

Parameter	Limitation
$h_{1}$	260 mm ≤ $h_{1}$ ≤ 389mm
$h_{2}$	30 mm ≤ $h_{2}$ ≤ 50mm
θ	0° ≤ θ≤ 11.3°
R	$h_{2}$ mm ≤ R ≤ 200mm

Optimization strategy

This study has developed a DMLC penumbra calculation and MOPSO program based on the aforementioned principles. The setup involved 100 initial sample points, 100 iterations, and an external archive size of 100, yielding a total of 100 Pareto optimal solutions. The initial velocity of the particles was set to zero, changing dynamically based on Equation (1). The neighborhood size was defined as 6. A linearly decreasing inertia weight (w) strategy was implemented to provide the algorithm with effective global search capabilities at the beginning of the run and improved local search capabilities later on. The initial value for the inertia weight was selected as 0.7 to balance avoiding premature convergence due to its small value and preventing a significant reduction in search capability due to its large value and the inertia weight decay factor is 0.95.

For the selection of $c_{1}$ and $c_{2}$ , which respectively represent considerations of individual experience and social experience, we investigated the impact of these two parameters within a certain range on the optimization time. The results are illustrated in Figure 2.

Figure 2.

Parameters optimization (a) Optimization time with $c_{1}$ (b) Optimization time with $c_{2}$ .

Parameters $c_{1}$ and $c_{2}$ were selected as 6.5 and 1, respectively, to enhance the efficiency of optimization. The chosen parameters and strategies were determined based on a thorough analysis of their impact on the algorithm's convergence and optimization efficiency.

The algorithmic programming and initialization strategies were carefully designed to ensure effective optimization outcomes throughout the iterative process. When defining variable ranges, using the central points of the variables to initialize the population helps avoid bias towards the boundaries, speeds up convergence by providing a balanced starting point, and increases population diversity, ensuring better exploration of the solution space, and the initial population data was directly included as a set of variables in the initial population. The optimization process continues until the iteration criteria are met or the convergence of the Pareto front is achieved. Figure 3 illustrates the optimization results at different stages, progressively approaching the Pareto front.

Figure 3.

Iterative process (a) Early stage (b) Mid-term stage.

3D modeling and experimental validation

Based on the dimensional parameters of the leaf structure, a set of DMLC-805 with 80 pairs of leaves, each 0.5 cm wide at the isocenter plane, was designed. The optimized DMLC leaf structure is depicted in Figure 4(a). This configuration offers a broad field and enables high-precision shaping effects. The overall three-dimensional structure of the DMLC is illustrated in Figure 4(b).

Figure 4.

Schematic diagram of DMLC-805 (a) Schematic diagram of leaf (b) Three-dimensional structure design of DMLC-805.

We fabricated an experimental prototype based on the three-dimensional structural diagram, as shown in the Figure 5.

Figure 5.

Prototype test.

Using the IBA Blue Phantom equipped with a CC13 ion chamber, as shown in Figure 6, the penumbra was measured from the scanned Off-axis dose distributions. Measurements were taken at a source-to-surface distance (SSD) of 100 cm with the water level 10 cm below the surface.

Figure 6.

Blue Phantom.

Optimization results and discussions

Figure 7 illustrates the Pareto front obtained through the MOPSO algorithm. Each coordinate point on the Pareto front represents a design solution, providing users with a wide range of optimization choices. The distribution of solutions on the Pareto front appears relatively uniform. The desired objectives inherently conflict and constrain each other: reducing the penumbra width typically results in decreased penumbra uniformity, whereas improving penumbra uniformity often leads to increased penumbra width.

Figure 7.

Pareto optimal front.

The extreme points on either end of the Pareto front represent the optimal penumbra width and uniformity achieved by the solutions. The solutions within the dashed line exhibit improved penumbra width and uniformity compared to the initial values. Users can selectively balance these two objectives based on practical considerations, choose a suitable solution, and derive the corresponding design parameters. Table 2 presents three representative solutions along with their initial structural parameters and corresponding computational outcomes.

Table 2.

Optimization results.

No.	Parameters				Result
No.	$h_{1}$ /mm	$h_{2}$ /mm	$θ$ /°	$R$ /mm	$Penumbra width$ /mm	Standard deviation/mm
0 (original)	280	36	6.3	80	2.451	0.036
1	389	32	1.2	70	2.586	0.027
2	389	39	11.3	120	2.239	0.033
3	389	50	10.5	200	2.161	0.039

The individual with the optimal penumbra width of 2.161 mm is superior to the initial model, accompanied by a reduction of 8.33 percentage points in the penumbra uniformity. In Solution 1, although the penumbra uniformity decreased from the initial 0.036 mm to 0.027 mm, the penumbra width is still greater than that of the initial model. Generally, a larger end radius (R) leads to a smaller penumbra width but worsens uniformity. Conversely, a smaller end radius R results in a larger penumbra width but improves uniformity.

We selected Scheme 2, which features a small penumbra width and improved uniformity, to process the experimental prototype based on the optimized parameters in Scheme 2. The optimized parameters were also integrated into the penumbra calculation program. The penumbra position curves before and after optimization are depicted in Figure 8. Position is expressed as the distance of the leaf from the central axis. As the leaf moves from one side of the radiation field to the other, the position of the tangent point between the radiation source and the end surface of the leaf changes with the leaf's position, meaning that $β$ constantly changes, causing the penumbra to continuously vary.

Figure 8.

Penumbra position curves.

Off-axis dose distributions were measured for field sizes of 40 × 40 mm², 60 × 60 mm², 80 × 80 mm², and 100 × 100 mm², as shown in Figure 9. The steep gradient in the transition region of the curve indicates a small distance between the 80% and 20% dose lines, suggesting that the structure has a small penumbra.

Figure 9.

Off axis curve.

The program developed in this article for optimizing DMLC structures is applicable across various application scenarios. The design presented above focuses on a large field. Adjusting the DMLC position constraints can yield an optimal DMLC structure design tailored for smaller fields. Figure 10(a) shows the optimized result for a 100 × 100 mm² small field, while Figure 10(b) illustrates the before-and-after comparison of the penumbra position curves.

Figure 10.

Optimization result for 100 × 100 mm² field (a) Pareto optimal front (b) Penumbra position curves.

Conclusion

Establishing a geometric analysis model of the penumbra for the DMLC elucidates the relationship between leaf structural parameters and penumbra performance. A mathematical model optimizing DMLC penumbra is developed based on this analytical model. This study highlights the innovative application of the MOPSO algorithm in optimizing the DMLC structure. Post-optimization, a significant and evenly distributed array of Pareto solutions is achieved, collectively forming the Pareto front. Each Pareto solution represents a viable DMLC design scheme. The significant improvements observed in the penumbra size and uniformity, compared to initial designs, underscore the algorithm's effectiveness. These advancements not only enhance the precision and uniformity of radiotherapy but also promise better patient outcomes. However, it is essential to recognize the inherent trade-off between penumbra width and uniformity: reducing penumbra width typically compromises uniformity.

The algorithm has proven effective across various scenarios, validated through numerical computations of the DMLC penumbra. The MOPSO algorithm provides an optimized solution set within a defined range, enabling designers to select a suitable DMLC structure based on practical needs. This approach represents a novel method for addressing penumbra optimization in DMLC structure design.

The experimental results indicate that the optimized structure has a good shaping effect on the X-ray. Consequently, the MOPSO algorithm emerges as a powerful tool for future advancements in radiotherapy technology, setting a new benchmark for treatment accuracy and effectiveness.

Footnotes

Acknowledgments

This work has received support from other members of the FDS consortium and is funded by Nanjing Life and Health Technology Special Project 2022SX00000303-202200303, Jiangsu Province Double Creation Talent Project, and the Zijinshan Yingcai Jiangbei Plan High Level Talent Project.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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