Novel Approach for Reducing Train Wheel and Brake Shoe Wear Using Speed Profile Optimization

Abstract

In this article, we propose a new approach for generating a train’s speed profile that takes into account the wear caused by the wheel–rail interaction and brake shoe wear. This method involves modeling the train and its path and simulating the train’s movement to calculate the wear caused by the wheel–rail interaction and brake shoe wear resulting from mechanical braking. Finally, using the particle swarm optimization algorithm, an optimal speed profile is produced, which minimizes the total wear. The results of implementing this method on the Shahriar–Roodshour route demonstrate that using an optimal speed profile results in a significant wear reduction of over 40%.

Keywords

speed profile optimization wear reduction railway PSO algorithm wheel and brake wear

Rail travel has gained widespread acceptance as a preferred mode of transportation for medium-length trips because of its higher level of safety, capacity, faster speed, and lower energy consumption in comparison with other modes. In the past two decades, different governments have prioritized the development of rail transportation to maintain these competitive advantages and create new ones. One way to achieve this is by improving the aforementioned parameters.

Reducing maintenance costs, fuel costs, and direct and indirect costs resulting from rail accidents can effectively lower the total costs of rail travel. Rail and train maintenance is a costly function of the rail network, and this article proposes a new approach for reducing these costs.

This research investigates the assumption that driver behavior, particularly unpredictable braking action and rate of braking, contributes considerably to wear and tear on wheels, rails, and brake shoes. To address this issue, a new approach for improving driver behavior is presented, which involves optimizing the train’s speed profile through the use of artificial intelligence. This approach is proposed to the train driver.

Wear Caused by Interaction between the Brake Shoes and Wheel

The friction mechanical brake is a type of brake commonly used in railway systems. This braking system involves pressing the brake shoes onto the wheel, which is in rolling contact with the rail, to initiate the braking action. During braking, the train’s kinetic energy is converted into heat, which is shared between the brake shoes and the wheel, leading to an increase in temperature. The heat generated is then transferred to the rail and subsequently dissipated through convection and radiation to the surrounding environment.

To calculate the heat generated from mechanical braking in trains, various models have been proposed by researchers. Wasilewski ( 1 ) conducted a comprehensive study on these models and provided detailed descriptions. One such model is a two-dimensional finite element model of the train’s mechanical brake with composite brake shoes, which was calibrated using data from a dynamometer test ( 2 ). The author also investigated the impact of thermal partition, contact pressure distribution, and specific heat capacity of the brake shoes on the calculated temperature fields.

Another researcher, Vernersson ( 3 ), used a two-dimensional axisymmetric finite element model to determine the mean temperature of the wheel and brake shoes and heat partitioning during tread braking.

Bosso et al. ( 4 ) developed a decoupled thermomechanical model that can calculate the contact pressure at the wheel–shoe interface and the resulting wheel temperature field generated during braking operations, taking into account frictional heating, air convection cooling, and the cold rail effect.

Yevtushenko et al. ( 5 ) presented finite element models for brake shoes to simulate repeated frictional heating. The simulations were conducted on a full-scale inertia dynamometer, and the brake shoes used in the study were made of two organic composite materials specifically developed for this research. The numerical calculations involved 2D axisymmetric and 3D finite element analyses, and the transient temperature changes obtained were compared with experimental data. The results showed good agreement between the simulations and the experimental observations over a continuous process of approximately 1,200 s.

The increase in temperature resulting from the heat generated between the wheel and brake shoes can cause damage to both components ( 6 ). One of the consequences of this damage is heat-dependent wear, which reduces the lifespan of the wheel and brake shoes, leading to increased maintenance costs and posing a potential safety risk to the train.

To calculate the wear caused by the increase in temperature, Chen et al. ( 7 ) presented a wheel wear model that considers the effects of temperature during tread braking based on the Archard wear model. In this model, the temperature of the wheel tread was calculated using a numerical algorithm based on the finite element method. In this article, it was shown that the contact patch area, wear depth, and worn area increase with the temperature caused by tread braking.

The wear caused to cast iron shoes and four composite shoes was also studied in Vakkalagadda et al. ( 8 ) using the rig wear test. Walia et al. ( 9 ) calculated the heat of and wear on the wheel and brake shoes of a city train using a thermal model, which they calibrated using measured values from a field test.

In recent years, predicting the lifespan of train wheels using wear rate calculation models has become a crucial area of research. The lifespan of train wheels is constrained by wear, emphasizing the importance of accurate knowledge of the contact between the wheel and rail to reduce the wear rate and increase the wheel’s lifespan, ultimately leading to a reduction in maintenance costs.

Wear Caused by Interaction between the Wheel and the Rail

The wheel–rail interaction is a complex mechanism that involves train–track dynamics, contact mechanics, and tribology. Initially, researchers attempted to calculate wheel wear and wear rates using laboratory and empirical methods. However, this approach was both time consuming and costly, and although it yielded highly accurate results, they were not generalizable to different conditions, such as varying axle loads and slip speeds. Consequently, several researchers have proposed various models for predicting wheel wear. One of the earliest attempts to simulate train wheel wear was made by Pearce and Sherratt ( 10 ). Their model was simple, involving the calculation of material lost from the wheel by using the friction coefficient after determining the contact forces and slip in the contact area. These calculations were performed after every 1,100 km of train movement.

Zobory ( 11 ) utilized the Hertzian theory and the Future Automotive Systems Technology Simulator (FASTSim) to resolve the problem of wheel–rail contact, employing various modeling approaches based primarily on the relationship between friction and energy dissipated in the contact area. Zobory introduced two friction coefficients, one for the mild regime on the tread and the other for the severe regime on the flange, depending on the material properties of the wheel and rail.

Jendel and Berg ( 12 ) developed a similar model using Genesys code to simulate the train’s dynamics. They employed local contact analysis using Hertzian theory and the FASTSim. The contact model used in this simulation identified up to two contact points simultaneously on a given wheel–rail pair. The Archards friction model was used to predict the wear. Whenever the maximum wear depth reached 0.1 mm, or the train had covered a maximum distance of 1,500 km, the wheel specifications were updated. The simulation results were compared with measurements taken from wheels on trains in service on the Stockholm railway network.

Braghin et al. ( 13 ) presented a wheel–rail contact prediction model for simulating train dynamics based on a multibody code, employing the CONTACT93 algorithm developed by Kalker to solve both the non-Hertzian normal contact and tangential contact problems. This friction model takes into account a direct correlation between the materials removed from the wheel and the work performed at the wheel–rail interface. They also examined the contact model proposed by Kik and Piotrowski and showed that this algorithm can be a valid alternative to CONTACT93 for acceleration simulations.

Mathematical Model of the Problem

In this section, the mathematical relationships governing the problem are described. The dynamic model of train movement and resistances to train movement are presented in Section 2.1. In Section 2.2, the mathematical model of the heat generated between the wheel and the brake shoe during braking is described. Then, in Section 2.3, the mathematical model for calculating the wear between the wheel and the brake shoe caused by mechanical braking and the wheel wear caused by the wheel–rail interaction is presented. Finally, in Section 2.4, the definitions and relationships with regard to the particle swarm optimization (PSO) algorithm are described.

Dynamic Model of the Train Movement

The movement of a train along a route is affected by various forces, including the traction force, resistance force, and braking force. The traction force creates the driving force necessary to overcome the resistances and accelerate the train. The resisting forces against the movement of the train are divided into two categories: main resisting forces; and resisting forces in relation to line construction. Brake force is also used to stop and reduce the speed of the train.

To model the movement of the train, a particle model can be used in which Newton’s second law is applied to each particle. The movement of the train can be modeled as follows:

m \frac{dv}{dt} = F_{T} - R_{e} - R_{x} - F_{B}

(1)

where $m$ is the mass of the train (kg), $v$ is the speed (m/s), $F_{T}$ is the traction force, $R_{e}$ is the main resistance, $R_{x}$ is the slope resistance, and $F_{B}$ is the braking force (all in Newton).

Accurately modeling the main resisting forces, including the aerodynamic, rolling, and gradient resistances, is essential for optimizing train operation and minimizing wear. The aerodynamic resistance depends on the train’s speed, frontal area, and shape, whereas the rolling resistance depends on the train’s weight, wheel diameter, and track conditions. The gradient resistance depends on the track gradient and the train’s weight.

The resisting forces in relation to line construction, including the curve resistance, switch resistance, and crossing resistance, also play a significant role in train operation. The curve resistance depends on the curve radius and the train speed, whereas the switch resistance depends on the switch type and the train speed. The crossing resistance depends on the crossing angle and the train speed.

When a train, both the locomotive and wagons/carriages, moves along a level track without any curves or slopes, the main resisting forces are applied. These forces include the friction force between the wheels and rails, the friction force in bearings and axle heads, air friction force, and other factors. The main resistive force can be calculated using the Davis equation as:

R_{e} = A + Bv + C v^{2}

(2)

where A, B, and C are Davis constants. $R_{x}$ shows the slope resistance of the track, which is defined as:

R_{x} = m . g . \sin θ

(3)

where m is the mass of the train (kg), g is the gravitational acceleration ( $\frac{m}{s^{2}}$ ), and θ is the slope of the track.

Thermal Model

The average heat flux generated at the shoe–wheel contact point can be determined as follows:

q^{brake} = \frac{Q^{brake}}{B L_{b}}

(4)

where $B$ is the contact width (m), $L_{b}$ is the length of the brake shoe (m), and $Q^{brake}$ is the braking power (w), which is defined by:

Q^{brake} = m v a

(5)

In this relationship, $m$ (kg) is the axial load, $v (m / s$ ) is the speed, and a is the braking acceleration ( $\frac{m}{s^{2}}$ ). The heat flux generated between the wheel and the brake shoe is divided by the thermal resistances between the wheel $q_{w}$ and the shoe $q_{b}$ as follows:

q_{w} = \frac{T_{b}^{cont} - T_{w}^{cont}}{R_{w} + R_{b}} + \frac{R_{b}}{R_{w} + R_{b}} q^{brake}

(6)

q_{b} = \frac{T_{w}^{cont} - T_{b}^{cont}}{R_{w} + R_{b}} + \frac{R_{w}}{R_{w} + R_{b}} q^{brake}

(7)

where $T_{w}^{cont}$ is the initial temperature of the wheel surface, $T_{b}^{cont}$ is the initial temperature of the shoe surface, $R_{w}$ is the thermal resistance of the wheel, and $R_{b}$ is the thermal resistance of the shoe. To consider the cold effect of the rail on the wheel, convection cooling $q^{c o n v} (\frac{W}{m^{2}})$ and radiation cooling $q^{r a d} (\frac{W}{m^{2}})$ have been modeled using classical relationships. The convection cooling model is:

q^{conv} = - h^{conv} (T - T_{\infty})

(8)

where $h^{conv}$ is the heat transfer coefficient, $T$ is the surface temperature, and $T_{\infty}$ is the ambient temperature. The radiation cooling model is:

q^{rad} = - ε σ (T^{4} - T_{\infty}^{4})

(9)

In this equation, $ε$ is the radiation emissivity and $σ$ is the Stefan Boltzmann constant. The negative sign in the above relationships indicates the exit of heat from the surface. According to the relationship $q^{conv}$ and $q^{rad}$ , the heat flux of a surface is equal to:

q^{cool} = q^{conv} + q^{rad}

(10)

Mechanical Model of Wear

In the following, wear resulting from the wheel–shoe contact and from the wheel–rail contact are modeled using Archard’s ( 14 ) wear model.

Wear on Brake Shoes

Archard’s wear model is used to model the wear on brake shoes and wheels, for which the wear rate is typically assumed to be proportional to the sliding speed V and the contact pressure P. However, in addition to these factors, the present study also considers temperature-dependent wear rate coefficients. The behavior of brake shoes and wheel wear during braking is dependent on the speed, axial load, and braking acceleration. The wear depth rate of the brake shoes, denoted as $\overset{\cdot}{w}$ (m/s), is calculated as follows:

{\dot{w}}_{b} = k_{f i e l d} k_{w}^{b l o c k} (T) μ p v

(11)

In this equation, $k_{w}^{block}$ is the temperature-dependent material wear property and $k_{field}$ is a dimensionless adaption factor. In addition, the amount of wheel wear is calculated by the following equation:

{\overset{\cdot}{w}}_{w} = k_{w}^{tread} (T) μ p v

(12)

where $k_{w}^{tread}$ is the temperature-dependent material wear property. The $k_{w}^{block}$ and $k_{w}^{tread}$ values are obtained from the pin-on-disk test performed in Vernersson et al. ( 15 ).

Wear at Wheel/Rail Contact Points

The Archard model is one of the most common wear models for estimating the wear depth caused by sliding, and it is expressed using the following equation ( 16 ):

\frac{V_{w}}{D} = K . \frac{N}{H}

(13)

where

is the volume of wear $(m^{3})$

is the sliding distance $(m)$

is the normal force $(N)$

is the Vickers hardness of the softer material $(N / m^{2})$

is a dimensionless wear coefficient, corresponding to the probability of extracting a wear particle by shear

The differential of Equation 13 with respect to time is equal to:

{\overset{\cdot}{V}}_{w} = K . \frac{N}{H} . \overset{\cdot}{D} = K . \frac{N}{H} . V_{v}

(14)

where $V_{v}$ is the total sliding speed and is obtained from the following equation:

V_{v} = \sqrt{V_{ξ}^{2} + V_{η}^{2}}

(15)

In this relationship, $V_{ξ}$ and $V_{η}$ are the longitudinal and lateral sliding speeds, respectively. The concept of sliding speed is defined as the difference between the actual linear speed of the train $v_{t}$ and the linear speed of the train wheel $v_{w}$ :

V_{ξ} = v_{w} - v_{t}

(16)

The linear speed of the train wheel $v_{w}$ is also obtained from the product of the angular speed of the wheel $ω_{w}$ by the wheel radius $r$ :

v_{w} = r ω_{w}

(17)

If the wear phenomenon is examined on an elementary surface $σ$ , we obtain the required local wear velocity:

{\overset{\cdot}{Z}}_{w} = \frac{{\overset{\cdot}{V}}_{w}}{σ} = \frac{K}{H} . p . V_{v}

(18)

where $p$ is the local Hertz pressure and can be calculated from the following equation:

p = \frac{F_{n}}{π ab}

(19)

In this relationship, $F_{n}$ is the normal force, and a and b are semi-axes of the elliptical contact patch in the longitudinal and lateral directions. By introducing the coefficient $K_{Archard}$ defined by:

K_{Archard} = \frac{K}{H}

(20)

the final Archard equation is obtained as follows:

{\overset{\cdot}{Z}}_{w} = K_{Archard} . p . V_{v}

(21)

in which the coefficient $K_{Archard}$ can be calculated using Figure 1.

PSO Algorithm

The PSO algorithm was introduced by Eberhart and Kennedy ( 17 ). This metaheuristic algorithm is particularly useful for optimizing nonlinear continuous functions. The concept of swarm intelligence, which is often observed in groups of animals, for example, herds, inspired the authors of the aforementioned article to develop the PSO algorithm. In general, the algorithm considers a group of n particles in an m-dimensional space:

X = (X_{1}, X_{2}, \dots, X_{n})

(22)

There is a position vector $X_{t}^{i}$ and a velocity vector $V_{t}^{i}$ in each iteration for each of the $i$ particles that creates this velocity, as follows:

X_{t}^{i} = (x_{i 1}, x_{i 2}, \dots, x_{in})^{T}

(23)

V_{t}^{i} = (v_{i 1}, v_{i 2}, \dots, v_{in})^{T}

(24)

These vectors are updated based on dimension j according to the following equations:

V_{ij} (t + 1) = W . V_{ij} (t) + c_{1} r_{1} (X_{best ij} - X_{ij}) + c_{2} r_{2} (X_{gbest ij} - X_{ij})

(25)

X_{ij} (t + 1) = X_{ij} (t) + V_{ij} (t)

(26)

where $i = 1, 2, 3, \dots, p$ and $j = 1, 2, 3, \dots, n$ , $W$ is the inertial weight constant, $c_{1}$ and $c_{2}$ are acceleration factors, $r_{1}$ and $r_{2}$ are two random values (0,1), $X_{best}$ is the best individual position of a particle, and $X_{gbest}$ is the best general position among all particles. Figure 2 shows the process of the PSO algorithm.

Implementation Results

This section presents the testing of the proposed method on a real route, namely, the Shahriar–Roodshour route in the Iranian railway network. As previously mentioned, the proposed method is capable of generating a speed profile for the train movement with the least travel time and the train movement with the least amount of wear.

First, this section outlines how to generate the speed profile. Then, technical information on the train and track is provided, and the technical details for the GT26CW locomotive are used to make the values obtained from the simulation more realistic. All calculations in the simulations of this section were conducted using the Python programming language on a PC with a 2.3 GHz processor speed and 8 GB RAM, running on the Windows 10 platform.

Speed Profile Generation

The proposed method for generating the speed profile involves designing a model to simulate the train’s dynamics. This model takes into account the train information, track information, and the set of notches to be applied. Notches are used to adjust the traction force level of the train. The system then calculates the traction force of each received notch based on the force–speed curve, and applies them in order. Using the relationships presented in Section 2, the train’s speed, position, and travel time are calculated.

The output of this model includes the train’s speed at any given moment, the time it takes to travel the route, the wear on the wheel caused by the wheel–rail interaction, and the wear on the wheel and brake shoes caused by mechanical braking.

In the next step, the PSO algorithm is used to generate the best optimum speed profile, which is represented by the set of notches required to achieve the desired speed profile. To obtain this profile, an initial population of particles is first created, in which each particle represents a set of notches that describe the train’s movement along the track. The following conditions are taken into account for the production of these particles:

The train’s speed must not exceed the maximum permissible speed of the line.

The travel time must not exceed the specified time.

The replacement time of each notch must be at least 3 s.

The distance traveled by the train must not exceed the length of the selected section of route.

Then, the cost function is calculated for each particle, and the velocity and position values are obtained for all particles. These speed and position values are then updated according to the particle with the lowest cost, and the process continues until the algorithm can no longer find a method with the lowest cost. Finally, the notches that generate the lowest cost are presented as the optimal train movement.

Simulation

In this section, we present the speed profile generated for the Shahriar–Roodshour route using the proposed method for both the approach considering train movement with the least travel time and the approach considering train movement with the least amount of wear.

The proposed method was tested on the Shahriar–Roodshour route, which is 32 km in length. The maximum speed allowed on this route is 120 km/h. Table 1 displays information about the train, and Table 2 presents the parameters of the PSO algorithm used for simulation. The gradient for this route is shown in Figure 3.

Table 1.

Train Information

Parameter	Value
Locomotive type	GT26CW
Weight of the locomotive	120 tn
Number of wagons/carriages	8
Weight of each wagon/carriage	50 tn
Number of locomotive axles	6
$Davis coefficients$	$R = 2.148 + 0.02 V + 0.0002 V^{2}$

Table 2.

Particle Swarm Optimization Algorithm Parameters

Parameter	Value
Number of particles	100
Inertia coefficient w	$0.4 \leq w \leq 0.7$
Parameter $c_{1}$	1.4962
Parameter $c_{2}$	1.4962
Maximum iteration	100

Minimum Travel Time Strategy

Figure 4 displays the speed profile acquired through the PSO algorithm for the movement mode that offers the shortest travel time. As depicted in the figure, the suggested approach for this strategy entails moving with maximum traction force (notch 8) until the maximum permitted speed (here, 120 km/h) is reached.

Once the train reaches the maximum speed permitted on the line, it endeavors to maintain this speed. During the braking phase, a combination of dynamic and pneumatic braking is applied to enable the train to stop in the shortest possible distance, utilizing its normal braking rates.

The Shahriar–Roodshour route features a smooth negative slope at the beginning, followed by several ups and downs that require the use of various notches to maintain the maximum speed. Figure 5 depicts the significant application of traction and braking forces required to operate the train within a specific speed range. This provides the speed profile with the minimum time to reach the end of the route, ensuring efficient operation of notches.

During downhill sections, dynamic braking is employed to prevent the train’s speed from exceeding the permitted limit of the line (120 km/h). However, this contributes to further wheel wear.

In the latter sections of the journey, the application of air brakes helps the train to maintain its braking rate until it reaches the stop point. At this stage, the brake shoes in the braking system are subject to wear.

Overall, this journey is estimated to result in a total amount of wheel wear of 1.53 μm and shoe wear of 2.46 μm, with a travel time of 1,041 s.

The traction and braking forces applied, wheel wear, and load profile are depicted in Figures 5 to 7.

Minimum Wear Strategy

Because of its 32-km length and steep slopes, the Shahryar–Roodshour route is suitable for testing the performance of the proposed method. As Figure 8 shows, the proposed method has made the most use of the slopes and ups and downs of the route and has tried to use the coasting mode on the slopes of the pathway as much as possible.

As shown in Figure 9, an important observation is that the brakes were not applied throughout most of the train’s movement, except in about the last 100 m. This is because the train was switched to coasting mode before entering any downhill sections, ensuring that the speed never exceeded the permitted limit of the line. This not only reduced wheel wear but also utilized the positive gradient at the end of the route to reduce the train’s speed significantly and also the braking duration, thereby reducing wear caused by braking.

Figure 10 displays the cumulative diagram of wheel wear, which shows the maximum rate of wear in the braking mode of the train. The total amount of wear is estimated to be no more than 0.83 μm. This indicates a 45.7% improvement compared with the minimum time strategy in the last section.

Figure 11 illustrates the load profile of the train movement, which shows the proportion of each notch and the coasting movement for the entire route. Comparing this figure with the load profile of the minimum time strategy in the last section reveals that the coasting movement increased from 38.3% to 59.4%, whereas the dynamic brake decreased from 9.3% to 0.9%, and the air brake decreased from 2.7% to 1.2%. The significant advantage of this approach is its ability to reduce braking duration. The simulation results are summarized in Table 3.

Table 3.

Simulation Results

Strategy	Travel time (s)	Wheel wear $(μ m)$	Shoe wear $(μ m)$	Air brake (%)	Dynamic brake (%)	Coasting movement (%)
Minimum time strategy	1,041	1.53	2.46	2.7	9.3	38.3
Minimum wear strategy	1,145	0.83	0.88	1.2	0.9	59.4
Changes (%)	+10%	−45.7%	−64.2%	−55.5%	−90.3%	+5.1%

Based on the findings, the proposed method can reduce wheel wear by 45.7% and brake shoe wear by 64.2% in the train movement with the minimum wear strategy, albeit at the expense of about 10% increase in travel time compared with the train movement with the minimum travel time strategy.

It is worth noting that the increase in travel time is typically accounted for in the design of train schedules. In addition to the buffer time allocated to each segment, train movements may be impeded by factors such as weather conditions, track maintenance, or other unforeseen circumstances. However, conflicts with other train movements are among the most important factors that can cause trains to operate at lower speeds than their maximum capability on certain parts of the route. These factors are carefully considered when optimizing train schedules to ensure safe and efficient train operations that balance travel time, reliability, and capacity.

Optimizing the speed profile under such circumstances in conjunction with other trains in the network could be a subject for further research in this area. Additionally, looking at energy consumption alongside the wear and tear of the wheels, rails, and brake shoes in the objective function would be an interesting consideration. This approach would ensure that train operations are not only efficient and safe but also sustainable and cost-effective.

Conclusion

This article has presented a novel approach for reducing train wheel and brake shoe wear through the optimization of a train’s speed profile. The multifaceted nature of our program, which integrates speed profile optimization, train and rail modeling based on real data, wheel and brake shoe wear modeling, and the application of PSO algorithms, has yielded promising results in significantly reducing wear on train critical components.

The core of our methodology lies in the intelligent generating of train speed profiles. By optimizing a train’s speed profile, we minimized the wear on its wheels and brake shoes. This work is pivotal for the development of strategies aimed at extending the lifespan of critical components, reducing maintenance costs, and promoting the overall sustainability of rail transportation.

As with any pioneering work, there are areas ripe for future exploration. Our study prompts further investigation into the adaptability of the proposed approach to diverse rail environments, varying operational conditions, and different types of rolling stock. Additionally, if the PSO algorithm were to be refined and advanced machine learning techniques incorporated, these alterations hold the potential for improving the precision and adaptability of our optimization strategy.

In conclusion, our research contributes a holistic framework for addressing the challenges posed by train wheel and brake shoe wear. The amalgamation of speed profile optimization and wheel and brake shoe wear represents a significant step toward providing sustainable and efficient rail transportation. It is our hope that this work will inspire continued research and innovation in the quest to optimize rail operations, reduce their environmental impact, and pave the way for a more resilient and resource-efficient future for rail transportation.

Footnotes

Acknowledgements

The authors gratefully acknowledge the assistance of the Iranian railway in providing the necessary information for this research.

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: H. Adeli, A. Mirabadi, S. Fazel, S. Yousefi; data collection: H. Adeli, S. Yousefi; analysis and interpretation of results: H. Adeli, A. Mirabadi, S. Fazel; draft manuscript preparation: H. Adeli, A. Mirabadi. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

The authors whose names are listed in this paper certify that they have no affiliations with or involvement in any organization or entity with any financial or other interest in the subject matter or materials discussed in this paper.

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