A 3D-printed phantom for stereotactic body radiation therapy simulation

In modern radiation therapy for lung cancer, examining the uncertainty between tumor motion and beam delivery is vitally important. To lower the radiation dose delivery to the patient’s normal tissue, narrowing the irradiation field margin to hit the tumor accurately is critical. Thus we proposed a phantom that simulates the thorax and lung tumor’s motions by employing a 3D printing technique. The lung tumor is controlled by a linear miniature Delta robot arm, with a maximum displacement of 20 mm in each direction. When we simulated the thoracic breathing movements at 12 mm in A–P (Anterior-Posterior), the control errors were within 10%. The average tracking errors of the prosthetic tumor were within 1.1 mm. Therefore, the 3D-printed phantom with a robot arm can provide a reliable simulation for training and dosimetry measurement before lung radiotherapy, especially SBRT.


Introduction
SBRT (Stereotactic Body Radiation Therapy) nowadays is one of the mainstream treatments for early stage lung cancer and is particularly beneficial for patients whose conditions may be unfit for surgery.In addition, this method is also effective in tumor control rates in combination with surgery and chemotherapy.For example, in the case of surgery, radiotherapy is administered before surgery to shrink the cancerous tumor, facilitating its removal during the subsequent surgical procedure (Rashtian et al 2012).Postoperative radiotherapy is then employed to eliminate any residual tumor that is inaccessible through conventional surgery.
SBRT utilizes high-energy x-rays, known as photons, to destroy cancerous cells and inhibit their growth and reproduction.However, functional movements of the body during the radiotherapy process can lead to overshooting, resulting in an overdose and increased side effects, e.g.acute (early), consequential, and late effects on normal tissues for the patient (Mageed and Gupta 2023).Therefore, to minimize positioning errors and overdosage injury, clinical implementation and testing of radiotherapy often rely on verifying the expected treatment effect through medical prosthesis models or 'phantoms' (Mann et al 2016).
Presently, two types of phantoms are used for radiotherapy on lung tumors: static and dynamic.Static phantoms primarily focus on verifying radiation dose calibration (Kinhikar et al 2009), while dynamic phantoms aim to simulate the chest movements and tumor motion caused by the patient's breathing during radiotherapy.The only feasible way to represent the patient's physical condition as closely as possible during treatment is through the design of dynamic phantoms (Lee et al 2020, Vásquez et al 2012).However, although phantoms for lung tumors are widely available in the market and can simulate respiratory motion, they cannot control the tumor's movement trajectory, resulting in only observations of fixed tumor paths during test studies.Additionally, the diversity of respiratory movements and tumor positions among individual patients can result in various tumor trajectories (Spoelstra et al 2008).Savanović et al (2022) reported that irregular breathing patterns would impact deliver time, especially in the amplitude-gated mode.Therefore, simulations of treatment plan that rely solely on monotonous tumor trajectories would be insufficient.
Regarding the types of respiratory simulations in phantom models, these are classified as pneumatic models driven by differential pressure through air or water; or models driven by motors and mechanical structures.Pneumatic phantoms utilize air control in an airbag to simulate a patient's breathing.Examples of such models include the RS-1500 and the LuCa, which pump air into silicone airbags (Vásquez et al 2012, Perrin et al 2017), and the Leutz-Schmidt et al (2020) model, which inflates a pig lung with air through an opening at the top of a plastic mold, causing the lung to stretch and contract.In the method conducted by Remmert et al (2007) and Lee et al (2020) water-pressure is applied to the pig lung, and in some cases, a sponge is used to simulate expansion and contraction.However, this method requires a high level of containment.If the airbag is damaged or the model cannot be kept airtight, the functionality of the prosthetic lung may deteriorate.Then there are the mechanical phantoms.Steidl et al conducted simulations by placing a motor beneath an artificial skeletal model and using a rope device to control the movement of its sternum section, thus simulating the breathing conditions of a patient (Steidl et al 2012).Another example is the QUA-SAR phantom, which uses a motor to raise and lower a black platform to simulate thoracic activity during breathing (Hurkmans et al 2011).Compared to airbag models, mechanical models are less prone to issues such as air or water leaks during simulations of patient thoracic activities, resulting in higher durability during system operations.
Meanwhile, the active tumor control system is achieved through a motor, powering the tumor model trajectory.A typical example is the QUASAR respiratory motion phantom, which utilizes a wooden or plastic tumor model tube inserted into a lung phantom.The motor will then power the tumor to simulate its trajectory (Hurkmans et al 2011, Lee et al 2015).However, the tumor trajectory generated by the QUA-SAR model mainly consists of spiral and anterior-posterior movements, which have limitations in replicating actual tumor movement patterns.A model developed by Steidl et al (2012) utilizes a cube detector to stand in as a tumor model, which is at the manipulator's end-effector to enable the tumor model to move in three dimensions.Even so, such a method is still unsuitable for 4D CT and radiotherapy equipment due to the large size of the industrial robotic arm.Thus, there have been attempts to improve tumor trajectory tracking in order to improve the chances of sparing normal tissue during radiotherapy.Sarkar et al (2020) employed Calypso lung radiofrequency beacons to track tumors during stereotactic body radiation therapy (SBRT) processes, improving tumor trajectory prediction.Capaldi et al (2021) presented an independent robotic quality assurance (QA)-phantom meant for lung QA with the commissioning using the Calypso system for lung treatments.Both studies highlight the potential of 3D printing in producing bioartificial organs that closely resemble their natural counterparts in structure, function, and biocompatibility.
Based on the identified design flaws above, we have developed a phantom model for SBRT lung tumor treatment to replicate a patient's breathing motion and tumor movement, serving as a simulation and training tool for healthcare professionals.SBRT employs respiratory gating and image guidance to minimize radiation dose exposure.Cone beam CT and fluoroscopy techniques determine tumor position before beam delivery (Cheung andSawanta 2014, Ehrbar et al 2016).Radiotherapy QA (Quality Assurance) is often limited by estimation errors due to the need for invasive dose inspection.Therefore, a suitable phantom can play a reliable testbed before therapy, allowing for confidence-building in radiotherapy.The characteristics of this phantom are as below: (i) Imitating the movement of the chest cavity when breathing using a mechanical structure.
(ii) Utilizing a minuscule 3D printed linear Delta robot arm to drive the prosthetic tumor in a complex three-dimensional motion.
(iii) Possessing the ability to replicate the patient's breathing pattern and tumor trajectory.
(iv) Eliminating the necessity for repetitive verification of the prosthetic tumorʼs position using 4D CT.
Considering the need for prosthetic tumors to resemble human tissue, these models must be nonmetal in origin to avoid CT artifact affects.We utilized PLA (Polylactic Acid) 3D printing technology to fabricate components of a lung phantom for SBRT simulation.
The study is organized as follows.Section 2 introduces the prosthesis, encompassing the hardware structure design, PIDK (Proportional-Integral-Derivative and Kalman Filter Control) breathing control, robot arm design, and prosthetic tumor design.Section 3 presents the experimental results, while section 4 provides discussions on the findings.Finally, section 5 concludes the study.

3D printing
In the past years, the advent of 3D printing with soft materials has enabled the production of phantoms with intricate geometries and realistic properties, making them valuable for evaluating medical technologies, training clinical practitioners, and simulating interventional procedures with novel devices (Wang et al 2020).Therefore, besides the commercial skeletal model from ENOVO, we have decided to use PLA 3D printing to construct the phantom prototype.These components include an FPGA (Field Programmable Gate Array) platform support structure, a digital camera lens hood, infrared light blocking plates and fixtures, a skeletal model fixture, a nylon rope axis, a servo motor fixture, a linear Delta mechanical arm, a prosthetic tumor, and its holder.All printed components are designed by ourselves, using Solidworks.This software outputs STL files, and then we use Ultimaker Cura to transform STL into gcode files for printing.The printing device is Infinity3DP X1Pro, which uses fused deposition modeling technology.

Respiratory platform design
The phantom system developed in this study focuses on two functions: respiratory actions and tumor trajectory replication.As shown in figure 1(a), the phantom utilizes a digital camera to monitor the LED spot on the sternum of the skeleton model.The dot trajectory is then transmitted to the PC to control the servo motor, enabling the desired up-and-down movement of the sternum.The position above the sternum is our patented shading plate, which blocks the infrared light from the hospital's SBRT respiratory gating system.This plate equips an infrared reflector for respiratory gating and a LED on both the front and rear sides (Yu 2022).Figure 1(b) illustrates the placement of the servo motor, which is beneath the ventral position of the ribcage section, connected to a spool with a nylon cable that directly interacts with the line plate pulley underneath.The rotation of the servomotor controls the retraction of the nylon rope.When the nylon rope contracts, it pulls down the shading plate and the model skeleton sternum, simulating exhalation, and then loosening the nylon rope allows the model skeleton sternum to return to its original position, simulating inhalation.To ensure stability and minimize displacement during operation, fixed bases for the skeleton model were designed using 3D printing technology.Figure 1(b) also shows that the FPGA platform is placed at the front of the phantom, with a digital camera attached to monitor the LED spot on the shading plate.The computer screen displays the position correction of the spot.Afterward, the FPGA platform will transmit the coordinates of the upwards and downwards movements of the LED spot to the computer for respiratory motion control.

PIDK breathing control
As the phantom must properly compress the sternum section to mimic the patientʼs respiratory motion, thus we implemented a closed-loop control system using PID (Proportional-Integral-Derivative) control with parameters determined using the Ziegler-Nichols method.The position of the LED spot observed by the digital camera and the desired breathing trajectory was used for error control and subsequent corrections.However, conventional PID control is not ideal for non-linear systems.Thus we added an improved linear Kalman Filter to create a PIDK controller and enhance the PID control results.
Unlike conventional closed-loop control designs where the Kalman filter is at the output stage of the system signal, we positioned the improved Kalman filter at the control stage of the closed-loop system.Such an arrangement allows us to estimate the closed-loop control error and apply additional control power to enhance the integral (I) and derivative (D) control capabilities.
Here the PIDK (Proportional-Integral-Derivative and Kalman Filter Control) controller utilizes only the Kalman Gain (K g ) and the Estimation of the state of the Kalman filter.
Among these, e Est(t) and e Est(t−1) respectively represent the current and previous control error estimation value.
After determining the current estimation value for the control error, we then decided that the additional control forces applied to I and D were set as ) is deemed as a constant.In addition, a default midline value (R avg ) is within the range of maximum compression (exhalation) and relaxation (inhalation) of the phantom.During the experiment, we found that the I control value was beneficial to the inspiratory action control, whereas the D control value was beneficial to the exhalation action control.Thus, we adopted different control strategies for the position of the LED spot projected on the screen (L) where D control is invariant, and, where I control is invariant.

Linear Delta manipulator
In contrast to Steidl et al (2012) 2(a), the Delta robot arm design incorporates a linear gearing system powered by servo motors, enabling the attached rod to move back and forth to drive the end-effector, as shown in figure 2(b).Figure 2(c) illustrates the 3D printed physical structure of the robotic arm and the links of the end-effector that are made of carbon fiber material to reduce weight.The prosthetic tumor is similarly 3D printed and attached to the end-effector.We also install a slide potentiometer at each link's base to detect the rodʼs movement.
In terms of the tumor position control, as the tumor model and the links are lightweight, we used an open-loop control method to control the displacement of the tumor.With inverse kinematics to control the amount of movement generated by the linear gears, the tumor model will move to the desired location.Additionally, the application of forward kinematics will aid in the verification of the tumorʼs current location, see appendix B. Considering the limited linear working area due to sliding resistance, we limit the displacement of the rod to 2.5 cm.Within this limit, we can calculate the achievable range of movement for the tumor model using forward kinematics, as demonstrated in figure 2(d).

Tumor
The prosthetic tumor is 3D printed using PLA material.Its outer section is a spherical object with a diameter of 3 cm, whereas its internal section is semihollow with a diameter of 2 cm to facilitate the placement of a Gafchromic film (EBT3), which is to inspect radiation doses and accuracy (Savanović et al 2021), as shown in figure 3(a).The example of beam delivery is in figure figure 3(b).
The printed infill density of the tumor is set from 40 to 85%.With the 4D CT, the tumor with 40% printing density is close to the lower detection threshold, while the 85% printing density is a practical upper threshold for 3D printers.The tumor model printed at 85% infill density results in radiation absorption between 90 and 140 HU measured at the solid lower section of the tumor model (Pallotta et al 2022).Such an infill approximates the HU of soft tissue, including an underused contrast medium.
As seen in figure 2(c), the tumor is fixed upon a semi-pivoted support, and due to the solid lower section of the tumor model, it maintains a horizontal position during movement.The tumor holder is fixed on the end-effector.As depicted in figure 3(c), 3D printed holders with varying lengths allow for easy simulation of the tumor model's placement in the upper, middle, and lower lung sections.Due to the symmetrical shape of the sternum, the working areas of the tumor on both the left and right sides are designed to be symmetrical.

Set-up
We conducted experiments using pre-recorded respiratory and actual tumor movement trajectories, where the computer simultaneously played back both actions to provide phantom imitation and error measurement.When the phantom was activated with 60 kg/cm servo motors, the sternum mimicked the breathing motion by compressing and expanding while simultaneously triggering the movement of the linear Delta robot arm to drive the prosthetic tumor inside the sternum.We first test with 2D and 3D

Respiratory tracking
During the respiratory simulation, the FPGA platform displayed the LED spot captured by the digital camera on the screen.The initial Y-axis coordinate of the LED spot's center position was 615 pixels vertically.When the servo motor pulled the phantom sternum downward by 8 mm to replicate exhalation, the Y-axis coordinate was 650 pixels.With a compression of 10 mm, the Y-axis coordinate was 660 pixels, and 668 pixels for a 12 mm compression.In the maximum downward range of 16 mm, the Y-axis coordinate was 680 pixels.The greater the compression of the phantom chest, the stronger the reactive force of the sternum, which in turn reduced the response speed of the servo motor and increased the difficulty in respiratory control.
Based on the respiratory motion waveform observed from the respiratory gating system, we decided to test the sine, triangular, and trapezoidal waveforms We determined the range of chest motion in A-P to be between 8 mm and 12 mm (Nøttrup et al 2007).During the experiment, we collected approximately 1500 data points between 1.4 s and 100 s and calculated the average accuracy as M represents the number of captured LED images, EE represents the position of the LED spot during exhalation, and EI represents the position during inhalation.
Tables 1 and 2 compares the control accuracy of PID and PIDK.In these cases, the control parameters for PID were based on a normal breathing cycle of 5 seconds (Pleil et al 2021) and observed for their control effectiveness in a 3-s breathing cycle.From table 1, it indicates that PID control can only achieve a control performance close to our PIDK control in the sine waveform, while PIDK control clearly outperforms PID control in the triangular and trapezoidal waveforms, see table 2. In all test scenarios, PIDK can maintain a control accuracy of over 94% in a normal breathing cycle, and even in the case of 3-s rapid breathing (Pleil et al 2021), it can still achieve a control accuracy of 90%-91%.
The reason PIDK can achieve higher control accuracy lies in its ability to add an appropriate amount of control force in the I and D controllers, as seen in equations (4) and (5) and can be observed in  figures 4 and 5, where the red line represents the desired respiratory waveform, and the blue line represents the tracking results of PID or PIDK.From the figures, PID tends to exhibit insufficient control force near the end of the respiratory signal, and excessive PID control force can lead to overshooting.After appropriately increasing the control force using the Kalman filter, it is evident from figure 5 that the tracking capability near the end of various respiratory signals has improved without any overshoot.

Tumor movement
Tumor trajectory error grows with the length of the moving path.Therefore, we focus on the most extended path length at the lower lung section.Before starting this experiment, we randomly inputted 13 test coordinates, including coordinates in all four quadrants of the xy plane and the depth axis z.Next, we instructed the linear Delta robot arm to move the prosthetic tumor to those positions.The acquisition of the tumor position was achieved by measuring the voltage division of the sliding resistor, which converted it into the elongation of the iron rod.The end position and the error with the target position were calculated through forward kinematics.As shown in table 3, x, y, and z represent the desired target positions, while x′, y′, and z′represent the actual positions of the tumor.It indicates that the tumor position and the error with the expected target are both within 0.63 mm.Next, we divided the evaluation of the tumorʼs movement capability into two main parts.In the first part, we plotted ideal 2D and 3D trajectories to test the tracking error of the tumor model.In the second part, we replicated and mapped patient lung tumor trajectories published by other researchers and added a spiral path with a diameter of 10 mm.The linear Delta robot arm was then instructed to mimic these trajectories to verify its practicality.The tumor moved along the fixed path for 15 cycles in each test scenario, and then the average error was collected.As shown in figures 6 and 7, we plotted the movement trajectories as points to provide guidance for tumor tracking.
Figures 6(a)-(c) respectively represents 2D triangular, square, and circular trajectories.After observing the moving trajectories of lung tumors in different positions from 4D CT, we found that the tumor movement patterns during respiration can be roughly categorized into three types in a three-dimensional space: diagonal line, triangle, and rectangle (Donnelly et al 2007).Therefore, we tracked these three types of trajectories in figures 7(a)-(c).The tumor tracking results are depicted in figures 6(d)-(f) and 7(d)-(f).These  results show that due to the absence of any filters or regression mechanisms, the movement trajectory of the prosthetic tumor exhibits some irregular jitter caused by noise and arm vibrations.Nevertheless, its movement trajectory still closely approximates the original 2D and 3D shapes.Finally, table 4 demonstrates the tracking errors in the 2D paths, ranging from 0.34 to 0.53 mm, with the worst error occurring in tracking the circular path.While the 3D trajectory tracking errors in the table slightly increased, ranging from 0.3 to 0.64 mm, with the worst error occurring in tracking the rectangle path.
After confirming that the average error in fixed point and ideal shape tracking is below the general error standard of 1.5 mm, we referred to the actual lung tumor trajectories collected by Nakagawa et al (2014) and Suh et al (2008) and plotted a more complex 3D tumor trajectory for tumor tracking, as shown in figure 8. From figure 8 and table 5, it is evident that the linear Delta robot arm we designed can effectively

Implementation
In figure 9, we demonstrate the implementation of SBRT.The proposed phantom is installed on a thick acrylic board, including an FPGA platform, a skeletal model, and a Delta manipulator.The therapist, therefore, can easily transit the phantom between 4D CT (figure 9 In this case, we simulate a beam delivery from the top side 0°where the physical field size is 1 × 1 cm 2 .We utilize EBT3 film to assess the tumor dose within the phantom.In film dosimetry, the initial step involves establishing a calibration curve function that converts the film's optical density (OD) value to radiation dose.Subsequently, the optical   the measured dose (film dosimetry) is less than 3%, and the deviation of beam delivery on the center point of the tumor is about 1.4 mm.

Discussion
Building a phantom to mimic the life-like behavior of patients has been the long-dreamed goal of  radiation therapy (IGRT) (Pallotta et al 2022).For the limitation of the proposed phantom, the respiratory control suffers a shortage of moving in the L-R (Left-Right) direction.We will resolve such a problem by using dual nylon ropes to control the retraction in the future.A moderate incline will be produced due to the unbalanced retracting force of nylon ropes.
To accurately simulate the patient's respiratory motion and tumor trajectory, this phantom model utilizes 60 kg/cm servo motors.Although the servo motors have internal closed-loop control, for respiratory control, pulling down the sternum to simulate breathing motion affects the rotational speed of the servo motors.Including the friction of the nylon rope, both introduce a time delay of approximately 20-40 ms.Therefore, we adopted a closed-loop control strategy for respiratory control.The experimental results indicate that the developed PIDK control method adds additional control force to the PID controller and does not decrease the tracking accuracy of the respiratory waveform, even under maximum chest breathing amplitude.For tumor motion control, as the tumor and arm links are lightweight, they do not degrade the rotational speed of the servo motors.Hence open-loop control is employed to drive the prosthetic tumor.However, the simulation of tumor movement trajectories is not smooth due to the vibration generated by the high-torque servo motors.We reinforced the stability of the structure by using a three-layer support structure designed through 3D printing to secure the servo motors, linear gears, bearings, and rods, see figure 2(c).The density of the 3D-printed supports is, therefore, increased.While metal supports can potentially overcome the vibration problem, it significantly increases the weight of the robot arm, which is unfavorable for medical personnel.In the future, we will further reduce the vibration by exploring different support structure designs to achieve more accurate tumor trajectories.
Additionally, comparing the proposed tumor control to the others, the current classifications for tumor movement in various phantom types are with passive and active control methods.The passive system buries the tumor or marker inside the phantom, and the simulated breathing movement of the phantom drives the tumor movement.The anthropomorphic lung phantom 'LuCa' demonstrated this approach by utilizing wood and radiochromic films to create a spherical tumor, inserting this prosthetic tumor into a plastic tube fixed inside the phantom.The movement of the airbag via expansion and contraction causes the tumor to move.However, this method does not control the tumor; it can only follow a fixed trajectory influenced by lung movement (Perrin et al 2017).ADAM (Anthropomorphic Dynamic breathing Model) uses a cylindrical phantom lung with multiple tumor models inserted at different locations.A motor attached to an iron plate compresses the phantom lung to simulate tumor movement (Pallotta et al 2018).However, this model can only simulate tumor movements in the Anterior-Posterior (A-P) and Superior-Inferior (S-I) directions.The location of the phantom tumors still needs to be verified using CT (Computed Tomography) radiography, which can be cumbersome for tumor trajectory formation and verification.Lee et al developed a sponge-type prosthesis to achieve tumor trajectories by adjusting the composition ratio of the internal sponge material.However, for each specific tumor trajectory, a corresponding sponge material ratio is indispensable, and the actual trajectory must be verified using CT imaging, making the simulation of tumor trajectories more complex (Lee et al 2020).
Finally, the significant advantage is its cost-effectiveness utilizing 3D printing technology for phantom development.The production cost of this phantom is approximately 2,100 USD, including FPGA, commercial skeletal models, and servo motors.The servo motors alone account for 1,300 USD of the total cost.Compared to commercially available radiation therapy phantoms that can cost tens of thousands of dollars but are limited to simulating simple tumor motion trajectories, our phantom design can replicate patient conditions more accurately.

Conclusion
In this paper, we have proposed a phantom that incorporates real-time respiratory motion control and tumor trajectory imitation to provide reproducibility of patient breathing motion and tumor positions.We focus on the advantages of versatility, cost-effectiveness, and flexibility using 3D printing.This phantom can assist in the training of radiation therapy professionals and the validation of treatment plans.For respiratory motion simulation, we tested sine, triangular, and trapezoidal waveforms using the PIDK control method to achieve control accuracy of over 94% under normal breathing conditions.Whereas during rapid breathing, the control accuracy was over 90%.For lung tumor trajectory simulation, we used a 3D-printed linear Delta robot arm to drive tumor movement and successfully achieved simulation of various complex tumor trajectories with a maximum average error of less than 1.1 mm.Compared to currently available phantoms with cumbersome devices designed with inconvenient operations and can only simulate simple respiratory or tumor movement, the proposed phantom has advantages such as easy setup, real-time monitoring, and high reproducibility of tumor positions.These advantages are beneficial to healthcare professionals in conducting treatment simulations of SBRT.Future works will focus on trajectory simulation for different organs with the delta robot arm.
(a)) and TrueBeam (figure 9(b)) without impacting the tumor's position.By tracking the infrared reflector on the sternum, see figure 9(a), the respiratory gating system can detect breathing amplitude and transform the data into waveforms (Figure 9(c)).Here, the computer simulates a trapezoid breathing mode with a 10 mm amplitude, and the gating system can accurately detect the amplitude of the reflector for 1.02 mm.When testing beam delivery, our sternum shading plate can also accommodate different reflectors, see figure 9(b).Besides, figure 9(d) indicates the viability to target the spherical tumor.It can be seen that the design of semi-hollow and -pivoted support helps the therapist to discriminate a 40% printed tumor (including the center point) and then successfully circle it.Finally, a shot of EBT3 film is shown in figure 9(e).
radiotherapy.The proposed mechanical phantom replicates both a controllable respiration and a programmable tumor trajectory of the patient.With a skeletal model, the therapist can conveniently rehearse a new treatment plan at most of the positions in the ribcage, using a realistic tumor trajectory.The outline of the ribcage is also viable for attaching artificial flesh or a heart model for future applications, and we can print the different shapes of tumor holders to achieve obstacle avoidance in the ribcage.These advantages enable the proposed phantom to play the patient role in a lung SBRT end-to-end (e2e) test, including dosimetry, motion management, target volume definition, patient positioning, CT image acquisitions to plan delivery verifications, and all image-guided
The modified Kalman gain is where e(t) accounts for the current control error value, whereas e(t − 1) accounts for the control error value of the previous moment.E Mea represents the pixel error (2 pixels) of the LED spot's center position as sensed by the digital camera.Thus, after determining the value of K g , the current estimate formula for the control error is (Wu et al 2020)2018)ustrial robotic arm to drive the tumor model in a multidimensional motion, we chose to use the PLA 3D printed linear Delta robot arm for its compact size and easy installation.Compared to the linear drive model(Alvares et al 2018), the typical Delta robot arm offers more working range but requires overcoming larger torques to operate(Wu et al 2020).Considering that the Delta robot arm has to work horizontally to drive the tumor model in the S-I direction, the choice of selection was made on the linear drive model.As shown in figure

Table 1 .
Comparisons of control performance between PID and PIDK with sine waveforms.

Table 2 .
Comparisons of control performance between PID and PIDK with triangular and trapezoidal waveforms.

Table 3 .
Errors between the desired targets and the position of tumor from 13 coordinates.
density (OD) value of the Region of Interest (ROI) is analyzed on the irradiated film to determine the measured dose value, which is then compared with the calculated dose in the treatment planning system (Niroomand-Rad et al 2020).The difference between the treatment planning system calculated dose versus