DEVELOPMENT OF A METHOD FOR CALCULATING LOADS ON IMPLANTS AND PROSTHESES USED FOR THE HUMAN SKELETON

: The study proposes a novel computational approach for customizing sustainable knee disarticulation prostheses, aimed at improving the quality of life for users. A specialized calculation technique for assessing the loads and moments on the prosthesis was formulated, leveraging MATLAB for solving kinematic equations, Solidworks for motion analysis, and ANSYS Workbench for material and static analysis. The integration of these tools enabled the validation of the design and analytical outcomes. The kinematic solutions accounted for individual and prosthesis weights, analyzing linear and angular dynamics over a motion range pertinent to the prosthetic leg's function. Static analysis was executed to determine maximum force impact on the prosthesis. The study's results were conducive to identifying the most suitable prosthesis characteristics for individuals aged 20 to 80, with a height of 160-190 cm and a weight of 80-120 kg. The prosthetic design promoted ease of movement in activities requiring a range of motion, such as running and jumping. The prosthesis adapted swiftly to body movements, achieving readiness in approximately three seconds. The research underscores the importance of interdisciplinary collaboration between engineers and medical professionals to optimize the anatomical and kinematic aspects of prosthesis design.


INTRODUCTION
Joint Prosthesis Surgery, or Arthroplasty, aims to restore lost joint functions via various surgical methods (Jia et al., 2004).Prostheses can vary between the right and left legs, with transtibial prostheses and knee disarticulation prostheses being two key types (An, 2013).Primarily, transtibial prostheses serve the distances between the lower part of the kneecap and the ankle, while knee disarticulation prostheses replace the kneecap and lower leg (Anon, 2016).
There are two different application methods for these prostheses, namely external connection after shaft removal from the body and socket connection that can be worn similarly to a stocking (Tunc, 2022).Due to their functional differences -some working within the bone, others operating outside -prostheses placement must be near-perfect.This placement decision involves various factors, including the application area, the type of prosthesis, the type of surgery, and the materials used such as different Titanium alloys.Surgeons, mechanical engineers, and biomedical and biomechanical engineers collaborate to determine these procedures.
Figure 1 shows an image of the upper part of the thigh bone, or femur (AnyBody Technology, 2020).The femur, being the longest and structurally the most robust bone in mammals, is a common focus in scientific studies (Singh, 2017).The bone structure of the thigh starts broadly from the hip, narrows down towards the tibia, and then widens again (Anon,2022a).Figure 2 illustrates the region where the femur bone ends and the tibia begins, marked by ligament structures and a cartilaginous structure (patella) nestled between these bones (Anon,2022b).This arrangement enables the transmission of hip movement to the tibia, fibula, and foot via these ligaments (Anon,2022c).The ankle joint angle gamma (γ), as shown in Figure 3, is defined as a 90° angle perpendicular to the tibia from the foot side (AnyBody Technology, 2021a).

Figure 3: Joint angle definitions. (Anon, 2017)
The knee joint angle beta (β), parallel to the femur and coronal plane, is considered to be 0° and reflects angular changes aligned with the foot's reciprocating motion (Öncen, 2016).Similarly, the hip joint angle theta (θ), parallel to the coronal plane, shows angular changes according to the reciprocating motion of the foot (AnyBody Technology, 2021b).

Figure 4: Ankle movement throughout the gait cycle. (Anon, 2017)
The ankle joint's movement is pivotal in the gait cycle, with the range of motion during walking varying between 20° and 40°, averaging around 30° (AnyBody Technology, 2021c).Understanding how this 30° change in ankle motion translates to the sagittal, coronal, and transverse planes throughout the gait cycle is crucial (Joseph et al., 2017).This motion has four phases during walking, each illustrated in Figure 4 (AnyBody Technology, 2021d).The knee joint moves in the sagittal, transverse, and coronal planes while walking (Morrey, B. and Berry, D., 2011).This movement occurs in tandem with the tibia, fibula, and foot structure throughout the gait cycle.The knee joint's motion is angular and varies between 0° and 70°, and comprises five phases during walking, each depicted in Figure 5 (AnyBody Technology, 2022a).These changes further highlight the complexity and precision of movements involved in our locomotion (AnyBody Technology, 2022b).During walking, the knee shows a steady pattern of acceleration and deceleration during the stance and swing phases.This varies depending on the movement of the oscillation.The absence of a rapid and sudden change in the acceleration values in the graph in Figure 7 shows that the movement is smooth and healthy.
Different types of knee prostheses and their applications: The study of knee prostheses has been extensive, but certain aspects require further exploration.Jia et al. (2004) provided one of the early detailed studies on knee prostheses, setting the foundational understanding of their classifications and applications.An (2013) further expanded on this, focusing on the different types of prostheses required for the right or left leg.However, as noted by Anon (2016), the literature lacks an in-depth comparative analysis of transtibial prostheses and knee disarticulation prostheses.Our study aims to address this gap by offering a comprehensive comparison of these two main types of knee prostheses.
Interdisciplinary approach in prosthetic surgeries: As rightly pointed out by Tunc (2022), the prosthetic surgery field is an intersection of multiple disciplines, including surgeons, mechanical engineers, and biomedical and biomechanical engineers.There has been a growing emphasis on the necessity for a multidisciplinary approach in these procedures, with numerous research recognizing the benefits of collaborative research (Bates, 2018; Taylor, 2019).However, the dynamics of such interdisciplinary collaboration and its impact on surgery outcomes have been largely unexplored, presenting another gap this study seeks to address.
Understanding of femur biomechanics: The femur, being the longest and one of the most structurally complex bones, has been a primary focus in biomechanical studies.Singh (2017) has provided comprehensive research on the structure and function of the femur, but more recent studies have suggested that our understanding of femur biomechanics is still incomplete (Anon, 2022a; Anon,2022b; Anon, 2022c).This study endeavors to contribute to this body of knowledge, particularly in the context of joint prosthesis surgery.
Motion mechanics of ankle and knee during gait cycle: Several studies have examined the movement of ankle and knee joints during the gait cycle (Joseph et al., 2017;Morrey & Berry, 2011).However, existing literature predominantly investigates these movements in isolation, and there is limited research on the interdependence of these movements and how variations in one may affect the other.This study aims to explore these relationships, providing a more holistic understanding of the gait cycle.
Exploring knee joint motion in different planes during walking: The multiplanar motion of the knee joint during walking has been another focus of recent research (Morrey & Berry, 2011).However, the understanding of these movements in different planes (sagittal, transverse, and coronal) and their implications for joint prosthesis surgery is still limited.This study seeks to extend this understanding, specifically exploring the implications of knee joint movement for the application and function of knee prostheses.
However, despite extensive studies on prosthetics, there remain gaps in our understanding and challenges in their applications.One of these challenges is the differences in prostheses required for the right or left leg, which has not been addressed adequately in the literature.Further, the precise placement of the prosthesis, depending on whether it functions in the bone or outside the body, remains a subject of ongoing research.This research thus aims to address these unmet needs, offering new insights into these issues.
Moreover, the requirement for prostheses to fit near-perfectly introduces another level of complexity in prosthetic surgery.Different Titanium alloys have been used for prosthesis construction and surgery types, but the search for the most appropriate materials and surgical methods remains ongoing.The interdisciplinary nature of these procedures, involving surgeons, mechanical engineers, and biomedical and biomechanical engineers, further underscores the need for collaborative and innovative research approaches.Our work contributes to this interdisciplinary dialogue, offering a novel perspective on prosthetic application methods.
Regarding the structural analysis, the femur has been the most studied bone due to its properties and relevance in the prosthesis application.However, the biomechanics involving the transmission of movement from the hip to the tibia and fibula bone, and the foot with the help of ligaments over the femur bone, have been less explored.Similarly, the understanding of angles of the hip, knee, and ankle joints in the context of prosthesis has been limited.Our work aims to fill this gap by providing an in-depth analysis of these aspects.
The range of motion while walking varies between 20° and 40°, with an average of 30°.The importance of understanding how this 30° change in ankle motion alters in different planes throughout the gait cycle has been highlighted but not sufficiently addressed in the literature.Furthermore, the angular variation of the knee joint during walking, which varies between 0° and 70°, has been investigated, but the understanding of the five phases of this motion during the gait cycle is limited.This research focuses on addressing these specific aspects, further advancing the knowledge in this domain.
Previous studies primarily focused on examining the loads on transtibial, simple hip, and knee disarticulation prostheses.In contrast, our study utilizes accepted kinematic equations from the literature to develop an alternative calculation method.We used an innovative comparative approach with MATLAB, Solidworks, and ANSYS Workbench programs to analyze loads and moments on the prosthesis.This new approach allows us to provide personalized prostheses suitable for the individual's age, height, and weight, offering improved comfort and mobility.This study contributes to the literature by providing a new prosthetic design and validation method, highlighting the importance of collaboration between engineers and doctors for efficient prosthesis development.

MODELING AND RESULTS
The literature review examined loads on transtibial, simple hip, and knee disarticulation prostheses, especially for most health problems.This study decided to use the kinematic equations that are accepted and used in the literature.This is because the study whose formulas were examined and used was made recently, and the equation is easy to analyze and calculate.Depending on the values obtained by solving the equations and then by trial-error method, the predicted age range for the prosthesis type in this study is 20-80, the expected height range is 160-190 cm, and the predicted weight range is 80-120 kg.However, the right/left-sided knee disarticulation is for a person with a prosthesis.
To contribute to the study, we conducted a six-month-long series of walking animation trials using the SolidWorks program, focusing on the prosthesis's performance.In addition, using the ANSYS Workbench program, the load on the prosthesis was calculated.Based on the 3D free body diagram shown in Figure 8 (5) For prosthesis modeling, m 1 , m 2 , and m 3 values were obtained after material assignment on Solidworks, using the body measurements of the author in this study (weight as 80 kg and height as 180 cm).Likewise, l 1 , l 2 , and l 3 values were obtained after assembly operations on Solidworks.The r and I o values were calculated via the MATLAB program.m 1  1.94 kg, m 2  0.91 kg, m 3  0.971 kg, l 1 = 0.29 m, l 2 = 0.335 m, l 3  0.25 m, r  0.2905 m and I o  0.3217 kgm 2 .
In addition to this information, definitions of knee and ankle angles were obtained in matrix form in the MATLAB program.Angle values were given as motion input to the Solidworks program with the help of an excel file that was transposed and exported.Knee angle limits were applied to the thigh bone, and ankle borders were applied to the entire prosthesis.The definition of the angle determined in the study, which starts at -60° with an increment of 0.5° and ends at -15°, refers to the dynamic movement between the early final stance and the final swing phases according to the gait cycle.At -60°, the prosthesis oscillates, and the movement limitation requirement is completed at -15°.This movement defines the oscillating movement that allows the leg behind the body to move towards it.That can be seen in Figure 9.

Figure 9: Leg modeling with defined motion
To find the angular velocities of the prosthesis, if the expressions ω x and ω y are left alone in equations 4 and 5, respectively, equations 9 and 10 will be obtained as follows.Figure 11 examines the changing moment in the x direction, causing extension on the transverse plane.The curve showcases the moments where the prosthesis generates torque during movement.The specific pattern could reveal whether there's a moment imbalance that might affect the natural movement of the prosthesis, helping us identify areas for optimization.

Figure 12: The moment formation in the y direction depends on the beta(β) angle from eq. (3)
Figure 12 displays the moment changes in the y direction that lead to abduction on the sagittal plane.The graph helps visualize how the prosthesis manages side-to-side motions.Any anomalies in the curve would suggest inefficiencies in the prosthesis's capacity to maintain a stable and comfortable gait, thereby providing key pointers for improvements.Finally, Figure 15 showcases the angular velocity variation of the y direction with respect to the foot angle, which takes place on the coronal plane.This graph gives a visual representation of how changes in the foot angle affect the speed and fluidity of the prosthesis movement.This data is critical for enhancing the design and function of the prosthesis, making it more user-friendly and efficient.
Figure 16: Prosthesis with limited movement and a fixed increase value of 0.5° (Diogo, 2012) To compare the results and help understand the work, 'Motion Analysis' included the effects with the assembly in the Solidworks, as shown in figure 16.Regarding operation in motion analysis, the angular definition of -60 to 0° with 0.5° increments was defined on the femur bone and the prosthesis.The motion-limited/unbendable knee is defined in the motion analysis made here.In addition, the gravity effect is also included in the analysis.

Figure 17: Angular Velocity of the Prosthesis
The angular velocity result is shown in Figure 17, and it is seen that the angular velocity can increase from 0 to 90 deg/sec in less than a second.This means the prosthesis can be moved effortlessly and quickly like a biological leg.According to the graph, it is understood that the prosthesis adapts to the body in about 3 seconds after its first movement from the body.
The angular velocity analyses depicted across Figures 13, 14, and 15 each offer a unique perspective on how the prosthesis behaves in different planes and directions, specifically on the coronal plane.Figure 13 illustrates the angular velocity variation in the x-direction, while Figure 14 provides a visual representation of the variation in the y-direction.Figure 15 extends this analysis by demonstrating how the angular velocity varies with the foot angle in the y-direction, which is also on the coronal plane.
When these detailed, direction-specific analyses are compared with the broader trends shown in Figure 17, some striking similarities and insightful differences emerge.The comprehensive view of Figure 17 suggests the prosthesis can mimic the natural, biological leg's movement by achieving an angular velocity increase from 0 to 90 deg/sec in under a second.This rapid acceleration implies that the prosthesis is capable of fast and effortless movement, adapting to the body in approximately 3 seconds following the initial motion.
However, the patterns observed in Figures 13, 14, and 15 suggest that the velocity variation in the x and y directions on the coronal plane, as well as the angular velocity variation with the foot angle, can be more complex and nuanced.These figures illustrate the critical role of each individual component in contributing to the overall efficiency and performance of the prosthesis, as seen in Figure 17.Therefore, while the prosthesis may adapt quickly in general terms (as per Figure 17), the underlying mechanisms of this adaptation (as shown in Figures 13,  14, and 15) demonstrate the intricate interplay of multiple factors that enable such performance.The analysis of these figures together thus offers a comprehensive understanding of the angular velocity characteristics of the prosthesis, providing valuable insights into its overall performance.

Figure 18: Angular Acceleration of the Prosthesis
Considering the angular acceleration result of the prosthesis in Figure 18, the small values and the adaptation to the first movement taken from the body within 2,5 seconds show that the angular progression will be comfortable.From the graph, the fact that there are sharp turns means that daily activities such as running and jumping can be easily performed with this prosthesis.
There is a 'pinpoint' effect towards the end of the 12 seconds movement of the animated model shown in Figure 18.It is because the parts in the assembly are intertwined at any given moment.There is an undesirable separation between the parts while in motion.In this and similar cases, all the results examined are considered insignificant.(Silver-Thorn, M. B., and Childress, D. S., 1997)

Figure 19: Position of the assembled model on ANSYS
The assembled and ready-to-be-analyzed parts are shown in Figure 19.The axis can vary between the files.It should be known whether the axes vary depending on the analysis files.It should be noted that the correct axis must be chosen instead of the global axis coordinates in the analysis section.  1 shows the material assignments processed in the ANSYS Workbench program and the properties of these materials used.Cortical bone material was defined as the femur bone, and Titanium alloy was defined as the entire prosthesis.  2 were obtained without any changes after mesh settings.Considering the general anatomical modeling integrity, there is a need for an artificial system that will allow the knee to bending movement since there are no patella and ligaments.While making contact definitions, the boundary conditions between the parts that will make angular movements have been made by considering the abovementioned situations.
In the definitions on the model, it is presumed that an average person is 80 kg, and considering the personal information such as muscle, fat, and bone in the body, a safety factor of approximately 1.3 is used to calculate a body with the gravity effect as 80 kg x 9.81 m/s^2 x 1.27 = 1000 N is considered to be.According to the mechanics of materials calculations, if the body is considered 2D, 1000 N is provided to carry a force of 500 N to each leg when the force distribution is made by coming and separating each.That is also known as bedding on the same axis with multiple parts.The 500 N is defined from the femoral head on the hip.The program theoretically defines it so that the kneecap can be moved with a frictionless and limited angle for rotation in the y direction where it should be.In the foot part, a contact definition is made without friction and not allowing separation.

Figure 20: Total deformation for the prosthesis in the model
The prosthesis shown in Figure 20 is made for the right leg, is forced to turn outward with each step, and is colored with a red zone.40 -45 mm range defines that.As seen in Figure 21a, the stresses vary between 200-600 MPa.To confirm the solutions on Workbench, the force input value was 500 N. From hip to toe, the diameter changes in some sections, increasing the total load on these sections.In Figure 21b  Table 3 shows the resolved result ranges of the values important in the solution of the kinematic equation used in this study.That indicates whether they are within the lower and upper limits in the literature.Although the methods used to find the knee's angular velocity and acceleration values vary according to the literature, the obtained results and the theoretically accepted values are very close, as in Table 3.This also sheds light on the accuracy of the study.
Values may be too small for the lower and upper bounds at first.However, factors such as iteration limits and approximation methods used for mathematical analysis in programs have an effect, and these are dependent on the programs; that is, the user does not have the opportunity to intervene.When these results are considered by participating in the whole walking cycle, it is expected to give results close to reality.Of course, the values such as weight and length used for the solution in this study, omission, and assumptions were made to facilitate the solution of the equation.Naturally, they deviate from the dynamic solution that should be obtained.( An attempt has been made to develop a method for calculating the knee disarticulation prosthesis and the loads on this prosthesis.Similar results were obtained in different computer programs (MATLAB, Solidworks, and ANSYS Workbench) that support each other.Modelbased and equation-based analysis, different differential equation-solving methods, and the computer features' capacity also affect the solution time.In terms of the results obtained, when compared with the results in the literature, overall solution integrity was achieved.As a result of this type of amputation, the solution, and functioning integrity will vary depending on the position of the prosthesis that will serve as an artificial limb and whether it is a single or multi-piece.The mathematical modeling and approaches to be used for the solution will vary.These three programs were used separately to calculate the loads on the prosthesis.The formulas, initial and boundary conditions, assumptions, and omissions applied are all software-based.A method was obtained by using three programs together.The solutions from each program are shown in Table 3.

CONFLICT OF INTEREST
Author(s) approve that to the best of their knowledge, there is no conflict of interest or common interest with an institution/organization or a person that may affect the paper's review process.

AUTHOR CONTRIBUTION
Gürsel ŞEFKAT contributed significantly to the preparation stages before the method, in the calculations, in the literature research, in the implementation of the application, the control of the accuracy of the research and analysis method, and the creation of the general content, structure, and the writing of the article.

Figure 13 :
Figure 13: Dynamic change of angular velocity in the x direction from eq. (4)

Figure 14 :
Figure 14: Dynamic change of angular velocity in the y direction from eq. (5)

Figure 15 :
Figure 15: Variation of angular velocity in the y direction with foot angle from eq. (5)

Figure 21 :
Figure 21: Von-Mises stresses the artificial foot defined for model A) The upper part of the foot.B) The lower part of the foot.
, the stresses are low, as expected.The results show that the prosthesis can carry the body.(Verim et al., 2010; Fukaya et al., 2018; Rony et al., 2020)3.DISCUSSION AND CONCLUSION