Design of a Lower Limb Prosthesis for Ballet Practice

Abstract

Ballet is a discipline that combines art and sport in a harmonious way. It is a practice that has high physical and mental demands to achieve the proper body precision. During this activity, numerous muscles, including those in the legs, need to be exercised. Therefore, individuals who have lost part of their lower limb due to amputation face numerous significant challenges when it comes to practicing ballet. Throughout this article, the key aspects that influence the design of a lower limb prosthesis specifically adapted for ballet practice will be analyzed. New materials will be explored with the goal of designing an optimal model that ensures maximum performance and comfort for the users. Additionally, the prosthesis will be customized using 3D-printing technology, and a prototype will be presented. This study will merge biomechanics, ergonomics, and design. Its goal is to find a solution that improves the quality of life for lower limb amputees whose passion is ballet. The aim is to overcome physical and emotional barriers and provide a way to reintegrate amputee dancers into the world of dance. It is important to highlight the novelty of this work: combining different disciplines to provide a solution for individuals who engage in dance as a hobby rather than professionally. The proposed methodology aims to enable users with disabilities to access a personalized, complex, and potentially costly product.

1. Introduction

A prosthesis is defined as a device, apparatus, or external piece used to partially or completely replace an organ or limb, aiming to replicate its original function in the body. These devices have been used since the beginning of humanity and have undergone a constant evolution that is still very much present today.

Within the field of lower limb prosthetics, different types of artificial limbs can be found according to the user’s needs and the desired functionality of the prosthesis. Depending on the type of amputation, different devices are designed. This study is focused on infracondylar amputation, located between the upper and middle thirds of the tibia, allowing the user to retain their knee joint for movement.

With the aim of obtaining a realistic and close perspective for the study, interviews were conducted with users involved in the field of prosthetics and sports. These interviews included an orthopedic technician with experience in university teaching, as well as a paralympic athlete employed by a renowned orthopedic company. The objective was to comprehend the design of prosthetics and identify the essential characteristics needed for a prosthesis suitable for dance. After analyzing the interviews and considering previous research on prosthetics and classical dance, it was concluded that the project would concentrate on redesigning a sports prosthesis.

Within the study of sports prosthetics, there are several available categories, but the most common one is the dynamic response foot. This type of prosthesis is characterized by sophisticated designs that absorb impact upon contact with the ground and utilize stored energy to propel the foot forward. During the toe-off phase, the spring action propels the prosthesis into the swing phase of gait, and the pattern is repeated [1]. An important benefit is the energy conservation achieved by using this type of prosthetic device, as it accumulates and returns energy during walking, mimicking the propulsion of the human foot [2].

To provide a more comprehensive understanding of the available categories in sports prosthetics, the following

Table 1. Available categories in sports prosthetics [3].

Prosthetic Category	Description	Example Use
Dynamic response foot	Absorbs impact and uses stored energy for propulsion	Running, fast walking
Articulated foot	Has joints for increased flexibility	Casual walking
Multi-axial response foot	Allows movement in multiple planes	Uneven terrain, outdoor activities
High-performance foot	Designed for specific high-performance activities	Athletics, competitive sports

is presented:

The choice of the dynamic response foot is due to its proven effectiveness in energy conservation and performance enhancement in sports activities. This type of prosthesis offers an optimal combination of functionality and comfort, making it the preferred choice for both amateur and professional athletes.

Comfort and responsiveness of the dynamic foot are essential in enabling prosthesis users to advance in their activities, from minimal to advanced levels, regardless of whether the athlete encounters a flat or uneven surface [1].

The following analysis studies the functioning of one of the most significant sports prostheses in the current market: the Cheetah range of prostheses by Össur [4].

These devices stand out for their innovative design and functionality for various physical activities. These employ dynamic response technology, absorbing impact upon contact with the ground and harnessing stored energy to propel forward movement. The most common prosthesis, the Flex-Foot Cheetah, has demonstrated the ability to return up to 90% of stored energy, surpassing conventional prostheses capabilities [5]. They are composed of optimized layers of carbon fiber through computational analysis and mechanical testing, ensuring proper flexion of the components in the heel and toe relative to the user’s weight and impact level. Additionally, they feature active tibial progression, reducing the need to actively push the body forward with the opposite foot and providing a more natural gait. With different variants designed for specific activities such as long sprints or jumps, the Cheetah range of sports prostheses offers tailored solutions to meet the needs of amputee athletes.

After conducting a thorough initial analysis of both the study discipline, classical dance, and sports prostheses, the following design requirements were defined, which will determine the essential characteristics to consider during the device creation process:

• The prosthesis should be lightweight, to avoid interfering with the required movements in ballet and maintaining the characteristic sensation of lightness in the discipline.
• The selected materials should provide sufficient strength and durability to withstand the intensity and physical demands of ballet, ensuring long-term functionality. Additionally, the material should guarantee user safety, avoiding potential failures.
• Materials will be preferably selected with a lower cost, if they meet the necessary characteristics, in order to obtain an affordable prosthetic device for all types of users.
• The device should ensure continued gait during activity, through dynamic response and active tibial progression.
• The design of the prosthesis should be proportional to the type of user. Dimensions, amount of material, and shape will depend on the user under study. Additionally, it should allow for personalized adaptation and adjustment to the individual’s body structure and the dancer’s preferences.

To be suitable for dance, this prosthesis should allow for a wide range of motion, freedom, elasticity, and flexibility similar to devices designed for running. The proposed methodology aims to enable users with disabilities to access a personalized, complex, and potentially costly product.

2. Materials and Methods

2.1. Analysis of the Foot in Ballet

Understanding the functioning of the foot requires a detailed study of its mechanics and principles. In the context of injuries sustained by ballet dancers, a common occurrence is the failure to meet the inherent requirements of joint movement. Consequently, this study will proceed to delve into an in-depth examination of the biomechanics behind two pivotal movements in classical dance: relevé, characterized by raising the heels off the ground through the flexion of the arch of the foot, and demi-plié, which involves partial knee flexion.

2.1.1. Biomechanical Analysis

Biomechanical Study of the Foot in Relevé

The foot comprises three arches, including the medial longitudinal arch supported by the plantar fascia and ligaments, and the lateral longitudinal arch stabilized by the proximal tuberosity of the fifth metatarsal. During relevé, the arches become more rigid to provide stability and leverage for pushing off. The foot’s biomechanics involve supination and pronation in various joints, with the medial arch becoming rigid through bone formation and plantar structure tension. The foot’s arches function as a truss-like structure not relying solely on muscles for support. Understanding these arches and plantar structures is crucial for designing prostheses and maintaining stability during relevé [6].

Using Kinovea software [7], version 2023.1, a specific tool for analyzing movements in sports, a study was conducted to analyze the angles formed by the foot during the relevé position. The study was based on photographs of a 21-year-old woman with dance experience as a hobby and an understanding of proper foot positioning in different steps. It is important to note that each individual has different flexibility, and the angles may vary. However, these data will be used as a reference for prosthetic design. In Figure 1a, the angle of toe flexion is observed to be approximately 80°. In Figure 1a,b, the angle formed between the foot and the horizontal support surface is studied. Taking the end of the first phalanx that connects to the metatarsal as the reference point, the angle formed between the horizontal and the heel is approximately 130°. In Figure 1c, the same measurement procedure is followed, but this time the angle formed between the phalanges and the ankle joint is observed. Furthermore, the angles formed between the plantar surface of the foot and the tibia are examined. Firstly, in Figure 1d, the measurement originates from the heel, resulting in an angle of 130°. If the measurement originates from the ankle, as shown in Figure 1e, the angle increases to 170°.

Biomechanical Study of the Foot in Plié

When performing a demi-plié, a significant talocrural flexion is necessary, which should be reflected in the prosthesis through compression. The compression should be extensive.

The angles formed between the foot and the leg during the plié are studied, following the same procedure as in the previous relevé study.

Figure 2a shows the angle formed between the plantar surface of the foot and the tibia during knee flexion in the plié, with the measurement originating from the heel. The angle formed is 50°. Comparing this angle with the corresponding one studied earlier for relevé, which was 130°, there is a compression of 80° between the two positions.

The dorsiflexion angle of the foot is 40°, which is suitable for absorbing the jump impact and generating propulsion (Figure 2b).

2.1.2. Kinetic Analysis of the Foot in Ballet

The analysis of forces during the relevé position is of great importance in the design of the prosthesis, as it is the most relevant position and the one on which the shape of the device will be based. This is the most common foot position for dancers, who use relevé both to move between different positions and to perform common steps.

When the foot is raised, using the toes as the only support, three main forces can be distinguished, as shown in Figure 3. Firstly, there is the ground reaction force (Frs), caused by the weight of the individual that the foot supports. Secondly, there is the tension generated by the soleus and gastrocnemius muscles on the calcaneus bone through the Achilles tendon (Fta). Lastly, there is the ankle reaction force generated by the tibia (Fti) [8].

A numerical study of the forces acting on the foot of an average user is conducted. It is assumed that the user is in a neutral and static position.

The analyzed forces are concurrent at a point, and the angles they form with the horizontal axis are known, so the force system can be represented in Cartesian axes as shown in Figure 4.

The equations of static equilibrium presented below (Equations (1)–(4)) were employed to determine the numerical values of these forces, using the average weight of a female population that is discussed later in this article as a reference for the final design.

$${\displaystyle \sum }{F}_x=0$$

(1)

$${\displaystyle \sum }{F}_x=F_{tax}-F_{tix}=0$$

(2)

$${\displaystyle \sum }{F}_y=0$$

(3)

$${\displaystyle \sum }{F}_y=F_{rs}+F_{tay}-F_{tiy}=0$$

(4)

The ground reaction force (Frs), caused by the weight of the individual supported by the foot, is calculated through the following equation:

$$F_{rs}=m\times g$$

(5)

where

m = body mass (kg);

g = gravity (m/s²).

Consequently, for an individual weighing 62.55 kg, the forces acting on the foot while standing on tiptoes are as follows:

$$F_{rs}=612.99 \mathrm{N}$$

(6)

$$F_{ti}=1674.72 \mathrm{N}$$

(7)

$$F_{ta}=1184.21 \mathrm{N}$$

(8)

This analysis assumes that the individual is supported by only one foot, which represents a worst-case scenario.

Through this numerical reasoning, the foot has been simplified as a mechanical element. However, it is important to note that the analyzed forces are those acting in a static moment, whereas, during the activity of ballet, the individual is in motion. Therefore, dynamic forces should also be considered.

To calculate the force that will act on the artificial limb, both the weight it will support (user’s weight) and the dynamic load factor will be taken into account.

The dynamic load factor quantifies how a static load is amplified when applied dynamically, reflecting increased stress during activities like ballet [9]. To obtain this factor, a study conducted at the University of Washington in 1981 on an elite ballerina is used. The study analyzes the vertical ground reaction force, measured in relation to time during the landing phase of a grand jeté [10], one of the steps that entails the most impact, where the prosthesis will be subjected to higher loads and tensions.

In Figure 5, it can be observed that the highest vertical force is 4.85 times the body weight, so this factor is rounded up to 5 in order to increase the safety factor.

Taking this information into account, it is concluded that the total force (FT) to which the prosthesis will be subjected is:

$$F_T=m\times g\times F_{CD}$$

(9)

where

m = body mass (kg)

g = gravity (m/s²);

$F_{CD}$ = dynamic load factor.

2.2. Prosthetic Design

It is important to consider certain characteristics related to the artificial limb being designed:

Energy Return: The designed artificial limb should act like a spring. When the user takes a step, the prosthesis should deform and then regain its original shape, converting elastic potential energy into kinetic energy. Therefore, the displacement produced in the prosthesis should be sufficient for the return force to have an appropriate value, allowing for the necessary energy return to enable the user to continue the movement.

Resilience: The chosen material for the prosthesis should be resilient, allowing the product to recover its original shape when the applied force is removed.

Ductility: It is a mechanical property of materials, opposite to fragility. It refers to the ability of a material to undergo plastic deformation without breaking when subjected to a sustained tensile force [11].

Material Failure Criteria: Static failure analysis aims to relate strength and static load, with the main objective of making decisions regarding the material to be used and the geometry of the designed element to satisfy design requirements such as safety and reliability, among others. The von Mises theory is the failure criterion used in the prosthetic design. It states that failure by yielding occurs when the total deformation energy per unit volume reaches or exceeds the deformation energy per unit volume corresponding to the material’s yield strength. In a simplified case of stress, yielding would occur if σ ≥ Sy, where σ is the equivalent von Mises stress and Sy is the yield strength of the material. Therefore, in prosthetic design, the following equation is used, introducing the safety factor, and ensuring that the resulting von Mises stress from the analysis does not exceed the selected material’s yield stress [12].

$$\sigma ={\displaystyle \frac{S_y}{n}}$$

(10)

where

σ = equivalent von Mises stress (Pa);

Sy = yield strength (Pa);

n = safety factor.

Another type of failure to be considered is fatigue failure. The action of fatigue loads causes a reduction in the mechanical strength of materials when these loads act repeatedly over a certain number of cycles. Knowing the fatigue limit of a material (S’n), which is the maximum stress it can withstand for an infinite number of cycles (typically considered infinite when the number of cycles is greater than 10⁶), tests are conducted where a load is applied repeatedly, and the cycles until the piece breaks are determined. These tests are represented by the S-N curve (stress–number of cycles) [13].

Geometry: The geometry of the designed product should align with the previously conducted biomechanical study. However, stress concentrations should also be considered.

Sustainability: A sustainable product is one in which materials and manufacturing processes do not have a negative impact on the environment. In the manufacturing of the prosthesis, processes should maximize the use of energy and avoid the release of toxic substances throughout the duration of the process.

Safety Factor: The safety factor is a measure intended to make a product safe by designing it to withstand conditions that are more adverse than normal, allowing it to continue functioning even under increased load or excessive use. As mentioned earlier, the theory of von Mises provides a way to calculate this factor, which will be used in the development of the prosthesis.

2.3. Population of the Study

The objective of designing this prosthesis is to make it accessible to a wide population of individuals who wish to enjoy ballet as a hobby and provide them with an artificial device that enables effective participation in this activity. As a result, anthropometric data obtained from a study based on a sample of 1723 Spanish individuals, including both males and females aged between 16 and 65 years, will be used as a reference, covering a broad range of potential users [14].

These data will be used for the generalized prototyping of the prosthesis. However, when the device is manufactured for an actual individual, it will be customized to the specific characteristics of each user, ensuring optimal and comfortable adaptation for ballet use.

Factors such as user weight, foot length, and width will be taken into account. Additionally, the optimal level of amputation, typically in the midshaft of the tibia, will be considered to ensure proper leverage and sufficient muscular coverage [15]. Therefore, tibia height data will be used as a reference, representing the distance from the feet resting on the ground to the highest point of the tibia when the subject is standing.

To encompass a high percentage of the population, data between the 5th and 95th percentiles will be considered, making the prosthesis suitable for 90% of the Spanish population.

The range of values between the two tibia heights provided by the percentiles will be utilized to determine the posterior height of the prosthesis to which the socket will be adjusted. The height of the socket can be adjusted higher or lower based on the amputation level of the individual using the prosthesis.

The data to be used are presented in

Table 2. Anthropometric data of the population of study.

Female Population
Measure	P5	P95	Average
Body mass (kg)	48.1	77	62.55
Tibia height (mm)	384	488	436
Foot length (mm)	215	257	236
Foot width (mm)	79	100	89.5
Male Population
Measure	P5	P95	Average
Body mass (kg)	58.6	95.8	77.2
Tibia height (mm)	409	520	464.5
Foot length (mm)	234	282	258
Foot width (mm)	87	112	99.5

.

2.4. Phases and Methodology

There are 6 main phases for applying this methodology:

• Phase 1. Defining geometry constraints. The main dimensions for both genders are defined in Table 2. Obtaining detailed dimensions for the studied case: transtibial amputation considering the difference between the percentile extremes in the studied population: 5% and 95%.
• Phase 2. Idea generation. Based on the previous phase, in this phase, preliminary sketches are generated. Best sketches should be defined in more detail.
• Phase 3. Stress analysis. Static stress analysis is performed for defined solutions. The simulation is carried out with Fusion 360 software [16]. Stress and deformation will be calculated and checked. Previous to the simulations, materials should be selected and also constraints and load force should be included in the finite element model.
• Phase 4. Results and review. Results from phase 3 should accomplish the displacement established range and safety factor considered for stress maximum values.
• Phase 5. Optimization. This phase tries to minimize prosthesis weight, by adjusting geometry (contours, thickness, and holes).
• Phase 6. Prototyping. Developing a prototype for checking and even improving the selected solution. In this case, to create a tangible representation of the model, a 9/20 scale prototype was fabricated using 3D printing with the 3D printer Onyx One by Markforged, which produces nylon parts reinforced with carbon or glass microfibers through fused filament fabrication (FFF). The printer’s proprietary software, Eiger, was used to convert and prepare the 3D model for printing, ensuring the necessary adjustments for the desired outcome.

3. Results and Discussion

3.1. Requirements of Prosthesis

The main dimensions of the prosthesis that will define its shape are determined by the conducted biomechanical study and the studied anthropometric data. For the design, the data from the female population collected in Table 2 will be used. In the case of designing a prosthesis oriented toward the male gender, the same procedure would be followed, but with the male data from Table 2.

The optimal level for a transtibial amputation is at the middle third of the tibia. Therefore, the minimum distance from the stump to the knee, ensuring proper use of the joint, must be one-third of the length of the tibia. It is thus considered that the distance from the stump to the ground will be two-thirds the tibial height, defined as the distance from the feet on the ground to the highest point of this bone, measured with the subject standing. To this length, an additional height is added, defined by the difference between the 5th and 95th percentiles of the population, which will serve as the adjustment surface for the socket. The following dimensions are obtained:

$$Prosthetic foot height={\displaystyle \frac{2}{3}}\times Tibial height={\displaystyle \frac{2}{3}}\times 436\cong 291 \mathrm{mm}$$

(11)

$$Fitting height=P95 Tibial height-P5 Tibial height=488-384=104 \mathrm{mm}$$

(12)

$$Total height=Prosthetic foot height+\mathrm{Fitting} \mathrm{height}=291+104=395 \mathrm{mm}$$

(13)

With all the necessary measurements defined, preliminary sketches are created using Autodesk Inventor software [17], an advanced 3D design tool, obtaining the following design (Figure 6).

Once the geometry of the design is defined, the force to which the prosthesis will be subjected is simulated in order to verify its feasibility and whether the designed shape with the applied material is suitable to withstand the maximum effort required for classical dance activity.

The force to which the model will be subjected during the experiment is calculated using Equation (14), as explained previously, to simplify the calculation of the force acting on the foot during movement. Considering that this specific model is designed for the female population, the resulting force value is:

F_T = m × g × F_CD = 62.55 × 9.8 × 5 ≅ 3065 N

(14)

To perform this simulation, a finite element analysis will be conducted using Fusion 360 software [16], version 2.7.0. A static stress analysis will be carried out, which examines the stress and deformation of the design based on a series of structural loads and constraints that need to be defined. The results of this study provide data regarding the stresses to which the model will be subjected, indicating areas of higher stress concentration, the safety factor of the prosthesis at different points, and the experienced displacement.

To initiate the finite element analysis, the first step is material selection, wherein all relevant material properties are specified as required by the software, along with their corresponding values. Subsequently, the fixed faces of the model are determined. In this instance, the upper face of the prosthesis is fixed, given its compression function, rendering this area immobile. The subsequent step involves applying the load acting upon the device. The load is applied to the ground contact area, which in this case is the flat lower face of the design. Hence, a force of 3065 N, previously calculated, is applied. The mesh was generated with an average element size of 5 mm to ensure a balance between computational efficiency and result accuracy, as smaller mesh sizes typically provide more precise results. Tetrahedral elements were used for a more accurate representation of complex geometries. The convergence criteria were set to a relative error tolerance of 1 × 10⁻⁶, ensuring that the residual forces and displacements between iterations were sufficiently minimized.

The maximum stress experienced by the design must be lower than the elastic limit of the material used for the prosthesis to prevent it from breaking.

A safety factor of 1.5–2 will be considered appropriate. When calculating the force that the prosthesis will be subjected to (Equation (9)), the highest dynamic load factor of 4.85 was taken into account, which was increased to five to further enhance the safety factor. Therefore, the finite element analysis will be performed using a higher force value than usual, applied at the moment of greatest impact during dance movements. Considering this increased load and the conditions of the simulation, a safety factor between 1.5 and 2 can be deemed appropriate. A higher safety factor reduces uncertainty in the design, so as long as the obtained value is greater than or equal to the mentioned range, the result will be considered acceptable.

The maximum displacement experienced by the model should not be too small or too large, as excessive flexion in the prosthesis can result in instability and misalignment of the impulse direction. A maximum displacement of 50 mm will be considered as the limit, as this value has been accepted in other research studies that examine the displacement of various sports prostheses during use [18].

3.2. Prosthesis Definition and Discussion

To select the prosthesis material, the different parameters mentioned above are considered, such as ductility, resilience, and energy storage, focusing primarily on composite materials with polymeric matrices, as they are the most commonly used for this type of application. The chosen material, optimal based on the established objectives, is the Epoxy/HS carbon fiber, UD prepreg, UD lay-up material. Other potential candidates include BMI/HS carbon fiber, UD prepreg, UD lay-up, and PEEK/IM carbon fiber, UD prepreg, UD lay-up. When comparing the properties of these three materials, it can be concluded that Epoxy/HS carbon fiber, UD prepreg, UD lay-up is the optimal choice as it has the lowest price, less than 40 EUR/kg, and a very high energy storage capacity, between 10,000 and 17,000 kJ/m³, making it the most suitable for the defined objectives.

Once the material is selected, finite element analysis is performed, obtaining the following results in

Table 3. Model with 20 mm thickness.

Model with 20 mm Thickness
Study Type	Results
	Max	Min
von Mises stress	116.4 MPa	0.416 MPa
Safety factor	15	15
Displacement	6.227 mm	0 mm

and

Table 4. Model with 10 mm thickness.

Model with 10 mm Thickness
Study Type	Results
	Max	Min
von Mises stress	465.2 MPa	1.029 MPa
Safety factor	15	4.202
Displacement	46.06 mm	0 mm

:

Based on the results, it can be concluded that the model with a thickness of 20 mm is over-designed. The minimum safety factor is equal to the maximum, at 15. As previously explained, a safety factor between 1.5 and 2 would be acceptable for this design, meaning a factor of 15 is excessively high. This indicates that the prosthesis is over-designed, and modifications can be made to optimize its performance. Specifically, the thickness of 20 mm results in a much higher safety margin than necessary, and reducing this thickness would allow for a lighter, more efficient design without compromising the structural integrity or safety of the prosthesis. However, the geometry designed with a 10 mm thickness using the Epoxy/HS carbon fiber, UD prepreg, UD lay-up material fulfills all the requirements to ensure appropriate usage during ballet.

The design weight is 0.808 kg, which contributes to the lightweight nature of the prosthesis. However, in classical dance, conveying a sense of weightlessness is crucial for achieving fluid and effortless movements. Therefore, it is crucial for prostheses to be as lightweight as possible, ensuring optimal design and greater comfort for the user. Furthermore, reducing weight decreases material usage, leading to a lower cost for the orthopedic device.

To reduce weight, a topological optimization was performed. This design approach aims to find the optimal material distribution in a product to enhance structural efficiency without compromising design strength. It relies on computational algorithms that analyze the distribution of stresses and loads in an object, determining which material areas are necessary and which can be eliminated or reduced.

The optimization utilized Fusion 360, the same software used for finite element analysis. The initial step involved establishing constraints, and selecting parts of the prosthesis that could not be modified, specifically the upper socket adjustment area and the lower surface contact area. Maintaining the integrity of the lower area is crucial since, as seen in the plantar pressure analysis in Figure 7, force accumulates mainly in the central area of the toes. Therefore, the part of the prosthesis replacing this area must be more robust. The design’s symmetry plane was also determined to facilitate the operation. Fixed faces were selected, and the corresponding load was applied. Additionally, to ensure precision, a 5 mm mesh was chosen for the optimization process.

The optimization resulted in a mass reduction of 50.82%, achieving a weight of 0.411 kg. The results appear favorable. However, the next step is to verify that this model possesses the necessary characteristics for proper functioning. To do so, the initial proposed geometry is modified by removing material from the areas indicated in the optimization. The amount of material removed may not be the same as that generated by the design tool, but it approximates that form without removing too much material to ensure that the design remains sufficiently robust. In Figure 8, the model prior to topological optimization is shown on the left, after selecting the fixed surfaces and applying the force, while the model after optimization is presented on the right.

Initially, the analyses were conducted with a thickness of 10 mm. However, the results were deemed inadequate due to the displacement exceeding 50 mm. Subsequently, the tests were repeated with a thickness of 12 mm, leading to the following favorable results summarized in

Table 5. Optimized model with 10 and 12 mm thickness.

Optimized Model with 10 mm Thickness
Study Type	Results
	Max	Min
von Mises stress	824.8 MPa	0.900 MPa
Safety factor	15	2.37
Displacement	79.48 mm	0 mm
Optimized Model with 12 mm Thickness
Study Type	Results
	Max	Min
von Mises stress	535.9 MPa	0.6475 MPa
Safety factor	15	3.648
Displacement	45.84 mm	0 mm

:

The final design has a mass of 0.674 kg, reflecting a reduction of approximately 16.6% compared to the initial model’s mass before optimization. Consequently, this outcome is deemed the most favorable, thus solidifying it as the ultimate design for the device.

After establishing the final geometry, a fatigue resistance analysis was conducted to evaluate the projected lifespan of the product, as illustrated in Figure 9.

The fatigue resistance study, like the previous studies, is conducted through finite element analysis, in this case using the Inventor Nastran software [19] (https://www.autodesk.com/products/inventor-nastran/overview, accessed on 17 October 2024), which allows for fatigue simulation on a given design.

The obtained results seen in

Table 6. Fatigue resistance analysis.

Study Type	Results
	Max	Min
Fatigue resistance	109 cycles	109 cycles

surpass the established cycle limit to deem a design as having an infinite lifespan (the value of 107 cycles has been considered for this purpose [20]). Therefore, it can be affirmed that the defined model will have an indefinitely solid lifespan, meaning it will not experience fatigue failure.

Consequently, it can be concluded that the selected material and the defined geometry align with the desired outcome.

The prosthetic design proposed in this study offers substantial advancements compared to existing sports prosthesis models available on the market [4]. The developed prosthesis, with a weight of only 0.674 kg, distinguishes itself by its lightness, surpassing models such as the Össur Flex-Foot Cheetah, which, although optimized for performance, is estimated to have a slightly higher weight due to its carbon fiber structure. Furthermore, this design allows for a maximum displacement of 45.84 mm, which is considerably high, yet within safe limits, compared to the typical displacement range of sports prostheses, which usually spans between 30 mm and 40 mm. This feature, along with a high safety coefficient, ensures reliable and durable use even under the most demanding conditions.

The versatility of this prosthesis’ measurements allows it to be adapted to a wide percentage of the population, thereby making it an inclusive and practical option. It is particularly noteworthy that, at present, there are no prostheses specifically designed for ballet practice, rendering this design especially innovative and unique. Additionally, owing to the topological optimization of the material, the design achieves not only a reduction in weight but also a distinct and aesthetically pleasing appearance. This visual appeal enhances its suitability for a discipline such as ballet, where aesthetics are of utmost importance.

In comparison, other models available on the market [4], such as dynamic response prostheses, may not offer the same level of customization and safety, particularly when considering the combination of lightness, maximum displacement, and adaptability to various body morphologies.

Finally, a prototype manufactured using the hereinbefore-described 3D printing was produced. The resulting prototype offers a scaled approximation of the final product. Although the printed material differs from the final design’s specified material, nylon’s strength and flexibility allow the model to mimic the actual design’s behavior under compressive force, as shown in Figure 10, thereby demonstrating its potential performance.

To fabricate this prosthesis on an industrial scale, it is essential to explore manufacturing processes that are compatible with both the selected material and the specified geometry. In the case of this model, characterized by a complex geometry and composed of a composite material, autoclave molding is a proposed manufacturing method. This process is particularly suitable for high-performance composites like Epoxy/HS carbon fiber, as it enables precise control of pressure, temperature, and vacuum, ensuring that the material achieves the desired mechanical properties. However, certain limitations must be taken into account, such as the need for specialized equipment, higher production costs, and longer manufacturing times, all of which could affect large-scale production feasibility.

4. Conclusions

Through a performance-focused and efficiency-driven approach to designing a prosthetic foot tailored for ballet practice, an innovative design for a lower limb prosthesis has been successfully developed to meet the needs of a classical ballet dancer.

The design process involved a comprehensive analysis of the demands of ballet, taking into consideration various aspects related to the physical requirements of the discipline, the impact exerted on the body during movement, the required flexibility, and more. By linking this analysis with the design and mechanical requirements that an artificial lower limb must fulfill in terms of dynamic response, a model has been developed whose results demonstrate the prosthesis’s ability to enable safe and efficient continuation of movement.

This study thus represents a new opportunity for those passionate about ballet, offering a basis for future research and development in the field with the aim of improving the quality of life and performance of amputee users.