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Original Research Article

Optimization of Important Relief Areas in Prosthetic Socket for Below-Knee Amputees Using Design of Experiment and Finite Element Model

Nehme, Gabi; Ghalambor, Saeed PhD

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Journal of Prosthetics and Orthotics: October 2014 - Volume 26 - Issue 4 - p 194-204
doi: 10.1097/JPO.0000000000000038
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A transtibial prosthesis is an artificial limb that replaces a leg missing below the knee. Transtibial amputees are usually able to regain movement more readily than does someone with a transfemoral amputation, owing, in large part, to retaining the knee, which gives more control to the user. A transtibial prosthetic leg is often referred to as a BK, or below-knee prosthesis. A prosthetic limb for a transtibial amputee consists of several components. First there is a custom-fit socket for each amputee. A fitting is attached to the lower end of the socket to attach a pylon. The pylon connects to a prosthetic foot and is sized to the proper length for the prosthetic limb. The most important aspect of a lower-limb prosthesis is socket design. The socket is the interface between the human and the mechanical support system. Ultimately, design and fit of the socket rely on patient acceptance, comfort, suspension, and energy expenditure. These factors determine the real utility of the final product as shown in Figure 1.1

Figure 1:
Lower-limb details from transtibial prostheses. A, Details from University of Texas, Southwestern Medical Center, 1988.1 B, Pressure relief points in the finite element model.

Several complex methodologies are available to model the geometry and material properties of bones; however, these models demonstrated a lack of accuracy when describing the physical behavior of the skeletal system.2 It is difficult to validate the outcome of any computational model that predicts muscle forces unless the boundary conditions are correct. It is generally believed that ischemia is related to the formation of pressure sores. Changes in local skin blood supply under various external loading conditions could compromise the health condition and integrity of the residual limb. An injured residual limb causes acute disruption in daily activities and can lead to immobility, further surgery, and even death.3 It is well known that each amputee’s residual limb is different and changes with time; it is essential to have a flexible socket made from flexible and wear-resistant materials to conform to different amputees.4 Most of patient irritations in prosthetic sockets occur when different applied loads result in large interface pressure and affect the amputee’s maneuverability. Flexible socket material and displacement can relieve pressure and enhance socket fitting to minimize irritation. Residual-limb skin under a pressure that could reach 220 kPa causes discomfort and accumulation of tissue fluid (edema) in the distal end, which leads to thicker and drier dead cells.5 Therefore, silicon liners that lock into place are used by all patients using traditional and new socket designs. The silicon liner materials have an impressive quality known as “memory,” which enables them to respond to varying degrees of pressure.

Bateni and Olney6 reported that the differences in motion between right- or left-side amputations were insignificant when investigating the role of the contralateral limb in amputees and determining lower-limb joint reaction forces and symmetry of motion in the amputee and nonamputee population. The biomechanics are also affected by the relative slippage between the subject’s skin, the prosthetic socket, and the deformation of the residual limb tissues.7 Major stress plays an important role at different parts of the amputee’s limb, according to Leon Bennett.8 Skin temperature may also be taken as a stress indicator, knowing that it has been shown that tissue temperature decreased as a direct consequence of applied loads. An increase in the temperature may be an early signal of the formation of pressure sores.

In this study, design of experiment (DOE) software was used in connection with FEM to study the interactions of three factors and 12 major responses, shown in Tables 2 and 3, and to optimize the conditions of the relief areas with respect to two different materials (ultra-high-molecular-weight polyethylene [UHMWPE] and Duraform PA6).

The purpose of this study was to determine the relevance of stresses and displacement on several created sockets and analyze relief areas with DOE to optimize the conditions of different medically graded materials. The five pressure relief areas in this study are medial tibia (MT), patellar tendon (PT), fibular head (FH), distal tibia (DT), and popliteal depression (PD). Clinical data showed that stresses were measured over eight sites, namely, PD, MT, lateral tibia, PT, kick point, medial gastrocnemius, lateral gastrocnemius, and medial supercondyle.3,9 In this work, sockets were divided into different categories and all FEM data collected and DOE responses were analyzed to check for relief to minimize patient discomfort.

The overall goal of this approach was to develop a framework that includes amputees’ limb information to define the overall socket design, performing shape using Canfit™ BK Design software (Vorum, Vancouver, British Columbia, Canada) and the analysis using FEM, then optimizing the conditions using DOE model and validating the DOE-optimized results with FEM before manufacturing.



Duraform PA6 provides strength and durability required for prosthetic practice. Medically graded Duraform PA6 yields among the most convenient materials when it comes to its broad range of chemical and thermal resistance, with a deflection temperature of 180°C and resistance to alkaline, hydrocarbon, fuels, and solvents. The Young modulus, yield strength, and Poisson ratio of Duraform PA6 are 1,600 MPa, 44 MPa, and 0.31, respectively.


High molecular weight makes UHMWPE a very tough material, and it is useful for many medical applications, especially those associated with prosthetic sockets. Because of its remarkable toughness, wear, and excellent chemical resistance (melting point ranges between 138°C and 142°C), UHMWPE is used in a diverse range of applications. The Young modulus, yield strength, and Poisson ratio of UHMWPE are 690 MPa, 22 MPa, and 0.44, respectively.



The finite element model (FEM) was established based on the three-dimensional (3D) geometry of the residual limb and the internal bone structure of a transtibial amputee. The drawing is a realistic representation of a refined residual limb using Canfit software. It was modeled and edited by Harika Center (Baabda Beirut, Lebanon) to conform a real patient’s residual limb, as shown in Figure 2. The drawing was then converted to .IGES via SOLIDWORKS 2010 to be properly imported into ANSYS 13. The principles regarding human subject experimentation outlined in the declaration of Helsinki were followed. Dr. Elie Dib, a registered sergeant, made sure that tests and experiments were achieved according to standards that respect patient and human dignity.

Figure 2:
Measurements of the amputee’s residual limb with scanning of the residual limb. Canfit digital modeling with a realistic representation of a refined residual limb (Harika Center).

The model consisted of the following steps:

  1. Import the. IGES geometry in ANSYS.
  2. Make all boundaries coincide, thus cutting line by line (large use of commands and memory).
  3. Divide the socket into six main areas, each having different elastic foundation stiffness (EFS).
  4. Assign each surface with a real constant on ANSYS, 1 for the regular zone and 2 to 6 for the ones with foundations.
  5. Create 3D surfaces to cut and divide the original geometry into the areas of interest. These cuts are needed to differentiate between surfaces with different elastic foundations, as shown in Figures 3 and 4.
Figure 3:
Three-dimensional surface creation to cut and divide the original geometry into the areas of interest.
Figure 4:
Socket views: 1, regular material (EFS = 0 kPa); 2, popliteal depression (EFS = 20 kPa); 3, distal tibia (EFS = 249 kPa); 4, patellar tendon (EFS = 30.23 kPa); 5, fibular head (EFS = 115.71 kPa); 6, medial tibia (EFS = 7.57 kPa). EFS, elastic foundation stiffness.


The design is based on the weight of an 80-kg man. The gait data imply that one leg of the human body is subjected to 93% to 110% of the total body weight.10 Esthetic considerations to produce a thin-walled shell socket that satisfies the design constraints and mimics the profile of the leg anatomy were also important.11–15 To ensure structural integrity, the resulting wall thickness t was designed such that the stresses do not exceed the material strength16,17 using the following equation:

The analysis sought the largest deformation while maintaining low stresses.


To have accurate results due to the complexity of the geometry and to include the EFS in all areas of interest, the meshing element SHELL63 was chosen. It has both bending and membrane capabilities. Both in-plane and normal loads are permitted.

The element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axes. Stress stiffening and large deflection capabilities are included. A consistent tangent stiffness matrix option is available for use in large deflection (finite rotation) analyses.

The elasticity of tissues is modeled in ANSYS by assigning an EFS value to the defined pressure relief areas (five designated areas: MT, PT, FH, DT, and PD) of the socket in contact with the residual limb. Therefore, it is a quantification of the elastic stiffness of the tissues that are in contact with the socket wall.5,17 It is assigned for the areas where relief is most commonly applied (Table 1).

where E is the Young modulus of elasticity of the soft tissues, δ is the unit normal deflection, and L is the thickness of the soft tissues. The meshing using the SHELL63 element was done for all cases, and it covers the entire socket, as presented in Figures 5 and 6.

Table 1:
Elastic foundation stiffness (EFS) for each of the areas of interest in a transtibial amputee patient17
Figure 5:
Shell63 meshing element (ANSYS).17
Figure 6:
Total meshed area of the socket using SHELL63 element and selected areas where pressure distribution and reliefs are applied.

The pressures applied on the socket areas were calculated by graduate students using transducers that can resolve the applied load. The transducers were mounted on the hard socket wall by an external mounting technique that was devised by the research team using different trial and errors and optimized by statistical techniques used in earlier studies.5,18 The values measured were, on average, 8% lower than those addressed by different authors.9,11,12 These pressures are as follows: (1) PT, 84 kPa; (2) FH, 48 kPa; (3) MT, 40 kPa; (4) tibial end, 99 kPa; (5) PD, 74 kPa; and (6) pressure on the bottom of the socket, 702.6 kPa. Six sets of real constants corresponding to the six area types that constitute the different areas of the socket were created as shown in Figure 3 and in the pressure distribution in Figure 6. Each set contains the thickness and EFS of its respective area.5,18


Table 2 presents the FEM data of eight socket cases with three factors each. The first four cases are for UHMWPE materials and the remaining four cases are for Duraform PA6. Each material includes two different overall thicknesses for the socket and two different relief areas that will be used to relieve pressure at the designated areas shown in Figure 4. The same setting will be applied for all FEM studies after choosing the proper material thickness. The bottom surface of the socket is fixed in all degrees of freedom; the upper surface of the socket is free. Figures 3–6 summarize the described setting for all cases and the distributed pressures.

Table 2:
Factors for materials, socket type, and relief areas considered in collecting stress and displacement using the FEM


The DOE software was used to properly explain the significant interactions of contact stresses and displacements at the designated socket areas. Design of experiment is the simultaneous study of combination of variables instead of studying each individually; the amount of finite element testing required would be drastically reduced, and greater process understanding would result. This is in direct contrast to the typical one-factor-at-a-time, which limits understanding and wastes data. Design of experiment uses standardized effects and analysis of variance (ANOVA) to reach meaningful results and optimize outcomes.


Model framework started with material properties and computer-aided design of patient model via the Canfit software system to construct FEM and calculate the values that will be analyzed using DOE to optimize the process of manufacturing a socket where designated pressure relief areas are sought. Flowchart and framework for a specific socket are shown in Figure 7. The first step is to acquire data from the patients’ residual limb using Canfit software (Harika Center), then to develop the finite element design and meshing; data collected from selected material types were used in a DOE model to optimize the different relief areas. Optimized design was rechecked by the FEM to establish the prosthetic socket structural integrity. After several months, the new manufactured socket was rechecked for sensitivity and wear to monitor patient comfort and was compared with a traditionally worn socket. Three patients were considered for the tests, and their integrities were preserved by the institutions. Patients’ information was kept secret by respecting their wishes, and data are used specifically to ensure that relieved sockets will perform better when used by transtibial amputees.

Figure 7:
Revised flowchart for socket design and analysis using several software applications.17



Finite element analysis of displacement and Von Mises stress was calculated for all the socket designs. These stress and displacement responses at different areas of the Duraform PA6 and UHMWPE sockets are displayed in Table 3, and selective socket conditions are presented in Figures 8 and 9. As a complement to clinical measurements, computational modeling and data provided by the FEM analysis were analyzed and optimized for different relief areas and were identified as a potential method for prediction and evaluation of pressure reliefs. Thus, reducing thicknesses at designated areas to increase relief without compromising structural integrity was optimized using DOE numerical techniques.

Table 3:
Stress (MPa) and displacement (mm) data for all socket cases collected from the finite element model to be analyzed by the design of experiment software
Figure 8:
Displacements (USUM) (mm) for 4-mm socket, 1-mm relief using Duraform.
Figure 9:
Von Mises equivalent stresses (SEQV) (MPa) for 4-mm socket, 1-mm relief using Duraform.


To observe the impact of the materials chosen, the thickness was fixed and the stress results based on material selection were compared. The 3-mm thickness for both materials (Duraform PA6 and UHMWPE) was selected. The values of the stresses in all five relief areas were directly compared; the stresses in the FH, MT, PT, and DT were slightly higher for Duraform PA6 than for UHMWPE. This was also true for the maximum and bottom stresses. However, in the PD area, the stresses of UHMWPE and Duraform PA6 were slightly similar (Table 3). In the case of the 4-mm socket, stresses of UHMWPE had higher values in the FH, MT, PT, and DT and lower values in the PD and bottom stress cases. Moreover, the maximum stress of Duraform PA6 was slightly higher than that of the UHMWPE (Table 3).


The same approach was used to observe the impact of displacement. The 3-mm thickness was fixed so that the displacements of Duraform PA6 and UHMWPE were directly compared. The displacement of Duraform PA6 was much less than that of UHMWPE for the same thickness except at the DT. Following the same procedure for the 4-mm thickness, a logical decrease in the displacement was noted for both materials. However, the observation remained the same with Duraform PA6 having a much smaller displacement than UHMWPE did in all designated areas (Table 3).


Areas were relieved by 1 and 2 mm in certain designated spots that show importance for transtibial amputees wearing prosthetic sockets.18 When the 3-mm overall thickness sockets were considered, the stresses increased in the relieved areas except the bottom stresses and the stresses at the PT for the 2-mm reduction at that location and for both materials (Table 3). Using the same analysis, the stresses were also higher at all relieved locations for the 4-mm sockets made of Duraform and UHMWPE, except for the bottom stresses. Following the same procedures, the displacement was also higher in the 1- and 2-mm relieved areas except for the 3-mm UHMWPE socket with 1- and 2-mm reliefs at the PT, and for the 4-mm UHMWPE socket with 2-mm relief at the DT (Table 3).

The data suggest that relief areas are very important for all designated sockets to reduce pressures at important locations. This analysis simulated the process of an 80-kg amputee when the pressures occur mainly over the areas where researchers at the University of Balamand found extreme stresses. Their data corroborated closely with the extreme pressure data of Zhang and Roberts,9 especially for the PT case. Therefore, DOE played an important role in optimizing relief areas without compromising the structural integrity of the design when using Duraform PA6 and UHMWPE.


Tests for three factors and 12 designated socket area responses are presented in Tables 2 and 3, where values of finite element stresses and displacements are shown. The FEM presented the events that occur during extreme pressure cycles, and the responses collected in Tables 3 were analyzed by DOE software.

The goal of the DOE model was to point out the interactions among the different factors involved in the FEM and to identify their mutual and relative influences. The DOE standardized effects and ANOVA for the stresses at the PD, DT, and MT presented clear evidence of their insignificant contribution to the model. Following the same procedures, the displacements at the PD, DT, and MT indicated clearly that these designated areas are insignificant and do not contribute positively to the overall FEM-DOE. Thus, these stresses and displacements were excluded from the model and optimization of stresses and displacements at PT and FH considered in the final DOE analysis.

According to Barrentine,19 DOE analysis uses desirability when multiobjective optimization is sought. According to this approach, all responses used in the analyses should be converted into corresponding desirability functions. The desirability is high when all responses approach their target values simultaneously. Optimum stresses and displacements at the PT and FH were obtained when optimized relief areas were considered without compromising the socket structural integrity. Each of these goals was linked to its own desirability function. The maximum value obtained for the desirability function was acceptable and is shown in Figure 10 for the Duraform PA6 and UHMWPE materials. Nevertheless, when probing maximum stresses for UHMWPE (4-mm socket, 1.45-mm optimum relief; Figure 10) and for Duraform PA6 (3-mm socket, 1.43-mm optimum relief; Figure 10) with respect to the optimized PT and FH designated areas in the model, it was indicated that these values, with a desirability approaching 90%, do not compromise the structural integrity of the sockets and were significantly less than the yield stress of both materials. Therefore, it can be concluded that reducing the thickness at these locations will increase benefits and comfort for amputees.

Figure 10:
Optimized relief area sockets at patellar tendon (PT) and fibular head (FH) for ultra-high-molecular-weight polyethylene and Duraform.

In this article, the contact stresses at the PT and FH were important in the optimization of relief areas in Duraform and UHMWPE. Because the model was robust over the design space with respect to these factors and responses, the relief area was optimized with a high desirability at 1.45 and 1.43 mm for both socket materials. Figure 10 shows desirability greater than 88%, which was very high for the stated goals of the responses. It can be concluded that both materials’ FEM data corroborated closely with the DOE optimization. Furthermore, finite element analysis data at the relief designated areas were very close to the DOE-optimized data. Evidence did not suggest the presence of large discrepancies in these tests.

An adequate FEM-DOE framework ensured the validation of DOE results, and a case study on a real person was performed. The socket was designed and manufactured with the optimized stress and displacement data, and relief areas were considered by reducing the material thickness at the PT and FH areas. All these factors made it highly probable that the dominant factor (relief area thickness) contributed positively to the comfort and safety of amputees.


Duraform socket A with 1.43-mm pressure relief areas was manufactured and tested by Harika Center (certified prosthetist). The prosthetist followed the FEM and DOE-optimized analysis. The socket was modeled after a real patient as stated previously. A positive plaster mold inside a plastic check socket was created. With the proper tools, the positive mold was finished following the patient pattern and corresponds to the proper relief areas (Figure 10). Patients agreed to wear the relieved and unrelieved sockets, and their wishes to keep their names anonymous were respected. After 12 months, the patient was more comfortable when compared with two other patients who used unrelieved sockets. The amputee wearing the optimized socket with pressure relief at the PT and FH did not complain about being especially uncomfortable after wearing it, whereas the two other people wearing the traditional sockets without relief complained about poor flexibility of the socket when using it for a long period. They also had skin problems, as shown in Figure 11, and the stability in which to control motion was very difficult. Figure 11A and B reveals the accumulation of red skin, which could lead to the accumulation of tissue fluid and cause thicker and drier dead cells according to Nehme et al.5,18Figure 11C indicates that socket relief at a specific location increases flexibility and gives more comfort to amputees. Comparison of stresses and displacement revealed the strength of our socket method in minimizing skin disease for transtibial amputees. It is easy to apply in underdeveloped countries because equipment to test people and manufactured sockets is not very expensive. Skillful people with FEM and DOE knowledge can figure out relief areas and thicknesses and manufacture sockets accordingly. Major differences in the unrelieved sockets can be noticed, such as redness and swelling (Figure 11A and B). This might be because of less flexibility at certain areas where the top layer of the materials is giving rise to areas of higher-stress concentrations and higher pressure on the residual limb, hence increasing skin irritation.

Figure 11:
Transtibial knee amputee patients’ frontal skin images: A, B, unrelieved sockets; C, relieved socket.


To improve prosthetic socket design and fitting, it is necessary to have a comprehensive understanding of the pressure distribution at the socket-limb interface and the residual limb’s strength to withstand pressure. Clinical experience and socket fit assessment can be validated only after prosthesis manufacturing and fitting. If the patient is not comfortable, the socket should be redesigned, requiring extra cost and money. Therefore, the Canfit-FEM-DOE system generated quantitative and objective information about residual limb modification and assessed the best possible ways to design and manufacture a socket. Consequently, it provided a reduction in time consumption or unnecessary complications for amputees, especially in poor countries. These simulations were used to obtain the residual limb deformed shape and to analyze its behavior and modifications. The methodology has been tested on patients wearing relieved and unrelieved sockets. The customized prosthetic socket where all phases are “computer aided” and all data involved in the process are digital proved extremely important for patient comfort. The minimized irritations might be caused by the flexibility added in the relief areas where skin redness was eliminated, as shown in Figure 11. By compiling all the data in underdeveloped countries, clinicians can classify their patients using a wide range of FEM-DOE analyses that could help them in future treatments. These computerized data could be used to assess prosthesis fit using common and inexpensive instruments.

In this study, the transtibial knee prosthetic socket model is controlled using FEM and DOE optimization techniques. The design was divided into several cases where Von Mises stresses and displacements were analyzed accordingly. The socket plays a very important role in load and energy transmission, as shown in the results. Stresses and displacements for an 80-kg man were optimized and statistically analyzed using ANOVA. All stresses are found to be below the yield strength of both materials. Finite element modeling and DOE gave an approximation for this complicated engineering problem and identified the best design to be chosen for these types of patients. It must be taken into consideration that minimal errors might have resulted in implementing these procedures. Also, the setting of the problem was simplified in terms of the number of patients available and the limited resources.

Results, validation, and optimization revealed that relief areas are extremely important in the design and manufacturing of a socket. The DOE analysis targeted relief areas and optimized displacements and stresses with a maximum desirability targeted value. The interactions between factors were directly proportional to the performance established by the DOE analysis. Significant improvement was shown during the optimization process when high desirability values were produced (Figure 10). Stresses and displacement responses were significant at the PT and FH of both materials. Important responses in the model played a strong role and indicated special relationships between FEM data and DOE optimization.

To examine the influence of factors and their interactions, ANOVA was carried out. Evaluation of each response and percentage probability were investigated for all factors. Relief areas at the PT and FH contributed largely to the model and established reliable data for socket manufacturing. Interactions between factors were increasingly significant at these areas.


Support provided by the University of Balamand and Harika Center is gratefully acknowledged. The authors to thank Dr. Saeed Ghalambor at the University of Texas at Arlington for useful discussion and for the free services that he provided.


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KEY INDEXING TERMS; amputees; design of experiment; finite element model; prosthetic socket; Canfit software

© 2014 by the American Academy of Orthotists and Prosthetists.