Design of a Single Layer Metamaterial for Pressure Offloading of Transtibial Amputees

Abstract

While using a prosthesis, transtibial amputees can experience pain and discomfort brought on by large pressure gradients at the interface between the residual limb and the prosthetic socket. Current prosthetic interface solutions attempt to alleviate these pressure gradients using soft homogenous liners to reduce and distribute pressures. This research investigates an additively manufactured metamaterial inlay with a tailored mechanical response to reduce peak pressure gradients around the limb. The inlay uses a hyperelastic behaving metamaterial (US10244818) comprised of triangular pattern unit cells, 3D printed with walls of various thicknesses controlled by draft angles. The hyperelastic material properties are modeled using a Yeoh third-order model. The third-order coefficients can be adjusted and optimized, which corresponds to a change in the unit cell wall thickness to create an inlay that can meet the unique offloading needs of an amputee. Finite element analysis simulations evaluated the pressure gradient reduction from (1) a standard homogenous silicone liner, (2) a prosthetist's inlay prescription that utilizes three variations of the metamaterial, and (3) a metamaterial solution with optimized Yeoh third-order coefficients. Compared to a traditional homogenous silicone liner for two unique limb loading scenarios, the prosthetist prescribed inlay and the optimized material inlay can achieve equal or greater pressure gradient reduction capabilities. These preliminary results show the potential feasibility of implementing this metamaterial as a method of personalized medicine for transtibial amputees by creating a customizable interface solution to meet the unique performance needs of an individual patient.

Introduction

A lower limb prosthesis provides an increased ambulatory function for an amputee. The amputee should feel confident that the prosthesis the comfort and functionality needed to complete routine tasks [1]. Given that 84% of lower limb amputees wear the prosthesis an average of 12 h a day, comfort is vital [2]. A prosthesis usually subjects the residual limb tissues to un-natural loading conditions, leading to discomfort, dermatological issues, deep tissue damage, and prolonged joint and muscle pain [3–7]. Understanding the pressure distributions around the limb is the first step to counteract the discomfort experienced by lower-limb amputees [8]. Most studies show the largest pressure concentrations can appear at the limb regions around the patellar tendon (PT), tibial crest, fibular head (FH), and tibial end (TE) [9–12].

The magnitudes of the peak stress (PS) found at the surface of the limb are often the focus of studies, as such stresses are key contributors to skin breakdown that can lead to many undesirable issues experienced by lower limb amputees. However, a determined range of stress magnitudes that can predict this skin breakdown does not exist because of variability between comfort and pain thresholds for individuals [13,14].

Peak pressure gradient (PPG), instead of PS, may play a more significant role in predicting residual limb issues. PPG is the spatial change in pressure at the limb surface, orthogonal to the normal vector pointing out from the residual limb. Preliminary studies suggest that PPG may be a better indicator of plantar skin injury than PS alone. The results show that PPG can better represent the high-stress concentrations within the soft tissue of the foot. These stress concentrations lead to skin breakdown, one of the root causes of plantar skin injury [14,15]. There is limited research directly targeting the reduction of PPG outside of plantar skin injury. The conclusions of Refs. [14] and [15] have prompted the desire to investigate how directly targeting the reduction of PPG on the residual limb of transtibial amputees can lead to a more comfortable experience for the amputee. This paper provides a preliminary investigation toward a prosthetic liner solution that would aim to directly target the reduction of PPG.

The first prosthetic liners were made from open and closed cell foams that were designed to encompass the residual limb [16,17]. Recently, silicone and elastomer liners replaced foams as the most commonly implemented pressure offloading solution because of increased durability and pressure distribution capabilities compared to foams [5,18].

Prosthetic liner materials have been a common focus of research. Sanders et al. [16] altered the mechanical properties of foam liners by weakening the cell structures during the vacuum-forming manufacturing of foam liners. By controlling the degree by which these cell structures are weakened, Sanders proposed that the foam liners could be altered to fit the performance needs of specific patients [16].

In another study, Sanders et al. [19] investigated the compression stiffness and coefficient of friction of different liner materials. Spenco®, Poron^®, silicone, soft pelite, medium pelite, firm Plastazote^®, regular Plastazote^®, and nickelplast were tested. The compression testing mimicked conditions the liner would experience within the prosthetic socket. The results suggested that softer materials like regular and firm Plastazote and soft and medium pelite would be most advantageous around boney prominences. In comparison, stiffer materials should be applied around the soft tissue.

Klute et al. [5] expressed concern that little scientific evidence is available to guide a prosthetist in the liner prescription and socket design processes. Research shows that liners can help distribute pressures and can lead to increased comfort [5–12]. However, there is limited evidence in the difference between the potential benefit of each material for specific patients.

Hafner et al. [20] aimed to understand the prosthetists' liner selection practices further. This survey-driven study showed that the liner manufacturers were the primary source for information on available liner products. Liner characteristics like durability, comfort, and suspension are often the driving factors in selecting a liner. Even though there are more than 70 available liner solutions on the market, the study's respondents reported only having prescribed 16 of the 70 liners to their patients. Of those 16, the respondents said they routinely only selected 2–3 liners to meet their patients' needs. The most common liner materials of these prescriptions were silicone, thermoplastic elastomer, and urethane. The study emphasized the need for an objective tool or resource to pair individual patients with liner solutions that meet each amputee's unique performance needs.

Nearly all transtibial amputee research is limited to the use of homogenous material liners. Boutwell et al. [20] investigated a form of the nonhomogenous liner by studying the effects of gel liner thickness on peak socket pressures around the fibular head. Most subjects perceived increased comfort with the thicker liner. The increased comfort was linked with a reduction in pressure around the limb. The authors proposed a thicker gel liner would reduce pressure around boney prominences as compared to thinner liners. However, the authors also noted that individual patients found the thicker liners to be significantly bulkier and increased heat and moisture capture. The variable thickness liner approach provides some immediate pressure relief but introduces an increased threat of heat and moisture build-up.

There is still a significant gap in the proper investigation of nonhomogenous liners. The research suggests that boney prominences should be offloaded by softer materials, while soft tissue benefits from the suspension capabilities of stiffer materials. A homogenous liner would not be able to satisfy this recommendation. One method of achieving variable stiffness within a single inlay body would be the implementation of metamaterials.

Metamaterial research first originated in the field of optics, acoustics, and mechanics [21–25]. Mechanical metamaterials are human-made materials constructed from unit cells with structural patterns, which result in the mechanical properties of the metamaterial being a function of the base constitutive material and the pattern design [26]. Metamaterials enable the customization of their mechanical response through the purposeful design of the unit cell parameters and selection of the base constitutive material. By varying the unit cell parameters as a function of the 3D position within a bulk metamaterial, the properties at each location can be customized to meet the design objectives of the metamaterial. Recent advances in additive manufacturing (also called 3D printing) enable printing a more comprehensive range of base materials (including elastomers) and fabrication of complex metamaterial geometries at a relatively low cost [27].

This paper investigates transtibial residual limb PPG reduction by implementing a metamaterial inlay for limb models targeting offloading at the FH, PT, and TE. Through finite element analysis (FEA), this investigation shows a potential reduction in the PPG when using a metamaterial liner compared to a traditional homogenous material liner and in the absence of a liner.

Clinically testing the limb stress distributions can be challenging due to cost, time, and patient recruitment and retention [28]. Therefore, a significant amount of prosthetic research is completed with the aid of FEA. FEA is a valid alternative to in vivo testing to determine the stress of a residual limb and deformations of liner and socket materials [10,11,29–34]. FEA is a potent tool when models are too complex for traditional analytical models. Dickinson et al. [33] provide a thorough review of the state of the art of using FEA as a tool to better understand the pressure distribution of lower limb amputees. The study found that FEA analysis of lower limb amputees can be broken into six distinct criteria: (1) Modeling liner and socket interface mechanics, (2) soft tissue internal mechanics, (3) identification of soft tissue characteristics, (4) proposals for incorporating FEA into prosthesis fitting and socket fit assessment, (5) design and analysis of individual prosthetic components, and (6) analysis of osseointegrated prosthesis concepts. These studies investigate a wide range of criteria, such as linear versus nonlinear material representations, static versus dynamic models, and variable loading conditions. Still, they all strive to improve the accuracy of FEA modeling and analysis of lower limb amputees.

The fourth topic is of vital interest to this paper. Goh et al. [35] developed an in-house FEA code that could iteratively design a socket using FEA pressure predictions. The code was dependent on a clinical understanding of the pressure-to-pain relationship for an individual patient. Colombo et al. [36] extended this idea by designing a methodology to design, test, and automatically optimize the socket for a particular patient. Sengeh et al. [37] and Nehme et al. [38] designed variable-stiffness sockets to actively modify stress distribution on a given patient's residual limb. FEA was used to obtain a pressure map of the residual limb to act as a guiding factor for material distribution.

The goal of lower-limb FEA models is to contribute to an on-demand prescription tool that prosthetists can use to help guide a prosthetic limb fitting. FEA may allow for rapid comparative analysis of different liner types and socket designs, which may lead to a more successful diagnosis in less time. The goal of using FEA as an on-demand tool to properly fit a user-specified biomechanical model was a significant inspiration for this study.

The main body of this paper is formatted into two main sections. The first section, Materials and Methods, describes the design of the novel metamaterial and the experimental methods used to test the mechanical response of each of the metamaterial variations. This section also includes a description of the finite element analysis and optimization methods used to provide a preliminary validation of the effectiveness of using this novel metamaterial as a liner material. The goal is to reduce PPG for two sample transtibial amputees requiring offloading at different locations around the residual limb. It should be noted that the methods presented in this paper do not convey a “one size fits all” model for the pressure offloading of amputees. The design of the limb models, metamaterial inlays, and loading conditions are highly susceptible to change between different patients.

The second section evaluates the results of the series of FEA's that compare the PPG reduction capabilities of the metamaterial inlay to a homogenous silicone liner and a socket without an interface liner. This section also includes the optimization results and how introducing optimization of the metamaterial may lead to a further reduction in PPG. This section concludes with a discussion of the limitations and clinical relevance of this work.

Materials and Methods

Metamaterial Material Properties.

Based on the recommendation from Sanders et al. [29], a method to implement an interface material between the residual limb and socket was sought, which could be soft at the boney prominences but stiffer around large concentrations of soft tissue. Members of the research team had previous experience working with a metamaterial with an adjustable mechanical response (Patent US10244818 B2) [39] used in orthotics to offload pressure for patients with diabetic foot ulcers. While the original application of this metamaterial was orthotics, the designers of this metamaterial saw the potential of implementation at other anatomical locations such as the hip, head, knee, hand, and chin, which sparked the interest in using this metamaterial for this study.

The metamaterial, seen in Fig. 1, is constructed by 3D printing walls of various thicknesses controlled by draft angles. These different wall thicknesses leave a triangular patterned unit cell indented into the top surface of the base material. The pattern is comprised of four equal-sized triangles spanning radially 180 deg. This four-triangle base unit is patterned orthogonally. The base material is made of 100% TangoPlus (Stratasys, Ltd, Eden Prairie, MN). All samples were additively manufactured using an Object Connex 350 3D printer. The base material is highly flexible and has a similar feel and appearance as a rubber. For this study, the draft angles were limited to 0, 1.9, 4.1, 6.6, 9.7, 14.5, and 27.5 deg. The material samples are referenced as DA₀₀, DA₀₂, DA₀₄, DA₀₆, DA₀₉, DA₁₄, and DA₂₇ with the two digits subscript to “DA” corresponding to the approximate draft angle associated with the patterned unit cell. The seven draft angles were selected to provide a set of materials with discernable shore O hardness values. The hardness values of DA₂₇ and the base material were similar enough to justify making 27.5 deg the largest draft angle. This metamaterial was initially designed for changes in hardness to mimic changes in the metamaterial's force–displacement response. The correlation between hardness and changes in force–displacement response is not directly proportional, but, in a relative sense, a low hardness value correlates to a smaller required force to reach the desired material deformation and vice-versa [39]. The change in draft angle alters the fill volumes of the voids, and thus the thickness of the walls between each cell and the relative density of the metamaterial, which in turn affects the mechanical response of the material as a whole.

Fig. 1.

(a) Triangular unit cell, (b) top view of patterned unit cells, and (c) isometric view

To develop material models for the metamaterial variations, 60 mm by 60 mm samples (n = 3) of each draft variant metamaterial were tested in uniaxial quasi-static compression, using an ADMET eXpert 5601 testing system (ADMET, Norwood, MA). The compression plate measured 203 mm by 203 mm and, therefore, covered the entirety of the testing sample. The testing followed ASTM D575 standards for all components of rubber material testing besides the thickness of the test specimen. ASTM D575 calls for a slab of approximately 13 mm in thickness. However, the metamaterial in this study is designed to be approximately 4.76 mm in thickness for use in the inlay. Therefore, the deflection rate of the plate was altered from 12 mm/min to 4.4 mm/min. Three samples of each material were tested to account for material property variability that can arise over time from this composition [40]. The bottom surface of each sample was carefully secured to the testing bed using adhesive tape to ensure the specimen laid flat and was parallel to the compression plate. The compression plate was set to apply a load to the sample until an upper-stress limit of 300 kPa was reached. Per Sanders et al. [19] testing recommendation, this upper-stress limit was deemed sufficient to account for any load the metamaterial may be subjected to while being used as a prosthetic liner. In future research, the metamaterials should be tested under an impact loading scenario to more accurately represent the concentrated loads the metamaterial liner may experience from the residual limb.

The engineering stress–strain curves, seen as the solid lines in Fig. 2, show the variable mechanical response that can be achieved through the altering of the draft angle. These curves were modeled under the assumption of incompressibility. Each material shows hyperelastic behaving material properties. The smaller wall thicknesses that arise from lower draft angles lead to buckling during compression. The buckling is represented by the noticeable reduction in slope between 40% and 60% strain of the stress–strain curves for DA₀₀, DA₀₂, DA₀₄, and DA₀₆.

Fig. 2.

Combined experimental (“E”) and material model (“MM”) stress–strain curves of metamaterial variations

A material model was needed to provide simplified representations of the experimental stress–strain curves. A range of hyperelastic material models can be used to model the nonlinear deformation of a material. The hyperelastic materials are described through a strain-energy function. The strain-energy density can derive the relationship between the stresses and strains of material during deformation. The nonlinear relationship between the stresses and strains is defined through a series of material parameters. High order material models have more material parameters that may more accurately describe the stress–strain relationship, but they also increase the complexity of the material model and may be subjected to overfitting. The material parameters are selected so that the model best matches the experimental stress–strain results [41].

This research required a material model that accurately represents the experimental data while limiting the number of material parameters. Using a tool within ansys Academic Research Mechanical, Release 19.2, the accuracy and simplicity of several material models were tested. Based on an initial visual comparison of how different material models represented the experimental data, the Yeoh third-order representation was selected to model the compression testing result. The strain energy density function of the Yeoh third Order model (Eq. (1)) can be used to derive the stress–strain relationship (Eq. (2)) needed for the model comparison, where $I_1$ is the first strain invariant. The three Yeoh material coefficients (Table 1) were found such that, when plugged into Eq. 2 [41], the model best matched the compression testing data of the given material. The third-order graphical representations of each material are the dashed lines in Fig. 2. The R² values found in Table 2 validate the selection of the Yeoh model. Under the incompressibility assumption, the three incompressibility parameters were set to 0 Pa⁻¹ to define the shear stress response of each of the metamaterials within ANSYS. Future research aims to couple the uniaxial compression test with a pure shear test to obtain a more accurate representation of how the metamaterials behave under pure shear

$$W=\sum _{i=1}^3{C}_i{(I_1-3)}^i$$ (1)

$$\sigma =\sum _{i=1}^{3}2i{C}_{i0}\left[\left(1+\epsilon \right)-{\left(1+\epsilon \right)}^{-2}\right]{[{\left(1+\epsilon \right)}^2+(2\left({1+\epsilon )}^{-1}-3\right)]}^{i-1}$$ (2)

Table 1.

Yeoh third-order material coefficients of each metamaterial and base composite material

	C10 (Pa)	C20 (Pa)	C30 (Pa)
DA00	3225	95,778	32987
DA02	646	1.599 × 105	42500
DA04	1846	1.877 × 105	2.855 × 105
DA06	1760	79,549	−7016
DA09	3260	66,206	−11,759
DA14	839	23,329	2455
DA27	1271	6326	9544
Base material	446	88,326	2.510 × 106

Table 2.

R² values for material coefficients with respect to compression testing results

DA00	DA02	DA04	DA06	DA09	DA14	DA27	Base material
0.908	0.904	0.921	0.986	0.990	0.978	0.974	0.947

Representing the buckling experience by lower draft angle materials is a limitation for this third-order material model, as seen in Fig. 2. The designs which experience buckling around 40–60% strain have a lower R² value because the third-order material model cannot correctly represent the buckling. Materials with larger draft angles, like DA₀₉ and DA₁₄, are not subject to any discernible wall buckling, which means they can be more accurately represented using third-order coefficients, which is apparent by the larger R² values. This limitation should be considered when the lower draft angle metamaterials are used in the inlay design, as the response may deviate from the material model's representation. Future research aims to address the limitation.

FEA Set Up.

Using ansys Academic Research Mechanical, Release 19.2, a nonlinear FEA was set up to mimic the in vivo conditions of a transtibial amputee. Two different limb shapes were evaluated. The limb models were taken from 3D surface scans of two transtibial limb shapes (IRB exempt category B4) using a structure sensor (Occipital Inc., San Francisco, CA). The limb shapes were obtained as part of an Institutional Review board-approved study. The first limb shape (L1) represents an approximately 180-pound, 5-ft 6-in transtibial amputee requiring pressure offloading at the FH. The second limb model (L2) represents an approximately 240-pound, 6-ft 4-in transtibial amputee, requiring offloading at the PT and TE. Both limb models had a conical shape with no abnormal protrusions.

Each limb model underwent a series of FEA to compare the PS and PPG on the surface of the residual limb under four conditions: (1) no inlay, (2) a homogenous silicone gel liner, (3) a practitioner prescribed inlay utilizing the metamaterials, (4) a metamaterial solution with optimized Yeoh third-order coefficients. The silicone gel liner is one of the most popular liner solutions for transtibial amputees and provides an appropriate comparison target for the metamaterial inlay [20].

Starting as generic 3D bone models, the tibia and fibula models were sized, formed, and placed within the limb models following surgical guidelines of transtibial amputations. These guidelines allow for a tibia and fibula pair to be adequately modeled for individual patients [42]. A bone cavity was created inside the limb. The inlay was constructed by isolating the exterior surface of the limb corresponding to the inlay shape and extruding it 4.76 mm in the normal direction to the residual limb. This method ensures that the inlay and limb remain flush against each other to mimic the interaction of an in vivo limb and inlay. The shapes of the inlays were derived from a clinical prosthetist's recommendation. The prosthetic socket was formed by scaling the limb to envelop the entire inlay and limb. A cavity was created to ensure that the inlay and limb remain flush against the socket, allowing for complete contact surfaces between the residual limb, liner, and socket. These contact surfaces can mimic a simplified version of a total surface bearing socket. Zheng et al. [43] implemented a similar method and yielded accurate results. This process was repeated for both limb shape simulations. Figure 3 shows the difference between models used for the two limb shapes. To simplify the analysis for L1, the model was reduced to target just the areas around the FH (Fig. 4). This simplification, referenced as the “reduced model,” drastically reduces the computational time of the simulation while still providing an accurate representation of the FH region stress distribution for L1. The full limb model was necessary for L2 due to specific offloading locations.

Fig. 3.

STL model for (a) L1 (b) L2 (Top surface of the limb and socket have been removed for clarity)

Fig. 4.

Isolated fibular head exploded view for L1

The tibia and fibula models were modeled with a linear elastic material with an elastic modulus of 15 GPa and Poisson's ratio of 0.3. The prosthetic socket was modeled as polypropylene with a linear elastic modulus of 1.5 GPa and Poisson's ratio of 0.3 [25]. The limb model was modeled homogenously as soft tissue. The hyperelastic material properties of soft tissue can be assumed to be incompressible and modeled using Yeoh third-order with C₁₀ = 0.004154, C₂₀ = 0.050753, C₃₀ = −0.013199 MPa [44]. In the Results and Discussion section, a comparison will be made between the prosthetist's recommended design, a homogenous silicone inlay design, an optimized metamaterial design, and a design with “no inlay.” For the no inlay conditions, the inlay was modeled as polypropylene. The silicon liner was assumed isotropic and linearly elastic with a modulus of elasticity of 380 kPa and Poisson's ratio of 0.3 [9,45].

Based on subject matter expertise and patient input, the prosthetist selected metamaterial variations and their corresponding layouts to design the patient inlays. The prosthetist's familiarity with these materials was limited to how each metamaterial felt compared to commonly used prosthetic liner materials. These comparisons were the guiding knowledge behind the prosthetist's prescriptions. For L1, the prosthetist prescribed an inlay (Fig. 5) that included three materials (DA₀₉, DA₁₄, and DA₂₇) with circular offloading of set radii around the fibular head. The prescription set DA₀₉ as an internal circular region that transitions to a region of DA₁₄ internally and externally bounded by the dimensions seen in Fig. 5. The remaining portions of the inlay are set as DA₂₇. The regions of DA₀₉, DA₁₄, and DA₂₇ set by the dimensions in Fig. 5 are referenced as the “FH inner material,” “FH middle material,” and “FH outer material,” respectively, in later optimization results for the FH.

Fig. 5.

Limb shape 1 prosthetists prescribed inlay (Prosthetic socket removed for clarity)

For L2, the practitioner prescription (Fig. 6) targeted the regions around the PT and TE as critical areas of offloading. The portion of the residual limb encompassing the PT requires only sections of DA₁₄ and DA₂₇. The DA₁₄ region is set by the parameters in Fig. 6(a). This offloading mainly targets the PT with an ellipse layout of DA₁₄ and extends down to the tibial crest. The regions of DA₁₄ and DA₂₇ set by the dimensions in Fig. 6(a) are referenced as the “PT inner material” and “PT outer material,” respectively, in later optimization results for the PT. The TE prescription includes DA₀₉, DA₁₄, and DA₂₇. The dimensions in Fig. 6(b) bound the internal and external ellipses. The internal and external ellipses were made of DA₀₉ and DA₁₄, respectively. DA₂₇ represents the remaining regions of the inlay. The regions of DA₀₉, DA₁₄, and DA₂₇ set by the dimensions in Fig. 6(b) are referenced as the “TE inner material,” “TE middle material,” and “TE outer material,” respectively, in later optimization results for the TE. It should be noted that, within the FEA, the metamaterials were represented by solid bodies assigned the Yeoh third-order coefficients presented in Table 1. This method ensures that the respective metamaterial regions of the inlay behave similarly to the metamaterials in a physical model.

Fig. 6.

Limb shape 2 prosthetists prescribed inlay (a) targeting the patellar tendon and (b) targeting the tibial end (prosthetic socket removed for clarity)

For all simulations, the exterior surface of the prosthetic socket was held as the fixed support. For L1, a load of 60 N was applied laterally on the isolated fibula of the reduced model. L2 was loaded with 1090 N of compression on the uppermost curved surfaces of the bone model and 205 N of anterior–posterior shear on the posterior curved surfaces of the bone model. These loading conditions are set to mimic a quasi-static loading representation in which these critical areas of the residual limb are being subjected to the largest loads experienced during the gait cycle, which would be the push-off phase for these given limb areas. The magnitudes of the loads are dependent on the weight of the patient and would be subject to change with new model investigations [46]. This quasi-static loading representation of the dynamic nature of an amputee in the gait cycle has been validated by Faustini et al. [47], who utilized quasi-static loading conditions derived from experimentally measured ground reaction forces to mimic in vivo testing within FEA.

The coefficient of static friction between the limb and socket, as well as the inlay and socket, were set to μ = 0.5. The coefficient of static friction between the inlay and limb is approximately 2.0, and, therefore, they were modeled as being bonded [22]. A mesh convergence study was run on the no inlay condition on L1 and L2, and a maximum mesh element size of 2.75 mm was deemed the appropriate selection for both models. This maximum element size produced 75,633 and 384,075 linear tetrahedral elements for L1 and L2, respectively.

Optimization Set Up.

Reductions of the PPG could likely be achieved through optimization of the three Yeoh material coefficients of the three selected materials within the inlay. Optimizing the material coefficients is meant to act as a representation of optimizing the draft angle of the metamaterial. The design approach of altering key parameters to guide the redesigning of unit cells was proposed by Satterfield et al. [48]. They determined that altering parameters that represent the physical response of a unit cell can lead to similar results as changing the unit cell directly.

When analytical gradient calculations are not possible or are too time costly, surrogate modeling is an effective strategy for optimization. The surrogate model, in this case, a response surface with corresponding sensitivity analysis, can be used to quickly evaluate the sensitivity of the output with respect to changes in the inputs. This prediction enables the identification of critical inputs that most influence the output. Isolating key inputs may help reduce the complexity of a given problem by eliminating the need to include low impact inputs within the model. Additionally, the response surface provides a simple analytic model approximation for the complex FEA model. It thereby enables rapid evaluation of the model and calculation of the gradients needed for gradient-based optimization.

A design of experiments (DOE) is required to produce a response surface and corresponding global sensitivity analysis. For L1, a 147-sample central composite DOE was run to create a standard second-order polynomial response surface. The bounds for each coefficient (Eq. (3)) were set based on the initial coefficient values in Table 1. Each DOE sample for L1 took approximately 5 min through the utilization of Clemson University's Palmetto Cluster. The response surface for the FH had a relative average error of 1.3%, which is sufficiently accurate for the current application.

A sensitivity analysis was performed using the response surface for the L1 model. Figure 7 shows the variables P9 and P10, which correspond to C₂₀ and C₃₀ of the inner material, respectively, which have the highest sensitivity and, therefore, the most effect on PPG.

Fig. 7.

Sensitivity analysis for a fibular head for L1

The reduction of PPG on the limb surface is the goal of the optimization. Finding an analytical equation that directly links reduction in PPG with design variables is difficult. Therefore, the optimization problem is reformulated to minimize an approximation of the PPG. An ideal liner under the given conditions would cause homogenous limb surface stress, meaning the maximum and average limb stress would be equal. Therefore, to minimize an approximate of the PPG, the objective of the optimization was set to minimize the difference between the maximum and average limb surface stress. The nonlinear programing by quadratic Lagrangian, a gradient-based algorithm, was used within ANSYS direction optimization. The nonlinear programing by quadratic Lagrangian method solves problems with a continuously differentiable objective function and constraints. The derivatives of the objective function were calculated using the central difference approach until the 0.1% convergence percentage was satisfied. The objective and constraints of the optimization are shown in the following equation:

$$\mathrm{Obj}=\min |\left({\mathrm{Stress}}_{\max }-{\mathrm{Stress}}_{\mathrm{avg}}\right)|$$ (3)

Design Variables: $C_{10-\mathrm{Inner}}$, $C_{10-\mathrm{Middle}}$, $C_{10-\mathrm{Outer}},\hspace{0.17em}{C}_{20-\mathrm{Inner}}$, $\hspace{0.17em}{C}_{20-\mathrm{Middle}},\hspace{0.17em}{C}_{20-\mathrm{Outer}}$, $\hspace{0.17em}{C}_{30-\mathrm{Inner}},\hspace{0.17em}{C}_{30-\mathrm{Middle}}$, $C_{30-\mathrm{Outer}}$

Subject to

$$\begin{array}{c}0\hspace{0.17em}\mathrm{Pa}<C_{10-\mathrm{Inner}},C_{10-\mathrm{Middle}},C_{10-\mathrm{Outer}}<10\hspace{0.17em}\mathrm{kPa}\\ 0\hspace{0.17em}\mathrm{Pa}<C_{20-\mathrm{Inner}},C_{20-\mathrm{Middle}},C_{20-\mathrm{Outer}}<500\hspace{0.17em}\mathrm{kPa}\\ 0\hspace{0.17em}\mathrm{Pa}<C_{30-\mathrm{Inner}},C_{30-\mathrm{Middle}},C_{30-\mathrm{Outer}}<500\hspace{0.17em}\mathrm{kPa}\\ {\delta }_{\max }\le 3\hspace{0.17em}\mathrm{mm}\end{array}$$

Constraining the maximum deformation (δ_max) to less than or equal to 3 mm ensures that the materials were stiff enough that the patient would have confidence that the inlay could stand up to the daily rigors required of an inlay. The lower angle (softer) unit cells of the metamaterial begin to bottom out when the deformation exceeds approximately 3 mm, causing the deformation to be entirely controlled by the base Tangoplus material and eliminating variance between the metamaterial variations. Experience has shown that patients can label materials that are too soft as “squishy” and, therefore, are unsatisfactory. The deformation constraint is an attempt to reduce the potential of this problem arising.

Changes to the L1 optimization method had to be made to produce analytical surrogate models to produce the optimization results for the TE and PT of L2. These changes come from the need to account for the different limb shape, loading conditions, areas of offloading, and prosthetist's prescription. Following the prosthetist's recommendation, the inlay for L2 was designed using two (six Yeoh coefficients) and three (nine Yeoh coefficients) metamaterial variations at the PT and TE, respectively. The two inlay areas were optimized individually. Therefore, the two limb areas required a 147 and 49 sample DOE for the TE and PT, respectively. Future research aims to investigate how changes in metamaterial selection may arise when the multiple critical areas are optimized simultaneously. Each FEA took approximately 28 min using the same resources from the Palmetto Cluster. The response surfaces for the TE and PT had a relative average error of 1.0% and 3.3%, respectively. The sensitivity analysis for the TE (Fig. 8(a)) shows that P2 and P9, corresponding to C₂₀ of the inner and outer materials, respectively, have the highest sensitivity magnitude and, therefore, have the most control over changes in the output. The sensitivity analysis for the PT (Fig. 8(b)) shows that P9, corresponding to C₂₀ of the outer material, has the largest sensitivity magnitude. The sensitivity is also negative in this analysis, meaning a negative correlation between P9 and the output.

Fig. 8.

Sensitivity analysis for L2 of (a) tibial end and (b) patellar tendon

Results and Discussion

Limb Shape 1 Results.

The equivalent Von-Mises pressure on the surface of the residual limb was determined for each condition. This equivalent stress is a combination of all the stresses subjected to a node on the surface of the limb. The effect normal versus shear stress has on pressure distribution around a residual limb is a thoroughly studied topic but is not of interest in this paper. Therefore, equivalent stress provides an adequate representation of pressure distributions. Results for each condition from L1 are summarized in Table 3. The PS and PPG values for the FH conditions were found using stress distribution images isolated to the area around the FH, like in Fig. 9. The stress distribution image for each liner type has not been included for the sake of space. Please contact the corresponding author for additional information. Using Fig. 9, the PS could be directly calculated as the maximum surface stress. The PPG was calculated by evaluating the distribution of stress values at the nodes around the maximum surface stress. The slope of the largest gradient was isolated and used as the PPG.

Table 3.

Peak stress and peak pressure gradient comparison for limb shape 1

Liner type	Peak stress (PS) (MPa)	Peak pressure gradient (PPG) (kPa/mm)	Gradient reduction compared to no inlay
No inlay	0.198	32.4	–
Silicone liner	0.134	22.0	32.1%
Prosthetists prescription	0.175	22.1	31.8%
Optimized material coefficients	0.075	10.4	67.9%

Fig. 9.

Surface stress distribution (MPa) for fibular head given (a) silicone liner and (b) optimized material inlay

The effectiveness of each liner type was judged based on its gradient reduction capabilities relative to the no inlay condition. The practitioner prescribed inlay and silicone liner conditions to provide a reduction in both the PS and PPG. The silicone liner and practitioner prescribed inlay resulted in nearly identical reductions in PPG, while the PS was reduced more using the silicone liner as compared to the practitioner prescription.

Previous research shows a range of accepted peak limb pressures in simulation and clinical settings [10,11,29–34]. When adjusted for the weight of the patient, the results of the no inlay condition simulation are within the range of accepted pressure values of the clinical findings on the FH [81–233 kPa], found by Yeung et al. [49], validating the assumptions and approximations made in this model to mimic in vivo conditions.

The optimized material coefficients in Table 4 were used to represent the optimized coefficient inlay condition. The “optimized material coefficients” results in Table 3 show a further reduction in PS and PPG are possible beyond the practitioner prescribed inlay condition. The optimized metamaterial layout stress distribution is shown in Fig. 9(b).

Table 4.

Optimized Yeoh third-order coefficients around the fibular head (FH)

	FH inner material	FH middle material	FH exterior material
C10 (Pa)	172	8099	8819
C20 (Pa)	1541	4.73 × 105	4.98 × 105
C30 (Pa)	11,148	3.91 × 104	2.29 × 105

To better understand the physical meaning of the optimized material coefficients, the stress–strain curves were produced utilizing Eq. (2) and compared to the stress–strain curves of the current third order representations of the metamaterial (DA₀₀-DA₂₇ and base material). These stress–strain curves allow for a visual comparison between the optimized material properties and how they behave relative to the third-order representations of the metamaterials with predetermined draft angles. This comparison will give insight into what draft angle the optimized metamaterial should be printed to achieve the desired pressure gradient reduction. The results indicated in Fig. 10 show that the internal material should have a draft angle smaller than DA₀₀, while the middle and exterior surfaces should be fabricated similarly to the base material.

Fig. 10.

Stress–strain comparison of optimized material properties and known metamaterials for fibular head

Limb Shape 1 Discussion.

Table 3 shows the results from each design type, and by all relevant measures, the optimized inlay outperforms all other designs. Based on the material properties selected for the models used, each of the interface solutions are more compliant than the prosthetic socket. The increased deformation allows the force to be distributed over a larger area, and, therefore, the PS is reduced [12]. The silicone liner has a smaller PS compared to the prosthetist's prescribed inlay because the silicone material around the FH was modeled as a softer material compared to the internal material of the prosthetist's prescribed inlay, DA₀₉. The prosthetist's prescription has a near-identical PPG as the silicone liner, which is interesting considering the PS is several kPa's larger. The similar PPG of the prosthetist's prescription and silicone liner can be accredited to the gradual mechanical response change of the metamaterial variations within the prosthetist's prescription. As Klute et al. Recommended, the softer DA₀₉ was located on the boney FH while DA₁₄ and DA₂₇ were located on the soft tissues, allowing for a less drastic reduction in pressure [5].

The optimized material coefficients make the internal material softer than silicone, allowing for a more significant deflection of the inlay around the boney prominence, reducing the PS and PPG. The maximum deflection of the inlay with the optimized material coefficients is 2.84 mm. The results of Fig. 10 suggest that the inner material should be softer than DA₀₀, which would not be possible using the current metamaterial design. However, these results indicate that given the current material availability, implementing DA₀₀ as the inner material would be the most beneficial to reduce PPG. It should be noted that the results of the FH optimization hover around the boundaries of the constraints set in Eq. (3), insinuating the possibility of a local minimum.

A further investigation should be run to evaluate new metamaterial designs with a broader range of mechanical responses that are still able to satisfy the multimaterial approach. As stated previously, the Yeoh third-order model begins to deviate from the experimental stress–strain results at lower draft angles due to the more severe buckling reaction of the metamaterial under compression. This deviation should be acknowledged for the inner material at the FH, and future research should investigate a material model that better captures the buckling reaction of lower draft angle materials to increase the accuracy of the material model.

Figure 9 shows the middle and exterior material properties are similar and both show a mechanical response similar to the base material, suggesting that proper offloading of the FH could be achieved with just two materials. The optimization results also show that the prosthetist's recommendation of DA₂₇ as the outer material was closer than the recommendation of DA₀₉ as the inner-most material.

Limb Shape 2 Results.

The PS and PPG comparisons were targeted at the PT and TE for L2. The results in Table 5 show that the introduction of an interface material around the TE will both reduce the PS and PPG. The PS and PPG were calculated similarly to L1 by using surface stress distribution images. The TE and PT were isolated and evaluated separately, as seen in Figs. 11 and 12. The practitioner prescribed inlay has greater gradient reduction capabilities compared to the silicone liner conditions at the TE. The results at the PT show that the silicone liner increased the PS and PPG while the practitioner prescribed inlay again reduced both components.

Table 5.

Peak stresses and peak pressure gradient comparison for limb shape 2 for the tibial end (TE) and patellar tendon (PT)

Limb area	Liner type	Peak stress (MPa)	Peak pressure gradient (kPa/mm)	Gradient reduction compared to no inlay
TE	No inlay	0.160	16.9	—
	Silicone liner	0.122	7.59	44.9%
	Prosthetists prescription	0.079	5.88	65.2%
	Optimized material coefficients	0.057	2.18	87.1%
PT	No inlay	0.053	2.33	—
	Silicone liner	0.076	3.23	−38.6%
	Prosthetists prescription	0.035	1.76	24.5%
	Optimized material coefficients	0.029	0.86	63.1%

Fig. 11.

Surface stress distribution (MPa) for tibial end given (a) silicone liner and (b) optimized material inlay

Fig. 12.

Surface stress distribution (MPa) for patellar tendon given (a) silicone liner and (b) optimized material inlay

Two separate optimization evaluations were run from the individual response surfaces to target the TE and PT individually. The objective and constraints of the optimizations were the same as those defined in Eq. (3). The limb surface area was broken into two regions to isolate the areas around the TE and PT. The coefficient results from Table 6 were used to represent the optimized condition. The optimized material coefficients inlay results from Table 5 show that the optimized material coefficients resulted in the smallest PS and the largest percent reduction in PPG. The PS and PPG values for the optimized material inlays were calculated using the stress distributions in Figs. 11 and 12.

Table 6.

Optimized Yeoh third-order coefficients for the tibial end (TE) and patellar tendon (PT)

Limb area	Coefficient	TE inner material	TE middle material	TE exterior material
TE	C10 (Pa)	1518	8857	9507
	C20 (Pa)	4975	4.18 × 105	4.78 × 105
	C30 (Pa)	27,704	2.56 × 105	2.39 × 105

		PT inner material		PT Exterior Material
PT	C10 (Pa)	9542	—	9959
	C20 (Pa)	4.94 × 105	—	4.94 × 105
	C30 (Pa)	2.63 × 105	—	4.84 × 105

The stress–strain curves were created to compare the physical behaviors of the optimized coefficients to the currently selected materials. Figure 13(a) shows that the internal material around the TE should have a draft angle much lower than the current DA₀₉. The draft angle should be between DA₀₀ and DA₀₂. The middle and outer materials should be designed similarly to the base material. Figure 13(b) indicates the two materials used to offload the PT should have similar mechanical responses compared to the base material.

Fig. 13.

Stress–strain curve comparisons of optimized material properties and known metamaterials around the (a) tibial end and (b) patellar tendon

Limb Shape 2 Discussion.

The L2 results again show that, within the scope of this research, the optimized metamaterial outperforms all other designs. The maximum pressure values around the PT and TE for the no inlay condition fall within the accepted pressure ranges of the clinical results found by Yeung et al. [49] of [50–372 kPa] and [95–295 kPa] for the PT and TE, respectively. The large ranges of accepted stress values highlight the caution that should be taken when comparing in vivo and computationally collected data.

Table 5 shows the results around the TE for L2 are like L1 in that all liner solutions lead to a predicted reduction in PS and PPG. The difference arises in that the practitioner prescribed inlay has a more significant reduction in both PS and PPG. This variance from the L1 results is expected because of the combined loading situation of L2. Whereas L1 only experiences a normal force, L2's loading situation was a combined loading of normal and shear force. At the TE, the larger 1090 N acts as the normal force, while the smaller 209 N acts as a shear force. The opposite occurs around the PT. With the introduction of a shear force, a softer material does not guarantee a smaller PS. Sanders et al. [19] proposed that a material with a lower shear stiffness allows the residual limb to move deeper into the socket upon weight-bearing, while one with a larger shear stiffness would not.

The increased deflection into the socket can lead to increased shear stress, increasing the magnitude of the equivalent stress. The elevated shear stresses cause the PS and PPG around the TE of the silicone liner with a lower shear stiffness to be greater than the practitioner prescribed inlay with a higher shear stiffness.

The effect of shear stress is even more apparent around the PT, where the PS and PPG values are increased with the introduction of the silicone liner compared to the no inlay condition. The larger shear force exposed to the PT caused the majority of the load to be taken on by the soft tissue. The soft tissue then experiences an increase in shear stress, which increases the magnitude of the PS. The softest material in the practitioner prescribed inlay around the PT is DA_14, which is significantly stiffer than silicone. This material difference reduces the deflection of the inlay and limits the shear stress on the limb, explaining the reduction in PS and PPG with the practitioner prescribed inlay condition.

The softest optimized coefficients for the TE are much softer than silicone. Additional FEA results show an inlay deflection of 2.5 mm around the “inner material” region of the optimized inlay, which is greater than the silicone liner. Table 5 shows that the optimized coefficient solution can achieve the smallest PS and PPG, even with an internal material softer than silicone. The explanation for this result may come from the multimaterial approach. The “middle material” and “external material” regions have higher shear stiffness than silicone and, in turn, can help limit the shear force experienced by the soft tissue at the TE. Additional FEA results show that the shear stress around the inner material region of the PT is lower for the optimized coefficient inlay than the silicone liner, which validates the assessment that the multimaterial solution can limit the total shear stress and, therefore, equivalent stress.

The optimized coefficient results for the PT do follow the claim that stiffer materials are better suited under high shear forces. The optimized coefficients cause the materials to be stiffer than the silicone liner and the prosthetist's prescribed inlay. These materials ensure that the maximum inlay deformation is limited to 1.4 mm to reduce the shear stress experienced on the limb, allowing the PS and PPG to be reduced.

Model Limitation Discussion.

Differences between the results presented in this paper and those found in the clinical literature may be attributed to specific assumptions and approximations that have been used in this methodology. The locations of the areas of loading were not identical to in vivo reactions of the residual limb. Lin et al. [11] determined that so long as the loaded nodes are far enough apart to cover most of the superior bone surface, the effect of the location variance between FEA and clinical testing could be reduced. The accuracy of this statement is highly dependent on the assumptions and loading conditions of individual models. Based on the simplicity of the bone models used in this study compared to Lin et al., this statement's accuracy as it pertains to this study is unknown.

The soft tissue was treated uniformly around the entire limb, which does not account for potential differences between muscle, fat, skin, and scar tissues present in the limb. These differing stiffnesses could alter the current results. The two limb models represent nontraumatic amputations, and, therefore, it was expected that tissue distribution would be anatomically similar in the limb, which may not be the case in evaluations of other amputees.

Another limitation is the prosthetist's initial design prescription was taken from a single prosthetist. It is common knowledge that there is considerable variation between prescriptions among different prosthetists. A more accurate model of the prosthetist's prescription inlay could have been derived from a survey across multiple prosthetists, allowing for a more widely accepted inlay design. Future work aims to broaden the resources used to derive the prosthetist's prescription.

This paper is a preliminary investigation into how limb pressure may be reduced when allowing a prosthetist to prescribe a custom nonhomogeneous metamaterial liner. For this reason, the simplicity of the limb models and loading conditions were deemed acceptable. To increase the potential for clinical implementation, the accuracy of the limb, bone, and socket models would need to be drastically improved. The loading conditions would need to more accurately represent the dynamic interaction between the residual limb, interface liner, and prosthetic socket. To better represent the incompressible behavior in the inlay and prevent volumetric locking, the finite elements should use a quadratic or higher basis mesh element. Finally, there is no precedent in the literature for approximating the PPG as the difference between the max and average surface stresses. Therefore, a method for more directly optimizing the PPG should be investigated. However, the approximation used in this study assumes that a more homogenous stress distribution on the limb surface will lead to a reduction in PPG. These changes will increase the complexity of the model but will also produce results that can better justify the clinical implementation of this novel prosthetic liner.

Clinical Relevance Discussion.

A difficult part of achieving comfort within the prosthetic is the patient reaction. A successful prosthetic fitting and liner selection can traditionally be judged based on a few key factors. These include the amputee's desire to wear and use the prosthetic, the amount of time per day the prosthetic is used, the comfortability, and the security that the prosthetic will hold up to the daily rigors of the amputee [50]. An inlay could be designed to minimize the PS and PPG, but if the patient does not feel comfortable, the solution is unviable. One area of potential concern could be the drastic change in mechanical response between materials. This change could be perceived as a rigid point on the inlay, potentially explaining why the prosthetists prescribed a three-material solution for the FH and TE for the respective limb shapes. The prosthetists anticipated that the transition between DA₀₉ to DA₁₄ and DA₁₄ to DA₂₇ is gradual enough to avoid an adverse reaction from the patient. Care must be taken when applying the optimized material property results in a clinical setting to ensure that the material changes are not too severe to warrant a patient complaint. Even so, socket comfort is highly subjective for each patient [51–53] and can be very difficult to mathematically determine if an optimized inlay solution will satisfy the patient.

While these patient concerns need to be considered, this research has begun a preliminary investigation into the use of nonhomogenous prosthetic liner materials to provide a more custom prescription for an amputee. Implementing optimization into the prescription process can potentially make it easier to meet the unique performance needs of an amputee. The design of patient-specific inlays will still require the prosthetist's input to provide an initial prescription, but employing methods used in this research may improve the chances that the amputee is fitted with a more comfortable solution in a shorter amount of time. Advances in the complexity of the models are required before being able to strive for clinical validation and, eventually, patient implementation.

Conclusion

The presented research has shown that a single layer metamaterial with geometric parameters changing as a function of position can reduce limb surface pressure in a transtibial prosthesis. The metamaterial representation was predicted to reduce peak stress and peak pressure gradients greater than a standard silicone liner solution around three key locations of the residual limb, suggesting that heterogeneous material property liners may better be equipped to increase comfort for amputees through reduction of peak stress and peak pressure gradient. The customizable material coefficients can represent the hyperelastic material properties of this single-layer metamaterial. The material properties can be optimized to meet the unique performance needs of an amputee. The resulting optimized coefficients offer insight into how the metamaterial geometric parameters could be designed for an individual patient. Instead of the traditional one-size-fits-all approach taken toward prosthetic prescriptions, we envision a future where a custom liner is designed through optimization and manufactured for each person, thereby enabling personally tailored treatment for each individual, leading to increased comfort and functionality for all.

Funding Data

This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. DOE's National Nuclear Security Administration under contract DE-NA-0003525. This work was partially funded by the SC TRIMH COBRE grant DHHS: NIH/NIGMIS P20GM121342, Hai Yao: PI, Clemson University. The views expressed in the article do not necessarily represent the views of the U.S DOE, NIH, or the United States Government.