Prioritizing biomaterials for spinal disc implants by a fuzzy AHP and TOPSIS decision making method

Considerable research has been focused on identifying the optimum biomaterial for spine implants. New technologies and materials have allowed surgeons to better grasp the biomechanical principles underpinning implant stability and function. An optimal biomaterial for total disc replacement (TDR) should include essential characteristics such as biocompatibility, long-term durability, the capacity to withstand mechanical stresses, and economic viability. Our research has focused on six biomaterials for TDR, including Ti–6Al–4V, CoCr alloy, stainless steel 316L, zirconia toughened alumina (ZTA), polyether ether ketone (PEEK) and ultra-high-molecular weight polyethylene (UHMWPE). Ten common properties, i.e., the Young’s modulus, density, tensile strength, the expense of the manufacturing process, the cost of raw material, wear rate, corrosion resistance, thermal conductivity, fracture toughness and compressive strength were utilized to assess these six different materials. The purpose of this study was to evaluate and rank the six alternative biomaterials proposed for use in the endplates and articulating surface of a spinal TDR. To accomplish this, a multi-criteria decision-making approach, namely the fuzzy analytic hierarchy process (fuzzy AHP) and the Technique of Order Preference by Similarity to Ideal Solution (TOPSIS) was adopted to solve the model. For validation and robustness of the proposed method, sensitivity analysis was performed, and comparison was performed with fuzzy-VIKOR and fuzzy-MOORA methods. In light of the study’s results, ZTA and Ti–6Al–4V were identified as the best suited materials for the articulating surface and endplates, respectively, in a spinal disc implant.


Fuzzy analytical hierarchy process (fuzzy AHP)
The AHP process is a decision-making technique designed to solve problems by decomposing them, grouping them, and then organizing them hierarchically.The method involves a comparison of criteria paired with a predetermined measuring scale to identify priority criteria.Since the primary input of the AHP approach is the perception of the experts, retrieval decisions are subjective.This technique additionally considers consistency of data with inconsistency bounds 38 .However, the accuracy of data and, consequently, the results will be impacted by a high degree of uncertainty and doubt in the evaluation process.Hence, the AHP process was extended based on the fuzzy logic theory.The fuzzy AHP approach is utilized similarly to the AHP method.The only difference is that the fuzzy AHP method transforms the AHP scale into a fuzzy triangle scale for priority access.The following stages were taken sequentially for the development of the fuzzy AHP method.

Define the problem
The problem was defined according to the criteria used to determine an appropriate material for endplates and articulating surfaces in TDR.In general, Fig. 1 illustrates the workflow of the decision-making process utilized in this paper.In this study, six biomaterials were considered as alternatives for TDR, including Ti-6Al-4V, CoCr alloy, stainless steel 316L, zirconia toughened alumina (ZTA), polyether ether ketone (PEEK), and ultra-highmolecular weight polyethylene (UHMWPE).Ten common properties, including the Young's modulus (GPa), density (g/cm 3 ), tensile strength (MPa), the expense of the manufacturing process (qualitative), the cost of raw material (qualitative), wear rate (qualitative), corrosion resistance (qualitative), thermal conductivity (W/mK), fracture toughness (Mpa√m), and compressive strength (MPa), were considered the main criteria for material selection in TDR applications (Table 1).The quantitative data was extracted from published sources 21,[39][40][41][42][43][44][45][46] .Due to a lack of comprehensive data and variations in measuring and testing methodologies for wear and corrosion studies, qualitative data was assigned on a numeric scale to these parameters [47][48][49][50][51][52][53] .According to Table 1, the lowest and highest values demonstrate the best wear rate and corrosion resistance performance for each material, respectively.Typically, manufacturers ascertain the costs associated with the manufacturing process.In addition to additive manufacturing, casting, extrusion, CNC, and injection molding are among the many procedures available.For instance, certain manufacturers might select the expensive injection molding method for production, whereas others might opt for the comparatively less expensive additive manufacturing method.The cost of raw materials is also established by the suppliers, and the quotations provided by different suppliers may differ, thereby complicating the task of determining an exact cost for each material.As a consequence, we consulted with some specialists at CeramTec and endeavored to assign a realistic metric value to these criteria.The highest value in Table 1 corresponds to the highest cost.It is worth noting that the material properties in Table 1 particularly for ZTA were based on published data and there was no evidence that this ZTA was used for implants.However, these information are useful for design guidance.

Create a comparison matrix
The pair-wise comparison matrix is simple, has a strong position for the consistency framework, obtains other information that may be required with all possible comparisons, and is able to analyze the overall priority sensitivity for changes in consideration.Equation ( 1) is utilized to define pair-wise comparisons.
where w i and w j are weights for the criterion i and the criterion j, n denotes the number of compared criteria, and a ij is the ratio of criterion i's weight to criterion j's weight.The scale of relative relevance for constructing a pairwise comparison matrix is depicted in Table 2. Table 3 indicates the pair-wise comparison matrix constructed in this paper.The following rules were taken into account when constructing the pair-wise comparison matrix: • If the element on the left of the pair-wise comparison matrix is more important than the element on the right, a positive integer (from 1 to 9) is placed in the cell; conversely, the reciprocal value of the integer is entered (Table 3).• The relative importance of each element relative to itself is one; hence, the diagonal of the matrix contains only ones (Table 3).
After determining the comparison of its criterion, each column is normalized into matrix form by dividing each value in column i and row j by the sum of its column (see Eq. 2). (1) Table 1.Objective data of the attributes of alternative biomaterials for TDR application 21,[39][40][41][42][43][44][45] .E and ρ represent the Young's modulus and density, respectively.For the qualitative parameters [47][48][49][50][51][52][53] , the highest intensity is 5, while the lowest intensity is 1.where w eigvec is the eigenvector, â i is the sum of the matrix normalization values in each row and is divided by the number of criterion (n).Then the largest eigenvalue (λ max ) is obtained by multiplying the number of columns with the main eigenvector (w eigvec ) (Eq. 4).Then the consistency index and consistency ratio are calculated by Eqs. ( 5) and (6).

Material E (GPa) Ρ (g/cm
where CI and CR denote the consistency index and consistency ratio, respectively.To obtain the consistency ratio (CR), CI is divided by the ratio index RI (see Table 4) for the same sized matrix.Saaty et al. 38 provided a comparison of the consistency index with a ratio index (RI) value (Table 4).This value relies on the order of the matrix (n).It is worth mentioning that CR should be around 10% or less to be acceptable.If the CR falls outside of this range, the participants' evaluations should be revised (Fig. 1).

Triangular fuzzy number (TFN)
In general, two primary types of fuzzy membership functions are commonly employed, namely triangular and trapezoidal.Among them, triangle membership functions are frequently favored.The main rationale for utilizing triangular fuzzy sets over trapezoidal ones is in their inherent simplicity and ease of computation 27,54 .Moreover, while dealing with subjective and imprecise information, the utilization of the triangular fuzzy set demonstrates its efficacy in developing decision-making problems 27,54 .
The fuzzy AHP scale has three values, namely, the lowest value (lower, L), the middle value (median, M), and the highest value (upper, U) (Fig. 2).The AHP comparison value is transformed to the fuzzy AHP scale value considering the scale of relative importance in Table 2 and according to the following equations:

Calculate the weight value of the fuzzy vector
According to the aggregated pair-wise comparison matrix, the geometric mean (r) for the i th criterion is calculated as follows: r i represents the fuzzy geometric mean value for the i th criterion, which is calculated by multiplying the fuzzy numbers ( a ij according to Eq. 7) in each row (i.e., lower values multiplied by lower values, middle values multi- plied by middle values, and higher values multiplied by higher values in each row).The mean value is then rooted by the number of criteria in each row (m).Then the fuzzy weight (W F ) for the i th criterion is calculated as follows: To calculate the weight factor of each criterion ( W Fi ), first all geometric mean values (r i ) are summed together (i.e., lower values summed together, middle values summed together, and upper values summed together).The resultant value is then reciprocated and multiplied by the fuzzy geometric mean value of each criterion ( r i ).To get a crisp numerical value for fuzzy weights ( W Fi ), the de-fuzzification approach is conducted by calculating the center of area (COA): Finally, the normalized weight is computed through dividing the weight factor of each criterion by the sum of the weight factors.

TOPSIS
The logic of TOPSIS is based on the concept that the selected option should have the shortest geometric distance from the best solution and the furthest geometric distance from the worst solution 55 .The TOPSIS process can be encapsulated in six steps as follows.
Step 1 The evaluation matrix (A) is created according to the M alternatives (biomaterials in Table 1) and N attributes (parameters in Table 1): Step 2 The evaluation matrix is normalized by dividing the value of each criterion ( a ij ) by the root sum squared of the criteria in each column as follows: Step 3 The weighted normalized decision matrix is obtained by multiplying the normalized decision matrix ( α ij ) by its associated normalized fuzzy AHP weights ( w j ): Step 4 Determine the best (b) and worst (w) alternative (A) for each attribute ( X): Step 5 The Euclidean distance between the target alternative ( X ij ) and the best ( X b j ) and worst ( X w j ) alternative is respectively calculated: i denotes the Euclidean distance between the target alternative and the best alternative, and D w i indicates the Euclidean distance between the target alternative and the worst alternative.
Step 6 The relative closeness (S i ) to the ideal solution is calculated and the performance order is ranked.The relative closeness of each alternative can be expressed as: S i lies between 0 and 1, and the greatest value means the better performance of the alternatives.Hence, the highest relative closeness value has been taken as the best alternative for endplate and articulating surface applications.

Sensitivity analysis and method validation
A sensitivity analysis of the results was conducted by varying the weights of each criterion in order to determine its impact on ranking.It also reveals whether the ranking orders remain consistent despite weight criteria variations.In addition to fuzzy AHP, four more possible ways were considered to identify the weight factors of criteria.Initially, equal weight factors were considered for all criteria.Afterwards, the criteria were divided into three groups (i.e., most beneficial, beneficial, and least beneficial criteria).These classifications were made as each material property has a unique effect on the performance of an articulating surface or endplate.Table 5 shows how both articulating surface and endplate case studies are grouped.Sequentially, the weight factors were assigned as follows: • 33% most beneficial, 33% beneficial, and 33% least beneficial criteria.
Vol:.( 1234567890) www.nature.com/scientificreports/ To validate the potential of hybrid fuzzy AHP-TOPSIS methodology in suitable biomaterial selection, the results were also compared with fuzzy-VIKOR 56 and fuzzy-MOORA 57 methodologies.

Postprocessing
Multi-criteria decision analysis (i.e., fuzzy AHP-TOPSIS) was used to determine the material for the design of endplate and bearing surfaces in TDR.This is a useful tool that may be applied to a variety of intricate decisions.Case studies determine the structure of the comparison matrix in the fuzzy AHP process.Wear rate and corrosion resistance are most relevant for articulating surfaces, while Young's modulus and corrosion resistance are most important for the endplates (Table 3).Since density and thermal conductivity were assessed to be of moderate importance to the designs, they were assigned the lowest relative weights (Table 3).The other attributes, which play a major role in TDR design, were assigned strong relative importance (Table 3).
In the TOPSIS process, the attributes including Young's modulus, density, expense of manufacturing process, the cost of raw material, and wear rate are considered beneficial with lower values, while the others, including tensile strength, corrosion resistance, thermal conductivity, fracture toughness, and compressive strength, are advantageous with higher values (Table 1).These specifications enable the determination of the best and worst alternatives for each attribute.
The calculations provided by the fuzzy AHP-TOPSIS methods were performed by developing a custom Python code (Python 3.8.5 programming language) in which all the data of materials and their properties were specified.The data can be introduced into the program manually or via a table file, depending on the preference.If the constructed pair-wise comparison matrix is inconsistent during the fuzzy AHP procedure, the program requests the adjustment of the pair-wise comparison matrix until consistency is achieved (Figs. 3, 4).

Fuzzy AHP weighting factors
To establish normalized weights, the fuzzy AHP method was implemented.In this process, first an AHP pair-wise comparison matrix was determined as depicted in Table 3 for the design of the articulating surface and endplate.
According to the AHP pair-wise comparison matrices (Table 3), the consistency ratios for the articulating surface and endplate were 0.077 and 0.094, respectively (Fig. 4).This means that the consistency of each opinion was considered acceptable, with lower than the inconsistency threshold value (0.1).
The AHP pair-wise comparison matrices were transformed into TFN scales (according to Eq. 7) upon acceptance of the consistency ratio, and then the normalized fuzzy AHP weight factor was determined for each attribute (Table 6).

Rank establishment by TOPSIS
All the criteria used for ranking the biomaterials had different units and dimensions.They need to be normalized by using Eq. ( 13).The normalized criteria values were then converted into normalized weighted values by multiplying with weights using Eq. ( 14).These normalized weight matrices for the design of the articulating surface and endplate are shown in Table 7.
After calculating the normalized weighted values, Eqs. ( 17)- (19) were used to calculate separation measures and relative closeness values (Table 8, Figs.5, 6, 7).From these relative closeness values, the ranking was given, and the alternatives were prioritized (Table 8, Fig. 7).Based on the ranking results shown in Fig. 7 and Table 8, ZTA had the highest relative closeness value for the design of an articulating surface, whereas Ti-6Al-4V had the highest relative closeness value for the design of an endplate in TDR applications.

Ranking validation and sensitivity analysis
The obtained ranking of proposed hybrid fuzzy AHP-TOPSIS was compared with fuzzy-VIKOR and fuzzy-MOORA approaches reported in Table 9. Accordingly, the ranking order for different alternative biomaterials was nearly similar when solved with other methods.Therefore, it can be concluded that proposed hybrid fuzzy AHP-TOPSIS method can be successfully implemented for selection of biomaterials based on given criteria with high accuracy.The slight observed variations can be attributed to data normalization techniques and mathematical foundations employed by each method.
The sensitivity analysis was performed to further improve and validate the results of biomaterial selection provided by the proposed fuzzy AHP-TOPSIS method (Fig. 8, Table 10).From Fig. 8 and Table 10, it was evident that criteria weight variation can alter the ranking of each alternative.For an articulating surface (Fig. 8a, Table 10a), it was found that the A1 alternative (ZTA) had the highest relative closeness in 4 trials out of 5 experiments.In contrast, the A5 alternative (PEEK) and A6 alternative (stainless steel 316L) had the lowest relative closeness across most trials.For the endplate (Fig. 8a, Table 10a), it was found that the A3 alternative (Ti-6Al-4V) had the highest relative closeness in 4 trials out of 5 experiments.In contrast, the A1 alternative (ZTA) and A6 alternative (stainless steel 316L) had the lowest relative closeness across most trials.Hence, the sensitivity analysis justifies that the alternative A1 (ZTA) and alternative A3 (Ti-6Al-4V) can be selected as the top priority (80%) material for the design of the articulating surface and endplate, respectively.The final ranking is based on the outcomes of fuzzy AHP-TOPSIS and can be expressed as: • A1 (ZTA) > A2 (CoCr alloy) > A3 (Ti-6Al-4V) > A4 (UHMWPE) > A5 (PEEK) > A6 (stainless steel 316 L) for articulating surface.• A3 (Ti-6Al-4V) > A4 (UHMWPE) > A5 (PEEK) > A2 (CoCr alloy) > A6 (stainless steel 316 L) > A1 (ZTA) for endplate.
According to the ranking validation and sensitivity analysis, it can be confirmed that hybrid fuzzy AHP-TOPSIS is effective and robust in material selection for spinal disc implants.

Discussion
TDR typically achieves motion preservation by articulating surfaces, which are susceptible to wear under repetitive motion and loading 58 .Polymeric debris generated by metal on polymer bearing surfaces can result in osteolysis and implant loosening 59 .While metal on metal bearing surfaces have superior wear properties, high metal concentrations and nanodebris can lead to local and systemic effects 59 .Ceramic on ceramic bearing surfaces showed less wear debris with lower biological reactivity due to their outstanding tribological properties and excellent biocompatibility 60 .In this study, ZTA also proved to be the optimal material for designing articulating surfaces in TDR applications.The most prevalent metallic biomaterials, namely CoCr alloy and Ti-6Al-4V, were afterwards ranked second and third, respectively.Even though ZTA has better biocompatibility and tribological qualities than CoCr alloy and Ti-6Al-4V, these materials were relatively close (Fig. 7, Table 8).It was justifiable that the technique offered a tradeoff between desired parameters.In this instance, ZTA's lower fracture toughness, greater Young's modulus, and more expensive manufacturing process cost contributed to a similar performance score for the design of articulating surfaces as compared to CoCr alloy and Ti-6Al-4V.The higher Young's modulus of biomaterials compared to cortical bone can lead to stress shielding, the remodeling-induced reduction of bone due to the removal of mechanical stress from the bone by an implant 61 .It is essential to achieve initial and long-term stability at the bone-implant interface to avoid these complications by considering materials with similar modulus to bone, adequate osseointegration and osseoconductivity, high fracture toughness, and corrosion resistance.In this investigation, Ti-6Al-4V was demonstrated to be the optimal alternative material for designing TDR endplates.Then, UHMWPE and PEEK followed closely behind Ti-6Al-4V in terms of performance.Although PEEK's elastic modulus was comparable to that of cortical bone, it was ranked third in this study.This placement was influenced by the PEEK's greater cost of raw materials, lower fracture toughness, compressive and tensile strengths.However, the difference in relative closeness between PEEK and Ti-6Al-4V was not considerable, and they can be utilized interchangeably (Fig. 7, Table 8).It was somewhat unexpected that ZTA came in last in the ranking.The ZTA ranking could be justified by the material's high Young's modulus, high cost, and low fracture toughness.
In addition to the factors outlined in Table 1, osseointegration, CT and MRI compatibility, and the risk of implant-related infections play key roles in the selection of biomaterials.Lee et al. 62 conducted an in-vitro Figure 6.The Euclidean distance between the target alternative and (a) the best alternative and (b) the worst alternative for endplates.The least distance to the best alternative and the greatest distance to the worst alternative are preferred, in that order.evaluation of the biological response of cells to different biomaterials, including ZTA, PEEK, silicon nitride (SN) and surface-textured silicon nitride (ST-SN) and Ti-6Al-4V.compared to Ti-6Al-4V, they found that all other materials generally demonstrated lower osteoclastic activity and inflammatory response.They also demonstrated that ZTA and SN enhanced osteogenic differentiation and actin length 62 .Moreover, ceramic biomaterials are more compatible with CT and MRI than metals, which can result in problematic artefacts, or frequently used polymers that are radiolucent.Consideration of these characteristics in the decision-making process will alter the relative closeness of biomaterials, particularly for ZTA.Notably, only a small number of these characteristics are available for several of the materials evaluated in this work.Therefore, it is recommended to undertake more in-vitro research addressing the evaluation of material biocompatibility and include the results in the process of material selection.The decision-making procedure revealed that the biomaterial ranking varies depending on the function.According to this approach, the combination of these materials could be effective in reducing clinical issues including inflammatory response, heterotopic ossification, dislocation, and migration.However, this combination must be technologically feasible for production.Recently, the Simplify® Cervical Artificial Disc has completed review for premarket approval for use in patients suffering from radiculopathy or myelopathy 63 .This implant was composed of two PEEK endplates and a ceramic core (ZTA).The external surface of the endplates was coated with titanium to promote bone formation and facilitate attachment to the vertebrae located above and below the implant.The materials employed in this implant were consistent with the outcomes of fuzzy AHP-TOPSIS.However, longitudinal clinical evidence is required for a complete knowledge of the efficacy of both the logical method and the implant.
Ghaleb et al. 67 evaluated numerous alternatives based on process agility, computing complexity, and the number of possible processes and criteria for manufacturing process application.Based on computational complexity, they determined that VIKOR outperformed TOPSIS and AHP.In terms of decision-making agility, the TOPSIS and VIKOR approaches were more applicable, although the rankings derived by AHP, TOPSIS, and VIKOR for the selection of manufacturing processes were nearly identical 67 .In our study, the proposed fuzzy AHP-TOP-SIS outcome was also compared with fuzzy-VIKOR and fuzzy-MOORA methods (Table 9).Accordingly, ZTA obtained the highest ranking across all the approaches for the selection of the articulating surface material.In terms of the endplate, Ti-6Al-4V exhibited the highest ranking according to the fuzzy AHP-TOPSIS and fuzzy MOORA methodologies.However, it obtained the second highest ranking according to fuzzy-VIKOR method, indicating a nearly same ranking across all approaches.Furthermore, the sensitivity analysis was performed to validate the outcomes obtained by the fuzzy AHP-TOPSIS method.When the results of the sensitivity analysis were evaluated collectively (Fig. 8, Table 10), they were overall consistent with one another.Nonetheless, a moderate difference was observed, revealing that the opinions of experts could affect the outcomes.
In the literature, several different approaches have been proposed for material selection, including MOORA, AHP, COPRAS, VIKOR, and TOPSIS.However, preference ranking organization method for enrichment evaluation (PROMETHEE) is one of a new ranking method which is considered as simple in conception and computation compared to many other MCDM methods 64 .PROMETHEE focuses on pair-wise comparisons to establish outranking relationships between alternatives and facilitates both partial and complete ranking of alternatives 65 .It also provides a superior visual representation during the concluding phase of the problem-solving process in comparison to alternative MCDM methods like AHP 66 .Several MCDM methods necessitate significantly more inputs than PROMETHEE 66 .For future research, it is advisable to employ the PROMETHEE approach to prioritize spinal implant biomaterials and subsequently compare the obtained findings with those derived from other MCDM methods, such as fuzzy AHP-TOPSIS.

Conclusion
This study emphasized the necessity of a unique and justified selection of biomaterials in accordance with their intended purpose during the early stage of development.The technique enables the consideration of multiple qualitative and quantitative criteria throughout the material selection process for a particular design.Moreover, it avoids later costs and delays and generates ideas through a systematic search of biomaterials.However, the assignment of acceptable numerical values for qualitative criteria necessitates a review of the relevant literature and consultation with industry sector experts in order to minimize bias or individual opinion in the results.Furthermore, it requires a thoughtful selection of criteria, as this selection also has a direct impact on the outcome.

Figure 3 .
Figure 3.The initial segment of the Python function requests the number of attributes and the relative importance of each attribute in relation to the others so that a pair-wise comparison matrix can be generated.

Figure 4 .Table 6 .
Figure 4.A part of the Python function produces a pair-wise comparison matrix, assesses consistency index, selects ratio index based on attribute count, and calculates consistency ratio for (a) articulating surfaces and (b) endplates.It compares the consistency ratio to the 0.1 threshold.If the values were below the threshold, a message says the pair-wise comparison matrix is reasonably consistent and evaluates fuzzy weight factors for each criterion.Otherwise, it requests an adjustment of the pair-wise comparison matrix until consistency is reached.

Figure 5 .
Figure 5.The Euclidean distance between the target alternative and (a) the best alternative, and (b) the worst alternative for articulating surfaces.The least distance to the best alternative and the greatest distance to the worst alternative are preferred, in that order.

Figure 7 .
Figure 7. Ranking graph which is derived from the result of relative closeness for design of (a) articulating surface and (b) endplate in TDR applications.

Table 2 .
Scale of relative importance in AHP scale and fuzzy AHP scale.

Table 5 .
Classification of criteria according to their level of importance for the function of the articulating surface and endplate.

Table 7 .
Normalized weighted matrix for design of (a) articulating surface and (b) endplate.

Table 8 .
The separation measures and the relative closeness for design of (a) articulating surface and (b) endplate.D b i , D w i and S i represent the best Euclidean distance, the worst Euclidean distance, and relative closeness respectively.Bold emphasizes best ranked biomaterials for (a) articulating surface and (b) endplate.'