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. Author manuscript; available in PMC: 2015 Aug 17.
Published in final edited form as: J Biomech. 2012 Oct 22;46(1):19–25. doi: 10.1016/j.jbiomech.2012.09.007

Clustering and Classification of Regional Peak Plantar Pressures of Diabetic Feet

Craig J Bennetts 1,2, Tammy M Owings 1, Ahmet Erdemir 1,2, Georgeanne Botek 3, Peter R Cavanagh 4
PMCID: PMC4538932  NIHMSID: NIHMS417438  PMID: 23089457

Abstract

High plantar pressures have been associated with foot ulceration in people with diabetes, who can experience loss of protective sensation due to peripheral neuropathy. Therefore, characterization of elevated plantar pressure distributions can provide a means of identifying diabetic patients at potential risk of foot ulceration. Plantar pressure distribution classification can also be used to determine suitable preventive interventions, such as the provision of an appropriately designed insole. In the past, emphasis has primarily been placed on the identification of individual focal areas of elevated pressure. The goal of this study was to utilize k-means clustering analysis to identify typical regional peak plantar pressure distributions in a group of 819 diabetic feet. The number of clusters was varied from 2 to 10 to examine the effect on the differentiation and classification of regional peak plantar pressure distributions. As the number of groups increased, so too did the specificity of their pressure distributions: starting with overall low or overall high peak pressure groups and extending to clusters exhibiting several focal peak pressures in different regions of the foot. However, as the number of clusters increased, the ability to accurately classify a given regional peak plantar pressure distribution decreased. The balance between these opposing constraints can be adjusted when assessing patients with feet that are potentially “at risk” or while prescribing footwear to reduce high regional pressures. This analysis provides an understanding of the variability of the regional peak plantar pressure distributions seen within the diabetic population and serves as a guide for the preemptive assessment and prevention of diabetic foot ulcers.

Keywords: diabetes, foot, ulcer, plantar pressure, biomechanics, k-means clustering, classification

Introduction

It has been estimated that the number of people worldwide with diabetes will surpass 365 million by 2030 (Wild et al., 2004). Foot ulceration in people with diabetes will continue to be a major public-health concern, considering the 15–25% lifetime risk of developing a foot ulcer (Abbott et al., 2005; Singh et al., 2005). At least 15% of these ulcers will lead to some form of foot amputation (http://www.diabetes.org). The burden of diabetic foot problems arises not only from the medical and economic effects, but also from the impact on the patient’s quality of life (Jaksa and Mahoney, 2010).

Plantar pressures, among other clinical measurements, have been used to assess foot conditions in patients with diabetes (Singh et al., 2005). Elevated pressures are believed to increase the risk of ulceration in the diabetic foot, particularly when combined with deformity and peripheral neuropathy (Lavery et al., 2003). Footwear design to relieve elevated plantar pressure is an ongoing research area (Mueller, 1999; Cavanagh and Owings, 2006) although the evidence base for the effectiveness of various offloading techniques remains limited (Cavanagh and Bus, 2010).

Given the wide variety of foot types (Ledoux et al., 2003) and foot biomechanics, a single footwear design cannot successfully decrease peak plantar pressures for all distributions. Ideally, a customized footwear design for each patient would be preferable, but such an approach is not always feasible. A practical solution that accommodates patient-specificity would be to identify groups of patients with similar peak pressure distributions and to establish footwear design guidelines that reduce high regional pressures for each group. New patients could then be classified into one of these groups and the group-specific footwear solution could be prescribed with the expectation that it would reduce corresponding elevated plantar pressures. Waldecker (2012) successfully used logistic regression with plantar pressure, force and pressure-time integral to predict the risk of ulceration although the validity of the pressure-time calculation in this and other studies has been challenged (Waaijman and Bus, 2012; Melai et al., 2011). Acharya et al. (2008, 2011) and See et al. (2010) used principal component analysis and artificial neural networks to classify patients as normal or diabetic, with and without neuropathy. The k-means clustering algorithm was chosen for its relative simplicity in comparison to these classification methods and its ability to independently classify data for different numbers of clusters, unlike hierarchical clustering algorithms (Vardaxis et al., 1998). Our goal was to use k-means cluster analysis to objectively and systematically determine characteristic regional peak plantar pressure distributions for patient classification that may be useful for footwear prescription.

Methods

The Diabetic Foot Clinic at the Cleveland Clinic has an established database of barefoot plantar pressures collected from patients diagnosed with diabetes. Measurements were collected using an EMED X pressure platform (Novel Inc., St. Paul, MN) on 819 different feet at a sampling frequency of 100 Hz. The subject group included 223 males and 215 females (mean age 59.5 +/− s.d. 12.6 years). Both feet were included in the analysis if they were available. Data collection and analysis protocols were approved by the Institutional Review Board of the Cleveland Clinic. Three barefoot, first-step trials were averaged from the stance phase of walking to generate a single peak pressure dataset. This protocol was found to provide reliable estimates of peak pressures (Bus and de Lange, 2005) and prevented time and patient burden of large number of trials in a clinical setting. Post-processing was conducted using Novel software (Novel Inc., St. Paul, MN) to divide the foot into seven regions: hallux, lesser toes (Toes 2–5), first metatarsal head (MTH 1), central metatarsal heads (MTH 2–4), fifth metatarsal head (MTH 5), midfoot, and heel (Figure 1). The magnitude of the peak pressure within each region was extracted to provide a data point, consisting of seven peak pressures for a given foot. These data points were then classified using k-means clustering.

Figure 1.

Figure 1

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Masking of foot regions to extract regional peak pressures. Illustration provided on sample plantar pressure data. MTH = metatarsal head.

K-means clustering (Duda et al., 2000) was performed using Spider (http://www.kyb.tuebingen.mpg.de/bs/people/spider/main.html), a freely available machine learning toolbox for Matlab (The Mathworks Inc., Natick, MA). The procedure used to perform k-means clustering is as follows:

  1. Randomly assign an equal number of data points to each of the k clusters

  2. Average the points in each cluster to determine the cluster centroid

  3. Reassign points to the nearest cluster by determining the Euclidean distance to all cluster centroids

  4. Return to step 2 if points were reassigned to a different cluster

The resulting set of k cluster centroids provides a model which can be used to subsequently classify data points by locating the nearest cluster centroid, by Euclidean distance. In this study, k-means clustering was performed from two to ten clusters (2 ≤ k ≤ 10) to explore the relationship between the specificity of the clusters and the accuracy of classification. It should be noted that number of clusters may be different than the number of regions analyzed, based on the distribution of the regional peak pressures in the data space. Exploring how different number of clusters divide the data space provides the ability to select an adequate number of clusters based on the constraints imposed by future applications and on specificity and accuracy.

K-means clustering does not ensure that the solution is either repeatable or at the global minimum since the final result may be dependent on the initial random assignment of data points to clusters (Duda et al., 2000). Running the k-means algorithm repeatedly provides a statistical estimate of the variability of cluster centroids due to the random initial cluster assignment. Each clustering analysis was performed 20 times to capture this variability. The analysis with the minimum sum of Euclidean distances between its centroids and the nearest averaged centroid from all 20 analyses was selected as the representative model.

The accuracy of clustering and classification of plantar pressure data was assessed by randomly dividing the data into two equally sized sets (dataA and dataB) and applying k-means clustering to each set. This process simultaneously assigned reference labels to the data points (labelsA and labelsB) and generated two independent models (modelA and modelB), defined by the cluster centroids. To ensure consistent labeling between the models, corresponding cluster centroids between the models needed to be determined. This was accomplished by repeatedly matching the closest pair of centroids between the two models until all centroids were matched. The labels associated with data points for modelB were relabeled to match those of modelA. The data from each set was then classified using the model of the other set, i.e. dataA against modelB and dataB against modelA, assigning each point a new label (labelAB and labelBA). A success rate was defined to be the percentage of points consistently labeled in both the original clustering and subsequent classification against the model of the other set. This metric was used to quantify the accuracy of the models, i.e. the consistency of the distribution of data between the two independent sets. This training and testing approach to establish the accuracy of plantar pressure clustering and classification was repeated 20 times for each number of clusters (2 ≤ k ≤ 10) to capture the variation due to initial random division of the whole data set into two groups.

After assessing the accuracy of the k-means clustering and classification on the plantar pressure data, the relative significance of the peak pressure distribution between clusters was examined by performing k-means clustering on the entire data set. In this case, the whole data set was used to obtain k clusters, again ranging from 2 to 10. In a given region of the foot, the influence of cluster definition on peak pressure was evaluated by a one-way analysis of variance (ANOVA) with a significance level set at 0.05. If ANOVA established statistically significant differences, follow-up comparisons using Tukey's Honestly Significant Difference criterion differentiated the clusters from each other by comparing regional peak pressures between clusters for a given number of clusters. The Matlab Statistical Toolbox (The Mathworks Inc., Natick, MA) was used for this analysis.

Results

The accuracy of clustering and classification between the two independent subsets of plantar pressure data decreased as the total number of clusters increased (Figure 2). For two clusters (k=2), the success rate was approximately 93%, which decreased to approximately 63% for ten clusters (k=10). The variability of the success rate (due to initial random assignment of data points to clusters) also changed as indicated by an increase in the standard deviation from 2.8% to 7.5%.

Figure 2.

Figure 2

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Success rate of clustering and classification, i.e. the percentage of feet consistently classified between two independent groups (training and testing sets). Upper and lower bounds represent one standard deviation above and below the average success rate for 20 trials.

K-means analysis successfully identified distinct pressure distributions for different numbers of clusters (Figure 3). For two clusters (k=2), an overall high pressure group and an overall low pressure group were identified. Increasing the number of clusters to three divided the high pressure group further into two groups with medially and laterally high forefoot pressures and an overall low pressure group. For four clusters (k=4), an additional group with high hallux pressures was identified; with five clusters (k=5), a new cluster with elevated pressures primarily at the first metatarsal head in the absence of high hallux pressure was differentiated. Further division of the low pressure group was observed for six clusters (k=6). For this new cluster, the pressures at the central metatarsal heads were higher than the overall low pressure group, but not as high as any high pressure clusters. With k set to 7, a small group with elevated peak pressures at the heel and midfoot was found. Further increase in the number of clusters resulted in a finer division of the low pressure group without further differentiating patients with elevated pressures.

Figure 3.

Figure 3

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Calculated cluster pressures (2 ≤ k ≤ 10, k = number of clusters), showing the relative average peak pressures for each foot region visualized with a linear color gradient, from yellow (low pressure) to red (high pressure). The number of feet grouped into each cluster is provided above the average peak pressure map for each cluster. Appendix provides actual mean and standard deviation for the clusters.

Statistical analysis was performed to find the mean and standard deviation of peak pressures for each cluster for a given number of clusters (2 ≤ k ≤ 10) (Appendix), and also to assess the statistical significance of the differences in the regional peak pressures between clusters (Figure 4). If a 2-way ANOVA found a statistically significant difference of the cluster definition on the regional peak pressure (p < 0.05), follow-up comparisons were performed to determine the significance of the statistical difference between regional peak pressures from cluster to cluster. Figure 4 summarizes this statistical comparison, which is also supported by the relative pressure maps of cluster centroids found in each k-means analysis provided in Figure 3.

Figure 4.

Figure 4

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Statistical comparison of regional peak pressures following one-way ANOVA tests between clusters. Comparisons tested the differences in regional peak pressures between clusters, at a prescribed number of clusters (2 ≤ k ≤ 10). Darker regions of the foot indicate statistically significant differences in the peak pressure between two clusters (at the intersection of the row and column for the clusters of interest). Refer to Figure 3 for cluster numbering (from 1 to k, going from left to right).

Discussion

K-means cluster analysis provided an objective differentiation of regional peak plantar pressure data, identifying not only major foot groupings, but also revealing several plantar pressure distributions that may require special footwear interventions. Although subjective classifications can and are made clinically, predominant plantar pressure distributions observed within a population can only be reliably identified using an objective and quantitative method, such as the k-means algorithm applied in this study. The initial distinction identified by the k-means algorithm when k=2 was the overall magnitude of peak pressures in all examined regions: there was a large group of feet with low-pressures in all 7 regions (592 total) and a smaller group with high-pressures in all regions (227 total) (Figure 3). As the number of clusters increased from for k=3 to k=5, the trend was that the high pressure feet were subdivided into groups with elevated pressure in different anatomical regions. For example, at k=3, in addition to the low pressure feet, groups in which pressure was elevated a) centrally and laterally and b) centrally and medially were identified. With k=4, the latter group was primarily subdivided into feet with and without substantial loading of the hallux. The group that exhibited elevated hallux loading tended not to load the first metatarsal head.

At a higher number of clusters, an additional regional peak pressure distribution was identified for a large number of feet with low pressure overall and a moderately elevated pressure concentration at the central metatarsal heads (k=6, Figure 3). This is not surprising as the second metatarsal is the often the longest in the foot and such feet are likely to exhibit relatively high pressures under the second metatarsal head region when compared to other areas of the foot. For a number of clusters greater than three, a hallux dominated pressure distribution persisted (Figure 3). It would be worth exploring whether or not feet from this cluster exhibit hallux limitus, a pathology that impairs mobility of the first metatarsophalangeal joint and increases pressures underneath the first ray (Van Gheluwe et al., 2006). Similarly, peak pressure profiles dominated by high midfoot and heel pressures were observed starting at k=7. These distributions may correspond to feet that have or are developing Charcot's neuroarthropathy, a complication of diabetes that can lead to severe deformities of the foot, particularly in the midfoot (Pinzur, 2007). The high heel pressures for this cluster may result from an abrupt heel strike. It should be noted that this cluster represents 4.3% of the tested diabetic feet population (35 of 819 feet), which may include outliers. When using fewer numbers of clusters, this small set of data points containing high pressures was averaged out by a relatively larger number of data points exhibiting lower peak pressures.

Random blinded assignment of feet to one of the k clusters has a theoretical success rate of 1/k × 100%. Though the quantified success rate of classification dropped to almost 60% as the number of clusters was increased to ten, it is still significantly higher when compared to a 10% chance of correct random assignment. It should be noted that these success rates are a lower bound, since comparative clustering and classification was performed using two groups, each containing half the data of the entire data set. The final clustering was performed on the whole data set, potentially representing a better differentiation of the regional peak plantar pressure data. The increase in the variability of the success rate from 2.8% to 7.5% as the number of clusters was increased from 2 to 10 is potentially a consequence of the k-means clustering algorithm, but it cannot be determined if this may also be due to physiological factors. As the number of clusters increases, so to does the number and the span of boundaries between clusters, increasing the likelihood that a greater number of data points lying near the boundaries may be misclassified between independently derived models. During classification, the models obtained from k-means were not updated; each new data point was simply assigned to the nearest cluster, based on its distance from all cluster centers. Incrementally updating cluster centroids as new data points become available would likely increase the success rate. A detailed investigation of data points where classification failure occurs may provide practical interpretation of these values. For example, when using six clusters, a new foot may be misclassified to a pressure group with high pressures underneath the central metatarsal heads instead of another with a similar relative regional peak pressure distribution but slightly lower central metatarsal head pressures (Figure 3; Appendix). Although this may be considered as a classification failure, it is not necessarily critical from a footwear prescription perspective. Designs to relieve pressures for both of these groups will likely be the same.

Footwear design based on plantar pressure clustering is a first step for establishing a decision support system for therapeutic prescription of insoles and shoe modifications. High pressure groups (overall or at particular regions), although a smaller subset of the population, are potentially at high-risk of ulceration (Lavery et al., 2003) and may require specialized solutions. For example, seven clusters summarize anecdotally common regional peak pressure distributions that may occur in diabetic feet (Figure 3) and examples of individual plantar pressure distributions (with regional peak pressures nearest to the cluster centroid for each group) are shown in Figure 5. The classification success rate for k=7 was around 65 % (Figure 2), but clustering provided relatively large and significantly different groups of feet at high risk of ulceration under individual regions such as the first metatarsal head, central metatarsal heads, fifth metatarsal head and hallux (Figures 3 and 4), which can be used to establish customized footwear solutions. This solution set also incorporates a smaller group of feet potentially at risk of midfoot ulceration that can be utilized to develop specialized orthotic or footwear solutions. The remaining two clusters have low peak pressures. In the absence of significant deformity or traumatic injury to the plantar surface, these feet are not likely to require the attention of an orthotist. The evidence linking the occurrence of elevated barefoot pressure to risk of ulceration still remains to be demonstrated by prospective studies. It is more likely that in-shoe pressure in the patient's own shoes will be better related to risk of ulceration. However, a recent multi-center randomized controlled study by Bus et al. (2012) failed to find a difference in outcome between patients whose shoes had specifically been designed to reduce plantar pressure and those who had not. Until such evidence is available, clinicians invariably use indications of high pressure (such as callus, redness, or prior tissue damage) as the basis for footwear design, and the data in this paper is intended to provide a quantitative basis for this approach.

Figure 5.

Figure 5

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Actual pressure distribution for a total number of clusters set to seven. A representative pressure distribution is shown for each cluster based on proximity to cluster centroid (by Euclidean distance). See Figure 3 for corresponding cluster centroids for k=7. This clustering will likely be useful for footwear design due to its specific identification of various high pressure patterns with reasonable accuracy (also see Figure 2).

It is likely that regional peak pressure distributions will be observed in practice that do not adequately fit into any of these groups, e.g. concentrated peak pressures under first and fifth metatarsals only. In the cluster analysis, such distributions were not observed, even for higher number of clusters. The smallest group for k=7 contained 35 feet, corresponding to 4.3% of the analyzed sample. Outlying peak pressure patterns should be expected less frequently than this, and when they are observed in the foot clinic, a patient-specific insole and/or shoe design would be needed.

A major limitation of the proposed approach is the assumption that plantar pressure distributions can be obtained for all patients. Currently, such measurements are available only in specialized clinics. As the necessary instrumentation becomes more widely available, this approach to classification can be used to guide a physician towards appropriate prescription of footwear for the patient. Other limitations are related to the selection of regions. There is presently no standardized method for determining the number of foot regions, and different masking patterns have been used in the literature (e.g. Cavanagh et al., 1987; Bennett and Duplock, 1993; Putti et al., 2008). We chose to summarize the whole time history of plantar pressure distribution by determining the peak pressures in seven regions. Using fewer regions, e.g. forefoot, midfoot, and rearfoot, would significantly reduce the practical utility of the data for footwear design. Increasing the number of regions would not likely identify additional peak pressure clusters. Nonetheless, the analysis can be repeated using different region definitions, if this approach is inadequate for prospective studies. Another concern with masking is related to regional peak pressures sometimes being at the border between two regions. This situation may cause variability of peak pressures in neighboring regions (Pataky et al., 2008) and may result in the peak pressure being assigned to the wrong region. Rather than using masking, high-resolution techniques using pixel level pressure analysis can be utilized (Pataky and Goulermas, 2008), followed by k-means analysis with data points for feet including peak pressures for each pixel. However, such techniques require image registration to associate points between pressure distributions at which the raw data is interpolated, increasing the computational cost and potentially weakening the relationship with measured pressure distributions, which may alter their classification.

Clustering analysis has been applied to classify other biomechanical data, including gait patterns (Giacomozzi et al., 2009; Sawacha et al., 2010; Watelain et al., 2000) and muscle function (Vardaxis et al., 1998; White and McNair, 2002). Specific to diabetic foot pressures, the prior work of Waldecker (2012) and See et al. (2010) focused on the identification of ulcer risk and patient classification. A previous study (De Cock et al., 2006) investigated plantar pressures during jogging on a limited number of foot regions with the number of clusters set to four. Our investigation is the first to apply a comprehensive clustering and classification analysis to regional peak plantar pressures of diabetic feet. K-means clustering provided independent classification for different numbers of clusters, which provides the reader the opportunity to select an appropriate number of clusters based on the constraints of their specific application. The k-means solutions provided in this study can be used in applied problems of foot biomechanics other than footwear design, such as those targeted at understanding population specific foot deformities (Schmiegel et al., 2008) when they result in a redistribution of regional peak pressures. It is also possible to extend the analysis to other regional pressure variables such as pressure-time integral (Melai et al., 2011) and to explore different machine learning methods such as support vector machines (Cristianini and Shawe-Taylor, 2000) if the outcome and accuracy of k-means clustering is found to be inadequate.

Supplementary Material

01

Acknowledgments

This study was supported by the National Institutes of Health (Grant # 5R01 HD037433). The authors would like to acknowledge efforts by Chris Borish for masking foot regions and the members of the Diabetic Foot Care Program at the Cleveland Clinic who assisted with the data collection.

Appendix

Cluster means (standard deviations) for a range of total number of clusters (2 ≤ k ≤ 10). The clusters numbers (provided in the # column) correspond to the order in Figure 3 (from left to right), which also includes a visualization of the means. All values are in kPa. MTH = metatarsal head.

k # Hallux Toes 2–5 MTH 1 MTH 2–4 MTH 5 Midfoot Heel
2 1 389 (259) 189 (130) 543 (293) 633 (245) 459 (278) 277 (204) 351 (223)
2 270 (174) 137 (90) 249 (131) 325 (123) 240 (145) 195 (110) 239 (111)
3 1 528 (270) 215 (131) 602 (300) 548 (251) 251 (130) 246 (176) 315 (191)
2 271 (184) 151 (113) 340 (196) 599 (241) 670 (218) 290 (216) 355 (225)
3 241 (128) 131 (84) 243 (125) 320 (125) 222 (114) 190 (100) 234 (107)
4 1 692 (199) 224 (141) 322 (164) 449 (204) 263 (133) 222 (121) 290 (154)
2 319 (194) 181 (110) 785 (217) 588 (275) 278 (172) 253 (204) 318 (208)
3 258 (157) 156 (114) 302 (162) 576 (234) 657 (225) 288 (207) 353 (217)
4 228 (107) 127 (81) 237 (113) 316 (123) 216 (108) 189 (101) 232 (107)
5 1 682 (193) 222 (141) 307 (145) 434 (182) 268 (142) 220 (121) 296 (160)
2 358 (223) 189 (106) 798 (209) 414 (181) 226 (154) 183 (86) 242 (108)
3 294 (198) 175 (110) 629 (269) 873 (222) 416 (240) 327 (256) 453 (285)
4 246 (132) 149 (113) 271 (138) 478 (155) 631 (209) 271 (193) 300 (159)
5 224 (105) 126 (79) 234 (107) 314 (124) 203 (93) 190 (103) 231 (109)
6 1 730 (180) 224 (154) 291 (137) 392 (166) 273 (138) 233 (124) 297 (162)
2 405 (226) 199 (104) 804 (209) 412 (187) 211 (128) 182 (83) 223 (92)
3 303 (205) 175 (110) 651 (263) 866 (222) 406 (217) 334 (259) 463 (287)
4 247 (112) 141 (90) 314 (117) 464 (137) 251 (112) 246 (167) 309 (143)
5 252 (155) 157 (126) 276 (165) 493 (189) 730 (197) 266 (181) 305 (166)
6 223 (107) 123 (74) 204 (93) 266 (82) 206 (106) 166 (56) 200 (70)
7 1 776 (176) 207 (110) 297 (165) 384 (154) 278 (141) 223 (97) 304 (171)
2 323 (185) 174 (103) 737 (207) 357 (127) 205 (130) 186 (90) 237 (109)
3 379 (223) 190 (119) 652 (286) 899 (180) 351 (192) 216 (115) 350 (214)
4 253 (117) 151 (97) 275 (112) 462 (121) 315 (129) 198 (87) 269 (110)
5 258 (169) 163 (134) 283 (177) 521 (222) 818 (181) 230 (123) 278 (115)
6 241 (149) 179 (184) 451 (278) 500 (188) 420 (242) 716 (246) 641 (284)
7 226 (113) 119 (73) 210 (92) 255 (75) 182 (84) 177 (73) 208 (85)
8 1 762 (173) 209 (110) 280 (137) 377 (147) 282 (138) 222 (97) 305 (170)
2 400 (252) 196 (120) 917 (192) 583 (280) 269 (171) 196 (86) 278 (156)
3 266 (122) 146 (94) 429 (125) 355 (117) 175 (75) 196 (101) 271 (127)
4 341 (178) 183 (136) 375 (162) 792 (170) 307 (156) 227 (132) 292 (116)
5 224 (104) 135 (94) 209 (83) 380 (92) 374 (108) 201 (84) 244 (87)
6 254 (167) 162 (131) 306 (183) 525 (209) 794 (190) 270 (163) 309 (135)
7 286 (197) 187 (136) 587 (288) 573 (259) 450 (276) 760 (279) 792 (303)
8 219 (109) 119 (67) 183 (65) 246 (79) 163 (59) 169 (72) 197 (78)
9 1 771 (182) 211 (108) 292 (163) 381 (145) 282 (139) 223 (97) 305 (170)
2 293 (171) 163 (90) 679 (181) 329 (113) 196 (126) 172 (76) 215 (88)
3 413 (235) 216 (136) 876 (227) 791 (236) 325 (173) 216 (107) 350 (209)
4 228 (102) 135 (92) 312 (105) 383 (102) 222 (94) 262 (122) 337 (125)
5 316 (156) 164 (91) 335 (130) 752 (175) 292 (145) 202 (106) 280 (111)
6 206 (99) 120 (82) 208 (100) 376 (97) 464 (112) 188 (88) 217 (68)
7 277 (183) 177 (143) 302 (182) 573 (219) 845 (186) 264 (154) 330 (145)
8 310 (204) 220 (223) 560 (285) 576 (262) 459 (279) 797 (257) 761 (317)
9 240 (118) 127 (76) 197 (78) 257 (84) 170 (64) 160 (51) 190 (53)
10 1 836 (165) 236 (132) 328 (187) 402 (155) 302 (149) 242 (114) 329 (189)
2 334 (188) 186 (103) 806 (193) 361 (132) 230 (149) 189 (89) 250 (136)
3 377 (232) 182 (111) 820 (229) 898 (177) 346 (181) 223 (118) 304 (151)
4 189 (77) 122 (79) 364 (118) 378 (119) 200 (90) 249 (114) 321 (133)
5 423 (98) 168 (99) 259 (109) 332 (109) 182 (73) 171 (79) 224 (75)
6 358 (163) 194 (140) 314 (126) 784 (167) 312 (167) 226 (141) 295 (120)
7 214 (102) 124 (86) 218 (99) 390 (101) 456 (107) 198 (102) 230 (82)
8 256 (163) 173 (145) 302 (182) 554 (217) 854 (184) 264 (157) 328 (147)
9 277 (202) 180 (143) 604 (303) 560 (256) 464 (283) 785 (289) 830 (305)
10 182 (78) 114 (60) 179 (67) 246 (82) 179 (69) 161 (51) 188 (54)
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Footnotes

Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Conflict of Interest

Ahmet Erdemir owns and operates innodof, LLC, a computational modeling and simulation company. Peter R. Cavanagh holds equity in DIApedia, LLC, and is a consultant for Langer, UK.

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