# Evaluating the Surface Topography of Pyrolytic Carbon Finger Prostheses through Measurement of Various Roughness Parameters

^{1}

^{2}

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## Abstract

**:**

_{a}); root mean-square roughness (S

_{q}); skewness (S

_{sk}); and kurtosis (S

_{ku}). The radii of the articulating surfaces were measured using a coordinate measuring machine, and were found to be: 2.5, 3.3, 4.2 and 4.7 mm for proximal, and 4.0, 5.1, 5.6 and 6.3 mm for medial components. ANOVA was used to assess the relationship between the component radii and each roughness parameter. S

_{a}, S

_{q}and S

_{sk}correlated negatively with radius (p = 0.001, 0.001, 0.023), whilst S

_{ku}correlated positively with radius (p = 0.03). Ergo, the surfaces with the largest radii possessed the better topographical characteristics: low roughness, negative skewness, high kurtosis. Conversely, the surfaces with the smallest radii had poorer topographical characteristics.

## 1. Introduction

_{a}) and root mean-square roughness (S

_{q}) have both been calculated from the topographical plots detailed in this manuscript, two additional parameters have been calculated: skewness (S

_{sk}) and kurtosis (S

_{ku}). S

_{sk}is a measure of height distribution about the profile line: positive S

_{sk}indicates the presence or more peaks; negative S

_{sk}indicates the presence of more valleys, and S

_{sk}≈ 0 indicates symmetrical height distribution about the profile line. S

_{ku}is a measure of how densely or sparsely peaks and valleys are distributed across the measured surface. Methods of calculation and further description of these parameters are covered in the experimental section of this manuscript. A previous tribological study with an industrial emphasis [16] has documented that all four of these topographical parameters have been obtained simultaneously from different grades of 100Cr6 steel samples. To the knowledge of the authors, this present study is the first of its kind to consider all four of these parameters concurrently, from the same topographical plots, to evaluate the articulating surfaces of joint prostheses.

_{a}; S

_{q}; S

_{sk}; and S

_{ku}.

## 2. Results and Discussion

#### 2.1. Topographical Results

_{a}values of: 69.2 nm (95% CI 63.4–74.8 nm) and 68.1 nm (95% CI 62.1–74 nm), respectively (Table 1), both significantly higher than the 50 nm threshold dictated by ISO 7206-2 [17] (p < 0.05). Conversely, the size 30 and 40 proximal components exhibited mean S

_{a}values of: 36.8 nm (95% CI 30.8–41.2 nm), and 25.9 nm (95% CI 23.6–28.3 nm), respectively, both significantly lower than the 50 nm threshold (p < 0.05). The mean S

_{q}values were higher than the S

_{a}values. The size 10 and 20 proximal components once again yielded results significantly higher than the 50 nm threshold (p < 0.05), with values of: 88 nm (95% CI 80.7–95.3 nm) and 87.9 nm (95% CI 80.5–95.2 nm), respectively. For the size 40 proximal component, the mean S

_{q}value also reflected the S

_{a}result, with a mean of 33.7 nm (95% CI 30.1–36.5 nm), significantly lower than the 50 nm threshold (p < 0.05). The size 30 proximal component, however, yielded a mean S

_{q}of 46.7 nm (95% CI 40.2–53.2 nm), close enough to the 50 nm threshold to accept the null hypothesis (p > 0.05). Mean S

_{a}and S

_{q}values for the medial components all fell below 50 nm. However, the size 10 and 20 medial components had mean S

_{q}values of: 46.4 nm (95% CI 34–58.7 nm), and 47.1 nm (95% CI 42.6–51.6 nm), respectively. These are close enough to 50 nm to accept the null hypothesis (p > 0.05).

_{sk}results fell within the recommended guidelines [18], specifying a skewness of no less than −1.5 and no greater than +1.5. The midpoint of this range was used as null hypothesis (S

_{sk}≈ 0) and was subsequently accepted in all but two instances (p > 0.05). The two instances where the null hypothesis was rejected (p < 0.05) was the size 40 proximal and medial components, which yielded S

_{sk}values of: −0.43 (95% CI −0.685 to −0.175); and −0.62 (95% CI −1.08 to −0.161) respectively. For all of the proximal and medial components, S

_{ku}significantly exceeded the recommended value of 3 [15] (p < 0.05). The size 40 proximal and medial components yielded considerably higher S

_{sk}values than the other proximal and medial components, with mean values of: 11.4 (95% CI 7.84–14.97); and 29.74 (95% CI 15.22 to 44.26) respectively. Furthermore, the medial components yielded higher S

_{sk}values than the proximal components.

_{a}and S

_{q}); and the higher observed S

_{sk}values. For the larger components, the topographical plots often exhibited valleys along the surface (Figure 3), providing an explanation for the low negative S

_{sk}values and high S

_{ku}values. In some cases, for the mid-size prostheses (size 20 and size 30); symmetrically distributed surfaces exhibiting a uniform Gaussian distribution were observed (Figure 4).

#### 2.2. Linear Regression and ANOVA

_{a}and S

_{q}with respect to the measured radii of the articulating surfaces. The average roughness model (Figure 5a) exhibited a negative correlation between S

_{a}and radius, with a R

^{2}value of 0.86. Similarly, the root mean-square roughness model (Figure 5b) also exhibited a negative correlation between S

_{q}and radius, with a R

^{2}value of 0.87. Furthermore, ANOVA (Table 2) showed these relationships to have a high significance (p = 0.001 for both models). The skewness model (Figure 6a) also exhibited a negative correlation between S

_{sk}and radius. Although this model had a weaker fit, with an R

^{2}value of 0.6, the trend was still significant (p = 0.023). The kurtosis model (Figure 6b) exhibited a positive correlation between S

_{ku}and radius. As with the skewness model, the fit was also weaker, with a R

^{2}value of 0.57, yet the trend still had a high level of significance (p = 0.03).

#### 2.3. Prior Assessment of Joint Prostheses of the Hand

#### 2.4. What Are the Most Appropriate Topographical Parameters?

_{a}and S

_{q}. In this present study, both parameters exhibited similar trends with respect to prosthesis size/radius. The main difference is the higher magnitude attributed to the S

_{q}parameter, on account of it being a root mean-square calculation. Both 2D (R

_{a}) and 3D (S

_{a}) average roughness values are accepted to provide a good gauge for the variation in heights obtained from a topographical plot; however, these parameters are not sensitive to small changes in profile on the surfaces [15,16]. Both the 2D (R

_{q}) and 3D (S

_{q}) root mean-square values are more sensitive to variation in heights, providing larger estimates for roughness than either R

_{a}or S

_{a}. Root mean-square roughness is more sensitive to heights than average roughness, yet still does not provide a detailed description of the surface [15,16]. Furthermore, the standard threshold of 50 nm is recommended only for average roughness, with no mention of root mean-square roughness [17]. From this, it can be argued that S

_{a}is a more appropriate way of quantifying surface roughness than S

_{q}.

_{ku}values documented in this article, with no comparative data available. All of the measured S

_{ku}values are significantly greater than the threshold of 3 (p < 0.05), which indicates that there are many sharp peaks or valleys formed on the surface. Furthermore, the kurtosis model (Figure 6b) indicated that S

_{ku}increased with prosthesis radius. It is also pertinent to mention that the S

_{sk}guidelines [18] (skewness of no less than −1.5 and no greater than +1.5), which was used to govern the null hypothesis for S

_{sk}z-tests (Table 1), was based on the assumption that too many peaks or valleys is detrimental to the surface, and that S

_{sk}= 0 ± 1.5 is the ideal. The null hypothesis was accepted for size 10; size 20; and size 30 measurements. The null hypothesis was rejected for the size 40 components on account of lower negative S

_{sk}values attributed to the components.

_{sk}(−3.11) and the highest S

_{ku}(23.2) yielded a very low coefficient of friction (0.1), which is indicative of a mixed of lubrication regime [20]. It has been proposed that the characteristic sharp valleys associated with high kurtosis and negative skewness (Figure 7) provide nanoscale reservoirs for lubricant [21]. It has also been noted that these reservoirs are too small to act as traps for typical wear particles ranging from 10 to 100 µm [16]. Hence, the combination of negative skewness and high kurtosis observed for the size 40 prosthesis (Figure 3) is advantageous.

## 3. Experimental Section

#### 3.1. Roughness Parameters

_{a}) (Equation (1)) is a good gauge for the variation in heights about the profile line. However, two highly cited studies, Gadelmawla et al. [15] and Sedlaček et al. [16], have stated that average roughness is not sensitive to small changes in profile:

_{q}) (Equation (3)) is more sensitive to variation in heights than average roughness, yet still does not provide a detailed description of the measured surface:

_{sk}) (Equation (3)) was also obtained. Skewness provides a more detailed description of the measured surface than both average roughness and root mean-square roughness. It is described as being sensitive to sporadic deep valleys or high peaks [15,16]. Positive skewness indicates the presence or more peaks, negative skewness indicates the presence of more valleys, and zero skewness indicates symmetrical height distribution about the profile line. Further to this, Gadelmawla et al. [15] has stated that skewness can be used to distinguish between dissimilar surfaces that have the same average roughness or the same root mean-square roughness values:

_{ku}) (Equation (4)) also provides a more detailed description of the measured surface than both average roughness and root mean-square roughness. Kurtosis describes the sharpness of the probability density of a surface [15,16]. If S

_{ku}< 3, then the surface has relatively few high peaks and few low valleys. Conversely, if S

_{ku}> 3, then the surface has relatively many high peaks and many low valleys:

#### 3.2. Samples

^{2}. Although it is possible to use larger measurement windows for flat surfaces, the small radii of the prostheses limited the surface area that could be successfully measured. To compensate for this, twenty topographical measurements were taken from each prosthesis pair using the white light interferometer, ten per proximal/medial component. For each condyle and plateau evaluated, five topographical plots were obtained: one measurement taken at the centre; then four peripheral measurements taken, equispaced at 90°. As there were two prostheses for each nominal size, 160 measurements were taken in total.

#### 3.3. Statistical Methods

_{a}and S

_{q}. For S

_{sk}, the hypothesised proportion was taken as the midpoint of −1.5 and +1.5 (zero), with care taken to note any outliers below −1.5 or above +1.5, as these scenarios indicate the presence of deep valleys and high peaks respectively [18]. For S

_{ku}, the hypothesised proportion was taken as 3—below this indicates the presence of relatively few high peaks and low valleys, above this indicates the presence of relatively many high peaks and low valleys [15,16]. Aside from sample statistics, linear regression and analysis of variance (ANOVA) were both used to determine the strength and validity of the relationships between the parameters and prosthesis radii.

## 4. Conclusions

_{a}values; the lowest S

_{q}values; low negative S

_{sk}values; and high S

_{ku}values. Conversely, prostheses with the smallest radii have exhibited inferior topographical properties. S

_{a}and S

_{q}exhibited very similar trends with respect to prosthesis radius. It is proposed here that the use of only one of these two parameters would be sufficient to gauge 3D roughness. S

_{a}would appear to be the better of the two as it is the standard value used to assess the roughness of joint prostheses [17]. This study has established relationships between prosthesis size (radius) and topographical properties (S

_{a}, S

_{q}, S

_{sk}, S

_{ku}). It is clear that the smaller pyrolytic carbon components are rougher than the larger components.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Topographical plot showing a surface with peaks (

**a**) and histogram exhibiting a positively skewed distribution (

**b**), obtained from medial component size 10.

**Figure 3.**Topographical plot showing a surface with valleys (

**a**) and histogram exhibiting a negatively skewed distribution (

**b**), obtained from proximal component size 40.

**Figure 4.**Topographical plot showing a flat surface (

**a**) and histogram exhibiting a uniform Gaussian distribution (

**b**), obtained from medial component size 30.

**Figure 5.**Average roughness linear regression model (

**a**) and root mean-square linear regression model (

**b**).

**Figure 7.**A 2D profile plot (

**a**) and histogram (

**b**) taken across a series of valleys demonstrating high 2D kurtosis (R

_{ku}= 27.69) and a negative skewness (R

_{sk}= −3.62).

**Table 1.**Data summary from statistical hypothesis test displaying: 95% confidence intervals; z-scores; and p-values for the respective sample means of S

_{a}, S

_{q}, S

_{sk}, and S

_{ku}.

Parameter | Size | Proximal Component | Medial Component | ||||
---|---|---|---|---|---|---|---|

Mean (95% CI) | z-Score | p-Value | Mean (95% CI) | z-Score | p-Value | ||

S_{a} (nm) | Size 10 | 69.2 (63.4–74.8) | 6.52 | 0.00 | 34.9 (25.6–44.1) | −3.20 | 0.00 |

Size 20 | 68.1 (62.1–74) | 5.96 | 0.00 | 37.4 (33.9–40.9) | −7.10 | 0.00 | |

Size 30 | 36.8 (30.8–41.2) | −5.30 | 0.00 | 18.1 (16.4–19.8) | −36.10 | 0.00 | |

Size 40 | 25.9 (23.6–28.3) | −20.10 | 0.00 | 13 (11.9–14) | −71.10 | 0.00 | |

S_{q} (nm) | Size 10 | 88 (80.7–95.3) | 10.26 | 0.00 | 46.4 (34–58.7) | −5.80 | 0.56 |

Size 20 | 87.9 (80.5–95.2) | 10.10 | 0.00 | 47.1 (42.6–51.6) | −1.27 | 0.20 | |

Size 30 | 46.7 (40.2–53.2) | −0.99 | 0.32 | 23.4 (21.3–25.5) | −24.87 | 0.00 | |

Size 40 | 33.7 (30.1–36.5) | −11.32 | 0.00 | 17 (15.7–18.3) | −48.77 | 0.00 | |

S_{sk} | Size 10 | −0.036 (−0.149–0.078) | −0.62 | 0.53 | −0.342 (−0.646 to −0.038) | −2.20 | 0.05 |

Size 20 | 0.024 (−0.176–0.128) | 0.31 | 0.76 | 0.036 (−0.126 to 0.182) | 0.44 | 0.66 | |

Size 30 | −0.038 (−0.148–0.072) | −0.68 | 0.50 | −0.222 (−0.486 to 0.046) | −1.62 | 0.10 | |

Size 40 | −0.43 (−0.685 to −0.175) | −3.31 | 0.00 | −0.62 (−1.08 to −0.161) | −2.65 | 0.01 | |

S_{ku} | Size 10 | 4.65 (4.23–5.06) | 8.20 | 0.00 | 11.86 (7.37–16.34) | 3.87 | 0.00 |

Size 20 | 4.77 (4.1–5.44) | 5.210 | 0.00 | 4.16 (3.73–4.59) | 5.340 | 0.00 | |

Size 30 | 4.16 (3.72–4.59) | 5.310 | 0.00 | 15.61 (8.94–22.28) | 3.710 | 0.00 | |

Size 40 | 11.4 (7.84–14.97) | 4.620 | 0.00 | 29.74 (15.22–44.26) | 3.610 | 0.00 |

Parameter | Source of Variation | Degrees of Freedom | Sum of Squares | Mean Square | F-Statistic | p-Value |
---|---|---|---|---|---|---|

S_{a} (nm) | Regression | 1 | 2632 | 2632 | 37 | 0.001 |

Error | 6 | 426 | 71 | – | – | |

Total | 7 | 3058 | – | – | – | |

S_{q} (nm) | Regression | 1 | 4303 | 4303 | 39 | 0.001 |

Error | 6 | 660 | 110 | – | – | |

Total | 7 | 4963 | – | – | – | |

S_{sk} | Regression | 1 | 0.215 | 0.215 | 9.1 | 0.023 |

Error | 6 | 0.142 | 0.204 | – | – | |

Total | 7 | 0.356 | – | – | – | |

S_{sk} | Regression | 1 | 311 | 311 | 7.8 | 0.03 |

Error | 6 | 243 | 39 | – | – | |

Total | 7 | 546 | – | – | – |

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**MDPI and ACS Style**

Naylor, A.; Talwalkar, S.C.; Trail, I.A.; Joyce, T.J.
Evaluating the Surface Topography of Pyrolytic Carbon Finger Prostheses through Measurement of Various Roughness Parameters. *J. Funct. Biomater.* **2016**, *7*, 9.
https://doi.org/10.3390/jfb7020009

**AMA Style**

Naylor A, Talwalkar SC, Trail IA, Joyce TJ.
Evaluating the Surface Topography of Pyrolytic Carbon Finger Prostheses through Measurement of Various Roughness Parameters. *Journal of Functional Biomaterials*. 2016; 7(2):9.
https://doi.org/10.3390/jfb7020009

**Chicago/Turabian Style**

Naylor, Andrew, Sumedh C. Talwalkar, Ian A. Trail, and Thomas J. Joyce.
2016. "Evaluating the Surface Topography of Pyrolytic Carbon Finger Prostheses through Measurement of Various Roughness Parameters" *Journal of Functional Biomaterials* 7, no. 2: 9.
https://doi.org/10.3390/jfb7020009