## Abstract

Total hip arthroplasty (THA) significantly improves the quality of life in majority of patients with severe osteoarthritis. However, long-term outcomes of THAs are compromised by aseptic loosening and periprosthetic osteolysis which needs revision surgery. Both of these are causally linked to a prosthetic wear deliberated from the prosthetic articulating surfaces. As a result, there is a need to measure the mode and magnitude of wear. The paper evaluates three optical methods proposed for construction of a device for the non-contact prosthetic wear measurement. Of them, the scanning profilometry achieved promising combination of accuracy and repeatability. Simultaneously, it is time efficient to enable the development of a sensor for wear measurement.

© 2009 Optical Society of America

## 1. Introduction

The increased rate of revision total hip arthroplasties is associated with great economic burden on any health care systems [1]. Previously it was found that polyethylene wear particles initiate cascades of events at the bone-implant interface that result in development of aseptic loosening and osteolysis [2]. There was observed that total hip prostheses with lower wear rates have had lower risk for failure from these reasons than those with higher wear rate [3]. This point is of central significance in developing and improving both the material features of bearing surfaces and methods for quantifying wear.

Need for wear measurement has encouraged the development of non-radiologic (*in vitro*) and radiologic (*in vivo*) methods for polyethylene wear quantification [4, 5]. The realization of any type of *in vivo* measurement is based on geometric principles and brings with it the necessity of many simplifications and compromising steps to acquire approximately precise values. In fact, the volumetric wear can be obtained only by mathematical conversion using the most linear shift of femoral head in the cup. Taken this into mind, these approaches can be considered as intrinsically insufficient. In addition, the determination of linear wear under low-wear condition such as is seen in hard prosthetic surfaces may be questionable.

As the first step to overcome these problems in retrieved cups the methodology based on determination of nine space coordinates on the surface of prosthetic balls captured inside the polyethylene cups was developed by us (See Sec.2.1). However, this procedure is time-consuming and does not fully address such problems of prosthetic wear measurement as low-wear condition, complex wear mode or wear anisotropy. Therefore, other approaches were requested to overcome these issues. The evaluation of the prosthetic wear can be performed employing some optical 3D methods. The principle of wear measurement is based on the determination of the 3D profile map of retrieved cup. The full wear map is obtained by comparing shape of the post-used cup with the shape of original one. Such approach gives primary the information about the magnitude of total wear but also about the contribution of wear what is useful to determine the various other parameters mentioned above. Moreover, the measurement is noncontact and time-sparing.

The purpose of this paper was to determine which of the three different optical methods is the most suitable for construction of a sensor for prosthetic wear measurement.

## 2. Description of methods

#### 2.1 Universal measuring microscope

Briefly, linear wear, defined as the maximum penetration of the prosthetic head into the polyethylene liner, was determined using a previously reported method that measures the shift of the center of the prosthetic head from the manufactured to the post-use position [4, 6]. Thus, the measurement reliability and accuracy strongly depends on the accuracy of that positioning.

With the help of the Universal-type measuring microscope (VEB, Carl Zeiss Jena, Germany) the 2×9 three-dimensional coordinates on the surface of the prosthetic ball in both positions are determined – Fig. 1(a). Assuming that the diameter of the cup and the head are known, it is possible to calculate the linear and volumetric wear using a geometrical “two-sphere model” and a computational algorithm. The influence of two different wear angles on the accuracy of measurement is shown on the Fig. 1(b).

The accuracy of the described method was previously assessed in a set of 30 retrieved cups, ranging between 1 to 4 µm and 1 to 9 mm^{3} for linear and volumetric wear, respectively [4]. Despite that accuracy and repeatability of this method seems to be satisfactory, it is very time-consuming and highly dependent on the magnitude and direction of measured prosthetic wear.

#### 2.2 Fourier profilometry

The ground of the first non-contact method, called Fourier (or phase-shifting sometimes) profilometry [7–9], is the sinusoidal grating projector which creates the optical pattern on the measured surface as follows from Fig. 2(a).

The grating is deformed by the surface 3D shape which is reflected in local changes of its phase. This pattern is recorded by the digital camera; resulting image is shown on the Fig. 2(b). Information about the profile of the object is computed using the standard image processing and special mathematical algorithm. The example of resulting profile map of the measured surface is presented on the Fig. 2(c).

To descibe the method qualitatively the geometry of experimental set-up is introduced on the Fig. 3. First, the sinusoidal grating is projected onto the reference plane placed in front of the measured object. This plane defines the topographic plane of measurement. The Fourier phase *Φ*
_{0}(*x,y*) of the recorded image can be computed using the standard mathematical algorithms. Replacing the reference plane the same sinusoidal grating is projected onto the measured surface and recorded again. The Fourier phase *Φ* (*x,y*) of the recorded image is computed again.

The change of phase *ΔΦ* (*x,y*)=*Φ* (*x,y*)-*Φ*
_{0}(*x,y*) is called the phase-shift. It contains the information about the topographic depth *W* (*x,y*) which is in fact the distance of the measured object surface and the reference plane in the given point (*x,y*). The relationship between the unwrapped phase-shift and topographic depth can be derived [9] in form

where *l* is the distance of the camera and reference plane, *p* is the projected sinusoidal grating period and *d* is the distance between the camera and projector according the Fig. 3.

The cup profile a rugged – very flat at the center and very steep around the edge. It is hard to project and record whole-field grating in sufficient quality. The grating around the center of cup is over-exposed and, simultaneously, the gratings near the edge of cup in low-contrasting (See Fig. 2(a) again). Therefore, the type of profile of the cup is the main limitation factor of usability of this method for the wear measurement.

#### 2.3 Scanning profilometry

With respect to the implant shape the one strip projection is better to use instead of the grating projection. Method based on this principle is called the scanning profilometry [10–12]. Its principle is clear from Fig. 4(a).

The projected linear strip is deformed by the measured surface again. The profile in one direction is computed from this strip deformation. Such created linear strip is deformed by the measured surface and observed by a camera. The example of recorded image of deformed strip is shown on the Fig. 4(b). The cup or strip is rotated to cover the whole area of the measured surface. The magnitude of step influences the resolution and sensitivity of the method. The record and the analysis are repeated consequently. As a result a whole-field profile is obtained connecting the individual linear profiles. The resulting wired model of the cup is shown on Fig. 4(c). The usual data format of the result profile map can be obtained using the standard conversion of this format.

The first step to create the whole profile map of object is the profile measurement in one direction. The geometry is introduced on the Fig. 5(a).

The deformation of strip in pixels *u* (*x*) is determined as shown on the Fig. 5(b). The depth of profile *W* (*x*) in the given point *x* can be derived in form

or, applying common mathematical approximations, in linear form

where *a,b* are parameters of mapping algorithm which depends on the geometrical parameters of the experimental setup introduced on Fig. 5(b). The parameters of sensitivity *c*
_{1} and *c*
_{2} can be determined from Eqs. (2), (3). The detailed derivation of these equations and its dependence on the geometry is given in [11].

The digital processing of only the one linear strip is not so complicated in comparison with the 2D grating analysis. On the other hand the measurement is not the real-time because of the positioning (rotating) the measured cup (or projected) and recording and processing the number of strips. But the total duration of measurement can be improved to the acceptable short using the automation and synchronization of the rotating, recording and data processing.

## 3. Model for validation of wear measurement method

As the total wear is not measured directly but it is computed from some measured quantities the validation of wear measurement process was tested using the mechanical simulation of wear. Several new PE cups of the same diameter (Chirulen, Aesculap, Tuttlingen, Germany) were milled to simulate the conditions similar to those observed *in vivo* with the various magnitude and direction of the wear. The principle of simulation is shown on the Fig. 6.

First, the milling parameters had to be chosen before the mechanical simulation starts. Because the direction of polyethylene wear depends *in vivo* on so-called abduction angle of the cup, we decided to test all measurement methods under the different angles of wear direction. A milling cutter was set to obtain wear perpendicular (α=0°) and oblique to the mouth of the cup (directions of 30° and 50°). For each of this direction, the deepness of milling was chosen to simulate three clinically important values of the total wear – low, medium and high wear conditions. The choice was carried out based on the estimation of resulting simulated values of wear using theirs computation. The chosen milling parameters and the expected computed values of wear are summarized in the Tab.1.

Cups prepared by such a way were measured separately employing the previously mentioned methods and the total wear values were determined. Validation was based on comparison of these values with gold standard. Here, gravimetry was considered as a gold standard. Briefly, removed mass of polyethylene was weighted before and after milling and the difference of weighs was equal to the wanted wear condition after involving the known density of polyethylene (944 kg per cubic metr).

## 4. Experimental data and results

All nine cups with various amount of simulated “wearing” were measured as described above. The results of gravimetric method are presented in the Tab.1. As one can see there are the discrepancies between expected computed values of total wear and ones measured by the gravimetric method. The reason for the difference is most probably the limited accuracy of the milling technique.

However, the computed values of total wear served just for the estimation of result values before milling. Only the gravimetric method data were taken into account as reference values for the comparison between wear data obtained for each of the methods under study. Absolute values of wear measured by all three methods under study are included in Tab.2. Relative ones compared with gravimetry values are introduced in Tab.3.

Two important conclusions follow from previous lists of results. First, the measuring microscope method gives worse or no results under the condition of small angle. From the principle of method, the lower the angle is the smaller part of original non-weared surface is available to capture the prosthetic ball inside the cup in one of the two positions. Second, the dispersion of results of both phase-shifting and scanning profilometry is in range of 10 percent. The relative differences can be taken as the estimation of accuracy of given method. Less than 10 percent accuracy seems to be sufficient for planed purpose of the measurement.

By a similar way, the repeatability can be be estimated using the cup with known wear. One of the cups was measured repeatedly under the various adjustments of experimental set-up. The starting position of rotation of the measured cup and the observation angle of the camera were changed. The results of measurement are shown in Tab.4. Results from 1 to 3 were obtained under the condition of the various starting cup rotation angle (randomly chosen) and the constant observation angle adjusted to the value of 40 degrees. In the others, the starting angle remained unchanged (the same as in the case 3) while the observation angle was settled to values of 25 and 55 degrees.

Concerning repeatability (here expressed in terms of estimation using the standard deviations), the scanning profilometry was more acceptable than the Fourier profilometry. Worse estimated value of repeatability of this method is explained by the whole-field grating projection to unsuitable surface profile (as described above). Despite the correcting algorithms applied for the image processing influence the resulting value of wear.

The suitability of scanning profilometry for the sensor construction is supported also by values of the total measurement duration. These involve the time for the preparation of measurement, actual measurement and data processing. The measurement by microscope was performed manually and the duration of main part – actual measurement – took approximately 30 minutes. As the process cannot be automated, the total duration cannot be decreased. In the case of Fourier profilometry, the data processing consumes the main part of total duration. The processing algorithm needs several decision parts which have to be performed manually. The total time of measurement is takes several minutes. Only the measurement process of scanning profilometry can be fully automated, so the maximum total time is about one minute.

## 5. Discussion and conclusion

There are two important findings routed from our study. The first is pertinent to the better accuracy and repeatability of the scanning profilometry over the other methods under study under all tested conditions. The second is related to the time need to wear measurement and data processing which was also much shorter in the case of the scanning profilometry. Taken together, we believe this approach opens the avenue to the automation of the whole measurement procedure which is needed before both starting any large clinical study (scientific reasons) and attest of new materials developed as a bearing surface in THA (technological and legal reasons).

The variations in wear mode and increased usage of contemporary hard bearing surfaces in clinical practice justifies further research in the field of wear measurement even under *in vitro* conditions. Previously, several approaches were developed to measure volumetric wear based on measuring of the residual thickness of the retrieved polyethylene liner [13], determining of coordinates on the surface of the prosthetic ball or cup [4], molding and use of the shadowgraph [14] or detection of fluid displacement [15]. Some of them rely on the determination of true position of the prosthetic ball in the post-use contour in the cup which should be different from the original one. Many of above procedures need mathematical apparatus to convert linear on volumetric data which may contribute to underestimation of the true wear [16]. The others are too impractical or time-consuming for practice to allow their wide-spread using (e.g., gravimetry and fluid-filling). For instance, in case of gravimetry, every liner had to be weighted before any surgery to enable precise weighting after the revision many years later. In comparison to all mentioned drawbacks, optical profilometry offers many advantages when combining high accuracy with low repeatability, rapidity and chance for automation.

In conclusion, the results of our study supports the choice of scanning profilometry as a suitable measurement tool for construction of the sensor for automated measurement of retrieved cups. The automation of rotation and the coordination with recording by the camera will be carried out to decrease the total duration of the whole measurement process. Parameters of the constructed sensor can be determined using the same procedure as described here. Based on it, higher number of retrieved or tested cups will be possible to measure which may be useful for clinicians and as well as for R&D researchers.

## Acknowledgment

These results of project OC168 was supported by the Min. of Education of the Czech Rep. These results of project 1M06002 was supported by the Min. of Education of the Czech Rep.

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