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Frequency Content of Prosthetic and Orthotic Shapes: A Requirement for CAD/CAM Digitizer Performance

Jeffrey A. Hastings
William M. Vannah, PhD
Joseph A. Stand III
David M. Harning, CPO
David M. Drvaric, MD

ABSTRACT

The objective of this study was to determine the frequency content of the surface bumpiness in shapes encountered in prosthetic and orthotic practice. This information is important in evaluating digitizers for use in prosthetic and orthotic CAD/CAM; a digitizer's frequency response should be capable of recording the intricacies of a limb's shape without degradation. A digitizing system without adequate frequency response has results analogous to a loudspeaker that muffles high frequency notes.

Of the shapes commonly encountered, below-knee (BK) residua were found to have surface undulations of the highest frequency. Fifty-five BK limbs were digitized, and evaluated to determine the spatial frequency spectra of the surface. Shapes with known frequencies were also tested, in order to ensure proper performance of the algorithm. The frequency content of the BK limbs was limited to 10 cycles per circumference or lower.

Key Words: Prosthesis; Orthosis; Artificial Limb; Computer-Aided Design; Frequency Analysis.

Introduction

A digitizer is a device that records the geometric shape of an object. These devices are used to create computer models representing physical objects; e.g., the shape of a residual limb. Data produced by digitizers are used in the fabrication of sockets for prosthetic limbs and orthoses. This process is currently used in O&P facilities, and is known as computer-aided design/computer-aided manufacturing (CAD/CAM). One of the factors responsible for how well a CAD/CAM-produced socket fits is the accuracy of the digitizer.

Previous tests of digitizing systems have examined their effects on segmental and total volume (1-3). However, these tests are not the most direct measure of surface bumpiness. In particular, volume tests do not reflect changes to the shape of the limb that occur without the volume of the limb changing. Volume conservation reflects the summed contributions of the surface contours, but does not address the conservation of individual contours.

Most CAD/CAM systems allow processing the shape data through smoothing functions. The purpose of smoothing functions is to remove unwanted bumpiness from the shape, while retaining the characteristic curves and surface features. Bumpiness may be thought of as high frequency oscillations in the shape, and is analogous to high-frequency noise, or static, in an electrical signal. A smoothing function may be thought of as analogous to the "scratch" or "hiss" filter found in some high-fidelity music amplifiers.

Smoothing may become more critical with the recent development of hand-held digitizers (4,5)a,b,c, which may be inherently noisier than traditional pivoting-arm digitizers. Noise may be generated with a hand-held digitizer from at least three sources. First, the operator's hand makes unintentional movements (for example, shaking, uneven pressure against the shape). Second, the patient's limb moves relative to the orienting transducer (6) or limb restraint device (7,8). Third, random error in the digitizing transducer itself will produce noise. Transducers commonly used in hand-held applications typically have lower accuracy and more noise than pivoting-arm and laser-shape digitizers (4,5). Smoothing functions can remove this noise.

An ideal smoothing routine is one that removes noise without affecting the actual contours of the residual limb's shape. Various smoothing routines--for example, averaging filters, low-pass filters, spline fitting (9-11)--have different performances in regard to this ideal. To select and tune a given smoothing routine, one must know what the cut-off frequency is that determines whether a given surface undulation is noise or an actual surface contour. Selecting a cut-off frequency is necessarily an imprecise art. It's common to choose a frequency slightly on the high side to be safe from removing actual contours of the shape. Removing actual contours of the shape results in subtle degradation of socket fit, with resultant poorer mobility. A better understanding of what frequencies exist in the surface of the residual limb is needed to design appropriate smoothing functions. The study described here was conducted to develop an understanding of the spatial frequency content of shapes encountered in orthotics and prosthetics.

Methods

Data

We have been digitizing shapes for above-knee, proximal femoral focal deficiency, and below-knee sockets, and for various torso orthoses. Inspection of the frequency spectra of these shapes revealed the below-knee (BK) shapes consistently exhibited the most high-frequency content. Therefore, 55 pediatric and adult BK shapes were used as the data source. These shapes were digitized at Newington Orthotic and Prosthetic Systems, in Hartford, Conn., and at the corresponding institution.

Equipment

Pivoting-arm digitizers^d,e were used to record the shapes of cast impressions of the residua. The shapes were recorded in cylindrical coordinates with five degree angular increments (72 points per circumference) and five mm slice thickness. A typical recorded shape is shown in Figure 1 .

This surface can be flattened to form a Cartesian field of radii f(x,y), where the axes are vertical height (x) and tangential position (y). Figure 2 shows such a Cartesian field with the axes labeled.

This Cartesian data field was evaluated using a 2-D discrete Fourier transform (DFT), producing a 2-D spatial frequency spectra. The principle behind a Fourier transform is that any curve shape can be represented by a collection of sine and cosine curves, if we choose the frequencies and amounts of the various sine and cosine curves correctly. In applying a Fourier transform to socket shapes, the intent is to generate a quantitative description of the bumpiness of the shape. If a shape's transform requires large amounts of high-frequency sine and cosine curves to represent it, then that shape is said to have a lot of high-frequency bumpiness. The amounts F(u,v) of the various frequencies to use are obtained by passing the data through the following equation:

Again, the values of F(u,v) give the coefficients, or relative power, present at the various vertical (u) and tangential (v) frequencies. For instance, F(3,4) gives the relative amount of surface bumpiness that had four oscillations about the circumference and three oscillations in an equal vertical distance. M and N are the total number of x and y data values respectively and j is the square root of -1. Figure 3 shows the spatial frequency spectrum of the residual limb in Figure 1 .

The zero frequency is equivalent to the mean radius of the surface. Because the evaluation was based on frequencies occurring at the surface, the zero frequency value was removed. To remove the zero frequency the average radial value is removed from each radial component before the 2-D DFT was performed. It also should be noted that the higher peaks in the range of one and two peaks per circumference represent the offset of the central axes of the residua.

The 2-D frequency spectrum F(u,v) was converted to cylindrical coordinates F(r,q,z); where r is equal to the magnitude from the point to the origin, q is the angle from the x-axis, and z is the height at each r and q. The average z value at a constant r was calculated, and divided by the square root of two, to produce an average frequency distribution plot, shown in Figure 4 .

After each sample's frequency distribution was determined, the average frequency distribution across the 55 residua was calculated.

Verification of Method

Various tests were conducted to verify the correct functioning of the computer programs. In one test, a lobed plaster column was used. This column was carved with sinusoidal oscillations running in the vertical and tangential directions. The column had a base diameter of 101.6 mm with a 3.33 mm amplitude sine wave superimposed to either side of the base diameter. The frequency of the sine wave was such that 10 cycles took place about the circumference (101.6p mm) of the column. The vertical spacing of the sine wave was identical--that is, 10 cycles in a distance equivalent to one circumference of the column. The 3.33 mm amplitude corresponded to the largest amplitude oscillations machinable with the 18 mm diameter round-end mill. The construction of this type of lobed test column has been more fully described elsewhere (5). This column was digitized in an identical manner as the BK socket molds. The shape record was then passed through the DFT algorithm. The resulting frequency spectrum in Figure 5 shows significant power only at 10 cycles/circumference.

Results

The average power spectrum is shown in Figure 6 and Figure 7 . The bulk of the surface undulations were below seven cycles/circumference, and there were virtually none above 10 cycles/circumference. The standard deviation (Figure 6) , which shows the variability of the individual spectra, becomes negligible above seven cycles/circumference.

Discussion

The objective of the study was to determine the maximum spatial frequency of common prosthetic and orthotic shapes. BK residual limbs were observed to possess the greatest amount of high-frequency content, and for these shapes it was found that virtually all the power in the spatial frequency spectrum occurred at or below 10 cycles per circumference. Thus, 10 cycles per circumference may serve as a cut-off frequency between real surface contours and noise for commonly-encountered shapes.

It should be noted that selection of this a cut-off frequency choice is not a precise matter but a reasoned judgment. The power spectrum tapers very gradually, and there is not a strong argument against choosing a somewhat lower cut-off frequency. Also, the shapes have differing frequency contents in different directions. For example, Figure 6 shows different spectra along the vertical and tangential axes. Ten cycles per circumference should be considered a conservative upper bound.

Plaster casts often carry additional roughness beyond that present in the limb itself. We deleted one sample from our database because a large vertical crease in the plaster was evident in the digitized shape record. Inclusion of this sample would have increased the high-frequency power, artificially beyond that present in the limb shape itself. A number of other casts which had less drastic artificial bumpiness were left in the database. Thus, the mean power spectrum presented here (Figure 6 and Figure 7 ) may overestimate the actual mean spectrum of the limb shapes themselves. Again, 10 cycles per circumference may be conservatively high.

This analysis suggests the accuracy of new digitizer designs should be verified, specifically including the effects of the accompanying smoothing function, before releasing them for clinical use. Excessive smoothing lessens the actual contours of the residuum; taken to an extreme, the eventual shape is an upright cylinder of uniform radius. The precise effects of small losses of contour in a prosthetic socket or orthosis are not known; it is possible that a subtle degradation in fit will occur. CAD/CAM systems already have been criticized as being capable of fast production of mediocre-fitting prosthetic sockets (12,13). If so, the burden on the clinician is increased. Evaluating a marginal limb fit is a gray area which is time-consuming, against a busy clinical schedule. Further, mediocre-fitting sockets may be accepted by wearers as "the way it is" accompanying limb loss, along with an unfortunately more limited lifestyle. Critical analysis of the performance of new automated limb fitting systems, including publication of frequency response, may need to be a part of their development.

There is a slight amount of power shown in the spectrum at vertical frequencies above 10 cycles/circumference. This power is all at low tangential frequencies--one or two cycles/circumference--and lies along the right-hand axis in Figure 6 . This power is due to the conical shape of the typical residual limb, and does not represent high-frequency surface bumpiness. In technical terms, representing a "saw-tooth" wave with a series of sine curves requires substantial amounts of high-frequency sine curves. The conical shape of the typical residual limb is seen by the Fourier transform as one segment of a repeating "saw tooth" wave. Again, the power observed at high vertical but low tangential frequencies is not actual surface bumpiness, but a consequence of the Fourier analysis technique.

Conclusion

The frequency evaluation determined the majority of the frequencies are at or below 10 cycles/circumference. This information will be helpful in designing smoothing functions for digitizing prosthetic and orthotic shapes. It also will assist in the evaluation of smoothing functions to verify the functions are performing at the appropriate settings, and not causing degradation of the shape record.

Acknowledgements

The authors gratefully acknowledge the support of the Shriners Hospital for Children Research Grant 9510 and the assistance of Jon Rice and Paul Macy of Newington Orthotic & Prosthetic Systems, Hartford, Conn.

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