P. Jacquot, P.K. Rastogi, L. Pflug
Swiss Federal Institute of Technology
Laboratory of Stress Analysis
CH-1015 Lausanne, Switzerland

Scanned, filtered by optical character recognition and adapted from Optical Engineering 24(5). 918-924 (September / October 1985). For further details please see the original article. This resource is intended for engineering students.

Abstract. In order to maintain fragments of fractured bones in a state of immobilization, the use of an external rigid frame has proved to be very advantageous. Confronted by contradictory requirements, the conception of external fixation has, however, been a difficult task. The present paper aims to show, through three examples of varied bearings, the interest of holographic interferometry in external osteosynthesis. The first example deals with the mechanical behavior of a key element of the fixation device-the ball joint-submitted to realistic loads. The last two examples compare two models of ball joints as to their characteristics of rigidity and of resistance to slipping. Whereas in the former case holographic interferometry primarily fulfills the function of a prelude to the modelization work, in the latter cases it serves to formulate an engineering diagnostic. The findings relate to the remarkable elastic behavior of the ball joint, to the effectiveness of a lightened bowl design, and to the fact that cousin models may behave quite differently as to their resistance to slipping rotations of the bar. In comparison with other experimental methods, holographic interferometry appears to be very competitive and result-oriented and, as such, is expected to multiply applications in similar evaluation tasks.

Subject terms: holographic Interferometry; mechanical testing, external fixation; orthopedics.

CONTENTS
1. Introduction
2. Examples treated
3. Evaluation of the mechanical behavior of the ball joint
3.1. Tightening of the screw
3.2. Axial force applied to the connecting bar
3.3. Vertical force applied to the connecting bar
3.4. Horizontal force applied to the connecting bar
3.5. Summary of ball joint behavior
4. Effectiveness of a lightened joint bowl
5. Resistance of two joint models to an axial rotation of the bar
6. Conclusion
7. Acknowledgment
8. References

1. INTRODUCTION

In external osteosynthesis, a number of pins interlinked by a frame go right through the fractured member (Fig. 1). Initially limited to long arm and leg bones, the external fixation models have been developed for nearly all parts of the human skeleton. Though considered at its advent as the last recourse before amputation, the external fixation has steadily found an important place in simple fracture management as well. The use of external fixation is, moreover, recommended in the treatment of both children's and adults' fractures.

Invited Paper 5170 received May 29, 1984; accepted for publication Feb. 13,1985; received by Managing Editor March 4,1985. This paper is a revision of Paper 398-19 which was presented at the SPIE conference on Industrial Application of Laser Technology, April 19-22, 1983, Geneva, Switzerland. The paper presented there appears (unrefereed) in SPIE Proceedings vol. 398.

(c)1985 Society of PhotoOptical Instrumentation Engineers.

OPTICAL ENGINEERING, September/October 1985, Vol. 24 No.5

Numerous advantages are linked to external osteosynthesis. 1~3 The surgical act mainly is concerned with managing healthy and un-traumatized regions. The reduction of fractures can be effected a posteriori, and a three-plane correction remains possible after application. The adjustable connecting bars permit functioning in conditions of compression (healing), distraction (bone lengthening), or neutralization (loss of osseous substance, infected pseudoarthroses). The frame does not darken x-ray radiographs or angiograms and leaves the operative field free for complementary surgical interventions. At the fracture site, the risk of infection is often found to be smaller than in the case of internal osteosynthesis. The external fixation facilitates the patient's early mobilization. The removal of the frame is a benign (ambulatory) operation.

However, an external fixation must meet a number of strict requirements if it is really to present all of these advantages. A long term rigidity and stability, for a minimal weight and number of elements, are required of the frame, which must remain very manageable. The notions of rigidity and stability should be understood differently depending upon the type of consolidation looked for. The primary consolidation, which promotes the direct joining of the fracture surfaces by autogenous union, only tolerates movements of less than 5 to 10 ~m on the fracture region. I The secondary consolidation, which aims at the formation of a periosteal callus, seems, on the contrary, to be stimulated by well-controlled micromovements about the fracture region. 3

Intolerance reactions from the tissues as well as the necessity for sterilizing at high temperatures limit the choice of materials. The frame architecture is imposed by purely medical considerations. Thus, according to the type of fracture, certain regions for the grafting of pins and certain pin spacings are prohibited so as not to impair the blood vessels and nerves in the region. It is not surprising with appropriate transducers. The objective so pursued is to establish a correlation between the indications furnished in permanence by the latter and the fracture healing rate.

Where does holographic interferometry stand vis-a'-vis these two tendencies? The structural models, which make the assumption of simple elastic beams for bone, pin, and bar and the hypothesis of rigid noneccentric joints between pins and bars, could appear in some instances to be too simplifying. Besides, it remains uncertain whether viscoelasticity of the bone, bone microcracking, and pin slipping at the pin/bone interface or the slipping of bars and pins in their connecting points are always negligible. 8 Holographic interferometry responds well to the need of displaying the real phenomena along with their degree of relative importance. 9

As for the tendency to equip the external fixation with measuring instruments, holographic interferometry is manifestly not well adapted. However, for the ensemble of operations as imperative as certifying a production, it can efficiently compete with gauges. Besides, in the health domain, it is an absolute necessity to ensure that the manufactured systems effectively fulfill the functions intended at their conception and do not suffer from any hidden malfunctioning.

Finally, one is naturally tempted to extrapolate in the field of external fixation the successes met by holographic interferometry in the neighboring branches of biomechanics or internal osteosynthesis. 10,11

2. EXAMPLES TREATED

In the context of the preceding section, we shall examine three particular questions: (I) the evaluation of the mechanical behavior of the ball joint under the effects of tightening and of differently oriented forces applied to the connecting bar; (2) the verification that the lightening of joint bowls is possible without impairing their performance; and (3) the comparison of two similar joint models with respect to slipping rotations of the bar around its axis.

Use was made of two classical holographic interferometers in the laboratory. The first yields the lines of equal out-of-plane displacement w with half optical wavelength sensitivity for collimated illumination and observation normal to the object surface:

The second is a holographic moire arrangement, providing the lines of equal in-plane displacement u or v, according to the relations

where 0x and are the half angles of the beams illuminating the object symmetrically with respect to its normal and contained in the horizontal or vertical planes.

These two optical arrangements are exploited in real-time by using either an immersion plate holder or an instant holocamera. The real time version of the second optical arrangement is extensively described in Ref. 12. In the first arrangement, a set of mirrors makes it possible to examine simultaneously the views of three ball joint faces lit and observed in the same geometry. By this means the Monge projections of the deformed system are directly obtained. The two optical systems are provided with micrometric translation and rotation movements in order to apply, if necessary, the fringe control techniques.

3. EVALUATION OF THE MECHANICAL BEHAVIOR OF THE BALL JOINT

The ball joint under examination has been in use for several years. It assures not only the connection between pins and bar under a wide range of angles, but also permits one to lock four degrees of freedom for the bar through tightening of a single wing screw.

Fig. 1 Schematic of the Hoflmann Vldai external fixation frame. The four components making up the frame are (1) transfixing pins, (2) universal ball joints, (3) adjustable connecting bars, and (4) articulation couplings. The inset shows a close-up view of a ball joint, which is composed of a wing screw, cheeks, (c) clips, and (d) a bowl.

The present status of the research in external fixation biomechanics can be subdivided into two tendencies. 5 In the first, structural analysis models are developed with a view to simulate the behavior of the system, both the fractured bone and the external frame. These models attempt in particular to predict the relative motions of the fractured fragments and the stresses occurring at the fracture site, since the two quantities are of prime importance in the search for optimal healing conditions. 6'7 This approach presents the great advantage that all the parameters involved can be varied at will. The existing computation programs provide, for a given frame architecture, the number of pins to be used, their diameter, their spacing, and the distance to the fracture seat as well as the distance between the connecting bars. Of course, it is the surgeon who finally decides upon the best solution to be adopted as a function of the fracture to be reduced.

The second tendency is characterized by the idea of conceiving the external fixation not only in its function as treatment, but also in its role as a measuring instrument and even as a research tool, in vivo and on the human body. The patients thus are called upon, more and more frequently, to carry the external fixations equipped

Fig. 2. Out of plane displacement fringes relative to a tightening torque Increment of 9 to 9.2 N m.

Although giving full satisfaction, it is nevertheless desirable to verify that the ball joint properly performs the assigned functions in addition to objectively defining its real mechanical behavior.

The response of the ball joint to realistic loads will be quantified by means of simplifying parameters, while avoiding the risk of disregarding a global trait of behavior.

3.1. Tightening of the screw

Figure 2 represents the out-of-plane displacement fringes as a function of an increment in the tightening torque S of the screw. This illustration is taken from a series that included the torque increments (9 to 9.2,9 to 9.4 9 to 10 N m) and (13 to 13.2,... , 13 to 14 N m). It is typical inasmuch as the fringes keep the same general aspect and only their densities change. Surgeons apply tightening torques ranging from 7 to II N m.

The front view fringes are characteristic of a flexion of the cheek, the most strained area being that between the axis of the connecting rod and the bowl. The maximum deflection is at the top part of the element. The slope along the y-axis at the top part is chosen as a significant parameter of this loading case. Over all the steps analyzed and per unit of tightening torque increment, this parameter, directly deduced from the fringe spacing, has a mean value. The fringe discontinuities at the periphery of the kneecap indicate that the latter slides in its seating, which is precisely its raison d'etre. From the front and side views, one might notice that the bowl becomes ovalized in a regular and progressive way.

On the top view, the fringes are nearly parallel and equidistant and therefore display a rigid-body rotation R1 of the upper side around the x-axis-rotation linked with the flexural deformation. However, the fringe density clearly varies from one cheek to another, indicating a differential rotation of the cheeks around the x-axis. From all the loading steps, it is apparent that parasitic microrotations take place around the three axes and fluctuate within the range 0 to 150 micro rad. These microrotations are likely to put the connecting bars in slight prestress merely by the action of tightening.

3.2. Axial force applied to the connecting bar

Loading increments always are exerted starting from an initial axial force of 50 N. Figure 3(a) shows the out of-plane displacement fringes for the maximum increment of 50 to 250 N. From the constant spacing and orientation of the fringes on the side and top views and from the absence of fringes on the front view, a very simple response is deduced: The cheeks undergo a rigid-body rotation R3 around the z-axis, while the bowl is moderately strained under the transmitted load. The relationship between the rotation R3 and the axial force increments is linear over the range examined. Per unit axial force increment, this rotation has the value given in the original.

-------- (4)

The reversion to the flat tint on the three views is noticed when the axial force is brought down to its initial value of 50 N. Over the range examined, the ball joint has an elastic behavior, like a monolithic block.

3.3. Vertical force applied to the connecting bar

Loading increments ranging from 0 to 32 N are exerted at 0.125 m from the y-axis. The fringes [Fig. 3(b)] present a remarkable similitude with the former case, in that the device responds with a pure rigid-body rotation R3 around the z-axis, proportional to the force exerted, while the bowl supports a moderate deformation. This rotation, converted to the unit torque due to the vertical force, has the value given in the original

------ (5)

In the range of the exerted forces, the device again has an elastic reversible behavior. The fringes disappear when the force is brought back to its initial level of 0 N [Fig. 4(a)].

3.4. Horizontal force applied to the connecting bar

Loading steps are examined ranging between 0 and 32 N for a horizontal force applied at 0.125 m from the y-axis. The front view [Fig. 4(b)] demonstrates that the cheek turns preponderantly around the y-axis (R2). However, since the fringes are more dense on top than at the bottom, a torsional deformation around the y-axis is also noticed. Moreover, the slight obliquity of the fringes means that rotation R2 is not pure. The redundant profile and top views make it possible to isolate component R2. On an average and converted to the unit torque increment exerted by the horizontal force, this parameter has the value given in the original.

(6)

(3) The side and top views confirm the existence of microrotations Rt and R3 around the x- and z-axes. These microrotations are found to vary in the range 0 to 125 ~rad proportionally to the vertical force in the interval 0 to 32 N.

As previously, the behavior of the ball joint is reversible. The maximum loading step was exerted for some 20 h without there being any noticeable evolution in the fringes. This illustrates the long-term stability of the locking device. The same remarks apply to the two preceding cases.

Fig. 3. Out of-plane displacement fringes corresponding to an increment of (a) axial force between 50 and 250 N and ~) vertical force between 0 and 32 N at 0.125 m from the V-axis. in both cases the tightening torque S Is maintained at 9 N.m.

Fig. 4. (a) Residual fringes obtained when the vertical force is brought down to its initial level of ON after an excursion to 32 N. ~) Horizontal force between 0 and 32 N at 0.125 m from the y axis. In both the cases S Is constant and equal to 1 N.m.

3.5. Summary of ball joint behavior

Holographic interferometry has permitted us to expose in a simple

and rapid manner the real behavior of the ball joint. The preponderant traits are summarized in Table 1. The observations relating to the displacement deformation reversibility when the loads are suppressed-except in the case of screw tightening-and to long-term stability are the more important findings. This elastic behavior is outstanding if one keeps in mind that the ball joint involves three connections through friction between four groups of independent elements (bowl, cheeks, clips, bar). These characteristics provide a rational basis for the modelization of the joint rigidity. In addition, the deduced parameters may serve as reference data in anticipation of a future evolution of the design.

Fig. 5. Out of plane displacement fringes relative to a tightening torque Increment of 9 to 9.8 N.m for the bowls made of (si steel and (b) aluminum alloy.

TABLE 1. Evaluation of the Mechanical Behavior of the Ball Joint

Reversible elastic behavior

The contrast of the fringes though relatively low is sufficient to trace them (Fig. 6). Considering the high fringe density, it seems improbable that the in-plane speckle interferometric technique will yield better results. Different methods for improving fringe contrast in holographic moire are discussed in Ref. 13.

The fringes keep the same general aspect for the two bowls, whose lateral edges tend to turn in opposite directions in their plane following a rotation R3 around the z-axis.

Over several cases of tightening varying between 9 to 9.2 and 9 to 9.8 N.m, the aluminum bowl presents an increased resistance to in-plane deformations in the proportion of 7 fringes against 15 fringes for steel per unit tightening torque increment.

An increased resistance of the aluminum bowl to in-plane and out-of-plane deformations therefore is observed-quantitatively in the ratio of the number of fringes-despite the reduction in weight mentioned. This example provides a simple verification of the validity of the new mechanical conception.

5. RESISTANCE OF TWO JOINT MODELS TO AN AXIAL ROTATION OF THE BAR

4. EFFECTIVENESS OF A LIGHTENED JOINT BOWL

The lightening of the frames is one of the permanent concerns of manufacturers. The patient must carry, sometimes for several months, frames often comprising more than six ball joints. His comfort thus depends upon any weight reduction, howsoever small, that can be technically imposed on the external fixation.

The two joint bowls being compared are made of a particular type of steel and of a special aluminum alloy. The latter is 35% lighter than the former despite an increase in volume due to the difference in modulus of elasticity of the two materials.

The two bowls are identically sectioned, as shown by the shaded regions in Fig. S. The plane cross section 50 arising facilitates the interpretation of the fringes. The fringes, in both the cases, indicate that the ovalization of the bowls is a result of opposite rotations of the lateral edges of the sections around the y-axis. Consequently, the comparison is simply done through fringe counting. Over a series of tightening torque increments varying from 9 to 9.2 N.m to 9 to 9.8 N.m, 9 fringes/N.m are found for the steel bowl and only

3 fringes/N.m for the aluminum bowl.

This last example refers to the problem of prototype certification. Just before the stage of mass production it sometimes happens that two or more versions of the same basic design still remain under consideration. Holographic interferometry may contribute to resolving the indecision.

Even for an ideal frame, the long-term stability is impaired by the bone creep at the pin/bone interface. This makes it necessary to renew the compression periodically and to check the locking of the joints frequently. Among other possible slipping rotations, we present here the case study of the resistance of two similar joint models to an axial rotation of the bar.

The two models, one of which is the standard model shown in Fig. 2, differ essentially in the conception of the clips. In the new model, one half of the clip, having a "C" shape, clasps the other half between its jaws, an arrangement that causes the two clip halves to rotate together and facilitates the introduction of the bar.

A tightening torque S (Fig. 7) is first applied to the locking screw. Increasing torques Cb are then exerted on the bar, step by step. At each loading step, the bar undergoes a rotation that can be split into an elastic torsion and an irreversible microrotation due to the slipping of the rod in the clips. In order to isolate the irreversible component, the bar is unloaded at the end of each step. The slipping microrotations R of the bar are measured using an instant holocamera. The reference hologram records a thin elongated plate in the state of zero torque Cb. The residual rotation fringes appearing

on this plate are counted after a return to zero torque Cb. Taking into account the length of the plate and Eq. (1), the sensitivity is found to be better than one second per fringe.

The autocollimator serves to control the elastic torsion of the bar and to determine the time taken by the bar to find a position of stable equilibrium after which it is possible to go back to zero torque. This autocollimator has a 2 5 sensitivity, which implies that the mirror has an excellent flatness and large diameter, i.e., is heavy and cumbersome. Hence, in practice it is impossible to place the mirror in the vicinity of the ball joint. The interval 0 to 600 5 is covered with the help of a single hologram by performing some 10 consecutive fringe compensations. Moreover, holographic interferometry allows a global control of the operations. The hologram of the plate covers simultaneously the ball joint and its support, as well as the guidance support of the bar. It is always noticed that the measurements are not tainted with parasitic movements or creeps of these supports.

On the whole a graph like the one in Fig. 8(a) is obtained, in

which the successive torques Cb are abscissa and slipping microrotation R ordinates. A set of curves of this type is obtained, by repetition, for the two models and for tightening torques S varying N~m by Nm between 4 and II N.m.

A last step of data reduction allows us to convert the 16 curves obtained into two curves, Cbt = f(S) and Pt = g(S), shown in Figs. 8(b) and 8(c). Cbt is the torque corresponding to a predetermined threshold t of rotations R, and Pt is the slope of the left curves at this threshold.

For a well-designed implement, the permissible torque Cbt should increase with the tightening torque S. The slope Pt should stabilize at as low a value as possible for high tightening torque, indicating that an abrupt disastrous sliding is impossible. From Figs. 8(b) and 8(c), it may be inferred that model I is correctly designed, unlike model 2. In particular for the latter, the change in tendency of the slope Pt to an upward rise soon after the tightening torque S enters the service load domain (7 to 11 N.m) is an obvious indication of its insufficiency. This amazing behavior of the second implement has been partly explained. At a certain torque S stage, the cheeks become so deformed that one of the extremities of the clip envelope "C" touches the opposite cheek. All additional tightening has the effect of deforming this "C," resulting in a progressive loosening of the connecting bar.

In the example presented, holographic interferometry shows up as a potential tool for analyzing the dry frictional effects starting from their early microscopic stages, which appear well before the free slipping state.

Fig. 8. in. plane displacement fringes relative to a tightening torque Increment of 9 to 9.8 N.m for the bowls made of (a) steel and (0) aluminum alloy.

Fig. 9. Schematic of the measurement of slipping microrotatlons of the bar around Its axis.

6. CONCLUSION

The three examples treated are intended to demonstrate the usefulness of holographic interferometry in the mechanics of external fixation. It is fitting to underscore that the technique performs multipurpose tasks. First of all, holographic interferometry serves as a preliminary to the elaboration of mathematical models, in order to identify the real phenomena and to assess their respective orders of magnitude. This aspect is present in the first example, related to the mechanical evaluation of the ball joint and for which dominant traits of behavior and the linearity domain of responses are emphasized. In addition, the technique is apt to render such engineering services as verification, comparison, and certification

Fig. 8. (a) Variation of the connecting bar rotation A as a function of the torque cb applied to the bar for a fixed tightening torque S acting at the clamping nut. cbt and Pt represent, respectively, the abscissa and the slope of the curve at any predetermined rotation value t. These curves, drawn for different values of 8, allow us to draw curves (0), Cbt - I(S), and (c), Pt - g(S), for the two models being compared

These points are illustrated in the last two examples.

From the practical point of view, the recent advent of the instant holocamera, combined with video cameras and monitor screens, amounts to a considerable progress. Finally, an adequate choice of the test conditions and the optical setups should lead, in many concrete problems, to sufficient qualitative and quantitative information while preserving the simplicity of interpretation and avoiding complicated postprocessing and lengthy calculation work.

7. ACKNOWLEDGMENT

The authors wish to thank M. Wagenknecht of Jaquet Orthopedic SA for his many helpful comments and advice and are grateful to him for having generously provided them with the relevant materials.

8. REFERENCES
1. D. C. Mears, External Fixation. The Current State of the Art, Chap. I, Williams and Wilkins Co., Baltimore (1979).
2. J. Vidal, Proc. 7th int. Conf on Hoffmann External Fixation, pp. 5-14, Diflinco, Geneva (19110).
3. F. Burny, Proc. 9th mt. Conj on Hoffmann External Fixation, pp. i-I 5, Clinique Universitaire d'Orthopedie, Geneva (1982).
4. All Type External Fixation Bibliography 1852-1982, Hoffmann External Fixation Information Center, Jaquet Orthopedie SA, Geneva,
(1982).
5. Abstracts of the 10th int. Conf on Hoffmann External Fixation, Clinique Universitaire Hopital Erasme, Brussels (1983).
6. E. Y. Chao, R. A. Kasman, and K. N. An, J. Biomechanics 15(12),
971 (1982).
7. F. Y. Chao and K. N. An, Finite Elements in Biomechanics, Chap. II, John Wiley and Sons, New York (1982).
8. F. J. Kip and R. Bosma, Eng. Med. 7(1), 43 (19711).
9. R. K. Erf, ed., Holographic Nondestructive Testing, Academic Press, New York (1974).
10. G. von Baily, ed., Holography in Medicine and Biology. Springer Verlag, New York (1979).
11. U. Hanser, Holography in Medicine and Biology. G. von Baily, ed., pp.27-33, Springer-Verlag, New York (1979).
12. C. A. Sciammareila, P. K. Rastogi, P. Jacquot, and R. Narayanan, Exp. Mech. 22(2), 52 (1982).
13. P. K. Rastogi, P. Jacquot, and L. Pfiug, Opt. Acta 30(8), 1067(1983).