Hip Replacement Project Update

Hip Replacement Project Update

Amos Winter

February 2, 2006

This document presents the most recent work on an analytical model to describe the dynamics of a hip replacement implant being hammered into a femur.

1.Impact force

The impact that occurs between the hammer and implant can be considered a collision between two rigid bodies, as each is made from a hard alloy. Stronge [1] defines the underlying premises of impact theory with the following statements:

• “In each of the colliding bodies the contact area remains small in comparison with both the cross-section dimensions and the depth of the body in the normal direction.”
• “The contact period is sufficiently brief that during contact the displacements are negligible and hence there are no changes in the system configuration; ie, the contact period can be considered instantaneous.”

These statements allow a single hammer blow to the implant to be viewed as two discrete events. The first is the impact, in which the implant is accelerated. The second is the movement of the implant within the bone, where it is decelerated to a stop.

During the impact, a large force is exerted on the implant over a very short period of time. Starting with Newton’s second law of motion, the applied force can be related to the acceleration of the body using Equation 1.1, assuming mass is constant.

/ (1.1)

Where F is force, p momentum, m is the mass of the implant, and v is the velocity of the implant.

Figure 1.1 shows the force-time relationship for a generic impact. A large peak force is achieved in a very short amount of time, which can be on the order of microseconds. The area under the curve is called the impulse, I. With the project’s current data acquisition setup of a force sensor attached to the implant, the impulse can be measured by summing the force at each increment of time.

Figure 1.1 Force-time relationship during an impact

The calculation of the impulse can be related to the change in velocity of the implant by rearranging Equation 1.1 and integrating over the time of the impact, as shown in Equation 1.2. With the results from the force sensor, and knowing the time over which the impact occurred, the final velocity of the implant can be calculated

/ (1.2)

An important note should be made about sampling frequency. If the data is sampled at too low a rate, the full force profile of the impact will not be determined. It is common practice to use at least a 10X faster sampling rate than the response being measured. An adequate sampling rate might be found by inspecting the speed of sound through the implant, C, which will be the same as the speed of the stress wave moving through the implant after impact. The speed of sound in the implant is found by knowing the modulus, E, and density, ρ, of the material [2]. This speed can be turned into an approximate frequency, f, by using a length scale, such as the length of the implant, l, as shown in Equation 1.3.

/ (1.3)

2.Implant slip and deceleration model

Knowing the impact force is valuable, but it only tells half the story of what happens to the implant during insertion. The force transducer does not sense how the implant decelerates in the bone after it is accelerated by the impact. It is the deceleration, not the impact acceleration, which will dictate the forces exerted on the bone from the implant. Figure 2.1 shows the first hypothesis for a model to describe the insertion of the implant into the femur.

Figure 2.1 Model of implant insertion into a femur

The model in Figure 2.1 captures the stick-slip relationship as the implant is being driven into the femur. In the model, k1 describes the increase in preload force between the bone and implant as the implant is driven deeper into the femur. The friction between the bone and implant is characterized by μ. The axial and shear compliance of the bone is described by k2. The total displacement of implant in respect to ground is x. The displacement of the implant in respect to the bone wall is x1. The displacement of the implant in respect to the ground, as only a result of compliance in the system is x2. The displacements are related to each other through Equation 2.1

/ (2.1)

Figure 2.2 shows a generic force-time curve during one cycle of hammering the implant into a femur. The first sharp jump in force corresponds to the impact between the hammer and implant, which accelerates the implant to a calculable velocity. The second phase, corresponding to a positive force, and thus a deceleration to the implant, represents the hypothesized response of the model in Figure 2.1.

Figure 2.2 Full acceleration-deceleration profile of implant during one hammer blow

As the implant first starts moving into the bone, the k2 spring begins absorbing kinetic energy from the implant. Equation 2.2 represents the dynamics of the implant before slip.

/ (2.2)

At some point, the reaction force being exerted by k2 exceeds the sticking limit between the bone and the implant, causing the implant to slip. During the slip, kinetic energy is dissipated in the friction between the bone and implant. The slip condition is represented by Equation 2.3.

/ (2.3)

When the slip stops, k2 recoils and may possibly oscillate. The model in Figure 2.1 neglects any damping terms. These terms are assumed to be much lower than the frictional terms, but may be necessary to add in future models.

After each blow of the hammer, the implant will be driven farther into the bone, compressing k1 and increasing the frictional force. Each successive blow will take more force to make the implant slip within the bone. At some point, the frictional force will exceed the reaction force from k2 and the implant will not slip at all. At this point, the implant is firmly contacted against the bone, and will not slip unless a greater impact force is applied. The movement of the bone will be purely from k2. The described slip trend is demonstrated in Figure 2.3. The maximum force exerted on the implant, corresponding to no slip, is Fmax.

Figure 2.3

A discrete force value at which the femur will shatter due to the insertion of an implant will be very difficult to quantify. There is a wide distribution in femur strength due to many factors, including age, gender, lifestyle, weight, etc. Thus, an absolute shattering force model may not be appropriate. Using an accelerometer placed inside the implant guide during insertion, an acceleration profile similar to the one in Figure 2.3 could be generated. A force transducer would not be required, as the accelerometer could capture the impact forces if the sampling rate is high enough. In the profile, the decreasing slip zone for successive hits could tell the surgeon when the implant is close to being firmly contacted against the bone. When “Hit n” is reached, the surgeon would know that one more blow could break the femur. A plot similar to Figure 2.3 would not tell the surgeon the correct impact force to use; this would still have to come from experience. What the plot would do is give feedback to the surgeon of how close the implant is to full contact with the bone.

A solution to the equation of motion for the model in Figure 2.1 is not fully possible without knowledge of the stiffness and frictional constants. These constants can be determined by inspecting the response returned by an accelerometer attached to the implant. A solution is most likely not even necessary, as each patient may have different constants for the equation. What will be most important is the decrease in slip zone with each hit, which can be represented visually from the accelerometer data.

3.Forces exerted on the bone from the implant

Inspecting the inside of a femur, as shown in Figure 3.1, the internal cavity has a slight taper from top to bottom. From inspection, this taper is approximately 1.5 degrees.

Figure 3.1 Internal taper of femur

Because the taper within the femur occurs at such a shallow angle, a large force magnification can occur when the implant is hammered into the bone. Figure 3.2 shows the reaction forces which act on the implant. As the bottom of the implant makes contact with the bone, and assuming the end of the implant is spherical, the reaction forces act normal to the bone’s internal surface.

Figure 3.2 Reaction forces on the implant from the bone.

Assuming that the implant makes contact at two points, the force exerted on the bone, Fbone, is given by Equation 3.1. At an angle of 1.5 degrees, Fbone = 19*Fmax, so the input force is magnified by a factor of nineteen.

/ (2.3)

References

[1] Sronge, W. J. Impact Mechanics. Cambridge: CambridgeUniversity Press, 2000.

[2] Macaulay, M. A. Impact Engineering. London: Chapman and Hall, 1987.