## Understanding the effect of the total mass and mass distribution within a pediatric prosthetic shank on the natural swing period of a transfemoral prosthesis.

### By Phil Stevens CPO, LO; David Baty CPO, LPO

### Abstract

A preference for reduced self-selected walking speed has been observed among ambulators with transfemoral amputations. Prolonged prosthetic swing phase periods have been identified as a potential cause, particularly among pediatric ambulators. Mathematic and physical modelings of prosthetic shanks predict a relationship between the total mass and mass distribution within a prosthetic shank and the resultant prosthetic swing periods. Analyses are performed based on a simple pendulum model and using second order differential equations. Swing periods of a sample pediatric shank were timed while mass and distribution of mass were varied to determine which of the two modeling methods more accurately predicted swing period trends. Data indicate that reasonable variations to both mass and distribution of mass fail to substantially effect the natural swing period of the shank.

### Introduction

Clinical observation reveals that transfemoral amputees walk more slowly than the normal population. This preference for slower self-selected walking speeds (SSWS) has been documented among both adult ^{2,5,9} and pediatric^{4} transfemoral amputees. Authors have further observed this is due in part to a prolonged prosthetic swing phase.^{1,3,8} When the prosthetic swing period is excessive or delayed, the amputee must either reduce his cadence to maintain gait symmetry, or revert to an asymmetrically timed gait.

The effect among the pediatric population may be even more pronounced as children rely greatly on a rapid cadence to obtain adequate speeds. ^{3,7,8} While friction mechanisms allow the clinician to match prosthetic heel rise to that of the sound side, such adjustments fail to impact the duration of the patient's prosthetic swing period.^{3} Authors have concluded that , "efforts to decrease prosthetic swing time might be better directed at decreasing the periodicity of the shank rather than the knee mechanism.^{3" }

When considering the prosthetic shank, the available variables are the total mass and the distribution of that mass within the system. Different modeling methods predict conflicting results. After evaluating two mathematical models and their predicted results, physical data were collected as swing periods were recorded throughout variations of total mass and mass distribution.

### Mathematical Models of Prosthetic Shanks

#### Pendulum Theory

One method used to predict the behavior of a prosthetic shank is to treat the system as a mechanical pendulum ^{3,8} The value of *m* and *d *can be readily determined by weighing the shank and determining its balance point (see Illustration 1 ).

If such an assumptions is made and friction is neglected, the period of the pendulum is governed by the following equation:

where *T* is the period of the pendulum, *I* is the rotational inertia of the shank, *m* is the mass of the shank, *g* is the acceleration due to gravity, and *d* is the distance from the pivot axis, or knee center, to the center of mass of the shank.

The rotational inertia of the shank, *I*, is defined by the following equation:

Reducing equation 1 by canceling like terms, we are left with the following, simplified variation of the pendulum equation:

Thus, modeling of prosthetic shank as a simple pendulum predicts that as the distance from the axis of rotation to the center of mass decreases, such as with the addition of mass proximally (see Illustration 2 ) the swing period will likewise decrease.

Further, it predicts that while the distribution of the mass within the shank is significant, the total value of the mass has no effect on swing duration. Finally, it predicts that assuming a constant value for d, the swing period will remain constant regardless of the angle traversed by the shank.

#### Differential Equations

A second method of predicting the behavior of a prosthetic shank is through the use of second order differential equations. In this method, predictions are based on the governing equation:

Where *m *and *d* are once again mass and distance to the center of mass respectively. ?*" *represents angular acceleration, *B* represents the coefficient of friction, ?*'* represents angular velocity, and ? represents the angle of the shank in radians with respect to gravity at a given time. As with the pendulum model, friction is neglected, therefore *B* is assumed to be zero. This leaves the simplified equation:

where the first term describes the moment of inertia acting on the pendulum and the second term describes that component of the force of gravity acting about the pivot, or knee center. At any given moment, the summation of these two terms must equal zero.

Unlike the pendulum model, there is no variable for time in the above equation. Rather, it must be derived from the angular acceleration. Using Mathcad or similar computing software it is possible to calculate the values of the angular velocities experienced at any given angle ?. Analysis of this plot also allows an appreciation of the time period required for the shank to move from its starting angle to zero radians, or vertical.

In Figure 1 , the following values were assigned to each variable as being representative of a standard endoskeletal prosthetic shank for a 10.5 year old ambulator (Hicks).

*m* = .975 kg

*d* = .29

? = 65 degrees or 1.13 radians (mean value for peak knee flexion in swing)

The top curve represents of the changing values of ? as the shank moves from peak knee flexion to a vertical shank, or full extension, as a function of time, along the *x axis. *

The lower of the curves represents the changing values of ?*'* or angular velocity with respect to time. It should be noticed that the peak angular velocity occurs when ? = zero. The duration of the "prosthetic swing descension" is identified as the time required for the shank to traverse the arc from peak knee flexion to terminal extension.

Using this method, the time periods can be predicted throughout variations of both *d *and *m*.

Figure 2 illustrates the predictions made when all variables are kept constant except the distribution of the shank mass, or *d. *Notice that the analysis predicts that swing times should decrease as the location of the center of mass from the pivot or knee center increases. This assumption directly contradicts that of the pendulum model.

Figure 3 illustrates the predictions made when the value of *d* is maintained while overall mass is varied. This modeling suggests that increases in mass will generally correlate with decreased swing times. Again, this directly contradicts the pendulum model which predicts that changes to the value of *m *will have no affect on swing periods.

Figure 4 is a three dimensional plot illustrating the cumulative trends towards a reduced natural periodicity of the shank as *m* and* d* are increased.

Table 1 summarizes the conflicting predictions with regard to the value of *T*, or the prosthetic swing time as *m* and *d* are varied. Note that a decrease in *T* implieas an increased shank speed.

Proposed variations |
Pendiulum Model |
Derivative Predictions |

Value of m is increased |
T will remain constant |
T will decrease |

Value of d is increased |
T will increase |
T will decrease |

### Physical Data Collection

Given the dramatically different predictions of the two modeling methods, a testing apparatus was fabricated and time data collected.

A representative shank was assembled meeting the criteria outlined above with regard to mass and mass distribution. This was attached to an Ottobock 3R36 constant friction knee, with no friction applied at the knee bolt. Time trials were collected as the shank was released from a consistent angle of 65 degrees and allowed to swing through vertical.

Two switches were assembled and positioned to detect the beginning and conclusion of the swing descention. Each switch consisted of a light sensitive detector positioned across from a light source. When the prosthetic shank was released, it immediately passed between the detector and the light source, signaling the timing device to begin. When the shank broke the light beam of the second detector, positioned at the prosthetic shank's vertical alignment, the second switch stopped the timing mechanism. The timing mechanism utilized accurately recorded time within .01 seconds. (see Figure 5 )

The first set of time trials maintained a constant value for *d* of 29 cm while increasing the value of *m *in 200 g increments. A minimum of 5 trials were recorded and the average of the recorded values is reported. It should be noted that time values never deviated by more than .02 seconds on a given trial. The results are listed in Table 1

Mass | Swing Period |
---|---|

975 g | .308 s |

1175 g | .300 s |

1375 g | .288 s |

1575 g | .292 s |

1775 g | .289 s |

The second series of time trials maintained a constant mass of 1775 g. The value of *d* varied from 29 cm to 19 cm. A minimum of 5 trial were collected with the average time period recorded. The results are listed in Table 3.

d |
Swing Period |

29 cm | .289 s |

19 cm | .294 s |

### Discussion

The different modeling techniques predicted contradictory results. While the pendulum based theory holds that decreased values of *d* should shorten prosthetic shank periodicity, the calculus based models predict the opposite. While the pendulum based theory holds that periods will be constant regardless of the value of *m*, the calculus based models predict that increases in *m *result in decreased swing periods (See Table 1).

The data obtained suggest that reasonable variations to the values for both *d* and *M *fail to validate the predictions of either modeling method. While modeling numbers predict that marginal adjustments to these values should affect swing periods by as much as 50%, very little change in swing times were observed. It should be emphasized that all modeling methods and data collections were performed to assess the effects on the natural period of the prosthetic shank. Values such as friction, and the impact of the prosthetic wearer on shank mechanics were ignored for reasons of simplification.

It should also be noted that the range of values of *d *and *m* were selected by the researcher on the basis of being reasonable. On the selected 42 cm prosthetic shank, it is simply not possible to decrease the value of *m* to any less than 19 cm without substantial increases in the value of *m*. Further, while increases of *m* beyond the range of the values collected may have produced significant changes in swing periods, such increases were deemed excessive and contradictory to the underlying purpose of improving prosthetic performance.

The results of the this study reinforce the findings of Selles et al, who in their review of the published reports on mass and mass distribution within prostheses concluded "models were not able to predict the outcomes of the experimental research. ^{6}" While it was anticipated that complete removal of external influences would validate modeling predictions, such was not found to be the case.

### Conclusion

There are several possible reasons for the lack of discernable differences in the time trials performed. In attempting to evaluate the only the natural periodicity of the shank, the actual event of shank dissension occurred in only .3 seconds. While the system allowed measurement discrepancies of .01 seconds, it is possible that the (Continued from page 18) event was to brief to appreciate subtle variations in the value of *T*.

Additionally, this values of *T* reported in this study are limited to the periods of shank dissension. The behavior of *T* might differ significantly if the timed event considered were a complete step duration, or complete stride duration. The addition of extension assists, or any human interface would further affect the behavior of *T*.

Further research is needed to better understand the mechanisms of varying the natural swing period of prosthetic shanks. Such an understanding would potentially benefit those patients for whom simple friction knees are the only option such as children and individuals with limited funding.

Additionally, improved understanding of the clinicians' limitations in affecting the natural swing period of a prosthesis may better validate the expense and weight of the more complex, variable cadence knee units.

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- Waters RL, Perry J, Antonelli D, Hislop H: The energy cost of walking lf amputees -- Influence of the level of amputation. J Bone Joint Surg 58(A):42-46, 1976

Phil Stevens is a staff clinician at Dynamic Orthotics and Prosthetics in Houston TX. David Baty is Director of Prosthetics at Dynamic Orthotics and Prosthetics