The use of a mechanical model to describe the stiffness and damping characteristics of the knee joint in healthy adults.

Although patients frequently complain of and clinicians try to relieve joint stiffness, there have been few methods of quantifying stiffness in a clinically feasible way. In addition, the concept of stiffness offers a means of describing motor behaviors such as rigidity or spasticity that are not well characterized by more conventional descriptors such as strength or range of motion (ROM). Many studies have defined stiffness slightly differently. The term stiffness," however, has generally been used to mean the resistance of a tissue, joint, or limb to a change in shape or position. For example, the rigidity associated with some central nervous system disorders may result in increased stiffness; that is, more force is required to flex or extend the knee. The literature in biomechanics, physiology, and motor learning is replete with studies examining the role of stiffness in controlling or hindering human movement.

The concept of stiffness of biological structures can be likened to, or modeled as, an elastic spring. The spring represents the behavior of the tissue or joint as it is deformed or bent. in many studies, the authors have noted a time-dependent nature of movement in which the response to the deforming force depends on the duration or velocity of the applied force. For example, cartilage continues to deform, or compress, as a constant compressive force is applied for a prolonged period.[1] Similarly, most connective tissue usually becomes stiffer as the force is applied more rapidly.[2] This time-dependent behavior can be represented by a damper, similar to the device put or a door to prevent its slamming shut.

Spring-damper models have been used extensively to model the behavior of the neuromusculoskeletal system, characterizing that behavior by stiffness and damping parameters. This study investigated the mechanical behavior of the knee joint. Consequently, only the literature related to mechanical models of joint complexes is reviewed. Such models represent the articular structures and surrounding musculature as mechanical analogs such as springs and dampers.

Several authors have measured the mechanical resistance to motion in relaxed, normal and diseased joints. Long et al[3] examined the time- and displacement-dependent resistance to a sinusoidal oscillation of the relaxed human metacarpophalangeal (MCP) joint. They noted that the joint's displacement-dependent (elastic) and time-dependent (viscous) behaviors could be described by a spring and damper in series with another spring. The mechanical behavior of the MCP joint was attributed to the passive elements of muscles. No attempt, however, was made to distinguish the contributions of muscle from those of noncontractile tissues such as ligaments and articular cartilage.

Similar approaches have been taken by Wright and Johns[4] and Backlund and Tiselius[5] to Study the stiffness of diseased MCP joints in humans. Wright and Johns reported that elastic and viscoelastic resistance to motion were the primary contributors to resistance in both normal and diseased joints. These authors noted an increased stiffness with age and a greater stiffness in male subjects than in female subjects, and they also found increases in stiffness in subjects with rheumatoid arthritis. A marked decrease in stiffness was reported following a synovectomy in one subject. Tiselius[6] also reported an increase in viscous behavior in cases of tenosynovitis but with no accompanying change in elastic stiffness.

To assess the contribution to stiffness made by the individual components of the joint, Johns and Wright[7] examined the stiffness of cat wrists after sequential resection of the skin, the flexor muscles, the extensor muscles, the tendons, and finally the joint capsule. The reduction of stiffness as the result of each transection was interpreted as the stiffness provided by that structure. They reported that the joint capsule was responsible for almost half (47%) of the overall resistance to motion and that another 41% was provided by the muscles. Tendons provided 10% of the resistance, and skin provided 2%. Barnett and Cobbold[8] also assessed the contribution to joint stiffness provided by relaxed muscles. The stiffness of the distal interphalangeal (DIP) joint of the long finger was measured with the MCP and proximal interphalangeal joints flexed and extended. The relaxed musculature was reported to provide approximately half of the elastic stiffness of the DIP joint.

Although these studies assessed some of the elastic and viscous characteristics of joint complexes, other studies have identified a more complicated relationship between joint stiffness and the surrounding tissues. For example, several author [9-l2] have reported that ankle joint stiffness is dependent on ankle joint position. An increase in stiffness was noted as the foot was moved toward the extremes of dorsiflexion[9,11] or plantar flexion.[9,11,12] Such increases were attributed to the passive elastic elements of the joint. Although Backlund and Tiselius[5] found increased stiffness in the human MCP joint with age, Chesworth and Vandervoort[l2] found no such age effect in the ankles of healthy women.

Muscle activity has been linked to changes in joint stiffness. Stiffness of the ankle joint complex was shown to increase with increased contractions of the surrounding musculature with little or no change in the damping characteristic.[13] Similarly, Pousson et al[14] demonstrated an increase in stiffness in the elbow joint with increasing activity of the elbow flexor muscles. These authors also reported an increase in stiffness for a given level of muscle activity following eccentric strength training.

Although investigators [4-6] have attempted to evaluate the effect of disease on joint stiffness, joint stiffness is not commonly measured in the clinic. There have been some attempts to assess the stiffness associated with rigidity and spasticity by evaluating the knee's ability to swing freely.[15-18] The first report of such an assessment was qualitative,[15] but other authors[16] have attempted to quantify the oscillations that result from releasing the relaxed limb from a supported extended position. The pattern of oscillation that was elicited was similar to that of a damped spring. The studies that quantified the oscillations used rather simplistic variables such as total swing time and the magnitude of the first flexion reversal without taking into account the dimensions of the swinging limb.[17,18]

Stiffness of joint complexes is frequently discussed in the biomedical literature, and the behavior of joints is often described by lumping the contributions of the surrounding tissue into variables of stiffness and damping, Even the effects of pathology on joint movement have been described in terms of stiffness.[16-18] With the exception of the pendulum tests first described by Wartenberg,[15] however, objective assessment of stiffness has not been implemented in the clinic, nor has the reliability of such measures been assessed. Thus, despite the discussions of the topic of stiffness in the literature, no study has yet presented a reliable, clinically feasible measure of joint stiffness.

The first purpose of this study was to assess the reliability of a pendulum test similar to that described by Wartenberg.[15] The second purpose of this study was to assess the effects of gender, age, and joint position on the stiffness and damping coefficients of the knee derived from that test. I hypothesized that the stiffness and damping coefficients would increase with age. I also hypothesized that these coefficients would differ between male and female subjects and would be altered by joint position.

Method

Model

If the displacement with respect to time of a damped, freely oscillating spring as well as its inertial properties are known, it is possible to calculate its damping and stiffness coefficients.[19] This is the theoretical basis for determining the coefficients of the damped spring model of the knee from an experimentally obtained displacement history. The model used in this study consists of two rigid links articulated by a viscously damped, torsional spring with linear stiffness (Fig. 1). During the experiments to measure the stiffness and damping coefficients of the model, the proximal link representing the thigh was assumed to be fixed. Therefore, the distal, or leg-foot, segment behaves like a compound pendulum. The inertial properties of the leg-foot segment were determined from a geometrical model of the leg-foot segment, with the dimensions of each subject's leg and foot as input data.[20]

Figure 2 presents a typical displacement history of a compound pendulum articulated by a damped spring. All the characteristics of the spring-damper system can be determined from these data.[19] Thus, the spring-damper system can be completely described from the motion history of that system during free vibrations.

Experimental Methodology

Subjects. Ninety-six healthy adult subjects (48 of each gender) were studied to determine experimentally the stiffness and damping coefficients of the knee during relaxed oscillations. Exclusion criteria consisted of any history of neuromuscular disease or any injury to the lower extremity that required medical or surgical intervention. In addition, each volunteer underwent a standard physical therapy assessment of the knee including measurements of ROM, manual muscle testing (MMT) of the quadriceps femoris and hamstring muscles, and tests of ligamentous laxity. A volunteer was excluded from the study if signs of excessive joint laxity, decreased ROM, or MMT scores of less than 3+/5 of either knee were detected. The ages of the subjects ranged from 20 to 79 years (X = 45.5, SD = 17.0). The age and gender distribution of the subjects is shown in Table 1.

[TABULAR DATA OMITTED]

Instrumentation. Angular displacement of the knee was monitored by a Triax triaxial electrogoniometer.(*) This device consisted of three potentiometers of conductive plastic with an independent linearity of 1%. The voltage output for movement in each axis was proportional to the angular displacement of the knee about a parallel axis. The electrogoniometer compensated for any gliding of the tibiofemoral joint by attaching to its axes of rotation three 2.54 x 2.54-cm (1 x 1-in) plastic parallelograms. These parallelograms were distorted as the knee joint axis moved linearly (Fig, 3). Henault et al,[21] however, reported that misalignment of an axis of up to 2.5 cm resulted in no error. Muscle activity was monitored by electromyographic (EMG) recordings from 11-mm surface electrodes.(dagger)

Testing procedure. To determine the stiffness and damping coefficients of the relaxed knee, each subject was seated on a table with the untested limb supported on a stool. The electrogoniometer was mounted on the subject's lower extremity to be tested with straps above and below the knee joint. The medial-lateral axis of the goniometer was aligned with the center of the lateral femoral condyle, which was considered to be the axis of flexion and extension of the knee.[22] The lower extremity to which the electrogoniometer was attached was supported by the examiner, with the knee completely extended. The thigh of that limb rested on a rigid support. The subject was asked to maintain a comfortable, upright sitting position.

For the first test, the thigh with its support was positioned in approximately 75 degrees of hip flexion so that when the leg-foot segment hung freely in a vertical position, the knee was flexed to 75 degrees (Fig. 4A), For the second test, the thigh and its support were repositioned with less hip flexion so that when the leg hung freely in a vertical position, the knee was flexed to 45 degrees (Fig. 4B). The position used in this test required that the subject sit somewhat off balance, with one hip flexed to approximately 75 degrees and the other to approximately 45 degrees. The order of testing was randomized for the two test positions.

The first test position was chosen in an attempt to reproduce the limits of motion of the knee during the swing phase of gait. Maximum knee flexion occurs during the first third of the swing phase and ranges from 55 to 80 degrees.[23] Thus, 75 degrees was chosen as the equilibrium position. The second test position of 45 degrees of knee flexion as the equilibrium point and with approximately 45 degrees of hip flexion was chosen to assess the effect of joint position on the mechanical behavior of the knee joint complex.

Three good trials were obtained in each position. A trial was considered good if the oscillations were smoothly decaying and if there was no significant muscle activity at the knee. To monitor muscle activity, surface electrodes were placed on the muscle bellies of the vastus medialis and biceps femoris muscles of the limb with the electrogoniometer. Prior to the application of the electrodes, the skin was prepared by rubbing it briskly with alcohol. The skin was shaved only when excessive hair was present. The vastus medialis and biceps femoris muscles were monitored simultaneously throughout the experiment.

At the beginning of each trial, the subject was asked to relax completely. The leg-foot segment was supported by the examiner with the knee in maximum extension. Care was taken to avoid raising the thigh from its support. Once the subject was completely relaxed, the leg-foot segment was released and fell into knee flexion. The limb was allowed to oscillate freely until it came to rest in the vertical position.

Data from the electrogoniometer and EMG data were collected at 500 Hz and recorded on a PDP 11/10 computer. [double dagger] Raw EMG data from the quadriceps femoris and hamstring muscles and the electrogoniometer output were also monitored on an oscilloscope so that any erratic motion or muscle activity could be identified immediately and the run repeated.

Electromyographic signals were filtered by a low-pass filter with a cutoff frequency of 250 Hz. A preamplifier was connected to each pair of electrodes by 30.5 cm (12 in) of cable and fastened to the limb being monitored. The EMG signals were passed through the preamplifier with a gain of 600. Additional gains of up to 10 times were available. The signal was monitored by an oscilloscope, [sections] and the gain was adjusted to yield a 1- to 5-V signal during a gentle isometric contraction. Any signal that was equal to or greater than one fifth of the submaximal signal was identified as significant muscle activity, and the trial was repeated. As the study progressed, it became evident that any EMG activity in the quadriceps femoris or hamstring muscles caused a visible interruption of the sinusoidal oscillation of the knee. The primary indicator of an unrelaxed and therefore unusable trial was a nonsinusoidal curve. The EMG activity became a secondary indicator of an unrelaxed oscillation. Figures 5A and 5B demonstrate "good" and "bad" oscillations, respectively. The amplification of the EMG signal in these figures was automatically twice the magnitude of the amplifier gain.

Calculation of the stiffness and damping coefficients was based on the first three full cycles of oscillation. Typical output from the relaxed oscillations about 75 and 45 degrees of knee flexion are presented in Figures 6A and 6B, respectively. As in Figures 5A and 5B, the amplification of the EMG signal in these figures was automatically twice the magnitude of the amplifier gain. These figures show that there was no perceptible EMG activity at either position. Thus, it was assumed that the measured stiffness and damping coefficients were a function of the mechanical properties of the relaxed musculature and inert tissue at the knee rather than any muscle contraction. This assumption was consistent with the data of Long et al,[3] which showed no difference in movement at the MCP joint with relaxed muscles and with the movement following a temporary nerve block.

To determine whether the electrogoniometer itself contributed to the damping or stiffness of the knee, the relaxed oscillations of one subject were also recorded using high-speed (50 frames per second) cinematography. The oscillations about 75 and 45 degrees of knee flexion were filmed with and without the electrogoniometer. Three trials at each position with and without the electrogoniometer were recorded. The film was then analyzed frame by frame in a Vanguard Motion Analyzer. [parallel] The peak displacements were measured, and the viscous damping factor of the knee was calculated for each trial with and without the electrogoniometer.

Data analysis. By monitoring the displacement history of the oscillation using an electrogoniometer, the stiffness and damping coefficients of the knee were determined from the following equations:

(1) k = I [multiplied by] [omega.sup.2]

where k is the stiffness coefficient, I is the moment of inertia of the leg-foot segment at its center of mass, and w is the natural frequency of the oscillation and

(2) c = 2 [multiplied by] [zeta] [multiplied by] [omega] [multiplied] I

where c is the damping coefficient and [zeta] is the viscous damping factor. [19,20]

The viscous damping factor and natural frequency were calculated directly from the oscillation data. The moment of inertia of the leg-foot segment was calculated for each subject from a geometrical model using anthropometric measurements from each subject.[20]

To assess the reliability of the measurements, 20 subjects (10 men and 10 women, aged 20-27 years) were tested on three separate days within 1 week. The mean age of this group was 21.6 years (SD=2.2). The average age and age range for each gender are shown in Table 1. Each test day included three test trials preceded by as many practice trials as were necessary to allow the subject to accomplish the relaxed oscillation. Intraclass correlation coefficients (ICC[2,1]) were calculated to determine within-day and across-day reliability.[24] This form of the ICC was chosen to treat the effects of the judges as random effects.

The effects of age and gender on the dependent variables, stiffness and damping coefficients, were assessed by a two-way analysis of variance ANOVA) by grouping the subjects into six age groups by decade. Interactions were ignored in the ANOVA because the number of subjects in each age group varied. The effects of joint position were evaluated by a t test for each coefficient.

Results

From the film data, the average viscous damping factor ([zeta]) was .054 and .057 (a unitless quantity) with and without the electrogoniometer, respectively. I concluded that the electrogoniometer had no measurable effect on the coefficients of the knee.

The ICCs for the stiffness and damping coefficients for the two positions between and across days are presented in Table 2. All of the coefficients showed high reliability (ICCs>.8), except the damping coefficient derived from the 45-degree test across the three test days, which showed moderate reliability (ICC>.67). The stiffness coefficients were slightly more reliable than the damping coefficients, and the coefficients from the 75-degree test were more reliable than those from the 45-degree test. Reliability of measurements taken on the same test day was greater than the reliability of measurements made across days.

Table 2. Intraclass Correlation
Coefficient (ICC[2,1]) Values for Stiffness
and Damping at Both Positions Across
Trials and Days

            Position ([degrees])   Days   Trials

Damping     45                     .679   .864
            75                     .805   .951
Stiffness   45                     .810   .930
            75                     .903   .959

To allow sufficient time for practice, the third trial of a single test day was considered to provide a reasonable representation of the stiffness and damping coefficients for each subject. The data used in the rest of the analyses were the third trials of the first day for subjects in the reliability study or of the only test day for the remaining subjects.

Analysis of variance revealed significant gender differences for both damping and stiffness coefficients at both positions (df = 1, P<.001). Both the stiffness and damping coefficients were consistency greater in the men than in the women. At 75 degrees, the mean stiffness was 4.07 N [multiplied by] m/rad (SD = 0.95) for the men as compared with 2.65 N [multiplied by] m/rad (SD = 0.64) for the women. At 45 degrees, the mean stiffness was 4.41 N [multiplied by] m/rad (SD = 1.11) for the men as compared with 2.78 N [multiplied by] m/rad (SD = 0.82) for the women. Similarly, at 75 degrees, the mean damping coefficient was 0.11 N [multiplied by] m [multiplied by] s/rad (SD = 0.05) for the men and 0.06 N [multiplied by] ms/rad (SD = 0.02) for the women: At 45 degrees, the mean damping coefficient was 0.13 N [multiplied by] ms/rad (SD = 0.06) for the men and 0.08 N [multiplied by] ms/rad (SD = 0.05) for the women.

Joint position also affected the stiffness and damping coefficients. To assess this effect, the data from both male and female subjects of all ages were combined. Both stiffness and damping coefficients were significantly reduced in the 75-degree position compared with the 45-degree position (t = 5.01, P<.001 for damping; t = 2.56, P<.05 for stiffness). The mean stiffness coefficient was 3.38 N [multiplied by] m/rad (SD = 1.08) at 75 degrees compared with 3.58 N [multiplied by] m/rad (SD = 1.27) at 45 degrees. The mean damping coefficient was 0.08 N [multiplied by] m [multiplied by] s/rad (SD = 0.04) at 75 degrees compared with 0.10 N [multiplied by] m [multiplied by] s/rad (SD = 0.05) at 45 degrees.

The results of the two-way ANOVA for the stiffness coefficient at 75 degrees did show a significant effect by age. The means of the stiffness and damping coefficients for each position for each age group are presented in Table 3. A post hoc Scheffe method of multiple correlations revealed that the means of the 30- to 39-year-old and 70+-year-old age groups were significantly different from the other means P<.05).

[TABULAR DATA OMITTED]

Discussion

The purpose of this study was to utilize the theories of spring mechanics to measure knee joint stiffness in a clinically feasible manner. Assessment of reliability revealed that the test yields moderately to highly reliable measurements. The greatest amount of variability was seen across days for the damping coefficient at 45 degrees (ICCs>.67). The variability in the measure quite possibly came from the difficulty in exactly reproducing the test position. More complete stabilization of the oscillating limb may have decreased this variability. Despite this variability, the tests were regarded as sufficiently reliable to use to evaluate a large population of healthy subjects.

Assessment of gender effects revealed significant differences between male and female subjects of all ages. These results are consistent with those reported by Wright and Johns.[4] Previous studies[13,14] demonstrated that stiffness increased with muscle activity and following a strengthening program. The gender differences seen in stiffness, therefore, may be related to the strength differences between the two genders. These differences may also reflect the generally larger limb size in men than in women because moment of inertia is a factor in the calculation of these coefficients.

I made no attempt to determine the source or sources of stiffness and damping at the knee. Studies of the wrist joints of cats revealed that almost 90% of the joint stiffness was due to the joint capsule and muscles.[7] The differences of both coefficients between the two genders, therefore, may reflect basic morphological differences in the two groups.

The differences between the two test positions may also be related to tissues of the joint. The hip was more extended in the 45-degree test, which may have stretched the rectus femoris muscle. This same test position, however, put the knee in more extension, which may have countered the changes on the rectus femoris muscle. The increased extension may have put more tension on the cruciate ligaments or joint capsule, or the articulating surfaces in contact may have had differences in thickness of the articular cartilage, which might account for the differences in mechanical characteristics. The importance of these differences for this study is not their cause, but their existence. These data suggest that there is a level of joint stiffness that is a result of the noncontracting tissues of the joint and that is modified by joint position. Active muscle contraction may then modify that inherent passive stiffness even more to alter the overall functional stiffness of the joint. If stiffness can be used as a basis for control in some movements, then the variability of joint stiffness resulting from joint position may provide some explanation for the variability of muscle activity that appears to modify limb stiffness still further.

The results of tests for assessing agerelated effects were more equivocal. The only significant age differences noted were in the stiffness coefficient at 75 degrees between the 70+-year-old and 30- to 39-year-old age groups. There were fewer subjects in the oldest age group, which may have affected the results, and even these subjects may not have been old enough to exhibit significant age effects. Despite these limitations, however, the decrease in stiffness in this age group, even at only one position, may reflect a loss of muscle mass, which could affect passive stiffness. On the other hand, it is more difficult to explain the increase in stiffness in the 30- to 39-year-old group. This group contained a higher percentage of male subjects (10 out of 17) than any other group. This distribution may have skewed the results toward increased stiffness. No effort was made to keep track of the level of physical activity of the participating subjects. The 30- to 39-year-old age group may have been more physically fit than the other groups, and their increased strength could have resulted in increased stiffness.

Wright and Johns[4] reported an increase in stiffness at the MCP joint of the finger with age, but Chesworth and Vandervoort[12] found no such relationship at the ankle. The tissue surrounding these two joints differs significantly from each other and may explain the differences demonstrated in the two studies. Although the intent of this study was not to identify the contribution of each tissue to the overall stiffness and damping characteristics of the knee, the actual function of an intact joint must reflect the mechanical nature of the entire joint. The purpose of this study was to relate this inherent mechanical character of the knee joint complex to its overall function. Because muscles are the only components of a joint that can actively alter their mechanical nature, their role may be to modulate joint stiffness during activity in order to optimize the stiffness of the joint for a given activity. Their activity thus may be dependent on the underlying stiffness of the inert components of the joint. Stiffness may then be a better measure of tissues' contribution to function than more traditional measures of tissue integrity such as strength and ROM, which have been shown to be poorly related to locomotor function.[25]

There is no known information about the magnitude of the torsional stiffness and damping coefficients of the knee. Therefore, the validity of the data from this study cannot be assessed by direct comparison with the data of other investigators. Wright and Johns[4] reported a range of approximately 0. 1 to 0.3 N [multiplied by] m/rad for the torsional elastic stiffness of the second MCP joint, approximately an order of magnitude smaller than those reported in this study for the knee. Other damped-spring models of the knee have used linear springs[26,27]; thus, the results again cannot be compared directly with the results of the torsional spring model presented in this study. Therefore, further investigations are essential to validate the results reported in my study.

Future studies are needed to relate this concept of stiffness to pathological conditions and to function. Such studies might compare the coefficients of patients with rheumatoid arthritis with those of age- and gender-matched healthy subjects. These ex- perimentally derived coefficients can then be used to mathematically predict the kinematic behavior of the knee during locomotion. Such studies may provide the necessary evidence to demonstrate the clinical relevance of stiffness measures for the knee joint.

Finally, the validity of a simple mechanical model with coefficients that behave linearly should be considered. The simple nature of the oscillation test presented in this study lends itself to a linear model. Preliminary mathematical tests of our data support their linearity. Expansion of this model to more complex functions such as locomotion, however, may require increasing the complexity of the model. For example, the transition from stance phase to swing phase may be more accurately modeled by a spring with nonlinear coefficients.

Conclusions

This report presented a method of reliably assessing knee joint stiffness by characterizing the relaxed oscillations of the leg-foot segment in terms of the stiffness and damping coefficients of a damped spring model. This model, while not isolating the effects of individual tissues or structures, demonstrates differences between male and female subjects that are consistent with morphological differences. Similarly, the model shows age-related effects that may reflect other physiological changes.

The oscillation test provides a rather simple and relatively quick measure of the mechanical properties of the entire joint complex and may prove to be a useful clinical tool for evaluating the integrity of the knee. The instrumentation needed to measure stiffness and damping is relatively inexpensive. The actual calculation of the parameters is straightforward and poses no significant technical difficulty. Further investigation is needed to validate the results of this study and to relate these stiffness and damping coefficients to functions such as gait.

References

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