ANALYTICAL APPROACH FOR EVLAUATION OF THE SENSITIVITY OF A HILL BASED MUSCLE MODEL

Transcription

1 ANALYTICAL APPROACH FOR EVLAUATION OF THE SENSITIVITY OF A HILL BASED MUSCLE MODEL Carol Scovil and Janet Ronsky Human Performance Laboratory, Department of Mechanical Engineering University of Calgary, Calgary, Alberta, Canada, scovil@kin.ucalgary.ca INTRODUCTION Component based Hill muscle s are commonly used in musculoskeletal s to predict muscle forces (Zajac, 1989) and have been recommended as the most practical and accurate approach for simulating human movement(van den Bogert et al., 1998; Winters et al., 1987). Hill based s can take several forms, each component of the representing the properties of one or more tissues in the musculo-tendenous unit. A simple Hill (Figure 1), can include three common elements. The contractile element (CE) s active muscle force and takes into account the force-length (FL) and force-velocity (FV) muscle properties. The series elastic element (SE) s the stretch in the tendon, aponeurosis, and other soft tissue during loading, while the parallel elastic element (PE) s the passive properties of the muscle fibers. In many studies, the PE is ignored, often considered irrelevant when ling physiological movements(winters, 1990). In other s, pennation angle or viscous elements of the muscle are considered while some s include these components in the WIDTH parameter of the FL relationship of the CE (Gerritsen, 1997). When ling movement in multi-body human systems, muscle force calculation is only one component of a larger. Generally muscle s are obtained from literature, often with little regard to the parameters used to define the muscle. To date, no single study has determined all the values necessary to describe the properties for a human muscle. Consequently, values must be compiled from several sources (Table 1). Some of these parameters are calculated in human muscles, while others have been taken from animal experiments. In addition, a wide range of values for these parameters have been reported in the literature. Due to the multifactorial nature of the muscle, only a qualitative evaluation of the sensitivity of such a to changes in muscle parameters has been done (Scovil & Ronsky, 2002). Figure 1: Hill muscle. (refer to text, Table 1 for abbreviations). A) Component. B) Force length properties of the CE (solid line) and PE (dotted line). C) Force length properties of the SE. D) Force velocity properties of the CE. This study presents a quantitative method to evaluate the sensitivity of a Hill to changes in its parameters using partial derivatives. A derivative calculation not only gives a quantitative evaluation, it also indicates the sensitivity of a given parameter over a continuous range of muscle input values. The results from this study will be compared to those presented in Scovil & Ronsky (2002). Table 1: Muscle parameters included in the muscle. Literature values for human muscle listed where possible. Values marked * are muscle specific, rectus femoris values shown. Value in Minimum Maximum Parameter Definition initial literature literature (24) value value A REL Hill s constant in FV equation, normalized to F MAX (5) 0.41 (20) B REL Hill s constant in FV equation, normalized to L CEopt 5.2 s s -1 (20) -1 (20) 5.2 s F ASYMP Force where the FV curve becomes asymptotic 1.5 F MAX 1.1 F (26) MAX 2.3 F (5) MAX F MAX Maximum isometric force for muscle group, based *1560 N σ MUS on physiological cross sectional area (PCSA) and 61 N/cm 2 25 N/cm 2 (15) 95 N/cm 2 (11) PCSA specific tension (σ MUS ), i.e. F MAX = PCSA σ MUS *25.6 cm 2 *8.9 cm 2 (21) *54.1 cm 2 (3) L CEopt length of fiber (the CE) at F MAX *0.084 m *0.055 m (9) *0.084 m (7) L PEsl Slack length of the PE 1.4 L CEopt 1 L (26) CEopt (24) 1.4 L CEopt L SEsl Slack length of the SE *0.346 m *0.346 m (6) *0.410 m (12) SF eccentric/concentric FV curve slopes as V (26) 2 (5) SL Slope of asymptote in FV curve 200 s s -1 (24) -1 (5) 300 s U PE Force in PE at L CEopt 0.5 F MAX 0.5 F (24) MAX 1.1 F (20) MAX U SE stretch in SE at F MAX (26) 0.09 (26) WIDTH width of parabola in FL curve, taking into account muscle fibre pennation and non-uniformities *1.44 *0.4 (8) *1.44 (10)

2 Sandercock and Heckman (1997) evaluated Hill predictions in the cat soleus muscle. They simulated walking experimentally and compared it to predictions from a Hill. In this in-situ experiment, direct measurements of the Hill parameters specific to the led muscle were possible. Interestingly, the measured optimal muscle fibre length (L CEopt ) was 3-8 mm shorter than what they had averaged from literature values. The Hill muscle was able to fit the experimental data to within 10%during muscle activation, and to within 30% during muscle relaxation. Replacing a linear SE with a non-linear SE had little influence on the resulting muscle force, but improved the muscle fibre length prediction significantly. They concluded that few, if any, studies assess Hill errors in practical applications. Hill-type s continue to receive widespread use, so knowledge of their associated errors is important. Lloyd and Buchanan (1996) created a knee for evaluating isometric varus/valgus moments with 13 muscles. To account for the range of parameter values reported in the literature, they selected the mean optimal fibre length, maximum isometric force, pennation angle and tendon slack length values. The parameters were optimized for each subject by varying their values by ± 50% to allow the to match experimental data. Optimal parameter values ranged from +2% to +24% of the literature mean. Several researchers (Brand et al., 1986,, Bobbert, 2001) have evaluated the sensitivity of muscle force sharing s in an overdetermined system. Muscle physiological crosssectional area (PCSA) was evaluated in these studies, and it is also a parameter used in the Hill muscle. As a result, the sensitivity of the parameters in the force-sharing can shed light on the sensitivity of Hill based musculoskeletal s to these same parameters. Raikova and Prilutsky (2001) investigated the sensitivity of a 3 degree of freedom, 9 muscle to PCSA and muscle moment arm parameters. These parameters were varied by factors ranging from 0.05 to 2. Predicted muscle force was found to be very sensitive to PCSA, but predicted activation patterns were not. Brand et al. (1986) also evaluated the sensitivity of a of muscle and joint forces at the hip during walking to PCSA values obtained from two different subjects, as well as from literature. They found predicted muscle forces were very sensitive to the PCSA, but no direct relationship between increasing PCSA and increasing predicted force in that muscle was identified. The joint forces resulting from the muscles were much less sensitive to muscle PCSA, not surprising given that the total resultant force in the hip was the same for each evaluation. Hoy et al. (1990) evaluated the sensitivity of a musculoskeletal to tendon slack length (L SEsl ), muscle moment arm and optimal muscle fibre length (L CEopt ). The joint angle at which maximum isometric muscle force (F MAX ) was produced was sensitive to changes of ±5% in L SEsl and L CEopt. The ratio of L SEsl or L CEopt to muscle moment arm altered the sensitivity of each muscle. A larger ratio in a given muscle for either L SEsl or L CEopt indicated resulting muscle forces had a greater sensitivity to that parameter. Most Hill muscle s include FL properties of the muscle (Winters, 1990; Zajac, 1989). Audu and Davy (1985) compared the effects of using different muscle s in a 3 joint, 5 muscle of kicking. They used Hill s of increasing complexity to evaluate the muscle forces and found that inclusion of the FL properties of the muscle provided much more accurate simulations of experimental muscle activations. Several authors (Chow et al., 1999; Winters, 1990) report that maximum shortening velocity, in addition to muscle force, changes with activation level. This is not considered in the current study as highspeed concentric contractions are rarely encountered during physiological movements due to the low forces produced at these speeds (Zajac, 1989). From these studies it is clear that musculoskeletal s are sensitive to their defining parameters, but that no single study has performed an integrated evaluation of all of the parameters in one. Given the sensitivities of several of the parameters reported by other authors, and the lack of a single experimental source of all parameter values, a quantified evaluation of the sensitivity of the Hill muscle is valuable to provide insights into these muscle s, and to direct further musculoskeletal ling research. METHODS A Hill based muscle (Figure 1) (mathematically described in Nagano and Gerritsen 2001) was used to muscle forces in the forward dynamics running of Wright et al. (1998). This is a 20 degree of freedom of the right leg, with 66 points for ground force interaction and 3 stimulation inputs that controlled the activation level of each of 14 Hill based muscle groups. These muscle groups in the leg are represented by gluteus maximus, gluteus medius, adductor magnus, illiopsoas, rectus femoris, hamstrings, the quadriceps muscles, gastrocnemius, soleus, flexor digitorum, tibialis posterior, tibialis anterior, extensor digitorum and the peroneals. The was able to match the kinematics and ground reaction forces from experimentally determined running data within two standard deviations. At each timestep, the inputs for the muscle were the lengths of the muscle (L MUS ) and the contractile element (L CE ). The outputs from the were the force in the muscle ( ), and the velocity of the contractile element ( ), to allow the prediction of L CE for the next timestep in the movement simulation. For sensitivity analysis, each of the initial muscle parameters (defined in Table 1) was varied individually by a factor of 0.5 to 1.5 and the was re-computed. A previous study evaluated the sensitivity of the muscle -to- parameters was assessed using two different methods (Scovil & Ronsky, 2002). Firstly, the influence of the altered parameters on the muscle outputs, and, was calculated (Figure 2A). Secondly, the sensitivity of the entire running simulation was evaluated, (Figure 2B) with changes measured in the ground reaction forces and joint angles in addition to and. This study presents a third evaluation, which involved taking the partial derivative of the Hill muscle equations with respect to each of its parameters (Figure 2C). The value of the partial derivative was calculated for the parameter values

3 Running simulation L MUS L CE L MUS L CE perturb parameters Calculate muscle Calculate muscle Comparison gives sensitivity of muscle to perturbations A perturb parameters Running simulation Re-compute running simulation joint angles ground forces joint angles ground forces Comparison gives sensitivity of entire forward dynamic to perturbations B X = The partial derivative gives the instantaneous rate of change in the output values as a function of the change in each parameter (X) C Figure 2: Model sensitivity evaluation methods. A) Muscle only B) Running simulation C) Partial derivative method.. used in this study over the range of L MUS and L CE characteristic of this running. Sensitivity of each was evaluated over an entire gait cycle for all muscles. The sensitivity of the first two s was evaluated qualitatively, based on the change observed in the output data. The output data plots were visually inspected and the change due to the initial perturbation were ranked into categories of: None: change less than 1% due to perturbation; Small: change of much less than the perturbation; Large: change of larger than or similar magnitude to the perturbation; Extreme: change in the resulting value by a factor of 25 or greater. (Figure 3) The derivative values were calculated from the muscle equations at each timepoint throughout the gait cycle (i.e. each L CE and L MUS ) and the mean and standard deviation of and over time were determined. The values in Table 2 represent the average mean for all the muscles together, and a pooled standard deviation to combine the variability from all the muscles for a given partial derivative calculation. The quantitative sensitivity evaluation of the partial derivatives was then ranked to match the qualitative ranking from the previous evaluation. Contour plots were created to reflect the continuity of the partial derivative over the range of L CE and L MUS in the running (Figure 4). A B Figure 3: Sensitivity of the running vertical ground reaction force. Legend: Thick line - unperturbed condition. Thin line parameter decreased 50%. Dashed line parameter increased by 50%. Examples of two sensitivity levels: A) Small: USE, B) Large: FMAX.

4 V A CE REL Figure 4: Example contour plot to demonstrate the continuity of the partial derivative method. This is the partial derivative of with respect to the value A REL over the values of L MUS and L CE encountered during running.. RESULTS The evaluation of the partial derivative of the muscle for each of the parameters gave results similar to those found for the muscle-only. Both outputs and were found to be extremely sensitive to changes in the series elastic parameters, L SEsl and U SE (Table 2). In addition the had a large sensitivity to changes in L CEopt, F MAX, F ASYMP and WIDTH. The sensitivity of the was only apparent in for the parameters L CEopt, F ASYMP and WIDTH, while the F MAX only affected. This can be attributed to the formation of the equations for use in the simulation; first force is calculated from muscle length parameters, and then the velocity of the muscle is evaluated based on the calculated force and the remaining parameters. As a result, sensitivities to some parameters will be apparent only in the muscle velocity at an individual timestep. With a numerical simulation over time, the changes would be propagated through to, as the altered velocity would alter the value of the subsequent muscle length. The derivative method has the advantage of being much more general, indicating the sensitivity over a continuous range of values not specific only to running. In addition it is a quantitative evaluation, giving concrete numerical results. One difference was found in the derivative method compared to the other evaluations - no effect was found for the PE parameters. In addition, the sensitivity of the derivative to the WIDTH parameter was found to be small using the derivative method, similar to simulation sensitivity evaluation, but different from the muscle alone. Overall the simulation of running was found to be less sensitive to perturbations in the muscle parameters than the muscle alone (Table 2). Large effects on the simulation outputs were limited to changes in F MAX, L CEopt, and L SEsl. Joint kinematics and GRF were found to be equally sensitive to perturbations in a given parameter. Of particular note is U SE, a parameter to which the muscle was extremely sensitive, but to which the dynamics simulation had only a small sensitivity. Both the muscle alone and the running simulation had small sensitivity to the parameters A REL, B REL, SF and SL. Changes in the PE parameters, L PEsl and U PE, both had a small effect on the running simulation. These changes were larger than the muscle alone, where the sensitivity was small and none respectively for the same parameters. DISCUSSION The partial derivatives of this Hill muscle were successfully evaluated. The techniques presented were evaluated for a particular running, but are generic, and can be applied to other motions, and other parameter ranges. The partial derivatives of the series elastic element parameters were very large for muscle L CE and L MUS values that occur during running. (Table 2) This implies that the constants L Sesl and U SE must be chosen with great care, as their values will strongly affect muscle force and velocity. This is also true for the partial derivatives with a large effect, L CEopt and F ASYMP. This analysis emphasizes the

5 Parameter Muscleonly (graphical) Multi-body dynamic Muscle-only - derivative calculation A REL Small Small Small: (0.28) B REL Small Small Small: 0.96 (0.15) F ASYMP Large Small Large: -6.5 (6.4) F MAX Large Large Large: 1.0 (0.0) L CEopt Large Large Large: 3.8 (17) L PEsl Small Small None: 8.5e -5 (3.5e -3 ) L SEsl Extreme Large Extreme: -1.1e 14 (1.2e 15 ) SF Small Small Small: (0.22) SL Small Small Small: 0.50 (2.5) U PE None Small None: 1.1e -4 (4.4e -4 ) U SE Extreme Small Extreme: -1.5e 4 (1.6e 5 ) WIDTH Large Small Small: (8.6) Table 2: Sensitivity of s to parameter changes. Qualitative evaluation of changes in outputs over entire gait cycle. The derivative values are a mean (standard deviation), calculated at the parameter values in the original. Sensitivity classified as: None: change of less than 1% due to perturbation; Small: change of less than the perturbation; Large: change of larger than or similar magnitude to the perturbation; Extreme: change in the resulting value by a factor of 25 or greater. importance of selecting or very accurately measuring the parameters the is most sensitive to, while the others can be estimated based on similar values reported in literature. Given the range of values that exist in the literature for the parameter values that the muscle is most sensitive to (Table 1), this study emphasizes the need for further research into quantifying these properties. These quantitative results are similar to the qualitative results found previously (Scovil & Ronsky, 2002). Two parameters, F MAX and WIDTH, had large qualitative effects on outputs, but have a small partial derivative. This difference between techniques may imply that the sensitivity levels for the qualitative study should be refined. This study presents a quantitative approach to Hill muscle evaluation that provides a generic, continuous equation that can be applied to other motions and s. The sensitivity of the was found to be similar for the qualitative muscle-only and the quantitative derivative calculation. This is reasonable, as both are calculated from the same muscle equations. The first evaluation shows the sensitivity of the due to ±50% change in the parameter value, the second sensitivity of the due to an infinitesimal change in the parameter value, or the slope of the curve. The first method has the advantage of giving a visual representation of the sensitivity, and does not require the input equations to be explicitly known to evaluate the sensitivity. The derivative method is more rigorous, and reveals how the sensitivity to a parameter changes for different input values of muscle and contractile element length. This provides a generic, continuous equation that can be applied to other motions and s, making the sensitivity found through the derivative method more generally relevant to the biomechanics community. The sensitivity of the entire simulated movement was greatly reduced compared to the muscle alone. The running had a large sensitivity to the value of tendon slack length (L SEsl ), maximum isometric force (F MAX ) and optimal muscle length (L CEopt ). The simulation had a small sensitivity to all the other parameters. It appears that the running was able to compensate for changes in muscle parameters better than the muscle alone. Three potential sources of the reduced sensitivity of the running include: 1) inertial or damping effects of the combined musculoskeletal system, 2) reduction of muscle force and velocity changes due to muscle interactions, and 3) evaluation of the running over time. This final source of reduced sensitivity arises because the running was evaluated dynamically over the entire running cycle, so changes in the previous timesteps were integrated into the next timestep. The muscle-only could not be numerically integrated (L MUS depended on joint angle) so no adjustment to changes over time was possible. Each timestep was evaluated based on L MUS and L CE output from the unperturbed, with no feedback due to the changes in and at the previous timestep. The ability of the running to "react" to the changes in and as it was evaluated may have caused it to be less sensitive than the muscle-only. Results from this study are similar to those found by Hoy et al., who reported that a Hill muscle could be sensitive to small variations (±5%) in L SEsl and L CEopt. The sensitivity of the muscle to changes in F MAX support previously reported findings of muscle sensitivity to physiological cross-sectional area (Lloyd et al., 1996, Brand et al., 1986) as PCSA is typically linearly related to F MAX by the specific tension of the muscle (Delp, 1990). In this study, the running was most sensitive to the same three parameters, L SEsl, F MAX and L CEopt, further supporting the need for accuracy in these values. Many s ignore the parallel elastic element (PE) (Zajac, 1989), an assumption that is supported by this study. This analysis showed that during the stance phase of running, the PE has only a small effect on the outcome. Likely, this is applicable to other physiological movements that do not take the muscle to extreme ranges of motion. It appears reasonable that the PE could be ignored in s of locomotion, further simplifying the Hill muscle. This study presents the use of partial derivative for an effective quantitative method to evaluate the sensitivity of a given to the muscle parameters used, and compared it to previous qualitative results (Scovil & Ronsky, 2002). The sensitivity of the muscle is dependent on the system being led, and parameters must be chosen accordingly. These parameters form the foundation for muscle s used for diverse movements, and should be selected with great care from literature, or measured specifically for the physiological movement being led. ACKNOWLEDGMENTS: W. Herzog. Funding: NSERC, AHFMR, CEMF, U of Calgary.

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