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A theoretical approach for modeling peripheral muscle fatigue and recovery
ARTICLE IN PRESS Journal of Biomechanics 41 (2008) 3046–3052
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A theoretical approach for modeling peripheral muscle fatigue and recovery Ting Xia a, Laura A. Frey Law a,b, a b
Center for Computer-Aided Design, College of Engineering, The University of Iowa, Iowa City, IA 52242, USA Graduate Program in Physical Therapy and Rehabilitation Science, College of Medicine, The University of Iowa, Iowa City, IA 52242, USA
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 11 July 2008
A three-compartment model is presented to describe muscle activation, fatigue, and recovery under a variety of loading conditions. Muscle is considered to be in one of three states: resting (MR), activated (MA), or fatigued (MF). A bounded proportional controller represents muscle activation–deactivation, the transfer between MR and MA. The fatigue and recovery rates determine the transfer to/from MF state. The model qualitatively demonstrates empirically based fatigue behavior, known as Rohmert’s curves, with isometric loading conditions. An expanded version of the model utilizes the properties of three muscle fiber types and a last-in-first-out stack mechanism to represent the known muscle recruitment hierarchy. Additionally, a novel yet practical approach is introduced to quantitatively evaluate taskrelated muscle fatigue for complex and/or dynamic movements at the joint level, encompassing the nonlinear influences of joint angle and velocity. This approach may have potential for digital human modeling, ergonomics, and other real-time applications due to its computational efficiency. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Compartmental model Proportional controller Muscle recruitment Joint strength Endurance time
1. Introduction Muscle fatigue, or failure to maintain the required or expected force (Edwards, 1981), is a complex phenomenon. Fatigue is the collective result of many physiological and neurological processes occurring simultaneously, making it impossible to single out a universal mechanism responsible for the loss of force (Fitts, 1994; Abbiss and Laursen, 2005; Barry and Enoka, 2007). Muscle fatigue is also task-related and can vary across muscles and joints (El ahrache et al., 2006; Enoka and Duchateau, 2007). This partially explains why muscle fatigue has been so challenging to represent analytically. Most existing muscle fatigue models adopt either empirical or theoretical approaches. Empirical fatigue models have made only limited advances since the work pioneered by Rohmert in the 1960s (Rohmert, 1960; El ahrache et al., 2006). Current standard fatigue tools used in ergonomic applications employ regression tools to describe the relationship between static task endurance times (ETs) and work intensity at specific body segment or whole body levels. Although straightforward, this approach lacks the ability to generalize to more complex tasks. Theoretical muscle fatigue models typically introduce one or more decay terms into existing muscle force models to represent
Corresponding author at: Graduate Program in Physical Therapy and Rehabilitation Science, The University of Iowa, 1-246 Medical Education Building, Iowa City, IA 52242-1190, USA. Tel.: +1 319 335 9804; fax: +1 319 335 9707. E-mail address: [email protected] (L.A. Frey Law).
0021-9290/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2008.07.013
fatigue (Giat et al., 1993; Hawkins and Hull, 1993; Giat et al., 1996; Riener et al., 1996; Ding et al., 2000). These models are relatively complex but useful at the single muscle level. However, they do not readily handle task-related biomechanical factors such as joint angle and velocity. For example, models can involve anywhere from 15 (Ding et al., 2000) to over 30 parameters (Giat et al., 1993), which must be fit individually. Although each has strengths, they are not easily implemented into multiple muscle or joint level applications requiring computational efficiency. More recently, a simple, yet elegant motor unit (MU)-based fatigue model was proposed, incorporating fatigue dynamics as its core mechanism rather than modifying of existing muscle force models (Liu et al., 2002). The model uses three muscle-activation states: resting, activated, and fatigued, with its behavior (i.e., muscle activation and fatigue dynamics) described using differential equations. The model predicts fatigue for a maximal static exertion (Liu et al., 2002), but cannot predict fatigue for submaximal or dynamic conditions. Thus, the purpose of this study was to develop a mathematical muscle fatigue model that (1) is consistent with known muscle physiology and joint biomechanical properties, (2) is capable of predicting peripheral fatigue for both simple and complex tasks at varying intensities, and (3) is computationally efficient. Using a unique combination of compartmental and control theory, we present a model of muscle fatigue and recovery that achieves these objectives. Furthermore, using a novel method of incorporating three-dimensional (3D) representations of maximum joint strength to normalize predicted task intensities, task complexity and joint mechanics are inherently included in this fatigue model.
ARTICLE IN PRESS T. Xia, L.A. Frey Law / Journal of Biomechanics 41 (2008) 3046–3052
In the following sections, the underlying theories in model construction and preliminary validation are presented. 2. Methods 2.1. Basic three-compartment fatigue model Compartment theory, an approach, which has been successfully applied to model substance transport and chemical reaction phenomena, provides an ideal mechanistic tool for muscle fatigue modeling. In particular, we propose to model fatigue at the whole muscle level, encompassing the excitation–contraction coupling chain within each MU. This can be easily adapted to synergistic muscles at the joint level, but model behavior is still described in terms of MU activation and fatigue. Previously, Liu et al. (2002) proposed an MU-based model that divides muscles into three activation states: resting, activated, and fatigued. We use a similar approach but combine compartment theory with control theory to rigorously define system behavior in accordance with muscle physiology. In theory, once an MU is activated, its force production begins to decay (fatigue) over time. Though the active tension of a single MU varies with the level of fatigue, the behavior of a group of activated MUs is mathematically equivalent to a mixture of ‘‘ideally activated’’ MUs in full tension and ‘‘ideally fatigued’’ MUs in zero tension. For this purpose, the collection of the hypothetical full-tensioned MUs is denoted as the activated compartment (MA), whereas the collection of the hypothetical zerotensioned MUs is denoted as the fatigued compartment (MF). Although MUs in vivo would never be expected to reach zero tension in reality, this simplified approach is used to represent the overall summation of MUs as they are activated, deactivated, and fatigued. Accordingly, the muscle force or tension generated by a group of MUs is proportional to the size of MA. The MUs in the resting state are denoted as MR. Fig. 1 shows the structure of this three-compartment system and the flows between the compartments, described mathematically by Eqs. (1). To maximize the flexibility of this model, we propose to use a relative unit-less measure of muscle force in percent of maximum voluntary contraction (%MVC); hence MA is equivalent to the task-specific muscle force in %MVC. dM R ¼ CðtÞ þ R M F dt dM A ¼ CðtÞ F M A dt dM F ¼ F MA R MF dt
(1)
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using a bounded proportional controller, dependent on whether MA achieves TL and the availability of MR. Eqs. (2) describe the mathematical representation of C(t). This fatigue model is relatively insensitive to changes in the muscle force development factor LD and relaxation factor LR, because the time course of muscle force development and relaxation is negligible compared to the time course of fatigue. For example, varying the values of LD and LR from 2 to 50, or a 2500% change, only alters ET by 10%. Thus, we simply assign the same arbitrary value to both. In this case, LD and LR can be viewed as tracking factors ensuring good system behavior: If M A oTL and MR 4TL MA ; CðtÞ ¼ LD ðTL M A Þ If M A oTL and MR oTL M A ; CðtÞ ¼ LD M R If M A XTL; CðtÞ ¼ LR ðTL M A Þ
(2)
Residual capacity (RC) is introduced to describe the remaining muscle strength capability due to fatigue, where 0% indicates no strength reserve (not physiological) and 100% indicates full non-fatigued strength (Eq. (3)). This time-varying term can be used as a multiplier to decay maximum strength capabilities. Additionally, the central drive necessary to perform a task is modeled as brain effort (BE, Eqs. (4)), which may be used as a simple estimate of perceived exertion: RCðtÞ ¼ M A þ M R ¼ 100% M F
(3)
TL 100% RC If TL4RC; BE ¼ 100%
(4)
If TLpRC; BE ¼
2.2. Incorporation of muscle recruitment hierarchy In humans, there are three primary types of skeletal muscle fibers: slow (S), fast fatigue-resistant (FR), and fast fatigable (FF). There exists a recruitment hierarchy in force generation, known as Henneman’s size principle (Henneman et al., 1965). In brief, S MUs are recruited first, followed by FR units during voluntary contractions. If needed, FF units are recruited last. Such muscle recruitment hierarchy can be modeled readily using a three-level last-in-first-out stack structure, where each level consists of a three-compartment sub-system shown in Fig. 1. Each MU sub-system has its unique fatigue and recovery properties in accordance with the fiber type it represents and its relative proportion determined by muscle composition. Fig. 2 shows the programming flowchart of the stack structure fulfilling recruitment hierarchy. The scheme also takes fatigue into account at each level (not shown in Fig. 2). For example, if S has 20% of MU in MF, 100% BE to S yields only 80% maximum force.
As shown in Fig. 1, the transfer rate between MA and MR is described by a bidirectional, time-varying muscle activation–deactivation drive, C(t). There are major differences between C(t) and conventional transfer rates. In traditional compartment theory, transfer rate is equal to the transfer coefficient multiplied by compartment size, such as that used for the fatigue and recovery rates (Eq. (1)). In contrast, C(t) is not proportional to compartment size nor is it transfer coefficient related. Furthermore, C(t) can be discontinuous. Such characteristics are desirable in describing complex in vivo activation–deactivation processes inherent to muscles. Further, it is assumed that the neuromuscular system can produce the required force (i.e., target load or TL) by controlling the size of MA. Hence, C(t) is modeled
target load
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Fig. 1. A three-compartment model representing the dynamic behavior of a homogeneous MU pool, where MR, MA, and MF denote the percentages of resting, activated, and fatigued MUs with respect to total MUs; F and R denote the fatigue and recovery coefficients, respectively, and C(t) denotes the muscle activation– deactivation drive.
end Fig. 2. Programming flowchart of a stack model representing muscle recruitment hierarchy.
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T. Xia, L.A. Frey Law / Journal of Biomechanics 41 (2008) 3046–3052
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Fig. 3. Determination of %MVC by mapping torque time history (solid line) to a theoretical 3D torque–velocity–angle strength surface. To illustrate, each dotted line in B connects a torque (J) and the corresponding maximum torques (}) on the 3D surface (removed for clarity) to map to, at specific joint angle and velocity (*).
2.3. Incorporation of task-driven capability While this and other models have focused on the muscle level for describing muscle force decay due to fatigue, this model can also predict fatigue at the joint level in terms of joint torque decay. In reality, modeling individual muscles in a group of synergistic muscles is relatively complex due to muscle redundancy and complex muscle force moment arm–joint angle relationships, and lacks of adequate techniques to assess individual muscle forces for validation. The only readily available data are net joint torques. Additionally, net joint torque is important with regard to the physical laws governing body movements. Fortunately, these difficulties can be avoided by modeling muscle fatigue at the joint level. In this case, all muscles contributing to a specific joint motion, e.g., elbow flexion, can be treated as a single entity. One means to represent joint strength is to use empirically based 3D surface representations, where maximum joint torque varies with joint angle and angular velocity. Such torque–velocity–angle (TVA) surfaces inherently include muscle length–tension, moment arm, and force–velocity properties. A static or dynamic task can then be transformed to a one-dimensional (1D) %MVC time history, by normalizing task-specific joint torques by the TVA surface. This produces the timevarying TL input for the model. For example, a dynamic joint torque, angle, and velocity time history for a particular task can be obtained through experimental and/or modeling methods (i.e., digital human modeling). Mapping each data triplet to the TVA surface provides the relative intensity required of the complex or dynamic task at each point in time (Fig. 3). This method, although simplistic, provides a novel means to incorporate task specificity into the model, and is very computationally efficient compared to models using explicit theoretical muscle force–velocity and length–tension relationships.
2.4. Preliminary validation MATLAB/SIMULINK (MathWorks, Natick, MA) was used to implement the basic three-compartment model (Fig. 1) and the advanced muscle recruitment hierarchy model (Fig. 2). For conceptual purposes, model parameters were assigned with arbitrary values (Table 1). For the base model, a homogeneous set of MUs was used (i.e., F ¼ 0.1 and R ¼ 0.02). LD and LR were set at 10, which were sufficient for the system to track TL quickly. To examine model response to sustained isometric loading conditions, TL was simulated by a step function with its amplitude set to a constant value between 0% and 100% MVC. The model started with zero fatigue (MF ¼ 0) and zero activation (MA ¼ 0). Additionally, the responses of the base model to a submaximal intermittent isometric loading condition (50% MVC, work cycle 50, 1:1 work rest ratio) and a submaximal dynamic loading condition (0.25 Hz sine wave, 20–80% peak-to-peak MVC) were examined. To examine the effect of rest period on ET, a submaximal intermittent isometric task (50% MVC, exertion duration ¼ 1) was simulated with work:rest ratios of 1:0, 1:1, 1:2, 1:5, and 1:10. The behavior of the advanced recruitment hierarchy model was simulated in response to an intermittent isometric loading condition, with TL varying between 10% and 40% MVC, work cycle ¼ 50, and work:rest in a 1:1 ratio. For simplicity, time is in arbitrary units in all simulations.
3. Results Fig. 4 shows the characteristic behavior of the basic threecompartment model, i.e., the time course of MA, MF, and MR, and
Table 1 Example muscle fatigue model parameters
S FR FF
Muscle composition (%)
Specific tension
F
R
LD
LR
50 25 25
1.0 2.0 3.0
0.01 0.05 0.1
0.002 0.01 0.02
10 10 10
10 10 10
ET in response to four representative sustained isometric loading conditions: 100% MVC, 70% MVC, 30% MVC, and 10% MVC. The simulated ET–%MVC relationship closely resembles a typical Rohmert’s curve (Fig. 5). Fig. 6 shows the behavior of the base model in response to intermittent isometric and dynamic loading conditions. The model clearly demonstrates the recovery effect (i.e., decrease in MF) during the resting periods under intermittent isometric loading conditions. The model can further predict the expected effect of rest periods on ET; greater rest results in greater ET (Fig. 7). Such resting effect persists after subtracting the rest periods from the total ET. The advanced recruitment hierarchy model behavior is demonstrated in response to an intermittent, isometric loading condition (Fig. 8). The behavior of each MU type (S, FR, and FF) and the predicted BE are also shown. As fatigue progresses, RC becomes the limiting factor, i.e., less than TL, and BE reaches 100%.
4. Discussion The compartment theory approach is capable of modeling complex and nonlinear peripheral muscle fatigue behavior. The addition of proportional controller provides the ability to predict fatigue and recovery for both simple and complex activation patterns. The model qualitatively reproduced expected ET behaviors under both sustained and intermittent isometric conditions, consistent with fatigue and recovery data published in the literature (Chaffin et al., 2006; El ahrache et al., 2006). Additionally, the model can provide theoretical predictions of remaining muscle strength (i.e., RC) and simplified estimates of perceived exertion (i.e., BE) in real-time. Although the current model is fundamentally a muscle activation-based model, and could be used at the single muscle level, we demonstrated its ability to function at the joint level using %MVC in torque. We applied a unique methodology to incorporate the inherent strength nonlinearities, i.e., force–velocity, length–tension, and moment arm–joint angle relationships, by normalizing target joint torque (i.e., TL) by empirically based
ARTICLE IN PRESS T. Xia, L.A. Frey Law / Journal of Biomechanics 41 (2008) 3046–3052
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Fig. 4. Distribution of muscle between compartments for four isometric loading conditions, where active (MA: solid line), fatigued (MF: dotted line), and resting (MR: dot–dash line) states are in percent of total motor units (%MU). Arrows demonstrate endurance time, where MA cannot maintain target load.
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%MVC Fig. 5. Endurance time under varying sustained isometric loading conditions. The vertical line represents the non-failure asymptote.
3D TVA joint strength surfaces. This novel approach allows the prediction of complex dynamic tasks to be represented simply as %MVC, thus the model can predict fatigue in tasks of any
complexity. This latest feature allows task-related muscle fatigue to be predicted in a computationally efficient manner. Moreover, initial fatigue conditions can be freely defined, offering the ability to predict fatigue and recovery for multiple tasks performed in sequence. Further, its simplicity would allow for fatigue assessment at multiple joints, thus ET for whole body tasks could be determined by the weakest link of all body joints involved. Future research is needed to determine the optimal model parameters to match realistic fatigue behavior, as well as the effects of age, sex, and training level. Currently, there are two types of muscle fatigue models, either empirically or theoretically based. Empirical fatigue models such as Rohmert’s curves (Rohmert, 1960) rely on statistical representations of measured task ET with respect to work intensity in %MVC. They are straightforward but with considerable drawbacks. Although frequently cited in ergonomics literature (El ahrache et al., 2006), Rohmert’s curves typically involve a sustained static contraction that does not translate readily to intermittent static or dynamic tasks. Attempts to replicate the original Rohmert curves have resulted in similar exponential decays, but with varying asymptotes (El ahrache et al., 2006). Differences in underlying study population, task specificity, or choice of regression protocol may explain the discrepancy. Additionally, regression curves from one task cannot be simply applied to another task. Thus, this statistical approach to modeling fatigue requires extensive data collection with new regression models for each new task. Lastly,
ARTICLE IN PRESS T. Xia, L.A. Frey Law / Journal of Biomechanics 41 (2008) 3046–3052
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Fig. 6. For (A) intermittent and (B) dynamic target loading conditions, the respective model responses are shown in panels (C) and (D), where active (MA: solid line), fatigued (MF: dotted line), and resting (MR: dot–dash line) states are in percent of total motor units (%MU).
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Fig. 7. The effects of resting on endurance time: (A) uncorrected endurance time and (B) pure exertion time with resting periods removed. Curves from left to right: work:rest ratios of 1:0, 1:1, 1:2, 1:5, and 1:10, respectively.
this approach does not provide a simple means to consider varying baseline conditions and/or the addition of multiple joints. Conversely, theoretical fatigue models are useful as they can be applied to a range of activities, and once validated do not require additional experimental data to predict physiological fatigue
behavior. Liu et al. (2002) proposed a three-state fatigue model that accurately predicted maximum isometric grip strength and fatigue. However, no control strategies were used, i.e., BE was defined explicitly as the transfer coefficient from MR to MA. While the approach appears sufficient for a maximum contraction, it
ARTICLE IN PRESS T. Xia, L.A. Frey Law / Journal of Biomechanics 41 (2008) 3046–3052
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Fig. 8. Demonstration of the fatigue dynamics with muscle recruitment hierarchy in response to an intermittent loading condition. Residual capacity (solid line), actual muscle force (dot line) for combined (A) and individual muscle types (B: S, C: FR, and D: FF), respectively, and the corresponding brain effort (dot–dash line, A).
does not apply to submaximal and complex dynamic tasks. In addition to activating muscle, BE also serves to prevent the activated muscles from returning to the resting state as in vivo muscles automatically deactivate without continuous central drive. However, this natural tendency in muscle deactivation is missing in Liu et al.’s model, but is accounted for in our approach. Moreover, the formulation of BE in our model (Eqs. (4)) is closer to the conventional usage of effort. While Liu et al. (2002) discussed muscle recruitment hierarchy in general terms, we provide a clear procedure to implement it mathematically. Our model provides the advantages of Liu et al.’s (2002) model (e.g., realistic peripheral muscle fatigue behavior, computationally efficient), as well as a practical means to address nonlinearities associated with joint angle, muscle length, and contraction velocity in complex tasks.
Other approaches to predicting muscle fatigue are based on single muscle and detailed muscle activation patterns (Giat et al., 1993, 1996; Ding et al., 2000), primarily for applications involving functional electrical stimulation for paralyzed muscle. However, these approaches typically incorporate fatigue as a modifier of relatively complex mathematical muscle models, e.g., simply inserting mathematical maneuvers such as fatigue decay terms (Ding et al., 2000). These approaches can provide realistic predictions of muscle force for isolated muscles, but are cumbersome for joint or whole body applications. Further, electrical stimulation does not activate muscles in a physiological manner, e.g., disrupted recruitment hierarchy (Kubiak et al., 1987; Trimble and Enoka, 1991); thus these models may not be valid to describe voluntary contractions.
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One limitation of the present fatigue model is it utilizes %MVC time history as input, thus is dependent on reasonable target torque profiles (task-specific joint torque) and maximum 3D strength surfaces. As dynamic strength is only recently gaining attention, these surfaces may not be readily available for all joints or populations. Additionally, the present model represents peripheral fatigue and does not include estimates for central fatigue (CF). However, CF could be represented as a scaling factor between 0 and 1, functionally reducing RC (Eqs. (5)). This would result in a modified BE: TL 100% RC CF If TL4RC CF; BE ¼ 100% If TLpRC CF; BE ¼
(5)
This approach is beyond the scope of this model, but could be investigated in future work. Although studies have demonstrated the effects of individual factors (e.g., body size), environmental factors (e.g., heat), task type (e.g., eccentric exercise), and psychological factors (e.g., motivation) on CF (Bigland-Ritchie et al., 1986; Crewe et al., 2008; Millet and Lepers, 2004; Noakes, 2000; Noakes et al., 2005; Tucker et al., 2006), the corresponding dose–response relationships, which are essential for modeling, are still under investigation. Because of these limitations, care must be taken when applying the current model. The model is best suited for low- to moderate-intensity tasks or short-term high-intensity tasks in a controlled environment. Prolonged high-intensity tasks, especially those involving significant amount of eccentric motion, and/ or concurrent exposure to harsh environment may pose additional stresses on the body, such as the cardiovascular system and thermal regulation, thus result in CF and earlier task failure. The accuracy of the model is also dependent on the accuracy of the %MVC task history, including all forces: external loads, limb weight, and co-contraction. In summary, this model provides a theoretical approach to mathematically predict joint torque decay associated with peripheral fatigue. This method is computationally efficient and may be useful for multiple disciplines: ergonomic applications, biomechanical models, and/or digital human modeling. Future work is needed to further validate the accuracy of the model for a variety of tasks under various conditions.
Conflict of interest statement None.
Acknowledgements The current project is supported in part by the United States Council for Automotive Research, Southfield, MI.
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A theoretical approach for modeling peripheral muscle fatigue and recovery Ting Xia a, Laura A. Frey Law a,b, a b
Center for Computer-Aided Design, College of Engineering, The University of Iowa, Iowa City, IA 52242, USA Graduate Program in Physical Therapy and Rehabilitation Science, College of Medicine, The University of Iowa, Iowa City, IA 52242, USA
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 11 July 2008
A three-compartment model is presented to describe muscle activation, fatigue, and recovery under a variety of loading conditions. Muscle is considered to be in one of three states: resting (MR), activated (MA), or fatigued (MF). A bounded proportional controller represents muscle activation–deactivation, the transfer between MR and MA. The fatigue and recovery rates determine the transfer to/from MF state. The model qualitatively demonstrates empirically based fatigue behavior, known as Rohmert’s curves, with isometric loading conditions. An expanded version of the model utilizes the properties of three muscle fiber types and a last-in-first-out stack mechanism to represent the known muscle recruitment hierarchy. Additionally, a novel yet practical approach is introduced to quantitatively evaluate taskrelated muscle fatigue for complex and/or dynamic movements at the joint level, encompassing the nonlinear influences of joint angle and velocity. This approach may have potential for digital human modeling, ergonomics, and other real-time applications due to its computational efficiency. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Compartmental model Proportional controller Muscle recruitment Joint strength Endurance time
1. Introduction Muscle fatigue, or failure to maintain the required or expected force (Edwards, 1981), is a complex phenomenon. Fatigue is the collective result of many physiological and neurological processes occurring simultaneously, making it impossible to single out a universal mechanism responsible for the loss of force (Fitts, 1994; Abbiss and Laursen, 2005; Barry and Enoka, 2007). Muscle fatigue is also task-related and can vary across muscles and joints (El ahrache et al., 2006; Enoka and Duchateau, 2007). This partially explains why muscle fatigue has been so challenging to represent analytically. Most existing muscle fatigue models adopt either empirical or theoretical approaches. Empirical fatigue models have made only limited advances since the work pioneered by Rohmert in the 1960s (Rohmert, 1960; El ahrache et al., 2006). Current standard fatigue tools used in ergonomic applications employ regression tools to describe the relationship between static task endurance times (ETs) and work intensity at specific body segment or whole body levels. Although straightforward, this approach lacks the ability to generalize to more complex tasks. Theoretical muscle fatigue models typically introduce one or more decay terms into existing muscle force models to represent
Corresponding author at: Graduate Program in Physical Therapy and Rehabilitation Science, The University of Iowa, 1-246 Medical Education Building, Iowa City, IA 52242-1190, USA. Tel.: +1 319 335 9804; fax: +1 319 335 9707. E-mail address: [email protected] (L.A. Frey Law).
0021-9290/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2008.07.013
fatigue (Giat et al., 1993; Hawkins and Hull, 1993; Giat et al., 1996; Riener et al., 1996; Ding et al., 2000). These models are relatively complex but useful at the single muscle level. However, they do not readily handle task-related biomechanical factors such as joint angle and velocity. For example, models can involve anywhere from 15 (Ding et al., 2000) to over 30 parameters (Giat et al., 1993), which must be fit individually. Although each has strengths, they are not easily implemented into multiple muscle or joint level applications requiring computational efficiency. More recently, a simple, yet elegant motor unit (MU)-based fatigue model was proposed, incorporating fatigue dynamics as its core mechanism rather than modifying of existing muscle force models (Liu et al., 2002). The model uses three muscle-activation states: resting, activated, and fatigued, with its behavior (i.e., muscle activation and fatigue dynamics) described using differential equations. The model predicts fatigue for a maximal static exertion (Liu et al., 2002), but cannot predict fatigue for submaximal or dynamic conditions. Thus, the purpose of this study was to develop a mathematical muscle fatigue model that (1) is consistent with known muscle physiology and joint biomechanical properties, (2) is capable of predicting peripheral fatigue for both simple and complex tasks at varying intensities, and (3) is computationally efficient. Using a unique combination of compartmental and control theory, we present a model of muscle fatigue and recovery that achieves these objectives. Furthermore, using a novel method of incorporating three-dimensional (3D) representations of maximum joint strength to normalize predicted task intensities, task complexity and joint mechanics are inherently included in this fatigue model.
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In the following sections, the underlying theories in model construction and preliminary validation are presented. 2. Methods 2.1. Basic three-compartment fatigue model Compartment theory, an approach, which has been successfully applied to model substance transport and chemical reaction phenomena, provides an ideal mechanistic tool for muscle fatigue modeling. In particular, we propose to model fatigue at the whole muscle level, encompassing the excitation–contraction coupling chain within each MU. This can be easily adapted to synergistic muscles at the joint level, but model behavior is still described in terms of MU activation and fatigue. Previously, Liu et al. (2002) proposed an MU-based model that divides muscles into three activation states: resting, activated, and fatigued. We use a similar approach but combine compartment theory with control theory to rigorously define system behavior in accordance with muscle physiology. In theory, once an MU is activated, its force production begins to decay (fatigue) over time. Though the active tension of a single MU varies with the level of fatigue, the behavior of a group of activated MUs is mathematically equivalent to a mixture of ‘‘ideally activated’’ MUs in full tension and ‘‘ideally fatigued’’ MUs in zero tension. For this purpose, the collection of the hypothetical full-tensioned MUs is denoted as the activated compartment (MA), whereas the collection of the hypothetical zerotensioned MUs is denoted as the fatigued compartment (MF). Although MUs in vivo would never be expected to reach zero tension in reality, this simplified approach is used to represent the overall summation of MUs as they are activated, deactivated, and fatigued. Accordingly, the muscle force or tension generated by a group of MUs is proportional to the size of MA. The MUs in the resting state are denoted as MR. Fig. 1 shows the structure of this three-compartment system and the flows between the compartments, described mathematically by Eqs. (1). To maximize the flexibility of this model, we propose to use a relative unit-less measure of muscle force in percent of maximum voluntary contraction (%MVC); hence MA is equivalent to the task-specific muscle force in %MVC. dM R ¼ CðtÞ þ R M F dt dM A ¼ CðtÞ F M A dt dM F ¼ F MA R MF dt
(1)
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using a bounded proportional controller, dependent on whether MA achieves TL and the availability of MR. Eqs. (2) describe the mathematical representation of C(t). This fatigue model is relatively insensitive to changes in the muscle force development factor LD and relaxation factor LR, because the time course of muscle force development and relaxation is negligible compared to the time course of fatigue. For example, varying the values of LD and LR from 2 to 50, or a 2500% change, only alters ET by 10%. Thus, we simply assign the same arbitrary value to both. In this case, LD and LR can be viewed as tracking factors ensuring good system behavior: If M A oTL and MR 4TL MA ; CðtÞ ¼ LD ðTL M A Þ If M A oTL and MR oTL M A ; CðtÞ ¼ LD M R If M A XTL; CðtÞ ¼ LR ðTL M A Þ
(2)
Residual capacity (RC) is introduced to describe the remaining muscle strength capability due to fatigue, where 0% indicates no strength reserve (not physiological) and 100% indicates full non-fatigued strength (Eq. (3)). This time-varying term can be used as a multiplier to decay maximum strength capabilities. Additionally, the central drive necessary to perform a task is modeled as brain effort (BE, Eqs. (4)), which may be used as a simple estimate of perceived exertion: RCðtÞ ¼ M A þ M R ¼ 100% M F
(3)
TL 100% RC If TL4RC; BE ¼ 100%
(4)
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2.2. Incorporation of muscle recruitment hierarchy In humans, there are three primary types of skeletal muscle fibers: slow (S), fast fatigue-resistant (FR), and fast fatigable (FF). There exists a recruitment hierarchy in force generation, known as Henneman’s size principle (Henneman et al., 1965). In brief, S MUs are recruited first, followed by FR units during voluntary contractions. If needed, FF units are recruited last. Such muscle recruitment hierarchy can be modeled readily using a three-level last-in-first-out stack structure, where each level consists of a three-compartment sub-system shown in Fig. 1. Each MU sub-system has its unique fatigue and recovery properties in accordance with the fiber type it represents and its relative proportion determined by muscle composition. Fig. 2 shows the programming flowchart of the stack structure fulfilling recruitment hierarchy. The scheme also takes fatigue into account at each level (not shown in Fig. 2). For example, if S has 20% of MU in MF, 100% BE to S yields only 80% maximum force.
As shown in Fig. 1, the transfer rate between MA and MR is described by a bidirectional, time-varying muscle activation–deactivation drive, C(t). There are major differences between C(t) and conventional transfer rates. In traditional compartment theory, transfer rate is equal to the transfer coefficient multiplied by compartment size, such as that used for the fatigue and recovery rates (Eq. (1)). In contrast, C(t) is not proportional to compartment size nor is it transfer coefficient related. Furthermore, C(t) can be discontinuous. Such characteristics are desirable in describing complex in vivo activation–deactivation processes inherent to muscles. Further, it is assumed that the neuromuscular system can produce the required force (i.e., target load or TL) by controlling the size of MA. Hence, C(t) is modeled
target load
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Fig. 1. A three-compartment model representing the dynamic behavior of a homogeneous MU pool, where MR, MA, and MF denote the percentages of resting, activated, and fatigued MUs with respect to total MUs; F and R denote the fatigue and recovery coefficients, respectively, and C(t) denotes the muscle activation– deactivation drive.
end Fig. 2. Programming flowchart of a stack model representing muscle recruitment hierarchy.
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Fig. 3. Determination of %MVC by mapping torque time history (solid line) to a theoretical 3D torque–velocity–angle strength surface. To illustrate, each dotted line in B connects a torque (J) and the corresponding maximum torques (}) on the 3D surface (removed for clarity) to map to, at specific joint angle and velocity (*).
2.3. Incorporation of task-driven capability While this and other models have focused on the muscle level for describing muscle force decay due to fatigue, this model can also predict fatigue at the joint level in terms of joint torque decay. In reality, modeling individual muscles in a group of synergistic muscles is relatively complex due to muscle redundancy and complex muscle force moment arm–joint angle relationships, and lacks of adequate techniques to assess individual muscle forces for validation. The only readily available data are net joint torques. Additionally, net joint torque is important with regard to the physical laws governing body movements. Fortunately, these difficulties can be avoided by modeling muscle fatigue at the joint level. In this case, all muscles contributing to a specific joint motion, e.g., elbow flexion, can be treated as a single entity. One means to represent joint strength is to use empirically based 3D surface representations, where maximum joint torque varies with joint angle and angular velocity. Such torque–velocity–angle (TVA) surfaces inherently include muscle length–tension, moment arm, and force–velocity properties. A static or dynamic task can then be transformed to a one-dimensional (1D) %MVC time history, by normalizing task-specific joint torques by the TVA surface. This produces the timevarying TL input for the model. For example, a dynamic joint torque, angle, and velocity time history for a particular task can be obtained through experimental and/or modeling methods (i.e., digital human modeling). Mapping each data triplet to the TVA surface provides the relative intensity required of the complex or dynamic task at each point in time (Fig. 3). This method, although simplistic, provides a novel means to incorporate task specificity into the model, and is very computationally efficient compared to models using explicit theoretical muscle force–velocity and length–tension relationships.
2.4. Preliminary validation MATLAB/SIMULINK (MathWorks, Natick, MA) was used to implement the basic three-compartment model (Fig. 1) and the advanced muscle recruitment hierarchy model (Fig. 2). For conceptual purposes, model parameters were assigned with arbitrary values (Table 1). For the base model, a homogeneous set of MUs was used (i.e., F ¼ 0.1 and R ¼ 0.02). LD and LR were set at 10, which were sufficient for the system to track TL quickly. To examine model response to sustained isometric loading conditions, TL was simulated by a step function with its amplitude set to a constant value between 0% and 100% MVC. The model started with zero fatigue (MF ¼ 0) and zero activation (MA ¼ 0). Additionally, the responses of the base model to a submaximal intermittent isometric loading condition (50% MVC, work cycle 50, 1:1 work rest ratio) and a submaximal dynamic loading condition (0.25 Hz sine wave, 20–80% peak-to-peak MVC) were examined. To examine the effect of rest period on ET, a submaximal intermittent isometric task (50% MVC, exertion duration ¼ 1) was simulated with work:rest ratios of 1:0, 1:1, 1:2, 1:5, and 1:10. The behavior of the advanced recruitment hierarchy model was simulated in response to an intermittent isometric loading condition, with TL varying between 10% and 40% MVC, work cycle ¼ 50, and work:rest in a 1:1 ratio. For simplicity, time is in arbitrary units in all simulations.
3. Results Fig. 4 shows the characteristic behavior of the basic threecompartment model, i.e., the time course of MA, MF, and MR, and
Table 1 Example muscle fatigue model parameters
S FR FF
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R
LD
LR
50 25 25
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0.002 0.01 0.02
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ET in response to four representative sustained isometric loading conditions: 100% MVC, 70% MVC, 30% MVC, and 10% MVC. The simulated ET–%MVC relationship closely resembles a typical Rohmert’s curve (Fig. 5). Fig. 6 shows the behavior of the base model in response to intermittent isometric and dynamic loading conditions. The model clearly demonstrates the recovery effect (i.e., decrease in MF) during the resting periods under intermittent isometric loading conditions. The model can further predict the expected effect of rest periods on ET; greater rest results in greater ET (Fig. 7). Such resting effect persists after subtracting the rest periods from the total ET. The advanced recruitment hierarchy model behavior is demonstrated in response to an intermittent, isometric loading condition (Fig. 8). The behavior of each MU type (S, FR, and FF) and the predicted BE are also shown. As fatigue progresses, RC becomes the limiting factor, i.e., less than TL, and BE reaches 100%.
4. Discussion The compartment theory approach is capable of modeling complex and nonlinear peripheral muscle fatigue behavior. The addition of proportional controller provides the ability to predict fatigue and recovery for both simple and complex activation patterns. The model qualitatively reproduced expected ET behaviors under both sustained and intermittent isometric conditions, consistent with fatigue and recovery data published in the literature (Chaffin et al., 2006; El ahrache et al., 2006). Additionally, the model can provide theoretical predictions of remaining muscle strength (i.e., RC) and simplified estimates of perceived exertion (i.e., BE) in real-time. Although the current model is fundamentally a muscle activation-based model, and could be used at the single muscle level, we demonstrated its ability to function at the joint level using %MVC in torque. We applied a unique methodology to incorporate the inherent strength nonlinearities, i.e., force–velocity, length–tension, and moment arm–joint angle relationships, by normalizing target joint torque (i.e., TL) by empirically based
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Fig. 4. Distribution of muscle between compartments for four isometric loading conditions, where active (MA: solid line), fatigued (MF: dotted line), and resting (MR: dot–dash line) states are in percent of total motor units (%MU). Arrows demonstrate endurance time, where MA cannot maintain target load.
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3D TVA joint strength surfaces. This novel approach allows the prediction of complex dynamic tasks to be represented simply as %MVC, thus the model can predict fatigue in tasks of any
complexity. This latest feature allows task-related muscle fatigue to be predicted in a computationally efficient manner. Moreover, initial fatigue conditions can be freely defined, offering the ability to predict fatigue and recovery for multiple tasks performed in sequence. Further, its simplicity would allow for fatigue assessment at multiple joints, thus ET for whole body tasks could be determined by the weakest link of all body joints involved. Future research is needed to determine the optimal model parameters to match realistic fatigue behavior, as well as the effects of age, sex, and training level. Currently, there are two types of muscle fatigue models, either empirically or theoretically based. Empirical fatigue models such as Rohmert’s curves (Rohmert, 1960) rely on statistical representations of measured task ET with respect to work intensity in %MVC. They are straightforward but with considerable drawbacks. Although frequently cited in ergonomics literature (El ahrache et al., 2006), Rohmert’s curves typically involve a sustained static contraction that does not translate readily to intermittent static or dynamic tasks. Attempts to replicate the original Rohmert curves have resulted in similar exponential decays, but with varying asymptotes (El ahrache et al., 2006). Differences in underlying study population, task specificity, or choice of regression protocol may explain the discrepancy. Additionally, regression curves from one task cannot be simply applied to another task. Thus, this statistical approach to modeling fatigue requires extensive data collection with new regression models for each new task. Lastly,
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Fig. 6. For (A) intermittent and (B) dynamic target loading conditions, the respective model responses are shown in panels (C) and (D), where active (MA: solid line), fatigued (MF: dotted line), and resting (MR: dot–dash line) states are in percent of total motor units (%MU).
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this approach does not provide a simple means to consider varying baseline conditions and/or the addition of multiple joints. Conversely, theoretical fatigue models are useful as they can be applied to a range of activities, and once validated do not require additional experimental data to predict physiological fatigue
behavior. Liu et al. (2002) proposed a three-state fatigue model that accurately predicted maximum isometric grip strength and fatigue. However, no control strategies were used, i.e., BE was defined explicitly as the transfer coefficient from MR to MA. While the approach appears sufficient for a maximum contraction, it
ARTICLE IN PRESS T. Xia, L.A. Frey Law / Journal of Biomechanics 41 (2008) 3046–3052
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Fig. 8. Demonstration of the fatigue dynamics with muscle recruitment hierarchy in response to an intermittent loading condition. Residual capacity (solid line), actual muscle force (dot line) for combined (A) and individual muscle types (B: S, C: FR, and D: FF), respectively, and the corresponding brain effort (dot–dash line, A).
does not apply to submaximal and complex dynamic tasks. In addition to activating muscle, BE also serves to prevent the activated muscles from returning to the resting state as in vivo muscles automatically deactivate without continuous central drive. However, this natural tendency in muscle deactivation is missing in Liu et al.’s model, but is accounted for in our approach. Moreover, the formulation of BE in our model (Eqs. (4)) is closer to the conventional usage of effort. While Liu et al. (2002) discussed muscle recruitment hierarchy in general terms, we provide a clear procedure to implement it mathematically. Our model provides the advantages of Liu et al.’s (2002) model (e.g., realistic peripheral muscle fatigue behavior, computationally efficient), as well as a practical means to address nonlinearities associated with joint angle, muscle length, and contraction velocity in complex tasks.
Other approaches to predicting muscle fatigue are based on single muscle and detailed muscle activation patterns (Giat et al., 1993, 1996; Ding et al., 2000), primarily for applications involving functional electrical stimulation for paralyzed muscle. However, these approaches typically incorporate fatigue as a modifier of relatively complex mathematical muscle models, e.g., simply inserting mathematical maneuvers such as fatigue decay terms (Ding et al., 2000). These approaches can provide realistic predictions of muscle force for isolated muscles, but are cumbersome for joint or whole body applications. Further, electrical stimulation does not activate muscles in a physiological manner, e.g., disrupted recruitment hierarchy (Kubiak et al., 1987; Trimble and Enoka, 1991); thus these models may not be valid to describe voluntary contractions.
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One limitation of the present fatigue model is it utilizes %MVC time history as input, thus is dependent on reasonable target torque profiles (task-specific joint torque) and maximum 3D strength surfaces. As dynamic strength is only recently gaining attention, these surfaces may not be readily available for all joints or populations. Additionally, the present model represents peripheral fatigue and does not include estimates for central fatigue (CF). However, CF could be represented as a scaling factor between 0 and 1, functionally reducing RC (Eqs. (5)). This would result in a modified BE: TL 100% RC CF If TL4RC CF; BE ¼ 100% If TLpRC CF; BE ¼
(5)
This approach is beyond the scope of this model, but could be investigated in future work. Although studies have demonstrated the effects of individual factors (e.g., body size), environmental factors (e.g., heat), task type (e.g., eccentric exercise), and psychological factors (e.g., motivation) on CF (Bigland-Ritchie et al., 1986; Crewe et al., 2008; Millet and Lepers, 2004; Noakes, 2000; Noakes et al., 2005; Tucker et al., 2006), the corresponding dose–response relationships, which are essential for modeling, are still under investigation. Because of these limitations, care must be taken when applying the current model. The model is best suited for low- to moderate-intensity tasks or short-term high-intensity tasks in a controlled environment. Prolonged high-intensity tasks, especially those involving significant amount of eccentric motion, and/ or concurrent exposure to harsh environment may pose additional stresses on the body, such as the cardiovascular system and thermal regulation, thus result in CF and earlier task failure. The accuracy of the model is also dependent on the accuracy of the %MVC task history, including all forces: external loads, limb weight, and co-contraction. In summary, this model provides a theoretical approach to mathematically predict joint torque decay associated with peripheral fatigue. This method is computationally efficient and may be useful for multiple disciplines: ergonomic applications, biomechanical models, and/or digital human modeling. Future work is needed to further validate the accuracy of the model for a variety of tasks under various conditions.
Conflict of interest statement None.
Acknowledgements The current project is supported in part by the United States Council for Automotive Research, Southfield, MI.
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