In theory, one way to increase this insufficient net joint torque at the ankle is to dorsiflex the ankle to the extent that the passive contribution to the ankle-joint torque becomes substantial. It was found, however (data not shown), that severe dorsiflexion was part of the optimal solution only when the hip was put much closer to the crank axis, resulting in a power output that was lower than that for the hip positions reported in this study. Irrespective of this finding, however, it is in our view undesirable to rely on passive structures to keep joint angles within the physiological range in reality, because this may lead to injury; in the optimal solutions discussed in this study, passive torques play no role.
Although not easily accomplished in reality, the suggestion that the plantarflexors have insufficient strength can be investigated in the context of this modeling study simply by increasing their maximal isometric force. Systematic increase of Fmax of the ankle muscles in steps of 25% of the original value indeed indicates that at 175% of the original strength, activation was tuned fairly well to the sign changes in MTC velocity once again, resulting in a power output of 83 W, which is 19% more than for the fixed-ankle setup. Still, at 175% ankle musculature strength, the contribution to the power output of the knee and hip muscles, together equaling 66 W, was lower than that found for the fixed-ankle setup (compare Table 3); however, the decrease in power by the knee and hip muscles is now more than offset by the power produced by the plantar flexors (individual muscle power data not shown).
A design variation that (in contrast to the variation in muscle strength) is feasible in reality concerns the position of the hinge between the foot sole and pedal. By moving this point backward, a given pedal force can be sustained with a smaller muscular torque at the ankle joint, due to a decreased moment arm. To investigate the potential of this design change, the hinge between foot and pedal was moved in the dorsal direction in steps of 0.025 m, and stimulation for the released-ankle setup was reoptimized for each of the hinge positions considered. Power output was highest at a hinge position 0.075 m dorsal of the default hinge position. With this hinge position, total power output was 80 W, which is 14% higher than the power output obtained for the fixed-ankle setup. Furthermore, the power output of hip and knee extensors was very close to their power output for the fixed-ankle setup, and thus the gain in power was largely produced by the additionally stimulated ankle-joint muscles (Table 4). In line with these results, the tight coordination between muscle activation and muscle-shortening velocity was now restored, as is illustrated for the gluteus maximus in Figure 7b. The resulting pedal force for optimal hinge position differed from the fixed-ankle setup in several respects (Fig. 4a). First of all, the radial component of the force was much smaller for the released-ankle setup. Second, the tangential component decreased less sharply around a 180° crank angle. Thus, the increase in average crank torque, which directly reflects the increase in power output, resulted primarily from an increase in total crank torque (for two legs) around a crank angle of 20 and 200° (Fig. 4b), which was where total crank torque was smallest. As a result, the minimal total crank torque for the released ankle joint was slightly positive, in contrast to the fixed-ankle setup, where it was slightly negative (Fig. 4b).
The simulation results presented indicate that the gain in power output as a result of additional stimulation of muscles spanning the ankle joint was smaller than expected on the basis of the additional muscle mass stimulated. The results strongly suggest that this was due to the constraint on coordination introduced by the accompanying release of the ankle joint. In fact, without tuning of the mechanical conditions (hinge position between foot and pedal), the power output was reduced rather than increased by release of the ankle joint and stimulation of the ankle muscles. An interesting side effect of releasing the ankle joint was that, while optimizing for power output, releasing the ankle joint resulted in reduction of the radial component of pedal force and an increase in the minimal value of the total torque exerted on the crank. With the released ankle joint, this total torque was found to be positive all through the crank cycle.
It must be kept in mind that these results were obtained on the basis of a highly simplified model of FES cycling. Several important issues related to FES cycling are not taken into account. First, pathological muscle responses such as spasms were not modeled, and muscle stimulation was not constrained by the need to prevent such responses; the only measure taken in this respect is the use of stimulation ramps, as is done experimentally (6). Even though for a ramp that covers 20° of crank angle, a stimulation frequency of 40 Hz (while cycling at 45 RPM) results in only three pulses during the ramp, using this discrete approximation of a ramp has been found experimentally to reduce the probability of spastic responses (6). Another important issue is that prevention of fatigue was not part of the optimization criterion used, whereas in reality the stimulation frequency is chosen such that fatigue is postponed, at the expense of initial power output (21). As a result, the optimal solution represented the mechanical behavior of the system under “sprinting conditions.” This may partly explain why, despite the fact that muscle forces were scaled to a mere 17% of those used in a previous study on able-bodied sprint cycling (19), the power output is still much higher than what is typically observed during FES cycling (5,6). Another factor contributing to this difference in power output may be that in available FES cycling ergometers, the crank-angle intervals over which muscles are stimulated are much shorter than those determined on the basis of optimization in this study. A final limitation is that most of the parameter values used in this study were based on a model of able-bodied cycling, and were thus not typical for subjects involved in FES cycling; it should be noted, on the other hand, that variability in muscle parameter values between paraplegics is so large (7) that a single model representative of paraplegics involved in FES cycling does not exist, as also argued by Sinclair (16). We expect that individualized models will be required to predict, for instance, the optimal hinge position between foot and pedal.
As a consequence of the issues mentioned above, the power output found in this study was higher than the sustainable power output achievable in reality. However, we see no reason why any of these issues should be more favorable (or less unfavorable) to the released-ankle setup than to the fixed-ankle setup. Consequently, we expect that our conclusion that releasing the ankle joint will not lead to a large increase in power output will be upheld in reality. It must be noted that this expectation does not imply that stimulation of the ankle muscles cannot serve any purpose. Stimulation of ankle muscles may well have positive effects on, for example, blood circulation, muscle volume, and bone mineralization; however, evaluation of such effects was beyond the scope of this study.
Apart from the fact that the gain in power output was predicted to be modest at best, releasing the ankle joint introduces injury risks not present in the fixed-ankle setup. With a released ankle joint, unpredictable behavior such as spasms, as well as time-dependent behavior (fatigue), can lead to situations where, for example, hyperextension of the knee occurs. Thus, extreme care must be taken during experimental validation of the predicted effect of releasing the ankle joint. In future work, we will first validate the model predictions for a fixed ankle joint, by measuring muscle characteristics in FES subjects and comparing individualized model-based predictions with experimental data for the fixed-ankle setup. As a second step, it will be attempted to experimentally evaluate the predictions on the effect of releasing the ankle joint.
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In this appendix we provide additional information about the functions described schematically in Figure 2.
Function f1 represents the activation dynamics model proposed by He et al. (9), slightly modified to ensure that active state q is never lower than q0:
When switching between no stimulation (STIM = 0) and maximal stimulation (STIM = 1), this equation represents a linear first-order dynamical system with an activation time constant equaling 1/(c1+c2) and a deactivation time constant equaling 1/c2. For continuously changing values of STIM, it is less straightforward to interpret this equation. Parameter values used in this study were q0 = 0.005, c1 = −6.2, and c2 = 15.5; these values result in the time constants given in the main text.
Function f2 describes how contractile element (CE) velocity is calculated from contractile element force Fce, contractile element length Lce, and q. This function depends both on the force–length and on the force–velocity characteristics that are presented graphically in Figure 8. The active force–length characteristic is modeled as a second-order polynomial where the zero-crossings for the normalized isometric force Fisom,rel occur at Lce,rel = Lce/Lce(opt) = 1 ± 0.56 for each of the muscles (see Fig. 8a). This characteristic is scaled by two muscle-specific parameters: the maximum isometric force Fmax and the CE optimum length Lce(opt). This active force–length characteristic is complemented by a passive force–length characteristic contributed by the parallel elastic element PE. The PE force is modeled to depend quadratically on Lce,rel, where PE starts to generate force at Lce,rel = 1.5, and where the value of PE force at Lce,rel = 1.56 equals 0.5·Fmax.
FIGURE 8— a. Isometric force–length characteristic as used in this study. Lce,rel is CE length relative to CE optimum length; Fisom,rel is isometric CE force relative to maximal isometric force; q is active state. See Appendix for details. b. Force–velocity characteristic as used in this study for Lce,rel (CE length relative to CE optimum length) = 0.6, 0.8, 1.0, and 1.4. Note that only for Lce,rel < 1 the maximal shortening velocity scales with Fisom,rel, the isometric CE force relative to maximal isometric force. See Appendix for details. c. Force–velocity characteristic as used in this study for active state q = 0.1, 0.4, 0.7, 1.0. Note that only for low values of q the maximal shortening velocity depends on q. See Appendix for details
The concentric part of the force–velocity characteristic is described using the classic Hill equation, which is solved for vce,rel, the time derivative of Lce,rel (note that vce,rel < 0 indicates CE shortening):
Here, Fce,rel indicates CE force relative to Fmax. Two modifications are incorporated: first, only for Lce,rel < 1, the maximum shortening velocity is scaled by Fisom,rel (22); this is implemented by setting a*rel = arel·Fisom,rel for Lce,rel > 1 and a*rel = arel otherwise; second, only for low values of q (q < qcrit), maximum shortening velocity is related to q, based on (13); in our implementation, when q < qcrit, maximum shortening velocity decreases quadratically with q in such a way that at q = q0, maximal shortening velocity is 10% of its original value. This is achieved by setting b*rel = brel for q ≥ qcrit and by setting:
for q < qcrit. For a graphical representation of the force–velocity characteristic at different values of q and Lce,rel, see Figures 8b and 8c. Parameter values used in this study were identical for all muscles: arel = 0.35, brel = 2.25, and qcrit = 0.3.
The eccentric part of the force–velocity characteristic is described by a slightly slanted hyperbola of the following general form:
We used a slanted hyperbola to prevent numerical problems in calculating vce,rel from Fce,rel at high eccentric velocities; the four parameters describing this hyperbola can be derived from the following four conditions. First, the eccentric and concentric functions are continuous; second, the first derivative of the eccentric curve in the isometric point is twice that of the concentric curve, based on (12); third, the slanted asymptote has a value of 1.5·q·Fisom,rel at vce,rel = 0; and fourth, the slanted asymptote has an arbitrarily chosen low slope dFce,rel/dvce,rel (see Figs. 8b and 8c).
Functions f3 and f5 both follow from the way in which we describe the relation between joint angles and muscle-tendon-complex length LMTC. In general terms, we described Loi as a polynomial function of a joint angle ϕjoint (defined as the difference between adjacent segment angles ϕ):
For biarticular muscles, LMTC depends on two joint angles. Values for A1 and A2 are based on cadaver data (Huijing, personal communication, 1990), obtained using the tendon excursion method (8). It was found that for most muscles the description of these data could not be improved significantly by using a polynomial of order higher than 1. As the moment arms are by definition equal to dLMTCdϕjoint, this implies that most of our moment arms do not vary with joint angle. A0, the length offset for each muscle, was estimated on the basis of origin and insertion locations.
Function f4 describes the force–length characteristic of the series elastic element (SE):
For each muscle, kse was set to a value that results in a relative elongation of 0.04 at maximal isometric force, that is:
SE slack length Lslack is a muscle-specific parameter that was chosen in such a way that the muscle generates active torque over the appropriate range of joint angles.
Function f6 represents the dynamics of the skeletal system. For both the fixed- and released-ankle setup, we derive these equations of motion using a Newton–Euler method that is extensively described in Casius et al. (3). In this method, kinematic constraints (e.g., to fix the position of the hip in space) are incorporated in terms of the second time derivative of the positional constraint equations. Although this method does not control for error accumulation at the positional level, it was found that violation of the kinematic constraints in terms of position were always below 10−6 m in this study.