**4.2. Running fatigue tests**

156 Injury and Skeletal Biomechanics

**4. Experimental setup** 

**4.1. Impact accelerations** 

back [13,15,22,26-29].

**Figure 2.** The force diagram for the model elements presented in Figure 1.

Information about the impulsive loading along the skeletal elements in long-distance running can be non-invasively obtained from the foot-ground reactive forces [25] and, more

Non-invasive in vivo measurements of acceleration and impact transmission along the human body were made by externally attaching light-weight, high-sensitivity accelerometers at strategic points including bony prominences, such as the tibial tuberosity below the knee area, the greater trochanter near hip level and the sacrum area at the lower

In this study, each subject was instrumented with two light-weight (4.2 grams) uniaxial (Kistler PiezoBeam, type 8634B50, Kistler, Winterthur, Switzerland), skin-mounted accelerometers connected to a coupler (Kistler Piezotron, type 5122). One was attached on the tibial tuberosity, and the second - on the sacrum. To achieve good reliability of the measurements by means of bone-mounted accelerometers, the accelerometers were pressed onto the skin in closest position to the bony prominences of the tibial tuberosity and the

directly, by measuring the transient accelerations on the shank caused by impact.

An overview of the experimental setup is shown in Figure 3. For examining the effect of global fatigue due to running, the subjects were asked to run on a Quinton Q55 treadmill.

**Figure 3.** Description of the experimental apparatus with a subject running on a treadmill. Two accelerometers are attached , one on the tibial tuberosity (Tib. Tub.) and the other on the sacrum (Sac.) ; the accelerometers' data are sampled through an amplifier and A/D converter to a PC. Likewise, the respiratory data are sampled and collected on-line.

Global, or metabolic fatigue is associated with the development of metabolic acidosis following an endurance exercise and is accompanied by a decrease in the end tidal carbon dioxide pressure (PETCO2) [31]. In long distance running metabolic fatigue is reached when the running speed exceeds the anaerobic threshold [31].

Running was thus for a duration of 30 min and at a speed exceeding the anaerobic threshold speed of each subject by 5%. Before the test a 15 min warming up running on the treadmill was performed. In this study, the average running speed for all 14 subjects was 3.53 m/s (SD, 0.19). It should be noted, however, that in addition to global fatigue, local fatigue in a muscle may also take place as a result of an intensive activity of this muscle. This type of fatigue is reflected by certain changes in its electromyogram (EMG) signal in the time and/or frequency domains [32]. Local fatigue was not considered in this study.

Modeling the Foot-Strike Event in Running Fatigue via Mechanical Impedances 159

Typical accelerometer traces for the tibial tuberosity (right leg) and sacrum are shown in Figure 5, for a complete running cycle, i.e., from heel-strike of the right foot till the next heelstrike of the same foot. Two major differences should be noted between the traces: (a) intensity of ~ 8 g in the tibial tuberosity versus less than 3 g in the sacrum; (b) while the tibial tuberosity exhibits one major peak within the first 50 ms of the running cycle from heel-strike and originating from the heel-strike of the right foot, the sacrum acceleration, due to its central location, exhibits two comparable positive peaks within the running cycle,

**Typical T.T. Acceleration - RUN I**

**0 100 200 300 400 500 600 700**

**Typical Sacrum Acceleration - RUN I**

**Figure 5.** Typical accelerometer traces for the tibial tuberosity (T.T.) (right leg) and sacrum, for a complete running cycle, i.e., from heel-strike of the right foot till the next heel-strike of the same foot.

The mechanical properties of biological material are, in general, multiple variabledependent. Specifically stiffness, in addition to its being non-linear e.g. strain dependent, often depends on the deformation rate. This is also the case with bones [33], tendons and ligaments [34], cartilage [35] and muscle [36]. Damping too may be position-dependent.

**0 100 200 300 400 500 600 700**

reflecting each of the right and left heel-strikes.

**Acceleration [g]**

**5. Parameter estimation of the model** 

**-2**

**-1**

**0**

**Acceleration [g]**

**1**

**2**

**3**

Respiratory data were collected from a Sensor-Medics 4400 device and included O2 V - Oxygen consumption, 2 VCO - CO2 production, VE - minute ventilation, PETCO2 - end-tidal CO2 pressure, <sup>2</sup> *VE VO* / - ventilatory equivalent for oxygen, <sup>2</sup> *VE VCO* / - ventilatory equivalent for CO2. The anaerobic threshold was determined as the point of initial increase of <sup>2</sup> *VE VO* / and <sup>2</sup> *VE VCO* / , which just precedes the initial decline of PETCO2 [31]. In a previous study, we have shown that 30 min running at a speed exceeding the anaerobic threshold is a sufficient time to induce general fatigue [8].

The respiratory data were evaluated at each of the 1st, 5th, 10th, 15th, 20th, 25th and 30th min of running and the accelerometer and force platform data were online sampled at 1667 Hz sampling rate. The model parameters were estimated, however, at the 1st, 15th and 30th min of running.

The dynamics of acceleration build-up at heel-strike is shown in Figure 4 where the simultaneous recordings of the tibial tuberosity acceleration and the ground reaction force (GRF) are shown in two time scales: complete running cycle (panel a) and zooming-in on the heel-strike event (panel b). In this case the build-up time to the tibial tuberosity peak acceleration was ~ 30 ms. It is also noted that the ground reaction force exhibits two peaks: a smaller one shortly after heel-strike and a larger one (~ 2.5 body weights), towards the middle of the running cycle.

**Figure 4.** Typical foot-ground reaction force (GRF) (transient curve) measured simultaneously with tibial tuberosity acceleration (spiky curve). **a.** complete stance phase; **b.** heel-strike period only.

Typical accelerometer traces for the tibial tuberosity (right leg) and sacrum are shown in Figure 5, for a complete running cycle, i.e., from heel-strike of the right foot till the next heelstrike of the same foot. Two major differences should be noted between the traces: (a) intensity of ~ 8 g in the tibial tuberosity versus less than 3 g in the sacrum; (b) while the tibial tuberosity exhibits one major peak within the first 50 ms of the running cycle from heel-strike and originating from the heel-strike of the right foot, the sacrum acceleration, due to its central location, exhibits two comparable positive peaks within the running cycle, reflecting each of the right and left heel-strikes.

**Figure 5.** Typical accelerometer traces for the tibial tuberosity (T.T.) (right leg) and sacrum, for a complete running cycle, i.e., from heel-strike of the right foot till the next heel-strike of the same foot.

### **5. Parameter estimation of the model**

158 Injury and Skeletal Biomechanics

Oxygen consumption,

time to induce general fatigue [8].

middle of the running cycle.

**<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>500</sup> -10**

**time [ms]**

**a. b.**

2 VCO

study.

of running.

**-5**

**0**

**5**

**Acceleration [g]**

**10**

**15**

Running was thus for a duration of 30 min and at a speed exceeding the anaerobic threshold speed of each subject by 5%. Before the test a 15 min warming up running on the treadmill was performed. In this study, the average running speed for all 14 subjects was 3.53 m/s (SD, 0.19). It should be noted, however, that in addition to global fatigue, local fatigue in a muscle may also take place as a result of an intensive activity of this muscle. This type of fatigue is reflected by certain changes in its electromyogram (EMG) signal in the time and/or frequency domains [32]. Local fatigue was not considered in this

Respiratory data were collected from a Sensor-Medics 4400 device and included O2

CO2 pressure, <sup>2</sup> *VE VO* / - ventilatory equivalent for oxygen, <sup>2</sup> *VE VCO* / - ventilatory equivalent for CO2. The anaerobic threshold was determined as the point of initial increase of <sup>2</sup> *VE VO* / and <sup>2</sup> *VE VCO* / , which just precedes the initial decline of PETCO2 [31]. In a previous study, we have shown that 30 min running at a speed exceeding the anaerobic threshold is a sufficient

The respiratory data were evaluated at each of the 1st, 5th, 10th, 15th, 20th, 25th and 30th min of running and the accelerometer and force platform data were online sampled at 1667 Hz sampling rate. The model parameters were estimated, however, at the 1st, 15th and 30th min

The dynamics of acceleration build-up at heel-strike is shown in Figure 4 where the simultaneous recordings of the tibial tuberosity acceleration and the ground reaction force (GRF) are shown in two time scales: complete running cycle (panel a) and zooming-in on the heel-strike event (panel b). In this case the build-up time to the tibial tuberosity peak acceleration was ~ 30 ms. It is also noted that the ground reaction force exhibits two peaks: a smaller one shortly after heel-strike and a larger one (~ 2.5 body weights), towards the

**Figure 4.** Typical foot-ground reaction force (GRF) (transient curve) measured simultaneously with tibial tuberosity acceleration (spiky curve). **a.** complete stance phase; **b.** heel-strike period only.

**-4**

**0**

**4**

**Acceleration [g]**

**8**

**12**

**0**

**400**

**800**

**1200**

**GRF [N]**

**1600**

**2000**


V -

**-200**

**5 15 25 35 45**

**time [ms]**

**200**

**600**

**G**

**R**

**F [N]**

**1000**

**1400**


The mechanical properties of biological material are, in general, multiple variabledependent. Specifically stiffness, in addition to its being non-linear e.g. strain dependent, often depends on the deformation rate. This is also the case with bones [33], tendons and ligaments [34], cartilage [35] and muscle [36]. Damping too may be position-dependent.

Due to nonlinearity of the stiffness/damping properties of the joints of the leg [e.g. 20,37], we were not generally able to estimate the model parameters while assuming that they remain constant over the heel-strike period. Thus, the heel-strike period was divided into two equal periods (22 ms each) and the parameters were estimated separately for each of these periods, using the Gauss-Marquardt [38-39] method of non-linear estimation. For the first period the initial conditions were as prescribed in equation (2), and for the second period the initial conditions used were the values reached at the end of the first period.

Modeling the Foot-Strike Event in Running Fatigue via Mechanical Impedances 161

**6. Model results and model sensitivity analysis** 

to parameters *k2* and *k3* and small sensitivity to parameters *k4* and *c3*.

Initial parameter value

Tested

Parameter

*ki* [N/m] *ci*[NS/m]

Table 1 shows the sample results of the stiffness and damping parameters for the two time zones. A sensitivity analysis can provide an indication to the quality of the estimation of parameters. Thus, sensitivity of the *m2* and *m4* model results to each of the estimated parameters was performed by two-fold multiplying and/or dividing each of the estimated parameters separately and for each time zone. The two-fold variation demonstrated a strong sensitivity of *m2* to *k1* and *k2* in the first time zone and to parameters *k1*, *k2*, *k3* and *c3* in the second time zone. Small to medium sensitivity was obtained in the second time zone to parameters *c1* and *k4*, repectively. The *m4* acceleration was not sensitive in the first time zone to none of the parameters, while in the second time zone it demonstrated strong sensitivity

sensitivity

sensitivity

*m2* acceleration

*m4* acceleration

ZONE 1 ZONE 2 ZONE 1 ZONE 2 ZONE 1 ZONE 2

*k1* 16720 265305 \*\*\* \*\*\* \* (\*\*) \* (\*\*\*)

*k2* 14276 31563 \*\*\* \*\*\* \* (\*) \*\*\*

*k3* 3946 16427 \* (\*\*) \*\*\* \* (\*) \*\*\*

*c1* 34.4 250.6 \* (\*\*\*) \*\* (\*\*\*) \* (\*\*) \* (\*)

*c2* 1.1 1.1 \* (\*) \* (\*) \* (\*) \* (\*)

*c3* 1.2 488 \* (\*) \*\*\* \* (\*) \*\* (\*\*)

*c4* 998 1000 \* (\*) \* (\*) \* (\*) \* (\*)

 *k4* 150000 150170 \* (\*) \*\* (\*\*) \* (\*) \*\* (\*\*\*)

**Table 1.** Summary of the sensitivity of the *m2* and *m4* accelerations to the model parameters. A twofold change up and down was checked in the two time solution-zones, separately for each parameter. Parameters for which the accelerations were not sensitive were checked also for a change in one order of magnitude (up and down). The results of this latter test are shown in parentheses (\* = negligibly low effect; \*\* = low to medium effect; \*\*\* = high effect, i.e. more than 10% change in peak acceleration of *m2* or *m4*).

In cases of low- or no-sensitivity the parameters were varied, up and down, by one order of magnitude with the results shown in parentheses. The one order of magnitude variation in the parameter values did not evoke sensitivity beyond that of the twofold variation, except

for *c1* on *m2* and *k1* on *m4*.

Figure 6 shows the model prediction of the shank mass (*m2* in Figure 1) acceleration (continuous line) versus the experimentally measured tibial tuberosity acceleration (denoted as +). The results of two different subjects are demonstrated (vertical partition of the Figure) for two time points of running (horizontal partition): 1st min, fatigue-free, and 15th min of the running test. The subtle, brief, discontinuity in the traces at mid-cycle represents the transition between the two parameter estimation periods.

**Figure 6.** The model prediction of the *m2* acceleration (continuous line) versus the experimentally measured tibial tuberosity acceleration (denoted as +). The upper panels represent the solution for the 1st min of running and the lower panels represent the solution for the 15th min of running. The subtle, short, discontinuity in the traces at mid-cycle represents the transition between the two parameter estimation periods, with piece-wise constant stiffness each. Each of the left and right panels represents a different subject.
