RESULTS AND DISCUSSION

In keeping with our hypothesis, our two-mass model was able to account for virtually all of the duration, amplitude and force–time pattern variability present in the vertical ground reaction force waveforms analyzed. Despite the large differences in waveform characteristics introduced by different speed, foot-strike and footwear conditions, our model accounted for an average of 97% of the individual force–time relationships (mean R²=0.97±0.01) and a minimum of 95% (Fig. 1). The accuracy of these fits across the heterogeneous waveforms tested suggests that two mechanical phenomena, acting in parallel, are sufficient to explain running ground reaction forces: (1) the collision of the lower limb with the running surface, and (2) the motion of the remainder of the body's mass throughout the stance phase.

The accuracy of the fits achieved using a model with only two mass components and with mass component values held constant across conditions differs from prevailing paradigms in several respects. First, while the sequential additions of third, fourth and fifth mass components to multi-mass models over the last two decades (Liu and Nigg, 2000; Ly et al., 2010; Nigg and Liu, 1999; Nikooyan and Zadpoor, 2011; Zadpoor and Nikooyan, 2010) may describe physical and mechanical reality as theorized (Nigg, 2010; Zadpoor and Nikooyan, 2010), these additional masses may also be unnecessary for waveform prediction. Second, the conclusion that the mass quantity decelerated upon foot–ground impact differs substantially for rear foot versus forefoot impacts (Lieberman et al., 2010; Nigg, 2010) should be reconsidered. The close fits we report here using a constant value of 8.0% of the body's mass across all foot-strike conditions indicates that a variable ‘effective mass’ may be unnecessary for accurate modeling and could be mechanically incorrect. For example, if we predict the sprint running waveform analyzed here (Fig. 1G,H) using the effective mass proportions suggested for a forefoot impact [m₁=1.7% and m₂=98.3% of total body mass M_b (Lieberman et al., 2010)] with a speed-specific foot collisional velocity (Mann and Herman, 1985) (Table 1), the rising edge of the sprint waveform is substantially under-predicted and the overall goodness of fit is considerably reduced (R²=0.95 to 0.82; see supplementary material Fig. S1). Third, the model's general features and simplifying assumptions permit impulse J₁ and J₂ durations and forces to be independent. In contrast, the dual ‘stacked spring-mass’ model-type (Alexander, 1990; Derrick et al., 2000; Ker et al., 1989) that Alexander originally introduced (Alexander, 1988) uses a serial, coupled configuration that may be incapable of predicting the brief simultaneous impulses responsible for the characteristic pattern of sprint running waveforms.

Indeed, the model's design was essential for achieving close fits to waveforms with variable rising edges, smooth falling edges and significantly different durations. Given the fixed-mass value of our lower-limb mass component, the close fits to the variable rising edges were achieved predominantly via the two model inputs (Table 1) responsible for the shape of collisional impulse J₁ (Fig. 1). Values for the first of the two, the vertical velocity of the lower limb at touchdown, are well-supported by the waveform-specific literature values available. Values for the second, the deceleration time of the lower limb upon touchdown, are well-supported by the detailed analysis of Nigg et al. (Nigg et al., 1987) for waveform 2, but are lacking for the other three. Fits along the smoother falling edges depend directly upon impulse J₂ because of the early conclusion of the J₁ collisional event. Given a known physical basis for determining total impulse J_T from contact and step times (Eqn 1; see Materials and methods), correctly quantifying impulse J₂ depends solely on the quantity subtracted for impulse J₁ (Eqn 2). While empirical validation clearly remains for several elements of our model, the fits achieved using anatomical mass inputs, realistic lower-limb velocities, and one mechanical explanation across conditions raise the possibility that the running force–motion relationship may be more general than previously recognized.

An additional factor in the accuracy of the fits we report was undoubtedly the model evaluation method adopted. The method chosen allowed us to assess a greater variety of waveforms than would have been possible via direct experimentation, but also involved two potential limitations. First, because the model fits were generated by varying the inputs, the goodness-of-fit values obtained should be regarded as the upper performance limits of the model. Second, the digitizing process enabling our approach might have transformed the literature waveforms into more model-conducive shapes. We were able to evaluate this second possibility empirically by applying the inputs used to fit two of the digitized waveforms (3 and 4) to the original waveform data. This process yielded fits that were the same or slightly greater for the original (respective R² values of 0.98 and 0.96) versus digitized versions because the original waveforms were so closely reproduced by digitizing (see supplementary material Tables S1–S4).

We close by providing respective, illustrative examples of the basic and applied advances made possible by the concise physical basis of our two-mass model. One basic insight provided by the framework of the model is the identification of a mechanical strategy that runners can adopt to achieve faster speeds. By simply increasing the lower limb's velocity prior to touchdown, and reducing deceleration time during impact, runners can elevate the collisional impulse (J₁) and total ground reaction forces as needed to attain faster speeds (Weyand et al., 2000; Weyand et al., 2009; Weyand et al., 2010). Both the existing literature data (Table 1) and our modeling results (Fig. 1) are consistent with this being a primary mechanism by which faster human runners do, in fact, attain faster sprint running speeds.

In application, the conciseness of the model could translate into practical techniques for determining ground reaction forces indirectly. At present, the lone indirect assessment method available (Bobbert et al., 1991) is scientifically rigorous, but impractical for broad usage. The existing technique involves the instantaneous summation of the accelerations of seven body segments based on high-frequency positional data from 10 bodily locations. In contrast, the scientific basis of our two-mass model (Eqns 1, 2, 3, 4, 5 and 6) reduces the data needed for indirect force determinations to three basic variables: aerial time, contact time and the vertical velocity of the lower limb. Thus, our model may allow video and other motion capture techniques to become practical tools for determining vertical ground reaction forces without direct measurement.