Ten varsity intercollegiate student athletes with mTBI or with recent diagnosis of concussion participated in this study. Three of the ten subjects had a history of a prior concussion (0.7+1.3, R = 0–3) and, utilizing the Cantu revised evidence based grading scale [28], eight of the ten subjects were classified as grade II concussions while two were classified as grade III. Further, 50% of the subjects reported post traumatic amnesia and 40% of the subjects reported loss of consciousness, however only one was lasted longer than a few seconds. All subjects denied current and past history of balance, neurological, metabolic, or vestibular disorders. All subjects provided written informed consent prior to participating as approved either by the University of Florida Institutional Review Board, for studies conducted by Dr. Hass, or by the Georgia Southern University Institutional Review Board, for studies conducted by Dr. Buckley. Analyses of these existing data, which are reported in the present paper, were also disclosed to the University of Florida and Georgia Southern University Institutional Review Boards, and to the US Army Research Office Human Research Protections Office, and were determined to be exempt.

Potential subjects were identified by the athletic training staff and the concussion diagnosis was confirmed by both the treating certified athletic trainer and the team physician. On the day following the concussion, Day 1, the experimental procedures were initiated. The subject reported to the biomechanics laboratory for testing each day until they were cleared to return to participation in accordance with university medical policies, seven days symptom free with progressive exertion (11.8+2.5 days). The last day of testing was the day the subject was cleared for return to full activity in their sport. One subject suffered a second concussion during the recovery from his initial concussion and did not return to participation that season. Upon arrival at the biomechanics laboratory, each subject performed one trial of static stance for 2 minutes. The subject was instructed to stand barefoot on the force platform and to remain as stationary as possible for the duration of the experiment. The trial was initiated and concluded with a verbal cue from the investigator. Ground reaction forces (A/P, M/L, and vertical) and center of pressure (A/P and M/L) data were collected from a single force platform (model OR-6, AMTI, Watertown, MA, USA) at 1000 Hz, where A/P and M/L denote Anterior/Posterior and Medial/Lateral balance, respectively. An example of a COP trajectory is shown in Fig. 1 for a subject on day 6 after concussion.

The area of the COP trajectory is a popular metric for characterizing postural sway. Consider a COP trajectory such as shown in Fig. 1. To compute the area the trajectory has traced out, one can partition a 2-dimensional plane that encompasses the trajectory into unit areas, and sum up all the non-empty unit areas covered by the COP trajectory.

Detrended fluctuation analysis (DFA) [29]–[31] characterizes the second order statistic – the correlation, in a time series. It can automatically remove certain trends or nonstationarity contained in the data under study. When applying DFA, one works on a random-walk-type process, and expects the process to have a power-law decaying spectral density. Denote the COP data by . The random-walk-type process can be obtained by first removing the mean value and then forming partial summation,(1)DFA works as follows. First, one divides the time series into non-overlapping segments (where the notation denotes the largest integer that is not greater than ), each containing points; then one calculates the local trend in each segment to be the ordinate of a linear least-squares fit for the random walk in that segment, and computes the “detrended walk”, denoted by , as the difference between the original walk and the local trend; finally, one examines if the following scaling behavior (i.e., fractal property) holds or not:(2)where the angle brackets denote ensemble average of all the segments. The parameter is often called the Hurst parameter [32]. When the scaling law described by Eq. (2) holds, the process under investigation is said to be a fractal process. The autocorrelation for the “increment” process, defined as , decays as a power-law,(3)When , the process is called memoryless or short range dependent. The most well-known example is the Brownian motion (Bm) process. In nature and in man-made systems, often a process is characterized by an . Prototypical models for such processes are fractional Brownian motion (fBm) processes. When , the process is said to have “anti-persistent” correlations [32]. For , the process has “persistent” correlations, or long memory properties [32]. The latter is justified by noticing that(4)In practice, quite often power-law relations are only valid for a finite range of . Unfortunately, some researchers try to estimate the parameter (or other scaling exponents such as the fractal dimension) by some optimization procedure without being concerned about the scaling region.

As we have discussed, to compute the area of the COP, one can count the number of non-empty grids/boxes covered by the COP trajectory. The idea can straightforwardly be extended to calculate Shannon and Renyi entropies, by the following procedure.

Assume a trajectory has visited unit areas, with the unit area being visited by times. This is schematically shown in Fig. 1. Note that an empty unit area, such as that denoted by in Fig. 1 is irrelevant. Let the trajectory be points long. Then the probability that the unit area being visited is . The Shannon entropy is defined by(5)where the unit for is a bit or baud corresponding to base 2 or in the logarithm. Without loss of generality, we shall choose base .

Renyi entropy is a generalization of Shannon entropy. It is defined by(6) has a number of interesting properties:

It is clear that so far as the unit areas are not visited with equal probability, the Renyi entropy provides a better and more comprehensive characterization than the area metric. In particular, we can envision two potential advantages:

A standard balance test may last only 20 s [26]. Would such short data be adequate for calculating nonlinear entropy metrics from postural sway data? To see the problem, we have first calculated a popular metric for characterizing postural sway, the area of the COP. Specifically, we have calculated the area of COP based on the first 20 s, first 40 s, first 60 s, first 80 s, first 100 s, and all 120 s data. The variation of the area vs. such data length for one subject is plotted in Fig. 2(a). We observe that on day 1 after concussion, the area increases with the data length rapidly. Actually, the growth rate is faster than linear! From other subjects' data, we have always observed that on day 1 after concussion, the area increases with data length at least linearly, and sometimes even exponentially.

The result shown in Fig. 2(a) compels us to ask: i) How long should the data be to make proper conclusions? Note that so far as area is the metric, no easy answer can be given, since area is a strictly non-decreasing function of data length. Note also that the work by Cavanaugh et al. [26] using approximate entropy was based on 20 s data. They commented “Paradoxically, the range of COP displacement after injury (approximately 4 cm) was less than at preseason (approximately 5 cm), suggesting that postural stability had improved, rather than become more impaired, after injury.” Their puzzling observation could just be due to the shortness of data they analyzed, noticing that the increase of area with data length is much slower on day 12 than on day 1 – in other words, postural instability would not be fully revealed by short data right after concussion. ii) How should comparisons be made among different injured subjects before traditional clinical measures such as GSC, SAC, or BESS scores return to baseline? This is a harder question to answer. Especially, depending on the severity of concussion, one subject's day 1 behavior could be similar to another subject's day 2, day 3, or even other day's behavior.

While a clean, definitive answer to both questions might be hard to obtain, later in Sec. 3, we shall develop an information theoretic approach so that we can gain important insights into these issues.

Numerous work has shown that gait dynamics can be modeled by processes, where is frequency and , where is called the Hurst parameter and characterizes the correlation structure of the process: depending on whether is smaller than, equal to, or larger than 1/2, the process is said to have anti-persistent, short-range, or persistent long-range correlations [29], [30]. Fractal analysis is also very promising for identifying differences in postural stability between control and elderly subjects [21]–[24]. This motivates us to ask whether the key parameter, , from fractal analysis, can be used to indicate postural instability after concussion.

To answer the above important question, we have systematically examined the frequency contents of sway data. Figs. 3–4 show the power spectral density (PSD) for COP on day 1 and day 10, where the 1st column is always for PSD in linear scale; the 2nd column for PSD is in log-log scale. At the first sight, the 2nd column is very interesting: we clearly observe a linear line in log-log scale for both A/P and M/L COP data, indicating that COP data may be modeled by processes.

To further confirm the scaling law of the COP data, we apply a more sophisticated method, DFA, which is a more reliable method for fractal analysis [29], [30]. The DFA curves for the COP data of one subject for day 1 and day 10 are shown in Figs. 3–4 as the 3rd column, where the quantities are plotted also in log-log scale – when the curve is linear, it means the process is a fractal process, with given by the slope of the linear curve. Indeed, the curves are quite linear. In fact, the 2nd and 3rd columns correspond well: the scaling is from about 1 Hz up for COP, amounting to samples in the DFA curve. Further, the estimated from PSD and DFA curves are consistent. Unfortunately, here is always greater than 1, and is not effective in indicating postural instability after concussion. This compels us to have a serious 2nd thought about the frequency contents of the data.

It turns out significant understanding can be found from the 1st column of Figs. 3–4: only low frequencies have appreciable power; the frequency range where fractal scaling is observed basically has negligible power. The 1st columns of Figs. 3–4 actually have indicated how long the sway data have to be for a meaningful analysis. For example, the frequency with the largest power in Figs. 3(a1,b1) is around 0.1 Hz. Thus, if one only uses 20 s data, as Cavanaugh et al. did [26], one basically only observes about 4 cycles of variations, if one assumes the frequency to be cut at 0.2 Hz. Since data here are not really periodic, but rather random, little can be inferred from data as short as 20 s.

We now check how Shannon entropy varies with data length. The result is shown in Fig. 2(b). We observe two interesting features: (1) Shannon entropy for data measured on day 1 after concussion keeps increasing with data length, while that on day 12 reaches saturation when data length is 40 s; with longer data length, it fluctuates slightly. The latter feature reflects that the trajectory visits different unit boxes with un-equal probability. (2) While overall, Shannon entropy for day 1 is much larger than that for day 12, when data length is as short as 20 s, the opposite is actually the case. Recall that a similar behavior has also been observed with approximate entropy with 20 s data right after concussion [26]. We now see that associating the value of approximate entropy with the complexity of postural sway based on such short data may not be justified.

The above discussions make it clear that we have to use all 120 s of the data, in order to reliably assess the effects of postural instability after concussion. The variations of entropies with the number of days after concussion are shown in Fig. 5 for two subjects. When such curves are averaged over the 10 subjects (up to day 10, which is the last day for some subjects), we obtain Fig. 6. While we identify that the general trend for the variation of the Renyi entropies is decreasing after concussion, suggesting recovery from concussion, we emphasize that the actual variation of the Renyi entropies is not simply monotonic, but quite complicated. This implies that the subjects might not have fully recovered from concussion-induced postural instability, even after a fairly long period of time (such as 1–2 weeks). This also means that these entropy measures can effectively indicate postural instability long after concussion has occurred. Note that there is a RTP protocol that lets concussed student athletes return to sports activity only 4–5 days after injury [33]. Our analysis indicates that such a clinical practice may be too aggressive.

Postural sway data are notoriously nonstationary. One motivation that the standard practice only collects 20–30 s data is to suppress nonstationarity. Now that we have shown that 20–30 s data are too short, the issue of nonstationarity becomes more acute. Therefore, an emerging challenge would be to properly deal with nonstationarity and remove noise from sway data, so that subsequent analysis is meaningful.

In this regard, a versatile adaptive algorithm for detrending, denoising, multiscale decomposition, and multifractal analysis recently developed by the authors may be very useful [34]–[37]. In fact, as far as denoising is concerned, the method is better than linear filters, wavelet and chaos-based approaches [36]. While we omit the details of the method here, we would like to show an example (Fig. 7) to illustrate the effectiveness of the method. Clearly, the long-term trend has been accurately determined.

It is important to note that the trend in Fig. 7(i.e., the red curve) may have its own significance; therefore, whether it should be filtered out or not depends on an understanding of the mechanism for the trend signal. Overall, we may conclude that nonstationarity associated with longer data will not pose a challenge in data analysis.