Background
The Mayo Clinic defines a concussion as a traumatic brain injury, resulting from direct (or indirect) force applied to the cranium. Concussion is typically accompanied by a miscellany of cognitive, emotional, and physiological symptoms, which tend to decrease in severity over time. However, long term repercussions of concussion may persist long after injury, especially if more than one concussion has been sustained.
In recent years, a growing number of retired football players have exhibited symptoms of lasting functional impairment, which has attracted attention from the scientific community. In 2013, many digital scanning techniques are used to evaluate concussion patients, and the technology continues to improve. However, this technology is not widely used in practice, except among professional athletes and those already showing severe symptoms. A football player who sustains a concussion may be required to suspend play, or may continue immediately if symptoms are not perceived as legitimate. Sophisticated analysis involving an MRI or similar scan is highly unlikely.
Before 2003, no football player was evaluated using digital technology. In 2003, researchers Paul Lauterbur and Peter Mansfield developed a mechanism for projecting Magnetic Resonance Imaging (MRI) quickly and accurately onto a two dimensional plane. This marked the beginning of clinical applications of the MRI, such as the evaluation of concussed football players.
Based on data collected from the 1999, 2000, and 2001 college football seasons, The NCAA Concussion Study was published to the Journal of the American Medical Association (JAMA) on November 19, 2003. As one of the original quantitative concussion investigations, the purpose of the study was to gather data on the effects of concussion, and to determine the necessary resting time before concussed players returned to the game. The cognitive, emotional, and physiological symptoms of concussion were rated by players and their coaches or trainers. The results were published in The NCAA Concussion Study, and in the following discussion, a mathematical model is constructed to quantitatively describe patterns of concussion recovery.
Summary of The NCAA Concussion Study
The participants of the study were college football players from a variety of universities throughout the United States, Players who had recently experienced concussions were targeted for the study (94 players), and a control group was also used (56 players), which included players at the same levels of competition who had not experienced concussion. Three specific measures were used to quantify the effects of concussions; Graded Symptom Checklist (GSC), Standardized Assessment of Concussion (SAC), and the Balance Error Scoring System (BESS).
The Graded Symptom Checklist is a composite score, ranked firsthand by the concussion participant regarding physical symptoms of well-being, to gauge functioning based on a physiological basis.
The Standardized Assessment of Concussion is administered by a coach or trainer, and is used to specifically determine the emotional and mental capacity of the concussed player.
In the Balance Error Scoring System test, participants are asked to perform a variety of basic balance exercises, such as standing in place with eyes closed, hands on the hips, and one foot off the ground, for 20 seconds. A variety of similar tasks are performed and the total score is the cumulative number of errors.
In general, a healthy person without concussion is more likely to score lower on the Graded Symptom Checklist, higher on the Standardized Assessment of Concussion, and lower on the Balance Error Scoring System.
Data Analysis and Model Fitting
Included in the concussion study were three figures indicating mean test scores for the GSC, SAC, and BESS tests for the concussed group and control group. The image of this figure was carefully studied, and by close inspection (counting and recording central pixel location of each data point), numerical data was extracted from this figure. This data was then used for the purpose of studying the mathematical trends underlying patterns of recovery. Statistical analysis was provided in the study, but not the raw data. If anything is lost by using this method of extraction, it is a degree of precision not necessary for this analysis. These original figures from the study are shown below.
Participants in the study were asked to perform the three tests on nine separate occasions; baseline (preseason), at the time of concussion (CC), postgame or post-practice (PG), and one, two, three, five, seven, and ninety days after sustaining the concussion. Although the data taken preseason, at time of concussion, and postgame can provide insight, there is no way to accurately assign them a time scale, so these data are excluded from the model. Similarly, the test scores taken ninety days post-concussion are excluded, because the time scale is more than an order of magnitude different from the other data.
The statistical analysis was based on mean test scores for concussed study participants one, two, three, five, and seven days after injury. The data were evaluated to determine which mathematical function provides the best approximation.
An exponential model of decay was ultimately determined most appropriate for modeling the given data.
Other model types were considered but ultimately discarded. The patterns for recovery were clearly not linear. Some of the data seemed to resemble a logarithmic curve, but long-term behavior indicated that recovery scores approached a limit, unlike a logarithmic curve. The data could be approximated with a polynomial, but there is no clear advantage of a second or third or fourth order polynomial over the others. More terms could always be added to a polynomial to fit the data more closely, but such an approach is not justified, and the model becomes wildly inaccurate at predicting trends outside the given interval. Furthermore, there is no explanation for injury recovery to follow a parabolic trajectory, or that of a higher order polynomial.
Results: Exponential Model and Linear Regression
In order to determine the accuracy of the constructed exponential models, an adjusted data series was evaluated using linear regression.
Original data for the three concussion tests (GSC, SAC, and BESS) each exhibited an exponential pattern of decay. So that linear regression could be used appropriately, a two-step adjustment was applied to the original data collected from The NCAA Concussion Study.
The first step was to position the data so that test scores approached "zero" instead of some nonzero equilibrium, and to make all of the test scores positive (recall patients reported higher SAC scores as they returned to health). These adjusted exponential models were used to determine coefficients A and B.
The second step was to take the natural logarithm of both dependent and independent variables. Because the recovery pattern is exponential, the resulting relationship is linear.
These two adjustments are mathematically legitimate and allow for more accurate analysis of the recovery pattern. Shifting the data series up or down does not change its inherent exponential distribution.
The linear relationship between the (logarithmic) data sets confirmed by linear regression, which produced low P-values, high t-statistics, and high F-statistics, for each of the three tests.', which in turn confirms an exponential pattern of decay for each of the original data sets.
Microsoft Excel's linear regression output is included in the appendix.
Conclusion
For all three tests, the exponential model proved to be a reasonable fit. For the BESS test, participants approached a long-term stability score of about seven errors per test, as balance and agility are not perfect even in most healthy persons. For the SAC test, participants approached a long-term cognitive recovery score of about twenty-nine, a reasonable functional level for healthy persons. SAC scores are also expected to increase as a person regains cognitive ability, following a negative exponential decay. Consequently, the data for these two tests was adjusted so that each model was a positive exponential curve decaying towards zero. Plotting these regular exponential functions on a logarithmic scale (both for dependent and independent axes) produced a linear relationship. Linear regression analysis was then performed on these transformed data, and suggests that the exponential pattern of recovery is indeed legitimate.
An exponential curve is a unique function in mathematics, as the rate of change of the exponential function is proportional to the original exponential function. In this study, participants are seen to recover at a rate proportional to the severity of their injury, ultimately approaching a long-term level of physiological and cognitive ability. It is not unreasonable to consider that in evolutionary terms, an injury or impairment should be dealt with in proportion to its severity or extent to which it prevents normal function. If a person is sick or tired or hungry, the body responds with an allocation of resources proportional to its perception of the importance of addressing such need. Recovery occurs most quickly just after injury, then gradually slows as the person regains normal functioning.
This study was performed at a time when little was known about brain injuries from a scientific perspective. The use of scanning technology and digital analysis in sports was in its infancy. Subsequent investigations have provided a wealth of information, confirming the long-term health detriments surrounding concussions, and stressing the importance of adequate recovery time following a traumatic brain injury.
Although sophisticated technology is now available to diagnose and quantify brain injury, it is rarely used in practice, except in cases of extreme severity of injury or for high-profile professional athletes. For high school and college and even professional athletes throughout America, concussions are still diagnosed using the GSC, SAC, and BESS tests. It is important for players and their families to understand the short- and long-term consequences of traumatic brain injury. The exponential trajectory of concussion recovery can be used as a helpful guideline for those without access to more sophisticated technology.
Appendices
Linear Regression Output
Graded Symptom Checklist (GSC)
Standardized Assessment of Concussion (SAC)
Balance Error Scoring System (BESS)
Concussion Evaluation Forms
Graded Symptom Checklist (GSC)
Standardized Assessment of Concussion (SAC)
Balance Error Scoring System (BESS)
References
Original Study
The NCAA Concussion Study: Acute Effects and Recovery Time Following Concussion in Collegiate Football Players (McCrea, Guskiewicz, Marshall, Barr, Randolph, Cantu, Onate, Yang, Kelly)
(Published in Journal of the American Medical Association)
http://jama.jamanetwork.com/article.aspx?articleid=197668
Medical Nobel Prize Awarded for MRI Applications
(National Institute of Health Article)
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC214010/
Definition of Concussion by the Mayo Clinic
http://www.mayoclinic.com/health/concussion/DS00320
Graded Symptom Checklist (GSC) test
http://www.bchmed.org/bec.nsf/Files/sportsinfo/$file/Graded%20Symptom%20Checklist%20(3).pdf
Standardized Assessment of Concussion (SAC) test
http://knowconcussion.org/wp-content/uploads/2011/06/SAC.pdf
Balance Error Scoring System (BESS) test
http://knowconcussion.org/wp-content/uploads/2011/06/BESS.pdf