Mathematical Model Analyzes the Severity of Disc Degeneration in College Students

By Lina Sorg

Anthropometry—the scientific study of the human body’s measurements—plays an important role in the engineering and creation of equipment like seatbelts, cars, and chairs. Unsurprisingly, designing chairs that can universally accommodate the entire population is quite difficult, as everyone has different proportions. Manufacturers exploit existing anthropometric databases that contain body dimension data in an attempt to produce chairs and other apparatuses that comfortably fit the human body. However, recent studies have revealed an absence of relevant data in these databases. For instance, certain types of body dimension data are limited, and their direct collection is prohibitively expensive. Therefore, measurements like hip breadth are predicted in many anthropometric data sets via outdated regression formulas that were derived in the 1960s or 70s. Human body dimensions have changed dramatically since this time, rendering some of this data unfeasible. “To what extent is there missing data in these databases, and how might this missing data be influencing the design of engineering equipment?”  Lisa M. Kuhn of Southeastern Louisiana University (SLU) asked.

Multiple factors—including hip breadth, sitting posture, discomfort, and non-ergonomic chair design—are correlated with lower back pain. Such pain is in turn correlated with the diagnosis of disc degeneration. And present-day MRI results provide evidence of early onset disc degeneration in adults as young as 19. During a minisymposium presentation at the 2021 SIAM Conference on Applications of Dynamical Systems, which is taking place virtually this week, Kuhn presented a mathematical model that accounts for risk factors and estimates how soon an individual might begin experiencing spinal disc degeneration. “The earlier you know about it, the sooner you can implement changes or hope for a cure,” she said. 

Kuhn’s work focuses specifically on the risk of disc degeneration for college students in the context of a typical student chair/desk. Her colleague noted that the design of student desks on campus are neither comfortable nor ergonomic. “They look like they were designed for children almost,” Kuhn said. “Students do not have healthy sitting postures whenever they are sitting in those chairs.” She and her colleagues surveyed 280 students at SLU with roughly 18 questions. The questions inquired about participants’ ages, heights, and weights, as well as general overall lifestyle/level of activeness. The survey also asked students whether they have ever been diagnosed with back pain, how often they use the desks on campus, how much time they spend in the chairs during a typical day or week, how (un)comfortable they feel while sitting in the chairs, and how quickly discomfort commences.

Kuhn then presented a population-dynamics model that was derived and validated by Chang-Jiang Zheng and James Chen  in 2015. Zheng and Chen verified the model—which predicts disc generation— using an independent data set that surveyed part of the Hungarian population on back pain prevalence; at the time of publication, it was the large data collection on back pain. Kuhn and her collaborators applied their own independent data to the model as well. The model has three compartments:

• Compartment 1 includes people of age \(t\) who do not have back pain or disc degeneration
• Compartment 2 includes people who do have disc degeneration (identified by an MRI) but do not experience symptoms
• Compartment 3 includes people who have both disc degeneration and symptoms.

Individuals in the latter two compartments can move back and forth because symptoms are not constant. These compartments yield a linear system of differential equations that contributed to the model’s development. Here, \(\alpha =\) the age-independent parameter, \(\beta =\) a symptom expression fraction, \(\lambda_1 =\) the disc degeneration rate, \(t_0 = 13.3 =\) the transition age at which discs generally lose their growth potential, \(t =\) age in years, and \(\bar{z}(t) =\) the likelihood of experiencing disc degeneration at age \(t\).

Zheng and Chen initially applied the method of least squares to the curve fitting to yield a low back pain model for the general population. The prevalence of pain ranged from less than 10 percent to almost 50 percent within people aged 15 to 40. During her analysis, Kuhn focused on the number of students who experience back pain—ranging from mild to severe—while using the campus chairs (see Figure 1). Ultimately, 226 of the 280 surveyed students reported varying degrees of back pain. Kuhn applied the results of the questionnaire to Zheng’s low back pain model, fit her team’s parameters accordingly, and generated a back pain prevalence model (see Figure 2). Here, the blue line indicates the Hungarian data from Zhang and Chen’s study and the purple line represents the SLU college students. 

“This shows substantially that these college students are at least reporting that they experience pain more frequently than the general population at the time of this study,” Kuhn said. “The amount that are saying they do experience pain in these chairs surprised us.” These results illustrate a significant increase in the prevalence of lower back pain for individuals who are using the desks, thus suggesting that the college-going population has more back pain than the general populace.

Kuhn and her collaborators are now building upon their findings to predict the diagnosis of disc degeneration in college students based on the survey responses. Because the survey directly inquired about the diagnosis of back pain or disc generation, diagnosis serves as the dependent variable. The team analyzed the survey data using Pearson’s chi-squared test with a significance level of five percent; for any value less than five percent, they rejected the null hypothesis of independence and identified an association between diagnosis and the variable in question. Results indicated evidence of association between diagnosis and five independent variables:

• Age group
• Whether an individual experiences pain while sitting in the chairs
• Sitting posture in the chairs
• How active an individual is in general, ranked on a scale of 1-10
• How often an individual takes breaks from sitting in the chairs.

Kuhn plans to utilize a multinomial logistic regression approach to predict the diagnosis between the independent variables. She would also like to modify Zheng and Chen’s model to include time dependence, as well as directly collect body dimension data from survey participants to rederive the outdated regression equations. All of these results are forthcoming.

Lina Sorg is the managing editor of SIAM News.